Shear Strength of Soil

Hsin-yu ShanDept. of Civil Engineering

National Chiao Tung University

Principal stress spaceaσ

bσ

cσ

Rendulic Plot

aσ

2 rσ

rσ

rσ

Rendulic PlotAxial Stress

'aσRadial Stress

rσ 2

aσ

We can plot stress paths of compression and extension tests without having them cross over one another

2 rσ

Redulic Diagram1. S-test: compression by increase of axial stress2. -test: compression by increase of axial stressR

2

1<fA

1>fA1

1

2 3

41=fA

0=fA 0<fA

ra σσ =Space diagonal

Stress path for isotropic consolidation

du∆

Compression test envelope (S.P.T.)aσ

2 rσ3. S-test: compression by decrease confining stress4. Elastic material: on octahedral plane

φφ

φφσσ

sin1cos2]

sin1sin1[31 −

+−+

= c

φφ

φφσσ

sin1cos2]

sin1sin1

21[231 −

+−+

⋅= c

φφ

φφσσ

sin1cos2]

sin1sin1

21[2

−+

−+

⋅= cra

ψtan d

Redulic Diagram2. S-test: extension by increase of radial stress3. -test: extension by decrease of axial stressR

2

1<fA1>fA

1

Extension test envelope (S.P.T.)

1

23 4

1=fA

0=fA

0<fA

ra σσ =

Space diagonal

Stress path for isotropic consolidation

aσ

2 rσ1. S-test: extension by decrease axial stress4. Elastic material: on octahedral plane

φφ

φφσσ

sin1cos2]

sin1sin1[

−+

−+

= car

φφ

φφσσ

sin1cos2]

sin1sin1[

−−

−+

= cra

φφ

φφσσ

sin1cos2]

sin1sin1

21[2

−−

−+

⋅= cra

ψtan d

e.g. °== 30 0 φc

For compression

°=−+

⋅= 65 sin1sin1

21tan ψ

φφψ

For extension

°=+−

⋅= 13 sin1sin1

21tan ψ

φφψ

For space diagonal

°=∠ − 35)2

1(tan 1

e.g. °== 30 0 φc

aσ

2 rσ

ra σσ =

Space diagonal

35°

13°

65°

Henkel (1967)

22.5°Undrained23°Undrained

22°Drained, σmean = const.

22°Drained, σmean = const.

21.5°Drained, σr ↑22°Drained, σr↓

22.5°Drained, σa ↓22°Drained, σa↑

TestsTests

Extension TestsCompression Tests

φ φ

Monitor the volume of water flowed into or out of the specimen during CD tests or trace effective stress path of CU tests

N.C. Clay

aσ

2 rσ

ra σσ =

Contour of constant water content (stress path of CU test)

w = 17%

w = 20%

Henkel (1960)

For c=0σa

σa = σb = σc

Octahedral plane

octcbam σσσσσ =++= )(31

σb

σc

Stress is constant for any point on the octahedral plane

.constoct =σ222 )()()(

31

accbbar σσσσσσ −+−+−=

octr τ3

1=

aσ

r

bσcσ

222 )()()(31

accbbaoct σσσσσστ −+−+−=

τoct = octahedral shear stress= maximum shear stress on

the octahedral plane

Failure Criteria

Mohr-CoulombTresca (Extended Tresca)von Mises

Mohr-Coulombφσσσσ sin)()( 3131 +=−For σ1 > σ2 > σ3

0]sin)()[(

]sin)()[(

]sin)()[(

213

213

232

232

221

221

=+−−

×+−−

×+−−

φσσσσ

φσσσσ

φσσσσ

T.E.

.)(31 constcbaoct =++= σσσσ

Independent of σ2σaT.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 columnσb

σc

Trescaoctασσσσασσ =

++=−

3)( 321

31

Often used for c=0 and φ=0α is the Tresca parameter equivalent to φExtended Tresca for φ > 0σaT.C.

T.E.T.E.

octac

octcb

octba

σασσ

σασσ

σασσ

⋅≤−

⋅≤−

⋅≤−

σb σcT.E.T.C. T.C.

