Simply Supported Beams

Simply Supported Beam with Uniform Load at One End: Reactions, Shear and Moment Formulas

A simply supported beam rests on a pin at one end and a roller at the other. It is statically determinate: the reactions follow from equilibrium alone, and the bending moment is zero at both supports. 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

Simply Supported Beam with Uniform Load at One End

Formulas

Reactions

R_A = \frac{wa}{2L}(2L-a)

R_B = \frac{wa^2}{2L}

Maximum shear force

V_1 = -R_A

V_2 = R_B

Maximum bending moment

M_{max} = \frac{R_A^2}{2w}

Shear and moment equations along the beam

Shear force, V(x)

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

Bending moment, M(x)

M_{(x)}= \left\{ \begin{array}{lcc} R_Ax-\frac{wx^2}{2} & \text{if} & x < a \\ \\ R_B(L-x) & \text{if} & x \geq a \end{array} \right.

Variables

Symbol	Meaning
$L$	span between supports
$a$	distributed load 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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