Structural Concrete - Hassoun

21.2 Uniformly Loaded Circular Beams 859
Figure 21.1
Circular beam.
column reactions will be equal, and each reaction will be equal to the total load on the beam divided
by the number of columns. Referring to Fig. 21.1, consider the part AB between two consecutive
columns of the ring beam. The length of the curve AB is r(2θ), and the total load on each column
is P u = w u r(2θ), where r is the radius of the ring beam and w u is the factored load on the beam per
unit length. The center of gravity of the load on AB lies at a distance
( ) r sin θ
x =
θ
from the center O. The moment of the load P u about AB is
M AB = P u × y = P u (x − r cos θ) =w u r(2θ)
( r sin θ
θ
)
− r cos θ
Consequently, the two reaction moments, M A and M B , are developed at supports A and B,
respectively. The component of the moment at support A about AB is M A sin θ = M B sin θ. Equating
the applied moment, M AB , to the reaction moments components at A and B,
( )
r sin θ
2M A sin θ = M AB = w u r(2θ) − r cos θ
θ
M A = M B = w u r 2 (1 − θ cot θ) (21.1)