Von Mises (φ = 0) 22222 )

3(2)()()( cba

accbbaσσσασσσσσσ ++

=−+−+−

σaT.C. Extended Tresca for φ > 0T.E.T.E.

c = 0 coneφ = 0 cyliner

.29 22 constoctoct == σατ

Circleσb σc

T.E.T.C. T.C.

222 )32()( ba

baσσασσ +

=−If σb = σc

σaT.C.

T.E.T.E.

σb σcT.E.T.C. T.C.

31

32

σσσσ

−−

=bBishop

φσσσσ sin

31

31 =+−

Mohr-Coulomb

)322

31(

1

31

31

b−+

=+−

ασσσσ

Extended Tresca

)3212

31(

1231

31

bbb −+−+=

+−

ασσσσExtended von Mises

)(sin31

311

σσσσφ

+−

= −f Extended von Mises

Extended Tresca

M-C T.E.M-C T.C.

b

Triaxial compression tests:Mohr-Coulomb criteria underestimate strength in plane strainVon Mises criteria overestimate strength in plane strain

Krey (1927)Consolidated-Undrained Direct Shear Tests (Actually, it’s difficult to control drainage in DS tests)

σmax

φ’

φ’c

Stress path

σ’m

Consolidate the specimen and then quickly reduce σ and then shear it

φστ ′′+= tanmf c

τ

c

Tiedemann (1933)

cm φσ tan

rcmf φσφστ tantan +=

σm

φrφ’c

σ

rφσ tan

Tiedemann failure envelope (only for σ < σm)

The maximum past pressure the soil was consolidatedc = f(σm)

cmf φσ tan=

NC clay failure envelope

τ

cIf σm changes, the envelope changes

If truly undrained, φr should be zero. τ

Gibson (1953) Ph.DDissertation

φr

40

30

Bentonite, P.I. = 5304°

20

10

P.I.

HvorslevDrained direct shear tests

ee Kc σ=It is a function of the void ratio

True cohesion

cσ σ

e

e Virgin consolidation curve

eφ True friction

eef K φσστ tan+=

eee

f K φσσ

στ

tan+=

eφ

eσσ

This translates into a SET of envelopes one envelope for each e or eσ

ee Kc σ=because

e

f

στ

K

He took the soil out of shear zone and study the distribution of stress carefully

10.0°0.14577%91127KleilBelt Ton

17.5°0.123%2547Wiener Tegel

K< 2 µwPIwLClay eφ

As wL, wPI, < 2µ ↑ K ↑ and eφ ↓

Bjerrum (1954) Ph.DDissertation

Basically unsuccessful in determining the Hvorslev coefficients

eef Kc φσστ tan0 ++= ee Kcc σ+= 0

Intercept cohesionτ

eφ

cφ

ecσ0c

Tests on specimens of the same void ratio tends to have same strength Hard to determine different τHard to determine and eKσ0c

Test by consolidating the specimens to different σmax, unload to the same e and shear

f3σ σ

Critical confining stress

During shear: no drainage e = const. e

e Virgin consolidation curve

Tests on specimens of the same void ratio tends to have same strength!

31 σσ −

31 σσ −sameS.P.T.

Critical state line

Hvorslev envelope

3σf3σNo change in effective stress for the specimen sheared at critical void ratio

Critical State Soil Mechanics

31Ip = I1 = first stress invariant

= σ1 + σ2 + σ3

octp σσσσ =++= )(31

321

For triaxial compression:

)2(31

31 σσ +=p

)(21

31 σσ +=p

σ2 = σ3

MIT people:

23 Jq ×=

J2 = deviatoric component of the second stress invariant

21332212 3 octJ σσσσσσσ +−−−=

22 2

3octJ τ=

octq τ2

3=

213

232

221 )()()(

31 σσσσσστ −+−+−=oct

For triaxial compression:

σ2 = σ3

231 )(2

31 σστ −=oct

)( 31 σσ −=q

)2(31

31 σσ +=p

)( 31 σσ −=q

For c=0σ1

Space diagonal

σ3

σ2

1σ

p32

2

z0 3 <σ

Tension zone

q32 y

Space diagonal

3 2 σ

py oct 33)(3

1321 ==++= σσσσ

qz oct 323 == τ

Tension zone

p 3

2

0 3 =σq

32

Tension zone

3q

p

State path: p, q, e (or other variables) path

N.C. Clay Undrained Loadingq

2p

R

1p

Envelope of NC clay

= Projection of critical void ratio (c.v.r.) line on p, q plane

S.P.T.

