Vertical curve design
In order to set out a vertical curve in the field, one requires levels along the curve at given chainage intervals. Before the levels can be computed, one must know the length L of the curve. The value of L is obtained from parameters supplied in Table of TD 9/93 and the appropriate parameters are K-values for specific design speeds and sight distances; then
where A = the difference between the two gradients (grade angle)
K = the design speed related coefficient
e.g. A 4% gradient is linked to a − 3% gradient by a crest curve. What length of curve is required for a design speed of 100 km/h?
A = (4% − (−3%)) = 7% (positive for crest)
C1 Desirable minimum crest K-value = 100
C2 One step below desirable minimum crest K-value = 55
∴ from L = KA
Desirable minimum length = L = 100 × 7 = 700m
One step below desirable minimum length = L = 55 × 7 = 385 m
Wherever possible the vertical and horizontal curves in the design process should be coordinated so that the sight distances are correlated and a more efficient overtaking provision is ensured.
The various design factors will now be dealt with in more detail.
Rate of change of gradient (r) is the rate at which the curve passes from one gradient (g1%) to the next (g2%) and is similar in concept to rate of change of radial acceleration in horizontal transitions. When linked to design speed it is termed rate of vertical acceleration and should never exceed 0.3m/s2. A typical example of a badly designed vertical curve with a high rate of change of grade is a humpbacked bridge where usually the two approaching gradients are quite steep and connected by a very short length of vertical curve. Thus one passes through a large grade angle A in a very short time, with the result that often a vehicle will leave the ground and/or cause great discomfort to its passengers. Fortunately, in the UK, few of these still exist. Commonly-used design values for r are:
3%/100 m on crest curves
1.5%/100 m on sag curves
thereby affording much larger curves to prevent rapid change of grade and provide adequate sight distances. Working from first principles if g1 = −2% and g2 = 4% (sag curve), then the change of grade from −2% to 4% = 6% (A), the grade angle. Thus, to provide for a rate of change of grade of 1.5%,
one would require 400 m (L) of curve. If the curve was a crest curve, then using 3% gives 200 m (L) of curve:
∴ L = 100A/r
and as shown previously, L = KA.