3.5.4 Definition and Types of Vertical Curves
The curve in a vertical alignment which is produced when two different gradients meet is known as vertical curves. It is provided to secure safety, safety, appearance and visibility. The most common practice has been to use parabolic curves in summit curves. This is because of the ease of setting it out on the field and the comfortable transition from one gradient to another. Furthermore, the use of parabolic curves gives excellent riding comfort. In the case of valley curves, the use of cubic parabola is preferred as it closely approximates the ideal transition requirements.
3.5.4.1 Types of vertical curves
Depending upon the shape of profile, a vertical curve may be divided into:
- Summit curve: When two grades meet at the summit and the curve will have convexity upwards, the curve is simply referred as summit curve.
- Valley (Sag) curve: When two grades meet at the valley (sag) and the curve will have convexity downwards, the curve is simply referred as the valley (sag) curve.
As in the case of horizontal curves, the different types of curves according to geometrical configuration are:
- Circular
- Quadratic parabola
- Cubic parabola and other forms of transition curves
3.5.4.2 Elements of vertical curve
The different elements of vertical curve are:
- Deviation angle (N) = n1 – n2
- Tangent length (T) = L/2
- Length of the curve (L) = 2*T = N*R
- Apex distance (E) = ±T2 / (2*R) = ±L2 / (8*R)
- Mid-ordinate (M) = R [1 – cos {(180*N) / (2*π)}]
Where, n is the side gradient and R is the radius of the curve.
3.5.5 Design of Vertical Summit Curve
Summit curves are those curves which have convexity upwards. They are formed under the four following conditions:
- When a positive gradient meets another mild positive gradient
- When a positive gradient meets a level zero gradient
- When a positive gradient meets with a negative gradient
- When a negative gradient meets another steeper negative gradient
During the design of vertical summit curve the comfort, appearance, and security of the driver should be considered. The sight distance must be considered in the design. All the types of sight distance should be considered during design as far as possible. During movement in a summit curve, there is less discomfort to the passengers because the centrifugal force developed by the movement of the vehicle on a summit curve act upwards which is opposite to the direction in which its weight acts. This relieves the load on the springs of the vehicle so stress developed will be less.
A simple parabolic curve is preferred in summit curve due to its easy implementation in the field, good riding comfort during driving and easy computation. The important part in summit curve design is the computation of the length of the summit curve which is done considering the sight distance parameters as shown below:
Two cases may arise during the design,
LENGTH OF SUMMIT CURVE IS GREATER THAN THE SIGHT DISTANCE
The equation of the parabolic curve is,
Or, y = a*x2
Where, a = N / (2*L)
From figure,
Heights (h1) = as12 and h2 = as22
Then, s1 + s2 = √ (h1 / a) + √ (h2 / a)
Or, s2 = (1/a) (√h1 + √h2)2
Or, s2 = (2*L) * (√h1 + √h2)2 / N
Therefore, L = (NS2) / [2 * (√h1 + √h2)2]
Where L is the length of the summit curve, s is the sight distance, N is the deviation angle, h1 is the height of the driver’s eye from the road surface, h2 is the height of obstruction from the road surface.
- In terms of stopping sight distance (SSD),
Height h1 = 1.2m and h2 = 0.15m
Substituting we get,
L = Ns2 / 4.4
- In terms of overtaking sight distance (OSD) and intermediate sight distance (ISD),
Height h1 = 1.2m and h2 = 1.2m
Substituting we get,
L = Ns2 / 9.6
LENGTH OF SUMMIT CURVE IS LESS THAN THE SIGHT DISTANCE
From basic geometry, one can write,
Or, s = L/2 + h1/n1 + h2/n2
Or, s = L/2 + h1/n1 + h2 / (N – n1)
For minimum value of s we differentiate the equation with respect to n1 and set it to zero,
Or, ds/dh1 = -h1/n12 + h2 / (N – n1)2 = 0
Or, h2n12 = h1 * (N2 + n12 – 2*N*n1)
Solving the equation we get,
Therefore, n1 = (N * √ (h1h2) – Nh1) / (h2 – h1)
Substituting value of n1 in s we get,
Or, s = L/2 + h1 / (N * √ (h1h2) – Nh1) / (h2 – h1) + h2 / [N – (N * √ (h1h2) – Nh1) / (h2 – h1)]
Solving for L we get,
L = 2s – [√ (2h1) + √ (2h2)] 2 / N
- In terms of stopping sight distance (SSD),
Height h1 = 1.2m and h2 = 0.15m
Substituting we get,
L = 2s – 4.4/N
- In terms of overtaking sight distance (OSD) and intermediate sight distance (ISD),
Height h1 = 1.2m and h2 = 1.2m
Substituting we get,
L = 2s – 9.6/N
It should be known that these values are the minimum lengths and greater sight distances should be used where there is economically and technically feasible.
