245 



the unity of area in fig. lb means in the next integration 



£1 



un^ts. 



The second pole distance //, therefore represents — 



units, if we suppose that 1 cm. of this distance represents ?h, cm' 

 (in the drawing 10 cm') of the area in (ig. 16. 



From all this it follows finally that 1 cm. I in fig. Ic represents 



m, m, m, n* H^H , 



em. 



EI 



Now the elastic curves y, and ?/„ must been drawn on the same 

 scale ; hence : 



wi, m, m, n* H, / 7, _ 



EI "" '" 



1 EI 



H^ = = 12,8 cm. 



25 TO, TO, TO, n* H^ 



The elastic curve y, once found, the drawing process is to be 

 repeated so many times, that the last approximations may be neglected. 

 By adding the different curves .v»,?/,, ,'/,••• • we obtain the elastic 

 curve //. The final result can be controlled as follows. We load the 

 beam at the one side by the well-known external forces, at the 

 other side by the continuous load ky, which follows from the elastic 

 curve y. Then we construct the elastic curve y. If the result?/ were 

 exact, the curves y and y must be identical. Fig. 1/, 9, A shows, that 

 a difference between the curves y and y cannot be observed. 



7. Considering fig. 1, it appears that the ordinates of the curves 

 y, and y, are proportional. If the factor of proportionality is called 

 — fi, so that .y, = — fJ?/i, it is easily seen that the ordinates of the 

 curve ?/, can be written as — (xy, and so on. 



The ordinates .v,,.v, , . . . y,, at any point can therefore been looked 

 upon as terms of a geometrical series and the curve y can be 

 obtained by adding y, to the sum of all the following approximations. 



Not only when the factor of proportionalitj' ft is <^1, but also 

 when fi^l, it may occur that the described drawing process is 

 useful to find the elastic curve. 



Supposing that the load — kyn gives rise to the deflexion — yyn 

 there can be found a factor r, such that the function ryn 



