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

Shear Flow Formula
The shear flow formula rewritten as
y
5 in.
(typ)
500 lb
q
VQ
= (b)
I z
and the fastener force–spacing relationship
qs ≤
nV f f
(c)
500 lb
x
A B C D
5 in.
1,000 lb
7 ft 7 ft
500 lb
will be employed to determine the shear force V f
produced in the lag screws of the built-up beam.
Appropriate values for the terms appearing in
these equations will now be developed.
Beam internal shear force V: The shear-force
and bending-moment diagrams for the simply
supported beam are shown. The V diagram
reveals that the internal shear force has a constant
magnitude of V = 500 lb throughout the entire
span of the beam.
V
M
−500 lb
−3,500
lb.ft
First moment of area, Q: Q is calculated for the
portion of the cross section connected by the lag screw. Consequently, Q is calculated
for the flange board in this situation:
8 in.
y
Q = (8 in.)(2 in.)(2.5 in.) = 40 in. 3
Fastener spacing interval s: The lag screws are installed at 5 in. intervals along the
span; therefore, s = 5 in.
2.5 in.
z
2 in.
8 in.
Shear flow q: The shear flow that must be transmitted from the stem to the flange
through the fastener can be calculated from the shear flow formula:
2 in.
3
VQ (500 lb)(40 in. )
q = = = 68.8 lb/in.
(d)
4
I 290.667 in.
z
Notice that the result obtained in Equation (d) from the shear flow formula is identical to
the result obtained in Equation (a). While the shear flow formula provides a convenient
format for calculation purposes, the underlying flexural behavior that it addresses may not
be readily evident. The preceding investigation using an FBD of the beam at C may help
to enhance one’s understanding of this behavior.
Fastener shear force V f : The shear force that must be provided by the fastener can be
calculated from the fastener force–spacing relationship. The beam is fabricated with one
lag screw installed in each 5 in. interval; it follows that n f = 1, and we have
qs ≤
nV f f
qs (68.8 lb/in.)(5 in.)
∴ Vf
= = = 344 lb perfastener
Ans.
n 1fastener
f
351