Cantilever Beams

Cantilever Beam with Uniform Distributed Load: Reactions, Shear and Moment Formulas

A cantilever beam is fixed at one end and free at the other. All load is carried back to the fixed support, where both the shear and the bending moment are largest. 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

Cantilever Beam with Uniform Distributed Load

Formulas

Reactions

R = wL

Maximum shear force

V_{max} = R

Maximum bending moment

M_{max} = \frac{wL^2}{2}

Shear and moment equations along the beam

Shear force, V(x)

V_{(x)}=wx

Bending moment, M(x)

M_{(x)}=\frac{wx^2}{2}

Variables

Symbol	Meaning
$L$	span between supports
$w$	distributed load value
$x$	distance along the beam measured from the left end
$R$	support reaction

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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Different supports, hinges or several loads? Use the full beam calculator.

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