dr. wa'el m. albawwab [email protected] ecgd4107 pavement engineering summer 2008 sat....

Dr. Wa'el M. Albawwab albawwab@Dr. Wa'el M. Albawwab [email protected]

11

ECGD4107ECGD4107Pavement EngineeringPavement Engineering

Summer 2008Summer 2008

Sat. 15:30-18:30 PMSat. 15:30-18:30 PMK004K004


22

Vertical AlignmentsVertical Alignments

Geometric CharacteristicsGeometric Characteristics

Sight Distances: Crests & SagsSight Distances: Crests & Sags

Lecture 3Lecture 3


33

Crest vertical curvesCrest vertical curves

Sag vertical curvesSag vertical curves

Types of Vertical AlignmentTypes of Vertical Alignment


44

Crest Vertical CurvesCrest Vertical Curves


55

Crest Vertical CurvesCrest Vertical Curves


66

Sag Vertical CurvesSag Vertical Curves


77

Sag Vertical CurvesSag Vertical Curves


88

GG11 = initial roadway grade in percent = initial roadway grade in percent

GG2 2 = final roadway grade in percent= final roadway grade in percent

A = Absolute value of difference in gradesA = Absolute value of difference in grades(initial minus final, usually in percent)(initial minus final, usually in percent)

Elements of Vertical CurvesElements of Vertical Curves


99

PVC = point of the vertical curve (the initial point of PVC = point of the vertical curve (the initial point of the curve)the curve)

PVI = point of vertical intersection (the intersection PVI = point of vertical intersection (the intersection of initial and final grades)of initial and final grades)

PVT = point of vertical tangent, (the final point of PVT = point of vertical tangent, (the final point of the vertical curve)the vertical curve)



1010

L = length of the curve measured in a constant L = length of the curve measured in a constant elevation horizontal plane elevation horizontal plane

Ascending = +ve gradingAscending = +ve grading

Descending = -ve gradingDescending = -ve grading



1111

Parabolic mathematical expressionParabolic mathematical expression

Provides a constant sloping rateProvides a constant sloping rate

y = axy = ax22 + bx + c + bx + c

Presentation of Vertical CurvesPresentation of Vertical Curves


1212

y = elevation at a corresponding distance x from they = elevation at a corresponding distance x from thebeginning of the vertical curve (PVC)beginning of the vertical curve (PVC)

x = distance from the beginning of the vertical curvex = distance from the beginning of the vertical curve

a & b = constant coefficientsa & b = constant coefficients

c = elevation at the beginning of the vertical curve c = elevation at the beginning of the vertical curve

Presentation of Vertical CurvesPresentation of Vertical Curves


1313

The first derivative is the slope at location xThe first derivative is the slope at location x

dy/dx = 2ax + bdy/dx = 2ax + b

At the PVC, x = 0 and slope = GAt the PVC, x = 0 and slope = G11

b = Gb = G11

Calibration of CoefficientsCalibration of Coefficients


1414

The second derivative is the rate of slope changeThe second derivative is the rate of slope change

dd22y/dxy/dx22 = 2a = (G = 2a = (G22 – G – G11)/L)/L

a = (Ga = (G22 – G – G11)/2L)/2L

At the PVC, x = 0 and y = yAt the PVC, x = 0 and y = y00

c = yc = y00



1515

Then the parabolic vertical curve can be definedThen the parabolic vertical curve can be definedin terms of Gin terms of G11, G, G22, L, and y, L, and y00 as: as:


01212 yxGx

L2)GG(

y


1616

SignificanceSignificance

For algebraic grade differences of 2% and For algebraic grade differences of 2% and greater, and design speeds equal to or greater greater, and design speeds equal to or greater than 60 km/h,than 60 km/h, the minimum length of vertical the minimum length of vertical curve in meters should not be less than twice the curve in meters should not be less than twice the design speed.design speed.

For algebraic grade differences of less than 2%, For algebraic grade differences of less than 2%, or design speeds less than 60 km/h, the vertical or design speeds less than 60 km/h, the vertical curve length should be a minimum of 60 m.curve length should be a minimum of 60 m.


1717

SignificanceSignificance

Vertical curves are not required where the Vertical curves are not required where the algebraic difference in grades is 0.5% or less. algebraic difference in grades is 0.5% or less.

Since flat vertical curves may develop poor Since flat vertical curves may develop poor drainage at the level section, adjusting the gutter drainage at the level section, adjusting the gutter grade or shortening the vertical curve may grade or shortening the vertical curve may overcome any drainage problems.overcome any drainage problems.


1818

A 600 ft long equal-tangent sag vertical curve has A 600 ft long equal-tangent sag vertical curve has

the PVC at station 170+00 with an elevation of the PVC at station 170+00 with an elevation of

1000 ft. The Initial grade is -3.5% and the final 1000 ft. The Initial grade is -3.5% and the final

grade is +0.5%. Determine the stationing and grade is +0.5%. Determine the stationing and

elevation of PVI, PVT and the lowest point on this elevation of PVI, PVT and the lowest point on this

curve.curve.

Example1Example1


1919

Since the curve is equal tangent, the PVI will be 300 Since the curve is equal tangent, the PVI will be 300 ft or 3 stations from the PVC, the PVT will be 600 ft ft or 3 stations from the PVC, the PVT will be 600 ft or 6 stations from PVC, therefore, the stationing of or 6 stations from PVC, therefore, the stationing of PVI and PVT are: 173 + 00 and 176 + 00 PVI and PVT are: 173 + 00 and 176 + 00 respectively.respectively.

