d - horizontal alignment and super elevation

Appendix D

Survey Manual - 2001 Appendix D-1

Appendix D

Horizonal Alignment andSuperelevation

1. General

The horizonal alignment of a new roadway is established initially by locating the points ofintersection (PI’s) at which the alignment must change driection. The PI’s are then con-nected by straight lines to create the tangent alignment of the roadway. Circular curvesappropriate for the design speed of the roadway, possibly with spirals, are then added to thetangent 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 ontheir existing alignments with minor changes in the alignment being made, as necessary, tomeet current design standards.

The primary goal of the designer should be to create a safe and functional roadway thatfacilitates travel. The horizontal alignment must provide adequate stopping sight distancethroughout its length and a desired level of comfort for drivers traversing horizontal curveswithin the set economic limits, right-of-way limitations, environmental constraints, etc.Although minimum allowable values are given herein, the designer is encouraged not todesign to the minimum standard.

Particular care should be taken to ensure that the final design results in an aestheticallypleasing road that fits the natural terrain along its alignment. The roadway should followthe 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:

(100 ft) (360°)2BD

R = = 5729.58 ftD

Horizontal Alignment and Superelevation

Survey Manual - 2001Appendix D-2

Minimum curve radius is a limiting design value that provides the desired degree of com-fort with respect to centrifugal force for drivers traversing horizontal curves at a givendesign speed. The minimum radius is determined by the maximum allowable rate ofsuperelevation and the maximum allowable side friction factor.

See the superelevation section of this appendix for a complete discussion of minimum curveradius for a particular design speed.

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

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

P.I. = Point of IntersectionP.C. = Point of CurvatureP.T. = Point of Tangency) = Deflection Angle between the TangentsT = Tangent DistanceE = External Distance

R = Radius of the Circular ArcM = Middle OrdinateL.C. = Long Chord (linear distance P.C. P.T.)C = Midpoint of Long ChordD = Degree of Curvature (arc definition)L = Length of Curve (arc distance P.C to P.T.)

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. Theformulas for the various elements of the curve are easily derived using the right triangles asshown in the figure and the geometric properties of circular curves.

))))) Deflation Angle Between the Tangents

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 = 1008.33 ft

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

T = R TAN

D =

L.C. = 2 R SIN

E = - R

L =

)2

)2

)2

R

COS

)R57.29578

5729.578R

)2

272



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) =

= = 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.

E = - R = - 4200 =119.34 ft

3. Spiral Curves

A circular curve has a radius that is constant, while a spiral curve has a radius that variesfrom 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 lineof constant curvature. When steering a vehicle into a circular curve, the driver naturallysteers the car in a spiral path by increasing the amount of curvature of the car’s path. Whenthe car attains the amount of curvature of the circular curve, the driver holds that positionthrough the curve until the car reaches the spiral at the end of the curve. The driver thensteers the vehicle back out to the tangent.

When traveling at low speeds or on curves with large radii, a driver can maneuver a vehiclefrom a line of straight travel into a circular curve without driving out of the lane. At highspeeds or on curves with small radii, this maneuver becomes more difficult, causing consid-erable movement of the vehicle within, and possibly outside, the lane. In such cases, spiralsare provided prior to and after the circular curve. This facilitates comfortable and safetravel thoughout the curve. Using the arc definition for horizontal curves:

R = feet

The particular spiral curve used in highway work has a degree of curvature that varieslinearly from 0 degrees (radius of curvature = infinity) to D (radius of curvature = R) overthe length of the spiral (LS).

As a vehicle traverses a circular curve of length Lc, the angle )c, between its initial directionof travel and its final direction of travel, is given by the relationship shown in Figure Appen-dix D-3. Note that )c can be visualized as a rectangular area created by plotting degree ofcurvature versus distance along a curve.

) R (100)5729.58

27(4200) (100)5729.58

5729.578D

)2cos

R272

4200

cos

Appendix D


Area = )c = D =Lc

10057.296(Lc)

R

Figure Appendix D-3

As a vehicle traverses a spiral curve of length Ls, the angle ∆s between its initial direction oftravel and its final direction of travel can be visualized (analogous to the circular curve) asthe 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 ofthe spiral.

Figure Appendix D-4

Area = )s =

= (D)

=

= 28.648

12

Ls

100

5729.58R

(Ls)100

12

Ls

R



Note that the ) of a spiral is one half of the ) for a circular curve of the same length anddegree of curvature.

As a vehicle travels a distance l along a spiral curve, the angle * between its initial directionof travel and its final direction of travel can be visualized as the shaded triangular areashown in Figure Appendix D-5.

Figure Appendix D-5

Using the following expressions, x (the distance along the tangent) and y (the tangentoffset) 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 radiansby multipying the angle in degrees by BBBBB/180 before being used in the followingequations:

Figure Appendix D-6

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

x = l 1- + - ...

y = l - + - ...*3

*3

42*5

1320

Area = * = (D) = )sl

Ls

l100

l2

Ls2

12

*2

10*4

216

Appendix D


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

Example Problem

Given: PI Sta. = 100 + 00)Total = 35°Design Speed = 65 miles/hrR = 2300 ft

From the WYDOT superelevation tables for emax = .08, the radius can be 1970 ft orgreater. To achieve a radius of curvature of 2300 ft, a spiral would have length of300 ft.

