nchrp_rpt_647AppendixA

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					                        Appendix A
Investigation of Suitable Soil Constitutive Models for 3-D
Finite Element Studies of Live Load Distribution Through
                    Fills Onto Culverts



   National Cooperative Highway Research Program
                   Project 15-29




                     Limited Use Document

               This Final Report is furnished only for review by
                members of the NCHRP project panel and is
                regarded as fully privileged. Dissemination of
             information included herein must be approved by the
                                   NCHRP.




                CNA Consulting Engineers
              Simpson, Gumpertz & Heger

                             April 2009
Table of Contents

1. INTRODUCTION                                                           1

2. REVIEW OF AVAILABLE SOIL MODELS                                        1
    2.1 Linear Elastic                                                    2
    2.2 Elasto-Plastic                                                    3
    2.3 Stress-Dependent Models                                           6
        2.3.1 Duncan-Selig Model                                          6
        2.3.2 Hardening Soil Model (Plaxis)                              10
    2.4 Findings from Soil Model Evaluation                              22

3. TWO-DIMENSIONAL MODELING OF CULVERTS                                 22
    3.1 Modeled Structures                                              22
    3.2 Material Models                                                 23
        3.2.1 Linear-Elastic Model                                      23
        3.2.2 Mohr-Coulomb Model with Perfect Plasticity in Plaxis      23
        3.2.3 Hardening-Soil Model in Plaxis                            24
        3.2.4 In-Situ Soil Material                                     24
        3.2.5 Other Materials                                           24
    3.3 Live Load                                                       24
    3.4 Finite Element Model                                            26
    3.5 Results of 2D Analysis                                          35
        3.5.1 Concrete Box                                              35
        3.5.2 Concrete Pipe                                             44
        3.5.3 Metal Pipe                                                49
        3.5.4 Thermoplastic Pipe                                        54
        3.5.5 Concrete Arch                                             59
        3.5.6 Metal Arch                                                64
        3.5.7 Summary of Results from 2D Preliminary Analyses           69
    3.6 Effect of Interface Strength                                    71
    3.7 Conclusion                                                      81

4. THREE DIMENSIONAL MODELING OF CULVERTS                                82
    4.1 Comparison of Responses to Factored and Unfactored Live Loads    82
        4.1.1 Introduction                                               82
        4.1.2 Method of Approach                                         82
        4.1.3 Results                                                    83
              4.1.3.1 HDPE Pipe in ABAQUS                                83
              4.1.3.2 Three-Sided Arch Top Culvert in CANDE              85
        4.1.4 Conclusion                                                 86
    4.2 Selected Field Tests for 3D Analysis                             86
        4.2.1 NCHRP Project 12-45                                        86
        4.2.2 Minnesota DOT Study                                        90
    4.3 Three-Dimensional Analysis                                       93
        4.3.1 General Information                                        93
        4.3.2 Long-Span Concrete Arch Culvert                            94
              4.3.2.1 Finite Element Model                               94
              4.3.2.2 Materials                                          95
              4.3.2.3 Loading and Boundary Condition                     96
              4.3.2.4 Results                                            97
        4.3.3 Long-Span Metal Arch Culvert                              101



NCHRP 15-29 Appendix A                                                    i
                4.3.3.1 Finite Element Model                                                                                       101
                4.3.3.2 Materials                                                                                                  102
                4.3.3.3 Loading and Boundary Condition                                                                             102
                4.3.3.4 Results                                                                                                    102
          4.3.4 60-in. Diameter HDPE Pipe                                                                                          108
                4.3.4.1 Finite Element Model                                                                                       108
                4.3.4.2 Materials                                                                                                  109
                4.3.4.3 Loading and Boundary Condition                                                                             109
                4.3.4.4 Results                                                                                                    111
          4.3.5 Discussion                                                                                                         126
      4.4 Comparison between the Mohr-Coulomb and Hardening-Soil Models in Three-
             Dimensional Analysis in PLAXIS                                                                                        129
          4.4.1 Method of Approach                                                                                                 130
          4.4.2 Results                                                                                                            130
                4.4.2.1 Metal Arch in Test 2 with 3 ft Cover                                                                       130
                4.4.2.2 HDPE Pipe with A2 Backfill and 2.8 ft Cover                                                                130
          4.4.3 Conclusion                                                                                                         135
      4.5 Three-Dimensional Analysis of Field Tests in ABAQUS                                                                      135
          4.5.1 Introduction                                                                                                       135
          4.5.2 Method of Approach                                                                                                 136
          4.5.3 Validation of ABAQUS Model                                                                                         139
          4.5.4 Results                                                                                                            140
                4.5.4.1 Metal Arch with 3 ft Cover                                                                                 140
                4.5.4.2 HDPE Pipe with A2 Backfill and 2.8 ft Cover                                                                142
          4.5.5 Conclusion                                                                                                         143

5. DISCUSSION                                                                                                                      144

6. CONCLUSIONS AND RECOMMENDATIONS                                                                                                 145

7. REFERENCES                                                                                                                      145

List of Tables

Table 1—Elastic soil properties for Backfill (Selig, 1990) ............................................................. 3
Table 2—Vertical Stresses and Estimated Horizontal Stresses under Gravity and Corresponding
Angle of Friction for SW85 ............................................................................................................ 6
Table 3—Parameters for Linear-Elastic and Mohr-Coulomb Models for SW85............................ 6
Table 4—Soil Properties of Backfill for Duncan-Selig Model (Selig, 1988) ................................... 9
Table 5—Input Parameters for Hardening-Soil Model for SW85, SW90, ML85, and CL85 ........ 15
Table 6—Structural Types and Cover Depths for 2D Analysis ................................................... 23
Table 7—Comparison of Bending Moments and Thrusts in Concrete Box Model ...................... 43
Table 8—Comparison of Bending Moments and Thrusts in Concrete Pipe Model ..................... 48
Table 9—Comparison of Bending Moments and Thrusts in Metal Pipe Model........................... 53
Table 10—Comparison of Bending Moments and Thrusts in Thermoplastic Pipe Model ........... 58
Table 11—Comparison of Bending Moments and Thrusts in Concrete Arch Model................... 63
Table 12—Comparison of Bending Moments and Thrusts in Metal Arch Model ........................ 68
Table 13—Ratios of Live Load Moments and Thrusts of Concrete Box ..................................... 69
Table 14—Ratios of Live Load Moments and Thrusts of Pipes with a Cover Depth of 2 ft ........ 70
Table 15—Ratios of Live Load Moments and Thrusts of Pipes with a Cover Depth of 6 ft ........ 70
Table 16—Ratios of Live Load Moments and Thrusts of Arches with a Cover Depth of 2 ft ...... 70
Table 17—Ratios of Live Load Moments and Thrusts of Arches with a Cover Depth of 6 ft ...... 70



NCHRP 15-29 Appendix A                                                                                                                 ii
Table 18—Comparison of Bending Moments and Thrusts between Concrete Pipe Models with
50% and 100% Interface Strength (Mohr-Coulomb Soil Model) ................................................. 76
Table 19—Comparison of Bending Moments and Thrusts Load between Thermoplastic Pipe
Models with 50% and 100% Interface Strength (Mohr-Coulomb Soil Model) ............................. 81
Table 20—Comparison of Structural Responses between Analyses with Factored and
Unfactored Live Loads (HDPE Pipe, A2 Backfill) ........................................................................ 85
Table 21—Comparison of Structural Responses between Analyses (Hanson Arch) ................. 86
Table 22—Properties of Reinforced Concrete Culvert ................................................................ 88
Table 23—Properties of Structural Steel Plate and Culvert ........................................................ 88
Table 24—Properties of Type S HDPE Pipe .............................................................................. 91
Table 25—Average Trench Measurements for Test Pipes in the MNDOT Study ....................... 92
Table 26—Soil Properties Used for the 3D Analyses of Long-Span Arches .............................. 96
Table 27—Concrete Properties Used for the 3D Analyses of Long-Span Arches ...................... 96
Table 28—Vertical Displacements at Crown of Concrete Arch due to Live Loads ..................... 99
Table 29—Chord Extension at Height of 88 in. of Concrete Arch Culvert due to Live Loads ..... 99
Table 30—Thrusts at Base of Concrete Arch Culvert due to Live Loads ................................... 99
Table 31—Axial and Bending Modulus of Metal Arch in Circumferential and Longitudinal
Directions (E=29,000 ksi) .......................................................................................................... 102
Table 32—Vertical Displacements at Crown of Metal Arch due to Live Loads ......................... 104
Table 33—Chord Extension at Height of 88 in. of Metal Arch Culvert due to Live Loads ......... 104
Table 34—Thrusts in Test 1 of Metal Arch Culvert due to Live Loads ...................................... 105
Table 35—Thrusts in Test 2 of Metal Arch Culvert due to Live Loads ...................................... 105
Table 36—Moments in Test 1 of Metal Arch Culvert due to Live Loads ................................... 106
Table 37—Moments in Test 2 of Metal Arch Culvert due to Live Loads ................................... 106
Table 38—Axial and Bending Modulus of HDPE Pipe in Circumferential and Longitudinal
Directions (E=100,000 psi)........................................................................................................ 109
Table 39—Soil Properties Used for the 3D Analyses of HDPE Pipes ...................................... 110
Table 40—Comparison of Vertical Displacements at Crown of HDPE Pipes under Heavy Truck
.................................................................................................................................................. 125
Table 41—Comparison of Vertical Displacements at Crown of HDPE Pipes under Light Truck
.................................................................................................................................................. 125
Table 42—Comparison of Diametrical Changes at Springline of HDPE Pipes under Heavy Truck
.................................................................................................................................................. 125
Table 43—Comparison of Diametrical Changes at Springline of HDPE Pipes under Light Truck
.................................................................................................................................................. 125
Table 44—Summary of Displacements under Wheel (Metal Arch, Test 2, 3 ft Cover) ............. 132
Table 45—Summary of Thrusts under Wheel (Metal Arch, Test 2, 3 ft Cover)......................... 132
Table 46—Summary of Moments under Wheel (Metal Arch, Test 2, 3 ft Cover) ...................... 133
Table 47—Summary of Vertical Displacements under Wheel (HDPE Pipe, A2 Soil, 2.8 ft Cover)
.................................................................................................................................................. 134
Table 48—Summary of Horizontal Chord Extensions under Wheel (HDPE Pipe, A2 Soil, 2.8 ft
Cover) ....................................................................................................................................... 135
Table 49—Summary of Force Results (HDPE Pipe, A2 Soil, 2.8 ft Cover) .............................. 135
Table 50—Orthotropic Properties Used in ABAQUS Analyses ................................................ 137
Table 51—Orthotropic Stiffness Properties .............................................................................. 137
Table 52—Soil Porperties Used for Soft Haunch and Void Areas ............................................ 138
Table 53—Summary of Displacements from ABAQUS Analyses with Orthotropic Properties
(Metal Arch, 3 ft Cover) ............................................................................................................. 141
Table 54—Summary of Displacements from ABAQUS Analyses with Orthotropic Properties
(HDPE Pipe, A2 Backfill)........................................................................................................... 143

List of Figures



NCHRP 15-29 Appendix A                                                                                                                             iii
Figure 1—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for SW85 in
Deviatoric Loading of Triaxial Test .............................................................................................. 16
Figure 2—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for SW90 in
Deviatoric Loading of Triaxial Test .............................................................................................. 17
Figure 3—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for ML85 in
Deviatoric Loading of Triaxial Test .............................................................................................. 18
Figure 4—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for CL85 in
Deviatoric Loading of Triaxial Test .............................................................................................. 19
Figure 5—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for SW85 in
Oedometer Loading .................................................................................................................... 20
Figure 6—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for SW90 in
Oedometer Loading .................................................................................................................... 20
Figure 7—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for ML85 in
Oedometer Loading .................................................................................................................... 21
Figure 8—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for CL85 in
Oedometer Loading .................................................................................................................... 21
Figure 9—Live Load per Unit Length of Culvert in 2D Analysis .................................................. 26
Figure 10—Conceptual Model for 2D Analysis of Pipes ............................................................. 27
Figure 11—Conceptual Model for 2D Analysis of Boxes ............................................................ 28
Figure 12—Conceptual Model for 2D Analysis of Arches ........................................................... 28
Figure 13—Finite Element Meshes of Concrete Box Model ....................................................... 29
Figure 14—Finite Element Meshes of Concrete Pipe Model ...................................................... 30
Figure 15—Finite Element Meshes of Metal Pipe Model ............................................................ 31
Figure 16—Finite Element Meshes of Plastic Pipe Model .......................................................... 32
Figure 17—Finite Element Meshes of Concrete Arch Model ...................................................... 33
Figure 18—Finite Element Meshes of Metal Arch Model ............................................................ 34
Figure 19—Deformation of Concrete Box due to Live Load ....................................................... 36
Figure 20—Bending Moments and Thrusts due to Live Load in Top Slab of Concrete Box Model
(0 ft Cover) .................................................................................................................................. 37
Figure 21—Bending Moments and Thrusts due to Live Load in Right Wall of Concrete Box
Model (0 ft Cover) ....................................................................................................................... 38
Figure 22—Bending Moments and Thrusts due to Live Load in Top Slab of Concrete Box Model
(2 ft Cover) .................................................................................................................................. 39
Figure 23—Bending Moments and Thrusts due to Live Load in Right Wall of Concrete Box
Model (2 ft Cover) ....................................................................................................................... 40
Figure 24—Bending Moments and Thrusts due to Live Load in Top Slab of Concrete Box Model
(6 ft Cover) .................................................................................................................................. 41
Figure 25—Bending Moments and Thrusts due to Live Load in Right Wall of Concrete Box
Model (6 ft Cover) ....................................................................................................................... 42
Figure 26—Deformation of Concrete Pipe due to Live Load ...................................................... 45
Figure 27—Bending Moments and Thrusts due to Live Load in Concrete Pipe Model (2 ft Cover)
.................................................................................................................................................... 46
Figure 28—Bending Moments and Thrusts due to Live Load in Concrete Pipe Model (6 ft Cover)
.................................................................................................................................................... 47
Figure 29—Deformation of Metal Pipe due to Live Load ............................................................ 50
Figure 30—Bending Moments and Thrusts due to Live Load in Metal Pipe Model (2 ft Cover) . 51
Figure 31—Bending Moments and Thrusts due to Live Load in Metal Pipe Model (6 ft Cover) . 52
Figure 32—Deformation of Thermoplastic Pipe due to Live Load .............................................. 55
Figure 33—Bending Moments and Thrusts due to Live Load in Thermoplastic Pipe Model (2 ft
Cover) ......................................................................................................................................... 56
Figure 34—Bending Moments and Thrusts due to Live Load in Thermoplastic Pipe Model (6 ft
Cover) ......................................................................................................................................... 57



NCHRP 15-29 Appendix A                                                                                                                              iv
Figure 35—Deformation of Concrete Arch due to Live Load ...................................................... 60
Figure 36—Bending Moments and Thrusts due to Live Load in Concrete Arch Model (2 ft Cover)
.................................................................................................................................................... 61
Figure 37—Bending Moments and Thrusts due to Live Load in Concrete Arch Model (6 ft Cover)
.................................................................................................................................................... 62
Figure 38—Deformation of Metal Arch due to Live Load ............................................................ 65
Figure 39—Bending Moments and Thrusts due to Live Load in Metal Arch Model (2 ft Cover) . 66
Figure 40—Bending Moments and Thrusts due to Live Load in Metal Arch Model (6 ft Cover) . 67
Figure 41—Plastic Points in Soil Elements of Concrete Pipe Models with 50% and 100%
Interface Strength (Mohr-Coulomb Soil Model, 2 ft Cover) ......................................................... 72
Figure 42—Plastic Points in Soil Elements of Concrete Pipe Models with 50% and 100%
Interface Strength (Mohr-Coulomb Soil Model, 6 ft Cover) ......................................................... 73
Figure 43—Comparison of Bending Moments and Thrusts due to Live Load between Concrete
Pipe Models with 50% and 100% Interface Strength (2 ft Cover) ............................................... 74
Figure 44—Comparison of Bending Moments and Thrusts due to Live Load between Concrete
Pipe Models with 50% and 100% Interface Strength (6 ft Cover) ............................................... 75
Figure 45—Plastic Points in Soil Elements of Thermoplastic Pipe Models with 50% and 100%
Interface Strength (Mohr-Coulomb Soil Model, 2 ft Cover) ......................................................... 77
Figure 46—Plastic Points in Soil Elements of Thermoplastic Pipe Models with 50% and 100%
Interface Strength (Mohr-Coulomb Soil Model, 6 ft Cover) ......................................................... 78
Figure 47—Comparison of Bending Moments and Thrusts due to Live Load between
Thermoplastic Pipe Models with 50% and 100% Interface Strength (2 ft Cover) ....................... 79
Figure 48—Comparison of Bending Moments and Thrusts due to Live Load between
Thermoplastic Pipe Models with 50% and 100% Interface Strength (6 ft Cover) ....................... 80
Figure 49—Finite Element Model of Three-Sided Arch Top Culvert with 3 ft Cover ................... 83
Figure 50—Comparison of Vertical and Horizontal Displacements from Factored Live Load with
1.75 times those from Unfactored Live Load (HDPE Pipe, A2 Backfill) ...................................... 84
Figure 51—Comparison of Thrusts and Moments from Factored Live Load with 1.75 times those
from Unfactored Live Load (HDPE Pipe, A2 Backfill) ................................................................. 84
Figure 52—Comparison of Thrusts and Moments from Factored Live Load with 1.75 times those
from Unfactored Live Load (Hanson Arch) ................................................................................. 85
Figure 53—Test Setup of NCHRP Project 12-45 ........................................................................ 87
Figure 54—Instrumentation for Deformation in Concrete Culvert ............................................... 89
Figure 55—Instrumentation for Deformation in Metal Culvert ..................................................... 89
Figure 56—Cross Sections of HDPE Pipes: Type D and Type S ............................................... 91
Figure 57—Typical Installation of PE Pipe .................................................................................. 91
Figure 58—Live Load Vehicle in the MNDOT Study ................................................................... 92
Figure 59—Typical Test Pipe Instrumentation in the MNDOT Study .......................................... 93
Figure 60—Typical Dimensions of Finite Element Models of Long-Span Arch and HDPE Pipe 94
Figure 61—Finite Element Model of Concrete Arch Culvert with a Cover Depth of 3 ft.............. 95
Figure 62—Live Load Position in the 3D Analysis of Long-Span Arches ................................... 97
Figure 63—Deformed Shapes of Concrete Arch in the Plane of Wheel Loads (Effects of Live
Loads only) ................................................................................................................................. 98
Figure 64—Displacements Due to Live Loads from the 3D Analyses of Concrete Arch Culvert 98
Figure 65—Thrusts and Moments due to Live Loads in the Plane of Wheel Loads from the 3D
Analyses of Concrete Arch Culvert ............................................................................................. 99
Figure 66—Plastic Points in Soil Elements in the Plane of Wheel Loads in Concrete Arch
Analysis..................................................................................................................................... 100
Figure 67—Finite Element Model of Metal Arch Culvert with a Cover Depth of 3 ft ................. 101
Figure 68—Soft Element to Match Longitudinal Stiffness of Metal Arch .................................. 102
Figure 69—Deformed Shapes of Metal Arch in the Plane of Wheel Loads (Effects of Live Loads
only) .......................................................................................................................................... 103



NCHRP 15-29 Appendix A                                                                                                                               v
Figure 70—Displacements due to Live Loads from the 3D Analyses of Metal Arch Culvert .... 104
Figure 71—Thrusts and Moments due to Live Loads in the Plane of Wheel Loads from the 3D
Analyses of Metal Arch Culvert ................................................................................................. 104
Figure 72—Plastic Points in Soil Elements in the Plane of Wheel Loads in Metal Arch Analysis
.................................................................................................................................................. 107
Figure 73—Finite Element Model of HDPE Pipe Culvert for Pipe Run 9 .................................. 108
Figure 74—Soft Element to Match Longitudinal Stiffness of HDPE Pipe ................................. 109
Figure 75—Positions of Live Load Vehicle Axles in the 3D Analyses of HDPE Pipes.............. 111
Figure 76—Deformed Shapes of Pipe Run 1 due to Live Loads in the Plane of Wheel Loads (A-
1, 1.4 ft Cover) .......................................................................................................................... 113
Figure 77—Vertical Crown Displacements of Pipe Run 1 due to Live Loads ........................... 113
Figure 78—Horizontal Displacements of Pipe Run 1 due to Live Loads (A-1, 1.4 ft Cover) ..... 114
Figure 79—Thrusts of Pipe Run 1 due to Live Loads in the Plane of Wheel Loads ................. 114
Figure 80—Moments of Pipe Run 1 due to Live Loads in Plane of Wheel Loads .................... 114
Figure 81—Plastic Points in Soil Elements of Pipe Run 1 in the Plane of Wheel Loads .......... 115
Figure 82—Deformed Shapes of Pipe Run 9 due to Live Loads in the Plane of Wheel Loads (A-
1, 2.5 ft Cover) .......................................................................................................................... 116
Figure 83—Vertical Crown Displacements of Pipe Run 9 due to Live Loads ........................... 116
Figure 84—Horizontal Displacements of Pipe Run 9 Due to Live Loads .................................. 117
Figure 85—Thrusts of Pipe Run 9 Due to Live Loads in the Plane of Wheel Loads ................ 117
Figure 86—Moments of Pipe Run 9 Due to Live Loads in Plane of Wheel Loads .................... 117
Figure 87—Plastic Points in Soil Elements of Pipe Run 9 in the Plane of Wheel Loads .......... 118
Figure 88—Deformed Shapes of Pipe Run 3 due to Live Loads in the Plane of Wheel Loads (A-
2, 1.6 ft Cover) .......................................................................................................................... 119
Figure 89—Vertical Crown Displacements of Pipe Run 3 due to Live Loads ........................... 119
Figure 90—Horizontal Displacements of Pipe Run 3 due to Live Loads (A-2, 1.6 ft Cover) ..... 120
Figure 91—Thrusts of Pipe Run 3 due to Live Loads in the Plane of Wheel Loads (A-2, 1.6 ft
Cover) ....................................................................................................................................... 120
Figure 92—Moments of Pipe Run 3 due to Live Loads in Plane of Wheel Loads .................... 120
Figure 93—Plastic Points in Soil Elements of Pipe Run 3 in the Plane of Wheel Loads .......... 121
Figure 94—Deformed Shapes of Pipe Run 7 due to Live Loads in the Plane of Wheel Loads (A-
2, 2.8 ft Cover) .......................................................................................................................... 122
Figure 95—Vertical Crown Displacements of Pipe Run 7 due to Live Loads (A-2, 2.8 ft Cover)
.................................................................................................................................................. 122
Figure 96—Horizontal Displacements of Pipe Run 7 due to Live Loads (A-2, 2.8 ft Cover) ..... 123
Figure 97—Thrusts of Pipe Run 7 due to Live Loads in the Plane of Wheel Loads (A-2, 2.8 ft
Cover) ....................................................................................................................................... 123
Figure 98—Moments of Pipe Run 7 due to Live Loads in Plane of Wheel Loads (A-2, 2.8 ft
Cover) ....................................................................................................................................... 123
Figure 99—Plastic Points in Soil Elements of Pipe Run 7 in the Plane of Wheel Loads (A-2, 2.8
ft Cover) .................................................................................................................................... 124
Figure 100—Ratios of 3D Analysis Results to Field Test Data for Displacements of Concrete
Arch........................................................................................................................................... 128
Figure 101—Ratios of 3D Analysis Results to Field Test Data for Displacements of Metal Arch
.................................................................................................................................................. 128
Figure 102—Ratios of 3D Analysis Results to Field Test Data for Displacements of HDPE Pipes
.................................................................................................................................................. 129
Figure 15—Comparison of Displacements between Cases with Mohr-Coulomb and Hardening-
Soil Models (Metal Arch, Test 2, 3 ft Cover) ............................................................................. 131
Figure 16—Comparison of Thrusts and Moments under Wheel between Cases with Mohr-
Coulomb and Hardening-Soil Models (Metal Arch, Test 2, 3 ft Cover) ..................................... 132




NCHRP 15-29 Appendix A                                                                                                                             vi
Figure 17—Comparison of Crown Vertical Displacements between Cases with Mohr-Coulomb
and Hardening-Soil Models (HDPE Pipe, A2 Soil, 2.8 ft Cover) ............................................... 133
Figure 18—Comparison of Horizontal Diameter Changes between Cases with Mohr-Coulomb
and Hardening-Soil Models (HDPE Pipe, A2 Soil, 2.8 ft Cover) ............................................... 134
Figure 19—Comparison of Thrusts between Cases with Mohr-Coulomb and Hardening-Soil
Models (HDPE Pipe, A2 Soil, 2.8 ft Cover) ............................................................................... 134
Figure 20—Comparison of Moments between Cases with Mohr-Coulomb and Hardening-Soil
Models (HDPE Pipe, A2 Soil, 2.8 ft Cover) ............................................................................... 134
Figure 21—Cross Section of Finite Element Model for HDPE Pipe in ABAQUS ...................... 138
Figure 22—ABAQUS Metal Arch Model with 3 ft Cover ........................................................... 139
Figure 23—ABAQUS HDPE Pipe Model with 3 ft Cover .......................................................... 139
Figure 24—Comparison of Vertical and Horizontal Displacements between PLAXIS 3D and
ABAQUS Analyses (Metal Arch, Test 2, 3 ft Cover) ................................................................. 140
Figure 25—Comparison of Thrusts and Moments under Wheel between PLAXIS 3D and
ABAQUS Analyses (Metal Arch, Test 2, 3 ft Cover) ................................................................. 140
Figure 26—Vertical and Horizontal Displacements from ABAQUS Analyses with Orthotropic
Properties (Metal Arch, 3 ft Cover) ........................................................................................... 141
Figure 27 –Thrusts and Moments under Wheel from ABAQUS Analyses with Orthotropic
Properties (Metal Arch, 3 ft Cover) ........................................................................................... 141
Figure 28—Vertical and Horizontal Displacements from ABAQUS Analyses with Orthotropic
Properties (HDPE Pipe, A2 Backfill, 2.8 ft Cover) ..................................................................... 142
Figure 29—Vertical and Horizontal Displacements from ABAQUS Analyses with Orthotropic
Properties (HDPE Pipe, A2 Backfill, 1.6 ft Cover) ..................................................................... 143




NCHRP 15-29 Appendix A                                                                                                          vii
1.       INTRODUCTION

NCHRP Project 15-29 was funded to investigate the distribution of live loads through fills and
onto culverts. The project is intended to improve AASHTO Specifications for design of buried
structures and to investigate differences between the AASHTO Standard Specifications for
Highway Bridges, 17th Edition (AASHTO, 2002) and the AASHTO LRFD Bridge Design
Specifications, 3th Edition (AASHTO, 2004).

The scope of project 15-29 is to conduct studies through three-dimensional (3-D) finite element
(FE) modeling of live loads on buried culverts and to develop new AASHTO Specifications
based on the findings. Results and proposed design methods will be evaluated against test
data available in the literature, but no new tests will be conducted as a part of this project.

This report presents an investigation of the available constitutive models for soils that could be
used in the 3-D analyses. Information presented includes initial review of available soil models
and important model features, preliminary two-dimensional FE studies of live load distribution,
and full 3-D FE studies.

2.       REVIEW OF AVAILABLE SOIL MODELS

Numerous soil constitutive models have been developed to date and are available for finite
element analysis. Lade (2005) prepared a summary of widely available soil constitutive models.
Each model has different capabilities and requires different experimental data for calibration.

Predicting the response of buried structures to surface live loads in a finite element analysis
requires a soil constitutive model that accurately captures culvert-soil interaction. Research has
been conducted with linear-elastic soil models (for example, Moore and Brachman (1994) and
Fernando and Carter (1998)); with nonlinear models including nonlinear elastic models,
perfectly plastic models, and plastic models with hardening (for example, Pang (1999)). For
typical culvert analysis, which has been historically conducted in 2-D, stress-dependent stiffness
and shear failure have been found to be important characteristics of suitable soil models. The
Duncan-Selig hyperbolic model (Duncan et al., 1980; Selig, 1988) has such features, and has
been implemented in the finite element programs CANDE (Musser, 1989) and SPIDA (Heger et
al., 1985) to analyze soil-structure interaction problems for culverts. Soil properties based on
these models have been used in the development of current AASHTO specifications for
reinforced concrete and thermoplastic pipe. The Duncan-Selig model, consists of the hyperbolic
Young's modulus model developed by Duncan (1980), and the hyperbolic bulk modulus




NCHRP 15-29 Appendix A                                                                            1
developed by Selig (1988). As discussed below, the soil properties used with this model were
developed by Selig (1988).

CANDE was developed by the FHWA, and has been widely used to design culverts, but
operates only in two dimensions.         For ease of computation and to allow comparison with
CANDE we conducted preliminary analyses in 2-D and then extended these models to 3-D for a
complete investigation of actual live load distribution.

3-D modeling is computationally intensive.        Because of this, it is important to select the
computationally simplest soil model that can accurately capture culvert/soil interaction as
resulting from live load.     We selected three levels of soil model with varying levels of
sophistication for use on the project:

•         linear-elastic (representing the simplest possible model),
•         Mohr-Coulomb (linear elastic model with post-failure plasticity), and
•         Plaxis 3D hardening-soil (stress dependence plus plasticity, similar to Duncan-Selig).

Features of the linear-elastic model, Mohr-Coulomb model, Hardening-Soil model, and Duncan-
Selig model are briefly discussed below. Compressive stresses are positive throughout this
report.

2.1       Linear Elastic

Modeling soil as linear elastic provides the most basic soil behavior, with no consideration of
non linear stress-strain behavior or plasticity at failure. Linear elastic soil behavior is described
by isotropic linear elasticity. Four elastic constants are used in analysis, but given any two of
the four, the other two can be calculated. The four parameters are: modulus of elasticity, E ,
Poisson’s ratio, ν , bulk modulus, B , and shear modulus, G .           In actual soil these elastic
constants vary with soil stress level, and some analysts use elastic properties that vary with
depth. One set of such properties, proposed by Selig (1990) are shown in Table 1. Selig
estimated Young's modulus of elasticity from the hyperbolic model for increasing values of
maximum principal stress, (σ 1 , typically vertical stress), with the minimum principal stress, (σ 3 ,
typically horizontal stress), equal to one-half to one- times the maximum principal stress. Elastic
constants can be selected by evaluating the soil vertical stress level, usually calculated as the
depth of fill times the soil density, ignoring the presence of a culvert. Procedures to select
elastic constants are described in detail in the following section on the Mohr-Coulomb model.
McGrath (1998) found that the proposed soil properties produced soil stiffnesses higher than



NCHRP 15-29 Appendix A                                                                              2
back-calculated from actual projects by Howard (1977), and thus concluded that the properties
are likely achievable but not suitable for routine design where backfill sources may be
undependable and soil gradations variable.

                    Table 1—Elastic soil properties for Backfill (Selig, 1990)
                                         Gravelly Sand (SW)
    Maximum                95% Standard Compaction                 85% Standard Compaction
 Principal Stress        E             B              ν          E             B           ν
    Level (psi)        (psi)         (psi)                     (psi)         (psi)
      0 to 1          1,600         2,800           0.40      1,300          900         0.26
      1 to 5          4,100         3,300           0.29      2,100         1,200        0.21
     5 to 10          6,000         3,900           0.24      2,600         1,400        0.19
     10 to 20         8,600         5,300           0.23      3,300         1,800        0.19
     20 to 40         13,000        8,700           0.25      4,100         2,500        0.23
     40 to 60         16,000       13,000           0.29      4,700         3,500        0.28
                                           Sandy Silt (ML)
    Maximum                95% Standard Compaction                 85% Standard Compaction
 Principal Stress        E             B              ν          E             B           ν
    Level (psi)        (psi)         (psi)                     (psi)         (psi)
      0 to 1          1,800         1,900           0.34       600           400         0.25
      1 to 5          2,500         2,000           0.29       700           450         0.24
     5 to 10          2,900         2,100           0.27       800           500         0.23
     10 to 20         3,200         2,500           0.29       850           700         0.30
     20 to 40         3,700         3,400           0.32       900          1,200        0.38
     40 to 60         4,100         4,500           0.35      1,000         1,800        0.41
                                           Silty Clay (CL)
    Maximum                95% Standard Compaction                85% Standard Compaction
 Principal Stress        E             B              ν         E             B           ν
    Level (psi)        (psi)         (psi)                    (psi)         (psi)
      0 to 1           400           800            0.42      100           100         0.33
      1 to 5           800           900            0.35      250           200         0.29
     5 to 10          1,100         1,000           0.32      400           300         0.28
     10 to 20         1,300         1,100           0.30      600           400         0.25
     20 to 40         1,400         1,600           0.35      700           800         0.35
        60            1,500         2,100           0.38      800          1,300        0.40



2.2      Elasto-Plastic

The Mohr-Coulomb failure criterion is used in many geotechnical engineering applications to
describe the shear strength of soil. The principal feature of the Mohr-Coulomb criterion is that
strength is dependent on confining stress. In common applications soil strength is described by
a friction angle and cohesion intercept. In this study, we report on the 3-D Mohr-Coulomb
model as implemented in Plaxis. This model uses an elastic perfectly-plastic constitutive model
(Brinkgreve and Broere, 2004). For stress states within the yield surface, the soil behavior is




NCHRP 15-29 Appendix A                                                                          3
elastic and is determined by isotropic linear elasticity, as described in Section 2.1. The Mohr-
Coulomb yield condition consists of six yield functions as shown below:


                                  σ2 −σ3        σ2 +σ3
                         f 1a =             +      sin φ − c cos φ ≤ 0
                                  2            2
                               σ −σ2        σ +σ2
                         f1b = 3           + 3     sin φ − c cos φ ≤ 0
                                  2            2
                               σ − σ1       σ + σ1
                         f 2a = 3          + 3     sin φ − c cos φ ≤ 0
                                  2            2
                               σ −σ3        σ +σ3                                        (1)
                         f 2b = 1          + 1     sin φ − c cos φ ≤ 0
                                  2            2
                               σ −σ2        σ +σ2
                         f 3a = 1          + 1     sin φ − c cos φ ≤ 0
                                  2            2
                               σ − σ1       σ + σ1
                         f 3b = 2          + 2     sin φ − c cos φ ≤ 0
                                  2            2

For these six yield functions, a non-associated flow rule is used, and six plastic potential
functions are introduced:


                                          σ2 −σ3        σ2 +σ3
                                  g1a =             +       sinψ
                                            2           2
                                         σ −σ2       σ +σ2
                                  g1b = 3           + 3     sinψ
                                            2           2
                                         σ − σ1      σ + σ1
                                  g 2a = 3          + 3     sinψ
                                            2           2
                                         σ1 − σ 3    σ +σ3                               (2)
                                  g 2b =            + 1     sinψ
                                            2           2
                                         σ1 − σ 2    σ +σ2
                                  g 3a =            + 1     sinψ
                                            2           2
                                         σ − σ1      σ + σ1
                                  g 3b = 2          + 2     sinψ
                                            2           2

where ψ is a dilatancy angle.


