Simply Supported Beam Deflection (UDL)
Simply Supported Beam Deflection Under a Uniform Load
This worksheet finds the maximum mid-span deflection of a simply supported beam carrying a uniformly distributed load (UDL), together with the peak bending stress. It is one of the most common serviceability checks in structural design: a beam can be strong enough yet still deflect too much to be acceptable.
From Euler–Bernoulli beam theory, the mid-span deflection of a simply supported prismatic beam under a UDL is δ = 5wL⁴ / (384EI), where w is the load per unit length, L the span, E Young’s modulus and I the second moment of area.
Inputs
A rectangular steel section of breadth b and depth h spanning L, carrying a service UDL w.
w := 12 kN/m=12000 kg/s^2
L := 6 m=6 m
E := 200 GPa=200000 MPa
b := 200 mm=200 mm
h := 400 mm=400 mm
Section properties
The second moment of area of a solid rectangle about its centroidal axis is I = b·h³/12.
I := b·h^3/12=1.07e-03 m^4
Deflection and stress
With I known, the mid-span deflection follows directly; the maximum bending moment for a UDL is M = wL²/8 and the extreme-fibre stress is σ = M·(h/2)/I.
delta := 5·w·L^4/(384·E·I)=0.95 mm
sigma_max := (w·L^2/8)·(h/2)/I=10.12 MPa
Results
The computed mid-span deflection should be compared with the span/deflection limit appropriate to the member — commonly L/250 for total load or L/360 for live load on floors. If the deflection exceeds the limit the section must be deepened or stiffened even when the bending stress is comfortably below yield. The peak bending stress shown should sit well under the steel’s design strength.
Assumptions & limitations
- Linear-elastic, small-deflection (Euler–Bernoulli) behaviour; no yielding.
- Prismatic section, constant EI along the span.
- Simple supports (pin/roller); self-weight not added separately — include it in w if required.
- Shear deflection neglected (valid for slender beams).