Steel Member Design — AISC 360-22

1. Pre-Sizing Rules of Thumb

2. Tension Members — Chapter D

2.1 Limit States (D2)

Tension members are governed by the lesser of two limit states. Both must be checked:

2.2 Shear Lag Factor U (Table D3.1)

U accounts for the fact that not all elements of a connected member carry load uniformly (shear lag). A_n = gross area minus hole areas (for bolt holes, add 1/16 in / 2 mm to hole diameter).

2.3 Block Shear (ACI §J4.3)

Block shear is a combined failure mode involving shear on one plane and tension on a perpendicular plane, governing gusset plates and coped beam ends:

A_nv = net area in shear; A_nt = net area in tension; A_gv = gross area in shear. U_bs = 1.0 for uniform tension stress; 0.5 for non-uniform.

2.4 Slenderness Limit

2.5 Net Area Calculation (§B4.3)

For bolted connections, the net area A_n is the gross area minus deductions for holes:

• Hole deduction: Deduct diameter = bolt diameter + 1/16 in (2 mm) for standard punched/drilled holes (damage from punching is included in the extra 1/16 in).
• Staggered holes: When holes are staggered, check all possible failure paths. For a diagonal path, add s²/(4g) to the net width for each stagger, where s = longitudinal spacing and g = transverse gauge.

A_n = w_n × t (plate) or sum of element net widths × thickness. A_e = U·A_n ≤ A_g. The critical path gives the minimum A_n.

3. Beam Flexure — Chapter F

3.1 Compact Section — Full Yielding (F2)

When the section is compact and bracing is adequate (L_b ≤ L_p), the beam reaches full plastic moment:

3.2 Compactness Classification

These ratios are dimensionless. For A992 steel (E = 29,000 ksi / 200,000 MPa, F_y = 50 ksi / 345 MPa): λ_pf = 9.15; λ_pw = 90.5. Most W shapes are compact.

3.3 Compression Element Classification — Table B4.1a

For members under uniform compression (columns, braces), only the slender/nonslender distinction matters. Slender elements reduce P_n via Q factor (§E7).

For A992 (F_y = 50 ksi): λ_r,flange = 13.5; λ_r,web = 35.9; λ_r,HSS-rect = 33.7. Most standard W-shapes are non-slender for compression.

3.4 Seismic Compactness — AISC 341-22 Table D1.1

Seismic systems require more stringent limits to sustain cyclic inelastic deformations without local buckling. Two tiers apply: highly ductile (λ_hd) and moderately ductile (λ_md).

C_a = P_u/(φ_cF_yA_g) ≤ 1.0 (LRFD). Higher axial load → more stringent web limit. Highly ductile required for SMF beams/columns, EBF members outside link, BRBF beams/columns. Moderately ductile for IMF, SCBF braces.

3.5 Noncompact and Slender Web I-Shapes (§F3–F5)

When flanges or webs are noncompact or slender, M_n is reduced below M_p:

• §F3 — Compact web, noncompact/slender flange: M_n reduced for flange local buckling (FLB). Linear interpolation between M_p (at λ_p) and 0.7F_yS_x (at λ_r); below 0.7F_yS_x for slender flanges.
• §F4–F5 (Plate girders) — Noncompact or slender web: Separate checks for compression flange yielding (CFY), LTB using Rpg factor, and tension flange yielding (TFY). The R_pg factor (≤1.0) accounts for web bend-buckling reducing the effective flange stress.

Practical note: For standard rolled W-shapes, §F2 governs for nearly all cases. §F3–F5 apply to welded plate girders or heavy built-up sections where the web is proportioned for shear efficiency over moment efficiency.

3.6 Other Cross-Section Types

4. Lateral-Torsional Buckling (LTB)

LTB reduces flexural capacity when the compression flange is inadequately braced. Three zones are defined by the unbraced length L_b.

4.1 Limiting Unbraced Lengths

4.2 LTB Zones

4.3 Moment Gradient Factor C_b

M_max = maximum moment; M_A, M_B, M_C = moments at quarter, half, and three-quarter points of the unbraced segment. C_b = 1.0 is conservative.

