張氏積分法在結構動力學上的應用

逐步積分法為動態歷時分析所廣泛運用的方法，而逐步積分法又分為內隱式積分法與外顯式積分法，前者雖具有無條件穩定的特色，但計算過程複雜繁瑣導致效率不佳，後者雖然每一步的計算非常的簡單與省時，卻往往為了滿足穩定條件的限制而被迫採用非常小的積分時間步長來進行逐步積分，因而需要較大的積分總步數，因此一個理想的積分法希望具有無條件穩定且計算簡單省時的優點，本文所採用的張氏積分法就有此特性。OpenSees是一個以C++編譯而成的有限元素軟體，為了探討張氏積分法的實用性以及計算效率上的優越性，特別將張氏積分法撰寫成C++的程式碼加入OpenSees當中，並且透過OpenSees建立各種不同結構型式的分析模型來進行動態歷時分析。除了張氏積分法以外，也利用等平均加速度積分法與Newmark外顯式積分法來進行動態歷時分析，並將分析所得結果相互比較，以證實張氏積分法能廣泛應用於求解各種不同線性及非線性的結構動力問題。同時，也可驗證張氏積分法的數值特性。最後則利用每次動力分析所使用的CPU時間比較，來進一步證實此積分法的計算效率。

關鍵字

無條件穩定；外顯式積分法； OpenSees

並列摘要

Step-by-step methods are widely used in the solution of dynamic problems, and they are classified as implicit and explicit methods. Implicit methods can have unconditional stability, however, the involvement of an iteration procedure lead to computationally inefficiency. Although the calculation of each time step of explicit method is simple, it can only have conditional stability and thus a small time step may be required to meet stability conditions. Therefore, an ideal integration method would like to have explicit formulation and unconditional stability simultaneously. Since Chang explicit method can integrate these two properties together, it is adopted in this study. OpenSees is a finite element software, where the code was written by C++ language. In order to study the feasibility and computational efficiency of Chang explicit method, its computing procedure is implemented into OpenSees for the dynamic analysis. Consequently, many structural dynamic problems are solved by Chang explicit method. The structural systems considered herein may be linear elastic or nonlinear. In addition, the structural nonlinearity includes both material nonlinearity and geometric nonlinearity. Both the Newmark explicit method and the constant average acceleration method are also used to solve all the structural dynamic problems for comparisons. As a result, the feasibility of using Chang explicit method to perform any dynamic analysis is verified. In addition, it is evident from the comparison of CPU time for each dynamic analysis that Chang explicit method is computationally efficient in the solution of an inertial type problem when compared to the Newmark explicit method and the constant average acceleration method.

並列關鍵字

Explicit Method ； Unconditional Stability ； OpenSees

參考文獻

[1] M. A. Dokainish and K. Subbaraj, "A survey of direct time-integration methods in computational structural dynamics—I. Explicit methods."Computers & Structures vol. 32, no. 6, 1989, pp. 1371-1386.

[2] K. Subbaraj and M. A. Dokainish, "A survey of direct time-integration methods in computational structural dynamics--II. Implicit methods."Computers & Structures vol. 32, no. 6, 1989, pp. 1387-1401.

[4] T. Belytschko and T. J. R. Hughes, Computational Methods for Transient Analysis, New York：North-Holland, 1983.

[5] M.W. Dobbs“Comments on ‘Stability and Accuracy Analysis of Direct Integration Method’by Bath and Wilson,”Earthquake Engineering and Structural Dynamics,Vol.2,pp.295-299,1974

[6] O.C. Zienkiewicz, The Finite Element Method, McGraw-Hill,Book Co(UK)Ltd.,Third Edition,1977.

國際替代計量

張氏積分法在結構動力學上的應用

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