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Merkezsel ve dışmerkezsel çapraz elemanlı çerçeve yapıların statik ve deprem yüküne göre optimum tasarımı

Merkezsel ve dışmerkezsel çapraz elemanlı çerçeve yapıların statik ve deprem yüküne göre optimum tasarımı

##### Dosyalar

##### Tarih

1985

##### Yazarlar

Gülay, F. Gülten

##### 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 dışmerkezsel çapraz elemanlı çerçeve yapıların deprem yü küne göre optimum tasarımı için bir çözümleme tekniği geliştirilmiş ve bu ko nuda genel bir bilgisayar programı hazırlanarak çeşitli yapı tipleri üzerin de sayısal uygulama sonuçları sunulmuştur. Matris yerdeğiştirme ve modla- rın süperpozisyonu yöntemleri kullanılarak formüle edilen tasarım proble mi, doğrusal olmayan matematik programlama tekniklerinden ardısıra doğru sal programlama yöntemine uygulanarak, optimum yapı hacmi ile tasarım de ğişkenlerinin hesabı için sistematik bir ardışık yaklaşım yöntemi öneril miştir. Birinci bölümde yapı optimizasyonunda kullanılan yöntemler tanıtıla rak, konu ile ilgili daha önce yapılan araştırmalar özetlenmiş ve çalış manın amacı açıklanmıştır. İkinci bölümde matris yerdeğiştirme ve modi arın süperpozisyonu yön temlerinin temel bağıntıları verildikten sonra, optimizasyon probleminde tasarım değişkenleri, amaç fonksiyonu ve kısıtlamalar açıklanarak, ardı sıra doğrusal programlama yönteminin esası; problemin bu yönteme uygulan ma düzeni ve bunun için doğrusal olmayan kısıtlamaların tasarım değişken lerine göre türev bağıntıları sunulmuştur. üçüncü bölümde konu ile ilgili sayısal uygulama yapmak üzere hazır lanan bilgisayar programının genel akış diyagramı ve işleme düzeni özet lenmiştir. Dördüncü bölümde önce tek katlı, tek açı ki ı ki ı normal, köşegeni i ve dışmerkez uzaklığı değişken çapraz el aman! ı çerçevelerin düşey yük ve dü şey +deprem yükü altında optimum yapı hacimleri hesaplanarak, sonuçlar karşılaştırmalı olarak incelenmiştir. Daha sonra iki katlı ve üç katlı ya pıların optimum tasarım problemi, yine hem düşey, hem de deprem +düşey yüklere göre, normal, köşegenli ve dışmerkezsel çapraz elemanlı çerçeve ler şeklinde alınarak çözülmüş, sonuçlar karşılaştırılmıştır. Yine bu bölümde literatürde haşka yöntemlerle çözülen tek katlı, iki katlı ve sekiz katlı yapı sistemlerinin tezde önerilen yöntemle çözümleri yapılarak, sonuçlar karsı Taştın İmi ştır. Beşinci ve son bölümde ise deprem yapılarının optimum tasarımı için önerilen çözümleme tekniğinin işlerliği ve dışmerkezsel çapraz elemanlı çerçevelerin optimum tasarıma etkileri ile ilgili sonuçlar verilerek, bu konuda ileride yapılabilecek araştırmalar için öneriler sunulmuştur.

