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Horizontal Alignment

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Horizontal Alignment

• Along circular path, vehicle undergoes centripetal acceleration towards center of curvature (lateral acceleration).

• Balanced by superelevation and weight of vehicle (friction between tire and roadway).

• Design based on appropriate relationship between design speed and curvature and their relationship with side friction and super elevation.

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Vehicle Cornering

Fcp

Fcn

Fc

W

Wp

Wn

Ff

Ff

1 ft

e

Rv

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• Figure illustrates the forces acting on a vehicle during cornering. In this figure, is the angle of inclination, W is the weight of the vehicle in pounds with Wn and Wp being the weight normal and parallel to the roadway surface respectively.

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• Ff is the side frictional force, Fc is the centrifugal force with Fcp being the centrifugal force acting parallel to the roadway surface, Fcn is the centrifugal force acting normal to the roadway surface, and Rv is the radius defined to the vehicle’s traveled path in ft.

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• Some basic horizontal curve relationships can be derived by summing forces parallel to the roadway surface.

Wp + Ff = Fcp

• From basic physics this equation can be written as

cos]sincos[sin22

vv gR

WV

gR

WVWfsW

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• Where fs is the coefficient of side friction, g is the gravitational constant and V is the vehicle speed (in ft per second).

• Dividing both the sides of the equation by W cos ;

cos

cos]

cos

sin

cos

cos[

cos

sin 22

vv gR

WV

gR

WVWfsW

)tan1(tan2

sv

s fgR

Vf

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• The term tan is referred to as the super elevation of the curve and is denoted by ‘e’.

• Super elevation is tilting the roadway to help offset centripetal forces developed as the vehicle goes around a curve.

• The term ‘fs’ is conservatively set equal to zero for practical applications due to small values that ‘fs’ and ‘’ typically assume.

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• With e = tan, equation can be rearranged as

)(

2

efg

VR

sv

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• In actual design of a horizontal curve, the engineer must select appropriate values of e and fs.

• Super-elevation value ‘e’ is critical since • high rates of super-elevation can cause vehicle

steering problems at exits on horizontal curves • and in cold climates, ice on road ways can reduce

fs and vehicles are forced inwardly off the curve by gravitational forces.

• Values of ‘e’ and ‘fs’ can be obtained from AASHTO standards.

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Horizontal Curve Fundamentals

• For connecting straight tangent sections of roadway with a curve, several options are available.

• The most obvious is the simple curve, which is just a standard curve with a single, constant radius.

• Other options include; • compound curve, which consists of two or

more simple curves in succession , • and spiral curves which are continuously

changing radius curves.

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Basic Geometry

TangentHorizontal

CurveTangent

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Tangent Vs. Horizontal Curve

• Predicting speeds for tangent and horizontal segments is different

• May actually be easier to predict speeds on curves than tangents• Speeds on curves are restricted to a few well

defined variables (e.g. radius, superelevation)• Speeds on tangents are not as restricted by design

variables (e.g. driver attitude)

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Elements of Horizontal Curves

R R

PT

MLT

PC

E

PI

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• Figure shows the basic elements of a simple horizontal curve. In this figure • R is the radius (measured to center line of the

road)• PC is the beginning point of horizontal curve• T is tangent length• PI is tangent intersection is the central angle of the curve• PT is end point of curve• M is the middle ordinate• E is the external distance • L is the length of the curve

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Degree of Curve

• It is the angle subtended by a 100-ft arc along the horizontal curve.

• Is a measure of the sharpness of curve and is frequently used instead of the radius in the actual construction of horizontal curve.

• The degree of curve is directly related to the radius of the horizontal curve by

RD

6.5729

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R R

PT

ML

T

PC

E

PI

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• A geometric and trigonometric analysis of figure, reveals the following relationships

DL

RM

RE

RT

100

)2

cos(1

1)

2cos(

1

2tan

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Stopping Sight Distance and Horizontal Curve Design

Sight Obstruction

s

Rv Critical inside lane

Highway Centerline

Ms

SSD

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• Adequate stopping sight distance must also be provided in the design of horizontal curves.

• Sight distance restrictions on horizontal curves occur when obstructions are present.

• Such obstructions are frequently encountered in highway design due to the cost of right of way acquisition and/or cost of moving earthen materials.

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• When such an obstruction exists, the stopping sight distance is measured along the horizontal curve from the center of the traveled lane.

• For a specified stopping sight distance, some distance, Ms, must be visually cleared, so that the line of sight is such that sufficient stopping sight distance is available.

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• Equations for computing SSD relationships for horizontal curves can be derived by first determining the central angle, s, for an arc equal to the required stopping sight distance.

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• Assuming that the length of the horizontal curve exceeds the required SSD, we have

Combining the above equation with following

we get;

DSSD s

100

vs R

SSD296.57

RD

6.5729

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• Rv is the radius to the vehicle’s traveled path, which is also assumed to be the location of the driver’s eye for sight distance, and is again taken as the radius to the middle of the innermost lane,

• and s is the angle subtended by an arc equal to SSD in length.

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• By substituting equation for s in equation of middle ordinate, we get the following equation for middle ordinate;

• Where Ms is the middle ordinate necessary to provide adequate stopping sight distance. Solving further we get;

vv R

SSDRMs

65.28cos1

v

svv

R

MRRSSD 1cos

65.28

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Max e

• Controlled by 4 factors:• Climate conditions (amount of ice and snow)• Terrain (flat, rolling, mountainous)• Frequency of slow moving vehicles which influenced

by high superelevation rates• Highest in common use = 10%, 12% with no ice and

snow on low volume gravel-surfaced roads• 8% is logical maximum to minimized slipping by

stopped vehicles

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Radius Calculation (Example)

Design radius example: assume a maximum e of 8% and design speed of 60 mph, what is the minimum radius?

fmax = 0.12 (from Green Book)

Rmin = _____602__________

15(0.08 + 0.12)

Rmin = 1200 feet

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Radius Calculation (Example)

For emax = 4%?