Tensile failure of soil is captured by specifying a tension cut-off. Three yield functions are
defined for the tension cut-off:




NCHRP 15-29 Appendix A                                                                         4
                                       f4 = σ1 − σ t ≤ 0
                                       f5 = σ 2 − σ t ≤ 0                                     (3)
                                       f6 = σ 3 − σ t ≤ 0

where σ t is allowable tensile stress. For these three yield functions for tension cut-off, an

associated flow rule is used.

Basic input parameters required for the Mohr-Coulomb model are modulus of elasticity, E ,
Poisson’s ratio, ν , cohesion, c , angle of friction, φ , dilatancy angle, ψ , and tensile strength,

σt .

In this study, c for the Mohr-Coulomb model is the same value as in the Duncan-Selig model,
which is given in Table 4. Although cohesion of SW85 is 0 psi for the Duncan-Selig model,
0.001 psi is assigned to cohesion in Plaxis for numerical stability. Elastic constants, E and ν ,
are selected from Table 1 based on the vertical stress at a given depth.            Table 2 shows
expected vertical and horizontal stresses in the soil at a given depth under gravity for SW85.
Table 2 also shows corresponding angle of friction at a given depth. In the Duncan-Selig model,
the angle of friction is also a function of confinement stress. To determine the angle of friction at
a certain depth, the horizontal stress was estimated by the empirical formula σ 3 = (1 − sin φ )σ 1

(Jaky, 1944), using the vertical stress as the first principal stress.

In this study, four sets of elastic constants for SW85 are identified for ranges from 0 ft to 1 ft,
from 1 ft to 6 ft, from 6 ft to 11 ft, and from 11 ft to 18 ft, which correspond to the stress ranges
in Table 1, and are shown in Table 3. Angle of friction in Table 3 is an average value of angle of
friction for each range of depth. Table 3 also gives dilatation angles that were estimated by
subtracting 30 deg from friction angles.




NCHRP 15-29 Appendix A                                                                              5
        Table 2—Vertical Stresses and Estimated Horizontal Stresses under Gravity and
                          Corresponding Angle of Friction for SW85
                      Depth          σ1             σ3             σ 3 /σ1          φ
                        (ft)         (psi)          (psi)           (psi)       (deg)
                       0.5           0.44           0.14           0.331        42.01
                       1.5           1.31           0.45           0.344        41.03
                       2.5           2.19           0.76           0.350        40.57
                       3.5           3.06           1.08           0.354        40.27
                       4.5           3.94           1.40           0.357        40.04
                       5.5           4.81           1.73           0.359        39.86
                       6.5           5.69           2.05           0.361        39.71
                       7.5           6.56           2.38           0.363        39.58
                       8.5           7.44           2.71           0.364        39.47
                       9.5           8.31           3.04           0.366        39.37
                       10.5          9.19           3.37           0.367        39.28
                       11.5         10.06           3.70           0.368        39.20
                       12.5         10.94           4.04           0.369        39.12
                       13.5         11.81           4.37           0.370        39.05
                       14.5         12.69           4.70           0.371        38.99
                       15.5         13.56           5.04           0.372        38.93
                       16.5         14.44           5.38           0.372        38.87
                       17.5         15.31           5.71           0.373        38.82



         Table 3—Parameters for Linear-Elastic and Mohr-Coulomb Models for SW85
                      Modulus of       Poisson’s            Angle of         Dilatation   Cohision
           Depth       Elasticity        Ratio              Friction          Angle
                          E                  ν                 φ                ψ            c
            (ft)         (psi)                               (deg)             (deg)       (psi)
           0 to 1       1,300                0.26            42.0              12.0        0.001
           1 to 6       2,100                0.21            40.4              10.4        0.001
          6 to 11       2,600                0.19            39.5               9.5        0.001
          11 to 18      3,300                0.19            39.0               9.0        0.001




2.3        Stress-Dependent Models

2.3.1      Duncan-Selig Model

The Duncan-Selig model is a composite of the Duncan hyperbolic Young's modulus (Duncan et
al, 1980) and the Selig hyperbolic bulk modulus (Selig, 1988). These models were developed in
2D to specifically address aspects of soil behavior that are important in culvert design.            Under


NCHRP 15-29 Appendix A                                                                                   6
this model Selig developed parameters for the Duncan Young's modulus model (Selig, 1988)
and two sets of parameters for the bulk modulus (Selig, 1988 and Selig, 1990). The 1988
properties for Young's and bulk modulus were used by AASHTO in the development of standard
designs for reinforced concrete pipe and later for the development of the one-dimensional
modulus values adopted by AASHTO for thermoplastic pipe design. The bulk modulus values
proposed by Selig in 1990 are higher and produce an overall soil stiffness about twice as stiff as
the 1988 values (McGrath, 1998).

The Duncan-Selig model provides non-linear behavior and includes the Mohr-Coulomb failure
criterion; however, the formulation is elastic and includes no plasticity. The soil stiffening or
softening is based largely on the confining stress, σ 3 , and the ratio of the deviator stress

relative to the ultimate stress.

The Duncan-Selig stress-strain relationship in the triaxial test during deviatoric loading can be
represented by a hyperbolic equation of the form


                                               ε1
                                        q=
                                             1 ε1
                                               +                                           (4)
                                             Ei qu

where

           q    : deviator stress (= σ 1 − σ 3 , σ 1 =maximum principal stress, and σ 3 =minimum
                  principal stress)
           ε1   : maximum principal strain

           Ei   : initial tangent modulus

          qu    : ultimate deviator stress at large strain

The initial tangent modulus, Ei , is assumed to increase with confining pressure as given by


                                                        n
                                                σ 
                                      E i = KPa  3 
                                                P                                        (5)
                                                 a

where

           Pa   : atmospheric pressure (=14.7 psi, used to non-dimensionalize the parameters
                  K and n )



NCHRP 15-29 Appendix A                                                                           7
           K    : non-dimensional parameter
           n    : non-dimensional parameter
The hyperbolic soil model is considered to be valid up to soil failure. Thus, the ultimate deviator
stress is defined in terms of the actual failure deviator stress by the failure ratio,


                                                     qf
                                              Rf =                                             (6)
                                                     qu

The failure envelope is expressed by


                                            2c cos φ + 2σ 3 sin φ
                                     qf =                                                      (7)
                                                 1 − sin φ

where

          qf    : deviator stress at failure

           c    : cohesion
           φ    : angle of friction
In this model, the angle of friction is a function of the confining stress and expressed as


                                                          σ3 
                                      φ = φ o − ∆φ log 10 
                                                              
                                                                                              (8)
                                                           Pa 

where

          φo    : value of φ when σ 3 = Pa

          ∆φ : reduction in φ for a ten-fold increase in σ 3

By differentiating Eq. 4 with respect to ε 1 , the tangent modulus, E , can be expressed as:


                              ∂q
                         E=
                              ∂ε 1
                                    R f (1 − sin φ )q 
                                                              2      n                         (9)
                                                            σ3 
                             = 1 −                     KPa  
                                                             P 
                                2C cos φ + 2σ 3 sin φ       a

The mean stress, σ m , can reasonably be represented by the hyperbolic equation:




NCHRP 15-29 Appendix A                                                                               8
                                                  Bi ε v
                                       σm =                                                          (10)
                                                1− εv /εu

where

           Bi   : initial tangent bulk modulus

           εv   : volumetric strain

           εu   : ultimate volumetric strain

Therefore, the tangent bulk modulus, B , is determined by


                                             ∂σ m
                                      B=
                                             ∂ε vol
                                                              2                                      (11)
                                                    σ 
                                            = Bi 1 + m 
                                                  Bi ε u 

Based on the theory of elasticity, Poisson’s ratio, ν , and shear modulus, G , can be expressed

by using E and B .

The Selig, 1988 parameters for the Duncan-Selig model for backfill are summarized in Table 4.

           Table 4—Soil Properties of Backfill for Duncan-Selig Model (Selig, 1988)
 Soil
 Type
            Standard
           Compaction
                        Density       K          n      Rf        Bi / Pa   εu      c       φ0     ∆φ
                (%)       (pcf)                                                    (psi)   (deg)   (deg)
                 95        141        950       0.60   0.70        74.8     0.02     0      48       8
Gravelly         90        134        640       0.43   0.75        40.8     0.05     0      42       4
 Sand            85        126        450       0.35   0.80        12.7     0.08     0      38       2
 (SW)            80        119        320       0.35   0.83        6.1      0.11     0      36       1
                 60        91          54       0.85   0.90        1.7      0.23     0      29       0
                 95        127        440       0.40   0.95        48.3     0.06    4.0     34       0
Sandy            90        120        200       0.26   0.89        18.4     0.10    3.5     32       0
  Silt           85        114        110       0.25   0.85        9.5      0.14    3.0     30       0
 (ML)            80        107         75       0.25   0.80        5.1      0.19    2.5     28       0
                 60        66          16       0.95   0.55        1.3      0.43     0      23       0
                 95        119        120       0.45   1.00        21.1     0.13    9.0     15       4
 Silty           90        112         75       0.54   0.94        10.2     0.17    7.0     17       7
 Clay            85        106         50       0.60   0.90        5.2      0.21    6.0     18       8
 (CL)            80        100         35       0.66   0.87        3.5      0.25    5.0     19      8.5
                 60        56          16       0.95   0.75        0.7      0.55     0      23      11




NCHRP 15-29 Appendix A                                                                                      9
2.3.2    Hardening Soil Model (Plaxis)

Two types of hardening can be modeled by the Hardening-Soil model (Brinkgreve and Broere,
2004): shear hardening due to primary deviatoric loading and compression hardening due to
primary compression.       A basic feature of the Hardening-Soil model in Plaxis is the stress
dependency of soil stiffness and the hyperbolic relationship between the vertical strain and the
deviatoric stress in primary triaxial loading. This model uses a yield function given below:


                                     f = f −γ    ≤0   p


                                          1      q       2q
                                     f =               −                                       (12)
                                         E50 1 − q / qu Eur
                                     γ p = 2ε 1p − ε vp

where


          γ   p
                  : plastic shear strain as a hardening parameter

         E50 : confining stress dependent secant modulus at 50% strength for primary loading

         Eur : confining stress dependent unloading/reloading modulus

          ε 1p : plastic strain in the 1-principal direction
          ε vp : plastic volumetric strain
E50 and Eur are dependent on confinement and evaluated by the following power lows:


                                                                            m
                                                     c cot φ + σ 3     
                                    E50 = E   ref
                                              50    
                                                     c cot φ + p ref   
                                                                                              (13)
                                                                       

and


                                                                            m
                                                     c cot φ + σ 3     
                                    Eur = E   ref
                                              ur    
                                                     c cot φ + p ref   
                                                                                              (14)
                                                                       

where


         p ref : reference confining pressure
          ref
         E50 : reference modulus for primary loading corresponding to the reference confining
                   pressure p ref



NCHRP 15-29 Appendix A                                                                            10
          ref
         Eur : reference modulus for unloading and reloading corresponding to the reference
                  confining pressure p ref

When the stress state is on the yield surface, f = 0 , and


                                                       1      q       2q
                           γ p = 2ε 1p − ε vp =                     −                           (15)
                                                      E50 1 − q / qu Eur

For hard soils, plastic volumetric strain tends to be small; therefore, ε 1p can be approximated by


                                                1       q       q
                                     ε 1p ≈                   −                                 (16)
                                              2 E50 1 − q / qu Eur

In the triaxial test stress path, Eur remains constant since confinement stress is constant, and

elastic strains during the deviatoric loading are evaluated by using Eur and ν ur as follows:


                                       q                         q
                            ε 1e =        and ε 2 = ε 3 = −ν ur
                                                e     e
                                                                                                (17)
                                      Eur                       Eur

When the plastic volumetric strain is small, the axial strain in the deviatoric loading of the triaxial
test can be expressed by a hyperbolic stress-strain curve as follows:


                                                         1       q
                                ε 1 = ε 1e + ε 1p ≈                                             (18)
                                                       2 E50 1 − q / qu

The relationship between the plastic shear strain rate and the plastic volumetric strain rate is
specified in the linear form:


                                              ε vp = sinψ m γ p
                                                                                               (19)

where ψ m is mobilized dilatancy angle. In the Hardening-Soil model, the following expression

is used for sinψ m .


                                                   sin φ m − sin φ cv
                                     sinψ m =                                                   (20)
                                                  1 − sin φ m sin φ cv




NCHRP 15-29 Appendix A                                                                              11
where φ cv and φ m are the critical state friction angle and the mobilized friction angle. sin φ m

and sin φ cv are evaluated by the following equations:


                                                          σ1 − σ 3
                                sin φ m =                                                    (21)
                                                    σ 1 + σ 3 + 2c cot φ

and


                                                         sin φ − sinψ
                                     sin φ cv =                                              (22)
                                                        1 − sin φ sinψ

When the failure criterion is satisfied ( q = q f ), the yield surface stops increasing in size, and

perfectly plastic yielding occurs.

The Hardening-Soil model used a cap type yield surface to account for plastic volumetric strain
due to primary compression in isotropic compression or oedometer loading. The cap yield
surface is defined by


                                                ~
                                                q2
                                      fc =              + p2 − p2 ≤ 0                        (23)
                                                α   2           p



where α                                                       nc
            is an auxiliary model parameter that relates to K 0 (= K 0 -value for normal
                                                                                       ~
consolidation), p is a mean stress, and p p is the isotropic pre-consolidation stress. q is a

special stress measure for deviatoric stresses and has the following expression:


                                     ~
                                     q = σ 1 + (δ − 1)σ 2 − δσ 3
                                            3 + sin φ
                                 δ=                                                          (24)
                                            3 − sin φ

The Hardening-Soil model used the following hardening law relating p p to volumetric cap strain

ε vpc (=plastic volumetric strain in isotropic compression):

                                                                      1− m
                                                        β  pp 
                                      ε    pc
                                                =                
                                                    1 − m  p ref 
                                          v                                                  (25)
                                                                 




NCHRP 15-29 Appendix A                                                                           12
where β is another model parameter that relates to E oed (= tangent modulus for primary
                                                     ref



oedometer loading at a vertical stress of σ 1 = p ref ).                       α and β are calculated internally in
                  nc      ref
Plaxis based on K 0 and E oed , respectively.


The tangent oedometer modulus is also defined by a power low:


                                                                               m
                                                        c cot φ + σ 1     
                                     E oed = E   ref
                                                 oed   
                                                        c cot φ + p ref   
                                                                                                            (26)
                                                                          

Tensile failure of soil is captured by specifying a tension cut-off as described for the Mohr-
Coulomb model.


Basic input parameters for the Hardening-Soil model are c , φ , ψ , E50 , E oed , m , Eur , ν ur ,
                                                                     ref    ref        ref



p ref , K 0nc , R f , and σ t . Some of the parameters are assigned to the default values as shown
below:

           ref    ref
          Eur : 3E50

          ν ur : 0.2
             p ref : 100 stress units
          K 0nc : 1 − sin φ

             Rf   : 0.9

             σt   : 0 stress units

To compare the Hardening-Soil model with the Duncan-Selig model, we conducted a set of
analyses in Plaxis 3D to investigate the actual 3D condition.                            The analyses consisted of
simulation of triaxial and oedometer tests for SW85, SW90, ML85, and CL85 soils. For the
triaxial test, we considered confining pressure of 1 psi, 2 psi, and 5 psi. The analysis model in
Plaxis 3D consists of a cube of soil (1 in. by 1 in. by 1 in.).

To determine input parameters of the Hardening-Soil model, we created general rules. Our goal
was to match the soil behavior between the two models at a confining pressure of 2 psi. The
rules are:


•        Use atmospheric pressure, Pa , as the reference pressure, p ref




NCHRP 15-29 Appendix A                                                                                          13
•                                              ref
         Use a value equal to 0.5 Ei for E50 to match the initial tangent modulus in the
         deviatoric loading of the triaxial test, instead of using the secant modulus of the
         Duncan-Selig model at a deviator stress of 50% of the failure stress
•                              ref      ref
         Use a value equal to E50 for E oed , instead of using the tangent modulus of the
         Duncan-Selig model in the oedometer loading at a vertical stress of p ref

•        Use the same friction angle as the Duncan-Selig model at a confining pressure of 2 psi
•        Use n for m
•        Use R f of the Duncan-Selig model, instead of using a default value

•        Use c of the Duncan-Selig model, but use 0.001 psi when it is 0 psi in the Duncan-
         Selig model
•        Use φ − 30 deg as ψ

•        Use default values for other parameters if possible

Table 5 shows input parameters determined by the general rules described above. For SW90,
Plaxis 3D did not allow us to use E oed = E50 and K 0 = 1 − sin φ .
                                    ref    ref      nc
                                                                              In this case only, we

increased K 0 from 0.287 (= 1 − sin φ ) to 0.310 to use E oed = E50 .
            nc                                            ref    ref




Figure 1 through Figure 4 show relationships between deviator stress and vertical strain during
the deviatoric loading of the triaxial test predicted by Plaxis 3D with the Hardening-Soil model
for SW85, SW90, ML85, and CL85. These figures also show the Duncan-Selig hyperbolic
model predictions by Eq. 4 for comparison. Note that failure stress of the Hardening-Soil model
is different from that of the Duncan-Selig for some cases because the angle of friction is not
dependent on confining pressure. For SW85 and SW90, results from the Plaxis Hardening-Soil
model closely match the Duncan-Selig hyperbolic soil model. For ML85, although stiffness
predicted by the Plaxis Hardening-Soil model is slightly lower, results match the Duncan-Selig
model relatively well. However, for CL85, the stiffness predicted by the Plaxis Hardening-Soil
model is significantly lower than that of Duncan-Selig. As explained above, ε vp is not small for

the soft soil, which leads to a larger vertical strain for the same stress and causes the stress-
strain relationship to deviate from the hyperbolic equation give in Eq. 18.

Figure 5 through Figure 8 show vertical stress and vertical strain relationships during the
oedometer loading predicted by Plaxis 3D with the Hardening-Soil model for SW85, SW90,
ML85, and CL85. These figures also show the stress-strain relationship of the Duncan-Selig
model as well as actual test data by Lin (1987).




NCHRP 15-29 Appendix A                                                                          14
The Duncan-Selig model always predicts a larger strain in the vertical stress range we
examined relative to actual test data, and SW85 is the poorest match among the four soil types
we examined. Lin (1987) also pointed out that the Duncan-Selig model for SW85 predicts the
oedometer stress-strain relationship much different from the actual test data.        The Plaxis
Hardening-Soil model predicts much lower strain when compared to the Duncan-Selig model
and the actual test data at the same stress level.

Strains predicted by the Plaxis Hardening-Soil model for ML85 lie between the Duncan-Selig
model and the actual test data. Strains predicted by the Plaxis Hardening-Soil model for CL85
are in good agreement with those of the Duncan-Selig model up to a vertical stress of 5 psi, and
they become significantly larger than those of either the Duncan-Selig model or the actual test
data.

 Table 5—Input Parameters for Hardening-Soil Model for SW85, SW90, ML85, and CL85
          Input Parameters          SW85             SW90       ML85          CL85
                c (psi)             0.001            0.001        3             6
               φ     (deg)          39.7              45.5       30            24.9
               ψ     (deg)           9.7              15.5        0             0
                   ref
               E   50     (psi)     3,308            4,704       633           161
                   ref
               E   oed    (psi)     3,308            4,704       633           161
                     m              0.35              0.75      0.25           0.6
                   ref
               E   ur     (psi)     9,924            14,112     1,900          482
                    ν ur             0.2              0.2        0.2           0.2
                    Rf               0.8              0.75      0.85           0.9
                    K 0nc           0.361            0.310      0.500         0.578
               σt        (psi)        0                0          0             0
                   ref
               p          (psi)     14.7              14.7      14.7           14.7




NCHRP 15-29 Appendix A                                                                       15
                                                    4.5

                                                    4.0

                                                    3.5




                            Deviator Stress (psi)
                                                    3.0

                                                    2.5
      (a) σ3 = 1 psi
                                                    2.0
                                                                                                     Duncan-Selig
                                                    1.5
                                                                                                     Failure in Duncan-Selig
                                                    1.0
                                                                                                     Plaxis Result
                                                    0.5
                                                                                                     Failure in Plaxis
                                                    0.0
                                                      0.000       0.002          0.004           0.006           0.008         0.010
                                                                               Vertical Strain (in/in)
                                                    9.0

                                                    8.0

                                                    7.0
                            Deviator Stress (psi)




                                                    6.0

                                                    5.0
      (b) σ3 = 2 psi
                                                    4.0
                                                                                                     Duncan-Selig
                                                    3.0
                                                                                                     Failure in Duncan-Selig
                                                    2.0
                                                                                                     Plaxis Result
                                                    1.0
                                                                                                     Failure in Plaxis
                                                    0.0
                                                      0.000           0.005              0.010             0.015               0.020
                                                                               Vertical Strain (in/in)
                                                    20

                                                    18

                                                    16
                            Deviator Stress (psi)




                                                    14

                                                    12

                                                    10
      (c) σ3 = 5 psi
                                                     8
                                                                                                     Duncan-Selig
                                                     6
                                                                                                     Failure in Duncan-Selig
                                                     4                                               Plaxis Result
                                                     2                                               Failure in Plaxis
                                                     0
                                                     0.000    0.005    0.010      0.015      0.020       0.025       0.030     0.035
                                                                              Vertical Strain (in/in)


Figure 1—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for SW85
                       in Deviatoric Loading of Triaxial Test




NCHRP 15-29 Appendix A                                                                                                                 16
                                                    7.0


                                                    6.0


                                                    5.0




                            Deviator Stress (psi)
                                                    4.0
      (a) σ3 = 1 psi
                                                    3.0
                                                                                                      Duncan-Selig
                                                    2.0                                               Failure in Duncan-Selig
                                                                                                      Plaxis Result
                                                    1.0
                                                                                                      Failure in Plaxis
                                                    0.0
                                                      0.000       0.002            0.004        0.006            0.008          0.010
                                                                                 Vertical Strain (in/in)
                                                    12


                                                    10
                            Deviator Stress (psi)




                                                     8


      (b) σ3 = 2 psi                                 6

                                                                                                      Duncan-Selig
                                                     4
                                                                                                      Failure in Duncan-Selig

                                                     2                                                Plaxis Result
                                                                                                      Failure in Plaxis
                                                     0
                                                     0.000    0.002      0.004      0.006     0.008      0.010        0.012     0.014
                                                                             Vertical Strain (in/in)
                                                    30


                                                    25
                            Deviator Stress (psi)




                                                    20


                                                    15
      (c) σ3 = 5 psi
                                                                                                      Duncan-Selig
                                                    10
                                                                                                      Failure in Duncan-Selig

                                                     5                                                Plaxis Result
                                                                                                      Failure in Plaxis
                                                     0
                                                     0.000       0.005             0.010        0.015            0.020          0.025
                                                                             Vertical Strain (in/in)


Figure 2—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for SW90
                       in Deviatoric Loading of Triaxial Test




NCHRP 15-29 Appendix A                                                                                                                  17
                                                    14


                                                    12


                                                    10




                            Deviator Stress (psi)
                                                     8
      (a) σ3 = 1 psi
                                                     6
                                                                                                       Duncan-Selig
                                                     4                                                 Failure in Duncan-Selig
                                                                                                       Plaxis Result
                                                     2
                                                                                                       Failure in Plaxis
                                                     0
                                                      0.00    0.02     0.04       0.06          0.08       0.10        0.12      0.14
                                                                              Vertical Strain (in/in)
                                                    18

                                                    16

                                                    14
                            Deviator Stress (psi)




                                                    12

                                                    10
      (b) σ3 = 2 psi
                                                     8
                                                                                                       Duncan-Selig
                                                     6
                                                                                                       Failure in Duncan-Selig
                                                     4
                                                                                                       Plaxis Result
                                                     2
                                                                                                       Failure in Plaxis
                                                     0
                                                      0.00   0.02    0.04      0.06      0.08      0.10       0.12      0.14     0.16
                                                                              Vertical Strain (in/in)
                                                    25



                                                    20
                            Deviator Stress (psi)




                                                    15

      (c) σ3 = 5 psi
                                                    10
                                                                                                       Duncan-Selig
                                                                                                       Failure in Duncan-Selig
                                                     5                                                 Plaxis Result
                                                                                                       Failure in Plaxis
                                                     0
                                                      0.00           0.05                0.10                 0.15               0.20
                                                                              Vertical Strain (in/in)


Figure 3—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for ML85
                       in Deviatoric Loading of Triaxial Test




NCHRP 15-29 Appendix A                                                                                                                  18
                                                    25



                                                    20




                            Deviator Stress (psi)
                                                    15

      (a) σ3 = 1 psi
                                                    10
                                                                                                      Duncan-Selig
                                                                                                      Failure in Duncan-Selig
                                                     5                                                Plaxis Result
                                                                                                      Failure in Plaxis
                                                     0
                                                      0.00          0.50                1.00                 1.50               2.00
                                                                             Vertical Strain (in/in)
                                                    25



                                                    20
                            Deviator Stress (psi)




                                                    15
      (b) σ3 = 2 psi
                                                    10
                                                                                                      Duncan-Selig
                                                                                                      Failure in Duncan-Selig
                                                     5                                                Plaxis Result
                                                                                                      Failure in Plaxis
                                                     0
                                                      0.00   0.20     0.40       0.60          0.80       1.00        1.20      1.40
                                                                             Vertical Strain (in/in)
                                                    30


                                                    25
                            Deviator Stress (psi)




                                                    20


                                                    15
      (c) σ3 = 5 psi
                                                                                                      Duncan-Selig
                                                    10
                                                                                                      Failure in Duncan-Selig

                                                     5                                                Plaxis Result
                                                                                                      Failure in Plaxis
                                                     0
                                                      0.00   0.20     0.40       0.60          0.80       1.00        1.20      1.40
                                                                             Vertical Strain (in/in)


Figure 4—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for CL85
                       in Deviatoric Loading of Triaxial Test




NCHRP 15-29 Appendix A                                                                                                                 19
                                      50



              Vertical Stress (psi)   40
                                                                                        Duncan-Selig
                                                                                        Plaxis Result
                                      30                                                Actual Test Data



                                      20



                                      10



                                       0
                                       0.000   0.005      0.010    0.015   0.020     0.025      0.030      0.035
                                                              Vertical Strain (in/in)


Figure 5—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for SW85
                               in Oedometer Loading




                                      50



                                      40
              Vertical Stress (psi)




                                      30



                                      20

                                                                               Duncan-Selig
                                      10                                       Plaxis Result
                                                                               Actual Test Data (SW85)
                                                                               Actual Test Data (SW95)
                                       0
                                       0.000      0.005           0.010      0.015           0.020         0.025
                                                              Vertical Strain (in/in)


Figure 6—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for SW90
                               in Oedometer Loading




NCHRP 15-29 Appendix A                                                                                             20
                                      50



              Vertical Stress (psi)   40



                                      30



                                      20


                                                                                Duncan-Selig
                                      10
                                                                                Plaxis Result
                                                                                Actual Test Data

                                       0
                                        0.00   0.02     0.04          0.06          0.08           0.10
                                                      Vertical Strain (in/in)


Figure 7—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for ML85
                              in Oedometer Loading




                                      50



                                      40
              Vertical Stress (psi)




                                      30



                                      20


                                                                                Duncan-Selig
                                      10
                                                                                Plaxis Result
                                                                                Actual Test Data

                                       0
                                        0.00   0.05     0.10          0.15          0.20           0.25
                                                      Vertical Strain (in/in)


Figure 8—Comparison of Plaxis Hardening-Soil Model and Duncan-Selig Model for CL85
                              in Oedometer Loading




NCHRP 15-29 Appendix A                                                                                    21
2.4      Findings from Soil Model Evaluation

The soil model comparison shows that the Young's modulus values of the Duncan-Selig and
Hardening Soil models are similar for SW85, SW90, ML85, and CL85 soils, especially SW soils.
The predicted one-dimensional stress-strain curves do not compare as well. We conclude that
the differences are a result of the parameters and not fundamental faults with the model. As
noted above, these differences have been observed previously. In general, having a lower
bound for the stiffness is an appropriate design decision, but may not be the best approach for
establishing design equations.    It is possible to modify input parameters of the Hardening-Soil
model to better match the soil behavior in the triaxial and oedometer tests for each soil type and
a given compaction level. However, we used the established design parameters given in Table
5 for the preliminary 2D analysis presented in the next section. Final design parametric study
may require modified parameters if we choose the Hardening-Soil model for the parametric
study.




3.       TWO-DIMENSIONAL MODELING OF CULVERTS

Soil models were initially tested in 2D to reduce computational time and allow a broader range
of structure types to be evaluated. We performed a set of preliminary 2D analyses in Plaxis 2D
Version 8 (Brinkgreve, 2002) on six structural types including concrete box, concrete pipe,
concrete arch, metal pipe, metal arch, and thermoplastic pipe for selected cover depths. By
using linear-elastic, Mohr-Coulomb (linear-elastic model with post-failure plasticity), and
Hardening-Soil models (stress-dependent stiffness (=shear hardening) with post-failure
plasticity and compression hardening) available in Plaxis 2D, we examined the effects of
different levels of sophistication of soil models on structural response to surface live loads.

3.1      Modeled Structures

We selected six structures as shown in Table 6. For each structure, we examined cases of
different cover depths as shown in Table 6.

Concrete box: Reinforced concrete box section (12 ft by 6 ft by 12 in.) with a compressive
strength of 5,000 psi specified in ASTM C1433 for HS20 live load conditions.

Concrete pipe: Class II reinforced concrete pipe with an internal diameter of 48 in., a wall
thickness of 5 in., and a compressive strength of 4,000 psi specified in ASTM C76 as Wall B.




NCHRP 15-29 Appendix A                                                                            22
Concrete arch: BEBO arch culvert with an inside span of 30 ft, an inside rise of 11 ft 4 in., a wall
thickness of 10 in., and a compressive strength of 4,200 psi, designated as BEBO Type E30/3.

Metal pipe: Corrugated steel pipe with an internal diameter of 48 in. Steel plates have 2-2/3 in.
by 1/2 in. corrugations with an uncoated thickness of 0.0598 in.

Metal arch: Corrugated steel arch culvert with a maximum span of 31 ft 7 in. and a total rise of
12 ft 1 in. (dimensions are to inside crests), designated as Type 108A30 by Contech. Steel
plates have 6 in. by 2 in. corrugations with an uncoated thickness of 0.215 in.

Thermoplastic pipe: Corrugated high-density polyethylene pipe with an internal diameter of 60
in. and Type S corrugation manufactured by Hancor, Inc. of Findlay, Ohio.

                Table 6—Structural Types and Cover Depths for 2D Analysis
                                                                           Availability of
              Structural Type          Span          Cover Depth
                                                                            Field Data
               Concrete Box             12 ft       0 ft, 2 ft, and 6 ft         No
               Concrete Pipe            4 ft           2 ft and 6 ft             No
               Concrete Arch            30 ft          2 ft and 6 ft            Yes
                 Metal Pipe             4 ft           2 ft and 6 ft             No
                 Metal Arch          31 ft 7 in.       2 ft and 6 ft            Yes
             Thermoplastic Pipe         5 ft           2 ft and 6 ft            Yes



3.2      Material Models

For each structural type and for each cover depth, we performed three analyses with the three
soil models: linear-elastic model, Mohr-Coulomb model, and Hardening-Soil model. In this set
of preliminary analyses, we used only SW85 as backfill material.                 For the linear-elastic
properties of backfill, we varied the soil modulus based on depth of fill.

3.2.1    Linear-Elastic Model

The linear elastic soil model was described in Section 2.1 above. As noted, linear elastic model
is the simplest constitutive soil models. Since elastic constants vary with soil stress level, we
used elastic constants of SW85 recommended by Selig (1990) as shown in Table 1.

3.2.2    Mohr-Coulomb Model with Perfect Plasticity in Plaxis

The Mohr-Coulomb model in Plaxis uses an elastic perfectly-plastic constitutive model as
described in Section 2.2. For stress states within the yield surface, the soil behavior is elastic


NCHRP 15-29 Appendix A                                                                              23
and is determined by isotropic linear elasticity.     The yield condition is expressed by Mohr-
Coulomb failure condition. Tensile failure of soil is captured by specifying a tension cut-off.
Input parameters of the Mohr-Coulomb model for SW85 are given in Table 3 except for tensile
strength. A tensile strength of 0 psi is used in this study.

3.2.3       Hardening-Soil Model in Plaxis

The Plaxis Hardening-Soil model was described in Section 2.3.2.            A basic feature of the
Hardening-Soil model is the stress dependency of soil stiffness and the hyperbolic relationship
between the vertical strain and the deviatoric stress in primary triaxial loading. Two types of
hardening are modeled: shear hardening and compression hardening. The shear yield surface
increases in size until the Mohr-Coulomb failure criterion is satisfied, at which point perfectly
plastic yielding occurs. A cap type yield surface is used to account for plastic volumetric strain
due to primary compression in isotropic compression or oedometer loading. Tensile failure of
soil is captured by specifying a tension cut-off. Basic input parameters of the Hardening-Soil
model for SW85 are given in Table 5.

3.2.4       In-Situ Soil Material

We used a linear-elastic model with E of 3,000 psi, ν of 0.25, and γ of 126 pcf for in-situ
material.

3.2.5       Other Materials

For other materials we used a linear-elastic model with properties as listed:

•           steel - E = 29,000,000 psi, ν of 0.3, and γ of 490 pcf,

•           concrete - E = 57,000     f c′ , ν of 0.17, and γ of 150 pcf, where f c′ is a specified
            compressive strength in psi, and

•           high-density polyethylene - E = 80,000 psi, ν of 0.35, and γ of 59.5 pcf.

3.3         Live Load

One of the key shortcomings of performing a 2D analysis to examine culvert structural response
is that the distribution of live load along the length of the culvert cannot be modeled. 3D
behavior must be addressed by modifying the load applied to the surface of the 2D finite
element mesh. While developing equations for this purpose is one of the goals of this project,
we have taken equations from codes and the literature that are suitable for the immediate




NCHRP 15-29 Appendix A                                                                          24
purpose of evaluating soil models. We used Eq. 27 for box culverts, Eq. 28 for pipes, and Eq.
29 for arches to calculate the live load per unit length of culvert.


