A new, simple and accurate transition curve type, for use in road and railway alignment design

Abstract

This paper evaluates all the available transition curve types related to road and railway alignments and proposes a new, well verified, transition curve type that combines the accuracy of clothoid curve and the simplicity of cubic parabola curve. A methodology similar to clothoid’s curve formation is used to introduce a new transition curve type called “of clothoid Symmetrically Projected Transition Curve (SPTC). All three transition curve types are being compared to each other, for a variety of transition length value versus Radius value combinations. The cubic parabola is a simple function of the form of y=f(x). Clothoid is a transition curve in the form of x=f(l), y=f(l), having as main characteristic the linearity of curvature variation versus its length. A new transition curve will be defined in the form of y=f(x) having also as main characteristic the linearity of curvature variation versus its projection length on axis X. By using the same calculation procedure as the clothoid, the new transition curve will be fully defined. A relation similar to (1) was used as base, by defining a parameter Α similar to the one used in the clothoid. The new curve will be called Symmetrically Projected Transition Curve (SPTC). Some remarkable results that derived from transition curves comparison are: There are no significant differences between the 3 curves in the area of short transition lengths. For long transition lengths, cubic parabola is diverging from the other 2. The deviation of the cubic parabola from the other curves for large values of Χ, ratios Χ/Α > 0.7, as well as the affinity of the clothoid with the SPTC are obvious. The most remarkable observation than can be made in the table is the fact that ΔΧ always zero for the SPTC (10terms). Thus, the SPTC curve is symmetrically projected on its basic tangent. This property contributes to the simplicity of the alignment design. That is another reason to prefer the SPTC curve. The use of cubic parabola in combination with approximate value of diversion can lead to design problems. The new transition curve can be used instead of cubic parabola especially when long transition lengths are required. The new transition curve can also be used successfully to join 2 homo-bending arcs. However, referring to cubic parabola calculations, for a ratio X/A ≥ 0.5 and taking in to account the approximate calculation procedure of ΔR it can lead to alignment design errors. Consequently, the usage limits for each transition curve should be well known. A new transition curve was also proposed in this work. The new curve is called Symmetrically Projected Transition Curve (SPTC). SPTC was found, in most cases, to have better performance than cubic parabola. Symmetry is an important characteristic of the SPTC and contributes to simplicity, accuracy and audit ability of the designed alignment. Finally SPTC can also be used as a transition curve between two adjacent circular arcs in the same direction.