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

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SHEAR STRESS IN bEAMS
Nail A
Nail B
(a) I-shaped wooden beam
cross section
z
Nail
z
(1)
(1)
(3)
(2)
FIGURE 9.15 Examples of
built-up flexural members.
y
(2)
y
(3)
Nail
(b) U-shaped wooden beam
cross section
z
(1)
Nail
y
(2)
(3)
Nail
(c) Alternative U-shaped
wooden beam cross section
Analysis and Design of Fasteners
Built-up cross sections use individual fasteners such as nails, screws, or bolts to connect
several components into a unified flexural member. One example of a built-up cross section
is shown in Figure 9.14a, and several other examples are shown in Figure 9.15. Although
these examples consist of wooden boards connected by nails, the principles are the same
regardless of the beam material or type of fastener.
A consideration of fasteners usually involves one of the following objectives:
• Given the internal shear force V in the beam and the shear force capacity of a
fastener, what is the proper spacing interval for fasteners along the beam span
(i.e., in the longitudinal x direction)?
• Given the diameter and spacing interval s of the fasteners, what is the shear stress
τ f produced in each fastener for a given shear force V in the beam?
• Given the diameter, spacing interval s, and allowable shear stress of the fasteners,
what is the maximum shear force V that is acceptable for the built-up member?
To address these objectives, an expression can be developed from Equation (9.12) that
relates the resistance of the fastener to the horizontal force F H required to keep an area A′ in
equilibrium. The length term ∆x in Equation (9.12) will be set equal to the fastener spacing
interval s along the x axis of the beam. In terms of the shear flow q, the total horizontal force
that must be transmitted between connected parts over a beam interval s can be expressed as
FH = qs
(a)
The internal horizontal force F H must be transmitted between the boards or plates by the
fasteners. (Note: The effect of friction between the connected parts is neglected.) The shear
force that can be transmitted by a single fastener (e.g., a nail, screw, or bolt) will be denoted
by V f . Since more than one fastener could be used within the spacing interval s, the number
of fasteners in the interval will be denoted by n f . The resistance provided by n f fasteners
must be greater than or equal to the horizontal force F H required to keep the connected part
in equilibrium horizontally:
F
≤ n V
(b)
H f f
Combining Equation (a) with Inequality (b) gives a relationship between the shear
flow q, the fastener spacing interval s, and the shear force that can be transmitted by a single
fastener V f . This relationship will be termed the fastener force–spacing relationship and
is given by the inequality
qs ≤ nV
(9.13)
The average shear stress τ f produced in a fastener can be expressed as
f
f
Vf
τ f = (c)
A
f
where the fastener is assumed to act in single shear and A f is the cross-sectional area of the
fastener. Using this relationship, we can rewrite Equation (9.13) in terms of the shear stress
in the fastener:
qs ≤ n τ A
(9.14)
f f f
This inequality will be termed the fastener stress–spacing relationship.