Highway Engineering

where
A is the algebraic difference between the two straight-line gradients.

Derivation of crest curve formulae

Case (1) S £L
Given that the curve is parabolic, the relevant offsets are equal to a constant
times the square of the distance from the point at which the crest curve is tan-
gential to the line of sight. Thus, with reference to Fig. 6.20:

H 1 =k(D 1 )^2 (6.51)

And:

H 2 =k(D 2 )^2 (6.52)

Since e =k(L/2)^2 :

(6.53)

Thus:

(6.54)

From Equation 6.46:

Therefore, substituting this expression into Equation 6.54:

(6.55)

And:

(6.56)

Bringing Lover to the RHS of the equation:

(6.57)

Since S, the required sight distance, equals D 1 +D 2 :

If the object is assumed to have zero height (H 2 =0), then Equation 6.49 is
reduced to:

LL== S

()+

m ()

2

12

2

A

2H 2H

see Equation 6.49

L= +

()

()+

AD D

2H 2H

12

2

12

2

D 2H

(^2) A

=^2 L

D

2H

(^1) A

=^1 L

e A

8

=L

DD H

4e

H

(^12) 4e

+=^1 LL^2 +^22

HH

e

12 4D 12 4D 22

2

+ = ()+ ()

L

Geometric Alignment and Design 185