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Yapı sistemlerinin hesap yöntemlerinin karşılaştırılması gelişigüzel kesitlerin bileşik eğilmeye göre ekonomik hesabı

Yapı sistemlerinin hesap yöntemlerinin karşılaştırılması gelişigüzel kesitlerin bileşik eğilmeye göre ekonomik hesabı

##### Dosyalar

##### Tarih

1991

##### Yazarlar

Kaya, Tuncay

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### Özet

Bu çalışma iki ana bölümden oluşmaktadır. Yapı Sistemlerinin Hesap Yöntemlerinin karşılaştırılması ve Gelişigüzel Kesitlerin Bileşik Eğilmeye Göre Ekonomik Hesabı. Yapı Sistemlerinin Hesap Yöntemlerinin Karşılaştırılması bölümünde, düzlemi içerisinde çeşitli yükler etkisinde bulunan üç açıklıklı bir düzlem çerçeve seçilmiş ve değişik yükleme durumları için farklı hesap yöntemleri kullanılarak statik hesapları yapılmıştır. - Çerçeve önce Yerdeğiştirme Yöntemi Kullanılarak yaklaşık olarak boyutlandırılmıştır. Daha sonra sırasıyla, sabit yükler için Matris- Yerdeğiştirme Yöntemi, Pl, P2 ve P3 hareketli yükleri için Moment dağıtma Yöntemi, W (Deprem) yükü için Matris Kuvvet Yöntemi, mesnet çökmeleri İçin Yerdeğiştirme Yöntemi kullanılarak kesit tesirleri he saplanmıştır. Böylece yöntemlerin birbirleriyle karşılaştırma olanağı doğmuştur. Ayrıca, elde edilen kesit tesirlerinin en elverişsiz kombinezonlarına göre kesit hesapları yapılmış ve Endirekt Deplasman Metodu yardımıyla iki kesite ait M, N, T tesir çizgileri çizilmiştir. Çalışmanın ikinci kısmında, betonarme yapı sistemlerinde karşılaşılabilecek yapı elemanlarının, çeşitli yüklemelerde hem negatif hem de pozitif momentlerin tesir etmeleri halinde, bu yüklemelerdeki Normal kuvvetlerle beraber, çekme ve basınç donatılarıyla ekonomik olarak donatılmasına ait bir çalışma yapılmıştır. özellikle deprem kuvvetlerinin etkili olduğu sistemlerde, yapı elemanlarına depremin yön değiştirmesi sonucu hem negatif hem de pozitif moment etkileyebilir. Kesitlerin çift donatılı tertiplenmesi halinde, hem alta hemde üste konulacak donatı ile verilen M, N ve -M, N kuvvet çiftinin karşılanabilmesi için, en ekonomik donatının bulunması gereklidir. Bu bölümde, bu tür kesitlerin iki yüzüne konulacak donatının hesabı için bir yöntem geliştirilerek, bir bilgisayar programı hazırlanmıştır. Basic Programlama Dilinde kodlanan bilgisayar programı, gelişigüzel kesitli, kiriş, kolon veya perde gibi bir yapı elemanına, iki doğrultuda etkiyen eğilme momenti ve normal kuvvet etkilerine göre, yönet melik gerekleri veya diğer sebeplerle yerleştirilen gövde donatılarının da katkısı alınarak konulması gerekli Esas Donatıları hesaplamak tadır. Taşıma gücü esaslarına göre geliştirilen hesap yönteminde; verilen kesitte önemli olan esas doğrultuda donatı hesabı yapıldıktan sonra mevcut bütün donatılar gözönünde bulundurularak esas doğrultuya dik yönde kesitin verilen normal kuvvetlere bağlı olarak moment taşıma kapasiteleri hesaplanmaktadır. Bu yönde hesaplanan taşıma güçlerinin yetersiz olması halinde bu kez bütün donatılara ek olarak konulması gereken ek donatılar için bu doğrultuda yeniden hesap yapılmakta ve kesit ekonomik bir biçimde donatılmaktadır.

