Unit IVDesign of Structures - II

Steel Beams

When the compression flange is held, the beam reaches its full plastic moment.

≈ 40 min + worked example

When a beam's compression flange is held — by the slab or the decking sitting on it — it cannot buckle sideways, and it reaches its full bending capacity. That is a laterally supported beam, and it is the friendly case to design. The work: classify the section, compute the bending and shear capacities, and check the deflection (which, for a long span, can quietly govern).

Learning objectives

By the end of this lesson, you will be able to — mapped to the course outcomes for Design of Structures I:

CO4 · Understand

Explain 'laterally supported' and why it lets a beam reach its full bending capacity.

CO4 · Understand

Classify a section as plastic, compact, semi-compact or slender and read off βb.

CO4 · Apply

Compute the design bending strength Md and the shear capacity Vd of a rolled beam.

CO6 · Apply

Check a beam for deflection and for web buckling/crippling at the supports.

Plastic to slender

Laterally supported, and the section classes

Restraining the compression flange rules out lateral-torsional buckling. The section is then classified — plastic, compact, semi-compact or slender — by its flange and web slenderness, and the class fixes whether the full plastic modulus can be used.^[1]

DiagramThe four IS 800 section classes as stress blocks: plastic/compact reach the full plastic block, semi-compact only first yield, slender buckles locally

DiagramBending stress at first yield (a triangle, giving the elastic modulus Ze) versus full plastification (a full rectangle, giving the larger plastic modulus Zp)

Holding the flange

When the compression flange is restrained against sideways movement — by a concrete slab, steel decking or close bracing — lateral-torsional buckling cannot occur and the beam reaches its full in-plane bending (plastic) capacity. An unrestrained beam, by contrast, must be reduced for that buckling.^{[1, 3]}

The checks

Bending, shear and deflection

Bending Md = βb·Zp·fy/γm0 (capped at 1.2·Ze·fy/γm0); shear Vd = Av·fyw/(√3·γm0) with Av ≈ D·tw; then deflection under service load, and web buckling and crippling at the supports.^[1]

DiagramA simply supported steel beam under load showing the three checks: bending at mid-span, shear at the supports and deflection limited to a fraction of the span

Md = βb·Zp·fy/γm0

For a laterally supported beam the design bending strength is Md = βb·Zp·fy/γm0, with βb = 1.0 for plastic and compact sections (using the plastic modulus Zp) and βb = Ze/Zp for semi-compact (using the elastic modulus Ze). γm0 = 1.10.^[1]

Drive the numbers

Steel-beam calculator

Enter the section's plastic and elastic moduli (from SP 6) and its depth and web; the tool returns the bending strength, the serviceability cap and the shear capacity. An ISMB 300 gives Md ≈ 148 kNm.^[1]

Steel beam · laterally supported (IS 800 cl. 8)

Plastic modulus Zp651.7 cm³Elastic modulus Ze573.6 cm³Section depth D300 mmWeb thickness tw7.5 mmSteel grade

Plastic section, βb = 1.0: Md = Zp·fy/γm0 (cap 1.2·Ze·fy/γm0); Vd = Av·fy/(√3·γm0), Av = D·tw. Read Zp, Ze from SP 6.

0.0 kNm

Bending strength Md

0.0 kNm

Serviceability cap

0.0 kN

Shear capacity Vd

Plastic moment is within the serviceability cap.

The contrasts

At a glance

Aspect	One	The other
Lateral restraint	Supported: Md = βb·Zp·fy/γm0 (full)	Unsupported: reduced for lateral-torsional buckling
Section modulus	Plastic Zp — full yielding across section	Elastic Ze — first yield only (Zp > Ze)
Section class	Plastic/compact: βb = 1.0 (use Zp)	Semi-compact: βb = Ze/Zp (local buckling limits it)
Two strength checks	Bending: Md ≤ 1.2·Ze·fy/γm0 cap	Shear: Vd = Av·fyw/(√3·γm0)
What can govern	Strength: bending or shear	Serviceability: deflection (often governs long spans)

Vocabulary

Key terms

Laterally supported

Compression flange restrained so lateral-torsional buckling does not govern.

Plastic section modulus (Zp)

Σ(area × distance from plastic neutral axis); used at full plastification.

Elastic section modulus (Ze)

I/y; the first-yield modulus. Zp > Ze; their ratio is the shape factor.

βb

Bending reduction factor: 1.0 for plastic/compact, Ze/Zp for semi-compact.

Section classification

Plastic / compact / semi-compact / slender, by flange and web slenderness vs ε.

Shear area (Av)

≈ D·tw for a rolled I-beam loaded in its plane.

Low-shear case

If V ≤ 0.6·Vd the bending capacity need not be reduced for shear.

Web buckling / crippling

Local failure of the web at supports/point loads — checked as a strut and in bearing.

Apply it

Worked example

An ISMB 300 (Zp = 651.7×10³ mm³, Ze = 573.6×10³ mm³, D = 300, tw = 7.5), Fe410, plastic section so βb = 1.0. Md = 1.0×651.7e3×250/1.10 = 148.1 kNm, within the cap 1.2×573.6e3×250/1.10 = 156.4 kNm. Shear Av = 300×7.5 = 2250 mm²; Vd = 2250×250/(√3×1.10) = 295.2 kN. Always finish with the deflection check.

Check your understanding

Self-assessment

1. For a plastic or compact laterally-supported section, βb equals —

2. The design shear strength of a rolled I-beam is —

3. The serviceability cap on the moment capacity of a simply-supported beam is —

In a nutshell

Recap

A laterally-supported beam (compression flange held) reaches its full bending capacity — no lateral-torsional buckling.

Classify the section first: plastic/compact use βb = 1.0 and Zp; semi-compact use βb = Ze/Zp.

Bending Md = βb·Zp·fy/γm0 (capped at 1.2·Ze·fy/γm0); shear Vd = Av·fyw/(√3·γm0) with Av ≈ D·tw.

Always check deflection (≈ span/300, unfactored) and, at supports, web buckling and crippling.

The evidence

References & further reading

[1]IS 800:2007 — General Construction in Steel, Code of Practice (Section 8: design of members subjected to bending; cl. 8.2, 8.4; Tables 2, 6). Bureau of Indian Standards.
[2]SP 6 (Part 1) — ISI Handbook for Structural Engineers: Structural Steel Sections. Bureau of Indian Standards.
[3]N. Subramanian, Design of Steel Structures (2nd ed.). Oxford University Press, 2016.
[4]S.K. Duggal, Limit State Design of Steel Structures (2nd ed.). McGraw-Hill Education, 2014.