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Chapter 5 of 5

Steel Beams and Plate Girders

In the RITES Manager & AM Civil syllabus under Steel Structures · 2 parts

Steel Beams — Laterally Supported & Unsupported · Plate Girders

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📑 Contents (15 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

Plate Girders

Last reviewed 16 Sept 2026 · 5 min read

Why plate girders

When the bending moment or span exceeds what the heaviest rolled beams can carry (roughly beyond 20–25 m spans or large concentrated loads), a plate girder is built up from plates (and sometimes angles) to exactly the required proportions. Uses: railway and road bridges, gantry girders for heavy cranes, transfer girders in buildings, industrial platforms.

Components

Web plate — deep and thin; carries shear and holds the flanges apart.
Flanges — flange plates (welded girders) or flange angles with cover plates (riveted/bolted girders); carry most of the bending moment.
Intermediate transverse stiffeners — prevent shear buckling of the web.
Longitudinal (horizontal) stiffeners — in very deep webs, in the compression zone, to prevent bending buckling.
Bearing (load-carrying) stiffeners — at supports and under concentrated loads.
Connections — flange-to-web welds or rivets/bolts designed for horizontal shear flow; splices in web and flanges.

Proportioning

Economical depth

For a girder of given moment $M$ with web slenderness $k = d / t_{w}$ , minimising the total steel area gives approximately

FormulaEconomical depth

d \approx (\frac{M k}{f _{y} / γ _{m 0}})^{1/3}

Common practice: $k$ = 100–200 for welded girders; depth about 1/10 to 1/12 of span for simply supported girders (deeper for bridges).

At the economical depth, the web area roughly equals the combined flange area (about half the girder's material in the web).

Flanges

Approximately, the flanges carry the moment as a couple:

A_{f} \approx \frac{M}{( f _{y} / γ _{m 0} ) d}

Flange outstand-to-thickness ratios must satisfy the section classification limits (plastic/compact/semi-compact) to avoid local buckling. For lateral stability the compression flange width is commonly about 1/40 to 1/30 of span (or restrained laterally).

Self-weight estimate

A common first estimate for the weight of a plate girder per metre is about $W /300$ of the total superimposed load $W$ (in consistent units) — refined after sizing.

Web slenderness limits (IS 800:2007)

Code ProvisionIS 800 cl. 8.6.1 — web thickness (selected)

Serviceability requirement:

Without transverse stiffeners: $d / t_{w} \leq 200 ε$
With transverse stiffeners only: $d / t_{w} \leq 200 ε$ when $c \geq d$ ; $c / t_{w} \leq 200 ε$ when $0.74 d \leq c < d$ ; $d / t_{w} \leq 270 ε$ when $c < 0.74 d$
With transverse and one longitudinal stiffener (at 0.2 $d$ from the compression flange): $d / t_{w} \leq 250 ε$
With a second longitudinal stiffener at the neutral axis: $d / t_{w} \leq 400 ε$

Compression flange buckling requirement (web must be stiff enough so the compression flange does not buckle into it):

Transverse stiffeners with $c \geq 1.5 d$ : $d / t_{w} \leq 345 ε^{2}$
Transverse stiffeners with $c < 1.5 d$ : $d / t_{w} \leq 345 ε$

Shear buckling of the web

A thin web buckles in diagonal compression before yielding in shear.

Simple post-critical method

V_{cr} = A_{v} τ_{b}

$τ_{b}$ depends on web slenderness $λ_{w} = \frac{f _{y w}}{3 τ _{cr, e}}$ :

$λ_{w}$	$τ_{b}$
≤ 0.8	$f_{y w} / 3$
0.8 to 1.2	$[1 - 0.8 (λ_{w} - 0.8)] f_{y w} / 3$
≥ 1.2	$f_{y w} / (3 λ_{w}^{2})$

Elastic critical shear stress:

τ_{cr, e} = \frac{k _{v} π ^{2} E}{12 ( 1 - μ ^{2} ) ( d / t _{w} ) ^{2}}, k_{v} = ⎩ ⎨ ⎧ 5.35 4 + \frac{5.35}{( c / d ) ^{2}} 5.35 + \frac{4}{( c / d ) ^{2}} no transverse stiffeners c / d < 1 c / d \geq 1

Closer stiffener spacing increases $k_{v}$ and shear strength.

Tension field method

After the web buckles, it continues to carry load through a diagonal tension field anchored by the flanges and stiffeners — the girder behaves like a Pratt truss with the web as tension diagonals and stiffeners as compression verticals. It gives higher shear capacity when stiffeners are provided at $c / d \geq 1$ and the end panels are properly anchored.

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