Structural & Fatigue

Fatigue — S-N Curve & Crack Growth

Basquin σ_a=σ_f'(2N_f)^b sets the S-N line; Paris da/dN=C(ΔK)^m integrates remaining life from an initial flaw.

Basquin S-N

σ_f' (MPa)

σ_a applied (MPa)

Paris crack growth

Δσ (MPa)

a_i (mm)

K_IC (MPa√m)

Pressure Vessel — Hoop Stress

σ_hoop=Pr/t; σ_long=Pr/2t (thin wall).

Pressure P (bar)

Radius r (mm)

Wall t (mm)

Yield Sy (MPa)

Euler Buckling

P_cr=π²EI/(KL)²; I=bh³/12.

E (GPa)

Length L (mm)

b (mm)

h (mm)

K (end)

Applied load (kN) — for MS

Von Mises Check

σ_vm=√(σ₁²−σ₁σ₂+σ₂²) ≤ Sy/SF.

σ₁ (MPa)

σ₂ (MPa)

Yield Sy (MPa)

Safety factor

Flat Plate Deflection

D=Et³/12(1−ν²); δ=q·a⁴/(D·C).

E (GPa)

t (mm)

a (mm)

q (kPa)

Boundary

Bolt Preload & Margin

F_pre=T/(K·d); MS=F_allow/F_applied−1.

Torque T (N·m)

K (nut factor)

d (mm)

Allowable (kN)

Applied (kN)

Stress Concentration & Fracture

K_t=1+2√(a/ρ); fracture when K=Yσ√(πa) ≥ K_IC.

Notch depth a (mm)

Tip radius ρ (mm)

σ nominal (MPa)

K_IC (MPa√m)

Miner's Cumulative Damage

D=Σ(nᵢ/Nᵢ); failure at D≥1.

Block	n applied	N to fail

Formula reference

σ_hoop=Pr/t · P_cr=π²EI/(KL)² · σ_vm=√(σ₁²−σ₁σ₂+σ₂²) · D=Et³/12(1−ν²), δ=qa⁴/DC · F_pre=T/Kd · MS=F_allow/F_applied−1.

Basquin σ_a=σ_f'(2N_f)^b → N_f=½(σ_a/σ_f')^(1/b) · Goodman σ_a/σ_e+σ_m/σ_u=1 · Miner Σnᵢ/Nᵢ=1.

K_t=1+2√(a/ρ) · K=Yσ√(πa), fracture at K≥K_IC · Paris da/dN=C(ΔK)^m, ΔK=YΔσ√(πa).