Geometric design of Highway Chapter-3-pptx

Chapter Three
Geometric Design of Highways
1

DESIGN CONTROLS AND CRITERIA
• The choice of design controls and criteria is influenced by the
following factors:
The functional classification of the road;
the nature of the terrain;
the design vehicle;
the traffic volumes expected on the road;
the design speed;
the density and character of the adjoining land use; and
economic and environmental considerations
3

Road Functional Classification
• As stated in previous chapters ERA classifies roads in the
country into five functional classes.
• These are Trunk roads, Link roads, Main Access roads,
Collector roads and Feeder roads.
4

• The geometric design elements of a road depend on the
transverse terrain through which the road passes.
• Transverse terrain properties are categorized into four classes
as follows:
Flat Terrain
Rolling Terrain
Mountainous Terrain
Escarpment Terrain
6

• Flat or gently rolling country, which offers few obstacles to the
construction of a road, having continuously unrestricted
horizontal and vertical alignment (transverse terrain slope up
to 5 percent).
7

• ROLLING: Rolling, hilly or foothill country where the slopes
generally rise and fall moderately and where occasional steep
slopes are encountered, resulting in some restrictions in
alignment (transverse terrain slope from 5 percent to 25
percent).
8

• MOUNTAINOUS: Rugged, hilly and mountainous country and
river gorges. This class of terrain imposes definite restrictions
on the standard of alignment obtainable and often involves
long steep grades and limited sight distance (transverse terrain
slope from 25 percent to 50 percent).
9

• ESCARPMENT: We refer to escarpment situations inclusive of
switchback roadway sections, or side hill transverse sections
where earthwork quantities are considerable, with transverse
terrain slope in excess of 50 percent).
10

• In general, construction costs will be greater as the terrain
becomes more difficult and higher standards will become less
justifiable or achievable in such situations than for roads in
either flat or rolling terrain.
• Drivers accept lower standards in such conditions and
therefore adjust their driving accordingly, so minimizing
accident risk. Design speed will therefore vary with transverse
terrain.
11

Design Vehicle
• Both the physical characteristics and turning capabilities of
vehicles are controls in geometric design.
• Vehicle characteristics and dimensions affecting design include
power to weight ratio, minimum turning radius and travel path
during a turn, and vehicle height and width.
• The road elements affected include the selection of maximum
gradient, lane width, horizontal curve widening, and junction
design.
12

• The present vehicle fleet in Ethiopia includes a high number of
four-wheel drive utility vehicles and overloaded trucks.
• Until more detailed information becomes available regarding
the makeup of the vehicle fleet in Ethiopia, the four design
vehicles indicated in Table below should be used in the control
of geometric design:
13

Minimum turning radius of DV1 passenger car
15

Minimum turning radius of DV2
16

• Roads conforming to Design Standards DS1 trough DS5 should
be designed to accommodate the most restrictive of the above
design vehicle.
• Standards DS6 and DS7, two lane roads should accommodate
all but the semi-trailer combination DV4. Standards DS8 and
DS9, for single lane roads should be designed similarly to DS6
and DS7; and Standard DS10 roads need only accommodate
the requirements for utility vehicle and passenger cars -DV1.
17

Density and Character of Adjoining Land Use
• For urban or peri-urban conditions, the design speed selection
is influenced by other factors.
• In such areas, speed controls are frequently included.
• Traffic speeds are in fact influenced by the presence of other
vehicles traveling in and across the through lanes, physical and
right of- way constraints, together with pedestrian and safety
considerations.
18

Design Traffic Volume
• A further factor influencing the development of road design
standards, and in particular the design speed, is the volume
and composition of traffic.
• Traffic indicates the need for improvement and directly affects
features of design such as widths, alignments, and gradients.
• Traffic data for a road or section of road, including traffic
trends, is generally available in terms of annual average daily
traffic (AADT).
19

• Design classes DS1 to DS10 have associated bands of traffic
flow as was shown in Table below
• The range of flows extends from less than 20 to 15,000
motorized vehicles per day (excluding motorcycles), and covers
the design conditions for all single and dual carriageway roads.
20

Design Speed
• The Design Speed is used as an index which links road function, traffic
flow and terrain to the design parameters of sight distance and
curvature to ensure that a driver is presented with a reasonably
consistent speed environment.
• Design elements such as lane and shoulder widths, horizontal radius,
super elevation, sight distance and gradient are directly related to, and
vary, with design speed.
• Thus all of the geometric design parameters of a road (such as curve
radius, gradient and tangent length which is between same direction
adjacent curves and reverse curves) are directly related to the selected
design speed.
22

