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Matris deplasman metodu ile statik analiz bilgisayar programı hazırlanması

Matris deplasman metodu ile statik analiz bilgisayar programı hazırlanması

##### Dosyalar

##### Tarih

1995

##### Yazarlar

Uşaklı, Hakan

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### Özet

Matris deplasman yönteminin genel esasları izah edildikten sonra gerekli rij itlik matrisi terimlerinin nasıl hesaplandığı ve matris denklemlerinin nasıl kurulduğu açıklanmıştır. Bilgisayar programının çalışma mantığını içeren akış şeması tanıtıldıktan sonra değişik problemleri çözebilmek için veri dosyalarının nasıl hazırlanması gerektiği anlatılmıştır. Hazırlanan bilgisayar programı kullanılan bant matris yöntemi sayesinde oldukça büyük sistemleri kısa sürelerde çözebilmektedir. Perdelere saplanan rijit uçlu kirişlerin ve her türlü değişken kesitlerin tek bir eleman ile tanımlanabilmesi en önemli özelliklerin arasındadır. Veri girişi SAP90 bilgisayar programına benzemektedir, analizi yapılacak sistemin geometrisi, sınır şartları ve dış yükleri verildikten sonra düğüm noktalarının yerdeğiştirmeleri ve çubuk uçlarında eğilme momenti, kesme kuvveti ve normal kuvvet değerleri program tarafından oluşturulan çıktı dosyasına kaydedilmektedir. Bölüm 3 xteki örneklerin sonuçları kaynaklarındaki değerler ile kıyaslandığında uyum içinde oldukları görülmektedir. SAP90 ile yapılan kıyaslamalar sonunda sonuçların çakıştığı gözlenmiştir.

