Elastik zemine oturan betonarme kirişlerin elasto-plastik davranışının incelenmesi yapı sistemlerinin hesap yöntemlerinin karşılaştırılması

Abstract

Bu çalışma iki ana bölümden oluşmaktadır: Elastik Zemine Oturan Betonarme Kirişlerin Elasto-Plastik Davranışının İncelenmesi ve Yapı Sistemlerinin Hesap Yöntemlerinin Karşılaştırılması. Birinci kısımda plastik mafsal hipotezi esas alınarak, elastik zemine oturan sonlu boydaki betonarme kirişlerin davranışı incelenmiş ve ekonomik olarak nasıl boyutlandırılacakları araştırılmıştır. Betonarme sistemlere plastik mafsal hipotezinin uygulanması halinde gözönünde tutulması gereken göçme kriterleri de ayrıca açıklanmıştır. Yapılan sayısal incelemeler, elastik zemine oturan betonarme kirişlerin elasto-plastik hesabı ile, toplam donatı oranının önemli miktarda azaltılabileceğini göstermektedir. İkinci kısımda Yapı Sistemlerinin Hesap Yöntemleri, seçilen iki açıklıklı düzlem çerçevede çeşitli yükleme durumları için farklı hesap yöntemleri kullanılarak karşılaştırılmıştır. önce açı yöntemine göre yapının ön boyutlandırması yapılmıştır. Daha sonra sırasıyla, sabit yükler için Matris Deplasman Yöntemi, P, ve P» ilave yükleri için Cross Yöntemi, W (Deprem) yükü için Açı Yöntemi, düz gün sıcaklık değişmesi için Matris Kuvvet Yöntemi, mesnet çökmeleri için de Cross Yöntemi kullanılarak kesit tesirleri hesaplanmıştır. Ayrıca, Endirekt Deplasman Yöntemi ne göre de iki kesitte M,N,T tesir çizgileri çizilmiştir.

