150 APPLIED MECHANICS 
An interesting case of great practical importance is s that of a flanged 
beam, which will now be considered, and for the sake 
of simplicity a numerical example will be taken. The 
I section shown in Fig. 203 has a total depth of 12 
inches, the flanges are 8 inches wide, and the web and 
flanges are all 2 inches thick. The formula which gives 
the shear stress at a distance h from the neutral axis 
Way FIG. 208. 
bl 
Consider the value of g at the neutral axis, where the shear stress is 
greatest. Here a= 24 square inches, y, = 4 inches, and ) = 2inches, therefore 
_ 48W 
I 
At the ee of the web and flanges, a= 16 square nea Yy=5 
40W 
I 
In the flanges at places indefinitely near to the junctions with the 
web, a=16 square inches, y,=5 inches, and b=8 inches, therefore g in 
10W 
ne 
For the dimensions given I= 896 in inch units, and if W=14 tons, 
the three values of g considered above are 1680, 1400, and 350 Ibs. per 
square inch. The diagram to the right 
in Fig. 203 shows the distribution of 
the shear stress. It will be seen that 
not only does the web take a large 
proportion of the whole of the shear rtd 
stress, but that the shear stress is 
nearly uniform over the section of the 
web. It may be noted that the 
curves in Fig. 203 are parabolas. 
In most practical cases the maxi- 
mum shear stress on a section of a 
beam is at its neutral axis, but this is not always the case. For example, 
consider a section which is a square (Fig. 204) with one diagonal vertical 
(in the plane of bending). Let d equal the length of a diagonal of the 
sae Using the same notation as before, B is easy to show that 
a 2M (2d - 8h?4d2), Differentiating, pe "W (ad -16h). Hence q is 
a maximum when 2d — 16h=0, or h=d/8. 
Putting h=d/8, the maximum value of ¢ is — tac 
has just been proved to be g= 
the flanges at these places is —— 
* te 
' 
' 
Fig. 204. 
4d?” 
Putting h=0, the value of ¢ at the neutral axis is ae ; 
The mean value of gis W + (5) = = , the same as the value at the 
neutral axis. 
The variation of the stress is shown plotted to the right in Fig. 204. 
The curved lines are portions of parabolas whose axes are horizontal, and 
at distances d/8 from the neutral axis of the section. 
