Civil Engineering Reference
In-Depth Information
d
l
dt
d
φ
r
d
l
l
A
t
r d
φ
dt
A
r
d
φ
T
t
dw
1
l
d
d
l
=
dw
1
-r
dt
(a) WARPING DISPLACEMENT (IN
l
-DIRECTION)
DUE TO ANGLE OF TWIST
d
l
t
A
l
γ
l
f
γ
l
t
dt
A
dw
2
T
t
l
=
γ
l
f
dt
dw
2
(b) WARPING DISPLACEMENT (IN
l
-DIRECTION)
DUE TO SHEAR DEFORMATION
Figure 7.2
Warping displacement in a tube
The warping displacement of a differential 2-D element A, shown in Figure 7.2, is composed
of two parts. The first part is induced by the rigid rotation d
φ
, as shown in Figure 7.2(a). The
second part is caused by the shear deformation
γ
t
, as shown in Figure 7.2(b). Under torsion,
the 2-D element A, d
t
plane. The symbol
r
is the distance from the center of twist to the centerline of the element.
The differential warping displacement d
by d
t
, in Figure 7.2(a) rotates through an angle
r
d
φ/
d
in the
−
w
1
is therefore:
r
d
d
d
w
1
=−
d
t
=−
r
θ
d
t
(7.12)
The differential warping displacement d
w
2
, of 2-D element A due to shear deformation in
Figure 7.2(b) is:
d
w
2
=
γ
t
d
t
(7.13)
Adding Equations (7.12) and (7.13) gives the total differential warping displacement d
w
due to both the rotation and the shear deformation:
d
w
=
d
w
1
+
d
w
2
=−
r
θ
d
t
+
γ
t
d
t
(7.14)