• The design speeds given in ERA manual have been determined
in accordance with the following guidelines:
i. Drivers on long-distance journeys are apt to travel at higher
speeds than local traffic.
ii. On local roads whose major function is to provide access,
high speeds are undesirable.
iii. Drivers usually adjust their speeds to physical limitations and
prevailing traffic conditions. Where a difficult location is
obvious to the driver, he/she is more apt to accept a lower
speed of operation.
23

iv. Economic considerations (road user savings vs. construction costs)
may justify a higher design speed for a road carrying large volumes
of traffic than for a less heavily trafficked road in similar
topography.
v. Change in design speed, if required due to a change in terrain class,
should not be effected abruptly, but over sufficient distances to
enable drivers to change speed gradually. The change in design
speed should not be greater than one design speed step, and the
section with the lower geometric standards should be long enough
to be clearly recognizable by drivers (not, for example, just one
single curve).
vi. It is often the case that the physical terrain changes two steps, i.e.-
from mountainous to flat terrain. Where possible in such
circumstances, a transition section of road shall be provided with
limiting parameters equivalent to the rolling terrain type. Where
this is not possible, i.e.- a Departure from Standards, special
attention shall be given to the application of warning signs and/or
rumble strips to alert the driver to the changing conditions.
24

CROSS SECTION ELEMENTS
• A cross-section will normally consist of the carriageway,
shoulders or curbs, drainage features, and earthwork profiles.
 Carriageway- the part of the road constructed for use by
moving traffic, including traffic lanes, auxiliary lanes such as
acceleration and deceleration lanes, climbing lanes, and
passing lanes, and bus bays and lay-byes.
 Roadway- consists of the carriageway and the shoulders,
parking lanes and viewing areas
 Earthwork profiles- includes side slopes and back slopes
26

• For urban cross-sections, cross-section elements may also
include facilities for pedestrians, cyclists, or other specialist
user groups. These include curbs, footpaths, and islands.
• It may also provide for parking lanes.
• For dual carriageways, the cross-section will also include
medians.
• Lane and shoulder widths should be adjusted to traffic
requirements and characteristics of the terrain.
28

Lane Widths
• A feature of a highway having great influence on safety and
comfort is the width of the carriageway.
• Lane widths of 3.65m are used for Design Classes DS1 and DS2.
• Narrower lanes are appropriate on lower volume roads.
29

Shoulders
• A shoulder is the portion of the roadway contiguous to the
carriageway for the accommodation of stopped vehicles;
traditional and intermediate non-motorized traffic, animals,
and pedestrians; emergency use; the recovery of errant
vehicles; and lateral support of the pavement courses.
• Where the carriageway is paved, the shoulder should also be
sealed with a single bituminous surface treatment. This has
several advantages. It would prevent edge raveling and
maintenance problems associated with parking on a gravel
shoulder.
30

Normal Cross fall
• Normal cross fall (or camber, crown) should be sufficient to provide
adequate surface drainage whilst not being so great as to make
steering difficult.
• On unpaved roads, the minimum acceptable value of cross fall
should be related to the need to carry surface water away from the
pavement structure effectively, with a maximum value above which
erosion of material starts to become a problem.
• The normal cross fall should be 2.5 percent on paved roads and 4
percent on unpaved roads.
• Shoulders having the same surface as the roadway should have the
same normal cross fall.
• Unpaved shoulders on a paved road should be 1.5 percent steeper
than the cross fall of the roadway.
31

• The precise choice of normal cross fall on unpaved roads will
vary with construction type and material rather than any
geometric design requirement.
• In most circumstances, cross falls of 4 percent should be used,
although the value will change throughout the maintenance
cycle.
32

Side Slopes and Back Slopes
• Side slopes should be designed to insure the stability of the
roadway and to provide a reasonable opportunity for recovery
of an out-of-control vehicle.
• Three regions of the roadside are important when evaluating
the safety aspects: the top of the slope (hinge point), the side
slope, and the toe of the slope (intersection of the fore slope
with level ground or with a back slope, forming a ditch).
• Research has found that rounding at the hinge point can
significantly reduce the hazard potential. Similarly, rounding at
the toe of the slope is also beneficial.
33

Designation of Roadside Regions
34

• Embankment or fill slopes parallel to the flow of traffic may be
defined as recoverable, non recoverable, or critical.
• Recoverable slopes include all embankment slopes 1:4 or
flatter.
• Motorists who encroach on recoverable slopes can generally
stop their vehicles or slow them enough to return to the
roadway safely. Fixed obstacles such as culvert head walls
should not extend above the embankment within the clear
zone distance.
35

• A non-recoverable slope is defined as one which is traversable,
but from which most motorists will be unable to stop or to
return to the roadway easily. Typically, vehicles on such slopes
typically can be expected to reach the bottom. Embankments
between 1:3 and 1:4 generally fall into this category.
• A critical slope is one on which a vehicle is likely to overturn.
Slopes steeper than 1:3 generally fall into this category.
• The selection of a side slope and back slope is dependent on
safety considerations, height of cut or fill, and economic
considerations.
36

