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Geomechanics 255 (610.255)Geomechanics 255 (610.255)

Geomechanics Group

School of Civil & Resource EngineeringThe University of Western Australia

Part 2: Soil StrengthProfessor Martin Fahey

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OutlineOutline• Shearing behaviour of sand (cohesionless soil)

– friction

– dilatancy

– concept of critical state (critical void ratio)

• Shearing behaviour of clays (cohesive soil)– critical state concept for clayey soils

– drained and undrained shear strength in triaxial tests

– relationship between pore pressure change in undrained tests, and volume change in drained tests

The aim is to show that the shearing behaviour of all soils (sands and clays) can be presented within the unified framework of Critical State Soil Mechanics. This links the volume change behaviour in drained shearing with the pore pressure changes that occur when drainage is not able to occur. For sands, undrained behaviour generally can only occur when the boundary conditions prevent – otherwise, shearing is generally slow enough to allow any pore pressures (positive or negative) that tend to occur to dissipate as the shearing progresses. (The exception may be very fast loading, as in an earthquake, or where the scale of the problem is very large, as with very large offshore gravity platforms). On the other hand, the permeability of clay soils is so low that it is very difficult to apply loads slowly enough for drained conditions to apply, and hence many problems involving applying loads to clayey soils deal with the undrained shear strength.

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Soil Strength: Angle of Internal Soil Strength: Angle of Internal Friction Friction ''

N

F

N

F

F

N '

': Angle of internal friction; : coefficient of frictiontan ' = = F/N

''

': Angle of repose of sand heap': Angle of plank when block slides

F

R

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Principle of Effective StressPrinciple of Effective Stress

N

F Water pressure u

At failure:

stress

F N u A

F

A

N u A

Au

u

n

n

. tan

.tan

tan

tan

effective

N

F

F

Note: As u (i.e. ' 0)strength () 0 (liquefaction)

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Direct Shear Box ApparatusDirect Shear Box Apparatus

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Other Versions at UWAOther Versions at UWA

Pneumatic jack (computer controlled) to apply vertical load

Load cells

Direct Via LeverHangers for load

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Behaviour of Sand in Direct Shear Behaviour of Sand in Direct Shear BoxBox

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Direct Shear Tests on SandDirect Shear Tests on Sand

D, M, L: Dense, Medium, Loose

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Direct Shear Box: Summary of ResultsDirect Shear Box: Summary of Results

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Absolute value of soil density not so important – what matters is how dense is the soil relative to its maximum possible value and its minimum possible value

Relative Density – Density Index (IRelative Density – Density Index (IDD))

Densest possible state (emin, or dmax)(obtained by vibration under load)

ID

1 or 100%

Loosest (stable) state (emax, or dmin)(obtained by pouring with funnel)

0

Density index ID (relative density) –

where density lies in the range min. to max. -

or rather where void ratio lies between loosest (emax) and

densest (emin) state

minmax

maxD ee

eeI

ID (%) 0 – 15 15 – 35 35 – 65 65 – 85 85 – 100

State Very loose Loose Medium Dense Very dense

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Apparent Cohesion in SandApparent Cohesion in Sand

• Failure surface is actually curved

• Straight line through tests results at ' of 40, 60 and 80 kPa implies a cohesion intercept (c') of 10 kPa

• This implies a strength at zero effective stress: NOT CORRECT

Mohr-Coulomb Failure Criterion: f = c' + ' tan '

“Apparent cohesion” c'

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"Saw-tooth" Model of Dilation"Saw-tooth" Model of Dilation

• Dilation has effect of increasing the apparent friction angle on interface above the true value ('cv)

• Apparent friction angle from sawtooth model:'peak = 'cv +

• Dilation angle =

• Observed relationship:

'peak 'cv + 0.8 (Bolton)

• Collapsing material (negative dilation) shows friction angle less than 'cv

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Stress-Ratio Dilation Relationship Stress-Ratio Dilation Relationship (Taylor)(Taylor)

dy/dx = 0

Point of max. slope (max)

Peak stress ratio (tan 'peak)

dy/dx negative, increasing towards zero

dy/dx = 0

"Constant volume" stress ratio (tan 'cv)

tan tan

tan tan

tan ; tan

max

max

cv

peak cv

dx

dx

dx dx

dy

dy

dy

dy

Str

ess

rati

o (

/' n

)V

erti

cal d

isp

lace

men

t y

(vol

. str

ain

) dx

dy

DENSE

LOOSE

DENSE

LOOSE

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Critical State ConceptCritical State Concept

• When sheared, state of soil tends to migrate to a unique line in - ' - e space. This is called the critical state line (CSL).

