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Uzay çubuk sistemlerde ikinci mertebe limit yük için yapı ağırlığını minimum yapan bir boyutlandırma yöntemi

Uzay çubuk sistemlerde ikinci mertebe limit yük için yapı ağırlığını minimum yapan bir boyutlandırma yöntemi

##### Dosyalar

##### Tarih

1994

##### Yazarlar

Orakdöğen, Engin

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Bu çalışmada, uzay çubuk sistemlerin ikinci mertebe limit yüke göre minimum ağırlıklı olarak boyutlandırılmaları için bir ardışık yaklaşım yöntemi geliştirilmiştir. Sekiz bölüm olarak sunulan bu çalışmanın birinci bölümünde konunun tanıtılması, konu ile ilgili çalışmaların gözden geçirilmesi, çalışmanın amacı ve kapsamı yer almaktadır. ikinci bölümde, çalışmada yapılan kabuller ve ardışık yaklaşım yönteminin esasları verilmiştir. Her adımı birbirini izleyen boyutlandırma ve sistem hesabı aşamalarından oluşan yöntemde, enkesit karakteristikleri arasındaki bağıntılar bir önceki adımın boyutlandırma aşamasından, ikinci mertebe limit yüke karşı gelen plastik kesit yerleri ile bu kesitlere ait akma vektörü doğrultuları ise bir önceki adımın sistem hesabı aşamasından alınmaktadır. Bu bölümde ayrıca, yaklaşık akma koşulu, çözümün sağlaması gereken koşullar ve kısıtlamalar ile minimum yapılması istenen ağırlık fonksiyonu ayrıntılı olarak açıklanmıştır. Üçüncü bölümde, ağırlık fonksiyonunun ve kısıtlamaların lineerleştirilmesi ile lineer programlama problemine dönüştürülen optimizasyon probleminin çÖzümündeki indirgeme ve değişken dönüşümü işlemleri, Simplex yönteminin bu problemin çözümüne uygulanması ve herhangi bir ardışık yaklaşım adımının boyutlandırma aşamasında izlenen yol açıklanmıştır. Dördüncü bölümde, uzay çubuk sistemlerde ikinci mertebe limit yükün hesabı için geliştirilen bir yük artımı yönteminin esasları ve ikinci mertebe limit yükün hesabında izlenen yol özetlenmiştir. Beşinci bölümde, yapı sistemlerinin, ikinci mertebe limit yüke göre ve göçme yüküne göre minimum ağırlıklı olarak boyutlandınlmalarında izlenen yol verilmiş, ayrıca, boyutlandırılan sistemde işletme yükleri altında sağlanması gereken koşullar açıklanmıştır. Altıncı bölüm, yöntemin sayısal uygulamaları için hazırlanan ve Fortran-77 programlama dilinde kodlanan bir bilgisayar programının ve programı oluşturan altprogramlann açıklanmasına ayırılmıştır. Yedinci bölümde, yöntemin sayısal uygulamaları için hazırlanan, bilgisayar programı kullanılarak çözülen üç örneğin sonuçları verilmiştir. Sekizinci bölümde, bu çalışmada elde edilen sonuçlar açıklanmış, Ek A ve Ek B de ise sırasıyla, kutu ve / kesitler için çıkarılan enkesit karakteristikleri arasındaki bağıntılar ve matris yerdeğiştirme yönteminde kullanılan eleman rijitlik matrisleri verilmiştir.