                              mmpf (1 + IM / 100) P
                                                        for H < 2.0 ft
                                    48 + 0.72 S
                      WLL   =
                                mmpf (1 + IM / 100) P                                        (27)
                                                        for H ≥ 2.0 ft
                              20.4 + 0.72 S + 1.15 H
                             

                                  mmpf (1 + IM / 100) P  0.7 Rt 
                         WLL =                          
                                                         L + 1.15 H 
                                                                                            (28)
                                      wt + 1.15 H        t          

                                  mmpf (1 + IM / 100) P  0.7 Rt 
                         WLL =                                    
                                    3( wt + 1.15 H )  Lt + 1.15 H 
                                                                  
                                                                                             (29)


where

         WLL : live load per unit length of culvert, lb/in.
         mmpf : multiple presence factor (=1.2, AASHTO LRFD 3.6.1.1.2)
          IM : dynamic load allowance (=33(1.0-0.125H/12)≥0%, AASHTO LRFD 3.6.2.2), %
           P   : wheel load magnitude (=16,000 lb, AASHTO LRFD 3.6.1.2.2), lb
           S   : clear span, ft
          H    : depth of cover from road surface to top of culvert, in.
          wt   : width of tire footprint at surface (=20 in., AASHTO LRFD 3.6.1.2.5), in.

          Lt   : length of tire footprint at surface (=10 in., AASHTO LRFD 3.6.1.2.5), in.

          Rt   : mean culvert radius (top radius for arches), in.



According to NCHRP 473 (McGrath, 2002), the live load calculated by Eq. 28 results in
reasonable thrusts but greater moments and deflections when compared to those from 3D
analysis, and the live load calculated by Eq. 29 results in reasonable moments but inaccurate
deflections and thrusts. For the purpose of comparison of different soil models, we used Eq. 28
for pipes and Eq. 29 for arches. Figure 9 shows live load per unit length of a culvert to be used
for 2D analyses of structures listed in Table 6.




NCHRP 15-29 Appendix A                                                                          25
                                                   900
                                                                                            Concrete Box




                  Live load per unit length of culvert
                                                   800
                                                                                            Concrete Pipe
                                                   700                                      Concrete Arch
                                                                                            Metal Pipe
                                                   600
                                                                                            Metal Arch
                                                   500

                                 (lb/in)
                                                                                            Thermoplastic Pipe

                                                   400

                                                   300

                                                   200

                                                   100

                                                         0
                                                             0   1   2   3      4      5       6         7       8
                                                                         Cover depth (ft)


                Figure 9—Live Load per Unit Length of Culvert in 2D Analysis



3.4        Finite Element Model

The finite element model includes the buried structure, in-situ soil, and backfill. Conceptual
models are shown in Figure 10 through Figure 12. Bedding thickness, H b , in this figure is

specified in Section 27.5 of AASHTO LRFD Bridge Construction Specifications (AASHTO,
2004). Thirty-nine models were created for six types of structures, 2 ft and 6 ft of cover depth
(in addition, 0 ft for concrete box culvert), and three soil constitutive models (linear-elastic,
Mohr-Coulomb, and Hardening-Soil models). The first 4 in. of backfill from the surface is always
modeled by linear-elastic soil model to prevent the soil from failing under the applied live loads.

The bottom of the model is restrained in the vertical and horizontal directions and the sides of
the model are restrained in the horizontal direction.

For the finite element models with either the Mohr-Coulomb soil model or the Hardening-Soil
model, soil was placed incrementally. In-situ soil was placed at once in the first stage. Backfill
soil was placed with about 1 ft increments.

For the finite element models with the linear-elastic soil model, full bonding was assumed at the
interface between the soil and the structure. For the models with either the Mohr-Coulomb soil
model or the Hardening-Soil model, the interface strength (friction and adhesion) was
considered. In Plaxis 2D, the interface strength is specified by a fraction of the soil strength as
follows:



NCHRP 15-29 Appendix A                                                                                               26
                           τ i = ci + σ n tan φi = Rint er (c + σ n tan φ )                (30)

where

          τi   : interface strength

          ci   : cohesion of the interface

          φi   : friction angle of the interface

          σ n : normal stress
         Rint er : strength reduction factor for the interface
           c   : cohesion of the soil
           φ   : friction angle of the soil

We used 0.5 for Rint er in this study.


Figure 13 through Figure 18 show finite element meshes for the six different structures.


                                                     10 in.



                                                      Wst



                                                                      Hst
                Backfill (SW85)         Hb



                                                                     1.5Hst

                In-situ soil


                                                      6Wst
                    Figure 10—Conceptual Model for 2D Analysis of Pipes




NCHRP 15-29 Appendix A                                                                        27
                                                   10 in.



                                                   Wst



                                                                 Hst
                 Backfill (SW85)
                                   Hb



                                                                 1.5Hst
                 In-situ soil


                                                    6Wst

                        Figure 11—Conceptual Model for 2D Analysis of Boxes
                                                      10 in.


                                              24
                                         77                    SW85
   28 ft 2 in.




                                                                          10 ft 2 in.

                                                                          18 ft
                 In-situ soil                          46 ft


                                                      192 ft

                       Figure 12—Conceptual Model for 2D Analysis of Arches




NCHRP 15-29 Appendix A                                                                  28
                                      72 ft

                                       7 ft
                                       13 ft


                                                                     9 ft


                               (a) 0 ft cover depth




                               (b) 2 ft cover depth




                               (c) 6 ft cover depth

             Figure 13—Finite Element Meshes of Concrete Box Model




NCHRP 15-29 Appendix A                                                      29
                                       24 ft




                                           26.5 in.




                                                                      6 ft



                                (a) 2 ft cover depth




                                (b) 6 ft cover depth


             Figure 14—Finite Element Meshes of Concrete Pipe Model




NCHRP 15-29 Appendix A                                                       30
                                      24 ft




                                          24.3 in.




                                                                    6 ft



                               (a) 2 ft cover depth




                               (b) 6 ft cover depth


              Figure 15—Finite Element Meshes of Metal Pipe Model




NCHRP 15-29 Appendix A                                                     31
                                       30 ft




                                               31.4 in.



                                                                      7.5 ft




                                (a) 2 ft cover depth




                                (b) 6 ft cover depth



              Figure 16—Finite Element Meshes of Plastic Pipe Model




NCHRP 15-29 Appendix A                                                         32
                                       192 ft




                                (a) 2 ft cover depth




                                          11 ft 9 in.                 10 ft 2 in.



                                    30 ft 10 in.
                                                                      18 ft
                                        46 ft
                                     83 ft 8 in.
                           (b) 2 ft cover depth (close-up)




                                (c) 6 ft cover depth




                           (d) 6 ft cover depth (close-up)



             Figure 17—Finite Element Meshes of Concrete Arch Model




NCHRP 15-29 Appendix A                                                        33
                                      192 ft




                               (a) 2 ft cover depth




                                         12 ft 2.5 in.              10 ft 2 in.



                                    31 ft 4 in.
                                                                    18 ft
                                       46 ft
                                    83 ft 8 in.
                          (b) 2 ft cover depth (close-up)




                               (c) 6 ft cover depth




                          (d) 6 ft cover depth (close-up)



              Figure 18—Finite Element Meshes of Metal Arch Model




NCHRP 15-29 Appendix A                                                      34
3.5        Results of 2D Analysis

3.5.1      Concrete Box

Figure 19 shows deformations of the concrete box due to live load. Figure 20 through Figure 25
show bending moments and thrusts in the concrete box due to the live load for different the soil
models and cover depths considered. Table 7 summarizes the results. The linear-elastic soil
model resulted in greater positive bending moments at the center of the top slab and greater
negative bending moments at the tip of the upper haunch in the wall than the other two soil
models except for a cover depth of 0 ft. The linear-elastic soil model resulted in thrusts in the
top slab and the walls that are between Mohr-Coulomb and Hardening-Soil models in size for a
cover depth of 0 ft, and resulted in greater thrusts in the top slab and the walls for cover depths
of 2 ft and 6ft. The Mohr-Coulomb model resulted in greater positive and negative moments
than the Hardening-Soil model, except for the positive moment in the top slab for a cover depth
of 2 ft.




NCHRP 15-29 Appendix A                                                                          35
                         Undeformed
                         Linear-Elastic Model
                         Mohr-Coulomb Model
                         Hardening-Soil Model            300X Deformation




                                       (a) 0 ft cover




                         Undeformed
                         Linear-Elastic Model
                         Mohr-Coulomb Model
                         Hardening-Soil Model            300X Deformation




                                       (b) 2 ft cover




                         Undeformed
                         Linear-Elastic Model
                         Mohr-Coulomb Model
                         Hardening-Soil Model            300X Deformation



                                        (c) 6 ft cover


             Figure 19—Deformation of Concrete Box due to Live Load




NCHRP 15-29 Appendix A                                                      36
                                                      12000

                                                      10000


          Bending Moment due to Live Load
                                                       8000

                                                       6000

                                                       4000
                    (lb*in/in)


                                                       2000

                                                              0

                                                       -2000

                                                       -4000                                  Linear-Elastic Model
                                                       -6000                                  Mohr-Coulomb Model

                                                       -8000                                  Hardening-Soil Model

                                                      -10000
                                                                  -80     -60    -40     -20         0      20       40     60      80
                                                                                        X-Coordinates (in.)

                                                                                  (a) Bending Moment

                                                       -88

                                                       -90
                    Thrust due to Live Load (lb/in)




                                                       -92

                                                       -94

                                                       -96
                                                                                                             Linear-Elastic Model
                                                       -98
                                                                                                             Mohr-Coulomb Model
                                                      -100                                                   Hardening-Soil Model

                                                      -102

                                                      -104

                                                      -106
                                                             -80        -60     -40     -20         0      20        40     60      80
                                                                                       X-Coordinates (in.)

                                                                                         (b) Thrust

Figure 20—Bending Moments and Thrusts due to Live Load in Top Slab of Concrete Box
                               Model (0 ft Cover)




NCHRP 15-29 Appendix A                                                                                                                   37
                                                        0




         Bending Moment due to Live Load
                                                     -1000

                                                     -2000

                                                     -3000
                   (lb*in/in)

                                                     -4000

                                                     -5000

                                                     -6000

                                                     -7000               Linear-Elastic Model
                                                                         Mohr-Coulomb Model
                                                     -8000
                                                                         Hardening-Soil Model
                                                     -9000
                                                             -50    -40     -30    -20     -10    0     10      20    30     40     50
                                                                                         Y-Coordinates (in.)

                                                                                  (a) Bending Moment

                                                     -150

                                                                                                             Linear-Elastic Model
                                                     -160
                   Thrust due to Live Load (lb/in)




                                                                                                             Mohr-Coulomb Model
                                                     -170                                                    Hardening-Soil Model

                                                     -180

                                                     -190

                                                     -200

                                                     -210

                                                     -220

                                                     -230
                                                            -50    -40     -30    -20     -10    0      10     20     30     40     50
                                                                                        Y-Coordinates (in.)

                                                                                           (b) Thrust


 Figure 21—Bending Moments and Thrusts due to Live Load in Right Wall of Concrete
                            Box Model (0 ft Cover)




NCHRP 15-29 Appendix A                                                                                                                   38
                                                      8000




          Bending Moment due to Live Load
                                                      6000

                                                      4000

                                                      2000
                    (lb*in/in)


                                                         0

                                                      -2000

                                                      -4000                         Linear-Elastic Model
                                                                                    Mohr-Coulomb Model
                                                      -6000                         Hardening-Soil Model

                                                      -8000
                                                              -80    -60   -40    -20      0      20         40       60      80
                                                                                 X-Coordinates (in.)

                                                                             (a) Bending Moment

                                                       80
                                                                                                           Linear-Elastic Model
                                                       60                                                  Mohr-Coulomb Model
                    Thrust due to Live Load (lb/in)




                                                                                                           Hardening-Soil Model
                                                       40

                                                       20

                                                        0

                                                       -20

                                                       -40

                                                       -60

                                                       -80

                                                      -100
                                                             -80    -60    -40    -20     0       20         40      60       80
                                                                                 X-Coordinates (in.)

                                                                                   (b) Thrust


Figure 22—Bending Moments and Thrusts due to Live Load in Top Slab of Concrete Box
                               Model (2 ft Cover)




NCHRP 15-29 Appendix A                                                                                                             39
                                                            0




         Bending Moment due to Live Load
                                                     -1000

                                                     -2000

                                                     -3000
                   (lb*in/in)


                                                     -4000

                                                     -5000

                                                     -6000                  Linear-Elastic Model
                                                                            Mohr-Coulomb Model
                                                     -7000
                                                                            Hardening-Soil Model
                                                     -8000
                                                                -50    -40     -30    -20     -10    0     10     20     30    40      50
                                                                                            Y-Coordinates (in.)

                                                                                     (a) Bending Moment

                                                     -120
                                                                                                                Linear-Elastic Model
                                                     -130
                                                                                                                Mohr-Coulomb Model
                   Thrust due to Live Load (lb/in)




                                                     -140                                                       Hardening-Soil Model
                                                     -150

                                                     -160

                                                     -170

                                                     -180

                                                     -190

                                                     -200

                                                     -210

                                                     -220
                                                            -50       -40     -30    -20     -10    0      10     20     30    40      50
                                                                                           Y-Coordinates (in.)

                                                                                              (b) Thrust


 Figure 23—Bending Moments and Thrusts due to Live Load in Right Wall of Concrete
                            Box Model (2 ft Cover)




NCHRP 15-29 Appendix A                                                                                                                      40
                                                      2500

                                                      2000


          Bending Moment due to Live Load
                                                      1500

                                                      1000

                                                       500
                    (lb*in/in)


                                                             0

                                                       -500

                                                      -1000

                                                      -1500                                   Linear-Elastic Model
                                                      -2000                                   Mohr-Coulomb Model

                                                      -2500                                   Hardening-Soil Model

                                                      -3000
                                                                 -80     -60    -40       -20        0      20       40   60   80
                                                                                         X-Coordinates (in.)

                                                                                     (a) Bending Moment

                                                      60

                                                      50
                    Thrust due to Live Load (lb/in)




                                                      40

                                                      30

                                                      20

                                                      10

                                                       0

                                                      -10                                 Linear-Elastic Model
                                                                                          Mohr-Coulomb Model
                                                      -20
                                                                                          Hardening-Soil Model
                                                      -30
                                                            -80        -60     -40      -20        0       20        40   60   80
                                                                                        X-Coordinates (in.)

                                                                                           (b) Thrust


Figure 24—Bending Moments and Thrusts due to Live Load in Top Slab of Concrete Box
                               Model (6 ft Cover)




NCHRP 15-29 Appendix A                                                                                                              41
                                                            0




         Bending Moment due to Live Load
                                                      -500



                                                     -1000
                   (lb*in/in)




                                                     -1500


                                                                            Linear-Elastic Model
                                                     -2000
                                                                            Mohr-Coulomb Model
                                                                            Hardening-Soil Model
                                                     -2500
                                                                -50     -40    -30    -20     -10       0   10       20    30     40    50
                                                                                            Y-Coordinates (in.)

                                                                                     (a) Bending Moment

                                                     -40

                                                     -45
                                                                                                                 Linear-Elastic Model
                   Thrust due to Live Load (lb/in)




                                                     -50                                                         Mohr-Coulomb Model

                                                     -55                                                         Hardening-Soil Model

                                                     -60

                                                     -65

                                                     -70

                                                     -75

                                                     -80

                                                     -85

                                                     -90
                                                           -50        -40     -30    -20     -10    0       10      20     30     40    50
                                                                                           Y-Coordinates (in.)

                                                                                              (b) Thrust


 Figure 25—Bending Moments and Thrusts due to Live Load in Right Wall of Concrete
                            Box Model (6 ft Cover)




NCHRP 15-29 Appendix A                                                                                                                       42
    Table 7—Comparison of Bending Moments and Thrusts in Concrete Box Model

                    (a) Bending Moments and Thrusts due to Earth Load
       Cover        Soil                     Moment (lb*in/in)             Thrust (lb/in)
       Depth       Model        Top Center   Tip of Haunch                 Top Center       Tip of Haunch
        (ft)                                      Top             Wall                          Wall
         0     Linear-Elastic     2,208           333            -2,408        84               -134
               Mohr-Coulomb       1,908            32            -1,956        33               -120
               Hardening-Soil     1,881            5             -1,908        29               -120
         2     Linear-Elastic     5,171           -532           -5,520        75               -341
               Mohr-Coulomb       4,940           -812           -5,215        12               -308
               Hardening-Soil     4,844           -909           -5,047        -1               -309
         6     Linear-Elastic     9,774          -1,829          -10,494       49               -685
               Mohr-Coulomb       9,748          -2,131          -10,474       -29              -618
               Hardening-Soil     9,567          -2,310          -10,134       -52              -621


       (b) Bending Moments and Thrusts due to Earth Load plus Surface Live Load

       Cover        Soil                     Moment (lb*in/in)             Thrust (lb/in)
       Depth       Model        Top Center   Tip of Haunch                 Top Center       Tip of Haunch
        (ft)                                      Top             Wall                          Wall
         0     Linear-Elastic    11,493          -3,344          -8,583        -12              -353
               Mohr-Coulomb      11,233          -3,605          -8,175        -57              -338
               Hardening-Soil    11,134          -3,703          -8,001        -75              -338
         2     Linear-Elastic    12,106          -3,661          -11,118       63               -542
               Mohr-Coulomb      11,536          -3,733          -10,635       78               -502
               Hardening-Soil    11,568          -3,947          -10,456       58               -499
         6     Linear-Elastic    11,734          -2,674          -12,425       55               -767
               Mohr-Coulomb      11,102          -2,602          -12,036       10               -690
               Hardening-Soil    10,815          -2,711          -11,516        2               -687



                (c) Bending Moments and Thrusts due to Surface Live Load

       Cover        Soil                     Moment (lb*in/in)             Thrust (lb/in)
       Depth       Model        Top Center   Tip of Haunch                 Top Center       Tip of Haunch
        (ft)                                      Top             Wall                          Wall
         0     Linear-Elastic     9,285          -3,677          -6,176        -96              -219
               Mohr-Coulomb       9,325          -3,637          -6,220        -90              -218
               Hardening-Soil     9,253          -3,708          -6,092       -103              -219
         2     Linear-Elastic     6,935          -3,130          -5,598        -13              -201
               Mohr-Coulomb       6,595          -2,921          -5,420        66               -195
               Hardening-Soil     6,724          -3,038          -5,409        59               -191
         6     Linear-Elastic     1,959           -844           -1,931         5                -82
               Mohr-Coulomb       1,354           -471           -1,562        39                -72
               Hardening-Soil     1,248           -400           -1,382        54                -66




NCHRP 15-29 Appendix A                                                                                      43
3.5.2    Concrete Pipe

Figure 26 shows deformations of the concrete pipe due to the live load. Figure 27 and Figure
28 show bending moments and thrusts in the concrete pipe due to the live load for different soil
models and cover depths, and Table 8 summarizes the results.            The linear-elastic model
resulted in greater positive moments at the crown and invert and greater negative moments at
springlines than did the other two soil models for both cover depths of 2 ft and 6 ft. The linear-
elastic model resulted in greater thrusts at springlines for a cover depth of 2 ft and everywhere
for a cover depth of 6 ft. Between the Mohr-Coulomb and Hardening-Soil models, the Mohr-
Coulomb model resulted in greater peak positive and negative moments for both cover depths
of 2 ft and 6 ft. The Mohr-Coulomb model resulted in less thrusts except at the crown for a
cover depth of 6 ft.




NCHRP 15-29 Appendix A                                                                         44
                                   Undeformed

                                   Linear-Elastic Model
                                   Mohr-Coulomb Model

                                   Hardening-Soil Model




                                                           150X Deformation



                                   (a) 2 ft cover




                                   Undeformed
                                   Linear-Elastic Model
                                   Mohr-Coulomb Model
                                   Hardening-Soil Model




                                                          150X Deformation

                                   (b) 6 ft cover



             Figure 26—Deformation of Concrete Pipe due to Live Load




NCHRP 15-29 Appendix A                                                        45
                                              1200
                                                                                                      Linear-Elastic Model


          Bending Moment due to Live Load
                                              1000
                                                                                                      Mohr-Coulomb Model
                                               800                                                    Hardening-Soil Model
                                               600

                                               400
                    (lb*in/in)


                                               200

                                                   0

                                              -200

                                              -400

                                              -600

                                              -800
                                                  -180     -135     -90     -45            0     45        90     135        180
                                                                          Degrees from Crown

                                                                        (a) Bending Moment

                                             20                 Linear-Elastic Model           Mohr-Coulomb Model
                                                                Hardening-Soil Model
                                               0
          Thrust due to Live Load (lb/in)




                                             -20


                                             -40


                                             -60


                                             -80


                                            -100


                                            -120
                                                -180     -135     -90      -45         0        45        90      135        180
                                                                          Degrees from Crown

                                                                                 (b) Thrust


Figure 27—Bending Moments and Thrusts due to Live Load in Concrete Pipe Model (2 ft
                                   Cover)




NCHRP 15-29 Appendix A                                                                                                             46
                                              100
                                                                                                      Linear-Elastic Model


          Bending Moment due to Live Load
                                                  80                                                  Mohr-Coulomb Model
                                                                                                      Hardening-Soil Model
                                                  60

                                                  40
                    (lb*in/in)

                                                  20

                                                   0

                                              -20

                                              -40

                                              -60

                                              -80
                                                 -180     -135       -90     -45            0    45        90     135        180
                                                                           Degrees from Crown

                                                                         (a) Bending Moment

                                             4                   Linear-Elastic Model           Mohr-Coulomb Model
                                                                 Hardening-Soil Model
                                             2
          Thrust due to Live Load (lb/in)




                                             0

                                             -2

                                             -4

                                             -6

                                             -8

                                            -10

                                            -12
                                               -180     -135       -90      -45         0       45        90      135        180
                                                                           Degrees from Crown

                                                                                  (b) Thrust


Figure 28—Bending Moments and Thrusts due to Live Load in Concrete Pipe Model (6 ft
                                   Cover)




NCHRP 15-29 Appendix A                                                                                                             47
    Table 8—Comparison of Bending Moments and Thrusts in Concrete Pipe Model

                      (a) Bending Moments and Thrusts due to Earth Load

           Cover             Soil        Moment (lb*in/in)     Thrust (lb/in)
         Depth (ft)         Model        Peak Pos Peak Neg       Crown      Springline
             2          Linear-Elastic      655        -615        -5           -115
                        Mohr-Coulomb        569        -503        -22           -90
                        Hardening-Soil      495        -427        -27           -91
             6          Linear-Elastic     1,282      -1,235       -49          -239
                        Mohr-Coulomb       1,158      -1,096       -59          -194
                        Hardening-Soil     1,020       -942        -72          -196



       (b) Bending Moments and Thrusts due to Earth Load plus Surface Live Load

           Cover             Soil        Moment (lb*in/in)     Thrust (lb/in)
         Depth (ft)         Model        Peak Pos Peak Neg       Crown      Springline
             2          Linear-Elastic     1,439      -1,266       -5           -208
                        Mohr-Coulomb       1,247      -1,113       -21          -160
                        Hardening-Soil     1,191      -1,029       -28          -164
             6          Linear-Elastic     1,344      -1,301       -50          -248
                        Mohr-Coulomb       1,196      -1,141       -57          -197
                        Hardening-Soil     1,053       -976        -71          -200



                 (c) Bending Moments and Thrusts due to Surface Live Load

           Cover             Soil        Moment (lb*in/in)     Thrust (lb/in)
         Depth (ft)         Model        Peak Pos Peak Neg       Crown      Springline
             2          Linear-Elastic      933        -693       0.68          -92.61
                        Mohr-Coulomb        856        -651       0.95          -69.52
                        Hardening-Soil      852        -640       -1.30         -72.98
             6          Linear-Elastic       77         -68       -0.30         -9.51
                        Mohr-Coulomb         39         -45       1.52          -2.78
                        Hardening-Soil       34         -35       1.57          -3.47




NCHRP 15-29 Appendix A                                                                   48
3.5.3    Metal Pipe

Figure 29 shows deformations of the metal pipe due to the live load. Figure 30 and Figure 31
show bending moments and thrusts in the metal pipe due to the live load for different soil
models and cover depths, and Table 9 summarizes the results.             The linear-elastic model
resulted in less peak positive and negative moments than did the other two soil models for both
cover depths of 2 ft and 6 ft. For a cover depth of 2 ft, the linear-elastic model resulted in less
thrusts at the crown and invert than did the other two models and greater thrusts at springlines
than did the Mohr-Coulomb model. For a cover depth of 6 ft, the linear-elastic model resulted in
greater thrusts at springlines than did the other two soil models and less thrusts at the crown
and invert than did the Mohr-Coulomb model. Between the Mohr-Coulomb and Hardening-Soil
models, the Mohr-Coulomb model resulted in greater peak positive and negative moments for
both cover depths of 2 ft and 6 ft. The Mohr-Coulomb model resulted in less thrusts everywhere
for a cover depth of 2 ft, whereas it resulted in greater thrusts everywhere for a cover depth of
6 ft.




NCHRP 15-29 Appendix A                                                                          49
                                  Undeformed
                                  Linear-Elastic Model
                                  Mohr-Coulomb Model

                                  Hardening-Soil Model




                                                         50X Deformation


                                  (a) 2 ft cover




                                  Undeformed

                                  Linear-Elastic Model
                                  Mohr-Coulomb Model
                                  Hardening-Soil Model




                                                         50X Deformation


                                  (b) 6 ft cover



              Figure 29—Deformation of Metal Pipe due to Live Load




NCHRP 15-29 Appendix A                                                     50
                                              200
                                                                                                     Linear-Elastic Model


          Bending Moment due to Live Load
                                              150                                                    Mohr-Coulomb Model
                                                                                                     Hardening-Soil Model
                    (lb*in/in)                100


                                               50


                                                   0


                                               -50


                                              -100


                                              -150
                                                  -180    -135      -90     -45        0        45        90      135       180
                                                                          Degrees from Crown

                                                                        (a) Bending Moment

                                               0
                                                                Linear-Elastic Model          Mohr-Coulomb Model
                                                                Hardening-Soil Model
          Thrust due to Live Load (lb/in)




                                             -20


                                             -40


                                             -60


                                             -80


                                            -100


                                            -120
                                                -180     -135     -90      -45         0       45         90      135       180
                                                                          Degrees from Crown

                                                                                 (b) Thrust


  Figure 30—Bending Moments and Thrusts due to Live Load in Metal Pipe Model (2 ft
                                    Cover)




NCHRP 15-29 Appendix A                                                                                                            51
                                               7.0
                                                                                                      Linear-Elastic Model


          Bending Moment due to Live Load
                                               6.0
                                                                                                      Mohr-Coulomb Model
                                               5.0                                                    Hardening-Soil Model
                                               4.0

                                               3.0
                    (lb*in/in)


                                               2.0

                                               1.0

                                               0.0

                                               -1.0

                                               -2.0

                                               -3.0
                                                   -180     -135      -90     -45           0    45        90      135       180
                                                                             Degrees from Crown

                                                                          (a) Bending Moment

                                            1.0                  Linear-Elastic Model           Mohr-Coulomb Model
                                                                 Hardening-Soil Model
                                            0.0
          Thrust due to Live Load (lb/in)




                                            -1.0

                                            -2.0

                                            -3.0

                                            -4.0

                                            -5.0

                                            -6.0

                                            -7.0

                                            -8.0

                                            -9.0
                                                -180      -135      -90      -45        0        45       90      135        180
                                                                            Degrees from Crown

                                                                                   (b) Thrust


  Figure 31—Bending Moments and Thrusts due to Live Load in Metal Pipe Model (6 ft
                                    Cover)




NCHRP 15-29 Appendix A                                                                                                             52
     Table 9—Comparison of Bending Moments and Thrusts in Metal Pipe Model

                      (a) Bending Moments and Thrusts due to Earth Load

           Cover             Soil        Moment (lb*in/in)    Thrust (lb/in)
         Depth (ft)         Model        Peak Pos Peak Neg      Crown      Springline
             2          Linear-Elastic       16         -15       -26           -78
                        Mohr-Coulomb         12         -6        -37           -57
                        Hardening-Soil       11         -9        -38           -57
             6          Linear-Elastic       30         -25       -77          -190
                        Mohr-Coulomb         25         -16       -95          -145
                        Hardening-Soil       22         -14       -97          -144



       (b) Bending Moments and Thrusts due to Earth Load plus Surface Live Load

           Cover             Soil        Moment (lb*in/in)    Thrust (lb/in)
         Depth (ft)         Model        Peak Pos Peak Neg      Crown      Springline
             2          Linear-Elastic       77         -35       -49          -160
                        Mohr-Coulomb        169         -98       -96          -114
                        Hardening-Soil      150         -90      -110          -125
             6          Linear-Elastic       32         -26       -79          -197
                        Mohr-Coulomb         31         -16       -97          -149
                        Hardening-Soil       24         -15       -97          -146



                 (c) Bending Moments and Thrusts due to Surface Live Load

           Cover             Soil        Moment (lb*in/in)    Thrust (lb/in)
         Depth (ft)         Model        Peak Pos Peak Neg      Crown      Springline
             2          Linear-Elastic       61         -27     -22.80         -82.59
                        Mohr-Coulomb        166         -99     -59.55         -57.03
                        Hardening-Soil      151         -91     -72.45         -68.27
             6          Linear-Elastic       2          -1       -1.74         -7.67
                        Mohr-Coulomb         6          -2       -2.10         -3.87
                        Hardening-Soil       4          -2       -0.01         -1.77




NCHRP 15-29 Appendix A                                                                  53
3.5.4    Thermoplastic Pipe

Figure 32 shows deformations of the thermoplastic pipe due to the live load. Figure 33 and
Figure 34 show bending moments and thrusts in the thermoplastic pipe due to the live load for
different soil models and cover depths, and Table 10 summarizes the results. The linear-elastic
model resulted in less peak positive and negative moments than did the other two soil models
for both cover depths of 2 ft and 6 ft. For a cover depth of 2 ft, the linear-elastic model resulted
in less thrusts except a small portion near the springlines. For a cover depth of 6 ft, the linear-
elastic model resulted in greater thrusts at springlines and less thrusts at the crown and invert
than did the other two soil models. Between the Mohr-Coulomb and Hardening-Soil models, the
Mohr-Coulomb model resulted in greater peak positive and negative moments except for the
peak positive moment at a cover depth of 6 ft. The Mohr-Coulomb model resulted in greater
thrusts except at the crown for a cover depth of 2 ft.




NCHRP 15-29 Appendix A                                                                           54
                                  Undeformed

                                  Linear-Elastic Model
                                  Mohr-Coulomb Model

                                  Hardening-Soil Model




                                                         30X Deformation

                                  (a) 2 ft cover




                                  Undeformed

                                  Linear-Elastic Model
                                  Mohr-Coulomb Model
                                  Hardening-Soil Model




                                                         30X Deformation

                                  (b) 6 ft cover



          Figure 32—Deformation of Thermoplastic Pipe due to Live Load




NCHRP 15-29 Appendix A                                                     55
                                              400
                                                                                                  Linear-Elastic Model


          Bending Moment due to Live Load
                                              300                                                 Mohr-Coulomb Model
                                                                                                  Hardening-Soil Model
                    (lb*in/in)                200


                                              100


                                                   0


                                              -100


                                              -200


                                              -300
                                                  -180    -135    -90      -45        0      45        90      135       180
                                                                         Degrees from Crown

                                                                       (a) Bending Moment

                                             20

                                               0
          Thrust due to Live Load (lb/in)




                                             -20

                                             -40

                                             -60

                                             -80

                                            -100

                                            -120

                                            -140           Linear-Elastic Model               Mohr-Coulomb Model
                                                           Hardening-Soil Model
                                            -160
                                                -180     -135    -90      -45        0       45       90      135        180
                                                                        Degrees from Crown

                                                                                (b) Thrust


Figure 33—Bending Moments and Thrusts due to Live Load in Thermoplastic Pipe Model
                                  (2 ft Cover)




NCHRP 15-29 Appendix A                                                                                                         56
                                               8.0
                                                                                                    Linear-Elastic Model


          Bending Moment due to Live Load
                                               6.0                                                  Mohr-Coulomb Model
                                                                                                    Hardening-Soil Model
                                               4.0

                                               2.0
                    (lb*in/in)


                                               0.0

                                               -2.0

                                               -4.0

                                               -6.0

                                               -8.0
                                                   -180     -135     -90     -45           0   45       90      135        180
                                                                            Degrees from Crown

                                                                         (a) Bending Moment

                                            1.0                                                 Linear-Elastic Model
                                                                                                Mohr-Coulomb Model
                                            0.0                                                 Hardening-Soil Model
          Thrust due to Live Load (lb/in)




                                            -1.0

                                            -2.0

                                            -3.0

                                            -4.0

                                            -5.0

                                            -6.0

                                            -7.0

                                            -8.0
                                                -180      -135     -90      -45        0       45      90      135         180
                                                                           Degrees from Crown

                                                                                  (b) Thrust


Figure 34—Bending Moments and Thrusts due to Live Load in Thermoplastic Pipe Model
                                  (6 ft Cover)




NCHRP 15-29 Appendix A                                                                                                           57
 Table 10—Comparison of Bending Moments and Thrusts in Thermoplastic Pipe Model

                      (a) Bending Moments and Thrusts due to Earth Load

           Cover             Soil         Moment (lb*in/in)     Thrust (lb/in)
         Depth (ft)         Model         Peak Pos Peak Neg       Crown         Springline
             2          Linear-Elastic        28         -29        -19            -76
                        Mohr-Coulomb          17         -20        -38            -59
                        Hardening-Soil        15         -19        -40            -60
             6          Linear-Elastic        45         -45        -38           -150
                        Mohr-Coulomb          28         -29        -75           -117
                        Hardening-Soil        23         -36        -84           -123



       (b) Bending Moments and Thrusts due to Earth Load plus Surface Live Load

           Cover             Soil         Moment (lb*in/in)     Thrust (lb/in)
         Depth (ft)         Model         Peak Pos Peak Neg       Crown         Springline
             2          Linear-Elastic       124         -57        -11           -150
                        Mohr-Coulomb         313        -209       -129           -130
                        Hardening-Soil       252        -175       -135           -131
             6          Linear-Elastic        48         -47        -37           -156
                        Mohr-Coulomb          29         -30        -79           -121
                        Hardening-Soil        23         -36        -87           -126



                 (c) Bending Moments and Thrusts due to Surface Live Load

          Cover             Soil         Moment (lb*in/in)     Thrust (lb/in)
        Depth (ft)         Model         Peak Pos Peak Neg       Crown      Springline
            2          Linear-Elastic        95         -43       7.80          -73.48
                       Mohr-Coulomb         323        -210      -91.42         -71.47
                       Hardening-Soil       269        -174      -95.22         -71.16
            6          Linear-Elastic        4          -2        0.60           -6.44
                       Mohr-Coulomb          5          -6        -3.42          -3.92
                       Hardening-Soil        7          -5        -2.67          -3.20




NCHRP 15-29 Appendix A                                                                       58
3.5.5    Concrete Arch

Figure 35 shows deformations of the concrete arch due to the live load. Figure 36 and Figure
37 show bending moments and thrusts in the concrete arch due to the live load for different soil
models and cover depths, and Table 11 summarizes the results.             The linear-elastic model
resulted in less peak positive and negative moments for a cover depth of 2 ft than did the other
two soil models. For a cover depth of 6 ft, it resulted in a greater peak moment than did the
other two soil models and an intermediate peak negative moment among the three soil models.
The linear-elastic model always resulted in greater thrusts for both cover depths of 2 ft and 6 ft.
Between the Mohr-Coulomb and Hardening-Soil models, the Mohr-Coulomb model resulted in
greater peak positive and negative moments for both cover depths of 2 ft and 6 ft, and it
resulted in less thrusts everywhere for both cover depths of 2 ft and 6 ft.




