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Mohr's Circle Calculator

Engineers use Mohr's circle to visualize how stresses transform across different orientations in a stressed material. Given the normal and shear stresses acting in the X and Y directions, this calculator determines the principal stresses (maximum and minimum), maximum shear stress, von Mises stress, and the orientation angle. Essential for mechanical design, structural analysis, and failure prediction.

Last updated:

Creators Davide Borchia, PhD
2,417 people find this calculator helpful

Understanding Stress State and Principal Stress

Every point in a loaded material experiences stress in multiple directions. A complete stress state includes three normal stresses (acting perpendicular to surfaces) and six shear stresses (acting along surfaces). In 2D plane stress problems, the analysis simplifies to two normal stresses and one shear stress pair.

Principal stresses represent the extreme normal stresses that occur when shear stresses vanish. At the principal orientation, the material experiences only tension or compression with no shear. Finding these principal values is critical because materials typically fail when principal stress exceeds strength limits, not when shear stress alone does.

By rotating the stress element through various angles, the stresses change according to transformation equations. The principal stresses represent the maximum and minimum values among all possible orientations—the absolute bounds of normal stress the material experiences.

Principal Stress and Shear Stress Equations

The calculator computes principal stresses from the applied normal stresses (σₓ and σᵧ) and shear stress (τₓᵧ) using these relationships:

σₘ = (σₓ + σᵧ) ÷ 2

σ₁ = σₘ + √[((σₓ − σᵧ) ÷ 2)² + τₓᵧ²]

σ₂ = σₘ − √[((σₓ − σᵧ) ÷ 2)² + τₓᵧ²]

τₘₐₓ = √[((σₓ − σᵧ) ÷ 2)² + τₓᵧ²]

σᴹⁱˢᵉˢ = √[σₓ² + σᵧ² − σₓσᵧ + 3τₓᵧ²]

θ = (arctan(2τₓᵧ ÷ (σₓ − σᵧ))) ÷ 2

  • σₓ, σᵧ — Normal stresses in the X and Y directions
  • τₓᵧ — Shear stress component
  • σₘ — Mean (average) stress
  • σ₁, σ₂ — Maximum and minimum principal stresses
  • τₘₐₓ — Maximum shear stress magnitude
  • σᴹⁱˢᵉˢ — von Mises equivalent stress
  • θ — Angle defining the principal plane orientation

Mohr's Circle: Graphical Stress Transformation

Mohr's circle is a geometric construction that maps all possible stress combinations across different orientations. On a graph with normal stress on the horizontal axis and shear stress on the vertical axis, each point represents the stresses on one particular plane through the material.

To construct the circle: plot two points representing the stress on perpendicular faces (σₓ, τₓᵧ) and (σᵧ, −τₓᵧ). The midpoint of the line joining these points is the circle's centre, located at (σₘ, 0). The circle's radius equals the maximum shear stress τₘₐₓ. The principal stresses appear where the circle intersects the horizontal axis—the rightmost and leftmost points.

This graphical approach reveals several insights: the difference between principal stresses equals twice the radius; rotating the stress element by angle θ rotates the corresponding point on the circle by 2θ; and any stress state is possible on the circle but nowhere outside it. Engineers use Mohr's circle to quickly identify critical orientations and verify stress calculations.

Critical Considerations for Mohr's Circle Analysis

Mohr's circle is powerful but requires careful interpretation to avoid common pitfalls.

  1. Sign Convention Matters — Shear stress sign conventions vary between disciplines and textbooks. Confirm whether positive shear stress indicates clockwise or counterclockwise rotation. Consistency is essential—reversing the sign of τₓᵧ flips the circle vertically and changes the calculated principal angle. Always document your sign convention at the start of any analysis.
  2. 2D vs 3D Stress States — Mohr's circle handles plane stress (σᵤ = 0) but cannot directly represent general 3D stress states. For 3D, construct three Mohr's circles for each pair of principal stresses. The largest circle's radius still gives the absolute maximum shear stress. Mistaking 2D results for a 3D problem can lead to dangerous underestimation of stress magnitudes.
  3. Principal Stress vs Principal Plane Orientation — The angle θ returned by the calculator locates the principal plane, not the principal stress value. A 90-degree rotation in θ switches which principal stress (σ₁ or σ₂) acts in each direction. Small errors in angle calculation accumulate in multi-step designs. Always verify which principal stress is σ₁ (larger) and which is σ₂ (smaller) before applying yield criteria.
  4. von Mises Stress is Not a Principal Stress — The von Mises stress combines all stress components into a single equivalent scalar. It predicts ductile material failure but is not an actual stress acting on any plane. Compare von Mises against material yield strength, not against principal stresses. Confusing these leads to incorrect failure predictions.

