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).

P-delta (local) vs P-Delta (global)

Second-order effects in beam-columns come from two distinct sources:

• P-delta (local, lowercase): Amplification of moments within the member due to axial force acting through the lateral deflection of the member itself. A column with 1 inch of mid-height deflection and 200 kips of axial force generates an additional moment of 200 × 1/12 = 16.7 kip-ft at mid-height. This is captured by the B1 amplifier.
• P-Delta (global, uppercase): Amplification of moments due to axial force acting through the lateral drift of the entire story. If a story drifts 0.5 inches laterally under a total story axial load of 2,000 kips, the resulting overturning moment is 2000 × 0.5/12 = 83 kip-ft, distributed among the columns in the story. This is captured by the B2 amplifier.

B1-B2 method

The amplified first-order analysis method (AISC Appendix 8) separates moments into non-sway (M_nt) and sway (M_lt) components:

Mr = B1 × M_nt + B2 × M_lt

B1 (member P-delta): B1 = Cm / (1 - Pr/Pe1) >= 1.0, where Pe1 is the Euler buckling load of the individual member. B1 amplifies only the non-translational (gravity) moments.

B2 (story P-Delta): B2 = 1 / (1 - sum(Pr) / sum(Pe_story)), where sum(Pe_story) is the sum of Euler loads for all columns in the story in the plane of bending. B2 amplifies only the sway (lateral) moments.

When second-order effects are significant

As a practical rule, second-order effects are significant when Pu/phiPn exceeds approximately 0.15. Below this threshold, the amplification is typically less than 5% and may be neglected for preliminary design. However, AISC Chapter C requires that second-order effects be included for all beam-column checks regardless of magnitude. The Direct Analysis Method (DAM) automatically captures both P-delta and P-Delta effects through a geometric nonlinear (P-Delta) analysis with reduced stiffness (0.80 tau_b × EI for all members).

Cm values

• Cm = 1.0 for members with transverse loads between joints (distributed or concentrated).
• Cm = 0.6 - 0.4*(M1/M2) for members with end moments only (no transverse loads). M1/M2 is positive for reverse curvature (double curvature bending). Minimum Cm = 0.4.
• Cm = 0.85 (approximate) for members with transverse loads and 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

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.

Worked example — Section H2 stress-based check for a channel in combined loading

Given: C12x20.7 channel (A36 steel, Fy = 36 ksi), length = 10 ft, pinned ends. The channel carries Pu = 30 kips (tension) and Mux = 25 kip-ft (strong-axis bending from self-weight and a small eccentric load). The eccentric load also produces a torsional moment Tu = 5 kip-in.

Step 1 — Section properties:

Ag = 6.09 in2, d = 12.0 in, tf = 0.501 in, tw = 0.282 in
Sx = 10.8 in3, Zx = 12.6 in3, Ix = 129 in4
J = 0.194 in4, Cw = 34.8 in6, y_bar = 0.698 in (shear center offset)

Step 2 — Available axial strength (tension):

phi_t x Pn = 0.90 x Fy x Ag = 0.90 x 36 x 6.09 = 197 kips

Step 3 — Available flexural strength: Since Lb = 10 ft is likely less than Lp for this section, the plastic moment controls. For a C12x20.7:

phi_b x Mnx = 0.90 x 36 x 12.6 / 12 = 34.0 kip-ft

Step 4 — Available torsional strength (warping + St Venant): The torsional strength is a stress-based check per Section H2. Compute the warping stress at the flange tip from Tu combined with the flexural stresses. The section H2 stress check sums the individual stresses at the critical point:

At the extreme fiber of the compression flange (point A, flange tip):

Axial stress: f_a = Pu / Ag = 30 / 6.09 = 4.93 ksi (tension, positive)
Bending stress: f_bw = Mux / Sx = 25 x 12 / 10.8 = 27.8 ksi
Warping normal stress from torsion: f_w = Tu x Q_w / (J x Cw) ~ 3.2 ksi at flange tip

