Plakların sonlu eleman ve sonlu farklar metodları ile çözümü ve iki metodun karşılaştırılması - Ergün, Ali (Fen Bilimleri Enstitüsü, 1996)
Bu yüksek lisans tezinde, uniform yayüı yük etkisinde çeşitli mesnetlenme tipine sahip plakların statik çözümü için nümerik yöntemler olan sonlu elemanlar ve sonlu farklar metodlan yardımı ile iki ayrı program BASIC dilinde düzenlenmiştir. Sonlu elemanlar metodunda farklı eleman sayışma göre, sonlu farklar metodunda ise farklı ağ nokta sayışma göre çeşitli plak çözümleri için uygulamalar yapılmış ve bu yolla bulunan plak çözümleri mevcut analitik çözümler ile karşılaştınimıştır. Ayrıca sonlu elemanlar ve sonlu farklar metodu da aynı plak probleminde çözüm elde etmek için gerekli olan denklem takımlanndaki bilinmeyen sayılan karşdaştırılmıştır. Bu çalışmada kullanılan blgisayar programının yapısı, verilerin girişi ve sonuçların alınmasıyla ilgili bilgiler verilmiştir. Tez çalışması dört ana bölümden oluşmaktadır. Birinci bölümde plak teorisi hakkında geniş bilgiler verilmiştir. Plak diferansiyel denkleminin çıkartılması için gerekli olan elastisite teorisindeki bağıntılar ve küçük sehimli ince plaklar için yapılan kabuller anlatılmıştır. İkinci bölümde sonlu elemanlar metodu ve metodun uygulanmasında gerekli olan eleman karakteristlikleri tanıtılmıştır. Eleman rijitlik matrisinin çıkartılması adım adım anlatılmıştır. Bu karakteristliklerden yararlanarak hazırlanan program tanıtılmıştır. Üçüncü bölümde ise, yine bir nümerik çözüm olan sonlu farklar metodu ve bu metodu teşkil eden sonlu farklar hakkında bilgi verilmiş ve plakların diferansiyel denkleminin sonlu farklar ile ifadesi için gerekli olan katsayılar şeması(şablonu) çıkartılmış ve bu esasa göre yapılan program hakkında bilgi verilmiştir. Son bölümde ise sonlu elemanlar ve sonlu farklar metodu ile programdan bulunan sonuçlar denklem takımındaki bilinmeyen sayışma göre karşılaştırılmış ve bu iki metodun yakınsama biçimi incelenmiştir. Ek kısmında programların yapısı ve çeşitli örnekler için alman çıktılar verilmiştir.

The solution of plate bending problems with clasical route is limited to relatively simple plate geometry, load and boundary conditions. If these conditions are more complex, the analysis becomes increasingly tedious and even impossible. In such cases numerical and approximate methods can be employed. Approximate methods are finite element method and finite differences method. After invention of high speed computer approximate methods have been begun to use in analysis of stuructural system. In this thesis, two computer programs were prepared in BASIC programming for solution of rectangular plates having uniformly loaded and different boundary conditions. One of the computer programs was prepared with the finite element method, other was prepared by the finite diffrence method. The analysis of the behaviour of plate bending problem by means of finite element and finite differences techniques require basic equations of elasticity theory. Set of equations are -the differential equations of equilibrium formulated in terms of the stresses acting on a body -the strain/displacement and compatibility differential equations -the stress/strain or material constitutive laws These equations are given in the first part of thesis. Furthermore, in the analysis of plate bending problems classical small-deflection theory is based on the following asssumptions. -deflections of plate are always small in relation to thickness of plate "h" -the points which lie on a normal to the mid-plane of the undeflected plate lie on a normal to the middle surface of the deflected plate -stresses normal to the mid plane oz is zero. The finite element method is firmly established as an engineering tool of wide applicability. For the structural stress analysis problem, the engineer seeks to determine displacements and stresses throughout the structure which is in equilibrium and is subjected to applied loads. Specially for many plates, it is difficult to determine displacements and stresses using classical route and thus the finite element method is necessary used There are two general approaches associated with the finite element method. First approach is called force or flexibility method uses internal forces as the unknowns of the problem. The second approach is called the displacement or stiffiiess method. This approach assumes the displacements of nodes as the unknowns of the problem. For example compatibility conditions requiring that elements connected at a common node, along a common edge before loading remain connected at that node, edge after deformation are initially satisfied. P Then the governing equations AAw=- are expressed in terms of nodal displacements using equations of equilibrium and an applicable law relating forces to displacements. These two approaches result in different unknows in the analysis and different matrices associated with their formulations. The displacement method is more desirable because its formulation is simpler. Consequently, the displacement method only will be used throughout this thesis. The finite element method involves modeling the structure using small interconnected elements called finite elements. A displacement function (shape function) is associated with each finite element. Every element is linked to every other element through common interforces, including nodes and boundary lines. By using known stress/strain proparties for the material making up plate, the behaviour of a given node can determine in terms of the properties of every other element in the plate. The total set of equations describing the behaviour of each node results in a series of algebraic equations best expressed in matrix notation. General steps of the finite element method to obtain solution of plate bending problems Step 1 Discretize and select element types Step 1 involves dividing the plate system into an equivalent system of rectangular finite elements with associated nodes called subsitute structure. The elements must be made small enough to give usable results and yet large enough to reduce computational effort. Small elements are generally desirable where the results are changing rapidly, whereas large elements can be used where results are relatively constant. Step 2 Select a displacement (shape) function. This step involves choosing a displacement function