Two-Span Beams

Two-Span Continuous Beam with Point Loads 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 Point Loads with Different Span Lengths

Formulas

Reactions

R1=M2a+P12R_1 = \frac{M_2}{a}+\frac{P_1}{2}
R2=P1+P2−R1−R3R_2 = P_1+P_2-R_1-R_3
R3=M2b+P22R_3 = \frac{M_2}{b}+\frac{P_2}{2}

Maximum shear force

V1=−R1V_1 = -R_1
V2=P1−R1V_2 = P_1-R_1
V3=−P2+R3V_3 = -P_2+R_3
V4=R3V_4 = R_3

Maximum bending moment

M1=R1a2M_1 = R_1\frac{a}{2}
M2=−316(P1a2+P2b2a+b)M_2 = -\frac{3}{16}(\frac{P_1 a^2+P_2 b^2}{a+b})
M3=R3b2M_3 = R_3\frac{b}{2}

Variables

SymbolMeaning
aafirst span between supports
bbsecond span between supports
P1P_1point load 01
P2P_2point load 02

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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