Stress Analysis Formula Sheet
Biaxial stress & strain Poisson’s ratio σx = E1-v2 (εx + vεy) εx = 1E (σx - vσy) Uniaxial: v = - εyεx σy = E1-v2 (εy + vεx) εy = 1E (σy - vσx) Biaxial: v = E2G - 1 εz = - vE (σx + σy) Triaxial stress & strain Uniaxial Stress & strain εx = σxE – v σyE – v σzE σx = E(1+v)(1-2v) [(1 – v) εx + v(εy + εz)] σx=E.εx εy = εz = - v σxE εy = σyE – v σxE – v σzE σy = E(1+v)(1-2v) [(1 – v) εy + v(εz + εx)] σx = PA εx = δlo εz = σzE – v σxE – v σyE σz = E(1+v)(1-2v) [(1 – v) εz + v(εx + εy)] F.S = Actual strength σutsRequired strength σw
Pressure vessels
Hoop stress (cyl.) Longitudinal (sph.) P=Int. pressure, t=thickness σ1 = Prt σ2 = Pr2t σ1 = 2σ2 Shear Stress & Strain Pure shear Mod of rigidity τ = shear force vx-area A εx= εy= εz = 0 Strain in rads. γxy= τxyG G =E2(1+v) G = τγ γxy = τxyG γyz = τyzG γzx = zxG
Torsion circular members Angle of twist ∅ γ = ρ ∅L γmax= c ∅L (c=ρ) γ = ρ cγmax (Radius ρ) ∅= T LJ G → T = J GL ∅ = L γminc1 τ = T ρJ τmax = T cJ → T = π2 c3τmax τ = ρ
Pressure vessels
Hoop stress (cyl.) Longitudinal (sph.) P=Int. pressure, t=thickness σ1 = Prt σ2 = Pr2t σ1 = 2σ2 Shear Stress & Strain Pure shear Mod of rigidity τ = shear force vx-area A εx= εy= εz = 0 Strain in rads. γxy= τxyG G =E2(1+v) G = τγ γxy = τxyG γyz = τyzG γzx = zxG
Torsion circular members Angle of twist ∅ γ = ρ ∅L γmax= c ∅L (c=ρ) γ = ρ cγmax (Radius ρ) ∅= T LJ G → T = J GL ∅ = L γminc1 τ = T ρJ τmax = T cJ → T = π2 c3τmax τ = ρ