2. Geometric Design & Vertical Alignment
Transportation Engineering - I

3. Geometric Design

4. Outline Concepts Vertical Alignment Horizontal Alignment
Fundamentals
Crest Vertical Curves
Sag Vertical Curves
Examples
Horizontal Alignment
Super elevation
Other Non-Testable Stuff

5. Concepts Alignment is a 3D problem broken down into two 2D problems
Horizontal Alignment (plan view)
Vertical Alignment (profile view)
Stationing
Along horizontal alignment
12+00 = 1,200 ft.

6. Stationing Horizontal Alignment Vertical Alignment
Therefore, roads will almost always be a bit longer than their stationing because of the vertical alignment
Draw in stationing on each of these curves and explain it

7. From Perteet Engineering
Typical set of road plans – one page only
From Perteet Engineering

8. Vertical Alignment

9. Vertical Alignment Objective: Primary challenge
Determine elevation to ensure
Proper drainage
Acceptable level of safety
Primary challenge
Transition between two grades
Vertical curves
Sag Vertical Curve G1 G2 G1 G2
Crest Vertical Curve

10. Vertical Curve Fundamentals
Parabolic function
Constant rate of change of slope
Implies equal curve tangents
y is the roadway elevation x stations (or feet) from the beginning of the curve

11. Vertical Curve Fundamentals
Choose Either:
G1, G2 in decimal form, L in feet
G1, G2 in percent, L in stations
PVI G1 δ
PVC G2
PVT
L/2 L x
Where: G1: Initial roadway grade( initial tangent grade)
G2: Final roadway grade
A: Absolute value of the difference in grades
L: Length of vertical curve measured in a horizontal plane
PVC: Initial point of the vertical curve
PVI: Point of vertical intersection ( intersection of initial and final grades)
PVT: Final point of the vertical curve

12. Vertical curves are almost arranged such that half of the curve length is positioned before the PVI and half after and are referred as equal tangent vertical curves.
A circular curve is used to connect the horizontal straight stretches of road, a parabolic curve is usually used to connect gradients in the profile alignment.

13. It provides a constant rate of change of slope and implies equal curve lengths.+ + + -
Level - + -
CREST VERTICAL CURVES

14. - - + - + -
Level +
SAG VERTICAL CURVES

15. Vertical Curve
For a vertical curve, the general form of the parabolic equation is;
Y = ax2 + bx + c
where, ‘y’ is the roadway elevation of the curve at a point ‘x’ along the curve from the beginning of the vertical curve (PVC).
‘C’ is the elevation of the PVC since x=0 corresponds the PVC 1

16. Slope of Curve
To define ‘a’ and ‘b’, first derivative of equation 1 gives the slope.
At PVC, x=0; 2

17. or Where G1 is the initial slope. 3

18. Taking second derivative of equation1, i.e. rate of change of slope;
The rate of change of slope can also be written as; 4 5

19. Equating equations 4 and 5or 6 7

20. Fundamentals of Vertical Curves
For vertical curve design and construction, offsets which are vertical distances from initial tangent to the curve are important for vertical curve design.

23. PVI
PVC
PVT
PVC
PVT
PVI
PVC
PVC
PVT
PVI
PVC
PVT
PVT

24. A vertical curve also simplifies the computation of the high and low points or crest and sag vertical curves respectively, since high or low point does not occur at the curve ends PVC or PVT.
Let ‘Y’ is the offset at any distance ‘x’ from PVC.

25. From an equal tangent parabola, it can be written as;
Ym is the mid curve offset & Yt is the offset at the end of the vertical curve.
From an equal tangent parabola, it can be written as;
where ‘y’ is the offset in feet and ‘A’ is the absolute value of the difference in grades(G2-G1, in %), ‘L’ is length of vertical curve in feet and ‘x’ is distance from the PVC in feet. 8

26. Putting the value of x=L in eq. 8

27. First derivative can be used to determine the location of the low point, the alternative to this is to use a k-value which is defined as
where ‘L’ is in feet and ‘A’ is in %.

