ADVANCES IN TRANSPORTATION STUDIES - · PDF fileCriteria for geometric design of highway alignments ... users could design the transition curves in the ... Advances in Transportation

Vol. XXXVI • July 2015

ADVANCES INTRANSPORTATION STUDIESAn International Journal

Editor-in-Chief: Andrea Benedetto

Section A & B

Contents

Section A

G. Bosurgi, A. D’Andrea, 5 An algorithm based on the PPC (Polynomial Parametric Curve)O. Pellegrino, G. Sollazzo for designing horizontal highway alignments

Y. Cong, X. Li, S. Zhang 21 Traffic flow forecasting by grey model and least squares support vector machine algorithm

L. Adacher, E. Cipriani, A. Gemma 35 The global optimization of signal settings and traffic assignment combined problem: a comparison between algorithms

S. Mitra, K. Utsav 49 A framework for in-vehicle warning system in reduced visibility using dynamic potential collision speed W. Fan, M. Kane, E. Haile 63 Predicting the severity of pedestrian crashes on highway-rail grade crossings

W. Li, Z. Sun, X. Hao, X. Feng, H. Zhao 75 Measuring method for asphalt pavement texture depth based on structured-light technology

Section B

F.E. Buitrago González, 87 Video-based approach to analyze driver decisions at signalizedA.M. Figueroa Medina intersections

S. Classen, A.K.A. Yarney, 99 Rater reliability to assess driving errors in a driving simulatorM. Monahan, S.M. Winter, K. Platek, A.L. Lutz

V. Ratanavaraha, S. Jomnonkwao 109 The efficiency testing of Shoulder Rumble Strips (SRS) on noise for alerting drivers in Thailand: a comparison among three types of SRS

L. Fan, L. Lu, W. Deng, J.J. Lu 119 Role of vehicle trajectory and lateral acceleration in designing horizontal curve radius of off-ramp: a driving simulator based study

J. Wang, Y. Li, X.J. Ji, S.E. Fang 133 Difference between motorcycle and electric bicycle accidents at arterial road access points in China

A. Tripodi, L. Persia 143 Impact of bike sharing system in an urban area

ADVANCES INTRANSPORTATION STUDIES

An International Journal

Section A

Advances in Transportation Studies an international Journal Section A 36 (2015)

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An algorithm based on the PPC (Polynomial Parametric Curve) for designing horizontal highway alignments

G. Bosurgi A. D’Andrea O. Pellegrino G. Sollazzo

Department of Civil, Computer, Construction and Environmental Engineering and of Applied Mathematics,

University of Messina (Italy), Vill. S. Agata, 98166, Messina (Italy) email: [email protected]

subm. 4th September 2014 approv. after rev. 26th January 2015

Abstract

Criteria for geometric design of highway alignments have been studied in various researches so far, providing new methods, as alternative to traditional approaches. They can be useful for obtaining better solutions, characterized by high performances in terms of motion safety and comfort. In particular, continuous curvature alignments, based on polynomial functions, are highly consistent solutions with users' driving behaviour and vehicle trajectories. Moreover, alignment geometry optimization allows engineers to minimize dynamic effects on drivers with considerable advantages in terms of comfort and safety. This paper carries on the research of a previous study, in which a polynomial parametric curve, called "PPC", useful for horizontal highway alignment design, has been proposed. Besides, this curve has been shown to be suitable for solving cases with complex geometry, such as the road interchanges. In these particular situations and in the design of horizontal curves in highway alignments, the computational advantages related to the use of a single curve, compared to traditional procedures based on a composition of different elements (clothoids and circular curves), are very clear. In particular, in this paper, an original algorithm, for simplifying the PPC introduction in the highway alignment design, is proposed. To verify the PPC improvements, an exhaustive numerical example, with several comparisons to traditional approach (characterized by straight sections, circular curves and clothoids as transition curves), has been performed. A both point by point and global examination of kinematic and dynamic variables directly involved while driving has attested the method validity. Keywords – highway design, polynomial curves, transition curves

1. Introduction

Driving safety and comfort are the main targets in highway alignment geometric design. A proper horizontal composition of the different geometric elements forming roads - such as tangent sections and curves - eliminates abrupt variations of driving behaviour, especially in terms of speed. This also handles in a right way the dynamic variables influencing safety and comfort.

