# Lecture 9

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

CEE 8442

Lecture 9
Selection and Effects of k

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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

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Dept. of Civil & Environmental Engineering

(CSR):
•   Standard application
•   How it typical values are determined
•   Modeling issues
•   Current research
•   Example

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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

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Dept. of Civil & Environmental Engineering

Standard Application
Estimate the settlement of a footing under a
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

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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
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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)

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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

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Dept. of Civil & Environmental Engineering

Modeling Issues
•   Foundation problems complex
•   Simplifications introduce approximations & error
•   Typical model = 2-D mathematical expression, k = psi
•   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

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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
• http://www.engineering.manhattan.edu/civil/CGT.html

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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
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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
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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

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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
• Presently, AREA, American Railway
Engineering Association, recommends the
use of the Cooper E-80 train for live load

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Dept. of Civil & Environmental Engineering

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

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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   

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Dept. of Civil & Environmental Engineering

Analysis continued

• Deflection and Moment

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

M ( x)   EIw( x)

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Dept. of Civil & Environmental Engineering

Typical Values

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

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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)

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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)

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Dept. of Civil & Environmental Engineering

Conclusions

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

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Dept. of Civil & Environmental Engineering

Characteristics and Analysis

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Dept. of Civil & Environmental Engineering

Track Types

• Ballasted Tracks
– Most Common Type
• Direct Fixation Tracks
• Embedded Tracks
– Turf Tracks

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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

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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
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

This is the standard method of construction for
tracks on aerial structures and tunnels.                                   26
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
– ballast, ties, and the associated maintenance are eliminated
– less maintenance, means less traffic disruptions
settlement is reduced
– when combined with welded rail, ride quality and operational
speeds improve
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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

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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
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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)
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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
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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
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
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Moment
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
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
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Moment
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

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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

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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
stiffer track.
-replace more
compressible subballast
with a stiffer base
-Reduced spacing
Elevation View                   between ties            38
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
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.
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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
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Dept. of Civil & Environmental Engineering

Conclusion

• Embedded Track
– Most commonly used for light rail in urban
– 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
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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

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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
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Dept. of Civil & Environmental Engineering

3 Modes of Failure

Opening

Sliding

Tearing

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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.

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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
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Dept. of Civil & Environmental Engineering

Analytical Model
P

a                  c

x

L

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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)
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Dept. of Civil & Environmental Engineering

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

• Keep working on your projects!

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