The geometric design of roadways deals with the portioning of the physical elements of the roadway according to standards and constraints. The basic objective in geometric design is to provide a smooth-flowing, crash-free facility. The American Association of State highway and Transportation Officials (AASHTO)has established guidelines relating to the design of roadways^[1]. Geometric roadway design can be broken into two main parts: vertical curves and horizontal curves. Combined, vertical and horizontal curves provide a three-dimensional layout for a roadway^[2]. Using guidelines provided by AASHTO, an engineer can design a roadway that is comfortable, safe, and appealing to the eye. The AASHTO guidelines take into account speed, vehicle type, road grade (slope), view obstructions, and stopping distance.

Vertical Curves

Vertical curves are used to provide a gradual change from one road slope to another, so that vehicles may smoothly navigate grade changes as they travel.

Sag vertical curves are those that have a tangent slope at the end of the curve that is higher than that of the beginning of the curve. When driving on a road, a sag curve would appear as a valley, with the vehicle first going downhill before reaching the bottom of the curve and continuing uphill or level.

Crest vertical curves are those that have a tangent slope at the end of the curve that is lower than that of the beginning of the curve. When driving on a crest curve, the road appears as a hill, with the vehicle first going uphill before reaching the top of the curve and continuing downhill.

Terminology

G1 = initial roadway (tangent)slope

G2 = final roadway (tangent)slope

A = absolute value of the difference in grades (initial minus final, expressed in percent)

L = curve length (along the x-axis)

BVC = begin of vertical curve

PVI = point of vertical interception (intersection of initial and final grades)

EVT = end of vertical tangent

x = distance from PTC/BVC

tangent elevation = elevation of a point along the initial tangent

Y (offset) = vertical distance from the initial tangent to the curve

Y’ = curve elevation = tangent elevation - offset^[3]

Sag Curves

Sag vertical curves are curves which have a negative A-value - that is, the tangent slope at the end of the curve is greater than the tangent slope at the beginning of the curve. The most important design criteria for these curves is headlight sight distance^[4]. When a driver is driving on a sag curve at night, the sight distance is limited by the higher grade in front of the vehicle. This distance must be be long enough that the driver can see an obstruction in the road and stop the vehicle within the headlight sight distance. The headlight sight distance (S) is determined by the angle of the headlight and angle of the tangent slope at the end of the curve. By first finding the headlight sight distance (S) and then solving for the curve length (L) in each of the equations below, the correct curve length can be determined. If the S<L curve length is greater than the headlight sight distance, then this number can be used. If it is smaller, this value cannot be used. Similarly, if the S>L curve length is smaller than the headlight sight distance, then this number can be used. If it is larger, this value cannot be used^[5].

Sight Distance < Curve Length (S<L)

$L=AS^{2}/(400+3.5S)$

Sight Distance > Curve Length (S>L)

$L=2S-(400+3.5S)/A$

Crest Curves

Crest vertical curves are curves which have a positive A-value - that is, the tangent slope at the end of the curve is less than the tangent slope at the beginning of the curve. The most important design criteria for these curves is stopping sight distance^[6]. As a vehicle comes over the top of a crest vertical curve, the driver must be able to see any obstructions that lie ahead within a safe stopping distance. If a car on the other side of the hill is stalled or there is an animal in the vehicle's lane, the driver must see the obstruction and be able to stop the car within the stopping sight distance. The stopping sight distance (S) is determined by the speed limit on a road. By first finding the stopping sight distance (S) and then solving for the curve length (L) in each of the equations below, the correct curve length can be determined. If the S<L curve length is greater than the stopping sight distance, then this number can be used. If it is smaller, this value cannot be used. Similarly, if the S>L curve length is smaller than the stopping sight distance, then this number can be used. If it is larger, this value cannot be used^[7].

Sight Distance < Curve Length (S<L)

$L=2S-(2158/A)$

Sight Distance > Curve Length (S>L)

$L=AS^{2}/2158$

Horizontal Curves

Horizontal alignment in road design consists of straight sections of road, known as tangents, connected by horizontal curves^[8]. The design of a horizontal curve entails the determination of a minimum radius (based on speed limit), curve length, and objects obstructing the view of the driver^[9]. Using AASHTO standards, an engineer works to design a road that is safe and comfortable. If a horizontal curve has a high speed and a small radius of curvature, an increased superelevation (bank) is needed in order to assure safety. If there is an object obstructing the view around a corner or curve, the engineer must work to ensure that drivers can see far enough to stop to avoid an accident or accelerate to join traffic.