NCHRP 15-29 Appendix A                                                                          59
                                  Undeformed

                                  Linear-Elastic Model
                                  Mohr-Coulomb Model

                                  Hardening-Soil Model   100X Deformation




                                   (a) 2 ft cover




                                  Undeformed

                                  Linear-Elastic Model
                                  Mohr-Coulomb Model
                                  Hardening-Soil Model   100X Deformation




                                   (b) 6 ft cover




             Figure 35—Deformation of Concrete Arch due to Live Load




NCHRP 15-29 Appendix A                                                      60
                                              25000
                                                                 Linear-Elastic Model


          Bending Moment due to Live Load
                                              20000
                                                                 Mohr-Coulomb Model

                                              15000              Hardening-Soil Model


                                              10000
                    (lb*in/in)


                                               5000

                                                       0

                                               -5000

                                             -10000

                                             -15000
                                                           0        25         50          75     100   125   150
                                                                                Y-Coordinate (in.)

                                                                         (a) Bending Moment

                                              0
                                                               Linear-Elastic Model
          Thrust due to Live Load (lb/in)




                                            -100               Mohr-Coulomb Model
                                                               Hardening-Soil Model
                                            -200


                                            -300


                                            -400


                                            -500


                                            -600
                                                   0           25         50             75      100    125   150
                                                                            Y-Coordinate (in.)

                                                                                    (b) Thrust


Figure 36—Bending Moments and Thrusts due to Live Load in Concrete Arch Model (2 ft
                                   Cover)




NCHRP 15-29 Appendix A                                                                                              61
                                              2500
                                                                   Linear-Elastic Model


          Bending Moment due to Live Load
                                              2000
                                                                   Mohr-Coulomb Model

                                              1500                 Hardening-Soil Model


                                              1000
                    (lb*in/in)


                                                  500

                                                      0

                                                  -500

                                             -1000

                                             -1500
                                                          0        25           50         75      100   125   150
                                                                                 Y-Coordinate (in.)

                                                                          (a) Bending Moment

                                             0
                                                              Linear-Elastic Model
                                            -10
          Thrust due to Live Load (lb/in)




                                                              Mohr-Coulomb Model
                                            -20               Hardening-Soil Model
                                            -30

                                            -40

                                            -50

                                            -60

                                            -70

                                            -80

                                            -90
                                                  0           25           50             75      100    125   150
                                                                             Y-Coordinate (in.)

                                                                                     (b) Thrust


Figure 37—Bending Moments and Thrusts due to Live Load in Concrete Arch Model (6 ft
                                   Cover)




NCHRP 15-29 Appendix A                                                                                               62
   Table 11—Comparison of Bending Moments and Thrusts in Concrete Arch Model

                      (a) Bending Moments and Thrusts due to Earth Load

           Cover             Soil         Moment (lb*in/in)       Thrust (lb/in)
         Depth (ft)         Model         Peak Pos Peak Neg         Crown         Springline
             2          Linear-Elastic      6,630      -5,541        -546          -1,363
                        Mohr-Coulomb        5,719      -4,125        -629          -1,088
                        Hardening-Soil      5,388      -3,376        -644          -1,105
             6          Linear-Elastic      14,343     -11,541       -973          -2,129
                        Mohr-Coulomb        13,802     -10,131      -1,104         -1,798
                        Hardening-Soil      13,744     -9,395       -1,118         -1,796



       (b) Bending Moments and Thrusts due to Earth Load plus Surface Live Load

           Cover             Soil         Moment (lb*in/in)       Thrust (lb/in)
         Depth (ft)         Model         Peak Pos Peak Neg         Crown         Springline
             2          Linear-Elastic      26,532     -12,969       -788          -1,786
                        Mohr-Coulomb        27,197     -11,849       -811          -1,457
                        Hardening-Soil      26,081     -10,084       -843          -1,498
             6          Linear-Elastic      16,522     -12,532      -1,001         -2,197
                        Mohr-Coulomb        15,765     -11,139      -1,118         -1,857
                        Hardening-Soil      15,572     -10,267      -1,132         -1,858



                 (c) Bending Moments and Thrusts due to Surface Live Load

          Cover             Soil         Moment (lb*in/in)       Thrust (lb/in)
        Depth (ft)         Model         Peak Pos Peak Neg         Crown      Springline
            2          Linear-Elastic     19,911      -8,961      -241.76         -423.16
                       Mohr-Coulomb       21,491      -9,891      -181.49         -368.86
                       Hardening-Soil     20,707      -8,989      -199.11         -392.64
            6          Linear-Elastic      2,180      -1,170       -28.15         -67.66
                       Mohr-Coulomb        1,964      -1,211       -14.02         -59.90
                       Hardening-Soil      1,829      -1,043       -14.02         -62.19




NCHRP 15-29 Appendix A                                                                         63
3.5.6    Metal Arch

Figure 38 shows deformations of the metal arch due to the live load. Figure 39 and Figure 40
show bending moments and thrusts in the metal arch due to the live load for different soil
models and cover depths, and Table 12 summarizes the results.            The linear-elastic model
resulted in less peak positive and negative moments for both cover depth of 2 ft and 6 ft than
did the other two soil models. The linear-elastic model resulted in less thrusts near the crown
and greater thrusts at other locations for both cover depths of 2 ft and 6 ft. Between the Mohr-
Coulomb and Hardening-Soil models, the Mohr-Coulomb model resulted in greater peak
positive and negative moments except for the peak negative moment for cover depths of 6 ft.
For a cover depth of 2 ft, the Mohr-Coulomb model resulted in greater thrusts except near the
crown, whereas for a cover depth of 6 ft, it resulted in less thrust except near the crown.




NCHRP 15-29 Appendix A                                                                        64
              Undeformed

              Linear-Elastic Model
              Mohr-Coulomb Model
                                                            20X Deformation
              Hardening-Soil Model




                                     (a) 2 ft cover




                                     Undeformed

                                     Linear-Elastic Model
                                     Mohr-Coulomb Model

                                     Hardening-Soil Model   20X Deformation




                                     (b) 6 ft cover




              Figure 38—Deformation of Metal Arch due to Live Load




NCHRP 15-29 Appendix A                                                        65
                                             12000
                                                            Linear-Elastic Model


          Bending Moment due to Live Load
                                             10000
                                                            Mohr-Coulomb Model
                                              8000          Hardening-Soil Model

                                              6000
                    (lb*in/in)

                                              4000

                                              2000

                                                    0

                                              -2000

                                              -4000

                                              -6000
                                                        0    25         50          75     100        125        150
                                                                           Y-Coordinate (in.)

                                                                   (a) Bending Moment

                                               0
                                                                                          Linear-Elastic Model
          Thrust due to Live Load (lb/in)




                                             -200                                         Mohr-Coulomb Model
                                                                                          Hardening-Soil Model

                                             -400


                                             -600


                                             -800


                                            -1000


                                            -1200
                                                    0       25        50           75     100        125         150
                                                                        Y-Coordinate (in.)

                                                                             (b) Thrust


  Figure 39—Bending Moments and Thrusts due to Live Load in Metal Arch Model (2 ft
                                    Cover)




NCHRP 15-29 Appendix A                                                                                                 66
                                              500
                                                           Linear-Elastic Model


          Bending Moment due to Live Load
                                              400
                                                           Mohr-Coulomb Model

                                              300          Hardening-Soil Model


                                              200
                    (lb*in/in)


                                              100

                                                   0

                                             -100

                                             -200

                                             -300
                                                       0    25          50        75       100   125   150
                                                                           Y-Coordinate (in.)

                                                                    (a) Bending Moment

                                              0
                                                           Linear-Elastic Model
          Thrust due to Live Load (lb/in)




                                             -20           Mohr-Coulomb Model
                                                           Hardening-Soil Model
                                             -40


                                             -60


                                             -80


                                            -100


                                            -120
                                                   0       25         50          75       100   125   150
                                                                        Y-Coordinate (in.)

                                                                             (b) Thrust


  Figure 40—Bending Moments and Thrusts due to Live Load in Metal Arch Model (6 ft
                                    Cover)




NCHRP 15-29 Appendix A                                                                                       67
     Table 12—Comparison of Bending Moments and Thrusts in Metal Arch Model

                      (a) Bending Moments and Thrusts due to Earth Load

           Cover             Soil        Moment (lb*in/in)     Thrust (lb/in)
         Depth (ft)         Model        Peak Pos Peak Neg       Crown      Springline
             2          Linear-Elastic      496       -1,040      -445          -1,043
                        Mohr-Coulomb       1,508      -1,727      -584           -771
                        Hardening-Soil     1,996      -2,034      -602           -802
             6          Linear-Elastic      489       -1,527      -901          -1,804
                        Mohr-Coulomb       1,541      -1,461     -1,199         -1,463
                        Hardening-Soil     1,452      -1,563     -1,210         -1,508



       (b) Bending Moments and Thrusts due to Earth Load plus Surface Live Load

           Cover             Soil        Moment (lb*in/in)     Thrust (lb/in)
         Depth (ft)         Model        Peak Pos Peak Neg       Crown      Springline
             2          Linear-Elastic     5,866      -1,528      -820          -1,417
                        Mohr-Coulomb       9,062      -4,294     -1,406         -1,118
                        Hardening-Soil     8,791      -4,358     -1,453         -1,132
             6          Linear-Elastic      605       -1,566      -924          -1,872
                        Mohr-Coulomb       1,432      -1,359     -1,266         -1,518
                        Hardening-Soil     1,376      -1,735     -1,275         -1,566



                 (c) Bending Moments and Thrusts due to Surface Live Load

           Cover             Soil        Moment (lb*in/in)     Thrust (lb/in)
         Depth (ft)         Model        Peak Pos Peak Neg       Crown      Springline
             2          Linear-Elastic     5,912      -1,733    -375.38         -373.62
                        Mohr-Coulomb      10,789      -4,581    -822.28         -346.96
                        Hardening-Soil    10,820      -4,474    -851.53         -330.03
             6          Linear-Elastic      190         -83      -22.87         -67.85
                        Mohr-Coulomb        411        -175      -66.11         -54.28
                        Hardening-Soil      357        -176      -64.47         -58.12




NCHRP 15-29 Appendix A                                                                    68
3.5.7       Summary of Results from 2D Preliminary Analyses

Table 13 compares bending moments and thrusts produced by different soil models for concrete
box culvert.         Table 14 and Table 15 compare bending moments and thrusts produced by
different soil models for pipe culverts. Table 16 and Table 17 compare bending moments and
thrusts produced by different soil models for long-span arches.

The linear-elastic soil model produced similar bending moments to those produced by the other
two soil models in concrete culverts for a shallow cover depth. However, thrusts in concrete
culverts produced with the linear-elastic model were slightly different from those with the other
soil models because the interface strength was not considered in the linear-elastic model.

Bending moments and thrusts in flexible culverts produced with the linear-elastic soil model
were not close to those with the other two soil models.

Comparing the Mohr-Coulomb and Hardening-Soil models, they produced very similar structural
responses. In many cases, but not always, the Mohr-Coulomb model produced greater peak
moments and less thrust, which resulted from stress-dependent stiffness in the Hardening-Soil
model.

               Table 13—Ratios of Live Load Moments and Thrusts of Concrete Box
 Cover        Soil         Ratio to Linear-Elastic Soil Model                            Ratio to Earth Load Case
 Depth       Model         Moment                                 Thrust                  Moment                              Thrust
  (ft)                    Top Center     Tip of Haunch          Top Center Tip of Haunch Top Center    Tip of Haunch        Top Center Tip of Haunch
                                          Top         Wall                     Wall                     Top         Wall                   Wall
   0     Linear-Elastic     1.000        1.000       1.000        1.000        1.000       4.205      -11.037       2.565     -1.139       1.636
         Mohr-Coulomb       1.004        0.989       1.007        0.943        0.996       4.887      -114.162      3.180     -2.730       1.824
         Hardening-Soil     0.997        1.008       0.987        1.079        0.999       4.920      -743.613      3.193     -3.587       1.827
   2     Linear-Elastic     1.000        1.000       1.000        1.000        1.000       1.341       5.888        1.014     -0.166       0.589
         Mohr-Coulomb       0.951        0.933       0.968        -5.311       0.970       1.335       3.595        1.039     5.775        0.633
         Hardening-Soil     0.970        0.971       0.966        -4.699       0.949       1.388       3.340        1.072    -82.777       0.617
   6     Linear-Elastic     1.000        1.000       1.000        1.000        1.000       0.200       0.462        0.184     0.106        0.120
         Mohr-Coulomb       0.691        0.558       0.809        7.454        0.876       0.139       0.221        0.149     -1.360       0.117
         Hardening-Soil     0.637        0.474       0.716        10.445       0.797       0.130       0.173        0.136     -1.042       0.105




NCHRP 15-29 Appendix A                                                                                                                             69
 Table 14—Ratios of Live Load Moments and Thrusts of Pipes with a Cover Depth of 2 ft
      Structural         Soil        Ratio to Linear-Elastic Soil Model             Ratio to Earth Load Case
      Material          Model        Moment                  Thrust                 Moment                Thrust
                                     Peak Pos Peak Neg       Crown        Springline Peak Pos Peak Neg    Crown    Springline
      Concrete      Linear-Elastic    1.000       1.000      1.000         1.000     1.423      1.128     -0.128     0.805
                    Mohr-Coulomb      0.917       0.939      1.398         0.751     1.504      1.293     -0.043     0.773
                    Hardening-Soil    0.913       0.923      -1.905        0.788     1.720      1.498     0.049      0.803
        Metal       Linear-Elastic    1.000       1.000      1.000         1.000     3.763      1.836     0.887      1.060
                    Mohr-Coulomb      2.717       3.692      2.612         0.691     13.481    15.526     1.614      1.009
                    Hardening-Soil    2.477       3.396      3.178         0.827     14.058    10.425     1.906      1.195
    Thermoplastic   Linear-Elastic    1.000       1.000      1.000         1.000     3.377      1.512     -0.414     0.965
                    Mohr-Coulomb      3.382       4.845     -11.728        0.973     18.700    10.748     2.417      1.220
                    Hardening-Soil    2.814       4.002     -12.214        0.968     18.333     9.398     2.379      1.185




 Table 15—Ratios of Live Load Moments and Thrusts of Pipes with a Cover Depth of 6 ft
      Structural         Soil        Ratio to Linear-Elastic Soil Model             Ratio to Earth Load Case
      Material          Model        Moment                  Thrust                 Moment                Thrust
                                     Peak Pos Peak Neg       Crown        Springline Peak Pos Peak Neg    Crown    Springline
      Concrete      Linear-Elastic    1.000       1.000      1.000         1.000     0.060      0.055     0.006      0.040
                    Mohr-Coulomb      0.500       0.667      -5.094        0.293     0.033      0.041     -0.026     0.014
                    Hardening-Soil    0.435       0.514      -5.264        0.365     0.033      0.037     -0.022     0.018
        Metal       Linear-Elastic    1.000       1.000      1.000         1.000     0.078      0.054     0.023      0.040
                    Mohr-Coulomb      2.626       1.847      1.204         0.505     0.246      0.158     0.022      0.027
                    Hardening-Soil    1.511       1.377      0.006         0.231     0.158      0.130     0.000      0.012
    Thermoplastic   Linear-Elastic    1.000       1.000      1.000         1.000     0.080      0.051     -0.016     0.043
                    Mohr-Coulomb      1.408       2.471      -5.711        0.610     0.181      0.195     0.045      0.034
                    Hardening-Soil    1.885       2.273      -4.454        0.497     0.299      0.147     0.032      0.026




Table 16—Ratios of Live Load Moments and Thrusts of Arches with a Cover Depth of 2 ft
      Structural         Soil        Ratio to Linear-Elastic Soil Model             Ratio to Earth Load Case
      Material          Model        Moment                  Thrust                 Moment                Thrust
                                     Peak Pos Peak Neg       Crown        Springline Peak Pos Peak Neg    Crown    Springline
      Concrete      Linear-Elastic    1.000       1.000      1.000         1.000     3.003      1.617     0.443      0.310
                    Mohr-Coulomb      1.079       1.104      0.751         0.872     3.758      2.398     0.288      0.339
                    Hardening-Soil    1.040       1.003      0.824         0.928     3.843      2.663     0.309      0.355
        Metal       Linear-Elastic    1.000       1.000      1.000         1.000     11.915     1.665     0.844      0.358
                    Mohr-Coulomb      1.825       2.644      2.191         0.929     7.152      2.653     1.409      0.450
                    Hardening-Soil    1.830       2.582      2.268         0.883     5.421      2.200     1.415      0.411




Table 17—Ratios of Live Load Moments and Thrusts of Arches with a Cover Depth of 6 ft
      Structural         Soil        Ratio to Linear-Elastic Soil Model             Ratio to Earth Load Case
      Material          Model        Moment                  Thrust                 Moment                Thrust
                                     Peak Pos Peak Neg       Crown        Springline Peak Pos Peak Neg    Crown    Springline
      Concrete      Linear-Elastic    1.000       1.000      1.000         1.000     0.152      0.101     0.029      0.032
                    Mohr-Coulomb      0.901       1.035      0.498         0.885     0.142      0.120     0.013      0.033
                    Hardening-Soil    0.839       0.892      0.498         0.919     0.133      0.111     0.013      0.035
        Metal       Linear-Elastic    1.000       1.000      1.000         1.000     0.389      0.055     0.025      0.038
                    Mohr-Coulomb      2.158       2.103      2.890         0.800     0.266      0.120     0.055      0.037
                    Hardening-Soil    1.877       2.111      2.819         0.857     0.246      0.113     0.053      0.039




NCHRP 15-29 Appendix A                                                                                                          70
3.6      Effect of Interface Strength

In the 2D preliminary analyses described above, the interface strength was set 50% of the soil
shear strength. To examine the effect of interface strength on structural response, we analyzed
the concrete and thermoplastic pipe with backfill modeled by the Mohr-Coulomb constitutive
model with the interface strength equal to 100% of the soil shear strength.

Figure 41 and Figure 42 show plastic points in soil elements modeled by Mohr-Coulomb
constitutive model in the concrete pipe models with interface strengths of 50% and 100% of the
soil shear strength. Figure 43 and Figure 44 compares thrusts and bending moments in the
concrete pipe model with an interface strength of 100% of the soil shear strength with those
from the previous analyses. Table 18 compares thrusts and bending moments in the concrete
pipe models with backfill modeled by Mohr-Coulomb model between the cases with interface
strengths of 50% and 100% of the soil shear strength. By changing the interface strength from
100% of the soil shear strength to 50%, peak live load moments decreased by a few percent for
the 2 ft cover case and by 15% for the 6 ft cover case. Live load thrusts were affected more by
this change. Peak thrusts decreased by 10% for the 2 ft cover case and by 21% for the 6 ft
cover case.

Results for the thermoplastic pipe are shown in Figure 45 to Figure 48 and Table 19. By
changing the interface strength from 100% of the soil shear strength to 50%, peak live load
moments in the thermoplastic pipe increased by 18% for the 2 ft cover case and decreased by
14% for the 6 ft cover case. Peak thrusts decreased by 10% for the 2 ft cover case and by 24%
for the 6 ft cover case.

In summary, structural responses to live loads did not change significantly when the interface
strength was changed from 50% of the soil shear strength to 100% although the cases with the
100% strength showed slightly larger peak responses than those with the 50% strength except
for moments of the thermoplastic pipe with 2 ft cover.      A change in the Interface strength
affected thrusts more than moments.       Structural responses of the thermoplastic pipe were
affected more by a change of interface strength than those of the concrete pipe. Structural
responses of the 6 ft cover cases were affected more by a change of interface strength than
those of the 2 ft cover cases; however, it should be noted that responses of the 6 ft cover cases
were much smaller than those for the 2 ft cover cases.




NCHRP 15-29 Appendix A                                                                        71
                                                         Without live load




                                                          With live load


                                   (a) 50% strength




                                                          Without live load




                                                          With live load


                                  (b) 100% strength
                                                               Tension cut-off point
                                                               Mohr-Coulomb point

 Figure 41—Plastic Points in Soil Elements of Concrete Pipe Models with 50% and 100%
              Interface Strength (Mohr-Coulomb Soil Model, 2 ft Cover)




NCHRP 15-29 Appendix A                                                                 72
                                                      Without live load




                                                       With live load


                                   (a) 50% strength




                                                      Without live load




                                                       With live load


                                                                        Tension cut-off point
                                  (b) 100% strength
                                                                        Mohr-Coulomb point

 Figure 42—Plastic Points in Soil Elements of Concrete Pipe Models with 50% and 100%
              Interface Strength (Mohr-Coulomb Soil Model, 6 ft Cover)




NCHRP 15-29 Appendix A                                                                          73
                                              1200                                                 Linear-Elastic Model
                                                                                                   (Full Bonding)


          Bending Moment due to Live Load
                                              1000                                                 Mohr-Coulomb Model
                                                                                                   (50% Strength)
                                               800
                                                                                                   Hardening-Soil Model
                                                                                                   (50% Strength)
                                               600
                                                                                                   Mohr-Coulomb Model
                                               400                                                 (100% Strength)
                    (lb*in/in)


                                               200

                                                   0

                                              -200

                                              -400

                                              -600

                                              -800
                                                  -180     -135    -90      -45           0   45        90     135        180
                                                                          Degrees from Crown

                                                                        (a) Bending Moment

                                             20


                                               0
          Thrust due to Live Load (lb/in)




                                             -20


                                             -40


                                             -60


                                             -80
                                                         Linear-Elastic Model (Full Bonding)
                                                         Mohr-Coulomb Model (50% Strength)
                                            -100
                                                         Hardening-Soil Model (50% Strength)
                                                         Mohr-Coulomb Model (100% Strength)
                                            -120
                                                -180     -135     -90      -45        0       45        90     135        180
                                                                         Degrees from Crown

                                                                                 (b) Thrust



  Figure 43—Comparison of Bending Moments and Thrusts due to Live Load between
        Concrete Pipe Models with 50% and 100% Interface Strength (2 ft Cover)




NCHRP 15-29 Appendix A                                                                                                          74
                                              100
                                                                       Linear-Elastic Model (Full Bonding)



          Bending Moment due to Live Load
                                                  80                   Mohr-Coulomb Model (50% Strength)
                                                                       Hardening-Soil Model (50% Strength)
                                                  60                   Mohr-Coulomb Model (100% Strength)

                                                  40
                    (lb*in/in)

                                                  20

                                                   0

                                              -20

                                              -40

                                              -60

                                              -80
                                                 -180     -135    -90        -45          0    45     90     135     180
                                                                           Degrees from Crown

                                                                        (a) Bending Moment

                                             4

                                             2
          Thrust due to Live Load (lb/in)




                                             0

                                             -2

                                             -4

                                             -6

                                             -8
                                                                                   Linear-Elastic Model (Full Bonding)
                                                                                   Mohr-Coulomb Model (50% Strength)
                                            -10
                                                                                   Hardening-Soil Model (50% Strength)
                                                                                   Mohr-Coulomb Model (100% Strength)
                                            -12
                                               -180     -135     -90       -45        0       45      90     135     180
                                                                         Degrees from Crown

                                                                                 (b) Thrust


  Figure 44—Comparison of Bending Moments and Thrusts due to Live Load between
        Concrete Pipe Models with 50% and 100% Interface Strength (6 ft Cover)




NCHRP 15-29 Appendix A                                                                                                     75
Table 18—Comparison of Bending Moments and Thrusts between Concrete Pipe Models
         with 50% and 100% Interface Strength (Mohr-Coulomb Soil Model)

                                       (a) Moments and thrusts

       Cover       Loads      Interface Moment (lb*in/in)       Thrust (lb/in)
     Depth (ft)               Strength   Peak Pos Peak Neg        Crown      Springline Peak Neg
         2          Dead        50%        568.8     -503.2        -22.1         -90.0    -94.1
                               100%        574.7     -507.9        -17.1         -97.5    -103.5
                  Dead plus     50%       1246.8     -1113.0       -21.1         -159.5   -159.5
                    Live       100%       1269.9     -1125.8       -9.6          -170.2   -170.3
                    Live        50%        855.6     -650.8         1.0          -69.5    -72.2
                               100%        865.4     -664.0         7.5          -72.7    -80.2
         6          Dead        50%       1158.2     -1096.3       -59.0         -194.4   -196.6
                               100%       1170.3     -1098.0       -51.7         -214.8   -218.6
                  Dead plus     50%       1196.3     -1141.0       -57.5         -197.2   -199.4
                    Live       100%       1215.8     -1146.7       -49.2         -218.5   -222.5
                    Live        50%        38.7       -45.5         1.5           -2.8     -3.1
                               100%        45.5       -49.4         2.5           -3.7     -3.9


             (b) Ratios of moments and thrusts of the model with 50% strength to
                            those of the model with 100% strength

                   Cover       Moment                  Thrust
                  Depth (ft) Peak Pos Peak Neg       Crown      Springline Peak Neg
                     2          0.99       0.98       0.13        0.96       0.90
                     6          0.85       0.92       0.60         0.75          0.79




NCHRP 15-29 Appendix A                                                                             76
                                                          Without live load




                                                          With live load


                                   (a) 50% strength




                                                          Without live load




                                                          With live load


                                  (b) 100% strength
                                                               Tension cut-off point
                                                               Mohr-Coulomb point

 Figure 45—Plastic Points in Soil Elements of Thermoplastic Pipe Models with 50% and
           100% Interface Strength (Mohr-Coulomb Soil Model, 2 ft Cover)




NCHRP 15-29 Appendix A                                                                 77
                                                      Without live load




                                                       With live load


                                   (a) 50% strength




                                                      Without live load




                                                       With live load


                                                                   Tension cut-off point
                                  (b) 100% strength
                                                                  Mohr-Coulomb point

 Figure 46—Plastic Points in Soil Elements of Thermoplastic Pipe Models with 50% and
           100% Interface Strength (Mohr-Coulomb Soil Model, 6 ft Cover)




NCHRP 15-29 Appendix A                                                                     78
                                              400
                                                                                                  Linear-Elastic Model



          Bending Moment due to Live Load
                                                                                                  (Full Bonding)
                                              300                                                 Mohr-Coulomb Model
                                                                                                  (50% Strength)
                                              200                                                 Hardening-Soil Model
                                                                                                  (50% Strength)
                                                                                                  Mohr-Coulomb Model
                    (lb*in/in)

                                              100                                                 (100% Strength)


                                                   0


                                              -100


                                              -200


                                              -300
                                                  -180    -135    -90      -45        0      45       90     135     180
                                                                         Degrees from Crown

                                                                       (a) Bending Moment

                                             20                    Linear-Elastic Model (Full Bonding)
                                                                   Mohr-Coulomb Model (50% Strength)
                                               0                   Hardening-Soil Model (50% Strength)
          Thrust due to Live Load (lb/in)




                                                                   Mohr-Coulomb Model (100% Strength)
                                             -20

                                             -40

                                             -60

                                             -80

                                            -100

                                            -120

                                            -140

                                            -160
                                                -180     -135    -90      -45        0       45      90      135     180
                                                                        Degrees from Crown

                                                                                (b) Thrust


  Figure 47—Comparison of Bending Moments and Thrusts due to Live Load between
     Thermoplastic Pipe Models with 50% and 100% Interface Strength (2 ft Cover)




NCHRP 15-29 Appendix A                                                                                                     79
                                               8.0                                                  Linear-Elastic Model
                                                                                                    (Full Bonding)


          Bending Moment due to Live Load
                                               6.0                                                  Mohr-Coulomb Model
                                                                                                    (50% Strength)
                                               4.0                                                  Hardening-Soil Model
                                                                                                    (50% Strength)
                                                                                                    Mohr-Coulomb Model
                                               2.0                                                  (100% Strength)
                    (lb*in/in)


                                               0.0

                                               -2.0

                                               -4.0

                                               -6.0

                                               -8.0
                                                   -180     -135     -90     -45           0   45     90     135     180
                                                                            Degrees from Crown

                                                                         (a) Bending Moment

                                            1.0
                                                                     Linear-Elastic Model (Full Bonding)
                                            0.0                      Mohr-Coulomb Model (50% Strength)
          Thrust due to Live Load (lb/in)




                                                                     Hardening-Soil Model (50% Strength)
                                            -1.0
                                                                     Mohr-Coulomb Model (100% Strength)
                                            -2.0

                                            -3.0

                                            -4.0

                                            -5.0

                                            -6.0

                                            -7.0

                                            -8.0
                                                -180      -135     -90      -45        0       45    90      135     180
                                                                           Degrees from Crown

                                                                                  (b) Thrust


  Figure 48—Comparison of Bending Moments and Thrusts due to Live Load between
     Thermoplastic Pipe Models with 50% and 100% Interface Strength (6 ft Cover)




NCHRP 15-29 Appendix A                                                                                                     80
 Table 19—Comparison of Bending Moments and Thrusts Load between Thermoplastic
    Pipe Models with 50% and 100% Interface Strength (Mohr-Coulomb Soil Model)

                                         (a) Moments and thrusts

         Cover       Loads      Interface Moment (lb*in/in)       Thrust (lb/in)
       Depth (ft)               Strength   Peak Pos Peak Neg        Crown      Springline Peak Neg
           2          Dead        50%        17.3       -19.6        -37.8         -58.6    -59.2
                                 100%        15.7       -24.0        -29.2         -68.5    -68.8
                    Dead plus     50%        313.4     -208.9       -129.2         -130.1   -181.9
                      Live       100%        275.4     -191.2       -112.3         -124.7   -198.6
                      Live        50%        322.8     -210.5        -91.4         -71.5    -135.1
                                 100%        273.2     -189.6        -83.0         -56.2    -150.7
           6          Dead        50%        28.0       -29.2        -75.5         -116.9   -118.3
                                 100%        24.7       -40.8        -59.8         -140.3   -141.2
                    Dead plus     50%        29.2       -29.9        -78.9         -120.8   -122.8
                      Live       100%        28.2       -42.4        -63.0         -145.7   -147.0
                      Live        50%         5.1       -5.7         -3.4           -3.9     -5.1
                                 100%         5.9       -6.5         -3.2           -5.4     -6.7


               (b) Ratios of moments and thrusts of the model with 50% strength to
                              those of the model with 100% strength

                     Cover       Moment                  Thrust
                    Depth (ft) Peak Pos Peak Neg       Crown      Springline Peak Neg
                       2          1.18       1.11       1.10        1.27       0.90
                       6          0.86       0.88       1.08         0.73          0.76




3.7      Conclusion

The Mohr-Coulomb and Hardening-Soil models produced similar structural responses of
culverts to surface live loads in the preliminary 2D analysis. Since the Hardening-Soil model is
more sophisticated model and it has more parameters to be determined, the Mohr-Coulomb
model is the best candidate soil model for the parametric study of NCHRP 15-29.

The interface strength did not significantly affect structural response of culverts to surface live
loads in the 2D analysis with the Mohr-Coulomb soil model, especially for the cases with a
shallow cover. This suggests that soil failure is most important effect in capturing live load
effects than slippage at the interface.




NCHRP 15-29 Appendix A                                                                               81
4.       THREE DIMENSIONAL MODELING OF CULVERTS

4.1      Comparison of Responses to Factored and Unfactored Live Loads

4.1.1    Introduction

In the panel comments on early reports, a few panel members showed their interest in a
comparison of structural responses to unfactored and factored live loads. If the structure and
surrounding soil have linear-elastic material properties, structural responses to the factored live
loads will differ from those to the unfactored live loads by a load factor. However, backfill
surrounding the structure is nonlinear, and the ratio of structural response to the factored load to
the response to the unfactored live load will not be exactly equal to the load factor. To examine
the effect of soil nonlinearity, SGH performed soil-structure interaction analyses of culverts
subjected to factored and unfactored live loads, and compared responses between the factored
and unfactored load cases.

4.1.2    Method of Approach

Three-dimensional soil-structure interaction analysis of HDPE pipe subjected to the surface live
load was performed using ABAQUS. HDPE pipes tested in the MNDOT study were selected
(Pipe Run 7 with A-2 backfill and 2.8 ft cover and Pipe Run 3 with A-2 backfill and 1.6 ft). For
each case, one analysis was performed with unfactored live load; another was performed with
factored live load. Design tandem (a pair of 25 kip axles spaced 4 ft apart with a transverse
spacing of 6 ft) specified in AASHTO LRFD specifications was used in the analysis with a tire
contact area of 20 in. x 10 in., a multiple presence factor of 1.2, and dynamic load allowance
calculated as 33% x (1.0—0.125 x cover height). A load factor of 1.75 was used for live load,
corresponding to Strength Limit I in AASHTO LRFD specifications. Only live load was factored
in the analysis. FEA models and soil properties used were described in Section 4.5.

We also performed 2D analyses of an three-sided arch top culvert (24 ft x 6 ft) with 3 ft cover.
SW95 properties were used for backfill soil, which was modeled by Duncan-Selig soil model.
The finite element model is shown in Figure 49. HS20 truck was assumed for live load truck,
and live load distribution along the length of the arch was calculated per AASHTO LRFD
specifications. A multiple presence factor of 1.2 and dynamic load allowance calculated as
33% x (1.0—0.125 x cover height) were also included. Equivalent 2D service live load was
calculated as 377 lb for a 1-in. thick slice as shown in Figure 49. With this model, we performed
three analyses for the following cases: Case 1 with unfactored earth load and unfactored live
load, Case 2 with unfactored earth load and factored live load, and Case 3 with factored earth



NCHRP 15-29 Appendix A                                                                           82
load and unfactored live load. Load factors of 1.35 and 1.75 were used for earth load and live
load, respectively.


