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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• 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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• Standard application• How it typical values are

determined• Modeling issues• Current research• Example

The Coefficient of Subgrade Reaction (CSR):

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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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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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• 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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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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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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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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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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• 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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• 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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• 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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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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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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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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Analysis continued

• Deflection and Moment

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1

)(n

npn xKPxw

)()( xwEIxM

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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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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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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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Conclusions

• Superposition makes this problem feasible

• Analysis required only Excel • Increasing k decreases

– deflection– moment

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Railroad Track Configurations

Characteristics and Analysis

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Track Types

• Ballasted Tracks– Most Common Type

• Direct Fixation Tracks– Long Island Railroad

• Embedded Tracks– Turf Tracks

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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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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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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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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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Embedded TrackTrack Modulus:

-difficult to determine

-rail deflections are extremely small

-field measurements estimate k = 2,000,000 psi

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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3 Modes of Failure

Opening

Sliding

Tearing

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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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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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Analytical Model

P

a c

L

x

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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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Next Week

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

• Keep working on your projects!

villanova university dept. of civil & environmental engineering advanced structural mechanics 1...

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