Structural Analysis Fundamentals / Beam deflection

Standard cases and serviceability

The formulae worth knowing, and why deflection often decides the size.

Lesson review: expert verification remains open

This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

Checks needed for project use

An independent structural reviewer must check remaining derivations, diagrams, sign conventions and the interactive models across their full input ranges.
Independently check deflection-lab implementations, principal-axis direction conventions and shear coefficients for non-rectangular sections; serviceability limits are project-specific.

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

Read the lesson register and outstanding review notes

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

Standard maximum deflections

\delta = \frac{PL^3}{48EI} \qquad \delta = \frac{5wL^4}{384EI} \qquad \delta = \frac{PL^3}{3EI}

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