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Beam deflection under a uniform load on a simply supported span has a closed-form answer. This article gives the formula, a worked value, and the reason the Calcs.com beam calculator solves the general case differently.

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The closed-form formula

For a simply supported span under a uniform load: δ = 5wL⁴ / 384EI where:
  • w is the load per unit length
  • L is the span
  • E is the elastic modulus
  • I is the second moment of area

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A worked value

A W14x68 carrying 1,000 plf over 20 ft deflects 0.17 in. Expressed as a span ratio, that is L/1,396. Span ratios are usually the more useful form, because deflection limits are normally written as L/360, L/250, or similar.

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Where the formula stops working

Closed-form results like this only hold for standard support and load cases. A simply supported beam with one uniform load is the textbook case. Real members often are not. The formula does not cover:
  • continuous beams over several spans
  • mixed point and distributed loads
  • fixed or spring supports
  • partial or linearly varying loads
  • internal hinges
For those, the Calcs.com beam calculator solves the general case with a finite element method rather than a standard formula. That is why the calculator handles any number of spans, supports, and loads.

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Sign convention

Take care when comparing a hand-calculated deflection with a calculator output, because the two may report opposite signs for the same physical movement. Until that is resolved, compare magnitudes and confirm the direction from the deflected shape on screen rather than from the sign alone.

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Checking against a limit

A deflection value on its own is not a check. Compare it against the limit that applies to the member, under the correct serviceability load case. Common limits are L/360 for floors and L/240 for roofs, but the governing value comes from your project requirements. Deflection is a serviceability result. Strength and stability checks against AISC, NDS, ACI 318, or AS 4100 are separate design checks, and the engineer remains responsible for the design.