Maximum Principal Stress or Tresca Criterion

The maximum principal stress criterion for failure simply states that failure (by yielding or by fracture) would occur when the maximum principal stress reaches a critical value (ie., the material s yield strength, ays, or fracture strength, a/, or tensile strength, aurs)- For a three-dimensional state of stress, given in terms of the Cartesian coordinates x, y, and z in Fig. 2.1 and represented by the left-hand matrix in Eqn. (2.1), a set of principal stresses (see Fig. 2.1) can be readily obtained by transformation  

Assume that the largest principal stress is ai, the failure criterion is then given by Eqn. (2.2). 

It is recognized that failure can also occur under compression. In that case, the strength properties in Eqn. (2.2) need to be replaced by the suitable ones for compression. 

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