Ce225 sm 16 mohr circle and failure theories

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Soil Mechanics ICE-225
Mohr circle, failure theories, and stress paths
Dr. ZafarMahmood
NUST Institute of Civil Engineering (NICE)
School of Civil & Env. Engineering (SCEE)

Normal and shear stresses on a plane
2
sn
sncosq
q
tnsinq
tn
sz
tzx
q
snsinq
C
D
tncosq
sn
sx
F
tn
F
sx
q
E
txz
q
txz
A
B
E
B
tzx
sz

3
sn
sncosq
q
tnsinq
tn
q
snsinq
tncosq
sn
Assume element
thickness is t
tn
F
Summing force components in horizontal dir.
sx
q
txz
E
B
tzx
Eq. 1
sz
Summing force components in vertical dir. and simplifying
Eq. 2

4
Eliminating tn from Eq. 1 and 2. Multiply Eq. 1 with sinq and Eq. 2 with cosq and add
and
since

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Eliminating tn from Eqs. 1 and 2. Multiply Eq. 1 with cosq and Eq. 2 with sinq and subtract

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sz
Stresses on a plane oriented at angle q to horizontal plane
tzx
sq
tq
sx
q
q
txz
Stresses on a plane oriented at angle q to major principal stress plane

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(i)
(ii)
Taking square
Taking square
Adding equations (i) and (ii)
The above is equation for a circle with a radius of (s1 – s3)/2 and its center at [(s1 + s3)/2 , 0]. When this circle is plotted in t-s space, it is known as the Mohr circle of stress.

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sz
tzx
sq
tq
sx
q
If we square and add these equations, we will obtain the equation for a circle with a radius of (s1 – s3)/2 and its center at [(s1 + s3)/2 , 0].
When this circle is plotted in t-s space, it is known as the Mohr circle of stress.
q
txz

Mohr’s circle for stress states
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sz
A (sz,txz)
tzx
sx
R
txz
2y
C
O
txz
D (s3,0)
E (s1,0)
A (sz,txz)
B (sx,-tzx)
B (sx,-tzx)
Assumption
sz > sx
Clockwise shear is +ive
(sx+sz)/2
(sz sx)/2

Mohr’s circle – principal stresses
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sz
A (sz,txz)
tzx
y
sx
R
txz
2y
C
O
txz
D (s3,0)
E (s1,0)
B (sx,-tzx)
(sx+sz)/2
(sz sx)/2
Principal stresses

Mohr’s circle – Pole
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sz
A (sz,txz)
Pole, P
tzx
R
txz
sx
y
2y
C
O
D (s3,0)
E (s1,0)
txz
B (sx,-tzx)
(sx+sz)/2
(sz sx)/2
The stress sz acts on horizontal plane & the stress sx acts on the vertical plane.
If we draw these planes in Mohr’s circle, they intersect at a point, P. Point P is called the pole of the stress circle.

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sz
A (sz,txz)
Pole, P
s1
tzx
s3
y
y
sx
R
txz
2y
C
O
txz
s1
D (s3,0)
E (s1,0)

s3
B (sx,-tzx)
(sx+sz)/2
(sz sx)/2
It is a special point because any line passing through the pole will intersect Mohr’s circle at a point that represents the stress on a plane parallel to the line.

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sq
sz
A (sz,0)
tq
sq
B (sx,0)
R
sx
q
q
C
O
q
tq
B (sx,0)
A (sz,0)
Pole, P

Example 1
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Stresses on an element are shown in the Figure below. Find the normal stress s and the shear stress t on the plane inclined at a = 35o from the horizontal reference plane.

Example 2
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The same element as shown in previous example is rotated 20o from the horizontal, as shown below. Find the normal stress s and the shear stress t on the plane inclined at a = 35o from the base of the element.