5. Shear Design — Chapter G

5.1 Shear Strength for Most W-Shapes (G2.1)

For hot-rolled I-shapes with h/t_w ≤ 2.24√(E/F_y), the shear strength reduction factor C_v1 = 1.0 and φ = 1.00:

A_w = d × t_w. For A992 W-shapes, most sections have h/t_w ≤ 2.24√(29000/50) = 54.0, confirming φ = 1.00 applies.

5.2 Stiffened Webs with Tension-Field Action (§G3)

Plate girders with intermediate transverse stiffeners can develop tension-field action (post-buckling diagonal tension) for higher shear capacity:

a = clear distance between stiffeners; C_v2 = shear buckling coefficient (function of h/t_w and a/h). Tension-field action is not permitted for: end panels, panels with large openings, or when 2A_w/(A_fc+A_ft) > 2.5.

5.3 Transverse Stiffener Design (§G2.2 / G3.3)

Intermediate stiffeners are required when h/t_w > 2.46√(E/F_y) (= 59.2 for A992). Requirements:

• Moment of inertia: I_st ≥ I_st1 (minimum for shear without TFA) and I_st2 (for TFA, §G3.3).
• Stiffeners may be single or double (one or both sides of web).
• Stiffener-to-web weld: termination ≥ 4t_w from flange fillet weld but ≤ 6t_w.
• Width-to-thickness ratio of stiffener: b_st/t_st ≤ 0.56√(E/F_y).

5.4 Shear in HSS and Box Sections (§G4)

For HSS round (§G5): V_n = F_cr·A_g/2, where F_cr is the shear buckling stress based on D/t. φ_v = 0.90 for both.

6. Column Compression — Chapter E

6.1 Euler Elastic Buckling Stress F_e (E3)

6.2 Critical Stress F_cr

Slenderness limit: KL/r ≤ 200 (recommended). For A992: 4.71√(E/F_y) = 4.71√(29000/50) = 113 (US) / 4.71√(200000/345) = 113 (SI).

6.3 Effective Length Factor K

6.4 Local Buckling Reduction — Q Factor (§E7)

When a section has slender elements (λ > λ_r from Table B4.1a), the nominal strength is reduced by Q = Q_s·Q_a:

• Q_s — for slender unstiffened elements (flange outstanding legs, angle legs): ranges from 1.0 at λ_r to a minimum value at high slenderness. Computed from §E7.1.
• Q_a — for slender stiffened elements (web, HSS walls): Q_a = A_eff/A_g, where A_eff uses the reduced effective width b_e per §E7.2.
• For non-slender sections: Q = 1.0, and the standard F_cr formulas apply directly.

With Q < 1.0: replace F_y with Q·F_y in the column curve, and the slenderness limit becomes 4.71√(E/(Q·F_y)).

6.5 Torsional and Flexural-Torsional Buckling (§E4)

For doubly symmetric W-shapes (typical columns), flexural buckling about the weak axis governs and §E4 rarely controls. However, §E4 must be checked for:

• Singly symmetric sections (channels, T-shapes, Tees, singly symmetric I-sections): Flexural-torsional buckling combines bending about the symmetric axis with torsion. Effective slenderness (KL/r)_eff is computed from the torsional buckling stress F_e (§E4-3 to E4-5).
• Asymmetric sections (single angles, unequal angles): All three buckling modes (x, y, z) interact — use the general procedure of §E4.
• Double angles and WT sections: F_e for flexural-torsional buckling per §E4-4/5, then enter the standard F_cr equations.

For wide-flange columns with KL_y governing, torsional buckling check is usually not critical because I_y·C_w/J is large for W-shapes. Always check when using channels or WT sections as compression members.

7. Combined Loading — Chapter H

Members subject to combined axial and bending loads are checked with the biaxial interaction equations of H1-1:

7.2 Unsymmetric or General Cross-Sections (§H2)

For sections without an axis of symmetry, or members with biaxial bending and axial load where §H1-1 does not apply, use the stress-based interaction:

f_ra = required axial stress; F_ca = available axial stress. Bending stresses computed at the critical (maximum stress) point including combined biaxial bending. Applies to asymmetric shapes, angles used as beams, and general unsymmetric built-up sections.

7.3 Combined Shear and Torsion (§H3)

For HSS and box sections subject to combined shear (V_r) and torsion (T_r):

V_c = φ·V_n from Chapter G; T_c = φ·F_cr·C, where F_cr is the torsional shear buckling stress and C is the torsional shear constant of the section (= 2·A_o·t for thin-walled closed sections, where A_o = enclosed area). φ = 0.90.

