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

ExAmpLE 9.6
An alternative cross section is proposed for the simply supported beam of Example 9.5.
In the alternative design, the beam is fabricated from two 1 in. by 10 in. wooden boards
nailed to a 2 in. by 6 in. flange board. 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 . If the allowable shear resistance of each nail is
80 lb, determine the maximum spacing interval s that is acceptable for the built-up beam.
6 in.
Nails
y
y
s
500 lb
A B C
x
3.50 in.
6.5 in.
z
2 in.
10 in.
7 ft 7 ft
1 in. 1 in.
2.5 in.
6 in.
y
2 in.
z
1 in. 1 in.
10 in.
Plan the Solution
The shear flow formula [Equation (9.12)] and the fastener force–spacing relationship
[Equation (9.13)] will be required to determine the maximum spacing
interval s. Since the 2 in. by 6 in. flange board is held in place by the nails, the
first moment of area, Q, as well as the shear flow q, will be based on this region
of the cross section.
SolutioN
Beam internal Shear Force V
The shear-force and bending-moment diagrams for the simply supported
beam are shown in Example 9.5. The shear force has a constant magnitude of
V = 500 lb throughout the entire beam span.
First moment of area, Q: Q is calculated for the 2 in. by 6 in. flange board, which is the
portion of the cross section held in place by the nails:
Q = (6 in.)(2 in.)(2.5 in.) = 30 in. 3
Shear flow q: The shear flow that must be transmitted through the pair of nails is
3
VQ (500 lb)(30 in. )
q = = = 51.6 lb/in.
4
I 290.667 in.
z
Maximum nail-spacing interval s: The maximum spacing interval for the nails can be
calculated from the fastener force–spacing relationship [Equation (9.13)]. The beam is
fabricated with two nails installed in each interval; consequently, n f = 2, and we have
qs ≤
nV f f
nV f f (2 nails)(80 lb/nail)
∴s
≤ = = 3.10 in.
Ans.
q 51.6 lb/in.
Thus, pairs of nails must be installed at intervals less than or equal to 3.10 in. In practice,
nails would be driven at 3 in. intervals.
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