Lecture 9

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					Villanova University                         Advanced Structural Mechanics
Dept. of Civil & Environmental Engineering




          CEE 8442
 Advanced Structural Mechanics

                   Lecture 9
           Selection and Effects of k



                                                                         1
  Villanova University                         Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



               Topics and Applications

• The Coefficient of Subgrade Reaction (CSR),
  k, foundation modulus
• Determination
• Application of Superposition, Cooper E-80
  Train
• Subgrade transition (like your homework)



                                                                           2
    Villanova University                         Advanced Structural Mechanics
    Dept. of Civil & Environmental Engineering

    The Coefficient of Subgrade Reaction
                   (CSR):
•   Standard application
•   How it typical values are determined
•   Modeling issues
•   Current research
•   Example



                                                                             3
  Villanova University                             Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                  Winkler Foundation
                                         Simplest analytical model
                                         of continuous elastic
                                         foundation
                                         When deflection, d is
                                         imposed on foundation, it
                                         resists with a pressure, q
                                         q (x) = ks d (x)

CSR parameter ( ks ) has units of force / length3

                                                                               4
  Villanova University                         Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                   Standard Application
Estimate the settlement of a footing under a
  concentrated load
Given the concentrated load, allowable bearing
  pressure and coefficient of subgrade reaction,
  settlement can be estimated as;

                   D = (4 * qall * B2) / (ks (B+1)2)

Quick and efficient way to estimate a fairly
  straightforward and common situation

                                                                           5
  Villanova University                         Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                How It Is Determined
• Plate-bearing test
• Testing method specified in ASTM D 1196-93
• Result of test is a plot of settlement vs.
  pressure
• Material CSR = slope of the elastic portion
• Field plate-bearing tests - time consuming
  and costly
• Representative values correlated with other
  soil properties
                                                                           6
                     Villanova University                                Advanced Structural Mechanics
                     Dept. of Civil & Environmental Engineering



                                             Plate-Bearing Test Data

                   0.1

                  0.08
                            Slope = ks
Settlement (ft)




                  0.06

                  0.04

                  0.02

                    0
                     0.00            2.00      4.00          6.00        8.00       10.00       12.00

                                                        Pressure (ksf)



                                                                                                        7
         Villanova University                                             Advanced Structural Mechanics
         Dept. of Civil & Environmental Engineering



                                  Typical Values

     Sand (dry to moist)                    Sand (saturated)                                   Clay
 Loose   29   -   92   lb / in3     Loose    38    -   55      lb / in3           Stiff   44    -     92   lb / in3
                             3                                       3                                             3
Medium   91   - 460    lb / in     Medium 128 - 147            lb / in       Very Stiff   92    - 184      lb / in
 Dense 460 - 1380 lb / in3          Dense 478 - 552            lb / in3           Hard          > 184      lb / in3




   Das, Principles of Foundation Engineering, 3rd Edition



                                                                                                               8
    Villanova University                         Advanced Structural Mechanics
    Dept. of Civil & Environmental Engineering



                     Modeling Issues
•   Foundation problems complex
•   Simplifications introduce approximations & error
•   Typical model = 2-D mathematical expression, k = psi
•   Real soils capable of load spreading
•   Stress at point results in settlement at many points
•   Real world models need to account for stress-
    deformation characteristics of soil, shape and size of
    loaded area and magnitude and position of nearby
    loaded areas


                                                                             9
  Villanova University                         Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                      Current Research

• Center for Geotechnology (CGT) at
  Manhattan College
• Soil Structure Interaction (SSI) Research
  Project
• Project has published a number of papers on
  modeling and obtaining more accurate
  subgrade models
• http://www.engineering.manhattan.edu/civil/CGT.html

                                                                          10
  Villanova University                         Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                      In-Class Example

