AISC 360 — Combined Forces (Chapter H)

Beam-column interaction per AISC 360-16: H1 equations, P-delta, Direct Analysis Method, B1-B2

Most columns in real structures are not loaded in pure compression — they experience simultaneous axial force and bending moments due to frame action, eccentric connections, or lateral loads. Chapter H of AISC 360-16 provides the interaction equations that combine the axial and flexural checks from Chapters E and F into a single utilization ratio. These equations, known informally as the "H1 equations," are the most frequently evaluated check in structural steel design.

1. The H1 Interaction Equations

For doubly symmetric and singly symmetric members subjected to flexure and compression:

H1-1a: When $P_r/P_c \geq 0.2$

\frac{P_r}{P_c} + \frac{8}{9}\left(\frac{M_{rx}}{M_{cx}} + \frac{M_{ry}}{M_{cy}}\right) \leq 1.0

H1-1b: When $P_r/P_c < 0.2$

\frac{P_r}{2 P_c} + \left(\frac{M_{rx}}{M_{cx}} + \frac{M_{ry}}{M_{cy}}\right) \leq 1.0

where:

$P_r$ = required axial strength (factored for LRFD, unfactored for ASD)
$P_c$ = available axial strength = $\phi_c P_n$ (LRFD) or $P_n/\Omega_c$ (ASD)
$M_{rx}, M_{ry}$ = required flexural strength about x and y axes, including second-order effects
$M_{cx}, M_{cy}$ = available flexural strength about x and y axes

The two-equation format avoids an excessively conservative interaction for members with low axial ratios. Equation H1-1b is more lenient on axial force (Pr/2Pc instead of Pr/Pc) but stricter on bending (full addition instead of 8/9 factor). The transition at Pr/Pc = 0.2 ensures continuity.

[DIAGRAM: Interaction diagram showing the bilinear curve. X-axis: Mr/Mc (from 0 to 1), Y-axis: Pr/Pc (from 0 to 1). Two lines: H1-1a from (0, 1.0) to (9/8, 0) and H1-1b from (0, 0.5) to (1.0, 0), meeting at the knee at Mr/Mc ≈ 0.44 and Pr/Pc = 0.2.]

2. Second-Order Analysis Requirements

The $M_{rx}$ and $M_{ry}$ in the H1 equations must include second-order effects — the additional moments caused by the axial force acting through the deflected shape. There are two components:

Effect	Symbol	Description
P-δ (member effect)	$P \cdot \delta$	Axial force × member deflection between ends. Amplifies moments in individual members. Always present.
P-Δ (frame effect)	$P \cdot \Delta$	Axial force × story drift (sway). Amplifies lateral moments. Only in unbraced/sway frames.

2.1. Direct Analysis Method (Chapter C)

The DAM is the preferred method for stability analysis. It directly models the second-order effects in the structural analysis and permits $K = 1.0$ for all members. Requirements:

1. Notional loads: $N_i = 0.002 \, Y_i$ at each level

2. Reduced stiffness: $EI^* = 0.8 \tau_b EI$ , $EA^* = 0.8 EA$

3. Second-order analysis: P-Δ and P-δ captured automatically

4. Column strength: Use $K = 1.0$ for all compression members

2.2. Approximate B1-B2 Method (Appendix 8)

When rigorous second-order analysis is not available, the B1-B2 method approximates second-order moments from first-order analysis results:

M_r = B_1 M_{nt} + B_2 M_{lt}

where $M_{nt}$ = moment from no-translation (gravity) analysis, $M_{lt}$ = moment from lateral translation (sway) analysis.

B1 — P-δ amplifier

B_1 = \frac{C_m}{1 - \alpha P_r / P_{e1}} \geq 1.0

where $C_m = 0.6 - 0.4(M_1/M_2)$ for members without transverse loads between supports ( $M_1/M_2$ is positive for single curvature), and $P_{e1} = \pi^2 EI / (KL)^2$ with $K = 1.0$ in the DAM. The factor $\alpha = 1.0$ for LRFD and $\alpha = 1.6$ for ASD.

B2 — P-Δ amplifier

B_2 = \frac{1}{1 - \alpha \sum P_r / \sum P_{e2}}

where $\sum P_r$ is the total required vertical load in the story and $\sum P_{e2}$ is the total story elastic buckling load. When $B_2 > 1.5$ , a rigorous second-order analysis is required — the B1-B2 approximation becomes inaccurate for high amplification.

α = 1.6 for ASD: This is a key difference between LRFD and ASD in second-order analysis. Since ASD uses unfactored loads, multiplying by α = 1.6 brings the loads to an equivalent factored level for the P-Δ check. Forgetting to use α = 1.6 in ASD underestimates the second-order effects and can lead to unconservative designs.

3. Other Sections in Chapter H

H2: Unsymmetric and Other Members

For members not covered by H1 (unsymmetric shapes, biaxial bending with tension):

\frac{f_a}{F_a} + \frac{f_{bx}}{F_{bx}} + \frac{f_{by}}{F_{by}} \leq 1.0

This stress-based interaction is more general but less refined than the H1 approach.

H3: Members Under Torsion

Section H3 provides interaction equations for members subjected to combined axial force, flexure, and torsion. For round and rectangular HSS:

\left(\frac{P_r}{P_c} + \frac{M_r}{M_c}\right)^2 + \left(\frac{V_r}{V_c} + \frac{T_r}{T_c}\right)^2 \leq 1.0

For open sections (W, C, L), torsion produces warping normal stresses that are added directly to the flexural stresses before checking H1. This typically requires specialized software.

