Chapter III Geometric design of Highways

Tewodros N. www.tnigatu.wordpress.com

tedynihe@gmail.com

• Introduction • Appropriate Geometric Standards • Design Controls and Criteria

• Design Class • Sight Distance • Design Vehicle • Traffic Volume and • Design Speed

• Geometric Design Elements – Horizontal Alignment

• Straights (Tangents) • Circular Curves • Super elevation • Transition Curves • Widening of Curves

– Vertical Alignment • Vertical Curves • Length Of Vertical Curves • Sight Distances At Underpass Structures: • Grades and Grade Control

– Cross-Section

Lecture Overview

Your Chosen Route

But, the road could not have such an alignment with sharp edges!

A

B

Introduction

• Horizontal Alignment consists of: – Straight lines – Curves:

» Simple circular curves » Compound Curves » Reverse Curves » Transitional Spirals

Considerations

A proper relation should be established between the design speed and curvature and their joint relations with super elevation and side friction. drainage

Simple Circular Curves - Terms

∆=deflection angle L=Length of Curve C=Chord Length R=Radius of Curvature M=Middle Ordinate E=External Distance T=Length of Tangent P.I.=Point of Intersection TC=Tangent to Circle CT=Circle to Tangent

Relations

[ ][ ])2/cos(1

1)2/sec(

)2/sin(2)2/tan(

∆−=−∆=

∆=∆=

∆=

RMRERL

RCRT

Compound Circular Curves - Terms

Reverse Curves

Stability of a VEH

a=radial acceleration; m=mass of vehicle; V=speed of vehicle; R=Radius of curvature; F=Frictional Resistance; R’=Reaction

When road has no camber and the VEH is on the verge of overturning

coefficient of friction

a

v

RFRmVF

mgR

′==

=′

//2

µRmV /2

mgF

R’

h

Stability of a VEH

To avoid overturning To avoid side slip

RmV /2

mgF

R’

h

gbRhVmgbRhmV <⇒< // 22

b

gRVmgRmV µµ <⇒< // 22

Resolving the Forces // and |to the road (// to the road) (| to the road)

Stability on Super-elevated Surface Forces & Equilibrium

θθ CosgR

WvWSinF2

=+

gRWv2

N

F

W

θ

Frictional force, F= µ N

NSingR

WvWCos =+ θθ2

e 1

Relations (cont.)

1

1

2

2

22

22

2

+

−=

−=

+

−=

+

−=

gRv

gRv

Tan

gRvCos

gRvSin

WSinCosgR

WvWCosSingR

Wv

WSinCosgR

WvN

µ

µθ

µθµθ

θθθθµ

θθµ

gRv2µ

But the term has a very small value and could be ignored for all practical purposes. Check using typical values like V=50km/hr; m=0.16; and R=100m.

Relations (cont.)

( )R

VR

VgRve

egRvTanThus

12781.96.3

,

222

2

===+⇒

=−=

µ

µθ

V=Km/hr R=m e=m/m µ =dimensionless

Safe Side Friction Factor, µ

• µ is low for high speed design than for high speed design

AASHTO side friction values

0.300.35

0.400.45

0.500.55

20 30 40 50 60 70 80 90 100 110

V, Design Speed (km/hr)

µ, c

oeffi

cient

of f

rictio

n

Design Speed (Km/hr) 50 65 80 100 120 130 Maximum f 0.16 0.15 0.14 0.13 0.12 0.11

Design Speed(Km/hr) 30 40 50 60 70 85 100 120 f 0.33 0.30 0.25 0.23 0.20 0.18 0.15 0.15

Coefficient of Lateral Friction as Recommended by AASHTO

Coefficient Of Lateral Friction as Recommended by TRRL Overseas Road Note 6

Example

1. Compute the minimum radius of a circular curve for a highway designed for 110 km/h. The maximum superelevation rate is 12%.

2. The design speed of asphalt concrete paved highway designed for construction is 80kph. During right-of-way reservation period it was found out that the space available for horizontal curve is only adequate for provision of maximum 200m radius. Can this speed be safely maintained on the road? If not, what should be done?

