Mohr's Circle Calculator

• •Use the Mohr's Circle Calculator when solving homework or exam problems that require quick numerical verification of your hand calculations — instant feedback helps identify arithmetic errors before they propagate.
• •Use it during the early design phase to rapidly iterate on parameters and narrow down feasible configurations before committing time to detailed finite element simulations or full design packages.
• •Use it when reviewing a colleague's calculation or checking a vendor's data sheet for plausibility — a quick sanity check can prevent costly downstream errors.
• •Use it to generate reference data for a technical report or presentation without manual computation, ensuring consistent, reproducible numbers throughout the document.
• •Use it in the field when a quick estimate is needed and a full engineering software package is not available.

The Mohr's Circle Calculator is a precision engineering calculation tool designed for students, engineers, and technical professionals. Calculate principal stresses, maximum shear stress, and principal angle from a 2D stress state All calculations are performed using established engineering formulas from the relevant scientific literature and standards. Inputs support both metric (SI) and imperial unit systems, with unit conversion handled automatically — simply select your preferred unit from the dropdown next to each field. Results are computed instantly in the browser without sending data to a server, ensuring both speed and privacy. This calculator is intended as a supplementary tool for learning and design exploration; always verify results against authoritative references for safety-critical applications.

Mohr's circle is a graphical representation of the stress state at a point, showing how normal and shear stresses vary as the reference coordinate system rotates. Plotting normal stress σ on the horizontal axis and shear stress τ on the vertical axis, the state of stress at a point traces out a circle as the coordinate system rotates. The center of the circle is at σ_avg = (σx + σy)/2, and the radius is R = √[((σx − σy)/2)² + τxy²]. The principal stresses σ₁ and σ₂ (the maximum and minimum normal stresses with zero shear) are at σ_avg ± R. The principal angle θ_p = (1/2)·atan2(2τxy, σx − σy) gives the orientation of the coordinate system where the principal stresses are aligned with the axes. The maximum in-plane shear stress τ_max equals the circle radius R = (σ₁ − σ₂)/2 and occurs on planes rotated 45° from the principal directions. Mohr's circle is one of the most useful tools in stress analysis because it makes the coordinate-transformation equations visual and intuitive. It reveals quickly: which planes are most highly stressed (the principal planes), the magnitude of maximum stresses, the presence or absence of tensile stresses (important for brittle material failure), and the full range of stress states available at a single point by rotating the reference frame. Otto Mohr introduced the graphical construction in 1882, and it remains the standard teaching tool for 2D stress analysis, though the analytical equations are now typically used for computer calculations. The calculator computes the principal stresses, maximum shear, and principal angles from the input stress components and displays the circle graphically for intuition and verification.

• •Principal stress analysis: given the stress state at a point (σx, σy, τxy) from finite-element analysis or hand calculation, compute the principal stresses σ₁ and σ₂. The principal stresses are what matter for failure prediction in most failure theories.
• •Failure theory evaluation: the maximum normal stress theory uses σ₁ vs tensile strength for brittle materials. The maximum shear stress theory uses τ_max vs shear strength for ductile materials. Both require Mohr's circle to find the maxima.
• •Pressure vessel analysis: a thin-walled cylinder has hoop stress σ_h = PR/t and longitudinal stress σ_l = PR/(2t), with zero shear. Mohr's circle shows the principal stresses are just σ_h and σ_l (no rotation needed), and the maximum shear is (σ_h − σ_l)/2 = PR/(4t).
• •Welded joint analysis: a weld carrying combined shear and normal loads produces a complex stress state. Mohr's circle gives the principal stresses used to check against weld allowables.
• •Shaft under combined bending and torsion: the outer surface of a shaft under combined loads has bending stress (normal) and torsional stress (shear). Mohr's circle gives the maximum principal stress and shear stress for strength checking.