MAXIMUM SHEAR STRESS THEORY OF FAILURE
A THEORY OF FAILURE APPLICABLE TO DUCTILE MATERIALS
Statement of the theory
When Yielding occurs in any material, the maximum shear stress at the point of failure
equals or exceeds the maximum shear stress when yielding occurs in the tension test
specimen.
The theory applies to ductile materials only, because it is based on yielding.
The three-dimensional (triaxial) stress situation.
In the three-dimensional stress situation, the state of stress at a particular location is fully
defined by three principal stresses  1 ,  2 ,  3 .
Maximum shear stress at a location of the element
The extreme values of shear stresses 12 , 13 ,  23 , in each of the three principal planes are
then given by the expressions:
 12 
1   2
 13 
,
1   3
,
 23 
 2  3
2
2
2
Expressing the principal stresses in the order of magnitude and sign  1   2   3
  3
Then the maximum shear stress is given by  13  1
2
THE CASE OF SIMPLE TENSION TEST WHEN YIELDING OCCURS
For the simple tension test specimen, the three principal stresses when yielding occurs are:
 3 =0
1 = S y ,
 2 =0,
 3 Sy  0 Sy


The maximum shear stress then becomes  13   max  1
2
2
2
THE CASE OF THREE DIMENSIONAL STRESS WHEN YIELDING OCCURS
The maximum shear stress theory of failure states:
When Yielding occurs in any material, the maximum shear stress at the point of failure
equals or exceeds the maximum shear stress when yielding occurs in the tension test
specimen.
 max 
1   3
2

Sy
2
University of Nairobi
Engineering Design II
The above equation implies that the shear yield strength of the material S sy 
Sy
2
But from analysis of plane stress situation, the maximum shear stress in plane stress is also
given in terms of plane stress elements
2
 x  y 
   xy 2
 max   

2


DESIGN EQUATION BASED ON THE MAXIMUM SHEAR STRESS THEORY
This is derived by adjusting the shear yield strength of the material with an appropriate factor
of safety f .s. . The design equation then becomes:
 max 
1   3
2

S sy
f .s.

Sy
2 * f .s.
OR
2
S
Sy
  x  y 
   xy 2  sy 
for plane stress situation
 max   

2
f .s. 2 * f .s.


APPLICATION OF THE DESIGN EQUATION
The principal stresses  1 ,  2 ,  3 are first determined by stress analysis. Such analysis
describes the principal stresses as a function of the load carried, and the geometry and
dimensions of the machine or structural element.
The maximum shear stress in the design equation is then expressed in terms of the
dimensions of the machine or structural element, while the right hand side is a function of
the tensile yield strength of the material. The tensile yield strength of the material becomes
useful because it more easily determined from laboratory experiments.
The factor of safety is simply a number chosen by the designer. The factor of safety together
with the strength of the material, gives the working1 (design, allowable) stress expected in the
machine part. The solution to the design equation then gives the minimum dimensions
required to avoid failure of the element by yielding.
Working Stress, page 527,Handbook, Metals Engineering –Design, American Society of Mechanical
Engineers (ASME)
1
Nyangasi
Page 2 of 2
30 January 2005