ROAD GEOMETRIC DESIGNTECHNICAL SECTIONNRAP/MoPW KABUL-AFGHANISTAN

PART : 03BY: MALYAR TALASH ROAD DESIGN ENGINEERmalyar@nrap.gov.af +93786575033

VERTICAL ALIGNMENT• The topography of the land traversed has

an influence on the alignment of roads and streets. Topography affects horizontal alignment, but has an even more pronounced effect on vertical alignment. To characterize variations in topography, engineers generally separate it into three classifications according to terrain—level, rolling, and mountainous.

VERTICAL ALIGNMENT• The two major aspects of vertical

alignment are vertical curvature, which is governed by sight distance and comfort criteria, and gradient which is related to vehicle performance and level of service.

• Vertical curves are applied to effect the transition between straight gradients. There are two types of vertical curves, namely crest vertical curves and sag vertical curves.

VERTICAL ALIGNMENTCrest and Sag Curves

PLAIN TERRAINThe plain terrain is defined as area which maintains between 0 and 10 No of Five meter contour lines per km plain or gently rolling tertian, this offers few obstacles to the construction of a road or in other words, the natural ground, cross slopes (ie. perpendicular to natural ground contours) in a flat terrain are generally below 3 %

PLAIN TERRAIN

ROLLING TERRAINBetween11 to 25 No Five meter contour lines per km. rolling, hilly or Foothill County where the slope is generally rise and fall moderately and where occasional steep slopes are encountered, resulting in some restrictions in alignment, in other words, the natural ground cross slopes in rolling terrain is generally between 3 – 25 %.

ROLLING TERRAIN

MOUNTAINOUS TERRAINGreater than 25 No .five meter contour lines per km rugged, hilly and mountainous county and river gorges this class of terrain imposes definite restrictions on the alignment and often involves long steep grades and limited sight distance. In other words, the naturals ground cross slopes in mountainous terrain are generally above 25 %

MOUNTAINOUS TERRAIN

VERTICAL ALIGNMENT• In level terrain, highway sight distances,

as governed by both horizontal and vertical restrictions, are generally long or can be made to be so without construction difficulty or major expense.

• In rolling terrain, natural slopes consistently rise above and fall below the road or street grade, and occasional steep slopes offer some restriction to normal horizontal and vertical roadway alignment.

VERTICAL ALIGNMENTIn mountainous terrain, longitudinal and transverse changes in the elevation of the ground with respect to the road or street are abrupt, and benching and side hill excavation are frequently needed to obtain acceptable horizontal and vertical alignment.

VERTICAL GRADES Grades for local residential streets should be as level as practical, consistent with the surrounding terrain. The gradient for local streets should be less than 15 %. Where grades of 4 % or steeper are necessary, the drainage design may become critical. On such grades special care should be taken to prevent erosion on slopes and open drainage facilities.

VERTICAL GRADES For streets in commercial and industrial areas, gradient design desirably should be less than 8 present, grades should be desirable less than 5 present, and a flatter grades should be encouraged. To provide for proper drainage, the desirable minimum grade for streets with outer curb should be 0.30 percent, but minimum grade of 0.20 percent maybe used.

VERTICAL GRADES The second issue to be considered for vertical alignment is the maximum gradient. Recommendations for maximum gradients using the design speed approach are given in below

VERTICAL GRADES

17

Descending Grades• Problem is increased speeds and loss of control

for heavy trucks• Runaway vehicle ramps are often designed and

included at critical locations along the grade• Ramps placed before each turn that cannot be

negotiated at runaway speeds• Ramps should also be placed along straight

stretches of roadway, wherever unreasonable speeds might be obtained

• Ramps located on the right side of the road when possible

18

Maximum Grades• Passenger cars – 4% to 5% no

problem• Upgrades: trucks average 7%

decrease in speed• Downgrades: trucks average speed

increase 5%

19

Vertical Curves• Crest – stopping, or passing sight

distance controls• Sag – headlight/SSD distance, comfort,

drainage and appearance control • Green Book vertical curves defined by

K = L/A = length of vertical curve/difference in grades (in percent) = length to change one percent in grade

20

Vertical Curve AASHTO Controls (Crest)

