Two-Span Beams

Two-Span Continuous Beam with Point Load: 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 Point Load

Formulas

Reactions

R_1 = \frac{Pb}{4L^3}(4L^2-a(L+a))

R_2 = \frac{Pa}{2L^3}(2L^2+b(L+a))

R_3 = -\frac{Pab}{4L^3}(L+a)

Maximum shear force

V_1 = -R_1

V_2 = \frac{Pa}{4L^3}(4L^2+b(L+a))

V_3 = R_3

Maximum bending moment

M_1 = -\frac{Pab}{4L^2}(L+a)

M_{max} = \frac{Pab}{4L^3}(4L^2-a(L+a))

Shear and moment equations along the beam

Shear force, V(x)

V_{(x)}= \left\{ \begin{array}{lcc} -R_1 & \text{if} & x < a \\ \\ -R_1+P & \text{if} & a \leq x \leq L \\ \\ R_3 & \text{if} & L < x \leq 2L \end{array} \right.

Bending moment, M(x)

M_{(x)}= \left\{ \begin{array}{lcc} R_1x & \text{if} & x < a \\ \\ (R_1-P)(x-a)+R_1a & \text{if} & a \leq x \leq L \\ \\ R_3(2L-x) & \text{if} & L < x \leq 2L \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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