The main objective of the structural engineering is to design structures which withstand external loads safely and at a minimum cost. During last decades, the developments in the optimization methods which attempt to find the most economical solutions to design problems by satisfying the required safety and rigidity constraints and minimizing the cost function as well as the developments in the non-linear analysis methods which aim to determine the real behavior of structures under external effects, give the structural engineer the opportunity to reach this objective. The structural systems made of ductile material demonstrate elastic-plastic behavior under increasing loads, i.e., the non-linear deformations are assumed to accumulate at certain sections which are defined as plastic sections while the remaining structure behaves linearly-elastic. As the loads reach a limiting value, either the complete structure or its part transforms into a mechanism or buckling failure occurs due to the lack of stability. The mechanism load is called as the first-order limit load and the buckling load is defined as the second-order limit load. The slender structural systems which demonstrate high ductility can be designed by equating the factored loads to the second-order limit load. The designed structure should also satisfy certain stress and displacement requirements under service loads. As the optimization problems gain great importance in the field of structural engineering, numerous research works have been carried out and various algorithms have been developed for solving these types of problems. These algorithms can be mainly classified as the optimality criteria approaches and the mathematical programming techniques. The objective of the mathematical programming techniques utilized for the Tninirmiin weight design of structural systems is to determine the design variables which minimize the objective function representing the cost of the structure and satisfy a set of constraints. The optimization problems in which all of the constraints and the objective function can be expressed linearly in terms of design variables are called as linear programming problems and those are expressed non-linearly in terms of design variables are called as non-linear programming problems. Cross-sectional properties of the members, displacements, internal force components, deformations etc. can be chosen as design variables. xvt In the minimum weight design problems where the cost of the structure is assumed to be represented by the weight of material, the objective function can be expressed in terms of the cross-sectional characteristics. The equilibrium and compatibility equations, the yield conditions, the stress and displacement limitations, constructive requirements constitute the constraints of the problem. When the effect of geometrical changes on the equilibrium equations is neglected, the structure can be designed for the first-order limit load at which the structure transforms into a complete mechanism. The internal forces corresponding the mechanism load are determined through the equilibrium equations. Therefore, the minimum weight design problem can be transformed into a linear programming problem, provided that the objective function and the yield conditions are expressed by linear functions of cross-sectional properties. The structural systems for which the effect of geometrical changes on the equilibrium equations is significant, should be designed for the second- order limit load. Since the equilibrium equations, the yield conditions and the relationships between the cross-sectional properties are non-linear and since the second-order limit load is not equal to the mechanism load, the minimum weight design of geometrically and materially non-linear structural systems generally requires the formulation and solution of a non-linear programming methods. However, if the axial forces caused by the factored loads, the relationships between the cross-sectional properties and the locations and deformation vectors of plastic sections corresponding the second-order limit load are estimated and if the yield surface is idealized as composed of planes, the minimum weight design problem can be reduced to successive solutions of a linear programming problem. In this study, a method of successive approximations is developed for the minimum weight design of non-linear framed space structures. Each step of the optimization process is composed of design and analysis phases. At each design phase of the method, a linear programming technique may be applied for structural optimization. The proposed optimum design method is independent of the characteristics of the structure and can be applied to all types of framed structures which comply the following assumptions. a)Bernoulli-Navier hypothesis is valid. b) Members forming the structure are considered to be made of ideal elastic-plastic material. c)It is assumed that the non-linear deformations accumulate at certain sections which are defined as plastic sections and the remaining structure behaves linearly-elastic. This assumption corresponds to the plastic hinge concept applied to planar framed structures. xvn d) It is considered that the yield condition can be idealized by assuming the yield surface to be composed of planes. e)The second-order theory is applied. In this theory, the effect of geometrical changes on the equilibrium equations is considered while their effect on the compatibility equations is neglected. f) In the application of the second-order theory, the second-order effects caused by the torsional displacement and the local deformations within the cross-section are ignored. Hence, only the second-order effects due to the geometrical changes within the principal planes of the members are considered. g) Members forming the structure are straight, prismatic and the axial force is constant along the member. In case of existence of members which do not satisfy these conditions, they can be approximated by dividing them into smaller straight and prismatic segments with constant axial force. In the proposed method, equilibrium equations are linearized by estimating the axial forces caused by the factored loads and yield condition is linearized by assuming the yield surface is composed of planes. In each design phase, the relationships between the cross-sectional properties are taken from the previous design phase and the plastic section locations and the corresponding deformation vector directions are taken from the non linear analysis of the system designed in the previous step. When the design variables, the weight function and the plastic sections and corresponding deformation vectors are same or close enough in two successive steps, the computation is terminated and the minimum weight solution is