Mohr's Circle for Plane Stress

In engineering design, components fail not necessarily due to the raw forces applied to them, but due to the internal stresses those forces create. Crucially, these stresses change depending on the orientation from which you look at the material element. Mohr's Circle is the powerful graphical technique that transforms the complex mathematics of stress transformation into an intuitive visual map, allowing engineers to instantly identify critical stresses like maximum normal and shear stress, which are essential for predicting failure.

The Foundation: Stress on an Inclined Plane

Consider a 2D square element of material under a state of plane stress. This means stresses act only in the x-y plane, with the stress components being the normal stress in the x-direction (${\sigma }_x$), the normal stress in the y-direction (${\sigma }_y$), and the shear stress (${\tau }_{xy}$). By sign convention, a normal stress is positive when tensile. For shear stress, a positive ${\tau }_{xy}$ acts upward on the right face of the element.

If you were to slice this element at an arbitrary angle $\theta$, the stresses on this new inclined plane (${\sigma }_{x^{\prime }}$ and ${\tau }_{x^{\prime }{y}^{\prime }}$) can be calculated using the stress transformation equations:

$${\sigma }_{x^{\prime }}=\frac{{\sigma }_x+{\sigma }_y}{2}+\frac{{\sigma }_x-{\sigma }_y}{2}\cos 2\theta +{\tau }_{xy}\sin 2\theta$$

$${\tau }_{x^{\prime }{y}^{\prime }}=-\frac{{\sigma }_x-{\sigma }_y}{2}\sin 2\theta +{\tau }_{xy}\cos 2\theta$$

While accurate, these equations are cumbersome for repeated calculations and don't provide an intuitive feel for how stresses vary with orientation. This is the problem Mohr's Circle solves.

Constructing Mohr's Circle: A Step-by-Step Guide

Mohr's Circle is a graphical representation of the transformation equations. It plots normal stress ($\sigma$) on the horizontal axis and shear stress ($\tau$) on the vertical axis. The circle's center and radius are derived directly from the known stress components on the original (x, y) faces.

Step 1: Plot the Reference Points. First, plot two key points using the sign convention for Mohr's Circle, which differs from the physical element convention. On Mohr's Circle, a shear stress that causes a clockwise rotation of the element is plotted as positive.

1. Point X: Coordinates (${\sigma }_x$, ${\tau }_{xy}$).
2. Point Y: Coordinates (${\sigma }_y$, ${\tau }_{yx}$). Remember, ${\tau }_{yx}=-{\tau }_{xy}$ due to equilibrium.

Step 2: Find the Center and Radius. Draw a line between Points X and Y. This line's midpoint intersects the $\sigma$-axis at the circle's center, $C$, located at the average normal stress: $$C={\sigma }_{avg}=\frac{{\sigma }_x+{\sigma }_y}{2}$$ The radius, $R$, of the circle is the distance from the center to either Point X or Point Y. It can be calculated using the geometry of the right triangle formed: $$R=\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$$ Draw the circle with center $C$ and radius $R$.

Interpreting the Circle: Principal and Maximum Shear Stresses

The power of Mohr's Circle becomes clear when extracting key information. The leftmost and rightmost points of the circle on the $\sigma$-axis are where shear stress is zero. These are the principal stresses, denoted ${\sigma }_1$ (maximum) and ${\sigma }_2$ (minimum). Their values are simply the center coordinate plus or minus the radius: $${\sigma }_1={\sigma }_{avg}+R$$ $${\sigma }_2={\sigma }_{avg}-R$$ The angle to rotate from Point X to the ${\sigma }_1$ point on the circle is $2{\theta }_p$, where ${\theta }_p$ is the physical orientation of the principal plane relative to the original x-face.

The top and bottom points of the circle represent the planes of maximum in-plane shear stress, ${\tau }_{max}$. The magnitude is equal to the radius of the circle: $${\tau }_{max}=R$$ At these points, the normal stress is not zero; it is equal to the average stress, ${\sigma }_{avg}$. The orientation of these planes is always 45 degrees away from the principal stress orientations on the physical element.

A Complete Worked Example

Let's apply this to a concrete case. Assume a stress element has ${\sigma }_x=100$ MPa, ${\sigma }_y=40$ MPa, and ${\tau }_{xy}=30$ MPa.

Analytical Method:

1. Calculate average stress: ${\sigma }_{avg}=(100+40)/2=70$ MPa.
2. Calculate radius: $R=\sqrt{((100-40)/2{)}^2+(30{)}^2}=\sqrt{(30{)}^2+(30{)}^2}=42.43$ MPa.
3. Principal Stresses: ${\sigma }_1=70+42.43=112.43$ MPa; ${\sigma }_2=70-42.43=27.57$ MPa.
4. Maximum Shear Stress: ${\tau }_{max}=R=42.43$ MPa.

Graphical Method (Mohr's Circle):

1. Plot Point X: (100, 30). Plot Point Y: (40, -30).
2. Find center at $\sigma =70$, draw circle through X and Y.
3. By inspection, the rightmost $\sigma$-intercept is near 112.4 MPa (${\sigma }_1$) and the top of the circle is near 42.4 MPa (${\tau }_{max}$), confirming our calculations instantly.

Common Pitfalls

1. Mixing Sign Conventions: The most frequent error is using the physical element sign convention when plotting on Mohr's Circle. Remember: on the circle, positive shear stress is plotted upward for a clockwise shear effect on the physical element. Always double-check Point Y's coordinates: it uses ${\tau }_{yx}=-{\tau }_{xy}$.
2. Confusing Angles: A rotation of $\theta$ on the physical element corresponds to a rotation of $2\theta$ on Mohr's Circle, and in the opposite direction. If you need to find stresses on a plane rotated 20° counterclockwise on the element, you rotate 40° clockwise on the circle from your reference point.
3. Forgetting the Third Principal Stress: Mohr's Circle for plane stress gives two principal stresses (${\sigma }_1$, ${\sigma }_2$). In a 3D world, the third principal stress (${\sigma }_3$) is often zero for plane stress conditions. However, the absolute maximum shear stress, critical for failure theories like Tresca, may involve ${\sigma }_3$ and be larger than the in-plane ${\tau }_{max}$ calculated by the circle.
4. Misidentifying Maximum Shear Stress Locations: The points of maximum in-plane shear stress on the circle have a non-zero normal stress (${\sigma }_{avg}$). This is a key distinction from simple shear states. Failure analysis requires knowing both the shear and normal stress on that potential failure plane.

Summary

• Mohr's Circle is a graphical calculator for stress transformation, plotting normal stress ($\sigma$) on the horizontal axis and shear stress ($\tau$) on the vertical axis.
• The circle's center is at the average normal stress, $({\sigma }_x+{\sigma }_y)/2$, and its radius is determined by $R=\sqrt{(({\sigma }_x-{\sigma }_y)/2{)}^2+{\tau }_{xy}^2}$.
• The principal stresses (${\sigma }_1$, ${\sigma }_2$) are the values where the circle intersects the $\sigma$-axis (where shear stress is zero).
• The maximum in-plane shear stress (${\tau }_{max}$) is equal to the radius of the circle and occurs on planes oriented 45° from the principal stress planes.
• Mastering Mohr's Circle requires strict adherence to its unique sign convention for shear stress and understanding the double-angle relationship ($2\theta$) between the physical element and the circle.