The combined normal stress at point A:

f_total = f_a + f_bw + f_w = 4.93 + 27.8 + 3.2 = 35.9 ksi
F_y = 36 ksi

Section H2 interaction (using ASD format for stress summation):

f_a / (phi_t x F_y) + f_bw / (phi_b x F_y) + f_w / (phi_b x F_y) <= 1.0
= 4.93/(0.90 x 36) + 27.8/(0.90 x 36) + 3.2/(0.90 x 36)
= 0.152 + 0.858 + 0.099 = 1.109 — FAILS

Step 5 — Redesign: The warping stress puts the section over the limit. Options: (1) reduce the eccentric load to reduce Tu, (2) brace the channel at midspan to reduce warping, or (3) use a heavier channel (C15x33.9) where the warping stress is a smaller proportion of capacity. With a C15x33.9:

Sx = 23.8 in3, f_bw = 25 x 12 / 23.8 = 12.6 ksi
f_total = 4.93 + 12.6 + 3.2 = 20.7 ksi
H2 check: 20.7 / (0.90 x 36) = 0.639 — OK

This example illustrates why Section H2 is essential for channels, tees, angles, and other singly-symmetric sections where torsion is present — the Section H1 equations do not capture warping stresses.

Where biaxial bending occurs

• Corner columns in buildings with moment frames in both directions. Wind or seismic forces from orthogonal directions produce simultaneous Mux and Muy.
• Crane columns in industrial buildings. The crane runway girder applies a strong-axis moment from vertical wheel loads and a weak-axis moment from lateral crane surge forces (bumper impact or skewing).
• Columns at re-entrant corners of L-shaped or U-shaped floor plans, where beams frame in from two perpendicular directions with unbalanced gravity loads.
• Pilaster columns in precast concrete or masonry buildings where lateral bracing connections impose eccentric axial loads in two directions.

Interaction surface concept

The bilinear interaction equations H1-1a and H1-1b are an approximation of the true three-dimensional interaction surface defined by the plastic behavior of the cross-section. For a wide-flange shape, this surface is defined by the fully-yielded condition at every point on the section, considering the combined stress state from axial force and biaxial bending.

The exact plastic interaction surface for a doubly symmetric I-shape is convex and can be constructed numerically. The AISC bilinear approximation (H1-1a / H1-1b) envelopes this surface conservatively — no point on the actual plastic surface lies outside the AISC interaction equations. For standard building design, the approximation is within 5-10% of the exact surface, which is acceptable given the resistance factors already applied to the capacity terms.

When biaxial moments are large relative to axial force, consider using a section with ry/rx closer to 1.0. Square HSS sections (HSS 10x10, HSS 12x12) and W14 shapes in heavier weights have nearly equal strong- and weak-axis properties, reducing the penalty from biaxial interaction.

Common mistakes

1. Forgetting second-order effects. Using first-order moments directly is unconservative.
2. Using the wrong equation. Check Pr/Pc first: >= 0.2 uses H1-1a, < 0.2 uses H1-1b.
3. Not checking both axes. Both Mrx and Mry terms must be included for biaxial bending.
4. Applying DAM stiffness reductions inconsistently. Reduced stiffness for analysis, nominal properties for capacity.
5. Not checking multiple points along the member. The critical section may be at an interior point.

AISC Chapter H interaction equations — detailed

AISC 360-22 Chapter H governs the design of members subject to combined axial force and flexure. Section H1 applies to doubly symmetric members (most standard W-shapes, HSS, and pipes), while Section H2 covers all other cases. The two primary interaction equations — H1-1a and H1-1b — are the backbone of beam-column design in the United States.

Full variable definitions

All demand values (Pu, Mux, Muy) must include second-order effects per Chapter C. All capacity values (φPn, φMnx, φMny) use nominal section properties with appropriate resistance factors.

Equation H1-1a — high axial ratio (Pu/φPn >= 0.2)

Pu/φPn + 8/9 × (Mux/φMnx + Muy/φMny) ≤ 1.0    [Eq. H1-1a]

This equation applies when axial compression dominates the design — specifically, when the axial demand-to-capacity ratio is 0.2 or greater. The 8/9 coefficient on the moment terms reflects the fact that the moment gradient across the member provides some relief; it is not simply a safety margin but a calibrated factor derived from plastic interaction surfaces for wide-flange sections.

