Structural Concrete - Hassoun

21.3 Semicircular Beam Fixed at End Supports 867
4. Torsional moment at support A. T A can be obtained by differentiating the strain energy of the
beam with respect to T A and equating it to 0. Considering that T A is acting clockwise at A,
then the bending moment at any section N is calculated as follows:
θ ( ) wr
M N = V A (r sin θ)−M A cos θ + T A sin θ − ∫
0 2 sin2 θ (rdα)×r sin (θ − α)
[ ( ) ]
M N = wr 3 π 1
8 sin θ − (1 + cos 2 θ) + T
6
A sin θ (21.13)
The torsional moment at any station N on the curved beam is equal to
T n =−V A r(1 − cos θ)+M A sin θ + T A cos θ + ∫
π∕2
× r[1 − cos(θ − a)]
0
( ) wr
2 sin2 α (rdα)
T N = wr 3 [ π
8 (cos θ − 1) + θ 4 + 1 24 sin2θ ]
+ T A cos θ (21.14)
Thestrainenergyis
where
U = ∫
ds = rdθ
G = modulus of rigidity
E = modulus of elasticity
I = moment of inertia of section
J = rotational constant of section
= polar moment of inertia
M 2 N ds
2 EI
+ ∫
T 2 N ds
2 GJ
(21.15)
To obtain T A , differentiate U with respect to T A :
δ U
δT A
= ∫
M N
EI × dM N
dT A
(rdθ)+ ∫
T N
GJ × dT N
dT A
×(rdθ) =0
Therefore,
dM N
dT A
= sin θ and
dT N
dT A
= cos θ
and
δ U
δT A
= r EI ∫
0
π∕2
+ r
GJ ∫
0
{ [
sin θ wr 2 π
8 sin θ − 1 ( 1 + cos 2 θ )] + T
6
A sin θ}
dθ
[ {wr 3 π
8 (cos θ − 1) + θ 4 + 1 ] }
24 sin2θ + T A cos θ cos θ × dθ = 0
π∕2
[ (
r
wr 3 π
2
EI 32 − 2 ) ( ) ] π
+ T
9 A + r
[ (
wr 3 π
2
4 GJ 32 − 2 )
+ T
9 A łeft( π )
= 0
4