SPECIAL TRANSPORTATION STRUCTURES (Notes for Guidance ) Highway Design Procedures/Route Geometric Design/Horizontal Alignment/Transition Curves

1
SPECIAL TRANSPORTATION STRUCTURES(Notes for
Guidance )Highway Design Procedures/Route
Geometric Design/Horizontal Alignment/Transition
Curves
Lecture Nine
Radu ANDREI, PhD, P.E., Professor of Civil
Engineering Technical University Gh. Asachi
IASI
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Highway design procedures Route Geometric
Design/Horizontal Alignment/Transition Curves

• The need for introduction of transition curves
• The clothoid curve used in horizontal alignment
• Criteria for the selection the length of the
  clothoid
• Methods for connection of tangents with circular
  arcs and two symmetrical spirals . Practical
  guide.
• Problems
• Additional Readings

3
The need for introduction of transition curves

• Transition spirals for railroad and highways are
  curves which provide gradual change in curvature
  from a straight to a circular path.
• Such easement curves have been necessary are
  high-speed railroads, from the stand point of
  comfortable operation and of gradually bringing
  the full supereleveation of the outer rail on
  curves.

4
The need for introduction of transition curves

• The use of transition spirals in the design of
  highways, possesses the same general advantages
  as for railroads, with the added "factor of
  safety", namely a transitional path is provided,
  thus reducing the tendency to deviate from the
  logical traffic lane.

5
The need for introduction of transition curves

• When a car, travelling on a straight of highway
  a circular path, the wheel must be set at a new
  angle, depending the radius of the curve. This
  movement cannot be done instantly but in
  measurable time interval, thus creating a demand
  for a transition curve, as shown in next slide,
  the length of such transition curve ( spiral)
  equals speed by time.

6
The need for introduction of transition curves
Principle of connection of two tangents with a
simple circular curve and two spirals
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The need for introduction of transition curves
Principle of connection of two tangents with a
simple circular curve and two spirals

• The role of transition curves can be better
  understood form the figure of the next slide
  where the variation of the value of curvature
  1/R, is described for two significant cases
• the connection of two tangents with a simple
  circle arc
• the same connection with a circle arc and two
  symmetrical transition curves

8
The need for introduction of transition curves
The variation of the curvature for a connection
of tangents made with a simple circular arc
with the variation of the curvature for the same
connection made with a circle arc and two
transition curves
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The need for introduction of transition curves
The variation of the curvature for a connection
of tangents made with a simple circular arc
with the variation of the curvature for the same
connection made with a circle arc and two
transition curves

• The following notations, and their Romanian
  equivalents , in relation with the figure from
  previous slide are usually used with the
  spirals
• PC(Ti) - point of curvature
• PT(Te) - point of tangency
• TS (Oi)- point of change from tangent to spiral
• SC (Si)- point of change from spiral to circle
• CS (Se)- point of change from circle to spiral
• ST (Oe)- point of change from spiral to tangent

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The need for introduction of transition curves
The variation of the curvature for a connection
of tangents made with a simple circular arc
with the variation of the curvature for the same
connection made with a circle arc and two
transition curves

• In relation with the figure from previous
  slide, in the first case, in the PC ( Ti),
  suddenly occurs the appearance of the centrifugal
  force CmV2/R , whose perception, by the driver
  is stronger as speed is higher and the radius (R)
  is smaller.
• In the second case, the transition from the
  curvature zero (1/? zero) in TS (Oi) along the
  spiral is made progressively to the value 1/?
  1/R in SC(Si), and C, the value of the
  centrifugal force, will follow the same
  variation, with a smooth transition of movement,
  leading to better comfort and safety.

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The clothoid curve used in horizontal alignment
12
The clothoid curve used in horizontal
alignmentThe equation of the clothoid

• In relation with the figure from prevoius slide
  , the equation defining the clothoid states that
  its curvature 1/? varies proportionally with its
  length s
• 1/? s/k1
• where k1 is a specific clothoid constant.

