Direct Solution Of The Deflection Angle Problem Of Horizontal Highway Circular Curves

Abstract

Determination of the five elements of highway horizontal circular curve, which are tangent distance, T, external distance, E, middle ordinate, M, chord length C, and length of curve, L, is directly established in terms of the given radius R and deflection angle ∆.

Field features may cause the infeasibility of the  intersection point PI. Consequently, the deflection angle ∆ and the radius R cannot be measured which means the arising of  non-linear problem.

This paper presents a simple and direct solution for the problem by remodeling the non-linear problem  as a rational function in terms of the deflection angle.  The model’s  function has been designated due to the similarity of the graphs of  both  non-linear function and the rational function.

Such function has three coefficients and two  asymptotes,  x= π/2  as a vertical asymptote and y = a, the first coefficient, as a horizontal asymptote.

Coefficients of the model were determined addressing the least squares technique employing  different values of the deflection angle along the entire  0 < Δ < π/2 interval. 

Such model has been verified through adopting different random locations along the interval  of J, and the output has proven an almost 100% accuracy.

Also, the applicapility of the model for generating deflection angle and curve radius with fair accuracy was investigated numerically adopting designated values of L and T. The results exhibited fair accuracy in comparative to the previous methods in the favour of the iterative techniques of such methods which lack necessary initial guess.

The highly precised output, dirct applicability and simplicity of the proposed model recommend such model for manipulation in practical purposes as well  in analytical analysis of the highway engineering problems.   Also, it may be considered as a preferable than  iterative approaches as such approached involving  absence of the  initial guess.