In this thesis a computer program has been prepared for the structural analysis of plane frame structures with combined shear walls and non-prismatic members. First, the stiffness matrix for an individual member is derived from basic principles. In a similar manner it is shown how the total stiffness matrix for an assemblage of individual members can be obtained. Finally, it is shown how the stiffness matrix for an assemblage can be obtained by superimposing the stiffness matrices of the individual members. This last step is basic to the application of the stiffness method to any structural problem. Then restraint and constraint conditions are imposed on the equation system. Having obtained the total structure stiffness matrix, it is next shown how the solution stems from a routine set of matrix calculations applied to the stiffness equation. As described in section 2.8 the matrix transformations can be carried out with standard solution techniques. A total solution is found; that is, the method yields all unknown displacements, reactions, and internal forces. The concept of a stress matrix is introduced for calculating the internal forces. Again the procedure demonstrated for the simple spring applies equally well to complex problems. Deflection and stiffness influence coefficients are of basic importance when studying complex composite structures. Such structures consist of an assemblage of many individual parts or members. They are too complex in geometry to be satisfactorily described in terms of the familiar differential equations of structural theory. As a result, basic structural principles like Equilibrium of forces, Compatibility of vii deformations, Hooke's law, relating forces and deformations are applied to such structures in a manner which leads to a mathematical formulation in terms of algebraic equations rather than differential equations. Influence coefficients, relating forces and displacements, enter prominently into these algebraic equations. Actual structures generally consist of basic structural components- stringers, beams, thin plates, etc.- properly fastened into an assemblage. For example, a truss is usually considered as an assemblage of axial force members, whereas a building frame consists of an assemblage of beams and columns. Consequently, it is vitally important to be able to form the total structure stiffness matrix from the stiffness matrices for the separate components. No matter what the actual may be, it can be idealized into an assemblage of members connected at node points. Assuming the stiffness matrix for each member to be known, the total structure stiffness matrix can be formed by superposition as described in section 2.8. Once boundary conditions and prescribed displacements are imposed into the equation, the resulting set of linear equations can be solved by standard solution techniques. Purpose of this thesis is to set up a computer program for the structural analysis of plane frame structures. As few as possible restricting assumptions have been made, it is possible to define every plane structure as long as the joint coordinates and member links are defined. Restraints can be specified independently at joints in three directions, two displacements and one rotation. Furthermore it is possible to define joint constraints to simulate high in-plane stiffness of beams and plates or to define rigid linked joints. These options are generally used for two dimensonal building frames, it is assumed that every node on the same level makes the same horizontal displacement because of the high stiffness of the floor diaphragm. But it is also viii possible to leave these nodes unconstraint, in this case the effects of axial deformations of the beams will also be taken into the stiffness equations. Much importance has been given to non-prismatic members which are usually not supported in structural analysis programs. By defining the variable moment of inertia as a shape function it has been possible to use the most common non- prismatic members with the same stiffness matrix. The supported member types are in figure 2.6. The member with linear changing end depth, the member with thick ends, the member with parabolic end depth, and the member with linear changing depth can be defined directly by substituting the appropriate values in the data file. The procedure of the shape function has the advantage that no additional nodes need to be defined at points with changing section properties. By defining the values shown in section 2.7 the program calculates the changing section properties, the moment of inertia and the section area, and applies the equations in section 2.9 to find the corresponding stiffness coefficients. The Simpson rule has been used for numerical integration of the shape functions. In addition to the predefined members shown in section 2.7 the program allows users to insert any non-prismatic member as long as the necessary stiffness coefficients are known. If the internal forces of a non-prismatic member are to be calculated, it is possible to set up a data file for this single member and divide it into parts. Then this single member can be solved under the original load conditions and the end forces found by the first solution. Another important type of member is the beam with rigid ends, these beams are used to model shear wall links. By a matrix transformation as explained in section 2.11 it is possible to assemble the rigid-end beam into the total structure without introducing very high stiffness terms to simulate the shear wall behavior, which causes difficulties in the stability of the equation system. The matrix transformation for this member is based on the relationship between ix the displacements at the center line and the support face. As shown in figure 2.10 the displacement values at the support face are defined in terms of the displacements at the center line. Rewriting this equations in matrix form and applying the principle of virtual work yields the modified stiffness matrix for the beam with rigid ends. Linear joint and member generation options are supported to ensure convenient data entry of repeating structural parts. The most common external span loads can be calculated by a subprogram as shown in section 2.6, whereas external joint loads can be defined directly in the data file. Every other type of external load can be included into the system as long as the inital bending moments and forces are known. More than one load condition can be specified in one execution, the corresponding displacements and internal forces are grouped in the output file by the program automatically. If the program runs out of memory it warns the user and requests him to reduce the number of load conditions. Another useful option is the opportunity to number the joints in a sequence defined by the user, to keep the band width of the total stiffness matrix as small as possible, saving memory and calculation time. Since the total stiffness matrix of the structure is banded and symmetrical a banded equation solver has been used, there are no joint or element number limitations, the program will attempt to solve the equation system for all specified load conditions as long as there is enough memory. The data file should be prepared in an ASCII text editor as described in section 2.13 and saved without extension. The output file will have the same name with. PRN extension. The output file will contain all structural information read by the program, formatted and grouped, making a post-check easier. x Generated joint numbers and coordinates, Restraints specifications, equation numbers of the joints assigned by the program, section property types and specifications, member connections or links, spanloading information, direct joint loading information will also be included in the output file. Next the resulting displacements for every joint and for every load condition are printed, finally the internal forces for all members are grouped and printed, these forces are two bending moments, two shear forces and the axial force. There are no prescribed units, all values entered by the user must be in a consistent unit system, whereas rotations are always in radians.