In structural engineering, both safety and economic factor are considered in the design of structures. As it is known, these two basic factors considerably effect each other. Before the use of computer technology in structural engineer ing, safety factor was the most important aspect in the design of structures. Due to the development of structural analysis methods and computer technology, the behaviour of structures is determined more precisely. Therefore, the problem of economical design becomes more important. Because of these reasons, structural engineers use the design methods which consider both material and geometrical non - linearities. This study consists of two major parts: Comparison of Methods of Structural Analysis and Economical Design of Random Reinforced Concrete Sections With Bending Moment and Axial Loads. In the first part, analysis of a three-span reinforced concrete plane frame subjected to various external effects is presented. Different analysis methods have been used for each external loading. Thus, the application and comparison of these methods have been illustrated. At the begining of the first part, the preliminary cross- sectional dimensions of frame have been determined through the utilization of the Slope-Deflection Method. By using the preliminary cross-section dimensions of the members, the main load combinations are effected on the frame; Dead loads, Live Loads and Earthquake Loads. In the analysis of dead load, the length-shortening of the member in the middle is neglected. Then, by using an approximate method, the exact internal forces are calculated. At the third step, internal forces are determined without neglecting the length- shortening of the middle-member. The results of these three seperate calculations are compared and negligence conditions of the length- - VI shortening for the frame are determined. To determine the effect of live loads, three seperate load cases are considered and internal forces are calculated. Finally, the effect of earthquake is analysed for the given sistem, by using W horizantal earthquake force which has been consantrated on the sistem joints. To determine earthquake forces, static equi valance-earthquake coefficent method is used. At the end of this chapter, a sufficient result can be obtained in predesigning of the structural system by decrasing the dimensions of the sections. In the second chapter of the first part, the structure is analysed by the Matrix Displacement Method for dead weight acting on the structure. In the Matrix Displacement Method, the unknowns are the joint translations and rotations. This method is more useful for the system having high degrees of statically indeterminacy. If the number of the members meeting at the the same joints are much, this method is more suitable for solving the system. In other words, the more members meeting at the same joints, the less unknowns. Although, the band width of simultaneous equation is limited and there is no elasticity in choosing the unknowns, generation of the stiffness matrix is usually not difficult because of localized effect. So a displacement of a joint effects only the members meeting at the given joint. Thus it is easy to formulate the Matrix Displacement Method and this method is more suitable for computer programming. In the third chapter of the first part, the structure is analysed by the Cross (Moment Distribution) Method for live loads, PI, P2 and P3 separetly. As it is known, the analysis of statically indeterminate structures, generally, requires the solution of linear simultaneous equations. In the application of the Cross Method to sidesway prevented systems, the unknowns are the end moments. These unknowns are determined through the iterative solution of the moment equilibrium equations of the joints. In the analysis of the sidesway permitted systems, however, additional unknowns such as, independent end displacements have to be considered. By using sidesway prevented system as a primary system, these unknowns are determined by means of the horizontal equilibrium equations. The other end displacement can be expressed in terms of the independent ones and are determined by the compatibility conditions. After PI, P2 and P3 load cases are obtained for each section, the most critical combinations of these load cases are calculated. In the fourth chapter, the structure subjected to lateral Earthquake loads is analysed by the Matrix Force Method. The effect of eartquake is determined by static equivalance earthquake Coefficent Method as consantrated on the joints by the weight of the structure. - vii - In the Matrix Force Method, the unknowns are the forces acting at the ends of the members which have formed the structure. In this method, among the conditions of end forces which satisfy the equilibrium equations, the solution is obtained so that the internal force-deformation relationships and the compatibility conditions are -satisfied. The internal forces and displacements are attained by means of this solution. In this method, first, a number of forces which are aqual to the number of unknowns called the degree of indeterminacy are released. Released forces are either support reactions or internal forces. Due to this property, analysis can be done with lesser unknowns for the systems having more members in a frame. In addition to this, it is possible to obtain equations in which the band width is kept small and system equation is stable, by means of the