Hvorslev’senvelopes

4p p3p

NC stress path

e Virgin isotropic consolidation curve

p2p 3p 4p1p

Overconsolidate Clay –Undrained loading

p

Increase OCR e

e Virgin consolidation curve

31 σσ −

31 σσ −same

S.P.T.

same void ratio tends to have same strength common point (at large strain)

points on critical void ratio (c.v.r.) line

Projection of critical void ratio (c.v.r.) line on p, q plane

NC

3σf3σNo change in effective stress for the specimen sheared at critical void ratio

Drained tests

p, q, e are changing throughout the test

Roscoe, S, and Wroth

They did not distinguish between (σ1 – σ3)maxand Useful in pile driving where the strain is very largeAs a building block for developing other stress-strain modelsThey adopted von Mises criteria in order to get from 2-D to 3-D

max3

1 )( σσ

Line of critical overconsolidation ratio

A line on the q = 0 plane that marks the boundary of soil tends to dilate or contract

3

On C.V.R.

Projection of C.V.R. line

Virgin consolidation curve

q

pCritical OCR line

C.V.R. line

Virgin consolidation curve

p

Drained test: no change in e

Drained tests

Dry

Wet

Start from the left, dilate, e↑Start from the right, contract, e↓

Critical OCR line

e (w)

Dry of critical: tends to dilate during shear, heavily overconsolidated clay (on the left of the critical OCR line)Wet of critical: tends to contract during shear, normally consolidated or slightly overconsolidated clay (on the right of the critical OCR line)

Undrained Tests

3

On C.V.R.∆u = 0

∆u

q

pCritical OCR lineThe only starting point with no pore water pressure change

2p1p 4p p3p

C.V.R. Line

Projection of stress path on the 3:1 plane

Undrained path

Drained path

Critical OCR line

e

Critical OCR line (R, S, & W)

Depends onThe nature of the testsSoil properties

May not be parallel to the virgin consolidation curve

Partly Saturated Soils

Concave downward

QφQQQ c φστ tan+=Q envelopes

τ

τ1

cQ

σ σ1

For partly saturated soils volume changes in Q tests

Q envelopes

Low Srτ

High Sr

Intermediate Sr

σ

Sr different effective stress is different

S envelopes – after saturation

Essentially one envelope regardless of Sr at compaction τ

σ

Capillary pressure

Whenever there is a curved air-water interface, there’s must be pressure drop

Capillary and the Capillary Fringe

Capillary: water molecules at the water table subject to an upward attraction due to surface tension of the air-water interface and the molecular attraction of the liquid and the solid phases.

Tension

If fluid pressures are measured above the water table, they will be found to be negative with respect to local atmospheric pressure

Air Pressure

If air in the pores is connected, the air pressure is equal to the local atmospheric pressureIf air in the pores is not connected and is in the forms of air bubbles such as in the capillary fringe, the air pressure is equal to the pressure of water, which is negative.

Capillary rise in a tube

= Ts

i f

s

λR

λσ

σcosλ

r

λ=0

gRh

wc ρ

λσ cos2=R=r

hc

uc = hc γw = 2 Ts / γw

water

Fig. 6.2 IDEALIZED pore diameter in a sediment with cubic packing. The equivalent capillary tube has a radius of 0.2 the diameter of the grains

Patm

0

0.4 (=0)hψ

ψψp g0.3

0.2

0.1

z (m)

Patm

water

y-4 -2 0 2 4 (J /kg)

-4 -2 0 2 4 yr l (kJ /cu.m = kP a)

-0.2 0 0.2 0.4 y /g (J /N = m)-0.4

z (m)

0.1

0.2

0.3

0.4ψp

ψh

ψg =gz

ψ

water

(J/Kg)-1 0 1 2 3 4

z (m)

0 0.2 0.4 0.6 0.8-0.2

-1.0

-0.2

-0.8

-0.4

h

ψg= gz

heads (m)

