# Deflection, slope and bending moment of beams

Notations and units:

$\delta$ = Deflection, $\theta$ = Slope, $E$ = Modulus of elasticity/youngs modulus, $I$ = Area moment of inertia

Units of the notations are in SI units.

1) CANTILEVER BEAM

Cantilever beam with point load at free end:

At free end : $\delta=\frac{PL^{3}}{3EI}$

At fixed end : Maximum bending moment = PL

$\theta_{max}=\frac{PL^{2}}{2EI}$

Cantilever beam with uniformly distributed load:

At free end : $\delta=\frac{PL^{4}}{8EI}$

At free end : $\theta=\frac{PL^{3}}{6EI}$

Maximum bending moment  = $\frac{PL^{2}}{2}$ at fixed end

Cantilever beam with one point load acting in between the ends:

At free end : $\delta=\frac{Pa^{2}(3L-a)}{6EI}$

Cantilever beam with moment acting at the free end:

At free end : $\delta=\frac{ML^{2}}{2EI}$

At free end : $\theta=\frac{ML}{EI}$

2) SIMPLY SUPPORTED BEAM

Simply supported beam with point load at centre:

$\delta=\frac{PL^{3}}{48EI}$ at centre

Maximum bending moment = $\frac{PL}{4}$ at centre

$\theta_{max}=\frac{PL^{2}}{16EI}$

Simply supported beam with uniformly distributed load:

$\delta=\frac{PL^{3}}{48EI}$ at centre

Maximum bending moment = $\frac{PL}{4}$

$\theta_{max}=\frac{PL^{2}}{16EI}$

Simply supported beam with only point load:

At point load : $\delta=\frac{Pa^{2}b^{2}}{3EIL}$

Maximum bending moment = $\frac{Pab}{L}$

Simply supported beam with two moments at each end:

At centre : $\delta=\frac{ML^{2}}{8EI}$

Simply supported beam with moment at centre:

Maximum bending moment = $\frac{M}{2}$

2) FIXED BEAM

Fixed beam with point load at centre:

At point load : $\delta=\frac{Pa^{3}b^{3}}{3EIL^{3}}$

Fixed beam with uniformly distributed load:

$\delta=\frac{PL^{4}}{384EI}$

Fixed beam with only point load:

At point load : $\delta=\frac{Pa^{3}b^{3}}{3EIL^{3}}$