Civil Engineering Reference
In-Depth Information
and the equation of equilibrium for torsion (in terms of warping and twisting
resistance) is
M
d
2
w(x)
d
x
2
d
4
GJ
d
2
(x)
d
x
4
φ
(x)
d
x
2
φ
=−
EI
w
+
(7.5)
with boundary conditions
d
2
w(
0
)
d
x
2
d
2
w(L)
d
x
2
d
2
d
2
(
0
)
d
x
2
φ
(L)
d
x
2
φ
w(
0
)
=
w(L)
=
=
= φ
(
0
)
= φ
(L)
=
=
=
0,
(7.6)
=
where
L
is the length of the beam or girder between lateral supports (
L
0 when
members are continuously laterally supported at compression flanges).
Equations 7.4 and 7.5 are satisfied when (Trahair, 1993)
w(L/
2
)
w(x)
=
φ
(L/
2
)
φ
1
=
(7.7)
(x)
sin
(
π
x/L)
and
2
E
2
I
w
I
y
L
4
π
EI
y
GJ
L
2
M
cr
= π
+
,
(7.8)
where
(x)
is the angle of twist about the shear center axis;
E
is the tensile modulus of
elasticity(
φ
∼
29,000 ksiforsteel)(seeChapter2);
G
istheshearmodulusofelasticity
=
E/
2
(
1
is Poisson's ratio (0.3 for steel); and
J
is the torsional constant
(depends on element dimensions). Equations for some common cross sections are
+ υ
)
, where
υ
lateral bending moment of inertia about the beam or girder vertical axis of symmetry;
and
I
w
=
ω
2
d
A
ω
is defined in terms of the position of the shear center and thickness of the member).
Warping constant values are available in many references (Roark and Young, 1982;
Seaburg and Carter, 1997). Equations for some common cross sections are given in
Table 7.1.
M
cr
is the critical lateral-torsional buckling moment.
For an I section with equal flanges (see Table 7.1)
=
C
w
is the torsional moment of inertia or warping constant (
2
t
f
b
3
Ar
y
≈
I
y
=
12
,
(7.9)
h
2
t
f
b
3
24
=
h
2
I
y
4
I
w
=
C
w
=
,
(7.10)
0.3
At
f
,
J
=
(7.11)
2
Ar
x
d
S
x
=
,
(7.12)
r
x
≈
0.4
d (
radius of gyration in the vertical direction
)
,
(7.13)