3.5.6 Design of vertical valley curve
Valley (Sag) curves are those curves which have convexity downwards. They are formed under the four following conditions:
- When a negative gradient meets another mild negative gradient
- When a negative gradient meets a level zero gradient
- When a negative gradient meets with a positive gradient
- When a positive gradient meets another steeper positive gradient
As compared to the design of summit curve, valley curve requires more consideration. During day time the visibility in valley curves are not hindered but during night time the only source of visibility becomes headlight in the absence of street lights. And in valley curves, the centrifugal force generated by the vehicle moving along a valley curve acts downwards along with the weight of the vehicle and this adds to the stress induced in the spring of the vehicle which causes jerking of the vehicle and discomfort to the passengers. Thus, the most important things to consider during valley curve design are:
- Impact and jerking free movement of vehicles at design speed
- Availability of stopping sight distance under headlight of vehicles during night driving
The best shape for a valley curve is transition curve. Some prefer to use the circular curve or quadratic parabola or combined circular spiral curve but mostly cubic parabola is generally preferred in vertical valley curves. Valley curve is made fully translational by providing two similar transition curves of equal length. It is set by cubic parabola y = bx3 where b = 2N / 3L2. The length of the valley transition curve is designed to the following two criteria:
COMFORT CRITERIA
This is used in design to provide impact-free movement of vehicles at design speed. In this criterion, the allowable rate of change of centrifugal acceleration (c) is limited to a comfortable level of 0.6m/s3.
The rate of change of centrifugal acceleration is given by,
Or, c = [(v2 / R) – 0] / t = (v2 / R) / (Ls / v)
Then, Ls = v3 / (c*R)
For a cubic parabola, the value of R is given by,
R = Ls / N
Substituting value of R we get,
Ls = (N*v3) / (c*Ls)
Or, Ls2 = (N*v3) / c
Therefore, Ls = √ [(N*v3) / c]
The total length of the valley curve is given by,
L = 2*Ls = 2* √ [(N*v3) / c]
Where, L is the total length of the valley curve, N is the deviation angle, v is the design speed and c is the rate of change of centrifugal acceleration which may be taken as 0.6 m/s3.
Applying value of c = 0.6 m/s3 we get,
Ls = 0.19 √ (N*v3)
L = 0.38 √ (N*v3)
SAFETY CRITERIA
Sight distance is highly reduced during headlight driving conditions during night time. The sight distance within which the head lights can illuminate is known as head light distance and it should be equal to the stopping sight distance. There is no problem of overtaking operation at night because of low traffic and the fact that other vehicles with headlights can be seen from a considerable distance. This design provides adequate stopping sight distance for vehicles under headlight at night time at any part of the curve. It may be determined from two conditions:
- Length of the valley curve is greater than the stopping sight distance
At the lowest point of the valley curve, the sight distance will be minimum because at the bottom of the curve where there is minimum radius which is a property of the transition curve. From the geometry of the figure, we have,
Or, h1 + s tan α = as2
Where, a = N / (2L)
Then, h1 + s tan α = Ns2 / (2L)
Therefore,
L = (Ns2) / (2h1 + 2s tan α)
Where N is the deviation angle, s is the sight distance, h1 is the height of the headlight beam from the road surface and α is the head beam inclination in degrees.
Taking, h1 = 0.75m [NRS] and α≈1 ÌŠ we get,
L = (Ns2) / (1.5 + 0.035s)
- Length of the valley curve is less than the stopping sight distance
In this case, the minimum sight distance will be at the beginning of the curve because the headlight beam at the beginning of the curve will just hit outside of the curve. But at the bottom of the curve, the headlight beam will reach far beyond the end point of the curve. Hence the length of the curve is determined to assume that the vehicle is at the beginning of the curve. From the geometry of the figure, we have,
Or, h1 + s tan α = (s – L/2) * N
Therefore,
L = 2*s – (2h1 + 2s tan α) / N
Where N is the deviation angle, s is the sight distance, h1 is the height of the headlight beam from the road surface and α is the head beam inclination in degrees.
Taking, h1 = 0.75m [NRS] and α≈1 ÌŠ we get,
L = 2*s – (1.5 + 0.035s) / N
The expression above is approximate but it is satisfactory in practice because the gradients are very small. During design, both cases need to be calculated because we will not know prior which case has to be adopted. After calculation, we adopt the greater length among the two during design.
According to the specification of NRS 2045, the criteria to be adopted are that the stopping sight distance shall be equal to the headlight sight distance and the centripetal acceleration is limited to 0.3 m/s2.
3.5.7 Lowest and highest point of vertical curve
From the element of a vertical curve, we can write,
Length of a curve (L) = R * N
Where, R is the radius of the curve and N is the deviation angle.
Then, R = L / N
The highest point is calculated in a summit curve whereas the lowest is calculated in a valley curve. It is formed near the smaller gradient among the tangents. If n1 = +2% and n2 = -1.5% then summit curve is formed where the highest point is formed near the n2 grade i.e. near the end of the vertical curve (EVC).
Then, the distance of highest point from beginning of the curve (x1) = R * n1 = (L * n1) / N
Similarly from end of the curve (x2) = R * n2 = (L * n2) / N
And tangent correction for the curve or the height of the highest point of the curve is given by,
Or, y = x2 / (2*R) = (N * x2) / (2*L)
The lowest point of the curve is calculated in the same manner in the case of valley curve.
In the case of valley curve, if the curve is cubic parabola then the lowest point of the valley curve lies on the side of the flatter grade and the distance of the lowest point from the flatter gradient is given by:
Or, x1 = L * √ [n1 / (2*N)]
And, tangent correction (y) = (x3 * 2N) / (3*L2)
Bibliography:
Marsani A. and Shrestha D.K. (2071), Transportation Engineering Volume – I, Divine Print Support, Lagan Tole, Kathmandu
Parajuli P.M. (1999), Course Manual of Transportation Engineering – I, IOE, Pulchowk Campus, Lalitpur, Nepal.
Nepal Road Standard 2070
Nepal Rural Road Standard 2071
CIVIL ENGINEERING 