D - Horizontal Alignment and Super Elevation

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Appendix D

Appendix D

Horizonal Alignment and Superelevation
1. General

The horizonal alignment of a new roadway is established initially by locating the points of intersection (PI’s) at which the alignment must change driection. The PI’s are then connected by straight lines to create the tangent alignment of the roadway. Circular curves appropriate for the design speed of the roadway, possibly with spirals, are then added to the tangent alignment to gradually effect the required change in direction at the PI’s. Construction on new alignments is relatively rare. More often, roadways are costructed on their existing alignments with minor changes in the alignment being made, as necessary, to meet current design standards. The primary goal of the designer should be to create a safe and functional roadway that facilitates travel. The horizontal alignment must provide adequate stopping sight distance throughout its length and a desired level of comfort for drivers traversing horizontal curves within the set economic limits, right-of-way limitations, environmental constraints, etc. Although minimum allowable values are given herein, the designer is encouraged not to design to the minimum standard. Particular care should be taken to ensure that the final design results in an aesthetically pleasing road that fits the natural terrain along its alignment. The roadway should follow the natural surroundings without sudden changes in directions.

2.

Circular Curves

WYDOT uses the arc definition of curvature. A one-degree curve has a central angle of 1° that is subtended by a 100 ft arc. There are 360° in a circle, therefore D(2BR) = 360 (100). It follows that R = 5729.578/D in feet. The relationship between the radius (R) and the degree of curvature (D) is: R= (100 ft) (360°) 5729.58 ft = D 2BD

Survey Manual - 2001

Appendix D-1

Horizontal Alignment and Superelevation

Figure Appendix D-1. 1 Degree Curve (Typical).

Minimum curve radius is a limiting design value that provides the desired degree of comfort with respect to centrifugal force for drivers traversing horizontal curves at a given design speed. The minimum radius is determined by the maximum allowable rate of superelevation and the maximum allowable side friction factor. See the superelevation section of this appendix for a complete discussion of minimum curve radius for a particular design speed.

P.I. = Point of Intersection P.C. = Point of Curvature P.T. = Point of Tangency ) = Deflection Angle between the Tangents T = Tangent Distance E = External Distance

R = Radius of the Circular Arc M = Middle Ordinate L.C. = Long Chord (linear distance P.C. P.T.) C = Midpoint of Long Chord D = Degree of Curvature (arc definition) L = Length of Curve (arc distance P.C to P.T.)

Figure Appendix D-2. Components of a Circular Curve.

Appendix D-2

Survey Manual - 2001

Appendix D

General Circular Curve Formulas for Arc Definition (S.I. Units) The standard nomenclature for a circular curve is shown in Figure Appendix D-2. The formulas for the various elements of the curve are easily derived using the right triangles as shown in the figure and the geometric properties of circular curves.

) Deflation Angle Between the Tangents T = R TAN ) 2

D=

5729.578 R

) L.C. = 2 R SIN 2 E= ) -R 2 )R L= 57.29578 COS Locating the P.C. and P.T. STA P.C. = STA P.I. - T STA P.T. = STA P.C. + L Example Problem Given: PI Sta. = 100+00 ) = 27° Radius = 4200 ft Calculation of P.C. Station: T = R Tan ) = 4200 Tan 27 = 1008.33 ft 2 2 Subtracting the distance T from the P.I. statio results in the P.C. station P.C. = P.I. - T = 10000 - 1008.33 = 89 + 91.67 R

Survey Manual - 2001

Appendix D-3

Horizontal Alignment and Superelevation

Calculation of P.T. Station: The P.T. station is found by adding the length of the actual curve to the P.C. station. L (length along curve) = 27(4200) (100) 5729.58 ) R (100) 5729.58

=

= 1979.20 ft

P.T. = P.C. + L = 8991.67 + 1979.20 = 109 + 70.87 Also useful is the E value, the shortest distance from the PI to the curve. 4200 R E= -R= ) 27 - 4200 =119.34 ft cos cos 2 2

3.