Rmin = _____602_________

15(0.04 + 0.12)

Rmin = 1,500 feet

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Sight Distance Example

A horizontal curve with R = 800 ft is part of a 2-lane highway with a posted speed limit of 35 mph. What is the minimum distance that a large billboard can be placed from the centerline of the inside lane of the curve without reducing required SSD? Assume p/r =2.5 and a = 11.2 ft/sec2

SSD = 1.47vt + _________v2____ = 246 ft 30( a___ G)

32.2

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Horizontal Curve Example

• Deflection angle of a 4º curve is 55º25’, PC at station 238 + 44.75. Find length of curve,T, and station of PC.

• D = 4º = 55º25’ = 55.417º

• D = _5729.58_ R = _5729.58_ = 1,432.4 ft

R 4

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Horizontal Curve Example

• D = 4º = 55.417º

• R = 1,432.4 ft

• L = 2R = 2(1,432.4 ft)(55.417º) = 1385.42ft

360 360

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Horizontal Curve Example

• D = 4º = 55.417º• R = 1,432.4 ft• L = 1385.42 ft• T = R tan = 1,432.4 ft tan (55.417) = 752.29 ft

2 2

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Stationing Example

Stationing goes around horizontal curve.

For previous example, what is station of PT?

PC = 238 + 44.75

L = 1385.42 ft = 13 + 85.42

Station at PT = (238 + 44.75) + (13 + 85.42) = 252 + 30.17

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Suggested Steps on Horizontal Design

1. Select tangents, PIs, and general curves make sure you meet minimum radii

2. Select specific curve radii/spiral and calculate important points (see lab) using formula or table (those needed for design, plans, and lab requirements)

3. Station alignment (as curves are encountered)4. Determine super and runoff for curves and put in

table (see next lecture for def.)5. Add information to plans

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Geometric Design – Horizontal Alignment (1)

• Horizontal curve – Plan view, profile, staking, stationing– type of horizontal curves– Characteristics of simple circular curve

• Stopping sight distance on horizontal

curves• Spiral curve

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18+

00

19+

00

20+

00

21+

00

200

300

400

500

600

700

700

15+00 16+00 17+00 18+00 19+00 20+00 21+00 22+00 23+00 24+00

Plan view and profile

plan

profile

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• Staking: route surveyors define the geometry of a highway by “staking” out the horizontal and vertical position of the route and by marking of the cross-section at intervals of 100 ft.

• Station: Start from an origin by stationing 0, regular stations are established every 100 ft., and numbered 0+00, 12 + 00 (=1200 ft), 20 + 45 (2000 ft + 45) etc.

Surveying and Stationing

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Horizontal Curve Types

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Curve Types

1. Simple curves with spirals

2. Broken Back – two curves same direction (avoid)

3. Compound curves: multiple curves connected directly together (use with caution) go from large radii to smaller radii and have R(large) < 1.5 R(small)

4. Reverse curves – two curves, opposite direction (require separation typically for superelevation attainment)

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1. Simple Curve

Straight road sections

Circular arc

R

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2. Compound Curve

Straight road sections

Circular arcsR1

R2

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3. Broken Back Curve

Straight road sections

Circular arc

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4. Reverse Curve

Straight road sections

Circular arcs

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Straight road section

R =

R = Rn

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(a) degree (b) Radian

Angle measurement

2957.57)/180(1

0174532.0180/1

radian

radiansradians

0

90

180

30

60

0

90

180

30

60

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As the subtended arc is proportional to the radius of the circle, then the radian measure of the angle.Is the ratio of the length of the subtended arc to the radius of the circle

lengtharcRmeasureradian

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• Define horizontal Curve:• Circular Horizontal Curve Definitions• Radius, usually measured to the centerline of the road, in ft.• = Central angle of the curve in degrees• PC = point of curve (the beginning point of the horizontal curve)• PI = point of tangent intersection• PT = Point of tangent (the ending point of the horizontal curve)• T = tangent length in ft.• M = middle ordinate from middle point of cord to middle point of

curve in ft.• E = External distance in ft.• L = length of curve• D = Degree of curvature (the angle subtended by a 100-ft arc* along

the horizontal curve)• C = chord length from PC to PT

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R R

2

2

M

E

PC PT

PI

L

T

2

2

290

2

90

C

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]1))2/cos(

1[(

RE

2957.57100

,

57.572918000)/180(100

RD

orRRR

D

RL180

Key measures of the curve

2tan

RT

2sin2

RC

Note converts from radians to degrees

/180

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Example:

A horizontal curve is designed with a 2000-ft radius, the curve has a tangent length of 400 ft and the PI is at station 103 + 00, determine the stationing of the PT.

Solution:

58.89106)58.897()0099(

0099)004()00103(

58.789180

62.22

2tan2000400

2tan

LPCPT

PC

ftRL

RT

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Spiral curves are curves with a continuously changing radii, they are sometimes used on high-speed roadways with sharp horizontal curves and are sometimes used to gradually introduce the super elevation of an upcoming horizontal curve

Spiral Curve:

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Length of Spiral

o AASHTO also provides recommended spiral length based on driver behavior rather than a specific equation.

o Superelevation runoff length is set equal to the spiral curve length when spirals are used.

o Design Note: For construction purposes, round your designs to a reasonable values; e.g. round

o Ls = 147 feet, to use Ls = 150 feet.

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Thanks