chapter 3 part ii st

7/30/2019 Chapter 3 Part II St

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CHAPTER 3

PART II

VERTICAL CURVES& HORIZONTAL SIGHT DISTANCE


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Vertical Alignment

Specifies the elevation of points along aroadway

Provides a transition between twogrades

Sag curves and crest curves

Equal-tangent curves - half the curvelength positioned before the PVI; halfafter


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Notation

Curve point naming is similar to horizontalcurves, with addition of V for vertical PVC: Point of Vertical Curvature

PVI: Point of Vertical Intersection(of initial and final tangents)

PVT: Point of Vertical Tangency

Curve positioning and length usually

referenced in stations Stations represent 1000 m or 100 ft

e.g., 1258.5 ft 12 + 58.5(i.e., 12 stations & 58.5 ft)


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Notation

G1 is initial roadway gradeAlso referred to as initial tangent grade

G2 is final roadway (tangent) grade

A is the absolute value of the difference ingrades (generally expressed in percent)A = |G2 G1|

L is the length of the vertical curve measuredin a horizontal plane (not along curve centerline, like horizontal curves)


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Fundamentals

Parabolic curves are generally used for design

Parabolic function y= ax2 + bx+ cy= roadway elevationx= distance from PVCc= elevation of PVC

Also usually design for equal-length tangents

i.e., half of curve length is before PVI and half after


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First Derivative

First derivative gives slope

At PVC, x = 0, so , by definition

G1 is initial slope (in ft/ft or m/m) aspreviously defined

bax

dx

dy 2

1Gdx

dyb


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Second Derivative

Second derivative gives rate of changeof slope

However, the average rate of change ofslope, by observation, can also bewritten as

Giving,

adx

yd2

2

2

L

GG

dx

yd 122

2

L

GGa

2

12


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Offsets are vertical distances from initialtangent to the curve

Offsets


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For an equal tangent parabola,

Y= offset (in m or ft) at any distance, x, fromthe PVC

A and L are as previously defined

It follows from the figure that,

2

200x

L

AY

200

800

ALY

AL

Y

f

m

Offset Formulas


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K Values

The rate of change of grade at successivepoints on the curve is a constant amount forequal increments of horizontal distance, and

Equals the algebraic difference betweenintersecting tangent grades divided by thelength of curve, or A/L in percent per ft (m)

The reciprocal L/A is the horizontal distancerequired to effect a 1% change in gradientand is, therefore, a measure of curvature

The quantity L/A is termed K


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K Values

The K-value can be used directly to computethe high/low points for crest/sag verticalcurves (provided the high/low point is not at

a curve end) by, xhl = K |G1|

Where x = distance from the PVC to the high/lowpoint

Additionally, K-values have importantapplications in the design of vertical curves,which we will see shortly


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Vertical Curves

Controlling factor: sight distance

Stopping sight distance should be provided asa minimum

Rate of change of grade should be keptwithin tolerable limits

Drainage of sag curves is important

consideration, grades not less than 0.5%needed for drainage to outer edge ofroadway


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Vertical Alignment Relationships

1

2

200

800

200

GKxA

LK

ALY

ALY

xL

AY

hl

f

m

L

GGa

a

dx

yd

Gdx

dyb

xatPVC

baxdx

dy

cbxaxy

2

2

:0,

2

12

2

2

1

2


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Example Problem: Vertical Curve

A vertical curve crosses a 4 diameter pipe atright angles. Pipe at sta 110+85 withcenterline elevation of 1091.60. PVI at sta110+00 elevation 1098.4. Equal tangentcurve, 600 long with initial and final gradesof +1.2% and -1.08%. Using offsets

determine the depth below the surface of thecurve the top of the pipe and determine thestation of the highest point of the curve.