Spiral Curves

A circular curve has a radius that is constant, while a spiral curve has a radius that varies from infinity to the radius of the circular curve it is intended to meet. A driver cannot suddenly change the path of his or her vehicle from a straight line to a line of constant curvature. When steering a vehicle into a circular curve, the driver naturally steers the car in a spiral path by increasing the amount of curvature of the car’s path. When the car attains the amount of curvature of the circular curve, the driver holds that position through the curve until the car reaches the spiral at the end of the curve. The driver then steers the vehicle back out to the tangent. When traveling at low speeds or on curves with large radii, a driver can maneuver a vehicle from a line of straight travel into a circular curve without driving out of the lane. At high speeds or on curves with small radii, this maneuver becomes more difficult, causing considerable movement of the vehicle within, and possibly outside, the lane. In such cases, spirals are provided prior to and after the circular curve. This facilitates comfortable and safe travel thoughout the curve. Using the arc definition for horizontal curves: R= 5729.578 feet D

The particular spiral curve used in highway work has a degree of curvature that varies linearly from 0 degrees (radius of curvature = infinity) to D (radius of curvature = R) over the length of the spiral (LS). As a vehicle traverses a circular curve of length Lc, the angle )c, between its initial direction of travel and its final direction of travel, is given by the relationship shown in Figure Appendix D-3. Note that )c can be visualized as a rectangular area created by plotting degree of curvature versus distance along a curve. Appendix D-4
Survey Manual - 2001

Appendix D

Area = )c = D

Lc 100

=

57.296(Lc) R

Figure Appendix D-3

As a vehicle traverses a spiral curve of length Ls, the angle ∆s between its initial direction of travel and its final direction of travel can be visualized (analogous to the circular curve) as the triangular area created by plotting degree of curvature versus distance along the spiral. Note that the slope of this line (change in degree of curvature per station) is the K value of the spiral. Area = )s = = Ls 1 (D) 2 100 5729.58 R Ls R (Ls) 100 1 2

=

= 28.648

Figure Appendix D-4

Survey Manual - 2001

Appendix D-5

Horizontal Alignment and Superelevation

Note that the ) of a spiral is one half of the ) for a circular curve of the same length and degree of curvature. As a vehicle travels a distance l along a spiral curve, the angle * between its initial direction of travel and its final direction of travel can be visualized as the shaded triangular area shown in Figure Appendix D-5.

l 1 Area = * = (D) L 2 s

l 100

l2 = )s L 2 s

Figure Appendix D-5

Using the following expressions, x (the distance along the tangent) and y (the tangent offset) can be obtained for any point located a distance l from the beginning of the spiral. Note that * is calculated from previous equations and then must be converted to radians by multipying the angle in degrees by B/180 before being used in the following equations:

x = l 1-

*2 *4 + - ... 10 216

* *3 *5 y=l 3 + - ... 42 1320

Figure Appendix D-6

Note that Xs (distance along the tangent to the end of the spiral) and Ys (tangent offset at the end of the spiral) can be obtained from the formulas given by using l = Ls and * = )s.

Appendix D-6

Survey Manual - 2001

Appendix D

See figures Appendix D-12 and Appendix D-13 that follow the example for the terminology used for a highway curve with symmetrical spirals.

Example Problem
Given: PI Sta. = 100 + 00 )Total = 35° Design Speed = 65 miles/hr R = 2300 ft From the WYDOT superelevation tables for emax = .08, the radius can be 1970 ft or greater. To achieve a radius of curvature of 2300 ft, a spiral would have length of 300 ft. Calculation of )s: 28.648(Ls) (28.648)(300) = = 3.7367° R 2300

)s =

Calculation of )c: As a vehicle traverses a curve with spirals, the angle between its initial direction of travel at the TS and its final direction of travel at the ST is )total. The part of this change in direction that occurs between the TS and the SC is )s, between the SC and CS is ∆c and between the CS and ST is )s. ˆ )total = )c + 2)s )c = )total - 2)s = 35° - 2 (3.7367°) = 27.5266° Calculation of Xs and Ys: B B )s (radians) = )s (degrees) 180 = 3.7367° 180 = .065218 radians Using l = Ls = 300 and * = )s = 0.065218 rad: 0.0652182 0.0652184 Xs = 300 1 + 10 216 = 299.872 ft 0.065218 0.0652183 0.0652185 Ys = 300 + 3 1320 42 = 6.520 ft

