Mohr's Circle Simulator

Modified: Jun. 9, 2026

Published: Jun. 5, 2026

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Real components rarely carry pure tension or pure shear. A shaft bends and twists at once, a bracket pulls on one face while shearing on another, and the plane that actually fails is almost never the plane you drew. Mohr’s circle is the tool that turns this mess into one picture: every orientation of the stress element is a single point riding around one circle, the points where the circle crosses the axis are the principal stresses, and the top of the circle is the worst shear. This simulator lets you rotate the element and watch that point move, read the principal stresses and their orientation, and check the state against von Mises and Tresca with a live safety factor. #MohrsCircle #StressTransformation #FailureAnalysis

What You Can Analyze

Key Features

1. Element synced to the circle The rotating stress element and the moving point on the Mohr circle update together, so the abstract construction is tied to a physical orientation at all times.

2. Live principal and shear readouts Principal stresses, principal angle, maximum shear, and the max-shear plane recompute as you drag any input.

3. Four analysis charts Normal stress and shear stress against rotation angle, plus the state placed in principal-stress space against the yield envelope and a load-margin chart.

4. von Mises and Tresca Switch criteria and enter a yield strength; the panel reports the equivalent stress and a safety factor for a plane-stress state.

5. Five stress-state presets Pure tension, pure shear, equal biaxial, general plane stress, and combined bending plus torsion, each a realistic starting point.

6. Live A/B comparison Save State A, change the stresses, and overlay state B on every chart to compare two loadings directly.

7. Downloadable resources A design-data sheet, the full transformation dataset as CSV (both states when you compare two), and a complete lab report with watermark-free diagram and charts.

Preset Stress States

Equations

Circle centre (average normal stress) and radius:

sigma_avg = (sx + sy) / 2
R = sqrt( ((sx - sy)/2)^2 + txy^2 )

Principal stresses and the orientation to them:

sigma_1,2 = sigma_avg +/- R
tan(2*theta_p) = 2*txy / (sx - sy)

Maximum in-plane shear and its plane:

tau_max = R
theta_s = theta_p - 45 degrees

Stresses on a face rotated by theta from the x-axis (the point that rides the circle):

sigma_x' = sigma_avg + ((sx - sy)/2)*cos(2*theta) + txy*sin(2*theta)
tau_x'y' = -((sx - sy)/2)*sin(2*theta) + txy*cos(2*theta)

Failure check for a plane-stress state (the out-of-plane principal stress is zero):

von Mises:  sigma_vm = sqrt( s1^2 - s1*s2 + s2^2 )
Tresca:     sigma_tr = max( |s1 - s2|, |s1|, |s2| )
safety factor = yield / equivalent stress

Guided Experiments

Work through the Stress Transformation Experiments lesson for structured, Python-verified exercises:

1. Build Mohr’s circle from a stress state
2. Stress on an inclined plane
3. Maximum shear and its plane
4. Pure shear and the forty-five-degree failure
5. Combined bending and torsion on a shaft surface
6. von Mises versus Tresca and the safety factor

Related Resources

• Stress Transformation Experiments: the hands-on lesson that pairs with this simulator.
• Principal Stresses and Failure Criteria: the lesson that owns the underlying theory.
• Solid Mechanics Analyzer: the full suite of stress-analysis simulators.
• Beam Analysis Simulator and Shaft Torsion Simulator: the stress states that feed a combined-loading Mohr’s analysis.