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

326
SHEAR STRESS IN bEAMS
a
d
A′
M
I z
y dA′
b
y
e
F H
A′
(M + ∆M)
I z
y dA′
x
FIGURE 9.7 Free-body diagram of area A′ (profile view).
∆x
axis, the internal bending moments produce compressive normal stresses, which act on the
free-body diagram in the directions shown.
If a beam is in equilibrium, then any portion of the beam that we choose to consider
must also be in equilibrium. We will consider a portion of the free-body diagram shown in
Figure 9.6, starting at section b–e (y = y 1 ) in Figure 9.5b and extending away from the neutral
axis (upward in this case) to the outermost boundary of the cross section (y = y 2 ) in
Figure 9.5b. This is the portion of the cross section designated as A′ in Figure 9.5b. A freebody
diagram of area A′ is shown in Figure 9.7.
The resultant force on sections a–b and d–e can be found by integrating the normal
stresses acting on area A′, which includes that portion of the cross-sectional area starting at
y = y 1 and extending vertically to the top of the cross section at y = y 2 . (See Figure 9.5b.)
No force exists on section a–d; however, we shall assume that an internal horizontal force
F H could be present on section b–e. The equilibrium equation for the sum of forces acting
on area A′ in the x direction can be written as
Σ M
Fx
= ydA
A I
′ -
∫ ∫ A
′ ′
z
( M +DM)
ydA′ + FH
= 0
I
z
(a)
where the signs of each term are determined by inspection of Figure 9.7. The integrals in
Equation (a) can be expanded to give
Σ M M
Fx
= ∫ ydA ydA
A I
′ - ∫A
I
′ -
′
′ A′
z
z
Canceling the leftmost two integrals and solving for F H gives
∫
DM
ydA′ + FH
= 0
I
z
(b)
F
H
=
∫ A
DM
ydA′
(c)
′ I
z
With respect to area A′, both DM and I z are constant; therefore, Equation (c) can be simplified to
The moment-of-inertia term
appearing in Equation (9.1)
stems from the flexure formula,
which was used to determine the
bending stresses acting over the
entire depth of the beam and
over area A′ in particular. For
that reason, I z is the moment of
inertia of the entire cross section
about the z axis.
F
H
M
= D ydA
I
∫ ′
A ′
z
The integral in Equation (d) is the first moment of area A′ about the neutral axis of the cross
section. This quantity will be designated Q. More details concerning the calculation of Q
will be presented in Section 9.4. By replacing the integral term with the designation Q,
Equation (d) can be rewritten as
F
H
(d)
= D MQ
I
(9.1)
z