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

Next, consider the web of the thin-walled cross section shown in Figure 9.22b. In the
web, the shear flow is
V ⎡ btd 1 ⎛ d ⎞ d
q = + + y y t
I 2 2⎝
⎜
2 ⎠
⎟ ⎛
⎝
⎜
2
- ⎞ ⎤
⎠
⎟
⎣
⎢
⎦
⎥
z
2
Vbtd Vt ⎛ d
2
⎞
= + - y
2I
2I
⎝
⎜
4 ⎠
⎟ (c)
z
z
361
SHEAR STRESS ANd SHEAR
FLOw IN THIN-wALLEd
MEMbERS
Using the expression for (q max ) f derived in Equation (b), we can rewrite Equation (c) as the
sum of the shear flows in the flange plus the change in shear flow over the depth of the web:
2
Vt ⎛ d
2
⎞
q = 2( qmax)
f + - y
2I
⎝
⎜
4 ⎠
⎟
The shear flow in the web increases parabolically from a minimum value of (q min ) w =
2(q max ) f at y = d/2 to a maximum value of
Vtd
( qmax) w = 2( qmax)
f +
8I
at y = 0.
Again, it should be noted that the shear flow expression here has been based on the
centerline dimensions of the cross section.
To determine the force in the web, Equation (c) must be integrated. Once more, with
the centerline bounds of y = ±d/2, the force in the web can be expressed as
z
z
2
F
w
d
2
= /2 Vt d
∫qdy
= ⎡
∫ bd y
d/2
2I
⎢ +
z ⎣ 4
-
-
3
Vt ⎡ 2
d ⎤
= bd +
2I
⎢
z ⎣ 6
⎥
⎦
d/2
3 ⎤
2
Vt ⎡ d 1
= bdy y y
2I
⎢ + -
z ⎣ 4 3
⎥
⎦
-d/2
2
⎤
⎥dy
⎦
or
F
w
=
V ⎡ 2 3
⎛ d⎞
td ⎤
⎢2bt⎜
⎟ + ⎥
Iz
⎣ ⎝ 2⎠
12 ⎦
(d)
The moment of inertia for the thin-walled flanged shape can be expressed as
3 2
⎡
3
bt ⎛ d⎞
⎤ td
Iz = Iflanges + Iweb
= 2 ⎢ + bt 12 ⎝
⎜ 2 ⎠
⎟ ⎥ +
⎣
⎦ 12
Since t is small, the first term in the brackets can be neglected, so
I
z
2 3
⎛ d⎞
td
= 2bt
⎝
⎜
2⎠
⎟ +
12
Substituting this expression into Equation (d) gives F w = V, which is as expected. (See
Figure 9.22d.)