Civil Engineering Reference
In-Depth Information
where
n
is the distance to any point in the section wall measured normally from its
midline. The distribution is therefore linear across the thickness as shown in Fig. 11.15
and is zero at the midline of the wall. An alternative expression for shear stress dis-
tribution is obtained, in terms of the applied torque, by substituting for d
θ
/d
x
in Eq.
(11.29) from Eq. (11.26). Thus
2
n
T
J
τ
=
(11.30)
It is clear from either of Eqs. (11.29) or (11.30) that the maximum value of shear stress
occurs at the outer surfaces of the wall when
n
=±
t
/2 . Hence
Gt
d
θ
Tt
J
τ
max
=±
d
x
=±
(11.31)
The positive and negative signs in Eq. (11.31) indicate the direction of the shear stress
in relation to the assumed direction for
s
.
The behaviour of closed and open section beams under torsional loads is similar in
that they twist and develop internal shear stress systems. However, the manner in
which each resists torsion is different. It is clear from the preceding discussion that
a pure torque applied to a beam section produces a closed, continuous shear stress
system since the resultant of any other shear stress system would generally be a shear
force unless, of course, the system were self-equilibrating. In a closed section beam
this closed loop system of shear stresses is allowed to develop in a continuous path
round the cross section, whereas in an open section beam it can only develop within
the thickness of the walls; examples of both systems are shown in Fig. 11.16. Here,
then, lies the basic difference in the manner in which torsion is resisted by closed and
open section beams and the reason for the comparatively low torsional stiffness of
thin-walled open sections. Clearly the development of a closed loop system of shear
stresses in an open section is restricted by the thinness of the walls.
Closed
section
Torque
Torque
Open
section
F
IGURE
11.16
Shear stress
development in closed and open
section beams subjected to torsion
E
XAMPLE
11.3
The thin-walled section shown in Fig. 11.17 is symmetrical about
a horizontal axis through O. The thickness
t
0
of the centre web CD is constant, while
the thickness of the other walls varies linearly from
t
0
at points C and D to zero at the
open ends A, F, G and H. Determine the torsion constant
J
for the section and also
the maximum shear stress produced by a torque
T
.
