Ce225 sm 16 mohr circle and failure theories

1

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


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


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


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

qq

sq tq


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.


2cos22

3131

2sin

231

sx

txz

sz

tzx

qq

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


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


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.

15

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.

17

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



DP


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


Ho

ro

Dr

Dz



DP


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.



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



DP



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

31

32

Ce225 sm 16 mohr circle and failure theories

Education

stress circle

mohr circle of stress

normal stress s

shear stress t

normal loading

stress sx acts

graph of stress

stress pathsdr