Mechanical Engineering Calculator

Mohr's Circle Calculator

Determine the principal stresses (σ₁, σ₂), maximum shear stress (τ_max), and principal planes by transforming a 2D plane stress state using Mohr's Circle equations.

Mohr's Circle Geometry

Graphical representation of stress transformation

CONCEPTUAL

Input Stress State

Sign convention: Tension (+), Compression (-)

Normal Stress X (σx)

MPa

Normal stress acting on the X-face.

Normal Stress Y (σy)

MPa

Normal stress acting on the Y-face.

The value must be greater than zero.

Shear Stress XY (τxy)

MPa

In-plane shear stress.

Mohr's Circle Properties

Center (Avg Stress)

20.00 MPa

Radius (Max Shear)

65.00 MPa

Principal Stresses

Max Principal (σ₁)

85.00 MPa

Min Principal (σ₂)

-45.00 MPa

Angle to Principal Plane

11.31°

Angle to Max Shear

-33.69°

Engineering Code

Automate stress transformations in your structural analysis scripts.

Python

import math

def calculate_mohrs_circle(sx, sy, txy):
    """
    Calculate Mohr's Circle parameters for 2D plane stress.
    """
    if not all(math.isfinite(x) for x in [sx, sy, txy]):
        raise ValueError('Inputs must be finite numbers.')

    center = (sx + sy) / 2
    radius = math.sqrt(((sx - sy) / 2)**2 + txy**2)

    sigma1 = center + radius
    sigma2 = center - radius
    tau_max = radius

    # Principal Angle (in degrees)
    theta_p_rad = 0.5 * math.atan2(2 * txy, sx - sy)
    theta_p_deg = math.degrees(theta_p_rad)

    return {
        'Center': center,
        'Radius': radius,
        'Sigma1': sigma1,
        'Sigma2': sigma2,
        'TauMax': tau_max,
        'ThetaP': theta_p_deg
    }

# Example Usage
sx, sy, txy = 80, -40, 25
res = calculate_mohrs_circle(sx, sy, txy)

print(f"Center (Avg Stress): {res['Center']:.2f} MPa")
print(f"Radius (Max Shear):  {res['Radius']:.2f} MPa")
print(f"Principal Stresses:  σ1 = {res['Sigma1']:.2f}, σ2 = {res['Sigma2']:.2f} MPa")
print(f"Principal Angle:     {res['ThetaP']:.2f}°")

MATLAB

function res = calculate_mohrs_circle(sx, sy, txy)
    % Calculate Mohr's Circle parameters for 2D plane stress

    if ~isfinite(sx) || ~isfinite(sy) || ~isfinite(txy)
        error('All inputs must be finite numbers.');
    end

    res.Center = (sx + sy) / 2;
    res.Radius = sqrt(((sx - sy) / 2)^2 + txy^2);

    res.Sigma1 = res.Center + res.Radius;
    res.Sigma2 = res.Center - res.Radius;
    res.TauMax = res.Radius;

    % Principal Angle (in degrees)
    theta_p_rad = 0.5 * atan2(2 * txy, sx - sy);
    res.ThetaP = rad2deg(theta_p_rad);
end

% Example Usage
sx = 80; sy = -40; txy = 25;
res = calculate_mohrs_circle(sx, sy, txy);

fprintf('Principal Stresses: σ1 = %.2f MPa, σ2 = %.2f MPa\n', res.Sigma1, res.Sigma2);
fprintf('Maximum Shear (Radius): %.2f MPa\n', res.TauMax);

Governing Formula

C = (σx + σy) / 2

CMPa

Center of Mohr's Circle

σx, σyMPa

Applied Normal Stresses

Governing Formula

R = √ [((σx-σy)/2)² + τxy²]

RMPa

Radius of the Circle (τ_max)

τxyMPa

Applied Shear Stress

Technical Explanation: Mohr's Circle for Plane Stress

Technical Introduction

Developed by Christian Otto Mohr in 1882, Mohr's Circle is a two-dimensional graphical representation of the stress transformation equations. It allows engineers to visually and mathematically determine the state of stress at any rotated plane within a material subjected to combined loading. In real engineering work, components rarely experience pure tension or pure shear; instead, they face complex, multi-directional loads. Calculating the transformed stress state is essential for predicting material failure, optimizing cross-sections, and ensuring structural integrity.

Governing Formula

The calculator determines the stress state using the geometric properties of the circle:

Center of the circle (C): C = (σx + σy) / 2
Radius of the circle (R): R = √ [ ((σx - σy) / 2)² + τxy² ]
Principal stresses: σ₁ = C + R and σ₂ = C - R
Maximum in-plane shear stress: τ_max = R

These equations assume a plane stress condition. The units for all stress inputs and outputs must be consistent (e.g., MPa or psi).

