Two-Span Beams

Two-Span Continuous Beam with Distributed Load with Different Span Lengths: Reactions, Shear and Moment Formulas

A two-span continuous beam runs over three supports. Continuity over the middle support produces a negative moment there and reduces the span moments compared with two separate simple spans. 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

Two-Span Continuous Beam with Distributed Load with Different Span Lengths

Formulas

Reactions

R_A = \frac{M_1}{a}+\frac{wa}{2}

R_B = wa+wb-R_A-R_C

R_C = \frac{M_1}{b}+\frac{wb}{2}

Maximum shear force

V_1 = -R_A

V_2 = wa-R_A

V_3 = -wb+R_C

V_4 = R_C

Maximum bending moment

M_1 = -\frac{wb^3+wa^3}{8(a+b)}

M_2 = R_Ax_1-\frac{wx_1^2}{2}

M_3 = R_Cx_2-\frac{wx_2^2}{2}

Shear and moment equations along the beam

Shear force, V(x)

V_{(x)}= \left\{ \begin{array}{lcc} -R_A+wx & \text{if} & x < a \\ \\ R_C-w(L-x) & \text{if} & x \geq a \end{array} \right.

Bending moment, M(x)

M_{(x)}= \left\{ \begin{array}{lcc} R_Ax-\frac{wx^2}{2} & \text{if} & x < a \\ \\ (R_A+R_B)x-\frac{wx^2}{2}-(R_A+R_B)L+\frac{wL^2}{2} & \text{if} & x \geq a \end{array} \right.

Variables

Symbol	Meaning
$a$	first span between supports
$b$	second span between supports
$w$	distributed load value
$x$	distance along the beam measured from the left end
$x_1$	R_A / w, position of the maximum moment in the first span
$x_2$	R_C / w, position of the maximum moment in the second span, measured from the right 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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