Beams Fixed at One End

Propped Cantilever Beam (Fixed at One End) with Central Point Load: Reactions, Shear and Moment Formulas

A propped cantilever is fixed at one end and simply supported (roller) at the other. It is statically indeterminate to the first degree, so the fixed end develops a moment and the reactions are not symmetric. 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

Propped Cantilever Beam (Fixed at One End) with Central Point Load

Formulas

Reactions

R_1 = \frac{5P}{16}

R_2 = \frac{11P}{16}

Maximum shear force

V_1 = -R_1

V_{max} = R_2

Maximum bending moment

M_1 = \frac{5PL}{32}

M_{max} = \frac{3PL}{16}

Shear and moment equations along the beam

Shear force, V(x)

V_{(x)}= \left\{ \begin{array}{lcc} R_1 & \text{if} & x < \frac{L}{2} \\ \\ -R_2 & \text{if} & x \geq \frac{L}{2} \end{array} \right.

Bending moment, M(x)

M_{(x)}= \left\{ \begin{array}{lcc} R_1x & \text{if} & x < \frac{L}{2} \\ \\ \frac{PL}{2}-R_2x & \text{if} & x \geq \frac{L}{2} \end{array} \right.

Variables

Symbol	Meaning
$L$	span between supports
$P$	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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