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

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

This equation can also be written as

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

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 center of gravity for the b.m. diagram for AB, x₂ is the distance from C to the c.g. for the b.m. diagram for BC.

The second equation is more general as it does not require that the weight of each segment be distributed uniformly.

External links

• CodeCogs: Continuous beams with more than one span