| 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)
- Yielding: φPn = 0.90×Fy×Ag ≥ Pu
- Net section fracture: φPn = 0.75×Fu×Ae ≥ Pu
- Block shear: φRn ≥ Pu
- 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.
