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Principal Stress Calculator

This principal stress calculator helps you calculate the amount of normal stress acting on a single major plane. Find the maximum, minimum, and angle of principal stress using this calculator.

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Maximum principal stress σ₁134.3398 MPa134.3398 MPa
Minimum principal stress σ₂-54.3398 MPa
Principal plane angle16.0027°rotate the element by this much to find the plane where the shear stress vanishes
Maximum in-plane shear94.3398 MPathe radius of Mohr's circle, at 45° to the principal planes
Absolute maximum shear94.3398 MPathe same as the in-plane value here
Mohr's circle centre40 MPathe average of the two normal stresses
Mohr's circle radius94.3398 MPa
von Mises equivalent stress168.226 MPathe single number to compare against yield for a ductile material
Tresca equivalent stress188.6796 MPatwice the maximum shear — the more conservative criterion, up to 15% higher than von Mises
Difference between the criteria12.16%Tresca never predicts a higher capacity than von Mises, so it is the safe choice when in doubt
Stress invariant σx + σy80 MPaequals σ₁ + σ₂ — invariant under rotation, which is a useful check on any transformation
Factor of safety, von Mises1.4861
Factor of safety, Tresca1.325always the lower of the two

The formula

σ₁,₂ = (σx+σy)/2 ± √(((σx−σy)/2)² + τxy²)

Rotate the element and the numbers change

Stress is not a single value but a tensor: the normal and shear components you measure depend on the orientation of the face you measure them on. The principal stresses are the values on the one orientation where the shear vanishes entirely, and they are the largest and smallest normal stresses any orientation can show.

Mohr's circle is the geometric picture of that. Its centre is the average normal stress, its radius the maximum shear, and rotating the element by θ moves you 2θ around the circle. The maximum shear always occurs at 45° to the principal planes, which is why a ductile bar in tension fails along a 45° cone.

Two failure criteria are in common use and they disagree by up to 15%. Von Mises matches experiment better for ductile metals; Tresca — simply twice the maximum shear — is more conservative and never predicts a higher capacity, so it is the safer default. Note the absolute maximum shear can exceed the in-plane value: in plane stress the third principal stress is zero, and if both in-plane stresses share a sign that zero widens the three-dimensional circle.