                                            P=377 lb




                                           Span = 24 ft
                                            Rise = 6 ft
        28 ft




                                              80 ft

                                                          Soil zones:
                                                             : In-situ soil (linear elastic)
                                                             : Concrete footing (linear elastic)
                                                             : Back fill (SW95 Duncan-Selig)

     Figure 49—Finite Element Model of Three-Sided Arch Top Culvert with 3 ft Cover



4.1.3           Results

4.1.3.1 HDPE Pipe in ABAQUS

Figure 50 and Figure 51 compare displacement and force results between the cases with
unfactored and factored live loads. To make the comparison easier, displacements and forces
in Figure 50 and Figure 51 for the unfactored live load are 1.75 times those from the analysis
with the unfactored live load.   Table 20 summarizes peak responses and shows ratios of
responses to the factored live load to responses to the unfactored live load. It is apparent in
Table 20 that ratios between the two cases are very close to 1.75. The maximum deviation
from 1.75 among the responses compared in Table 20 is 1.805, which is only 3 percent greater
than 1.75.




NCHRP 15-29 Appendix A                                                                             83
                                            0.25                                                                                                                                       0.07




                                                                                                                                        Diametrical Change between Springlines (in.)
                                                                                        2.8 ft Cover Unfactored LL X 1.75                                                                                                                 2.8 ft Cover Unfactored LL X 1.75
     Vertical Displacement at Crown (in.)


                                                                                        2.8 ft Cover Factored LL                                                                       0.06                                               2.8 ft Cover Factored LL
                                            0.20                                        1.6 ft Cover Unfactored LL X 1.75                                                                                                                 1.6 ft Cover Unfactored LL X 1.75
                                                                                        1.6 ft Cover Factored LL                                                                       0.05                                               1.6 ft Cover Factored LL

                                            0.15
                                                                                                                                                                                       0.04


                                                                                                                                                                                       0.03
                                            0.10

                                                                                                                                                                                       0.02
                                            0.05
                                                                                                                                                                                       0.01


                                            0.00                                                                                                                                       0.00
                                                   0         2          4           6              8         10         12                                                                    0         2             4               6             8           10        12
                                                        Distance from Symmetry Line of Tandem Axles (ft)                                                                                             Distance from Symmetry Line of Tandem Axles (ft)


                                                       (a) Vertical crown displacement                                                                                                              (b) Horizontal chord extension

                                            Figure 50—Comparison of Vertical and Horizontal Displacements from Factored Live
                                               Load with 1.75 times those from Unfactored Live Load (HDPE Pipe, A2 Backfill)



                                                                                                                                                                                                  2.8 ft Cover Unfactored LL X 1.75           2.8 ft Cover Factored LL
                                            100                                                                                                                                        30
                                                                                                                                                                                                  1.6 ft Cover Unfactored LL X 1.75           1.6 ft Cover Factored LL
                                             90
                                                                                                                                                                                       20
                                                                                                                             Bending Moment (lb-in./in.)




                                             80

                                             70
                                                                                                                                                                                       10
Thrust (lb/in.)




                                             60

                                             50                                                                                                                                         0

                                             40
                                                                                                                                                                                       -10
                                             30                       2.8 ft Cover Unfactored LL X 1.75
                                             20                       2.8 ft Cover Factored LL
                                                                      1.6 ft Cover Unfactored LL X 1.75                                                                                -20
                                             10
                                                                      1.6 ft Cover Factored LL
                                              0                                                                                                                                        -30
                                               180     135       90      45        0         -45       -90   -135     -180                                                                180     135       90        45         0            -45       -90      -135    -180
                                                                       Degrees from Crown                                                                                                                          Degrees from Crown

                                                                            (a) Thrust                                                                                                                                    (b) Moment

Figure 51—Comparison of Thrusts and Moments from Factored Live Load with 1.75 times
             those from Unfactored Live Load (HDPE Pipe, A2 Backfill)




NCHRP 15-29 Appendix A                                                                                                                                                                                                                                                        84
                  Table 20—Comparison of Structural Responses between Analyses with Factored and
                                  Unfactored Live Loads (HDPE Pipe, A2 Backfill)
         Case                             Displacement under wheel (in.)                                                    Thrust (lb/in.)                       Moment (lb-in./in.)
    Cover                   Live Load          Vertical               Horizontal                Crown                           Springline         Peak        Peak Pos.           Peak Neg.
                            Unfactored          0.096                     0.025                 27.3                                30.6           35.5                14.0          -10.9
          2.8 ft             Factored           0.169                     0.044                 49.3                                53.6           63.3                24.2          -19.1
                              Ratio:
                            Factored /          1.757                     1.744                 1.805                               1.752          1.781           1.730             1.750
                            Unfactored
                            Unfactored          0.126                     0.033                 43.2                                43.8           51.7                13.9          -14.8
          1.6 ft             Factored           0.220                     0.057                 76.2                                75.8           91.4                24.6          -25.8
                              Ratio:
                            Factored /          1.749                     1.750                 1.764                               1.731          1.769           1.769             1.740
                            Unfactored




4.1.3.2 Three-Sided Arch Top Culvert in CANDE

Figure 52 compares force results from the three cases examined for the Hanson arch. To make
the comparison easier, forces in Figure 52 for the unfactored live load are 1.75 times those from
the analysis with the unfactored live load. Table 21 summarizes peak responses and shows
ratios of responses to the factored live load to responses to the unfactored live load. As is the
case with HDPE pipes analyzed in ABAQUS, ratios between factored and unfactored live load
cases are very close to 1.75.                                          The maximum deviation from 1.75 among the responses
compared in Table 21 is 1.792, which is only 2 percent greater than 1.75.




                  600                                                                                                  15000


                  500                                                                                                  10000
                                                                                                 Moment (in.-lb/in.)




                  400                                                                                                   5000
Thrust (lb/in.)




                  300                                                                                                      0


                  200                                                                                                   -5000

                                                               Factored DL & LL                                                                Factored DL & LL
                  100                                          Factored LL & Unfactored DL                             -10000                  Factored LL & Unfactored DL
                                                               Unfactored DL & LL X 1.75                                                       Unfactored DL & LL X 1.75
                   0                                                                                                   -15000
                        0      10        20     30        40         50           60       70                                   0      10     20       30         40          50    60       70
                                              Element Number                                                                                        Element Number

                                              (a) Thrust                                                                                           (b) Moment

Figure 52—Comparison of Thrusts and Moments from Factored Live Load with 1.75 times
                  those from Unfactored Live Load (Hanson Arch)




NCHRP 15-29 Appendix A                                                                                                                                                                       85
      Table 21—Comparison of Structural Responses between Analyses (Hanson Arch)
      Case                                   Vertical Crown      Peak Thrust    Moment (lb-in./in.)
                                            Displacement (in.)     (lb/in.)    Peak Pos.   Peak Neg.
      Case 1: Unfactored DL & LL                  0.28              304         -6,180        6,298
      Case 2: Unfactored DL & Factored LL         0.48              527         -10,640      11,175
      Case 3: Factored DL & LL                    0.49              527         -10,750      11,287
      Ratio: Case 2 / Case 1                      1.741             1.735        1.722        1.774
      Ratio: Case 3 / Case 1                      1.755             1.735        1.739        1.792




4.1.4     Conclusion

Based on the limited cases we examined in this study, structural responses to the factored live
load can be estimated by scaling those to the unfactored live load by a load factor.

4.2       Selected Field Tests for 3D Analysis

As one final validation of the suitability of the Mohr-Coulomb model for evaluating structural
response of culverts subjected to surface live loads, we conducted a set of 3D soil-structure
interaction analyses using Plaxis 3D Tunnel Version 2 (Brinkgreve, 2004) to simulate selected
field tests and compared predicted responses with the existing field test data.

We selected two studies: NCHRP Project 12-45 (Webb, 1999; Taleb, 2000; McGrath et al.,
2002) and Minnesota DOT study (McGrath et al., 2002; McGrath and Beaver, 2005). These
studies include a reinforced concrete arch culvert, a corrugated structural plate metal culvert,
and a corrugated polyethylene pipe. Details of each culvert are provided below.

4.2.1     NCHRP Project 12-45

Long-span reinforced concrete and metal arch culverts were tested at the University of
Massachusetts, Amherst to investigate the structural behavior when subjected to live loads with
shallow fills (1 ft to 3 ft cover). A 30-ft span x 11-ft 4-in. rise x 42-ft long reinforced concrete
arch culvert and a 31-ft 2-in. span at the footing x 12-ft 1-in. rise x 40-ft long structural plate
metal arch culvert were installed end to end in a pre-excavated wide trench as shown in Figure
53. The trench was backfilled with existing site material, a well-graded sand with gravel.

The concrete arch culvert was the BEBO type arch, designated BEBO Type E30/3. Properties
of the concrete arch culvert are summarized in Table 22. The metal culvert was a Contech
Construction Products Type 108A30 nongalvanized corrugated steel arch culvert. Properties of




NCHRP 15-29 Appendix A                                                                                 86
the metal arch culvert are summarized in Table 23. Each culvert was supported on 4.9 ft wide x
2 ft deep continuous reinforced concrete spread footings.

Live load testing was conducted with a tandem-axle truck with 70,000 lb on the tandem axles at
depths of fill of 3 ft, 2 ft, and 1 ft. Center-to-center spacing of the tandem axles was 4 ft 7 in.,
and center-to-center spacing between the wheels of the tandem axles was 6 ft 5 in. Live load
testing was conducted twice: once with backfill compacted to 92 percent of maximum density
(Test 1) and once with backfill compacted to 87 percent of maximum density (Test 2).

Figure 54 and Figure 55 summarize locations of displacement measurements in the concrete
and metal arch culverts, respectively.




                      In-Situ Soil




                                      (a) Cross-sectional view




                                         (b) Elevation view


                        Figure 53—Test Setup of NCHRP Project 12-45




NCHRP 15-29 Appendix A                                                                           87
               Table 22—Properties of Reinforced Concrete Culvert
                                    Culvert Properties
                                                     30 ft (inside)
                  Span
                                                     31 ft 8 in. (outside)
                                                     11 ft 4 in. (inside)
                  Rise
                                                     12 ft 2 in. (outside)
                  Wall Thickness                     10 in.
                  Cross Sectional Area per Unit
                                                     10 in.2/in.
                  Length
                  Moment of Inertia per Unit
                                                     83.3 in.4/in.
                  Length
                  Specified Compressive Strength 4,200 psi
                  Poisson’s Ratio                    0.17
                  Density                            150 pcf
                         Circumferential Reinforcement Details
                  Area of Inside Steel               0.0451 in.2/in.
                  Area of Outside Steel              0.0451 in.2/in.
                  Specified Yield Strength           70 ksi
                  Inside Cover                       1.5 in.
                  Outside Cover                      2.0 in.


             Table 23—Properties of Structural Steel Plate and Culvert
                                   Culvert Properties
                  Bottom Span                              31 ft 2 in.
                  Maximum Span                             31 ft 7 in.
                  Total Rise                               12 ft 1 in.
                  Top Radius                               20 ft 7 in.
                  Side Radius                              7 ft 3 in.
                  Angle below Horizontal                   14˚ 3’
                                     Plate Properties
                  Corrugation Pitch and Depth              6 in. x 2 in.
                  Uncoated Plate Thickness                 0.215 in.
                  Nominal Uncoated Section Depth           2.215 in.
                  Cross Sectional Area per Unit Length     0.267 in.2/in.
                  Moment of Inertia per Unit Length        0.127 in.4/in.
                  Section Modulus                          0.115 in.3/in.
                  Modulus of Elasticity                    29,000 ksi
                  Poisson’s Ratio                          0.3
                  Yield Strength                           40.9 ksi
                  Ultimate Strength                        55.0 ksi
                  Density                                  490 pcf




NCHRP 15-29 Appendix A                                                       88
          Figure 54—Instrumentation for Deformation in Concrete Culvert




            Figure 55—Instrumentation for Deformation in Metal Culvert




NCHRP 15-29 Appendix A                                                    89
4.2.2     Minnesota DOT Study

Corrugated high density polyethylene (HDPE) pipes with a diameter of 60 in. were tested at the
MnRoad Research Center to investigate performance of PE pipes under live loads with shallow
fills (1 ft to 3 ft). The PE pipes were either Type S or Type D corrugation as shown in Figure 56.
They both met the requirements of AASHTO M294. The Type S pipes were manufactured by
Hancor, Inc. of Findlay, Ohio, and The Type D pipes were manufactured by Advanced Drainage
Systems, Inc. of Hilliard, Ohio. As discussed in Section 4.3.4, only pipes with the Type S
corrugation were modeled in the 3D analysis. Properties of the Type S pipe are summarized in
Table 24.

Figure 57 shows typical installation of a test pipe. The test pipes were installed in a rectangular
trench.     They were then backfilled with soils meeting the requirements of either Group
Classification A-1 or A-2 per AASHTO M145. The target compaction of backfill was 90 percent
of the maximum standard Proctor density. The road surface consisted of 8 in. of gravel base
and 4 in. of asphalt pavement. The average trench width, cover depth, backfill material, and
percent compaction of backfill for each test pipe are shown in Table 25.

Two test vehicles were used: a truck with a maximum axle load of 24,000 lb (heavy truck) and a
truck with a maximum axle load of 18,000 lb (light truck). Axle loads of the test vehicles are
shown in Figure 58.

Figure 59 shows typical test pipe instrumentation for extensively instrumented sections. The
accuracy of LVDTs was 0.005 in. Some pipe sections were instrumented to collect static data,
and others were instrumented to collect dynamic data. Static tests were conducted by placing
the truck wheels over the instrumented pipe cross sections. Dynamic tests were conducted by
recording data at a sampling rate of 200 Hz while the test vehicle passed over the pipes. Only
static tests were modeled in the 3D analysis in the current study.




NCHRP 15-29 Appendix A                                                                          90
                                      Type D




                                      Type S
           Figure 56—Cross Sections of HDPE Pipes: Type D and Type S



                    Table 24—Properties of Type S HDPE Pipe
                  Inside Diameter                          60 in.
                  Outside Diameter                         67.3 in.
                  Cross Sectional Area per Unit Length     0.538 in.2/in.
                  Moment of Inertia per Unit Length        0.798 in.4/in.
                  Distance from Inside Diameter to
                                                           1.37 in.
                  Neutral Axis
                  Modulus of Elasticity                    100 ksi
                  Poisson’s Ratio                          0.35
                  Density                                  0.0344 pci



                                                        CL of Truck



                      Depth of Fill
                                                                      Open End (typ.)




                                                 Pipe        60 in. nominal




                                       ~ 65 ft

                    Figure 57—Typical Installation of PE Pipe



NCHRP 15-29 Appendix A                                                                  91
     Table 25—Average Trench Measurements for Test Pipes in the MNDOT Study
                                                 Average           Average          % of Maximum
       Pipe                     Backfill
               Pipe Type                          Cover             Trench       Standard Proctor Dry
       Run                      Material
                                                 Depth (ft)        Width (ft)     Density (Average)
       1         Type S             A-1            1.4                8.0                 97
       2         Type D                            1.4                9.2                 93
       3         Type S             A-2            1.6                8.8                 91
       4         Type D                            1.7                8.8                 88
       7         Type S                            2.8                9.5                 82
       8         Type D                            2.8                8.5                 85
       9         Type S             A-1            2.5                9.2                 85
       10        Type D                            2.4                9.0                 87




                                          19’-0 3/4”   4’-4”        34’-7 5/8”     4’-0”




                        Axle No:
         Light Truck (80,000 lb):
       Heavy Truck (102,000 lb):

                                                               Reference Axle

                      Figure 58—Live Load Vehicle in the MNDOT Study




NCHRP 15-29 Appendix A                                                                                  92
                   Pavement
                   Surface
                                                                                   OCP OCV

                                              Settlement Gage

                   Soil Pressure Cell                                                           Type
                                                                                                ADS D
                                                                                                4 gages
                                                                                                4 gages
                                                          Strain
                                                          Gages                     ICP   ICV

                                                                                     OTC OTE

                                                                                                  OW
                                        30°
                                                                                                  Hancor S
                                                                                                    Type
                                                                                                    gages
                                                                                                  5 5 gages



                                                                             IF1                   IF2
                                LVDTs
                                                                          OCP=outside center pipe
                                                                          OCV=outside center valley
                                                                          ICP=inside center pipe
                                                          Thermocouples   ICV=inside center valley

                                                                          OTC=outside top center
                                                                          OTE=outside top edge
                                                                          OW=outside web
                                                                          IF1=inside foot 1
                                                                          IF2=inside foot 2




            Figure 59—Typical Test Pipe Instrumentation in the MNDOT Study



4.3      Three-Dimensional Analysis

4.3.1    General Information

The 3D finite element analysis was performed using a commercial soil-structure interaction finite
element software, Plaxis 3D Tunnel Version 2 (Plaxis 3D). Plaxis 3D uses 15-node wedge
elements for soils and 8-node plate elements for structures.

Figure 60 shows typical dimensions of finite element models of long-span arches and HDPE
pipes. By using symmetry conditions, only one side of axles was modeled. The long-span
concrete and metal arch models had a length of 20 ft in the longitudinal direction, and the HDPE
pipe model had a length of 12 ft.



NCHRP 15-29 Appendix A                                                                                        93
Linear-elastic properties were assigned to structures and in-situ soils. The Mohr-Coulomb soil
model was used to describe constitutive models of backfill soils. Parameters for Mohr-Coulomb
model were determined based on Duncan-Selig parameters (Duncan et al., 1980; Selig, 1988)
and elastic parameter recommended by Selig (1990). Procedures to determine Mohr-Coulomb
soil parameters were described in Section 2.2. A tension cut-off stress of 0 psi was used for all
backfill soil types. Actual soil parameters used in the analysis are described in the sections
below.

                                                    Cover Depth
                                                      (varies)
                                                           24             Embankment
                                                      77
                                                                                Backfill
          28 ft 2 in.




                                                                                                     10 ft 2 in.


                        In-Situ Soil                                46 ft                            18 ft


                                                                  83 ft 8 in.

                                                                    192 ft

                                                           (a) Long-span arch

                                   4 in. Pavement      8 in. Gravel


                                                                                                   12 in.
                                       Cover Depth
                                         (varies)                                          AASHTO Backfill
                                                                                             (A-1 or A-2)


                                                                                             6 in. Bedding
                                                                    60 in.
                                                                   Nominal

                                  In-Situ Soil                                         84 in.
                                                                   8 ft 8 in.




                                                                      30 ft

                                                            (b) HDPE pipe
 Figure 60—Typical Dimensions of Finite Element Models of Long-Span Arch and HDPE
                                        Pipe



4.3.2    Long-Span Concrete Arch Culvert

4.3.2.1 Finite Element Model

Figure 61 shows a finite element model of the concrete arch culvert with a cover depth of 3 ft.
The coordinate system of the model was oriented so that the x-axis aligns with the transverse
direction of the culvert (parallel to span), the y-axis aligns with the vertical direction, and the z-



NCHRP 15-29 Appendix A                                                                                             94
axis aligns with the longitudinal direction. The model had a length of 20 ft in the longitudinal
direction of the culvert. The model with a cover depth of 3 ft had 7,623 elements and 23,086
nodes.

Four models were created in total: (1) Test 1 with a cover depth of 3 ft, (2) Test 1 with a cover
depth of 1 ft, (3) Test 2 with a cover depth of 3 ft, and (4) Test 2 with a cover depth of 1 ft.




                                                                                          28 ft 2 in.

                y
                                         192 ft
                                                                                      20 ft
                        x
          z                                                         Plane of symmetry
                                             (a) 3D model




      y


              x


                                           (b) Cross section

   Figure 61—Finite Element Model of Concrete Arch Culvert with a Cover Depth of 3 ft



4.3.2.2 Materials

Table 26 shows soil properties used in the 3D analyses of the concrete culvert. Backfill and
embankment of Test 1 were assigned properties of SW95, and those of Test 2 were assigned to
properties of SW85. Densities of the backfill soils were from those used in the computational
models of the NCHRP project 12-45 (McGrath et al., 2002).              Table 27 gives properties of
concrete used in the 3D analyses.


NCHRP 15-29 Appendix A                                                                                  95
         Table 26—Soil Properties Used for the 3D Analyses of Long-Span Arches
                                          Modulus
                    Density    Depth          of       Poisson’s      Angle of    Angle of    Cohesion
   Soil Type
                     (pcf)      (ft)      Elasticity     Ratio        Friction    Dilatancy     (psi)
                                            (psi)
    Backfill         121        0 to 1      1,600         0.40            57.8       27.8      0.001
    SW95                        1 to 5      4,100         0.29            54.3       24.3
(Mohr-Coulomb)                 5 to 10      6,000         0.24            53.2       23.2
                               10 to 18     8,600         0.23            52.2       22.2
    Backfill         111        0 to 1      1,300         0.26            42.0       12.0      0.001
    SW85                        1 to 6      2,100         0.21            40.4       10.4
(Mohr-Coulomb)                 6 to 11      2,600         0.19            39.5       9.5
                               11 to 18     3,300         0.19            39.0       9.0
     In-Situ
                     127         any        6,000         0.25
 (Linear-Elastic)



      Table 27—Concrete Properties Used for the 3D Analyses of Long-Span Arches
                                       Density          Modulus of
                                                                            Poisson’s Ratio
                                        (pcf)          Elasticity (ksi)
               Concrete Arch             150               3,694                 0.17
                 Footing                 150               3,916                 0.17



4.3.2.3 Loading and Boundary Condition

Side planes of the model (y-z planes) were fixed in the x-direction. The bottom plane of the
model (x-z plane) was fixed in the y-direction. The front and rear planes of the model (x-z
planes) were fixed in the z-direction. Plate elements that extended to the front and rear planes
were fixed about rotations around the x- and y-axes.

Loading steps in the analysis were: (1) in-situ soil under gravity, (2) in-situ soil and the culvert
under gravity, (3) in-situ soil, the culvert, and backfill under gravity, and (4) gravity and live
loads. Effects of construction sequence during backfilling were not considered. We did not
assign horizontal stresses in the backfill as initial conditions, but let the horizontal stresses be
those due to gravity effects; therefore, the horizontal stresses in the analysis may be different
from those in the field tests after compaction.

In the analysis, the tandem axles were placed symmetrically over the crown of the arch as
shown in Figure 62. The footprint of two wheels was assumed to be 12 in. x 24 in.




NCHRP 15-29 Appendix A                                                                              96
                                                                                          Longitudinal
                                  14 ft (assumed)             4 ft 7 in.                   direction
                   20 ft


                                                                      6 ft
           North                                                                                 South

                                         24 in.
                                                                             Plane of symmetry
                                                  12 in.


              Axle Load =   14.4 kip        35.1 kip       33.7 kip


                                             Crown of culvert

          Figure 62—Live Load Position in the 3D Analysis of Long-Span Arches



4.3.2.4 Results

Figure 63 shows deformed shapes of the concrete culvert due to live loads for the four different
cases. Figure 64 shows vertical and horizontal displacement results due to live loads along the
length of the concrete culvert for the four cases. Figure 65 shows thrusts and moments in the
concrete culvert due to live loads for the four cases. Table 28, Table 29, and Table 30 compare
vertical displacements at crown, chord extensions, thrusts at base between the field tests and
the 3D analyses. In the presentation of results, the downward displacement of the crown is
taken as positive, and the vertical displacement of the crown is the relative displacement
between the crown and footings.         Compressive force is taken as positive for thrusts, and
moment that produces tension on the inside of the culvert wall is taken as positive.

Vertical crown displacements of the concrete arch estimated by the 3D analysis were much
larger than those measured in the field tests. Horizontal chord extensions of the concrete arch
estimated by the 3D analysis were also larger than those measured in the field test except for
the case of Test 1 with a cover depth of 1 ft. Thrusts at the base of the concrete arch from the
3D analyses were significantly smaller than those measured in the field test, especially for
Test 2.

Figure 66 shows plastic points in the soil elements after the surface live loads were applied.
Plastic points are the integration points in a plastic state.                   Two types of plastic points are



NCHRP 15-29 Appendix A                                                                                      97
defined: tension cut-off point and Mohr-Coulomb point. Tension cut-off point indicates that the
tension cut-off criterion was applied to the integration point. Mohr-Coulomb point indicates that
the integration point lies on the Mohr-Coulomb failure surface. Tension failure occurred near
the wheels at the surface. Since Test 2 was conducted with less compacted backfill (SW85)
than Test 1, more Mohr-Coulomb plastic points were found in Test 2 cases than in Test 1 cases.




                                                                                                         Undeformed
                                                                                                         Test 1 (Cover = 3 ft)
                                                                                                         Test 1 (Cover = 1 ft)
                                                                                                         Test 2 (Cover = 3 ft)
                                                                                                         Test 2 (Cover = 1 ft)                                                              400X Deformation


                                               Figure 63—Deformed Shapes of Concrete Arch in the Plane of Wheel Loads
                                                                    (Effects of Live Loads only)



                                        0.06                                                                                                                              0.045
                                                                                                                     Horiz. Extension of Chord at Height of 88 in. (in)




                                                                                                                                                                          0.040
  Vertical Displacement at Crown (in)




                                        0.05
                                                                                                                                                                          0.035

                                        0.04                                                                                                                              0.030

                                                                                                                                                                          0.025
                                        0.03
                                                                                                                                                                          0.020

                                        0.02                                            Test 1 (Cover = 3 ft)                                                             0.015                                          Test 1 (Cover = 3 ft)
                                                                                        Test 1 (Cover = 1 ft)                                                                                                            Test 1 (Cover = 1 ft)
                                                                                                                                                                          0.010
                                        0.01                                            Test 2 (Cover = 3 ft)                                                                                                            Test 2 (Cover = 3 ft)
                                                                                        Test 2 (Cover = 1 ft)                                                             0.005                                          Test 2 (Cover = 1 ft)
                                                           Wheel Load Location                                                                                                               Wheel Load Location
                                        0.00                                                                                                                              0.000
                                               0            5                    10        15                   20                                                                0           5                10            15                  20
                                                     Distance from Symmetry Line of Tandem Axles (ft)                                                                                  Distance from Symmetry Line of Tandem Axles (ft)


                                                   (a) Vertical displacement at crown                                                                                             (b) Chord extension at a height of 88 in.

               Figure 64—Displacements Due to Live Loads from the 3D Analyses of Concrete Arch
                                                   Culvert




NCHRP 15-29 Appendix A                                                                                                                                                                                                                           98
                        kip/ft
                        kip/ft                                                     kip-in/ft
                                                                                    kip*in/ft
                            -4                                                           -40




                            8                                                             80
                    Test 1 (Cover = 3 ft)                                        Test 1 (Cover = 3 ft)
                    Test 1 (Cover = 1 ft)                                        Test 1 (Cover = 1 ft)
                    Test 2 (Cover = 3 ft)                                        Test 2 (Cover = 3 ft)
                    Test 2 (Cover = 1 ft)                                        Test 2 (Cover = 1 ft)



                  (a) Thrust                                                   (b) Moment

Figure 65—Thrusts and Moments due to Live Loads in the Plane of Wheel Loads from the
                       3D Analyses of Concrete Arch Culvert




    Table 28—Vertical Displacements at Crown of Concrete Arch due to Live Loads
       Cover      Test 1                                           Test 2
                Field Test           Plaxis 3D        Ratio:      Field Test   Plaxis 3D                 Ratio:
         (ft)                                       Plaxis 3D /                                    Plaxis 3D /
                   (in.)                (in.)       Field Test
                                                                     (in.)        (in.)            Field Test
          3       0.008                     0.046      5.83         0.024       0.052                    2.19
          1       0.028                     0.050      1.83         0.012       0.056                    4.71



Table 29—Chord Extension at Height of 88 in. of Concrete Arch Culvert due to Live Loads
       Cover      Test 1                                           Test 2
                Field Test           Plaxis 3D        Ratio:      Field Test   Plaxis 3D                 Ratio:
         (ft)                                       Plaxis 3D /                                    Plaxis 3D /
                   (in.)                (in.)       Field Test
                                                                     (in.)        (in.)            Field Test
          3       0.016                     0.031      1.98         0.008       0.037                    4.66
          1       0.035                     0.034      0.96         0.016       0.039                    2.46



         Table 30—Thrusts at Base of Concrete Arch Culvert due to Live Loads
       Cover      Test 1                                           Test 2
                Field Test           Plaxis 3D        Ratio:      Field Test   Plaxis 3D                 Ratio:
         (ft)                                       Plaxis 3D /                                    Plaxis 3D /
                  (kip/ft)            (kip/ft)      Field Test
                                                                    (kip/ft)    (kip/ft)           Field Test
          3     1.37 - 2.40                 0.923   0.38 - 0.67     2.672       0.843                    0.32
          1     1.37 - 2.40                 0.938   0.39 - 0.68     6.715       0.845                    0.13




NCHRP 15-29 Appendix A                                                                                            99
                                 (a) Test 1 and 3 ft cover




                                 (b) Test 1 and 1 ft cover




                                 (c) Test 2 and 3 ft cover




                                 (d) Test 2 and 1 ft cover
                                                                Tension cut-off point
                                                                Mohr-Coulomb point
Figure 66—Plastic Points in Soil Elements in the Plane of Wheel Loads in Concrete Arch
                                       Analysis




NCHRP 15-29 Appendix A                                                                  100
4.3.3    Long-Span Metal Arch Culvert

4.3.3.1 Finite Element Model

Figure 67 shows a finite element model of the metal arch culvert with a cover depth of 3 ft. The
coordinate system of the model was oriented as reported above for the concrete culvert. The
model had a length of 20 ft in the longitudinal direction of the culvert. The model with a cover
depth of 3 ft had 17,512 elements and 49,928 nodes.

The metal arch made of corrugated structural metal plates has different axial and bending
stiffnesses in the circumferential and longitudinal directions. EA and EI are summarized in
Table 31 for the circumferential and longitudinal directions. However, since orthotropic material
properties could not be specified for plate elements in Plaxis 3D, short elements with low
section properties were inserted between elements with section properties close to
circumferential properties to match stiffnesses as shown in Figure 68. A 1 , I 1 , A 2 , and I 2 for
L 2 /L 1 =1/11 are given in Figure 68.

Four models were created in total: (1) Test 1 with a cover depth of 3 ft, (2) Test 1 with a cover
depth of 1 ft, (3) Test 2 with a cover depth of 3 ft, and (4) Test 2 with a cover depth of 1 ft.




                                                                                          28 ft 2 in.
                  y
                                           192 ft
                                                                                       20 ft
                           x
             z                                                        Plane of symmetry
                                             (a) 3D model




         y


                 x
                                           (b) Cross section

     Figure 67—Finite Element Model of Metal Arch Culvert with a Cover Depth of 3 ft



NCHRP 15-29 Appendix A                                                                             101
Table 31—Axial and Bending Modulus of Metal Arch in Circumferential and Longitudinal
                            Directions (E=29,000 ksi)
                             Direction                 EA (lb/in.)        EI (lb-in.2/in.)
                          Circumferential              7,731,400            3,680,100
                           Longitudinal                  47,908              201,057




                                                                        L2/L1=1/11

                                   A1          I1               A2          I2
                                (in2/in)    (in4/in)         (in2/in)    (in4/in)

                                0.2908      0.1384         1.384x10-4   1.112x10-4


          Figure 68—Soft Element to Match Longitudinal Stiffness of Metal Arch



4.3.3.2 Materials

Table 26 shows soil properties used in the 3D analyses of the metal culvert. Backfill and
embankment of Test 1 were assigned to properties of SW95, and those of Test 2 were assigned
to properties of SW85. For the metal arch, modulus of elasticity of 29,000 ksi and Poisson’s
ratio of 0.3 were used.

4.3.3.3 Loading and Boundary Condition

Loading and boundary conditions for the 3D analyses of the metal culvert were the same as
those used for the concrete culvert, which were described in Section 4.3.2.3.

4.3.3.4 Results

For the 4 cases analyzed: Figure 69 shows deformed shapes of the metal culvert due to live
loads, Figure 70 shows vertical and horizontal displacement results due to live loads along the
length of the concrete culvert, and Figure 71 shows thrusts and moments in the concrete culvert
due to live loads. Table 32 and Table 33 compare vertical displacements at crown and chord
extensions between the field tests and the 3D analyses. Table 34 through Table 37 compare
thrusts and moments at various locations of the metal culvert between the field tests and the 3D



NCHRP 15-29 Appendix A                                                                       102
analyses. Designations of measurement locations are shown in Figure 55. In the presentation
of results, the downward displacement of the crown is taken as positive, and the vertical
displacement of the crown is the relative displacement between the crown and footings.
Compressive force is taken as positive for thrust, and a moment that produces tension on the
inside of the culvert wall is taken as positive.

Vertical crown displacements of the metal arch estimated by the 3D analysis were larger than
those measured in the field test except for Test 1 with a cover depth of 1 ft. The 3D analysis
estimates of vertical displacements of the metal culvert were much closer to the field test data
than those of the concrete culvert.        However, the 3D analysis significantly overestimated
horizontal chord extensions of the metal arch.

Thrusts at the springlines of the metal arch from the 3D analyses were significantly larger than
those of the field test. Thrusts at the shoulders of the metal arch from the 3D analyses were
significantly smaller than those measured in the field test. Moments of the metal arch from the
3D analyses were in good agreement with those measured in the field test except for the crown.

Figure 72 shows plastic points in the soil elements after the surface live loads were applied.
Tension failure occurred near the wheels at the surface. More Mohr-Coulomb plastic points
were found in Test 2 cases than in Test 1 cases. Mohr-Coulomb plastic points spread in a
wider area in the metal arch analysis when compared to the concrete arch analysis.




                                              Undeformed
                                              Test 1 (Cover = 3 ft)
                                              Test 1 (Cover = 1 ft)
                                              Test 2 (Cover = 3 ft)
                                              Test 2 (Cover = 1 ft)   30X Deformation


          Figure 69—Deformed Shapes of Metal Arch in the Plane of Wheel Loads
                             (Effects of Live Loads only)




NCHRP 15-29 Appendix A                                                                      103
                                       1.8                                                                                                                                      0.7




                                                                                                                           Horiz. Extension of Chord at Height of 88 in. (in)
                                       1.6                                                    Test 1 (Cover = 3 ft)                                                                                                                    Test 1 (Cover = 3 ft)
 Vertical Displacement at Crown (in)

                                                                                                                                                                                0.6
                                                                                              Test 1 (Cover = 1 ft)                                                                                                                    Test 1 (Cover = 1 ft)
                                       1.4
                                                                                              Test 2 (Cover = 3 ft)                                                             0.5                                                    Test 2 (Cover = 3 ft)
                                       1.2                                                    Test 2 (Cover = 1 ft)                                                                                                                    Test 2 (Cover = 1 ft)
                                       1.0                                                                                                                                      0.4

                                       0.8                                                                                                                                      0.3

                                       0.6
                                                                                                                                                                                0.2
                                       0.4
                                                                                                                                                                                0.1
                                       0.2
                                                           Wheel Load Location                                                                                                                      Wheel Load Location
                                       0.0                                                                                                                                      0.0
                                             0              5                     10              15                  20                                                              0              5                    10               15                  20
                                                     Distance from Symmetry Line of Tandem Axles (ft)                                                                                         Distance from Symmetry Line of Tandem Axles (ft)


                                                 (a) Vertical displacement at crown                                                                                                       (b) Chord extension at a height of 88 in.

