Mohr's Circle — Stress Analysis

Mohr's circle is a graph for a two-dimensional stress state. In the usual plane-stress setting, it lets you read the principal stresses, the maximum in-plane shear stress, and the stress on a rotated plane without recomputing the full transformation each time.

Start with the plane-stress components $\sigma_x$, $\sigma_y$, and $\tau_{xy}$. The circle has center

$$C = \left(\frac{\sigma_x + \sigma_y}{2}, 0\right)$$

and radius

$$R = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$$

If all you need is principal stress and maximum in-plane shear stress, these two expressions already give almost everything.

What Mohr's Circle Shows

The horizontal axis is normal stress $\sigma$, and the vertical axis is shear stress $\tau$. Every point on the circle represents the stress state on some plane through the same material point.

The center is the average normal stress. Moving left or right changes the normal stress on the plane, and moving up or down changes the shear stress on that plane.

The two horizontal intercepts are the principal stresses because the shear stress is zero there. The top and bottom points give the maximum in-plane shear stress, and its magnitude is the radius.

Formulas You Read From The Circle

Once you know $C$ and $R$, the main results are

$$\sigma_1 = \frac{\sigma_x + \sigma_y}{2} + R$$ $$\sigma_2 = \frac{\sigma_x + \sigma_y}{2} - R$$ $$\tau_{max,\ in\text{-}plane} = R$$

These results are for the plane-stress picture shown here. If the stress state is fully three-dimensional, the overall maximum shear stress depends on the full set of principal stresses, not just this single circle.

If you also want the principal-plane angle, use the stress-transformation relation

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

when $\sigma_x \ne \sigma_y$ and your sign convention matches the circle you drew.

Worked Example: Find The Principal Stresses

Suppose a point in a plate has

$$\sigma_x = 80\ \mathrm{MPa}, \quad \sigma_y = 20\ \mathrm{MPa}, \quad \tau_{xy} = 30\ \mathrm{MPa}$$

First find the center:

$$C = \left(\frac{80 + 20}{2}, 0\right) = (50, 0)$$

Then find the radius:

$$R = \sqrt{\left(\frac{80 - 20}{2}\right)^2 + 30^2} = \sqrt{30^2 + 30^2} = \sqrt{1800} \approx 42.4\ \mathrm{MPa}$$

Now read the principal stresses:

$$\sigma_1 = 50 + 42.4 = 92.4\ \mathrm{MPa}$$ $$\sigma_2 = 50 - 42.4 = 7.6\ \mathrm{MPa}$$

And the maximum in-plane shear stress is

$$\tau_{max,\ in\text{-}plane} = 42.4\ \mathrm{MPa}$$

So the stress state has one large tensile principal stress, one much smaller tensile principal stress, and a maximum in-plane shear stress of $42.4\ \mathrm{MPa}$. That is the practical payoff of Mohr's circle: one sketch shows the important extremes immediately.

Why The Angle On The Circle Is Doubled

When the physical element rotates by an angle $\theta$, the point on Mohr's circle moves by $2\theta$. That factor of two is why angle questions often feel inconsistent until you remember that the circle uses doubled angles.

The shear sign convention also matters. Different textbooks place positive shear in opposite vertical directions, so the circle may appear reflected. If the convention is used consistently, the principal stress values still agree.

Common Mistakes In Mohr's Circle

Do not use the standard classroom circle blindly. The construction here assumes plane stress, so it is not the complete answer for a general 3D stress state.

Do not confuse the center with one of the original stresses. The center is the average normal stress, so it only matches $\sigma_x$ or $\sigma_y$ in special cases.

Do not mix up the radius and the intercepts. The radius gives the maximum in-plane shear stress, while the principal stresses are $C + R$ and $C - R$.

If you use stress-transformation formulas and a circle sketch together, keep the same sign convention for shear in both places. Otherwise the angle or the plotted points can come out mirrored.

Where Mohr's Circle Is Used

Mohr's circle appears in mechanics of materials, machine design, structural analysis, and failure analysis. It becomes especially useful when a part carries combined loading, such as bending plus torsion or tension plus shear.

Even when software does the arithmetic, the circle still helps you see what is large, what drops to zero, and which planes are most critical.

Try Your Own Version

Set $\tau_{xy} = 0$ or make $\sigma_x = \sigma_y$, then predict the circle before you calculate anything. If you want to go one step further, try your own stress-transformation case with a new set of numbers and check whether the circle matches your intuition.