Worked examples

Tutorial 3 Q1: what the four section classes mean in bending

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Describe the bending behaviour of Classes 1,2,3 and 4: plastic, compact, semi-compact and slender.

Original source: LectureNotes/Ch 3_Beam.pdf — p. 35. Values tagged given are in the question or diagram; lookup values come from a named table; calculated values follow from the working; assumptions are stated explicitly.

Q1 is a written question with no member geometry drawing. The Ch 1 p.21 source image is a classification-limit table; select its row by compression element and stress pattern.

b

T

d

t

are plate-element dimensions for local buckling, not beam span.Open full-size image

Source-table labels and selectable formulas:Table 7.1: width/thickness limits for sections other than circular hollow sections (CHS) and rectangular hollow sections (RHS). Headings identify compression element, ratio and limiting value. The three limits below correspond respectively to Class 1 plastic, Class 2 compact and Class 3 semi-compact.

Outstand flange in compression due to bending, hot-rolled section: $\frac{b}{T}$ ， $9 ε$ , $10 ε$ , $15 ε$ ; welded section, same ratio: $8 ε$ , $9 ε$ , $13 ε$ . Outstand flange under axial compression: $\frac{b}{T}$ ; Classes 1/2 are not applicable; Class 3 is $13 ε$ .
Internal flange element in compression due to bending: $\frac{b}{T}$ ， $28 ε$ , $32 ε$ , $40 ε$ ; under axial compression, Classes 1/2 are not applicable and the Class 3 cell is merged with the above $40 ε$ .
I-, H- or box-section web with neutral axis at mid-depth: $\frac{d}{t}$ ， $80 ε$ , $100 ε$ , $120 ε$ .
General I/H/box web: whether $r_{1}$ is negative or positive, Class 1 is $\frac{80 ε}{1 + r_{1}}$ , but the limit must be $\geq 40 ε$ . For Class 2, when $r_{1}$ is negative use $\frac{100 ε}{1 + r_{1}}$ ; when positive use $\frac{100 ε}{1 + 1.5 r_{1}}$ , but the limit must be $\geq 40 ε$ . The merged Class 3 cell is $\frac{120 ε}{1 + 2 r_{2}}$ , but the limit must be $\geq 40 ε$ and extends to the axial-compression row. Every row uses ratio $\frac{d}{t}$ ; Classes 1/2 are not applicable in the axial-compression row.
Channel web: $\frac{d}{t}$ ， $40 ε$ , $40 ε$ , $40 ε$ .
Angle in compression due to bending: $\frac{b}{t}$ and $\frac{d}{t}$ must both satisfy their limits; each limit is $9 ε$ , $10 ε$ , $15 ε$ .
Single angle, or two separated angles, under axial compression: $\frac{b}{t}$ , $\frac{d}{t}$ , $\frac{b + d}{t}$ must all satisfy their limits. Classes 1/2 are not applicable; the three Class 3 limits are respectively $15 ε$ , $15 ε$ , $24 ε$ .
Outstanding legs of angles in back-to-back contact, or of an angle whose back is in continuous contact with another component: $\frac{b}{t}$ , with common limit $9 ε$ , $10 ε$ , $15 ε$ .
Stem of a hot-rolled tee or a tee cut from a hot-rolled I/H-section: $\frac{D}{t}$ ， $8 ε$ , $9 ε$ , $18 ε$ .

Source note a: $b$ , $D$ , $d$ , $T$ , $t$ is defined by Figure 7.1. For box sections, $b, T$ is the flange dimension and $d, t$ the web dimension; web/flange identification depends on major- or minor-axis bending (clause 7.1). Note b: $ε = \sqrt{\frac{275}{p_{y}}}$ ， $p_{y}$ has units $\frac{N}{{mm}^{2}}$ . Note c: stress ratio $r_{1}, r_{2}$ is defined in clause 7.3. Note d: for a hybrid section web, calculate $ε$ using flange design strength $p_{yf}$ . The slide footer says “Table 8 — Extract of Table 7.1 of HK Code”: course Table 8 extracts code Table 7.1. They are different levels of table numbering.