Figure 70—Displacements due to Live Loads from the 3D Analyses of Metal Arch Culvert



                                                                        kip/ft
                                                                         kip/ft                                                                                                                                 kip-in/ft
                                                                                                                                                                                                                  kip*in/ft
                                                                             -5                                                                                                                                        -30




                                                                             10                                                                                                                                        60

                                                                    Test 1 (Cover = 3 ft)                                                                                                                     Test 1 (Cover = 3 ft)
                                                                    Test 1 (Cover = 1 ft)                                                                                                                     Test 1 (Cover = 1 ft)
                                                                    Test 2 (Cover = 3 ft)                                                                                                                     Test 2 (Cover = 3 ft)
                                                                    Test 2 (Cover = 1 ft)                                                                                                                     Test 2 (Cover = 1 ft)


                                                                 (a) Thrust                                                                                                                               (b) Moment

Figure 71—Thrusts and Moments due to Live Loads in the Plane of Wheel Loads from the
                        3D Analyses of Metal Arch Culvert


                                                 Table 32—Vertical Displacements at Crown of Metal Arch due to Live Loads
                                                 Cover          Test 1                                                                                                                    Test 2
                                                             Field Test                Plaxis 3D                Ratio:                                                                Field Test         Plaxis 3D                    Ratio:
                                                   (ft)                                                     Plaxis 3D /                                                                                                         Plaxis 3D /
                                                                (in.)                     (in.)             Field Test
                                                                                                                                                                                         (in.)              (in.)               Field Test
                                                    3            0.413                      0.509                1.23                                                                     0.453             0.756                     1.67
                                                    1            1.130                      1.104                0.98                                                                     1.055             1.540                     1.46


 Table 33—Chord Extension at Height of 88 in. of Metal Arch Culvert due to Live Loads
                                                 Cover          Test 1                                                                                                                    Test 2
                                                             Field Test                Plaxis 3D                Ratio:                                                                Field Test         Plaxis 3D                    Ratio:
                                                   (ft)                                                     Plaxis 3D /                                                                                                         Plaxis 3D /
                                                                (in.)                     (in.)             Field Test
                                                                                                                                                                                         (in.)              (in.)               Field Test
                                                    3            0.035                      0.225                6.36                                                                     0.118             0.383                     3.24
                                                    1            0.091                      0.375                4.14                                                                     0.094             0.645                     6.83



NCHRP 15-29 Appendix A                                                                                                                                                                                                                                     104
          Table 34—Thrusts in Test 1 of Metal Arch Culvert due to Live Loads
                               Field Test       Plaxis 3D           Ratio:
      Cover    Location
                                 (kip/ft)        (kip/ft)   Plaxis 3D / Field Test
        (ft)                 P1          P2                     P1           P2
         3       NS         0.71        0.35      2.04         2.87         5.86
                 NC         1.22        0.82      2.48         2.04         3.03
                 NH         8.61        7.99      2.55         0.30         0.32
                 CR                    -1.23      1.90                     -1.55
                 SH        9.07        9.51       2.60         0.29         0.27
                 SC        3.69        3.83       2.41         0.65         0.63
                 SS         1.60       -1.68      1.80         1.13        -1.07
         1       NS         0.42        0.57      2.82         6.69         4.91
                 NC        -2.58       -3.02      3.74        -1.45        -1.24
                 NH        32.64       24.69      5.22         0.16         0.21
                 CR                    10.33      8.48                      0.82
                 SH        26.48       17.72      5.19         0.20         0.29
                 SC        -0.08       3.66       3.63       -45.06         0.99
                 SS         0.28       -0.55      2.49         8.87        -4.51



          Table 35—Thrusts in Test 2 of Metal Arch Culvert due to Live Loads
                               Field Test       Plaxis 3D           Ratio:
      Cover    Location
                                 (kip/ft)        (kip/ft)   Plaxis 3D / Field Test
        (ft)                P1           P2                     P1           P2
         3       NS        0.00         0.69      2.05                      2.98
                 NC        1.47         1.37      2.23         1.52         1.63
                 NH        10.62        7.52      2.06         0.19         0.27
                 CR                     3.08      1.33                      0.43
                 SH         5.49       6.19       2.16         0.39         0.35
                 SC         0.48       4.43       2.23         4.65         0.50
                 SS                    -1.48      1.87                     -1.26
         1       NS         0.00        0.18      2.68                     14.81
                 NC        -3.98       -3.79      3.15        -0.79        -0.83
                 NH        27.32       19.99      4.57         0.17         0.23
                 CR                     8.97      8.01                      0.89
                 SH        24.75       16.62      4.56         0.18         0.27
                 SC        -1.80       3.66       3.11        -1.73         0.85
                 SS                    -2.30      2.36                     -1.03




NCHRP 15-29 Appendix A                                                               105
         Table 36—Moments in Test 1 of Metal Arch Culvert due to Live Loads
                              Field Test        Plaxis 3D              Ratio:
      Cover    Location
                              (kip-in./ft)      (kip-in./ft)   Plaxis 3D / Field Test
        (ft)                P1            P2                       P1           P2
         3       NS        -0.76        -0.52      -1.25          1.64         2.40
                 NC        -0.21        -0.76      -1.83         8.74          2.40
                 NH        -5.91        -5.80      -4.36         0.74          0.75
                 CR                      4.88       9.68                       1.98
                 SH        -5.90        -4.85      -4.91         0.83          1.01
                 SC        -1.39        -2.26      -1.62         1.17          0.72
                 SS        -1.78        0.10       -0.97         0.54         -9.67
         1       NS        -0.12         0.00      -0.81         6.53
                 NC         2.84         2.24      -1.01         -0.36        -0.45
                 NH       -19.77       -20.25     -14.39         0.73          0.71
                 CR                      3.05      17.18                       5.63
                 SH       -16.65       -13.16     -14.51         0.87          1.10
                 SC        -0.54        -0.63      -0.96         1.80          1.53
                 SS        -0.01        0.15       -0.45        49.42         -3.09



         Table 37—Moments in Test 2 of Metal Arch Culvert due to Live Loads
                              Field Test        Plaxis 3D              Ratio:
      Cover    Location
                              (kip-in./ft)      (kip-in./ft)   Plaxis 3D / Field Test
        (ft)                 P1           P2                      P1            P2
         3       NS         0.00        -0.03      -3.22                      96.92
                 NC        -0.24        -0.51      -2.58        10.66          5.03
                 NH        -5.52        -4.96      -5.65         1.02          1.14
                 CR                      2.04      13.39                       6.58
                 SH        -3.64        -2.91      -6.18         1.70          2.12
                 SC        -0.53        -3.40      -2.46         4.63          0.72
                 SS                      0.52      -2.46                      -4.72
         1       NS        0.00          0.26      -1.55                      -5.96
                 NC        3.39          2.78      -2.68         -0.79        -0.96
                 NH       -16.79       -17.41     -18.80         1.12          1.08
                 CR                      4.28      22.39                       5.24
                 SH       -15.15       -10.34     -19.24         1.27          1.86
                 SC        -0.95        -4.26      -2.71         2.86          0.64
                 SS                      1.29      -0.77                      -0.60




NCHRP 15-29 Appendix A                                                                  106
                                 (a) Test 1 and 3 ft cover




                                 (b) Test 1 and 1 ft cover




                                 (c) Test 2 and 3 ft cover




                                 (d) Test 2 and 1 ft cover
                                                                 Tension cut-off point
                                                                 Mohr-Coulomb point

  Figure 72—Plastic Points in Soil Elements in the Plane of Wheel Loads in Metal Arch
                                       Analysis




NCHRP 15-29 Appendix A                                                                   107
4.3.4     60-in. Diameter HDPE Pipe

4.3.4.1 Finite Element Model

Four pipe runs with Type S pipes were modeled: (1) Pipe Run 1 (A-1 backfill and 1.4 ft cover),
(2) Pipe Run 9 (A-1 backfill and 2.5 ft cover), (3) Pipe Run 3 (A-2 backfill and 1.6 ft cover), and
(4) Pipe Run 7 (A-2 backfill and 2.8 ft cover). An average trench width varied from pipe to pipe
as shown in Table 25. A trench width of 8 ft 8 in. was used for all four models as shown in
Figure 60.

Figure 73 shows a finite element model of the HDPE pipe culvert for Pipe Run 9 (A-1 backfill
and 2.5 ft cover). Soils and pavement were modeled by wedge elements, and the concrete
culvert was modeled by plate elements. The coordinate system of the model was oriented so
that the x-axis aligns with the transverse direction of the culvert, the y-axis aligns with the
vertical direction, and the z-axis aligns with the longitudinal direction. The model had a length of
12 ft in the longitudinal direction of the culvert. The model for Pipe Run 9 had 19,888 elements
and 55,993 nodes.

HDPE pipes of Type S corrugation have different axial and bending stiffnesses in the
circumferential and longitudinal directions. EA and EI are summarized in Table 38 for the
circumferential and longitudinal directions. As discussed in Section 4.3.3.1 for the metal arch
culvert, short elements with low section properties were inserted between elements with section
properties close to circumferential properties to match stiffnesses as shown in Figure 74. A 1 , I 1 ,
A 2 , and I 2 for L 2 /L 1 =1/20 are given in Figure 74.




                                                                                    15 ft 7 in.



         y



  z               x
                         Plane of symmetry        30 ft                  12 ft

             Figure 73—Finite Element Model of HDPE Pipe Culvert for Pipe Run 9


NCHRP 15-29 Appendix A                                                                            108
Table 38—Axial and Bending Modulus of HDPE Pipe in Circumferential and Longitudinal
                            Directions (E=100,000 psi)
                           Direction                  EA (lb/in.)           EI (lb-in.2/in.)
                        Circumferential                53,600                   79,800
                         Longitudinal                  32,700                    1,020



                           M                          A2, I2                          M

                                          A1, I1




                                  L1                    L2                   L2/L1=1/20

                                  A1          I1                  A2          I2
                               (in2/in)    (in4/in)            (in2/in)    (in4/in)

                               0.562        0.840              0.0365     1.005x10-3


          Figure 74—Soft Element to Match Longitudinal Stiffness of HDPE Pipe



4.3.4.2 Materials

Table 39 shows properties of soils and pavement used in the 3D analyses of the HDPE pipe
culverts. The A-1 backfill was assigned properties of SW95, and the A-2 backfill was assigned
properties of ML95. A target compaction of the backfill was 90 percent of maximum standard
Proctor density. As shown in Table 25, backfill densities ranged from 82 percent to 97 percent
of the maximum standard Proctor density. Since the backfill was compacted more when the
pavement was placed, 95 percent compaction was selected for the 3D analysis. Properties of
pavement were those used in the MNDOT study (McGrath, 2005).                              Gravel was assigned
properties of SW95.

For the HDPE pipes, modulus of elasticity of 100,000 psi and Poisson’s ratio of 0.35 were used.

4.3.4.3 Loading and Boundary Condition

Boundary conditions of the model were the same as those described in Section 4.3.2.3.

Loading steps in the analysis were: (1) in-situ soil under gravity, (2) in-situ soil, bedding, and the
culvert under gravity, (3) in-situ soil, bedding, the culvert, and backfill under gravity, (4) in-situ
soil, bedding, the culvert, backfill, gravel, and pavement under gravity, and (5) all gravity and


NCHRP 15-29 Appendix A                                                                                   109
live loads. Effects of construction sequence during backfilling were not considered. We did not
assign horizontal stresses in the backfill as initial conditions, but let the horizontal stresses be
those due to gravity effects; therefore, the horizontal stresses in the analysis may be different
from those in the field tests after compaction.

In the 3D analysis, two positions of a live load vehicle were examined as shown in Figure 75: (1)
the reference axle of the tandem axles was placed over the crown, and (2) the tandem axles
were placed symmetrically over the crown. These two positions were designated as Position 3
and Position 4 in the field test. The footprint of two wheels was assumed to be 10 in. x 20 in.

For each of the four pipe runs, four cases of live loads were analyzed: (1) heavy truck at
Position 3, (2) heavy truck at Position 4, (3) light truck at Position 3, and (4) light truck at
Position 4. Differences of axle loads between the heavy and light trucks are shown in Figure
58.

              Table 39—Soil Properties Used for the 3D Analyses of HDPE Pipes
                                         Modulus
                    Density   Depth          of       Poisson’s   Angle of   Angle of     Cohesion
      Soil Type
                     (pcf)     (ft)      Elasticity     Ratio     Friction   Dilatancy      (psi)
                                           (psi)
  A-1 Backfill       141       0 to 1      1,600        0.40        57.8        27.8        0.001
    SW95                       1 to 5      4,100        0.29        54.3        24.3
(Mohr-Coulomb)                5 to 10      6,000        0.24        53.2        23.2
                              10 to 18     8,600        0.23        52.2        22.2
  A-2 Backfill       127       0 to 1      1,800        0.34        34.0        4.0          4.0
    ML95                       1 to 6      2,500        0.29
(Mohr-Coulomb)                6 to 11      2,900        0.27
                              11 to 18     3,200        0.29
     In-Situ
                     145        any       15,000        0.30
 (Linear-Elastic)
    Pavement                   0 to
                     150                 400,000        0.35
 (Linear-Elastic)              0.33
     Gravel
                              0.33 to
      SW95           141                   1,600        0.40        57.8        27.8        0.001
                                1.0
(Mohr-Coulomb)




NCHRP 15-29 Appendix A                                                                          110
                 0’-0”                                              Symmetric = 2’-2” each side

              (a) Position 3                                               (b) Position 4

                         Note: Axles colored in pink were modeled in the 3D analysis.

   Figure 75—Positions of Live Load Vehicle Axles in the 3D Analyses of HDPE Pipes



4.3.4.4 Results

Figure 76 shows deformed shapes of Pipe Run 1 due to live loads for the four different cases.
Figure 77 shows vertical crown displacements due to live loads along the length of Pipe Run 1
for the four cases. Figure 78 shows horizontal displacements of culvert (diametrical change at
the springline) due to live loads along the length of Pipe Run 1 for the four cases. Figure 79
shows thrusts of Pipe Run 1 in the plane of wheel loads due to live loads for the four cases.
Figure 80 shows moments of Pipe Run 1 in the plane of wheel loads due to live loads for the
four cases. Figure 81 shows plastic points in the soil elements in the analysis of Pipe Run 1. In
these figures for analysis results, field test data are also shown whenever available. The field
test data are designated by the month when the tests were conducted, such as Oct 00, May 01,
and Aug 02. These three tests were conducted right after the installation and seven months
and 22 months after the installation. Figure 82 through Figure 87 show results of Pipe Run 9.
Figure 88 though Figure 93 show results of Pipe Run 3. Figure 94 through Figure 99 show
results of Pipe Run 7. Table 40 and Table 41 compare vertical displacements at the crown
between the field tests and the 3D analyses for the heavy and light trucks. Table 42 and Table
43 compare horizontal displacements (diametrical changes at the springline) between the field
tests and the 3D analyses for the heavy and light trucks. In the presentation of results, the
downward displacement of the crown is taken as positive, and the vertical displacement of the
crown is the relative displacement between the crown and the invert. A diametrical change is
positive when there is an extension. Compressive force is taken as positive for thrusts, and
moment that produces tension on the inside of the culvert wall is taken as positive.




NCHRP 15-29 Appendix A                                                                            111
Responses of the pipes with a nominal cover of 1 ft were estimated by the 3D analyses better
than those of the pipes with a nominal cover of 2 ft. The 3D analysis tends to overestimate as
the cover height increases. The 3D analysis estimated the vertical displacements better than
the horizontal displacements.

In general, moments were estimated by the 3D analyses better than thrusts.

For the A-1 backfill cases (Pipe Runs 1 and 9), shear failure indicated by Mohr-Coulomb plastic
points occurred at the invert and at the soil-structure interface below the springlines. Shear
failure was also observed at boundaries between in-situ soil and backfill up to a depth of about
3 ft. However, for the A-2 backfill cases (Pipe Runs 3 and 7), not many Mohr-Coulomb points
were found. Instead, tension failure occurred at the invert and at the boundaries of in-situ soil
and backfill up to a depth of 3 ft. The difference in soil failure profile between A-1 and A-2
backfills stems from cohesion assigned to each soil models. A cohesion value of SW95 for A-1
back fill was 0.001 psi, and that of ML95 for A-2 backfill was 4.0 psi. Owing to the cohesion of
ML95, tension failure occurred prior to shear failure at the invert and at the boundaries of in-situ
soil and backfill up to a depth of 3 ft for the A-2 backfill cases.




NCHRP 15-29 Appendix A                                                                          112
                                                                                                                                                         Undeformed
                                                                                                                                                         Heavy P3
                                                                                                                                                         Heavy P4
                                                                                                                                                         Light P3
                                                                                                                                                         Light P4




                                                                                                                                                                  100X deformation


                                        Figure 76—Deformed Shapes of Pipe Run 1 due to Live Loads in the Plane of Wheel
                                                                  Loads (A-1, 1.4 ft Cover)



                                        0.14                                                                                                           0.14
                                                                                       Heavy P3 (Plaxis)                                                                                               Light P3 (Plaxis)
                                                                                       Heavy P4 (Plaxis)                                                                                               Light P4 (Plaxis)
 Vertical Displacement at Crown (in.)




                                                                                                                Vertical Displacement at Crown (in.)




                                        0.12                                                                                                           0.12
                                                                                       Heavy P3 (Oct-00)                                                                                               Light P3 (Oct-00)
                                                                                       Heavy P3 (May-01)                                                                                               Light P3 (May-01)
                                        0.10                                                                                                           0.10
                                                                                       Heavy P3 (Aug-02)                                                                                               Light P3 (Aug-02)
                                                                                       Heavy P4 (Oct-00)                                                                                               Light P4 (Oct-00)
                                        0.08                                                                                                           0.08
                                                                                       Heavy P4 (May-01)                                                                                               Light P4 (May-01)
                                                                                       Heavy P4 (Aug-02)                                                                                               Light P4 (Aug-02)
                                        0.06                                                                                                           0.06


                                        0.04                                                                                                           0.04


                                        0.02                                                                                                           0.02


                                        0.00                                                                                                           0.00
                                               0       2          4         6         8         10         12                                                 0     2          4         6         8           10          12
                                                     Distance from Symmetry Line of Tandem Axles (ft)                                                             Distance from Symmetry Line of Tandem Axles (ft)

                                                                 (a) Heavy truck                                                                                               (b) Light truck

                                                   Figure 77—Vertical Crown Displacements of Pipe Run 1 due to Live Loads
                                                                             (A-1, 1.4 ft Cover)




NCHRP 15-29 Appendix A                                                                                                                                                                                                     113
                                                          0.020                                                    Heavy P3 (Plaxis)                                                                                  0.020                                                    Light P3 (Plaxis)
          Diametrical Change between Springlines (in.)




                                                                                                                                                                      Diametrical Change between Springlines (in.)
                                                                                                                   Heavy P4 (Plaxis)                                                                                                                                           Light P4 (Plaxis)
                                                                                                                   Heavy P3 (Oct-00)                                                                                                                                           Light P3 (Oct-00)
                                                          0.015                                                    Heavy P3 (May-01)                                                                                  0.015
                                                                                                                                                                                                                                                                               Light P3 (May-01)
                                                                                                                   Heavy P3 (Aug-02)                                                                                                                                           Light P3 (Aug-02)
                                                                                                                   Heavy P4 (Oct-00)
                                                          0.010                                                                                                                                                       0.010                                                    Light P4 (Oct-00)
                                                                                                                   Heavy P4 (May-01)
                                                                                                                                                                                                                                                                               Light P4 (May-01)
                                                                                                                   Heavy P4 (Aug-02)
                                                                                                                                                                                                                                                                               Light P4 (Aug-02)
                                                          0.005                                                                                                                                                       0.005


                                                          0.000                                                                                                                                                       0.000


                                                         -0.005                                                                                                                                                      -0.005


                                                         -0.010                                                                                                                                                      -0.010
                                                                  0            2          4          6         8           10        12                                                                                       0            2         4          6         8          10           12
                                                                            Distance from Symmetry Line of Tandem Axles (ft)                                                                                                            Distance from Symmetry Line of Tandem Axles (ft)


                                                                                        (a) Heavy truck                                                                                                                                            (b) Light truck
Figure 78—Horizontal Displacements of Pipe Run 1 due to Live Loads (A-1, 1.4 ft Cover)
                                                         70                                                              Heavy P3 (Plaxis)                                              70
                                                                                                                         Heavy P4 (Plaxis)
                                                                                                                         Heavy P3 (Oct-00)
                                                                                                                         Heavy P3 (May-01)                                                                                                                                    Light P3 (Plaxis)
                                                         60                                                                                                                             60
                                                                                                                         Heavy P3 (Aug-02)                                                                                                                                    Light P4 (Plaxis)
                                                                                                                         Heavy P4 (Oct-00)
                                                         50                                                              Heavy P4 (May-01)                                              50
                                                                                                                         Heavy P4 (Aug-02)
Thrust (lb/in.)




                                                                                                                                             Thrust (lb/in.)




                                                         40                                                                                                                             40


                                                         30                                                                                                                             30


                                                         20                                                                                                                             20


                                                         10                                                                                                                             10


                                                          0                                                                                                                                                          0
                                                           180        135          90    45      0       -45       -90      -135    -180                                                                              180         135      90      45       0       -45       -90    -135     -180
                                                                                        Degrees from Crown                                                                                                                                       Degrees from Crown

                                                                                        (a) Heavy truck                                                                                                                                           (b) Light truck

                                                               Figure 79—Thrusts of Pipe Run 1 due to Live Loads in the Plane of Wheel Loads
                                                                                            (A-1, 1.4 ft Cover)
                                                         40                                                                                                                                              40
                                                                                                                   Heavy P3 (Plaxis)
                                                                                                                   Heavy P4 (Plaxis)                                                                                                                                          Light P3 (Plaxis)
                                                         30                                                        Heavy P3 (Oct-00)                                                                     30
Bending Moment (lb-in./in.)




                                                                                                                                             Bending Moment (lb-in./in.)




                                                                                                                   Heavy P3 (May-01)                                                                                                                                          Light P4 (Plaxis)
                                                         20                                                        Heavy P3 (Aug-02)                                                                     20
                                                                                                                   Heavy P4 (Oct-00)
                                                                                                                   Heavy P4 (May-01)
                                                         10                                                        Heavy P4 (Aug-02)                                                                     10


                                                          0                                                                                                                                                          0


                                                         -10                                                                                                                             -10


                                                         -20                                                                                                                             -20


                                                         -30                                                                                                                             -30
                                                            180       135          90     45     0       -45       -90      -135    -180                                                    180                                   135       90      45      0       -45       -90    -135     -180
                                                                                        Degrees from Crown                                                                                                                                       Degrees from Crown

                                                                                        (a) Heavy truck                                                                                                                                           (b) Light truck

                                                                 Figure 80—Moments of Pipe Run 1 due to Live Loads in Plane of Wheel Loads
                                                                                            (A-1, 1.4 ft Cover)


NCHRP 15-29 Appendix A                                                                                                                                                                                                                                                                            114
         (a) Heavy Truck Position 3                      (b) Heavy Truck Position 4




          (c) Light Truck Position 3                      (c) Light Truck Position 4

                         Tension cut-off point   Mohr-Coulomb point

  Figure 81—Plastic Points in Soil Elements of Pipe Run 1 in the Plane of Wheel Loads
                                    (A-1, 1.4 ft Cover)




NCHRP 15-29 Appendix A                                                                 115
                                                                                                                                                         Undeformed
                                                                                                                                                         Heavy P3
                                                                                                                                                         Heavy P4
                                                                                                                                                         Light P3
                                                                                                                                                         Light P4




                                                                                                                                                                  100X deformation


                                        Figure 82—Deformed Shapes of Pipe Run 9 due to Live Loads in the Plane of Wheel
                                                                  Loads (A-1, 2.5 ft Cover)



                                        0.08                                                                                                           0.08
                                                                                       Heavy P3 (Plaxis)                                                                                               Light P3 (Plaxis)
                                        0.07                                           Heavy P4 (Plaxis)                                               0.07                                            Light P4 (Plaxis)
 Vertical Displacement at Crown (in.)




                                                                                                                Vertical Displacement at Crown (in.)




                                                                                       Heavy P3 (Oct-00)                                                                                               Light P3 (Oct-00)
                                        0.06                                           Heavy P3 (May-01)                                               0.06                                            Light P3 (May-01)
                                                                                       Heavy P3 (Aug-02)                                                                                               Light P3 (Aug-02)
                                        0.05                                           Heavy P4 (Oct-00)                                               0.05                                            Light P4 (Oct-00)
                                                                                       Heavy P4 (May-01)                                                                                               Light P4 (May-01)
                                        0.04                                                                                                           0.04
                                                                                       Heavy P4 (Aug-02)                                                                                               Light P4 (Aug-02)

                                        0.03                                                                                                           0.03

                                        0.02                                                                                                           0.02

                                        0.01                                                                                                           0.01

                                        0.00                                                                                                           0.00
                                               0       2          4         6         8         10         12                                                 0     2          4         6         8           10          12
                                                     Distance from Symmetry Line of Tandem Axles (ft)                                                             Distance from Symmetry Line of Tandem Axles (ft)

                                                                 (a) Heavy truck                                                                                               (b) Light truck

                                                   Figure 83—Vertical Crown Displacements of Pipe Run 9 due to Live Loads
                                                                             (A-1, 2.5 ft Cover)




NCHRP 15-29 Appendix A                                                                                                                                                                                                     116
                                                          0.016                                                                                                                                         0.016
           Diametrical Change between Springlines (in.)




                                                                                                                                                         Diametrical Change between Springlines (in.)
                                                                                                                    Heavy P3 (Plaxis)                                                                                                                              Light P3 (Plaxis)
                                                          0.014                                                     Heavy P4 (Plaxis)                                                                   0.014                                                      Light P4 (Plaxis)
                                                                                                                    Heavy P3 (Oct-00)                                                                                                                              Light P3 (Oct-00)
                                                          0.012                                                     Heavy P3 (May-01)                                                                   0.012                                                      Light P3 (May-01)
                                                                                                                    Heavy P3 (Aug-02)                                                                                                                              Light P3 (Aug-02)
                                                          0.010                                                     Heavy P4 (Oct-00)                                                                   0.010                                                      Light P4 (Oct-00)
                                                                                                                    Heavy P4 (May-01)                                                                                                                              Light P4 (May-01)
                                                          0.008                                                     Heavy P4 (Aug-02)                                                                   0.008                                                      Light P4 (Aug-02)

                                                          0.006                                                                                                                                         0.006

                                                          0.004                                                                                                                                         0.004

                                                          0.002                                                                                                                                         0.002

                                                          0.000                                                                                                                                         0.000
                                                                   0            2          4          6         8           10           12                                                                      0            2          4          6          8           10           12
                                                                             Distance from Symmetry Line of Tandem Axles (ft)                                                                                              Distance from Symmetry Line of Tandem Axles (ft)

                                                                                           (a) Heavy truck                                                                                                                               (b) Light truck

                                                                         Figure 84—Horizontal Displacements of Pipe Run 9 Due to Live Loads
                                                                                                  (A-1, 2.5 ft Cover)
                                                          35                                                                                                                                            35

                                                          30                                                                                                                                            30

                                                          25                                                                                                                                            25
Thrust (lb/in.)




                                                                                                                                              Thrust (lb/in.)




                                                          20                                                                                                                                            20

                                                          15                                                                                                                                            15

                                                          10                                                                                                                                            10                                                    Light P3 (Plaxis)
                                                                                                                                                                                                                                                              Light P4 (Plaxis)
                                                                                                                                                                                                                                                              Light P3 (Oct-00)
                                                           5                                              Heavy P3 (Plaxis)                                                                              5                                                    Light P3 (May-01)
                                                                                                                                                                                                                                                              Light P3 (Aug-02)
                                                           0                                              Heavy P4 (Plaxis)                                                                              0                                                    Light P4 (Oct-00)
                                                                                                                                                                                                                                                              Light P4 (May-01)
                                                                                                                                                                                                                                                              Light P4 (Aug-02)
                                                          -5                                                                                                                                            -5
                                                            180        135          90    45      0       -45        -90    -135     -180                                                                 180        135          90    45      0       -45         -90    -135        -180
                                                                                         Degrees from Crown                                                                                                                            Degrees from Crown
                                                                                         (a) Heavy truck                                                                                                                                 (b) Light truck

                                                                Figure 85—Thrusts of Pipe Run 9 Due to Live Loads in the Plane of Wheel Loads
                                                                                             (A-1, 2.5 ft Cover)
                                                          15                                                                                                                                            15                                                         Light P3 (Plaxis)
                                                                                                                                                                                                                                                                   Light P4 (Plaxis)
                                                                                                                     Heavy P3 (Plaxis)                                                                                                                             Light P3 (Oct-00)
                                                          10                                                                                                                                            10                                                         Light P3 (May-01)
Bending Moment (lb-in./in.)




                                                                                                                                              Bending Moment (lb-in./in.)




                                                                                                                     Heavy P4 (Plaxis)
                                                                                                                                                                                                                                                                   Light P3 (Aug-02)
                                                                                                                                                                                                                                                                   Light P4 (Oct-00)
                                                           5                                                                                                                                             5
                                                                                                                                                                                                                                                                   Light P4 (May-01)
                                                                                                                                                                                                                                                                   Light P4 (Aug-02)
                                                           0                                                                                                                                             0


                                                           -5                                                                                                                                            -5


                                                          -10                                                                                                                                           -10


                                                          -15                                                                                                                                           -15
                                                             180       135          90     45     0       -45        -90    -135     -180                                                                  180       135          90    45      0       -45         -90    -135        -180
                                                                                         Degrees from Crown                                                                                                                            Degrees from Crown
                                                                                         (a) Heavy truck                                                                                                                                 (b) Light truck

                                                                  Figure 86—Moments of Pipe Run 9 Due to Live Loads in Plane of Wheel Loads
                                                                                             (A-1, 2.5 ft Cover)



NCHRP 15-29 Appendix A                                                                                                                                                                                                                                                                        117
          (a) Heavy Truck Position 3                    (b) Heavy Truck Position 4




           (c) Light Truck Position 3                    (c) Light Truck Position 4

                        Tension cut-off point   Mohr-Coulomb point


  Figure 87—Plastic Points in Soil Elements of Pipe Run 9 in the Plane of Wheel Loads
                                    (A-1, 2.5 ft Cover)




NCHRP 15-29 Appendix A                                                                118
                                                                                                                                                         Undeformed
                                                                                                                                                         Heavy P3
                                                                                                                                                         Heavy P4
                                                                                                                                                         Light P3
                                                                                                                                                         Light P4




                                                                                                                                                                  100X deformation


                                        Figure 88—Deformed Shapes of Pipe Run 3 due to Live Loads in the Plane of Wheel
                                                                  Loads (A-2, 1.6 ft Cover)



                                        0.18                                                                                                           0.18
                                                                                       Heavy P3 (Plaxis)                                                                                               Light P3 (Plaxis)
                                        0.16                                           Heavy P4 (Plaxis)                                               0.16                                            Light P4 (Plaxis)
 Vertical Displacement at Crown (in.)




                                                                                                                Vertical Displacement at Crown (in.)




                                                                                       Heavy P3 (Oct-00)                                                                                               Light P3 (Oct-00)
                                        0.14                                                                                                           0.14
                                                                                       Heavy P3 (May-01)                                                                                               Light P3 (May-01)
                                        0.12                                           Heavy P3 (Aug-02)                                               0.12                                            Light P3 (Aug-02)
                                                                                       Heavy P4 (Oct-00)                                                                                               Light P4 (Oct-00)
                                        0.10                                           Heavy P4 (May-01)                                               0.10                                            Light P4 (May-01)
                                                                                       Heavy P4 (Aug-02)                                                                                               Light P4 (Aug-02)
                                        0.08                                                                                                           0.08

                                        0.06                                                                                                           0.06

                                        0.04                                                                                                           0.04

                                        0.02                                                                                                           0.02

                                        0.00                                                                                                           0.00
                                               0       2          4         6         8         10         12                                                 0     2          4         6         8           10          12
                                                     Distance from Symmetry Line of Tandem Axles (ft)                                                             Distance from Symmetry Line of Tandem Axles (ft)

                                                                 (a) Heavy truck                                                                                               (b) Light truck

                                                   Figure 89—Vertical Crown Displacements of Pipe Run 3 due to Live Loads
                                                                             (A-2, 1.6 ft Cover)




NCHRP 15-29 Appendix A                                                                                                                                                                                                     119
                                                          0.030                                                                                                                                         0.030
           Diametrical Change between Springlines (in.)




                                                                                                                                                         Diametrical Change between Springlines (in.)
                                                                                                                    Heavy P3 (Plaxis)                                                                                                                             Light P3 (Plaxis)
                                                                                                                    Heavy P4 (Plaxis)                                                                                                                             Light P4 (Plaxis)
                                                          0.025                                                     Heavy P3 (Oct-00)                                                                   0.025
                                                                                                                                                                                                                                                                  Light P3 (Oct-00)
                                                                                                                    Heavy P3 (May-01)                                                                                                                             Light P3 (May-01)
                                                          0.020                                                     Heavy P3 (Aug-02)                                                                   0.020                                                     Light P3 (Aug-02)
                                                                                                                    Heavy P4 (Oct-00)                                                                                                                             Light P4 (Oct-00)
                                                                                                                    Heavy P4 (May-01)                                                                                                                             Light P4 (May-01)
                                                          0.015                                                     Heavy P4 (Aug-02)                                                                   0.015                                                     Light P4 (Aug-02)


                                                          0.010                                                                                                                                         0.010


                                                          0.005                                                                                                                                         0.005


                                                          0.000                                                                                                                                         0.000
                                                                   0            2          4          6         8           10           12                                                                      0            2          4          6         8            10           12
                                                                             Distance from Symmetry Line of Tandem Axles (ft)                                                                                              Distance from Symmetry Line of Tandem Axles (ft)

                                                                                         (a) Heavy truck                                                                                                                                 (b) Light truck
Figure 90—Horizontal Displacements of Pipe Run 3 due to Live Loads (A-2, 1.6 ft Cover)
                                                          50                                                                                                                                            50

                                                          45                                                                                                                                            45                                                          Light P3 (Plaxis)
                                                                                                                                                                                                                                                                    Light P4 (Plaxis)
                                                          40                                                                                                                                            40

                                                          35                                                                                                                                            35
Thrust (lb/in.)