In recent years, numerous studies have confirmed the assembly of elements with a constant curvature (tangents and circular curves) disagrees, above all in curves, with the actual trajectories chosen by drivers [39]. Users, in fact, gradually adapt while steering on curves to balance the dynamic effects of the radial acceleration. The "optimal trajectory" simulation, depending on the design speed of the curve, was the main aim in the analysis of analytic problems regarding the introduction of the transition curve between tangents and circular curves [30].

ISBN 978-88-548-8566-0 – ISSN 1824-5463-15002 – DOI 10.4399/97888548856601 – pag. 5-20


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The purpose of the transition curves (clothoid, cubic parabola, sinusoidal curve, etc.), whose the clothoid is the most widespread example, is to make gradual the radial acceleration and its variation, guarantee a better visual perception of the curve and assure a gradation of the steering manoeuvre in accord to the actual drivers' behaviour [13]. In the first steps of the analysis, several researches provided appreciable results using generic polynomial curves [8, 17, 28] and the least-square method to optimize their equations [10]. Considering the best trajectories for robots and automated vehicles [27, 31] or particular visual techniques or infrastructure safety and driving comfort examinations [1, 6, 9, 12, 33, 34], different attempts to improve transitions curves have been performed. The attention has also been focused on spline functions and other polynomial curves, comparing them to traditional transition curves [18, 19, 25, 29, 35]. In several papers, in order to define a completely curvilinear road alignment, many authors developed and presented interesting specific algorithms [11, 20]. Furthermore, other authors studied multispiral and G2 spiral characteristics, obtaining several solutions to handle transition between straights sections and circular curves [7, 14, 15, 40]. Similar studies have been carried out in the railway field, for optimizing the riding quality in high-speed lines; in this regard, various authors proposed suitable geometric solutions compatible with kinematic and dynamic models related to train motion [21, 22, 23, 24, 26, 36, 37, 38]. Other researchers adopted fifth degree polynomial curves and proved their geometric and computational versatility also in resolving some design cases with very complex geometry [2, 3, 4, 32]. In particular, in a previous paper [4] a specific polynomial parametric curve, called "PPC", was suggested and it has been showed to be appropriate for defining alignment elements with complex geometry (i.e. egg-shaped transition, transition between two reversing circular curves, semi-direct and inner-loop connections). Adopting the PPC, the study underlined higher potential and more advantages in comparison with traditional solutions. From this, significant computational advantages emerged in numerous applications. In particular, considering all the imposed geometric and dynamic constraints, especially the most critical situations can be solved in an easy way using the PPC.

In this study, the authors developed an automatic and high-performance procedure to define an entire horizontal highway alignment using the PPC. An original algorithm and a computer software was developed, for simplifying the whole procedure. This study is useful to evaluate deeply the main geometric and dynamic characteristics of the PPC and the related advantages. Despite some observations and considerations about the effects of the PPC parameters are discussed in this paper, its main goal is to compare a whole PPC horizontal alignment with a traditional one, for assuring the better performance of the PPC. Moreover, this algorithm could be very useful to improve automatic procedures of highway alignment optimization [5]: in a very easy way, in fact, users could design the transition curves in the alignment as single elements, without any analytic and geometric complication due to a composition of clothoid and circular curves. Therefore, using this approach, optimization can quickly take into account dynamic aspects and provide a better and more proper alignment.

In the following sections, the geometric characteristics of the PPC will be briefly provided and, then, the computational model will be presented.

