Abstract
The five elements of any horizontal circular curve, which are tangent distance, T, external distance, E, middle ordinate, M, chord length C, and length of curve, L, can be directly determined if the radius R and deflection angle Delta are given. Practically there are some cases where the deflection angle Delta and the radius R cannot be measured due to the inaccessibility, of the intersection point PI. Thus, two other curve elements have to be known in order to layout the horizontal circler curve.
This paper presents a direct method to extract the unknown elements exploiting a cubic polynomial equation which characterizes the curve deflection angle. The least squares technique is considered in order to facilitate the solution with fair accuracy.
Particularly, the cubic polynomial is selected as it generates an adequate desired accuracy. If higher accuracy is aspired, higher order polynomials, i.e. quartics, quintics,..., may be adopted.
On contrast to other methods, which adopt iteration process, the presented method is direct, non-iterative, accurate, and easy to handle values of deflection angle, Delta, ranging front 0 to pi without imposing any derivatives.
The major benefits of the proposed method are illustrated as: firstly it is direct method which may easily be executed using the pocket calculator with high accuracy, and secondly, it avoids the uncertainty of the convergence due to the lack of pre-estimation of the initial value of deflection angle and number of iteration trials required when iteration solution is considered
The applicability and accuracy of the presented method is justified through a numerical application. Using the same values of L and T, it has been found that the pep-cent of error for the previous methods in five digits values vary from 0 to 0.00133% for Delta and 0.00084 % to 0.00252 % for R. The proposed method gives an error of 0.00533 % for Delta and 0.00644 % for R. The effect of such errors in roadway geometry design is negligible.