Read the diagram and collect the data

Given: a bending section is assigned one of four local-buckling classes. Required: describe moment resistance and rotation capacity. No force, length, numerical section or steel grade is given.

Before calculating: recognition and strategy

Think about the order of three events: the first fibre yields, plastic stress spreads through the section, and a thin plate locally buckles. Rotation capacity matters when a plastic hinge must keep turning while still carrying moment.

The four behaviours and their exam consequences

Simple explanation: A thin part can wrinkle first

A thin plate may wrinkle before the whole steel member reaches its intended resistance.

Check flange and web slenderness using their own definitions.
Compare each ratio with the correct class limits.
The less favourable element determines the section class.

Remember: Bending limits and uniform-compression limits are different.

Related concept and full method

Class	Behaviour before local buckling	Permitted course bending approach
1 — plastic	Develops full plastic resistance and enough plastic rotation for a plastic hinge to redistribute moment.	Plastic resistance, with the course 1.2pᵧZ ceiling; suitable rotation capacity must still be relevant to the analysis.
2 — compact	Develops plastic resistance, but local buckling limits the rotation available afterward.	Same capped plastic section resistance in the elastic-analysis examples; do not claim adequate plastic-hinge rotation.
3 — semi-compact	Extreme compression fibre can reach yield, but local buckling prevents full plastic stress redistribution.	Use elastic resistance pᵧZ for low-shear bending.
4 — slender	Local plate buckling occurs before gross-section elastic yield resistance is reached.	Use effective section properties/local-buckling treatment; neither gross pᵧS nor gross pᵧZ is automatically valid.

The most slender relevant compression element governs the whole section class. A stocky flange does not rescue a slender web. Classification concerns local plates; an otherwise Class 1 beam can still fail global lateral-torsional buckling.

Elastic first-yield reference:

M_{el} = p_{y} Z

Plastic reference:

M_{pl} = p_{y} S

Supplied Class 1/2 design formula:

M_{c} = \min (p_{y} S, 1.2 p_{y} Z)

Here pᵧ is design strength in $\frac{N}{{mm}^{2}}$ , $Z$ elastic modulus and $S$ plastic modulus in ${mm}^{3}$ ; multiplication gives $N \cdot mm$ . Divide by $10^{6}$ for $kN \cdot m$ . The two modulus symbols are deliberately different.

Animation labWhy thin elements buckle locally3 concepts · 1 source expressions

Supplement to the original lesson. Enable JavaScript to play, step through calculations and rotate 3D models. The following explanation remains readable offline.

The flange outstand and web have different widths, thicknesses and edge support conditions.
A thinner plate can wrinkle locally before the complete member loses stability.
Class 1 allows plastic rotation; Class 2 reaches plastic resistance; Class 3 reaches elastic resistance; Class 4 requires effective properties.
Check every relevant compression element with the supplied limits and stress distribution. The deformation shown is qualitative.

Compact exam answer

Class 1 reaches plastic resistance with adequate plastic rotation; Class 2 reaches plastic resistance with limited rotation; Class 3 reaches elastic first-yield resistance but not full plastic resistance; Class 4 buckles locally before gross elastic resistance and needs effective properties.

Mistakes to avoid

Do not describe Class 2 as elastic-only.
Do not equate section class with lateral restraint.
Do not say Class 4 has zero strength.

Procedure for an unfamiliar variant

Identify the compression plate elements.
Compare relevant width/thickness ratios with the correct stress-pattern limits.
Take the governing class.
State its moment and rotation consequences separately.

Independent self-check

Try it yourself. Invented variant: a beam is Class 1 but its compression flange is unrestrained over $9 m$ . Can you omit LTB?

Reveal answer and reasoning

No. Class 1 prevents premature local plate buckling for the assumed stress pattern. It does not prevent the whole beam twisting and moving sideways. LTB is a separate member check.