within each element. The function is defined within the element using the nodal values of the element. Polynomials are used as shape functions because they are simple to work within finite element formulation. For plate element, the shape function is a function of coordinates in its plane (x - y plane) Step 3 Define the strain/displacement and stress/strain relationships Strain/displacement and stress/strain relationships are necessary for deriving the equations for each finite element. This step requires basic equations of elasticity theory. Step 4 Derive the element stiffness matrix and equations Using the principle of virtual work will produce the equations to describe the behaviour of an element. Step 5 Assemble the element equations to obtain the global equations and introduce boundary conditions. The individual element equations can be added together using a method of superposition to obtain total equations for the whole plate. Step 6 Solve for the unknown degrees of freedom Total equations modified to account for the boundary conditions is a set of simultaneous algebraic equations. These equations are solved to obtain w, 0X, 0y using an elimination method a Step 7 Solve for the element strains and stresses Stresses and strains can be obtained in terms of displacements determined using strain/displacement and stress/strain relationships. In part two, for rectangular plate elements, shape functions, element behaviour matrix and element node action matrix was seperately derived by using finite element method steps. For solution of rectangular plate bending problems which have uniformly loaded and different boundary conditions, the program was written in BASIC programming using above steps. In BASIC programming with finite element method dimensions of plate in X and Y directions "a, b ", the number of slice in X and Y directions "N, M " to constitute plate elements, properties for the material making up plate, "E, v", thickness of plate "h" and boundary conditions for each nodal points are given as input data. Then the results of solution for a node, coordinate of node, w deflection, 0X rotation about X axis, 9y rotation about Y axis, Mx, My, M^ moments are written on a line by program. Part appendix provides solution of rectangular plate with finite element method program. Numerical treatment of differential equations can yield approximate results, acceptable for most pratical purposes. Among the numerical techniques presently available, the finite difference method is one of the most general. In applying this P method, the derivatives in the differantial equation under considiration AAw=- are replaced by difference quantities at some selected points. These points are located at joints of rectangular, triangular or other network called finite difference mesh. We describe the deflected plate surface w=w(x, y) by determining approximate values for the deflections at these mesh points. In this method the governing differential equations (and the equations of the boundary conditions) are transformed into a set of simultaneous equations by using stencils for interior mesh points of plate. The finite difference method is simple, reusable formulas (stencils), versatility, suitability for computer ( or in many cases even for hand computation), acceptable accurracy for most tecnical purposes provided that a relatively fine mesh is required to obtain an acceptable accuracy ( + 5 %) especially for internal forces. In part three, the finite difference expressions for the fourth-order derivatives '*V> a4- a4 N W f and mixed fourth derivative w,dK28y were derived by using the finite differences for one dimensional case. P The finite difference representation of AAw=- at pivotal point m, n was found by using stencils for interior mesh points. For solution of rectangular plate bending problems which have uniformly loaded and simple support or fixed boundary conditions, the program was writen in BASIC programming using the finite differences method. In program, dimensions of plate in X and Y directions "a, b ", the number of slice in X and Y directions "N,M " to constitute mesh, properties for the material making up plate, "E, v", thickness of plate "h" and boundary conditions for each edge of plate are given as input data. Then the results of solution for a interior mesh point: the number and coordinate of interior mesh point, w deflection and Mx, My, Mxy moments are written on a line by xu program. Part appendix provides solution of rectangular plate with finite difference method program. In part four present the comparasion of the finite element method and the finite difference method for plate bending problems. For example determining the maximum deflection of a uniformly loaded square plate with fixed boundary conditions. Assume the following values v = 0.3 =/)"> E = 2850000 t/nr h = 0. 15m, dimensions of plate a=4m, b=4' a) Solution with the finite element method A comparsion the finite element method having 363 unknowns with the analytic solution of problem indicates an 2.0 % discrepancy b) Solution with the finite differences metod A comparsion of the finite differences method having 289 unknowns with the analytic solution of problem indicates an 2.7 % discrepancy The finite element method converges to exact solution with less unknown than the finite differences method for fixed boudary conditions. An other example, determining the maximum deflection of a uniformly loaded square plate with simple suport boundary conditions. Assume the following values v = 0.3, E = 2850000 t/m2 h = 0. 15m, dimensions of plate a=4ra, b=4m a) Solution with the finite element method max (w)=0.895.10"3 m with 363 unknowns. A comparsion with the analytic solution of problem indicates an 1.4 % discrepancy. Xlll b) Solution with the finite differences metod max(w)=0.885.10"3 m with 121 unknowns. A comparsion with the analytic solution of problem indicates an 0 % discrepancy. Comparasion of these tecniques indicates that the finite differences method converges to exact solution with less unknown than the finite element method for simple support boundary conditions.
Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1996
Keywords: Karşılaştırmalı analiz; Levhalar; Sonlu elemanlar yöntemi; Sonlu farklar yöntemi; Comparative analysis; Plates; Finite element method; Finite differences method