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

Nail
A
(1)
z
y
(2)
Nail C
Nail
B
(3)
M
z
(2)
(1)
(4)
y
(3)
M+
∆M
x
M
A′
I z
y dA′
F H
(3)
Area A′
(4)
(a) Cross section made from
four wooden boards
∆x
(b) Built-up beam subjected to
nonuniform bending
∆ x
(M + ∆M)
y dA′
A′
I z
(c) Free-body diagram of board (3)
FIGURE 9.14 Horizontal equilibrium of a built-up beam.
the forces that act in the longitudinal direction (i.e., the x direction). In this instance, we will
consider board (3) as area A′. A free-body diagram of board (3) is shown in Figure 9.14c.
Using an approach similar to that employed in the derivation presented in Section 9.3,
Equation (9.1) relates the change in internal bending moment ∆M over a length ∆x to the
horizontal force required to satisfy equilibrium for area A′:
F
H
MQ
= ∆
I
The change in internal bending moment, ∆M, can be expressed as ∆M = (dM/dx) ∆x = V ∆x,
thereby allowing Equation (9.1) to be rewritten in terms of the internal shear force V:
z
The term I z appearing in
Equations (9.1), (9.11), and
(9.12) is always the moment of
inertia of the entire cross section
about the z centroidal axis.
F
H
VQ
= D x
(9.11)
I
z
Equation (9.11) relates the internal shear force V in a beam to the horizontal force F H required
to keep a specific portion of the cross section (area A′) in equilibrium. The term Q is
the first moment of area A′ about the neutral axis, and I z is the moment of inertia of the
entire cross section about the neutral axis.
The force F H required to keep board (3) (i.e., area A′) in equilibrium must be supplied
by nail B shown in Figure 9.14a, and it is the presence of individual fasteners (such as
nails) that is unique to the design of built-up flexural members. In addition to using the
flexure formula and the shear stress formula to consider bending stresses and shear
stresses, the designer of a built-up flexural member must ensure that the fasteners which
will connect the pieces together are adequate to transmit the horizontal forces required for
equilibrium.
To facilitate this type of analysis, it is convenient to introduce a quantity known
as shear flow. If both sides of Equation (9.11) are divided by Dx, then the shear flow q
can be defined as
FH
VQ
q
D x
= = (9.12)
I
The shear flow q is the shear force per unit length of beam span required to satisfy horizontal
equilibrium for a specific portion of the cross section. Equation (9.12) is called the
shear flow formula.
z
It is important to understand that
shear flow stems from normal
stresses created by internal
bending moments that vary
along the span of the beam. The
term V appears in Equation (9.12)
as a substitute for dM/dx. Shear
flow acts parallel to the longitudinal
axis of the beam—that is,
in the same direction as the
bending stresses.
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