Mohr’s Circle Stress Calculator

Calculator form

Enter plane stress values

Use any consistent stress unit. The unit label is printed in results, downloads, and chart labels.

σx

Normal stress on the x-face.

σy

Normal stress on the y-face.

τxy

In-plane shear stress.

Rotation angle, θ (degrees)

Plane angle measured from the x-face.

Stress unit label

Examples: MPa, psi, ksi, Pa.

Decimal places

Allowed range: 0 to 6.

Example data table

Sample plane stress case

σx	σy	τxy	θ	σ1	σ2	τmax	σθ	τθ	Unit
80	20	30	20°	92.426	7.574	42.426	92.265	3.698	MPa

This example matches the prefilled values in the calculator. Click “Load Example” to restore it instantly.

Formula used

Mohr’s Circle equations

Circle center: C = (σx + σy) / 2

Circle radius: R = √[ ((σx - σy) / 2)² + τxy² ]

Principal stresses: σ1 = C + R and σ2 = C - R

Maximum in-plane shear: τmax = R

Principal plane angle: θp = 0.5 × atan2(2τxy, σx - σy)

Stress transformation: σθ = C + ((σx - σy)/2)cos(2θ) + τxy sin(2θ)

Shear transformation: τθ = -((σx - σy)/2)sin(2θ) + τxy cos(2θ)

Optional equivalent stress: σvm = √(σx² - σxσy + σy² + 3τxy²)

Angles in the equations use radians internally, but the form accepts degrees for convenience. The plotted circle uses the computed center and radius directly.

How to use this calculator

Simple workflow

Enter the normal stresses σx and σy.
Enter the in-plane shear stress τxy.
Provide the plane rotation angle θ in degrees.
Choose a unit label and desired decimal precision.
Press Calculate Stress State.
Review the result block shown below the header and above the form.
Inspect the circle plot, transformed point, and principal values.
Download the result summary as CSV or PDF when needed.

FAQs

Common questions

1) What does Mohr’s Circle show?

Mohr’s Circle gives a geometric view of plane stress transformation. It shows principal stresses, maximum in-plane shear stress, the average normal stress, and the stress components on any rotated plane.

2) Why are the principal stresses important?

Principal stresses are the extreme normal stresses that occur when shear becomes zero. Engineers use them to assess failure risk, crack tendency, safety margin, and how a material behaves under combined loading.

3) What does the radius of the circle represent?

The radius equals the maximum in-plane shear stress. A larger radius means the current stress state has a stronger spread between the average normal stress and the extreme transformed stresses.

4) Why is the angle on Mohr’s Circle doubled?

A physical plane rotation of θ corresponds to a movement of 2θ on Mohr’s Circle. That doubling comes directly from the trigonometric stress-transformation equations using cos(2θ) and sin(2θ).

5) Can I use negative stresses?

Yes. Negative normal stresses can represent compression if that matches your sign convention. Negative shear is also allowed. The calculator keeps the signs and transforms the stress state consistently.

6) What unit should I enter?

Use any consistent unit such as MPa, Pa, psi, or ksi. The equations are unit-consistent, so the same unit is carried through the results, graph labels, and export files.

7) What is the difference between τxy and τmax?

τxy is the shear stress on your original coordinate faces. τmax is the maximum in-plane shear stress obtained after rotating to a special orientation where the shear reaches its largest magnitude.

8) Why is von Mises stress included?

Von Mises stress helps compare a combined plane-stress state with yield strength in ductile materials. It does not replace principal stresses, but it adds a practical design check for many engineering problems.