Mohr's Circle Calculator

Mohr's Circle is a graphical technique used in mechanics to visualize the transformation of a 2D stress state at a point in a material. By plotting normal stress on the horizontal axis and shear stress on the vertical axis, the circle's geometry directly reveals principal stresses, maximum shear stress, and the orientation of principal planes.

A stress state describes all the normal and shear stresses acting on an infinitesimal element at a specific point in a loaded body. For a 2D (plane stress) problem, the state is fully defined by σx, σy, and τxy — the three inputs used in this calculator.

Principal stresses (σ₁ and σ₂) are the maximum and minimum normal stresses at a point, occurring on planes where shear stress is zero. They are calculated as: σ₁,₂ = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²]. The angle between the x-axis and the principal plane is θp = 0.5 × arctan(2τxy / (σx − σy)).

Maximum shear stress (τmax) is the largest shear stress at a point, occurring on planes oriented 45° from the principal planes. It equals τmax = √[((σx − σy)/2)² + τxy²], which is also the radius of Mohr's Circle. It's critical for failure analysis — many ductile materials fail when τmax exceeds the shear strength.

Von Mises stress is a scalar measure used to predict yielding of ductile materials under multiaxial loading. For plane stress, it equals σ' = √(σ₁² − σ₁σ₂ + σ₂²). If von Mises stress exceeds the material's yield strength, plastic deformation is expected.

The rotation angle θ lets you compute the transformed stresses (σx', σy', τx'y') on an element rotated by that angle from the original orientation. This corresponds to moving along Mohr's Circle by an angle of 2θ, and is useful for evaluating stress on a specific inclined plane.

Mean stress, also called hydrostatic stress, is the average of the principal stresses: σm = (σ₁ + σ₂)/2 = (σx + σy)/2. It represents the center of Mohr's Circle and is important in fatigue analysis and soil mechanics.

Tensile normal stresses are positive; compressive normal stresses are negative. Shear stress τxy is positive when it acts in the positive y-direction on the positive x-face (counterclockwise convention). Rotation angles follow the standard mathematical convention — positive angles are counterclockwise.