Calculation of )s:

)s = = = 3.7367°

Calculation of )c:

As a vehicle traverses a curve with spirals, the angle between its initial direction oftravel at the TS and its final direction of travel at the ST is )total. The part of thischange in direction that occurs between the TS and the SC is )s, between the SCand 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:

)s (radians) = )s (degrees) = 3.7367°

= .065218 radians

Using l = Ls = 300 and * = )s = 0.065218 rad:

Xs = 300 1 - +

= 299.872 ft

Ys = 300 - +

= 6.520 ft

(28.648)(300)2300

28.648(Ls)R

B180

B180

0.0652182

100.0652184

216

0.0652183

0.0652183

420.0652185

1320



Certain quantities associated with the circular part of the total curve must now be deter-mined for use in later calculations. Note that )c, not )total, is used in all calculationsdealing with the circular portion of the curve.

Figure Appendix D-7

Lc = =

= 1104.98 ft

Tc = (R) Tan

= (2300) Tan

= 563.37 ft

Ec = - R = - 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 symmetricalabout a line passing through point D and the center of the circular curve.

AE = Tc = 563.37 ftAB = AE (cos )s) = 563.37 cos (3.7367°) = 562.17 ftBE = AE sin )s = 563.37 sin (3.7367°) = 36.72 ft

Figure Appendix D-8

)c2

27.52662

R)c2

cos

2300

cos 27.52662

Example Problem, continued

)c(R)57.296

27.5266(2300)57.296

Appendix D


CD is calculated using Triangle II found in Figure Appendix D-10. The total curve issymmetrical about DE therefore:

pCDE = =

= 72.5°

CE = BE + Ys = 36.72 + 6.52 = 43.24 ft

CD = = = 13.63 ft

DE = = = 45.25 ft

Ts = Xs + AB + CD = 299.872 + 562.17 + 13.63= 875.67 ft

180° - )Total2

180° - 35°2

***Calculation of Stations at the TS, SC, CS, and ST:

TS Sta. = P.I. Sta. - Ts = 100+00 - 875.67 ft = 91+24.33SC Sta. = Ts Sta. + Ls = 91 + 24.33 + 300 = 94.24.33CS Sta. = SC Sta. + Lc = 94 + 24.33 + 1104.98 = 10 + 529.31ST 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 ftEs = DE + Ec = 45.25 + 67.99 = 113.24 ft

***Calculation of x and y for a point 150 feet from TS:

* = )S = 3.7967 = 0.934175°

* radians = * degrees = 0.934175°

= 0.016304 radians

Figure Appendix D-9


B180

B180

CEtan pCDE

43.16tan (72.5°)

CEsin pCDE

43.16sin (72.5°)

l2

Ls21502

3002



x = 150 1 - +

= 149.99 ft

y = 150 - +

= .815 ft

Figure Appendix D-10


0.0163043

0.0163043

420.0163045

1320

0.0163042

100.0163044

216

Ys

Appendix D






***Approximate Method of Calculating Xs and Ys***

This method will yield values for Xs and Ys that are sufficiently accurate for use in prelimi-nary design, for field checking curves staked by coordinates, etc., and is based on the followingapproximations.

1) The chord length between the TS and the SC is approximately equal to Ls.

2) The deflection angle to the SC is approximately equal to )S3


Xs = (Ls) cos (approximate)

Xs = (300) cos = 299.929 ft

Compared to an exact value of 299.87 ft.

Ys = (Ls) sin (approximate)

Ys = (300) sin = 6.521 ft

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 lfrom the beginning of the spiral by using l for Ls and * for )s.

)S3

)s3

3.73670°3

3.73670°3

Example Problem

Appendix D


Circular Curvature Formula Summary

R =

R (min) =

T = R TAN

P.C. = P.I. - T

L =

P.T. = P.C. + L

E = - R

5729.58D

)2

)R57.29578

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

V2

15 (.01e + f)

R

cos)2



U.S. Customary Curvature Formula Summary

)s =

* = )s

x = l 1 - + . . .

y = l - + - . . .

