TRANSITION CURVE. 147 



Assuming V as constant in equation (2) : 



e ^ — — 



32.2 R 



it is seen that for various degrees of curvature the supereleva- 

 tion varies inversely as the radius, or what is the same thing, 

 the superelevation varies directly as the curvature. Hence, 

 in connecting the tangent M'A with the circular curve BC by 

 means of a transition curve we must adopt a form of transi- 

 tion curve such that the curvature and superelevation at all 

 points maintain the relationship expressed by equation (2). 

 Since the simplest and most natural thing is to have the sup- 



gV 



erelevation increase uniformly from o to — ^ it follows 



32.2 R 



that the logical form of transition curve to adopt is that 



curve whose curvature increases (and radius decreases) 



directly as the distance measured along the curve. Since the 



angular deviation of the transition curve from M'AX is 



small, this condition may be expressed by the equation 



dV 



Curvature == _iL ^ — ^ = kl (3) 



dl dl^' 



dy kP 



Integratmg, Tr = — + ^^i 



dl 2 



dy 

 Since — - =^ o when 1 = o, kj = o ; 



dl 



dy kl 



2 



whence — := — ^^'' 



dl 2 



Integrating again, y = — + k 



2 



6 



and since y = o when 1 =; o, k^ = o ; 



whence, finally 



kl* 



y^ _ ....... 



which is the equation of the Transition Curve. 



(5) 



