Mechanics of Materials

125/3337–11FE Reference Handbook 10.4 · Mechanics of Materials · Beam Deflections

Handbook formula

Cantilever Tip Deflection and Slope

δ_{end} = \frac{P L ^{3}}{3 E I}, θ_{end} = \frac{P L ^{2}}{2 E I} (end load)

Handbook beam tables: end load, uniform load (wL⁴/8EI), and moment load (ML²/2EI) are the FE Civil workhorses. Superpose if several loads act. Signs: usually down positive in the table — match the figure.

$P$: End load
$E I$: Flexural rigidity

Step-by-step solved example

Cantilever L=2 m, P=3 kN, EI=6×10^6 N·m². Tip δ?

End load: δ = PL³/3EI, θ = PL²/2EI.

1. Formula
δ=PL³/3EI.
2. Numbers
(3000)(8)/(3×6e6)=24000/18e6=0.00133 m=1.33 mm.

Answer: δ = 1.33 mm

10 practice questions

0/10 correct

1.Uniform w on a cantilever: δ_end =

2.If E doubles, δ

3.Simple beam midspan δ for P at mid is

4.Slope θ is in

5.Superposition is valid while

6.End moment M on a cantilever: δ =

7.L doubled → end-load δ scales by

8.I for a rectangle about bending axis is

9.A propped cantilever is

10.P=1 kN, L=1 m, E=200 GPa, I=1e7 mm⁴: convert I to m⁴ first (1e−5 m⁴). δ=