Stress transformation
Plane stress at a point: the stresses on a rotated plane, the principal stresses and their angle, the largest shear in the plane and in three dimensions, the von Mises stress, and Mohr's circle. Use it for a point in a plate, a web, a weld throat or a connection element where you have the stress components. It is mechanics of materials, in closed form.
Inputs of the stress state #
- σx, σy, τxy: the stress components, tension positive; τxy positive on the positive x face in the positive y direction (the convention of Gere and Goodno). For the von Mises check they are the required stresses of the method chosen: factored for LRFD, service for ASD.
- Angle θ: of the rotated plane, counterclockwise from x to the plane's normal, in degrees.
- Limit stress Fy, φ and Ω: the yield stress for first yield, or the limit you choose. 0.90 and 1.67 are prefilled, the AISC 360-22 factors for yielding; no code sets this check for a general stress state, so they are yours to set.
This is a check of one stress state: it has no auto-design.
Checks performed #
| Check | Basis |
|---|---|
| Stresses on the rotated plane | Transformation equations, σx1 + σy1 = σx + σy |
| Principal stresses and angle | σ1,2 = σavg ± R, tan 2θp = 2τxy/(σx − σy) |
| Largest in-plane shear and its plane | R, at 45° from the principal planes |
| Largest shear in three dimensions | σz = 0: half the largest difference of σ1, σ2 and 0 |
| Von Mises stress against the limit | √(σ1² − σ1σ2 + σ2²) ≤ φ Fy or Fy / Ω |
On Mohr's circle the normal stress is drawn to the right and the shear downward, so the diameter turns in the same sense as the element, by twice its angle.
Limitations of the stress transformation #
- Plane stress (σz = 0) at one point; no three-dimensional state, no strain transformation.
- Von Mises is a first-yield criterion for ductile metals: for brittle materials, concrete or wood use the criterion of their code.
- The stresses are entered: the calculation does not find them from forces and a section.
Verification examples #
Verification status: 25 values compared in 5 examples (1 published, 4 hand calculations from the cited equations): 25 within tolerance, 0 recorded deviations. Thin: only 1 published example. Code basis: written without the code document on hand, and confirmed only in part through published examples.
| Example | Source | Values |
|---|---|---|
| Element rotated 45° | Gere and Goodno, Mechanics of Materials, Ch. 7, first transformation example | Rotated stresses, principal stresses and angle, shears, von Mises |
| Metric input, element rotated 30° | Hand calculation, transformation equations | Rotated stresses in MPa |
| Pure shear, mixed signs, uniaxial compression | Hand calculation | Principal stresses, angles, largest shears, von Mises |
Gere and Goodno is not on hand: the numbers of its example are as recalled and are exactly what the transformation equations give.