Solved Example — W12×65 Beam-Column

Given: W12×65 (W310×97), A992 ( $F_y = 50$ ksi), in a braced frame. Unbraced length: $KL_x = 24$ ft, $KL_y = 12$ ft (braced at midheight for weak axis). Factored forces (LRFD): $P_u = 267$ kips, $M_{ux} = 147$ ft·kips, $M_{uy} = 24$ ft·kips. Members with no transverse loads between supports (Cm applies).

Properties: $A_g = 19.1$ in², $r_x = 5.28$ in, $r_y = 3.02$ in, $Z_x = 96.8$ in³, $Z_y = 44.1$ in³, $S_x = 87.9$ in³, $r_y = 3.02$ in, $r_{ts} = 3.40$ in, $J = 2.18$ in&sup4;, $h_o = 11.9$ in.

Step 1 — Available Compressive Strength (Pc)

\frac{KL}{r_x} = \frac{24 \times 12}{5.28} = 54.5

\frac{KL}{r_y} = \frac{12 \times 12}{3.02} = 47.7

Strong axis governs (54.5). Transition: 113.4 → inelastic.

F_e = \frac{\pi^2 \times 29{,}000}{54.5^2} = 96.3 \text{ ksi}

F_{cr} = 0.658^{50/96.3} \times 50 = 0.658^{0.519} \times 50 = 0.799 \times 50 = 40.0 \text{ ksi}

P_c = \phi_c P_n = 0.90 \times 40.0 \times 19.1 = 688 \text{ kips}

Step 2 — Available Flexural Strength (Mcx, Mcy)

Strong axis (x):

M_p = 50 \times 96.8 = 4{,}840 \text{ in·kips} = 403.3 \text{ ft·kips}

L_p = 1.76 \times 3.02 \times 24.08 = 128 \text{ in} = 10.7 \text{ ft}

With $L_b = 12$ ft for strong axis (assuming lateral bracing at ends only for a 24 ft column with midheight bracing for weak axis), we need to check LTB. Since $L_b = 24$ ft > $L_p = 10.7$ ft, LTB applies. For a column with end moments (no transverse loads), $C_b = 1.0$ (conservative).

$L_r \approx 35.3$ ft (from AISC tables). Since $L_p < L_b = 24 < L_r$ → inelastic LTB:

M_{nx} = 1.0\left[403.3 - (403.3 - 0.7 \times 50 \times 87.9/12)\left(\frac{24 - 10.7}{35.3 - 10.7}\right)\right]

= 403.3 - (403.3 - 256.3) \times 0.541 = 403.3 - 79.4 = 323.9 \text{ ft·kips}

M_{cx} = \phi_b M_{nx} = 0.90 \times 323.9 = 291.5 \text{ ft·kips}

Weak axis (y):

No LTB for minor-axis bending (Section F6). Yielding governs:

M_{cy} = \phi_b \cdot F_y \cdot Z_y = 0.90 \times 50 \times 44.1 / 12 = 165.4 \text{ ft·kips}

Step 3 — Check Axial Ratio

\frac{P_r}{P_c} = \frac{267}{688} = 0.388 \geq 0.2 \quad \rightarrow \text{use H1-1a}

Step 4 — Interaction Check (H1-1a)

\frac{P_r}{P_c} + \frac{8}{9}\left(\frac{M_{rx}}{M_{cx}} + \frac{M_{ry}}{M_{cy}}\right) = 0.388 + \frac{8}{9}\left(\frac{147}{291.5} + \frac{24}{165.4}\right)

= 0.388 + 0.889 \times (0.504 + 0.145) = 0.388 + 0.889 \times 0.649

= 0.388 + 0.577 = 0.965 \leq 1.0 \quad \checkmark \;\; (96.5\%)

Result: The W12×65 passes at 96.5% utilization — very efficient. The strong-axis bending dominates the interaction (Mrx/Mcx = 0.504 contributes 45% of the total utilization). If the weak-axis moment increased to 40 ft·kips, the interaction ratio would be: 0.388 + 0.889 × (0.504 + 0.242) = 1.05 → FAILS. The engineer would then need to add weak-axis bracing, increase the column size, or reduce the moment.

Verify automatically in CalcSteel — Open the editor with AISC 360 →

5. International Comparison — Combined Forces

Aspect	AISC 360	Eurocode 3	NBR 8800
Interaction equations	H1-1a/b (simple, bilinear)	k_ij method (Annex A or B) — complex	H1-1a/b (= AISC)
Number of equations	2	2 (but with k_yy, k_yz, k_zy, k_zz factors)	2 (= AISC)
Second-order	DAM or B1-B2	GNA or amplification factors	DAM or B1-B2
Complexity	Low — suitable for hand calculation	High — typically requires software	Low (= AISC)

The AISC H1 equations are elegant in their simplicity: two equations, clear transition, easily programmable. The Eurocode k_ij method is more precise (accounting for the shape of the moment diagram along the member length) but requires computing 4 interaction factors, each with its own sub-formula — practically impossible by hand. CalcSteel implements both approaches and typically shows 5–15% higher utilization with EC3 for members governed by LTB.

← AISC 360 Overview|Shear (Ch. G)|Serviceability (Ch. L) →

Open the editor with AISC 360 →