Combined Axial & Bending Design — AISC 360 Interaction Equations Worked Example

Every column in a real building carries both axial load and bending moment. Pure axial compression exists only in textbook problems. Wind, seismic drift, beam end reactions, and eccentric connections all introduce moments — and when those moments combine with axial compression, the column must be checked using interaction equations rather than standalone axial or flexural checks. This is the domain of AISC 360-22 Chapter H1, which provides the interaction framework that governs combined loading for all doubly symmetric steel members.

In this guide: We work through the complete AISC 360-22 H1 combined axial and bending design procedure. We cover the H1-1a and H1-1b interaction equations, second-order moment amplification (B1 and B2), the P-M interaction diagram, a full worked example with a W12x65 column under 440 kips axial + 85 kip-ft moment, a four-code comparison table (AISC 360 vs AS 4100 vs EN 1993 vs CSA S16), and the most common mistakes engineers make when checking combined loading.

PRELIMINARY — NOT FOR CONSTRUCTION. All results discussed are for educational and reference use only. Must be independently verified by a licensed Professional Engineer or Structural Engineer before use in any project.

What You Will Learn

By the end of this guide, you will understand:

• The two H1 interaction equations and when to use each (the Pr/Pc > 0.2 threshold)
• How to calculate second-order amplified moments using B1 and B2 factors
• How to construct and read a P-M interaction diagram
• How AISC 360, AS 4100, EN 1993, and CSA S16 differ in their approach to combined loading
• The seven most common combined-loading mistakes that cause design failures
• How to verify your results using a free column capacity calculator

The H1 Interaction Equations — AISC 360-22 Section H1.1

AISC 360 provides two interaction equations for doubly symmetric members subject to combined axial force and flexure. The choice of equation depends on the magnitude of the axial load relative to the column's axial capacity.

Equation H1-1a — Axial-Dominated (Pr/Pc >= 0.2)

When the required axial strength exceeds 20% of the available axial strength, use:

Pr/Pc + 8/9 x (Mrx/Mcx + Mry/Mcy) <= 1.0

Where:

• Pr = required axial strength (Pu for LRFD, Pa for ASD)
• Pc = available axial strength (phi_c x Pn for LRFD, Pn/Omega_c for ASD)
• Mrx, Mry = required flexural strength about x and y axes, including second-order effects
• Mcx, Mcy = available flexural strength about x and y axes
• 8/9 = amplification factor on the moment terms

The 8/9 factor is not a safety factor — it is a mathematical consequence of the linear approximation to the true interaction curve. When the section is under high axial load, the moment capacity degrades faster than a simple linear reduction. The 8/9 factor accounts for this nonlinearity on the compression-dominated side of the interaction diagram.

Equation H1-1b — Moment-Dominated (Pr/Pc < 0.2)

When the axial load is small relative to capacity, the section behaves more like a beam and a simpler linear interaction applies:

Pr/(2 x Pc) + (Mrx/Mcx + Mry/Mcy) <= 1.0

The factor of 2 in the denominator means that the axial term is effectively half of the capacity check — because at low axial loads, the moment capacity is nearly the full plastic moment and axial effects are secondary.

The 0.2 Threshold — Why It Matters

The boundary between H1-1a and H1-1b at Pr/Pc = 0.2 is not arbitrary. At approximately 20% of the axial capacity, the P-M interaction curve transitions from compression-governed behavior (where the plastic neutral axis is deep in the web) to flexure-governed behavior (where the neutral axis is near the flange). Using the wrong equation at this boundary can be unconservative by up to 12%.

Practical rule: If your column has any significant axial load (dead + live from multiple floors), you will almost certainly be above 0.2. Use H1-1a by default; only use H1-1b for very lightly loaded members like wind posts or bracing elements.

Second-Order Moment Amplification — B1 and B2 (Appendix 8)

Before entering the H1 interaction equation, the required flexural strengths Mrx and Mry must include second-order (P-Delta) effects. These amplify first-order moments by a factor that depends on how close the axial load is to the elastic buckling load.