• CSL has same gradient as NC line ()

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Dilation depends on density Dilation depends on density andand stress stress levellevel

LOOSE

DENSE

Critical State Line (CSL)

Voi

d r

atio

e

Normal effective stress 'n (or mean effective stress p')

At high stress, even dense samples may contract

At low stress, even loose samples may dilate

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Relative Density Corrected for Stress Relative Density Corrected for Stress LevelLevel

• For plane strain, Bolton found that:– max (º) = 6 IR for plane strain

– ´max – ´cv = 0.8 max

– ´max – ´cv 5 Irº

• For triaxial conditions– Must define 'dilatancy' in general as

– where v is volumetric strain = 1 + 2 + 3.

1 is the major principal strain (generally a in triaxial tests)

– (negative sign, because expansion - I.e. dilation - is negative by normal sign convention, but want 'dilatancy' to be positive)

minmax

maxD

D

eDR

ee

eeI

as definedindex density the is I

(kPa) stress effectives mean the is p'

1plog10II

1

v

d

d

R1

v I3.0d

d

p' (kPa) º

10 100 1,000 10,000

0.2 (20%) 3.2º 0.5º -2.3 (?)

0.5 (50%) 17.1º 10.2º 3.3º ID

0.8 (80%) (30.9º ?) 19.9º 8.8º

-2.2º (?)

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Drained & Undrained Shear StrengthDrained & Undrained Shear Strength

Loose states

Dense states

CSL

Void ratio e

Normal effective stress 'n (or mean effective stress p')

Shear stress

Drained strength sd

Undrained strength su

Drained strength sd

Undrained strength su

Suction increases effective stress

Positive pore pressure reduces effective stress

Dil

atio

n

Con

trac

tion

Undrained test

no volume change allowed

'n

eo

'cv

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Triaxial TestTriaxial Test

The triaxial test enables a variety of stress or strain controlled tests to be carried out on cylindrical soil specimens.

Sample

Membrane

Loading ram

Top cell

External LVDT

Top cap

Top "O" rings

Cell shroud

Triaxial pedestal

Bottom "O" rings

To air-water interface cylinderBottom drainage

Loading frame

Top porous disc

Bottom porous disc

Internal load cell

Phosphor bronze springs

Strain gauges

Top drainage

Fa

cell

cell

cell

Area, A

u

Fa

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One of the UWA Triaxial SystemsOne of the UWA Triaxial Systems

Cell cover lowered once sample in place

Axial motor drive system

Sample goes here

Sample, enclosed in rubber membrane, with axial strain measuring devices attached

Cell pressure controller

Control and data logging system

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Triaxial Test: BackgroundTriaxial Test: Background• Direct shear test useful, but limited

– Know only 1 normal stress ('n), don't know horizontal normal stresses

– Failure plane pre-defined - must coincide with the shear box

• Triaxial test still limited:– vertical and horizontal directions still principal directions

– horizontal stress equal in all directions

– “true triaxial” test would allow different '1, '2, '3 on three faces of cubical sample

– even more general - allow shear stresses to be applied to the three faces

'v (='1)

'h

(='2)'h

(='3)

'1

'2

'3

“True triaxial” ('1'2 '3)Triaxial

'v

'h 'h

hv

“Simple shear”

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Triaxial Test: Conduct of TestTriaxial Test: Conduct of Test• Almost always use saturated samples (using high

backpressure uo to achieve full saturation)

• Almost always consolidate the sample to some stress state (in situ stresses often) before carrying out the strength test