Idealized stress-strain response
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Material response to normal loading and unloading
DP
Lets apply incremental vertical load DP to a deformable cylinder of area A, the cylinder will compress by Dz and the radius will increase by Dr. This is called uniaxial loading. The change in vertical stress is
Dz
Original configuration
Ho
Forces and displacements on a cylinder
Deformed configuration
The vertical and radial strains are,
ro
Dr
The ratio of the lateral (radial) strain to axial (vertical) strain is called Poisson’s ratio, m, defined as

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DP
Linearly elastic material:For equal increments of DP, we get the same value of Dz. We get straight line OA in graph of stress-strain. Upon unloading cylinder returns to its original configuration.
Dz
Ho
Linearly elastic
A
ro
E
Dr
B
1
Stress (sz)
Nonlinearly elastic
O
Strain (ez)

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DP
Nonlinearly elastic material:For equal increments of DP, we get the different values of Dz, but on unloading the cylinder returns to its original configuration. The plot of strain-strain relationship is curve OB.
Dz
Ho
Linearly elastic
A
ro
E
Dr
B
1
Stress (sz)
Nonlinearly elastic
O
Strain (ez)

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DP
Elastoplastic material: Soils do not return to their original configuration after unloading. OA is the loading response. AB is the unloading response and BC is the reloading response. Strain during loading OA consists of two parts – elastic (recoverable), BD, and plastic (unrecoverable), OB.
A
C
Dz
E
Elastic response during unloading
1
Ho
Stress (sz)
D
B
O
Strain (ez)
ro
Dr
Plastic
Elastic

Friction
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 = angle of obliquity.  is the angle that reaction on the plane of sliding makes with normal to that plane. When sliding is imminent  reaches its limiting value . tan is called coeff. of friction.
Note: maximum sliding resistance is observed when angle of obliquity  reaches its limiting value .

Mohr failure criterion
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Otto Mohr (1900) hypothesized a criterion of failure for real materials. “The materials fail when the shear stress on the failure plane at failure reaches some unique function of the normal stress on that plane or tff = f(sff)”, where t is the shear stress and s is the normal stress. The first subscript f refers to the plane on which the stress acts (in this case the failure plane) and the second f means at the failure. tff is called the shear strength of the material.
t
tff = f(sff)
s

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If we know principal stresses at failure, we can construct a Mohr circle to represent the state of stress.
If we conduct several tests to failure, and construct Mohr circle for each state of stress, we can draw failure envelope.
This envelope expresses functional relationship between shear stress tffand normal stress sffat failure.
Not possible
Stable condition, since it does not touch failure envelope

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Using pole method, we can determine angle of the failure plane from the point of tangency of the Mohr circle and Mohr failure envelope.
The hypothesis that the point of tangency defines the angle of the failure plane in the element or test specimen, is the Mohr failure hypothesis.

Coulomb strength equation
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M Coulomb (1776) studied lateral earth pressure exerted against retaining walls. He observed that there was a stress-independent component shear strength and a stress-dependent component.
The stress-dependent component is similar to sliding friction in solids, so he called this component the angle of internal friction, f.
The other component seemed to be related to the intrinsic cohesion of the material and is commonly denoted by symbol c.

Mohr-Coulomb Failure Criterion
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If we combine Coulomb equation with the Mohr failure criterion, it becomes Mohr-Coulomb failure criterion.

Stress condition before failure
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t
tff is the shear strength available (shear stress on the failure plane at failure).
tff
tf is mobilized shear resistance on potential failure plane
tf
s
Since we haven’t reached failure yet, there is some reserve strength remaining, and this is the factor of safety of the material.

Stress conditions at failure
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Note: shear stress on the failure plane at failure tff is not the largest of maximum shear stress in the element. The maximum shear stress tmax acts on the plane inclined at 45o and is equal to tmax = (s1f – s3f)/2 >tff

Maximum obliquity
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Maximum shearing resistance is observed when angle of obliquity  reaches its limiting value . For this condition line OD becomes tangent to the stress circle at angle  to axis OX (see fig. below).
Note: Failure plane is not the plane subjected to the maximum value of shear stress. The criterion of failure is maximum obliquity, not maximum shear stress.
Although plane AE is subjected to greater shear stress than plane AD, it is also subjected to a larger normal stress & therefore the angle of obliquity is less than on AD which is plane of failure
tmax

Ce225 sm 16 mohr circle and failure theories

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