Three-Span Beams

Three-Span Continuous Beam with Distributed Load: Reactions, Shear and Moment Formulas

A three-span continuous beam runs over four supports. The interior supports carry negative moments, and pattern loading (loading some spans only) can govern the design. 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

Three-Span Continuous Beam with Distributed Load

Formulas

Reactions

R_1 = 0.400wL

R_2 = 1.100wL

Maximum shear force

V_1 = R_1

V_2 = 0.600wL

V_3 = 0.500wL

Maximum bending moment

M_1 = 0.080wL^2

M_2 = 0.100wL^2

M_3 = 0.025wL^2

Shear and moment equations along the beam

Shear force, V(x)

V_{(x)}= \left\{ \begin{array}{lcc} R_1-wx & \text{if} & x < L \\ \\ R_1+R_2-wx & \text{if} & L \leq x \leq 2L \\ \\ -R_1+w(3L-x) & \text{if} & 2L < x \leq 3L \end{array} \right.

Bending moment, M(x)

M_{(x)}= \left\{ \begin{array}{lcc} R_1x-\frac{wx^2}{2} & \text{if} & x < L \\ \\ (R_1+R_2)(x-L)-\frac{wx^2}{2}+R_1L & \text{if} & L \leq x \leq 2L \\ \\ R_1(3L-x)-\frac{w(3L-x)^2}{2} & \text{if} & 2L < x \leq 3L \end{array} \right.

Variables

Symbol	Meaning
$L$	distance between supports
$w$	distributed 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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