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Düzlemi içinde ve düzlemine dik yüklü taşıyıcı sistemlerin çubuk sistemlerle modellenmesi

Düzlemi içinde ve düzlemine dik yüklü taşıyıcı sistemlerin çubuk sistemlerle modellenmesi

##### Dosyalar

##### Tarih

1993

##### Yazarlar

Türk, H. Ersan

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### Özet

Matris deplasman metodu, taşıyıcı sistemlerin dış etkiler altında, uç deplasman ve uç kuvvetlerinin bulunmasında ; bilgisayarlara kolaylıkla uyarlanabilen matris formunda formülasyonlar içermesi ve değişik tipte sistemlerin çözümlerinde de kolaylıkla uygulana bilmesi nedenleriyle, inşaat mühendisliğinde yaygın bir kullanım alanı bulmaktadır. Beş bölümden oluşan bu çalışmada; düzlemi içinde ve düzlemine dik yüklü taşıyıcı sistemlerin, çubuk elemanlar kullanılarak modellemeleri yapılmış ve matris deplasman yöntemi kullanılarak yapılan bilgisayar programlan yardımıyla, sistemi oluşturan bu çubuk elemanların uç deplasman ve uç kuvvetleri bulunmuştur. Bu sonuçlar, kesin çözümler kullanılarak veya sonlu elemanlar yöntemi kullanılarak bulunan sonuçlarla karşılaştırmıştır. Birinci bölümde, çalışmanın genel kapsamı üzerinde durulmuştur. İkinci bölümde, plak teorisinde yapılan kabuller ve kartezyen koordinatlarda plak diferansiyel denkleminin bulunması anlatılmıştır. Üçüncü bölümde, levha denkleminin bulunması anlatılmıştır. Dördüncü bölümde, matris deplasman metodunun esası anlatılmış ; düzlemi içinde ve düz lemine dik yüklü sistemlere, bu metodun uygulanması açıklanmıştır. Bu bölümde, ayrıca, matris deplasman metodunun bir uygulama sahası olan " sistem ortamına eşdeğer bir ortamda sonlu elemanlar alınması " yöntemine de değinilmiştir. Beşinci bölümde ; düz lemi içinde ve düzlemine dik yüklü sistemler için yapılan uygulamalar yer almaktadır. Her uygulamada, giriş ve çıkış bilgileri basılmış ve sistemin özellikleri verilmiştir.

