Mohr's Circle Calculator — Principal Stress & Shear Stress Analysis

Compute principal stresses, maximum shear stress, and plane orientation angles using Mohr's circle method. Free online plane stress calculator with step-by-step formula breakdown for engineers and students.

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Mohr's Circle Calculator

Enter plane stress components to compute principal stresses, maximum shear stress, and orientation angles.

Unit:

Sign convention: Tension (+) / Compression (−). Shear stress τxy positive on +x face in +y direction.

Enter stress values and click Calculate Mohr's Circle to see the results.

Mohr's Circle Formulas Explained

Mohr's Circle is a graphical representation of the state of plane stress at a point. It relates normal stresses, shear stress, principal stresses, and the orientation of principal planes through a set of fundamental equations.

σavg = (σx + σy) / 2
R = √[((σx − σy) / 2)² + τxy²]
σ1 = σavg + R   |   σ2 = σavg − R
τmax = R   |   θp = ½ · arctan(2τxy / (σx − σy))

Variable Definitions

  • σx, σy — Normal stresses in the x and y directions (tension positive, compression negative)
  • τxy — Shear stress acting on the x-face in the y-direction
  • σavg — Average normal stress (center of Mohr's circle)
  • R — Radius of Mohr's circle (equal to maximum shear stress τmax)
  • σ1, σ2 — Maximum and minimum principal stresses
  • θp — Principal plane angle (orientation of σ1 from x-axis)

How to Use Mohr's Circle Calculator

Using this Mohr's Circle calculator is straightforward. Follow these steps for accurate plane stress analysis:

  1. Enter normal stress σx — Input the normal stress acting on the x-face. Use positive values for tension.
  2. Enter normal stress σy — Input the normal stress acting on the y-face.
  3. Enter shear stress τxy — Input the shear stress component. Follow the standard sign convention.
  4. Select your unit — Choose MPa, psi, or ksi. All calculations are unit-independent.
  5. Click Calculate — The calculator computes σ1, σ2, τmax, σavg, R, and θp.
  6. Review the breakdown — Examine each calculation step and verify against your engineering judgment.

Mohr's Circle Worked Examples

Example 1: Biaxial Tension with Shear

Given σx = 100 MPa, σy = 50 MPa, τxy = 25 MPa.

σavg = (100 + 50) / 2 = 75 MPa
R = √[((100−50)/2)² + 25²] = √[25² + 25²] = 35.36 MPa
σ1 = 75 + 35.36 = 110.36 MPa
σ2 = 75 − 35.36 = 39.64 MPa
τmax = 35.36 MPa
θp = 0.5 × arctan(50/50) = 22.5°

Example 2: Pure Shear

Given σx = 0, σy = 0, τxy = 40 MPa.

σavg = 0
R = √[0 + 40²] = 40 MPa
σ1 = 40 MPa, σ2 = −40 MPa
τmax = 40 MPa
θp = 45° (principal planes at 45° to x-axis)

Example 3: Uniaxial Tension

Given σx = 200 MPa, σy = 0, τxy = 0.

σavg = 100 MPa
R = 100 MPa
σ1 = 200 MPa, σ2 = 0 MPa
τmax = 100 MPa
θp = 0° (principal direction aligned with x-axis)

Real-World Mohr's Circle Applications

  • Structural Engineering: Analyzing stress states in beams, columns, and connections to ensure designs meet safety codes.
  • Mechanical Design: Evaluating failure criteria (Tresca, von Mises) for shafts, pressure vessels, and machine components under combined loading.
  • Geotechnical Engineering: Determining principal stresses in soil mechanics for foundation design and slope stability analysis.
  • Aerospace Structures: Assessing multi-axial stress states in aircraft skins, fuselage panels, and turbine components.
  • Materials Testing: Interpreting strain gauge rosette data to find principal strains and stresses in experimental mechanics.
  • Civil Infrastructure: Evaluating stress conditions in bridges, dams, and retaining walls under service loads.
  • Biomechanics: Analyzing stress distributions in bones and implants to predict failure and optimize prosthetic design.

