Engineering
How do I calculate beam deflection?
Short answer
Beam deflection comes from Euler–Bernoulli beam theory. Identify the support conditions and load, then apply the standard formula for that case, such as 5wL^4/(384EI) for the midspan deflection of a simply supported beam under uniform load. E is the elastic modulus and I the second moment of area. Keep units consistent and compare the result with a limit such as L/360.
For a slender beam, deflection follows from Euler–Bernoulli beam theory, which relates the bending moment M(x) along the beam to its curvature: E * I * y'' = M(x). Integrating twice and applying the support conditions gives closed-form results for the common cases, so most hand calculations are a table lookup plus arithmetic.
The quantities you need
L, the span between supports (or the length of a cantilever).E, the modulus of elasticity. Structural steel is about 200 GPa (29,000 ksi) and aluminum about 70 GPa.I, the second moment of area of the cross-section about the bending axis. For a rectangle it isb * h^3 / 12. For rolled shapes, read it from the section table.- The load: a point load
P(force) or a uniformly distributed loadw(force per unit length).
Standard formulas
| Case | Maximum deflection | Where |
|---|---|---|
Simply supported, point load P at midspan |
P*L^3/(48*E*I) |
midspan |
Simply supported, uniform load w |
5*w*L^4/(384*E*I) |
midspan |
Cantilever, point load P at the free end |
P*L^3/(3*E*I) |
free end |
Cantilever, uniform load w |
w*L^4/(8*E*I) |
free end |
Fixed at both ends, uniform load w |
w*L^4/(384*E*I) |
midspan |
Fixed at both ends, point load P at midspan |
P*L^3/(192*E*I) |
midspan |
When a beam carries more than one load, calculate each load case separately and add the deflections. This superposition works as long as the material stays elastic and deflections are small.
Worked example
A simply supported steel beam spans 6 m and carries a uniform service load of 10 kN/m, self-weight included. The section has I = 80,000,000 mm⁴ (8.0 × 10⁻⁵ m⁴) and steel has E = 200 GPa (200,000,000 kN/m²).
E * I= 200,000,000 × 0.00008 = 16,000 kN·m².- Maximum deflection =
5 * 10 * 6^4 / (384 * 16,000)= 64,800 / 6,144,000 = 0.01055 m, or 10.5 mm at midspan. - Compare with the limit. L/360 for a 6 m span is 6,000 / 360 = 16.7 mm, so the beam passes. The actual deflection is about L/569.
In a spreadsheet this is one cell: =5*B2*B3^4/(384*B4*B5), with load, span, E and I in B2:B5. Convert everything to one unit system first. The most common error is mixing millimeters with meters, or newtons with kilonewtons, which is off by a factor of 1,000 or more.
Deflection limits
Deflection is a serviceability check, not a strength check. Use unfactored service loads, and compare with the limit your code or specification sets. Building codes typically express it as a fraction of span. IBC Table 1604.3, for example, gives L/360 for floor members under live load and L/240 for dead plus live load. Other limits apply to roofs, cantilevers and members that support brittle finishes, so check the code and material standard that governs your project.
Limits of the formulas
Euler–Bernoulli theory ignores shear deformation, so it underestimates deflection in short, deep beams. It also assumes a constant cross-section along the span, a linear elastic material and small rotations. For stepped sections, mixed supports or unusual loading, use a structural analysis program or a numerical method, and treat the hand formulas as a check.
The beam deflection and bending calculator sheet applies these formulas with example data, and the blog post on beam deflection in a spreadsheet covers building the calculation yourself.