Civil Engineering Reference
In-Depth Information
where
t
is the maximum torsional moment applied at the end support,
L
z
is
the distance from the end support with maximum torsional moment,
L
is the length
of the span.
The angle of twist,
−
, is provided by the solution of Equations 8.16, 8.17, or 8.18
and depends on both loading and boundary conditions. The general form of the angle
of twist,
θ
θ
, can be expressed as (Kuzmanovic and Willems, 1983)
A
sinh
z
a
B
cosh
z
a
θ =
+
+
C
+
D(z)
,
(8.19)
where
A
,
B
,
C
, and
D
are constants depending on boundary conditions
and load-
ing,
D(z)
is an expression in terms of
z
, depending on loading, and
a
=
√
EC
w
/GJ
(a characteristic length).
The equations for the angle of twist and its derivatives for a concentrated tor-
sional moment,
T
, applied at the center of a simply supported span are (Salmon and
Johnson, 1980)
θ =
A
sinh
z
a
z
a
−
, (8.20a)
+
B
cosh
z
a
T
z
2
GJ
=
T
a
2
GJ
sinh
(z/a)
cosh
(L/
2
a)
+
C
+
1
,
T
2
GJ
d
d
z
=
cosh
(z/a)
cosh
(L/
2
a)
−
(8.20b)
,
d
2
T
2
GJa
θ
d
z
2
sinh
(z/a)
cosh
(L/
2
a)
=
−
(8.20c)
.
d
3
T
2
GJa
2
θ
d
z
3
cosh
(z/a)
cosh
(L/
2
a)
=
−
(8.20d)
Example 8.1 illustrates the use of Equations 8.20a-d for the determination of
combined stresses due to torsion and flexure.
The equations for the angle of twist and its derivatives for a uniformly distributed
torsionalmoment,
t
,appliedatthecenterofasimplysupportedspanare(Kuzmanovic
and Willems, 1983)
1
,
tanh
L
2
a
sinh
z
a
cosh
z
a
t
a
2
GJ
z
2
2
a
2
+
zL
2
a
2
−
θ =
−
+
−
(8.21a)
tanh
L
2
a
cosh
z
a
,
sinh
z
a
t
a
GJ
d
d
z
=
z
a
+
L
2
a
−
+
−
(8.21b)
tanh
L
2
a
sinh
z
a
1
,
cosh
z
a
d
2
t
GJ
θ
d
z
2
=
−
+
−
(8.21c)
tanh
L
2
a
cosh
z
a
.
sinh
z
a
d
3
t
GJa
θ
d
z
3
=
−
+
(8.21d)
Example 8.3 illustrates the use of Equations 8.21a-d for the determination of
combined stresses due to torsion and flexure.
