• Practically the mechanical components are subjected to several types of loads simultaneously. When the component is subjected to several types of loads, combined stresses are induced.
• The design of machine parts subjected to these loads should be related to experimentally determined properties of material under similar conditions. But it's impossible to determine the mechanical properties for all possible combination of loads. Actually, the mechanical properties are determined by simple tension test. With this information, the designer is expected to design components subjected to bi-axial or tri-axial stresses under different condition of loads.

• So, the relationship between the strength of a mechanical component -subjected to a complex state of stresses and the mechanical properties of simple tension test is obtained by the theories of failure. By using these theories the dimensions of the component are determined, irrespective of the nature of the stresses induced in the component due to the complex load.

• The different theories of failure are as follows:

  I. Maximum principal stress theory (Rankine’s theory.)

  II. Maximum shear stress theory (Coulomb, Tresca & Guest’s theory)

  III. Distortion Energy Theory (Huber or Von Mises & Heneky's theory)

  IV. Maximum strain theory (St. Venant's theory)

  V. Maximum total strain energy theory (Haigh's theory)

1. Maximum Principal stress theory

The theory states that the failure of the mechanical component, subjected to bi-axial or tri-axial stresses, occurs when the maximum principal stress reaches the yield or ultimate strength of the material.

If $\sigma_1,\sigma_2,\sigma_3$ - Three principal stresses

& $\sigma_1\gt\sigma_2\gt\sigma_3$

then according to this theory, the failure occurs whenever

$\sigma_1 = S_{yt} \ \text{or} \ \sigma_1=S_{ut}$

The dimensions of the component - are determined by using factor of safety.

$\therefore \sigma_1 = \frac{S_{yt}}{fs}$ or $\sigma_1 = \frac{S_{ut}}{fs}$

• It is used for brittle materials.

2. Maximum Shear stress theory:

The theory states that the failure of a mechanical component subjected to bi-axial or tri-axial stresses occurs when the maximum shear stress at any point in the component becomes equal to the maximum shear stress in the standard specimen of the simple tension test when yielding starts.

$\therefore$ According to this theory, $S_{sy}=0.5 \ S_{yt}$

$\therefore$ Dimensions of the component are determined by:

$\frac{\sigma_1-\sigma_2}{2}=\frac{S_{yt}}{2(fs)}$

OR $\sigma_1-\sigma_2 = \frac{S_{yt}}{(fs)}$

It is used for ductile materials.

3. Distortion Energy Theory:

The theory states that the failure of the mechanical component subjected to bi-axial or tri-axial stresses occur when the strain energy of distortion per unit volume at any point in the component becomes equal to the strain energy of distortion per unit volume in a standard tension test specimen when yielding starts.

According to this theory, for tri-axial stresses $\frac{S_{yt}}{(fs)}=\sqrt{\frac{1}{2}[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2]}$

For bi-axial stresses,

$\frac{S_{yt}}{(fs)}=\sqrt{(\sigma_1^2-\sigma_1\sigma_2+\sigma_2^2)}$

and

$S_{sy}=0.577 \ S_{yt}$

It is used for ductile materials.