Beam-Columns Notes

Last reviewed 16 Sept 2026 · 4 min read

Where beam-columns occur

A beam-column carries axial load together with bending moment:

columns of rigid frames (moments from beam connections and sway),
columns carrying eccentric loads (brackets, crane girders),
top chords of trusses with purlins between panel points,
columns under wind or earthquake lateral load,
rafters of portal frames (axial compression plus bending).

Behaviour

Axial compression amplifies bending: the lateral deflection $δ$ caused by moments adds an extra moment $P δ$ (the P-δ effect within a member; P-Δ for sway of the whole storey). The amplified moment is approximately

M_{ma x} \approx \frac{C _{m} M _{0}}{1 - P / P _{cr}}

where $C_{m}$ accounts for the moment distribution (1.0 for uniform moment; less when end moments are unequal or of opposite sign) and $P_{cr}$ is the Euler load in the plane of bending.

Failure modes: in-plane instability (bending about the major axis with axial load), flexural–torsional (lateral–torsional) buckling out of plane, or local yielding at the most stressed section.

IS 800:2007 — two checks

1. Cross-section strength (no buckling)

FormulaSection check (cl. 9.3.1)

For plastic and compact sections, the reduced plastic moment capacity in presence of axial force is used; a conservative simple check is

\frac{N}{N _{d}} + \frac{M _{y}}{M _{d y}} + \frac{M _{z}}{M _{d z}} \leq 1.0

$N_{d} = A_{g} f_{y} / γ_{m 0}$ ; $M_{d y}$ , $M_{d z}$ = design moment capacities about the minor and major axes (without axial force).

More refined: $(\frac{M _{y}}{M _{n d y}})^{α_{1}} + (\frac{M _{z}}{M _{n d z}})^{α_{2}} \leq 1.0$ with moments reduced for axial force, e.g. for I-sections $M_{n d z} = 1.11 M_{d z} (1 - n) \leq M_{d z}$ , $n = N / N_{d}$ .

2. Member buckling resistance

FormulaOverall member check (cl. 9.3.2.2)

\frac{P}{P _{d y}} + \frac{K _{y} C _{m y} M _{y}}{M _{d y}} + \frac{K _{L T} M _{z}}{M _{d z}} \leq 1.0

\frac{P}{P _{d z}} + \frac{0.6 K _{y} C _{m y} M _{y}}{M _{d y}} + \frac{K _{z} C _{m z} M _{z}}{M _{d z}} \leq 1.0

$P_{d y}$ , $P_{d z}$ = design compressive strengths for buckling about the minor and major axes; $M_{d z}$ includes lateral–torsional buckling where relevant.

$K_{y} = 1 + (λ_{y} - 0.2) n_{y} \leq 1 + 0.8 n_{y}$ , $K_{z} = 1 + (λ_{z} - 0.2) n_{z} \leq 1 + 0.8 n_{z}$ with $n = P / P_{d}$ about each axis.

$K_{L T} = 1 - \frac{0.1 λ _{L T} n _{y}}{C _{m L T} - 0.25} \geq 1 - \frac{0.1 n _{y}}{C _{m L T} - 0.25}$ .

$C_{m y}$ , $C_{m z}$ , $C_{m L T}$ = equivalent uniform moment factors from the shape of the moment diagram (for end moments $M$ and $ψ M$ : $C_{m} = 0.6 + 0.4 ψ \geq 0.4$ ).

The equations read: axial utilisation + amplified bending utilisations ≤ 1.

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Beam-Columns

Where beam-columns occur

Behaviour

IS 800:2007 — two checks

1. Cross-section strength (no buckling)

2. Member buckling resistance

This chapter is in the syllabus of

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