In this thesis, a solution technique is presented for the optimi zation of concentrically and eccentrically braced frames under static and earthquake loading and a general computer program is developed for numerical solutions of various types of steel frames. Optimum results for several example problems are also presented and compared with those available in the literature. The optimization problem has been formulated with the matrix disp lacement method and the earthquake loads are calculated by modal super position technique. For the solution of the nonlinear mathematical prog ramming problem, the sequential linear programming technique is utilized which is based on the linearization of nonlinear constraints with Taylor series expansions. An iterative design alogrithm is offered to minimize the total weight or the volume of the structure under equilibrium and behavioral constraints as well as side constraints put on the cross- sectional areas. The structural system is assumed to behave linearly and elastically for the calculation of lateral earthquake loads that are superimposed on static loading for the optimum design at each iteration step. While the static loads of a structure is generally known before design proce dure, the probabilistic and random nature of earthquake loads brings out some compexity to the design problem in addition to its stiffness and time dependent character. VII The actual continuous structural system is idealized as discrete mass system in which the masses are assumed to be concentrated at each story. The overall stiffness matrix of the system is reduced by static condensation, expressing the rotational displacements in terms of trans lations. Undamped free vibrational periods of the structure are compu ted with the solution of the eigenvalue problem and the mode shapes are obtained. The Housner average accelation spectrum curves are employed for avoiding the time dependency of the problem to get maximum values over the time interval for the calculated modal periods and the selected damping ratio. The probable maximum values of lateral seismic load are obtained by superposing the modal contributions of desired modes uti lizing the root-mean-square method. Dynamic analysis is performed for the calculation of lateral earthquake load at each cycle as the stiffness distributions of the structure changes. The design variables are taken to be the cross- sectional areas as well as free nodal displacements of the structure. The moments of inter- tia and section modulus of the elements are assumed to be continiously related to the cross-sectional areas. The expressions for available profile sections used in example problems are derived by least-mean- square method in power functions of area. The design objective is to minimize the total weight or the volume of the structure which is accepted to be a good approximation in the studies of the minimum cost design of steel structures. The displacement and side constraints as well as the objective function can be expressed linearly while the stiffness and stress const raints are nonlinear, so the design problem mathematically turns out to be a nonlinear programming problem. The nonlinear constraints are linearized by taking the first order derivatives of their Taylor Series expansions about the initial design point so as to reduce it to a linear programming problem. Move limits VIII on areas are added to reduce the linearization errors. Then, the result ing linear problem is solved by the Simplex routine. The computed optimum values are used as the new initial design point and the whole process is repeated until the convergence criterion is satisfied which is chosen to be the relative change of objective function in two subsequent iterative cycles should be less than a specified tolerence e. The dynamic analysis is performed at each iteration and the new earthquake loads are computed as the stiffness distribution of the struc ture changes. With this approach the complex procedure of taking the de rivatives of the eigenvalues and eigenvectors is avoided. It is observed that the computed earthquake loads and the natural periods of the struc tural system are also converged as the convergence criterion is statis- fled. The rate of convergence is quite fast compared with the other met hods used in the literature. It is also noticed that the necessary num bers of iteration generally increases with the number of story, but dec reases with lateral rigidity of the structural system, i.e. for the unbraced system 7-8 iterations for single, 12-13 iterations for two, 15- 16 iterations for three story buildings while for the braced systems 5-6 iterations for one, 9-10 iterations for two and three story buildings. In the first chapter of the thesis, the structural optimization problem is introduced, the optimization techniques developed by the earlier researchers are reviewed and the describtion of the present status of the seismic structural optimization is presented. The scope and the purpose of the presented study is also given in this chapter. The current literature survey shows that the sequential linear programming technique is quite successful for the optimum design of fra med structures under static loading but the application of this method to the optimization of dynamically analized seismic frames has not been searched yet. It is also observed in the literature that the optimum design of eccentrically braced frames under earthquake excitation is lacking which offers a perfect system for both being economical over tra ditional moment resisting frames and also satisfying ideal ductility and stiffness requirements in seismic areas. IX For these reasons, the present study is focused on the development of a solution technique for the optimum design of eccentrically and con centrically braced frames under static and earthquake loading and to investigate the effect of change of e/g. ratio (eccentricity to span length ratio) on the optimum structural volume. In the second chapter, the basic equations of the matrix displace ment method is derived and the main theory of the modal superposition technique is outlined for the computation of earthquake loads. Then, the structural optimization problem is formulated and the objective function, stiffness, stress, displacement constraints as well as move limits are explained. The gradient expressions of nonlinear constraints are also given in this chapter. In the third chapter, the computer program developed on IBM 4341, in Fortran language is described and the general flowchart is given. The program consists of a main program called OPTIM and a series of subprog rams attached to it. In the first part of the computation, the initial data is read and stored. In the second part, the lateral earthquake loads are computed by dynamic analysis and modal superposition technique. In the third part of the program, the static analysis is carried out and the no dal displacements are calculated. In the fourth part, the objective func tion and the constraints are derived and after the nonlinear stiffness and stress constraints are linearized, the coefficients are arranged in a sui table format to be applicable for Simplex routine. In the fifth part of the program, the linear programming problem is solved by Simplex method. In the fourth chapter, the results of the numerical solutions are presented in a comparative manner. One bay single-story frames of unbra ced, concentrically and eccentrically braced frames with different eccen tricity to span length ratio are solved for static and earthquake loads and the results are compared. 20.8% economy in material is shown to be ob tained by adding the diagonal elements to the one story frame under com bined earthquake loads. This economy over the normal moment resisting frame is about 19.8% for e/g=0.15, 13.5% for e/£=0.20, 10.5% for e/£=0.30 and 2.83% for e/j=0.40 which is noticed to decrease as the eccentricity length ratio (e/g ). increases. One bay two and three story unbraced, con centrically and eccentrically braced frames for static and seismic ex- citation are also solved and the optimum volumes of the structures are compared in this chapter. The example problems available in literature are also solved with the proposed algorithm and the results are compared. One story one-bay, one story three-bay structures and two story one-bay structures /designed by other methods are solved with the presented design technique. The two and eight story shear frames solved by method of feasible direction 1n the literature are also solved by the proposed method. All of the results obtained are found to be statisfactory (within 10%). In the fifth and last chapter the final results of the research work and the recommendations for further studies are presented. In conclusion: 1) The proposed methodology for the optimum design of seismic frames is proved to be an efficient and reliable technique. 2) The optimum volumes computed in this work are all very close to the ones obtained in the earlier studies and the number of iterations are in most cases lower than those which proves the efficiency of the proposed design technique. Also the results being irrespective of the choice of the initial design point displays another practical advantage over some other optimization techniques. 3) The braced frames are much more economical than the normal mo ment resisting frames under earthquake loading condition. This economy is being 21% for single,52.7% for two and 51.7% for three story concentri cally braced frames of the same span length. 4) The eccentrically braced frame provides an efficient structural system for seismic load with its high moment absorbing capacity and duc tile behaviour in addition to the amount of economy gained in comparision with normal frames. The use of this type of frames gives the opportunity to the design engineer to select a suitable structural system which is stiff enough to carry static and dynamic loads and to minimize structu- ral damage ana also ductile enough so as ncrfc-fco-ccrHapse under strong ground motion.

