Maximum Normal Stress Theory

The maximum-normal-stress (MNS) theory for brittle materials states that failure occurs whenever one of the three principal stresses equals or exceeds the strength.

General Case

In the general 3D case, we have three principal stresses ${\sigma }_1\ge {\sigma }_2\ge {\sigma }_3$. Fracture occurs when:

$${\sigma }_1\ge S_{ut}\text{or}{\sigma }_3\le -S_{uc}$$

For design purposes with a factor of safety $n$, we use:

$${\sigma }_1=\frac{S_{ut}}{n}\text{or}{\sigma }_3=-\frac{S_{uc}}{n}$$

Plane Stress

For plane stress, we have two principal stresses in the plane of analysis, such that ${\sigma }_A\ge {\sigma }_B$. The third principal stress is zero. Fracture occurs when:

$${\sigma }_A\ge S_{ut}\text{or}{\sigma }_B\le -S_{uc}$$

When designing with a factor of safety, we write:

$${\sigma }_A=\frac{S_{ut}}{n}\text{or}-\frac{S_{u\mathring{a}c}}{n}$$