In this work, the matrix displacement method is applied to the systems loaded in or perpendicular to their planes for determining the nodal forces, displacements and the stresses of the beam elements which form the structural system. The results of using this method are compared with the exact solutions of the systems. The aim is to find out the approximation of using the matrix displacement method to the exact solutions or to the other appropriate methods, such as finite element method, by dividing the structures into the grids. The Appropriate Methods In the past decates, the numerical and approximate methods in the solution of problems of applied mechanics have gained considerably importance as the application areas are a lot and the computers are widely used. Exact solutions for planes are available only for certain shapes, boundaries and loading conditions ( Timoshenko and Woinowsky-Krieger 1959 ). When solutions for arbitrary shaped plates supported by complex boundary conditions are needed, a numerical method must therefore be used. Several numerical methods are usually adopted for the analysis, such as finite element, finite difference, matrix displacement or matrix force techniques. Using these methods require preparing the appropriate element mesh, and they require huge memory space for computational purposes when large number of degrees of freedom involved. Rayleigh-R.tz method can be thought in order to avoid the mesh generation and the computer memory space problem as only one single superelement is used in the whole process. The difficulty of using the Rayleigh-Ritz method is the choice of suitable functions to approximate the deflected shape, which must satisfy the prescribed boundan/ conditions of plates Bhat ( 1985a) proposed a set of beam-characteristic orthogonal polynomials to study the bending deflection of rectangular plates under static loading. The variational method, as formulated by Galerkin and Vlaslov, is a valuable method for the approximate analysis of plates of various boundary conditions, subjected to arbitrary loads. This metfiod is recommended for hand computation. The accuracy of the method depends considerably on the choice of shape functions. The disadvantages of the variational method are : (1 ) it requires the knowledge of higher mathematics; (2) although the required computations are simple, they can occasionally be quite lengthy; (3) it does not lend itself to computer application; (4) its use is not recommended for plates with holes or for cases where the boundary condition along an edge changes arbitrarily; (5) It can not be applied directly when the integral of the edge forces is not zero. For the finite difference method, the solution of the generated algebraic equations can easily be achieved both computers and by hand computation, if the numbers of unknowns is not large. When high accuracy is required, the use of this method may not be available because of the slow convergence characteristics. The disadvantages of the finite difference method are : (1) it requires mathematically trained operators; (2) it requires more work to achieve complete automation of the procedure; (3) concentrated load combined with a fine grid may create singularity problems; (4) certain boundary conditions such as elastic restrain or support, for instance, may be difficult to handle. The critical operation in the finite element methods is the generation of element- stiffness matrices, which are intimately linked to the compatibility of the deformations within the element as well as between the adjacent elements. The derivation of conforming shape functions for generation of element stiffness properties, is a difficult task and should be left to researchers. On the other hand, when suitable stiffness coefficients are avilable, the procedure can be completely automated. The disadvantages of finite element method are: (1) It cannot be used without a readily available well-tested element-stiffness matrix, the use of which results in a (preferably monotonic) convergence to the exact solution; (2) a considerable number of elements may be required, in some instances, to obtain high accuracy and working with large matrices may create problems if the memory is restricted. Matrix-Displacement Analysis of Gridworks Although a set of algebraic equations can be presented in matrix form resulting from energy or finite difference methods, such an approach cannot be classified as one of the matrix methods of structural mechanics. The general approach to solution of specific structural problems by matrix-computer analysis remains the same, irrespective of whether the loac,s supported by beam, rigit frame, plate, shell, or even by three-dimensional solid. In the classical methods of structural analysis, either matrix-displacement or matrix-force techniques can be applied. Using the displacement method, nodal displacements are unknowns. This approach corresponds to the matrix formulation of slope deflection equations. Extension of the classical methods of structural analysis results in the force method for matrix analysis of structures. For some types of structural systems, successful application of the matrix-force method may require a high degree of structural engineering competence and more matrix operations than the displacement method. VI Determining the Element Force-Displacements Relationships A number of alternative methods are available for the calculation of force-displacement relationships describing the stiffness characteristics of structural elements and the choice of a particular method depends mainly on the type of the element. The following methods can be used : 1. Unit-displacement theorem 2. Castigliano's theorem ( part I ) 3. Solution of differential equations for the element displacements 4. Inversion of the displacement-force relationships The first one of these methods, the application of the unit-displacement theorem, is undoubtedly the most convenient since it leads directly to the required matrix equation relating element forces to their correponding displacements. In the second method, based on a direct application of Castigliano's theorem ( part I ), the strain energy is first calculated in terms of element displacements and temperature and is differentiated with respect to a selected displacement The differentiation