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Continuous Beam Analysis

Three-Moment Equation · 2–3 Spans · Uniform Loads

When to use: Analyze a continuous beam spanning 2 or 3 bays over simple supports, each span carrying its own uniform load w. The three-moment (Clapeyron) equation solves the interior support moments; reactions follow from span-end shears. Assumes constant EI and pinned ends (M₀ = Mₙ = 0).

Spans & Loading

Number of Spans

Span L₁

Load w₁

kip/ft

Span L₂

Load w₂

kip/ft

Key Formulas

Mᵢ₋₁Lᵢ + 2Mᵢ(Lᵢ+Lᵢ₊₁) + Mᵢ₊₁Lᵢ₊₁

  = -(wᵢLᵢ³/4 + wᵢ₊₁Lᵢ₊₁³/4)

V_left = wL/2 - (Mr-Ml)/L

V_right = wL/2 + (Mr-Ml)/L

R_interior = V_right(i-1) + V_left(i)

M₀ = Mₙ = 0 (simple ends)

Max Support Moment

75.00

kip·ft (hogging)

Support Moments

M0 (end)0.00 kip·ft

M1 (interior)-75.00 kip·ft

M2 (end)0.00 kip·ft

Reactions

R018.75 kip

R122.50 kip

R218.75 kip

References

Clapeyron Three-Moment Theorem

AISC Manual — continuous beam coefficients

Hibbeler Structural Analysis Ch. 11