2. PPC characteristics and dynamic parameters

2.1. Brief theoretical notes

In this section, the main geometric and analytic aspects of the PPC are provided. However, further details can be found in a previous paper [4].


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-degree polynomial, as evidenced in equation (1):

)(1 2345 feldlclblalR

l [1/m] (1)

where l = s/L normalized abscissa (L= total length of the curve and s = curvilinear abscissa); R =

minimum value of the radius of the osculating circle [m]. The coefficients a, b, c, d, e, f are obtained by imposing the boundary conditions for the curvature and its derivative; = parameter that changes the shape and the length of the curve.

This study, whose goal is to develop a whole alignment using the PPC, focuses only on the condition of transition between two tangent sections, considering the following boundary conditions:

�(0) = 0 �(1) = 0

�(�) = �� =1

R��

��(0) = ��(0)�� = 0

��(1) = ��(1)�� = 0

��(�) = ��(�)�� = 0

where osculating circle

with R = Rmin. In detail, using the first two conditions, the authors fix the continuity of the curvature at the

tangent points A and B. The third condition defines the position of the maximum curvature point, while the last conditions prthe curvature function. expression (1) is changed as follows:

)2345(1

)( dlclblalR

l (2)

Referring to Figure 1, the deviation angle of the PPC results in: l

dllLl0

[rad] (3)

a is the deviation angle of the tangents, setting (l , the total length is:

1

0)( dll

L [m] (4)

F a is equal to 0 and, therefore, the curvature results in:

)234(1

)( dlclblR

l (5)


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Fig. 1 - Principal geometric parameters of the curve

Considering the boundary conditions, b, c, and d are respectively 16, -32, and 16. Thus, solving the integral, L becomes:

min875.1 RL (6)

Because and , the parametric equations of the curve are obtained by the Taylor series expansion:

dlllRlxl

0..4

2412

21

1031.1)( min [m] (7)

dlllRlyl

0..3

61

031.1)( min [m] (8)

Figure 1 shows the geometrical sizes of the curve. The authors indicated the coordinates of the Rmin Rmin.

The evaluation of the distances Di and Du is helpful to calculate residual tangent length and obtain the coordinates of point A (i.e. the origin of local reference system) and B in the global reference system.

cotff YXDi [m] (9)

sinfY

Du [m] (10)

The coordinates of the point A in the global reference system can be calculated using expressions (11) and (12):

23** sin

3DiXX VA [m] (11)

23** cos3

DiYY VA [m] (12)

where X*

A = X coordinate of point A in the global reference system [m]; Y*A = Y coordinate of point

A in the global reference system [m]; X*V3 = X coordinate of vertex V3 in the global reference

system [m]; Y*V2 = Y coordinate of vertex V3 in the global reference system [m]; 23 = direction

angle of straight V2V3 [rad].


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2.2. Dynamic variables

With reference to the checks on the motion and on the effects over the driver, the authors adopted the following dynamic variables, considering a hypothesis of constant speed along the curve:

- rate of change of radial acceleration

dlvd

Llv

lz)2()( [m/s3] (13)

- the steering speed

dld

Lvp

l [rad/s] (14)

- the roll speed

dld

Luv

lVrmax

max)( [rad/s] (15)

where p = wheelbase of the vehicle [m]; umax = maximum value of the superelevation rate [%/100];

v= speed [m/s]. For the superelevation rate, using the equation of the curvature, expression (16) is adopted:

lulu 1maxmax [%/100] (16)

Thus, the non-compensated rate of change of radial acceleration is represented by equation (17):