0

-0.4-0.6-0.8

H

-0.6

-1.2

1 2

Height of Capillary Rise

1.50.1000.50Fine gravel40.0400.20Very coarse sand150.0100.05Coarse sand250.0060.03Medium sand500.0030.0150Fine sand1000.00150.0075Very fine sand3000.00050.0025Coarse silt7500.00020.0008Fine silt

Capillary Rise (cm)

Pore Radius (cm)

Grain Diameter

(cm)

Sediments

Water Potential

The potential energy, or force potential of ground water consists of two parts: elevation and pressure (velocity related kinetic energy is neglected)

Suction of Water in Soils

Fluid pressures in the vadose zone are negative, owing to tension of the soil-surface-water contactThe negative pressure head is measured in the field with a tensiometer

a

c∆z2

d

b∆z1 Soil Surface

Suction – Calibrate Before Use

Suction

Matric suctionElevation headPressure head

Osmotic suction

Head (Water Potential)

Gravity potential, ZElevation head

Moisture potential, ψSuction headCan be several orders of magnitude greater than the gravity potential1 bar ≈ 10 m of water column

Soil Water Characteristics

Defines the relationship between water content and water potential (suction)

0 0.1 0.2 0.3 0.40

-0.2

-0.4

-0.6

-0.8

-1.0

hm (m)

θ

SWCC of a Coarse Sand

Soil Water Characteristic Curve

SWCC of various soils

Absorption and desorption characteristics with primary scanning curves

-1 0 6

-1 0 4

-1 0 2

-1 0 0

0 0 .1 0 .2 0 .3 0 .4

W a te r c o n te n t,θ

P re ssu re h e a d(cm )

h b

-1 0 7

-1 0 5

-1 0 3

-1 0 1

P o o r ly so r ted

W ell so r ted

Entry pressure hb

Volumetric water content,

ψMain drying curve

Main wetting curve

Drying scanning curve

Wetting scanning curve

θ

Absorption and desorption characteristics with primary scanning curves

Hysteresis

Ink bottle effectTrapping of airAdvancing and receding contact angle

Determination of SWCCLaboratory

Pressure plate apparatusFilter paperThermocouple PsychrometerCentrifuge

FieldTensiometerWater content measurement

z (m)

0 0.4-0.4

-1.0

-0.2

-0.8

-0.4

hψ g= gz

heads (m)

0

-0.8-1.2-1.6

H

-0.6

-1.2

Effluent port that can vary elevation

water

Ceramic plate

soil

Pychrometers

Measure relative humidity in close proximity to air-water interface

)%100

ln()( HMgRTuh

w

cc −=

γ

H = relative humidityR = the gas constantT = absolute temp.M = molecular wt. of water vaporg = acceleration due to gravity

48.4314200.0

93.01420.0

99.28142.0

99.9214.2

99.991.42

H (%)Typical values of uc (psi)

98116

4658

2229

9.012

4.66

2.33

0.81

Osmotic suction at 20°C (bars)

Amount of NaCl in water (g/L)

Most negative pore pressure is measured from:Highly plastic fine-grained soil

土壤中空氣存在的四種狀態：(A)空氣完全連續階段，(B)空氣部份連續階段，(C)空氣內部連續階段，(D)空氣完全封閉階段

土壤中可能存在之飽和狀態：(a)鐘擺狀飽和，(b)絲狀飽和， (c)島狀飽和

Effective stressTerzaghi (1923) :

u−= σσ

Skempton，1960

ku−= σσ

k= (1-a tanϕ/tanφ) – for shear strengthk = 1- Cs/C – for compressibilitya: contact area of solids on a cross sectionϕ: friction angle of solidsφ: friction angle of soilCs: compressibility of solidsC: compressibility of soil skeleton

Stress/Pressure

ua – pore air pressureuw – pore water pressureuc = ua - uw

Partly saturated soilBishop’s “chi” theory (1955)

)()( waa uuu −+−= χσσ

χ = 0χ = 0

Sr = 100%Sr = 0%

Saturated soilOven-dried soil

au−= σσ wu−= σσ

Bishop(1960) and Aitchison(1960)

aw ukuk 21 −−= σσ

k2 =1- k1 , if k1 = χ

χ depends on Sr and void ratio (e)

