Cantilever Beams

Cantilever Beam with End Point 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 End Point Load

Formulas

Reactions

R=PR = P

Maximum shear force

V=RV = R

Maximum bending moment

Mmax=−PLM_{max} = -PL

Shear and moment equations along the beam

Shear force, V(x)

V(x)=PV_{(x)}= P

Bending moment, M(x)

M(x)=PxM_{(x)}= Px

Variables

SymbolMeaning
LLspan between supports
PPpoint load value
xxdistance along the beam measured from the left end
RRsupport 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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