Formulas for maximum deflection of cantilever and simply supported beams under concentrated loads, including the role of EI and span.
What Deflection Measures
Under load, beams deform elastically. Deflection is the transverse displacement of the neutral axis. Design codes limit deflection for serviceability (comfort, finishes, vibration) even when strength is adequate.
Factors That Control Deflection
For linear elastic beams:
- Load (P) — larger force increases deflection.
- Span (L) — deflection scales with L³ for many common cases. Doubling length increases deflection by a factor of 8.
- Flexural rigidity (EI)
- E — Young's modulus (material stiffness)
- I — second moment of area (section geometry; deeper sections resist bending more)
Common Formulas (Point Load)
Simply Supported Beam (load at midspan)
\delta_{max} = \frac{P L^3}{48 E I}Typical of floor joists and simple bridge spans.
Cantilever Beam (load at free end)
\delta_{max} = \frac{P L^3}{3 E I}Typical of balconies and diving boards.
For the same $P$, $L$, $E$, and $I$, the cantilever deflects 16× more than the midspan-loaded simply supported beam ($48/3 = 16$).
Units Checklist
Keep SI consistent: $P$ in N, $L$ in m, $E$ in Pa (N/m²), $I$ in m⁴ → $\delta$ in m. Convert GPa → Pa and cm⁴ → m⁴ before substituting.
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