Theorem of three moments

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-D be the three consecutive points of support, and denote by- l the length of AB and ${\displaystyle l'}$ the length of BC, by w and ${\displaystyle w'}$ the weight per unit of length in these segments. Then^[1] the bending moments ${\displaystyle M_{A},\,M_{B},\,M_{C}}$ at the three points are related by:

${\displaystyle M_{A}l+2M_{B}(l+l')+M_{C}l'={\frac {1}{4}}wl^{3}+{\frac {1}{4}}w'(l')^{3}.}$

This equation can also be written as ^[2]

${\displaystyle M_{A}l+2M_{B}(l+l')+M_{C}l'={\frac {6a_{1}x_{1}}{l}}+{\frac {6a_{2}x_{2}}{l'}}}$

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.