5.3 Combined Stresses and Mohr's Circle

Combined stresses occur when multiple types of loading act on a material at the same time. Mohr's circle is a graphical tool for visualizing and analyzing these stress states, making it much easier to find principal stresses and maximum shear stress without relying solely on equations. Mastering these concepts is essential for designing components that survive complex, real-world loading.

This section builds on your earlier stress-strain knowledge. You'll learn how to determine principal stresses, construct Mohr's circle, apply stress transformation equations, and connect everything to failure theories used in actual design decisions.

Principal Stresses and Mohr's Circle

Determining Principal Stresses

At any point in a loaded material, the stress state changes depending on the orientation of the plane you're looking at. Principal stresses are the maximum and minimum normal stresses that exist at that point, and they occur on planes where the shear stress is exactly zero. They're denoted $\sigma_1$ (maximum) and $\sigma_2$ (minimum).

Why do these matter? Because principal stresses represent the extreme values of normal stress at a point. If you're checking whether a material will fail, these are often the numbers you need.

You can find principal stresses two ways:

• Analytically, using the stress transformation equations and solving for the angle $\theta_p$ at which shear stress equals zero:

$\tan 2\theta_p = \frac{2\tau_{xy}}{\sigma_x - \sigma_y}$

• Graphically, by constructing Mohr's circle (covered below), where the principal stresses are simply the points where the circle crosses the horizontal axis.

Once you have $\theta_p$, the principal stress values are:

$\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$

The "+" gives $\sigma_1$ and the "−" gives $\sigma_2$.

Maximum Shear Stress

The maximum in-plane shear stress ($\tau_{max}$) occurs on planes oriented 45° from the principal stress planes. It's calculated as:

$\tau_{max} = \frac{\sigma_1 - \sigma_2}{2}$

Notice this is also equal to the radius of Mohr's circle. On the planes of maximum shear stress, the normal stress is not zero; it equals the average normal stress:

$\sigma_{avg} = \frac{\sigma_1 + \sigma_2}{2}$

This is a detail students often miss on exams. The planes of max shear carry both shear and normal stress.

Understanding $\tau_{max}$ is critical for components subjected to torsion or direct shear, such as shafts, gears, and bolted joints.

Mohr's Circle for Plane Stress

Mohr's circle is a graphical representation of the stress state at a point. It lets you read off the normal and shear stresses on any oriented plane, and it makes the relationship between those stresses intuitive.

How to construct Mohr's circle (step by step):

1. Establish your known stresses: $\sigma_x$, $\sigma_y$, and $\tau_{xy}$ from the given loading.

2. Plot the center of the circle on the $\sigma$-axis at: $C = \left(\frac{\sigma_x + \sigma_y}{2},\ 0\right)$

3. Calculate the radius: $R = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$

4. Plot two reference points:

   • Point $X$: $(\sigma_x,\ \tau_{xy})$ representing the x-face of the element
   • Point $Y$: $(\sigma_y,\ -\tau_{xy})$ representing the y-face

5. Draw the circle through these points with center $C$ and radius $R$.

6. Read off results:

   • Principal stresses $\sigma_1$ and $\sigma_2$ are where the circle intersects the $\sigma$-axis.
   • $\tau_{max}$ is the top (and bottom) of the circle.
   • Stresses on any rotated plane at angle $\theta$ correspond to rotating $2\theta$ around the circle.

The transformation equations that Mohr's circle represents graphically are:

• $\sigma_{\theta} = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2}\cos 2\theta + \tau_{xy}\sin 2\theta$
• $\tau_{\theta} = -\frac{\sigma_x - \sigma_y}{2}\sin 2\theta + \tau_{xy}\cos 2\theta$

A key thing to remember: angles on Mohr's circle are doubled. A 45° physical rotation corresponds to a 90° rotation on the circle. That's why the principal stress planes and maximum shear stress planes, which are 45° apart physically, appear 90° apart on the circle.

Stress Transformation and Failure Theories

Stress Transformation

Stress transformation is the process of determining the stress components on a plane oriented at some angle $\theta$ relative to your original coordinate system. You need this whenever you're analyzing stresses on inclined planes, weld lines, joints, or any feature that doesn't align with your reference axes.

The full set of transformation equations:

• $\sigma_{x'} = \sigma_x\cos^2\theta + \sigma_y\sin^2\theta + 2\tau_{xy}\sin\theta\cos\theta$
• $\sigma_{y'} = \sigma_x\sin^2\theta + \sigma_y\cos^2\theta - 2\tau_{xy}\sin\theta\cos\theta$
• $\tau_{x'y'} = (\sigma_y - \sigma_x)\sin\theta\cos\theta + \tau_{xy}(\cos^2\theta - \sin^2\theta)$

These can also be written using double-angle identities, which is exactly what gives you the Mohr's circle form. The two representations are mathematically identical; Mohr's circle just makes the relationships visual.

A useful check: $\sigma_{x'} + \sigma_{y'} = \sigma_x + \sigma_y$. The sum of normal stresses is invariant under rotation. If your transformed stresses don't satisfy this, you've made an error somewhere.

Von Mises Stress and Failure Theories

Once you know the stress state at a critical point, the next question is: will the material fail? That's where failure theories come in. Different theories apply to different material types.

Von Mises Stress ($\sigma_{VM}$) combines all stress components into a single equivalent scalar value. For plane stress (where $\sigma_3 = 0$):

$\sigma_{VM} = \sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2}$

The physical idea behind Von Mises stress is that it represents the distortion energy in the material. Yielding occurs when this distortion energy reaches the same level it would under a simple uniaxial tension test. That's why you compare $\sigma_{VM}$ directly to the uniaxial yield strength $S_y$.

The three main failure theories for static loading:

How to choose:

• For ductile materials (most metals, many polymers), use Von Mises or Tresca. Von Mises is more accurate and less conservative. Tresca is simpler and gives a slightly more conservative prediction, which can be useful as a quick check.
• For brittle materials (ceramics, cast iron, glass), use the Maximum Normal Stress theory, since brittle materials fail by fracture rather than yielding, and fracture is governed by the largest tensile stress.