villanova university dept. of civil & environmental engineering advanced structural mechanics 1...
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
CEE 8442Advanced Structural Mechanics
Lecture 9Selection and Effects of k
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
• The Coefficient of Subgrade Reaction (CSR), k, foundation modulus
• Determination• Application of Superposition, Cooper E-
80 Train• Subgrade transition (like your
homework)
Topics and Applications
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
• Standard application• How it typical values are
determined• Modeling issues• Current research• Example
The Coefficient of Subgrade Reaction (CSR):
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
CSR parameter ( ks ) has units of force / length3
Simplest analytical model of continuous elastic foundation
When deflection, is imposed on foundation, it resists with a pressure, q
q (x) = ks (x)
Winkler Foundation
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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;
= (4 * qall * B2) / (ks (B+1)2)
Quick and efficient way to estimate a fairly straightforward and common situation
Standard Application
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
• 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
How It Is Determined
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Plate-Bearing Test Data
0
0.02
0.04
0.06
0.08
0.1
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Pressure (ksf)
Set
tlem
ent (
ft) Slope = ks
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Das, Principles of Foundation Engineering, 3rd Edition
Loose 29 - 92 lb / in3 Loose 38 - 55 lb / in3 Stiff 44 - 92 lb / in3
Medium 91 - 460 lb / in3 Medium 128 - 147 lb / in3 Very Stiff 92 - 184 lb / in3
Dense 460 - 1380 lb / in3 Dense 478 - 552 lb / in3 Hard > 184 lb / in3
Sand (dry to moist) Sand (saturated) Clay
Typical Values
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
• Long strip footing with column load at end
• Solution– w(x) = ( 2 P / k ) e-x cos x– M(x) = ( -P / ) e-x sin x
• Concentrate on M(x) equation
• Vary ks values and check results
In-Class Example
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
• 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
Example
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
• 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
Conclusion
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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
xxe
kK
x
p sincos2
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Analysis continued
• Deflection and Moment
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1
)(n
npn xKPxw
)()( xwEIxM
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Typical Values
• Concrete Sub-grade; k=4000 lbs/in2
• Crushed Stone Sub-grade; k= 1800 lbs/in2
• Soil Sub-grade; k = 300 lbs/in2
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Results: Deflection
DEFLECTIONS
-0.500
0.000
0.500
1.000
1.500
2.000
2.500
-100 -50 0 50 100
length (ft)
defl
ecti
on
t (i
n)
Concrete Sub-base
Crushed Stone Sub-base
Soil Sub-base
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Results: Moment
MOMENT
-600
-400
-200
0
200
400
600
-100 -50 0 50 100
distance (ft)
mo
men
t (k
-in
)
Concrete Sub-base
Crushed Stone Sub-base
Soil Sub-base
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Conclusions
• Superposition makes this problem feasible
• Analysis required only Excel • Increasing k decreases
– deflection– moment
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Railroad Track Configurations
Characteristics and Analysis
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Track Types
• Ballasted Tracks– Most Common Type
• Direct Fixation Tracks– Long Island Railroad
• Embedded Tracks– Turf Tracks
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Ballasted Track
Ties:
-Constructed from wood or concrete
-(k wood < k concrete)
Ballast:
-Constructed from limestone, heavy stone, or granite
-(k limestone<k h.stone<k granite)
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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.
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Embedded TrackTrack Modulus:
-difficult to determine
-rail deflections are extremely small
-field measurements estimate k = 2,000,000 psi
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Embedded TrackTurf 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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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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)=wr
i(0)
7.) wlii(0)=wr
ii(0) 8.)wliii(0)+wr
iii(0)= (P/EI)
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Transition from Low Modulus to High Modulus Track
w(x) vs. x
-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
-250 -200 -150 -100 -50 0 50 100 150 200 250
x
w(x
)
Low Modulus
High Modulus
w(x) vs. x
-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
-250 -200 -150 -100 -50 0 50 100 150 200 250
x
w(x
)
Low Modulus
High Modulus
- damage can be done to both track and vehicle in areas of abrupt track modulus change
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Transition from Ballasted to Embedded Track
Displacement
w(x) vs. x
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
-250 -200 -150 -100 -50 0 50 100 150 200 250
x
w(x
)
Low Modulus
High Modulus
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Transition from Ballasted to Embedded Track
Moment
M(x) vs. x
-400000
-300000
-200000
-100000
0
100000
200000
300000
-250 -200 -150 -100 -50 0 50 100 150 200 250
x
M(x
)
LowModulusHighModulus
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Transition from Ballasted to Direct Fixation Track
w(x) vs. x
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
-250 -200 -150 -100 -50 0 50 100 150 200 250
x
w(x
)
Low Modulus
High Modulus
Displacement
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Transition from Ballasted to Direct Fixation Track
M(x) vs. x
-2000000
-1000000
0
1000000
2000000
3000000
4000000
5000000
-250 -200 -150 -100 -50 0 50 100 150 200 250
x
M(x
)
LowModulusHighModulus
Moment
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Cooper Results for Deflection Varying k
DEFLECTION Varying K
-1
0
1
1
2
2
3
-100 -50 0 50 100
distance ft
defl
ecti
on
in
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Cooper Results for Moment Varying k
MOMENT Varying K
-600
-400
-200
0
200
400
600
-150 -100 -50 0 50 100 150
distance ft
mo
men
t k-i
n
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Transition Zones
Plan View
Elevation View
Approach Slabs:-extend from embedded track slab a min. of 20ft. into ballasted section-slab typically located @ 1ft. below the ties immediately adjacent to stiffer track. -replace more compressible subballast with a stiffer base-Reduced spacing between ties
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Transition Zones
• Direct Fixation to Ballasted Track– Fastener design
continues to improve. – New fastener spring
rates allow modulus to decrease
– Lower track modulus allows easier transition
• Embedded to Ballasted Track– Continues to evolve and
improve– Rail deflections are hard
to match to ballasted track
– Differential in track modulus may be too large to 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.
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
3 Modes of Failure
Opening
Sliding
Tearing
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Specimen Used in Composite Materials
P
P
a
Comparable to a fracture toughness specimen
This type of specimen is used to determine the interlaminar fracture toughness
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Analytical Model
P
a c
L
x
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
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. wI
c(0) = wIa(0)
3. wIIc(0) = wII
a(0)4. wIII
c(0) = wIIIa(0)
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Villanova UniversityDept. of Civil & Environmental Engineering
Advanced Structural Mechanics
Next Week
• Introduction to Fracture• Factors Effecting Fracture• Material Toughness• Linear Elastic Fracture Mechanics
• Keep working on your projects!