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

9.6 Shear Stresses in Beams of circular cross Section
z
r
y
V
πr
2
A′=
2
4r
3π
In beams with circular cross sections, transverse shear stress acts parallel to the y axis only
over some, but not all, of the entire depth of the shape. Consequently, the shear stress formula
is not applicable in general for a circular cross section. However, Equation (9.2) can
be used to determine the shear stress acting at the neutral axis.
A solid circular cross section of radius r is shown in Figure 9.11. To use the shear
stress formula, the value of Q for the highlighted semicircular area must be determined.
T,′Æhe area of the semicircle is A′ = π r 2 /2. The distance from the neutral axis to the centroid
of the semicircle is given by – y ′= 4r/3π . Thus,
or, in terms of the diameter d = 2r,
2
4r
πr
2
3
Q = y′ A′= = r
(9.7)
3π
2 3
FIGURE 9.11 Solid circular
cross section.
z
V
y
FIGURE 9.12 Hollow circular
cross section.
R
r
1
3
Q = d
(9.8)
12
The width of the circular cross section at the neutral axis is t = 2r, and the moment of inertia
about the z axis is I z = π r4 /4 = π d4 /64. Substituting these relationships into the shear
stress formula gives the following expression for τ max at the neutral axis of a solid circular
cross section:
τ
= VQ V
r
It
= 2
3
1 4V
4V
2
πr
/4
× 3
× 2r
= 3πr
= 3A
(9.9)
max 4
z
A hollow circular cross section having outside radius R and inside radius r is shown
in Figure 9.12. The results from Equations (9.7) and (9.8) can be used to determine Q for
the highlighted area above the neutral axis:
2
Q R r D d
3 [ 3 3 ] 1
12 [ 3 3
= - = - ]
(9.10)
The width t of the hollow circular cross section at the neutral axis is two times the wall
thickness, or t = 2(R - r) = D - d. The moment of inertia of the hollow circular shape about
the z axis is
π
I R r D d
4 [ 4 4
z
] π
64 [ 4 4
= - = - ]
9.7 Shear Stresses in Webs of Flanged Beams
The elementary theory used to derive the shear stress formula is suitable for determining
only the shear stress developed in the web of a flanged beam (if it is assumed that the beam
is bent about its strong axis). A wide-flange beam shape is shown in Figure 9.13. To determine
the shear stress at a point a located in the web of the cross section, the calculation for
Q consists of finding the first moment of the two highlighted areas (1) and (2) about the
neutral axis z (Figure 9.13b). A substantial portion of the total area of a flanged shape is
concentrated in the flanges, so the first moment of area (1) about the z axis makes up a large
percentage of Q. While Q increases as the value of y decreases, the change is not as
pronounced in a flanged shape as it would be for a rectangular cross section. Consequently,
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