Mohr’s Circle

** **

Consider
the state of plane-stress shown in the figure.

If negative
the shear stress _{} is plotted against
the normal stress _{} for each angle _{} of the incline, the
resulting plot will be a circle of radius R and with its center located at _{}. This can be directly shown by examining the equations for _{} and _{}.

This
circle is called Mohr’s circle and is a powerful tool for visualizing the
possible states of stress at a material point. For example, at a glance one can
see that the maximum normal stress at the point is the average stress plus the
radius, the minimum normal stress at the point is the average stress minus the
radius and the maximum shear stress at the point is equal to the radius.

The equations to calculate these stresses are

_{}

where

_{}

As can be seen the maximum and minimum normal stresses
occur on surfaces that have zero shear stress, but the maximum shear stress
occurs on a surface that has a normal stress equal to the average normal
stress.

The convention for positive and negative shear stress on
the Mohr’s circle is given on the axis and in terms of the direction of the
shear stress on the stress element is shown in the following figure.

One
can locate points on Mohr’s circle by locating the point representing the
surface normal to the *x*-axis, which has the coordinates _{}, and then going along the circle an angle of _{} to get to the state
of stress on the surface with an incline angle of _{}.

On
Mohr’s circle, point *X* represent
the surface with normal along the *x*-direction and point *Y*
represents the surface with normal along the *y*-direction. Since these
surfaces are 90^{o} apart on the stress element, on Mohr’s circle they
are 180^{o} apart (i.e., they end up on opposite sides of the
circle).

** **

**Principal stresses:**

Principal
stresses referrer to the maximum and minimum normal stresses. As can be seen on
Mohr’s circle, the principal normal stresses occur on surfaces which have no
shear stress. Also, the maximum shear stress is 90^{o} away from the
maximum normal stress on Mohr’s circle so that it is on a surface oriented 45^{o}
away from the surface on which the maximum normal stress occurs.

**Out of plane stresses:**

Consider
the stress element shown where upon a state of plane stress a normal stress _{} is added in the third
direction. Since this added stress is along a direction perpendicular to the
plane stress, it does not interact with the in-plane stress (the stresses in
the *x-y* plane) and all the results obtained for plane stress still hold.
We can now rotate the stress element around the *z*-axis to get an element
along the principal directions (only normal stresses remain).

Each
two of the resulting directions now represents a simple state of plane stress
for which shear stresses are zero. One can draw the three Mohr’s circles which
result from each pair of directions to get the following Mohr’s circles. One
can show that all states of stress fall between the three circles (the shaded
area).

The
absolute maximum and minimum normal stresses depend on the relative values of
the three principal stresses _{}, _{}, and _{} (and need not be as
shown in the figure). Also the maximum shear stress will equal the radius of
the largest circle.

ã Mehrdad Negahban and the University of Nebraska, 1996-2000.

All rights reserved

Copy and distribute freely for personal use only

Department of Engineering Mechanics, University of Nebraska, Lincoln, NE 68588-0526

Last modified at: 9:42
AM,
Monday, March 13, 2000