shear strengthb

Learning outcomes

Understand the assumptions and limitations of four

soil models: Coulomb, Mohr-Coulomb, Tresca and

Taylor.

Know how to select the appropriate soil model to

interpret soil test data.

1

Importance

All models make assumptions. You must understand these

assumptions to know the limitations of a selected model.

The response of soils depends on many factors including the

drainage condition, the history of loading and the stress path.

You must be able to select and use the appropriate model

that best represents the expected soil condition. Poor choice

and use could lead to misrepresentation and failure.

2

Key terms

Shear strength of a soil is the maximum internal resistance to applied shearing

forces.

Effective friction angle, , is a measure of the shear strength of soils due to friction.

Cementation, ccm, is a measure of the shear strength of a soil from forces that

cement the particles.

Soil tension, ct, is a measure of the apparent shear strength of a soil from soil suction

(negative pore-water pressures or capillary stresses).

Cohesion, co, is a measure of the intermolecular forces.

3

Key terms

Undrained shear strength, su, is the shear strength of a soil when sheared at constant

volume.

Apparent cohesion, C, is the apparent shear strength at zero normal effective stress.

Critical state is a stress state reached in a soil when continuous shearing occurs at

constant shear stress to normal effective stress ratio and constant volume.

Dilation is a measure of the change in volume of a soil when the soil is distorted by

shearing.

4

MODELS TO INTERPRET SHEAR STRENGTH

A soil model is an idealized representation of the soil to allow us to understand its response to loading and other external events.

A soil model should not be expected to capture all the intricacies of real soil behavior.

Each soil model may have a different set of assumptions and may only represent one or more aspects of soil behavior.

Popular soil models

Coulomb

Mohr-Coulomb

Tresca

Some other soil models

Taylor

Critical state

5

Simple

COULOMBS SOIL MODEL

Soils, in particular granular soils, are endowed by nature with slip planes.

Each contact of one soil particle with another is a potential micro-slip plane.

Loadings can cause a number of these micro slip planes to align in the direction of least resistance.

Thus, we can speculate that a possible mode of soil failure is slip on a plane of least resistance.

6

COULOMBS SOIL MODEL FOR UNCEMENTED,SOILS

LINEAR FAILURE ENVELOPE

Soils at critical state: = cs, = p = 0

CURVED FAILURE ENVELOPE

Soils at peak state: = p, = p > 0

is the dilation angle (a measure of the soils

ability to expand > increase in volume)

7

( ) tan ( ) tanf n f p n f cs p

( ) tanf n f cs

cs is a constant for a given soil and is a fundamental soil property; p is not a constant for a given soil it depends on the normal effective stress and the ability of the soil to dilate.

Soil fails by impending frictional sliding on a plane

p cs

WHAT IS DILATANCY?

Dilation is not a peculiarity of soils,

but occurs in many other materials,

for example, rice and wheat.

The ancient traders of grains were

well aware of the phenomenon of

volume expansion of grains.

However, it was Osborne Reynolds

(1885) who described the

phenomenon of dilatancy and

brought it to the attention of the

scientific community..

Dilation can be seen in action at

a beach.

8

COULOMBS SOIL MODEL FOR CEMENTED SOILS

ccm is the cementation strength and o is the apparent friction angle.

Neither ccm nor o is a fundamental soil parameter.

Adding the cementation strength to the apparent frictional strength is not strictly correct since they are not mobilized at the same shear strains.

9

( ) tanf cm n f oc

ISSUES WITH AND USE OF THE COULOMBS MODEL

USE

It can be used for failures that occur along a slip plane, such as a joint or the interface of two soils or the interface between a structure and a soil.

Stratified soil deposits such as overconsolidated varved clays (regular layered soils that depict seasonal variations in deposition) and fissured clays are likely candidates for failure following Coulombs model, especially if the direction of shearing is parallel to the direction of the bedding plane.

ISSUES

Coulombs model applies strictly to two rigid bodies with a common potential sliding plane.

It is a limiting force model (force at impending frictional sliding )

It does not consider soil deformation.

It is independent of the loading history of the soil.

10

KEY POINTS REGARDING COULOMBS MODEL

11

tan ,f n cs pf

n tan f cm ofc

MOHRCOULOMB (MC) FAILURE CRITERION

MC failure criterion defines failure when the maximum principal effective stress ratio,

called the maximum effective stress obliquity, is achieved and not when the maximum shear stress

is achieved.

The failure shear stress is then less than the maximum shear stress.

12

Soil fails by frictional sliding on a plane of maximum

stress obliquity

1

3

( ),

( )

f

f

1 3[( )/2]max


Friction angle

Inclination of failure plane to the

plane of the major principal

stress

Maximum shear stress

13

1 3[( )/2]max

1 3

1 3

1 3 1 3

( ) ( )2sin( ) ( )

2

f f

f f

ff f

OB

OA

452 4 2


Failure stresses for uncemented

soils

14

1 3 1 3( ) sin2 2

n f

1 3 cos2

f

MC FAILURE CRITERION

Uncemented soils

at critical state

At peak state

15

1 3

1 3

sin =cscs

1 3 cos

2cs cs

1 3

1 3

sin =pp

1 3= cos2

p p

p

MC FAILURE CRITERION

Unsaturated, cemented, cohesive

soils

16

o C

Sh

ear

stre

ss

Normal effective

stress, n

1 3

1 3

sin 2 cot +

o

oC

1 31

= tan 1 sin 1 sin2

f o o oC

ISSUES WITH AND USE OF THE MC MODEL

USE

It can be used for long term

(drained condition) stability

calculations and to interpret

the long term strength of

overconsolidated fine-

grained and dense coarse-

grained soils.

