Beams Fixed at Both Ends

Fixed-Fixed Beam with Central Point Load: Reactions, Shear and Moment Formulas

A fixed-fixed beam is clamped at both ends. Both ends resist rotation, so negative (hogging) moments appear at the supports and the midspan moment is smaller than in a simply supported beam. 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

Fixed-Fixed Beam with Central Point Load

Formulas

Reactions

R = \frac{P}{2}

Maximum shear force

V = R

Maximum bending moment

M_{max} = \frac{PL}{8}

Shear and moment equations along the beam

Shear force, V(x)

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

Bending moment, M(x)

M_{(x)}= \left\{ \begin{array}{lcc} \frac{P}{8}(4x-L) & \text{if} & x < \frac{L}{2} \\ \\ \frac{P}{8}(4(L-x)-L) & \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
$R$	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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