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Betonarme yapılarda daha güçlü kolon tasarımının lineer olmayan sistem davranışına etkisi

Betonarme yapılarda daha güçlü kolon tasarımının lineer olmayan sistem davranışına etkisi

##### Dosyalar

##### Tarih

1997

##### Yazarlar

Aksoylu, M. Günhan

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### Özet

Yüksek lisans tezi olarak sunulan bu çalışmada betonarme yapılarda daha güçlü kolon tasarımının lineer olmayan sistem davranışına etkisi araştırılmıştır. Altı bölüm halinde sunulan çalışmanın birinci bölümünde konunun tanıtılması, konu ile ilgili çalışmaların ve yönetmeliklerin gözden geçirilmesi, çalışmanın amacı ve kapsamı yer almaktadır. İkinci bölüm, betonarme çubukların lineer olmayan davranışlarının idealleştirilmesine ayrılmıştır. Bu bölümde, önce betonarmenin temel varsayımları verilmiş, daha sonra bileşik eğilme etkisindeki betonarme kesitlerde iç kuvvet-şekildeğiştirme bağıntıları ve bileşik iç kuvvet durumuna ait taşıma güçlerini ifade eden akma koşullan incelenerek bu bağıntı ve koşulların nasıl idealleştirilebileceği açıklanmıştır. Üçüncü bölümde, betonarme uzay çubuk sistemlerde ikinci mertebe limit yükün hesabı ve göçme güvenhğinin belirlenmesi amacıyla geliştirilen ve bu çalışmada yararlanılan bir yük artımı yönteminin dayandığı varsayımlar, yöntemin esasları, formülasyonu ve yöntemin uygulanmasında izlenen yol açıklanmıştır. Yöntemde, düşey işletme yüklerinin bu yükler için öngörülen güvenlik katsayısı ile çarpımından oluşan belirli değerleri altında, aralarındaki oran sabit kalacak şekilde monoton olarak artan yatay yüklere göre hesap yapılarak incelenen sistemin limit ve göçme yükleri hesaplanmaktadır. Dördüncü bölümde, çeşitli boyutlandırma kriterleri tanımlanmıştır. Daha güçlü kolon tasarımını öngören ve bir bölümü deprem yönetmeliklerinde yeralan bu kriterlerin yanında, boyutlandırmada daha güçlü kolon ilkesinin gözönünde tutulmaması ve yapının sadece bir bölüm kolonlarının güçlendirilmesi halleri de ayrıca incelenmiştir. Beşinci bölüm sayısal incelemelere ayrılmıştır. Sayısal incelemeler betonarme konut binalarını temsil etmek üzere seçilen bir taşıyıcı sistem modeli üzerinde gerçekleştirilmiştir. Önceki bölümde açıklanan boyutlandırma kriterlerine göre boyutlandırılan taşıyıcı sistem modelinin malzeme ve geometri değişimleri bakımından lineer olmayan teoriye göre hesabı ile elde edilen sayısal sonuçlar bu bölümde verilerek tartışılmıştır. Altıncı bölümde, bu çalışmada elde edilen sonuçlar açıklanmıştır.