Lvluglv

dldlznc

2 [m/s3] (17)

where g = gravity acceleration [m/s2]. As underlined in the previous expressions, the particular geometric characteristics of the PPC

guarantee the continuity of the dynamic variables involved while driving that directly influence the driver's behaviour and the comfort. For instance, the gradual trend of the steering speed is certainly consistent with the actual behaviour of the driver along the curve, where the actual trajectory depends not only on the speed and on the driver’s experience, but also on an increasing adaptation to the curve geometry (including or not transition curves). Thus, a more gradual variation of the curvature, of the steering speed, and of the rate of change of radial acceleration helps users to drive better and in a safer and more comfortable way. Consequently, the PPC seems to be more appropriate than the clothoids in the design of horizontal curves, as proved by the numerical examples provided in the following sections. expressions (13), (14), (15), and (17) become the following ones (18), (19), (20), and (21):

lR

vlz 2min

3

875.132 (18)

lRvpl 2

min875.132 (19)


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lRaulVr

min

max

875.132 (20)

2minmax1)(

dnc v

Ruglzlz (21)

where

llll 23 32 (22)

The maximum and minimum values of the previously listed variables can be obtained for , and they are respectively:

2min

3

minmax, 642.1Rvz (23)

2min

minmax, 642.1R

vp (24)

min

maxminmax, 642.1

RauVr

(25)

3. Script architecture of the proposed algorithm

In Figure 2, a flowchart depicting the original script architecture is represented. In the following, the whole procedure for defining highway horizontal alignments using PPCs is explained and the different optimizing choices are separately analysed.

Fig. 2 - Flowchart of the script


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In detail, the procedure can be divided in the following different phases:

Data inputThe first step is the definition of the general design data and the X* and Y* coordinates of the

vertices Vi. The origin of the global reference system is placed in the point V1 (i.e. the first vertex). Therefore, for each curve, the values of Rmin ust be set. The choice of Rmin must be related to specific design criteria to guarantee geometric consistency of the road alignment, in

appropriate to use 0.5 so as to determine a symmetrical curve. Anyway, in some specific situations, it may be necessary to vary this coefficient to obtain different solutions that are best

s

inappropriate variations of the curve modifying its shape and its analytic structure.

PPC calculation and optimization of the designThrough the expression (2) it

PPC. Therefore, the algorithm evaluates and checks the dynamic variables (steering speed, roll speed and rate of change of radial acceleration), in order to verify the compliance with the limitvalues related to safety and comfort needs defined by road standards Regarding the rate of change of radial acceleration z, for instance, the authors have considered the following expression taken by the Italian standard [16]:

Vz 4.50

lim (V [km/h]) (26)

By way of example, some figures about the representation of the PPC geometry (Figure 3) and the trends of the related dynamic variable (Figure 4), for a specific curve (Rmin

below.In the input phase, user can choose among the following criteria:a) design of PPCs using the imposed osculating circle radii – i.e. “no control” side in Fig. 2;b) correction of the imposed osculating circle radii, in accordance with limitations on rate

of change of radial acceleration z – i.e. “z control” side in Fig. 2;c) correction of the imposed osculating circle radii, to obtain a geometrically correct and

continuous alignment – i.e. “geometric control” side in Fig. 2.In case a), the analysis is executed using the R values set in the input phase. The PPCs are

only calculated and represented, without any optimization. Following this procedure, the user could guess the final alignment is not continuous: in particular, it cannot guarantee the path geometric congruence (e.g. intersection points between two following curves could be found).

Fig. 3 - Representation of a single PPC


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Fig. 4 - Diagrams of geometric and dynamic parameters of a single PPC