Slope Ratio Table – Vertical to Horizontal
37

Roadside Ditches
• Will be discussed in drainage design in last chapter
Clear Zone
• Once a vehicle has left the roadway, an accident may occur. The end
result of an encroachment depends upon the physical characteristics
of the roadside environment.
• Flat, traversable, stable slopes will minimize overturning accidents,
which are usually severe.
• Elimination of roadside furniture or its relocation to less vulnerable
areas are options in the development of safer roadsides.
• For adequate safety, it is desirable to provide an unencumbered
roadside recovery area that is as wide as practical on a specific
highway section.
• The cleared width should be a minimum of 15 meters each side from
the edge of the roadway for the higher road standards.
38

• For lower standard roads, the clear zone can be reduced as
practical. It should extend beyond the toe of the slope. Lateral
clearances between roadside objects and obstructions and the
edge of the carriageway should normally be not less than 1.5
meters.
• At existing pipe culverts, box culverts and bridges, the
clearance cannot be less than the carriageway width; if this
clearance is not met, the structure must be widened. New pipe
and box culvert installations, and extensions to same, must be
designed with a 1.5-meter clearance from the edge of the
shoulder.
39

Elements of Geometric Design
• The principal elements of geometric design include:-
 Sight distance
 Horizontal alignments and
 vertical alignments
40

Sight Distances
• The path and speed of motor vehicles on highways and streets
are subject to the control of drivers whose ability, training, and
experience are quite varied.
• For safety on highways, the designer should provide sight
distance of sufficient length that drivers can control the
operation of their vehicles to avoid striking an unexpected
object in the traveled way.
• Certain two-lane highways should also have sufficient sight
distance to enable drivers to occupy the opposing traffic lane
for passing other vehicles without risk of a crash.
41

• There are basically three types of sight distances
Stopping Sight distances: the sight distances needed for
stopping, which are applicable on all highways
Passing Sight Distances: the sight distances needed for the
passing of overtaken vehicles, applicable only on two-lane
highways
Decision Sight Distances: the sight distances needed for
decisions at complex locations
42

Stopping Sight Distance
• It is the available sight distance on a roadway that should be
sufficiently long to enable a vehicle traveling at or near the
design speed to stop before reaching a stationary object in its
path.
• Stopping sight distance is the sum of two distances:
1. Brake reaction distance: the distance traversed by the
vehicle from the instant the driver sights an object
necessitating a stop to the instant the brakes are applied;
and
2. Braking distance : the distance needed to stop the vehicle
from the instant brake application begins.
43

• ERA provides the following equation to determine stopping
sight distance
where
d = distance (meter)
t = driver reaction time, generally taken to be 2.5 seconds
V = initial speed (km/h)
f = coefficient of friction between tires and roadway
44

Coefficient of friction f by ERA is given as:-
45

• When the road is not flat and has some grade then the
stopping sight distance equation is modified as:
where,
d = distance (meters)
t = driver reaction time, generally taken to be 2.5 seconds
V = initial speed (km/h)
f = coefficient of friction between tyres and roadway
g = gradient of road as a percentage (downhill is negative)
46

Ex-1 Determine the stopping sight distance for a design of
30km/hr for a flat terrain
Ex-2 Determine the stopping sight distance for a design speed of
40km/hr and an uphill slope of 20%.
47

Control of Sight Distance
• Sight distances should be checked during design and
adjustments made to meet the minimum requirements.
• The following values should be used for the determination of
sight lines.
(a) Driver's eye height: 1.05 meters
(b) Object height for stopping sight distance: 0.2 meters
(c) Object height for passing sight distance: 1.30 meters
(d) Object height for decision sight distance 0.00 meters
48

• On the inside of horizontal curves, it may be necessary to
remove buildings, trees or other sight obstructions or widen
cuts on the insides of curves to obtain the required sight
distance
Sight Distance for Horizontal Curves
49

• Length of Sight Line (S) = 2R sin(Δ /2)
where Δ =Deflection angle
• Length of Middle Ordinate (M) = R(1-cos(Δ/2)
Ex-3 If the radius of an alignment is 1000m and the deflection
angle is 20o determine the required sight distance and the
required clearance (Middle Ordinate)
50

Stopping Sight Distance: Single Lane Roads
• Certain classes of roads only have a single lane, with passing
pullouts. In these circumstances, a stopping sight distance is
required to enable both approaching drivers to stop.
• This distance is the sum of the stopping sight distance for the
two vehicles, plus a 30- meter safety distance. The resultant
distance is that shown in Table 7-1, doubled, plus 10meters.
• This distance is the sum of the stopping sight distance for the
two vehicles, plus a 30- meter safety distance.
• Design speed = 50 km/hr.
SSD = (55 x 2) + 30 = 140 meters
52

Passing Sight Distance
• Passing Sight Distance is the minimum sight distance on two-
way single roadway roads that must be available to enable the
driver of one vehicle to pass another vehicle safely without
interfering with the speed of an oncoming vehicle traveling at
the design speed.
• Within the sight area the terrain should be the same level or a
level lower than the roadway. Otherwise, for horizontal curves,
it may be necessary to remove obstructions and widen cuttings
on the insides of curves to obtain the required sight distance.
53

