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

Identifying the proper Area for Q
In analyzing the shear flow q for a particular application, the most confusing decision often
concerns which portion of the cross section to include in the calculation of Q, the first moment
of area A′. The key to identifying the proper area A′ is to determine which portion of
the cross section is being held in place by the fastener.
Several built-up wooden beam cross sections are shown in Figure 9.14a and Figure 9.15.
In each case, nails are used to connect the boards together into a unified flexural member. A
vertical internal shear force V is assumed to act in the beam for each cross section.
For the tee shape shown in Figure 9.14a, board (1) is held in place by nail A. To analyze
nail A, the designer must determine the shear flow q transmitted between board (1) and
the remainder of the cross section. The proper Q for this purpose is the first moment of
board (1)’s area about the z centroidal axis. Similarly, the shear flow associated with nail B
requires Q for board (3) about the neutral axis. Nail C must transmit the shear flow arising
from boards (1), (2), and (3) to the stem of the tee shape. Consequently, the proper Q for
nail C involves boards (1), (2), and (3).
Figure 9.15a shows an I-shaped cross section that is fabricated by nailing flange boards
(1) and (3) to web board (2). Nail A connects board (1) to the remainder of the cross section;
therefore, the shear flow q associated with nail A is based on the first moment of area, Q, for
board (1) about the z axis. Nail B connects board (3) to the remainder of the cross section.
Since board (3) is smaller than board (1) and more distant from the z axis, a different value of
Q will be calculated, resulting in a different value of q for board (3). Consequently, it is likely
that the spacing interval s for nail B will be different from s for nail A. In both instances, I z is
the moment of inertia of the entire cross section about the z centroidal axis.
Figures 9.15b and 9.15c show alternative configurations for U-shaped cross sections
in which board (2) is connected to the remainder of the cross section by two nails. The part
held in place by the nails is board (2) in both configurations. Both alternatives have the
same dimensions, the same cross-sectional area, and the same moment of inertia. However,
the value of Q calculated for board (2) in Figure 9.15b will be smaller than that of Q for
board (2) in Figure 9.15c. Consequently, the shear flow for the first configuration will be
smaller than q for the alternative configuration.
349
SHEAR FLOw IN
buILT-uP MEMbERS
ExAmpLE 9.5
A simply supported beam with an overhang supports a concentrated load of 500 lb at D.
The beam is fabricated from two 2 in. by 8 in. wooden boards that are fastened together
with lag screws spaced at 5 in. intervals along the length of the beam. The centroid location
of the fabricated cross section is shown in the sketch, and the moment of inertia of
the cross section about the z centroidal axis is I z = 290.667 in. 4 . Determine the shear force
acting in the lag screws.
8 in.
y
y
5 in.
(typ)
500 lb
3.50 in.
z
2 in.
A B C D
x
6.5 in.
8 in.
5 in.
7 ft 7 ft
2 in.