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

p16.41 A simple pin-connected wood truss is loaded and supported
as shown in Figure P16.41. The members of the truss are
square Douglas fir timbers (finished dimensions = 3.5 in. by 3.5 in.)
with F c = 1,350 psi and E′ min = 580,000 psi.
(a) For the loads shown, determine the axial forces produced in
chord members AF, FG, GH, and EH and in web members
BG and DG.
(b) Use the AWC NDS column design formula to determine the
allowable compressive load P allow for each of these members.
(c) Report the ratio P allow /P actual for each of these members.
F G H
A B C D E
1,200 lb 2,400 lb 3,200 lb
4 spaces @ 6 ft = 24 ft
FIGURE p16.41
8 ft
16.6 Eccentrically Loaded columns
Although a given column will support its maximum load when the load is applied centrically,
it is sometimes necessary to apply an eccentric load to a column. For example, a floor beam
in a building may in turn be supported by an angle bolted or welded to the side of a column
as shown in Figure 16.12. Since the reaction force from the beam acts at some eccentricity e
from the centroid of the column, a bending moment is created in the column in addition to a
compressive axial load. The bending moment applied to the column will increase the stress
in the column and, in turn, decrease its load-carrying capacity. Three methods will be presented
here for analyzing columns that are subjected to an eccentric axial load.
The Secant Formula
The secant formula [Equation (16.20)] was derived on the assumption that the applied load
had an initial eccentricity e. If e is known, then its value can be substituted into the secant
formula to determine the failure load (i.e., the load that causes incipient inelastic action).
As mentioned previously, there is usually a small amount of unavoidable eccentricity that
must be approximated when this formula is used for centric loads. The form of the secant
formula makes it somewhat difficult to solve for the value of P/A that produces a specific
maximum compression stress value; however, a number of equation-solving computer
programs are available that can readily produce this sort of numerical solution.
z
y
x
FIGURE 16.12 Column
subjected to eccentric load P.
c
Eccentricity e
P
Allowable-Stress method
The topic of bending due to an eccentric axial load was discussed in Section 8.7. Figure 8.14
depicted the stress distributions caused by axial loads and by bending moments, and it
illustrated the resulting stress distribution from the combined effects. Equation (8.19) was
used to calculate the normal stress produced by the combination of an axial force and a
bending moment. Buckling was not considered in Section 8.7; however, the approach taken
in Equation (8.19) can be adapted for use in that context.
The allowable-stress method simply requires that the sum of the compressive axial
stress and the compressive bending stress be less than the allowable compression stress
prescribed by the pertinent column formula for centric loading. In this regard, Equation (8.19)
can be restated as
σ
x
P Mc
= + ≤ σ
A I
allow
(16.30)
707