freedom in choosing unknowns. These equatinos can be written systematically and derived automatically. In the fifth chapter of the first part, the structure subjected to different support settlements is analysed by the Slope- Deflection Method. The unknowns in this method are rotations of the joints and independent translations of the members end. In solving the structure for different support settlements, first, the fixed end moments due to the different support settlements are determined in order to express the equilibrium equations. Therefore,.independent end displacements are taken as zero and the fixed end moments caused by different support settlements are found by means of the compatibility conditions through which the end displacement in the members are determined. At the end of these calculations, the dimensions of the critical cross-sections obtained by the preliminary analysis are checked under the most unsuitable loading conditions. These loading conditions are some combinations which consider different external effects actions in certain proportions according to Turkish Design Code. In this study, it is observed that the most unsuitable loading combinations for critical cross-sections are; 1.0 G+1.2 Q+1.2 T 1.0 G+1.0 Q+1.0 E' 1.4 G+1.6 Q Where, D= Dead loads Q> Live loads T= Different Support settlement loads E= Lateral Earthquake loads - Vlll Considering the most critical combination, each cross-section is reinforced with respect to the Turkish Design Code. In the sixth chapter of the first part, finally, the infuluence lines for bending moment, axial force and shear force of two given sections are obtained by means of the Indirect Displacement Method which is an efficient and reliable method. In the second part of this study, an economical design method for irregular shaped reinforced concrete sections ffected by positi ve and negative bending moments together with axial loads is studied. Especially in our country, sufficent using of resources gets much more important day by day. Developments in computer technology provide structural engineers to solve structural systems easily and find more economical solutions according to the same approximate methods used before, without wasting time. In addition to this, the safety factor for structures can be determined more exactly and this makes the prices lower. Since the price of the structure are proportional to the material used, some economic solutions for structural analysis have to be developed. For the structures to be built in earthquake zones, the sign of bending moments can be changed on the same sections, for the same or different load cases. In this case, reinforcement has to be placed to the both sides of the sections. The rate of the reinforcement to be placed to the both sides of the sections is very important as economical solution. In a reinforced concrete section, without depending on the size of the internal forces acting on the section, there should be minimum reinforcements of which the amount is defined by The Design Codes. These reinforcements which are expressed as a certain percentage of the section area or a certain reinforcemets in a certain space, have to be placed in the section. In ordinary calcu lations, the contributions of these reinforcements to the bearing capacity of the section may be neglected. But in reality, these reinforcements contribute to the bearing capacity of the section. In an economical solution, these reinforcements should be taken into account. If the- earthquake effects are notable, for the sections exposed to the bending moments in both directions, the calculations which are completed considering the most unfavourable effects, may not give an economical result's in many times. Furthermore, the rates of the reinforcements to be placed on the both sides of the section are not known, these rates have to be estimated in the beginning. IX - The structure members effected by axial loads and bending moments in two directions are generally rectangular, "I" or "L" shaped and tubular sections that surround the openning of lift or stair. Especially, in a one axis calculation, the main reinforcements which are placed on the farest end points get close to the neutral axis and get web reinforcements which they can not be neglected in the other direction. In this investigation, an economical reinforcement design method for random shaped reinforced sections exposed to the axial loads and bending moments in both directions are developed taking into consideration the contributions to the bearing capacity of the section of existing reinforcement in the section. Depending on the method proposed, a computer programme is developed in BASIC Programming Language given in Appendix 1. The computer programme first calculates the main reinforcement for important direction depending on the axial loads and two different signed bending moments which affect the section. Then, taking the contributions of these main reinforcements and all existing reinforcements in the section, the program calculates two moment bearing capacities depending on the axial loads in the second direction. If the moment bearing capacity is lesser than given bending moments, program calculates additional main reinforcement for the second direction. All the reinforcements either required by codes or placed for other purpose are being utilized by means of the proposed method staying very close to the safety factors as far as the ultimate strength concerned.

