Chapter 3 of 6

Beams & Beam-Columns

In the GATE Civil syllabus under Steel Structures · 2 parts

Steel Beams — Laterally Supported & Unsupported · Beam-Columns

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📑 Contents (13 sections)

Part 1 of 2

Steel Beams — Laterally Supported & Unsupported

Last reviewed 16 Sept 2026 · 5 min read

Types and behaviour

Steel beams (joists, girders, lintels, purlins, stringers, spandrel beams) are usually rolled I-sections (ISMB, ISWB, parallel-flange beams), channels for light purlins, or built-up/plate girders for heavy loads.

A beam can fail by:

Flexural yielding (formation of a plastic hinge),
Shear yielding of the web,
Local buckling of the flange or web (thin elements),
Web buckling or web crippling under concentrated loads,
Lateral–torsional buckling (LTB) — the compression flange buckles sideways and the section twists, when it is not laterally restrained.
Excessive deflection (serviceability).

Laterally supported beams

A beam is laterally supported when its compression flange is restrained against lateral movement and twisting continuously (e.g. embedded in or connected to a concrete slab) or at close spacing. LTB then cannot occur.

FormulaDesign bending strength (IS 800 cl. 8.2.1.2)

When the factored shear $V \leq 0.6 V_{d}$ (low shear):

M_{d} = \frac{β _{b} Z _{p} f _{y}}{γ _{m 0}}

$β_{b}$ = 1.0 for plastic and compact sections; $β_{b} = Z_{e} / Z_{p}$ for semi-compact sections.

To avoid irreversible deformation at service load:

simply supported beams: $M_{d} \leq \frac{1.2 Z _{e} f _{y}}{γ _{m 0}}$
cantilever beams: $M_{d} \leq \frac{1.5 Z _{e} f _{y}}{γ _{m 0}}$

Shear

FormulaShear strength

V_{d} = \frac{A _{v} f _{y w}}{3 γ _{m 0}}

$A_{v}$ = shear area: $h t_{w}$ for rolled I and channel sections bent about the major axis; $d t_{w}$ for welded sections.

If $V > 0.6 V_{d}$ (high shear), the design moment is reduced: $M_{d v} = M_{d} - β (M_{d} - M_{f d})$ , with $β = (\frac{2 V}{V _{d}} - 1)^{2}$ and $M_{f d}$ = plastic moment of the flanges alone.

Shear buckling of the web need not be checked if $d / t_{w} \leq 67 ε$ (unstiffened web).

Web buckling and web crippling

At supports and under concentrated loads, the thin web may buckle like a column or cripple (yield locally).

Web crippling (bearing): $F_{w} = \frac{( b _{1} + n _{2} ) t _{w} f _{y w}}{γ _{m 0}}$ , where $b_{1}$ = stiff bearing length and $n_{2}$ = dispersion length through the flange and root at a slope of 1 : 2.5 ( $n_{2} = 2.5 (t_{f} + r)$ ).
Web buckling: the web below the load acts as a strut of effective length about $0.7 d$ (for flanges restrained against rotation), with width $b_{1} + n_{1}$ (load dispersed at 45° to the mid-depth); strength $= f_{c d} \times$ area, using buckling class c.

If either check fails, provide bearing (load-carrying) stiffeners.

Deflection

Check deflection under unfactored imposed loads against IS 800 Table 6 limits (e.g. span/300 for floor members supporting elements not susceptible to cracking; span/360 where elements are susceptible to cracking).

Laterally unsupported beams — lateral–torsional buckling

When the compression flange is free between supports (gantry girders, beams without slab, cantilevers), the beam can buckle laterally and twist at a moment below $M_{p}$ .

FormulaIS 800 cl. 8.2.2 — design bending strength with LTB

M_{d} = β_{b} Z_{p} f_{b d}, f_{b d} = χ_{L T} \frac{f _{y}}{γ _{m 0}}

χ_{L T} = \frac{1}{ϕ _{L T} + ϕ _{L T}^{2} - λ _{L T}^{2}} \leq 1, ϕ_{L T} = 0.5 [1 + α_{L T} (λ_{L T} - 0.2) + λ_{L T}^{2}]

λ_{L T} = \frac{β _{b} Z _{p} f _{y}}{M _{cr}}

$α_{L T}$ = 0.21 for rolled sections, 0.49 for welded sections.

Elastic critical moment for a doubly symmetric I-section (simplified):

M_{cr} = \frac{π ^{2} E I _{y} h}{2 L _{L T}^{2}} [1 + \frac{1}{20} (\frac{L _{L T} / r _{y}}{h / t _{f}})^{2}]^{0.5}

$M_{cr}$ increases with lateral stiffness ( $I_{y}$ ), torsional stiffness and shorter unbraced length. LTB is not a concern if $λ_{L T} \leq 0.4$ .

Effective length for LTB (simply supported, loads not destabilising)

Restraint at supports	$L_{L T}$
Torsionally restrained, compression flange fully restrained against rotation in plan	$0.70 L$
Torsionally restrained, compression flange partially restrained against rotation in plan	$0.85 L$
Torsionally restrained, both flanges free to rotate in plan	$1.00 L$

Increase effective length (e.g. by 20%) where the load acts on the top flange and is free to move laterally with it (destabilising load). Cantilevers have their own table.

Ways to improve LTB resistance: lateral bracing of the compression flange at intervals, choosing wider flanges or box/tubular sections, top-flange channels (as in gantry girders), torsional restraints at supports.

Part 2 of 2

Beam-Columns

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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