Equation H1-1b — low axial ratio (Pu/φPn < 0.2)

Pu/(2×φPn) + (Mux/φMnx + Muy/φMny) ≤ 1.0       [Eq. H1-1b]

When the axial component is small (less than 20% of capacity), the member behaves more like a beam than a column. Equation H1-1b halves the axial contribution and removes the 8/9 coefficient from the moment terms, which is appropriate because moment-dominated members are less sensitive to the combined effect.

Why two equations?

The bilinear formulation exists because a single linear equation cannot accurately represent the true plastic interaction surface of a steel cross-section across all axial-to-moment ratios. Equation H1-1a is more restrictive for axial-dominated members and correctly penalizes simultaneous high axial force and bending. Equation H1-1b relaxes the axial term for moment-dominated members where the axial force does not significantly reduce the plastic moment capacity. The transition at Pu/φPn = 0.2 is the point where the two equations produce the same result, ensuring a smooth and continuous transition between the two regimes. This bilinear approximation was calibrated against exact plastic interaction surfaces and provides a conservative but not overly conservative envelope.

Worked example — W12x65 beam-column (AISC 360-22 LRFD)

This example demonstrates the full interaction check for a W12x65 column in a moment frame. The calculation follows AISC 360-22 LRFD provisions.

Given: W12x65, A992 steel (Fy = 50 ksi, Fu = 65 ksi), KL = 12 ft (K = 1.0, pinned-pinned), Pu = 200 kips, Mux = 120 kip-ft, Muy = 0 (uniaxial bending only). Second-order effects are assumed to be included in the given moments.

Step 1: Section properties

The W12x65 is a compact section per AISC Table B4.1b, so the plastic modulus Zx may be used for flexural capacity.

Step 2: Axial capacity (Chapter E)

The critical slenderness ratio governs the compressive strength. Since ry < rx, the weak axis controls:

KL/ry = 12 × 12 / 2.68 = 53.7

Elastic critical stress:

Fe = π² × 29,000 / 53.7² = 907,920 / 2,884 = 99.1 ksi

Check the inelastic buckling limit:

4.71 × √(E/Fy) = 4.71 × √(29,000/50) = 4.71 × 24.08 = 113.4

Since KL/ry = 53.7 < 113.4, use the inelastic buckling formula:

Fcr = 0.658^(Fy/Fe) × Fy = 0.658^(50/99.1) × 50 = 0.658^0.505 × 50 = 0.811 × 50 = 40.5 ksi

Compressive capacity:

φPn = 0.90 × 40.5 × 19.1 = 696 kips

Step 3: Flexural capacity (Chapter F)

For a compact W-shape with adequate lateral bracing, the plastic moment controls. Assuming the unbraced length Lb is short enough that Lb ≤ Lp (which is typical for columns braced by floor beams):

φMnx = φ × Fy × Zx / 12 = 0.90 × 50 × 96.8 / 12 = 363 kip-ft

Step 4: Interaction check

Determine which equation applies:

Pu/φPn = 200/696 = 0.287 ≥ 0.2  →  Use Equation H1-1a

Evaluate H1-1a:

0.287 + 8/9 × (120/363 + 0) = 0.287 + 8/9 × 0.331 = 0.287 + 0.294 = 0.581 ≤ 1.0  →  PASS

Result: The demand-to-capacity (D/C) ratio is 0.581. The W12x65 section has 42% reserve capacity, meaning this column could carry significantly more load before reaching its interaction limit. A smaller section (such as a W12x53) might also work and should be checked for economy.

P-M interaction curve for W12x65

The P-M interaction curve shows all combinations of axial load and moment that produce failure. For the W12x65, the key points on the interaction curve are:

*Note: Point D is not precisely the balanced failure point — it is the point on the interaction curve at Pn = 200 kips, which happens to be the Pu value. At this exact point, the interaction ratio equals 1.0, representing the maximum moment this column can carry at Pu = 200 kips.

