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Horizont

al Curves

Introduction•Use of curves, horizontal and vertical.•Types of horizontal curves: Circular and spiral.

We will cover circular curves only, spiral curves are given for future reference.

Definitions:Horizontal Curves: curves used in horizontal planes to

connect two straight tangent sections.Simple Curve: circular arc connecting two tangents. The

most commonSpiral Curve: a curve whose radius decreases uniformly

from infinity at the tangent to that of the curve it meets.

Compound Curve: a curve which is composed of two or more circular arcs of different radii tangent to each other, with centers on the same side of the alignment.

Broken-Back Curve: the combination of short length of tangent (less than 100 ft) connecting two circular arcs that have centers on the same side.

Reverse Curve: Two circular arcs tangent to each other, with their centers on opposite sides of the alignment.

Easement Curves: curves used to lessen the effect of the sudden change in curvature at the junction of either a tangent and a curve, or of two curves.

Super elevation: a difference of elevation between the edges of the cross section, to overcome the effect of centrifugal force. Changes gradually in a spiral curve, inversely proportion to the radius.

When to Use WhatSimple circular curves are the most common type.Spirals are used at highway exits, sometimes, and all the times in railroad curves.

The rest of curves are used when the designer has to.

Degree of Circular CurveCurves are identified either by:Radius: 1000-m radiusDegree of Curve (D): central angle subtended by a 100ft: circular arc (in arc definition), or chord in chord definition.

The arc definition is commonly used in highways, chord definition is commonly used in railroads.

In arc definition, D = 5729.58R

In chord definition, D =50

R 2 sin-1( )

degrees, R in ft.

Pay attention to units, we will use ft for length, how about angles?

Circular Curves NotationsDefinitions: Point of intersection (vertex) PI, back and forward tangents.Point of Curvature PC, beginning of the curvePoint of Tangency PT, end of the Curve.Tangent Distance T: Distance from PC, or PT to PILong Chord LC: the line connecting PC and PTLength of the Curve L: distance for PC to PT:

measured along the curve, arc definition.measured along the 100 chords, chord definition

External Distance E: The length from PI to curve midpoint.Middle ordinate M: the radial distance between the midpoints of the

long chord and curve.POC: any point on the curve.POT: any point on tangentIntersection Angle I: the change of direction of the two tangents,

equal to the central angle subtended by the curve.

Circular Curves FormulasRemember that : R is radius, perpendicular to the tangents at PC, and PT D is the curve degree, use ARC definition.

L = 100 ID

ft, D and I in same units. = I

D(sta) = R I I in rad

R = 5729.58

D(ft)

LC = 2R sin( I2)T = R tan( I

2)

E = R[ 1cos (I/2) - 1] M = R(1 - cos I

2 )E = T tan( I

4) M = E cos I2

Circular Curve Stationing--Route Survey are usually staked out as a series of tangents having continuous stations. The station of the PI and the value of (I) are determined.Stations of PC, and PT are computed from PI, R is given by designers:

Compute

Station of PC = station of PI - T Station of PT = station of PC + L

Station equations at PT: the route considering the curve is shorter than it was computed considering the tangents.= (station of PI + T) - (station of PC + L)This amount should be subtracted from stations of all the points after PT.

T = R tan( I2

)

Circular Curves Layout by Deflection Angles with a Total Station or an EDM

All stations will be positioned from PC. Compute the chord length and the deflection angle from the direction

PC-PI as follows: (see fig 25-6)

Example

a= Sa D200 (degrees)

Where: da = DSa 100

or, da = Sa D100

Theory; the angle between the tangent and a chord is equal to half the central angle subtended by the chord, so get a

Also, sin a = Ca

2Rfrom which Ca = 2R sin a

Ca = 2R sin a

In a curve whose I = 8° 24’, station of PC is 62+ 17.08, D = 2° 00’, calculate the necessary information to stake out points at stations 63+00, 64+00, and at the PT.Answer: .. δa= Sa D/200 deg, and Ca = 2R sin δa .. At station 63+00, Sa = 6300 – 6217.08 = 82.92 ft

then, δ = (82.92) (2)/200 = 0.8292 = 00° 49’ 45”C= 2 (5729.58/2) sin(00° 49’ 45”) = 82.92 ft

At station 64+00, Sa = 182.92 ftThen δ = (182.92) (2)/200 = 1.8292 = 1° 49’ 45” C = 2 (5729.58/2) sin(1° 49’ 45”) = 182.89 ftAt the PT: Sa = ?, and Ca = ?

