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

358
SHEAR STRESS IN bEAMS
C
F
dx
t
C′
(2)
dF
t
C
τ
zx
xz
(c) Horizontal shear
stress at C in
flange element (1)
τ
z
y
F + dF
x
M
C′
V
(2)
z
C
y
V + dV
M + dM
x
(b) FBD of flange element (2)
dx
FIGURE 9.17 Thin-walled wide-flange beam.
(a) Thin-walled beam section
The shear stress τ xy acting on an x face in the vertical y direction at point B of the
flange element is shown in Figure 9.16d. The top and bottom surfaces of the flange are free
surfaces; thus, τ yx = 0. Since the flange is thin and the shear stresses on the top and bottom
of the flange element are zero, the shear stress τ xy through the thickness of the flange will
be very small and thus can be neglected. Consequently, only the shear stresses (and shear
flows) that act parallel to the free surfaces of the thin-walled section will be significant.
Next, consider point C on the upper flange of the beam segment shown in Figure 9.17a.
An FBD of flange element (2) is shown in Figure 9.17b. With the same approach used for
point B, it can be demonstrated that the shear stress τ xz must act in the direction shown in
Figure 9.17c. Similar analyses for points D and E on the lower flange of the cross section
reveal that the shear stress τ xz acts in the directions shown in Figure 9.18.
Equation (9.16) can be used to determine the shear stress in the flanges (Figure 9.19a)
and the web (Figure 9.19b) of wide-flange shapes, in box beams (Figures 9.20a and 9.20b),
in half-pipes (Figure 9.21), and in other thin-walled shapes, provided that the shear force V
y
C
B
V
t
y
τ xz
Cut
Free surface
Free surface
Free surface
τ
xy
y
Cut Free surface
z
D
FIGURE 9.18 Shear stress
directions at various locations
in the cross section.
E
x
z z N.A.
N.A.
V
V
t
(a) Shear stress in flange
(b) Shear stress in web
FIGURE 9.19 Shear stresses in a wide-flange shape.