In this thesis, a solution technique is presented for the optimi zation of concentrically and eccentrically braced frames under static and earthquake loading and a general computer program is developed for numerical solutions of various types of steel frames. Optimum results for several example problems are also presented and compared with those available in the literature. The optimization problem has been formulated with the matrix disp lacement method and the earthquake loads are calculated by modal super position technique. For the solution of the nonlinear mathematical prog ramming problem, the sequential linear programming technique is utilized which is based on the linearization of nonlinear constraints with Taylor series expansions. An iterative design alogrithm is offered to minimize the total weight or the volume of the structure under equilibrium and behavioral constraints as well as side constraints put on the cross- sectional areas. The structural system is assumed to behave linearly and elastically for the calculation of lateral earthquake loads that are superimposed on static loading for the optimum design at each iteration step. While the static loads of a structure is generally known before design proce dure, the probabilistic and random nature of earthquake loads brings out some compexity to the design problem in addition to its stiffness and time dependent character. VII The actual continuous structural system is idealized as discrete mass system in which the masses are assumed to be concentrated at each story. The overall stiffness matrix of the system is reduced by static condensation, expressing the rotational displacements in terms of trans lations. Undamped free vibrational periods of the structure are compu ted with the solution of the eigenvalue problem and the mode shapes are obtained. The Housner average accelation spectrum curves are employed for avoiding the time dependency of the problem to get maximum values over the time interval for the calculated modal periods and the selected damping ratio. The probable maximum values of lateral seismic load are obtained by superposing the modal contributions of desired modes uti lizing the root-mean-square method. Dynamic analysis is performed for the calculation of lateral earthquake load at each cycle as the stiffness distributions of the structure changes. The design variables are taken to be the cross- sectional areas as well as free nodal displacements of the structure. The moments of inter- tia and section modulus of the elements are assumed to be continiously related to the cross-sectional areas. The expressions for available profile sections used in example problems are derived by least-mean- square method in power functions of area. The design objective is to minimize the total weight or the volume of the structure which is accepted to be a good approximation in the studies of the minimum cost design of steel structures. The displacement and side constraints as well as the objective function can be expressed linearly while the stiffness and stress const raints are nonlinear, so the design problem mathematically turns out to be a nonlinear programming problem. The nonlinear constraints are linearized by taking the first order derivatives of their Taylor Series expansions about the initial design point so as to reduce it to a linear programming problem. Move limits VIII on areas are added to reduce the linearization errors. Then, the result ing linear problem is solved by the Simplex routine. The computed optimum values are used as the new initial design point and the whole process is repeated until the convergence criterion is satisfied which is chosen to be the relative change of objective function in two subsequent iterative cycles should be less than a specified tolerence e. The dynamic analysis is performed at each iteration and the new earthquake loads are computed as the stiffness distribution of the struc ture changes. With this approach the complex procedure of taking the de rivatives of the eigenvalues and eigenvectors is avoided. It is observed that the computed earthquake loads and the natural periods of the struc tural system are also converged as the convergence criterion is statis- fled. The rate of convergence is quite fast compared with the other met hods used in the literature. It is also noticed that the necessary num bers of iteration generally increases with the number of story, but dec reases with lateral rigidity of the structural system, i.e. for the unbraced system 7-8 iterations for single, 12-13 iterations for two, 15- 16 iterations for three story buildings while for the braced systems 5-6 iterations for one, 9-10 iterations for two and three story buildings. In the first chapter of the thesis, the structural optimization problem is introduced, the optimization techniques developed by the earlier researchers are reviewed and the describtion of the present status of the seismic structural optimization is presented. The scope and the purpose of the presented study is also given in this chapter. The current literature survey shows that the sequential linear programming technique is quite successful