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

2. 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

7. 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

9. 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

10. 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 C=mV2/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.

11. The clothoid curve used in horizontal alignment

12. where k1 is a specific clothoid constant.
The clothoid curve used in horizontal alignment The 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.

13. The clothoid curve used in horizontal alignment The 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.

14. The clothoid curve used in horizontal alignment The 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

15. The clothoid curve used in horizontal alignment The 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.

16. The clothoid curve used in horizontal alignment The 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  <0, the curve is situated in the third quarter of the trigonometric circle and in the first quarter for  >0. 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

17. The clothoid curve used in horizontal alignment The 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..

18. The clothoid curve used in horizontal alignment The omothety of Clothoids

19. The clothoid curve used in horizontal alignment The 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.

20. The clothoid curve used in horizontal alignment The 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

21. The clothoid curve used in horizontal alignment The 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

22. The clothoid curve used in horizontal alignment Criteria 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 = V, where v is expressed in m/sec. and V is considered in Km/h

23. The clothoid curve used in horizontal alignment Criteria 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.

24. The clothoid curve used in horizontal alignment Criteria 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 > = 2R/18 ; L > = 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 > =  24R * R

25. The clothoid curve used in horizontal alignment Criteria 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 > =  24R * R

26. The clothoid curve used in horizontal alignment Criteria 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 > = 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.

27. 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)

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

29. The clothoid curve used in horizontal alignment Methods 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;

30. The clothoid curve used in horizontal alignment Methods 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;

31. 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 Road Works, Geometrical Elements of Lay Out> 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);

32. 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.

33. The clothoid curve used in horizontal alignment
The Romanian norm : STAS Road Works, Geometrical Elements of Lay Out> 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;

34. 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);

35. The clothoid curve used in horizontal alignment Problems
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

36. The clothoid curve used in horizontal alignment Problems
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.

37. 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

38. 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 Road works. Geometrical elements of Lay out. Design specifications

39. 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