Civil Engineering Reference
In-Depth Information
and substitution of Equation 8.9 into Equation 8.11 yields
Gt
d
,
T
t
t
J
=
d
z
τ
t
=
(8.12)
where
t
is the thickness of the element.
The shear stresses from warping effects of the applied torsional moments on an
I-shaped section are
3
2
T
w
A
f
h
τ
w
=
(8.13)
and substitution of Equation 8.10 into Equation 8.13 yields
d
3
d
3
,
Eb
f
h
16
3
2
EC
w
A
f
h
θ
d
z
3
θ
d
z
3
τ
w
=−
=−
(8.14)
where
A
f
is the area of the flange and is equal to
b
f
t
f
,
I
y
h
2
4
=
t
f
b
f
h
2
24
(I
f
)h
2
2
C
w
=
=
,
h
is the distance between centroids of flanges for I-shaped members.
The normal stresses from warping effects of the applied torsional moments on
an I-shaped section are determined by considering the normal stress from the lateral
bending of the flanges
d
2
d
2
,
M
l
x
I
f
EI
f
h
2
θ
d
z
2
EC
w
h
θ
d
z
2
σ
w
=
=
=
(8.15)
where
x
isthedistanceontheflangefromtheneutralaxisofflexuralstressdistribution
in the flange (maximum at
x
b/
2),
M
l
is the lateral bending moment on one flange,
I
f
is the moment of inertia of the flange.
The differential equation of torsion is (from Equations 8.8 through 8.10)
=
d
3
GJ
d
d
z
−
θ
d
z
3
.
T
=
EC
w
(8.16)
For torsional moments that vary uniformly along the length (
z
axis) of a member,
d
T
/d
z
is
GJ
d
2
d
4
d
T
d
z
=
θ
d
z
2
−
θ
d
z
4
.
t
=
EC
w
(8.17)
Fortorsionalmomentsthatvarylinearlyalongthelength(
z
axis)ofamember,d
T
/d
z
is
t
z
L
=
GJ
d
2
d
4
d
T
d
z
=
θ
d
z
2
−
θ
d
z
4
,
EC
w
(8.18)