– isotropic consolidation: vertical and horizontal stresses equal (increase cell pressure only, allowing drainage against constant back pressure)

– 'h = '3 = c - uo, and '1 = 'v = 'h = '3 in this stage

– anisotropic consolidation: generally vertical stress greather than horizontal stress: increase cell pressure and apply additional vertical load

– 'h = '3 = c - uo, and '1 = 'v > 'h = '3 in this stage

• “Shearing” phase (in the simplest test): increase the vertical load (stress) until the sample fails

– other “stress paths” also possible - see later

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Stress Paths in Triaxial TestsStress Paths in Triaxial Tests• Different stress paths in “shearing” phase:

1. keep cell pressure constant (h = 0) and increase vertical stress (v +)

2. keep vertical stress constant (v = 0) and reduce cell pressure (h -)

3. keep vertical stress constant (v = 0) and increase cell pressure (h +)

4. keep cell pressure constant (h = 0) and reduce vertical stress (v -)

5. vary both cell pressure and vertical stress in some predetermined way, to produce any type of stress path

• Stress path in q-p space:q = v - h p = (v + 2h)/31. h = 0 and v = + q = +v and p = +v/3 q/p =3

q

p

3

1

Stress path: a plot showing how the stresses vary during a test.In this case, this is a Total Stress Path (TSP).In this case, shearing starts from an isotropic stress state, following isotropic consolidation.

q

p

3

1

Anisotropic consolidation phase

Shearing phase

In this case, shearing starts from an anisotropic stress state, following anisotropic consolidation.

Anisotropic consolidation phase

Shearing phase

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Total and Effective Stress Paths (TSP, Total and Effective Stress Paths (TSP, ESP)ESP)

Stress Parameters:

Deviator stress: q

Mean effective stress:

v h

v hp'2

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1

q

p, p'

"Standard" stress path: h constantv increased to failure

v increasing

h constant

q = v

p = v/3q/ p = 3

TSP: Total stress path (imposed by apparatus)

u (+)

pp'

p' = p - u

ESP: Effective stress path (soil response)

q = q'

u may be negative)

A

B(b')B'

(ESP)

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Drained & Undrained Strength Drained & Undrained Strength (Clays)(Clays)

"Wet of critical"

"Dry of critical"CSL

Void ratio e

Mean effective stress p'

Deviator stress q

Drained strength sd

Undrained strength su

Drained strength sd

Undrained strength su

Dil

atio

n

Con

trac

tion

Undrained testno volume change allowed

mean effective stress p'

eo

3

1

u +u -

TSP

ESP

NC line

OC line

Undrained strength depends on p'o and OCR

CSL

A

Ad

B

Bd

Au , Bu

A

Ad

Bd

B

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Initial and Final Undrained StrengthInitial and Final Undrained Strength

CSL

e

p'

q

su after consolidation

p'

In situ eo

NCL

su for NC soil increases

after consolidation

In situ su

e after consolidation

su

In situ su

su = k.z (k = 1 to 2 kPa/m)

(or su = suo + k.z)

su after consolidation

Dep

th (

m)

How long for strength increase to occur ???

Tank or GBS v

NC soil

GBS v p'

CSL suo

k

1

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Staged Loading (Undrained)Staged Loading (Undrained)

CSL

e

p'

q

su after two increments

p'

In situ eo

NCL

In situ su

Fully drained sd

e after two increments

CSL

q due to total load > in situ su

failure if applied in 1 incrementTSP

ESP in undrained

loading

Consolidation between increments

1

1

2

2

3

3

45

4

56

6

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Drained Tx Tests: Silica & Calc. Drained Tx Tests: Silica & Calc. SandsSands

Silica sandSilica sand Calc. sand (Dog's Bay)

Calc. sand (Dog's Bay)

Dila

tion

Dila

tion

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Drained & Undrained Tx Tests, Calc. Drained & Undrained Tx Tests, Calc. SandSand

Dog's BayDog's Bay Dog's BayDog's BayTSP

DrainedDrained

UndrainedUndrained