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The clothoid curve used in horizontal
alignmentThe equation and Modulus of clothoid

• The clothoid represents a mechanical curve by
  excellence , because it corresponds to the trace
  described by the vehicle wheels during its
  transition from tangent to the circle arc, in
  the conditions when the vehicle speed V, is kept
  constant while the angle of rotation of the
  steering wheel is increased uniformly by the
  driver, so that the angular acceleration of the
  entire vehicle ? v/? is kept constant, (
  d?/dt c)
• Integrating this equation and replacing w v/?
  and t s/v, where s is the space travelled
  by the vehicle and v is its speed, one may
  derive the equation of the clothoid ?s A2. In
  this relation the value A2 , which is a constant
  factor, is called Modulus of the clothoid.

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The clothoid curve used in horizontal
alignmentThe curvature (1/?) and the length
(L) of the clothoid

• From the equation ?s A2 one may derive, the
  curvature of the clothoid 1/? s/ A2
• As in the connection point SC between the
  clothoid and the circular arc, both curves has
  the radius R and in the same point the length of
  the clothoid arc "s "becomes " L" so that the
  relation for calculation of the length of the
  clothoid arc , L, may be derived from the
  equation ?s RL A2

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The clothoid curve used in horizontal
alignmentThe equation for calculation of the
Modulus of the clothoid

• If in relation RL A2, we replace L with
  its value obtained from the condition imposed
  for the minimum length of the clothoid arc L
  V3/47Rj, where "V" is the vehicle speed
  expressed in Km/h and "j" is a comfort
  coefficient, one may obtain the relation for
  the calculation of the for the modulus of the
  clothoid A, A ? (V3/47 j)
• As this relation states that the modulus of the
  clothoid is a function of the design speed V,
  and one may conclude that for every design
  speed corresponds only a unic modulus and
  consequently, only one spiral.

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The clothoid curve used in horizontal
alignmentThe equation of the clothoid

• The only independent variable of a clothoid is
  its angle ? formed by the tangent with the
  positive sense of the abscissa, as shown in the
  figure from the next slide.When this angle ? ,
  varies from zero to infinite for ? lt0, the
  curve is situated in the third quarter of the
  trigonometric circle and in the first quarter for
  ? gt0. The clothoid has two asymptotic points
  placed symmetrically from the origin TS which
  is also an inflexion point for the curve. The
  value s of the useful clothoid arc, having an
  unitary modulus, A 1, may be derived from the
  same figure, for ? ?/2 , as follows
• s A ? ? 1.733

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The clothoid curve used in horizontal
alignmentThe omothety of Clothoids

• Clothoids are omothetical curves, their omothety
  consisting in having similar geometric figures
  keeping the homologous elements parallel and
  also the congruence of the angles, as shown in
  the figure on th next slide
• In relation with Fig.4.5, for the two points M?
  M? situated on the same line passing through
  the origin of the axes, which represents also
  the centre of omothety for the two considered
  curves, one may derive a set of equalities for
  the different ratios between their corresponding
  geometrical elements, as follows
• r?/r? x?/ x? y?/y? ??/?? s?/s? x?/x?
  x?/x? ??/?? n?/n? b?/ b? A/A? A
  ? (4.7.)
• In relation with this set of relations 4.7., and
  for practical reasons, a basic clothoid having
  its modulus A? 1 and its specific elements r?,
  x?, .. b? and any other clothoid characterised
  by its modulus A and by its elements r?, x?, ..
  b? have been considered for applying the
  homothety criterion.
• In these conditions, ? is defined as omothety
  coefficient, and by taking into consideration the
  basic clothoid having the modulus A? 1, the
  following new set of relationships 4.8., used for
  the practical calculation of the elements of a
  real clothoid of a known modulus A, as functions
  of the homologous elements of the basic clothoid
  
• r? A r? x? A x? ... b? A b? (4.8.)
• As we have been mentioned above, to each
  specific design speed V corresponds an unique
  clothoid, defined by its modulus A. This modulus
  can be determined if we know one of its elements,
  for example its length, previously determined
  from geometric or mechanical criteria,
  established for clothoids. The main elements of
  the basic clothoid may be extracted from special
  design tables and then the main functions of the
  real clothoid may be calculated using the set of
  relationships 4.8..

