tailieunhanh - the boundary element method with programming for engineers and scientists - phần 10

discretised vào bước thời gian n t kích thước. Giả sử một gia tốc trung bình trong thời gian bước hệ thống phương trình vi phân có thể được chuyển đổi thành một hệ thống các phương trình đại số (Newmark method4) | 446 The Boundary Element Method with Programming Figure Distribution of maximum compressive stress comparison with theory DYNAMICS Here we extend the coupling method to dynamics. The dynamic equilibrium equations which arise from finite element discretisation see Bathe4 can be written as M fi C U K u F where M c K are the assembled mass damping and stiffness matrices and u u u are the acceleration velocity and displacement vectors. The time may be discretised into n time steps of size At. Assuming an average acceleration within the time step the system of differential equations can be transformed into a system of algebraic equations Newmark method4 k u t f where t nAt and t_ n -1 Az COUPLED BOUNDARY ELEMENT FINITE ELEMENT ANALYSIS 447 The dynamic stiffness matrix is given by r 4 - 2 K 74rlMl V lC K t I and F M T t- b t- u t_ c H2t ủ t- t- J F t_ Since we have already worked out a dynamic stiffness matrix of the boundary element region in Chapter 14 the coupling procedure is now straightforward. For a fully coupled problem the system of equations is given by FnK T Tk 1 ì ủ t ÍFÌ ÍFÌ L J BE L J FE BE FE Example The example is that of a concrete column embedded in a semi-infinite soil mass. The description of the problem can be seen in Figure . The top of the column is subjected to a suddenly applied loadp t of 1 MN m2 . The material properties for the column are spec. weight 2500 kg m3 E 30 000 MN m2 v . For the soil we have spec. Weight 2000 kg m3 E 100 MN m2 v . p t Figure Description of example p t i k t Figure shows the mesh used for the analysis it consists of a finite element region that describes the column and a boundary element region that describes the semi-infinite ground. The mesh has 1500 degrees of freedom. Figure shows the time-dependent 448 The Boundary Element Method with Programming displacements at the top of the column obtained from the analysis. The results compare well .