Interaction Formula⚠ unverified

Understanding the Parameters

• Axial demand and capacity $P$, $P_{\text{allow}}$ — the first term is the fraction of the column's compression capacity in use. $P_{\text{allow}}$ is itself the buckling- or yield-governed axial capacity (from the Euler/Johnson formulas divided by a factor of safety), so this term already carries the column's slenderness behaviour.

• Bending demand and capacity $M$, $M_{\text{allow}}$ — the second term is the fraction of the section's moment capacity in use. $M_{\text{allow}}$ comes from the flexure/section-modulus check (see Section Modulus, Beam Bending Stress). For rigour, $M$ should be the amplified (second-order) moment from Beam Column Moment or Secant Formula Stress, not the first-order moment.

• The sum — a linear interaction. It treats the two effects as simply additive, which is conservative and simple. Real steel codes use a bilinear interaction (AISC: $P/P_c + \tfrac{8}{9}M/M_c$ when axial dominates) that better matches test data; this calculator uses the simpler straight-line version.

Note: this is the simplified linear interaction. The AISC/Eurocode design equations add moment- amplification factors and a bilinear form; use those for code-compliant checks. Documented so users don't mistake the ratio for a full code check.

Derivation (Approaching a Proof)

The linear interaction equation follows from a superposition-of-stresses argument at the extreme fibre. The maximum combined normal stress in a beam-column is the sum of the axial and bending stresses:

$$\sigma_{\max} = \frac{P}{A} + \frac{M c}{I}.$$

Require this to stay below the allowable stress $\sigma_{\text{allow}}$ and divide through by it:

$$\frac{P/A}{\sigma_{\text{allow}}} + \frac{Mc/I}{\sigma_{\text{allow}}} \le 1.$$

Identify $P_{\text{allow}} = \sigma_{\text{allow}} A$ (axial capacity) and $M_{\text{allow}} = \sigma_{\text{allow}} I/c = \sigma_{\text{allow}} S$ (moment capacity via the section modulus $S=I/c$). Substituting gives the interaction form:

$$\frac{P}{P_{\text{allow}}} + \frac{M}{M_{\text{allow}}} \le 1.$$

So the linear interaction is exactly the extreme-fibre stress check re-expressed in terms of capacities. Its limitation is visible in the derivation: it assumes the same allowable governs both, and it uses whatever $M$ is supplied — which is why a proper check first amplifies $M$ for P-delta effects and why codes replace the straight line with a fitted curve to reflect partial plastification and stability interaction.

Dimensional check. Each term is (force/force) or (moment/moment) — dimensionless; their sum is dimensionless. ✓

History and Development

Interaction equations for combined axial and bending have been the backbone of beam-column design since the mid-20th century. The linear form is the oldest and simplest; AISC introduced amplified, bilinear interaction equations (the $B_1/B_2$ moment magnifiers and the $8/9$-slope curve) as second-order analysis and plastic-design research matured. Eurocode 3 uses interaction factors $k_{ij}$ derived from extensive column-test calibration. All descend from the same superposition idea made rigorous here.

Related Concepts: Beam Column Moment, Secant Formula Stress, Euler Buckling Load column, Section Modulus, Beam Bending Stress, Factor of Safety

Notes: Simplified linear interaction — conservative and not a substitute for the code bilinear/ amplified equations. Use the amplified (second-order) moment for $M$. Ratio $\le 1.0$ passes.