Civil Engineering Reference
In-Depth Information
7
Torsion
7.1 Analysis of Torsion
In Chapter 5, we have applied the rotating angle theory to 2-D elements subjected to shear
and normal stresses. In this chapter we will apply the rotating angle softened truss model
(RA-STM), presented in Section 5.4, to torsion (Hsu, 1984; Hsu and Mo, 1985a, 1985b,
1985c; Hsu, 1988, 1990, 1991a, 1991b, 1993). Section 7.1 will be devoted to the analysis of
torsion, while Section 7.2 will include several important topics in the design for torsion.
The softened membrane model (SMM) studied in Chapter 6, Section 6.1, can also be applied
to torsion (Jeng and Hsu, 2009). However, this analytical model will not be presented in
this topic.
7.1.1 Equilibrium Equations
7.1.1.1 Shear Elements in Shear Flow Zone
A reinforced concrete prismatic member is subjected to an external torque
T
as shown in
Figure 7.1(a). This external torque is resisted by an internal torque formed by the circulatory
shear flow
q
along the periphery of the cross-section. This shear flow
q
occupies a zone,
called the shear flow zone, which has a thickness denoted as
t
d
. This thickness
t
d
is a variable
determined from the equilibrium and compatibility conditions. It is not the same as the given
wall thickness
h
of a hollow member.
A 2-D element
A
in the shear flow zone (Figure 7.1a), is subjected to a shear stress
τ
t
=
q
t
d
, as shown in Figure 7.1(b). The in-plane equilibrium of this element should satisfy the
three equations, (5.94)-(5-96), in Section 5.4.2:
/
σ
d
sin
2
σ
−
ρ
f
−
ρ
p
f
p
=
α
r
(7.1) or [1]
σ
d
cos
2
σ
t
−
ρ
t
f
t
−
ρ
tp
f
tp
=
α
r
(7.2) or [2]
τ
t
=
(
−
σ
d
)sin
α
r
cos
α
r
(7.3) or [3]