• Long strip footing with column load at
  end
• Solution
  – w(x) = ( 2 P b / k ) e-bx cos bx
  – M(x) = ( -P / b ) e-bx sin bx
• Concentrate on M(x) equation
• Vary ks values and check results
                                                                          11
  Villanova University                         Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                             Example
• Dense, dry sand w/ ks = 460 lb / in3
• Dense, dry sand w/ ks = 1380 lb / in3
• Moments for low and high values of ks
  – Mlow = 160,500 lb-in
  – Mhigh = 145,100 lb-in
• Conclusion: Approximately a 10%
  difference for moment
                                                                          12
   Villanova University                         Advanced Structural Mechanics
   Dept. of Civil & Environmental Engineering



                             Conclusion
• Winkler Foundation Model is approximately
  130 years old
• Exceptions have been taken with the model
  for approximately 60 years
• Means of improvement are not new!
• Simplicity is the appeal
• Previous results (i.e. structure performance)
  acceptable, or it would have been discarded
  long ago

                                                                           13
   Villanova University                         Advanced Structural Mechanics
   Dept. of Civil & Environmental Engineering




       Analysis of a Cooper E-80 Train
       on a Single Rail for Different k’s

• Theodore Cooper was one of the first
  engineers to establish live loads for railroads
• Presently, AREA, American Railway
  Engineering Association, recommends the
  use of the Cooper E-80 train for live load


                                                                           14
  Villanova University                         Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                          Investigation
• Solve for the displacement and
  moment on an elastic foundation,
  constant k, due to the Cooper E-80
  Train loading
• Determine the effects on displacement
  and moment caused by different
  values of k

                                                                          15
  Villanova University                         Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                                Analysis
• Infinitely long elastic foundations
• Center the train about the origin
• Use Superposition and a Green’s function, Kp for a
  point load on an infinite beam



Kp 
           b
          2k
                e
                     b x 
                               cosb x    sin b x   

                                                                          16
  Villanova University                             Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                     Analysis continued

• Deflection and Moment

                                                     
                            18
            w( x)   Pn K p x   n
                           n 1



            M ( x)   EIw( x)


                                                                              17
  Villanova University                         Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                         Typical Values

• Concrete Sub-grade; k=4000 lbs/in2
• Crushed Stone Sub-grade; k= 1800 lbs/in2
• Soil Sub-grade; k = 300 lbs/in2




                                                                          18
                   Villanova University                          Advanced Structural Mechanics
                   Dept. of Civil & Environmental Engineering



                                      Results: Deflection

                                DEFLECTIONS

                                  2.500

                                  2.000
                                                                  Concrete Sub-base
deflectiont (in)




                                  1.500

                                  1.000                           Crushed Stone Sub-
                                                                  base
                                  0.500                           Soil Sub-base
                                  0.000
                   -100       -50       0         50       100
                                 -0.500
                                    length (ft)




                                                                                            19
      Villanova University                               Advanced Structural Mechanics
      Dept. of Civil & Environmental Engineering



                             Results: Moment

                                         MOMENT

                             600

                             400
                                                                Concrete Sub-base
moment (k-in)




                             200
                                                                Crushed Stone Sub-
                                0
                                                                base
                -100   -50          0        50    100
                             -200                               Soil Sub-base

                             -400

                             -600
                             distance (ft)



                                                                                     20
  Villanova University                         Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                            Conclusions

• Superposition makes this problem
  feasible
• Analysis required only Excel
• Increasing k decreases
  – deflection
  – moment


                                                                          21
Villanova University                         Advanced Structural Mechanics
Dept. of Civil & Environmental Engineering




        Railroad Track Configurations


            Characteristics and Analysis




                                                                        22
  Villanova University                         Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                            Track Types

• Ballasted Tracks
  – Most Common Type
• Direct Fixation Tracks
  – Long Island Railroad
• Embedded Tracks
  – Turf Tracks


                                                                          23
Villanova University                          Advanced Structural Mechanics
Dept. of Civil & Environmental Engineering



                      Ballasted Track
                                             Track Modulus is
                                             estimated considering:
                                             -crosstie size
                                             -depth of ballast and
                                             sub ballast
                                             -type of ballast rock or
                                             stone
                                             -crosstie spacing


