Combined Loading — AISC 360 Chapter H Interaction Equations
Most structural steel members carry axial force and bending simultaneously. Columns in moment frames resist compression and bending; bracing members carry tension and self-weight bending. AISC 360-22 Chapter H provides interaction equations that check these combined conditions.
AISC 360-22 Section H1 — doubly symmetric members
When Pr/Pc >= 0.2 (high axial ratio)
Pr/Pc + (8/9) * (Mrx/Mcx + Mry/Mcy) <= 1.0 [Eq. H1-1a]
When Pr/Pc < 0.2 (low axial ratio)
Pr/(2*Pc) + (Mrx/Mcx + Mry/Mcy) <= 1.0 [Eq. H1-1b]
Where Pr = required axial strength, Pc = available axial strength (phiPn), Mrx/Mry = required flexural strength about x/y axis, Mcx/Mcy = available flexural strength (phiMn).
The "kink" at Pr/Pc = 0.2 transitions from an axial-dominated regime to a flexure-dominated regime.
Second-order effects (P-delta)
Critical: Required strengths Pr and Mr must include second-order effects. Use AISC Chapter C (Direct Analysis Method) or Appendix 8 (B1-B2 amplification).
B1-B2 method: Mr = B1M_nt + B2M_lt, where B1 = Cm/(1 - Pr/Pe1) >= 1.0 (member P-delta) and B2 = 1/(1 - sum(Pr)/sum(Pe_story)) (frame P-Delta).
Cm values: 1.0 for members with transverse loads. 0.6 - 0.4*(M1/M2) for end moments only (positive for reverse curvature). 0.85 approximate for transverse loads with end restraints.
Worked example — W14x82 beam-column
Given: W14x82, Fy = 50 ksi, KL = 14 ft (weak axis, braced). Pu = 350 kips, Mux = 200 kip-ft, Muy = 0.
Axial: KL/ry = 67.7. Fcr = 35.7 ksi. phiPn = 0.9035.724.0 = 771 kips.
Flexural: Lb = 7 ft < Lp = 8.8 ft, so Mn = Mp. phiMnx = 0.9050148/12 = 555 kip-ft.
Interaction: Pr/Pc = 350/771 = 0.454 > 0.2, use H1-1a. 0.454 + (8/9)*(200/555) = 0.454 + 0.320 = 0.774 <= 1.0 OK. Utilization: 77.4%.
Tension + bending
Same equations apply with Pc = tensile capacity (phiPn from Chapter D). No P-delta amplification needed (tension reduces second-order effects). B1 = 1.0.
Code comparison — beam-column interaction
| Feature | AISC 360 (H1) | AS 4100 (Sec. 8) | EN 1993-1-1 (6.3.3) | CSA S16 (13.8) |
|---|---|---|---|---|
| Interaction form | Bilinear (H1-1a/1b) | Linear (N* + M*x + M*y) | Two equations with k factors | Linear with U1 amplifiers |
| Transition point | Pr/Pc = 0.2 | No transition | No transition | No transition |
| Bending coefficient | 8/9 = 0.889 (for H1-1a) | 1.0 | kyy, kzy, etc. from Annex A/B | 0.85 (approximate) |
| Second-order method | DAM (Ch. C) or B1-B2 | delta_b, delta_s amplifiers | EN 1993-1-1 Cl. 5.2.2 | U1 amplifier |
| phi / gamma | phi_c = 0.90, phi_b = 0.90 | phi = 0.90 | gamma_M1 = 1.00 | phi = 0.90 |
AS 4100 Section 8.4: Uses a linear interaction N*/phi_N_c + M*_x/(phi_M_sx) + M*_y/(phi_M_sy) <= 1.0, with amplification via delta_b (member) and delta_s (sway) factors applied to the moments before entering the interaction equation.
EN 1993-1-1 Section 6.3.3: Uses two interaction equations with k-factors from Annex A (exact) or Annex B (simplified). The k-factors (kyy, kyz, kzy, kzz) account for moment gradient, axial force level, and member slenderness. Both equations must be satisfied simultaneously.
CSA S16 Section 13.8: Cf/Cr + 0.85 x U1x x Mfx/Mrx + beta x U1y x Mfy/Mry <= 1.0, where U1 is the amplification factor similar to AISC's B1 and beta = 0.6 for Class 1/2 sections.
Section H2 — unsymmetric and other members
For singly symmetric members (channels, tees) and members subject to torsion in addition to flexure, AISC H2 provides:
f_ra/F_ca + f_rbw/F_cbw + f_rbz/F_cbz <= 1.0
where f_ra, f_rbw, f_rbz are the required axial and bending stresses at the critical point on the cross-section, and F_ca, F_cbw, F_cbz are the corresponding available stresses. This equation is checked at individual points on the cross-section (typically at flange tips) rather than using the section-level interaction of H1.
Biaxial bending considerations
When a column is subjected to bending about both axes simultaneously (biaxial bending), both Mrx and Mry terms appear in the interaction equation. This occurs at corner columns, columns at re-entrant corners, and any column where lateral loads act in two directions simultaneously. The interaction penalty for biaxial bending can be severe — a column that is 60% utilized in uniaxial bending about each axis independently may be 100%+ utilized when both moments act simultaneously.
Common mistakes
- Forgetting second-order effects. Using first-order moments directly is unconservative.
- Using the wrong equation. Check Pr/Pc first: >= 0.2 uses H1-1a, < 0.2 uses H1-1b.
- Not checking both axes. Both Mrx and Mry terms must be included for biaxial bending.
- Applying DAM stiffness reductions inconsistently. Reduced stiffness for analysis, nominal properties for capacity.
- Not checking multiple points along the member. The critical section may be at an interior point.
Frequently asked questions
What is the interaction equation? It checks whether combined axial and bending demands exceed member capacity. If the sum of demand/capacity ratios (with appropriate coefficients) is <= 1.0, the member is adequate.
When do I need to check combined loading? Whenever a member carries both axial force and bending simultaneously: moment frame columns, bracing with self-weight, beams with axial restraint, truss members with secondary bending.
What is Cm? The equivalent uniform moment factor in the B1 amplifier. Cm = 1.0 is conservative. For end moments only: Cm = 0.6 - 0.4*(M1/M2), which can be as low as 0.4 for reverse curvature.
Run this calculation
Related references
- Column Buckling Equations
- Lateral-Torsional Buckling
- Effective Length Factor K
- How to Verify Calculations
Disclaimer
This page is for educational and reference use only. It does not constitute professional engineering advice. All design values must be verified against AISC 360-22 Chapter H and the governing project specification. The site operator disclaims liability for any loss arising from the use of this information.