Beam deflection is the bending displacement of a structural member when loads act on it. Engineers calculate deflection to keep floors, bridges, machine frames, and shelves safe and comfortable to use. A beam can have stresses that are below the failure limit but still bend too much for the design.
Deflection control helps prevent cracking, vibration problems, misalignment, and serviceability failures.
The amount a beam bends depends on the load, span length, support type, material stiffness, and cross-sectional shape. These effects combine through the flexural rigidity EI, where E is the elastic modulus and I is the second moment of area. Longer beams deflect much more because many common formulas include L cubed or L to the fourth power.
Standard beam formulas let engineers quickly estimate maximum deflection for point loads, distributed loads, cantilevers, and simply supported beams.
Key Facts
- Flexural rigidity is EI, where E is Young's modulus and I is the second moment of area.
- Euler-Bernoulli beam relation: M(x) = EI d2y/dx2 for small deflections.
- Simply supported beam with center point load: delta_max = P L^3 / (48 E I).
- Simply supported beam with uniform load: delta_max = 5 w L^4 / (384 E I).
- Cantilever beam with end point load: delta_max = P L^3 / (3 E I).
- Cantilever beam with uniform load over full length: delta_max = w L^4 / (8 E I).
Vocabulary
- Deflection
- Deflection is the displacement of a beam from its original straight position under load.
- Neutral axis
- The neutral axis is the line within a bent beam where the longitudinal strain is zero.
- Young's modulus
- Young's modulus is a material property that measures stiffness in tension or compression.
- Second moment of area
- The second moment of area describes how a cross section distributes material relative to an axis and strongly affects bending resistance.
- Distributed load
- A distributed load is a load spread over a length of the beam, often measured in newtons per meter.
Common Mistakes to Avoid
- Using the wrong support condition is wrong because a cantilever, simply supported beam, and fixed-ended beam have different boundary conditions and different deflection formulas.
- Forgetting to convert units is wrong because E, I, L, P, and w must be in consistent units for the deflection result to be meaningful.
- Treating I as only a size measurement is wrong because the shape and orientation of the cross section can change I dramatically even if the area stays the same.
- Ignoring the strong effect of span length is wrong because deflection often scales with L^3 or L^4, so a small increase in length can cause a large increase in bending.
Practice Questions
- 1 A simply supported steel beam has L = 4.0 m, E = 200 GPa, I = 8.0 x 10^-6 m^4, and a center point load P = 12 kN. Use delta_max = P L^3 / (48 E I) to find the maximum deflection in millimeters.
- 2 A cantilever beam has L = 2.5 m, E = 70 GPa, I = 3.0 x 10^-6 m^4, and an end point load P = 800 N. Use delta_max = P L^3 / (3 E I) to find the tip deflection in millimeters.
- 3 Two simply supported beams have the same material, span, and load, but one has twice the second moment of area of the other. Explain which beam deflects less and by what factor.