The passing sight distance is generally determined by a formula
with four components, as follows:
• d1 = initial maneuver distance, including a time for
perception and reaction
• d2 = distance during which passing vehicle is in the opposing
lane
• d3 = clearance distance between vehicles at the end of the
maneuver
• d4 = distance traversed by the opposing vehicle
54

• The formulae for these components are as indicated below:
d1 = 0.278 t1 (v – m + a.t1/2)
Where
t1 = time of initial manoeuvre, s
a = average acceleration, km/h/s
v = average speed of passing vehicle, km/h
m = difference in speed of passed vehicle and passing vehicle,
km/h
55

d2 = 0.278 v.t2.
Where
t2 = time passing vehicle occupies left lane, s
v = average speed of passing vehicle, km/h
d3 = safe clearance distance between vehicles at the end of the
manoeuvre, and is dependent on ambient speeds as per Table
7.5:
d4 = distance traversed by the opposing vehicle, which is
approximately equal to d2 minus the portion of d2 whereby the
passing vehicle is entering the left lane, estimated as:
d4 = 2d2/3
56

The minimum Passing Sight Distance (PSD) for design is
therefore:
PSD = d1+ d2 + d3 + d4
57

Ex-4 Determine minimum passing sight distance based on the following study
on a certain proposed road alignment
Overtaken vehicle
Speed= 55km/hr
Passing Vehicle
Initial Maneuver
Average Speed = 70km/hr
Average acceleration = 2.3km/h/s
T1= 4sec
d1=?
Occupation of left turn
T2= 10sec
d2=?
Clearance length
d3=?
Opposing Vehicle
d4= ?
58

• The usual values resulting from application of the formulae are
reduced in ERA manual, as it is deemed appropriate to address
the distances covered by twice the d4 distance and the
clearance distance d3.
• A driver finding that he has insufficient distance after initiating
the passing maneuver can choose to abort the maneuver
• The reduced passing sight distance is shown on the fifth
column in the following table.
59

ERA provides the following table for quick determination of SSD
and PSD
60

Decision Sight Distances
• Decision sight distance, sometimes termed ‘anticipatory sight
distance’, is the distance required for a driver to:
detect an unexpected or otherwise ‘difficult-to-perceive’
information source or hazard in a roadway environment that
may be visually cluttered;
recognize the hazard or its potential threat;
select an appropriate speed and path; and
initiate and complete the required safety manoeuvre safely
and efficiently.
62

• Because decision sight distance gives drivers additional margin for
error and affords them sufficient length to manoeuvre their vehicles
at the same or reduced speed rather than to just stop, it is
substantially longer than stopping sight distance.
• Drivers need decision sight distances whenever there is likelihood
for error in information reception, decision-making, or control
actions. Critical locations where these kinds of errors are likely to
occur, and where it is desirable to provide decision sight distance
include:
Approaches to interchanges and intersections;
Changes in cross-section such as at toll plazas and lane drops;
Design speed reductions; and
Areas of concentrated demand where there is likely to be ‘visual
noise’, e.g. where
sources of information, such as roadway elements, opposing
traffic, traffic control devices, advertising signs and construction
zones, compete for attention.
63

ERA provides the following decision sight distances in meters for
different design speeds
64

Horizontal Alignment
• The horizontal alignment consists of a series of straight sections (tangents),
circular curves, transition curves (spirals) and super-elevation.
Tangent Sections
• From an aesthetic point of view, tangent sections may often be beneficial
in flat country but are less so in rolling or mountainous terrain.
• From a safety standpoint, they provide better visibility and more passing
opportunities.
• However, long tangent sections increase the danger from headlight glare
and usually lead to excessive speeding.
• Hence ERA states the maximum length of a tangent section not to exceed
4.0 kilometers.
• ERA 2013 states the length of tangent sections should not exceed 20 times
the design speed. Thus if the design speed is 100km/hr the length should
not exceed 2000 meters or 2 kilometers.
65

Circular Curve
• When a vehicle moves in a circular path, it is forced radially
outward by centrifugal force.
• The centrifugal force is counter balanced by super elevation of
the roadway and/or the side friction developed between the
tires and the road surface.
66

• For calculation of the minimum horizontal radius, R min, for a
particular design speed, ERA provides the following equation:
Where
VD = Design Speed (km/h)
e = Maximum superelevation (%/100)
f = Side friction coefficient
67

Side friction coefficients are dependent on
i) vehicle speed;
ii) type, condition and texture of roadway surface;
iii) weather conditions; and
iv) type and condition of tyres.
68

Ex-5 Determine the minimum radius that should be provided in a
circular alignment for a design speed of 50 km/hr and super-
elevation of 8% with a side friction of o.17
69

• ERA provides the following side friction factors and calculated
minimum radius to different super-elevations for paved and
unpaved roads.
Table: Minimum Turning radius for paved roads
Table: Minimum Turning radius for unpaved roads
70