People Also Ask

Mohr's Circle is used to determine principal stresses, maximum shear stress, and stress transformation on any arbitrary plane. It provides a visual method to analyze the state of plane stress at a point, which is essential for failure analysis using criteria like Tresca (maximum shear stress) and von Mises (distortion energy). Engineers use it to design safe structural and mechanical components under combined loading.
The radius R of Mohr's Circle is calculated using: R = √[((σx - σy)/2)² + τxy²]. This radius equals the maximum shear stress τmax. Graphically, it's the distance from the center point (σavg, 0) to any point on the circle. The radius represents the magnitude of the maximum shear stress acting on planes oriented 45° from the principal planes.
Normal stresses (σx, σy) are the stresses acting perpendicular to the original coordinate axes. Principal stresses (σ1, σ2) are the maximum and minimum normal stresses that occur on planes where shear stress is zero. They are found by rotating the coordinate system by the principal angle θp. Principal stresses are critical for failure prediction because they represent the extreme values of normal stress at a point.
In the standard engineering sign convention: tensile normal stresses are positive, compressive negative. For shear stress τxy, it is positive when acting on the positive x-face in the positive y-direction. In Mohr's Circle plotting, positive shear is typically plotted downward on the τ-axis. This convention ensures consistency with the direction of principal stress rotation.
The principal angle θp is derived from: θp = 0.5 × arctan(2τxy / (σx - σy)). In Mohr's Circle, the angle between the x-axis point and the σ1 point on the circle is 2θp (twice the physical angle). The factor of 1/2 in the formula accounts for this relationship. When σx = σy, the denominator is zero, requiring special handling using arctan2 or limiting cases.

Frequently Asked Questions

This calculator focuses on 2D plane stress analysis, which covers the vast majority of practical engineering cases. For 3D stress states, three Mohr's circles are constructed from the three principal stresses. This tool provides the foundation; 3D analysis extends the same principles to triaxial stress states.
When σx = σy, the center of Mohr's circle lies on the σ-axis at σavg = σx = σy. If τxy > 0, the principal angle θp = 45°. If τxy < 0, θp = -45°. If τxy = 0 as well, the stress state is hydrostatic (equal stress in all directions), and Mohr's circle reduces to a single point with zero shear stress on all planes.
The von Mises equivalent stress can be expressed in terms of principal stresses: σvm = √(σ1² - σ1σ2 + σ2²). It's also related to Mohr's Circle parameters: σvm = √(3R² + σavg² - σ1σ2). The von Mises criterion predicts yielding in ductile materials and is complementary to the maximum shear stress (Tresca) criterion derived directly from Mohr's Circle.
Yes. Negative normal stresses represent compression, and negative shear stresses indicate reversed direction. The calculator handles all real-number inputs correctly. Compressive stresses shift Mohr's circle to the left on the σ-axis, and the principal stress calculations remain valid regardless of sign.
The maximum shear stress τmax = R is the largest shear stress that exists at the point, acting on planes oriented 45° from the principal planes. It's critical for ductile material failure prediction via the Tresca criterion: yielding occurs when τmax reaches half the yield strength (τmax ≥ σy/2).
In Mohr's Circle construction, physical plane rotations of θ correspond to 2θ rotations on the circle. This doubling arises from the trigonometric identities in the stress transformation equations: σn = σavg + R·cos(2θ - 2θp) and τn = R·sin(2θ - 2θp). The doubling allows a complete 360° traverse of the circle to correspond to a 180° rotation of the physical plane, which returns to the original stress state.

Mohr's Circle Glossary

Principal Stress

The normal stress acting on a plane where shear stress is zero. σ1 is the maximum, σ2 the minimum principal stress.

Plane Stress

A stress state where all stress components in one direction (typically z) are zero. Common in thin plates and surfaces.

Maximum Shear Stress

The largest shear stress magnitude at a point, equal to the radius R of Mohr's circle. Acts on planes 45° from principal planes.

Principal Angle (θp)

The angle from the x-axis to the direction of σ1. Calculated as θp = ½ arctan(2τxy / (σx − σy)).

Tresca Criterion

A failure theory stating yielding occurs when τmax reaches half the material's yield strength. Directly uses Mohr's Circle results.

von Mises Stress

An equivalent stress combining all stress components: σvm = √(σ1² − σ1σ2 + σ2²). Used for ductile failure prediction.

Stress Invariants

Quantities that remain unchanged under coordinate rotation. I1 = σx + σy (twice the center of Mohr's circle) is the first invariant.

Pure Shear

A stress state where normal stresses are zero and only shear stress exists. Mohr's circle is centered at the origin with radius |τxy|.

Editorial Review & Methodology

This Mohr's Circle calculator was built and reviewed by the NumbrWiz Editorial Team with input from practicing mechanical and structural engineers. The formulas and methodology are verified against standard mechanics of materials textbooks including Hibbeler, Beer & Johnston, and Timoshenko references.

  • Formula verification: All equations cross-checked against multiple authoritative sources in solid mechanics and strength of materials.
  • Edge case testing: Validated with pure shear, uniaxial tension, hydrostatic stress, and equal biaxial stress scenarios against analytical solutions.
  • Sign convention: Follows the standard engineering sign convention (tension positive) consistent with major textbooks and FEA software.
  • UX review: Designed for intuitive input with clear labeling, unit selection, and comprehensive step-by-step breakdown.

Transparency note: All calculations run client-side in your browser. No data is ever collected, stored, or transmitted. Results are for educational and preliminary design purposes; always verify critical calculations with appropriate engineering judgment and professional standards.

Page last reviewed: May 2026 · NumbrWiz Editorial Team