Mohr’s Circle Calculator

Related Calculators

Plane stress: A 2D stress case where only the in-plane stresses are analyzed here: Normal stress in x direction, sigma_x, Normal stress in y direction, sigma_y, and Shear stress, tau_xy.

Average normal stress, sigma_avg: The midpoint of sigma_x and sigma_y. It is the center of Mohr's circle on the normal-stress axis.

Circle radius, R: The distance from the circle center to the stress points. In this calculator, it is also Maximum in-plane shear stress, tau_max.

Major principal stress, sigma_1: The larger principal normal stress. On its plane, shear stress is zero.

Minor principal stress, sigma_2: The smaller principal normal stress. On its plane, shear stress is zero.

Principal plane angle, theta_p: The physical angle of the plane where a principal stress occurs. It is not the same as the Mohr-circle angle.

Mohr-circle angle, 2theta: Twice the physical rotation angle. This doubled angle is why Mohr's circle often causes angle confusion.

Shear sign convention: The rule that decides whether a shear stress value is positive or negative. Keep one convention throughout the whole problem.

Mistake: Entering Normal stress in x direction, sigma_x in MPa and Normal stress in y direction, sigma_y in psi while leaving Stress unit on one label.
Fix: Convert all stress inputs first, then choose one Stress unit for every value.

Mistake: Typing compression as a positive value in Normal stress in y direction, sigma_y when your class uses tension-positive normal stress.
Fix: Enter compression as negative if that matches your course convention, and keep that same convention for Normal stress in x direction, sigma_x .

Mistake: Treating Plane angle, theta as the Mohr-circle rotation instead of the physical plane rotation.
Fix: Enter the real physical angle in Plane angle, theta and use Mohr-circle angle, 2theta only as the doubled output.

Mistake: Comparing Principal plane angle, theta_p directly to a sketch that uses a different angle range.
Fix: Change Angle output convention to match your class format before interpreting Principal plane angle, theta_p or Maximum shear plane angle, theta_s .

Mistake: Entering the wrong sign for Shear stress, tau_xy and then assuming Shear stress on selected plane, tau_theta is wrong.
Fix: Check the chosen Shear sign convention note and keep the same shear sign convention for both input and interpretation.

Mistake: Expecting transformed-plane results when Plane angle, theta is left blank.
Fix: Enter a numeric angle in Plane angle, theta if you need Normal stress on selected plane, sigma_theta and Shear stress on selected plane, tau_theta .

• This calculator is for 2D plane stress only. It does not compute full 3D stress states or strain-based Mohr circles.
• All stress inputs must use the same selected stress unit. The tool labels results with that unit but does not convert mixed-unit inputs for you.
• Angle outputs depend on the selected Angle output convention, so the same physical direction can be displayed in different but equivalent angle ranges.
• If Circle radius, R equals 0, then the principal and maximum-shear plane orientations are not unique even though the stress values are still valid.
• If Plane angle, theta is blank, the calculator can still return circle and principal-stress results, but transformed-plane outputs cannot be computed.
• Shear-related signs depend on the chosen Shear sign convention note. If your class uses a different convention, stress values may look sign-reversed even when the physics is consistent.

What the calculator computes

This tool starts from a 2D plane-stress state with sigma_x, sigma_y, and tau_xy. It then finds the circle center, circle radius, principal stresses, maximum in-plane shear stress, and, if a plane angle is provided, the transformed stresses on that rotated plane.

Core equations

σ_avg = (σ_x + σ_y) / 2

R = sqrt(((σ_x - σ_y) / 2)^2 + τ_xy^2)

σ_1 = σ_avg + R

σ_2 = σ_avg - R

θ_p = 0.5 * atan2(2 * τ_xy, σ_x - σ_y)

θ_s = θ_p + 45 deg

σ_θ = σ_avg + ((σ_x - σ_y) / 2) * cos(2θ) + τ_xy * sin(2θ)

τ_θ = -((σ_x - σ_y) / 2) * sin(2θ) + τ_xy * cos(2θ)

The calculator uses atan2 instead of plain atan so the principal-plane angle lands in the correct quadrant. It also reports the physical plane angles separately from the Mohr-circle angle 2theta, which is twice the physical rotation.

Mini example

Using sigma_x = 80, sigma_y = 20, and tau_xy = 30, the average normal stress is 50 and the circle radius is about 42.4264. That gives sigma_1 about 92.4264 and sigma_2 about 7.5736. The principal plane angle is 22.5 deg, so the maximum shear plane angle is 67.5 deg. If theta = 25 deg, then 2theta = 50 deg, sigma_theta is about 91.3209, and tau_theta is about 4.5077.

How to interpret the results

A larger Circle radius, R means the stress state changes more from one plane orientation to another. Major principal stress, sigma_1 is the largest normal stress and Minor principal stress, sigma_2 is the smallest. Maximum in-plane shear stress, tau_max equals the radius for this 2D plane-stress case. On a selected plane, the sign of Shear stress on selected plane, tau_theta depends on the chosen shear convention.

Special cases and assumptions

If sigma_x, sigma_y, and tau_xy are all zero, all stress outputs are zero and the angle orientation is not unique, so 0 deg is only a convenient display choice. If R = 0, the principal stresses are equal and the principal and shear plane angles are not unique. This method assumes plane stress and does not include out-of-plane stresses, material behavior, or failure criteria [1][2].