Algebraic method applied to solving inversion problem of single layered rigid pavements

Abstract

Surface waves can be used to determine the mechanical and physical properties of the materials of a layered structure such as soil sites and highway pavements. The elastic properties and thicknesses of the component materials are derived from measurements of the frequency and the phase velocity of vibrations generated along the surface of the ground. The phase velocity is not constant, but varies with the frequency, enabling an experimental dispersion curve of the system to be plotted. The experimental dispersion data is then used to back calculate the unknown parameters of the system through an inversion process by employing one of the available inverse methods. For many years, the numerical inverse procedure has been extensively used for determination of the layers moduli and thicknesses of a layered structure. However, this technique is complex, time consuming and requires experienced person and engineering judgment. Ideally, the inverse should be obtained by means of a true mathematical inverse. The frequency equation for the system is expanded as a polynomial, and is solved for the unknown parameters of the system. In this research a direct symbolic solution for determination of the thickness of a free plate model, using ”Mathematica” computer software, has been developed. It is used to calculate the thickness of the uniform free plate corresponding with pairs of the values of phase velocity and frequency as data. The wave propagation in elastic plates is analogous to propagation in layered spaces and therefore the free plate system can be used to model the surface layer of a single layered rigid pavement structure. Single layered rigid pavements are extensively and increasingly used in highways and airport runways. The characteristic equation of the system which corresponds with surface waves can be represented by power series. By making power series, the equation is expanded symbolically in terms of the layer thickness parameter. Inverse series performs reversion of the series, which gives a series for the inverse of the function. The normal expression now is a polynomial representing the thickness of the surface layer in terms of the velocities of waves in the surface layer and of the measured values of the frequency and the corresponding phase velocity. The frequency and the phase velocity are obtained from the results of the measurements made in the field on the ground surface of the structure. In order to evaluate the effectiveness and reliability of the proposed technique, the application of the method to a number of published set of data obtained in the field has been investigated and assessed. The comparison between the experimental results and the developed algebraic solutions show very good agreement, indicating that the proposed method can be used as an economic and effective technique to determine the thickness of the rigid pavements surface layer.