)()( waa uuu −+−= χσσ

Fredlund&Morgenstern(1977)

Control s, ua, uw and observe the volume change of soils (fix two and let one of them change at a time)If any one of them were kept constant, volume remained the same

Mongiovi及Tarantino

Keep the volume of the soil specimen constant observe the change of stress variablesVery small changes of the stress variablesThe effective stress of unsaturated soil is controlled by two stress variables: (σ – ua) and (ua – uw)

Allam & Sridharan (1987)

πγχσσ −+−+−= Tuuu waa )()(

γ = perimeter of referenced air-water interfaceT = air-water interfacial tension make solids get closer to each otherπ = tension of dissolved material

飽和度Sr與結構張力、收縮表面張力γＴ變化圖

Allam & Sridharan (1987)

KaoliniteSr = 0.3, γT accounts for 13 – 36% of the suctionEffect of π was not significant

Fredlund(1979)

Regina clay:Water content: 24% 30%π accounts for 40% 80% of the total suction

Tensor expression

−−

−

azzyzx

yzayyx

xzxyax

uu

u

στττστττσ

−−

−

wa

wa

wa

uuuu

uu

000000

When Sr 100%, uw ua, (ua – uw) 0

−−

−

wzzyzx

yzwyyx

xzxyWx

uu

u

στττστττσ

Pore pressure coefficient χ

Bishop, Blight, and Donald(1960): χ is between 0 and 1Sparks(1963) based on force equilibrium:

−+

−

−+

=θ

θθπθ

θθπχcos

1cossin2cos

1cossin4

2

rpΤ

p = pore water pressure =T = surface tension of waterr = radius of soil grainsθ = angle between surface of water and center of soil grains,

When Sr 100% and q is 45° χ = 1.57

( ) ( )

−+−

−Τ

1cossin1

cos11cos

θθθθ

r

Blight(1967)Get χ by CD and CU testsWhen close to saturation, c may become > 1Blight thought it might due to lack of lab data, and this was not real

However, he did not consider the effect of air-water interface

If consider γT in the formulation c = Sr; but not experimental evidence to support this

( )

( )wa

a

uu

u

−

−+−

′+′

=3131 2

121 σσσσ

χ

Shear strength of unsaturated soil

According to Bishop, the generalized Mohr-Coulomb failure criteria is:

This equation is difficult to verify, because it is difficult to determine χ

( ) ( )[ ] φχφστ tantan waanf uuuc −+−+=

Fredlund

Suggests that the shear strength is controlled by two independent variables: (σ - ua) and (ua- uw) He and Morgenstern suggest:

( ) ( ) bwaanf uuuc φφστ tantan −+−+=

(σn – ua) and (ua – uw) is the normal stress and matricsuction on the failure surfaceφb is the friction angle associated with the matricsuction; generally it is not a constant, but decreases as the matric tension increases

Relationship between φb and matricsuction (Gan and Fredlund,1988)

Lu (1992)

Suggests the following criteria for expansive soils:

( ) φφστ tantan sanf Puc +−+=

Ps: expansive pressure of soil

Xu (1998)

Based on the concept of fractal dimension on expansive soil:

( ) ( ) φφστ 1+−+−+= mnwaanf

n tantan uukuc

k: fitting coefficientm, n: fractal dimension parameters

The introduced theories and criteria have a common problem: difficult to measure the parametersVanapalli And Fredlund(1996) developed an empirical equation relating the shear strength and the soil water characteristic curve

Fredlund et al. (1978

The shear strength provided by the matricsuction, τus, is:

bwaus uu φτ tan)( −=

φb varies with the water content of soil, i.e., it varies with the matric suction of soil

( )( )φτ tan)( waus uu Θ−= [ ]k

( ) ( ) ( ) φτφ tan)(

)(tan

−Θ

−+Θ=−

=wa

k

wawa

b uudduu

uudd k

sθθ

=ΘΘ is the normalize volumetric water contentθs is the volumetric water content at saturationk is a fitting parameter has to be determined by experiments

Vanapalli et al. (1996)

Obtain shear strength based on SWCC

( ) ( ) φθθθθφστ tantan

−−

−+−+=rs

rwaanf uuuc