The Study of Geometry of the Selected Transition Curves in the Design of Circular Roads Advances in Science and Technology Research Journal

Transition curves connecting straight sections with circular curves and vice versa are present on all types of roads. They are designed to smooth the transition from rectilinear to circular motion. In numerous instances, they are hardly noticeable by road use. Although adequately selected, they increase the safety and comfort of driving. The research aimed to investigate the effectiveness of three most popular curves, in the design of road routes: clothoid, a parabola of the 3rd-degree and Bernoulli lemniscate. Starting from the permissible values of the increment of centripetal acceleration on the transition curves, the minimum lengths of the transition curves for the assumed speeds of the wheeled vehicle were determined. Then, based on the formulas of individual curves, their lengths were calculated, depending on the design speed and the radius of the circular arc. As a result of the research, the differences between the individual curves were obtained. Finally, it was found that due to the uncon strained movement of the vehicle on the road, all curves could be used as transition curves. By slightly adjusting their lengths, we can achieve the appropriate value of the increment of centripetal acceleration. The legitimacy of using different curves as transition curves lies primarily in their different geometric appearance. The choice of the curve to be used is based primarily on the terrain conditions and the type of the planned section of the route (serpentines, turnouts, etc.).


INTRODUCTION
Public roads are places where the transition curves are applied [1,2]. The requirements to be met by the transition curves in Poland can be found in the "Regulation of the Minister of Transport and Maritime Economy of March 2, 1999, on technical conditions to be met by appropriate public roads and their location" [3] and in the "Regulation of the Minister of Infrastructure of January 16, 2002, on technical and construction regulations for toll motorways" [4].
According to the aforementioned legal acts, transition curves should be used when connecting two road sections with different, constant curvature values. It is allowed to replace the transition curve with the transition straight line only in justified cases on L and D class roads and Z class streets [3].
The transition curve has a variable radius, from r = ∞ (k = 0) at contact with the straight line to the value of R equal to the size of the arc radius [5,6]. Thanks to this, the continuity of the route is maintained [7]. A vehicle moving along a curve is influenced by centrifugal force, the value of which increases with increasing curvature. In the transition curve, there is a combination of translational and rotary motion [8,9]. There have been many studies verifying the suitability of the curves for the routing of circular roads [10][11][12][13], in which emphasis was placed on the dynamics of the vehicle movement along this curve and on their compliance with the boundary conditions [14]. The track-vehicle system is often considered crucial for road safety [15].
This study focuses on the appropriate selection of the transition curve length.. According to The Study of Geometry of the Selected Transition Curves in the Design of Circular Roads the literature the three transition curves are most frequently used in the design of circular roads: clothoid [16,17], the 3rd-degree parabola [18] and Bernoulli lemniscate [18]. The above curves are also the most frequently used in designing and building roads in Poland [19,20]. The study aimed to investigate whether the selected transition curves will not exceeding the allowable increase in centripetal acceleration and the resulting geometric condition of reaching at least the minimum length.

Research methodology
The primary condition for the selection of the transition curve is the maximum increase in centripetal acceleration acting on the vehicle moving along this curve, which depends on the speed on a given road section. The turning angle along the transition curve should be in the range of 3°-30°. It does not apply when the angle of the route change is less than 9° and the serpentine (Table 1) [3,4].
The roadway should have a transverse slope enabling the free flow of water, and its value, except for a few exceptions, depends on the surface type and amounts to 2.0%, 3.0% or 4.0% [3]. The transverse slope of the motorway roadway should not be less than 2% [4]. There are three cases when the transition curve may not be used [3]: • the radius of the arc in the plane is higher than 2,000 m on the road outside the built-up area at a design speed of 33.33 m/s and 27.78 m/s; • the radius of the arc in the plane is higher than 1,000 m at a design speed of 22.22 m/s and less; • the road on the building site has a transverse slope on the curve in the plane, as on a straight section.
On the basis of this regulation, 37 cases were selected for which the lengths of the transition curves were tested: Based on the said regulation [3], using the formula [21]: Seven minimum values for the parameter a min were calculated (Table 2), applicable to G class roads [3].
Using the formula [14]: The minimum transition curve lengths L min have been calculated depending on the adopted length of the radii of the circular arc R.

RESULTS
First, graphs showed the transition curve's length variability depending on the circular arc's assumed radius.
From Figures 1 and 2 we can see that the minimum length of the transition curve always increases with the increase in the design speed. We observe an inverse relationship between a circular arc's speed and projected radius. The larger the circular arc radius, we can use a shorter transition curve for a given vehicle speed. Although the towards which the transition curve tends, the more fl attened the diagram in this relationship. As expected, the diff erences in the curve lengths for the successive design speeds are the smallest for the lowest speed.

The clothoid
Based on the formula [22]: where The parameter values a min presented in Table 3, the lengths of the clothoid were calculated depending on the speed and the adopted radius of the circular arc. We can see that the minimum length of the transition curve obtained for the clothoid coincide. It results from the fact that the   calculated parameter a min is the same as the proportionality coefficient of the clothoid [22].
where: l -length of transition curve [m]; k -curvature of the transition curve [ 1 ] .
Substituting for k with the following relationship defining the curvature: where: k -curvature of the transition curve [ 1 ] ; we get: and thus: Therefore, the clothoid was not analysed further.