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The clothoid curve used in horizontal
alignmentThe omothety of Clothoids
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The clothoid curve used in horizontal
alignmentThe basic clothoid A 1

• In relation with the figure from the previous
  slide , for the two points M? M? situated on
  the same line passing through the origin of the
  axes, which represents also the centre of
  omothety for the two considered curves, one may
  derive a set of equalities for the different
  ratios between their corresponding geometrical
  elements, as follows r?/r? x?/ x? y?/y?
  ??/?? s?/s? x?/x? x?/x? ??/??
  n?/n? b?/ b? A/A? A ?
  In relation with this set of relations,
  and for practical reasons, a basic clothoid
  having its modulus A? 1 and its specific
  elements r?, x?, .. b? and any other clothoid
  characterised by its modulus A and by its
  elements r?, x?, .. b? have been considered for
  applying the homothety criterion.

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The clothoid curve used in horizontal
alignmentThe basic clothoid A 1

• In these conditions, ? is defined as omothety
  coefficient, and by taking into consideration the
  basic clothoid having the modulus A? 1, the
  following new set of relationships , used for the
  practical calculation of the elements of a real
  clothoid of a known modulus A, as functions of
  the homologous elements of the basic clothoid
  r? A r? x? A x? ... b? A b?

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The clothoid curve used in horizontal
alignmentThe main elements of the basic
clothoid A 1

• To each specific design speed V corresponds an
  unique clothoid, defined by its modulus A.
• This modulus can be determined if we know one of
  its elements, for example its length, previously
  determined from geometric or mechanical criteria,
  established for clothoids.
• The main elements of the basic clothoid may be
  extracted from special design tables and then
  the main functions of the real clothoid may be
  calculated using the existing set of
  relationships

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The clothoid curve used in horizontal
alignmentCriteria for the selection the length
of the clothoid

• The first criterion is an empirical one , stating
  that the length of a spiral curve has to be
  selected in such a way that its route will be
  travelled by the vehicle in a limited time of
  two or three seconds, this time being considered
  in accordance with the importance of the road.
  In these conditions, the total length of the
  spiral may be calculated with the simple
  relation , as space as function of speed and
  time, as follows L vt or L 2V/3.6
  0.556V, where v is expressed in m/sec. and V is
  considered in Km/h

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The clothoid curve used in horizontal
alignmentCriteria for the selection the length
of the clothoid

• The second criterion is stating that the
  variation of the normal acceleration of the
  vehicle a v2/R during the travel of the spiral
  has to vary proportionally with time t, in the
  condition that travel is made with an uniform
  speed (v constant), in comfort and safe
  conditions described by a comfort factor j. This
  criteria may be written as follows v2/R j
  (L/v)
• From this relation, the minimum length L of the
  spiral may be derived L v3/Rj or L V3/
  47 Rj where v is expressed in m/sec. and V is
  considered in Km/h, and the comfort coefficient
  j has a vale ranging from 0.3 to 0.5 for roads
  and from 0.5 to 0.7 for railroads.