In structural engineering, both safety and economic factor are considered in the design of structures. As it is known, these two basic factors considerably effect each other. Before the use of computer technology in structural engineer ing, safety factor was the most important aspect in the design of structures. Due to the development of structural analysis methods and computer technology, the behaviour of structures is determined more precisely. Therefore, the problem of economical design becomes more important. Because of these reasons, structural engineers use the design methods which consider both material and geometrical non - linearities. This study consists of two major parts: Comparison of Methods of Structural Analysis and Economical Design of Random Reinforced Concrete Sections With Bending Moment and Axial Loads. In the first part, analysis of a three-span reinforced concrete plane frame subjected to various external effects is presented. Different analysis methods have been used for each external loading. Thus, the application and comparison of these methods have been illustrated. At the begining of the first part, the preliminary cross- sectional dimensions of frame have been determined through the utilization of the Slope-Deflection Method. By using the preliminary cross-section dimensions of the members, the main load combinations are effected on the frame; Dead loads, Live Loads and Earthquake Loads. In the analysis of dead load, the length-shortening of the member in the middle is neglected. Then, by using an approximate method, the exact internal forces are calculated. At the third step, internal forces are determined without neglecting the length- shortening of the middle-member. The results of these three seperate calculations are compared and negligence conditions of the length- - VI shortening for the frame are determined. To determine the effect of live loads, three seperate load cases are considered and internal forces are calculated. Finally, the effect of earthquake is analysed for the given sistem, by using W horizantal earthquake force which has been consantrated on the sistem joints. To determine earthquake forces, static equi valance-earthquake coefficent method is used. At the end of this chapter, a sufficient result can be obtained in predesigning of the structural system by decrasing the dimensions of the sections. In the second chapter of the first part, the structure is analysed by the Matrix Displacement Method for dead weight acting on the structure. In the Matrix Displacement Method, the unknowns are the joint translations and rotations. This method is more useful for the system having high degrees of statically indeterminacy. If the number of the members meeting at the the same joints are much, this method is more suitable for solving the system. In other words, the more members meeting at the same joints, the less unknowns. Although, the band width of simultaneous equation is limited and there is no elasticity in choosing the unknowns, generation of the stiffness matrix is usually not difficult because of localized effect. So a displacement of a joint effects only the members meeting at the given joint. Thus it is easy to formulate the Matrix Displacement Method and this method is more suitable for computer programming. In the third chapter of the first part, the structure is analysed by the Cross (Moment Distribution) Method for live loads, PI, P2 and P3 separetly. As it is known, the analysis of statically indeterminate structures, generally, requires the solution of linear simultaneous equations. In the application of the Cross Method to sidesway prevented systems, the unknowns are the end moments. These unknowns are determined through the iterative solution of the moment equilibrium equations of the joints. In the analysis of the sidesway permitted systems, however, additional unknowns such as, independent end displacements have to be considered. By using sidesway prevented system as a primary system, these unknowns are determined by means of the horizontal equilibrium equations. The other end displacement can be expressed in terms of the independent ones and are determined by the compatibility conditions. After PI, P2 and P3 load cases are obtained for each section, the most critical combinations of these load cases are calculated. In the fourth chapter, the structure subjected to lateral Earthquake loads is analysed by the Matrix Force Method. The effect of eartquake is determined by static equivalance earthquake Coefficent Method as consantrated on the joints by the weight of the structure. - vii - In the Matrix Force Method, the unknowns are the forces acting at the ends of the members which have formed the structure. In this method, among the conditions of end forces which satisfy the equilibrium equations, the solution is obtained so that the internal force-deformation relationships and the compatibility conditions are -satisfied. The internal forces and displacements are attained by means of this solution. In this method, first, a number of forces which are aqual to the number of unknowns called the degree of indeterminacy are released. Released forces are either support reactions or internal forces. Due to this property, analysis can be done with lesser unknowns for the systems having more members in a frame. In addition to this, it is possible to obtain equations in which the band width is kept small and system equation is stable, by means of the freedom in choosing unknowns. These equatinos can be written systematically and derived automatically. In the