Practical Applications in Design

Mohr's circle analysis appears in pressure vessel design, where hoop and longitudinal stresses create a non-principal stress state. Welded structures also experience complex stresses; Mohr's circle reveals the most damaging orientations for crack growth. In composite material design, fibre orientation is often chosen to align with principal stresses and avoid transverse shear.

Soil mechanics uses Mohr-Coulomb failure criteria based on Mohr's circle to predict when slopes fail or underground structures collapse. Fatigue analysis employs Mohr's circle to track mean and alternating stress components across cycles. Finite element analysis output stresses at element centres must be interpreted via Mohr's circle to identify critical planes and verify the mesh orientation does not artificially skew results.

Frequently Asked Questions

What does 'plane stress' mean and when does it apply?

Plane stress assumes one normal stress component (typically σᵤ) is negligible or zero. This approximation holds for thin-walled structures, flat plates under in-plane loading, and surface stresses in large solid bodies. When the thickness perpendicular to the X–Y plane is very small compared to other dimensions, the stress through that thickness equilibrates to zero, justifying the 2D analysis. If thickness is comparable to other dimensions, treat the problem as 3D.

How does the orientation angle θ relate to failure?

The orientation angle θ tells you the direction of the principal planes relative to your reference axes. Material failure depends on which principal stress acts perpendicular to existing flaws or cracks. If a defect is oriented perpendicular to σ₂ (the smaller principal stress), it may not propagate even if σ₁ is high. Conversely, a defect aligned with σ₁ is most dangerous. Use θ to assess whether existing geometry (welds, material grain) aligns with or opposes the maximum tensile stress.

Why is von Mises stress useful if it isn't a real stress?

Von Mises stress represents the total distortional energy in the stress state. For ductile metals, failure occurs when this energy exceeds the material's capacity, regardless of the balance between normal and shear stresses. It combines all stress components into one comparable number. Compare this number directly to the material's yield strength from a tensile test (which is a principal stress state but one dimensional, making von Mises more general). For brittle materials or when shear failure is primary, use principal stresses instead.

Can I use Mohr's circle for 3D stress states?

Not directly in the 2D form. For 3D, you construct three separate Mohr's circles—one for each pair of the three principal stresses. The largest circle reveals the absolute maximum shear stress. However, the 3D case is complex; the angle between principal axes cannot be captured in 2D circles alone. For full 3D analysis, use eigenvalue computation (finding principal stresses and their directions algebraically) or specialized 3D stress visualization software.

What if my two normal stresses are equal (σₓ = σᵧ)?

When σₓ = σᵧ, the difference term vanishes, and the Mohr's circle becomes smaller. Both principal stresses equal (σₓ + σᵧ) ÷ 2, and the maximum shear stress depends only on τₓᵧ. This is called a hydrostatic-like condition in 2D. The principal angle θ becomes undefined or ±45° depending on shear direction. The circle shrinks to radius |τₓᵧ|. Geometrically, this means all normal stress components are identical; the stress state is less severe than when a large difference exists between σₓ and σᵧ.

How do I use Mohr's circle to check if my FEA results are correct?

Extract the stress tensor components from your FEA output at a critical node. Construct Mohr's circle from σₓ, σᵧ, and τₓᵧ. Compare the principal stresses and their orientations to your FEA-reported principal stresses. If they match, your stress element interpretation is correct. Also check whether the element orientation in your mesh coincides with your reference axes; rotated elements give different σₓ and σᵧ but the same principal stresses. Use Mohr's circle as a sanity check against numerical errors or misunderstood output conventions.

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