Example 6.2: Find the maximum value of  if a state of stress of   and  is applied to the  lamina using a) maximum stress theory b) maximum strain theory and c) Tsai-Hill theory.

Solution:
           
The strain transformation matrix and reduced stiffness matrix are as given above. The stress transformation matrix  is given as

The stresses in principal direction are

The strains in principal direction are

Maximum Stress Theory:

Using the inequalities for this theory, we have

Thus, we see that for  will cause failure as  will exceed the limiting value.

Maximum Strain Theory:

Using the inequalities for this theory, we get

Thus, we see that for  will cause failure as  will exceed the limiting value in tension.

Tsai-Hill Theory:
           
The expression of failure envelope of this theory for planar stresses becomes

This gives . For this value of  the lamina fails by Tsai-Hill theory.