The use of elastic-plastic analysis and design methods, which consider the non-linear behavior of reinforced concrete as well as the non-linearity caused by geometrical changes may result in both more realistic and more economical solutions for reinforced concrete structures. Furthermore, by the use of these methods, the effects of various design philosophies on the non-linear behavior of reinforced concrete structures can also be investigated. In this study, the non-linear behavior of reinforced concrete building frames designed by various strong column-weak beam approaches is examined in detail. Both material and geometrical non-linearities are considered. The thesis consists of six chapters. In the first chapter, after introducing the subject and the related works, the scope and objectives of the study are explained. The objective of this study is to investigate the non-linear behavior of reinforced concrete building frames which are designed according to different strong column- weak beam philosophies imposed by various codes and to determine their safety factors against earthquake loads. For this purpose, a six-story model structure which represents the residential reinforced concrete buildings is designed according to various design philosophies. Then, the designed frames are analyzed by the second-order elastic-plastic theory, under constant gravity loads and monotonically increasing lateral earthquake loads. Finally, by evaluating the Numerical results, strong column-weak beam design philosophies given in various earthquake codes are discussed and compared. The second chapter outlines the non-linear behavior of reinforced concrete frame elements. The investigation covers the actual internal force-deformation relationships, the yield (failure) conditions and the idealization of non-linear behavior. This investigation is based on three basic assumptions made for reinforced concrete, such as a- plane sections remain plane after bending, b- full bond exists between concrete and reinforcing steel, c- tensile strength of concrete is negligible after cracking. The non-linear behavior of reinforced concrete frame elements under biaxial bending combined with axial force is idealized by the ideal elastic-plastic internal force- deformation relationships. This idealization corresponds to the plastic section XV concept. When the state of internal forces at a critical section reaches the ultimate value defined by the yield (failure) condition, plastic deformations occur. The plastic deformations are limited to the rotational capacity. The rotational capacity may be expressed in terms of the length of plastic region and the ultimate plastic curvature. In this study, an approximated yield surface which is composed of 24 linear regions is used for reinforced concrete elements subjected to biaxial bending combined with axial force. In the third chapter, the assumptions, the basic principles and mathematical formulation of the load increment method used in the investigation are presented and the corresponding analysis procedure is explained. The following assumptions and limitations are imposed in the development of the method. a- The internal force-deformation relationships for reinforced concrete frame elements under biaxial bending combined with axial force are assumed to be ideal elastic- plastic. b- Non-linear deformations are assumed to be accumulated at plastic sections while the remaining part of the structure behaves linearly elastic. This assumption is the extension of classical plastic hinge hypothesis which is limited to planar elements subjected to simple bending. c- Yield (failure) conditions may be expressed in terms of bending moments and axial force. In this study, the effects of shear forces and torsional moment on the yield conditions are neglected. d- The plastic deformation vector is assumed to be normal to the yield surface, for the case of biaxial bending combined with axial force. e- The second-order theory may be applied to the analysis of slender structures with high axial forces. In the second-order theory, the equilibrium equations are formulated for the deformed configuration while the effect of geometrical changes on the compatibility equations is ignored. f- Changes in the direction of loads due to deflections are assumed to be negligible. g- The structure is composed of straight prismatic members with constant axial forces. The members which do not meet these requirements can be divided into smaller straight and prismatic segments with constant axial forces. h- Distributed loads may be approximated by sufficient number of statically equivalent concentrated loads. In the load increments method used in this study, the structure is analyzed under factored constant gravity loads and monotonically increasing lateral loads. Thus, at XVI the end of this analysis, the factor of safety against lateral earthquake loads is determined under the anticipated safety factor for gravity loads. When the gravity loads are known, the member axial forces can be easily estimated through equilibrium equations. Thus, the second-order effects are linearized by calculating the elements of stiffness and loading matrices for the estimated axial forces. In this method the structure is analyzed for successive lateral load increments. At the end of each load increment, the state of internal forces at a certain critical section reaches the limit state defined by the yield condition, that is, a plastic section forms. Since the yield vector is assumed to be normal to the yield curve, the plastic deformation components may be represented by a single plastic deformation parameter which is introduced as a new unknown for the next load increment. Besides, an equation is added to the system of equations to express the incremental yield condition for the last formed plastic section. This equation is linear, because the yield surface is approximated to be composed of linear regions. Since the system of equations corresponding to the previous load increment has already been solved, the solution for the current load increment is