Principal Stresses • Max Shear • Stress Transformation • Rotation — Simulate • Explore • Practice • Quiz
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.
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.
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.
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.
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.
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.
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.
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.