Mohr Circle Calculator
This Mohr circle calculator finds the principal stresses, the maximum shear stress and the orientation of the principal planes for a two-dimensional state of stress. Mohr's circle is the algebraic device engineers use to transform stress between axes and to read off the extreme values that govern failure. For a plane element loaded by normal stresses in two perpendicular directions and a shear stress, the center of the circle sits at the average of the two normal stresses, and its radius is the root of the squared half-difference plus the squared shear. The principal stresses are the center plus and minus the radius, the maximum shear equals the radius, and the principal angle locates the rotation that removes shear. These relations underpin the stress analysis behind structural and vehicle safety work at agencies such as the US National Highway Traffic Safety Administration. Enter the two normal stresses and the shear stress, and the calculator returns the center, radius, both principal stresses, the maximum shear and the principal angle. Every figure is computed deterministically from the Mohr's circle formulas shown in full below, with a worked example that reconciles exactly to the calculator so you can follow each step.
Mohr's circle gives the extremes: for normal stresses 100 and 40 with shear 30, the principal stresses are 112.43 and 27.57, the maximum shear is 42.43, and the principal angle is 22.50 degrees. The circle's center is the average stress.
Mohr's circle formulas
center = ( sigma_x + sigma_y ) / 2
radius = sqrt( ( ( sigma_x - sigma_y ) / 2 )^2 + tau_xy^2 )
sigma_1, sigma_2 = center +/- radius
tau_max = radius, theta_p = 0.5 x atan2( 2 tau_xy, sigma_x - sigma_y )
The center is the average normal stress, the point about which the circle is drawn. The radius captures the combined effect of the difference in normal stresses and the shear. The principal stresses are the ends of the horizontal diameter, the maximum shear is the radius, and the principal angle rotates the element so shear vanishes.
Worked example
Find the principal stresses for sigma_x = 100, sigma_y = 40 and tau_xy = 30.
- center = (100 + 40) / 2 = 70.00
- radius = sqrt(((100 - 40)/2)^2 + 30^2) = sqrt(30^2 + 30^2) = sqrt(1,800) = 42.43
- sigma_1 = 70.00 + 42.43 = 112.43; sigma_2 = 70.00 - 42.43 = 27.57
- tau_max = 42.43; theta_p = 0.5 x atan2(60, 60) = 22.50 deg
The principal stresses are 112.43 and 27.57, the maximum shear is 42.43, and the principal angle is 22.50 degrees. These are the calculator's default inputs, so the result above matches the widget exactly.
Key Mohr's circle quantities
Each output is read directly from the geometry of the circle.
| Quantity | Meaning | Value |
|---|---|---|
| Center | Average normal stress | 70.00 |
| Radius | Half the principal stress range | 42.43 |
| sigma_1 | Maximum principal stress | 112.43 |
| sigma_2 | Minimum principal stress | 27.57 |
Stress analysis and transportation safety context: US National Highway Traffic Safety Administration (NHTSA).
Mohr Circle Calculator: frequently asked questions
What are principal stresses?
Principal stresses are the maximum and minimum normal stresses acting on an element, found on the planes where the shear stress is zero. They are the extreme values among all possible orientations of the element, so they are what you compare against a material's strength. On Mohr's circle they are the two points where the circle crosses the normal-stress axis.
What is the maximum shear stress?
The maximum in-plane shear stress equals the radius of Mohr's circle, the largest shear that occurs on any plane through the element. It acts on planes oriented 45 degrees from the principal planes. Ductile materials often fail in shear, so this value drives the maximum-shear-stress yield criterion.
What does the principal angle tell me?
The principal angle is the rotation from the original x-axis to the plane carrying the maximum principal stress. The calculator reports half the arctangent of twice the shear over the difference in normal stresses, which is the angle that turns the element so the shear vanishes. Adding 45 degrees gives the orientation of maximum shear.
Does sign matter for the shear input?
Yes. The sign of the shear stress follows your chosen convention and affects the principal angle, though the magnitudes of the principal stresses and the maximum shear are unchanged because they depend on the square of the shear. Keep a consistent sign convention so the reported angle matches your element diagram.
Is this plane stress or plane strain?
The formulas give the in-plane principal stresses for a two-dimensional stress state, which is the plane-stress transformation. For a full three-dimensional assessment you would also consider the out-of-plane principal stress, which can change the absolute maximum shear. This tool reports the in-plane results.
Official sources
- Vehicle and transportation structural safety reference: US National Highway Traffic Safety Administration (NHTSA). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.