Kiriş problemleri sonlu elemanlar yöntemi ile çözümü
The Solution of beam problems with finite element method
- Tez No: 45957
- Danışmanlar: YRD. DOÇ. DR. İSMAİL BİNİCİ
- Tez Türü: Yüksek Lisans
- Konular: Makine Mühendisliği, Mechanical Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1995
- Dil: Türkçe
- Üniversite: Marmara Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 182
Özet
ÖZET Yük altında yapıların davranışlarının incelenebilmesi büyük ve karmaşık denklemlerin teşkil edilip bunların çözülmesi gerekmektedir. Bu tür mühendislik problemlerinin çözümü için yapılacak en iyi yaklaşık çözüm şekli nümerik çözüm metodları olmaktadır. Nümerik metodlardan, Sonlu Elemanlar Metodu (SEM)
Özet (Çeviri)
THE SOLUTION OF BEAM PROBLEMS WITH FINITE ELEMENT PROGRAMMING SUMMARY The Finite Element Method (FEM) is a numerical analysis technique for obtaining approximate solutions to a wide variety of engineering problems. Although first developed for structural analysis, it now solves problems in heat transfer, fluid mechanics, acoustics, electromagnetic, and other specialized disciplines. The Finite Element Method is the unifying approach it offers to the solution of diverse engineering problems, so the method has attracted a wide variety of theoreticians and practitioners for various disciplines, including engineering, mathematics and computer science. In more and more engineering situations today, we find that it is necessary to obtain approximate numerical solutions to problems rather than exact closed - form solutions. For example, we may want to find the load capacity of a plate that has several stiffeners and odd-shaped holes, we can write down the governing equations and boundary conditions for these problems, but we see immediately that no simple analytical solution can be found. The Finite Element Method is a numerical method for solving a system of governing equations over the domain of continuous physical system. FEM applies to many fields of science and engineering, but this text focuses on its application to structural analysis of linear elastic system. The field of continuum mechanics and theory of elasticity provide the governing equations. The basis of the Finite Element Method for analysis of solid structures is summarized in the following steps. Small parts called elements subdivide the domain of the solid structure illustrated in Figure S-1. These elements assemble through interconnection at a finite number of points on each element called nodes.This assembly provides a model of the soiled structure. Within the domain of each element we assume a simple general solution to the governing equations. The specific solution for each element becomes a function of unknown solution values at the nodes. Application of the general solution for to all the elements results in a finite set of algebraic equations to be solved for the unknown nodal values. By subdividing a structure in this manner, one can formulate equations for each separate finite element which are then combined to obtain the solution of the whole physical system. If the structure response is linear elastic, the algebraic equation are linear and are solved with common numerical procedures. Figure S-1 Two-Dimensional Continuum Domain Since the continuum domain is divided into finite elements with nodal values as solution unknowns, the structure loads and displacement boundary conditions must translate to nodal quantities. Single forces like F apply to nodes directly while distributed loads like P are converted to equivalent nodal values. At least two sources of error are now apparent. The assumed solution within the element is rarely the exact solutions. The error is the difference between assumed and exact solutions. The magnitude of this error depends on the size of the elements in the subdivision relative to the solution variation. Fortunately, most element formulations converge to the correct solution as the element size reduces.The second error source is the precision of the algebraic equation. This is a function of the computer accuracy, the computer algorithm, the number of equation, and the element subdivision. Both error sources are reduced with good modeling practices. In theory all solid structures could be modeled with three-dimensional soiled continuum elements. However, this is impractical since many structures are simplified with correct assumptions without any loss of accuracy, and to do so greatly reduces the effort required to reach a solution. Different types of elements are formulated to address each class of structure. Elements are broadly grouped into two categories, structural element and continuum elements. Structural elements are trusses, beams, plates, and shells. Their formulation uses the same general assumption about behavior as in their respective structural theories. Finite Element solutions using structural elements are then no more accurate than a valid solution using conventional beam or plate theory for example. Continuum elements are the two-and three-dimensional solid elements. Their formulation basis comes form the theory of elasticity. The theory of elasticity provides the governing equations for the deformation and stress response of a linear elastic continuum subjected to external loads. Few closed form or numerical solutions exist for two-dimensional continuum problems, and almost none exist for three-dimensional problems which make the Finite Element Method invaluable. In the past, the economic limitations imposed by computer costs have restricted the general use of such techniques. However this barrier is being rapidly removed and the finite element solutions of such problems is already economically acceptable for selected industrial applications.Such developments along with future enhancements in the element characteristics, equation solution techniques etc., suggest that the finite element method will play a major role in engineering design for many years to come. As you will see, in this thesis not only the finite element program but also the finite element method theory which will help any user to understand the structure of the program is presented. Therefore in Chapter 2 classifications of engineering structural components include trusses, beams, plates, shells, plane solids, axisymmethc solids, and torsion bars. Each classification has its corresponding mechanics theory of behavior. Chapter 3 introduction and the basic expressions of the finite element method for structural applications, the kinds of finite elements and the structure of the program are presented. The major portion of this chapter is written for readers who have never programmed digital computer to solve a continuum problem by the finite element method. Having an understanding mathematics of the finite element method and knowing how to derive element equations for a given problem are not enough to solve the problem; it is