Structural Concrete - Hassoun

882 Chapter 21 Beams Curved in Plan
where
λ = EI/GJ
a = half total length of beam (part AB or BC)
θ = half angle between two sides of V-shape beam
The torsional moment at the centerline section is
T c =
M c
sin θ cos θ = M c cot θ (21.46)
2. The bending and torsional moments at any section N along half the beam AC or BC at a
distance x measured from C are calculated as follows:
M N = M c − Px
(21.47)
1
2
T N = T c = M c cot θ (21.48)
The moments at the supports are determined by assuming x = a:
M A = M c − Pa
(21.49)
1
2
T A = T c = M c cot θ (21.50)
Example 21.6
Determine the bending and torsional moments in a V-shape beam subjected to a concentrated load P =
30 K acting at the centerline of the beam. Given: θ = π/4, y/x = 2.0, and a = 12 ft.
Solution
1. For a rectangular section with y/x = 2.0, λ = 3.39.
2.
M C = Pa
(
sin 2 )
π∕4
= 0.057(Pa)
4 sin 2 π∕4 + 3.39 cos 2 π∕4
= 0.057 × 30 × 12 = 20.5K⋅ ft
M A = M c − Pa =(0.057 − 0.5)Pa =−0.443(Pa)
1
2
=−0.0443 × 360 =−159.5K⋅ ft
M N = 0 when M c − Px = 0
1
2
Hence, 0.057Pa–0.5Px = 0andx = 0.114a = 0.114 × 12 = 1.37 ft measured from c.
3. T A = T c = T N = M c cot π = 0.057(Pa)=20.5K⋅ ft
4
Example 21.7
Determine the bending and torsional moments in the beam of Example 21.6 if the angle θ is π/ 2(a
straight beam fixed at both ends).