This study which is presented as Master Thesis is composed of two parts: 1- Investigation of elastic-plastic behaviour of concrete beams on elastic foundation. 2- Comparison of Methods of Structural Analysis. The first part of the study consists of six chapters. In the first chapter the subject is introduced and the aim of the study is explained. Design of concrete beams supported by elastic foundation is encountered in many applications of structural engineering. Continuous foundations of structures, foundations of portal cranes, concrete piles under horizontal loads and sheet piles can be given as examples. On the other hand, the idealization of beams on elastic foundation is also used for the analysis of grid systems and rotational shells. As the loads acting on a beam on elastic foundation increases, maximum bending moments, generally occuring under the concentrated, would also increase and after reaching the limiting values, plastic hinges and elastic- plastic deformations would occur in these sections. Since behaviour of beams on elastic foundation under bigger loads is equivalent to small span beams on elastic foundation, after development of plastic hinges, negative bending moments would not increase significantly. This situation indicates that the design methods which consider the elastic-plastic behaviour of the system may result in mo re s o lut ions. In the second chapter, the analysis method of beams on elastic foundation is summarized. In the third chapter, the basis of plastic hinge hypothesis are given by describing the plastic moment, plastic hinge and the determination of limit loads of structural systems are explained. VI The maximum load that a system, can resist according to plastic hinge theory, is called as "limit load". When loads acting on the system reach the limit load, because of plastic hinges occured, the system, partially or totally, goes into mechanism and collapses. As it will be explained later, system may collapse before the limit load is exceeded, because of several reasons, such as buckling, rupture of critical sections, large deformations, displacements and cracks. There are two basic methods which are applied for the calculation of limit load. a- Direct calculation of limit load b- Calculation of the limit load by load increment method. In the fourth chapter, the application of plastic hinge hypothesis to the analysis of concrete beams on elastic foundation is explained. A structural system supported by an elastic foundation does not collapse through mechanism as long as the soil does not collapse. In this situation, a beam on elastic foundation may collapse due to of one of the following reasons : a- Exceeding the bearing capacity of the soil b- Exceeding rotational capacity in plastic hinges c- Occurence of chained plastic hinges d- Occurence of large cracks and displacements. According to load parameter-displacement relation of a beam on elastic foundation, under increasing loads, first plastic hinge is occurred at a critical section for the value of load parameter. Behaviour of the system for bigger values of load.parameter is equal to the behaviour of the system on elastic foundation obtained by introducing a real hinge at the location of plastic hinge and by applying the plastic moment to this hinge as an external load. When one of the collapse mode is reached, the system collapses. This load is called as collapse load of system on elastic foundation. In order to determine the collapse load of the system, analysis of the hinged beam on elastic foundation, is needed. vxx For the calculation of hinged system supported by the elastic foundation. Force method is used and two hinged beams on elastic foundation are taken as sub-systems, unknowns are chosen as shear forces at hinge locations. Internal forces and displacements of the hinged system are determined by superposition equations using the values of unknowns and external loads acting on the main system. The unknowns are determined through the geometrical compatibility equations. For the calculation of displacements and internal forces of the sub-systems, caused by the external loads and unit loadings, a computer programme prepared for the analysis of finite length beams on elastic foundation is used. In the fifth chapter, design methods of concrete beams on elastic foundation are summarized and proposed design approach is explained. Design by elastic analysis of the systems In this method, which is generally used in pratice, statical analysis of the system is carried out according to the elastic theory, but in design calculations ultimate strength theory is used. Top and bottom reinforcement of the beam are determined to resist the positive and negative moments obtained from the statical analysis. Desing by considering the redistribution of bending moments: In continuous foundations where the theory of beams on elastic foundation is applied, positive bending moments under concentrated loads generally have large values relative to negative moments. However, if redistribution of bending moments is considered, it is observed that the increase in negative moments is relatively smaller in magnitude than the decrease in positive moments. This result can be explained by the existence of the elastic soil. As explained above, the consideration of redistribution of bending moments presents more economical solutions for the design of continuous foundations on elastic soil. According to TS 500 Item 7.2.7, in redistribution, maximum bending moment is decreased by 15% and bending moment diagram is re-determined in a way that equilibrium conditions are satisfied. Bending moment diagram corresponding to this situation is determined by the calculation of the system obtained by introducing hinges to point or points where the maximum moments occur, and by applying the external bending moments of (l-0.15)x M max to these points. viii Design of the system according to the elastic-plastic theory: The numerical investigations introduced within the content of this study have shown that a bigger redistribution ratio than 15%, presented by TS 500 conservatively, can be taken when elastic-plastic calculations of concrete beams on elastic foundation are carried out. By this way, more economical solutions for the design of foundations can be made possible. When a concrete foundation on elastic soil is designed according to the elastic-plastic theory, the following proposed steps are applied: 1- Under service loads, system is analyzed according to the elastic theory. Under these loads plastic hinges are required not to develop in the system. Therefore, the sections where maximum bending moments occur, are reinforced in such a way that the first plastic hinge would develop after service loads are exceeded. In continuous foundations subjected to concentrated loads, maximum bending moments usually occur under these loads. Therefore, at the first step of the design, the bottom reinforcement of beam on elastic foundation is determined to satisfy the above condition. 2- The load under which the first plastic hinge is developed is determined by equating the maximum bending moment to M, plastic moment. P * 3- The behaviour of the system with plastic hinges under additional AP loads is equivalent to that of the system obtained by replacing the plastic hinges with real hinges. This system is solved for AP.=1 unit loading and the potential plastic hinge or hinges are searched. 4- The other sections are designed by equating the design loads to the collapse load of the structure. The sixth chapter is devoted to numerical illustrations. Two examples have been carried out. The result of the examples have indicated that the total reinforcement can be reduced significantly when the structure is designed by the elastic-plastic theory as proposed hereby. In the second part, the analysis of a two-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. ix The preliminary cross-sectional dimensions of the frame have been determined through the utilization of the Slope-Deflection Method. At the end of this chapter, a sufficient result can be obtained in the preliminary- design of the structural system by decreasing the characteristic strengths of material in some proportion since only the dead weight and live loads are considered. In the second chapter of the second 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 systems having high degrees of statical indeterminacy. In other words, if systems having more members meeting at joints of the systems, this method supply to operate with lesser unknowns. Although, the band width of simultaneous exuations 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, at is easy to formulate the Matrix Displacement Method and this method is more suitable for computer programming. In the third chapter of this part, the structure is an analysed by the Moment distribution (Cross) Method for live loads PI and P2. As it is known, the analysis of statically indeterminate structures, generally, requires the solution of linear simultaneous equations. In the Moment Distribution (Cross) Method, the unknowns are rotations and translations of the joints. In this method, a part the simultaneous equations which correspond to the joint rotations are solved by using successive iterations. In the fourth chapter, the structure subjected to lateral load is analysed by the Slope-Deflection Method. The unknowns, in this method, are rotations of joints and independent translations of the members ends, as in the Cross Method. The linear simultaneous equations are obtained automatically. In the fifth part, the uniform temperature changes have been taken into account, as an external effect on the structure. Uniform temperature change is the temperature change at centerline of the members. Because of this effect, some internal forces acting on the cross-sections of statically indeterminate structure occur. To determine these forces the sttucture is analysed by the Matrix Force Method. x 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, first, a number of forces released which are equal to number of unknowns (the degree of indeterminacy). Each release can be made by the removal of either support reactions or internal forces. Due to this property, analysis can be made with lesser unknowns for the systems having more mumbers in a frame. In addition, 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 equations, however, are written systematically even they can be derived automatically. The last analysis method to determine the structure subjected to different support settlements is Moment Distribution (Cross) Method. At the end of these calculations, the dimensions of the critical cross-sections obtained from the preliminary analysis are checked under the most unsuitable loading conditions. These loading conditions are some combinations which consider different external effects acting in certain proportions according to Turkish Design Code. In this study, it is observed that the most unsuitable loading condition is obtained from the following combination: 1.4 x G + 1.6 x P where G : Dead Weight P : Live Load In the sixth part of the first part, finally, the influence 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.