Elements of a Circular Curve
1) Δ is the Deflection Angle (in degrees).
2) R is the Radius of the curve.
3) T is the Tangent Distance PC to PI;
T = R.tan(Δ /2)
4) E is the External Distance:
E = R.[sec(Δ /2) – 1]
5) L is the Curve Length. L
= 2.π.R. Δ /360
6) M is the Middle Ordinate
M = R.[1 – cos (Δ /2)]
7) C is the Chord Length from PC to PT
C = 2.R.sin(Δ /2)
8) Point PC is the Point-of-Curvature Station
PC = PI – T
9) PT is the Point-of-Tangency Station PT = PC+ L
71

Ex-6 A curve has a deflection angle of = 23o 18’ 02”, and a radius
of 1432.6m. The Point of Intersection (PI) is 5+053.87. Calculate
the tangent distance (T), external distance (E),curve length (L),
Point of Curvature (PC), and Point of Tangent (PT).
72

• 4 x 4 utility vehicles, buses and trucks, and trucks with trailers
(DV1, DV2/3,and DV4) require minimum design turning radii of
7.3, 12.8, and 13.7m, respectively.
• As it is not practically possible to exclude any of these
categories from the lower standard roads, and as a certain
amount of tolerance is required for safe operations, the
minimum horizontal curve radius of 15m is specified in ERA
manual for all design standards.
73

Isolated Curves
• The horizontal curvature over a particular road section should be as
consistent as possible.
• Long tangent roadway segments joined by an isolated curve
designed at or near the minimum radius are unsafe.
• Long straight sections encourage drivers to drive at speeds in excess
of the design speed, hence sudden and unexpected sharp curves are
dangerous.
• Good design practice is to avoid the use of minimum standards in
such conditions.
• For isolated curves, the minimum horizontal curve radius should be
increased by 50 percent.
• This will usually result in the ability to negotiate the curve at a speed
approximately 10 km/hr higher than the design speed.
74

Reverse Curves, Broken-Back Curves, and Compound Curves
• Curves are more frequent in rugged terrain. Tangent sections
are shortened, and a stage may be reached where successive
curves can no longer be dealt with in isolation.
• Three cases of successive curves are
i) Reverse Curve: a curve followed by another curve in the
opposite direction.
ii) Broken-back Curve: a curve followed by another curve in
the same direction but with only a short tangent in-
between.
iii) Compound curve: curves in the same direction but of
different radii, and without any intervening tangent section.
76

Reverse Curves, Broken-Back Curves, and Compound Curves
77

• The occurrence of abrupt reverse curves (having a short tangent
between two curves in opposite directions) should be avoided.
• Such geometrics make it difficult for the driver to remain within his
lane. It is also difficult to super-elevate both curves adequately, and
this may result in erratic operation.
• The broken-back arrangement of curves (having a short tangent
between two curves in the same direction) should also be avoided
except where very unusual topographical or right of way conditions
dictate otherwise.
• Drivers do not generally anticipate successive curves in the same
direction hence safety is compromised.
• Problems also arise associated with super-elevation and drainage.
78

• The use of compound curves affords flexibility in fitting the road to
the terrain and other controls. Caution should, however, be
exercised in the use of compound curves because the driver does
not expect to be confronted by a change in radius once he has
entered a curve, hence safety is compromised. Their use should also
be avoided where curves are sharp.
• Compound curves with large differences in curvature introduce the
same problems as are found at the transition from a tangent to a
small-radius curve.
• Where the use of compound curves cannot be avoided, the radius of
the flatter circular arc should not be more than 50 percent greater
than the radius of the sharper arc; ie. R1 should not exceed 1.5.R2.
79

Minimum Length of Curve
• For small changes of direction it is desirable to use large radius
curves.
• This improves the appearance of the road by removing rapid
changes in edge profile.
• It also reduces the tendency for drivers to cut the corners of
small radius curves.
• Providing the curve radii are sufficiently large, it may be
possible to maintain a passing zone through a curve.
• The minimum length of curve for a deflection angle of 5o or
less is 300 meters.
80

Widening on Curves and Embankments
• The use of long curves of tight radii should be avoided where
possible because drivers following the design speed will find it
difficult to remain in the traffic lane.
• Widening ofthe carriageway where the horizontal curve is tight
is usually necessary to ensure that the rear wheels of the
largest vehicles remain on the road when negotiating the
curve; and, on two lane roads, to ensure that the front
overhang of the vehicle does not encroach on the opposite
lane.
• Widening is therefore also important for safety reasons.
81

• Curve widening is required on all standards of roads and should
be sufficient to cater for the design vehicle.
• Widening is also required for Design Standards DS1 through
DS5 at high fills for the psychological comfort of the driver.
• The steep drops from high embankments un nerve some
drivers and the widening is primarily for psychological comfort
although it also has a positive effect on safety.
• Widening for curvature and for high embankments should be
added where both cases apply.
• Widening is provided to make driving on a curve comparable
with that on a tangent.
• Where a road has to be rehabilitated and it is not possible to
increase the radius of curvature, the designer could consider
the need for curve widening.
82

ERA provides the following table for curve widening:-
83
• Widening should transition gradually on the approaches to the curve so that
the full additional width is available at the start of the curve.