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Solution

curveofsurfacebelow3ft1093.6-1096.6ispipeMeaning,

6.109326.1091diameterpipe1/2pipeofCLofelevationpipeof

6.109682.242.1099

82.2)385()600(200

)08.1(2.1

200tangent)andcurvebetween(distancepipe?theaboveoffsettheisWhat

42.1099)/2.185.3(8.1094pipetheabove

8.1094)/2.13(4.1098

pipeat

2

2

f ttop

ftCurve

f tY

xL

AY

f tstaftstaTangentInitial

f tstaftstaPVC

elevation

elevation

elevation


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Solution Continued

'79.3152.116.263

16.236)08.1(2.1

600

3.11Eqand3.10EqusingfoundbecanCurveonPtHighestofLocation

1

GKx

K

hl


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Stopping Sight Distance &Crest Curves

Two different factors are important forcrest curves

The drivers eye height in vehicle, H1

Height of a roadway obstruction object, H2


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SSD & Curve Design

It is necessary, when designing verticalcurves, to provide adequate stopping-

sight distance (SSD)

Because curve construction is

expensive, we want to minimize curvelength, subject to adequate SSD


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SSD and Curve Design

SSD formulation was given in Chapter2, i.e., ds= d+ dr(Eq. 2.50)

It is repeated in Chapter 3 as Eq. 3.12

rtV

Gg

ag

VSSD 1

2

1

))((2

Table 3.1 gives SSD values in 5mph increments based onthis equation and a=11.2ft/s2 and tr = 2.5s


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Minimum Curve Length

By using the properties of a parabolafor an equal tangent curve, it can be

shown that the minimum length ofcurve, Lm, for a required SSD is

LSfor3.14Eq)(200

2

LSfor3.13Eq

)(2002

21

2

21

2

A

HHSL

HH

ASL

m

m


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For the sight distance required to provideadequate SSD, current AASHTO designstandards use the following specifications:

H1(drivers eye height) = 3.5 ft (1080 mm)

H2 (object height) = 2.0 ft (600 mm)


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Substituting these values into previoustwo equations yields:

LSSDfor2158

2

LSSDfor2158

2

ASSDL

SSDAL

m

m

Since using these equations can be cumbersome, tables have beendeveloped, utilizing K=L/A (discussed earlier)


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Example 3.5

A highway is being designed to AASHTOguidelines with a 70-mph design speed

and, at one section, an equal tangentvertical curve must be designed toconnect grades of +1.0% and2.0%.

Determine the minimum length ofvertical curve necessary to meet SSDrequirements.


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3.5 Solution

okis3.15eq.ofuseso73082.740

82.740

21587303

2158

LSSDfor3.15eqUsing

22

f tf t

f tL

SSDAL

m

m


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US Customary Metric

Rate of verticalcurvature, Ka

Rate of verticalcurvature, Ka

esignspeedmi/h)

Stoppingsight

distance(ft)

Calculated Design

Designspeed(km/h)

Stoppingsight

distance(m)

Calculated Design

15 80 3.0 3 20 20 0.6 1

20 115 6.1 7 30 35 1.9 225 155 11.1 12 40 50 3.8 4

30 200 18.5 19 50 65 6.4 735 250 29.0 29 60 85 11.0 11

40 305 43.1 44 70 105 16.8 17

45 360 60.1 61 80 130 25.7 2650 425 83.7 84 90 160 38.9 39

55 495 113.5 114 100 185 52.0 52

60 570 150.6 151 110 220 73.6 74

65 645 192.8 193 120 250 95.0 9570 730 246.9 247 130 285 123.4 124

75 820 311.6 312

80 910 383.7 384a Rate of vertical curvature,K, is the length of curve per percent algebraic difference in

intersecting grades (A). K= L/A

K-values for adequate SSD

Design Controls for Crest Vertical Curves Based on SSD


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Example 3.6

Solve Example Problem 5 using the K-values in Table 3.2.

f t

KALm

00.741

3247


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Sag Vertical Curves

Four criteria for establishing length ofsag curves

Headlight sight distance Passenger comfort

Drainage control

General appearance


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Headlight Sight Distance

At night, the portion of highway that is visibleto the driver is dependent on the position ofthe headlights and the direction of the lightbeam

Headlights are assumed to be 2 ft (600 mm)and 1-degree upward divergence of the light

beam from the longitudinal axis of the vehicle Equations 3-19 through 3-23 describe the

required sight distance for sag curves


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Sag Vertical Curve Length

The most controlling factor is headlightsight distance

If for economic reasons such lengthscannot be provided, fixed sourcelighting should be provided to assist the

driver.