In this thesis a computer program has been prepared for the structural analysis of plane frame structures with combined shear walls and non-prismatic members. First, the stiffness matrix for an individual member is derived from basic principles. In a similar manner it is shown how the total stiffness matrix for an assemblage of individual members can be obtained. Finally, it is shown how the stiffness matrix for an assemblage can be obtained by superimposing the stiffness matrices of the individual members. This last step is basic to the application of the stiffness method to any structural problem. Then restraint and constraint conditions are imposed on the equation system. Having obtained the total structure stiffness matrix, it is next shown how the solution stems from a routine set of matrix calculations applied to the stiffness equation. As described in section 2.8 the matrix transformations can be carried out with standard solution techniques. A total solution is found; that is, the method yields all unknown displacements, reactions, and internal forces. The concept of a stress matrix is introduced for calculating the internal forces. Again the procedure demonstrated for the simple spring applies equally well to complex problems. Deflection and stiffness influence coefficients are of basic importance when studying complex composite structures. Such structures consist of an assemblage of many individual parts or members. They are too complex in geometry to be satisfactorily described in terms of the familiar differential equations of structural theory. As a result, basic structural principles like Equilibrium of forces, Compatibility of vii deformations, Hooke's law, relating forces and deformations are applied to such structures in a manner which leads to a mathematical formulation in terms of algebraic equations rather than differential equations. Influence coefficients, relating forces and displacements, enter prominently into these algebraic equations. Actual structures generally consist of basic structural components- stringers, beams, thin plates, etc.- properly fastened into an assemblage. For example, a truss is usually considered as an assemblage of axial force members, whereas a building frame consists of an assemblage of beams and columns. Consequently, it is vitally important to be able to form the total structure stiffness matrix from the stiffness matrices for the separate components. No matter what the actual may be, it can be idealized into an assemblage of members connected at node points. Assuming the stiffness matrix for each member to be known, the total structure stiffness matrix can be formed by superposition as described in section 2.8. Once boundary conditions and prescribed displacements are imposed into the equation, the resulting set of linear equations can be solved by standard solution techniques. Purpose of this thesis is to set up a computer program for the structural analysis of plane frame structures. As few as possible restricting assumptions have been made, it is possible to define every plane structure as long as the joint coordinates and member links are defined. Restraints can be specified independently at joints in three directions, two displacements and one rotation. Furthermore it is possible to define joint constraints to simulate high in-plane stiffness of beams and plates or to define rigid linked joints. These options are generally used for two dimensonal building frames, it is assumed that every node on the same level makes the same horizontal displacement because of the high stiffness of the floor diaphragm. But it is also viii possible to leave these nodes unconstraint, in this case the effects of axial deformations of the beams will also be taken into the stiffness equations. Much importance has been given to non-prismatic members which are usually not supported in structural analysis programs. By defining the variable moment of inertia as a shape function it has been possible to use the most common non- prismatic members with the same stiffness matrix. The supported member types are in figure 2.6. The member with linear changing end depth, the member with thick ends, the member with parabolic end depth, and the member with linear changing depth can be defined directly by substituting the appropriate values in the data file. The procedure of the shape function has the advantage that no additional nodes need to be defined at points with changing section properties. By defining the values shown in section 2.7 the program calculates the changing section properties, the moment of inertia and the section area, and applies the equations in section 2.9 to find the corresponding stiffness coefficients. The Simpson rule has been used for numerical integration of the shape functions. In addition to the predefined members shown in section 2.7 the program allows users to insert any non-prismatic member as long as the necessary stiffness coefficients are known. If the internal forces of a non-prismatic member are to be calculated, it is possible to set up a data file for this single member and divide it into parts. Then this single member can be solved under the original load conditions and the end forces found by the first solution. Another important type of member is the beam with rigid ends, these beams are used to model shear wall links. By a matrix transformation as explained in section 2.11 it is possible to assemble the rigid-end beam into the total structure without introducing very high stiffness terms to simulate the shear wall behavior, which causes difficulties in the stability of the equation system. The matrix transformation for this member is based on the relationship between ix the displacements at the center line and the support face. As shown in figure 2.10 the displacement values at the support face are defined in terms of the displacements at the center line. Rewriting this equations in matrix form and applying the principle of virtual work yields the modified stiffness matrix for the beam with rigid ends. Linear joint and member generation options are supported to ensure convenient data entry of repeating structural parts. The most common external span loads can be calculated by a subprogram as shown in section 2.6, whereas external joint loads can be defined directly in the data file. Every other type of external load can be included into the system as long as the inital bending moments and forces are known. More than one load condition can be specified in one execution, the corresponding displacements and internal forces are grouped in the output file by the program automatically. If the program runs out of memory it warns the user and requests him to reduce the number of load conditions. Another useful option is the opportunity to number the joints in a sequence defined by the user, to keep the band width of the total stiffness matrix as small as possible, saving memory and calculation time. Since the total stiffness matrix of the structure is banded and symmetrical a banded equation solver has been used, there are no joint or element number limitations, the program will attempt to solve the equation system for all specified load conditions as long as there is enough memory. The data file should be prepared in an ASCII text editor as described in section 2.13 and saved without extension. The output file will have the same name with. PRN extension. The output file will contain all structural information read by the program, formatted and grouped, making a post-check easier. x Generated joint numbers and coordinates, Restraints specifications, equation numbers of the joints assigned by the program, section property types and specifications, member connections or links, spanloading information, direct joint loading information will also be included in the output file. Next the resulting displacements for every joint and for every load condition are printed, finally the internal forces for all members are grouped and printed, these forces are two bending moments, two shear forces and the axial force. There are no prescribed units, all values entered by the user must be in a consistent unit system, whereas rotations are always in radians.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Sosyal Bilimler Enstitüsü, 1995

##### Anahtar kelimeler

Bilgisayar destekli programlama,
Matrisler,
Computer aided programming,
Matrices