Civil Engineering Reference
In-Depth Information
r
y
≈
0.2
b (
radius of gyration in the lateral direction on the
compression side of the neutral axis
)
,
(7.14)
h
≈
d
,
(7.15)
E
G
=
)
=
0.38
E
.
(7.16)
2
(
1
+ υ
Substitution of Equations 7.9 through 7.16 into Equation 7.8 yields
15.4
E
(L/r
y
)
2
2
0.67
E
Ld/bt
f
2
,
f
cr
=
+
(7.17)
where
f
cr
=
M
cr
/S
x
is the critical lateral-torsional buckling stress.
The first term in Equation 7.17 represents the warping torsion effects and the
secondtermdescribesthepuretorsioneffects.Fortorsionallystrongsections(shallow
sections with thick flanges) the warping effects are negligible and
0.67
E
Ld
/
bt
f
0.21
0
.24
.
E
Ld
/
bt
f
π
π
E
f
cr
=
=
=
Ld(
√
1
(7.18)
+ υ
)/bt
f
For torsionally weak sections (deep sections with thin flanges and web, typical of
railway plate girders) the pure torsion effects are negligible and
1.56
,
15.4
E
(L
/
r
y
)
2
2
E
(L/r
y
)
2
π
f
cr
=
=
(7.19)
which is analogous to determining the elastic (Euler) column strength of the flanges.
Using a factor of safety of 9
/
5
1.80, Equations 7.18 and 7.19 for torsionally
strong and weak sections, respectively, are
=
0.13
f
cr
π
E
F
cr
=
1.80
=
Ld(
√
1
(7.20)
+ υ
)/bt
f
and
0.87
.
f
cr
2
E
(L
/
r
y
)
2
π
F
cr
=
1.80
=
(7.21)
However, due to residual stress, unintended load eccentricity, and fabrication
imperfections, axial compression member strength is based on an inelastic buck-
ling strength parabola when
F
cr
≥
F
y
/2 and an Euler (elastic) buckling curve for
F
cr
≤
When
F
cr
=
(
0.55
F
y
)/
2, Equation 7.21 (elastic buckling curve) is equal and tan-
gent to the inelastic buckling strength parabola (transition curve). From Equation 7.21
the slenderness,
L
/
r
y
,at
(
0.55
F
y
)/
2is
5.55
E
F
y
L
r
y
=
.
(7.22)
