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

2. Highway design procedures Route Geometric Design/Horizontal Alignment/Circular Curves
Circular curves .Simple curves. Main Functions. Compound curves. Revere curves
Criteria for the selection of a radius for a circular connection curve
Problems
Additional Readings

3. Route Geometric Design
Geometric design of highways deals with the dimensions of the visible features of the highway such as the alignment, site distances, widths, slopes, grades ,etc.
The geometric design is neatly distinguished from the structural design which deals with thickness, composition of materials, load carrying capacity, etc.

4. Route Geometric Design
. When economy is necessary, it is recommended that the main geometric features of the alignment, grade and sight distance do not be affected, because such features, once moulded into the landscape and tied by the right of way and surfacing, are the most expensive and difficult to correct in the future life of the highway.

5. Route Geometric Design
In order to achieve a flexible and in the same time an efficient design, appropriate road design standards have been drafted and intended to apply as general design controls regardless of the system of which the highway is a part. The use of more liberal values than the indicate minimum values from standards is recommended where conditions are favourable and the costs are not excessive.

6. Route Geometric Design
The following documents are recommended to be consulted, in conjunction with this course, for a better understanding of the geometric design criteria for highways:
The American Association for State Highway and Transportation Officials (AASHTO) geometric design standards;
The British Highway Link Design standard TD 9/93
The Romanian standard STAS Road Works. Geometrical elements of lay outs/ Design specifications

7. Route Geometric Design/Horizontal Alignment
The design of a road or of a railway is usually done , in office or in terrain, by working with the projections of this axis on both horizontal and vertical planes, and than by coordinating these both projections, in order to get a three dimensional efficient and high technical level design.
By projecting the axis of a highway or of a railroad on a horizontal plane , one may get the the route in plane or the so called horizontal alignment, as shown in the next slide

8. Route Geometric Design/Horizontal Alignment

9. Route Geometric Design/Horizontal Alignment
In relation with the previous slide , technically, the straight lines are called tangents, and a curve uniting two intersecting tangents is known by its radius R or by the angle subtended at the centre of the arc.
The usual connection curves used in horizontal alignments are classified as simple, composed and reverse curves.

10. Route Geometric Design/Horizontal Alignment /Circular Curves Simple Curves
A simple curve is a circular arc joining two tangents. The next slide shows a connection curve with all its related parts or functions, are explained, as follows:
1) If the tangents be produced, they will meet at a point of intersection PI (or the vertex ,V)

11. The relation for the calculation of the long chord: LC = 2R sin /2
Route Geometric Design/Horizontal Alignment/Circular Curves Simple Curves
2) Proceeding from left to right around the curve, point A, the beginning of curve is called point of curvature PC, while point B, the end of the curve is called PT point of tangency
3) The external angle of deflection between tangents is called the intersection angle or angle. This angle  is equal to the central angle subtended by the arc AB.
4) The distance VA=VB from the PI to the PC or to the PT is called tangent distance T. The relation for the calculation of the tangent distance :T = R tan 1/2 
5) The straight line AB from the beginning to the end of the curve is called the long chord or LC.
The relation for the calculation of the long chord: LC = 2R sin /2
6) The external distance E is the distance from PI to the middle of the curve. The relation for calculation of the external distance: E = R( sec /2 - 1)
7) The middle ordinate M is the length of the ordinate from the middle of the long chord LC to the middle of the curve and its value is caluclated with the relation: M = R( 1 - cos /2)

12. Route Geometric Design/Horizontal Alignment/Circular Curves Simple Curves/Main Functions

13. Route Geometric Design/Horizontal Alignment/Circular Curves Simple Curves
4) The distance VA=VB from the PI to the PC or to the PT is called tangent distance T.
T = R tan 1/2 
5) The straight line AB from the beginning to the end of the curve is called the long chord or LC.
LC = 2R sin /2

