Module 13 · Lesson 13.3
Standard cases and serviceability
The formulae worth knowing, and why deflection often decides the size.
Why this matters
Plenty of beams are chosen not because they would break, but because a thinner one would bounce, crack the ceiling below, or simply look wrong. That is serviceability, and it is a real design decision.
By the end of this lesson you should be able to
- Apply standard deflection formulae
- Check a span/deflection ratio
- Explain why deflection often governs
What it calculates: Maximum deflection for three common cases: simply supported with a central point load, simply supported with a full UDL, and a cantilever with a tip load.
- δ
- Maximum deflection (mm)
- P
- Point load (N)
- w
- Distributed load intensity (N/mm)
- L
- Span (mm)
- EI
- Flexural stiffness (N·mm²)
This assumes
- Elastic behaviour
- Uniform section along the span
- Small deflections
- The exact support conditions stated
In plain terms: Span appears cubed or to the fourth power. Doubling the span of a UDL-loaded beam multiplies the deflection by sixteen. Nothing else in the formula comes close to that sensitivity, which is why long spans get deep quickly.
Worked example
Deflection of a simply supported steel beam
Given
- Span 5.0 m, simply supported
- Central point load of 30 kN
- Steel, E = 205 000 N/mm²
- Section I = 1.2 × 10⁸ mm⁴
Find
The maximum deflection, and whether it is acceptable against a span/360 limit.
Assumptions
- Elastic behaviour
- Uniform section
- Load applied exactly at midspan
Practice
A cantilever 2.0 m long carries a 10 kN point load at its tip. E = 205 000 N/mm² and I = 4.0 × 10⁷ mm⁴. What is the tip deflection?
Practice
A simply supported beam spans 6.0 m under a UDL of 12 kN/m. With E = 205 000 N/mm² and I = 2.0 × 10⁸ mm⁴, what is the maximum deflection? (Note 12 kN/m = 12 N/mm.)
Practice
A cantilever 3.0 m long carries a UDL of 5 N/mm. With E = 205 000 N/mm² and I = 8.0 × 10⁷ mm⁴, what is the tip deflection?
Practice
A simply supported beam deflects 18 mm over a 7.2 m span. What is the span-to-deflection ratio? (Enter the number after 'span/'.)
Summary
- δ = PL³/48EI, 5wL⁴/384EI and PL³/3EI cover most hand checks
- Span dominates: it appears cubed or to the fourth power
- Compare against a span/deflection limit such as span/360
- Deflection frequently governs long spans even when strength is fine
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint