ce 404 lecture 5,4(vertical alighnment)

LECTURE #4,5

Transi t ion curve

vert ica l a l ignment

2016CE 404. Highway Engineering

1

HIGHWAY ENGINEERING

(CE 404)

بسم هللا الرحمن الرحيم

By

ALGHRAFY

Transition curve


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Transition curve is provided to change the horizontal alignment from straight to circular curve gradually and has a radius which decreases from infinity at the straight end (tangent point) to the desired radius of the circular curve at the other end (curve point)

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Transition curve


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point of intersection (PI)

tangent to spiral (TS)

spiral to curve (SC)

curve to spiral (CS)

spiral to tangent (ST)

point of curvature (PC)

point of tangency (PT)

The length of the spiral (LS)

shift distance or throw distance (T)

tangent distance (TS)

the long chord (LCS) of the spiral

LT and ST long tangent and the short tangent of the spiral.

spiral point of intersection (SPI).

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Transition curve


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Transition curve is not necessary when :

o In low speed environments where drivers regulate their travel speed from their judgment of the apparent curvature of the road ahead.

o On Large radii curves R > 900𝑚

o On small radii curves in low speed environments where pavement widening to accommodate the tracking widths required by heavy vehicles is necessary.

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Transition curve objectives


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o To introduce gradually the centrifugal force between the tangent point and the beginning of the circular curve, avoiding sudden jerk on the vehicle. This increases the comfort of passengers.

o To enable the driver turn the steering gradually for his own comfort and security,

o To provide gradual introduction of super elevation, . And

o To provide gradual introduction of extra widening

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Transition curve


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o Spiral or clothoid,o Cubic parabola, ando Lemniscate.

Type of transition curve

Case-1:Rate of change of centrifugal acceleration

Length of transition curve

𝐿𝑆 =0.0215𝑉3

𝐶 × 𝑅

𝐶 =80

75 + 𝑉

0.5 < C < 0.8 IR0.3 < C < 0.9 AASHTO0.3 < C < 0.9 BC

Where, Ls= length of transition curve in ‘m’ C= allowable rate of change of centrifugal accleration, m/ sec² R= Radius of the circular cu rve in ‘m’ V= speed kmph

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Transition curve


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case-2:Rate of introduction of super-elevation

𝑳𝑺 =𝒆 ×𝒘 × 𝒏𝟏 × 𝒃𝒘

𝑮𝒎𝒂𝒙

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Transition curve


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case-3:Minimum Spiral Curve Constant

𝑳𝑺 =𝑨𝒎𝒊𝒏𝟐

𝑹

By empirical formulacase-4:

𝑳𝑺 =𝟐. 𝟕𝑽𝟐

𝑹For plane and rolling terrain:

For mountainous and steep terrain: 𝑳𝑺 =𝑽𝟐

𝑹

The design length of transition curve(Ls) will be the highest value of case-1,2,3 and 4

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Transition curve


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Transition curve


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∆𝑪=𝐷 × 𝐿𝐶30.5

General Spiral Equations

∆𝑺=𝐷 × 𝐿𝑆61

∆= ∆𝑪 + 2∆𝑆

𝐿 = 𝐿𝐶 + 2𝐿𝑆

𝑇𝑆 = 𝑄 + 𝑅 + 𝑇 𝑡𝑎𝑛∆

2

𝐸𝑆 =(𝑅 + 𝑇)

𝑐𝑜𝑠∆2

− 𝑅

𝑇 =𝐿𝑆2

24𝑅

∆𝑺=𝐿𝑆2𝑅

𝐿𝐶 = 𝑹(∆ − 2∆𝑆)

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Vertical Alignment


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The vertical alignment is the elevation or profile of the centerline of the road.Consists of straight sections of the highway known as grades, or tangentsconnected by vertical curves.The design of vertical alignment involves selection of suitable grades for the tangent sections and the design of the vertical curves.Topography of the area which the road traverses has a significant impact on the design of the vertical alignment.

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Vertical Alignment


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Vertical Alignment


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Vertical Alignment

Geometric Elements of Vertical Curves

Vertical Grades

Passing Lanes

Sight Distance

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Vertical Alignment


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The design of the vertical alignment is influenced by

o consideration of terrain (Topography) ,

o Type of roads

o Vehicle Operational Characteristics

o Design speed

o cost, and

o safety.

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Vertical Alignment


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Highway engineers generally separate the characteristics of variations in typography according to the terrain:

Level terrain: highway sight distances, as governed by both horizontal and vertical restrictions, are usually long or can be made without construction difficulty.Rolling terrain: natural slopes consistently rise above or fall below the road grade, and occasional steep slopes offer some restriction to normal alignment.Mountainous terrain: longitudinal and transverse changes in the elevation of the ground are usually abrupt, and benching and side hill excavation are frequently needed.

