Beam Deflection (Distributed Load) Calculator
This calculator finds the maximum mid-span deflection of a simply supported beam carrying a uniform distributed load (UDL) across its full length. The classic formula is delta = 5wL^4 / (384EI), where w is the load intensity per unit length, L is the span, E is the modulus of elasticity, and I is the second moment of area about the bending axis. Distributed loads arise from self-weight, floor slab loads, and snow loads, making this one of the most commonly applied beam deflection cases in structural engineering practice. Use consistent units throughout: for example, kips/in, inches, ksi, and in^4 will return a deflection in inches.
Distributed load deflection formula
delta = 5 × w × L⁴ / (384 × E × I)
Where: w = uniform load per unit length, L = span, E = modulus of elasticity, I = second moment of area. The deflection is at mid-span and is the maximum for this loading case.
Comparing point load and distributed load deflections
- For the same total load W = wL, the distributed load produces 5/8 of the deflection caused by a central point load of equal magnitude.
- Deflection increases with the fourth power of span: a 20% increase in span raises deflection by (1.2)^4 = 2.07 times.
- Increasing the moment of inertia I (e.g. by using a deeper section) reduces deflection proportionally.
- Camber is often specified for long steel beams to offset dead-load deflection so the beam is level under full service loading.
Frequently asked questions
What is the formula for deflection under a uniform distributed load?
For a simply supported beam carrying a uniform distributed load w (force per unit length), the maximum mid-span deflection is delta = 5wL^4 / (384EI), where L is the span, E is the modulus of elasticity, and I is the moment of inertia.
How do I convert a total load to a distributed load w?
Divide the total load W by the span L: w = W/L. If the total load is 20 kips on a 10-ft beam, w = 2 kips/ft. Ensure units are consistent with E and I.
Why is the coefficient 5/384 rather than 1/48?
The coefficient depends on the loading pattern. For a central point load the coefficient is 1/48; for a full uniform load it is 5/384, reflecting how the distributed load spreads bending moment and curvature along the entire span.
Is this formula valid for non-uniform loads?
No. The 5wL^4/(384EI) formula assumes a perfectly uniform load across the full span. For trapezoidal or partially distributed loads, use superposition or numerical integration.
What deflection limit should I use for a floor beam?
AISC and IBC commonly require live-load deflection to stay within L/360 for floors supporting brittle finishes (e.g. tile), and L/240 for other floor beams. Total load deflection is often limited to L/240.
Official sources
- American Institute of Steel Construction (AISC): AISC Steel Construction Manual, Part 3 Design of Flexural Members.
- American Society of Civil Engineers: ASCE 7 Minimum Design Loads for Buildings.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.