axial-stress

name	axial-stress
description	Axial stress analysis — tension/compression, net section, bearing, tearout, block shear, statically indeterminate axial, thermal expansion, residual stress, plastic limit load.
metadata	{"priority":8,"promptSignals":{"phrases":["axial stress","axial load","tension member","compression member","net section","bearing stress","tearout"],"minScore":4}}

Axial Stress Analysis — Complete Skill

Basic Axial Stress

σ = P/A (normal stress, uniform over cross-section) Strain: ε = σ/E = P/(AE) (Hooke's law, elastic range) Deformation: δ = PL/AE = σL/E

Sign convention: tension positive (+), compression negative (-)

Net Section (Tension Members with Holes)

Net area: A_n = A_g - Σ(d_h × t) [d_h = hole diameter = d_bolt + 1/16" AISC] Effective net area: A_e = U × A_n [U = shear lag factor] U = 1 - x̄/L [x̄ = distance from weld/connection to centroid of connected element] U = 0.75 min for single bolt row

AISC strength: Yield on gross: P_n = F_y × A_g Fracture on net: P_n = F_u × A_e (often governs for bolted connections) LRFD: φ = 0.90 (yield), φ = 0.75 (fracture)

Bearing Stress (at pins, bolts)

σ_b = P/(d × t) [projected area = diameter × thickness] AISC limit: σ_b ≤ φ × 2.4Fu (single bolt) or 3.0Fu (long slotted, bolts in a row) φ = 0.75 (bearing)

Shear Tearout (Bolt Shear-Out of Material)

τ = P/(A_v) [A_v = net shear area = 2 × Lv × t for double shear tearout] AISC limit: R_n = 0.6 × Fu × A_nv (shear fracture)

Block Shear (Combined Shear + Tension)

R_n = 0.6Fu × A_nv + Ubs × Fu × A_nt ≤ 0.6Fy × A_gv + Ubs × Fu × A_nt A_nv = net shear area, A_nt = net tension area Ubs = 1.0 (uniform tension), 0.5 (non-uniform) φ = 0.75

Saint-Venant's Principle

Stress concentration near load application dissipates within ~1 × dimension from the point At distances > 1D from application point: σ = P/A (uniform — design conservatively)

Stress Concentration (Axial)

Kt from charts/equations (Peterson's Stress Concentration Factors):

• Circular hole in infinite plate: Kt = 3.0 (regardless of hole size)
• Circular hole in finite plate width W: Kt increases as d/W increases
• Shoulder fillet r/d=0.1: Kt ≈ 1.65 (axial)
• Shoulder fillet r/d=0.01: Kt ≈ 2.7 σ_max = Kt × P/A_net (for static, ductile: often ignored if Sy/Sut > 0.7) σ_max = Kf × P/A_net (for fatigue: Kf = 1 + q(Kt-1))

Statically Indeterminate Axial

Example: bar fixed at both ends with load P at point C Compatibility: δ_AC = -δ_CB (deformations must be consistent) Force method: remove one support as redundant R Compatibility equation: δ_0 + R × f = 0 Where δ_0 = tip deflection (without R), f = flexibility coefficient (unit load displacement)

Thermal loading: δ_thermal = α × ΔT × L (free thermal expansion) If constrained: σ_thermal = -E × α × ΔT (compressive if heated, tensile if cooled) Combined thermal + mechanical: δ_total = PL/AE + α×ΔT×L

Multi-material bar: Same deformation compatibility applies P = ΣP_i (equilibrium), δ₁ = δ₂ (compatibility for parallel arrangement) P₁/P₂ = (A₁E₁)/(A₂E₂) (stiffness sharing)

Plastic Analysis (Limit Load)

Plastic limit: all fibers yield simultaneously P_yield = A × Sy (first fiber yields — elastic limit) P_plastic = A × Sy (for uniform cross-section — same as yield) For non-uniform cross-section: P_limit based on section that yields first, then redistribution

Residual Stresses

After plastic deformation, elastic springback leaves residual stress: σ_residual = σ_applied - σ_springback (= P/A - E×ε_elastic) Residual stress can be beneficial (compressive) or detrimental (tensile) Shot peening, prestressing: intentional beneficial compressive residuals

Design Checks (AISC Axial Tension, LRFD)

1. Yielding: φPn = 0.90×Fy×Ag ≥ Pu
2. Net section fracture: φPn = 0.75×Fu×Ae ≥ Pu
3. Block shear: φRn ≥ Pu
4. Slenderness (preferred): L/r ≤ 300 (tension), L/r ≤ 200 (compression)

Output

Provide: σ_axial [MPa/ksi], δ [mm/in], governing failure mode, factor of safety, net section and block shear checks if connections present.