Simply Supported Beam Deflection (UDL) Calculator
The maximum deflection of a simply supported beam under a uniformly distributed load is a classic result of structural engineering, and it governs whether a floor feels solid or bounces underfoot. A uniformly distributed load, or UDL, spreads weight evenly along the span, and the beam sags most at the middle. The deflection grows with the fourth power of the span length, so doubling the span makes the beam sag sixteen times as much, all else being equal. It shrinks as the material stiffness and the cross-sectional moment of inertia rise. This calculator takes the distributed load per unit length, the span, the modulus of elasticity and the second moment of area, then returns the maximum mid-span deflection in meters using consistent SI units. Because the values can be very small, the result is shown with adaptive precision. The formula assumes a prismatic, linearly elastic beam on simple supports, the standard textbook case. Structural loading conventions and the related engineering safety standards are published by US federal agencies including the National Highway Traffic Safety Administration. Every figure is computed deterministically from the standard deflection formula, shown in full below, with a worked example that reconciles exactly to the calculator so you can check each step yourself.
Maximum mid-span deflection equals 5 w L^4 / (384 E I). For a 4 m beam with a 2,000 N/m load, E of 200 GPa and I of 8.0e-6 m4, the deflection is 0.004167 m (about 4.17 mm). Deflection grows with the fourth power of span.
Beam deflection formula (UDL)
delta = 5 w L^4 / (384 E I)
delta = maximum mid-span deflection (m)
w = distributed load (N/m), L = span (m)
E = modulus of elasticity (Pa), I = second moment of area (m4)
The numerator 5 w L^4 captures how load and span drive deflection, with span entering to the fourth power. The denominator 384 E I captures stiffness: a higher modulus or a larger moment of inertia reduces sag. Use consistent SI units throughout so the answer comes out in meters.
Worked example
A simply supported steel beam spans 4 m and carries a uniformly distributed load of 2,000 N/m. Steel has E = 200 GPa = 200,000,000,000 Pa and the section has I = 0.000008 m4.
- Numerator: 5 x 2,000 x 4^4 = 5 x 2,000 x 256 = 2,560,000
- Denominator: 384 x 200,000,000,000 x 0.000008 = 614,400,000
- Divide: delta = 2,560,000 / 614,400,000 = 0.004167 m
- That is about 4.17 mm of sag at mid-span
So the beam deflects about 4.17 mm at its center. These are the calculator's default inputs, so the result above matches the widget exactly.
Simply Supported Beam Deflection (UDL) Calculator: frequently asked questions
How do you calculate maximum beam deflection under a UDL?
For a simply supported beam with a uniformly distributed load, the maximum deflection at mid-span is 5 w L^4 / (384 E I), where w is the load per unit length, L the span, E the modulus of elasticity and I the second moment of area. Use consistent SI units and the answer is in meters.
Why is span raised to the fourth power?
Deflection depends on how bending moments accumulate along the beam and on integrating curvature twice over the length. Both effects scale strongly with span, and the algebra works out to L^4. The practical result is that longer spans deflect dramatically more for the same load.
What is the second moment of area?
The second moment of area, often written I, measures how a cross-section's material is distributed about the bending axis. Deeper sections place material further from the neutral axis and so resist bending far better. A rectangle has I = b h^3 / 12.
Does this formula include the beam's own weight?
Not unless you add it. If self-weight matters, include it in w as a load per unit length alongside any applied distributed load, since a uniform self-weight behaves exactly like a UDL.
What is the beam deflection formula?
For a simply supported beam under a uniformly distributed load, the maximum deflection is delta = 5 w L^4 / (384 E I).
Official sources
- Structural loading and engineering safety standards reference: US National Highway Traffic Safety Administration (NHTSA). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.