Mohr Circle Calculator

Example data table

Formula used

For plane stress, the Mohr circle center and radius are: C = (σx + σy)/2 and R = √[ ((σx − σy)/2)² + τxy² ].

Principal stresses: σ₁ = C + R, σ₂ = C − R. Maximum shear: τmax = R.

Principal plane rotation: θp = ½ atan2(2τxy, σx − σy). For a rotated plane by θ: σ(θ) = C + ((σx−σy)/2)cos(2θ) + τxy sin(2θ), τ(θ) = −((σx−σy)/2)sin(2θ) + τxy cos(2θ).

Plane-stress von Mises: σv = √(σx² − σxσy + σy² + 3τxy²).

How to use this calculator

1. Enter σx, σy, and τxy with consistent units.
2. Choose θ to evaluate stresses on a rotated plane.
3. Optional: add yield strength and a target safety factor.
4. Click Calculate to view results above the form.
5. Use CSV or PDF buttons to export your report.

Mohr circle guide

1) What the calculator represents

This tool models a two-dimensional stress state using Mohr’s circle, a graphical method that links normal stress and shear stress on all possible plane orientations. By entering σx, σy, and τxy, the calculator builds the circle defined by its center C and radius R. From these values, you immediately obtain principal stresses, maximum shear, and the physical angles associated with each condition.

2) Inputs and sign convention

Use consistent units (Pa, MPa, psi, or ksi) for σx, σy, and τxy. Positive τxy follows the standard plane-stress transformation convention used in mechanics of materials. If your source uses an alternate sign convention, keep the same convention for all calculations and interpret θ accordingly.

3) Center and radius with practical meaning

The center C = (σx + σy)/2 represents the average normal stress, while the radius R = √[ ((σx − σy)/2)² + τxy² ] measures the intensity of stress variation with orientation. When τxy = 0, the radius reduces to |σx − σy|/2, and the circle becomes purely normal-stress driven.

4) Principal stresses and their angles

Principal stresses are the extreme normal stresses: σ₁ = C + R and σ₂ = C − R. The principal plane angle θp is computed with θp = ½ atan2(2τxy, σx − σy). In design checks, σ₁ often governs tension cracking, while σ₂ can govern compression limits.

5) Maximum shear and critical planes

Maximum in-plane shear equals τmax = R and occurs on planes rotated about 45° from the principal planes. For many ductile materials, shear-dominated conditions can drive yielding, so τmax is a key diagnostic output, especially when σx and σy are similar but τxy is significant.

6) Stresses on a rotated plane

If you specify a rotation θ, the calculator returns σ(θ) and τ(θ) using the standard transformation equations. This is useful for bolts, welds, adhesive joints, composite plies, and any scenario where the load path is not aligned with the global axes. You can sweep θ values to identify orientations with low shear or low normal stress.

7) Von Mises check for plane stress

For plane stress (σz = 0), the von Mises equivalent stress is σv = √(σx² − σxσy + σy² + 3τxy²). Provide an optional yield strength to compute utilization (σv/σyld) and an estimated factor of safety (σyld/σv). This supports quick screening before a more detailed finite element model.

8) Data quality and reporting

Validate input magnitudes, units, and sign before exporting. The CSV format is convenient for lab reports and spreadsheets, while the PDF summary is suitable for submission or record keeping. For repeated studies, keep a consistent unit system and document the assumed sign convention alongside your results.

FAQs

1) What is Mohr’s circle used for?

It visualizes how normal and shear stresses change with plane orientation, and it provides principal stresses, maximum shear stress, and associated angles for a 2D stress state.

2) Can I use negative stresses?

Yes. Compression is commonly negative and tension positive, but the choice must remain consistent across σx, σy, and τxy. The outputs remain correct under a consistent convention.

3) What does the angle θ mean here?

θ is the physical rotation of the plane relative to the x-axis. The circle uses 2θ internally because stress transformation depends on double-angle relationships.

4) Why is τmax equal to the circle radius?

On Mohr’s circle, shear stress corresponds to the vertical coordinate. The largest vertical distance from the center is the radius, so the maximum shear magnitude equals R.

5) How do I interpret σ₁ and σ₂?

They are the extreme normal stresses on planes where shear is zero. σ₁ is the maximum normal stress and σ₂ is the minimum normal stress for the given state.

6) Is von Mises valid for all materials?

It is widely used for ductile metals under yielding. Brittle materials often require other criteria, such as maximum normal stress or Mohr–Coulomb style limits, depending on application.

7) What unit should I choose?

Any stress unit works if all inputs share it. Pick the unit that matches your data source (MPa, psi, or ksi), then export so reports stay consistent and unambiguous.