Structural Concrete - Hassoun

21.6 Circular Beam Subjected to a Concentrated Load at Midspan 875
Example 21.3
A curved beam has a quarter-circle shape in plan with a 10 ft radius. The beam has a rectangular section
with the ratio of the long to the short side of 2.0 and is subjected to a factored load of 8 K/ft. Determine
the bending and torsional moments at the centerline of the beam, supports, and maximum values.
Solution
1. For a rectangular section with y/x = 2, λ = EI/GJ = 3.39 (Table 21.2).
2. The bending and torsional moments can be calculated using Eqs. 21.31 through 21.35 for θ = π/4.
From Eq. 21.31,
(
K 1 = 2 2sin π 4 − π )
= 1.2576
4
K 2 = sin π 2 = 1.0
( ) π
K 3 = 4 cos π 4 4 = 2.2214
( ) π
K 4 = 2 (3.39 + 1)−(3.39 − 1)sin π 4 2 = 4.506
M c = wr2 [3.39(1.2576 + 1.0 − 2.2214)+(1.2576 − 1.0)]
4.506
= 0.0844wr 2
For w = 8K⋅ft and r = 10 ft, M c = 64 K⋅ft; T c = 0
3. M N = M c cos α−wr 2 (1−cos α) = wr 2 (1.08 cos α−1)
T N = M c sin α − wr 2 (α − sin α) =wr 2 (1.08 sin α − α)
For the moments at the supports, α = θ = π/4.
(
M A = wr 2 1.08 cos π )
4 − 1 =−0.236wr 2
=−0.236 × 8 ×(10) 2 =−189 K ⋅ ft
(
T A = wr 2 1.08 sin π 4 − π )
= 0.022 wr 2 =−17.4K⋅ ft
4
For M N = 0, 1.08 cos α−1 = 0, or cos α = 0.926 and α = 22.2 ∘ = 0.387 rad. To calculate T N,max ,
let dT N /dα = 0, or 1.08 cos α–1 = 0. Then cos α = 0.926 and α = 22.2 ∘ .
T N (max) =wr 2 (1.08 sin 22.2 − 0.387) =0.0211 wr 2
T N,max = 0.0211 − 800 = 16.85 K ⋅ ft
21.6 CIRCULAR BEAM SUBJECTED TO A CONCENTRATED LOAD AT MIDSPAN
If a concentrated load is applied at the midspan of a circular beam, the resulting moments vary with
the magnitude of the load, the span, and the coefficient λ = EI/GJ. Considering the general case of
a circular beam fixed at both ends and subjected to a concentrated load P at midspan (Fig. 21.8),
the bending and torsional moments can be calculated from the following expressions: