494 Mr. L. N. G. Filon. Approximate Solution for [June 12, 



The various cases, which are separately dealt with, are as follows : — 

 (1.) When the external stresses upon the top and bottom faces 



y = ± h are purely normal and are symmetrical about the mid-section, 



x = 0. 



In the first place, making first a large, but not infinite, the various 

 terms of the sine and cosine series (1) may be expanded in terms of 

 y/a. Approximate expressions are then obtained for the displacements 

 and stresses in a very long beam, at a distance from the regions near 

 the points where the loads are applied. The validity of such expan- 

 sions has been discussed in a paper by the author "On the Elastic 

 Equilibrium of Circular Cylinders under Certain Practical Systems of 

 Load."* 



The results in the present case show that to this approximation the 

 stresses P, S are given in terms of the total bending moment and total 

 shear by the formulae given by de Saint- Venant for a beam terminally 

 loaded, but otherwise free. 



In the case of the displacements, however, it is found that, for a 

 doubly supported beam under a central isolated load, the vertical 

 deflection of the central axis contains a term - kx where x is positive, 

 and + kx where x is negative. Such a term was put in by de Saint- 

 Venant for a built-in beam. Professor Love, starting from different 

 conditions for a built-in end, arrived at the conclusion that the term 

 should be zero. 



As a matter of fact the term is found to exist, but the coefficient k 

 is only 0*74 of de Saint- Venant's value, showing that in passing an 

 isolated load the slope of the elastic line varies fairly abruptly, but 

 only to about three-fourths of the extent anticipated by de Saint- 

 Venant. 



The variations in the central deflection, as the supports are brought 

 closer and closer together, are also investigated. It is found that the 

 excess of the actual over the Euler-Bernoulli deflection (which excess 

 is sometimes referred to by engineers as the " deflection due to shear ") 

 decreases eventually as the span decreases and, for exceedingly small 

 spans, may even become negative. 



The series in powers of r, deduced from the other expressions when a is 

 made infinite, are used to show the variations of stress in the mid-section 

 and the results are compared with those obtained by Sir G. Stokesf and 

 Boussinesqj from an empirical formula. It is shown that, though the 

 empirical formula gives an approximation to the stress in some places, 

 it is by no means to be relied upon. 



The case of a beam under two opposite isolated loads, which leads at 

 once to the more interesting problem, of a beam carrying an isolate 



* * Phil. Trans.,' A, vol. 198, pp. 147—233. 

 f ' Phil. Mag.,' ser. 5, vol. 32, pp. 500-503. 

 X Loc. cit. 



