# THEORIES OF FAILURE: Maximum principal (or normal) stress theory (also known as Rankine’s theory), Maximum shear stress theory (also known as Guest’s or Tresca’s theory), Maximum principal (or normal) strain theory (also known as Saint Venant theory), Maximum strain energy theory (also known as Haigh’s theory), Maximum distortion energy theory (also known as Hencky and Von Mises theory)

It has already been discussed in the previous chapter that 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 tests, therefore, predicting failure in members subjected to uniaxial 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 biaxial stress are as follows:

Maximum principal (or normal) stress theory (also known as Rankine’s theory).

Maximum shear stress theory (also known as Guest’s or Tresca’s theory).

Maximum principal (or normal) strain theory (also known as Saint Venant theory).

Maximum strain energy theory (also known as Haigh’s theory).

Maximum distortion energy theory (also known as Hencky and Von Mises theory).

Since ductile materials usually fail by yielding i.e. when permanent deformations occur in the material and brittle materials fail by fracture, therefore the limiting strength for these two classes of materials is normally measured by different mechanical properties. For ductile materials, the limiting strength is the stress at yield point as determined from simple tension test and it is, assumed to be equal in tension or compression. For brittle materials, the limiting strength is the ultimate stress in tension or compression.

## 1. Maximum principal stress theory (Rankine’s theory)

According this theory 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. Since the limiting strength for ductile materials is yield point stress and for brittle materials (which do not have well defined yield point) the limiting strength is ultimate stress, therefore according to the above theory, taking factor of safety (F.S.) into consideration, the maximum principal or normal stress in a bi-axial stress system is given by

Since the maximum principal or normal stress theory is based on failure in tension or compression and ignores the possibility of failure due to shearing stress, therefore it is not used for ductile materials. However, for brittle materials which are relatively strong in shear but weak in tension or compression, this theory is generally used.

## 2. Maximum shear stress theory (Guest’s or Tresca’s theory)

According to this 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. Mathematically,

## 3. Maximum principal strain theory (Saint Venant theory)

According to this 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 (i.e. strain at yield point) as determined from a simple tensile test. The maximum principal (or normal) strain in a bi-axial stress system is given by

## 4. Maximum strain energy theory (Haigh’s theory)

According to this 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 (i.e. strain energy at the yield point ) per unit volume as determined from simple tension test.

We know that strain energy per unit volume in a bi-axial stress system,

## 5. Maximum distortion energy theory (also known as Hencky and Von Mises theory)

According to this theory, the failure or yielding occurs at a point in a member when the distortion strain energy (also called shear strain energy) per unit volume in a bi-axial stress system reaches the limiting distortion energy (i.e. distortion energy at yield point) per unit volume as determined from a simple tension test. Mathematically, the maximum distortion energy theory for yielding is expressed as