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Min Sag Curve Length

Like crest curves, we need expressionsfor determining the minimum length of

crest curve required for adequate SSD

LSfortan(200

2

3.20Eq

LSfortan(200

3.19Eq.

2

A

SHSL

SH

SSDAL

m

m


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For the sight distance required to provide

adequate SSD, current AASHTO designstandards use the following specifications:

H(headlight height) = 2.0 ft (600 mm) (headlight angle) = 1


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Minimum Sag Curve Length

US Customary Metric

or SSD L

A

.+Lm

SSD53400SSD2

A

.+Lm

SSD53120SSD2

(3.22)

Substituting the recommended values for beta and Hgives:

If not sure which equation to use, assumeSSD < L first (for either sag or crestcurves)


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K Values for Adequate SSD

US Customary Metric

Rate of vertical

curvature, Ka

Rate of vertical

curvature, Ka

esign

speed

mi/h)

Stopping

sight

distance

(ft)Calculated Design

Design

speed

(km/h)

Stopping

sight

distance

(m)Calculated Design

15 80 9.4 10 20 20 2.1 3

20 115 16.5 17 30 35 5.1 6

25 155 25.5 26 40 50 8.5 9

30 200 36.4 37 50 65 12.2 13

35 250 49.0 49 60 85 17.3 18

40 305 63.4 64 70 105 22.6 23

45 360 78.1 79 80 130 29.4 30

50 425 95.7 96 90 160 37.6 38

55 495 114.9 115 100 185 44.6 45

60 570 135.7 136 110 220 54.4 55

65 645 156.5 157 120 250 62.8 63

70 730 180.3 181 130 285 72.7 73

75 820 205.6 206

80 910 231.0 231

Rate of vertical curvature, K, is the length of curve per percent algebraic difference in

intersecting grades (A). K=L/A

Design Controls for Sag Vertical Curves Based on SSD

Table 3.3


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Passing Sight Distance & CrestVertical Curve Design

Only a factor for vertical curves

A consideration for two-lane highways

Sag curves have unobstructed sightdistance

Assume driver eye height and height of

object on roadway surface both 3.5


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Stopping Sight Distance &Horizontal Curve Design

Adequate sight distance must be provided inthe design of horizontal curves

Cost of right of way or the cost of movingearthen materials often restrict designoptions

When such obstructions exist, stopping sight

distance is checked and measured along thehorizontal curve from the center of thetraveled lane


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Sight Distance Relationships

)(cos90

SSDforsolving),90cos1(

Mforsolvecan3.38)(eqcurvehorizontalsimpleof

ordinatemiddleforequationgeneralintongSubstituti

180

angle)central(not thedistancesightstoppingrequired

thetoequallengtharcanforangleThe?isWhat

*180

1

s

s

v

svv

v

vs

v

s

sv

R

MRRSSD

RSSDRM

RSSD

RSSD


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Sight Distance Example

Horizontal curve with 2000 radius;12lanes; 60mph design speed.

Determine the distance that must becleared from the inside edge of theinside lane to provide sufficientstopping sight distance.

Si h Di E l


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Sight Distance ExampleContinued

f tM

RR

R

SSD

RM

s

v

vvs

33.20))1994(1417.3

)570(90cos1(1994

1994620002/12

)

90

cos1(

*SSD is determined from Table 3.1 for 60mph design speed

chapter 3 part ii st

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