Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Module 4: Lecture 5 on Stress-strain relationship
and Shear strength of soils
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr Circles & Failure Envelope:
A Bσ
τ
A = Failure
B = Stable
AB
Slip surface
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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 soils
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr-Coulomb failure envelope for shear strength of soils
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.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr Circles & Failure 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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr Circles & Failure 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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Mohr Circles in terms of effective stresses:
σ
τEffective stresses
Total stresses
σh' σv' σh σv
u
σhY
σv
=σh'
Y
σv'
uY
u
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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:
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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:
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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.
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