How to use this calculator

Enter the two normal stresses and the shear stress acting on an element aligned with x and y. Tension is positive. Shear τ_xy is positive when it acts in the +y direction on the +x face. Angles are measured counterclockwise from the x-axis. Change θ to read the stresses on any rotated plane; the red dashed line on the circle moves with it.

Formulas

Center and radius of the circleσ_avg = σ_x + σ_y2 R = √((σ_x − σ_y)/2)² + τ_xy²

Principal stresses and maximum in-plane shearσ_1,2 = σ_avg ± R τ_max = R

Principal angle (to the σ₁ plane)θ_p = ½ atan2(2τ_xy, σ_x − σ_y) θ_s = θ_p − 45°

Stresses on a plane rotated by θσ_x′ = σ_avg + σ_x − σ_y2 cos 2θ + τ_xy sin 2θ τ_x′y′ = −σ_x − σ_y2 sin 2θ + τ_xy cos 2θ

von Mises stress for plane stressσ_vm = √σ₁² − σ₁σ₂ + σ₂²

On the circle, a rotation of θ on the element is a rotation of 2θ around the circle. The drawing plots point X at (σ_x, τ_xy) and Y at (σ_y, −τ_xy) with shear positive downward, the convention used in Hibbeler’s Mechanics of Materials, so both rotate in the same direction.

Worked example

σ_x = 80 MPa, σ_y = −40 MPa, τ_xy = 30 MPa (the default)

σ_avg = (80 − 40)/2 = 20 MPa. R = √(60² + 30²) = 67.08 MPa, which is also τ_max.
σ₁ = 20 + 67.08 = 87.08 MPa and σ₂ = 20 − 67.08 = −47.08 MPa.
θ_p = ½ atan2(60, 120) = 13.28° counterclockwise from x.
At θ = 30°: σ_x′ = 20 + 60 cos 60° + 30 sin 60° = 75.98 MPa and τ_x′y′ = −60 sin 60° + 30 cos 60° = −36.96 MPa.
σ_vm = √(87.08² + 87.08 × 47.08 + 47.08²) = 117.9 MPa.

In-plane versus absolute maximum shear

For plane stress the third principal stress is σ₃ = 0. When σ₁ and σ₂ have opposite signs, the in-plane maximum shear R is also the absolute maximum. When they have the same sign, the largest shear acts on a plane tilted out of the x–y plane and equals half the larger principal stress. The Tresca yield criterion uses this absolute value.

Assumptions and limits

Plane stress: σ_z, τ_xz and τ_yz are zero, as on a free surface or in a thin plate.
Stresses are at a single point; they come from your own analysis, such as σ = Mc/I and τ = Tc/J.
Linear elastic behavior is assumed when the von Mises stress is compared with yield.

Frequently asked questions

Why does my textbook draw the circle upside down?

Some books plot shear positive upward and place Y at (σ_y, τ_xy), which makes the circle rotate opposite to the element. Both conventions give the same principal stresses and angles when used consistently.

How do I use the von Mises stress?

Compare it with the yield strength. For ductile metals, yielding is predicted when σ_vm reaches σ_y, so the factor of safety is σ_y/σ_vm.

Where do combined stresses come from in practice?

A shaft carrying bending and torque at the same time has σ_x from bending and τ_xy from torsion. Get them from the bending stress and shaft torsion calculators, then combine them here.

For education and preliminary checks only. Have a qualified engineer verify any design before it is built.