Principal stresses and Mohr’s circle

From a plane stress state σ_x, σ_y, τ_xy: the centre and radius of Mohr’s circle, the principal stresses, the angle of the principal planes, the maximum shear stress and the von Mises equivalent stress.

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The calculation

sigma_x := 80 MPa
sigma_y := -20 MPa
tau_xy := 30 MPa
sigma_c := (sigma_x + sigma_y)/2 = MPa
R := sqrt(((sigma_x - sigma_y)/2)^2 + tau_xy^2) = MPa
sigma_1 := sigma_c + R = MPa
sigma_2 := sigma_c - R = MPa
theta_p := atan2(2*tau_xy/(1 MPa), (sigma_x - sigma_y)/(1 MPa))/2*180/pi =
tau_max := R = MPa
sigma_vm := sqrt(sigma_1^2 - sigma_1*sigma_2 + sigma_2^2) = MPa

Results for these inputs

• Centre 30 MPa, radius 58.3 MPa.
• Principal stresses 88.3 MPa and −28.3 MPa, on planes at 15.5°.
• Maximum shear stress 58.3 MPa.
• Von Mises stress 105.4 MPa.

How it works

σ_1,2 = (σ_x + σ_y)/2 ± √(((σ_x − σ_y)/2)² + τ_xy²), tan 2θ_p = 2τ_xy/(σ_x − σ_y), and σ_vm = √(σ₁² − σ₁σ₂ + σ₂²). atan2 takes plain numbers, so the stresses are divided by 1 MPa there.

These calculations use classical textbook mechanics, not a design code. For design, use the method, factors and checks that apply to your work and have the result checked by a competent person.

Draw the circle

Plot Mohr’s circle as a parametric graph: plot([sigma_c/(1 MPa) + R/(1 MPa)*cos(t), R/(1 MPa)*sin(t)], t, 0, 2*pi) with the type set to Parametric. See the graph plotter.

Questions

How do I calculate principal stresses?

Take the average normal stress (σx + σy)/2 as the centre of Mohr’s circle and √(((σx − σy)/2)² + τxy²) as its radius; the principal stresses are centre ± radius.

What is the von Mises stress?

An equivalent stress that combines the principal stresses, σvm = √(σ1² − σ1σ2 + σ2²) in plane stress, often compared with the yield strength.

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