Mohr’s Circle Stress Calculator

Article

1) Practical role of Mohr’s Circle

Mohr’s Circle is a fast method to interpret plane stress in plates, shafts under combined loading, and thin-walled parts. It converts (σx, σy, τxy) into a circle on the σ–τ plane, so extreme normal stress and extreme shear stress become clear.

2) Inputs and unit consistency

Enter σx and σy as normal stresses and τxy as the in-plane shear stress from your stress element. Keep one sign convention for τxy across the problem. Choose Pa, kPa, MPa, GPa, or psi and use that same unit for all three inputs.

3) Center and radius as key data

The circle center is C = (σx + σy)/2, the average normal stress. The radius is R = √[ ((σx − σy)/2)² + τxy² ], which controls how far the stress state can vary with orientation. In this tool, C and R are reported directly for quick checking.

4) Principal stresses for reporting

Principal stresses occur where shear on the plane is zero. They are σ1 = C + R and σ2 = C − R, and they are useful for brittle failure checks, documentation, and comparing load cases. A quick validation is σ1 + σ2 = σx + σy.

5) Maximum shear and safety margins

The maximum in-plane shear stress equals τmax = R. This value is often used in shear-driven assessments and as a sanity check when τxy is large. The calculator also reports θs = θp + 45° to identify shear-critical plane orientations.

6) Angle outputs and interpretation

The principal plane angle is θp = ½·atan2(2τxy, σx − σy). Here θ is measured from the x-axis to the plane’s normal. A second principal plane is θp + 90°. Because Mohr’s Circle uses 2θ, the rotation on the circle is double the physical rotation.

7) Stress on a chosen plane (optional θ)

If you enter θ, the calculator evaluates σ(θ) and τ(θ) using the standard transformation equations. This helps when you must verify a specific cut, weld direction, laminate angle, or measurement orientation. Compare multiple θ values to see how close a plane is to principal or max shear conditions.

8) Plot, CSV, and PDF for traceability

The plot shows the circle and the points (σx, τxy) and (σy, −τxy), plus the σ1 and σ2 intercepts. Visual agreement is a strong sign that signs and units are correct. Export CSV for spreadsheets and PDF for reports, audits, or QA records.

FAQs

1) What is the main output of Mohr’s Circle?

It provides principal stresses (σ1, σ2), maximum shear stress (τmax), and the angles where these occur, all derived from σx, σy, and τxy.

2) Why does the circle use (σ, τ) axes?

The σ–τ plane maps normal stress and shear stress on any rotated plane. A single circle captures how stress components change with orientation.

3) How should I choose the sign of τxy?

Use the sign convention from your textbook or standard and apply it consistently. If your τxy sign is flipped, the circle mirrors about the σ-axis and angles change sign.

4) What does θp represent in this calculator?

θp is the angle from the x-axis to the normal of a principal plane. At that orientation, shear stress on the plane is zero and normal stress is extreme.

5) Why is τmax equal to the radius R?

On Mohr’s Circle, the greatest |τ| occurs at the top and bottom of the circle. The distance from center to any point is R, so τmax equals R.

6) When should I use the optional θ input?

Use θ when you need stress on a specific physical plane, such as a weld, interface, or cut orientation. The tool returns σ(θ) and τ(θ) for that plane.

7) Does this tool handle 3D stress states?

No. This version is for plane stress using σx, σy, and τxy. For 3D, you need a full stress tensor and three principal stresses from eigenvalues.