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  Module 3: Geometric design of highways
Lecture 18 Vertical alignment II
  

Length of the valley curve

The valley curve is made fully transitional by providing two similar transition curves of equal length The transitional curve is set out by a cubic parabola $y=bx^3$ where $b=\frac{2N}{3L^2}$ The length of the valley transition curve is designed based on two criteria:
  1. comfort criteria; that is allowable rate of change of centrifugal acceleration is limited to a comfortable level of about $0.6 m/{sec^3}$.
  2. safety criteria; that is the driver should have adequate headlight sight distance at any part of the country.

Comfort criteria

The length of the valley curve based on the rate of change of centrifugal acceleration that will ensure comfort: Let $c$ is the rate of change of acceleration, $R$ the minimum radius of the curve, $v$ is the design speed and $t$ is the time, then $c$ is given as:
$\displaystyle c$ $\textstyle =$ $\displaystyle \frac{\frac{v^2}{R}-0}{t}$  
  $\textstyle =$ $\displaystyle \frac{\frac{v^2}{R}-0}{\frac{L}{v}}$  
  $\textstyle =$ $\displaystyle \frac{v^3}{LR}$  
$\displaystyle L$ $\textstyle =$ $\displaystyle \frac{v^3}{cR}$  

For a cubic parabola, the value of $R$ for length $L_s$ is given by:
$\displaystyle R$ $\textstyle =$ $\displaystyle \frac{L}{N}$  

Therefore,
$\displaystyle L_s$ $\textstyle =$ $\displaystyle \frac{v^3}{\frac{cL_s}{N}}$  
$\displaystyle L_s$ $\textstyle =$ $\displaystyle \sqrt[2]{\frac{Nv^3}{c}}$  
$\displaystyle L$ $\textstyle =$ $\displaystyle 2\sqrt[2]{\frac{Nv^3}{c}}$ (1)

where $L$ is the total length of valley curve, $N$ is the deviation angle in radians or tangent of the deviation angle or the algebraic difference in grades, and $c$ is the allowable rate of change of centrifugal acceleration which may be taken as $0.6 m/{sec^3}$.