Switchback Curves
• Switchback or hairpin curves are used where necessary in
traversing mountainous and escarpment terrain.
• Employing a radius of 20m or less, with a minimum of 10m,
they are generally outside of the standards for all road designs
• Switchback curves require careful design to ensure that all
design vehicles can travel through the curve.
• They must therefore provide for the tracking widths of the
design vehicles
84

• Switchback requirements can be determined which allow for:
 Passage of two opposing DV4 vehicles. This is recommended
for Design Standards DS1- DS3
 Passage of a single DV4 and a DV1. This is recommended for
Design Standards DS4- DS5
 Passage of only a single DV4. This is recommended to Design
Standards DS6- DS9
86

Transition Curves
• The characteristic of a transition curve is that it has a constantly
changing radius.
• Transition curves may be inserted between tangents and circular
curves to reduce the abrupt introduction of lateral acceleration.
• They may also be used between two circular curves.
• For large radius curves, the rate of change of lateral acceleration is
small and transition curves are not normally required.
• If the choice is made to employ a transition curve, the Euler(An Euler
spiral is a curve whose curvature changes linearly with its curve
length (the curvature of a circular curve is equal to the reciprocal of
the radius). Euler spirals are also commonly referred to as
spiros, clothoids, or Cornu spirals.) spiral, which is also known as the
clothoid, shall be used.
87

• The equation of clothoid is in the form of
RL = A2
where
A2 is a constant that controls the scale of the clothoid
R is the radius of the horizontal curve
L is the length of the clothoid
88

• Two formulae are required for the analysis of transition curves:
S = L2/24R
L = V3/(3.63 x C x R)
where
S is the shift (m)
L is the length of the transition curve (m)
R is the radius of the circular curve (m)
V is the design speed (km/hr)
C is the rate of change of radial acceleration (m/s3) c should
normally not be less than 0.3 m/s3 for unrestricted design,
although in urban areas it may be necessary to increase it to 0.6
m/s3 or even higher, for sharp curves in tight locations.
The length of transition should normally be limited to
Lmax =√(24R)
89

• From the geometry of the above figure:
IB = (R + S)tan(ϴ/2)
• It has been proved that B is the mid-point of the transition
Therefore:
BT = L/2
• Combining these two equations, the length of the line IT is obtained:
IT = (R + S)tan(ϴ /2) + L/2
• If a series of angles and chord lengths are used, the spiral is the
preferred form.
• If, as is the case here, x and y co-ordinates are being used, then any
point on the transition curve can be estimated using the following
equation of the curve
• x = y3÷6RL
• When y attains its maximum value of L (the length of the transition
curve), then the maximum offset is calculated as follows:
• x = L3 ÷6RL = L2 ÷ 6R
91

Generation of offset values for plotting a transition curve.
92

ERA provides the following requirement of transition curves for
different design speeds
93

Ex A transition curve is required for a single carriageway road
with a design speed of 85 km/hr. The bearings of the two
straights in question are 17° and 59° . Assume a value of 0.3
m/s3 for C. Calculate the following:
(1) The transition length, L
(2) The shift, S
(3) The length along the tangent required from the
intersection point to the start of the transition, IT
(4) The form of the cubic parabola and the co-ordinates of the
point at which the transition becomes the circular arc of
radius R.
94

Bonus Assignment
• For the above example drive an equation for the three
alignments tangent, transition and circular curve in terms of x
and y form and draw the alignment on graph paper in scale.
95

Super-elevation
• A tighter curve can be designed if higher values of super-
elevation are used, but high values of super-elevation are not
recommended if the friction is low, such as in locations where
mud is likely to contaminate the road surface regularly.
• High values are also not recommended where mixed traffic
and/or roadside development severely limit the speed of
vehicles.
• In urban areas an upper limit of 4 percent should be used.
Similarly, either a low maximum rate of super-elevation or no
super-elevation is employed within important intersection areas
or where there is a tendency to drive slowly because of turning
and crossing movements, warning devices, and signals.
96

Super-elevation Runoff
• In alignment design with spirals, the super-elevation runoff is
provided over the whole of the transition curve.
• The length of runoff is the spiral length, with the tangent to
spiral (TS) transition point at the beginning and the spiral to
curve (SC) transition point at the end.
• The change in cross slope begins by removing the adverse cross
slope from the lane or lanes on the outside of the curve on a
length of tangent just ahead of TS (the tangent runout).
• Between the TS and SC (the super-elevation runoff) the
travelled way is rotated to reach the full super-elevation at the
SC.
• The procedure is reversed on leaving the curve.
97

