Structural Analysis Fundamentals / module 13
Beam deflection
How much a beam moves, why that often governs, and how to picture the shape first.
What you will understand
- Distinguish deflection from rotation
- Relate bending moment to curvature
- Sketch a deflected shape from the supports and loading
- Use standard deflection formulae and check span/deflection limits
- Explain why serviceability often decides the beam size
Lessons
Deriving the beam deflection equation
Euler–Bernoulli theory: from exact curvature to EI d²v/dx² = M, and where it fails.
Curvature and deflected shapes
Picture the shape before you calculate the number.
Standard cases and serviceability
The formulae worth knowing, and why deflection often decides the size.
Euler–Bernoulli and Timoshenko
Euler–Bernoulli neglects shear deformation. Timoshenko retains it by treating the section rotation and centreline slope as independent fields. This lesson compares their equations and the size of the shear contribution.
Macaulay's method and moment–area
Two ways to get a deflection when the loading is not in any table.
Deflection under unsymmetrical bending, and deflection due to shear
Beams that move sideways, and the deflection that bending theory leaves out.
Module summary
- Curvature is proportional to bending moment and inversely proportional to EI
- Boundary conditions fix the deflected shape: a pin holds position, a fixed end holds position and slope
- Standard cases cover most hand checks: PL³/48EI, 5wL⁴/384EI, PL³/3EI, wL⁴/8EI
- Deflection frequently governs design even when the beam is nowhere near failing
Teaching basis and review status
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 notesCheck your understanding
Practice
A uniformly loaded steel beam has a span-to-depth ratio of 4. Taking the shear share as 2.5(d/L)², what percentage of the bending deflection does shear add?
Practice
A beam's shear deflection is 4% of its bending deflection. The span is doubled and the depth left alone. What is the shear share now, as a percentage?
Practice
A simply supported beam of 6.0 m carries a UDL of 10 kN/m. E = 205 000 N/mm² and I = 3.0 × 10⁸ mm⁴. What is the maximum deflection? (Note 10 kN/m = 10 N/mm.)
Check yourself
A simply supported beam's span is doubled, with the same UDL intensity. What happens to the maximum deflection?
Check yourself
At a simple support, which is zero?