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

ExAmpLE 16.5
2 in.
z
Spacer block
y
2 in.
0.375 in.
Double angles
1
L3 × 2 ×
4
Long legs back-to-back
A compression chord of a small truss consists of two L3 × 2 × 1/4
steel angles arranged with long legs back-to-back as shown. The
angles are separated at intervals by spacer blocks that are 0.375 in.
thick. Determine the allowable axial load P allow that may be supported
by the compression chord if the effective length is
(a) KL = 8 ft.
(b) KL = 12 ft.
Use the AISC equations, and assume that E = 29,000 ksi and σ Y = 36 ksi.
Plan the Solution
After computing the section properties for the built-up shape, we will
use the AISC ASD formulas [i.e, Equations (16.22) through (16.25)]
to determine the allowable axial loads.
SolutioN
Section Properties
The following section properties can be obtained from Appendix B for the L3 × 2 × 1/4
structural steel shape:
A = 1.19 in. I = 1.09 in. r = 0.953 in. I = 0.390 in.
2 4 4
z z y
The subscripts for these properties have been adapted to correspond to the axes shown on
the cross section. In addition, the distance from the back of the 3 in. leg to the centroid of
the angle shape is given in Appendix B as x = 0.487 in. For the coordinate system defined
here, this distance is measured in the z direction; therefore, we will denote the distance
from the back of the 3 in. leg to the centroid of the angle shape as z = 0.487 in.
The double-angle shape is fabricated from two angles oriented back-to-back with a
distance of 0.375 in. between them. The area of the double-angle shape is the sum of the
areas of two angles; that is, A = 2(1.19 in. 2 ) = 2.38 in. 2 . Additional section properties for
this built-up shape must be determined.
Properties about the z axis for the double-angle shape: The z centroidal axis for the
double-angle shape coincides with the centroidal axis of a single-angle shape. Therefore,
the moment of inertia about the z centroidal axis for the double-angle shape is simply two
times the single angle moment of inertia: I z = 2(1.09 in. 4 ) = 2.18 in. 4 . The radius of gyration
about the z centroidal axis is the same as that for the single angle; therefore, r z =
0.953 in. for the double-angle shape.
Properties about the y axis for the double-angle shape: The y centroidal axis for the doubleangle
shape can be located by symmetry. Since the y centroids of the two individual angles
do not coincide with the y centroidal axis for the double-angle shape, the moment of inertia
about the vertical centroidal axis must be calculated with the parallel-axis theorem:
I
y
⎡
2
4
⎛ 0.375 in. ⎞ ⎤
= 2⎢0.390 in. + ⎜ + 0.487 in. ⎟ (1.19 in. ) ⎥ = 1.8628 in.
⎣
⎝ 2
⎠ ⎦
2 4
The radius of gyration about the y centroidal axis is computed from the double-angle
moment of inertia I y and area A:
r
y
I
4
y 1.8628 in.
= = = 0.885 in.
2
A 2.38 in.
700