Mohr Circle - 2D Principal Stress Solver

A mohr circle calculator turns three plane-stress components (sigma x, sigma y, and tau xy) into the geometric picture that drives every 2D stress transformation. Plot the input state on a normal-shear diagram, draw a circle through it, and the principal stresses, max in-plane shear, and principal angle drop out of the geometry. Use this mohr circle calculator to check homework, verify a hand solution, or scan a parameter study in seconds.

• • Find principal stresses for a 2D stress element: Enter sigma x, sigma y, and tau xy and read off sigma 1, sigma 2, the principal angle, and the maximum in-plane shear.
• • Resolve stress on an inclined plane: Set theta to a weld, fiber, or fatigue plane orientation and read the normal and shear stresses directly from the output panel.
• • Compute the 2D von Mises stress: Use the in-plane principal stresses for a yield check in thin plates, gaskets, and pressure-vessel walls.
• • Compare absolute 3D maximum shear: Provide an out-of-plane sigma 3 to evaluate the absolute maximum shear across all three principal stresses.

Mohr's circle is a 2D graphical construction, not a stress-balance law. Every point on the circle is a valid (normal, shear) pair on a rotated orientation of the same element; the principal stresses are the two points where the circle crosses the normal-stress axis.

Inputs are MPa. Convert ksi or psi to MPa first; the circle geometry is identical because it is dimensionless in stress.

When a beam load creates the plane-stress state, the Beam Bending Stress Calculator is the natural next stop to resolve sigma 1 against the flexure formula at the same cross section.

The mohr circle calculator uses three standard formulas. The center is the average of the two normal stresses. The radius is the square root of the squared half-difference plus the squared shear stress. The principal stresses are the center plus and minus the radius.

• sigma_x: Normal stress on the x-face of the element (MPa, positive in tension).
• sigma_y: Normal stress on the y-face of the element (MPa, positive in tension).
• tau_xy: Shear stress on the x-face in the y-direction (MPa).
• sigma_avg: Mohr's circle center: the only normal stress that does not change with rotation.
• R: Mohr's circle radius: the maximum in-plane shear stress, and half the principal stress difference.

The principal angle comes from 2*theta_p = atan2(2*tau_xy, sigma_x - sigma_y). The factor of two trips students up: rotating the physical element by theta moves the point on the circle by 2*theta, so the angle on the diagram is double the physical rotation you apply in the lab.

For the rotated plane, sigma_theta = sigma_avg + (sigma_x - sigma_y) / 2 * cos(2*theta) + tau_xy * sin(2*theta); tau_theta uses the complementary cos and sin. The tool evaluates both expressions so you can verify an inclined-plane stress without redrawing the circle.

According to Wikipedia (Mohr's circle), the center equals (sigma_x + sigma_y) / 2 and the radius equals sqrt(((sigma_x - sigma_y) / 2)^2 + tau_xy^2), with the principal angle at 2*theta_p = atan2(2*tau_xy, sigma_x - sigma_y).

If you start from an internal load diagram, the Shear Force & Bending Moment Calculator gives the tau xy component that feeds directly into the input row.

Four ideas carry 90% of Mohr's circle. Lock these in and the calculator is a quick check, not a crutch.

Every other property on the circle (any rotated-plane normal stress, any rotated-plane shear stress, the maximum shear orientation) follows from these four. Once you can read the center, the radius, the principal angle, and the von Mises form, the tool is a fast check on screen.

The radius from this Mohr's circle construction is the same stress amplitude that drives the S-N curve, so feeding the principal stress difference into the Fatigue Life Estimation (S-N Curve) Calculator closes the fatigue loop.

Run the mohr circle calculator in four short steps and use the result to make a structural decision.

1. 1 Enter the three plane-stress components: Type sigma x, sigma y, and tau xy in MPa. Positive is tension; tau xy follows the sign convention in your textbook.
2. 2 Add sigma 3 only if the plate is not in plane stress: Leave sigma 3 = 0 for a thin plate. Set it to a measured out-of-plane stress for a thick section or 3D yield check.
3. 3 Set theta to the orientation to inspect: Use 0 for the input state, the principal angle to read sigma 1 or sigma 2, or 45 degrees plus the principal angle for the maximum in-plane shear.
4. 4 Read the results and act on them: Compare sigma 1 and the von Mises stress to the material yield. Compare the absolute max shear to half the yield for a Tresca check. Use sigma_theta and tau_theta to size a weld or fastener.

For a steel plate with sigma x = 220, sigma y = 40, tau xy = 60 MPa, set theta = 0 and the mohr circle calculator returns sigma 1 = 246.4 MPa, sigma 2 = 13.6 MPa, principal angle = 25.1 degrees, and von Mises = 235.8 MPa. That tells you the highest tensile stress is 246.4 MPa at 25.1 degrees from the x-axis, and the equivalent stress is 235.8 MPa, the number to compare against the steel's yield strength.

When the sigma x and sigma y inputs come from an applied load, the Newton's Laws / Forces Calculator resolves the supporting force system that produces them.

This tool is a quick sanity check for hand solutions and a fast what-if aid for design.

• • Skip the redraw: No need to re-plot the circle for every parameter change. The center, radius, and principal stresses update as soon as you change an input.
• • Catch sign errors fast: A wrong sign in tau xy flips the principal angle sign. The tool surfaces the error in seconds because the expected radius is stable.
• • Full plane stress story: Principal stresses, principal angle, max in-plane shear, rotated-plane normal and shear, von Mises, and absolute 3D max shear on one screen.
• • Ready for inclined-plane problems: Set theta to a weld, ply, grain, or fatigue plane and read the local stress directly.
• • Two cross-checks in one tool: Compare the von Mises number to the yield strength and the absolute max shear to half the yield (Tresca) to bracket failure load.

Use it as a teaching aid first: type the textbook example, confirm the principal angle matches, then try a non-symmetric case. Once the numbers feel right, the tool is fast enough for design iterations.

Five factors shape the result. Most are inputs, but two are physical caveats worth knowing before you trust the number.

• • This is a 2D plane-stress tool. It does not handle 3D Mohr's circle, where three principal stresses produce three concentric circles on the same normal-shear diagram.
• • The principal angle is undefined when the circle degenerates to a point (sigma_x = sigma_y and tau_xy = 0). The tool returns 0 in that case; check the radius first to confirm the state is isotropic.

If you need a full 3D check, take the two in-plane principal stresses, combine them with sigma 3, and run a separate 3D von Mises or principal-stress evaluation.

According to Wikipedia (von Mises yield criterion), the 2D plane-stress form of the von Mises criterion is sigma_v^2 = sigma_1^2 - sigma_1*sigma_2 + sigma_2^2, which reduces to sigma_v = sqrt(sigma_1^2 + sigma_2^2 - sigma_1*sigma_2) when sigma_3 = 0.

For a deformable element, the tau_xy that drives this analysis often comes from a contact stress, which the Spring Constant & Deflection Calculator can estimate before Mohr's circle is applied.