obtained. Since the plastic deformation vector, is normal to the yield surface, the finite plastic deformations at a plastic section can be represented by a single parameter called as plastic deformation parameter. Utilizing the matrix displacement method and considering the plastic deformation parameters at plastic sections as structural variables along with the nodal displacement components, the equilibrium equations corresponding the second-order limit load can be written directly and systematically. These equilibrium equations also cover the geometrical compatibility conditions for the second-order limit load. The yield condition for members subjected to combined biaxial bending and axial force can be expressed in a general form, as in the following: K(Mz,My,N)<0 where K(Mz,My,N) represents a non-linear function of internal forces. If the yield surface representing function K is idealized as composed of planes, the linear yield condition XV1I1 is obtained. In this relationship, Mz and Afv are the bending moments about the z and y local axes respectively, N is the axial force and A^A,,/!,,/? are the constant coefficients defining the planes forming the yield surface. The displacements, deformations, internal forces and the corresponding cross-sectional properties which belong to the solution of the TninİTmım weight design problem should satisfy the following conditions and constraints: a) equilibrium equations and compatibility conditions corresponding the second-order limit load, b) yield condition constraints at all sections of the structure, c) displacement and deformation constraints for the second-order limit load, d) constructive restrictions related to the cross-sectional dimensions. In designing a structural system as the second-order limit load equals to factored loads, the axial forces due to the factored loads can be estimated easily. The second-order equilibrium conditions are expressed as linear equations by calculating the stability functions for the estimated axial forces. As mentioned above these equations also cover the geometrical compatibility conditions. The yield condition constraints are expressed in terms of the nodal displacement components, the plastic deformation parameters and the characteristic plastic moments of members. These constraints are linear inequalities since the yield surface is assumed to be composed of planes. The linear displacement and plastic deformation limitations and the limitations on the cross-sectional properties of members constitute the other constraints of the optimization problem. By expressing the structural weight in terms of a linear function of member plastic moments, the minimum weight design problem transforms into a linear programming problem. In this study Simplex method is utilized for the solution of the linear programming problem. Before the solution of the minn-mrm weight design problem by the Simplex method, some pre-calculations are required. These pre-calculations are explained below. i) The determination of the initial feasible solution of the Simplex method requires the calculation of plastic moment variables when the nodal displacements and the plastic deformation parameters are equal to XIX zero. However, the equilibrium equations cannot be satisfied when these variables are equal to zero. Therefore, new plastic deformation parameters which satisfy the equilibrium equations are defined. Thus, by equating these variables to zero, the initial feasible solution is obtained. ii) The variables in the Simplex method must be positive. As the nodal displacements and plastic deformation parameters may have positive or negative values, these variables are expressed as the difference of two positive variables. As explained above, the initial feasible solution or so called initial tableau in the Simplex method is obtained in a simple way which is appropriate to the phenomena of the structural optimization problem, instead of mathematical techniques. After the initial feasible solution is obtained, Simplex method is applied to determine the values of design variables which minimize the objective function.. In certain cases, the structural system may collapse before the second- order limit load due to the excessive displacements, plastic deformations and the large cracks in reinforced concrete structures. The load parameter which causes the structural collapse is defined as the collapse load. The minimum weight design of structures for the collapse load can be performed by the proposed method with certain modifications. In the design phase of the method the equilibrium equations are written by considering the plastic deformation parameters corresponding the collapse mode. Furthermore, displacement and plastic deformation constraints related to the collapse load are added to the constraints of the optimization problem. The proposed optimum design method is also be applied to the minimum weight design of materially and geometrically non-linear plate and shell structures which can be idealized by equivalent frame elements. A computer program called as MWDOS has been developed and coded in FORTRAN-77 programming language for the numerical applications of the proposed method. The minimum weight design of plane and space framed structures for the second-order limit load (or for the collapse load) can be performed by the use of this computer program. As MWDOS computer program is executed in a batch program structure, it can also be utilized for optimum design of large structural systems. Three numerical examples are given to illustrate the minimum weight design method developed herein. In the first example a single story, single bay plane frame is designed by the proposed method and the numerical results are compared with those obtained by a previous study. In the second example, a single story, single bay space frame is designed under different design criteria and the numerical results are compared. In this example the results of minimum weight design for the second-order limit load are given in detail. The third example is devoted to the minimum weight design of a single bay, ten story space frame. XX Numerical examples have shown that the effect of geometrical changes on the equilibrium equations and compatibility conditions corresponding the second-order limit load must be considered in the optimization problem. It is observed that, if the effect of geometrical changes on the equilibrium equations is not considered the factor of safety of the structure decreases 20.3%. Numerical examples have also shown that the optimum solutions obtained with using different initial values are extremely close or almost the same. Based on these results, it can be concluded that, the number of the local optimums in the domain of the feasible solutions are few and their values are close to each other.