obtained by the elimination of the new unknown. In the second order elastic-plastic theory, the structure generally collapses at the second order limit load due to the lack of stability. This situation is checked by testing the determinant value of the extended system of equations. If the magnitude of determinant is less than or equal to zero, the second order limit load is reached. Hence, the computational procedure is terminated. In some cases, the structure may be considered as being collapsed due to large deflections and excessive plastic rotations. At each step of the load increments method, a structural system with several plastic sections is analyzed for a lateral load increment. In the mathematical formulation of the method, two groups of unknowns are considered, such as a- nodal displacement components, b- plastic deformation parameters at plastic sections. The equations are also considered in two groups. a- The equilibrium equations of nodes in the directions of nodal displacement components. b- The incremental yield conditions of plastic sections, which express that the state of internal forces at a plastic section remains on the yield surface during a load increment. XVU In the fourth chapter several design criteria, most of them aim various strong column- weak beam approaches, are defined. These are as follows. In the first criterion, the strong column-weak beam design philosophy is not considered in order to obtain a reference solution for the comparison of remaining design criteria. In the second criterion, it is imposed that total moment carrying capacities of the columns must be greater than 1.2 times of total moment carrying capacities of the beams at every joint of the structure for the considered earthquake direction, that is, (Mrt + MA)>1.2(Mrl + Mff) (1) In this relationship, Mrt and M* are the top and bottom column moment carrying capacities, Mri and M^ are the left and right beam moment carrying capacities, respectively. In the third criterion, a part of the story columns are designed by the strong column philosophy for each earthquake direction. The following conditions should be met when choosing the columns which will be strengthened. a- The portion of the lateral loads resisted by the story columns which satisfy equation (1) must be at least 60-70 % of the story shear. b- The eccentricity between storey mass center and the resultant shear force of the columns which satisfy equation (1) must be 7 < 0.20 (2) b in which, b is the building width perpendicular to the earthquake direction and e is the eccentricity mentioned. c- For more economical solution, the strengthening priority must be given to the columns which have bigger rigidity in the earthquake direction considered. The fourth criterion, which is stated in the Turkish Earthquake Code published in 1996, is applied to building frames for which the following condition is satisfied in each earthquake direction and at all stories. Oi=-f£0.70 (3) "ik In this relationship, V^ is the total shear force resisted by the columns which satisfy equation (1) at both ends, and Vjk is the total shear force of the /' th story. XVUl In this approach, the seismic internal forces of columns which satisfy equation (1) are increased by (1/cti) and the remaining columns are designed to resist twice the seismic internal forces. In the fifth criterion, the structural system for which condition (1) is not satisfied at all joints, is considered as having normal ductility level and the corresponding response factor R is applied for earthquake analysis. In the sixth criterion, the contribution of columns which do not meet condition (1) is omitted in the earthquake analysis. This is realized by placing hinges at the column ends for which condition (1) is not satisfied. The fifth chapter is devoted to the numerical investigations. Numerical investigations are carried out on a six-story reinforced concrete model space frame which is selected to represent the low-rise residential buildings. The model structure is designed according to various criteria mentioned in chapter four. Then, the designed frames are analyzed by the second-order, elastic-plastic theory under constant factored gravity loads and monotonically increasing lateral earthquake forces. The analysis is carried out by means of a computer program developed for practical applications of the load increments method given in chapter three. In each analysis, a- the second-order limit loads and collapse loads, b- lateral load parameter versus lateral displacement diagrams, c- total number of plastic sections and their distribution among beams and columns, d- lateral load parameters for the first plastic sections developed in each type structural element are obtained. Besides, the distribution of plastic sections among beams and columns is investigated for each joint. At the end of this chapter, the analytical results obtained for each design approach are compared and discussed. The following conclusions can be drawn from the comparison of the numerical results. a- The safety factor against lateral earthquake loads varies between 1.573 and 2.337 for different design approaches. Considering the load and resistance factors imposed by the codes, these values are found to be higher than required. b- When the amount of total column reinforcement for Design 1 is taken as unity, relative total column reinforcement varies between 1.179 and 1.903 for remaining design approaches. c- Lateral load carrying capacity is reduced by 10.9-13.6 % when second-order effects are considered. d- The plastic rotation capacities are exceeded before the limit load is attained. The ratio of collapse loads to limit loads varies between 0.922 and 0.999. XIX e- First plastic section develops either at or above the working lateral load level. f- The number of column plastic sections is reduced substantially when strong column-weak beam design is adopted. The sixth chapter covers the conclusions. The basic features of the investigation performed in the study and the general evaluation of numerical results are presented in this chapter.