14. Route Geometric Design/Horizontal Alignment/Circular Curves Simple Curves
6) The external distance E is the distance from PI to the middle of the curve.
E = R( sec /2 - 1)
7) The middle ordinate M is the length of the ordinate from the middle of the long chord LC to the middle of the curve
M = R( 1 - cos /2)

15. Route Geometric Design/Horizontal Alignment/Circular Curves Simple Curves
8. The length of the curve C is calculated with the following relations:
C = 2R /360
C = 2R g/ 400g

16. Route Geometric Design/Horizontal Alignment/Circular Curves/Compound Curves
Compound curves : Two concentric circular curves constitute a compound curve if they join at a point of tangency where both curves are on the same side of the common tangent. The radius of the two curves are different but in the same directions at their junction. The point of tangency is called PCC, meaning point of compound curvature, as shown in the next slide

17. Route Geometric Design/Horizontal Alignment/Circular Curves Compund Curves

18. Route Geometric Design/Horizontal Alignment/Circular Curves Reverse curves
When two such curves have their centres on appropriate sides of the common tangent, as shown in the next slide, this combination is known as a reverse curve.
In relation with the figure from the next slide, the two circular curves AC and CB located on opposite sides of a common tangent constitute a reverse curve. The common point of reverse curvature is called PRC.

19. Route Geometric Design/Horizontal Alignment/Circular Curves Reverse curves

20. Route Geometric Design/Horizontal Alignment/Circular Curves Reverse curves
Such curves are used frequently in mountainous country, in cities ,and in the layout of railway spur tracks and crossovers. When used, it is desirable that the two curves to be separated by an intermediate tangent distance of sufficient length, in order to expedite the super-elevation approach and runoff.
The reverse curves should be avoided on major high- speed highways and railroads, as there is no possibility to elevate the outer bank at PRC.

21. Route Geometric Design/Horizontal Alignment/Circular Curves Criteria for the selection of a radius for a circular connection curve
On curves, super-elevation is necessary to overcome the tendency of vehicles to slide away from the centre of the curve or to overturn, due to the centrifugal force.
Centrifugal forces acts above the roadway surface through the centre of gravity of the vehicle and creates an overturning moment about the points of contact between the outer wheels and the road pavement. ( see the figure in the next slide)
In relation with this figure, opposing overturning is the stability moment created by the weight of vehicle downward its centre of gravity. Overturn can occur only when the overturning moment exceeds the stabilising moment

22. Route Geometric Design/Horizontal Alignment/Circular Curves Criteria for the selection of a radius for a circular connection curve The main forces involved in the interaction between the running vehicle and the road pavement on a flat circular curve.

23. Route Geometric Design/Horizontal Alignment/Circular Curves Criteria for the selection of a radius for a circular connection curve
On a superelevated curve, both the tendency to slide and the tendency to overturn can be completely eliminated , if friction is neglected, by superelvating sufficiently, so that the component of the weight parallel to the road surface ( G sin ) equals the component of centrifugal force ( Fc cos ), parallel to the roadway surface, as shown in the next slide.

24. Route Geometric Design/Horizontal Alignment/Circular Curves Criteria for the selection of a radius for a circular connection curve The main forces involved in the interaction between the running vehicle and the road pavement on a superelevated curve

25. Route Geometric Design/Horizontal Alignment/Circular Curves Criteria for the selection of a radius for a circular connection curve
These two cases presented above are extreme situations, and their studies conducted with the aim to combine both friction and super- elevation effects for road design purposes , lead to the deriving of so called design relations , which permit to calculate the minimum radius ( Rmin) of a curve function of a given design speed (V), maximum permitted superelevation(ps)and a comfort coefficient (k)

26. Rmin >=V2/ 13ps( k + g) (5.1)
Route Geometric Design/Horizontal Alignment/Circular Curves Criteria for the selection of a radius for a circular connection curve
Rmin >=V2/ 13ps( k + g) (5.1)
V is the selected design speed in Km/h,
ps is the maximum permitted superelevation expressed in percentage ,
the value of k ranges between 10 to 15 for motorways and between 15 and 25 for the other public roads
and g= 9,8 m/s2

27. Route Geometric Design/Horizontal Alignment/Circular Curves Criteria for the selection of a radius for a circular connection curve
If the cross profile of a carriage way is converted from the normal shape ( two turnovers of a slope pa), to a cross profile with a unique slope (positive turnover), we can obtain the current radius (Rc) by using the same relation, just by replacing ps with pa, as follows :
Rc = V2 / 13pa ( k + g) ( 5.2).

