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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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