Civil Engineering Reference
In-Depth Information
For a closed section, the total differential warping displacement integrating around the
whole perimeter must be equal to zero, giving
d
w
=−
θ
r
d
t
+
γ
t
d
t
=
0
(7.15)
Recalling
r
d
t
=
2
A
o
from Equation (2.45) gives
γ
t
d
t
=
2
A
o
θ
(7.16)
When the wall thickness of the tube is uniform, the shear stress
τ
t
is a constant, resulting
in a uniform shear strain of
γ
t
. Then
γ
t
could be taken out of the integral in Equation (7.16),
giving
γ
t
d
t
=
2
A
o
θ
(7.17)
Since
d
t
is the perimeter of the centerline of the shear flow,
p
o
,wehave
p
o
2
A
o
γ
t
θ
=
(7.18) or [8]
It is clear from Equation [8] that the angle of twist
θ
will produce a shear strain
γ
t
in
the 2-D elements of the shear flow zone. This shear strain
γ
t
will induce the steel strains
ε
and
ε
t
in the
−
t
coordinate and the concrete strains
ε
r
and
ε
d
in the
r
−
d
coordinate. The
relationship between the strains in the
−
t
coordinate (
ε
,
ε
t
, and
γ
t
) and the strains in the
r
−
d
coordinate (
ε
r
and
ε
d
) is described by the three compatibility equations, [5]-[7].
7.1.2.3 Bending of Diagonal Concrete Struts
θ
In a torsional member, the angle of twist
also produces warping in the wall of the member,
which, in turn, causes bending in the concrete struts. In other words, the concrete struts are
not only subjected to compression due to the circulatory shear, but also subjected to bending
due to the warping of the wall. The relationship between the angle of twist,
θ
and the bending
will now be studied.
A box member with four walls of thickness
t
d
and subjected to a torsional moment
T
is
shown in Figure 7.3(a). The length of the member is taken to be
L
. The centerline of shear flow
q
along the top wall has a width of
L
cot
curvature of concrete struts
ψ
α
r
, so that the diagonal line along the center plane of
top wall OABC has an angle of
α
r
with respect to the shear flow
q
. When this member receives
an angle of twist
, this center plane will become a hyperbolic paraboloid surface OADC, as
shown in Figure 7.3(b). The edge CB of the plane rotates to the position CD through an angle
θ
θ
L
. The curve OD will have a curvature of
to be determined.
A 3-D coordinate system with axes,
x
,
y
and
ψ
, is imposed on the center plane OABC, as
shown in Figure 7.3(b). The axes
x
and
y
are along the edges OC and OA, while the axis
w
is
normal to the center plane. The hyperbolic paraboloid surface OADC can then be expressed by:
w
w
=
θ
xy
(7.19)
where
w
is the displacement perpendicular to the
x
−
y
plane.