                                                                         24
   Villanova University                             Advanced Structural Mechanics
   Dept. of Civil & Environmental Engineering



                         Ballasted Track




Ties:                                    Ballast:
-Constructed from wood                   -Constructed from
or concrete                              limestone, heavy stone,
                                         or granite
-(k wood < k concrete)
                                         -(k limestone<k h.stone<k
                                         granite)                              25
Villanova University                            Advanced Structural Mechanics
Dept. of Civil & Environmental Engineering



                 Direct Fixation Track

                                             Track Modulus
                                             estimated considering
                                             the vertical deflection,
                                             which can occur in:
                                             -rail bending
                                             -flexure of slab at
                                             subbase materials for
                                             at-grade installations

This is the standard method of construction for
tracks on aerial structures and tunnels.                                   26
   Villanova University                         Advanced Structural Mechanics
   Dept. of Civil & Environmental Engineering



Long Island Railroad Concrete Slab Track
• During the 1980’s the LIRR undertook the nine year
  task of replacing all of its ballasted track with direct
  fixation track.
• The slab track system consists of a concrete slab
  supported on a subgrade of sand and a subbase of
  asphalt.
• The change from a ballasted track to direct fixation
  track presented several advantages:
   – ballast, ties, and the associated maintenance are eliminated
   – less maintenance, means less traffic disruptions
   – load is distributed more uniformly on the subgrade, thus
     settlement is reduced
   – when combined with welded rail, ride quality and operational
     speeds improve
                                                                           27
Villanova University                         Advanced Structural Mechanics
Dept. of Civil & Environmental Engineering



                     Embedded Track
                                             Track Modulus:
                                             -difficult to determine
                                             -rail deflections are
                                             extremely small
                                             -field measurements
                                             estimate k = 2,000,000
                                             psi




                                                                        28
     Villanova University                         Advanced Structural Mechanics
     Dept. of Civil & Environmental Engineering



                          Embedded Track
    Turf Track: Another Type of Embedded Track




•Spawned from European light rail systems desire to blend
the transit track and system into the landscape.
•Developed for selected purposes:
       -reduce the visual effect of ballasted track
       -reduce the noise from trams as much as possible
       -provide year-round greenery in the vicinity of the track
                                                                             29
  Villanova University                                  Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering

            Transition from Low Modulus to
                 High Modulus Track



Winkler Base Analysis of Track Transition
        2 D.E.’s: 1.) EI*wliv(x)-kl*wl(x), (-inf. < x < 0)
                      2.) EI* wriv(x)-kr*wr(x), (0 < x < inf.)
        2 B.C’s: 1.&2.) LIM(x->-inf.) {wl,wli} -> finite
                      3.&4.) LIM(x->inf.) {wr, wri} -> finite

        4 M.C’s: 5.) wl(0)=wr(0)               6.)wli(0)=wri(0)
                      7.) wlii(0)=wrii(0)      8.)wliii(0)+wriii(0)= (P/EI)
                                                                                   30
        Villanova University                                     Advanced Structural Mechanics
        Dept. of Civil & Environmental Engineering

                      Transition from Low Modulus to
                           High Modulus Track
                                            w (x) vs. x
                                            w (x) vs. x
                                             0.01
                                             0.01

                                                0
                                                0
        -250
        -250   -200
               -200     -150
                        -150   -100
                               -100   -50
                                      -50           0
                                                    0     50
                                                          50   100
                                                               100    150
                                                                      150       200
                                                                                200        250
                                                                                           250
                                            -0.01
                                            -0.01

                                            -0.02
                                            -0.02

                                            -0.03
                                            -0.03

                                            -0.04
                                            -0.04
w (x)
w (x)




                                            -0.05
                                            -0.05

                                            -0.06
                                            -0.06

                                            -0.07
                                            -0.07
                                                                            Low Modulus
                                                                            Low Modulus
                                            -0.08
                                            -0.08
                                                                            High Modulus
                                                                            High Modulus
                                            -0.09
                                            -0.09

