Highway Engineering

must be shifted inwards from its initial position by the value Sso that the curves
can meet tangentially. This is the same as having a circular curve of radius (R
+S) joining the tangents replaced by a circular curve (radius R) and two tran-
sition curves. The tangent points are, however, not the same. In the case of the
circular curve of radius (R+S), the tangent occurs at B, while for the circu-
lar/transition curves, it occurs at T (see Fig. 6.14).
From the geometry of the above figure:

(6.27)

It has been proved that B is the mid-point of the transition (see Bannister and
Raymond, 1984 for details).
Therefore:
BT =L/2 (6.28)

Combining these two equations, the length of the line IT is obtained:

(6.29)

If a series of angles and chord lengths are used, the spiral is the preferred form.
If, as is the case here, x and y co-ordinates are being used, then any point on
the transition curve can be estimated using the following equation of the curve
which takes the form of a cubic parabola (see Fig. 6.15):

x =y^3 ∏ 6 RL (6.30)

When y attains its maximum value ofL(the length of the transition curve), then
the maximum offset is calculated as follows:

x =L^3 ∏ 6 RL=L^2 ∏ 6 R (6.31)

IT=+()RStan /()q 22 +L/

IB=+()RStan /()q 2

Geometric Alignment and Design 175

y

x

Straight line

Transition curve

Tangent T

Figure 6.15

Generation of offset

values for plotting a

transition curve.