Moreover, the max values of the evaluated dynamic variables could reach and exceed the limit threshold. However, the algorithm provides useful information for helping the designer in the definition of radius values and vertex positions. Otherwise, the method b) performs a preliminary check of the max value of the rate of change of radial acceleration z for each curve: in this case, in order not to exceed the limit value zlim in any point of the curve, a specific procedure for radius optimization is adopted. This is an iterative procedure, because the radius value influences the design speed of the curve, modifying its geometry and the related dynamic variables. Obviously, by increasing the osculating circle radius values, the PPCs become longer and thus the geometric congruence between following curves cannot be ensured. However, in order to optimize design inputs (radius choice and vertex positioning), in the planning phases, this method can be very useful. On the contrary, the definition of radius values providing geometric continuity and congruence between following curves is the target of the last procedure, c). In this case, the initial radii are reduced and, as a consequence, even if intersection points between curves are avoided by decreasing their length, the dynamic prescriptions can easily be violated. However, this procedure, as the previous one, is useful to determine and design the input values, providing other strategic information about radii and residual straights. After choosing a specific procedure, the algorithm analyses the alignment and evaluates each PPC, providing both local and global geometric representations and several diagrams related to the main dynamic variables.

4. Numerical example

A specific numerical example is presented: in particular, the authors compared the results obtained by using the PPC with those produced through traditional design procedures based on a composition of transition (clothoids in this particular case) and circular curves.

In this paper, the several considerations about the optimization of the geometric characteristics of this specific numerical application are not provided, and the authors chose to report only the final input values determining a proper alignment, for discussing the main benefits due to the PPC application.


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Tab. 1 - Vertex coordinates

Vertex X*[m] Y*[m]1 0.00 0.002 86.10 281.983 536.48 414.244 783.53 2.155 1310.61 220.636 1452.98 0.36

Tab. 2 - Values of Rmin, Vp and parameter A

Vertex Rmin [m] Vp[km/h] A[m]1 --- --- ---2 150 66.50 121.503 160 68.18 136.004 165 68.99 150.005 175 70.56 136.006 --- --- ---

The considered road path is made of 6 different vertices. The coordinates X* and Y* of each vertex in the global reference system are listed in Table 1. The PPCs have been designed in a

been defined using monoparametric clothoids and their parameters A. In table 2, the initial design data of the different curves are provided (minimum radii, design speeds and parameters of clothoids). Moreover, in Figures 5, 6, 7, 8, and 9, the results of the elaboration are represented, comparing the two different solutions in terms of geometric characteristics (alignment and curvature) and trend of dynamic variables (rate of change of radial acceleration and steering speed). In Fig. 5, the tangents have been evidenced using little circles.

Fig. 5 - Final alignments Fig. 6 - Curvature diagrams


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Fig. 7 - Rate of change of radial acceleration diagrams

Fig. 8 - Steering speed diagrams

Fig. 9 - Design speed diagrams

The variable s* represents the curvilinear abscissa of the whole alignment.In Figure 9, according to Italian Road Standard definition of design speed, the related trends

for both the alignments have been plotted. Obviously, this is not an operative speed, but only a design speed using which each geometric element of the road (superelevation rate, sight distance, lengths of crest and sag vertical curves, etc.) can be defined. This diagram can help engineers to design a proper and consistent alignment, eliminating abrupt changes of speed between continuous elements and permit them to evaluate the geometric consistency of the proposed alignment. Finally, the most representative data of the two different solutions are listed in Table 3.


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Tab. 3 - Final results

Vertex Rmin [m]

Vp [km/h]

[°]

zlim [m/s3]

zmax [m/s3]

Ltot [m]

dVi [m]

Res. Str. [m]

1 PPC --- --- --- --- --- --- 294.83 147.63 Clot --- --- 156.92

2 PPC 150 66.45 56.65 0.758 0.464 278.10 469.4 102.39 Clot 0.427 259.26 146.62

3 PPC 160 68.18 75.42 0.739 0.330 394.91 480.47 10.20 Clot 0.367 327.93 81.44

4 PPC 165 68.99 81.57 0.730 0.297 440.45 570.57 62.58 Clot 0.313 371.27 150.46

5 PPC 175 70.56 79.64 0.714 0.290 456.08 262.27 4.75 Clot 0.407 358.46 56.35

Total length – PPC[m] 1897.10 Total length – clothoid [m] 1908.71

5. Results and discussion

Comparing the two solutions, first of all, it is easy to note main differences in the variation of the dynamic variables along the curve (Fig. 7, 8). In particular, using PPCs, the dynamic trends of both variables show more gradual variations and, consequently, this produces increasing comfort levels for users and improves driving behaviour. This progressive tendency strictly depends on the

