Two-Span Beams

Two-Span Continuous Beam with Central Point Load in One Zone: 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.

Calculate with your values

Load case and diagrams

Two-Span Continuous Beam with Central Point Load in One Zone

Formulas

Reactions

R_1 = \frac{13P}{32}

R_2 = \frac{11P}{16}

R_3 = -\frac{3P}{32}

Maximum shear force

V_1 = -\frac{13P}{32}

V_2 = \frac{19P}{32}

V_3 = -\frac{3P}{32}

Maximum bending moment

M_{max} = \frac{13PL}{64}

M_1 = -\frac{3PL}{32}

Shear and moment equations along the beam

Shear force, V(x)

V_{(x)}= \left\{ \begin{array}{lcc} -R_1 & \text{if} & x < \frac{L}{2} \\ \\ -R_1+P & \text{if} & \frac{L}{2} \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 < \frac{L}{2} \\ \\ (R_1-P)(x-\frac{L}{2})+\frac{R_1L}{2} & \text{if} & \frac{L}{2} \leq x \leq L \\ \\ -R_3(x-2L) & \text{if} & L < x \leq 2L \end{array} \right.

Variables

Symbol	Meaning
$L$	span between supports
$P$	point 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.

Calculate this case with your own values

Enter the span and loads and get the reactions and the shear and bending moment diagrams instantly, in metric or imperial units. Free with a Google account.

Calculate with your values

Different supports, hinges or several loads? Use the full beam calculator.

Other two-span beams load cases

See all 37 beam formula cases