is repeated for each element displacement in turn, and this generates a complete set of force-displacement equations, which can be formulated in matrix notation. In the third method, the solutions of the differential equations for displacements are used to derive the required stiffness relation ships. Naturally, the application of this method is limited to structural elements for which solutions for displacements are available. In the fourth method, the equations for the displacement-force relationships are determined first, and then these equations are inverted to find force-displacements relationships. Matrix Formulation of the Displacement Method and the Order of the Process The fundamental assumption used in the analysis is that the structure can be satisfactorily represented by an assembly of discrete elements having simplified elastic properties and that these elements are interconnected so as to represent the actual continuous structure. The boundary displacements are compatible at least at the node points, where the elements are joined, and the stresses within each element are equilibrated by a set of element forces Pi in the directions of element displacements Ui where superscript i denotes the i.th element Under the external loads which applied directly to the nodes, the element forces are related to the corresponding displacements by the matrix equation : R=ki di Computer programs which are written for both of the plane systems consist of the following order : VII 1. The transformation matrix T is formed by using the coordinates, as expressed in the related chapters. 2. For each element, the stiffness matrix is formed in terms of element coordinates and converted to the system coordinates with the following matrix form : kxyz= TTk*yzT 3. Using the code numbers method, the common stiffness matrix of the system is formed as follows : K = Ekxyz 4. Using the code numbers of the nodes, the Q external and nodal load vector is formed. 5. By solving the equation of K D = Q, the nodal displacement vector O is calculated in global coordinates. Q nodal Toad vector must be given at the same direction of D displacement vector. 6. By multiplying the stiffness matrix and the displacement matrix of the element as the following, the Pxyz element forces for global coordinates are calculated. xyz : global coordinates Px 2=kxyz dxyz »yz : local coordinates 7. The element forces, in common coordinates is converted to the element coordinates as the following form : - -xyz = - -xyz -xyz = J !îxyz I I -xyz ixyz = _xyz I -XyZ For two-dimensional problems, the stiffness matrix of a beam element has the order of 6x 6, as there exists six forces and six displacements. For the systems loaded in or perpendicular to their planes, the element stiffness matrices are formed according to the force-displacement relationships of the elements. VII I This study for applying the matrix displacement method by using the beam elements consists of five main parts. The first part is the introduction part of the study. In the second part, the differential equation of the plate systems in cartesian coordinate system is expressed and formulated. In the third part, the differential equation of the systems loaded perpendicularly to their planes in cartesian coordinate system is expressed and formulated. The fourth part includes the application of the matrix-displacement method and the introduction of the systems loaded in or perpendicular to their planes. The fourth part also includes the introduction of the framework method which is created by Hrennikoff, using the equality of the potantial energy variations of the analysed systems and the substituting systems. The framework method is also expressed in this part. The fifth part consists of eight particular problems which are solved by using the computer programs for the application of the matrix-displacement method. Framework Method The familiar concept of frame analysis can be extended to two-dimensional continua. Hrennikoff has approximated such elastic bodies by various assemblages of bars and beams endowed with certain elastic properties for representation of the actual continuum, and has arranged them in definite patterns. Essentially, the framework method replaces the continuous surface structure by an idealized discrete system, the elements of which are interconnected only at finite nodal or joint points. In such a substitute structure, having its size of subdivisions decreased, the state of stress approaches that of the corresponding continuum. In this way, the analysis of stretching and bending of plates is reduced to the solution of an equivalent truss or gridwork, respectively, which can be conviniently accomplished by the matrix displacement method. Although such a finite framework cannot be completely equivalent to the replaced continuum, an accuracy sufficient for most practical purposes, can be achieved. The simple lattice analogy can be used to introduce the basic concepts involved in the generation of mambran and bending stiffness coefficients for finite plate elements. The framework method, based on an equivalent lattice system, uses concepts and the methods familiar to structural engineers for the solution of complex plate problems. A special advantage of this approach is that the results are obtained without formulating the governing differential equations. The treatment of boundary value problems, which are often very difficult in the case of classical and other numerical methods, can also be accomplished in the simplest manner With a sufficiently large number of elements any arbitrary geometrical shape, load and boundary condition can be treated with relative ease. The accuracy obtainable is sufficient for most practical purposes. In the case of lateral bending of plates, the solution converges quickly, even with a crude load representation The framework method appears to have the following advantages over most finite element approaches : 1. The generation of element stiffness coefficients and their analytical representation are extremely simple. 2. The resulting stiffness matrices are conforming and they satisfy all continuity requirements between individual cells. 3. Extension of the method to more complex problems, such as elastic stability analysis, for instance, can be accomplished with relative ease.

