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

of Q. However, what is the negative sign on Q for this FBD telling us? Consider the direction
that the shear flows on the two faces exposed by the vertical cut. On the right-hand
portion, shear stresses must flow in the positive x direction on the exposed z face to satisfy
equilibrium. On the left-hand portion, shear stresses must flow in the negative x direction
for equilibrium. Thus, the magnitudes of Q must be the same, but the directions that shear
stresses flow on the two exposed z faces must be opposite to satisfy equilibrium.
For the second FBD, plainly only one vertical cut is made through the cross section
and, consequently, t = 6 mm. The shear stress magnitude is calculated as
VQ
τ = =
It
3
(65,000 N)( ⎢-8,250 mm ⎢)
4
(2,532,789 mm )(6 mm)
= 35.29 MPa
Approach 3
For the third approach, we will take advantage of the cross section’s
symmetry with respect to the y axis. We will make one vertical cut
at point a and a second vertical cut at point a′, which is located at
the mirror image of point a on the opposite side of the plane of symmetry.
We will consider the portion of the cross section that begins
at point a and extends rightward to point a′. From symmetry, we
know that the shear stress at points a and a′ must be equal.
For this FBD,
Q = 2 × (27.5 mm)(50 mm × 6mm)
= 16,500 mm 3
50 mm 50 mm
27.5 mm b a a'
z
c
d
y
V
6 mm
We are making two cuts through the section; therefore, t = 2 × (6 mm) = 12 mm. The
shear stress at a is thus
3
VQ (65,000 N)(16,500 mm )
τ = = = 35.29 MPa
4
It (2,532,789 mm )(12 mm)
Conclusions
What have we shown in this example? Clearly, there are several ways to calculate shear
stress in a cross section.
In Approach 1, we made two vertical cuts through the cross section. The first vertical
cut was at the point of interest (i.e., point a), and the second vertical cut was on the plane
of symmetry. To use the first approach, we need to understand that shear stresses are zero
on vertical planes of symmetry (i.e., planes of symmetry that are parallel to the direction of
the shear force V). We must recognize that the term t should consider only thicknesses of
surfaces upon which shear stresses act. If the shear stress on a surface is zero, then its
thickness is not included in the value of t. All in all, this approach is not recommended for
novices because of its potential for misinterpretation.
In Approach 2, we calculated Q from portions of the cross section both above and
below the neutral axis. This approach gives a negative value for Q, which can be confusing
at first glance. However, we need only remember to use the absolute value of Q in our
stress calculations to use the second approach successfully.
With Approach 3, we take advantage of symmetry in choosing the portion of the cross
section needed for our calculation of Q. The process of calculating Q is simplified, since the
FBD is entirely on one side of the neutral axis and the sign of Q will be positive, thus avoiding
the confusion that can occur in Approach 2. The thicknesses of both cut surfaces are included
in the value of t, thereby avoiding the misunderstanding that could occur in Approach 1.
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