Dreiemoment

Dreiemoment, in structural engineering often called the three-moment theorem, is a relation that connects the bending moments at the supports of a continuous beam with the geometry of its spans and the external loads on those spans. It is a key tool for analyzing indeterminate beam structures, especially when a beam has more than two supports.

Origin and terminology: The result is traditionally attributed to Clapeyron (1833) and is widely known in English as Clapeyron’s theorem of three moments. In German literature it is referred to as the Dreimomentensatz or Dreimomentequation.

Scope and assumptions: The theorem applies to prismatic beams (constant EI) with small deflections, linear elastic behavior, and supports arranged collinearly. It is useful for continuous beams with three consecutive supports, though it can be extended to longer systems by applying the relation to successive triplets of supports. The end moments at the supports are related through a linear equation that depends on the span lengths and the external loading of the spans.

Statement (conceptual): For three consecutive supports A, B, C with spans AB = L1 and BC = L2, the end moments M_A, M_B, M_C satisfy a linear relation that involves L1, L2 and the bending moment effects produced by the external loads on each span. The end moments at simple supports (where the ends are not fixed) may be known or unknown and are determined by solving the three-moment equation together with boundary conditions.

Applications and methods: The theorem is used to analyze two-span continuous beams and forms the basis for classical methods such as slope-deflection and moment distribution. It remains a foundational concept in structural analysis and design.

See also: Clapeyron’s theorem, three-moment equation, slope-deflection method, moment distribution.