Stress and Strains in Soil and Rock

Hsin-yu ShanDept. of Civil Engineering

National Chiao Tung University

Sensitivity

Strength of the soil (in an undisturbed state) divided by the strength in a completely remolded state “at the same water content”For most soil, sensitivity, st, ranges between 1.5 ~ 10

Six Factors Affecting Sensitivity

Metastable soil structureCementationWeatheringThixotropic hardeningLeaching and ion exchangeEffect of addition of dispersive agents

1000S

ensi

tivity

, St(lo

g)

1Liquidity Index, L.I.

Effect of Salt Concentration

Effect on diffuse double layer

She

ar s

treng

thP.L.

L.L.

w %w %

Salt concentration Salt concentration

Thixotropy

An isothermal, reversible, time-dependent increase in strength at a constant water content

Disturbance,Remold

She

ar s

treng

th

Aging

Remolded strength

Time, t

Por

e pr

essu

re, u

shear

Time, t

Cementation

Effect of removal of the cementation bonds in the soil

4,000 psfEDTA(disodium salt of

ethylene diamenetetra acetic acid)

5

12,000 psfSea water3

11,000 psfOriginal pore liquid

4

Max. shear strength

Leaching solutionTest No.

Residual Strength

Peak strengthS

hear

stre

ngth

Residual strength

εa

Residual Strength Occurs:

At large shear strain/displacementUnder drained condition

S tests are appropriate tests for measuring the residual strength

Especially for clay

We should not use peak strength for design involving high-sensitivity clay

For overconsolidated clays, usually pr φφ >

τ

σ0≈rc

pφ

rφ

pc

Measuring Residual Strength

Direct shear (allowed displacement has to be large enough)Ring shearConsolidated-drained triaxial test

Strain-Rate Effect

Mainly for undrained loading

Equilibrium of pore water pressureCreep of soil structure under load

Undrained creep testTime, t

Stra

in

cycle log/% 203log

)( 31 −≈∆

−∆

ftσσ

f)( 31 σσ −

1 10 100 1000 10000Time, logtf (min)

Anisotropy

Lean sensitive clays are more affected by “rotation of principal planes” than highly plastic clays of low sensitivity

σ1f σ1f

Ladd and Foott (1974)

0.19Plane strain “passive” (σ1fhorizontal)

0.20Direct simple shear

0.16Triaxial extension(σ1fhorizontal)

0.33Triaxial compression (σ1fvertical)

0.34Plane strain “active” (σ1fvertical)

τf/σ’vcType of test/Loading condition

σ1f

Factors Influencing Undrained Shear Strength

Initial effective stressEffective stress shear strength parameters

c and φ of N.C. clay show no anisotropyc and φ of O.C. clay has anisotropic effect

Pore water pressure generated during shearFor N.C. clay, the change of pore pressure is not affected by the orientation of principal stressThe pore water pressure of O.C. clay is dependent on the orientation of principal stress

Triaxial Extension Test

Decrease vertical stress (∆σvf) to induce failure

∆σ1f∆σ3f

∆σvf ∆σf

∆σ1f = 0

u0

σvc

σhc

033 uvcfvcfvf −−=−=∆ σσσσσ

)0()( 0303 uAuBu vcffvcff ++−+−−=∆ σσσσ

)()( 3103113 ffvcoffff uK σσσσσσσ −−+=−−=

if B =1

])1())[(1(

))(1(

31

03

vcofff

vcfff

KA

uAu

σσσ

σσ

−+−−=

++−−=∆

Mohr-Coulomb Equation:

φσσ

φσσσσsin]

2[sin

22)(

0313131

ffffff uu ∆−−

+=

+=

−

03131

33131

)(2

22

uK vcoffff

fffff

++−−−

=

+−

=+

σσσσσ

σσσσσ

pc

AKA

f

of

vc

fff

vc

f =−−

−−=

−=

]sin)21(1[sin)]1(1[2/)( 31

φφ

σσσ

στ

For N.C. clay, the parameters in the above equations are somehow independent of consolidation pressure

constant≈pc

Triaxial Compression Test

Increase vertical stress (∆σvf) to induce failure

∆σ1f∆σ3f

∆σvf ∆σf

∆σ3f = 00uvcvc += σσ

0uhchc += σσu0

011 uvcfvf −−=∆=∆ σσσσ

03 =∆ fσ

if B =1

)( 01 uAu vcfff −−=∆ σσ

Mohr-Coulomb Equation:

φσσ

φσσσσsin]

2[sin

22)(

0313131

ffffff uu ∆−−

+=

+=

−

0313311 )()( uK vcoffffff ++−=+−= σσσσσσσ

031

33131

2

22

uK vcoff

fffff

++−

=

+−

=+

σσσ

σσσσσ

pc

AKAKf

ofo

vc

fff

vc

f =−−

−+=

−=

]sin)21(1[sin)]1([2/)( 31

φφ

σσσ

στ

For N.C. clay, the parameters in the above equations are somehow independent of consolidation pressure

constant≈pc

5.0 9.0 32 ==°= of KAφ

35.0≈vc

f

στ

Triaxial compression

20.0≈vc

f

στ

Triaxial extension

This is due to stress-induced anisotropy instead of inherent anisotropySpecimens of triaxial extension tests will experience larger shear deformationThe direction of major principal stress has to rotate 90°