In this work, the matrix displacement method is applied to the systems loaded in or perpendicular to their planes for determining the nodal forces, displacements and the stresses of the beam elements which form the structural system. The results of using this method are compared with the exact solutions of the systems. The aim is to find out the approximation of using the matrix displacement method to the exact solutions or to the other appropriate methods, such as finite element method, by dividing the structures into the grids. The Appropriate Methods In the past decates, the numerical and approximate methods in the solution of problems of applied mechanics have gained considerably importance as the application areas are a lot and the computers are widely used. Exact solutions for planes are available only for certain shapes, boundaries and loading conditions ( Timoshenko and Woinowsky-Krieger 1959 ). When solutions for arbitrary shaped plates supported by complex boundary conditions are needed, a numerical method must therefore be used. Several numerical methods are usually adopted for the analysis, such as finite element, finite difference, matrix displacement or matrix force techniques. Using these methods require preparing the appropriate element mesh, and they require huge memory space for computational purposes when large number of degrees of freedom involved. Rayleigh-R.tz method can be thought in order to avoid the mesh generation and the computer memory space problem as only one single superelement is used in the whole process. The difficulty of using the Rayleigh-Ritz method is the choice of suitable functions to approximate the deflected shape, which must satisfy the prescribed boundan/ conditions of plates Bhat ( 1985a) proposed a set of beam-characteristic orthogonal polynomials to study the bending deflection of rectangular plates under static loading. The variational method, as formulated by Galerkin and Vlaslov, is a valuable method for the approximate analysis of plates of various boundary conditions, subjected to arbitrary loads. This metfiod is recommended for hand computation. The accuracy of the method depends considerably on the choice of shape functions. The disadvantages of the variational method are : (1 ) it requires the knowledge of higher mathematics; (2) although the required computations are simple, they can occasionally be quite lengthy; (3) it does not lend itself to computer application; (4) its use is not recommended for plates with holes or for cases where the boundary condition along an edge changes arbitrarily; (5) It can not be applied directly when the integral of the edge forces is not zero. For the finite difference method, the solution of the generated algebraic equations can easily be achieved both computers and by hand computation, if the numbers of unknowns is not large. When high accuracy is required, the use of this method may not be available because of the slow convergence characteristics. The disadvantages of the finite difference method are : (1) it requires mathematically trained operators; (2) it requires more work to achieve complete automation of the procedure; (3) concentrated load combined with a fine grid may create singularity problems; (4) certain boundary conditions such as elastic restrain or support, for instance, may be difficult to handle. The critical operation in the finite element methods is the generation of element- stiffness matrices, which are intimately linked to the compatibility of the deformations within the element as well as between the adjacent elements. The derivation of conforming shape functions for generation of element stiffness properties, is a difficult task and should be left to researchers. On the other hand, when suitable stiffness coefficients are avilable, the procedure can be completely automated. The disadvantages of finite element method are: (1) It cannot be used without a readily available well-tested element-stiffness matrix, the use of which results in a (preferably monotonic) convergence to the exact solution; (2) a considerable number of elements may be required, in some instances, to obtain high accuracy and working with large matrices may create problems if the memory is restricted. Matrix-Displacement Analysis of Gridworks Although a set of algebraic equations can be presented in matrix form resulting from energy or finite difference methods, such an approach cannot be classified as one of the matrix methods of structural mechanics. The general approach to solution of specific structural problems by matrix-computer analysis remains the same, irrespective of whether the loac,s supported by beam, rigit frame, plate, shell, or even by three-dimensional solid. In the classical methods of structural analysis, either matrix-displacement or matrix-force techniques can be applied. Using the displacement method, nodal displacements are unknowns. This approach corresponds to the matrix formulation of slope deflection equations. Extension of the classical methods of structural analysis results in the force method for matrix analysis of structures. For some types of structural systems, successful application of the matrix-force method may require a high degree of structural engineering competence and more matrix operations than the displacement method. VI Determining the Element Force-Displacements Relationships A number of alternative methods are available for the calculation of force-displacement relationships describing the stiffness characteristics of structural elements and the choice of a particular method depends mainly on the type of the element. The following methods can be used : 1. Unit-displacement theorem 2. Castigliano's theorem ( part I ) 3. Solution of differential equations for the element displacements 4. Inversion of the displacement-force relationships The first one of these methods, the application of the unit-displacement theorem, is undoubtedly the most convenient since it leads directly to the required matrix equation relating element forces to their correponding displacements. In the second method, based on a direct application of Castigliano's theorem ( part I ), the strain energy is first calculated in terms of element displacements and temperature and is differentiated with respect to a selected displacement The differentiation is repeated for each element displacement in turn, and this generates a complete set of