Survey Manual - 2001

Appendix D-7

Horizontal Alignment and Superelevation

Example Problem, continued
Certain quantities associated with the circular part of the total curve must now be determined for use in later calculations. Note that )c, not )total, is used in all calculations dealing with the circular portion of the curve. )c(R) 27.5266(2300) = 57.296 57.296 = 1104.98 ft Tc = (R) Tan )c 2 Lc = = (2300) Tan = 563.37 ft R Ec = )c cos 2
Figure Appendix D-7

27.5266 2 2300 -R= cos 27.5266 2 - 2300

= 67.99 ft

As shown in Figure Appendix D-10, Ts is the sum of Xs, AB, and CD. AB is calculated using triangle I (see Figure Appendix D-10). AE is tangent to the curve at the SC. pBAE = )s. The entering and exiting spirals are of equal length; therefore, the total curve is symmetrical about a line passing through point D and the center of the circular curve. AE = Tc = 563.37 ft AB = AE (cos )s) = 563.37 cos (3.7367°) = 562.17 ft BE = AE sin )s = 563.37 sin (3.7367°) = 36.72 ft

Figure Appendix D-8

Appendix D-8

Survey Manual - 2001

Appendix D

Example Problem, continued
CD is calculated using Triangle II found in Figure Appendix D-10. The total curve is symmetrical about DE therefore: pCDE = = 72.5° 180° - )Total 180° - 35° = 2 2

CE = BE + Ys = 36.72 + 6.52 = 43.24 ft CE 43.16 = = 13.63 ft tan pCDE tan (72.5°) 43.16 CE DE = = = 45.25 ft sin (72.5°) sin pCDE CD = Ts = Xs + AB + CD = 299.872 + 562.17 + 13.63 = 875.67 ft

Figure Appendix D-9

***Calculation of Stations at the TS, SC, CS, and ST: TS Sta. = P.I. Sta. - Ts = 100+00 - 875.67 ft = 91+24.33 SC Sta. = Ts Sta. + Ls = 91 + 24.33 + 300 = 94.24.33 CS Sta. = SC Sta. + Lc = 94 + 24.33 + 1104.98 = 10 + 529.31 ST Sta. = CS Sta. + Ls = 10 + 529.31 + 300 = 10 + 829.31 ***Calculation of Es: Es is the sum of DE and EF (see Figure Appendix D-10). EF = Ec = 67.99 ft Es = DE + Ec = 45.25 + 67.99 = 113.24 ft ***Calculation of x and y for a point 150 feet from TS: * = )S l2 = 3.7967 1502 = 0.934175° 3002 Ls2 B 180 = 0.934175° B 180

* radians = * degrees

= 0.016304 radians

Survey Manual - 2001

Appendix D-9

Horizontal Alignment and Superelevation

Example Problem, continued
2 4 x = 150 1 - 0.016304 + 0.016304 10 216

= 149.99 ft
5 3 y = 150 0.016304 - 0.016304 + 0.016304 1320 3 42

= .815 ft

Ys

Figure Appendix D-10

Appendix D-10

Survey Manual - 2001

Appendix D

Example Problem, continued

Figure Appendix D-11

Survey Manual - 2001

Appendix D-11

Horizontal Alignment and Superelevation

Example Problem
***Approximate Method of Calculating Xs and Ys*** This method will yield values for Xs and Ys that are sufficiently accurate for use in preliminary design, for field checking curves staked by coordinates, etc., and is based on the following approximations. 1) The chord length between the TS and the SC is approximately equal to Ls. )S 2) The deflection angle to the SC is approximately equal to 3 )S Xs = (Ls) cos 3 (approximate) Xs = (300) cos 3.73670° = 299.929 ft 3 Compared to an exact value of 299.87 ft. )s Ys = (Ls) sin 3 (approximate) Ys = (300) sin 3.73670° = 6.521 ft 3

Compared to an exact value of 6.52 ft. The approximate values of x and y can be determined for any point located any distance l from the beginning of the spiral by using l for Ls and * for )s.

