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

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SHEAR STRESS IN bEAMS
It is useful to visualize shear flow in the same manner that one might visualize fluid
flow in a network of pipes. In Figure 9.22c, the shear flows q in the two top flange elements
are directed from the outermost edges toward the web. At the junction of the web
and the flange, these shear flows turn the corner and flow down through the web. At the
bottom flange, the flows split again and move outward toward the flange tips. Because the
flow is always continuous in any structural section, it serves as a convenient method for
determining the directions of shear stresses. For instance, if the shear force acts downward
on the beam section of Figure 9.22a, then we can recognize immediately that the
shear flow in the web must act downward as well. Also, since the shear flow must be
continuous through the section, we can infer that (a) the shear flows in the upper flange
must move toward the web and (b) the shear flows in the bottom flange must move away
from the web. Using this simple technique to ascertain the directions of shear flows and
shear stresses is easier than visualizing the directions of the forces acting on elements
such as those in Figures 9.16b and 9.17b.
The preceding analysis demonstrates how shear stresses and shear flow in a thinwalled
cross section can be calculated. The results offer a more complete understanding
of how shear stresses are distributed throughout a beam that is subjected to shear forces.
(Recall that, in Section 9.7, shear stresses in a wide-flange cross section were determined
for the web only.) Three important conclusions should be drawn from this analysis:
1. The shear flow q is dependent on the value of Q, and Q will vary throughout the cross section.
For beam cross-sectional elements that are perpendicular to the direction of the shear
force V, q and hence τ will vary linearly in magnitude. Both q and τ will vary parabolically
in cross-sectional elements that are parallel to or inclined toward the direction of V.
2. Shear flow will always act parallel to the free surfaces of the cross-sectional elements.
3. Shear flow is always continuous in any cross-sectional shape subjected to a shear force.
Visualization of this flow pattern can be used to establish the direction of both q and τ
in a shape. The flow is such that the shear flows in the various cross-sectional elements
contribute to V while satisfying both horizontal and vertical equilibrium.
closed Thin-Walled Sections
Flanged shapes such as wide-flange shapes (Figure 9.19) and tee shapes are classified as
open sections, whereas box shapes (Figure 9.20) and circular pipe shapes are classified as
closed sections. The distinction between open and closed sections is that closed shapes
have a continuous periphery in which the shear flow is uninterrupted while open shapes do
not. Now, consider beam cross sections that satisfy two conditions: (a) The cross section
has at least one longitudinal plane of symmetry, and (b) the beam loads act in this plane of
symmetry. For open sections, such as flanged shapes, satisfying these conditions, the shear
flow and shear stress clearly must be zero at the tips of the flanges. For closed sections,
such as box or pipe shapes, the locations at which the shear flow and the shear stress vanish
are not so readily apparent.
A thin-walled box section subjected to a shear force V is shown in Figure 9.23a. The
section is split vertically along its longitudinal plane of symmetry in Figure 9.23b. The shear
flow in vertical walls of the box must flow parallel to the internal shear force V; thus, the shear
flow in the top and bottom walls of the box must act in the directions shown. On the plane of
symmetry, the shear stress at points B and B′ must be equal; however, the shear flows act in
opposite directions. Similarly, the shear stress at points C and C′ must be equal, but they, too,
act in opposing directions. Consequently, the only possible value of shear stress that can satisfy
these constraints is τ = 0. Since q = τt, the shear flow must also be zero at these points. From
this analysis, we can conclude that the shear flow and the shear stress for a closed thin-walled
beam section must be zero on a longitudinal plane of symmetry.