page_title: Theorem of three moments
section_title: The sign convention
hierarchical_section_title: Theorem of three moments / Derivation of three moments equations / The sign convention
caption_attribution_description: English: This Figure used for derive the Three Moment Equation.
context_page_description: In civil engineering and structural analysis Clapeyron's theorem of three moments is a relationship among the bending moments at three consecutive supports of a horizontal beam.
  Let A, B, C be the three consecutive points of support, and denote by- l the length of AB and the length of BC, by w and the weight per unit of length in these segments. Then the bending moments at the three points are related by:
  This equation can also be written as
  where a₁ is the area on the bending moment diagram due to vertical loads on AB, a₂ is the area due to loads on BC, x₁ is the distance from A to the centroid of the bending moment diagram of beam AB, x₂ is the distance from C to the centroid of the area of the bending moment diagram of beam BC.
  The second equation is more general as it does not require that the weight of each segment be distributed uniformly.
context_section_description: According to the Figure 04,
  The moment M1, M2, and M3 be positive if they cause compression in the upper part of the beam. ([:wikt:sagging|sagging]] positive)
  The deflection downward positive. (Downward settlement positive)
  Let ABC is a continuous beam with support at A,B, and C. Then moment at A,B, and C are M1, M2, and M3, respectively.
  Let A' B' and C' be the final positions of the beam ABC due to support settlements.
caption: Figure 04-Deflection Curve of a Continuous Beam Under Settlement
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