Consideration of terrain, Type of roads, and Design speed

Vertical Alignment


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Passenger cars: Grades as steep as 4% to 5% generally do not affect speed of most vehicles (may affect some compact/subcompact vehicles)Trucks: Effects on speed much more important

Vehicle Operational Characteristics

The term “critical length of grade” is used to indicate the maximum length of a specified ascending gradient upon which a loaded truck can operate without an unreasonable reduction in speed (commonly 10 mph [15 km/h])

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Vertical Grades


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Maximum Grades

o Maximum grades have been established, based on operating characteristics of design vehicle.

o Max grade vary from 5% for DS of 70 mi/h to 12% for DS of 30 mi/h.o See Table for recommended values of max grades.o Max grades in the Table should not be used frequently, particularly when grades

are long and traffic include high % of trucks.o on low volume rural highways, maximum grades may be increase by 2% When :o grade length (< 500 ft) and o roads are one-way in the downgrade direction,

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Vertical Grades


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Maximum Grades

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Vertical Grades


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Vertical Grades


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Minimum Grades

o Minimum grades depend on the drainage condition of the highway.o 0% grades may be used on uncurbed pavements with adequate cross-slopes to

laterally drain the surface water.o When pavements are curbed, longitudinal flow should be provided to facilitate

the longitudinal flow of surface water.o Min of (0.5%).

Grade Change Without Vertical Curves

Designing a sag or crest vertical point of intersection without a vertical curve is generally acceptable where the grade difference (A) is:o 1.0 percent or less for design speeds equal to or less than 45 mph [70 km/h]o 0.5 percent or less for design speeds greater than 45 mph [70 km/h].

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Vertical Curve


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Used to provide gradual change from one tangent grade to another so that vehicle may run smoothly as they traverse the highway.o Classes or types

Change in grade: A = G1 – G2

where G is expressed as % (positive /, negative \)

G2

G1

G2

G1

G2

G1

G2

G1

Crest vertical curves, A is positiveSag vertical curves, A is negative

G1

G2

G2

G2

G2

G1G1

G1

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Vertical Curve


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Properties of Vertical Curves

For a vertical curve, the general form of the parabolic equation is;

y = ax2 + bx + c

Y

X

From an equal tangent parabola, it can be written as;

y = ax2

adx

dy2

2

2

But The rate of change of slope can also be written as;

L

GG

dx

dy 21

2

2

L

GGa

2

21

The rate of change of slope

L

Aa

2

2

2x

L

Ay

2

2

4L

xey

842

2 ALL

L

Ae

2

22

8x

L

ey

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Design Of Vertical Curve


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Elevation and station

Crest vertical curves

H1= HPVC + g1 x - (A)x2 / 2L

H2= HPVT + g2 x - (A)x2 / 2L

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29Elevation on Vertical Curve

Sag vertical curves

H1= HPVC - g1 x + (A)x2 / 2L

H2= HPVT - g2 x + (A)x2 / 2L

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The elevation and location of the high or low point

crest vertical curves

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Elevation on Vertical Curve

Sag vertical curves

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32Length of Crest (summit) Vertical Curves

G1G2

PVI

PVTPVC

h2h1

L

SSD

221

2

22100 hh

SSDAL

A

hhSSDL

2

212002

For SSD < L For SSD > L

Line of Sight

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Assumptions for design h1 = driver’s eye height = 1.07 m.

h2 = tail light height = 0.61 m.

Simplified Equations

658

2SSDA

L A

SSDL658

2


SSD= d1+d2

d1 = 0.278 x v x t

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658

2SSDA

L

For SSD < L

KAL

658

2SSDK

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Length of Crest (summit) Vertical Curves

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Length of Crest (summit) Vertical Curves

passing sight distance

864

2PSDA

L

A

PSDL864

2

For PSD < L

For PSD > L

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37Length of Sag Vertical Curves

The selection of the minimum length of a sag vertical curve is controlled by (1) comfort while driving on the curve, (2) general appearance of the curve, and (3) SSD provided by the headlight,

(1) comfort while driving on the curve,

(2) general appearance of the curve

395

2AvL

Minimum Length based on Comfort Criterion.

Minimum length for the general appearance criterion

AL 5.30

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G1 G2

PVI

PVTPVC

h2=0h1

L

Light Beam Distance (SSD)

tan200 1

2

Sh

SSDAL

A

SSDhSSDL

tan2002 1


headlight beam (diverging from LOS by β degrees)

Length of Sag Vertical Curves

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Assumptions for design h1 = headlight height = 0.61 m.

β = 1 degree

Simplified Equations

SSD

SSDAL

5.3120

2

A

SSDSSDL

5.31202


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40Length of Sag Vertical Curves

For SSD < L

KAL

SSD

SSDAL

5.3120

2

SSD

SSDK

5.3120

2

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Length of Sag Vertical Curves



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Length of Sag Vertical Curves Undercrossing

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Thank You


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ce 404 lecture 5,4(vertical alighnment)

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