• Minimum length must provide stopping sight distance (SSD) or “S” in equations

• Two situations (both assume h1=3.5’ and h2=2.0’)

Source: Transportation Engineering On-line Lab Manual, http://www.its.uidaho.edu/niatt_labmanual/

21

Assistant with Target Rod (2ft object height)

Observer with Sighting Rod (3.5 ft)

Notation• Curve point naming is similar to horizontal

curves, 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)

Notation• G1 is initial roadway grade

– Also referred to as initial tangent grade• G2 is final roadway (tangent) grade• A is the absolute value of the difference in

grades (generally expressed in percent)– A = |G2 – G1|

• L is the length of the vertical curve measured in a horizontal plane (not along curve center line, like horizontal curves)

Fundamentals• Parabolic curves are generally used for

design– Parabolic function y = ax 2 + bx + c

y = 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

First Derivative• First derivative gives slope

–

• At PVC, x = 0, so , by definition

• G1 is initial slope (in ft/ft or m/m) as previously defined

baxdxdy

2

1Gdxdyb

Second Derivative• Second derivative gives rate of

change of slope

• However, the average rate of change of slope, by observation, can also be written as

• Giving,

adx

yd 22

2

LGG

dxyd 122

2

LGGa

212

Offsets are vertical distances from initial tangent to the curve

Offsets

• For an equal tangent parabola,

– Y = offset (in m or ft) at any distance, x, from the PVC

– A and L are as previously defined• It follows from the figure that,

2

200x

LAY

200

800ALY

ALY

f

m

Offset Formulas

“K” Values• The rate of change of grade at successive

points on the curve is a constant amount for equal increments of horizontal distance, and

• Equals the algebraic difference between intersecting tangent grades divided by the length of curve, or A/L in percent per ft (m)

• The reciprocal L/A is the horizontal distance required to effect a 1% change in gradient and is, therefore, a measure of curvature

• The quantity L/A is termed ‘K’

“K” Values• The K-value can be used directly to

compute the high/low points for crest/sag vertical curves (provided the high/low point is not at a curve end) by,– xhl = K |G1|– Where x = distance from the PVC to the

high/low point• Additionally, K-values have important

applications in the design of vertical curves, which we will see shortly

Vertical Curves• Controlling factor: sight distance• Stopping sight distance should be

provided as a minimum• Rate of change of grade should be kept

within tolerable limits• Drainage of sag curves is important

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

Vertical Alignment Relationships

1

2

200

800

200

GKxALK

ALY

ALY

xL

AY

hl

f

m

LGGa

adx

yd

Gdxdyb

xatPVC

baxdxdy

cbxaxy

2

2

:0,

2

12

2

2

1

2

Stopping Sight Distance & Crest Curves• Two different factors are important

for crest curves– The driver’s eye height in vehicle, H1

– Height of a roadway obstruction object, H2

SSD & Curve Design• It is necessary, when designing

vertical curves, to provide adequate stopping-sight distance (SSD)

• Because curve construction is expensive, we want to minimize curve length, subject to adequate SSD

Minimum Curve Length• By using the properties of a

parabola for an equal tangent curve, it can be shown that the minimum length of curve, Lm, for a required SSD is

LSfor 3.14 Eq )(200

2

LSfor 3.13 Eq )(200

221

221

2

AHH

SL

HHASL

m

m

Minimum Curve Length• For the sight distance required to

provide adequate SSD, current AASHTO design standards use the following specifications:

– H1 (driver’s eye height) = 3.5 ft (1080 mm)

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

Minimum Curve Length• Substituting these values into

previous two equations yields:

LSSDfor 4042

LSSDfor 404

2

ASSDL

SSDAL

m

m

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

Example

Sag Vertical Curves• Four criteria for establishing length

of sag curves– Headlight sight distance– Passenger comfort– Drainage control– General appearance

Headlight Sight Distance• At night, the portion of highway that is

visible to the driver is dependent on the position of the headlights and the direction of the light beam

• 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

Sag Vertical Curve Length• The most controlling factor is

headlight sight distance• If for economic reasons such

lengths cannot be provided, fixed source lighting should be provided to assist the driver.