ISSUES

MC model applies strictly to two rigid bodies with a common potential sliding plane.

It is a limiting stress model.

It does not consider soil deformation. Soil deformation is important in real soils.

It is independent of the loading history of the soil. The strength of real soils is dependent on loading history.

The shear strength in compression and extension is the same. Real soils show different strengths in compression and extension. Usually, the extension strength is lower than the compressive strength.

17

KEY POINTS: MC FAILURE CRITERION

Coupling Mohrs circle with

Coulombs frictional law allows us

to define shear failure based on

the stress state of the soil.

Failure occurs, according to the

MohrCoulomb failure criterion,

when the soil reaches the

maximum principal effective

stress obliquity.

The maximum shear stress is not

the failure shear stress.

Information on the deformation

or the initial stress state of the

soil is not needed to interpret soil

strength using the MC failure

criterion.

18

TRESCAS MODEL

Trescas failure criterion is used to interpret the undrained shear strength.

The shear strength under undrained loading depends only on the initial void ratio or the initial water content or initial confining pressure.

An increase in initial normal effective stress, sometimes called confining pressure, causes a decrease in initial void ratio and a larger change in excess porewater pressure when a soil is sheared under undrained condition.

The result is that the Mohrs circle of total

stress expands and the undrained shear

strength increases. Thus, su is not a

fundamental soil property.

19

Soil fails when the shear stress is one-half the principal stress difference

1 3 1 3( ) ( ) ( ) ( )

2 2

f f f f

us

TRESCAS MODEL

The value of su depends on the

magnitude of the initial confining

pressure or the initial void ratio

(or initial water content).

Analyses of soil strength and soil

stability problems using su are

called total stress analyses (TSA).

20

ISSUES WITH AND USE OF THE TRESCAS MODEL

USE

Short term (undrained

condition) stability

calculations and to interpret

the undrained shear strength

of fine-grained soils.

ISSUES

It is a yield criterion for solid bodies that has

been adopted as a failure criterion for soils

(a deformable body).

It is a limiting stress criterion.

It does not consider soil deformation. Soil

deformation is important in real soils.

It is independent of the loading history of the

soil. The strength of real soils is dependent

on loading history.

Compression and expansion strength is the

same. Real soils show different strengths in

compression and in expansion

21

KEY POINTS TRESCAS FAILURE CRITERION

For a total stress analysis, which

applies to fine-grained soils, the

shear strength parameter is the

undrained shear strength, su.

Tresca failure criterion is used to

interpret the undrained shear

strength of fine grained soils

The undrained shear strength

depends on the initial void ratio

or initial water content or initial

confining pressure. It is not a

fundamental soil shear strength

parameter.

Information on the deformation

of the soil is not needed to

interpret soil strength using

Tresca failure criterion.

22

TAYLORS FAILURE CRITERION

Taylor (1948) used an energy method to derive a simple soil model.

He showed that the shear strength of soil is due to sliding friction from shearing and the interlocking of soil particles.

23

The shear strength comes from sliding friction and the interlocking of soil particles

Unlike Coulomb failure criterion, Taylor failure criterion does not require the assumption of any physical mechanism of failure, such as a plane of sliding.

It can be applied at every stage of loading for soils that are homogeneous and deform under plane strain conditions similar to simple shear.

TAYLORS FAILURE CRITERION: FORMULATION

Equilibrium:

Simplification:

Critical state:

Peak:

24

The shear strength comes from sliding friction and the interlocking of soil particles

f z z zd d d

z

f

z

d

d

tan ;f cs 0.zd

d

tan cs

z cs

tan tancs pz p

ISSUES WITH AND USE OF THE TAYLORS MODEL

USE

Long term stability

calculations of homogeneous

soils.

Cannot be applied to soils

that fail along a joint or an

interface between two soils.

ISSUES

Applies to two-dimensional stress systems.

An extension of Taylor failure criterion

to account for three-dimensional stress is

presented in Chapter 11.

Neither Taylor nor Coulomb failure

criterion explicitly considers the rotation of

the soil particles during shearing.

Gives a higher peak dilation angle than

Coulomb failure criterion.

25

DIFFERENCES AMONG THE THREE POPULAR FAILURE CRITERIA

Name Failure criteria Soil treated as Best used for

Test data

interpretation*

Coulomb Failure occurs by

impending, frictional

sliding on a slip plane.