Figure Appendix D-12

Appendix D-12

Survey Manual - 2001

Appendix D

Circular Curvature Formula Summary

R=

5729.58 D

R (min) =

V2 15 (.01e + f) ) 2

T = R TAN

P.C. = P.I. - T )R 57.29578

L=

P.T. = P.C. + L R ) cos 2

E=

-R

R = Curve Radius e = Superelevation Rate f = Side Friction Factor V = Design Speed (mph) D = Degree of Curve ) = Deflection Angle T = Tangent Distance

P.C. = Point of Curvature P.I. = Point of Inflection P.T. = Point of Tangency L = Length of Curve

Survey Manual - 2001

Appendix D-13

Horizontal Alignment and Superelevation

U.S. Customary Curvature Formula Summary

)s =

28.64788 Ls R l2 Ls2
4 *2 + * ... 10 216

* = )s

x=l 1-

y=l

* *3 *5 + -... 3 42 1320

)c = )total - 2)s )s (radians) = )s (degrees) )c (R) 57.29578 B 180

Lc =

Tc = R tan )c 2 Ec = R cos ) 2 -R

AE = Tc AB = AE (cos )s) BE = AE (sin )s) pCDE = 180° - )total 2

CE = BE + Ys CD = CE tan pCDE

Appendix D-14

Survey Manual - 2001

Appendix D

DE =

CE sinpCDE

TS = Xs + AB + CD TS Sta. = P.I. Sta. - Ts SC Sta. = TS Sta. + Ls CS Sta. = SC Sta. + Lc ST Sta. = CS Sta. + Ls EF = Ec Es = DE + Ec Xs = Ls cos )s (approximate) 3 Ys = Ls sin )s (approximate) 3 P = Y - R (1 - cos)) P = L (.00145444 )s - 1.582315 x 10-8 ()s)3) R = Circular curve radius )s = Change in direction of travel from beginning to end of spiral * = Change in direction of travel to a point a distance l from beginning of spiral Ls = Length of spiral l = Length along spiral to an intermediate point )c = Change in direction of travel from beginning to end of a circular curve P = Throw distance

Survey Manual - 2001

Appendix D-15

Horizontal Alignment and Superelevation

Point of Rotation

Figure Appendix D-13

The roadway template pivots about the point of rotation when applying superelevation. The profile gradeline is usually established at the point of rotation since this point is not affected by superelevation. It is important to delineate this point on the typical sections. WYDOT applies superelevation by first rotating the lane(s) that are crowned in the opposite direction of the intended superelevation (i.e. adverse crown). The lane(s) crowned in the direction of the intended superelevation do not rotate until the other lanes achieve reverse crown (R.C.). This occurs at a distance of C + C from the beginning of crown runoff (as illustrated above). At this point, all lanes rotate simultaneously until full superelevation is achieved. For undivided highways, the point of rotation is usually located at the design centerline. For divided highways, the point of rotation is usually located at the closest to the median edge of traveled way.

Appendix D-16

Survey Manual - 2001

Appendix D

Location of S and C on a Simple Curve

Figure Appendix D-14

Superelevation is applied such that one-third of the superelevation runoff distance is located on the curve (i.e. between the P.C. and the P.T.) at each end of the curve. Thus, two-thirds of the length of S is located off the curve, as well as the entire length of C. The method of applying superelevation is illustrated for a two-lane roadway in the diagram above.

Survey Manual - 2001

Appendix D-17

Horizontal Alignment and Superelevation

Figure Appendix D-15—Location of S and C on a Curve with Spiral Transitions

C is applied prior to reaching the spiral and ends at the T.S. (tangent to spiral). S starts at the T.S. and ends at the S.C. (spiral to curve). On the other side of the curve, S starts at the C.S. (curve to spiral) and ends at the S.T. (spiral to tangent). C begins at the S.T. and extends beyond the spiral. The method of applying superelevation is illustrated for a two-lane roadway in Figure Appendix D-15.

