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

332
SHEAR STRESS IN bEAMS
The shear stress τ as a function of the vertical coordinate y can now be determined from the
shear formula:
τ =
2
2
VQ V 1 ⎛ h
2
⎞ 6V
⎛ h
2
⎞
= × -
⎛ ⎞ ⎝
⎜ y
⎠
⎟ b = -
It
bh ⎝
⎜ y
⎠
⎟ (9.5)
3
z 1
3
2 4
4
bh b
⎝ 12 ⎠
The accuracy of Equation (9.6)
depends on the depth-to-width
ratio of the cross section. For
beams in which the depth is
much greater than the width,
Equation (9.6) can be considered
exact. As the cross section
approaches a square shape, the
true maximum horizontal shear
stress is somewhat greater than
that given by Equation (9.6).
Equation (9.5) is a second-order equation, which indicates that τ is distributed parabolically
with respect to y (Figure 9.10c). At y = ± h/2, τ = 0. The shear stress vanishes at the
extreme fibers of the cross section, since A′ = 0; consequently, Q = 0 at these locations.
There is no shear stress on a free surface of the beam. The maximum horizontal shear stress
occurs at y = 0, which is the neutral-axis location. At the neutral axis, the maximum horizontal
shear stress in a rectangular cross section is given by
VQ V 1
τ = = × ⎛
⎛ ⎞ ⎝ ⎜ h⎞
bh 3V
max
⎠
⎟ = =
It z 1
3
2 2 2 2bh
bh b
⎝ 12 ⎠
3
2
V
A
(9.6)
It is important to emphasize that Equation (9.6) gives the maximum horizontal shear stress
only for rectangular cross sections. Note that the maximum horizontal shear stress at the
neutral axis is 50 percent greater than the overall average shear stress given by τ = V/A.
To summarize, the shear stress intensity associated with an internal shear force V in
a rectangular cross section is distributed parabolically in the direction perpendicular to the
neutral axis (i.e., in the y direction) and uniformly in the direction parallel to the neutral
axis (i.e., in the z direction) (Figure 9.10d). The shear stress vanishes at the upper and
lower edges of the rectangular cross section and peaks at the neutral-axis location. It is
important to remember that shear stress acts on both transverse and longitudinal planes
(Figure 9.10e).
The expression “maximum shear stress” in the context of beam shear stresses is problematic.
In Chapter 12, the discussion of stress transformations will show that the state of
stress existing at any point can be expressed by many different combinations of normal and
shear stress, depending on the orientation of the plane surface upon which the stresses act.
(This notion was introduced previously in Section 1.5, pertaining to axial members, and
in Section 6.4, regarding torsion members.) Consequently, the expression “maximum
shear stress,” when applied to beams, could be interpreted to mean either
(a) the maximum value of τ = VQ/It for any coordinate y in the cross section, or
(b) the maximum shear stress at a particular point in the cross section when all possible
plane surfaces that pass through the point are considered.
In this chapter, to preclude ambiguity, the expression “maximum horizontal shear
stress” will be used in a context in which the maximum value of τ = VQ/It for any coordinate
y in the cross section is to be determined. Since shear stresses on perpendicular
planes must be equal in magnitude, it would be equally proper to use the expression
“maximum transverse shear stress” for this purpose. In Chapter 12, the notion of stress
transformations will be used to determine the maximum shear stress at a particular
point. In Chapter 15, maximum normal and shear stresses at specific points in beams
will be discussed in more detail.