                                                                                                                                              Thrust (lb/in.)




                                                          30                                                                                                                                            30

                                                          25                                                                                                                                            25

                                                          20                                                                                                                                            20

                                                          15                                                                                                                                            15

                                                          10                                              Heavy P3 (Plaxis)                                                                             10

                                                           5                                              Heavy P4 (Plaxis)                                                                              5

                                                           0                                                                                                                                             0
                                                            180        135          90    45      0       -45        -90    -135     -180                                                                 180        135          90    45      0       -45        -90     -135       -180
                                                                                         Degrees from Crown                                                                                                                            Degrees from Crown

                                                                                         (a) Heavy truck                                                                                                                               (b) Light truck

Figure 91—Thrusts of Pipe Run 3 due to Live Loads in the Plane of Wheel Loads (A-2, 1.6
                                      ft Cover)
                                                          30                                                                                                                                            30

                                                          25                                                         Heavy P3 (Plaxis)                                                                  25                                                          Light P3 (Plaxis)
                                                                                                                     Heavy P4 (Plaxis)                                                                                                                              Light P4 (Plaxis)
Bending Moment (lb-in./in.)




                                                                                                                                              Bending Moment (lb-in./in.)




                                                          20                                                                                                                                            20

                                                          15                                                                                                                                            15

                                                          10                                                                                                                                            10

                                                           5                                                                                                                                             5

                                                           0                                                                                                                                             0

                                                           -5                                                                                                                                            -5

                                                          -10                                                                                                                                           -10

                                                          -15                                                                                                                                           -15

                                                          -20                                                                                                                                           -20
                                                             180       135          90     45     0       -45        -90    -135     -180                                                                  180       135          90     45     0       -45        -90     -135       -180
                                                                                         Degrees from Crown                                                                                                                            Degrees from Crown

                                                                                         (a) Heavy truck                                                                                                                                (b) Light truck

                                                                  Figure 92—Moments of Pipe Run 3 due to Live Loads in Plane of Wheel Loads
                                                                                             (A-2, 1.6 ft Cover)


NCHRP 15-29 Appendix A                                                                                                                                                                                                                                                                  120
           (a) Heavy Truck Position 3                   (b) Heavy Truck Position 4




           (c) Light Truck Position 3                    (c) Light Truck Position 4

                        Tension cut-off point   Mohr-Coulomb point

  Figure 93—Plastic Points in Soil Elements of Pipe Run 3 in the Plane of Wheel Loads
                                    (A-2, 1.6 ft Cover)




NCHRP 15-29 Appendix A                                                                121
                                                                                                                                                       Undeformed
                                                                                                                                                       Heavy P3
                                                                                                                                                       Heavy P4
                                                                                                                                                       Light P3
                                                                                                                                                       Light P4




                                                                                                                                                                100X deformation


                                        Figure 94—Deformed Shapes of Pipe Run 7 due to Live Loads in the Plane of Wheel
                                                                  Loads (A-2, 2.8 ft Cover)



                                        0.09                                                                                                         0.09
                                                                                     Heavy P3 (Plaxis)                                                                                               Light P3 (Plaxis)
                                        0.08                                         Heavy P4 (Plaxis)                                               0.08                                            Light P4 (Plaxis)
 Vertical Displacement at Crown (in.)




                                                                                                              Vertical Displacement at Crown (in.)




                                                                                     Heavy P3 (Oct-00)                                                                                               Light P3 (Oct-00)
                                        0.07                                                                                                         0.07
                                                                                     Heavy P3 (May-01)                                                                                               Light P3 (May-01)
                                        0.06                                         Heavy P3 (Aug-02)                                               0.06                                            Light P3 (Aug-02)
                                                                                     Heavy P4 (Oct-00)                                                                                               Light P4 (Oct-00)
                                        0.05                                         Heavy P4 (May-01)                                               0.05                                            Light P4 (May-01)
                                                                                     Heavy P4 (Aug-02)                                                                                               Light P4 (Aug-02)
                                        0.04                                                                                                         0.04

                                        0.03                                                                                                         0.03

                                        0.02                                                                                                         0.02

                                        0.01                                                                                                         0.01

                                        0.00                                                                                                         0.00
                                               0     2          4         6         8         10         12                                                 0     2          4         6         8           10          12
                                                   Distance from Symmetry Line of Tandem Axles (ft)                                                             Distance from Symmetry Line of Tandem Axles (ft)

                                                               (a) Heavy truck                                                                                               (b) Light truck

                               Figure 95—Vertical Crown Displacements of Pipe Run 7 due to Live Loads (A-2, 2.8 ft
                                                                  Cover)




NCHRP 15-29 Appendix A                                                                                                                                                                                                   122
           Diametrical Change between Springlines (in.)   0.025                                                                                                                                          0.025




                                                                                                                                                          Diametrical Change between Springlines (in.)
                                                                                                                    Heavy P3 (Plaxis)                                                                                                                                Light P3 (Plaxis)
                                                                                                                    Heavy P4 (Plaxis)                                                                                                                                Light P4 (Plaxis)
                                                          0.020                                                     Heavy P3 (Oct-00)                                                                    0.020                                                       Light P3 (Oct-00)
                                                                                                                    Heavy P3 (May-01)                                                                                                                                Light P3 (May-01)
                                                                                                                    Heavy P3 (Aug-02)                                                                                                                                Light P3 (Aug-02)
                                                                                                                    Heavy P4 (Oct-00)                                                                                                                                Light P4 (Oct-00)
                                                          0.015                                                                                                                                          0.015
                                                                                                                    Heavy P4 (May-01)                                                                                                                                Light P4 (May-01)
                                                                                                                    Heavy P4 (Aug-02)                                                                                                                                Light P4 (Aug-02)
                                                          0.010                                                                                                                                          0.010



                                                          0.005                                                                                                                                          0.005



                                                          0.000                                                                                                                                          0.000
                                                                   0            2          4          6         8           10           12                                                                       0            2          4          6           8            10           12
                                                                             Distance from Symmetry Line of Tandem Axles (ft)                                                                                               Distance from Symmetry Line of Tandem Axles (ft)

                                                                                         (a) Heavy truck                                                                                                                                  (b) Light truck

Figure 96—Horizontal Displacements of Pipe Run 7 due to Live Loads (A-2, 2.8 ft Cover)
                                                          35                                                                                                                                             35


                                                          30                                                                                                                                             30


                                                          25                                                                                                                                             25
Thrust (lb/in.)




                                                                                                                                               Thrust (lb/in.)




                                                          20                                                                                                                                             20


                                                          15                                                                                                                                             15


                                                          10                                                                                                                                             10

                                                                                                                Heavy P3 (Plaxis)                                                                                                                              Light P3 (Plaxis)
                                                           5                                                                                                                                              5
                                                                                                                Heavy P4 (Plaxis)                                                                                                                              Light P4 (Plaxis)

                                                           0                                                                                                                                              0
                                                            180        135          90    45      0       -45        -90     -135       -180                                                               180        135          90    45      0       -45          -90      -135       -180
                                                                                         Degrees from Crown                                                                                                                             Degrees from Crown

                                                                                         (a) Heavy truck                                                                                                                                  (b) Light truck

Figure 97—Thrusts of Pipe Run 7 due to Live Loads in the Plane of Wheel Loads (A-2, 2.8
                                      ft Cover)
                                                          20                                                                                                                                             20
                                                                                                                    Heavy P3 (Plaxis)                                                                                                                                 Light P3 (Plaxis)
                                                          15                                                        Heavy P4 (Plaxis)                                                                    15                                                           Light P4 (Plaxis)
Bending Moment (lb-in./in.)




                                                                                                                                               Bending Moment (lb-in./in.)




                                                          10                                                                                                                                             10


                                                           5                                                                                                                                              5


                                                           0                                                                                                                                              0


                                                           -5                                                                                                                                             -5


                                                          -10                                                                                                                                            -10


                                                          -15                                                                                                                                            -15
                                                             180       135          90     45     0       -45        -90     -135       -180                                                                180       135          90     45     0       -45          -90      -135       -180
                                                                                         Degrees from Crown                                                                                                                             Degrees from Crown

                                                                                         (a) Heavy truck                                                                                                                                  (b) Light truck

Figure 98—Moments of Pipe Run 7 due to Live Loads in Plane of Wheel Loads (A-2, 2.8 ft
                                      Cover)


NCHRP 15-29 Appendix A                                                                                                                                                                                                                                                                     123
          (a) Heavy Truck Position 3                     (b) Heavy Truck Position 4




           (c) Light Truck Position 3                    (c) Light Truck Position 4

                         Tension cut-off point   Mohr-Coulomb point

Figure 99—Plastic Points in Soil Elements of Pipe Run 7 in the Plane of Wheel Loads (A-
                                    2, 2.8 ft Cover)




NCHRP 15-29 Appendix A                                                                124
 Table 40—Comparison of Vertical Displacements at Crown of HDPE Pipes under Heavy
                                       Truck
  Pipe   Backfill   Average       Truck     Plaxis 3D            Field Test (in.)           Ratio: Plaxis 3D / Field Test
  Run               Cover (ft)   Position     (in.)     Oct-00      May-01      Aug-02      Oct-00    May-01     Aug-02
   1       A-1         1.4          3        0.092      0.106         0.117         0.119    0.87       0.79       0.78
                                    4        0.085      0.074         0.095         0.088    1.15       0.90       0.97
   3       A-2         1.6          3        0.101      0.174         0.122         0.091    0.58       0.83       1.11
                                    4        0.100      0.137         0.097         0.072    0.73       1.03       1.39
   7       A-2         2.8          3        0.069      0.061         0.036         0.039    1.13       1.91       1.76
                                    4        0.076      0.065         0.040         0.039    1.18       1.91       1.96
   9       A-1         2.5          3        0.060      0.075         0.051         0.031    0.81       1.19       1.95
                                    4        0.065      0.071         0.050         0.037    0.91       1.30       1.75

 Table 41—Comparison of Vertical Displacements at Crown of HDPE Pipes under Light
                                       Truck
  Pipe   Backfill   Average       Truck     Plaxis 3D            Field Test (in.)           Ratio: Plaxis 3D / Field Test
  Run               Cover (ft)   Position     (in.)     Oct-00      May-01      Aug-02      Oct-00    May-01     Aug-02
   1       A-1         1.4          3        0.071      0.093         0.103         0.088    0.76       0.69       0.81
                                    4        0.065      0.101         0.122         0.063    0.64       0.53       1.02
   3       A-2         1.6          3        0.078      0.075         0.050         0.056    1.04       1.55       1.39
                                    4        0.076      0.059         0.045         0.045    1.28       1.68       1.68
   7       A-2         2.8          3        0.053      0.049         0.018         0.022    1.07       2.92       2.39
                                    4        0.058      0.049         0.026         0.026    1.18       2.23       2.23
   9       A-1         2.5          3        0.046      0.030         0.024         0.024    1.53       1.92       1.92
                                    4        0.049      0.031         0.023         0.023    1.58       2.13       2.13

Table 42—Comparison of Diametrical Changes at Springline of HDPE Pipes under Heavy
                                      Truck
  Pipe   Backfill   Average       Truck     Plaxis 3D            Field Test (in.)           Ratio: Plaxis 3D / Field Test
  Run               Cover (ft)   Position     (in.)     Oct-00      May-01      Aug-02      Oct-00    May-01     Aug-02
   1       A-1         1.4          3        0.014      -0.009        0.015         0.013    -1.58      0.95       1.10
                                    4        0.017      -0.006        0.017         0.014    -2.91      1.03       1.25
   3       A-2         1.6          3        0.021      0.014         0.006         0.012    1.48       3.44       1.72
                                    4        0.026      0.027         0.016         0.014    0.95       1.60       1.83
   7       A-2         2.8          3        0.016      0.009         0.004         0.005    1.74       3.91       3.13
                                    4        0.019      0.016         0.008         0.006    1.21       2.43       3.24
   9       A-1         2.5          3        0.011      0.005         0.005         0.002    2.13       2.13       5.33
                                    4        0.013      0.007         0.007         0.003    1.89       1.89       4.40

Table 43—Comparison of Diametrical Changes at Springline of HDPE Pipes under Light
                                     Truck
  Pipe   Backfill   Average       Truck     Plaxis 3D            Field Test (in.)           Ratio: Plaxis 3D / Field Test
  Run               Cover (ft)   Position     (in.)     Oct-00      May-01      Aug-02      Oct-00    May-01     Aug-02
   1       A-1         1.4          3        0.011      0.012         0.013         0.002    0.91       0.84       5.49
                                    4        0.013      0.016         0.017         0.005    0.83       0.78       2.66
   3       A-2         1.6          3        0.016      0.010         0.009         0.005    1.59       1.77       3.18
                                    4        0.019      0.011         0.011         0.005    1.77       1.77       3.89
   7       A-2         2.8          3        0.012      0.005         0.003         0.003    2.40       4.00       4.00
                                    4        0.015      0.007         0.006         0.004    2.11       2.46       3.69
   9       A-1         2.5          3        0.008      0.001         0.000         0.002    8.16        ---       4.08
                                    4        0.010      0.001         0.001         0.002   10.03      10.03       5.02



NCHRP 15-29 Appendix A                                                                                                    125
4.3.5    Discussion

Figure 100 shows ratios of the 3D analysis results to the field test data of displacements of the
concrete arch culvert due to live load. These ratios are tabulated in Table 28 and Table 29. For
the concrete arch culvert, a ratio of the vertical displacement at the crown of the 3D analysis to
that of the field test ranged from 1.83 to 5.83. A ratio of the chord extension of the 3D analysis
to that of the field test ranged from 0.96 to 4.66. The 3D analysis overestimated displacements
in most cases. Between the 3 ft and 1 ft cover cases, the 3D analysis estimated displacements
of the 1 ft cover case better. A ratio of thrust at base of the 3D analysis to that of the field test
ranged from 0.13 to 0.68. The 3D analysis underestimated the thrusts at base.

Figure 101 shows ratios of the 3D analysis results to the field test data of displacements of the
metal arch culvert due to live load. These ratios are tabulated in Table 32 and Table 33. For
the metal arch culvert, a ratio of the vertical displacement at the crown of the 3D analysis to that
of the field test ranged from 0.98 to 1.67. A ratio of the chord extension of the 3D analysis to
that of the field test ranged from 3.24 to 6.83.         The 3D analysis estimated the vertical
displacements of the metal arch much better than those of the concrete arch although there is
still a tendency of overestimation by the 3D analysis.             The 3D analysis significantly
overestimated the chord extension of the metal arch. The 3D analysis overestimated thrusts
below the points where curvature changes (NC and SC), and underestimated thrusts above
those points especially at shoulders (NH and SH). Thrusts at the crown for 1 ft cover were
estimated by the 3D analysis relatively well. The 3D analysis overestimated moments in many
cases; especially, crown moments were significantly overestimated. The 3D analysis estimated
moments at the shoulders and curvature points relatively well.

Figure 102 shows ratios of the 3D analysis results to the field test data of displacements of the
HDPE pipe culverts due to live load. These ratios are tabulated in Table 40 through Table 43.
For the HDPE pipe culverts, a ratio of the vertical displacement at the crown of the 3D analysis
to that of the field test ranged from 0.78 to 2.13. A ratio of the diametrical change at springline
of the 3D analysis to that of the field test data ranged from 0.78 to 10.03 except for a few data
points where shortening of diameter was found in the field tests. The 3D analysis slightly
underestimated displacements of the most of 1 ft cover cases, and the 3D analysis
overestimated displacements of the most of 3 ft cover cases.           Thrusts at the crown were
overestimated by the 3D analysis when compared to the field test data of May 2001. Moments
at the crown were underestimated by the 3D analysis for Pipe Run 1 with the heavy truck, and
they were overestimated Pipe Run 9 with the light truck when compared to the field test data of



NCHRP 15-29 Appendix A                                                                           126
May 2001. However, moments were estimated by the 3D analysis relatively well especially for
Pipe Run 9.

There seems to be a tendency that the 3D analysis prediction becomes better as the response
becomes larger. The 3D analyses estimated responses of the metal arch culvert and the HDPE
pipe culvert better than those of the concrete culvert. The vertical displacements at the crown
were estimated better than the horizontal displacements in most cases. The difference in small
structural responses between the 3D analysis and the field tests may have stemmed from the
accuracy of the measurements in the field.       For example, the 3D analyses estimated the
diametrical change at the springline of the HDPE pipe due to live loads to be an order of 0.01 to
0.02 in. This diametrical change translates to 0.005 to 0.01 in. of horizontal displacement at the
springline on each side. The accuracy of the LVDTs that were used in the MNDOT study was
0.005 in., which is the same order as the horizontal displacement of the springline. Therefore, it
is likely that the measured horizontal displacements were less accurate than the measured
vertical displacements.

Since the vertical crown displacements were estimated by the 3D analyses relatively well
especially for the metal arch and the HDPE pipes, we conclude that the Mohr-Coulomb model is
an appropriate soil model for backfill.




NCHRP 15-29 Appendix A                                                                        127
                                               6.0                                                                                                                   5.0
Ratio of Analysis Results to Field Test Data




                                                                                                                      Ratio of Analysis Results to Field Test Data
                                                                                       3 ft Cover        1 ft Cover                                                        3 ft Cover       1 ft Cover
                                               5.0
                                                                                                                                                                     4.0




                                                                                                                              for Horizontal Displacement
         for Vertical Displacement




                                               4.0
                                                                                                                                                                     3.0

                                               3.0

                                                                                                                                                                     2.0
                                               2.0

                                                                                                                                                                     1.0
                                               1.0


                                               0.0                                                                                                                   0.0
                                                            Test 1                              Test 2                                                                             Test 1                               Test 2
                                                                           Test Case                                                                                                                Test Case

                                                     (a) Crown vertical displacement                                                                                         (b) Horizontal chord extension

                                               Figure 100—Ratios of 3D Analysis Results to Field Test Data for Displacements of
                                                                               Concrete Arch



                                               1.8                                                                                                                   7.0
Ratio of Analysis Results to Field Test Data




                                                                                                                      Ratio of Analysis Results to Field Test Data




                                               1.6   3 ft Cover      1 ft Cover                                                                                                              3 ft Cover    1 ft Cover
                                                                                                                                                                     6.0
                                                                                                                              for Horizontal Displacement




                                               1.4
         for Vertical Displacement




                                                                                                                                                                     5.0
                                               1.2

                                               1.0                                                                                                                   4.0

                                               0.8                                                                                                                   3.0

                                               0.6
                                                                                                                                                                     2.0
                                               0.4
                                                                                                                                                                     1.0
                                               0.2

                                               0.0                                                                                                                   0.0
                                                            Test 1                              Test 2                                                                             Test 1                               Test 2
                                                                           Test Case                                                                                                                Test Case

                                                     (a) Crown vertical displacement                                                                                         (b) Horizontal chord extension

Figure 101—Ratios of 3D Analysis Results to Field Test Data for Displacements of Metal
                                       Arch




NCHRP 15-29 Appendix A                                                                                                                                                                                                           128
 Ratio of Analysis Results to Field Test Data   3.0                                                                                                               3.0




                                                                                                                   Ratio of Analysis Results to Field Test Data
                                                       Position 3 (Oct-00)                                                                                               Position 3 (Oct-00)
                                                       Position 3 (May-01)                                                                                               Position 3 (May-01)
                                                2.5                                                                                                               2.5
                                                       Position 3 (Aug-02)                                                                                               Position 3 (Aug-02)
          for Vertical Displacement




                                                                                                                            for Vertical Displacement
                                                       Position 4 (Oct-00)                                                                                               Position 4 (Oct-00)
                                                2.0                                                                                                               2.0
                                                       Position 4 (May-01)                                                                                               Position 4 (May-01)
                                                       Position 4 (Aug-02)                                                                                               Position 4 (Aug-02)
                                                1.5                                                                                                               1.5


                                                1.0                                                                                                               1.0


                                                0.5                                                                                                               0.5


                                                0.0                                                                                                               0.0
                                                          1                  3                7         9                                                                   1                   3              7   9
                                                                                   Pipe Run                                                                                                         Pipe Run


                             (a) Crown vertical displacement (Heavy truck)                                                                                        (b) Crown vertical displacement (Light truck)


                                                12.0                                                                                                              12.0
                                                         Position 3 (Oct-00)                                                                                              Position 3 (Oct-00)
 Ratio of Analysis Results to Field Test Data




                                                                                                                   Ratio of Analysis Results to Field Test Data
                                                10.0     Position 3 (May-01)                                                                                      10.0    Position 3 (May-01)
                                                                                                                                                                          Position 3 (Aug-02)
                                                         Position 3 (Aug-02)
         for Horizontal Displacement




                                                 8.0
                                                         Position 4 (Oct-00)                                               for Horizontal Displacement             8.0    Position 4 (Oct-00)
                                                                                                                                                                          Position 4 (May-01)
                                                         Position 4 (May-01)                                                                                              Position 4 (Aug-02)
                                                 6.0                                                                                                               6.0
                                                         Position 4 (Aug-02)
                                                 4.0                                                                                                               4.0

                                                 2.0                                                                                                               2.0

                                                 0.0                                                                                                               0.0
                                                           1                   3              7         9                                                                    1                  3              7   9
                                                -2.0                                                                                                              -2.0

                                                -4.0                                                                                                              -4.0
                                                                                   Pipe Run                                                                                                         Pipe Run

                                 (c) Horizontal diameter change (Heavy truck)                                                                                     (d) Horizontal diameter change (Light truck)


                                                                                              Pipe Run 1: A-1 backfill and 1.4 ft cover
                                                                                              Pipe Run 3: A-2 backfill and 1.6 ft cover
                                                                                              Pipe Run 7: A-2 backfill and 2.8 ft cover
                                                                                              Pipe Run 9: A-1 backfill and 2.5 ft cover



Figure 102—Ratios of 3D Analysis Results to Field Test Data for Displacements of HDPE
                                       Pipes

4.4                                                    Comparison between the Mohr-Coulomb and Hardening-Soil Models in Three-
                                                       Dimensional Analysis in PLAXIS

Initial investigation of culvert responses to live loads from 2D analyses with linear-elastic, Mohr-
Coulomb, and Hardening-soil models showed that responses from the Mohr-Coulomb and
Hardening-soil models were very close to each other whereas responses from the linear-elastic
model were significantly different from other models. As a result, the Mohr-Coulomb model was
selected to be used in the 3D analysis of field tests. Subsequent Panel comments suggested



NCHRP 15-29 Appendix A                                                                                                                                                                                                 129
comparison of the Mohr-Coulomb and Hardening-soil models in the 3D analysis as a
confirmation of selection of an appropriate soil model. To compare culvert responses from
these two soil models, 3D analyses were performed of long-span metal arch from NCHRP
Project 12-45 (McGrath et al. 2002) and HDPE pipe from the MNDOT study (McGrath et al.
2005).

4.4.1    Method of Approach

Soil-structure interaction analysis of culverts subjected to the surface live load is performed
using Plaxis 3D. Two structural models were selected as described above: (1) Long-span metal
arch, Test 2, 3 ft cover (NCHRP Project 12-45); and (2) HDPE pipe, Pipe Run 7, A-2 backfill, 2.8
ft cover (MNDOT study).      These structures were analyzed with both Mohr-Coulomb and
Hardening-Soil models, and structural responses were compared. In the metal arch model,
backfill was assumed to have properties of SW85, and the soil above the crown of arch was
assumed to have properties of SW95. In the HDPE pipe model, backfill was assumed to have
properties of ML95. The interface strength was assumed to be 50% of strength of surrounding
soil..

4.4.2    Results

4.4.2.1 Metal Arch in Test 2 with 3 ft Cover

Figure 103 compares vertical crown displacements and horizontal chord extensions along the
culvert between the two cases of the metal arch analysis: the case with the Mohr-Coulomb
model and the case with the Hardening-soil model. Figure 104 compares thrusts and moments
under the wheel load between the two cases. These figures also show displacements and
forces measured in the field tests. Measured thrusts and moments in Figure 104 are average
values of measurements under the left and right wheels. Table 44 through Table 46 also
compare displacements and forces between the two case. Differences in moments and thrusts
between the two soil models were insignificant. Displacements were slightly smaller with the
Hardening-soil model than the Mohr-Coulomb model: by 9 percent for the vertical crown
displacement and by 16 percent for the horizontal displacement.        Therefore, displacement
results were closer to the filed measurements with the Hardening-soil model than with the Mohr-
Coulomb soil model in this case.

4.4.2.2 HDPE Pipe with A2 Backfill and 2.8 ft Cover

Figure 105 and Figure 106 compare vertical crown displacements and horizontal diameter
extensions along the culvert between the two cases of HDPE pipe analysis: the case with the


NCHRP 15-29 Appendix A                                                                      130
Mohr-Coulomb model and the case with the Hardening-soil model. These figures also show
displacements measured in the field tests. Figure 107 and Figure 108 compares thrusts and
moments under the wheel load between the two cases.                                                                                                                                  Table 47 through Table 49 also
compare displacements and forces between the two case. Calculated displacements were
larger with the Hardening-soil model than with the Mohr-Coulomb model: by about 30 percent
for the vertical crown displacement and by about 55 percent for the horizontal displacement.
Therefore, displacement results from the Mohr-Coulomb model were closer to the measured
displacements in the field tests in this particular case.                                                                                                                           Due to the larger displacements,
moments and thrusts were also larger with the Hardening-soil model.                                                                                                                                                 Thrusts from the
Hardening-soil model were up to 20 percent higher than those from the Mohr-Coulomb model,
and moments were up to 44 percent higher. Softer soil responses obtained for the Hardening-
soil model can be explained by the lower stiffness of the Hardening soil model when compared
to that of the Duncan-Selig model.                                                                  By selecting higher stiffness values for Hardening-soil
properties of ML95, it will be possible to bring force results closer to those from the Mohr-
Coulomb model.

                                        0.9                                                                                                                             0.50
                                                                                                                   Horiz. Extension of Chord at Height of 88 in. (in)




                                                                  Mohr-Coulomb Model (Test 2, 3 ft Cover)                                                               0.45                       Mohr-Coulomb Model (Test 2, 3 ft Cover)
                                        0.8
  Vertical Displacement at Crown (in)




                                                                  Hardening Soil Model (Test 2, 3 ft Cover)                                                                                        Hardening Soil Model (Test 2, 3 ft Cover)
                                        0.7                                                                                                                             0.40
                                                                  Field Test (Test 2, 3 ft Cover)                                                                                                  Field Test (Test 2, 3 ft Cover)
                                                                                                                                                                        0.35
                                        0.6
                                                                                                                                                                        0.30
                                        0.5
                                                                                                                                                                        0.25
                                        0.4
                                                                                                                                                                        0.20
                                        0.3
                                                                                                                                                                        0.15
                                        0.2                                                                                                                             0.10
                                        0.1                                                                                                                             0.05
                                                         Wheel Load Location                                                                                                             Wheel Load Location
                                        0.0                                                                                                                             0.00
                                              0           5                10                15               20                                                               0          5                    10             15                20
                                                   Distance from Symmetry Line of Tandem Axles (ft)                                                                                Distance from Symmetry Line of Tandem Axles (ft)


                                                  (a) Vertical crown displacement                                                                                                  (b) Horizontal chord extension

                                        Figure 103—Comparison of Displacements between Cases with Mohr-Coulomb and
                                                     Hardening-Soil Models (Metal Arch, Test 2, 3 ft Cover)




NCHRP 15-29 Appendix A                                                                                                                                                                                                                         131
                                                                                                        kip-in/ft
                                                                                                             -30
                            kip/ft
                                -5




                                25                                                                           60



            Mohr-Coulomb Model (Test 2, 3 ft Cover)                                      Mohr-Coulomb Model (Test 2, 3 ft Cover)
            Hardening Soil Model (Test 2, 3 ft Cover)                                    Hardening Soil Model (Test 2, 3 ft Cover)
            Field Test (Test 2, 3 ft Cover)                                              Field Test (Test 2, 3 ft Cover)



                   (a) Thrust                                                                  (b) Moment

  Figure 104—Comparison of Thrusts and Moments under Wheel between Cases with
       Mohr-Coulomb and Hardening-Soil Models (Metal Arch, Test 2, 3 ft Cover)


   Table 44—Summary of Displacements under Wheel (Metal Arch, Test 2, 3 ft Cover)
                                                                                                      Ratio:
                                                                  Plaxis 3D (in.)
          Vertical or                    Field                                                Plaxis 3D / Field Test
          Horizontal                   Test (in.)               Mohr-      Hardening-          Mohr-      Hardening-
                                                               Coulomb        Soil           Coulomb          Soil
         Vertical crown
                                           0.45                 0.79            0.72              1.74                     1.58
         displacement
        Horizontal chord
                                           0.25                 0.40            0.34              1.60                     1.34
           extension



      Table 45—Summary of Thrusts under Wheel (Metal Arch, Test 2, 3 ft Cover)
       Location Field Test                              Plaxis 3D (kip/ft)             Ratio: Plaxis 3D/Field Test
                         (kip/ft)                  Mohr-               Hardening          Mohr-                     Hardening
                                                Coulomb                  Soil            Coulomb                        Soil
          NS               0.69                         3.12             3.54               4.54                        5.15
          NC               1.42                         4.41             4.70               3.10                        3.31
          NH               9.07                         5.92             6.27               0.65                        0.69
          CR               3.08                         7.05             7.75               2.29                        2.51
          SH               5.84                         5.80             6.18               0.99                        1.06
          SC               2.46                         4.08             4.46               1.66                        1.81
          SS              -0.34                         2.75             3.25              -7.98                       -9.44




NCHRP 15-29 Appendix A                                                                                                               132
                                                           Table 46—Summary of Moments under Wheel (Metal Arch, Test 2, 3 ft Cover)
                                                               Location Field Test                    Plaxis 3D (kip-in./ft)                                                                      Ratio: Plaxis 3D/Field Test
                                                                              (kip-in./ft)              Mohr-                    Hardening                                                               Mohr-                Hardening
                                                                                                      Coulomb                      Soil                                                              Coulomb                      Soil
                                                                  NS              -0.03                  -3.14                     -3.61                                                                 94.73                  108.90
                                                                 NC               -0.38                  -2.75                     -2.13                                                                 7.27                     5.63
                                                                 NH               -5.24                  -5.75                     -5.42                                                                 1.10                     1.03
                                                                 CR                2.04                 13.71                      13.38                                                                 6.74                     6.57
                                                                  SH              -3.28                  -6.43                     -6.26                                                                 1.96                     1.91
                                                                  SC              -1.97                  -2.62                     -1.78                                                                 1.33                     0.91
                                                                  SS               0.07                  -2.66                     -3.31                                                                 -35.61                 -44.31




                                                0.12                                                                                                                               0.12
                                                                                               Mohr-Coulomb Model (Heavy P3)                                                                                                   Mohr-Coulomb Model (Light P3)
                                                                                               Mohr-Coulomb Model (Heavy P4)       Vertical Displacement at Crown (in.)                                                        Mohr-Coulomb Model (Light P4)
 Vertical Displacement at Crown (in.)




                                                                                               Hardening Soil Model (Heavy P3)                                                                                                 Hardening Soil Model (Light P3)
                                                0.10                                                                                                                               0.10
                                                                                               Hardening Soil Model (Heavy P4)                                                                                                 Hardening Soil Model (Light P4)
                                                                                               Heavy P3 (Oct-00)                                                                                                               Light P3 (Oct-00)
                                                                                               Heavy P3 (May-01)                                                                                                               Light P3 (May-01)
                                                0.08                                           Heavy P3 (Aug-02)                                                                   0.08
                                                                                                                                                                                                                               Light P3 (Aug-02)
                                                                                               Heavy P4 (Oct-00)                                                                                                               Light P4 (Oct-00)
                                                                                               Heavy P4 (May-01)                                                                                                               Light P4 (May-01)
                                                0.06                                           Heavy P4 (Aug-02)                                                                   0.06
                                                                                                                                                                                                                               Light P4 (Aug-02)


                                                0.04                                                                                                                               0.04


                                                0.02                                                                                                                               0.02


                                                0.00                                                                                                                               0.00
                                                       0           2         4         6          8          10           12                                                              0         2          4         6           8           10              12
                                                                Distance from Symmetry Line of Tandem Axles (ft)                                                                                  Distance from Symmetry Line of Tandem Axles (ft)

                                                                           (a) Heavy truck                                                                                                                   (b) Light truck

                                Figure 105—Comparison of Crown Vertical Displacements between Cases with Mohr-
                                       Coulomb and Hardening-Soil Models (HDPE Pipe, A2 Soil, 2.8 ft Cover)


                                                0.035                                         Mohr-Coulomb Model (Heavy P3)                                                        0.035
 Diametrical Change between Springlines (in.)




                                                                                                                                    Diametrical Change between Springlines (in.)




                                                                                              Mohr-Coulomb Model (Heavy P4)                                                                                                    Mohr-Coulomb Model (Light P3)
                                                                                              Hardening Soil Model (Heavy P3)                                                                                                  Mohr-Coulomb Model (Light P4)
                                                0.030                                         Hardening Soil Model (Heavy P4)                                                      0.030                                       Hardening Soil Model (Light P3)
                                                                                              Heavy P3 (Oct-00)                                                                                                                Hardening Soil Model (Light P4)
                                                                                              Heavy P3 (May-01)                                                                                                                Light P3 (Oct-00)
                                                0.025                                                                                                                              0.025                                       Light P3 (May-01)
                                                                                              Heavy P3 (Aug-02)
                                                                                              Heavy P4 (Oct-00)                                                                                                                Light P3 (Aug-02)
                                                                                              Heavy P4 (May-01)                                                                                                                Light P4 (Oct-00)
                                                0.020                                                                                                                              0.020                                       Light P4 (May-01)
                                                                                              Heavy P4 (Aug-02)
                                                                                                                                                                                                                               Light P4 (Aug-02)
                                                0.015                                                                                                                              0.015


                                                0.010                                                                                                                              0.010


                                                0.005                                                                                                                              0.005


                                                0.000                                                                                                                              0.000
                                                           0        2         4         6         8           10          12                                                                  0      2         4          6          8           10              12
                                                                 Distance from Symmetry Line of Tandem Axles (ft)                                                                                 Distance from Symmetry Line of Tandem Axles (ft)

                                                                           (a) Heavy truck                                                                                                                   (b) Light truck



NCHRP 15-29 Appendix A                                                                                                                                                                                                                                           133
                              Figure 106—Comparison of Horizontal Diameter Changes between Cases with Mohr-
                                     Coulomb and Hardening-Soil Models (HDPE Pipe, A2 Soil, 2.8 ft Cover)


                              45                                                                                                                45
                                                                                                                                                                                         Mohr-Coulomb Model (Light P3)
                              40                                                                                                                40                                       Mohr-Coulomb Model (Light P4)
                                                                                                                                                                                         Hardening Soil Model (Light P3)
                              35                                                                                                                35
                                                                                                                                                                                         Hardening Soil Model (Light P4)
                              30                                                                                                                30
Thrust (lb/in.)




