Single Span and Cantilever Beams

Overhanging Beam with Point Load: Reactions, Shear and Moment Formulas

An overhanging beam rests on two supports and extends past one or both of them. Loads on the overhang produce negative moments over the support and can reduce the positive moment in the span. Below are the closed-form formulas for this load case: support reactions, maximum shear, maximum bending moment and the shear and moment equations along the beam.

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Load case and diagrams

Formulas

Reactions

R_1 = \frac{Pb}{L}

R_2 = \frac{Pa}{L}

Maximum shear force

V_1 = -R_1

V_2 = R_2

Maximum bending moment

M_{max} = \frac{Pab}{L}

Shear and moment equations along the beam

Shear force, V(x)

V_{(x)}= \left\{ \begin{array}{lcc} -R_1 & \text{if} & x < a \\ \\ R_2 & \text{if} & a \leq x < L \\ \\ 0 & \text{if} & L \leq x \end{array} \right.

Bending moment, M(x)

M_{(x)}= \left\{ \begin{array}{lcc} R_1x & \text{if} & x < a \\ \\ R_2(L-x) & \text{if} & a \leq x < L \\ \\ 0 & \text{if} & L \leq x \end{array} \right.

Variables

Symbol	Meaning
$L$	span between supports
$a$	point load position
$P$	point load value
$x$	distance along the beam measured from the left end

Sign convention: positive moment is sagging (tension at the bottom); negative moment is hogging (tension at the top), as in cantilevers and over interior supports.

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