REACTIONS OF DRAW BRIDGES i? 1 



Fig. 9, assuming that the segments are loads acting through their 

 centers of gravity. 



The pole distance H may be taken as any convenient length, and 

 the pole O may be taken at any point [in (c) the pole has been selected 

 to bring the closing line horizontal for convenience only]. 

 Then in (c) 



R l a = P l b P 2 d 

 and 



p b.p d 



*!= (19) 



a 



R 2 c = P 1 m + P 2 n 

 and 



and 





Proof. The ordinates to the equilibrium polygon in (c) are pro- 

 portional to the ordinates to the true elastic curve of the beam in (b) 

 when it is loaded with a given load at 2. 



Now in (e) Fig. 9, if the deflection at 2 due to a load P at I 

 is d, then if the load P be moved to 2, the deflection at I will be d. 

 This can be proved by calculating the bending moments at 2 and I 

 for the conditions, since the deflections are directly proportional to 

 the bending moments. With P at I, the bending moment at 2 is 



Pab Pba 



- ; and with P at 2, the bending moment at I is - , and the 



L-, J^, 



proposition is proved. 



Now in (c), if the deflection due to a load unity at 2 is m at P lt 

 then the deflection at 2 due to a load unity at P will be m. If load 

 R 2 is applied at 2, the work done in making the elastic curve pass 

 through 2 will be R 2 c ; while the resistance due to a load P will be 

 P! times the deflection at 2 due to the load P lf which is equal to P x m. 

 In like manner the resistance due to P 2 will be P 2 n, and 



