Art. 81. 



EQUATION OF THE ELASTIC LINE. 



120 



line. Expressing M in terms of x and integrating twice, an equa- 

 tion giving y in terms of x the equation of the eslatic line is 



gotten. The first integration gives -^, the slope of the elastic line. 



dx 



A differential equation must be written for each segment 

 of a beam into which the loads divide it, because the expression 

 for M is different in each segment. Thus in a beam supported 

 at its ends and carrying a single load, the terms in x, of the 

 values of M, for the two segments are different ; the curve of the 

 elastic line is continuous past the load, but two equations are 

 required to express it. When there are a number of loads on a 

 beam, the evaluation of the constants of integration becomes 

 burdensome. 



The constants of integration are evaluated by means of the 



limiting conditions. At the supports y = ; at fixed ends -^ =0; 

 at the juncture of two segments, y and -/- from one equation are 



(J 3u 



equal to y and -j^ from the other equation. 



The deflection at any point due to several loads is the 

 resultant of those due to the loads considered separately. The 

 equations of the elastic line for all the usual cases of simple 

 loading accompany Figs. 94 to 105. The manner of their 

 derivation is given below, for several cases. 



82. Deflection of a Cantilever Beam for a Load at its 

 End. Fig. 94 shows a beam fixed at the left end and carrying 

 a load P at the free end. The maximum moment is evidently 

 at the support, and is equal to PL (concave on lower side). 

 From ' ' the sum of the vertical components equals zero, " JK = P. 

 Taking a section at a distance x from the support, the bending 

 moment is P(Lx), considering the part to the right of the 

 section. The same result is, of course, obtained by considering 

 the left-hand part ; in this case , x = - PL+Px = -P(Lx). 



From equation (39) 



Since ^_ o when x = 0, C must also be zero. If the load 

 were not at the end of the beam, the part to the right of the load 