Rigid, frictional

material

Layered or fissured

overconsolidated soils or a

soil where a prefailure

plane exists

Direct shear

Mohr

Coulomb

Failure occurs by

impending, frictional

sliding on the plane of

maximum principal

effective stress obliquity.

Rigid, frictional

material

Long term (drained

condition) strength of

overconsolidated fine-

grained and dense coarse-

grained soils

Triaxial

Tresca Failure occurs when one-

half the maximum

principal stress

difference is achieved.

Homogeneous solid Short term (undrained

condition) strength of fine-

grained soils

Triaxial

* will discuss later

26

SUMMARY OF EQUATIONS FOR THE THREE POPULAR FAILURE CRITERIA

27

Name Peak Critical state

Coulomb ( ) tan ( ) ( ) tan p n f cs p n f p

unsaturated, cemented soils: ( ) tanf n f oC o t cmC c c c

( ) tan cs n f cs

MohrCoulomb 1 3

1 3

sin pp

1 3

1 3

sin cscs

3 2

1

( ) 1 sintan 45

( ) 1 sin 2

p p p

p p

23

1

( ) 1 sintan 45

( ) 1 sin 2

cs cs cs

cs cs

1 3

1 3

Cemented soils: sin 2 cot +

o

oC

o t cmC c c c

Inclination of the failure plane to the plane on which

the major principal effective stress acts.

=45 +2

po

p

Inclination of the failure plane to the plane

on which the major principal effective stress

acts.

452

o cs

cs

Tresca

1 3( )

2

p

u ps

1 3( )

2

cs

u css

RANGES OF FRICTION ANGLES AND DILATION ANGLES FOR SOILS

Ranges of Friction Angles for Soils (degrees)

Soil type

Gravel 3035 3550

Mixtures of gravel and

sand with fine-grained

soils 2833 3040

Sand 2737* 3250

Silt or silty sand 2432 2735

Clays 1530 2030 515

*Higher values (3237) in the range are for sands

with significant amount of feldspar (Bolton, 1986).

Lower values (2732) in the range are for quartz

sands.

Typical Ranges of Dilation Angles for Soils

Soil type

(degrees)

Dense sand 1015

Loose sand

TYPICAL VALUES OF SU FOR SATURATED FINE-GRAINED SOILS

Description su (kPa( su (psf)

Very soft < 10 200 > 4000

29

Quiz 1 30

Which failure criterion (model) is best suited to analyze

the potential failure of the soil mass shown?

1. Mohr-Coulomb

2. Coulomb

3. Tresca

4. None of the above

Dense sand

Stiff overconsolidated clay

Loose silty sand

Quiz 2 31

The critical state friction angle of a soil is 30 degrees.

If the normal effective stress imposed by a building is

100 kPa, the shear stress (kPa) to cause failure is most

nearly

1. 86.6

2. 100

3. 50

4. 57.7

Quiz 3 32

The critical state friction angle of a soil is 30 degrees.

The ratio of the major principal effective stress to the

minor principal effective stress to cause failure is most

nearly

1. 0.5

2. 1

3. 2

4. 3

PRACTICAL IMPLICATIONS OF FAILURE CRITERIA

Region I. Impossible soil

states. A soil cannot have soil

states above the boundary

AEFB.

33


Region II. Impending instability (risky

design).

Region AEFA is characteristic of dilating

soils that show peak shear strength and

are associated with the formation of

shear bands. The shear bands consist of

soils that have reached the critical state

and are embedded within soil zones with

high interlocking stresses due to particle

rearrangement. These shear bands grow

as the peak shear strength is mobilized

and as the soil strain-softens subsequent

to the critical state.

34


Region III. Stable soil states (safe design).

One of your aims as a geotechnical engineer is to design geotechnical systems on the basis that if the failure state were to occur, the soil would not collapse suddenly but would continuously deform under constant load. This is called ductility. Soil states that are below the failure line or failure envelope AB would lead to safe design. Soil states on AB are failure (critical) states

35

KEY POINTS

There are several failure criteria for soils.

Each criterion has application to certain soil conditions.

The three popular failure criteria (Coulomb, Mohr-Coulomb and Tresca) assume that soil is a rigid-plastic material with no deformation prior to failure.

The Coulomb and Mohr-Coulomb failure criteria are applicable to estimate long term failure.

The Mohr-Coulomb failure criterion also assume that failure shear strength of soil in compression and extension is the same. In reality, the shear strength at failure in extension is less than in compression.

36

KEY POINTS

Soil states above the peak shear strength boundary are impossible.

Soil states within the peak shear strength boundary and the failure line (critical state) are associ-ated with brittle, discontinuous soil responses and risky design.

Soil states below the failure line lead to ductile responses and are safe.

You should not rely on p in geotechnical design, because the amount of dilation one measures in laboratory or field tests may not be mobilized by the soil under construction loads. You should use cs unless experience dictates otherwise.

A higher factor of safety is warranted if p rather than cs is used in design.

37

KEY POINTS

The undrained shear strength, su, applies only to fine-grained soils.

The undrained shear strength is not a fundamental soil parameter.

The undrained shear strength depends on the initial void ratio or initial confining pressure (consolidation pressure).

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shear strengthb

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