The interaction curve for a W-shape is convex outward when plotted as Pn vs Mn. Points inside the curve are safe; points outside are overstressed. The curve reaches its widest moment capacity at approximately Pn = 0.15 x Pno (where adding a small compressive force actually increases the moment capacity by shifting the neutral axis slightly into the web, engaging more of the compression flange).

Construction of the curve: Each point on the interaction curve represents a plastic stress distribution at a different neutral axis location:

• When the NA is at the infinity (pure compression), the full section is at Fy in compression — Point A.
• As the NA moves downward through the section, more of the section goes into tension, creating a moment couple while the net axial force decreases.
• When the NA is at the plastic centroid (mid-depth for a symmetrical section), the net axial force is zero and the moment equals Mp — Point E.
• Moving the NA further into the tension zone produces combined tension and moment — Point F.

For a compact W-shape with Lb <= Lp, the plastic interaction surface is defined by the full yielding of the cross-section at every point along the depth, providing the theoretical upper bound of combined strength.

Design procedure for beam-columns

The following step-by-step procedure provides a systematic approach to beam-column design per AISC 360-22. This procedure applies to doubly symmetric members governed by Chapter H, Section H1.

Step 1 — Determine factored loads. Obtain the factored axial force Pu and factored moments Mux, Muy from structural analysis using LRFD load combinations (ASCE 7-22). These are the first-order demands.

Step 2 — Amplify moments for second-order effects. Use either the Direct Analysis Method (AISC Chapter C) with reduced stiffness, or the Amplified First-Order Analysis method (Appendix 8) with B1 and B2 amplifiers. The B1 factor captures member P-delta effects (deformation within the member length), while B2 captures story P-Delta effects (lateral drift of the entire story). Never use first-order moments directly for beam-column checks.

Step 3 — Calculate axial capacity (φPn). Determine the compressive strength per AISC Chapter E using the appropriate effective length KL and the controlling slenderness ratio. For columns in moment frames, K is typically determined from the alignment charts or taken as 1.0 when the Direct Analysis Method is used.

Step 4 — Calculate flexural capacities (φMnx, φMny). Determine the flexural strength per AISC Chapter F. Consider lateral-torsional buckling (LTB) limits — the unbraced length Lb relative to Lp and Lr determines whether the plastic moment, inelastic buckling, or elastic buckling capacity controls. For weak-axis bending, Chapter F6 applies and the plastic moment typically controls for compact sections.

Step 5 — Determine which interaction equation applies. Calculate the ratio Pu/φPn. If this ratio is 0.2 or greater, use Equation H1-1a (the high-axial equation). If less than 0.2, use Equation H1-1b (the low-axial equation). This check must be performed first because the two equations have different forms.

Step 6 — Evaluate the interaction equation. Substitute all values into the appropriate equation and compute the result. The sum represents the overall demand-to-capacity ratio for the member under combined loading.

Step 7 — Verify adequacy. Confirm that the interaction result is less than or equal to 1.0. Values exceeding 1.0 indicate the member is overstressed and must be redesigned.

Step 8 — Resize if overstressed. If the D/C ratio exceeds 1.0, select a larger section and repeat from Step 3. Prioritize increasing the section in the direction that contributes most to the overstress — if the axial term is dominant, choose a section with more area; if the moment term is dominant, choose a section with a larger modulus. Iteration is typically required only once or twice to converge on an acceptable section.

Common beam-column sections

The following table lists W-shapes commonly selected for beam-column applications. These sections offer a good balance of axial area and flexural section moduli, making them efficient for combined loading conditions.

The typical Pu ranges assume KL = 10-14 ft, Fy = 50 ksi (A992), and Pu/phiPn in the range of 0.4-0.7. Actual capacity varies with unbraced length and end conditions.

