Mohr's Circle Calculator
Transform any plane-stress state (σx, σy, τxy) into principal stresses, maximum shear, and rotated stress components — the fastest way to check yield, fracture, or orientation.
σ₁,₂ = (σx+σy)/2 ± √[ ((σx−σy)/2)² + τxy² ]
Mohr Master transforms any plane-stress state (σx, σy, τxy) into principal stresses, maximum shear, and rotated stress components with a live interactive circle and step-by-step solutions.
Sign Conventions & Geometry
Mohr’s circle turns the algebra of stress transformation into pure geometry. Two conventions must be fixed before drawing: the sign of the stresses and the direction of the rotation angle.
Normal & Shear Stress
Tensile normal stress is positive; on Mohr’s circle, clockwise shears (positive τxy) plot upward. σ₁ is the algebraically larger principal stress.
Angle θ from the x-axis
Counterclockwise physical rotation (CCW θ) is positive. A CCW element rotation moves counterclockwise through 2θ on the circle — the principal planes are found at θp = ½·atan2(2τxy, σx − σy).
Verified Example States
σx = 80 MPa · σy = 40 MPa · τxy = 20 MPa
σx = σy = 0 · τxy = 50 MPa
σx = 100 MPa · σy = 0 · τxy = 0
σx = σy = 75 MPa · τxy = 0
In case (a): σ_avg = (80 + 40)/2 = 60 MPa, R = √(20² + 20²) = 28.28 MPa, so σ₁ = 88.28, σ₂ = 31.72 MPa, and 2θp = atan2(40, 40) = 45° → θp = 22.5°. In case (b) the circle is centered at the origin with radius 50, so pure shear is equivalent to equal-and-opposite principal stresses. In case (d) the radius vanishes and the circle collapses to a single point — every plane sees σ = 75 MPa and τ = 0.
Engineering Formulas & Governing Equations
Center of Mohr’s Circle (Average Stress)
Equation 01The circle center sits at the mean normal stress on the σ-axis.
Radius of Mohr’s Circle
Equation 02The radius measures the maximum shear stress available in the stress state.
Principal Stresses
Equation 03The two in-plane principal stresses occur where the circle crosses the σ-axis (τ = 0).
Principal Plane Angle
Equation 04The doubled angle 2θp appears because the circle spans 2θ in stress space. atan2 keeps the correct quadrant.
Maximum In-Plane Shear
Equation 05The largest shear acts on planes midway between the principal planes, where the normal stress equals σ_avg.
Stress at an Arbitrary Angle θ
Equation 06Rotating the element by θ in physical space rotates the point by 2θ on the circle.
Invariants of Plane Stress
Equation 07The sum and product of the two faces’ normal/shear terms are independent of the rotation angle θ.
Calculation Details & Clarifications
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