Mohr’s Circle Calculator
Visualize & Calculate Stress Transformations in 5 Minutes
Struggling to determine principal stresses or maximum shear stress in your mechanics class or design project? Mohr’s Circle is a powerful graphical and computational tool that transforms complex stress analysis into visual clarity. Our interactive calculator provides instant results with visual circle representation, making stress transformation analysis accessible for students, engineers, and designers alike.
Why Mohr’s Circle Matters for Mechanical Engineers
Mohr’s Circle is a graphical representation of the stress transformation equations at a point within a material. Named after German civil engineer Christian Otto Mohr, this elegant geometric construction converts algebraic stress transformation equations into an intuitive visual format that reveals critical information about stress states at any orientation.
Rather than solving trigonometric equations repeatedly for different angles, Mohr’s Circle allows engineers to visualize all possible stress states on a material element simultaneously. Every point on the circle represents the normal and shear stress on a plane at a specific orientation—making it an indispensable tool for stress analysis.
Design Applications
Determine principal stresses for evaluating failure criteria in machine elements, pressure vessels, and structural components. Identify critical stress states that govern yielding, fracture, or fatigue life.
Stress Transformation
Calculate normal and shear stresses on any oriented plane without complex trigonometric calculations. Essential for analyzing welds, joints, and inclined sections in structural analysis.
Failure Analysis
Identify maximum shear stress planes for ductile material failure prediction and principal stress planes for brittle fracture assessment. Critical for safe design of mechanical systems.
Academic Excellence
Master mechanics of materials coursework by visualizing abstract concepts. Verify hand calculations, check homework solutions, and develop intuitive understanding of stress behavior.
Real-World Engineering Impact
From designing shafts subjected to combined bending and torsion to analyzing welded joints under complex loading, Mohr’s Circle bridges theoretical mechanics and practical engineering. It supports students mastering fundamental concepts while providing working engineers with rapid stress analysis capabilities essential for design validation, failure investigation, and finite element analysis verification.
Theory & Underlying Concepts
Stress Components & Coordinate Systems
Understanding stress components is fundamental to applying Mohr’s Circle. Consider a small material element subjected to forces creating internal stresses. These stresses act on the element’s faces and are characterized by:
Normal Stresses (σ)
Stresses acting perpendicular to the face of an element. Denoted as σₓ (acting in x-direction on x-face) and σᵧ (acting in y-direction on y-face).
- Tensile stress: Positive sign convention—material is being pulled apart
- Compressive stress: Negative sign convention—material is being squeezed
Shear Stresses (τ)
Stresses acting parallel to the face of an element, creating a sliding or shearing action. Denoted as τₓᵧ (acts on x-face in y-direction) and τᵧₓ (acts on y-face in x-direction).
- Complementary shear: For equilibrium, τₓᵧ = τᵧₓ in magnitude
- Sign convention: Positive when creating clockwise rotation on the element
Stress Transformation Equations
When a material element is rotated by an angle θ from the original x-y coordinate system, the stress components on the new faces change. The transformation equations relate original stresses to stresses on the rotated element:
Normal Stress on Rotated Plane:
Shear Stress on Rotated Plane:
Understanding the Variables
- θ (theta): Rotation angle of the element measured counterclockwise from the x-axis
- 2θ (two-theta): Appears in equations due to double-angle trigonometric identities—this is why angles on Mohr’s Circle are doubled
- σₙ: Normal stress on the rotated plane
- τₙ: Shear stress on the rotated plane
Constructing Mohr’s Circle: Step-by-Step
Mohr’s Circle is constructed in stress space where the horizontal axis represents normal stress (σ) and the vertical axis represents shear stress (τ). Follow these systematic steps:
Plot the Initial Stress Points
Plot point A with coordinates (σₓ, τₓᵧ) representing the stress state on the x-face of the original element.
Plot point B with coordinates (σᵧ, -τₓᵧ) representing the stress state on the y-face. Note the negative sign on shear stress for the y-face.
Determine the Circle Center
The center C of Mohr’s Circle lies on the σ-axis (where τ = 0) at:
This represents the average normal stress, which remains constant regardless of element orientation—an invariant property of the stress state.
Calculate the Circle Radius
The radius R is the distance from the center to either point A or B:
This radius equals the maximum shear stress that can occur at any orientation of the element.
Draw the Complete Circle
Using center C and radius R, draw the complete circle. Every point on this circle represents a valid stress state at some angle θ. The circle encompasses all possible combinations of normal and shear stress for the given loading condition.
Important Geometric Properties
- Points A and B are diametrically opposite on the circle, representing perpendicular faces
- Moving around the circle by angle 2θ corresponds to rotating the element by angle θ
- The circle always intersects the σ-axis at two points—these are the principal stresses
- The topmost and bottommost points of the circle represent maximum shear stress states
Finding Principal Stresses & Maximum Shear Stress
The most valuable information extracted from Mohr’s Circle includes the principal stresses and maximum shear stress—critical values for failure analysis and design optimization.
Principal Stresses
Principal stresses are the maximum and minimum normal stresses that occur at specific orientations where shear stress equals zero. These planes are called principal planes.
Maximum Principal Stress (σ₁)
The rightmost point where the circle intersects the σ-axis.
Minimum Principal Stress (σ₂)
The leftmost point where the circle intersects the σ-axis.
Principal Plane Orientation
The angle to the principal planes from the x-axis is:
This equation yields two angles 90° apart, corresponding to the two principal planes. Remember: the angle on the circle is 2θ, so divide by 2 to get the actual element rotation angle θp.
Maximum Shear Stress
Maximum shear stress occurs at planes oriented 45° from the principal planes. This value is critical for ductile material failure prediction using the Tresca yield criterion.
Maximum In-Plane Shear Stress
Equals the radius of Mohr’s Circle. Found at the highest and lowest points on the circle.
Maximum Shear Plane Orientation
The planes of maximum shear stress are oriented at θs = θp ± 45° from the x-axis. On Mohr’s Circle, these points are 90° from the principal stress points.
Normal Stress on Maximum Shear Planes
Importantly, planes of maximum shear stress are not stress-free. They experience normal stress equal to the average normal stress:
Extension to 3-D Stress States
While our calculator focuses on plane stress (2-D) for accessibility, real engineering components often experience three-dimensional stress states with three principal stresses: σ₁, σ₂, and σ₃.
Mohr’s Circles for 3-D Stress
A complete 3-D stress state requires three Mohr’s Circles:
- Large circle: Between σ₁ (maximum) and σ₃ (minimum principal stresses)
- Medium circle: Between σ₁ and σ₂
- Small circle: Between σ₂ and σ₃
The absolute maximum shear stress is the radius of the largest circle:
Advanced Mohr’s Circle Calculator
Professional stress transformation analysis with interactive visualization
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