Formula Library

Searchable database of governing structural design equations, each with variables, assumptions, code reference, and links to worked examples.

Axial Tension

Yielding on gross section

phi_t P_n = 0.90 · F_y · A_g
Variables
  • F_y — specified minimum yield stress (ksi)
  • A_g — gross cross-sectional area (in²)
Assumptions
  • Load applied along member axis
  • No initial imperfections considered
AISC 360-22 §D2(a)Worked example → wx-4-1
Axial Tension

Rupture on effective net area

phi_t P_n = 0.75 · F_u · A_e
Variables
  • F_u — specified minimum tensile strength (ksi)
  • A_e — effective net area = U·A_n (in²)
Assumptions
  • A_n uses d_h = d_b + 1/8
  • Shear-lag factor U per Table D3.1
AISC 360-22 §D2(b)
Compression

Compressive strength (flexural buckling)

phi_c P_n = 0.90 · F_cr · A_g
Variables
  • F_cr — critical stress (Eq E3-2 or E3-3) (ksi)
  • A_g — gross area (in²)
Assumptions
  • Members without slender elements
  • Effective length factor K assumed known
AISC 360-22 §E3
Flexure

Plastic moment capacity

phi_b M_p = 0.90 · F_y · Z_x
Variables
  • Z_x — plastic section modulus about x-axis (in³)
Assumptions
  • Compact section (λ ≤ λ_p)
  • Full lateral bracing (L_b ≤ L_p)
AISC 360-22 §F2.1
Shear

Shear strength of W-shape webs

phi_v V_n = 1.00 · 0.6 · F_y · A_w · C_v1
Variables
  • A_w — web area = d · t_w (in²)
  • C_v1 — web shear strength coefficient (—)
Assumptions
  • Rolled I-shapes with h/t_w ≤ 2.24√(E/Fy) give C_v1 = 1.0
AISC 360-22 §G2.1
Torsion

Torsional strength of HSS

phi_T T_n = 0.90 · F_cr · C
Variables
  • C — HSS torsional constant (in³)
  • F_cr — critical torsional stress (ksi)
Assumptions
  • Closed HSS section
  • Ends free to warp
AISC 360-22 §H3.1
Combined Loading

Beam-column interaction (H1-1a)

P_r/P_c + 8/9 · (M_rx/M_cx + M_ry/M_cy) ≤ 1.0 (P_r/P_c ≥ 0.2)
Variables
  • P_r — required axial (kips)
  • P_c — available axial (kips)
  • M_r — required moment (k·ft)
  • M_c — available moment (k·ft)
Assumptions
  • Doubly symmetric section
  • Elastic second-order effects included in P_r, M_r
AISC 360-22 §H1.1
Connections

Bolt shear strength

phi R_n = 0.75 · F_nv · A_b
Variables
  • F_nv — nominal shear stress (Table J3.2) (ksi)
  • A_b — nominal bolt area (in²)
Assumptions
  • Threads in shear plane → F_nv = 54 ksi for A325
AISC 360-22 §J3.6
Deflection

Midspan deflection (simple beam, UDL)

Δ_max = 5·w·L^4 / (384·E·I)
Variables
  • w — uniform load (k/in)
  • L — span (in)
  • E — modulus of elasticity (ksi)
  • I — moment of inertia (in⁴)
Assumptions
  • Simply supported
  • Elastic behaviour
Mechanics of Materials
Stability

Euler buckling load

P_e = π² · E · I / (K · L)²
Variables
  • K — effective length factor (—)
  • L — unbraced length (in)
Assumptions
  • Perfectly straight, elastic, pin-ended column
AISC 360-22 §E3 Commentary