Statics & Mechanics of Materials — Cheat Sheet
Equilibrium, stress, bending, torsion, deflection, buckling, combined loading.
Equilibrium
| Force balance | ΣFₓ = 0, ΣF_y = 0 |
| Moment balance | ΣM = 0 |
| Resultant | R = √(Fₓ² + F_y²), θ = atan(F_y/Fₓ) |
Section properties
| Centroid | x̄ = ΣAx / ΣA |
| 2nd moment of area | I = ∫y² dA |
| Parallel axis | I = I_c + A·d² |
| Section modulus | Z = I / c |
| Radius of gyration | r = √(I / A) |
Axial stress & strain
| Direct stress | σ = P / A |
| Strain | ε = δ / L |
| Hooke's law | σ = E·ε |
| Elongation | δ = PL / AE |
| Shear modulus | G = E / [2(1+ν)] |
Bending & shear
| Bending stress | σ = M·c / I = M / Z |
| Transverse shear | τ = VQ / (I·b) |
| Radius of curvature | ρ = EI / M |
Torsion
| Shear stress | τ = T·r / J |
| Angle of twist | θ = TL / (GJ) |
| Polar moment (solid) | J = πd⁴ / 32 |
Stability & combined
| Euler buckling | P_cr = π²EI / (KL)² |
| Critical stress | σ_cr = π²E / (KL/r)² |
| Principal stress | σ₁,₂ = (σₓ+σ_y)/2 ± √[((σₓ−σ_y)/2)² + τₓy²] |
| Max shear | τ_max = √[((σₓ−σ_y)/2)² + τₓy²] |
| von Mises | σ′ = √(σ₁² − σ₁σ₂ + σ₂²) |
| Factor of safety | n = S_y / σ |
Thin pressure vessel
| Hoop stress | σ_θ = p·r / t |
| Longitudinal | σ_L = p·r / (2t) |
Formula reference — verify against the governing standard. Not a substitute for engineering judgment.