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).
Analyze 2- or 3-span continuous beams over simple supports using the three-moment (Clapeyron) equation. Enter span lengths and uniform loads per span to compute interior support moments, end reactions, and the maximum hogging moment at interior supports — no matrix analysis required.
The three-moment (Clapeyron) theorem relates the moments at three consecutive supports through span lengths and loading. For 2 spans a single equation yields M₁; for 3 spans a 2×2 linear system is solved directly. Once interior moments are known, span-end shears and support reactions follow from simple statics: V_left = wL/2 − (Mr − Ml)/L, V_right = wL/2 + (Mr − Ml)/L.
Three-moment equation: Mᵢ₋₁Lᵢ + 2Mᵢ(Lᵢ+Lᵢ₊₁) + Mᵢ₊₁Lᵢ₊₁ = −(wᵢLᵢ³/4 + wᵢ₊₁Lᵢ₊₁³/4). End conditions: M₀ = Mₙ = 0 (simply supported ends). Interior reaction: Rᵢ = VR(i-1) + VL(i). Assumes constant EI throughout.
Use for continuous floor beams, bridge girders, or any beam that spans over three or four supports with uniform gravity loading. The tool accurately captures the hogging moment reduction at interior supports (negative moment at supports, positive midspan) that is the defining characteristic of continuous construction. For point loads or non-uniform EI, use the Beam Diagram Simulator.
Interior support moments are hogging (negative by convention), meaning the top fiber is in tension. This is the opposite of the sagging (positive) midspan moment in a simply supported beam. Continuous beams redistribute moment so that peak midspan moments are lower than an equivalent simply supported span.
The three-moment (Clapeyron) theorem is a compatibility equation requiring that the slope of the elastic curve is continuous at interior supports. It expresses the relationship between three consecutive support moments in terms of span geometry and load — producing one equation per interior support.
No — this tool assumes constant EI throughout. Variable-EI continuous beams require a more general moment-distribution or stiffness-matrix approach.
For span i with UDL wᵢ and end moments Ml, Mr, the midspan moment is Mmid = wᵢLᵢ²/8 − (Ml + Mr)/2. The positive midspan moment is always less than the simply supported value wL²/8 due to the restraining effect of support continuity.