B1 — P-Delta Amplification Between Brace Points

For braced frames where sidesway is prevented, second-order effects are captured by the B1 amplifier:

B1 = Cm / (1 - Pr/Pe1)

Where:

• Cm = equivalent moment factor (0.60 for braced frames with transverse loads, 1.0 for uniform moment)
• Pe1 = Euler buckling load in the plane of bending = pi^2 x E x I / (K1 x L)^2

The amplified moment is:

Mr = B1 x Mnt (Mnt = first-order moment, no translation)

Critical insight: As Pr approaches Pe1, the denominator (1 - Pr/Pe1) approaches zero and B1 grows without bound. For a column loaded at 80% of its Euler load, B1 = Cm / (1 - 0.8) = 5 x Cm — the design moment is amplified fivefold. This is not a flaw in the code; it is the physical reality of elastic instability. The column cannot be loaded above Pe1 without bracing.

B2 — P-Delta0 Sidesway Amplification

For moment frames where sidesway occurs:

B2 = 1 / (1 - sum(Pr) / sum(Pe2))

Where sum(Pr) and sum(Pe2) are taken over all columns in the storey. For braced frames, B2 = 1.0 and only B1 amplification applies.

Worked Cm Values

Using Cm = 1.0 is always conservative. Using the actual Cm from the moment diagram can reduce the amplified moment by 20-40%.

Worked Example — W12x65 Column, Axial + Strong-Axis Bending

Design Parameters

Section Properties — W12x65

For a complete W-shape properties database, see the Section Properties Directory.

Step 1: Factored Loads (LRFD)

Pu = 1.2 x 180 + 1.6 x 140 = 216 + 224 = 440 kips

First-order moment (already factored): Mu_nt = 85 kip-ft = 1020 kip-in

Step 2: Axial Compression Capacity — AISC 360 Chapter E

Section classification (Table B4.1b): Flange bf/2tf = 9.92 > compact limit of 9.15, but < noncompact limit of 24.1. Flange is noncompact; apply Qs reduction. Q = Qs x Qa = 0.95.

Weak-axis flexural buckling governs. KLy = 0.80 x 15 x 12 = 144 in. KLy/ry = 144 / 3.02 = 47.7.

Euler stress: Fe = pi^2 x 29,000 / 47.7^2 = 125.8 ksi.

Inelastic buckling limit: 4.71 x sqrt(E / (Q x Fy)) = 4.71 x sqrt(29,000 / (0.95 x 50)) = 116.4.

Since KL/r = 47.7 < 116.4, inelastic buckling applies:

Fcr = Q x (0.658^(Q x Fy / Fe)) x Fy = 0.95 x (0.658^(0.95 x 50 / 125.8)) x 50 = 40.7 ksi.

Pn = Fcr x A = 40.7 x 19.1 = 777 kips.

LRFD: phi_c x Pn = 0.90 x 777 = 699 kips.

Axial utilization: Pr/Pc = 440 / 699 = 0.630.

Since 0.630 > 0.2, use Equation H1-1a.

Step 3: Second-Order Moment Amplification (B1)

Strong-axis effective length for bending: K1 = 1.0 (pinned top in the plane of bending for the braced frame).

Pe1x = pi^2 x E x Ix / (K1 x L)^2 = pi^2 x 29,000 x 533 / (1.0 x 180)^2 = 152,600,000 / 32,400 = 4,710 kips.

Cm = 0.60 (braced frame, transverse load between supports, ends restrained — conservative for a column with end moment).

B1 = Cm / (1 - Pu/Pe1x) = 0.60 / (1 - 440 / 4,710) = 0.60 / (1 - 0.0934) = 0.60 / 0.9066 = 0.662.

Amplified moment: Mrx = B1 x Mnt = 0.662 x 85 = 56.3 kip-ft = 676 kip-in.

Note that B1 < 1.0 in this case because Cm = 0.60 and Pu/Pe1 is small. For a column loaded near its buckling capacity, B1 amplifies rather than reduces.

Step 4: Strong-Axis Flexural Strength

For column application with continuous bracing in the plane of the frame, Lb = 0 for strong-axis buckling. Section is noncompact (flange bf/2tf = 9.92 > 9.15), so Mn is between Mp and My.

Mp = Zx x Fy = 96.8 x 50 = 4,840 kip-in = 403.3 kip-ft. My = Sx x Fy = 87.9 x 50 = 4,395 kip-in = 366.3 kip-ft.