The use of elastic-plastic analysis and design methods, which consider the non-linear behavior of reinforced concrete as well as the non-linearity caused by geometrical changes may result in both more realistic and more economical solutions for reinforced concrete structures. Furthermore, by the use of these methods, the effects of various design philosophies on the non-linear behavior of reinforced concrete structures can also be investigated. In this study, the non-linear behavior of reinforced concrete building frames designed by various strong column-weak beam approaches is examined in detail. Both material and geometrical non-linearities are considered. The thesis consists of six chapters. In the first chapter, after introducing the subject and the related works, the scope and objectives of the study are explained. The objective of this study is to investigate the non-linear behavior of reinforced concrete building frames which are designed according to different strong column- weak beam philosophies imposed by various codes and to determine their safety factors against earthquake loads. For this purpose, a six-story model structure which represents the residential reinforced concrete buildings is designed according to various design philosophies. Then, the designed frames are analyzed by the second-order elastic-plastic theory, under constant gravity loads and monotonically increasing lateral earthquake loads. Finally, by evaluating the Numerical results, strong column-weak beam design philosophies given in various earthquake codes are discussed and compared. The second chapter outlines the non-linear behavior of reinforced concrete frame elements. The investigation covers the actual internal force-deformation relationships, the yield (failure) conditions and the idealization of non-linear behavior. This investigation is based on three basic assumptions made for reinforced concrete, such as a- plane sections remain plane after bending, b- full bond exists between concrete and reinforcing steel, c- tensile strength of concrete is negligible after cracking. The non-linear behavior of reinforced concrete frame elements under biaxial bending combined with axial force is idealized by the ideal elastic-plastic internal force- deformation relationships. This idealization corresponds to the plastic section XV concept. When the state of internal forces at a critical section reaches the ultimate value defined by the yield (failure) condition, plastic deformations occur. The plastic deformations are limited to the rotational capacity. The rotational capacity may be expressed in terms of the length of plastic region and the ultimate plastic curvature. In this study, an approximated yield surface which is composed of 24 linear regions is used for reinforced concrete elements subjected to biaxial bending combined with axial force. In the third chapter, the assumptions, the basic principles and mathematical formulation of the load increment method used in the investigation are presented and the corresponding analysis procedure is explained. The following assumptions and limitations are imposed in the development of the method. a- The internal force-deformation relationships for reinforced concrete frame elements under biaxial bending combined with axial force are assumed to be ideal elastic- plastic. b- Non-linear deformations are assumed to be accumulated at plastic sections while the remaining part of the structure behaves linearly elastic. This assumption is the extension of classical plastic hinge hypothesis which is limited to planar elements subjected to simple bending. c- Yield (failure) conditions may be expressed in terms of bending moments and axial force. In this study, the effects of shear forces and torsional moment on the yield conditions are neglected. d- The plastic deformation vector is assumed to be normal to the yield surface, for the case of biaxial bending combined with axial force. e- The second-order theory may be applied to the analysis of slender structures with high axial forces. In the second-order theory, the equilibrium equations are formulated for the deformed configuration while the effect of geometrical changes on the compatibility equations is ignored. f- Changes in the direction of loads due to deflections are assumed to be negligible. g- The structure is composed of straight prismatic members with constant axial forces. The members which do not meet these requirements can be divided into smaller straight and prismatic segments with constant axial forces. h- Distributed loads may be approximated by sufficient number of statically equivalent concentrated loads. In the load increments method used in this study, the structure is analyzed under factored constant gravity loads and monotonically increasing lateral loads. Thus, at XVI the end of this analysis, the factor of safety against lateral earthquake loads is determined under the anticipated safety factor for gravity loads. When the gravity loads are known, the member axial forces can be easily estimated through equilibrium equations. Thus, the second-order effects are linearized by calculating the elements of stiffness and loading matrices for the estimated axial forces. In this method the structure is analyzed for successive lateral load increments. At the end of each load increment, the state of internal forces at a certain critical section reaches the limit state defined by the yield condition, that is, a plastic section forms. Since the yield vector is assumed to be normal to the yield curve, the plastic deformation components may be represented by a single plastic deformation parameter which is introduced as a new unknown for the next load increment. Besides, an equation is added to the system of equations to express the incremental yield condition for the last formed plastic section. This equation is linear, because the yield surface is approximated to be composed of linear regions. Since the system of equations corresponding to the previous load increment has already been solved, the solution for the current load increment is obtained by the elimination of the new unknown. In the second order elastic-plastic theory, the