54 - nptel.ac.in · mohr-coulomb failure envelope for shear strength of soils circle a well below...

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

54


Module 4: Lecture 5 on Stress-strain relationship

and Shear strength of soils


Stress state, Mohr’s circle analysis and Pole, Principalstress space, Stress paths in p-q space;

Mohr-Coulomb failure criteria and its limitations,correlation with p-q space;

Stress-strain behavior; Isotropic compression andpressure dependency, confined compression, large stresscompression, Definition of failure, Interlocking conceptand its interpretations, Drainage conditions;

Triaxial behaviour, stress state and analysis of UC, UU, CU,CD, and other special tests, Stress paths in triaxial andoctahedral plane; Elastic modulus from triaxial tests.

Contents


Stress-strain relationships and Failure criteria The little hump in the stress‐strain curve for mild steel after

yield is an example of work hardening. Many soils are also work‐hardening, for example,

compacted clays and loose sands. Sensitive clay soils anddense sands are examples of work‐softening materials.

At what point on the stress‐strain curve do we have failure? In some situations, if a material is stressed to its yield point,

the strains or deflections are so large that for all practicalpurposes the material has failed.

This means that the material cannot satisfactorily continueto carry the applied loads. The stress at “failure” is oftenvery arbitrary, especially for nonlinear materials.


With these materials, we usually define failure at somearbitrary percent strain, i.e. 15% or 20%, or at a strain ordeformation at which the function of the structure might beimpaired.

Now we can also define the strength of a material. It is the maximum or yield stress or the stress at some strain

which we have defined as “failure.”

Stress-strain relationships and Failure criteria

There are many ways of defining failure in materials; or putanother way, there are many failure criteria.

Most of the criteria don’t work for soils. The most common failure criterion applied to soils is the

Mohr‐ Coulomb failure criterion.


Mohr-Coulomb Failure CriterionCharles Augustin de COULOMB (1736‐1806) is well known from hisstudies on friction, electrostatic attraction and repulsion.

Christian Otto MOHR (1835‐1918) hypothesized (1900) a criterion offailure for real materials in which he stated that materials fail when theshear stress on the failure plane at failure reaches some uniquefunction of the normal stress on that plane:

τff = f (σff)where τ is the shear stress and σ is the normal stress.The first subscript f refers to the plane on which the stress acts (in thiscase the failure plane) and the second f means “at failure.” τff is theshear the material.


Mohr-Coulomb Failure Criterion A failure theory is required to relate the available

strength of a soil as a function of measurableproperties and the imposed stress conditions.

The Mohr-Coulomb failure criterion is commonly usedto describe the strength of soils.

Its main hypothesis is based on the premise that acombination of normal and shear stresses creates amore critical limiting state than would be found if onlythe major principal stress or maximum shear stresswere to be considered individually.


Mohr-Coulomb Failure Criterion

σ

τ

τff = f (σff)

An element at failure with theprincipal stresses that caused failureand the resulting normal and shearstresses on the failure plane.

We will assume that a failureplane exists, which is not a badassumption for soils, rocks, andmany other materials.

If we know the principal stressesat failure, we can draw a Mohrcircle to represent this state ofstress for this particular element.


Mohr hypothesis: theMOHR‐COULOMB FAILURECRITERION failure point of tangencydefines the angle of the failure plane inthe element or test specimen.The Mohr failure hypothesis isillustrated for the element at failure shown.

Stated another way: the Mohrfailure hypothesis states that the pointof tangency of the Mohr failureenvelope with the Mohr circle atfailure determines the inclination ofthe failure plane.

Mohr-Coulomb Failure Criterion


Mohr-Coulomb failure criteria

τf

σ σ

τ

φ

c

τf = c + σ tanφ

Cohesion Friction angle

τf is the maximum shear stress the soil can take just before failure, under normal stress of σ.

Failure will thus occur atany point in the soil where acritical combination of shearstress and effective normalstress develops


Mohr-Coulomb failure criteria

σf tanφ

σfσ

τ

φ

c

τf = c + σf tanφ

Higher the values of c and φ , Higher the shear strength of soil

Cohesive component

Frictional component

Shear strength consists of two components : Cohesion and Friction


Mohr Circles & Failure Envelope:

A Bσ

τ

A = Failure

B = Stable

AB

Slip surface


Mohr-Coulomb failure envelope for shear strength of soils

Circle A well belowthe Mohr-Coulombenvelope (safe stateof stress)

Circle B is tangential to the Mohr-Coulomb envelope (critical stressconditions corresponds to failure)

So even though thestress combination, σnand τmax, for circle A isobviously greater thanthat of circle B, it iscircle B that is on theverge of failure.

State of stressrepresented by Mohrcircles that existbeyond the Mohr-Coulomb envelopecan not exist.