28. Route Geometric Design/Horizontal Alignment/Circular Curves Criteria for the selection of a radius for a circular connection curve
If no change of cross profile on the curved section are affected, the outer lane has a slope which contributes to the skidding, because the component of the vehicle weight parallel to the pavement surface is oriented outside the centre of the curve, as the centrifugal force is oriented, as shown in the next slide
In such a case , we may derive and get the recommendable radius ( Rrec), with the relation 5..3:
Rrec = V2 / 13pa ( k - g ) (5..3.)

29. Route Geometric Design/Horizontal Alignment/Circular Curves Criteria for the selection of a radius for a circular connection curve The interaction of forces between the running vehicle and the road pavement on circular section without super-elevation.

30. Route Geometric Design/Horizontal Alignment/Circular Curves Criteria for the selection of a radius for a circular connection curve
The radii ranging between Rmin and Rc are called minimum radii, and the arrangement of such a curve consists of : spiral insertion, curve widening, full super-elevation and sight distance along the inside of the curves.
The radii ranging between Rc and Rrec are called current radii and the arrangement of such curves includes: spiral insertion, curve widening, if R<= 300m, and sight distance along the inside of curves, if necessary.
The radii higher the R rec, are called recommendable radii, and such curve do not need any special arrangements

31. Problems
1. For a given contour map, drawn at the scale 1:1000, undertake the appropriate route study for a two-way road link, between the terminal points A & B , by using the method of the ax of zero level. Start your route study on the map with a maximum permitted gradient ga = 4%.
2. Produce at least three alternatives and then proceed with the compensation of the polygonal lines thus obtained, by drawing the appropriate intersecting tangents.

32. Problems
3.Determine the values of the angles formed in each vertex in order to make the necessary tangent connections with simple circle arcs, of appropriate radii.
4.Derive the values of the minimum (Rmin), current (Rc) and recommended ( Rrec) by using the specific design relations, for the following given parameters:
-the design speed, V = 40Km/h
-the gravity acceleration : g = 9,8 m/s2
- the comfort coefficient k = 30
-the normal cross slope on the tangent: pa =2,5%
-the maximum permitted slope for superelevation on curves: ps = 6%;

33. Problems
5. Select and introduce circular arcs of appropriate radii and calculate their main functions.
6. Specify the necessary arrangements for each connection curve used in your horizontal alignment in terms of the need for introducing transition curves, curve widening, curve superelevations, etc.
7. At this stage calculate the length of tangents and the length of circle arcs used for connections and derive the total length of your route, in horizontal alignment.

34. Additional Readings
.Woods K. B., Highway Engineering Handbook, McGRAW- HILL Book Company, First edition, 1960
.Hikerson F.T. RouteLocation and Design, Mc GRAW-HILL, Fifth Edition, 1967
. Zarojanu Gh.H. Popovici D., Drumuri- Trasee, Editura VENUS, Iasi,1999
.Civil Engineer's Reference Book, 3-rd Edition, Butterworths, London, 1975

35. Additional Readings
.Civil Engineer's Reference Book, 3-rd Edition, Butterworths, London, 1975
.Dorobantu si al. Drumuri. Calcul si Proiectare, Editura tehnica bucuresti, 1980
.STAS Road works. Geometrical elements of Lay out. Design specifications.
.British Standard : BS:TD.9/93 Geometrical Design of Roads, Design Specifications

36. Additional Readings
American Association for State Highway and Transportation Officials: AASHTO Geometric Design Standards for Highways and Free Ways, 1975
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