Structures: stress analysis, thin-walled sections, buckling and failure criteria
Structures covers stress and strain transformations, bending and shear in beams, torsion of thin-walled sections, buckling and yield/fatigue failure criteria - the section where units and section properties decide the answer.
Core rules
- Principal stresses: σ1,2 = (σx+σy)/2 ± √[((σx−σy)/2)² + τxy²]; Mohr's circle gives the same result graphically, with maximum shear radius.
- Bending: σ = My/I, τ = VQ/(It); for a rectangle I = bh³/12 and maximum shear is 1.5 V/A.
- Thin-walled torsion: closed section shear flow q = T/(2A) with J = 4A²/∮(ds/t) (Bredt–Batho); open section J ≈ Σbt³/3, which is orders of magnitude smaller.
- Shear centre: the load point producing no twist; it coincides with the centroid only for doubly symmetric sections.
- Buckling: Pcr = π²EI/(KL)², with K = 1 pinned–pinned, 0.5 fixed–fixed, 2 fixed–free.
- Failure: von Mises (distortion energy) for ductile metals, maximum principal stress for brittle materials.
Worked example
A rectangular beam 50 mm wide, 100 mm deep carries M = 10 kN·m. I = 50 × 100³/12 = 4.167 × 10⁶ mm⁴ = 4.167 × 10⁻⁶ m⁴. With c = 0.05 m, σ = Mc/I = (10⁴ × 0.05)/(4.167 × 10⁻⁶) = 1.2 × 10⁸ Pa = 120 MPa.
Traps
- Mixing millimetres and metres in I; a factor of 10¹² error is typical.
- Using Σbt³/3 for a closed box: the closed-section torsion constant is vastly larger.
- Applying Euler buckling outside the elastic range, where the tangent-modulus or Johnson parabola applies.
- Assuming the shear centre equals the centroid for a channel or angle section.