ME. L. N. G. FILON ON AN APPROXIMATE SOLUTION FOR BENDING A 
I‘28 
sketched, for this case. It is easy to see that the centre corresponds to a maximum 
rZ A ' 
irl 4 : s :. 
2\Y S 3 ' 
fu’ the shear, for the next higher terms in the expansion of S are — — ~ 
We have therefore a numerical maximum if S4. < iSg, and a rough numerical 
calculation enables us to verify that this is tlie case. 
The shear is therefore greatest at the centre, l)ut decreases extremely slovdy and 
remains constant over nearly half the section. 
Another case of interest presents itself v'hen the shear at the centre is exactly 
equal to its mean value over the section. 
Tins occurs when Sq = -7854 = 7r/4. 
If we write So = (^A>) A — ,1 (//6 )Wt,. = 2-818 //h - 4-138 we find that this 
roughly corresponds to l;h = - 32 . 
Measured on the diagram for S^ on fig. vii. the value of Ijh corresponding to 
Sq = 7r/4 would lie afiout - 3 . 5 . Tliis latter value is probably the more correct, as for 
values of//& > -3 the abo^-e ap})roximation for Sq is hardly sufficient. 
In this case it is found that 8^/2 — S^ = -4 roughly. The shear is therefore 
a minimum at the centre. It increases as we proceed outwards, but not very rapidly, 
and decreases down to zero at tlie edges. The curve is shown as {h) on fig. vi. The 
total area of the curve reckoned from a horizontal tangent at the middle point as 
liase is zero, fie., there is as much above as lielow. 
Finally, curve [d) on fig. vi. shows the disti'iliution of shear when the arm of the 
couple is indefinitely increased. This is the parabola 
It is striking how very early tins limiting distribution is reached. Fig. vii. already 
shows that the coefficients of the series reacli their limiting values with great 
rapidity. For an arm of tlie couple equal to twice the lieight of the lieam, the 
parabolic distribution of shear, corresponding to a long cantilever, will, at the mid¬ 
section, be practically undisturl)ed. 
§ 32 . Practicrd Imporfanec offline Problem. 
The problem which has been investigated in tin’s part of the paper is one of 
considerable importance in practice. Tlie only way in which we can apply a shearing- 
force to materials is liy means of two ojiposite asymmeti'ically situated pressures, 
such as we have dealt Avith in this case. The case of material cut through by 
scissors, whicli is frequently rpioted as an example of the application of shearing- 
stress, really corresponds to a stress-distrilnition of this kind. Similarly, a rivet 
which fastens together tAvo plates is sulijected to stress-systems of this type 
AvheneA’er the comjiound plate undergoes strain in its OAvn plane. In nearly eA’ery 