28. This value ‘k’ can be used directly to compute the high / low points for crest/ sag vertical curves by
x=kG1
where ‘x’ is the distance from the PVC to the high/ low point. ‘k’ can also be defined as the horizontal distance in feet required to affect a 1% change in the slope.

29. A 500-meter equal-tangent sag vertical curve has the PVC at station with an elevation of 1000 m. The initial grade is -4% and the final grade is +2%. Determine the stationing and elevation of the PVI, the PVT, and the lowest point on the
Solution : The curve length is stated to be 500 meters. Therefore, the PVT is at station ( ) and the PVI is in the very middle at , since it is an equal tangent curve. For the parabolic formulation, equals the elevation at the PVC, which is stated as 1000 m. The value of b equals the initial grade, which in decimal is The value of a can then be found as
Basic Equation of the parabola: y = ax2+bx+c
At x= 0 y= C= b= G1= , a = G2-G1/2L =( ))/2x500=
Using the general parabolic formula, the elevation of the PVT can be found:
y = x2+ (-0.04x)+c = (500)+(-0.04*500)+1000= 995m
Since the PVI is the intersect of the two tangents, the slope of either tangent and the elevation of the PVC or PVT, depending, can be used as reference. The elevation of the PVI can then be found as : y = -0.04(250)+1000= 990 m
To find the lowest part of the curve, the first derivative of the parabolic formula
can be found. The lowest point has a slope of zero, and thus the low point location can be found: dy/dx = x-0.04 = 0 x = 0.04/ = m
Using the parabolic formula, the elevation can be computed for that location. It
turns out to be at an elevation of m, which is the lowest point along the curve.

30. Sight Distances
Sight Distance is a length of road surface which a particular driver can see with an acceptable level of clarity. Sight distance plays an important role in geometric highway design because it establishes an acceptable design speed, based on a driver's ability to visually identify and stop for a particular, unforeseen roadway hazard or pass a slower vehicle without being in conflict with opposing traffic.
As velocities on a roadway are increased, the design must be catered to allowing additional viewing distances to allow for adequate time to stop. The two types of sight distance are:
(1) stopping sight distance and (2) passing sight distance.

31. Stopping Sight Distance
At every point on the roadway, the minimum sight distance provided should be sufficient to enable a vehicle traveling at the design speed to stop before reaching a stationary object in its path. Stopping sight distance is the aggregate of two distances:
brake reaction distance and braking distance.
Brake reaction time is the interval between the instant that the driver recognizes the existence of an object or hazard ahead and the instant that the brakes are actually applied. Extensive studies have been conducted to ascertain brake reaction time. Minimum reaction times can be as little as 1.64 seconds: 0.64 for alerted drivers plus 1 second for the unexpected signal.
Some drivers may take over 3.5 seconds to respond under similar circumstances. For approximately 90% of drivers, including older drivers, a reaction time of 2.5 see is considered adequate. This value is therefore used in Table on next page

33. Vehicle Stopping Distance
Vehicle stopping distance is calculated by the following formula
where V1 initial speed of vehicle
f friction
G percent grade

34. Distance Traveled During Perception/ Reaction Time
It is calculated by the following formula
dr = V1* tr
where V1 Initial Velocity of vehicle
tr time required to perceive and react to the need to stop

35. Hence formula for the Stopping sight distance will be;

36. SSD and Crest Vertical Curve
In providing the sufficient SSD on a vertical curve, the length of curve ‘L’ is the critical concern.
Longer lengths of curve provide more SSD, all else being equal, but are most costly to construct.
Shorter curve lengths are relatively inexpensive to construct but may not provide adequate SSD.
In developing such an expression, crest and sag vertical curves are considered separately.
For the crest vertical curve case, consider the diagram.

37. SSD and Crest Vertical Curve
PVI S L
PVC
PVT H2 H1
S = Sight distance (ft)
H1= height of driver’s eye above road-way surface (ft)
L = length of the curve (ft), H2= height of roadway object (ft) ,
A = difference in grade
Lm= Minimum length required for sight distance.