8. Composite Beam Design — Chapter I

Composite beams combine a steel section with a concrete slab via steel headed stud anchors, increasing flexural stiffness and strength.

8.1 Effective Slab Width

8.2 Full Composite Action

8.3 Stud Shear Anchor Strength Q_n (I8.2a)

8.4 Effective Moment of Inertia (Partial Composite)

I_s = bare steel moment of inertia; I_t = full composite moment of inertia (transformed section).

9. Connections — Chapter J

Connection design covers welds, bolts, and the plates, gussets, and angles that transmit forces between members.

9.1 Bolt Types and Strengths

N = threads included in shear plane (lower F_nv); X = threads excluded. φ = 0.75 for tension and shear.

9.2 Bearing-Type Connection (J3.6–J3.7)

9.3 Slip-Critical Connection (J3.8)

Required when joint movement would impair function (dynamic loads, fatigue, joints with oversized holes). Slip resistance is a serviceability limit state.

μ = mean slip coefficient (Class A: 0.35, Class B: 0.50); D_u = 1.13; h_f = 1.0 (no fillers); T_b = bolt pretension from Table J3.1; n_s = number of slip planes.

9.4 Fillet Weld Strength (J2.4)

a = weld size (leg); effective throat = 0.707a. Common electrodes: E70XX (F_EXX = 70 ksi / 482 MPa). Minimum weld size governs thin material; maximum weld size = t − 1/16 in (t − 2 mm) along edges ≥ 1/4 in (6 mm) thick.

9.5 Bolt Spacing and Edge Distance (§J3.3–J3.5)

These limits ensure proper fabrication clearance and reliable bearing/tearout behavior. Edge distance below the minimum reduces bearing strength proportionally.

9.6 Complete and Partial Joint Penetration Welds (§J2)

CJP (Complete Joint Penetration) — §J2.1a: Full weld throat equals the connected plate thickness. Design strength governed by the base metal: φR_n = φ·F_BM·A_BM. Pre-qualified CJP joints per AWS D1.1/AISC. Required for primary tension moment connections in SMF/IMF beam-column joints.

PJP (Partial Joint Penetration) — §J2.1b: Effective throat is less than the plate thickness. R_n = 0.60·F_EXX·A_we (same form as fillet weld). PJP welds are not permitted for primary tension members in SMF connections; CJP is required there.

Transverse (90°) fillet welds are ~50% stronger than longitudinal (0°) welds: R_n multiplied by (1 + 0.50·sin¹·⁵θ), capped at M_n = M_p.

9.7 Column Base Plate Design (AISC Design Guide 1)

Base plates transfer column loads to the concrete foundation via bearing pressure. Anchor rods (ASTM F1554 Gr. 36, 55, or 105) resist uplift and shear.

Plate dimensions N × B chosen from A_req and constructability. Overhang dimensions n = (N−0.80d)/2 and m = (B−0.95b_f)/2. Plate thickness from cantilever bending at critical sections:

λn' accounts for the plastic hinge in the column footprint zone. For columns with large moments, use AISC DG1 procedures for combined axial + moment base plates including anchor rod tension design.

10. Seismic Steel Design — AISC 341-22

AISC 341-22 (Seismic Provisions for Structural Steel Buildings) supplements AISC 360-22 for Seismic Design Categories C–F. The system response modification factor R is from ASCE 7-22 Table 12.2-1.

10.1 R Factor Summary — ASCE 7-22 Table 12.2-1

NL = Not Limited (height); NP = Not Permitted. Ω₀ = overstrength factor (for capacity-protected elements); C_d = deflection amplification factor (for drift check). Verify current ASCE 7-22 Table 12.2-1 for full height limits by SDC.

