F-04 — combined bending + axial

two loads, one section, one equation.

interaction equation tool

drag the operating point. watch the interaction.

Drag the point on the interaction diagram to explore how axial load and moment combine.

applied loads
P (k)120
M (k-in)600
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utilization
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live equation
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Pu = Applied axial demand with load factors
Mu = Applied bending demand with load factors
φPn = Factored axial strength of the member
φMn = Factored bending strength of the member
explained
When axial load and bending act simultaneously, neither can be checked alone. The interaction equation combines both demands into a single utilization ratio. Below 1.0 the member passes; above 1.0 it fails. The curve on the diagram shows every possible passing combination.
key concepts
overview Combined stresses add: σ = P/A ± My/I

When P and M act together, their stresses add: σ(y) = P/A ± My/I. The cross-section doesn't care which load caused it — it just sees total stress. The AISC H1-1 interaction equation treats each load as a fraction of the section's capacity: P/P_n + (8/9)·M/M_n ≤ 1.0. If the point is inside the boundary on the interaction diagram, the section works. Outside, it doesn't.

when two loads act together Axial and bending stresses superpose on the cross-section

Most real columns carry both axial load (P from gravity) and bending moment (M from lateral loads, eccentricity, or frame action). The stresses add: σ = P/A ± My/I. One side of the section gets compression from both effects; the other gets partial relief. The section doesn't care which load caused the stress — it just sees the total. Design must check the combined effect, not each load independently.

the interaction equation AISC H1-1 checks utilization as a fraction of capacity

AISC H1-1 expresses this as a utilization check. When axial demand is significant (Pu/φPn ≥ 0.2): Pu/φPn + (8/9)(Mu/φMn) ≤ 1.0. When axial is light (Pu/φPn < 0.2): Pu/(2φPn) + Mu/φMn ≤ 1.0. The kink at Pu/φPn = 0.2 is visible in the interaction diagram — the tool below plots this boundary. Any point inside the curve passes; outside fails.