162 



DEFLECTION OF BEAMS 



loaded with a uniform load of w per lineal foot. The bending mo- 

 ment parabola is shown in (b), and the beam is loaded with the 

 bending moment parabola in (c) Fig. 3. To find the equation of the 

 elastic curve, take moments of forces to the left of a point at a dis- 

 tance x from the left support. 



The equation of the bending moment parabola with the origin 

 of co-ordinates at the left support is 3; = y 2 w L x y 2 w x*, the area 

 of a segment of the parabola is A = % w L x 2 1-6 w x z , and the 

 center of gravity measured back from x is 



Taking moments of forces to the left of a point x, and reducing, 

 we have 



24 E I y = w ( x* + 2 L x 3 L 3 x) ( 10) 



The deflection is a maximum when x = l /2 L, and may be found 

 directly by taking moments, or may be found from (10), and is 



(11) 



Cantilever Beam Concentrated Load. The cantilever beam in 

 (a) Fig. 4, has a concentrated load, P, at its extreme end. It will 

 be seen that the cantilever beam may be considered as one-half of a 

 simple beam with a span 2 L, and a load 2 P, at the center. The 

 equation of the elastic curve may be found as in Fig. 2. Load the 

 beam with the bending moment diagram as in (b) Fig. 4, and consider- 

 ing the cantilever as one-half of the simple beam we have, after re- 

 ducing, 



$PL 2 x P. V s (12) 



PL* K-* - 



>x o5 



(t>) 



FIG. 4. 



-=0 