3. Prepare a table giving chords and deflection angles for staking out a 450 m radius circular curve with a total deflection angle of 17°. The TC point is at station 22 +40. Give deflection angles and chords at 20 m intervals, including full stations.

Super-elevation rate, e

• Is the raising of the outer edge of the road along a curve in-order to counteract the effect of radial centrifugal force in combination with the friction between the surface and tyres developed in the lateral direction

• Maximum value is controlled by: – Climatic conditions: frequency & amount of snow/icing – Terrain condition: flat vs. mountainous – Area type: rural vs. urban – Frequency of very slow moving vehicles

• 0.1m/m is a logical maximum super-elevation • Minimum super-elevation rate is determined by drainage

requirements • UK emax: 0.07 (rural) & 0.05 (urban)

Maximum Degree of Curvature

• Minimum radius for safety (veh. stability) • Limiting value for a given design speed (given emax &

µmax)

• The respective maximum Degree of Curvature is:

• Sharper Curve might justify use of e>emax or a higher dependence on tyre friction or both

( )µ+=

eVR

127

2

min

( )( )22

min

143240127

92.114592.1145maxV

eeVR

D µµ

+=

+==

Application of Super-elevation

Is done in two stages: 1. Neutralizing the camber of the road gradually,

bringing it in to a straight line slope 2. Increasing the slope gradually until design super-

elevation is attained

Application of S. E.

C L

Normal Cross-section

Complete elimination of crown

X-section becomes a straight fall from one side to another

Rotation about the crown

Rotation about the inner edge

C L

h H

h2

h1

h

Where: H=ew w=width of roadway

H

Application of S. E. (curve with transition spirals)

Application of S. E.

for curves with no spirals Super-elevation is started 1/2 to 1/3 into the tangent with the balance being applied with in the curve

Transition Design • Tangent to curve transition

Application of S. E.

w – width of a single lane n – number of lanes

∆××

=

→××=×∆→

××=→=

×∆=→=∆

nweL

nweL

nwexwnxe

LxLx

r

r

rr

Determine the Length of superelevation runoff

Inside edge

outside edgeNormal crown Relative gradient

Tangent Run-outRunoff

Supperelevation runoff

4%

0%

8%

∆

e x

Lr

Application of S. E.

)()( 1w

dr bewnL

∆=

Lr = minimum length of superelevation runoff, ft; = maximum relative gradient, percent; n1 = number of lanes rotated bw = adjustment factor for number of lane rotated w= width of one traffic lane (12 ft.) ed = design superelevation rate, percent;

∆

The length of superelevation runoff is determined based on a maximum acceptable

Difference between longitudinal grades of the axis of rotation

Application of S. E.

Source: pp 170 of Green Book

Application of S. E.

Inside edge

outside edgeNormal crown Relative gradient

Tangent Run-outRunoff

Supperelevation runoff

4%

0%

8%

∆

The length of tangent runout is determined by the length of superelveation run-off and slope Of normal crown and superelevation rate e.

Lr

Lr + Lt

e e×w×n

enc×w×n

rnc

t

nctr

r

nctr

r

Le

eL

eee

LLL

nwenwe

LLL

=

→+

=+

→××

××=

+

Application of S. E.

rd

NCt L

eeL =

Lt = minimum length of tangent runout, ft; eNC = normal cross slope rate, percent ed = design superelevation rate, percent;

Green Book:

Application of S. E.

Transition Curves

• Transition curves provide a gradual change from the tangent section to the circular curve and vice versa.

