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

Similarly, the moment of inertia of the built-up column about the z axis is equal to
twice that of a single channel shape about its strong axis (i.e., the z′ axis):
I = 2(67.3 in. ) = 134.6 in.
z
4 4
The horizontal distance from the y centroidal axis for the entire cross section to
the back of one channel is 4.25 in. The distance from the back of the channel to
its y′ centroidal axis is given in Appendix B as 0.634 in. Therefore, the distance
between the centroidal axis for the entire cross section and the centroidal axis
for a single channel shape is equal to the difference in these two numbers:
4.25 in. − 0.634 in. = 3.616 in. This distance is shown in the figure to the right.
From the parallel-axis theorem, the moment of inertia of the built-up shape
about its y centroidal axis is
I = 2[2.27 in. + (3.616 in.) (4.48 in.) ] = 121.6961 in.
y
4 2 2 4
Since I y < I z , the built-up column will buckle about its y axis.
The Euler buckling load is calculated from Equation (16.5):
10 in.
0.634 in.
z′
y
y′
3.616 in.
z
C10 × 15.3
4.25 in.
8.5 in.
P
cr
2 2 4
π EI π (29,000 ksi)(121.6961 in. )
= = = 151.2 kips Ans.
2
2
L [(40 ft)(12 in./ft)]
pRoBLEmS
p16.1 Determine the slenderness ratio and the Euler buckling
load for round wooden dowels that are 1 m long and have a diameter
of (a) 16 mm and (b) 25 mm. Assume that E = 10 GPa.
p16.2 An aluminum alloy tube with an outside diameter of
3.50 in. and a wall thickness of 0.30 in. is used as a 14 ft long
column. Assume that E = 10,000 ksi and that pinned connections
are used at each end of the column. Determine the slenderness ratio
and the Euler buckling load for the column.
p16.3 A WT205 × 30 structural steel section (see Appendix B
for cross-sectional properties) is used for a 6.5 m column. Assume
pinned connections at each end of the column. Determine
(a) the slenderness ratio.
(b) the Euler buckling load. Use E = 200 GPa for the steel.
(c) the axial stress in the column when the Euler load is applied.
p16.4 Determine the maximum compressive load that an
HSS6 × 4 × 1/4 structural steel column (see Appendix B for crosssectional
properties) can support if it is 24 ft long and a factor of
safety of 1.92 is specified. Use E = 29,000 ksi for the steel.
p16.5 Two C12 × 25 structural steel channels (see Appendix B
for cross-sectional properties) are used for a column that is 35 ft
long. Assume pinned connections at each end of the column, and
use E = 29,000 ksi for the steel. Determine the total compressive
load required to buckle the two members if
(a) they act independently of each other.
(b) they are latticed back-to-back as shown in Figure P16.5.
p16.6 Two L102 × 76 × 9.5 structural steel angles (see Appendix B
for cross-sectional properties) are used as a compression member
that is 4.5 m long. The angles are separated at intervals by spacer
blocks and connected by bolts (as shown in Figure P16.6), which
ensure that the double-angle shape acts as a unified structural member.
Assume pinned connections at each end of the column, and use E =
200 GPa for the steel. Determine the Euler buckling load for the doubleangle
column if the spacer block thickness is
(a) 5 mm
(b) 20 mm.
FIGURE p16.5
Lacing
bars
C12 × 25
6 in.
C12 × 25
677