also necessary to know how to translate the equation into computer instructions so that the element equation can be evaluated, assembled, and solved. A reader familiar with the FORTRAN language should be able to follow all of the coding with the aid of the many comment statement appearing throughout the program. Chapter 4 deals with the problems associated with input and output. The input data required for finite element analysis with isoparametric elements is discussed and subroutines for data assimilation are presented. Chapter 5 is devoted entirely to the development of specific expressions for the isoparametric beam element it is this stage that problems specifically associatedwith the isoparametric element concept are first encountered and the structural elements serves as a convenient introductory vehicle in view of it is relative simplicity. In particular the Jacobean matrix, which enables transformation between local and global quantities to be made, is introduced and numerical integration techniques essential to isoparametric elements are discussed. The subroutines performing the standard steps, such as shape function and stiffness formulation, equivalent nodal force generation and stress resultant evaluation, are developed at this stage. These subroutines are essentially of the same form as for more sophisticated applications and Chapter 5 therefore allows the reader to familiarize himself with the general structure of isoparametric element programs. The solution of equation systems by the frontal method is dealt with in Chapter 6 and a sophisticated subroutine is developed which can be employed as a general purpose finite element solver. The Frontal equation technique is described in detail and it is advantages outlined. In Chapter 7 deals with the problems associated with input and output. The input data required for finite element analysis with bar element is discussed and subroutines for data assimilation are presented. For checking input values three subroutines are presented. The data is checked in stages and if any errors are detected, appropriate diagnostic messages are printed and the remainder of the input data is echoed by lineprinter. The subroutines developed in Chapters 4 to 7 for finite element analysis are assembled in Chapter 8 to form complete program which can be employed for beam problems analysis. Numerical example for beam problems are also presented demonstrating the efficiency of the parabolic isoparametric element. The construction of a finite element programs employing the displacement approach falls naturally into three stages. The preprocessing stage creates the model of the structure from inputs provided by the analyst. A preprocessor thenassembles the data into a format suitable for execution by the processor in the next stage. The processor is the computer code that generates and solves the system equations. The third stage is postprocessing. The solution in numeric form is very difficult to evaluate except in the most simple cases. The postprocessor accepts the numeric solution, presents selected data, and produces graphic displays of the data that are easier to understand and evaluate. Figure S-2 draws a block diagram of a finite element computer program. Before entering the program's preprocessor, the user should have planned the model and gathered necessary data. In the preprocessor block, the user defines the model through the commands available in the preprocessor. The definition includes input and generation of all node point coordinates, input and generation of node connectivity to define all elements, input of material properties, and specifying all displacement boundary conditions, load and load cases. The copulation of the preprocessing stage results in creation of an input data file for the analysis processor. The prqcessor read from the input data file each element definition, calculates terms of the element stiffness matrix, and stores them in a data file. The element type selection determines the form of the element stiffness matrix. The next step is to assemble the structure stiffness matrix by matrix addition of all element stiffness matrices. The application of enough displacement boundary conditions to prevent rigid body motion reduces the structure stiffness matrix to a nonsingular form. Then the equation solution with Gaussian Elimination. The solution here yields values for all node point displacement components in the model. The node displacements associated with each element combined with the element formulation matrix yield the element strains. The element strains with the material properties yield the stresses in each element. The processor then produces an output listing file with data files for postprocessing.PREPROCESSOR Main Program INPUT DATA Node and Element Definition, Materials, Boundary Conditions, Loads Element File Load File Check Data* Element Transfer Mat mul FEM ANALYSIS FORM ELEMENT [k] Read Element data. Calculate Element Stiffness Matrix,[kl Element File FORM SYSTEM [KJ Assemble Element [k]s to Form the System Stiffness Matrixjkl APPLY DISPLACEMENT BOUNDARY CONDITION BC COMPUTE DISPLACEMENTS Solve the System Equations |K]{D}={F} for the Displacements {D}=[hT{F} Load File COMPUTE STRESSES Calculate Stresses and Output Files for Postprocessor Plotting INT FORCE ^Displacement Stress Files. Figer S-2 Finite Element Compter Program Block DiagramPostprocessing takes the results file and allows the user to create graphic displays of the structural deformation and stress components. The node displacements are usually very small for most engineering structures so they are magnified to show an exaggerated shape. Node displacements are single-valued, but node values of stress are multi-valued if more than one element is attached to a given node. Node stress values are usually reached by extrapolation form internal element values and then averaged for all elements attached to the node. Contour plots or other stress plots desired by the user are created from the node values. The engineer is then responsible for interpreting the results and taking whatever action is proper. The user must estimate the validity of the results first. This is very important because the tendency is to accept the results without question. Experience, thorough checking of the modeling assumptions and resulting predicted behavior, and correlation with other engineering calculations or experimental results all contribute to estimating the validity of the results. In this thesis all program was written using FORTRAN-77 programming language in about 1100 statement on an IBM PC/DX compatible machine. The basic aim of this thesis is that this programs should help the reader to take the painful step from theory to program, thus enabling him to develop programs for his own particular applications in his own environment.
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