Now, if the stresses increase so that failure occurs, thenthe Mohr circle becomes tangent to the Mohr failureenvelope.

According to the Mohr failure hypothesis, failure occurson the plane inclined at αf and with shear stress thatplane of τff.

This is not the largest or maximum shear stress in theelement!!!

The maximum shear stress acts on the plane inclined at45° and is equal to:



Mohr-Coulomb failure envelope for shear strength of soilsWhy does not failure occur on 45° plane? It cannot because on that plane the shear strength

available is greater than τmax. So failure cannot occur.



This condition is represented by the distance fromthe maximum point on the Mohr circle up to theMohr failure envelope

That would be the shear strength availablewhen the normal stress available on the 45°plane was (σ1f + σ3f)/2.


Mohr-Coulomb failure envelope for shear strength of soils The only exception would be when shear strength is

independent of normal stress, i.e., when Mohr failureenvelope is horizontal and φ = 0.

Such materials are called purely cohesive forobvious reasons or this may result in completelysaturated and un-drained conditions.


Mohr Circles & Failure Envelope

Bσ

τ

σc σc+∆σ

∆σ

Y

σc

σc

σc

∆σ

σc

Initially Mohr’s circle is a point

Soil element does not fail if the Mohr’s circle is contained within the envelope



Bσ

τ

σcσc+∆σ++

Y

σc

σc

σc

∆σ++

σc

As loading progresses, Mohr’s circle becomes larger

Failure occurs when Mohr’s circle touches the envelope



σ

τ

σc σc+∆σY

σc

σc

σc

∆σ++

σc

Failure plane oriented at (90+φ)/2 to horizontal. i.e. (45+φ/2) with horizontal

φ 90+φ

Loading plane orientation


Mohr Circles in terms of effective stresses:

σ

τEffective stresses

Total stresses

σh' σv' σh σv

u

σhY

σv

=σh'

Y

σv'

uY

u


Mohr Circles failure envelope in terms of effective stresses:

σ

τ c – φ in terms of σ

σ '

τ c′ – φ′ in terms of σ′

Yσc

σc

Yσc

σc

∆σf

At failure,σ3 = σc; σ1 = σc + ∆σfσ3′ = σ3 – uf ; σ1′ = σ1 - uf


Principal stress relations at failure The relationship between the shear strength parameters

and the effective principal stresses at failure at aparticular point can be deduced.

θ is thetheoreticalanglebetweenthe majorprincipalplane andthe planeof failure.

c′cotφ′

For c′ = 0


Principal stress relations at failure

Now

Therefore

The following equation is referred to as the Mohr-Coulombfailure criterion:

With c′ = 0

In the special case, when φ = 0:


Principal stress relations at failureThe essential points are:

1. Coupling Mohr’s circle with Coulomb’s frictional lawallows us to define shear failure based on the stressstate of the soil.

2. The Mohr-Coulomb criterion is:


Principal stress relations at failure

σ

τ

σ1σ3

φ 90+φ

Xσ3

σ1

)2/45tan(2)2/45(tan)2/45tan(2)2/45(tan

213

231

φφσσ

φφσσ

′−′−

′−=

′+′+

′+=

c

c


q

p

ΨKf

a

τ

σ

φ

c

Relationship between Kf line and Mohr-Coulomb failure envelope

qf = a + pf tanΨ τf = c + σf tan φ

From geometries of the two circles, it can be shown that: sinφ = tan Ψ

c = a/cosφ

⇒ So, from a p-q diagram the shearstrength parameters φ and c mayreadily be computed.


Mohr circles for three dimensional state of stress

Effect of intermediate principal stress σ2 on condition at failure.

Since by definition σ2 liessomewhere between the major and minor principal stresses, the Mohr circles for the three principalStresses look like those shown herein.


Mohr circles for three dimensional state of stressEffect of intermediate principal stress σ2 on condition at failure. It is obvious that σ2 can have no influence on the

conditions at failure for the Mohr failure criterion, nomatter what magnitude it has.

The intermediate principal stress σ2 probably doeshave an influence in real soil, but theMohr‐Coulomb failure theory does not consider it.


Usual experimental range in the laboratory

Limitations of Mohr-Coulomb theory:1. Linearization of the limit stress envelope

φ, c

• Possible overestimation of the safety factor in slope stability calculations,• Difficulties in calibration because of linearization

τ

σ


2. Mohr-coulomb failure criterion is well proven for most of the geomaterials, but data for clays is still contradictory.

3. Soils on shearing exhibit variable volume change characteristics depending on pre-consolidation pressure which cannot be accounted with Mohr-Coulomb theory.

4. In soft soils volumetric plastic strains on shearing are compressive (negative dilation) whilst the Mohr-Coulomb model will predict continuous dilation.

54 - nptel.ac.in · mohr-coulomb failure envelope for shear strength of soils circle a well below...

Documents