### Theories of Failure under Static Load

We know the strength of machine members is based upon the mechanical properties of the materials used. Since these properties are usually determined from simple tension or compression test, therefore, predicting failure in members subjected to uni-axial stress is both simple and straight forward. But the problem of predicting the failure stresses for members subjected to bi-axial or tri-axial stresses is much more complicated. In fact, the problem is so complicated that a large number of different theories have been formulated. The principal theories of failure for a member subjected to bi-axial stress are as follows.

## Types of Theories of Failure under Static Load are

- Maximum principal or normal stress theory (Rankine’s Theory)
- Maximum shear stress theory (Guest’s theory)
- Maximum Principal strain theory (Saint Venant’s theory)
- Maximum strain energy theory (Haigh’s theory)
- Maximum distortion energy theory (Hecky and Von Mises theory)

**Maximum principal or normal stress theory (Rankine’s Theory)**: According to Maximum principal stress theory or normal stress theory or Rankine’s theory, the failure or yielding occurs at a point in a member when the maximum principal or normal stress in a bi-axial stress system reaches the limiting strength of the material in a simple tension test.**Maximum shear stress theory (Guest’s theory)**: According to Maximum shear stress theory or Guest’s Theory, the failure or yielding occurs at a point in a member when the maximum shear stress in a bi-axial stress system reaches a value equal to the shear stress at yield point in a simple tension test.**Maximum Principal strain theory (Saint Venant’s theory)**: According to Maximum Principal strain theory or Saint Venant’s theory , the failure or yielding occurs at a point in a member when the maximum principal or normal strain in a bi-axial stress system reaches the limiting value of strain as determined from a simple tension test.**Maximum strain energy theory (Haigh’s theory)**: According to Maximum strain energy theory or Haigh’s theory , the failure or yielding occurs at a point in a member when the strain energy per unit volume in a bi-axial stress system reaches the limiting strain energy per unit volume as determined from a simple tension test.**Maximum distortion energy theory (Hecky and Von Mises theory)**: According to Maximum distortion energy theory or Hecky and Von Mises theory , the failure or yielding occurs at a point in a member when the distortion strain energy (Shear strain energy) per unit volume in a bi-axial stress system reaches the limiting distortion energy per unit volume as determined from a simple tension test.