Vertical Alignment of Road (Vertical Curves)

introducti onAn arc with infinite radius , provided b/w intersecting straights to avoid the abrupt change of direction both in horizontal & vertical plane.

CLASSIFICATIONCURVES

HORIZONT AL CURVES

VERTICAL CURVES

VERTICAL CURVES

Objective

Curves are need to provide smooth transitions b/w grade lines of highways &rail roads bcoz these curve exists in vertical planes they are called vertical curves. The function of vertical curves is to provide a gradual change in grade from the initial (back) tangent to the grade of second (forward) tangent.

DEFINITIO NA curve used to connect two different grade liens of railways or highways is called a vertical curves.

As shown in figure below:

Types

Two basic types of vertical curves exists. Crest vertical curves. Sag vertical curves.

Crest vertical curve

Sag vertical curve

It under goes a negative change in grade, that is the curve turns downward. (as shown)

The curve in which change in grade is positive and curve turns upward. (as shown)

upward grade (+) downward grade(-)

upward

down ward grade (-)

grade(+)

Factors effect the design of grade lines

There are several factors that must be taken in to account when designing a grade line of tangents and curves on highway or railroad project.

They include:i.

Providing good fit with existing ground profile. Maintaining the fixed elevations like intersections with roads.(intersection vs interchange) Not exceeding maximum specified grades. Maintaining the adequate drainage. Provide sufficient sight distances for safe vehicle operation.

ii.

iii.

iv.

v.

Expression of grade /gradient

The grade or gradient of a road or a railway is expressed in two ways, As a percentage,(e.g.) 2% or 4% As 1 in n,(e.g.) 1 in 80, 1 in 40.

i.

ii.

Sign of gradesPlus sign Negative sign

An ascending or up grade rising to the right being denoted by plus(+) sign.

A descending or down grade falling to the right is denoted by a minus sign.

Setting out a vertical curve

Vertical curve is generally staked out by computing the elevations of the selected points on the tangents & curves .

A circular curve would be feasible, but it has the weakness that for equal increments along the horizontal axis (a, b, c, & d) the corresponding vertical distances are unequal (w, x, y, & z) and inconvenient to calculate.

However, unlike a circular curve a vertical curve does not have a constant radius at all points along the curve but the most significant strength of the vertical curve is its constant rate of change. In fact, this constant rate of change is the key to the calculation of a vertical curve. This quality is expressed by the general equation of a parabola:

y = ax2

In a vertical curve the offsets from a tangent to the curve always vary as the square of the distance along the tangent from the point of tangency.

Vertical Curve Notation

Here the notation used in Setting out of Vertical Curve will include: The rate of change is r. It can either be The distance the of of twoexpressed pointthe L as curve. of The elevation of length theagrade isthrough Itlines The vertex, change thefeetverticalthe stations. The beginning vertical curve100-fty. in the along the beginning The end thewhere ofin vertical curveof the The totalof grades are vertical curve to the The two in the ofstation desired is grade the elevation straight is can calculated terms of directiontangency, VPT Theintersection, VPI. is be expressed curvature, vertical curvestationing. stations or in feetis x.is station the by , (g2 VPC ) C entering grade meet foundwhich of verticalVPC. desired but point of for point the elevation is change in vertical verticalin 100-ft is vertical curve dividing the total is of is the - g1 It is g1 and the exiting grade is horizontal. always measured along the g2 grade by the length of the vertical curve.

Calculation MethodsThere are two methods for determining elevations along vertical curves that will be presented here. Either method is applicable to both highway and railroad profiles. They are:Tangent offset Method Equation of Parabola Method

Information needed for these methods: Grade or slope on each side of curve Elevation and station of VPC Curve length (Horizontal distance (VPC VPT) L )