gradual variation of the dynamic variables. This constitutes the main advantage of this curve, since it guarantees higher performances in terms of users’ comfort. By analyzing the numerical example, it is easy to remark PPCs are slightly longer than the traditional curves (clothoid, circular curve, clothoid) for each vertex: on average, this difference varies between 7% and 21% (Table 3). Although longer curves may induce wider right-of-way costs and more design troubles, this is not a remarkable difference and, moreover, the total length of the alignment is shorter in the PPC-case. Anyway, these particular conditions and, in detail, the length of the different horizontal elements may be handled properly, in specific critical scenarios, by varying the parameter values. This can be done in order to adapt the curve to the different design needs, and to design it in compliance with all the several constraints. The effects of the two parameters, evidenced in Figures 10, 11 and in Table 4, are discussed in the following. Besides, for proving the better performances of the PPC is worthwhile to check the peak values assumed by the rate of change of radial acceleration z, widely adopted as the main representative variable for describing user’s comfort. Considering dynamic limitations, although in both procedures zmax values are smaller than the threshold value, in three different curves (3, 4, 5), the outcomes produced by the PPC are smaller than those in the clothoid solution. In detail, examining the numerical values, the differences in terms of zmax are between 10% and 30% and they become greater with increasing of R and . Then, using PPCs improves horizontal alignment characteristics, providing benefits in terms of users’ comfort. This is due not only to a definition of more realistic trajectories, but also to a more advantageous variation of dynamic variables and to a mitigation of their critic peaks, as evidenced by the numerical outcomes and the previous figures

Furthermore, in actual design problems, specific advantages, in terms of length and shape of

solutions, useful to solve several practical design situation, can be reached by modifying them in a right way.


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Fig. 10 -

Fig. 11 -

For underlining the actual advantages related to the parameter variation, other interesting numerical considerations have been done on a specific curve. In particular, the first curve (vertex 2) has been analysed.

solutions hav(0.8, 0.9, 1.0, 1.1, 1.2); the main results are provided in Fig. 10, 11 and in Tab. 4. This is very worthwhile in order to understand the influence of these parameters on the geometry of the curve and, thus, on the dynamic effects on users.

The final alignments of specific solutions are represented in the following diagrams. In detail,

Although parameter effects on the alignments are provided separately in different figures, the important differences produced are not clearly visible. However, there are interesting dissimilarities and it is very precious to analyse them deeply. In detail, some numerical outcomes

maximum curvature point and, thus, the PPC becomes asymmetric. As a consequence, the curve stretches and, while a tangent point approaches to the vertex, the other one moves away from it. Because of these features, the rate of change of radial acceleration z is subject to a quicker changealong the curve and it shows a higher peak.


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Tab. 4 - L, di and zmax for different parameter values

L [m] 280.7 293.2 305.2 316.7 327.8 di [m] 129.6 134.3 138.8 143.2 147.4

zmax [m/s3] 0.580 0.556 0.540 0.528 0.519

L [m] 255.3 265.8 275.9 285.7 295.2 di [m] 135.5 140.8 145.9 150.9 155.6

zmax [m/s3] 0.494 0.479 0.489 0.462 0.457

L [m] 280.7 293.2 305.2 316.7 327.8 di [m] 167.2 175.1 182.6 189.8 196.8

zmax [m/s3] 0.580 0.556 0.540 0.528 0.519 By

the vertex. The curve becomes longer and the tangent points (on both sides) diverge from the urve (due to the