• In the design of curves without spirals the super-elevation
runoff is considered to be that length beyond the tangent run-
out.
• Empirical methods are employed to locate the super-elevation
runoff length with respect to the point of curvature (PC).
• Current design practice is to place approximately two-thirds of
the runoff on the tangent approach and one-third on the curve
99

Shoulder Super-elevation
• Shoulder super-elevation rates corresponding to carriageway
super-elevation rates.
• On the low side (inner shoulder) of super-elevated curves, the
shoulder super-elevation matches the roadway super-
elevation.
• On the high side (outer shoulder), the super-elevation is set
such that the grade break between the
• An exception to this occurs at a maximum super-elevation of 8
percent, where the resultant shoulder super-elevation would
be an undesirable flat configuration. Here the super-elevation
is set at -1% to drain the shoulder.
103

VERTICAL ALIGNMENT
• Vertical alignment is the combination of parabolic vertical
curves and tangent sections of a particular slope.
• The selection of rates of grade and lengths of vertical curves is
based on assumptions about characteristics of the driver, the
vehicle and the roadway.
• Vertical curvature may impose limitations on sight distance,
particularly when combined with horizontal curvature.
105

• Thus the two major aspects of vertical alignment are
 Vertical curvature, which is governed by sight distance
criteria, and
 Gradient, which is related to vehicle performance and level
of service.
106

Vertical Curve Formula
• Vertical curves are required to provide smooth transitions
between consecutive gradients.
• The simple parabola is specified for these because the parabola
provides a constant rate of change of curvature and, hence,
acceleration and visibility, along its length.
107

• Equations relating the various aspects of the vertical curve are as follows
Y(x) = r.X2/200 + X.g1/100 + YBVC
r = (g2 – g1)/L = G/L
Where
BVC = Beginning of the vertical curve. The coordinates are normally
(0,Y(0)),
EVC = End of the vertical curve. The coordinates are normally (L, Y(L)),
Y(X) = Elevation of a point on the curve (metres)
X = Horizontal distance from the (BVC) (metres)
g1 = Starting gradient (%),
g2 = Ending gradient (%),
r = Rate of change of grade per section (% per metre),
L = Length of curve (horizontal distance) in metres,
G = g1 - g2 (%),
108

• K = L/G = horizontal distance required to achieve a 1% change in grade
(metres),
• Z = vertical distance from the tangent to the curve (metres)
• Some useful relationships are;
Equation of tangent g1 is Y(X) = Y(0) +g1.X/100
Equation of tangent g2 is Y(X) = Y(L) + g2.(X-L)/100
The y coordinate of the EVC is Y(L) = (g1+g2)L/200+Y(0)
The Intersection Point always occurs at an x coordinate of 0.5L hence the elevation
is always;
Y(IP) = (g2+3.g1)L/800 + Y(0)
109

• There are two types of vertical curve crest and sag vertical
curves
110
Crest Curve

Ex.Two grade lines intersect at Station 2+200 where the
point of vertical intersection(PVI) elevation is 239.5 m. The
starting grade is –6 percent and the ending grade is +2
percent. The length of curve is 400 m. Compute the
elevation at station 2+200.
112

Ex. For the crest curve two tangent grade lines are +6% and -
3%. The Beginning of the Vertical Curve is at chainage
0.000 and its elevation 100.0m. The length of the vertical
curve is 400m. Compute the End of Vertical Curve and the
coordinates of the Intersection Point.
113

Crest curves
• Two conditions exist when considering the minimum sight
distance criteria on vertical curves.
• The first is where the sight distance (S) is less than the length of
the vertical curve (L), and
• The second is where sight distance extends beyond the vertical
curve.
114

• Consideration of the properties of the parabola results in the following
relationships for minimum curve length to achieve the required sight distances:
• For S < L (the most common situation in practice):
Lm = (G.S2)/[200(h1
0.5 + h2
0.5)2]
Lm = K.G
• where
Lm = minimum length of vertical crest curve (metres)
S = required sight distance (metres)
h1 = driver eye height (metres)
h2 = object height (metres)
G= g1-g2
K = is a constant for given values of h1 and h2 and stopping sight distance (S) and
therefore speed and surface friction.
• For S > L
Lm = 2S - [200.(h1
0.5 + h2
0.5)2]/G ]
• Eye height (h1) has been taken as 1.05 metres, and object heights h2 of 0.6
metres above the road surface.
115

• Ex A vertical crest curve on a single carriageway
road with a design speed of 85 km/hr is to be built
in order to join an ascending grade of 4% with a
descending grade of 2.5%. The motorist’s eye height
is assumed to be 1.05m while the object height is
assumed to be 0.6m. Calculate the minimum curve
length required in order to satisfy the requirements
of minimum sight stopping distance
116

• ERA provides the following minimum values of K values for
crest curves
117
Minimum K values for Unpaved roads