                                             -0.1
                                             -0.1
                                                    x
                                                    x


- damage can be done to both track and vehicle in
areas of abrupt track modulus change
                                                                                                 31
               Villanova University                                  Advanced Structural Mechanics
               Dept. of Civil & Environmental Engineering


               Transition from Ballasted to Embedded Track
                                                  w (x) vs. x
                                                  0.005


                                                      0
        -250    -200     -150     -100      -50            0    50      100     150       200        250

                                                  -0.005


                                                   -0.01
w (x)




                                                  -0.015


                                                   -0.02


                                                  -0.025                              Low Modulus

                                                   -0.03
                                                                                      High Modulus


                                                  -0.035
                                                           x

                                            Displacement                                             32
              Villanova University                                   Advanced Structural Mechanics
              Dept. of Civil & Environmental Engineering


              Transition from Ballasted to Embedded Track
                                                   M(x) vs. x
                                                 300000
                                                                                      Low
                                                                                      Modulus
                                                                                      High
                                                 200000                               Modulus


                                                 100000



                                                      0
       -250    -200     -150     -100      -50             0    50      100     150      200    250
M(x)




                                                 -100000



                                                 -200000



                                                 -300000



                                                 -400000
                                                           x
                                                                                                33
                                                 Moment
               Villanova University                                  Advanced Structural Mechanics
               Dept. of Civil & Environmental Engineering

                                Transition from Ballasted to
                                   Direct Fixation Track
                                                  w (x) vs. x
                                                    0.2



                                                      0
        -250    -200     -150      -100     -50           0     50      100     150       200        250

                                                   -0.2



                                                   -0.4
w (x)




                                                   -0.6



                                                   -0.8
                                                                                      Low Modulus

                                                     -1                               High Modulus


                                                   -1.2
                                                   x
                                             Displacement                                            34
              Villanova University                                    Advanced Structural Mechanics
              Dept. of Civil & Environmental Engineering

                               Transition from Ballasted to
                                  Direct Fixation Track
                                                    M(x) vs. x
                                                 5000000
                                                                                       Low
                                                                                       Modulus
                                                                                       High
                                                 4000000                               Modulus


                                                 3000000



                                                 2000000
M(x)




                                                 1000000



                                                       0
       -250   -200      -150      -100     -50              0    50      100     150      200     250

                                                 -1000000



                                                 -2000000
                                                            x
                                                                                                 35
                                                  Moment
                Villanova University                           Advanced Structural Mechanics
                Dept. of Civil & Environmental Engineering



   Cooper Results for Deflection Varying k

                                      DEFLECTION Varying K

                                                    3

                                                    2
deflection in




                                                    2

                                                    1

                                                    1

                                                    0
                 -100              -50                   0       50              100
                                                    -1
                                                 distance ft



                                                                                          36
     Villanova University                               Advanced Structural Mechanics
     Dept. of Civil & Environmental Engineering



       Cooper Results for Moment Varying k

                             MOMENT Varying K

                                      600

                                      400
moment k-in




                                      200

                                        0
              -150   -100     -50           0      50       100       150
                                     -200

                                     -400

                                     -600
                                     distance ft




                                                                                   37
Villanova University                           Advanced Structural Mechanics
Dept. of Civil & Environmental Engineering



                     Transition Zones
                                             Approach Slabs:
                                             -extend from embedded
                                             track slab a min. of 20ft.
                                             into ballasted section
                                             -slab typically located @
                                             1ft. below the ties
             Plan View                       immediately adjacent to
                                             stiffer track.
                                             -replace more
                                             compressible subballast
                                             with a stiffer base
                                             -Reduced spacing
            Elevation View                   between ties            38
   Villanova University                               Advanced Structural Mechanics
   Dept. of Civil & Environmental Engineering



                        Transition Zones

• Direct Fixation to                       • Embedded to
  Ballasted Track                            Ballasted Track
   – Fastener design continues                  – Continues to evolve and
                                                  improve
     to improve.
                                                – Rail deflections are hard to
   – New fastener spring rates                    match to ballasted track
     allow modulus to decrease                  – Differential in track modulus
   – Lower track modulus allows                   may be too large to
     easier transition                            overcome simply through
                                                  flexible rail


