Von Mises & Principal Stress Calculator (Mohr’s circle)

Understanding the von Mises and principal stress calculator — concept and principle

In a real part, a point is usually loaded by normal stresses in two directions and a shear stress at the same time. This calculator converts that plane-stress state into principal stresses, the maximum in-plane shear, the principal angle and the von Mises equivalent stress, then compares it with the yield strength to give safety factors by the von Mises and Tresca criteria.

How it is calculated

• Centre and radius of Mohr's circle: C = (σx + σy)/2 and R = √(((σx − σy)/2)² + τxy²).
• Principal stresses: σ₁ = C + R, σ₂ = C − R. R is also the maximum in-plane shear stress.
• Von Mises stress: σ_vm = √(σx² − σxσy + σy² + 3τxy²).
• Safety factors: von Mises n = f_y ÷ σ_vm; Tresca n = f_y ÷ max(|σ₁|, |σ₂|, |σ₁ − σ₂|), which includes the out-of-plane stress σ₃ = 0.
• Principal angle: θp = ½·atan2(2τxy, σx − σy), measured from the x axis.

The principle

Rotating the reference axes changes the stress components, and Mohr's circle draws all those combinations as one circle. Its two ends on the horizontal axis are the principal stresses, the directions where shear vanishes. Ductile metals yield by shear slip, not by normal stress alone. The von Mises criterion says yielding starts when the distortion energy reaches its value in a simple tensile test; the Tresca criterion says it starts when the largest shear stress reaches half the yield stress. Tresca is slightly more conservative.

Worked example

1. Inputs: σx = 80 MPa, σy = 20 MPa, τxy = 30 MPa, f_y = 250 MPa.
2. C = 50 MPa; R = √(30² + 30²) = 42.42641 MPa (max in-plane shear).
3. σ₁ = 92.42641 MPa, σ₂ = 7.573593 MPa.
4. σ_vm = √(6,400 − 1,600 + 400 + 2,700) = √7,900 = 88.88194 MPa.
5. Von Mises safety factor: 250 ÷ 88.88194 = 2.81272; Tresca: 250 ÷ 92.42641 = 2.704855.
6. Principal angle: ½ × atan2(60, 60) = 22.5°.

How to read the result

A safety factor above 1 means the point is predicted not to yield under static load. Note that the absolute maximum shear here is σ₁/2 ≈ 46.2 MPa, larger than R, because σ₃ = 0 lies outside the in-plane circle.

Common mistakes

• Applying von Mises to brittle materials such as cast iron; use maximum normal stress or Mohr–Coulomb instead.
• Forgetting the sign of compressive stresses.
• Using the static safety factor for fatigue loading.

Frequently asked questions

Is von Mises stress a real stress?

It is an equivalent scalar that lets a combined stress state be compared with one tensile test value.

Why are Tresca and von Mises different?

They model yield differently. The largest difference is in pure shear, where Tresca predicts yielding at 0.5 f_y and von Mises at about 0.577 f_y.

What does the principal angle tell me?

It is the rotation from the x axis to the direction of σ₁, where the material sees no shear.

Sources: R. G. Budynas and J. K. Nisbett, Shigley's Mechanical Engineering Design, McGraw-Hill · R. von Mises, "Mechanik der festen Körper im plastisch-deformablen Zustand", Göttinger Nachrichten (1913)