Tools · Strength of materials
Mohr’s circle for stresses
A guided walk-through: plot the two faces, find the centre and radius, then read off principal stresses, principal planes and maximum shear. Try each calculation yourself before the answer is shown, then rotate the element freely.
Guided · step 1 of 9
Read the element
Look at the stresses on the element. Tension is positive. τxy is positive when it acts upward on the right face (and so to the right on the top face). Every face of the element will become one point on the circle.
σx = 80 MPa, σy = −40 MPa, τxy = 30 MPa
Stresses on the element (MPa)
Examples
σx′ 80 · σy′ −40 · τ 30
Full solution, all steps at once
- Find the centre CC = σx + σy2
C = 80 + (−40)2 = 20 MPa
- Find the radius R and draw the circleR = √(σx − σy2)² + τxy²
R = √(80 − (−40)2)² + 30² = √60² + 30² = 67.082 MPa
- Read the principal stressesσ1,2 = C ± R
σ1 = 20 + 67.082 = 87.082 MPa
σ2 = 20 − 67.082 = −47.082 MPa
- Find the principal planestan 2θp = 2τxyσx − σy
tan 2θp = 2 × 3080 − (−40) = 0.5
2θp = 26.57°, so θp = 13.28° (plane of σ1); the plane of σ2 is at −76.72°
- Find the maximum shearτmax = R, at θs = θp − 45°
τmax = R = 67.082 MPa at θs = −31.72°, with σ = C = 20 MPa on those planes
Absolute maximum shear (σz = 0): largest of |σ1 − σ2|2, |σ1|2, |σ2|2 = 67.082 MPa
Conventions: tension positive; τxy positive when it acts upward on the right face; θ anticlockwise from x. On the circle, shear is plotted positive downward so that the circle and the element turn the same way (2θ on the circle for θ on the element).