Cubic parabola
Starting from the equations for curvature [23] and curve length [24] written in the explicit form: where: k -curvature of the transition curve [ 1 ] ; x -value of the OX axis in the and using the fundamental relationship for curvature [22]: where: k -curvature of the transition curve [ 1 ] ; r -radius of curvature [m]. Based on the above formulas, using the bisection and the trapezoidal method, the lengths of a cubic parabola were calculated depending on the speed and the adopted radius of the circular arc (Table 4).
During the calculations, it turned out that for two cases: R = 300 m and v = 27.78 m/s and for R = 300 m and v = 33.33 m/s, the function: where: a -transition curve parameter [m] x -value of the OX axis in the Cartesian coordinate system [m] R -radius of a circular arc [m], is positive in the range x ∈ (0, R⟩ . Therefore, it was necessary to find the minimum radius of the circular arc so that it would be possible to use a cubic parabola as the transition curve, with the assumed values of the minimum curve length from the equation a -being 267.292 and 351.364, respectively. After using the bisection method again, the desired values were read out graphically and were determined with an accuracy of 1 m. The results are presented in Table 5.
After modifying the length of circular arcs, a cubic parabola always meets the essential condition when designing a road route L calc ≥ L min . It should also be noted that the lengths obtained for the cubic parabola are usually very close to the minimum sizes. From the road designer's perspective, that proportionally increases the radius of the circular curve in relation to the designed vehicle speed, we are most interested in the values located on the main diagonal of Table 6. The most considerable differences are marked in blue.
When calculating the length of the parabola, it was first necessary to obtain the abscissa of x. Compared to L, it turned out that the most significant difference obtained for v = 33.33 m/s and R = 300 m does not exceed 5.7 m (Table 7). It shows the correctness of the simplification. For road construction, the formulas for a cubic parabola, where the equality L = x is assumed.
A lemniscate, unlike a cubic parabola, does not satisfy the condition at all: L calc ≥ L min . However, due to the differences L calc -L min (presented in Table 9) in the vast majority of cases, do not exceed 1 meter (this value is exceeded in three cases: v = 27.78 m/s and R = 300 m, v = 33.33 m/s and R = 300 m and v = 33.33 m/s and R = 500 m) consider that it can be used as a transition curve due to the accuracy of construction works and the specificity of the movement of a wheeled vehicle on the road (unconstrained trajectory). As in the case of the cubic parabola, the most considerable difference was obtained for v = 33.33 m/s, R = 300 m.
Satisfying the condition L calc ≥ L min is motivated by the requirement that the permissible value of centripetal acceleration should not be exceeded along the entire length of the transition curve [3,4]. Therefore, the influence of the transition  Table 10 were calculated based on the formula: where From the Table 11 we can see that the difference Ψ lemniscate -Ψ d in most cases, does not exceed 0.001 m/s 2 . For v = 13.89 m/s it is 0.002 m/s 2 , the same as for the design speed of 60.80 and 25.00 m/s with a circular arc radius of 300 m. The most significant differences are noticeable where the calculated length of the transition curve differed the most from the minimum value: for v = 27.78 m/s and v = 33.33 m/s for the smallest analysed radius of the circular curve, respectively 0.004 m/s 2 and 0.009 m/s 2 . Table 12 showing the values from the main diagonals for the 3rd-degree parabola and the lemniscate.   It is easy to notice from Table 12 that with the optimal adaptation of the vehicle speed to the radius of the circular arc, the length of the transition curve, depending on the equation chosen for it (clothoid, 3rd-degree parabola, Bernoulli lemniscate), will vary by up to ±20 cm, compared to the minimum value. As shown on the example of a lemniscate, the mentioned difference does not significantly affect the driving comfort, the more so as in the case of roads, we deal with a free path.

CONCLUSIONS
The primary condition that transition curves in road engineering should meet is: Ψ calc ≤ Ψ d , from which the condition results: L calc ≥ L min . The calculations presented in this paper show that both the clothoid and the 3rd-degree parabola meet these conditions. If we compare the minimum lengths of the spiral curves to those calculated for the 3rd order parabola, the calculated length of the parabola is always higher, and almost 80% of the   differences fall within the range < 0,000; 1,000 >. The lengths obtained for the lemniscate do not meet the condition in question at all. Received differences L calc -L min are between -0.008 m for v = 13.89 m/s and R = 900 m, and -12.588 m for v = 33.33 m/s and R = 300 m. This difference exceeds 1 m for only three cases, which, due to the accuracy of construction works and the specificity of the movement of a wheeled vehicle on the road (unconstrained trajectory), does not disqualify a lemniscate as a transition curve.
Looking at the obtained results of calculations, it can be noticed that when designing roads, it is crucial to properly adjust the vehicle's speed to the circular curve's radius. With the increase in the design speed, the circular arc radius towards which the transition curve tends should be increased.
Based on results, it should be stated that the choice of the curves in question as the transition curve is of no great importance. However, due to their different geometrical course, they are applicable in various terrain conditions. For example, lemniscate is very commonly used in serpentine and road junctions. Clothoid, on the other hand, is popular due to the ease of calculations.