應用SCEM於拱形梁之分析研究

Abstract

摘 要 一般結構上多求解受橫向載重直梁問題，其控制微分方程具有一定的規律利於方便求解，但若因考量多項因素而必須分析拱形梁，發現其變為一複雜的變係數常微分控制方程式求解問題，因而並非容易求得解析解。 本文將以先行對拱形梁模型的假設，配合修正後的Spline Collocation Method及予以元素化之Spline Collocation Element Method (簡稱SCM與SCEM)，進行對拱形梁之分析。利用SCM與SCEM直接模擬拱形梁之控制方程式，並以查表的方式取代原本複雜之積分過程，進而轉為求解聯立代數式而得出結構物各項數值解的方法，讓我們可以看到其求解過程的便利性。且其能不侷限於任何形式的載重或邊界支承條件，僅需結構之反應控制方程式，即能求得令人滿意之近似解，並由於其計算過程方便於電子運算，若以編撰程式加以分析，則將更為省時。 最後以多個數值實例闡釋SCM與SCEM理論分析步驟及其分析所獲致之結果，並與其它精確數值解進行結果分析比較，藉以凸顯SCM與SCEM應用於拱形梁之實質優點，俾提供結構工程設計之參考。

Abstract In general, the governing differential equation problem of an initial straight elastic beam subjected to a lateral load has certain rule to be solved easily. Sometimes we must analyze all kinds of questions of arch structure in order to consider some reasons, and the questions have been complicated problems of solving ordinary differential equations with various coefficients. Therefore they are not easy to find analytic solutions. In this paper, analyses of arch assumed by establishing models are conducted using the modified spline collocation method and spline collocation element method (SCM&SCEM). SCM and SCEM can be applied to arch by direct use of governing equation, and it is more convenient that the coefficient matrix for the weighting coefficients can be assembled simply by finding from the table for the values of quintic spline function at knots without process of complicated calculation. In addition, the reasonable results of the arch undergoing various types of loadings and boundary conditions can be obtained just needing the governing equation, and it is more economical and time-saving for writing computer programs which use less arithmetic operators. By comparing with other accurate numerical solutions at last, it is shown that the analyses of the arch problems by the SCM and SCEM are stable and convergent, and the excellent accuracy in results is achieved. Besides, the ease of using this method has been clearly shown.