Elevation of PVI is:Elevation of PVI is:

1000 + (-3.5/100)(3)(100) = 989.5 ft 1000 + (-3.5/100)(3)(100) = 989.5 ft

Example1 / SolutionExample1 / Solution


2020

The elevation of the PVT is:The elevation of the PVT is:

989.5 + (0.5/100)(3)(100) = 991.0 ft989.5 + (0.5/100)(3)(100) = 991.0 ft

Coefficients of the curve:Coefficients of the curve:

b = G1 = -3.5/100 = -0.035b = G1 = -3.5/100 = -0.035

a = (Ga = (G22 – G – G11)/2L = [0.5-(-3.5)]/[(100)(2)(600)] = 1/30000)/2L = [0.5-(-3.5)]/[(100)(2)(600)] = 1/30000

yy00 = 1000 ft = 1000 ft

Example1 / ContinuedExample1 / Continued


2121

The function of the curve:The function of the curve:

y = xy = x22/30000 – 0.03.5x + 1000/30000 – 0.03.5x + 1000

The lowest point on the curve:The lowest point on the curve:

dy/dx = 0 at x = 525.0 ftdy/dx = 0 at x = 525.0 ft

The level of this point is:The level of this point is:

y = 990.8125 fty = 990.8125 ft

Example1 / ContinuedExample1 / Continued


2222

Offsetting of Vertical CurvesOffsetting of Vertical Curves


2323

12

2212

GGA where

xL200

Ax

)L2)(100(

GGY

Offset (Y): the vertical distance from the initial Offset (Y): the vertical distance from the initial tangent to the curve linetangent to the curve line



2424

800AL

Ym

200AL

Yf

Substituting for the middle offset at x = L/2Substituting for the middle offset at x = L/2


Substituting for the final offset at x = LSubstituting for the final offset at x = L


2525

K: the horizontal distance required to cause a 1% K: the horizontal distance required to cause a 1% change in the slope of the vertical curvechange in the slope of the vertical curve

AL

K

K-value can be used to calculate the highest and K-value can be used to calculate the highest and lowest point location in crest and sag curveslowest point location in crest and sag curves

1max GKx



2626

SSD for Vertical CurvesSSD for Vertical Curves


2727

s = SSDs = SSD

HH11 = height of driver’s eye above road surface = height of driver’s eye above road surface

HH22 = height of object above road surface = height of object above road surface



2828

From the property of parabola for an equal From the property of parabola for an equal tangent curve, tangent curve,

for S < L:for S < L:

For S > L:For S > L:

221

2

)HH(200

AsL

A

)HH(200s2L

221



2929

To determine the minimum length of curve To determine the minimum length of curve required to provide adequate SSD (recommended required to provide adequate SSD (recommended by AASHTO’s Green Book)by AASHTO’s Green Book)

Use: HUse: H11 = 3.5 ft and H = 3.5 ft and H22 = 2.0 ft = 2.0 ft

Use:Use: )G01.0348.0(30v

vt47.1SSD2



3030

For L > SSDFor L > SSD

For L < SSDFor L < SSD

2158SSDA

L2

m

A2158

SSD2Lm



3131



3232

The critical concern for sag vertical curve design The critical concern for sag vertical curve design is the actual length of roadway illuminated by the is the actual length of roadway illuminated by the vehicle headlights during nighttimevehicle headlights during nighttime

In day light, the driver's sight distance on a sag In day light, the driver's sight distance on a sag vertical curve is unrestricted.vertical curve is unrestricted.



3333

L can be expressed by the parabola properties as:L can be expressed by the parabola properties as:

For L > SFor L > S

For L < SFor L < S

)tanSH(200

SAL

2

A)tanSH(200

S2L



3434

To determine the minimum length of curve To determine the minimum length of curve required to provide adequate SSD (recommended required to provide adequate SSD (recommended by AASHTO’s Green Book)by AASHTO’s Green Book)

Use H = 2.0 ft and Use H = 2.0 ft and = 1.0 degrees = 1.0 degrees

UseUse )G01.0348.0(30v

vt47.1SSD2



3535

For L > SSDFor L > SSD

For L < SSDFor L < SSD


SSD5.3400SSDA

L2

m

ASSD5.3400

SSD2Lm


3636

SSD5.3400

SSDK

2

2158SSD

K2

For crest vertical curves:For crest vertical curves:

For sag vertical curves:For sag vertical curves:



3737

In practical calculations of LIn practical calculations of Lmm for both crest and for both crest and

sag vertical curvessag vertical curves

The assumption that L > SSD is always the The assumption that L > SSD is always the dominant decision madedominant decision made

When computing SSD, G is always ignoredWhen computing SSD, G is always ignored



3838

Passing Sight Distances on crest vertical curves:Passing Sight Distances on crest vertical curves:

For L > PSD:For L > PSD:

For L< PSD:For L< PSD:

864PSDA

L2

m

A864

PSD2Lm

PSD for Vertical CurvesPSD for Vertical Curves


3939

As was the case for stopping sight distance,As was the case for stopping sight distance,

It is typically assumed L > PSDIt is typically assumed L > PSD

864

PSDK

2



4040

)5H(800ASSD

Lc

2

m

A)5H(800

SSD2L cm


Underpass distance on sag vertical curves:Underpass distance on sag vertical curves:

For L > S:For L > S:

For L < S:For L < S:

HHcc: is the net clearance height of overpass : is the net clearance height of overpass

structure above road surfacestructure above road surface


4141

Study this lecture (AASHTO 2001 – Ch. 3)Study this lecture (AASHTO 2001 – Ch. 3)

Review geometric alignments (Practical)Review geometric alignments (Practical)

Due to next lecture:Due to next lecture:

dr. wa'el m. albawwab [email protected] ecgd4107 pavement engineering summer 2008 sat....

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