For the noncompact flange in flexure:

Mn = Mp - (Mp - 0.7 x Fy x Sx) x (lambda_f - lambda_pf) / (lambda_rf - lambda_pf) = 4,840 - (4,840 - 3,077) x (9.92 - 9.15) / (24.1 - 9.15) = 4,840 - 1,763 x 0.77 / 14.95 = 4,840 - 90.8 = 4,749 kip-in = 395.8 kip-ft.

LRFD: phi_b x Mnx = 0.90 x 395.8 = 356.2 kip-ft.

Step 5: H1-1a Interaction Check

Pr/Pc + 8/9 x (Mrx/Mcx) <= 1.0

= 0.630 + 8/9 x (56.3 / 356.2)

= 0.630 + 0.889 x 0.158

= 0.630 + 0.140

= 0.770

Result: 0.770 <= 1.0 — PASS. The W12x65 has 23% reserve capacity under combined loading.

Interpretation

The axial term (0.630) dominates the interaction — 82% of the total 0.770 comes from compression. The moment term contributes only 18%. This is typical for interior columns in braced frames, where axial loads from multiple floors are high while bending moments from gravity framing are modest.

If the same column carried a 150 kip-ft moment (e.g., from wind in a moment frame), the interaction would be:

= 0.630 + 0.889 x (150 / 356.2) = 0.630 + 0.374 = 1.004 — FAILS.

Lesson: Combined loading is sensitive to small changes in moment. A column that passes axial and flexural checks independently can still fail the interaction equation.

The P-M Interaction Diagram

The P-M interaction diagram (also called a column interaction curve) is the graphical representation of all acceptable (P, M) combinations for a given section. Understanding this diagram is essential for interpreting interaction equation results.

How the Curve Is Built

The interaction curve is constructed from four key points:

The curve between points A and B is compression-controlled: the section crushes before the tension flange yields. Between B and C, the section is tension-controlled: yielding governs. The balanced point B represents simultaneous crushing of the compression fiber and yielding of the tension flange.

Reading the W12x65 Interaction Curve

For our W12x65 at 15 ft:

• Point A (pure compression): P = 699 kips, M = 0
• Point C (pure flexure): P = 0, M = 356.2 kip-ft
• Our design point: P = 440 kips, M = 56.3 kip-ft

A radial line from the origin through our design point hits the curve at approximately (P_failure, M_failure) = (571 kips, 73 kip-ft). Since our point (440, 56.3) is inside the curve, the section passes. The interaction ratio 0.77 means we are at 77% of the distance from the origin to the failure surface.

What the Curve Tells You

• Above the balanced point: Adding more moment rapidly reduces axial capacity. Small moment increases produce large interaction ratio increases.
• Below the balanced point: Adding more axial load has relatively little effect on moment capacity — the section behaves more like a beam.
• Steep curve near failure: The near-vertical region above the balanced point means that columns under high axial load are very sensitive to incidental moments from connection eccentricity or frame action.

Code Comparison — Combined Axial & Bending Across Standards

Every major steel design code addresses combined loading, but the mathematical treatment differs significantly. The table below compares the approach of each standard.

Key Differences Explained

EN 1993 interaction factors (k_yy, k_yz, k_zy, k_zz): The Eurocode approach is the most mathematically complex. Instead of a single amplification factor, it uses four interaction factors that account for cross-section class, buckling curve, and moment distribution. Annex B provides two alternative methods (Method 1 and Method 2) with different factor derivations. This produces more accurate results for non-standard sections but requires significantly more computation.

AS 4100 threshold at 0.15: The Australian standard uses a lower threshold (Pr/Pc = 0.15 vs AISC's 0.20), meaning more members are checked with the axial-governing equation. This is slightly more conservative for members with intermediate axial loads.

CSA S16 moment factor of 0.85: Instead of 8/9 (0.889), CSA S16 uses 0.85 on the moment terms in its axial-dominated equation. The Canadian code is slightly more conservative on the moment contribution for compression-governed sections.

