A linear relation among bending moments at three consecutive supports of an elastic, continuous beam obtained from compatibility of deflections and linear Euler–Bernoulli bending theory; it reduces static indeterminacy by coupling unknown support moments through span lengths and flexural rigidities, allowing solution of continuous-beam bending under small‑deflection, linear‑elastic assumptions.

Principle

Continuity of deflection and slope across supports together with linear bending kinematics produces a homogeneous linear relation between adjacent support moments weighted by span geometries and flexural stiffnesses; assembling such relations over all supports yields a solvable linear system for unknown moments.

Demonstration

Illustrative scenario: a continuous three‑span beam with equal EI and uniform spans. Recognition: supports are continuous so moments at intermediate supports are unknown. Action: write the three‑moment relations for adjacent triplets and solve the resulting linear system. Consequence: the computed support moments satisfy compatibility and equilibrium and determine bending moment diagrams and reactions.

Misapplication

Applying the three‑moment relation when Euler–Bernoulli assumptions fail (e.g., very deep beams where shear deformation is significant, large deflections introducing geometric nonlinearity, or when supports include hinges that break moment continuity), or using it as a universal formula for determinate single spans.

Consequence

Transforms a statically indeterminate continuous-beam problem into a linear algebra problem whose solution yields internal bending moments and reactions under the stated elastic, small‑deflection assumptions; incorrect application produces quantitatively wrong moment distributions and unsafe or overconservative designs.

Reversal

The theorem is invalid if material or geometric nonlinearity dominates (plasticity, large deflection), if shear deformation or rotary inertia must be included (Timoshenko or dynamic regimes), or if continuity at a support is intentionally removed (hinged support), in which case moment compatibility relations change.

Boundary

Valid for linear, elastic, small‑deflection bending of slender beams following Euler–Bernoulli assumptions, with supports that enforce continuity of slope and displacement; excludes plates, deep beams requiring shear correction, and situations with non‑elastic behavior or discontinuous kinematics.

Semantic Tension

Analytical convenience versus model fidelity: the three‑moment relation simplifies indeterminate bending to linear algebra but can mislead when higher‑order effects matter or when numerical methods could model more complex behavior directly.

Synthesis

Clapeyron’s three‑moment relation is a compatibility-derived linear shortcut: it encodes elastic continuity across supports into algebraic constraints that solve indeterminacy only within the narrow regime of linear Euler–Bernoulli beam theory.