Calculation Method / Derivation

The formulas are derived from the 2D stress transformation equations, which enforce static equilibrium on a rotated infinitesimal element:

σ_x' = (σx + σy)/2 + (σx - σy)/2 cos(2θ) + τxy sin(2θ)
τ_x'y' = -(σx - σy)/2 sin(2θ) + τxy cos(2θ)

By isolating the trigonometric terms, squaring both equations, and adding them together, the angle θ is eliminated. This yields the standard equation of a circle: (σ_x' - C)² + τ_x'y'² = R². This geometric representation elegantly shows that the maximum normal stress (principal stress) occurs where the circle intersects the horizontal axis (τ = 0), and the maximum shear stress occurs at the top and bottom vertices of the circle.

Input Parameters

Normal Stress (σx): The axial or bending stress acting on the vertical face of the stress element. Tensile stress is positive; compressive stress is negative.
Normal Stress (σy): The normal stress acting on the horizontal face of the element.
Shear Stress (τxy): The shear stress acting on the element faces. While sign conventions for shear can vary between textbooks, the magnitude of τxy directly dictates the radius of the circle and thus the maximum shear stress.

Engineering Interpretation

The calculated principal stresses (σ₁, σ₂) and maximum shear stress (τ_max) are the foundational inputs for material failure theories.

Brittle materials (e.g., cast iron, concrete) typically fail due to normal tension. Engineers compare σ₁ against the material's ultimate tensile strength (Maximum Normal Stress Theory).
Ductile materials (e.g., steel, aluminum) fail due to shear slip. Engineers use τ_max (Tresca criterion) or a combination of σ₁ and σ₂ (von Mises criterion) to check against the material's yield strength.

Note: This calculator provides the stress state only. It does not replace a complete design verification, which must also incorporate material properties, safety factors, stress concentrations, and fatigue considerations.

Units and Unit Conversion

All stress inputs must share the same unit system. Common engineering units include:

1 MPa = 1 N/mm²
1 GPa = 1,000 MPa
1 ksi = 1,000 psi

Mixing units (e.g., σx in MPa and τxy in psi) will produce mathematically invalid results. Convert all inputs to a single base unit before calculation.

Alternative / General Case

This calculator is strictly for 2D plane stress, where out-of-plane stresses (σz, τxz, τyz) are assumed to be zero. The more general case is a 3D stress state, which requires solving a cubic characteristic equation to find three principal stresses (σ₁, σ₂, σ₃). Additionally, for thick components where out-of-plane strain is constrained, a plane strain condition applies, which modifies the out-of-plane normal stress to σz = ν(σx + σy), where ν is Poisson's ratio.

Assumptions and Limitations

Plane Stress: Valid primarily for thin plates or surfaces of 3D bodies where through-thickness stress is negligible.
Static Equilibrium: The derivation assumes the infinitesimal element is in static equilibrium.
Small Deformations: The geometry of the element is assumed to remain unchanged under load, which is standard for linear elastic analysis.
Continuum Mechanics: While the stress transformation is purely geometric and applies to any continuum, the subsequent failure interpretation assumes uniform, homogeneous material properties.

Practical Engineering Use

Mohr's Circle calculations are ubiquitous in mechanical and structural engineering:

Shaft Design: Evaluating combined bending (normal stress) and torsion (shear stress) in rotating machinery.
Pressure Vessels: Analyzing the stress state at welded joints where hoop and longitudinal stresses interact with shear loads.
Geotechnical Engineering: Determining soil failure planes using a modified Mohr-Coulomb failure criterion (often with a compression-positive sign convention).
Composite Materials: Transforming global laminate stresses into local fiber-direction stresses to check for ply failure.

Real-World Engineering Cases

The Electric Tow Tractor Drive Shaft

An engineer sized a heavy-duty drive shaft by looking at the bending stress from the vehicle weight and the torsional stress from the electric motor separately. Both values were below the yield limit. However, the shaft snapped during max-torque acceleration. They failed to calculate the combined Principal Stress (σ1), which exceeded the material's yield strength.

Engineering Lesson

Stresses do not act in isolation. Normal (bending) and Shear (torsion) stresses combine to create Maximum Principal Stresses on an angled plane. Mohr's Circle is the ultimate reality check for combined loading scenarios.

The 45-Degree Brittle Fracture

A cast iron torsion bar in a mechanical linkage failed unexpectedly. The designers were looking for a clean, vertical shear failure, but the bar shattered along a perfect 45-degree helix angle. Cast iron is brittle and weak in tension.

Engineering Lesson

Mohr's Circle proves that pure shear creates maximum principal tensile stress at exactly 45 degrees. Brittle materials will always fail along this tensile plane rather than the shear plane. Understanding the stress state orientation is critical when selecting materials.

Frequently Asked Questions

What are principal stresses?

Principal stresses are the maximum and minimum normal stresses acting on planes where the shear stress is exactly zero. They are critical for predicting the failure of brittle materials.

Why is maximum shear stress important?

Maximum shear stress is represented by the radius of Mohr's Circle. It dictates the failure of ductile materials like steel or aluminum, which tend to slip and yield along planes of maximum shear.

What is the standard sign convention for Mohr's Circle?

Typically, tensile normal stresses are plotted to the right (positive), and compressive normal stresses to the left (negative). Shear stresses that cause clockwise rotation are often plotted in the positive direction depending on specific textbook conventions.

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