Numerical Comparison — Same Column, Four Codes

Using our W12x65 at 15 ft with Pu = 440 kips, Mu = 56.3 kip-ft (amplified):

All four codes agree that the W12x65 is acceptable. AISC, AS 4100, and CSA S16 produce similar ratios (0.77-0.80). EN 1993 typically produces a slightly higher interaction ratio due to its more conservative treatment of combined effects through interaction factors.

Common Mistakes in Combined Axial & Bending Design

1. Forgetting Second-Order Amplification

The single most common error is using first-order moments Mnt directly in the H1 equation without B1 and B2 amplification. For columns with Pu/Pe1 > 0.5, this can underestimate the design moment by a factor of 2 or more. Always calculate B1 and B2 before entering the interaction equation.

2. Using H1-1b When Pr/Pc > 0.2

The H1-1b equation divides the axial term by 2, making it less conservative. Using H1-1b above the 0.2 threshold can produce interaction ratios that are 5-10% unconservative. If in doubt, use H1-1a — it is the more conservative equation.

3. Applying B1 to the Wrong Moment

B1 amplifies Mnt (the first-order moment with no translation, typically from gravity loads in a braced frame). B2 amplifies Mlt (the first-order moment from sidesway, typically from wind or seismic lateral loads). Swapping these produces nonsense results.

4. Ignoring Weak-Axis Bending

Many engineers check only strong-axis bending when the column has weak-axis moments from cladding rails, girts, or eccentric connections. The H1 equation includes both Mry and Mrx, and the weak-axis flexural capacity (phi_b x Mny) is typically much lower than the strong-axis capacity. Even a small weak-axis moment — say, 5 kip-ft from girt eccentricity — can push a borderline design over 1.0.

5. Using Gross Section Modulus Sx Instead of Zx

For compact sections in flexure, the plastic modulus Zx applies. Using Sx instead of Zx underestimates flexural capacity by the shape factor (typically 1.10-1.16 for W-shapes), leading to an unnecessarily conservative interaction ratio.

6. Failing to Re-Check After Section Upgrade

When an interaction check fails and you upgrade the section, the new section has different Ix, A, ry, and Zx values. The axial capacity, Euler load, and B1 factor all change. Re-running only the flexural check with the new section while keeping the old B1 value is a mistake.

7. Mixing LRFD and ASD Values in the Same Equation

Pu (LRFD) and Pa (ASD) produce different interaction ratios even for the same nominal strength. Do not divide an LRFD Pu by an ASD Pn/Omega_c. The phi factor and Omega factor are not reciprocals of each other for all limit states.

Frequently Asked Questions

What is the AISC 360 H1 interaction equation for combined axial and bending?

The AISC 360-22 H1 interaction equations check combined axial force and flexure in doubly symmetric members. When Pr/Pc >= 0.2, use Equation H1-1a: Pr/Pc + 8/9 x (Mrx/Mcx + Mry/Mcy) <= 1.0. When Pr/Pc < 0.2, use Equation H1-1b: Pr/(2 x Pc) + (Mrx/Mcx + Mry/Mcy) <= 1.0. Second-order effects (B1 and B2) must be included in Mr before entering the equation.

What is second-order moment amplification (B1 and B2)?

B1 amplifies moments between brace points (P-delta effect): B1 = Cm / (1 - Pr/Pe1). B2 amplifies moments from sidesway (P-delta0 effect): B2 = 1 / (1 - sum(Pr) / sum(Pe2)). When Pr approaches Pe1, B1 can exceed 2.0 — doubling the design moment. For braced frames, B2 = 1.0 and only B1 amplification applies.

How does the P-M interaction diagram work?

A P-M interaction diagram plots acceptable (P, M) combinations. The curve is compression-controlled above the balanced point and tension-controlled below it. The AISC H1 equations approximate the curve with linear segments. Any point inside the curve is acceptable. The interaction ratio Pr/Pc + 8/9 x Mr/Mc gives the distance from the origin to the failure surface along a radial line.

When is combined axial and bending the governing limit state?

Combined loading governs when (1) a column carries significant moment from frame action, (2) beam-column connections introduce eccentric moments, (3) lateral loads (wind, seismic) produce overturning moments, or (4) the axial utilization alone is above 0.7 and any incidental moment is present. For corner columns in moment frames and columns in braced frames with girt/purlin eccentricity, combined loading is almost always the governing check.

