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

τ xz
Cut
Free
surface
y
Free surface
Cut
τ xz
τ xy
y
Free surfaces
Cut Cut
τ xy
359
SHEAR STRESS ANd SHEAR
FLOw IN THIN-wALLEd
MEMbERS
z N.A.
z N.A.
V
V
t
t
(a) Shear stress in upper wall (b) Shear stress in side walls
FIGURE 9.20 Shear stresses in a box-shaped cross section.
acts along an axis of symmetry of the cross section. For each shape, the cutting plane of the
free-body diagram must be perpendicular to the free surface of the member. The shear
stress acting parallel to the free surface can be calculated from Equation (9.16). (As discussed
previously, the shear stress acting perpendicular to the free surface is negligible
because of the thinness of the element and the proximity of the adjacent free surface.)
z N.A.
y
Free surfaces
Cut
τ t
Shear Flow in Thin-Walled Sections
The shear flow along the top flange of the wide-flange shape shown in Figure 9.22a will be
studied here. The product of the shear stress at any point in a thin-walled shape and the
thickness t at that point is equal to the shear flow q:
V
FIGURE 9.21 Shear stresses in
a half-pipe cross section.
VQ VQ
τ t = ⎛ ⎞
t q
⎝ ⎜ It⎠
⎟ = = (9.17)
I
z
For a given cross section, the shear force V and the moment of inertia I z in Equation (9.17)
are constant. Thus, the shear flow at any location in the thin-walled shape depends only on
the first moment of area, Q. Consider the shear flow acting on the shaded area, which is
located a horizontal distance s from the tip of the flange. The shear flow acting at s can be
calculated as
z
VQ V ⎛
q
st d ⎞
= = ⎜ ⎟ = Vtd s
(a)
I I ⎝ 2⎠
2I
z z z
Note that Q is the first moment of the shaded area about the neutral axis. From inspection
of Equation (a), the distribution of shear flow along the top flange is a linear function of s.
The maximum shear flow in the flange occurs at s = b/2:
( q )
max f =
Vtd ⎛ b⎞
Vbtd
2I
⎝
⎜
2⎠
⎟ = (b)
4I
z
z
Note that s = b/2 is the centerline of the section. Since the cross section is assumed to be
thin walled, centerline dimensions for the flange and web can be used in the calculation.
This approximation procedure simplifies the calculations and is satisfactory for thin-walled
cross sections. Owing to symmetry, similar analyses of the other three flange elements