The procedure of the the midpoint O ofmethod is Find the elevation of tangent offset the vertical Compute find the elevation ofdesired station. Find below: Finally, tangent offset Y for the midpoint the shown theto the elevation of the the station M of curve: the tangent offset the station the VPI to VPI alculate thecurve theof the that iscurve. and VPT. And l Find elevation line, of Forthe chordtangentverticaldesiredthe VPT: the of the From the from the VPC toU from on grade a Compute the elevation curve the elevation tangent sag relationship is: desired station on the vertical the straight In a on Desired sincesymmetrical Station on the curve: this is a the tangent symmetrical vertical curve = evationvertical curve.must verticaltangent = the line, symmetrical vertical curve thissubtracted The to the offset the be at any or elevation n of desired O of vertical of the PVI is: midway between station on curve added station, In a midpointThe elevation the vertical PI elevation+difference the curve is square tangent, tangent elevation of between the the of the of V vary with Desiredtangent It desiredoffsetsthe on that tangentVPCStation offset Y is the (g1) (Distance of the on elevation from the station station + on of simply VPIof average VPC the elevation of from the midpoint of the is elevation straight grade line. The vertical curve the of elevation - VPT chord the distance the elevation = VPT. of the + (g1) which is from the VPC or * (Distance of Desired Station fromO the the elevation of elevation VPI) midpoint Tangent offset Y = Tangent offset PVI. and VPI and the the U VPC(distance VPC andisthe VPT. VPC)2 / (Distance of VPI from VPC)2 in the vertical curve. In this case: of Figure a crest curve, therefore the relationship tion of the midpoint O = (VPI elevation + Chord Midpoint elevati The elevation of the VPT is: is: the midpoint M = (VPC elevation + VPT elevat ation of Elevation of desired station on VPI vertical curve =(g2) VPT elevation = the elevation + gent (ElevationU = VPIstation on the tangent - tangent offset Y)elevati offset of desired(distance VPI-- Curve Midpoint elevation VPT)

Tangent Offset Method

Equation of Parabola MethodThe general equation of parabola applied to vertical curves is given by: y = ax2 + bx + c (1)

Where; y = Roadway elevation at distance x x = Distance from VPC a, b = Coefficients that define shape c = Elevation of VPC

Now; At the VPC:From Equation (1)

x=0

y=c

dy / dx = 2ax + b dy / dx = b = g1 d2y / dx2 = 2aa = (g2 - g1) / 2L a=r/2

So; And;

2a = (g2 - g1) / L So finally;

y = r/2 x2 + g1 x + c

The objective is to find the elevation of desired station on the vertical curve illustratedThe value for the rate of change, r, is found by subtracting the first, ordata to grade (g1) fromequation and entering the second, or Insertr, which isthe above change along Find the distance by the the Find x, which isthe that difference alongtotal exiting grade (g2) and dividing rate of Compute the elevation ofwhich the elevation compute vertical curve elevation and VPT. lengthcurve. curve the the place for the VPI of desired the of to final curve.

PROCEDU RE

point. is desired.

r = (g2 - g1) / L

AdvantageThis helps in investigating drainage conditions, in computing clearance beneath over head structures or cover sight distance.

Crest Vertical CurvesSSD

PVI

Line of SightPVC

G1

PVT

G2

h1 L

h2

For SSD < L

For SSD > L

L=

100 2h1 + 2h2

(

A( SSD )

2

)

2

200 h1 + h2 L = 2( SSD ) A

(

)

2

Crest Vertical CurvesAssumptions for design h1 = drivers eye height = 3.5 ft. h2 = tail light height = 2.0 ft. Simplified EquationsFor SSD < L2

For SSD > L

A( SSD ) L= 2158

2158 L = 2( SSD ) A

Crest Vertical CurvesAssuming L > SSDSSD K= 21582

Design Controls for Crest Vertical Curves

from AASHTOs A Policy on Geometric Design of Highways and Streets 2001

Design Controls for Crest Vertical Curves

Sag Vertical CurvesLight Beam Distance (SSD)

G1

headlight beam (diverging from LOS by degrees)PVC PVT PVI

G2

h1

h2=0

For SSD < L

L

For SSD > L

( SSD ) 2 A L= 200( h1 + S tan )

200( h1 + ( SSD ) tan ) L = 2( SSD ) A

Sag Vertical CurvesAssumptions for design h1 = headlight height = 2.0 ft. = 1 degree Simplified EquationsFor SSD < L

For SSD > L

A( SSD ) L= 400 + 3.5( SSD )2

400 + 3.5( SSD ) L = 2( SSD ) A

Sag Vertical CurvesAssuming L > SSDSSD K= 400 + 3.5SSD2

Design Controls for Sag Vertical Curves

from AASHTOs A Policy on Geometric Design of Highways and Streets 2001

Design Controls for Sag Vertical Curves

Thanks