Mohr's Circle Calculator - Principal Stresses and Stress Transformation
Enter the normal stresses and shear stress at a point in a loaded material to get the principal stresses, maximum in-plane shear stress, von Mises stress, mean stress, and the orientation angle of the principal planes. You also get the transformed stresses on a rotated element at any angle you choose. All results update instantly as you type.
What is Mohr's Circle?
Mohr's Circle is a graphical representation of the two-dimensional stress state at a point in a loaded body. Developed by the German civil engineer Christian Otto Mohr in 1882, it maps all possible normal and shear stress combinations on differently oriented planes through that point onto a circle in the (sigma, tau) plane. The horizontal axis represents normal stress and the vertical axis represents shear stress. The center of the circle sits at the mean (hydrostatic) stress, and the radius equals the maximum in-plane shear stress. Reading off the circle at a given angle gives the stress components on any rotated plane, making stress transformation visual and intuitive rather than purely algebraic.
How to read the inputs and outputs
Enter the three independent stress components that describe the 2D stress state at the point of interest: sigma_x (normal stress on the vertical face), sigma_y (normal stress on the horizontal face), and tau_xy (in-plane shear stress). Tensile normal stresses are positive; compressive stresses are negative. The shear stress sign follows the standard mechanics-of-materials convention: positive tau_xy acts upward on the right face. The calculator returns the maximum principal stress (sigma1), the minimum principal stress (sigma2), the maximum in-plane shear stress (tau_max), the mean stress, the von Mises stress, and the angles of the principal and maximum-shear planes. The rotation angle input lets you find the stress components on any rotated element.
Principal stresses and why they matter
Principal stresses are the normal stresses that act on planes where the shear stress is exactly zero. Every stress state has two such planes (in 2D), oriented at right angles to each other. The larger principal stress is sigma1 and the smaller is sigma2. These are the extreme values of normal stress at the point, so they govern failure by fracture in brittle materials (maximum normal stress criterion) and are also the inputs to the von Mises and Tresca yield criteria for ductile materials. The angle theta_p between the x-axis and the sigma1 plane is found from arctan(2*tau_xy / (sigma_x - sigma_y)) / 2. The maximum in-plane shear stress equals the radius of the circle and acts on planes inclined 45 degrees from the principal planes.
Von Mises stress and yield criteria
The von Mises (equivalent or effective) stress combines the principal stresses into a single scalar that predicts yielding in ductile metals. It equals sqrt(sigma1^2 - sigma1*sigma2 + sigma2^2) in plane stress. A material yields when this value reaches its uniaxial yield strength (Sy). The von Mises criterion is preferred for ductile materials like steel and aluminium because it better matches experimental yield data than the simpler maximum-shear-stress (Tresca) criterion. To use the result: if the von Mises stress is below the yield strength, the material remains elastic; if it equals or exceeds the yield strength, yielding begins. For a safety factor, divide the yield strength by the von Mises stress.
Stress state classifications using principal stresses
| Stress state | Condition | Typical application |
|---|---|---|
| Uniaxial tension | sigma2 = 0, sigma1 > 0 | Tensile rod, axially loaded bar |
| Uniaxial compression | sigma1 = 0, sigma2 < 0 | Column, compressed strut |
| Biaxial tension | sigma1 > 0, sigma2 > 0 | Pressurized thin-walled vessel |
| Biaxial compression | sigma1 < 0, sigma2 < 0 | Concrete under 2D load |
| Pure shear | sigma1 = -sigma2, sigmaM = 0 | Torsion of circular shaft |
| General plane stress | sigma1 and sigma2 any sign | Combined bending and torsion |
Common stress state types encountered in structural and mechanical engineering.
Frequently asked questions
What units does the Mohr's circle calculator use?
The calculator works in whatever stress unit you enter. The default is MPa (megapascals), which is the standard SI unit for stress in structural and mechanical engineering. If you enter values in psi, kPa, ksi, or any other unit, all outputs will be in that same unit because the formulas are dimensionally consistent. Just make sure all three input stresses use the same unit.
What is the sign convention for shear stress?
This calculator uses the standard mechanics-of-materials (Hibbeler) convention: tau_xy is positive when it acts upward on the positive X face (right face) of the element, and downward on the positive Y face (top face). This is also called the mathematical positive convention. Some structural analysis textbooks use the opposite sign for one face, so check which convention your course or software uses before comparing results.
How is the principal plane angle theta_p defined?
Theta_p is the counter-clockwise angle, measured in degrees, from the original x-axis to the plane on which the maximum principal stress (sigma1) acts. The formula is theta_p = 0.5 * arctan(tau_xy / ((sigma_x - sigma_y)/2)). Because arctan has a period of 180 degrees, the two principal planes are always 90 degrees apart. If theta_p is 25 degrees, then sigma1 acts on a plane 25 degrees counter-clockwise from the x-face, and sigma2 acts on the perpendicular plane at 115 degrees.
What does a Mohr's circle of zero radius mean?
A circle with radius zero means sigma_x = sigma_y and tau_xy = 0. All planes through that point experience the same normal stress and zero shear stress. This is called a hydrostatic (or isotropic) stress state. Physically, this occurs at the centre of a uniformly pressurized sphere, or under equal biaxial loading. The principal stresses are both equal to sigma_x, and there is no preferred principal direction because every direction is a principal direction.
What is the difference between in-plane and absolute maximum shear stress?
The in-plane maximum shear stress (tau_max) is the radius of the 2D Mohr's circle and acts on planes tilted 45 degrees from the principal planes, all within the plane of the stress state. In a true 3D analysis, there is also an out-of-plane shear stress to consider: the absolute maximum shear stress is half the difference between the largest and smallest of the three principal stresses (sigma1, sigma2, and zero for plane stress). If sigma1 and sigma2 have opposite signs, tau_abs_max = (sigma1 - sigma2)/2 = R, which equals the in-plane value. If both principal stresses are the same sign, tau_abs_max = sigma1/2, which is larger than R. For ductile materials, always check the Tresca criterion against the absolute maximum shear stress.
How do I use these results to check if a part will yield?
Compare the von Mises stress to the material yield strength Sy. If von Mises < Sy, the part stays elastic. The safety factor is n = Sy / (von Mises stress). For the Tresca criterion instead, compare 2 * tau_max (the in-plane maximum shear stress) to Sy: if 2 * tau_max >= Sy, the material yields. For brittle materials, apply the maximum normal stress criterion: if sigma1 >= Su (ultimate tensile strength) or |sigma2| >= Suc (ultimate compressive strength), fracture occurs.