Art. 85. 



DEFLECTION OF A BEAM. 



133 



form load (Fig. 100), y, to P, (Fig. 94), y 2 to P 2 , and ?/s to P 3 . 

 The deflection at C due to P 2 is a case similar to that of 1\ at the 

 end; the effect of P 2 at the end is found from the slope of the 

 elastic line at C (82). In a similar way, the effect of P 3 is found. 

 The total deflection at the end is y+y^+y z -\-y z . By adding the 

 ordinates of the four curves, at any section, the ordinate of the 

 elastic line is obtained. 



The deflection at a 

 particular section may 

 be gotten l)y a single in- 

 tegration. If in Fig. 89 

 tangents be drawn to 

 the elastic line, NN, at 

 the two sections, the an- 

 gle between them will 

 be d<f>, because they are 

 perpendicular to AB and CD. This is illustrated in Fig. 91 fot 



. 91. 



a particular case. xd<j> = y' and 



Substituting the 



value of d(f> from equation (&), Art. 81 



JMxdx ,, m 



-ET (40) 



When E and / are constant, they may be placed outside of 



the integral sign, j Mdx is evidently the area of the moment 



diagram, and I Mdx x is the statical moment of this area about 

 an axis passing vertically through the origin. 



The advantage of equation (40) lies in its applicability to 

 beams carrying more than one load, for the statical moment of 

 the area of any part of the moment diagram is easily gotten. It 

 should be noted that equation (40), in the case of Fig. 91, really 

 gives the deflection of the support from the tangent. 



The application of equation (40) to the case shown in 

 Fig. 91 is as follows: From Art. 83 M=y 2 w(x ~\, and 

 I are constant. 