Do I need to check biaxial bending if my moment is only in one axis?

Strictly, no — if Mry = 0, the Mry/Mcy term is zero. But you should verify that there is truly no weak-axis moment. Eccentric connections, girt rails that frame into the column web, and construction tolerances can all introduce small weak-axis moments that are often neglected. If the column web carries a connection with even 2 inches of eccentricity, the resulting weak-axis moment may be non-zero.

Is this calculator a replacement for professional engineering judgment?

No — this is an educational reference only. All combined axial and bending designs must be independently verified by a licensed Professional Engineer before use in any project. Results are PRELIMINARY — NOT FOR CONSTRUCTION.

Key Takeaways

• Use H1-1a when Pr/Pc >= 0.2, H1-1b otherwise. For most real columns, H1-1a governs.
• Always amplify first-order moments with B1 (and B2 for unbraced frames) before entering the H1 equation. Pu/Pe1 is the governing parameter for amplification.
• The P-M interaction diagram shows why columns under high axial load are highly sensitive to even small incidental moments — the curve is steepest above the balanced point.
• Weak-axis bending matters. Even 5 kip-ft of minor-axis moment from connection eccentricity can push a borderline design past 1.0 because Mcy is much smaller than Mcx.
• The 8/9 factor is not arbitrary — it accounts for the nonlinear degradation of moment capacity under high axial compression.
• Code differences are real but small for standard W-shapes. AISC 360, AS 4100, and CSA S16 produce similar interaction ratios (within ~5%). EN 1993 may be more conservative due to interaction factors.
• Section classification drives capacity. A noncompact flange (like the W12x65) reduces both Pc and Mc, compounding the interaction effect.

Run This Calculation

Use the Steel Calculator tools to verify the worked example above with your own inputs. Each tool handles multiple design codes and shows the full calculation breakdown.

Column Capacity Calculator — Axial compression and buckling check per AISC 360 Chapter E, AS 4100 Section 6, EN 1993-1-1 Clause 6.3, and CSA S16 Clause 13.3. K-factor selection, effective length, and slenderness checks.

Beam Capacity Calculator — W-shape moment and shear capacity with LTB, section classification, and deflection checks across all four codes. Required for the Mc terms in the interaction equation.

Load Combinations Calculator — ASCE 7-22, EN 1990, AS/NZS 1170, and NBC load combinations. Generate factored load cases for your P and M values.

Section Properties Database — Browse 500+ W, HSS, C, L, and WT sections. Look up A, Ix, Sx, Zx, rx, ry, J, Cw, and classification limits for any section used in your combined loading checks.

Further Reading

• Steel Column Design Example — AISC 360-22 LRFD Worked Solution
• AISC 360 Design Examples — Beam, Column & Connection Worked Solutions
• AS 4100 Worked Examples — Beam, Column & Bolt Group Design
• EN 1993 Design Examples — Beam, Column & Connection Worked Solutions
• CSA S16 Design Examples — Beam, Column & Connection Worked Solutions
• Steel Frame Analysis Tutorial — Portal Method to Matrix Stiffness
• How to Read Steel Section Tables — W, HSS, C, L Properties Decoded
• Steel Section Properties Guide — Advanced Guide to A, Ix, Sx, Zx, rx, ry, J, Cw
• Structural Steel Connection Design Guide — Types, Checks & Worked Examples
• Complete Guide to Wind Load Calculation per ASCE 7-22
• Column K-Factor Table — 6 End Conditions, AISC Values
• Steel Fy & Fu Reference — Yield and Tensile Strength by Grade

Disclaimer (educational use only)

This page is provided for general technical information and educational use only. It does not constitute professional engineering advice, a design service, or a substitute for an independent review by a qualified structural engineer. Any calculations, outputs, examples, and workflows discussed here are simplified descriptions intended to support understanding and preliminary estimation.

All real-world structural design depends on project-specific factors (loads, combinations, stability, detailing, fabrication, erection, tolerances, site conditions, and the governing standard and project specification). You are responsible for verifying inputs, validating results with an independent method, checking constructability and code compliance, and obtaining professional sign-off where required.

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