Stiffness Matrix And Equivalent Node Load Formulae Or Varying Cross-section Spatial Beam Elements

Piles with some variable parameters over all or part of their length are generically called varying cross-section elements. The benefit of using varying cross-section elements is elegant in design and economic, because they can match the loading condition perfectly. So far, varying cross-section elements have been widespread used in projects and their analysis have become more and more important. Finite element analysis will be discussed in the dissertation, combining with the technical project of "Water-cube", national Swimming Center of China for Beijing 2008 Olympic Games.The forms and types of varying cross-section elements are introduced first, while the application and research history are reviewed.Then, the methods of obtaining element stiffness matrices and equivalent node loads are discussed.Comparing with other methods, Transfer Matrix Method is adopted to deduce element stiffness matrices because of its advantage. Then, the principle of Transfer Matrix Method is introduced, and the relative equation between the stiffness matrix and the transfer matrix is obtained. In order to derive transfer matrices, two methods which are called solid method and sub-structure method are extracted, and the flexibility of sub-structure method is approved through the derivation process. And then the spatial elastic element stiffness matrix equation is given. The general equations about special varying cross-section elements are obtained, and thesimplified method of complex elements is listed.Based on the method and equations above, the element stiffness matrices of several varying cross-section elements are deduced and the accuracy is approved by examples. Stiffness modified parameters are discussed for more study about elements with two tapered ends.Equivalent node loads for varying cross-section elements are derived by Transfer Matrix Method too. Some basic concepts, such as load state vectors of nodes and etc, are introduced, and solid method and sub-structure method are adopted to deduce load state vectors of nodes. Then, the general equations for equivalent node loads and the expression of load state vectors of nodes under different external forces are given.According to the method and equations, the Equivalent node loads under different external forces for several varying cross-section elements are deduced and the accuracy is approved by Force Method, besides, the examples are given too. For elements with tapered ends, the equivalent node load equations can be derived based on those of same cross-section elements and tapered elements according to sub-structure method.For the purpose that the equation derived in the dissertation can be used in projects, the program follows the spatial structure software MSTCAD is programmed. The program blocks are introduced first, and then part of the interfaces are listed.At the end of this article, the main counlusions of my work are addressed, followed by the future work to be done.