在單層土壤中基礎之阻抗矩陣

Abstract

本篇論文將解析發展沉埋基礎之扭轉、垂直、水平、翻轉與偶合阻抗矩陣分析程式。在分析沉埋基礎之阻抗矩陣時，所分析層狀土壤可分成内域與外域兩部分，内域與外域的土壤可分別以未知模態係數表示，未知模態係數將透過變分技巧及利用內外域之間位移與應力的連續性來求得，土層的形狀函數亦可用求得的模態系數來表示。 分析整體的層狀土壤形狀函數時,土壤可分成内域與外域兩部分, 内域的解析解為特解加上齊性解, 而外域部份僅存在齊性解。內外域的超越函數齊性解將透過數值方法發展出一套程式可求得複數彈性波數值。 層狀土壤的阻抗矩陣可藉由彈性波數的值求得來建立，完成後進行沉埋基礎數值的阻抗矩陣程式開發，並發展出一套沉埋基礎之扭轉、垂直、水平、翻轉與偶合阻抗矩陣分析程式。 在分析半無限域中沉埋基礎之阻抗矩陣時,係利用有限域解析沉埋基礎的阻抗矩陣並調整土層之深度來近似，為了求得半無限域中沉埋基礎的阻抗矩陣，在有限域的沉埋基礎之阻抗矩陣中,增加最下層土層之厚度來近似半無限域沉埋基礎的阻抗矩陣，但增加下層土壤的厚度會發生數值不穩定現象，此時可透過所提出的數值方法解決數值不穩定現象，最後發展出一套半無限域沉埋基礎之扭轉、垂直、水平、翻轉與偶合阻抗矩陣分析程式。

A computer program is developed in the thesis for calculating torsional, vertical, horizontal, coupling and rocking impedances in frequency domain for axial-symmetric foundations embedded in layered medium. In this process of formulating the impedances, the soil medium is divided into interior and exterior domains. The analytical solutions are formed separately with unknown coefficients for both domains. In order to find the unknown coefficients for both domains, the variational principle is employed using the continuity conditions (both displacements and stresses) at the interfaces between interior and exterior domains, interior domain and foundation, and exterior domain and foundation to find impedance functions. To solve those problems, the analytic solution for the interior domain is the combination of a homogeneous solution and a particular solution, the exterior domain is described by a homogeneous solution only. To obtain the homogeneous solution, one has to solve the complex root of the transcendental equations. A numerical scheme has been proposed. The wave numbers of transcendental equations have been employed for finding impedance matrices. Some numerical results of torsional, vertical, horizontal, coupling and rocking impedances with different embedded depths will be presented in layered medium and comments on the numerical scheme will be given. The impedance matrices of axial-symmetric foundations embedded in an elastic half-space medium approximated using analytical solutions in layer. To approximate the situation of half-space medium, the thickness of one layer medium gradually increased to see if the impedance function is approaching those for the case of half-space medium. However, as the thickness increases the numerical instability problem will be arisen. To overcome this numerical problem, a new numerical technique will be developed. Some numerical results of torsional, vertical, horizontal, coupling and rocking impedances with different embedded depths will be presented in an elastic half-space medium and comments on the numerical scheme will be given.