)c = )total - 2)s

)s (radians) = )s (degrees)

Lc =

Tc = R tan

Ec = - R

AE = TcAB = AE (cos )s)BE = AE (sin )s)

pCDE =

CE = BE + Ys

CD =

28.64788 LsR

l2

Ls2

*2

10*4

216

*3

*3

42*5

1320

B180

)c (R)57.29578

)c2

CEtan pCDE

180° - )total2

)2

cosR

Appendix D


DE =

TS = Xs + AB + CD

TS Sta. = P.I. Sta. - TsSC Sta. = TS Sta. + LsCS Sta. = SC Sta. + LcST Sta. = CS Sta. + Ls

EF = EcEs = DE + Ec

Xs = Ls cos (approximate)

Ys = Ls sin (approximate)

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 spiralLs = Length of spirall = Length along spiral to an intermediate point)c = Change in direction of travel from beginning to end of a circular curveP = Throw distance

CEsinpCDE

)s3

)s3



Point of Rotation

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 isnot affected by superelevation. It is important to delineate this point on the typicalsections. WYDOT applies superelevation by first rotating the lane(s) that are crownedin the opposite direction of the intended superelevation (i.e. adverse crown). Thelane(s) crowned in the direction of the intended superelevation do not rotate untilthe other lanes achieve reverse crown (R.C.). This occurs at a distance of C + Cfrom the beginning of crown runoff (as illustrated above). At this point, all lanesrotate simultaneously until full superelevation is achieved.

For undivided highways, the point of rotation is usually located at the designcenterline. For divided highways, the point of rotation is usually located at theclosest to the median edge of traveled way.


Appendix D


Location of S and C on a Simple Curve

Superelevation is applied such that one-third of the superelevation runoff distance islocated 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 entirelength of C. The method of applying superelevation is illustrated for a two-laneroadway in the diagram above.




C is applied prior to reaching the spiral and ends at the T.S. (tangent to spiral). Sstarts 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 beginsat the S.T. and extends beyond the spiral. The method of applying superelevation isillustrated for a two-lane roadway in Figure Appendix D-15.

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

Appendix D


Spiral Transitions

A vehicle entering a curve must transition from a curve of infinite radius (i.e. astraight line) to a fixed radius (i.e. the given radius for that particular curve). Toaccomplish this, the vehicle traverses a spiral path. A spiral path has a continuallychanging radius. If the curvature of the alignment is not excessively sharp, thevehicle can usually traverse this spiral within the width of the travel lane. If thecurvature is relatively sharp for a given design speed, it becomes desirable to placea spiral transition at the beginning and end of the curve so that the vehicle moreeasily transitions into and out of the circular curve while staying within the travellane. Consequently, the alignment with spirals will more closely duplicate the natu-ral path of the vehicle. Using spiral transitions will shorten the circular portion ofthe 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 spiralthrow distance can be thought of as the amount that a vehicle will depart from acompletely circular path while transitioning to the circular path. WYDOT recom-mends using spiral transitions if T is equal to or greater than 1.5 feet.

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




Length of Spiral

The length of the spiral that each vehicle requires varies. The minimum length ofthe spiral can be calculated using the law of mechanics, but this approach generallyyields lengths much shorter than the superelevation runoff lengths. It is thereforeconvenient to use the superelevation runoff length as the length of spiral, since thelengths are conservative and the superelevation gradually increases as the radiusdecreases (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 certaindesign situations. Restrict their use to cases in which nonconsecutive curves with orwithout 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 greaterthan 1.5/1 for highways with design speeds in excess of 30 mph. This figure isbased on the assumption that the direction of travel is in the direction of the largerradius curve to the smaller radius curve. If the converse is the case, with the smallerradius curve coming first, then the 1.5/1 ratio is not as critical but should be less than2.0/1.0.

Superelevation runoff should be carefully considered for compound and reversecurves.

Appendix D







Compound Curvature Design Considerations

Superelevation and crown runoff areas demand careful considerations forconsecutive curves in the same or opposite directions. WYDOT considersthe desirable length of normal crown tangent to be a minimum of 200 ftbetween consecutive curve sections.

If there is not room for 200 ft of normal crown tangent, or if the total tangentlength between curves is less than two-thirds S1 + two-thirds S2 (see FigureAppendix D-19), the designer will want to consider alternate means of pro-viding superelevation runoff distance.

Although there are several ways to design these areas, there is no one bestapproach, so the several methods that can be used are covered herein. Thedesigner will need to decide which method or combination of methods is themost adequate and appropriate for the given situation.

1. Two-thirds of superelevation runoff is typically off the curve whileone-third is typically on the curve. Consider running off up to one-half of the superelevation on the curve.

2. The designer can use a tangent section with .01 crown between curvesto reduce the crown runoff length.

3. If consecutive curves are curbing the same direction (their centersare on the same side), then consider holding .02 reverse crown be-tween the superelevation runoff areas.

4. The designer can have the superelevation of the first curve transitiondirectly into the superelevation rate of the second consecutive curve,taking care to avoid drainage problems on long, flat curves and alsoon the high/low point on vertical curves.

5. For two-lane undivided highways rotated about the centerline, thedesigner may consider eliminating the 1.5 factor from the edgelinegradient formula used in the superelevation tables, allowing thesuperelevation rotation rate of the roadway to increase, thereby short-ening the runoff length.

d - horizontal alignment and super elevation

Documents

normal crown tangent

smaller radius curve

larger radius curve

curvature versus distance

spiral throw distance

minimum curve radius

superelevation runoff

circular portion