The main objective of the structural engineering is to design structures which withstand external loads safely and at a minimum cost. During last decades, the developments in the optimization methods which attempt to find the most economical solutions to design problems by satisfying the required safety and rigidity constraints and minimizing the cost function as well as the developments in the non-linear analysis methods which aim to determine the real behavior of structures under external effects, give the structural engineer the opportunity to reach this objective. The structural systems made of ductile material demonstrate elastic-plastic behavior under increasing loads, i.e., the non-linear deformations are assumed to accumulate at certain sections which are defined as plastic sections while the remaining structure behaves linearly-elastic. As the loads reach a limiting value, either the complete structure or its part transforms into a mechanism or buckling failure occurs due to the lack of stability. The mechanism load is called as the first-order limit load and the buckling load is defined as the second-order limit load. The slender structural systems which demonstrate high ductility can be designed by equating the factored loads to the second-order limit load. The designed structure should also satisfy certain stress and displacement requirements under service loads. As the optimization problems gain great importance in the field of structural engineering, numerous research works have been carried out and various algorithms have been developed for solving these types of problems. These algorithms can be mainly classified as the optimality criteria approaches and the mathematical programming techniques. The objective of the mathematical programming techniques utilized for the Tninirmiin weight design of structural systems is to determine the design variables which minimize the objective function representing the cost of the structure and satisfy a set of constraints. The optimization problems in which all of the constraints and the objective function can be expressed linearly in terms of design variables are called as linear programming problems and those are expressed non-linearly in terms of design variables are called as non-linear programming problems. Cross-sectional properties of the members, displacements, internal force components, deformations etc. can be chosen as design variables. xvt In the minimum weight design problems where the cost of the structure is assumed to be represented by the weight of material, the objective function can be expressed in terms of the cross-sectional characteristics. The equilibrium and compatibility equations, the yield conditions, the stress and displacement limitations, constructive requirements constitute the constraints of the problem. When the effect of geometrical changes on the equilibrium equations is neglected, the structure can be designed for the first-order limit load at which the structure transforms into a complete mechanism. The internal forces corresponding the mechanism load are determined through the equilibrium equations. Therefore, the minimum weight design problem can be transformed into a linear programming problem, provided that the objective function and the yield conditions are expressed by linear functions of cross-sectional properties. The structural systems for which the effect of geometrical changes on the equilibrium equations is significant, should be designed for the second- order limit load. Since the equilibrium equations, the yield conditions and the relationships between the cross-sectional properties are non-linear and since the second-order limit load is not equal to the mechanism load, the minimum weight design of geometrically and materially non-linear structural systems generally requires the formulation and solution of a non-linear programming methods. However, if the axial forces caused by the factored loads, the relationships between the cross-sectional properties and the locations and deformation vectors of plastic sections corresponding the second-order limit load are estimated and if the yield surface is idealized as composed of planes, the minimum weight design problem can be reduced to successive solutions of a linear programming problem. In this study, a method of successive approximations is developed for the minimum weight design of non-linear framed space structures. Each step of the optimization process is composed of design and analysis phases. At each design phase of the method, a linear programming technique may be applied for structural optimization. The proposed optimum design method is independent of the characteristics of the structure and can be applied to all types of framed structures which comply the following assumptions. a)Bernoulli-Navier hypothesis is valid. b) Members forming the structure are considered to be made of ideal elastic-plastic material. c)It is assumed that the non-linear deformations accumulate at certain sections which are defined as plastic sections and the remaining structure behaves linearly-elastic. This assumption corresponds to the plastic hinge concept applied to planar framed structures. xvn d) It is considered that the yield condition can be idealized by assuming the yield surface to be composed of planes. e)The second-order theory is applied. In this theory, the effect of geometrical changes on the equilibrium equations is considered while their effect on the compatibility equations is neglected. f) In the application of the second-order theory, the second-order effects caused by the torsional displacement and the local deformations within the cross-section are ignored. Hence, only the second-order effects due to the geometrical changes within the principal planes of the members are considered. g) Members forming the structure are straight, prismatic and the axial force is constant along the member. In case of existence of members which do not satisfy these conditions, they can be approximated by dividing them into smaller straight and prismatic segments with constant axial force. In the proposed method, equilibrium equations are linearized by estimating the axial forces caused by the factored loads and yield condition is linearized by assuming the yield surface is composed of planes. In each design phase, the relationships between the cross-sectional properties are taken from the previous design phase and the plastic section locations and the corresponding deformation vector directions are taken from the non linear analysis of the system designed in the previous step. When the design variables, the weight function and the plastic sections and corresponding deformation vectors are same or close enough in two successive steps, the computation is terminated and the minimum weight solution is obtained. Since the plastic deformation vector, is normal to the yield surface, the finite plastic deformations at a plastic section can be represented by a single parameter called as