fifth chapter of the first part, the structure subjected to different support settlements is analysed by the Slope- Deflection Method. The unknowns in this method are rotations of the joints and independent translations of the members end. In solving the structure for different support settlements, first, the fixed end moments due to the different support settlements are determined in order to express the equilibrium equations. Therefore,.independent end displacements are taken as zero and the fixed end moments caused by different support settlements are found by means of the compatibility conditions through which the end displacement in the members are determined. At the end of these calculations, the dimensions of the critical cross-sections obtained by the preliminary analysis are checked under the most unsuitable loading conditions. These loading conditions are some combinations which consider different external effects actions in certain proportions according to Turkish Design Code. In this study, it is observed that the most unsuitable loading combinations for critical cross-sections are; 1.0 G+1.2 Q+1.2 T 1.0 G+1.0 Q+1.0 E' 1.4 G+1.6 Q Where, D= Dead loads Q> Live loads T= Different Support settlement loads E= Lateral Earthquake loads - Vlll Considering the most critical combination, each cross-section is reinforced with respect to the Turkish Design Code. In the sixth chapter of the first part, finally, the infuluence lines for bending moment, axial force and shear force of two given sections are obtained by means of the Indirect Displacement Method which is an efficient and reliable method. In the second part of this study, an economical design method for irregular shaped reinforced concrete sections ffected by positi ve and negative bending moments together with axial loads is studied. Especially in our country, sufficent using of resources gets much more important day by day. Developments in computer technology provide structural engineers to solve structural systems easily and find more economical solutions according to the same approximate methods used before, without wasting time. In addition to this, the safety factor for structures can be determined more exactly and this makes the prices lower. Since the price of the structure are proportional to the material used, some economic solutions for structural analysis have to be developed. For the structures to be built in earthquake zones, the sign of bending moments can be changed on the same sections, for the same or different load cases. In this case, reinforcement has to be placed to the both sides of the sections. The rate of the reinforcement to be placed to the both sides of the sections is very important as economical solution. In a reinforced concrete section, without depending on the size of the internal forces acting on the section, there should be minimum reinforcements of which the amount is defined by The Design Codes. These reinforcements which are expressed as a certain percentage of the section area or a certain reinforcemets in a certain space, have to be placed in the section. In ordinary calcu lations, the contributions of these reinforcements to the bearing capacity of the section may be neglected. But in reality, these reinforcements contribute to the bearing capacity of the section. In an economical solution, these reinforcements should be taken into account. If the- earthquake effects are notable, for the sections exposed to the bending moments in both directions, the calculations which are completed considering the most unfavourable effects, may not give an economical result's in many times. Furthermore, the rates of the reinforcements to be placed on the both sides of the section are not known, these rates have to be estimated in the beginning. IX - The structure members effected by axial loads and bending moments in two directions are generally rectangular, "I" or "L" shaped and tubular sections that surround the openning of lift or stair. Especially, in a one axis calculation, the main reinforcements which are placed on the farest end points get close to the neutral axis and get web reinforcements which they can not be neglected in the other direction. In this investigation, an economical reinforcement design method for random shaped reinforced sections exposed to the axial loads and bending moments in both directions are developed taking into consideration the contributions to the bearing capacity of the section of existing reinforcement in the section. Depending on the method proposed, a computer programme is developed in BASIC Programming Language given in Appendix 1. The computer programme first calculates the main reinforcement for important direction depending on the axial loads and two different signed bending moments which affect the section. Then, taking the contributions of these main reinforcements and all existing reinforcements in the section, the program calculates two moment bearing capacities depending on the axial loads in the second direction. If the moment bearing capacity is lesser than given bending moments, program calculates additional main reinforcement for the second direction. All the reinforcements either required by codes or placed for other purpose are being utilized by means of the proposed method staying very close to the safety factors as far as the ultimate strength concerned.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1991

##### Anahtar kelimeler

Taşıma gücü,
Yapı sistemleri,
Yük analizi,
İki eksenli eğilme,
Bearing capacity,
Structure systems,
Load analysis,
Biaxial bending