10.2 Special Moment Frame (SMF) — Chapter E

SMF provides ductility through plastic hinging in beams away from columns (protected zones). Key requirements:

• Beam-to-column connections: Must be prequalified per AISC 358, or qualify via testing. Plastic hinge location moved away from column face (e.g., RBS — reduced beam section).
• Expected plastic moment: M_pe = R_y · F_y · Z_x, where R_y = 1.1 for A992; 1.5 for A36.
• Strong column — weak beam: ΣM*_pc / ΣM*_pb ≥ 1.0, where M*_pc and M*_pb are projected onto the joint centerline accounting for axial load.
• Continuity plates (column web stiffeners) required when beam flanges deliver concentrated force exceeding column web/flange capacity.
• Panel zone shear: φR_n = 0.60·F_y·d_c·t_w·(1 + 3b_cf·t_cf² / (d_b·d_c·t_w))
• Protected zone: No attachments by notching, drilling, or welding within the plastic hinge region.

10.3 Special Concentrically Braced Frame (SCBF) — Chapter F

SCBF relies on both tension and compression braces for ductility. Post-buckling behavior must be accommodated:

• Width-to-thickness limits: All members must be highly ductile (seismically compact) per Table D1.1.
• Brace slenderness: KL/r ≤ 200 (preferred ≤ 4√(E/F_y) ≈ 96 for A992).
• Brace-to-gusset: Gusset plate must accommodate out-of-plane brace rotation via 2t_p offset from the brace work point.
• Required strength of columns and beams: Designed for the amplified seismic load using R_yF_yA_g of braces (capacity design).

10.4 Eccentrically Braced Frame (EBF) Link — Chapter F

The ductile "link" beam segment between brace connections is the designated yield element:

V_p = 0.6F_y(d − 2t_f)t_w; M_p = F_yZ_x. Link rotation angle γ_p ≤ 0.08 rad (shear link) or 0.02 rad (moment link). Link stiffeners at each end; intermediate stiffeners for shear links with e ≤ 5d.

10.5 Buckling-Restrained Braced Frame (BRBF) — AISC 341-22 §F4

BRBs eliminate compression buckling by encasing the steel core in a mortar-filled steel tube, creating a member with equal tension and compression yield capacity.

• BRB elements must be qualified by testing per §K3 or use pre-qualified products.
• Adjusted brace strength for capacity design: ω·β·R_y·F_y·A_sc (compression) and ω·R_y·F_y·A_sc (tension). Typical values: ω ≈ 1.5 (strain hardening), β ≈ 1.1 (compression overstrength).
• Beams and columns: highly ductile λ_hd compactness required (§D1.1).
• A_sc = area of the steel core (yielding segment only — not the connection plates).
• BRBF allows R = 8 with significantly simpler connection details than SMF — a practical advantage for buildings with large lateral demands but limited connection access.

10.6 Capacity Design Principle

Capacity design ensures that yielding occurs in the intended ductile element (brace, link, beam plastic hinge) before non-ductile elements (connections, columns) are stressed. For capacity-protected elements, forces are computed from the expected (probable) material strengths:

These amplified forces are used to design connections, columns, and other elements that must remain elastic. The overstrength factor Ω₀ from ASCE 7-22 Table 12.2-1 provides an alternative simplified amplification for certain elements per ASCE 7-22 §12.4.3.

11. Second-Order Effects — Direct Analysis Method (DAM)

Real structures experience additional moments due to displaced geometry (P-Δ at story level, P-δ at member level). AISC 360-22 Chapter C mandates accounting for these effects; the Direct Analysis Method (DAM) is the primary approach.

11.1 DAM Procedure (§C2)

• Reduced stiffness: Use E* = 0.8τ_b·E and EI* = 0.8τ_b·EI in analysis. τ_b = 1.0 when α·P_r/P_y ≤ 0.5; otherwise τ_b = 4·(α·P_r/P_y)·(1 − α·P_r/P_y) (where α = 1.0 for LRFD, 1.6 for ASD).
• Notional loads: Apply N_i = 0.002·Y_i as a lateral load at each floor level (Y_i = gravity load at level i). These represent initial geometric imperfections.
• K = 1.0: After applying reduced stiffness and notional loads, K = 1.0 may be used for compression member design (no alignment chart needed).

11.2 Alternative Methods

11.3 B₁–B₂ Amplifiers (§C2.1b — Simplified 2nd-Order)

When a first-order analysis is used, moments are amplified by:

C_m = equivalent uniform moment factor (0.6–0.4·M₁/M₂ for braced members); P_e1 = π²EI/(KL)² about the bending axis; P_e,story = story elastic buckling load. M_r = B₁·M_nt + B₂·M_lt (combined amplified moment).