Examples

In the following examples the strength parameters for various composites used are given (properties from Soden et al [9]). In these examples AS4/3501-6 Epoxy material is used.

Example 6.1: A ply of  fibre orientation is in the planar state of stress. The strains are . Check that whether lamina will fail if a) maximum stress theory b) maximum strain theory and c) Tsai-Hill theory is used.

Solution:
           
The strains given in global direction need to be transformed in principal directions. The state of strain and stress given here is planar. Hence, we need to use the planar transformations. For , we have

and

First, we find the strains and stresses in principal material directions as:

Maximum Stress Theory:
From stresses we see that ,  and . Hence, according to this theory the lamina fails as  has exceeded .

Maximum Strain Theory:
From the strains in principal direction we see that ,   and . Thus, according to this theory the  strain component has exceeded the limiting value . Hence, this lamina fails according to this theory.

Tsai-Hill Theory:
For planar stress the Tsai-Hill theory is

Now we have to check whether left hand side exceeds unity or not. Here, the normal stresses are positive (tensile) hence we use tensile strength parameters in respective mode.

Putting the values of stresses and strength parameters

The failure index is more than unity. Hence, according to this theory lamina will fail. Further, it can be seen that the contribution to failure index due to  is significant compared to other terms. Hence, the major mode of failure is tension in 2-direction of lamina.