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

360
SHEAR STRESS IN bEAMS
t
b
2
s
t
b
ds
d
2
y
dy
d
2
N.A.
t
V
d
2
N.A.
t
V
d
2
(a) Calculating shear flow in flanges
(b) Calculating shear flow in web
(qmax)f
(qmax)f
F f
F f
(q max) w
F w = V
(qmax)f
(qmax)f
2(q max) f
2
Vtd
8I
F f
F f
(c) Shear flow distribution in flanges and web
(d) Forces and their directions
in flanges and web
FIGURE 9.22 Wide-flange shape with equal flange and web thicknesses.
produce the same result for (q max ) f . The linear variation of shear flow in the flanges is
shown in Figure 9.22c.
The total force developed in the upper left flange of Figure 9.22a can be determined
by integration of Equation (a). The force on the differential element ds is dF = q ds. The
total force acting on the upper left flange element is, therefore,
F
f
b 2
= /2Vtd
Vb td
∫ qds = ∫ sds
0 2I
= 16I
This same result can also be obtained by calculating the area under the triangular distribution
in Figure 9.22c, since q is a distribution of force per length:
F
f
1 b
= q
2 ( max ) f =
2
z
1 ⎛ Vbtd ⎞
2
b Vbtd
2⎝
⎜ 4I
⎠
⎟ =
2 16I
Again on the basis of symmetry, the force F f in each flange element will be the same. The
flange forces are shown in their proper directions in Figure 9.22d. From the direction of these
forces, it is evident that horizontal force equilibrium of the cross section is maintained.
z
z
z