Mohr's Circle

Mohr’s circle provides a graphical description of stress at a single point in a structure; each location on the circle represents the stress state on a specific surface point on the structure.

Conventions:

• If the shear stress tends to rotate the element clockwise, it’s plotted as positive on Mohr’s circle.
• If the shear stress tends to rotate the element counterclockwise, it’s plotted as negative.

We saw that principal stress, average stress, and extreme-value shear stresses can be found with plane stress transformation. Specifically, we had:

$$\begin{array}{rl}\sigma & =\frac{{\sigma }_x+{\sigma }_y}{2}+\frac{{\sigma }_x-{\sigma }_y}{2}\cos 2\varphi +{\tau }_{xy}\sin 2\varphi \\ \tau & =-\frac{{\sigma }_x-{\sigma }_y}{2}\sin 2\varphi +{\tau }_{xy}\cos (2\varphi )\end{array}$$

Mohr’s circle expresses these ideas graphically, with a circle defined by:

$${\left(\sigma -\frac{{\sigma }_x+{\sigma }_y}{2}\right)}^2+{\tau }^2={\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2$$

The center of the circle is given by:

$$C=\left[\frac{{\sigma }_x+{\sigma }_y}{2},0\right]$$

The radius is given by:

$$R=\sqrt{\left[\frac{({\sigma }_x-{\sigma }_y{)}^2}{2}\right]+{\tau }_{xy}^2}={\tau }_{\text{max}}$$

The principal stresses can then be easily obtained with:

$${\sigma }_{\text{max, min}}={\sigma }_{\text{avg}}\pm R$$

Procedure

1. Locate the center $C=\left[\frac{({\sigma }_x+{\sigma }_y)}{2},0\right]$.
2. Locate face $A$ at $({\sigma }_x,-{\tau }_{xy})$.
3. Locate face $B$ at $({\sigma }_y,{\tau }_{xy})$.
4. Connects $A,B,C$ to form the circle diameter and draw the circle around it.
5. Use geometry to compute the principal stresses, extreme-value shear stresses, principle angles, etc.