for the optimum design of fra med structures under static loading but the application of this method to the optimization of dynamically analized seismic frames has not been searched yet. It is also observed in the literature that the optimum design of eccentrically braced frames under earthquake excitation is lacking which offers a perfect system for both being economical over tra ditional moment resisting frames and also satisfying ideal ductility and stiffness requirements in seismic areas. IX For these reasons, the present study is focused on the development of a solution technique for the optimum design of eccentrically and con centrically braced frames under static and earthquake loading and to investigate the effect of change of e/g. ratio (eccentricity to span length ratio) on the optimum structural volume. In the second chapter, the basic equations of the matrix displace ment method is derived and the main theory of the modal superposition technique is outlined for the computation of earthquake loads. Then, the structural optimization problem is formulated and the objective function, stiffness, stress, displacement constraints as well as move limits are explained. The gradient expressions of nonlinear constraints are also given in this chapter. In the third chapter, the computer program developed on IBM 4341, in Fortran language is described and the general flowchart is given. The program consists of a main program called OPTIM and a series of subprog rams attached to it. In the first part of the computation, the initial data is read and stored. In the second part, the lateral earthquake loads are computed by dynamic analysis and modal superposition technique. In the third part of the program, the static analysis is carried out and the no dal displacements are calculated. In the fourth part, the objective func tion and the constraints are derived and after the nonlinear stiffness and stress constraints are linearized, the coefficients are arranged in a sui table format to be applicable for Simplex routine. In the fifth part of the program, the linear programming problem is solved by Simplex method. In the fourth chapter, the results of the numerical solutions are presented in a comparative manner. One bay single-story frames of unbra ced, concentrically and eccentrically braced frames with different eccen tricity to span length ratio are solved for static and earthquake loads and the results are compared. 20.8% economy in material is shown to be ob tained by adding the diagonal elements to the one story frame under com bined earthquake loads. This economy over the normal moment resisting frame is about 19.8% for e/g=0.15, 13.5% for e/£=0.20, 10.5% for e/£=0.30 and 2.83% for e/j=0.40 which is noticed to decrease as the eccentricity length ratio (e/g ). increases. One bay two and three story unbraced, con centrically and eccentrically braced frames for static and seismic ex- citation are also solved and the optimum volumes of the structures are compared in this chapter. The example problems available in literature are also solved with the proposed algorithm and the results are compared. One story one-bay, one story three-bay structures and two story one-bay structures /designed by other methods are solved with the presented design technique. The two and eight story shear frames solved by method of feasible direction 1n the literature are also solved by the proposed method. All of the results obtained are found to be statisfactory (within 10%). In the fifth and last chapter the final results of the research work and the recommendations for further studies are presented. In conclusion: 1) The proposed methodology for the optimum design of seismic frames is proved to be an efficient and reliable technique. 2) The optimum volumes computed in this work are all very close to the ones obtained in the earlier studies and the number of iterations are in most cases lower than those which proves the efficiency of the proposed design technique. Also the results being irrespective of the choice of the initial design point displays another practical advantage over some other optimization techniques. 3) The braced frames are much more economical than the normal mo ment resisting frames under earthquake loading condition. This economy is being 21% for single,52.7% for two and 51.7% for three story concentri cally braced frames of the same span length. 4) The eccentrically braced frame provides an efficient structural system for seismic load with its high moment absorbing capacity and duc tile behaviour in addition to the amount of economy gained in comparision with normal frames. The use of this type of frames gives the opportunity to the design engineer to select a suitable structural system which is stiff enough to carry static and dynamic loads and to minimize structu- ral damage ana also ductile enough so as ncrfc-fco-ccrHapse under strong ground motion.

##### Açıklama

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1985

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1985

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1985

##### Anahtar kelimeler

İnşaat Mühendisliği,
Civil Engineering