structure generally collapses at the second order limit load due to the lack of stability. This situation is checked by testing the determinant value of the extended system of equations. If the magnitude of determinant is less than or equal to zero, the second order limit load is reached. Hence, the computational procedure is terminated. In some cases, the structure may be considered as being collapsed due to large deflections and excessive plastic rotations. At each step of the load increments method, a structural system with several plastic sections is analyzed for a lateral load increment. In the mathematical formulation of the method, two groups of unknowns are considered, such as a- nodal displacement components, b- plastic deformation parameters at plastic sections. The equations are also considered in two groups. a- The equilibrium equations of nodes in the directions of nodal displacement components. b- The incremental yield conditions of plastic sections, which express that the state of internal forces at a plastic section remains on the yield surface during a load increment. XVU In the fourth chapter several design criteria, most of them aim various strong column- weak beam approaches, are defined. These are as follows. In the first criterion, the strong column-weak beam design philosophy is not considered in order to obtain a reference solution for the comparison of remaining design criteria. In the second criterion, it is imposed that total moment carrying capacities of the columns must be greater than 1.2 times of total moment carrying capacities of the beams at every joint of the structure for the considered earthquake direction, that is, (Mrt + MA)>1.2(Mrl + Mff) (1) In this relationship, Mrt and M* are the top and bottom column moment carrying capacities, Mri and M^ are the left and right beam moment carrying capacities, respectively. In the third criterion, a part of the story columns are designed by the strong column philosophy for each earthquake direction. The following conditions should be met when choosing the columns which will be strengthened. a- The portion of the lateral loads resisted by the story columns which satisfy equation (1) must be at least 60-70 % of the story shear. b- The eccentricity between storey mass center and the resultant shear force of the columns which satisfy equation (1) must be 7 < 0.20 (2) b in which, b is the building width perpendicular to the earthquake direction and e is the eccentricity mentioned. c- For more economical solution, the strengthening priority must be given to the columns which have bigger rigidity in the earthquake direction considered. The fourth criterion, which is stated in the Turkish Earthquake Code published in 1996, is applied to building frames for which the following condition is satisfied in each earthquake direction and at all stories. Oi=-f£0.70 (3) "ik In this relationship, V^ is the total shear force resisted by the columns which satisfy equation (1) at both ends, and Vjk is the total shear force of the /' th story. XVUl In this approach, the seismic internal forces of columns which satisfy equation (1) are increased by (1/cti) and the remaining columns are designed to resist twice the seismic internal forces. In the fifth criterion, the structural system for which condition (1) is not satisfied at all joints, is considered as having normal ductility level and the corresponding response factor R is applied for earthquake analysis. In the sixth criterion, the contribution of columns which do not meet condition (1) is omitted in the earthquake analysis. This is realized by placing hinges at the column ends for which condition (1) is not satisfied. The fifth chapter is devoted to the numerical investigations. Numerical investigations are carried out on a six-story reinforced concrete model space frame which is selected to represent the low-rise residential buildings. The model structure is designed according to various criteria mentioned in chapter four. Then, the designed frames are analyzed by the second-order, elastic-plastic theory under constant factored gravity loads and monotonically increasing lateral earthquake forces. The analysis is carried out by means of a computer program developed for practical applications of the load increments method given in chapter three. In each analysis, a- the second-order limit loads and collapse loads, b- lateral load parameter versus lateral displacement diagrams, c- total number of plastic sections and their distribution among beams and columns, d- lateral load parameters for the first plastic sections developed in each type structural element are obtained. Besides, the distribution of plastic sections among beams and columns is investigated for each joint. At the end of this chapter, the analytical results obtained for each design approach are compared and discussed. The following conclusions can be drawn from the comparison of the numerical results. a- The safety factor against lateral earthquake loads varies between 1.573 and 2.337 for different design approaches. Considering the load and resistance factors imposed by the codes, these values are found to be higher than required. b- When the amount of total column reinforcement for Design 1 is taken as unity, relative total column reinforcement varies between 1.179 and 1.903 for remaining design approaches. c- Lateral load carrying capacity is reduced by 10.9-13.6 % when second-order effects are considered. d- The plastic rotation capacities are exceeded before the limit load is attained. The ratio of collapse loads to limit loads varies between 0.922 and 0.999. XIX e- First plastic section develops either at or above the working lateral load level. f- The number of column plastic sections is reduced substantially when strong column-weak beam design is adopted. The sixth chapter covers the conclusions. The basic features of the investigation performed in the study and the general evaluation of numerical results are presented in this chapter.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1997

##### Anahtar kelimeler

Betonarme yapılar,
Kolon tasarımı,
Kolonlar,
Reinforced concrete structures,
Column design,
Columns