ON STRESS DISTRIBUTION IN ENGINEERING MATERIALS. 



469 



generally made by engineers. It has been shown (L. N. G. Filon, Phil. 

 Trans. A, vol. 201, pp. 84-86) to be mathematically incorrect; a certain 

 definite amount of change of direction is introduced in a continuous 

 beam as we pass a point of concentrated load, over and above that due 

 to the Euler-Bernouilli curvature, but the deflection thus introduced is 

 not equal to the so-called ' deflection due to shear ' which is sometimes 

 appealed to by engineers. 



Neglecting, however, such changes in the slope of the central line and 

 assuming the nodes fixed in position, consider a bay A. B (fig. 2), of 

 length /, between two nodes. 



M+cLM 



Let P be the thrust in this bay and let S, M be the shear, and bending 

 moment at any point Q of the spar, x units to the right of A, at which 

 the deflection is y. 



Then, E being the Young's modulus and I the moment of inertia of 

 the cross-section of the spar, w the load in lbs. weight per foot run, which 

 will be assumed uniform, we have the following fundamental equations : 



EI^=M 

 dx' 



dx dx 



dx 



whence 



dm 



dx" 



+«'M=w 



(1) 

 (2) 

 (3) 



(4) 



where P/EI = n^ 



The solution of (1) and (4) is 



M= 



w 



:asin»ja;-l-;8cos «a;+ „ 



11 . 



EI 



y-- 



--, sm nx- 



fi 



wx 



■;Ccos nx+~,^yx+h, 



where a, ^, y, S are arbitrary constants : these are determined from the 

 conditions that M = M^ when x= 0, M = Mb when x = /, y = Q when 

 X = and x = /. 



1919. 



M M 



