Geometric Design & Vertical Alignment Transportation Engineering - I
Outline • Concepts • Vertical Alignment • Fundamentals • Crest Vertical Curves • Sag Vertical Curves • Examples • Horizontal Alignment • Fundamentals • Super elevation • Other Non-Testable Stuff
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.
Stationing Horizontal Alignment Vertical Alignment
Vertical Alignment • Objective: • Determine elevation to ensure • Proper drainage • Acceptable level of safety • Primary challenge • Transition between two grades • Vertical curves Sag Vertical Curve G1 G2 G2 G1 Crest Vertical Curve
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
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
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.
It provides a constant rate of change of slope and implies equal curve lengths. + + + - Level - + - CREST VERTICAL CURVES
- - + - + - Level + SAG VERTICAL CURVES
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
Slope of Curve • To define ‘a’ and ‘b’, first derivative of equation 1 gives the slope. • At PVC, x=0; 2
or Where G1 is the initial slope. 3
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
Equating equations 4 and 5 • or 6 7
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.
PVI PVC PVT PVC PVT PVI PVC PVC PVT PVI PVC PVT PVT
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.
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
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 %.
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.
A 500-meter equal-tangent sag vertical curve has the PVC at station 100+00 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 105+00 (100+00 + 5+00) and the PVI is in the very middle at 102+50, 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 -0.04. The value of a can then be found as 0.00006. • Basic Equation of the parabola: y = ax2+bx+c • At x= 0 y= C=1000 b= G1= -0.04 , a = G2-G1/2L =(0.02-0.04))/2x500=0.00006 • Using the general parabolic formula, the elevation of the PVT can be found: • y = 0.00006x2+ (-0.04x)+c = 0.00006(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 = 0.00012x-0.04 = 0 x = 0.04/0.00012= 333.333 m Using the parabolic formula, the elevation can be computed for that location. It turns out to be at an elevation of 993.33 m, which is the lowest point along the curve.
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.
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
Vehicle Stopping Distance • Vehicle stopping distance is calculated by the following formula where V1 initial speed of vehicle f friction G percent grade
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
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.
S PVI PVT PVC H2 L SSD and Crest Vertical Curve 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. H1
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)
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
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.
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.
S β H PVT PVC PVI L • The sag vertical curve sight distance problem is illustrated in the following figure.
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
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
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.
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.
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?
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: