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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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Minimum Curve Length
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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Minimum Curve Length
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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Minimum Curve Length
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