                                                                                                                  Thrust (lb/in.)
                              25                                                                                                                25

                              20                                                                                                                20

                              15                                                                                                                15
                                                         Mohr-Coulomb Model (Heavy P3)
                              10                                                                                                                10
                                                         Mohr-Coulomb Model (Heavy P4)
                               5                         Hardening Soil Model (Heavy P3)                                                         5
                                                         Hardening Soil Model (Heavy P4)
                               0                                                                                                                 0
                                180     135       90       45        0      -45       -90    -135     -180                                        180       135    90        45          0     -45      -90     -135       -180
                                                         Degrees from Crown                                                                                                Degrees from Crown

                                                         (a) Heavy truck                                                                                                    (b) Light truck

Figure 107—Comparison of Thrusts between Cases with Mohr-Coulomb and Hardening-
                  Soil Models (HDPE Pipe, A2 Soil, 2.8 ft Cover)


                              25                                                                                                                25
                                                                                   Mohr-Coulomb Model                                                                                                Mohr-Coulomb Model
                              20                                                   (Heavy P3)                                                   20                                                   (Light P3)
                                                                                   Mohr-Coulomb Model                                                                                                Mohr-Coulomb Model
Bending Moment (lb-in./in.)




                                                                                                                  Bending Moment (lb-in./in.)




                                                                                   (Heavy P4)                                                                                                        (Light P4)
                              15                                                   Hardening Soil Model                                         15
                                                                                                                                                                                                     Hardening Soil Model
                                                                                   (Heavy P3)                                                                                                        (Light P3)
                              10                                                   Hardening Soil Model                                         10
                                                                                                                                                                                                     Hardening Soil Model
                                                                                   (Heavy P4)                                                                                                        (Light P4)
                               5                                                                                                                 5

                               0                                                                                                                 0

                               -5                                                                                                                -5

                              -10                                                                                                               -10

                              -15                                                                                                               -15

                              -20                                                                                                               -20
                                 180    135       90       45        0      -45       -90     -135    -180                                         180      135     90       45          0     -45      -90     -135       -180
                                                         Degrees from Crown                                                                                                Degrees from Crown

                                                         (a) Heavy truck                                                                                                    (b) Light truck

Figure 108—Comparison of Moments between Cases with Mohr-Coulomb and Hardening-
                   Soil Models (HDPE Pipe, A2 Soil, 2.8 ft Cover)


                  Table 47—Summary of Vertical Displacements under Wheel (HDPE Pipe, A2 Soil, 2.8 ft
                                                       Cover)
                Truck                  Position                Field Test (in.)                Plaxis 3D (in.)                                                           Ratio: Plaxis 3D / Field Test
                                                                                              Mohr-          Hardening                                    Mohr-Coulomb Model                    Hardening Soil Model
                                                       Oct-00     May-01      Aug-02        Coulomb            Soil                                   Oct-00      May-01      Aug-02         Oct-00      May-01        Aug-02
               Heavy                      3            0.061        0.036         0.039       0.069            0.089                                     1.13      1.91           1.76        1.47         2.48            2.29
                                          4            0.065        0.040         0.039       0.076            0.103                                     1.18      1.91           1.96        1.59         2.58            2.65
                         Light            3            0.049        0.018         0.022       0.053            0.068                                     1.07      2.92           2.39        1.38         3.75            3.07
                                          4            0.049        0.026         0.026       0.058            0.077                                     1.18      2.23           2.23        1.58         2.98            2.98




NCHRP 15-29 Appendix A                                                                                                                                                                                                      134
 Table 48—Summary of Horizontal Chord Extensions under Wheel (HDPE Pipe, A2 Soil,
                                  2.8 ft Cover)
 Truck     Position               Field Test (in.)             Plaxis 3D (in.)                          Ratio: Plaxis 3D / Field Test
                                                              Mohr-      Hardening         Mohr-Coulomb Model              Hardening Soil Model
                          Oct-00     May-01      Aug-02      Coulomb        Soil        Oct-00    May-01     Aug-02     Oct-00      May-01      Aug-02
 Heavy        3           0.009        0.004         0.005    0.016        0.024          1.74     3.91        3.13       2.69          6.06     4.85
              4           0.016        0.008         0.006    0.019        0.031          1.21     2.43        3.24       1.95          3.90     5.20
  Light       3           0.005        0.003         0.003    0.012        0.018          2.40     4.00        4.00       3.67          6.11     6.11
              4           0.007        0.006         0.004    0.015        0.023          2.11     2.46        3.69       3.35          3.91     5.87




             Table 49—Summary of Force Results (HDPE Pipe, A2 Soil, 2.8 ft Cover)
  Truck       Position                                 Thrust (lb/in.)                                      Moment (lb-in./in.)
                                         Crown                           Peak                    Peak Positive               Peak Negative
                                      MC               HS          MC              HS            MC              HS              MC            HS
  Heavy               3              19.5             24.6        28.7             32.1          15.7           20.5          -11.1            -15.4
                      4              24.5             32.8        33.2             39.8          11.7           14.7          -11.5            -16.5
   Light              3              14.9             18.8        22.0             24.3          12.0           15.1             -8.6          -11.7
                      4              18.5             24.9        25.2             30.2          8.8            10.5             -8.7          -12.4




4.4.3       Conclusion

With the soil properties reported used, there was not a consistent trend in structural responses
when the soil model was changed from the Mohr-Coulomb model to the Hardening-soil model.
However, for both structures (the metal arch and the HDPE pipe), changing the soil constitutive
model from the Mohr-Coulomb model to the Hardening-soil model affected displacement results
more than moments and thrusts.

We will examine appropriateness of the Mohr-Coulomb model by simulating the field tests in
ABAQUS as described in the next section.

4.5         Three-Dimensional Analysis of Field Tests in ABAQUS

4.5.1       Introduction

In Task 2 of NCHRP Project 15-29, SGH performed 3D analyses of field live load tests (NCHRP
Project 12-45 and MNDOT study) using PLAXIS 3D with the Mohr-Coulomb constitutive model
for backfill. A few panel members showed their concern about appropriateness of the Mohr-
Coulomb model because calculated structural responses were not close to the measured
responses in some cases.                             SGH believes that there are two reasons for the deviation of




NCHRP 15-29 Appendix A                                                                                                                            135
calculated responses from the measured responses: (1) orthotropic section properties for metal
arch and HDPE pipe and (2) level of backfill compaction.

Corrugated metal plate arch and HDPE pipe have different stiffnesses in the circumferential and
longitudinal directions due to their corrugation or plate profile.     Since orthotropic properties
cannot be assigned to plate elements in PLAXIS 3D, we inserted strips of very thin elements to
match circumferential and longitudinal stiffnesses.     However, due to these thin elements,
effective shear stiffness of the metal arch or HDPE pipe becomes lower than the actual shear
stiffness, which resulted in a concentration of displacement near the wheel loads.

In the cases analyzed for the metal arch in PLAXIS 3D, crown vertical displacements in Test 1
were predicted well by the analysis whereas those calculated for Test 2 were significantly
greater than measured displacements.       In Test 1, backfill soil was compacted for a target
compaction of 95 percent of the maximum standard proctor density and resulted in 92 percent.
In Test 2, backfill soils above and below the crown were compacted for target compaction of 95
percent and 85 percent, respectively, and resulted in 96 percent and 87 percent. SGH believes
that the backfill soil below the crown was further compacted when the backfill soil above the
crown was compacted to 96 percent of the standard proctor density in Test 2. The PLAXIS 3D
analysis reported in the earlier did not consider this additional compaction effort, and soil
properties for SW85 were used for the backfill soil below the crown.

To examine the two items discussed above, SGH performed two cases of 3D soil-structure
interaction analysis in ABAQUS: (1) long-span metal arch, Test 2, 3 ft cover (NCHRP Project
12-45); and (2) HDPE pipe, Pipe Run 7, A-2 backfill, 2.8 ft cover (MNDOT study).

4.5.2    Method of Approach

Soil-structure interaction analysis of culverts subjected to the surface live load was performed
using ABAQUS. Two structural types were selected as described above: (1) Long-span metal
arch (Test 1 and Test 2 with 3 ft cover in NCHRP Project 12-45); and (2) HDPE pipe (Pipe Run
7 with A-2 backfill and 2.8 ft cover and Pipe Run 3 with A-2 backfill and 1.6 ft cover in the
MNDOT study).

Although longitudinal and circumferential stiffnesses for axial force and bending are known, we
can match only three of the four stiffnesses exactly because shell elements have uniform
thickness and only orthotropic material properties are defined.          We calculated moduli of
elasticity of the longitudinal and circumferential directions and thickness of the shell element to



NCHRP 15-29 Appendix A                                                                         136
match longitudinal and circumferential bending stiffnesses and circumferential axial stiffness.
Shear modulus in the orthotropic properties used in the analysis was determined by multiplying
actual shear modulus by a ratio of the actual plate thickness to the effective plate thickness
used in the analysis. Table 50 shows orthotropic properties used in ABAQUS analyses, and
Table 51 compares stiffnesses calculated from the orthotropic properties given in Table 50 with
actual stiffnesses. Although axial stiffness of the metal arch used in ABAQUS turned out to be
close to the actual stiffness, axial stiffness of the HDPE pipe used in ABAQUS was significantly
smaller than the actual stiffness.

                  Table 50—Orthotropic Properties Used in ABAQUS Analyses
                        Modulus of elasticity (psi)     Poisson’s      Shear          Shell element
      Structure
                    Circumferential    Longitudinal       ratio      modulus (psi)    thickness (in.)
     Metal arch        3,234,000          17,670           0.3         1,001,000           2.390
     HDPE pipe          12,680               162           0.35          2,865             4.227


                           Table 51—Orthotropic Stiffness Properties
                    Actual or           Axial, EA, (lb/in.)              Bending, EI, (lb-in.2/in.)
      Structure
                    ABAQUS      Circumferential     Longitudinal    Circumferential     Longitudinal
                     Actual        7,731,000           47,917          3,681,000           20,106
     Metal arch      Used in
                                  7,731,000           42,231          3,681,000           20,106
                    ABAQUS
                     Actual          53,600           32,700            79,800            1,019
     HDPE pipe       Used in
                                     53,600            684              79,800            1,019
                    ABAQUS



Analyses in ABAQUS were performed with the Mohr-Coulomb model, and structural responses
were compared with those measured in the field tests. In the metal arch model for Test 2, two
sets of backfill properties were examined for backfill below the crown of arch: SW85 and SW90.
Properties of SW95 were used for the soil above the crown in Test 1 and Test 2 and for backfill
in Test 1. For the HDPE pipe, two sets of backfill properties were used: ML90 and ML95. Also,
soft haunch and void areas were modeled by using very soft soil properties as shown in Figure
109. These areas were not modeled in PLAXIS 3D. Heavy live load truck was positioned in a
symmetric manner over crown (Position P4) for the HDPE pipe. The interface was introduced
between the structure and surrounding soil using contact elements. Coefficient of friction was
assumed to be 50 percent of angle of friction of surrounding soil. Structures, live load tests, and
soil properties of SW85, SW95, ML90, and ML95 were described in detail earlier. Figure 110
and Figure 111 show ABAQUS models of the metal arch and the HDPE pipe.




NCHRP 15-29 Appendix A                                                                                  137
                     4 in. Pavement    8 in. Gravel


                                                                                   12 in.
                         Cover Depth
                           (varies)                                      AASHTO Backfill
                                                                             (A-2)
                      Soft Haunch
                                                                             6 in. Bedding
                                                  60 in.
                               Void              Nominal

                    In-Situ Soil                                         84 in.
                                                 8 ft 8 in.




                                                      30 ft

    Figure 109—Cross Section of Finite Element Model for HDPE Pipe in ABAQUS



              Table 52—Soil Porperties Used for Soft Haunch and Void Areas
                    Modulus of         Poisson’s              Angle of            Dilatation   Cohesion
       Area          Elasticity          Ratio                Friction             Angle
                          E               ν                      φ                   ψ            c
                       (psi)                                   (deg)                (deg)        (psi)
    Soft haunch        1,000             0.35                   28                    0           1.0
        Void             50              0.30                   23                    0           2.5




NCHRP 15-29 Appendix A                                                                                    138
                  Figure 110—ABAQUS Metal Arch Model with 3 ft Cover




                 Figure 111—ABAQUS HDPE Pipe Model with 3 ft Cover



4.5.3   Validation of ABAQUS Model

Before performing analyses with orthotropic material properties for structure in ABAQUS, we a
performed analysis of the metal arch in ABAQUS with strips of thin elements to examine
whether ABAQUS can produce the same results as PLAXIS 3D did.            The same material
properties were used in both models. However, finite element meshes in the cross section were
not the same because 15-noded wedge elements were used for soil in PLAXIS 3D while 8-
noded brick elements were used in ABAQUS.            Figure 112 and Figure 113 compare
displacement and force results from PLAXIS 3D and ABAQUS, respectively. Vertical crown
displacement and horizontal chord extension under wheel in ABAQUS differ from those of
PLAXIS 3D by only 2 percent and 7 percent, respectively. Maximum thrust and moment in
ABAQUS differ from those of PLAXIS 3D by only 4 percent and 3 percent, respectively. We
concluded that ABAQUS can produce essentially the same results as PLAXIS 3D when the
same problem is analyzed.




NCHRP 15-29 Appendix A                                                                   139
                                        1.0                                                                                                                                             0.50




                                                                                                                                        Horiz. Extension of Chord at Height of 88 in.
                                        0.9                                                    Test 2, 3 ft Cover (PLAXIS, SW85)                                                        0.45                                             Test 2, 3 ft Cover (PLAXIS, SW85)
  Vertical Displacement at Crown (in)


                                                                                                                                                                                                                                         Test 2, 3 ft Cover (Field Test)
                                        0.8                                                    Test 2, 3 ft Cover (Field Test)                                                          0.40
                                                                                                                                                                                                                                         Test 2, 3 ft Cover (ABAQUS, SW85)
                                        0.7                                                    Test 2, 3 ft Cover (ABAQUS, SW85)                                                        0.35

                                        0.6                                                                                                                                             0.30




                                                                                                                                                            (in)
                                        0.5                                                                                                                                             0.25

                                        0.4                                                                                                                                             0.20

                                        0.3                                                                                                                                             0.15

                                        0.2       Wheel Load Location                                                                                                                   0.10       Wheel Load Location

                                        0.1                                                                                                                                             0.05

                                        0.0                                                                                                                                             0.00
                                              0                 5                         10                    15                 20                                                          0                 5                    10                    15                20
                                                       Distance from Symmetry Line of Tandem Axles (ft)                                                                                                 Distance from Symmetry Line of Tandem Axles (ft)


                                                    (a) Vertical crown displacement                                                                                                                   (b) Horizontal chord extension

 Figure 112—Comparison of Vertical and Horizontal Displacements between PLAXIS 3D
              and ABAQUS Analyses (Metal Arch, Test 2, 3 ft Cover)
                                                                                                                                                                                                                             kip-in/ft
                                                                                 kip/ft                                                                                                                                            -15

                                                                                     -2.5




                                                                                      12.5                                                                                                                                          30

                                                                    Test 2, 3 ft Cover (PLAXIS, SW85)                                                                                                             Test 2, 3 ft Cover (PLAXIS, SW85)
                                                                    Test 2, 3 ft Cover (Field Test)                                                                                                               Test 2, 3 ft Cover (Field Test)
                                                                    Test 2, 3 ft Cover (ABAQUS, SW85)                                                                                                             Test 2, 3 ft Cover (ABAQUS, SW85)



                                                                       (a) Thrust                                                                                                                                    (b) Moment

Figure 113—Comparison of Thrusts and Moments under Wheel between PLAXIS 3D and
               ABAQUS Analyses (Metal Arch, Test 2, 3 ft Cover)

4.5.4                                              Results

4.5.4.1 Metal Arch with 3 ft Cover

Figure 114 and Figure 115 show displacement and force results from ABAQUS analyses with
orthotropic properties for the metal arch. Table 53 compares displacement results between
ABAQUS analyses and field test data. By comparing displacements between the ABAQUS
model with orthotropic properties and SW85 properties and the PLAXIS model, it is obvious that
both vertical and horizontal displacements were significantly reduced in the ABAQUS model,
and that horizontal displacements were affected more than vertical displacements by modeling
the metal arch with orthotropic properties. These results confirm that modeling technique used
in PLAXIS 3D for the corrugated metal arch caused unrealistically large horizontal
displacements. In Figure 114, we can also see that using SW90 properties for backfill in Test 2
case brought calculated displacements much closer to the measured data, which shows that the
backfill soil in Test 2 that was compacted to 87 percent of the maximum standard proctor


NCHRP 15-29 Appendix A                                                                                                                                                                                                                                                       140
density was compacted to about 90 percent of the maximum standard proctor density when the
soil above the crown was compacted.




                                       0.9                                                                                                                                          0.45




                                                                                                                                    Horiz. Extension of Chord at Height of 88 in.
                                                                                                                                                                                                                                    Test 1, 3 ft Cover (Field Test)
                                                                                         Test 1, 3 ft Cover (Field Test)
                                       0.8                                                                                                                                          0.40                                            Test 1, 3 ft Cover (ABAQUS, SW95)
 Vertical Displacement at Crown (in)




                                                                                         Test 1, 3 ft Cover (ABAQUS, SW95)
                                                                                                                                                                                                                                    Test 2, 3 ft Cover (Field Test)
                                                                                         Test 2, 3 ft Cover (Field Test)
                                       0.7                                                                                                                                          0.35                                            Test 2, 3 ft Cover (ABAQUS, SW85)
                                                                                         Test 2, 3 ft Cover (ABAQUS, SW85)
                                                                                                                                                                                                                                    Test 2, 3 ft Cover (ABAQUS, SW90)
                                       0.6                                               Test 2, 3 ft Cover (ABAQUS, SW90)                                                          0.30                                            Test 1, 3 ft Cover (PLAXIS, SW95)
                                                                                         Test 1, 3 ft Cover (PLAXIS, SW95)
                                                                                                                                                                                                                                    Test 2, 3 ft Cover (PLAXIS, SW85)
                                       0.5                                               Test 2, 3 ft Cover (PLAXIS, SW85)                                                          0.25




                                                                                                                                                        (in)
                                       0.4                                                                                                                                          0.20

                                       0.3                                                                                                                                          0.15

                                       0.2                                                                                                                                          0.10
                                                 Wheel Load Location
                                                                                                                                                                                                         Wheel Load Location
                                       0.1                                                                                                                                          0.05

                                       0.0                                                                                                                                          0.00
                                             0                5                         10               15                  20                                                            0             5                         10               15                  20
                                                      Distance from Symmetry Line of Tandem Axles (ft)                                                                                            Distance from Symmetry Line of Tandem Axles (ft)


                                                   (a) Vertical crown displacement                                                                                                               (b) Horizontal chord extension

                                        Figure 114—Vertical and Horizontal Displacements from ABAQUS Analyses with
                                                        Orthotropic Properties (Metal Arch, 3 ft Cover)



                                                                                                                                                                                                                        kip-in/ft
                                                                                                                                                                                                                             -15
                                                                               kip/ft
                                                                                  -2.5




                                                                                  12.5                                                                                                                                        30

                                                                  Test 1, 3 ft Cover (Field Test)                                                                                                            Test 1, 3 ft Cover (Field Test)
                                                                  Test 1, 3 ft Cover (ABAQUS, SW95)                                                                                                          Test 1, 3 ft Cover (ABAQUS, SW95)
                                                                  Test 2, 3 ft Cover (Field Test)                                                                                                            Test 2, 3 ft Cover (Field Test)
                                                                  Test 2, 3 ft Cover (ABAQUS, SW85)                                                                                                          Test 2, 3 ft Cover (ABAQUS, SW85)
                                                                  Test 2, 3 ft Cover (ABAQUS, SW90)                                                                                                          Test 2, 3 ft Cover (ABAQUS, SW90)



                                                                    (a) Thrust                                                                                                                               (b) Moment

Figure 115 –Thrusts and Moments under Wheel from ABAQUS Analyses with Orthotropic
                          Properties (Metal Arch, 3 ft Cover)



                                         Table 53—Summary of Displacements from ABAQUS Analyses with Orthotropic
                                                            Properties (Metal Arch, 3 ft Cover)
                                                                                                                                                                                                                                Ratio:
                                                                                  Field Test (in.)                                  ABAQUS (in.)
                                                                                                                                                                                                                       ABAQUS / Field Test
                                                                           Vertical                 Horizontal                Vertical                                                         Horizontal              Vertical     Horizontal
                                        Test 1 (SW95
                                                                             0.413                      0.083                     0.397                                                          0.163                   0.961                     1.970
                                        in ABAQUS)
                                        Test 2 (SW90
                                                                             0.453                      0.252                     0.489                                                          0.208                   1.080                     0.824
                                        in ABAQUS)



NCHRP 15-29 Appendix A                                                                                                                                                                                                                                              141
4.5.4.2 HDPE Pipe with A2 Backfill and 2.8 ft Cover

Figure 116 and Figure 117 show displacement and force results from ABAQUS analyses with
orthotropic properties for the HDPE pipe. Table 54 compares displacement results between
ABAQUS analyses and field test data measured in October 2000, which were the first
measurements after the pipe installation.                                                                      In the case of 2.8 ft of cover, calculated
displacements with ML 95 properties were greater than the data measured in October 2000 by
18 percent and 25 percent for vertical and horizontal displacements, respectively, while in the
case of 1.6 ft of cover, calculated displacements with ML95 properties were smaller than the
data measured in October 2000 by 27 percent and 7 percent for vertical and horizontal
displacements, respectively. These results show that displacements calculated in ABAQUS are
in good agreement with those measured in the first live load tests after the pipe installation.

                                        0.10                                                                                                                   0.040
                                                                                                                Diametrical Change between Springlines (in.)



                                                                                Heavy P4 (Oct-00)                                                                                                     Heavy P4 (Oct-00)
                                        0.09
                                                                                                                                                               0.035
 Vertical Displacement at Crown (in.)




                                                                                Heavy P4 (May-01)                                                                                                     Heavy P4 (May-01)
                                        0.08                                    Heavy P4 (Aug-02)                                                                                                     Heavy P4 (Aug-02)
                                                                                                                                                               0.030
                                        0.07                                    Heavy P4 (ABAQUS, ML90)                                                                                               Heavy P4 (ABAQUS, ML90)
                                                                                Heavy P4 (ABAQUS, ML95)                                                        0.025                                  Heavy P4 (ABAQUS, ML95)
                                        0.06

                                        0.05                                                                                                                   0.020

                                        0.04
                                                                                                                                                               0.015
                                        0.03
                                                                                                                                                               0.010
                                        0.02
                                                                                                                                                               0.005
                                        0.01

                                        0.00                                                                                                                   0.000
                                               0      2          4         6         8         10         12                                                           0      2         4         6         8         10        12
                                                    Distance from Symmetry Line of Tandem Axles (ft)                                                                       Distance from Symmetry Line of Tandem Axles (ft)


                                                   (a) Vertical crown displacement                                                                                         (b) Horizontal chord extension

                                          Figure 116—Vertical and Horizontal Displacements from ABAQUS Analyses with
                                                    Orthotropic Properties (HDPE Pipe, A2 Backfill, 2.8 ft Cover)




NCHRP 15-29 Appendix A                                                                                                                                                                                                          142
                                        0.18                                                                                                                   0.040




                                                                                                                Diametrical Change between Springlines (in.)
                                                                               Heavy P4 (Oct-00)                                                                                                       Heavy P4 (Oct-00)
                                        0.16                                                                                                                   0.035                                   Heavy P4 (May-01)
 Vertical Displacement at Crown (in.)




                                                                               Heavy P4 (May-01)
                                                                               Heavy P4 (Aug-02)                                                                                                       Heavy P4 (Aug-02)
                                        0.14
                                                                               Heavy P4 (ABAQUS, ML90)                                                         0.030                                   Heavy P4 (ABAQUS, ML90)
                                        0.12                                   Heavy P4 (ABAQUS, ML95)                                                                                                 Heavy P4 (ABAQUS, ML95)
                                                                                                                                                               0.025
                                        0.10
                                                                                                                                                               0.020
                                        0.08
                                                                                                                                                               0.015
                                        0.06
                                                                                                                                                               0.010
                                        0.04

                                        0.02                                                                                                                   0.005

                                        0.00                                                                                                                   0.000
                                               0      2          4         6           8       10        12                                                            0       2         4         6         8         10        12
                                                    Distance from Symmetry Line of Tandem Axles (ft)                                                                        Distance from Symmetry Line of Tandem Axles (ft)


                                              (a) Vertical crown displacement              (b) Horizontal chord extension
                                          Figure 117—Vertical and Horizontal Displacements from ABAQUS Analyses with
                                                    Orthotropic Properties (HDPE Pipe, A2 Backfill, 1.6 ft Cover)


                                           Table 54—Summary of Displacements from ABAQUS Analyses with Orthotropic
                                                              Properties (HDPE Pipe, A2 Backfill)
                                                                           Field Test on Oct. 00                                                                                                     Ratio:
                                                                                                                                                               ABAQUS (in.)
                                                                                    (in.)                                                                                                     ABAQUS / Field Test
                                                                           Vertical     Horizontal            Vertical                                                     Horizontal         Vertical   Horizontal
                                           2.8 ft cover (ML95
                                                                               0.065            0.016          0.076                                                         0.020              1.175              1.248
                                              in ABAQUS)
                                           1.6 ft cover (ML95
                                                                               0.137            0.027          0.099                                                         0.025              0.725              0.927
                                              in ABAQUS)



4.5.5                                              Conclusion

It is important to model orthotropic properties of structures to accurately calculate structural
response.                                             The modeling technique that SGH used in PLAXIS 3D underestimated shear
stiffness of the corrugated plate and resulted in concentrated displacements under wheel loads.

To accurately simulate field tests, the actual achieved density of backfill should be estimated,
and backfill properties for the corresponding density should be used in the analysis.

Because structural responses from ABAQUS analyses with the Mohr-Coulomb soil model are in
good agreement with those measured in the field tests for the cases examined in this study, we
conclude that the Mohr-Coulomb model can be an appropriate model to produce structural
response of culvert subjected to live loads with sufficient accuracy.




NCHRP 15-29 Appendix A                                                                                                                                                                                                           143
Since the Mohr-Coulomb model is the soil model of our choice and orthotropic properties of
corrugated plates were found to be important, ABAQUS is the better software to use for the
problem we have than PLAXIS 3D.

5.       DISCUSSION

The preliminary 2D analyses showed that a soil-structure interaction analysis of buried culverts
subjected to live loads with the linear-elastic soil model could produce significantly different
structural response than that with the Mohr-Coulomb soil model and the Hardening-Soil model.
Since a difference between the linear-elastic model and the Mohr-Coulomb model is whether or
not soil failure is modeled by plasticity, plasticity is one of the key aspects of soil models that are
suitable for this project.

The analyses presented here indicate some of the difficulties in predicting structural response of
buried culverts subjected to live loads. The soil parameters currently used in design appear to
yield soil behavior that is softer than achieved in the field tests. As noted above, given the
variability of real world soils and in field compaction effort, this conservatism is justified in
design. Soil parameters could be developed just for the current study that match the soil test
data (Section 2), which in turn produce better estimates of live load response of buried culverts.
However, the same question will arise, that is how should the parameters be modified for design
of actual structures which will experience all of the variability noted.   Given the success of the
Duncan-Selig model and the Selig (1988) properties, it is appropriate to continue with design
parameters that are conservative.

In addition to the soil parameters of a specific soil type, there are other uncertainties in the field
tests, which made matching field data in the analysis difficult. Backfill densities are reported as
the density measured at the time of backfilling, however, there is considerable activity over the
pipes after the backfilling is completed, and this activity likely densifies the soil. For example:

•        In the long span study, the soil surface was compacted with a large vibratory roller prior
         to the live load tests to assure that the surface soil could carry the heavily loaded truck
         without significant rutting. This likely densified the clean gravel backfill.

•        In the MN/DOT study, the backfill was overlaid with 8 in. of gravel, and 4 in. of
         pavement. Thus again the backfill over the top crown of the test pipes was likely
         densified prior to live load testing.

•        In the MN/DOT study, after the construction was complete, there was still considerable
         variability in the data as a result of seasonal differences, temperature variations and
         perhaps other parameters.



NCHRP 15-29 Appendix A                                                                            144
6.       CONCLUSIONS AND RECOMMENDATIONS

This report documents the investigation of soil models for analysis of live load effects on buried
structures. We recommend that the additional studies carried out in this project be conducted
with a linearly-elastic, perfectly-plastic model with a Mohr-Coulomb failure criterion.        This
selection offers the best mix of capturing the important aspects of soil behavior in transmitting
live loads to structures and of offering simplicity in modeling that will allow the team to complete
the most analyses in the least amount of time.          In implementing this soil model, we do
recommend that the elastic soil properties be selected based on depth of fill as in this study.
This technique does not offer all of the benefits of the Duncan-Selig/Hardening Soil models in
capturing stress-dependent stiffness behavior of soil, but for the purposes of a live load study, it
appears to provide sufficient accuracy.

Parameters for the soil model should be those reported above based on the Selig 1988 and
1990 properties. The bulk modulus values in Selig 1990 should be considered suitable for
analysis when justified by data, but may not be a lower bound. The proposed properties will
prove to be somewhat conservative relative to field data, but will represent a lower bound of
behavior.

7.       REFERENCES

Brinkgreve, R.B.J., Ed. (2002), Plaxis 2D Tunnel Version 8, Plaxis, Netherlands.

Brinkgreve, R.B.J. and Broere, W., Eds. (2004), Plaxis 3D Tunnel Version 2, Plaxis,
     Netherlands.

Duncan, J.M., Byrne, P., Wong, K.S., and Mabry, P. (1980), Strength, Stress-Strain and Bulk
     Modulus Parameters for Finite Element Analysis of Stress and Movements in Soil Masses,
     Report No. UCB/GT/80-01, University of California, Berkeley, Berkeley, CA.

Fernando, N. S. M. and Carter, J. P. (1998), “Elastic Analysis of Buried Pipes under Surface
     Patch Loadings,” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 124,
     No. 8, pp. 720-728.

Heger, F. J., Liepins, A. A., and Selig, E. T. (1985), “SPIDA: Analysis and Design System for
     Buried Concrete Pipe,” Advances in Underground Pipeline Engineering, Proceedings of
     the International Conference, American Society of Civil Engineers, pp. 143-154.




NCHRP 15-29 Appendix A                                                                          145
Howard, A. K. (1977), “Modulus of Soil Reaction Values for Buried Flexible Pipe,” Journal of the
     Geotechnical Engineering, Vol. 103, No. GT1, New York, NY.

Jaky, J. (1944), “The Coefficient of Earth Pressure at Rest,” Journal of The Society of Hungarian
     Architects and Engineers, Budapest, pp. 355-358 (in Hungarian).

Lade, P. V. (2005), “Overview of Constitutive Models for Soils,” Soil Constitutive Models—
     Evaluation, Selection, and Calibration, J. A. Yamamuro and V. N. Kaliakin, Eds., ASCE,
     Reston, VA, pp. 1-34.

Lin, R. D. (1987), Direct Determination of Bulk Modulus of Partially Saturated Soils, Master
     Thesis, University of Massachusetts, Amherst, MA.

McGrath, T. J. (1998), “Replacing E′ with the Constrained Modulus in Flexible Pipe Design,”
     Pipelines in the Constructed Environment, Proceedings of the 1998 Pipeline Division
     Conference, American Society of Civil Engineers, pp. 28-40.

McGrath, T.J. and Beaver J.L. (2005), Performance of Thermoplastic Pipe Under Highway
     Vehicle Loading, Project Report Prepared for Minnesota Department of Transportation,
     Simpson Gumpertz & Heger Inc., Waltham, MA.

McGrath, T.J., DelloRusso, S.J., and Boynton, J. (2002), “Performance of Thermoplastic Culvert
     Pipe Under Highway Vehcle Loading,” Pipelines 2002, G. Kurz, Ed., American Society of
     Civil Engineers.

McGrath. T.J., Moore, I.D., Selig, E.T., Webb, M. C., and Taleb, B. (2002), Recommended
     Specifications for Large-Span Culverts, NCHRP Report 473, Transportation Research
     Board, Washington, D.C.

Moore, I. D. and Brachman, R. W. (1994), “Three-Dimensional Analyses of Flexible Circular
     Culverts,” Journal of Geotechnical Engineering, Vol. 120, No. 10, pp. 1829-1844.

Musser, S. C. (1989), CANDE-89 User Manual, FHWA-RD-89-169, Federal Highway
     Administration, McLean, VA.

Pang, S. (1999), “Discussion: Elastic Analysis of Buried Pipes under Surface Patch Loadings by
     N. S. M. Fernando and J. P. Carter,” Journal of Geotechnical and Geoenvironmental
     Engineering, Vol. 125, No. 12, p. 1104.




NCHRP 15-29 Appendix A                                                                       146
Selig, E.T. (1988), “Soil Parameters for Design of Buried Pipelines,” Pipeline Infrastructure, B.
     A. Bennett, Ed., ASCE, New York, NY, pp. 99-116.

Selig, E.T. (1990), “Soil Properties for Plastic Pipe Installations,” Buried Plastic Pipe
     Technology, STP1093, G.S. Buczala and M.J. Cassady, Eds., ASTM, Philadelphia, PA,
     pp. 141-158.

Taleb, B. (2000), Behavior of Large-Span Metal and Reinforced Concrete Culverts under Earth
     and Live Loadings, Ph.D. Dissertation, University of Massachusetts Amherst, Amherst,
     MA.

Webb, M.C. (1999), Improved Design and Construction of Large-Span Culverts, Ph.D.
     Dissertation, University of Massachusetts Amherst, Amherst, MA.




NCHRP 15-29 Appendix A                                                                       147

				
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