Selecting beam-column sections

Heavier sections within the same depth group provide more axial capacity without increasing the floor-to-floor height, which is critical for architectural coordination. The W12 and W14 series are the most common beam-column choices in building construction for several reasons: they offer wide flanges that resist strong-axis bending effectively, they have sufficient cross-sectional area for axial loads, and they are readily available from service centers.

W12 sections are preferred when floor-to-floor heights are limited, as the 12-inch nominal depth fits within typical ceiling plenums. W14 sections are favored for taller buildings and heavier loads because their deeper profile provides greater moment resistance and their wider flanges offer better weak-axis properties for biaxial bending conditions.

For corner columns subjected to significant biaxial bending, consider W14 sections or even built-up sections. The W14x120 and heavier shapes in that series have ry/rx ratios closer to 1.0, meaning they are more equally resistant about both axes — an advantage when both Mux and Muy are substantial.

For preliminary sizing, a common rule of thumb is to start with a section where the axial demand-to-capacity ratio (Pu/φPn) is approximately 0.4 to 0.6 when acting alone. This leaves sufficient reserve for the moment terms in the interaction equation. If the column carries significant biaxial moments, start with a ratio closer to 0.3 to 0.4.

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.

How much does biaxial bending reduce capacity? Significantly. A column at 60% utilization about each axis independently has an interaction value of 0.6 + (8/9)*(0.6) = 1.13, which fails. Biaxial bending always reduces available axial capacity.

When can I ignore P-delta effects? When Pr/Pe < 0.05 approximately (AISC C2.2b). For most practical columns, second-order effects are significant and must be included.

Do I need to check combined loading for tension members? Yes, when the tension member also carries bending from self-weight, connection eccentricity, or lateral loads. The same interaction equations apply, but P-delta amplification is not needed (tension is stabilizing).

What is the difference between Section H1 and Section H2 in AISC 360?

Section H1 applies to doubly symmetric members (W-shapes, HSS, pipes) where combined axial and bending forces can be checked using the section-level interaction moment. Section H2 applies to all other cases: singly symmetric members (channels, tees, angles) and members subject to torsion. H1 uses the bilinear equations H1-1a/b with 8/9 coefficient and the 0.2 transition. H2 uses a stress-based summation: f_ra/F_ca + f_rbw/F_cbw + f_rbz/F_cbz <= 1.0, where each term is the ratio of required stress to available stress at the critical point on the cross-section. H2 requires checking individual points on the cross-section (flange tips, web toe of fillet) because the stress distribution is not uniform — warping torsion and unsymmetric bending produce stress concentrations that the section-level H1 check would miss.

How do LRFD and ASD interaction checks differ for combined loading?

LRFD uses H1-1a/b directly with factored loads Pu and Mu: Pu/phi_c_Pn + 8/9 x (Mux/phi_b_Mnx) <= 1.0. The capacities phiPn and phiMn already include the resistance factors. ASD uses the same equations but with service-level loads Pa and Ma and the safety factors omega_c and omega_b: (Pa/Pn/omega_c) + 8/9 x (Ma/Mn/omega_b) <= 1.0. The practical difference: for a given member and loading, the LRFD and ASD interaction ratios will be similar but not identical because the load combinations differ (LRFD has higher loads, ASD uses service loads). For the W12x65 example with Pu = 200 kips and Mu = 120 kip-ft (LRFD factored), the ASD equivalent would use Pa ~ 133 kips and Ma ~ 80 kip-ft (service level, assuming 1.5 load factor), giving an ASD interaction ratio of approximately 0.55 — slightly lower than the LRFD ratio of 0.581.

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Related Pages

• Column Buckling Equations — Euler and Fcr formulas
• Lateral-Torsional Buckling — Lp, Lr, Cb reference
• Effective Length Factor K — alignment chart and DAM
• K-Factor Reference — braced and unbraced frames
• Load Combinations — ASCE 7, EN 1990, AS 1170, NBCC
• Compact Section Limits — section classification
• Beam Sizes Chart — 148 W-shapes
• Steel Fy & Fu Reference — material properties
• Cb Factor — moment gradient modifier
• Tools Directory
• Reference Tables Directory
• How to Verify Calculator Results

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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.