Civil Engineering Reference
In-Depth Information
t
T
q
q
δ
x
δ
s
s
T
q
x
q
δ
x
x
F
IGURE
11.11
Torsion of a
thin-walled closed
section beam
q
s
q
δ
s
(a)
(b)
section. The torque
T
induces a stress system in the walls of the beam which consists
solely of shear stresses if the applied loading comprises only a pure torque. In some
cases structural or loading discontinuities or the method of support produce a system
of direct stresses in the walls of the beam even though the loading consists of torsion
only. These effects, known as axial constraint effects, are considered inmore advanced
texts.
The shear stress system on an element of the beamwall may be represented in terms of
the shear flow,
q
, (see Section 10.4) as shown in Fig. 11.11(b). Again we are assuming
that the variation of
t
over the side
s
of the element may be neglected. For equilibrium
of the element in the
x
direction we have
q
δ
s
∂
q
∂
s
δ
+
δ
x
−
q
δ
x
=
0
which gives
∂
q
∂
s
=
0
(11.17)
Considering equilibrium in the
s
direction
q
x
∂
q
∂
x
δ
+
δ
s
−
q
δ
s
=
0
from which
∂
q
∂
x
=
0
(11.18)
Equations (11.17) and (11.18) may only be satisfied simultaneously by a constant value
of
q
. We deduce, therefore, that the application of a pure torque to a thin-walled closed
section beam results in the development of a constant shear flow in the beam wall.
However, the shear stress,
τ
, may vary round the cross section since we allow the wall
thickness,
t
, to be a function of
s
.
The relationship between the applied torque and this constant shear flow may be
derived by considering the torsional equilibriumof the section shown in Fig. 11.12. The
torque produced by the shear flow acting on the element,
δ
s
, of the beam wall is
q
δ
sp
.
Hence
pq
d
s
T
=