PI (V)

PC

TC

Circular Curve Layout by Coordinates with a Total Station

Given: Coordinates and station of PI, a point from which the curve could be observed, a direction (azimuth) from that point, AZPI-PC , and curve info.Required: coordinates of curve points (stations or parts of stations)

and the data to lay them out.

{this topic and all the following until “sight distance” is mentioned for future reference and will not be covered.}

Solution: - from XPI, YPI, T, AZPI-PC, compute XPC, YPC

- compute the length of chords and the deflection angles.

- use the deflection angles and AZPI-PC, compute the azimuth

of each chord.

- knowing the azimuth and the length of each chord, compute

the coordinates of curve points.

- for each curve point, knowing it’s coordinates and the total

station point, compute the azimuth and the length of the

line connecting them.

- at the total station point, subtract the given direction from the

azimuth to each curve point, get the orientation angle.

Special Circular curve ProblemsPassing a curve through a certain point:- When?-The problem: fig(24-15) Given PI, point (P) that should be on the curve, and the tangents.

Required: R. Solution:

1-Establish an arbitrary coordinate system, origin is at PI, X axis is the line PC-PI. In that system we know the coordinates of PI, PC.

In that system the coordinates of the origin O is:Xo = -T = -R tan (I/2) Yo = -R

2- Measure the angle and the distance PI-P3- Compute the coordinates of P: Xp = - d cosθ Yp = - d sin θ

4- Substitute in the general equation of a circle: R2 = (XP – Xo )2 +(YP – Yo)2

Solve the equation to compute R:

R2 = (XP + R tanI2)

2

+ (YP+ R)2

Intersection of a circular curve and a straight line

• Form the line and the circle equations, solve them simultaneously to get the intersection point.

Intersection of two Circular Curvessimultaneously solve the two circle equations.

Sight distance on Horizontal Curves{Required topic}• What is the problem?

• Stopping distance depends on: speed, perception and reaction time, coefficient of friction, and pavement condition.

• Available sight distance = C = 8mRWhere m is the distance from the obstruction to the center of the road, along a radius.

Two solutions if C is less than the minimum safe sight distance:- Move the obstruction- Reduce the speed.

PI (V)

PC

TC

Spiral Curves

{This topic will not be covered}

• Used to provide gradual transition in horizontal curvature, and hence superelevation.

• Definitions:– Back and forward tangents.– Entrance and exit spirals. Geometrically identical.– TS, SC, CS, ST. What is in between?– SPI: the angle beteen the tangents at TS and SC.

– Spiral Angle S: the angle between the two tangents.

– Spiral Length LS: the arc length of the spiral.

Spiral Geometry• Basic spiral properties:

– Radius changes uniformly from infinity at TS to the radius of the circular curve at the SC. So, it’s degree of curve DS changes uniformly from 0o to D at the SC.

• Average degree of curve is D/2.

• In circular curves, L = (I/D) 100 ft, or I = LD stations

• similarly, S= Ls (D/2) S and D in deg, L in stations

– Spiral angles at any point is proportional to the square of the distance Lp from TS to the point. P = S

• In Fig 25-15, M is the mid point of the spiral, Lp = Ls/2 but M is not = (S /2).Since D changes uniformly, degree of the curve = D/2 at M. But D changes uniformly, so the average degree of curvature between TS and M is (D/2)/2 = D/4

• Then, M = ( Ls/2) (D/4) = (Ls D/8) = S /4

LP

LS)2(

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