EFFECT OF BRIDGE PIEKS 327 



Also vi = i % 



Ui III 



Q* Q* 



(c 6 2 /> 2 ) 2 (bi /ii) 2 



CM 1 1 



/y. ^ j 



2 ^r ( c 2 6 2 2 /i 2 2 l>i 2 





= ^!I 



20 c 2 6 2 2 (/n - a;) 2 



The value of c varies with the form of pier, but with pointed cutwaters 

 is about '95 (Eytelwein), diminishing to "85 for a bridge having square 

 or rectangular piers. By considering the problem as one of flow through 

 a weir or notch having a submerged crest (this crest being level with the 

 bed of the stream), and under an effective head x, we may obtain a second 

 expression for Q in terms of x, and by equating these two expressions the 

 value of hi may be obtained in terms of Q and of bi and 62. From this, 

 an application of equation (9) (p. 319) will give the depth at any point 

 up stream, and the entire up-stream profile may then be plotted. On the 

 down-stream side of the obstacle there is a gradual rise of the surface 

 level as the depth increases to a uniform value H. 



ART. 92. 



With radial outward flow over a horizontal bed, such as occurs when a 

 vertical stream impinges on such a surface, we have, if h is the depth at 

 a radius r, Q the quantity per second, and v the velocity at radius r, 

 Q = v X 2 TT ? h = const. 



... vr ** + r h*Z + *h = 



d r ' d r ' 



(1) 



.. 



d r ( hdr r ) 



Substituting this value in equation (2) (p. 309), we have 

 dh_ ij?dh. v* I f* P 

 ~T~r~ ' \gh dl^ J~r\ f ^7' ~A' 

 But with a horizontal bed i = o. 



P 27rr 1 



Also -r = p. -- r = T, and I = r 



A 2 TT r h h' 



j? fv* tMl _/_) 



< *f* = 9\ r * h ) 



v 2 .. v 2 



r i j- 



gh g h 



(2) becomes = <LL _ f = 9 r . (8) 



dr t v 2 .. v 2 



i r i 



