Civil Engineering Reference
In-Depth Information
F
IGURE
11.13
Examples of solid beam
sections
Rectangular
block
'Thick'
I-section
'Thick'
channel
section
Thin-walled
Z-section
t
f
D
r
t
W
d
t
f
F
IGURE
11.14
Torsion constant for a 'thick'
I-section beam
b
obvious exception is the hollow circular section bar which is, however, a special case
of the solid circular section bar. The prediction of stress distributions and angles of
twist produced by the torsion of such sections is complex and relies on the St. Venant
warping function or Prandtl stress functionmethods of solution. Both of thesemethods
are based on the theory of elasticity which may be found in advanced texts devoted
solely to this topic. Even so, exact solutions exist for only a few practical cases, one of
which is the circular section bar.
In all torsion problems, however, it is found that the torque,
T
, and the rate of twist,
d
θ/
d
x
, are related by the equation
GJ
d
θ
d
x
T
=
(11.26)
where
G
is the shear modulus and
J
is the
torsion constant
. For a circular section bar
J
is
the polar second moment of area of the section (see Eq. (11.3)) while for a thin-walled
closed section beam
J
, from Eq. (11.25), is seen to be equal to 4
A
2
/
(d
s
/
t
). It is
J
,in
fact, that distinguishes one torsion problem from another.
For 'thick' sections of the type shown in Fig. 11.13
J
is obtained empirically in terms
of the dimensions of the particular section. For example, the torsion constant of the
'thick' I-section shown in Fig. 11.14 is given by
2
α
D
4
J
=
2
J
1
+
J
2
+
