Structural Concrete - Hassoun

15.6 Torsion Theories for Concrete Members 533
An approximate evaluation of the torsional capacity of a cracked section may be expressed
as follows:
( ) At f y
T n = 2 x
s 1 y 1 (15.11)
where
A t = area of one leg of stirrups
s = spacing of stirrups
x 1 and y 1 = short and long distances, center to center of closed rectangular stirrups or corner bars
The preceding expression neglects the torsional capacity due to concrete. Mitchell and Collins
[12] presented the following expression to evaluate the angle of twist per unit length ψ:
( ) [ ( (
P0
( ε1
) Ph εh tan α ) )
ψ =
+
+ 2ε ]
d
(15.12)
2A 0 tan α P 0 sin α
where
ε 1 = strain in longitudinal reinforcing steel
ε h = strain in hoop steel (stirrups)
ε d = concrete diagonal strain at position of resultant shear flow
P h = hoop centerline perimeter
α = angle of diagonal compression = (ε d + ε 1 )/[ ε d + ε h (P h /P 0 )]
A 0 = area enclosed by shear, or
= torque/2q where q = shear flow
P 0 = perimeter of shear flow path (perimeter of A 0 )
The preceding twist expression is analogous to the curvature expression in flexure
(Fig. 15.10):
φ = curvature = ε c + ε s
(15.13)
d t
Figure 15.10
(a) Torsion and (b) flexure.