force-displacement equations, which can be formulated in matrix notation. In the third method, the solutions of the differential equations for displacements are used to derive the required stiffness relation ships. Naturally, the application of this method is limited to structural elements for which solutions for displacements are available. In the fourth method, the equations for the displacement-force relationships are determined first, and then these equations are inverted to find force-displacements relationships. Matrix Formulation of the Displacement Method and the Order of the Process The fundamental assumption used in the analysis is that the structure can be satisfactorily represented by an assembly of discrete elements having simplified elastic properties and that these elements are interconnected so as to represent the actual continuous structure. The boundary displacements are compatible at least at the node points, where the elements are joined, and the stresses within each element are equilibrated by a set of element forces Pi in the directions of element displacements Ui where superscript i denotes the i.th element Under the external loads which applied directly to the nodes, the element forces are related to the corresponding displacements by the matrix equation : R=ki di Computer programs which are written for both of the plane systems consist of the following order : VII 1. The transformation matrix T is formed by using the coordinates, as expressed in the related chapters. 2. For each element, the stiffness matrix is formed in terms of element coordinates and converted to the system coordinates with the following matrix form : kxyz= TTk*yzT 3. Using the code numbers method, the common stiffness matrix of the system is formed as follows : K = Ekxyz 4. Using the code numbers of the nodes, the Q external and nodal load vector is formed. 5. By solving the equation of K D = Q, the nodal displacement vector O is calculated in global coordinates. Q nodal Toad vector must be given at the same direction of D displacement vector. 6. By multiplying the stiffness matrix and the displacement matrix of the element as the following, the Pxyz element forces for global coordinates are calculated. xyz : global coordinates Px 2=kxyz dxyz »yz : local coordinates 7. The element forces, in common coordinates is converted to the element coordinates as the following form : - -xyz = - -xyz -xyz = J !îxyz I I -xyz ixyz = _xyz I -XyZ For two-dimensional problems, the stiffness matrix of a beam element has the order of 6x 6, as there exists six forces and six displacements. For the systems loaded in or perpendicular to their planes, the element stiffness matrices are formed according to the force-displacement relationships of the elements. VII I This study for applying the matrix displacement method by using the beam elements consists of five main parts. The first part is the introduction part of the study. In the second part, the differential equation of the plate systems in cartesian coordinate system is expressed and formulated. In the third part, the differential equation of the systems loaded perpendicularly to their planes in cartesian coordinate system is expressed and formulated. The fourth part includes the application of the matrix-displacement method and the introduction of the systems loaded in or perpendicular to their planes. The fourth part also includes the introduction of the framework method which is created by Hrennikoff, using the equality of the potantial energy variations of the analysed systems and the substituting systems. The framework method is also expressed in this part. The fifth part consists of eight particular problems which are solved by using the computer programs for the application of the matrix-displacement method. Framework Method The familiar concept of frame analysis can be extended to two-dimensional continua. Hrennikoff has approximated such elastic bodies by various assemblages of bars and beams endowed with certain elastic properties for representation of the actual continuum, and has arranged them in definite patterns. Essentially, the framework method replaces the continuous surface structure by an idealized discrete system, the elements of which are interconnected only at finite nodal or joint points. In such a substitute structure, having its size of subdivisions decreased, the state of stress approaches that of the corresponding continuum. In this way, the analysis of stretching and bending of plates is reduced to the solution of an equivalent truss or gridwork, respectively, which can be conviniently accomplished by the matrix displacement method. Although such a finite framework cannot be completely equivalent to the replaced continuum, an accuracy sufficient for most practical purposes, can be achieved. The simple lattice analogy can be used to introduce the basic concepts involved in the generation of mambran and bending stiffness coefficients for finite plate elements. The framework method, based on an equivalent lattice system, uses concepts and the methods familiar to structural engineers for the solution of complex plate problems. A special advantage of this approach is that the results are obtained without formulating the governing differential equations. The treatment of boundary value problems, which are often very difficult in the case of classical and other numerical methods, can also be accomplished in the simplest manner With a sufficiently large number of elements any arbitrary geometrical shape, load and boundary condition can be treated with relative ease. The accuracy obtainable is sufficient for most practical purposes. In the case of lateral bending of plates, the solution converges quickly, even with a crude load representation The framework method appears to have the following advantages over most finite element approaches : 1. The generation of element stiffness coefficients and their analytical representation are extremely simple. 2. The resulting stiffness matrices are conforming and they satisfy all continuity requirements between individual cells. 3. Extension of the method to more complex problems, such as elastic stability analysis, for instance, can be accomplished with relative ease.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1993

##### Anahtar kelimeler

Diferensiyel denklemler,
Matris deplasman yöntemi,
Differential equations,
Maytrix displacement method