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Mohr circle, failure theories, and stress paths
Dr. Zafar MahmoodNUST Institute of Civil Engineering (NICE)School of Civil & Env. Engineering (SCEE)
Soil Mechanics ICE-225
2Normal and shear stresses on a plane
sn sncosqq
snsinq
tn
tncosq
q
tnsinq
sx
txz
sz
tzx
qA B
CD
E
F
q
sn tn
sx
txz
sz
tzxBE
F
cosEFEB sinEFFB
3Normal and shear stresses on a plane
q
sn tn
sx
txz
sz
tzxBE
Fsn sncosqq
snsinq
tn
tncosq
q
tnsinq
Summing force components in horizontal dir.Assume element
thickness is t
0..cos..sin.... tEFtEFtFBtEB nnxzx
0.cos.sinsin.cos. EFEFEFEF nnxzx
sincoscossin xzxnn Eq. 1
Summing force components in vertical dir. and simplifying
sincossincos zxznn Eq. 2
4Normal and shear stresses on a planeEliminating tn from Eq. 1 and 2. Multiply Eq. 1 with sinq and Eq. 2 with cosq and add
22 sinsincossincossin xzxnn
sincoscossincoscos 22zxznn
sincos2sincossincos 2222zxxzn
2sinsincos 22zxxzn
2
2cos1cos2
since and2
2cos1sin 2
2sin2
2cos1
2
2cos1zxxzn
2sin2cos22 zx
xzxzn
5Normal and shear stresses on a plane
sincoscoscossincos 22xzxnn
22 sinsincossinsincos zxznn
2222 sincossincossincos zxzxn
2cos2
2sinzxzxn
2cos2sin2 zx
xzn
Eliminating tn from Eqs. 1 and 2. Multiply Eq. 1 with cosq and Eq. 2 with sinq and subtract
6Normal and shear stresses on a plane
Stresses on a plane oriented at angle q to horizontal plane
Stresses on a plane oriented at angle q to major principal stress plane
2sin2cos22 zx
xzxz
2cos2sin2 zx
xz
2cos22
3131
2sin2
31
sx
txz
sz
tzx
sq tq
7Normal and shear stresses on a plane
2cos22
3131
2sin2
31
2cos22
3131
Taking square 2cos22
22
31
2
31
Taking square 2sin2
22
312
(i)
(ii)
Adding equations (i) and (ii)
2
3122
31
20
2
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.
8Normal and shear stresses on a plane
2cos22
3131
2sin
231
sx
txz
sz
tzx
sq tqIf 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.
9Mohr’s circle for stress states
sx
txz
sz
tzx
2
zxOC
A (sz,txz)
B (sx,-tzx)
A (sz,txz)
B (sx,-tzx)
E (s1,0)
(sx+sz)/2 (sz sx)/2
R txz
C 2yOD (s3,0)
Assumption• sz > sx
• Clockwise shear is +ive
22
2 xzxzACR
10Mohr’s circle – principal stresses
xz
zx
2
2tan
22
1 22 zxxzxzCEOCOE
sx
txz
sz
tzx
y
A (sz,txz)
B (sx,-tzx)
E (s1,0)
(sx+sz)/2 (sz sx)/2
R txz
C 2yOD (s3,0)
22
3 22 zxxzxzDCOCOE
Principal stresses
11Mohr’s circle – Pole
xz
zx
2
2tan
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.
A (sz,txz)
B (sx,-tzx)
E (s1,0)
(sx+sz)/2 (sz sx)/2
R txz
C 2yOD (s3,0)
Pole, P
sx
txz
sz
tzx
y
12Mohr’s circle – Pole
sx
txz
sz
tzx
y
A (sz,txz)
B (sx,-tzx)
E (s1,0)
(sx+sz)/2 (sz sx)/2
R txz
C 2yOD (s3,0)
Pole, Py
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.
s1
s3
s1
x
zx
1
tan
s3
13Mohr’s circle – Pole
sx
sz
q
A (sz,0)
B (sx,0)
A (sz,0)
RCqO
Pole, P
B (sx,0)
q
sqtq
sq
tq
14Example 1
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.
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16Example 2
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.
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18Idealized stress-strain responseMaterial response to normal loading and unloading
Ho
ro
Dr
Dz
Original configuration
Deformed configuration
DP
Forces and displacements on a cylinder
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
A
Pz
The vertical and radial strains are,
or
oz r
r
H
z
,
The ratio of the lateral (radial) strain to axial (vertical) strain is called Poisson’s ratio, m, defined as
z
r
19Idealized stress-strain response
Ho
ro
Dr
Dz
Original configuration
Deformed configuration
DP
Forces and displacements on a cylinder
Strain (ez)
Str
ess
(sz)
Linearly elastic
Nonlinearly elastic
O
A
BE
1
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.
Material response to normal loading and unloading
20Idealized stress-strain response
Ho
ro
Dr
Dz
Original configuration
Deformed configuration
DP
Forces and displacements on a cylinder
Strain (ez)
Str
ess
(sz)
Linearly elastic
Nonlinearly elastic
O
A
BE
1
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.
Material response to normal loading and unloading
21Idealized stress-strain response
Strain (ez)
Str
ess
(sz) Elastic response
during unloading
OB
E
1
A C
Plastic Elastic
D
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.
Ho
ro
Dr
Dz
Original configuration
Deformed configuration
DP
Forces and displacements on a cylinder
Material response to normal loading and unloading
22Friction
= 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 .
23Mohr failure criterionOtto 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.
tff = f(sff)
s
t
24Mohr failure criterionIf 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 tff and normal stress sff at failure.
Stable condition, since it does not touch failure envelope
Not possible
25Mohr failure criterionUsing 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.
26Coulomb strength equation 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.
27Mohr-Coulomb Failure CriterionIf we combine Coulomb equation with the Mohr failure criterion, it becomes Mohr-Coulomb failure criterion.
28Stress condition before failure
tf is mobilized shear resistance on potential failure plane
tff is the shear strength available (shear stress on the failure plane at failure).
Since we haven’t reached failure yet, there is some reserve strength remaining, and this is the factor of safety of the material.
s
t
tf
tff
29Stress conditions at failure
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
30Maximum obliquityMaximum 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.
tmax
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
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