Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

Lateral
bracing
x
C
B
P
17.5 ft
17.5 ft
z
Strong
axis
y
Weak
axis
W8 × 24
Plan the Solution
If the W8 × 24 column were supported only at its ends, then
buckling about the weak axis of the cross section would be
anticipated. However, additional lateral support is provided to
this column, so the effective length with respect to buckling of
the weak axis is reduced. For this reason, both the effective
length and the radius of gyration with respect to both the strong
and weak axes of the column must be considered. The critical
buckling load will be dictated by the larger of the two effectiveslenderness
ratios.
SolutioN
The following section properties can be obtained from Appendix B
for the W8 × 24 structural steel shape:
z
y
A
x
P
C
I
r
z
z
= 82.7 in. I = 18.3 in.
4 4
y
= 3.42 in. r = 1.61 in.
The subscripts for these properties have been modified to correspond
to the axes shown on the cross section.
Buckling About the Strong Axis
The column could buckle about its strong axis, resulting in the
buckled shape shown in which the column deflects in the x–y
plane. For this manner of failure, the effective length of the
column is KL = 35 ft. The critical buckling load is therefore
y
P
cr
2
2 4
π EIz
π (29,000 ksi)(82.7 in. )
= = = 134.2 kips
2
2
( KL)
[(35 ft)(12 in./ft)]
z
Lateral
bracing
y
B
35 ft
Although we are not required to determine P cr , it is instructive to
calculate the effective-slenderness ratio for buckling about the
strong axis:
(35 ft)(12in./ft)
( KL/ r)
z = = 122.8
3.42 in.
z
A
Buckling about the strong axis.
Buckling About the Weak Axis
Alternatively, the column could buckle about its weak axis. In this case, the column
deflection would occur in the x–z plane as shown in the accompanying figure. For this
manner of failure, the effective length of the column is KL = 17.5 ft. The critical buckling
load about the weak axis is therefore
P
cr
2
π EI 2 4
y π (29,000 ksi)(18.3 in. )
= = = 118.8 kips
2
2
( KL)
[(17.5 ft)(12 in./ft)]
y
The effective-slenderness ratio for buckling about the weak axis is
(17.5 ft)(12 in./ft)
( KL/ r)
y = = 130.4
1.61 in.
The critical load for the column is the smaller of the two load values:
686
P
cr =
118.8 kips