PPC extension) and its max value decreases progressively. Then, a proper choice of the parameters allows engineers to adapt the PPC to all specific design constraints, keeping constant the radii (and thus the speed of the curve) and the vertex position. This guarantees high geometric flexibility of the PPC, assuring to engineers such a design freedom to provide the best alignments in every practical situation. However, this curve could seem improper for practical application, since the minimum radius is realized for one point only. Actually, despite some guidelines, in order to guarantee a better users’ comfort, recommend to maintain this value for a certain length, by adopting the PPC the comfort standards are higher along the whole curve. This is due to the smoother trends of the curvature and thus of the dynamic variables along the curve. The authors believe more researches are needed for solving properly this doubt, but they think the more gradual trend dynamic variables can help drivers to avoid misperception of the curve evolution. Anyway, in order to design the curve in compliance with this specific prescription, considering the high flexibility of the curve, it is possible to draw a different PPC, used as a common transition element, by simply changing the boundary conditions of the curvature equation. In this way, the curve can be designed in a traditional way, maintaining a residual circular curve. This permits to realize two new PPCs as common transition element alternative to the clothoid for each curve, assuring at the same time higher and higher dynamic performances.This solution can be selected also for solving another possible drawback of the PPC. Since the curvature is smoother than the clothoid, the superelevation rate could increase more slowly. At the beginning of the curve, where observations demonstrated the operative speed may be higher than the design one, there could be problems for safety. Although many other examinations are need also for this issue, by decreasing the available length for handling the superelevation variation, this specific PPC can resolve this question. In the same way, considering the whole PPC, designers can easily avoid this critical scenario adopting the max superelevation for a certain length. This length must be established in a proper way, in order to reach higher values of superelevation at the beginning of the curve also, even in compliance with the specific prescription about the relative grade of superelevation runoff. Furthermore, as a direct consequence, the points closer to the minimum radius point will be designed with a higher safety coefficient. Finally, in Figures 12 and 13, two original different diagrams, useful to quickly optimize the design solutions in general cases of symmetry ( = are related to the minimum radius and to the deviation angle. They can be useful in planning phases for optimizing vertex position and their mutual distance.


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Fig. 12 - Rate of change of radial acceleration for the PPC ( = 0.5)

Fig. 13 - Total length of the PPC ( = 0.5)

By way of example, using the diagram presented in Figure 12, the zmax value for a PPC for different values of Rminchoosing the best Rmin max.

The diagram of Figure 13, instead, allows designers to evaluate the total length of the PPC for several values of Rmin

6. Conclusions

In this paper, the authors proposed a particular algorithm for designing highway horizontal alignments using the PPC as transition curve. As evidenced by the numerical examples, the PPC guarantees a more proper transition between straights and circular curves than traditional solutions (clothoids), in terms of both geometric and dynamic aspects.

Furthermore, even if more improvements of the curve may be useful in order to simplify its practical adoption, the results and comparisons provided in this paper are very interesting, in terms of flexibility and computational performances of the PPC. This derives from the opportunity to define a unique element for the whole curve, reducing analytic complications and decision time. Moreover, for designing roads in compliance with all the possible constraints, additional flexibility can be obtained by varying the two parameters of the PPC, or by selecting different boundary conditions for the curvature equation.

The algorithm presented in this paper further underlined the significant potentialities of the PPC in the geometric design of the horizontal highway alignments. Indeed, through few and quick operations, using this curve, it can be possible to define specific design solutions, actually optimized and in compliance with several criteria provided by the road standards.


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Besides, as underlined in a previous paper, other cases characterized by a very complex geometry, such as for instance the semi-direct and inner-loop connections, can be quickly solved and designed. In this regard, it could be very useful to study in future researches the actual geometric, dynamic and computational advantages related to the use of PPCs in the interconnection design, and to properly analyze in a deep way some possible drawbacks appeared in this research. Finally, this algorithm could simplify the introduction of the transition curves in highway alignment optimization procedures, in order to improve the searching method and evaluate and consider dynamic variables in the path choice.

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