Terminology

R = Radius

PC = Point of Curvature (point at which the curve begins)

PT = Point of Tangent (point at which the curve ends)

PI = Point of Intersection (point at which the two tangents intersect)

T = Tangent Length

C = Long Chord Length (straight line between PC and PT)

L = Curve Length

M = Middle Ordinate (distance from sight-obstructing object to the middle of the outside lane)

e = Rate of Superelevation

fs= Coefficient of Side Friction

u = Vehicle Speed

$\Delta$ = Deflection Angle^[10]

Geometry

$T=Rtan(\Delta /2)$

$C=2Rsin(\Delta /2)$

$L=R\Delta \pi /180$ ^[11]

Sight Distance

$M=R(1-cos((28.65S)/R))$

Super Elevation

$R=u^{2}/(15(e+fs))$

References

^ Garber, N.J., and Hoel, L., A., Traffic and Highway Engineering, 3rd Edition. Brooks/Cole Publishing, 2001
^ Homburger, W.S., Hall, J.W., reilly, W.R. and Sullivan, E.C., Fundamentals of Traffic Engineering (15th ed), ITS Course Notes UCB-ITS-CN-01-1, 2001
^ Garber, N.J., and Hoel, L., A., Traffic and Highway Engineering, 3rd Edition. Brooks/Cole Publishing, 2001
^ Garber, N.J., and Hoel, L., A., Traffic and Highway Engineering, 3rd Edition. Brooks/Cole Publishing, 2001
^ Homburger, W.S., Hall, J.W., reilly, W.R. and Sullivan, E.C., Fundamentals of Traffic Engineering (15th ed), ITS Course Notes UCB-ITS-CN-01-1, 2001
^ Garber, N.J., and Hoel, L., A., Traffic and Highway Engineering, 3rd Edition. Brooks/Cole Publishing, 2001
^ Homburger, W.S., Hall, J.W., reilly, W.R. and Sullivan, E.C., Fundamentals of Traffic Engineering (15th ed), ITS Course Notes UCB-ITS-CN-01-1, 2001
^ Garber, N.J., and Hoel, L., A., Traffic and Highway Engineering, 3rd Edition. Brooks/Cole Publishing, 2001
^ Homburger, W.S., Hall, J.W., reilly, W.R. and Sullivan, E.C., Fundamentals of Traffic Engineering (15th ed), ITS Course Notes UCB-ITS-CN-01-1, 2001
^ Garber, N.J., and Hoel, L., A., Traffic and Highway Engineering, 3rd Edition. Brooks/Cole Publishing, 2001
^ Garber, N.J., and Hoel, L., A., Traffic and Highway Engineering, 3rd Edition. Brooks/Cole Publishing, 2001

[1] Garber, N.J., and Hoel, L., A., Traffic and Highway Engineering, 3rd Edition. Brooks/Cole Publishing, 2001

[2] Homburger, W.S., Hall, J.W., reilly, W.R. and Sullivan, E.C., Fundamentals of Traffic Engineering (15th ed), ITS Course Notes UCB-ITS-CN-01-1, 2001

[3] Garber, N.J., and Hoel, L., A., Traffic and Highway Engineering, 3rd Edition. Brooks/Cole Publishing, 2001

[4] Garber, N.J., and Hoel, L., A., Traffic and Highway Engineering, 3rd Edition. Brooks/Cole Publishing, 2001

[5] Homburger, W.S., Hall, J.W., reilly, W.R. and Sullivan, E.C., Fundamentals of Traffic Engineering (15th ed), ITS Course Notes UCB-ITS-CN-01-1, 2001

[6] Garber, N.J., and Hoel, L., A., Traffic and Highway Engineering, 3rd Edition. Brooks/Cole Publishing, 2001

[7] Homburger, W.S., Hall, J.W., reilly, W.R. and Sullivan, E.C., Fundamentals of Traffic Engineering (15th ed), ITS Course Notes UCB-ITS-CN-01-1, 2001

[8] Garber, N.J., and Hoel, L., A., Traffic and Highway Engineering, 3rd Edition. Brooks/Cole Publishing, 2001

[9] Homburger, W.S., Hall, J.W., reilly, W.R. and Sullivan, E.C., Fundamentals of Traffic Engineering (15th ed), ITS Course Notes UCB-ITS-CN-01-1, 2001

[10] Garber, N.J., and Hoel, L., A., Traffic and Highway Engineering, 3rd Edition. Brooks/Cole Publishing, 2001

[11] Garber, N.J., and Hoel, L., A., Traffic and Highway Engineering, 3rd Edition. Brooks/Cole Publishing, 2001

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