Mohr's Circle — Stress Analysis
Principal Stresses • Max Shear • Stress Transformation • Rotation — Simulate • Explore • Practice • Quiz
1 Overview
The Mohr’s Circle Simulator is an interactive visualisation tool for 2D plane stress analysis. It lets you input a stress state (σx, σy, τxy) and instantly see the corresponding Mohr’s Circle, stress element, and all key results: principal stresses σ1 and σ2, maximum shear stress τmax, the principal angle θp, and the stress transformation at any arbitrary angle θ.
Mohr’s Circle converts the stress transformation equations into a simple geometric construction. A rotation of θ on the physical element corresponds to 2θ on the circle — this fundamental 2:1 relationship is the key to interpreting every result.
2 Getting Started
The simulator opens in Simulate mode with a default stress state (σx = 80 MPa, σy = −40 MPa, τxy = 50 MPa). The canvas shows both the stress element (left) and Mohr’s Circle (right), with the current stress point highlighted. Below the canvas are four sliders: σx, σy, τxy, and θ (rotation angle).
Try the Presets to load common stress states instantly: Pure Tension, Pure Shear, Biaxial Equal, and General Stress. The readout cards below the sliders display σ1, σ2, τmax, θp, σavg (circle centre), and the radius R.
3 Simulate Mode
In Simulate mode, drag the sliders to define your plane stress state. The σx and σy sliders range from −200 to +200 MPa; positive values are tension, negative values are compression. The τxy slider ranges from −150 to +150 MPa.
As you change the stress inputs, the circle redraws in real time. The centre moves to σavg = (σx + σy) / 2 and the radius becomes R = √[((σx − σy) / 2)² + τxy²]. The principal stresses are σ1 = σavg + R and σ2 = σavg − R.
Use the θ slider (0° to 180°) to rotate the stress element. The corresponding point traces around the circle at 2θ, and the transformed normal and shear stress values update on the element diagram. This visually demonstrates the stress transformation equations.
4 Explore Mode
Switch to Explore mode to study 12 concepts organised into three categories: Stress Basics (normal stress, shear stress, sign conventions), Mohr’s Circle (construction steps, principal stresses, maximum shear, rotation), and Applications (Von Mises criterion, failure theories, combined loading).
Each concept includes a detailed explanation with diagrams drawn on the canvas. Use these as a structured study guide to build your understanding before attempting Practice and Quiz modes.
5 Practice & Quiz
Practice mode generates random stress states and asks you to calculate a specific quantity — for example, “Find σ1 for σx = 60, σy = −30, τxy = 40 MPa.” Enter your numeric answer, click Check, and receive instant feedback with your running score. Step-by-step solutions are shown after an incorrect attempt.
Quiz mode presents 5 randomised questions mixing multiple-choice and numeric formats. Topics include principal stress calculation, maximum shear stress, principal angle θp, and Von Mises equivalent stress. Your final score is displayed with a detailed review of each question.
6 Tips & Best Practices
- Remember the 2:1 rule: a physical rotation of θ corresponds to 2θ on Mohr’s Circle. The principal angle θp = ½ arctan(2τxy / (σx − σy)).
- The maximum shear stress always equals the circle’s radius: τmax = R = (σ1 − σ2) / 2.
- For pure shear (σx = σy = 0), the principal stresses are equal and opposite: σ1 = +τxy and σ2 = −τxy.
- For biaxial equal stress (σx = σy, τxy = 0), Mohr’s Circle collapses to a point — no shear on any plane.
- Use the Von Mises formula σv = √(σ1² − σ1·σ2 + σ2²) to check ductile yielding after finding principal stresses.
- Compare your hand-calculated θp value with the simulator’s readout to build confidence in your trigonometric approach.
- Set θ to θp and verify that the shear stress on the element becomes zero — confirming the principal plane definition.
Mohr's Circle — Stress Analysis and Transformation
Mohr's Circle is one of the most important graphical tools in mechanics of materials and solid mechanics. Developed by the German civil engineer Christian Otto Mohr in 1882, it provides an elegant graphical method for determining the state of stress at a point on a body subjected to plane stress conditions. By plotting normal stress (σ) on the horizontal axis and shear stress (τ) on the vertical axis, engineers can instantly visualise how stresses transform as the orientation of the plane changes. This simulator lets you explore Mohr's Circle interactively — adjusting σx, σy, τxy, and the rotation angle θ to see how the stress element and the corresponding point on the circle change simultaneously.
Understanding Plane Stress and the Stress Element
In plane stress analysis, we consider a thin element where all stresses act in one plane. The state of stress at a point is defined by three components: the normal stress σx acting in the x-direction, the normal stress σy acting in the y-direction, and the shear stress τxy acting on the x- and y-faces. A positive normal stress indicates tension (pulling the element apart), while a negative value indicates compression (pushing it together). Shear stress follows a sign convention where positive τxy causes clockwise rotation of the element. The stress element is a small square drawn at the material point showing all these stress components with arrows on each face. When the element is rotated by an angle θ, the stress components transform according to the stress transformation equations, and Mohr's Circle provides a graphical representation of these equations.
