Civil Engineering Reference
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Euler K = 0.5
f y
Euler K = 1.0
f all
Euler K = 2.0
Fixed K = 0.5
Pinned K = 1.0
Cantilever K = 2.0
C c /2.0 C c /1.0 C c /0.5
L / r
FIGURE 6.6 Effect of end restraint on allowable stresses and slenderness values at elastic
(Euler) buckling.
FS, of 1.95 to arrive at the allowable compressive strength, C all ,of
2 EI
(KL) 2
P cr
1.95 =
0.514
π
C all =
(6.20)
or
2 E
(KL/r) 2 .
C all
A g =
F cr
1.95 =
0.514 π
F all =
(6.21)
Elastic buckling, described by Equation 6.21, will occur at values of KL/r
C c . C c
is defined by the intersection of the Euler buckling curve (Figure 6.6) with a transition
curve from compressive yielding, f y , as shown by the vertical lines in Figure 6.6 for
members with K
0.5, 1.0, and 2.0. The transition curve represents the effects of
eccentricities, initial imperfections, and residual stresses introduced during fabrica-
tion and erection of steel railway bridge compression members. For elastic buckling
(at large KL / r) , the degree of member end restraint, expressed in terms of the effective
length factor, K , greatly affects the allowable compressive stress, f all , as shown within
the shaded area in Figure 6.6.
If they exist, explicit consideration of relatively large load eccentricities
and/or geometric imperfections must be made for long and slender compression
members.
=
6.3.1.1.2 Elastic Buckling with Load Applied Eccentric to the Centroidal
Axis of the Member
Assuming that
• The member has no geometric imperfections (perfectly straight)
• Plane sections remain plane after deformation
• Flexural deflection is considered only (shear deflection is neglected)
The transition curve describes the inelastic buckling of members with KL / r less than that for elastic
buckling but greater than the maximum KL / r value for compressive yielding.
 
 
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