Single Span and Cantilever Beams

Overhanging Beam with Uniform Distributed Load Cantilever on Both Sides: 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

Overhanging Beam with Uniform Distributed Load Cantilever on Both Sides

Formulas

Reactions

R_A = \frac{wL(L-2c)}{2b}

R_B = \frac{wL(L-2a)}{2b}

Maximum shear force

V_1 = wa

V_2 = -R_A+wa

V_3 = R_B-wc

V_4 = -wc

Maximum bending moment

M_1 = -\frac{wa^2}{2}

M_2 = R_A(\frac{R_A}{2w}-a)

M_3 = -\frac{wc^2}{2}

Shear and moment equations along the beam

Shear force, V(x)

V_{(x)}= \left\{ \begin{array}{lcc} wx & \text{if} & x < a \\ \\ -R_A+wx & \text{if} & a \leq x < a+b \\ \\ -w(L-x) & \text{if} & a+b \leq x \leq L\end{array} \right.

Bending moment, M(x)

M_{(x)}= \left\{ \begin{array}{lcc} \frac{-wx^2}{2} & \text{if} & x < a \\ \\ R_Ax-\frac{wx^2}{2}-R_Aa & \text{if} & a \leq x < a+b \\ \\ -\frac{w(L-x)^2}{2} & \text{if} & a+b \leq x \leq L\end{array} \right.

Variables

Symbol	Meaning
$a$	left cantilever length
$b$	length between supports
$c$	right cantilever length
$w$	distributed 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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