Computer-aided computation of deflection and bending moment of regular and irregular plates using finite difference approach / Pedro Ordonio Gelilio. 6

Thesis (M.A.) -- Pamantasan ng Lungsod ng Maynila, 1993.;A directed study presented to the faculty of Graduate School of Engineering in partial fulfillment of the requirements for the degree Master of Engineering (MEng) with specialization in Structural Engineering. 56

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ABSTRACT: Computers are widely used in the solution of the problems of science, engineering, and business. They have the ability to operate at great speed, and to carry out long and complex sequences of operations without human interventions. It is on this premise that this study was conceptualized and put to realization by the author. The deflection and bending moments of plates are analyzed by the method of finite difference in an attempt to come up with a better alternative in solving difficult problems in thin plates. Furthermore, this study treats not only simple shapes and loadings, but also irregular ones, in contrast to the somewhat limited variety of shapes and loadings amenable to simple analytical solution. Plates are classified in two general shapes: regular and irregular. Under the regular shapes are plates with square and rectangular boundaries while irregular shapes have circular, skew, triangular, hexagonal, elliptical pr composite boundaries. Load cases are categorized as uniform, triangular or hydrostatic and point load. The boundary conditions for regular plates are composed of simply supported at all edges, fixed supported at all edges, simple-fixed combinations and simple-fixed-free combinations. However, the boundary conditions for irregular plates are simply supported at all edges only. The program language used in the study is the Advanced Basic and all the programs are compiled for an easy setting of menu and developed in a user-friendly interactive mode. Program operations can be best described thus; as the input data is completed it will pass through the address identifier, then it goes to the matrix generator and is subsequently evaluated by some finite techniques for the solution of simultaneous linear equations. The output data will be displayed on the screen and then store the data files in the diskette for future use or verification. For regular shape, plates the Lapplace operator for deflection is V4w=qo/D and for bending moment V²M=Ø for moment and operator such as V²M=qo. But for free edge boundary conditions; additional operator such as V²Mx=Ø for moment and V³x=Ø for shear are employed at the free edge. It should also be noted that only one free edge in combination with other boundary conditions in a plate is considered in this study. The operator for elliptical plate is composed of V²w=-qo / D from regular shape and irregular star pattern provided by Ugural (1981). Moreover, the remaining plates such as skew, triangular, hexagonal, and circular utilize single operator thus derived for each plates. In the manual procedure, the operator or stencil of the formula is placed at the nodes of the pivot grid from which the coefficient of the stencil automatically becomes array variables while the nodes represent the location or address in the square matrix. In contrasting fashion for computer input, the number of each node from the pivot grid will be fed into the computer beginning from the top left of the pivot grid going to the right. The program determines the address by interpreting the number code and correspondingly assigns the coefficient of the operator/ stencil at that particular node. These techniques eventually generate a system of linear algebraic equations converted to matrix form. It is then analyzed and solved by Gauss-Jordan with partial pivoting as the one matrix method employed here. Since the method could only generally provide exact figures as compared to the series solution, the study had by far established approximate number of subdivisions of the plates that yields results of acceptable accuracy for practical applications. The results obtained in this study using finite difference method (FDM) were compared to finite element method (FEM) and exact solution. An extrapolation procedure was used near the fixed boundary, because the moments produced by the MicroFeap-P2 (FEM) software are located at the middle of the plate elements while the moments of the FDM are obtained at the nodes of the grid. Based from the FEM and FDM results, one could easily see the close correlation between these values which must be expected, since the methods are closely related.

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