Beams Fixed at Both Ends

Fixed-Fixed Beam with 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

Formulas

Reactions

R_A = \frac{Pb^2}{L^3}(3a+b)

R_B = \frac{Pa^2}{L^3}(a+3b)

Maximum shear force

V_1 = -R_A

V_2 = +R_B

Maximum bending moment

M_1 = -\frac{Pab^2}{L^2}

M_2 = \frac{2Pa^2b^2}{L^3}

M_3 = -\frac{Pa^2b}{L^2}

Shear and moment equations along the beam

Shear force, V(x)

V_{(x)}= \left\{ \begin{array}{lcc} -R_A & \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{Pab^2}{L^2} & \text{if} & x < a \\ \\ R_B(L-x)-\frac{Pa^2b}{L^2} & \text{if} & x \geq a \end{array} \right.

Variables

Symbol	Meaning
$L$	span between supports
$a$	point load position
$P$	point load value
$b$	L − a, the distance from the load to the far end
$x$	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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