Appendix D-18

Survey Manual - 2001

Appendix D

Spiral Transitions A vehicle entering a curve must transition from a curve of infinite radius (i.e. a straight line) to a fixed radius (i.e. the given radius for that particular curve). To accomplish this, the vehicle traverses a spiral path. A spiral path has a continually changing radius. If the curvature of the alignment is not excessively sharp, the vehicle can usually traverse this spiral within the width of the travel lane. If the curvature is relatively sharp for a given design speed, it becomes desirable to place a spiral transition at the beginning and end of the curve so that the vehicle more easily transitions into and out of the circular curve while staying within the travel lane. Consequently, the alignment with spirals will more closely duplicate the natural path of the vehicle. Using spiral transitions will shorten the circular portion of the curve and offset the circular curve laterally, as seen in Figure Appendix D-15. The lateral offset distance is known as the spiral throw distance (T). The spiral throw distance can be thought of as the amount that a vehicle will depart from a completely circular path while transitioning to the circular path. WYDOT recommends using spiral transitions if T is equal to or greater than 1.5 feet.

Figure Appendix D-16

In the diagram above, the vehicle is assumed to be traveling a path flush with the centerline stripe (dashed line) prior to reaching the curve. As the vehicle enters the simple curve, the vehicle assumes a spiral path illustrated by the solid line. The maximum lateral offset distance is the spiral throw distance (T). Most vehicles travel down the middle of the travel lane. Therefore, the maneuvering room is in reality one-half that shown in the above example. If T becomes too large, the vehicle may drift out of its travel lane. To prevent this, spiral transitions are used to accommodate the natural vehicle path.

Survey Manual - 2001

Appendix D-19

Horizontal Alignment and Superelevation

Length of Spiral The length of the spiral that each vehicle requires varies. The minimum length of the spiral can be calculated using the law of mechanics, but this approach generally yields lengths much shorter than the superelevation runoff lengths. It is therefore convenient to use the superelevation runoff length as the length of spiral, since the lengths are conservative and the superelevation gradually increases as the radius decreases (i.e. as the curvature of the spiral transition gets sharper).

5.

Compound and Reverse Curves
Compound and reverse curves can sometimes be used advantageously in certain design situations. Restrict their use to cases in which nonconsecutive curves with or without spirals are not effective and do not fit the terrain and proposed alignment. The ratio of the larger radius curve to the smaller radius curve should not be greater than 1.5/1 for highways with design speeds in excess of 30 mph. This figure is based on the assumption that the direction of travel is in the direction of the larger radius curve to the smaller radius curve. If the converse is the case, with the smaller radius curve coming first, then the 1.5/1 ratio is not as critical but should be less than 2.0/1.0. Superelevation runoff should be carefully considered for compound and reverse curves.

Appendix D-20

Survey Manual - 2001

Appendix D

Figure Appendix D-17

Figure Appendix D-18

Survey Manual - 2001

Appendix D-21

Horizontal Alignment and Superelevation

Compound Curvature Design Considerations Superelevation and crown runoff areas demand careful considerations for consecutive curves in the same or opposite directions. WYDOT considers the desirable length of normal crown tangent to be a minimum of 200 ft between consecutive curve sections. If there is not room for 200 ft of normal crown tangent, or if the total tangent length between curves is less than two-thirds S1 + two-thirds S2 (see Figure Appendix D-19), the designer will want to consider alternate means of providing superelevation runoff distance. Although there are several ways to design these areas, there is no one best approach, so the several methods that can be used are covered herein. The designer will need to decide which method or combination of methods is the most adequate and appropriate for the given situation. 1. Two-thirds of superelevation runoff is typically off the curve while one-third is typically on the curve. Consider running off up to onehalf of the superelevation on the curve. The designer can use a tangent section with .01 crown between curves to reduce the crown runoff length. If consecutive curves are curbing the same direction (their centers are on the same side), then consider holding .02 reverse crown between the superelevation runoff areas. The designer can have the superelevation of the first curve transition directly into the superelevation rate of the second consecutive curve, taking care to avoid drainage problems on long, flat curves and also on the high/low point on vertical curves. For two-lane undivided highways rotated about the centerline, the designer may consider eliminating the 1.5 factor from the edgeline gradient formula used in the superelevation tables, allowing the superelevation rotation rate of the roadway to increase, thereby shortening the runoff length.

2. 3.

4.

5.

Figure Appendix D-19

Appendix D-22

Survey Manual - 2001

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