Civil Engineering Reference
In-Depth Information
where
is a numerical factor to correct for nonuniform stress distribution across the
cross section of the compression member,
A
g
is the gross cross-sectional area of the
compression member,
G
eff
is the effective shear modulus equal to
E
eff
/(
2
(
1
β
+ υ
))
,
and
0.3 for steel.
The inclusion of Equation 6.41 into Equation 6.14 with
E
υ
is the Poisson's ratio
=
=
E
eff
yields the
differential equation
d
2
y(x)
d
x
2
k
2
y(x)
U
+
P/A
g
G
eff
))
=
(6.42)
(
1
+
(
β
(
1
+
(
β
P/A
g
G
eff
))
with solution analogous to Equation 6.15 of
2
E
eff
I
P
cr
=
π
P
cr
α
=
2
,
(6.43)
KL)
2
(
α
where
P
cr
is the critical buckling load for compression member with gross cross-
sectional area,
A
g
, moment of inertia,
I
, and length,
L
(see Equation 6.15), and
1
.
P
cr
A
g
G
eff
β
α =
+
(6.44)
Equation 6.43 may be written as
P
cr
P
cr
=
+
(
β
P
cr
/A
g
G
eff
))
.
(6.45)
(
1
Equation 6.45 illustrates that the critical buckling load,
P
cr
, for built-up compression
members can be readily determined based on the critical buckling load,
P
cr
, for closed
members of the same cross-sectional area,
A
g
.
The majority of steel railway superstructure compression members are slender and
connected with modern fasteners and are assumed to be pin connected at each end
(
K
0.75).Therefore,thecriticalbucklingstrengthofbuilt-upcompressionmembers
of various configurations (using lacing and batten bars, and perforated cover plates)
with pinned ends will be considered further.
Equation 6.44 may be written as
=
1
1
P
cr
A
g
G
eff
β
α =
+
=
+ Ω
P
cr
,
(6.46)
where
β
A
g
G
eff
=
2
β
(
1
+ ν
)
1
P
Ω =
=
.
(6.47)
A
g
E
eff
Ω
The value of
is determined through investigation of the deformations of the lacing
bars, batten plates, and/or perforated cover plates caused by lateral displacements
from shear force,
V
. The results of such investigations for various built-up com-
pression members are presented in the next sections (see e.g., Timoshenko and
Gere, 1961).
Ω
