Clapeyrons Three Moment Equation- Aircraft Structure

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[1] the bending moments M_A,\, M_B,\, M_C at the three points are related by:

M_A l + 2 M_B (l+l') +M_C l' = \frac{1}{4} w l^3 + \frac{1}{4} w' (l')^3.

This equation can also be written as [2]

M_A l + 2 M_B (l+l') +M_C l' = \frac{6 a_1 x_1}{l} + \frac{6 a_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.