Minimum K values for Unpaved roads
118

Sag Curves
• It is assumed that adequate sight distance will be available on sag
curves in daylight.
• However, at night, visibility is limited by the distance illuminated by
the headlamp beams.
• Assumptions concerning the brightness of the headlights, their
height above the road and the divergence of the beams have been
made and minimum sag curve lengths for this condition have been
computed.
• However the results lead to unrealistically long vertical curves,
especially at higher speeds, and the required sight distances may be
in excess of the effective range of the headlamp beam.
• Thus, the only likely situation when the calculations are useful is on
the approaches to fords and drifts and other similar locations where
flowing or standing water may be present on the road surface.
119

• It is therefore recommended that, for most situations, sag
curves are designed using a driver comfort criterion of vertical
acceleration.
• A maximum acceleration of 0.3m/sec2 is often used.
• K > V2/395
120

Minimum Lengths of Vertical Curves
• Especially for trunk and link roads, where the algebraic
difference between successive gradients is often small, the
intervening minimum vertical curve, obtained by applying the
above formulae in previous slides, becomes very short. This
can create the impression of a kink in the grade-line.
• If the vertical alignment is allowed to contain many curves of
short length, the result can be a ‘hidden dip’ profile, and/or a
‘roller coaster’ type profile.
• For this reason, where the algebraic difference in gradient is
less than 0.5 percent, a minimum curve length is
recommended for purely aesthetic reasons.
• The minimum length should not be less than twice the design
speed in km/h and, for preference, should be 400 metres or
longer, except in mountainous or escarpment terrain.
121

Hidden Dip and Roller Coaster Profiles
122

• Where a crest curve and a succeeding sag curve
have a common beginning and end, the visual
effect created is that the road has suddenly
dropped away.
• In the reverse case, the illusion of a hump is
created. Either effect is removed by inserting a
short length of straight grade between the two
curves.
• Typically, 60 m to 100 m is adequate for this
purpose
123

Maximum Gradients
• Vehicle operations on gradients are complex and
depend on a number of factors:
severity and length of gradient;
 level and composition of traffic; and
the number of overtaking opportunities on the
gradient and in its vicinity.
• ERA provides the following table of maximum
gradient for different terrains
124

PHASING OF HORIZONTAL AND VERTICAL ALIGNMENT
• Phasing of the vertical and horizontal curves of a road implies
their coordination so that the line of the road appears to a
driver to flow smoothly, avoiding the creation of hazards and
visual defects.
• It is particularly important in the design of high-speed roads on
which a driver must be able to anticipate changes in both
horizontal and vertical alignment well within the safe stopping
distance.
• It becomes more important with small radius curves than with
large.
• Defects may arise if an alignment is mis-phased.
• Defects may be purely visual and do no more than present the
driver with an aesthetically displeasing impression of the road.
126

• When the horizontal and vertical curves are
adequately separated or when they are coincident, no
phasing problem occurs and no corrective action is
required.
• Where defects occur, phasing may be achieved either
by separating the curves or by adjusting their lengths
such that vertical and horizontal curves begin at a
common station and end at a common station
127

Types of Mis-phasing and Corrective Action
Vertical Curve Overlaps One End of the Horizontal Curve :
• If a vertical curve overlaps either the beginning or the end of a
horizontal curve, a driver’s perception of the change of
direction at the start of the horizontal curve may be delayed
because his sight distance is reduced by the vertical curve. This
defect is hazardous.
• The defect may be corrected in both cases by completely
separating the curves. If this is uneconomic, the curves must be
adjusted so that if the horizontal curve is of short radius they
are coincident at both ends, or if the horizontal curve is of
longer radius they need be coincident at only one end.
128

Deformation: End of horizontal curve follows vertical curve
129

Insufficient Separation between the Curves:
• If there is insufficient separation between the ends of the horizontal
and vertical curves, a false reverse curve may appear on the outside
edge-line at the beginning of the horizontal curve.
• Corrective action consists of increasing the separation between the
curves, or making the curves concurrent
130

Both Ends of the Vertical Curve Lie on the Horizontal Curve
• If both ends of a crest curve lie on a sharp horizontal curve, the radius of
the horizontal curve may appear to the driver to decrease abruptly over
the length of the crest curve.
• If the vertical curve is a sag curve, the radius of the horizontal curve may
appear to increase.
• The corrective action is to make both ends of the curves coincident or to
separate them.
131

Vertical Curve Overlaps Both Ends of the Horizontal Curve
• If a vertical crest curve overlaps both ends of a sharp
horizontal curve, a hazard may be created because a
vehicle has to undergo a sudden change of direction
during the passage of the vertical curve while sight
distance is reduced.
• The corrective action is to make both ends of the curves
coincident.
132

The Economic Penalty Due to Phasing
• The correct phasing of vertical curves restricts the
designer in fitting the road to the topography at the lowest
cost.
• Therefore, phasing is usually bought at the cost of extra
earthworks and the designer must decide at what point it
becomes uneconomic.
133

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