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}}

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