Mohr's Circle Stress Analyzer

The core of Mohr's Circle is transforming stresses from an (x,y) coordinate system to any rotated angle θ. The circle is constructed from the known stress state.

Here, C is the average normal stress (the circle's center on the σ-axis), and R is the circle's radius, determined by the difference in normal stresses and the shear stress.

Using the center and radius, we can find the principal stresses (maximum and minimum normal stress) and the maximum shear stress.

σ₁ and σ₂ are the principal stresses. They act on planes where the shear stress τ is zero. τ_max is the largest possible shear stress at that point, equal to the circle's radius.

For a plane stress state ($\sigma_x, \sigma_y, \tau_{xy}$), the principal stresses (normal stresses on the planes where shear is zero) and the maximum shear stress are:

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

$\tau_{max} = R = \dfrac{\sigma_1-\sigma_2}{2}, \qquad \tan 2\theta_p = \dfrac{2\tau_{xy}}{\sigma_x-\sigma_y}$

$\sigma_1$ is the maximum and $\sigma_2$ the minimum principal stress, and $\theta_p$ the principal direction. The plane of maximum shear is 45° from the principal planes, and its magnitude equals half the difference of the principal stresses (the radius $R$ of Mohr's circle). This simulator computes $\sigma_1, \sigma_2, \tau_{max}, \theta_p$ instantly from the input stresses.

The stresses on a plane rotated by angle $\theta$ are given by the stress transformation equations.

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

$\tau_\theta = -\dfrac{\sigma_x-\sigma_y}{2}\sin 2\theta + \tau_{xy}\cos 2\theta$

The locus of $(\sigma_\theta, \tau_\theta)$ in the $\sigma$–$\tau$ plane is a circle — Mohr's circle. To draw it:

1. Center: $C=\left(\dfrac{\sigma_x+\sigma_y}{2},\ 0\right)$ on the $\sigma$ axis.
2. Reference points: plot $X(\sigma_x,\ \tau_{xy})$ and $Y(\sigma_y,\ -\tau_{xy})$.
3. Circle: draw the circle with $XY$ as diameter (radius $R$).
4. Principal stresses: intersections with the $\sigma$ axis are $\sigma_1, \sigma_2$; the top of the circle is $\tau_{max}$.
5. The 2θ rule: rotating the physical plane by $\theta$ moves the point by $2\theta$ in the same direction on the circle.

The principal and maximum shear stresses from Mohr's circle feed directly into yield/failure criteria.

Ductile materials yield in shear (use Tresca or von Mises); brittle materials fail under maximum tension (use the maximum principal stress criterion). Mohr's circle extends to 3D ($\sigma_1,\sigma_2,\sigma_3$), where the maximum shear stress is the radius of the largest circle, $(\sigma_1-\sigma_3)/2$.

Structural Engineering & Beam Design: When designing steel I-beams or concrete columns, engineers use Mohr's Circle to find the principal stresses from complex loading. This helps determine if the material will fail in tension, compression, or shear, ensuring the structure can handle real-world forces like heavy loads or wind.

Geotechnical Engineering & Slope Stability: Analyzing soil or rock slopes for landslides involves calculating stresses within the ground. Mohr's Circle is fundamental for determining the shear stress on potential failure planes, which is critical for designing safe embankments, tunnels, and open-pit mines.

Pressure Vessel & Pipeline Design: Tanks holding pressurized gases or liquids experience biaxial stress states (hoop and longitudinal stress). Mohr's Circle is used to find the maximum shear stress, which is key for applying failure theories (like Tresca or von Mises) to select the right wall thickness and material.

Composite Material Analysis: Materials like carbon fiber are made of layers (laminae) oriented in different directions. Mohr's Circle is used to transform stresses from the global structure coordinates to the fiber direction, predicting if a layer will fail under load, which is essential for designing lightweight aircraft and automotive parts.

First, understand that "Mohr's stress circle assumes a two-dimensional (plane stress) state." For example, it's perfect for analyzing plates or thin-walled structures, but to fully represent the complex stress state inside a solid (3D) component, three Mohr's circles are needed. You can think of the circle displayed by this tool as one of them (the circle determined by the maximum and minimum principal stresses).

Next, errors in setting the sign (direction) of shear stress are very common. While the center and radius of the circle remain unchanged by the positive or negative value of $\tau_{xy}$, whether the circle is drawn in the upper or lower half changes, resulting in a 180° shift in the angle (2θ) of the plane on which the principal stresses act. For example, compare $\sigma_x=100$, $\sigma_y=0$, $\tau_{xy}=30$ and $\tau_{xy}=-30$. The principal stress values are the same, but the circle is inverted vertically. In practice, if you don't strictly align your coordinate system definitions, you might reinforce in the wrong direction.

Also, "Von Mises stress cannot be read directly from Mohr's circle" is another key point. This tool calculates and displays both, but Von Mises stress is an equivalent stress based on shear strain energy, a concept that also includes three-dimensional effects. Under a 2D plane stress state, it is calculated from the principal stresses as $\sigma_{vM} = \sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2}$. Mohr's circle is a means to find principal stresses; you cannot plot Von Mises stress itself on the circle.