Direct Simple Shear

Under the condition of the applied stresses, it can assumed that:

Pure shear applied to horizontal and vertical planesThe failure plane is not horizontal, α=φ/2The horizontal plane is the plane of maximum shear stress at failure

∆τ

τffτmax,f

φ/2τ

∆σv=∆σh=0

σ

Direct Shearσvc

σ1fσ3f

22.02/)(

19.0 31 =−

=∆

vc

f

vc

hf

σσσ

στ

vc

f

vc

hf

σσσ

στ 2/)(

25.0 31 −==

∆

32.02/)(

19.0 31 =−

=∆

vc

f

vc

hf

σσσ

στ

DSS RoscoeFour platesPure shear is applied to horizontal and vertical plane

DSS NGIRubber membrane and circular ringsHorizontal plane is the plane of maximum shear stressFailure plane is not horizontal(Most reasonable)

Determination of Undrained Shear Strength

Take undisturbed samplesSubject specimens to all-around confining pressureShear the specimens to failure with no drainage

τff τmax,f

φ/2τ

su

φττ cosmax, fff =

σ

Lab. Strength is Probably Lower Than the Field Strength Because:

Specimens tested in the lab are “disturbed”Lab confining pressure is less than that in the fieldSome drainage will occur in the field (higher effective stress)

Lab. Strength is Probably Higher Than the Field Strength Because:

Strain rates in the lab are much higher than the strain rate in the fieldLab’s Q strengths based on triaxialcompression (sDSS < sT.C.)su (= τmax,f) > τff

Effect of SamplingN.C. clay, OCR=1

e or w%

Swelling The sample swells and takes in water

Field consolidation (before sampling)

No swellingStress relief only (in sampling tube)

σlog

0uvcvc += σσ0uvc +=∆ σσ

0uK vcohc += σσ0uK vco +=∆ σσ

u0 + ∆u

In the field During sampling

01 σσ =

03 σσ ==

)]1(1[

)1(

)]()([

)(

0

0

0000

0031

KA

KA

KAuuuu

uu

vc

vcvc

vcvcvc

vcvc

−−=

−−=

+−++−−−=∆−=

∆−+−===

σ

σσ

σσσ

σσσσσ

In the lab. Before setup.

vco AK σσσσ 5.0 ,0.1 ,5.0For 031 =====

vco AK σσσσ 32 ,3

2 ,5.0For 031 ===== (Elastic)

After N.C. clay goes through the sampling process, it may behaves like O.C. clay

Virgin consolidation curve(actually, we don’t have it)

e or w%

fieldv,σ cσ σlog

To Obtain “Field” Undrained Shear Strength of N.C. clay

ComputeMeasure shear strength in the labCompute field strength

vc

f

στfieldv,σ

fieldvlabvc

fus ,)( σ

στ

×=

O.C. clays

Virgin consolidation curve(actually, we don’t have it)e or w%

labmax,σfieldv,σ fieldmax,σ labv,σσlog

To Obtain “Field” Undrained Shear Strength of O.C. clay

ComputeMeasure shear strength in the lab for the field OCR

Compute field strengthvc

f

στ

fieldv,σ

fieldvlabvc

fus ,)( σ

στ

×=

fieldmax,σ

constant≈vc

f

στ

For a given OCR

constant≈vc

f

στ

For a given OCR

vc

f

στ

OCR

Failure Criteria

Mohr-CoulombTresca (Extended Tresca)von Mises

Mohr-Coulomb

For σ1 > σ2 > σ3φσσσσ sin)()( 3131 +=−

Independent of σ2σa

T.E.

0]sin)()[(

]sin)()[(

]sin)()[(

213

213

232

232

221

221

=+−−

×+−−

×+−−

φσσσσ

φσσσσ

φσσσσ

T.C.

T.E.

σb σcT.E.T.C. T.C.

For c=0σa

If c is not 0, but φ is, the shape would be a hexagonal column

σb

σc

Tresca

octασσσσασσ =++

=−3

)( 32131

Often used for c=0 and φ=0αis the Trescaparameterequivalent to φExtended Tresca for φ > 0

σaT.C.

T.E.

octac

octcb

octba

σασσ

σασσ

σασσ

⋅≤−

⋅≤−

⋅≤−

T.E.

σb σcT.E.T.C. T.C.

Von Mises (φ = 0) 22222 )

3(2)()()( cba

accbbaσσσασσσσσσ ++

=−+−+−

σaT.C.c = 0 coneφ = 0 cylinerT.E.T.E.

σb σcT.E.T.C. T.C.

σaT.C.

T.E.T.E.

σb σcT.E.T.C. T.C.