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The clothoid curve used in horizontal
alignmentCriteria for the selection the length
of the clothoid

• The so called optical comfort criterion states
  that in order to get a smoother transition from
  the tangent to the circular arc and to fit
  harmoniously the curve in the existing landscape,
  the length of the clothoid has to be of such
  value, as to provide a change of the route
  direction of at least three degrees, so that the
  driver to be capable to perceive the conditions
  of curve.
• This condition imposes that the common tangent of
  both curves has to make with the positive sense
  of the abscissa , a angle ? of at least 3? or of
  1/18 radian ? L/2R 1/18
• L gt 2R/18 L gt R/9 From this relation
  one may derive the value of the modulus of the
  clothoid as follows A ? RL R/3 This
  optical criteria may be completed with the
  condition imposed for the shifting of the
  circle ?R, necessary for a curvature to be
  sensed by the drivers, its usual recommended
  values ranging between 0.5m to 1m , the maximum
  admitted value for ?R being 2.5m. The minimum
  length of the spiral is derived from the relation
  giving the value of this shifting, as follows
• ?R L2/24 R
• L gt ? 24R ?R

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The clothoid curve used in horizontal
alignmentCriteria for the selection the length
of the clothoid

• This optical criteria may be completed with the
  condition imposed for the shifting of the
  circle ?R, necessary for a curvature to be
  sensed by the drivers, its usual recommended
  values ranging between 0.5m to 1m , the maximum
  admitted value for ?R being 2.5m. The minimum
  length of the spiral is derived from the relation
  giving the value of this shifting, as follows
• ?R L2/24 R
• L gt ? 24R ?R

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The clothoid curve used in horizontal
alignmentCriteria for the selection the length
of the clothoid

• A final criterion states that the length of the
  central circle arc has to be of such value, so
  that to be travelled in a time of the least one
  second, this condition being written as follows
  C gt V/3,6 If this condition is not
  satisfied, one may not use at all the central
  circular arc and instead , he may use only two
  progressive curves, with special conditions
  imposed for their radiuses and for their lengths.
• Based on the described criteria, finally one
  should adopt the maximum value obtained for the
  length of the clothoid.

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The clothoid curve used in horizontal
alignment Methods for introducing of two
symmetrical spirals between the tangents and the
circle arc

• In order to introduce transition curves between
  tangents and circle arc, it is necessary to
  slightly shift the circle toward the interior
  of the curve with an offset ?R, this shifting may
  be achieved in two ways, as follows
• by keeping unchanged the radius of curvature and
  shifting the whole circle along the bisector
  of the angle between the tangents
• by keeping the centre of the circle unmoved and
  reducing the radius of curvature R ?R to the
  value R, this being one of the solution
  recommended by the Romanian standards( see next
  slide)

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The clothoid curve used in horizontal
alignment Methods for introducing of two
symmetrical spirals between the tangents and the
circle arc
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The clothoid curve used in horizontal
alignmentMethods for introducing of two
symmetrical spirals between the tangents and the
circle arc Practical guide

• According Romanian practice 4, in relation with
  the figure from the previous slide, a practical
  guide for introducing two symmetrical spirals
  between two tangents and a circle arc had been
  derived, this guide involving the following
  recommended steps
• )First, fix on the tangent the point PC, of the
  theoretical circle of radius R ?R
• )Maintain the centre of this theoretical circle
  and reduce its radius uith the quantity ?R, so
  that the radius of the effective circle becomes
  R

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The clothoid curve used in horizontal
alignmentMethods for introducing of two
symmetrical spirals between the tangents and the
circle arc Practical guide

• )From the PC of the theoretical circle having
  the radius R ?R, along the direction of the
  tangent and opposed to the vertex V, measure the
  distance X', in order to get the origin of the
  clothoid
• )On the same point, PC, along the direction of
  the radius of the circle the ordinate Y' is
  measured.
• )To obtain the end point SC, of the clothoid,
  respectively the beginning of the circle arc,
  measure the distance X, and from there,
  perpendicular on the tangent measure the
  ordinate Y

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The clothoid curve used in horizontal
alignment

