Lab · Structural Mechanics

Stress Transformation Lab

The same state of stress at a point, seen from every orientation at once. Sweep the rotation angle and watch the element and Mohr's circle move together — they are two drawings of a single engine result, never two separate calculations.

Stress state

Normal stress σx[MPa]

Normal stress σy[MPa]

Shear stress τxy[MPa]

Rotation θ (counter-clockwise)

30°

Major principal σ₁

88.31 MPa

at 15.482° CCW from +x

Minor principal σ₂

-28.31 MPa

at 105.48° CCW from +x

Maximum in-plane shear

58.31 MPa

on planes at -29.518°

Elements

Reference element. Tensile normal stress pulls outward, compressive pushes inward; a positive τ acts on the +x face in the +y direction.

Rotated 30° counter-clockwise. Tensile normal stress pulls outward, compressive pushes inward; a positive τ acts on the +x′ face in the +y′ direction.

Mohr's circle

τ is positive upward and both axes share one scale. Rotating the element counter-clockwise by θ moves the plotted face clockwise around the circle by 2θ.

Rotated state

Rotation θ	30°
σx′	80.981 MPa
σy′	-20.981 MPa
τx′y′	-28.301 MPa
Centre σavg	30 MPa
Radius R	58.31 MPa
Von Mises equivalent σᵥ	105.36 MPa
Invariant σx + σy	60 MPa → 60 MPa

What this Lab models

The state of stress at a single point of a body whose third face is free, so σz = 0. One engine evaluation produces the rotated components, the principal values, the Mohr geometry and the von Mises equivalent stress; the element drawings and the circle are two renderings of that one result and cannot disagree.

Model scope

· Plane stress only: the out-of-plane principal stress is zero and is not drawn on the in-plane circle.
· σ is tensile positive; τxy acts on the +x face in the +y direction; θ is a counter-clockwise rotation of the axes.
· A counter-clockwise physical rotation θ moves the plotted point clockwise by 2θ around Mohr's circle.
· Transformation is pure equilibrium and holds for any material; the von Mises value additionally assumes a ductile metal.