These transitions require more maintenance than most
track sections. The bending forces in each transition will
not eliminate all damage.                                                         39
  Villanova University                         Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                             Conclusion

• Ballasted Track
  – Most commonly used Track type
  – Composed of Crosstie, Ballast, and
    Fastenings
  – k value based mainly on Crosstie spacing
    and Ballast composition
  – k value for can range from 1500 – 5000 psi
    for wood X-ties and 5000 – 8000 psi for
    concrete.
                                                                          40
  Villanova University                         Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                             Conclusion

• Direct Fixation Track
  – Most commonly used on bridges and in
    tunnels
  – Ballastless track in which rail is directly
    fastened to a concrete slab
  – k value is easily determined from the
    amount of vertical deflection within the
    fasteners.
  – k value commonly around 15,000 psi
                                                                          41
 Villanova University                         Advanced Structural Mechanics
 Dept. of Civil & Environmental Engineering



                            Conclusion

• Embedded Track
  – Most commonly used for light rail in urban
    business centers
  – Track is embedded within a concrete slab
    and only portion showing is the rail head
  – k value is very large, though very hard to
    establish, because deflections are so small
  – k value is assumed to be 2,000,000 psi
                                                                         42
  Villanova University                         Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                             Conclusion

• Transition Zone:
  – Immediate transition causes damage
  – This damage is greater in the transition
    from ballast to embedded than it is in
    ballast to direct fixation due to the large
    variation in k
  – Approach slabs are used to ease transition
    from low to high modulus track

                                                                          43
Villanova University                         Advanced Structural Mechanics
Dept. of Civil & Environmental Engineering




      The Topic Transition Zone:
    How can Winkler Foundations Be
    Related to Fracture Mechanics?

  Fracture Specimen for Composite
              Material
  Application of a Beam on an Elastic
Foundation to a Double Cantilever Beam
                                                                        44
Villanova University                            Advanced Structural Mechanics
Dept. of Civil & Environmental Engineering



                   3 Modes of Failure

                                             Opening




                                             Sliding




                                             Tearing

                                                                           45
  Villanova University                         Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering



                   Critical Configuration

• Mode I fracture has been found to have
  the lowest critical strain energy release
  rate
• If load, P, is applied to a specimen in
  Mode I, Mode II, and Mode III … the
  Mode I case will govern.


                                                                          46
Villanova University                           Advanced Structural Mechanics
Dept. of Civil & Environmental Engineering



Specimen Used in Composite Materials
             P
                                     Comparable to a fracture
                                     toughness specimen




                                     This type of specimen is
                                     used to determine the
                                     interlaminar fracture
            P
                                     toughness
                 a
                                                                          47
Villanova University                                 Advanced Structural Mechanics
Dept. of Civil & Environmental Engineering



                      Analytical Model
                  P




                              a                  c

                                             x

                                        L

                                                                                48
  Villanova University                             Advanced Structural Mechanics
  Dept. of Civil & Environmental Engineering


                           Formulation
• 2 DE’s: EI wcIV + k wc = 0                    0<x<c
          EI waIV = 0                          -a < x < c

• 4 BC’s:         1.   waII(-a) = 0
                  2.   waIII(-a) = P/EI
                  3.   wcII(c) = 0
                  4.   wcIII(c) = 0
• 4 MC’s:         1.   wc(0) = wa(0)
                  2.   wIc(0) = wIa(0)
                  3.   wIIc(0) = wIIa(0)
                  4.   wIIIc(0) = wIIIa(0)
                                                                              49
    Villanova University                         Advanced Structural Mechanics
    Dept. of Civil & Environmental Engineering


                              Next Week
•   Introduction to Fracture
•   Factors Effecting Fracture
•   Material Toughness
•   Linear Elastic Fracture Mechanics

• Keep working on your projects!



                                                                            50

				
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