Min Sag Curve Length• Like crest curves, we need

expressions for determining the minimum length of crest curve required for adequate SSD

LSfor tan(2002

3.20 Eq

LSfor tan(200

3.19 Eq.2

ASHSL

SHSSDAL

m

m

45

Vertical Curve AASHTO Controls (Crest)

Since you do not at first know L, try one of these equations and compare to requirement, or use L = KA (see tables and graphs in Green Book for a given A and design speed)

46

Chart vs computedFrom chart V = 60 mph K = 151 ft / % change

For g1 = 3 g2 = - 1 A = |g2 – g1| = |-1 – 3| = 4

L = ( K * A) = 151 * 4 = 604

47

Sag Vertical Curves• Sight distance is governed by

night- time conditions– Distance on curve illuminated by

headlights need to be considered• Driver comfort• Drainage• General appearance

48

Vertical Curve AASHTO Controls (Sag) Headlight Illumination sight distance

S < L: L = AS2

400 + (3.5 * S)

S > L: L = 2S – (400 + 3.5S) A

49

Vertical Curve AASHTO Controls (Sag)

• For driver comfort use: L > AV2

46.5 (limits g force to 1 fps/s)

• To consider general appearance use:

L > 100 A

50

Sag Vertical Curve: ExampleA sag vertical curve is to be designed to join a –3% to a +3% grade. Design speed is 40 mph. What is L?Skipping steps: SSD = 313.67 feet S > LDetermine whether S<L or S>L

L = 2(313.67 ft) – (400 + 2.5 x 313.67) = 377.70 ft [3 – (-3)]313.67 < 377.70, so condition does not apply

51

Sag Vertical Curve: ExampleA sag vertical curve is to be designed to join a –3% to a +3% grade. Design speed is 40 mph. What is L?Skipping steps: SSD = 313.67 feet

L = 6 x (313.67)2 = 394.12 ft 400 + 3.5 x 313.67 313.67 < 394.12, so condition applies

52

Sag Vertical Curve: ExampleA sag vertical curve is to be designed to join a –3% to a +3% grade. Design speed is 40 mph. What is L?Skipping steps: SSD = 313.67 feetTesting for comfort:L = AV2 = (6 x [40 mph]2) = 206.5 feet 46.5 46.5Testing for appearance:L = 100A = (100 x 6) = 600 feet

Minimum Curve Length

For the sight distance required to provide adequate SSD, current AASHTO design standards use the following specifications:

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

Minimum Sag Curve Length

U S C u s t o m a r y M e t r i c

F o r S S D < L

SSD53400SSD 2

. +

AL m

SSD53120

SSD 2

. +

AL m ( 3 .2 1 )

F o r S S D > L

A. + L m

SSD53400SSD2

A. + L m

SSD53120SSD2 ( 3 .2 2 )

Substituting the recommended values for beta and Hgives:

If not sure which equation to use, assume SSD < L first (for either sag or crest curves)

K Values for Adequate SSDUS Customary Metric

Rate of vertical curvature, Ka

Rate of vertical curvature, Ka Design

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

a 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

Passing Sight Distance & Crest Vertical Curve Design

• Only a factor for vertical curves• A consideration for two-lane

highways• Sag curves have unobstructed sight

distance• Assume driver eye height and height

of object on roadway surface both 3.5’

Stopping Sight Distance & Horizontal Curve Design

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

• Cost of right of way or the cost of moving earthen materials often restrict design options

• When such obstructions exist, stopping sight distance is checked and measured along the horizontal curve from the center of the traveled lane

Sight Distance Relationships

)(cos90

SSDfor solving),90cos1(

Mfor solvecan 3.38) (eq curve horizontal simple of ordinate middlefor equation general into ngSubstituti

180angle) central (not the distancesight stopping required

the toequallength arcan for angle The ? isWhat

*180

1

s

s

v

svv

vvs

v

s

sv

RMRRSSD

RSSDRM

RSSD

RSSD

Sight Distance ExampleHorizontal curve with 2000’ radius;

12’lanes; 60mph design speed. Determine the distance that must be cleared from the inside edge of the inside lane to provide sufficient stopping sight distance.

Sight Distance Example Continued

ftM

RRRSSDRM

s

v

vvs

33.20))1994(1417.3

)570(90cos1(1994

1994620002/12

)90cos1(

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