High-speed roadways with sharp curvature needs transition curves to prevent drivers from encroaching into

adjoining lanes. Advantages:

• Provides an easy-to-follow path so that centrifugal force increases and decreases gradually; lesser danger of overturning/ side-slipping

• Vehicle could keep to the middle of lane while traversing a curve

• Is convenient for the application of super-elevation

• Improved visual appearance, no “kinks”

Transition Curves - Geometry PI: Point of Intersection TS: Tangent to spiral SC: Spiral to Circle CS: Circle to Spiral ST: Spiral to tangent Ls: Total length of spiral Lc: Length of circular curve qs: Central angle of spiral arc of length Ls ∆=total deflection angle of the curve Ys=tangent offset at SC Xs= K=abscissa of shifted PC with reference to TS

Length of Transition Curves

1. Length required for super-elevation runoff • Super-elevation runoff: length of highway needed to accomplish

the change in cross slope from a normal crown section to a fully super-elevated section (or, vice versa)

• The rate of raising the outer edge above the centre line should be:

Design Speed (kph) Ratio V:H 80 1:200 64 1:175 48 1:150 32 1:125

Length of Transition Curves

2. Length required for driver comfort – Rate of change of radial acceleration mustn’t exceed a certain

value which is acceptable to drivers

– Transition curve must, therefore, be long enough to ensure that the radius can be changed at a slow rate

Rva

RmvP r

22

=⇒=

discomfortpassengergreaterforceradialinchangefasterRinchangefaster

Rofchangeofrateaofchangeofratetconsv

r

⇒⇒⇒

⇒≈

1tan

α

Length of Transition Curves

cRV

cRVls

cRvls

lsRv

vlsRv

taCaccradialofchangeofrate

vlst

Rva

r

SCfromTS

r SCfromTS

66.46)()28.0(

.,

33

3

32

)(

2)(

==

=

==∆

∆=

=∆

=∆

−

−

The value of c lie in the range 0.2 to 0.6m/sec3; c=0.3m/sec3 is often used.

Example • A two-lane highway (one 3.6m lane in each direction)

goes from normal crown with2% cross-slopes to 8% superelevation by means of a spiral transition curve. Determine the minimum length of the transition if the difference in grade between the centerline and edge of traveled way is limited to 1/200. Round up to the next largest 20 m interval.

• Draw the superelevation diagram for the transition described in part a. The station of the TS is 120 + 00.

2% 2%

3.6m 3.6m C L

Example

• A roadway goes from tangent alignment to a 250 m circular curve by means of an 80 m long spiral transition curve. The deflection angle between the tangents is 45°. Use formulas to compute Xs, Ys, p, and k. Assume that the station of the P.I., measured along the back tangent, is 250 +00, and compute the stations of the TS, SC, CS, and ST.

Sight Distance on Horizontal Curves

Considering a vehicle at A and an object at O, sight distance should at-least equal to safe stopping distance.

If the angle subtended at the centre of the circle is 2q, then

Centre line of Road

M

Sight distance a measured along centreline of inside lane

Sight line A O

))65.28(1(

)23.57(

,23.571802

RCosRm

RSCosCosR

mRRS

RSD

−=

==−=

=

θ

θθπ

Widening of Highway Curves

Need – Rear wheels don’t follow front wheels, – Trailers fitted on trucks, don’t follow path of trucks

wheels – To have adequate sight-distances – Drivers tend to keep greater clearances with vehicles

coming from the opposite direction and might thus move out of a lane when traversing a curve

– Widening is also required for Design Standards DS1 through DS5 at high fills for the psychological comfort of the driver.

R2

R1 B f

L

Amount of Extra Widening

Let R1=radius of inner rear-wheel on a curved truck (m) R2=radius of outer front-wheel (m) B=width of vehicle f=widening (m) L=Length of vehicle (m)

( )

2

2

2

22

2

22

22

22

222

222

2

222

21

22)2(

2

RL

fRLffRfL

ffRRfRLR

fRLR

fRLRBR

≈−=⇒−=⇒

+−=−=−⇒

−=−

−=−=+

Widening - Methods

• On a simple curve (i.e. with no spirals) widening should be applied on the inside edge of a pavement only. For curves with spirals, widening could be applied on the inside (only) or could be equally divided b/n the inside and outside

• Widening should be attained gradually over the s.e. runoff length but shorter lengths are sometimes used (usually this length is 30 – 60m).

• Widening is costly and very little is gained from a small amount of widening.

Widening on Curves and High Fills (ERA Manual)

Thank You!