Constructing Mohr's Circle
To construct Mohr's Circle: (1) Plot point X = (σx, τxy) and point Y = (σy, −τxy) on the σ–τ plane. (2) Connect X and Y with a straight line — this is the diameter of the circle. (3) The centre of the circle is at (σavg, 0) where σavg = (σx + σy) / 2. (4) The radius of the circle is R = √(((σx − σy) / 2)² + τxy²). The rightmost point on the σ-axis gives the maximum principal stress σ1 = σavg + R, and the leftmost gives the minimum principal stress σ2 = σavg − R. The top and bottom of the circle give the maximum shear stress τmax = R. The angle from the diameter line to the σ-axis is 2θp, where θp is the principal angle — the rotation needed to align the element with the principal stress directions.
Stress Transformation Equations
The transformed normal stress on an inclined plane at angle θ is given by σn = σavg + R · cos(2θ − 2θp), and the transformed shear stress is τn = R · sin(2θ − 2θp). These equations correspond to tracing a point around Mohr's Circle. As θ varies from 0° to 180°, the point travels a full 360° around the circle. This 2:1 relationship between the rotation on the physical element and the angle on Mohr's Circle is a fundamental property: an angle θ on the element corresponds to 2θ on the circle. Understanding this relationship is essential for correctly interpreting Mohr's Circle and applying it to real engineering problems.
Applications in Engineering Design
Mohr's Circle is widely used in structural, mechanical, and aerospace engineering for failure analysis and design. The Von Mises equivalent stress σv = √(σ1² − σ1·σ2 + σ2²) is derived from principal stresses obtained via Mohr's Circle and is compared against the material yield strength to determine whether yielding occurs. Engineers use Mohr's Circle to identify critical stress states in pressure vessels, shafts under combined loading, welded joints, and composite materials. This simulator's Simulate mode lets you change all stress inputs and see both the stress element and Mohr's Circle update simultaneously. Use Explore mode to study 12 concepts across stress basics, circle construction, and applications. Practice mode generates random problems, and Quiz tests your knowledge with randomised questions.
Key Formulas at a Glance
For quick reference, the essential Mohr's Circle formulas are: Centre: σavg = (σx + σy) / 2. Radius: R = √(((σx − σy) / 2)² + τxy²). Principal stresses: σ1 = σavg + R and σ2 = σavg − R. Maximum shear: τmax = R = (σ1 − σ2) / 2. Principal angle: θp = ½ · arctan(2τxy / (σx − σy)). Transformation: σn = σavg + R·cos(2θ − 2θp) and τn = R·sin(2θ − 2θp). Von Mises: σv = √(σ1² − σ1·σ2 + σ2²). These formulas form the complete toolset for 2D stress analysis using Mohr's Circle.
Failure Theories and Mohr's Circle
Mohr's Circle provides the principal stresses needed for all major failure theories. The Maximum Normal Stress Theory (Rankine) predicts failure when σ1 exceeds the ultimate tensile strength — applicable to brittle materials. The Maximum Shear Stress Theory (Tresca) predicts yielding when τmax = (σ1 − σ2)/2 exceeds the shear yield strength — slightly conservative for ductile metals. The Distortion Energy Theory (Von Mises) is the most accurate for ductile materials, using the equivalent stress σv. By computing principal stresses from Mohr's Circle, engineers can quickly evaluate factor of safety under any combined loading condition.
Who Uses This Simulator?
This tool is designed for mechanical and civil engineering students, strength of materials learners, design engineers, and instructors teaching stress analysis. It provides visual, interactive understanding of Mohr's Circle without requiring laboratory equipment or complex FEA software. Whether you are preparing for exams, solving homework problems, or verifying hand calculations, this simulator offers an intuitive way to master 2D stress analysis and transformation.
Mohr’s Circle Formulas — 2D Stress Transformation
| Parameter | Formula | Description |
|---|---|---|
| Centre of Circle | C = (σx + σy) / 2 | Average normal stress |
| Radius of Circle | R = √[((σx−σy)/2)² + τxy²] | Half-range of principal stresses |
| Maximum Principal Stress | σ1 = C + R | Largest normal stress on any plane |
| Minimum Principal Stress | σ2 = C − R | Smallest normal stress on any plane |
| Maximum Shear Stress | τmax = R | Occurs at 45° to principal planes |
| Principal Angle | 2θp = arctan(2τxy / (σx−σy)) | Orientation of principal planes |
Explore Related Simulators
If you found this Mohr’s Circle simulator helpful, explore our Thin-Walled Pressure Vessel simulator, Stress–Strain Curve simulator, Beam Bending simulator, and Shaft Torsion simulator for more hands-on practice.