plastic deformation parameter. Utilizing the matrix displacement method and considering the plastic deformation parameters at plastic sections as structural variables along with the nodal displacement components, the equilibrium equations corresponding the second-order limit load can be written directly and systematically. These equilibrium equations also cover the geometrical compatibility conditions for the second-order limit load. The yield condition for members subjected to combined biaxial bending and axial force can be expressed in a general form, as in the following: K(Mz,My,N)<0 where K(Mz,My,N) represents a non-linear function of internal forces. If the yield surface representing function K is idealized as composed of planes, the linear yield condition XV1I1 is obtained. In this relationship, Mz and Afv are the bending moments about the z and y local axes respectively, N is the axial force and A^A,,/!,,/? are the constant coefficients defining the planes forming the yield surface. The displacements, deformations, internal forces and the corresponding cross-sectional properties which belong to the solution of the TninİTmım weight design problem should satisfy the following conditions and constraints: a) equilibrium equations and compatibility conditions corresponding the second-order limit load, b) yield condition constraints at all sections of the structure, c) displacement and deformation constraints for the second-order limit load, d) constructive restrictions related to the cross-sectional dimensions. In designing a structural system as the second-order limit load equals to factored loads, the axial forces due to the factored loads can be estimated easily. The second-order equilibrium conditions are expressed as linear equations by calculating the stability functions for the estimated axial forces. As mentioned above these equations also cover the geometrical compatibility conditions. The yield condition constraints are expressed in terms of the nodal displacement components, the plastic deformation parameters and the characteristic plastic moments of members. These constraints are linear inequalities since the yield surface is assumed to be composed of planes. The linear displacement and plastic deformation limitations and the limitations on the cross-sectional properties of members constitute the other constraints of the optimization problem. By expressing the structural weight in terms of a linear function of member plastic moments, the minimum weight design problem transforms into a linear programming problem. In this study Simplex method is utilized for the solution of the linear programming problem. Before the solution of the minn-mrm weight design problem by the Simplex method, some pre-calculations are required. These pre-calculations are explained below. i) The determination of the initial feasible solution of the Simplex method requires the calculation of plastic moment variables when the nodal displacements and the plastic deformation parameters are equal to XIX zero. However, the equilibrium equations cannot be satisfied when these variables are equal to zero. Therefore, new plastic deformation parameters which satisfy the equilibrium equations are defined. Thus, by equating these variables to zero, the initial feasible solution is obtained. ii) The variables in the Simplex method must be positive. As the nodal displacements and plastic deformation parameters may have positive or negative values, these variables are expressed as the difference of two positive variables. As explained above, the initial feasible solution or so called initial tableau in the Simplex method is obtained in a simple way which is appropriate to the phenomena of the structural optimization problem, instead of mathematical techniques. After the initial feasible solution is obtained, Simplex method is applied to determine the values of design variables which minimize the objective function.. In certain cases, the structural system may collapse before the second- order limit load due to the excessive displacements, plastic deformations and the large cracks in reinforced concrete structures. The load parameter which causes the structural collapse is defined as the collapse load. The minimum weight design of structures for the collapse load can be performed by the proposed method with certain modifications. In the design phase of the method the equilibrium equations are written by considering the plastic deformation parameters corresponding the collapse mode. Furthermore, displacement and plastic deformation constraints related to the collapse load are added to the constraints of the optimization problem. The proposed optimum design method is also be applied to the minimum weight design of materially and geometrically non-linear plate and shell structures which can be idealized by equivalent frame elements. A computer program called as MWDOS has been developed and coded in FORTRAN-77 programming language for the numerical applications of the proposed method. The minimum weight design of plane and space framed structures for the second-order limit load (or for the collapse load) can be performed by the use of this computer program. As MWDOS computer program is executed in a batch program structure, it can also be utilized for optimum design of large structural systems. Three numerical examples are given to illustrate the minimum weight design method developed herein. In the first example a single story, single bay plane frame is designed by the proposed method and the numerical results are compared with those obtained by a previous study. In the second example, a single story, single bay space frame is designed under different design criteria and the numerical results are compared. In this example the results of minimum weight design for the second-order limit load are given in detail. The third example is devoted to the minimum weight design of a single bay, ten story space frame. XX Numerical examples have shown that the effect of geometrical changes on the equilibrium equations and compatibility conditions corresponding the second-order limit load must be considered in the optimization problem. It is observed that, if the effect of geometrical changes on the equilibrium equations is not considered the factor of safety of the structure decreases 20.3%. Numerical examples have also shown that the optimum solutions obtained with using different initial values are extremely close or almost the same. Based on these results, it can be concluded that, the number of the local optimums in the domain of the feasible solutions are few and their values are close to each other.

##### Açıklama

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1994

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1994

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1994

##### Anahtar kelimeler

Ardışık yaklaşımlar,
Bina bilgisi,
Uzay çubuk sistemleri,
Yükler,
Successive approximations,
Building information,
Space stick systems,
Loads