Learning topic
Stress Analysis at a Point (Principal Stresses)
How to calculate principal stresses and use Mohr's Circle for 2D/3D stress analysis.
Stress components at a point depend on the orientation of the plane, while the physical stress state itself remains unchanged. Stress analysis determines the normal and shear stresses on rotated planes and identifies their extreme values.
Principal planes and principal stresses
Principal planes are planes on which shear stress is zero. The normal stresses acting on them are the principal stresses. For plane stress, they can be calculated directly from $\sigma_x$, $\sigma_y$, and $\tau_{xy}$.
For a plane stress state defined by $\sigma_x$, $\sigma_y$, and $\tau_{xy}$, the principal stresses are:
$$\sigma_{1,2}=\frac{\sigma_x+\sigma_y}{2}\pm\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}.$$
Shear stress is zero on the principal planes. Their orientation can be determined from:
$$\tan 2\theta_p=\frac{2\tau_{xy}}{\sigma_x-\sigma_y},$$
with the quadrant and sign convention handled consistently.
- $\sigma_1$, $\sigma_2$ — principal normal stresses;
- $\theta_p$ — orientation angle of a principal plane;
- $\sigma_x$, $\sigma_y$, $\tau_{xy}$ — components of the original plane stress state.
Stress transformation
When the coordinate axes are rotated through an angle $\theta$, the components of plane stress transform according to trigonometric relations. With a commonly used sign convention:
$$\sigma_{x'}=\frac{\sigma_x+\sigma_y}{2}+\frac{\sigma_x-\sigma_y}{2}\cos2\theta+\tau_{xy}\sin2\theta,$$
$$\tau_{x'y'}=-\frac{\sigma_x-\sigma_y}{2}\sin2\theta+\tau_{xy}\cos2\theta.$$
The sign of the shear component must always be interpreted consistently with the selected convention.
Mohr's circle
For a plane stress state, the center and radius of Mohr's circle are:
$$C=\frac{\sigma_x+\sigma_y}{2},\qquad R=\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}.$$
The principal stresses are:
$$\sigma_1=C+R,\qquad \sigma_2=C-R,$$
and the maximum absolute in-plane shear stress is:
$$|\tau_{\max,\,\mathrm{in\text{-}plane}}|=R.$$
A physical rotation of a plane by $\theta$ corresponds to a rotation of the radius on Mohr's circle by $2\theta$. The direction of rotation depends on the adopted sign convention.
Mohr's circle is a graphical representation of the same transformation equations. It provides a convenient way to identify principal stresses, maximum shear stresses, and the stresses acting on a plane of specified orientation.
Maximum shear stress
In the plane representation, the radius of Mohr's circle equals the maximum absolute shear stress among planes whose normals lie in the $x$–$y$ plane. For a complete three-dimensional assessment, all three principal stresses must be considered; the absolute maximum shear stress equals half the largest difference between them.
Example
Let $\sigma_x=80\ \text{MPa}$, $\sigma_y=20\ \text{MPa}$, and $\tau_{xy}=30\ \text{MPa}$. The circle center is $C=50\ \text{MPa}$ and its radius is:
$$R=\sqrt{30^2+30^2}\approx42.43\ \text{MPa}.$$
Therefore, $\sigma_1\approx92.43\ \text{MPa}$, $\sigma_2\approx7.57\ \text{MPa}$, and the maximum in-plane shear stress is approximately $42.43\ \text{MPa}$.
Verification
The sum of the two principal stresses must equal $\sigma_x+\sigma_y$, and their average must equal the center coordinate of Mohr's circle. These relations provide useful checks on the calculation.
About this topic
Determining extreme values of normal and shear stresses is a critical step in structural strength evaluation. This section presents analytical formulas for principal stress calculations as well as Mohr's Circle—a graphical method for stress state transformation. Mohr's Circle provides visual insight into stress variation relative to plane inclination angles, simplifying peak shear stress determination.