# Maximum principal stress theory of failure with equations

Sunday, February 5, 2012 8:57**Maximum principal stress theory :**

** This theory states that “**A structural component will fail when maximum principal stress of the system will become equal to the yield strength of same material in a simple tension test.”

**3-d equations for ductile materials:**

σ_{1}=σ_{yt}

σ_{3}=σ_{yc}

**2-d equations for ductile materials:**

σ_{1}=σ_{yt}

σ_{2}=σ_{yc}

**3-d Equation for brittle materials:**

σ_{1}= σ _{ut }

σ_{3}= σ _{uc}

**2-d equation for brittle materials**:

σ_{1}= σ _{ut }

σ_{2}= σ _{uc}

where

- σ
_{1 }= Major principal stress in x direction. - σ
_{2 }= Minor principal stress in y direction. - σ
_{3 }= Minor principal stress in z direction. - σ
_{yt}= Maximum Tensile strength in yielding. - σ
_{yc }=Maximum Compressive strength in yielding. - σ
_{ut}= Ultimate tensile strength. - σ
_{ut }= Ultimate compressive strength.

This theory gives satisfactory results for brittle materials because brittle materials fails in tension. It gives unsatisfactory results for ductile materials because ductile materials fail in maximum shear stress.

This hypothesis, proposed by Rankine, which was also intended for use to predict yielding of a ductile material.

**Merits :**

- Calculations are easy.
- Gives satisfactory results for brittle materials.

**De-merits **:

- It computes failure load based only on values of principal stress.
- Gives unsatisfactory results for ductile materials.