• )The common tangent to the clothoid and the
  circular arc makes with the tangent the angle ?,
  and the intersection of this line with the
  tangent is located at the distance N, measured
  from the origin of the clothoid. The Romanian
  norm STAS 863-85 . Road Works, Geometrical
  Elements of Lay Outgt Design Specifications,
  contains all these main elements( functions) of
  clothoid arcs, having a lengths given as
  function of the Design Speed ( V) and radius of
  curvature R, as follows
• R- radius of curvature
• l,L -the length of the clothoid from the origin
  to the common point with the circular arc
• A-the modulus of the clothoid
• ?R- the shifting of the circle done in order
  to accommodate clothoids
• X- the abscissa of the end point of the
  clothoid
• Y-the ordinate of the end point of the
  clothoid
• X'- abscissa of the centre of the circular arc
• Y'- the ordinate of the point M(X'Y') of the
  clothoid
• N- the abscissa of the point of intersection
  between the common tangent and the tangent
• ?- the angle made by the common tangent with
  the tangent
• -the slope in transverse profile or the interior
  slope of the superelevated curves ()
• d- the maximum gradient permitted in longitudinal
  profile ()
• e- the widening of the traffic lane ( cm)

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The clothoid curve used in horizontal
alignment

• )The common tangent to the clothoid and the
  circular arc makes with the tangent the angle ?,
  and the intersection of this line with the
  tangent is located at the distance N, measured
  from the origin of the clothoid.

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The clothoid curve used in horizontal
alignment

• The Romanian norm STAS 863-85 . Road Works,
  Geometrical Elements of Lay Outgt Design
  Specifications, contains all these main
  elements( functions) of clothoid arcs, having a
  lengths given as function of the Design Speed (
  V) and radius of curvature R, as follows
• R- radius of curvature
• l,L -the length of the clothoid from the origin
  to the common point with the circular arc
• A-the modulus of the clothoid
• ?R- the shifting of the circle done in order
  to accommodate clothoids

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The clothoid curve used in horizontal
alignment

• X- the abscissa of the end point of the
  clothoid
• Y-the ordinate of the end point of the
  clothoid
• X'- abscissa of the centre of the circular arc
• Y'- the ordinate of the point M(X'Y') of the
  clothoid
• N- the abscissa of the point of intersection
  between the common tangent and the tangent
• ?- the angle made by the common tangent with the
  tangent
• -the slope in transverse profile or the interior
  slope of the superelevated curves ()
• d- the maximum gradient permitted in longitudinal
  profile ()
• e- the widening of the traffic lane ( cm)

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The clothoid curve used in horizontal
alignmentProblems

• WORKSHOP No.2
• For the best selected route in the frame of the
  Workshop no.1, and in accordance with the
  necessary arrangements specified for each
  connection curve , used in horizontal alignment,
  proceed as follows
• 1. Introduce transition curves and calculate
  their main functions and then, the appropriate
  functions of the remaining circle arcs

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The clothoid curve used in horizontal
alignmentProblems

• WORKSHOP No.2
• 2. After introducing symmetrical transition
  curves, calculate the length of the new tangents
  and the length of all connection curves (
  spirals and remaining circle arcs) and derive
  the total length of your route, in horizontal
  alignment, at this stage.

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Additional Readings

• Andrei R. Land Transportation Engineering,
  Technical Publishers, Chisinau, 2002
• Garber j.N., Hoel A.,L, Traffic and Highway
  Engineering, revised second edition, PWS
  Publishing,1999
• Woods K. B., Highway Engineering Handbook,
  McGRAW- HILL Book Company, First edition, 1960

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Additional Readings

• Zarojanu Gh.H. Popovici D., Drumuri- Trasee,
  Editura VENUS, Iasi,1999
• Belc F. Cai de comunicatie terestre. Elemente de
  proiectare, Editura Orizonturi Universitare,
  Timisoara, 1999
• STAS 863-85 Road works. Geometrical elements of
  Lay out. Design specifications

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Additional Readings

• Hikerson F.T. RouteLocation and Design, Mc
  GRAW-HILL, Fifth Edition, 1967
• Civil Engineer's Reference Book, 3-rd Edition,
  Butterworths, London, 1975
• Dorobantu si al. Drumuri. Calcul si Proiectare,
  Editura tehnica bucuresti, 1980