38. Minimum Length of the Curve
For a required sight distance S is calculated as follows;
If the sight distance is found to be less than the curve length (S>L)
for sight distances that are greater than the curve length (S<L)

39. For the sight distance required to provide adequate SSD, standard define driver eye height H1 is 3.5 ft and object height H2 is 0.5 ft. S is assumed is equal to SSD. We get
SSD > L
SSD < L

40. Working with the above equations can be cumbersome.
To simplify matters on crest curves computations, K- values, are used.
L = K*A
where k is the horizontal distance in feet, required to affect 1 percent change in slope.

42. SSD and Sag Vertical Curve
Sag vertical curve design differs from crest vertical curve design in the sense that sight distance is governed by night time conditions, because in daylight, sight distance on a sag vertical curve is unrestricted.
The critical concern for sag vertical curve is the headlight sight distance which is a function of the height of the head light above the road way, H, and the inclined upward angle of the head light beam, relative to the horizontal plane of the car, β.

43. The sag vertical curve sight distance problem is illustrated in the following figure.β H
PVC
PVI
PVT S L

44. By using the properties of parabola for an equal tangent curve, it can be shown that minimum length of the curve, Lm for a required sight distance is ;
For S>L
For S<L

45. For the sight distance required to provide adequate SSD, use a head light height of 2.0 ft and an upward angle of 1 degree.
Substituting these design standards and S = SSD in the above equations;
For SSD>L
For SSD<L

46. As was the case for crest vertical curves, K-values can also be computed for sag vertical curves.
Caution should be exercised in using the k-values in this table since the assumption of G=0 percent is used for SSD computations.

48. Engineering vs. Politics
A current roadway has a design speed of 100 km/hr, a coefficient of friction of 0.1, and carries drivers with perception-reaction times of 2.5 seconds. The drivers use cars that allows their eyes to be 1 meter above the road. Because of ample road kill in the area, the road has been designed for car cases that are 0.5 meters in height. All curves along that road have been designed accordingly.
The local government, seeing the potential of tourism in the area and the boost to the local economy, wants to increase the speed limit to 110 km/hr to attract summer drivers. Residents along the route claim that this is a horrible idea, as a particular curve called "Dead Man's Hill" would earn its name because of sight distance problems. "Dead Man's Hill" is a crest curve that is roughly 600 meters in length. It starts with a grade of +1.0% and ends with (-1.0)%. There has never been an accident on "Dead Man's Hill" as of yet, but residents truly believe one will come about in the near future.

49. A local politician who knows little to nothing about engineering (but thinks he does) states that the 600-meter length is a long distance and more than sufficient to handle the transition of eager big-city drivers. Still, the residents push back, saying that 600 meters is not nearly the distance required for the speed. The politician begins a lengthy campaign to "Bring Tourism to Town", saying that the residents are trying to stop "progress". As an engineer, determine if these residents are indeed making a valid point or if they are simply trying to stop progress?

50. Using sight distance formulas from other sections, it is found that 100 km/hr has an SSD of 465 meters and 110 km/hr has an SSD of 555 meters,
Given the criteria stated above. Since both 465 meters and 555 meters are less than the 600-meter curve length, the correct formula to use would be:

52. For flash animation please visit the following site”

53. Problem for Pondering Problem:
To help prevent future collisions between cars and trains, an at-grade crossing of a rail road by a country road is being redesigned so that the county road will pass underneath the tracks. Currently the vertical alignment of the county road consists of an equal tangents crest vertical curve joining a 4% upgrade to a 3% downgrade. The existing vertical curve is 450 feet long, the PVC of this curve is at station , and the elevation of the PVC is feet. The centerline of the train tracks is at station Your job is to find the shortest vertical curve that provides 20 feet of clearance between the new county road and the train tracks, and to make a preliminary estimate of the cut at the PVI that will be needed to construct the new curve.

55. Study the solution and resolve the question by making the sketches etc