Transcript LAND TRANSPORTATION ENGINEERING (Notes for Guidance )

SPECIAL TRANSPORTATION
STRUCTURES
(Notes for Guidance )
Highway Design Procedures/Route Geometric
Design/Horizontal Alignment/Transition Curves
Radu ANDREI, PhD, P.E.,
Professor of Civil Engineering
Technical University “Gh. Asachi” IASI
Lecture
Nine
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
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.
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.
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.
The need for introduction of transition curves
Principle of connection of two tangents with a simple
circular curve and two spirals
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
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 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
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.
The clothoid curve used in horizontal alignment
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.
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.
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
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.
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
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
The clothoid curve used in horizontal alignment
The omothety of Clothoids
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.
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
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
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 =
0.556V, where v is expressed in m/sec. and V is
considered in Km/h
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.
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
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
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.
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)
The clothoid curve used in horizontal alignment
Methods for introducing of two symmetrical spirals
between the tangents and the circle arc
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;
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;
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 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;
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 clothoid curve used in horizontal alignment
( The Romanian norm : STAS 863-85 . 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
accommodate clothoids;
circle
done in order to
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);
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
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.
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
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
Additional Readings
• Hikerson F.T. RouteLocation and Design, Mc GRAWHILL, 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