Bazı düzensizlikler içeren üç boyutlu büyük yapı sistemlerinin doğrusal olmayan çözümlemesi

Abstract

Bazı düzensizlikler içeren üç boyutlu büyük yapı sistemlerinin malzeme ve geometri değişimi bakımından doğrusal olmayan çözümlemesi ve gerçekleşen sistem sünekliğinin belirlenmesi için bir algoritmanın geliştirildiği bu çalışma, üç ana bölüm ile yapılan deneyleri konu alan ekleri içermektedir. Yapı sisteminin boyutlandırılmasında kullanılan deprem yükleri sistem süreklik düzeyiyle ilişkilidir. Hesabın başlangıcında öngörülen sistem süneklik düzeyinin ne ölçüde gerçekleştiğim belirlemek için, doğrusal olmayan çözümlemeye gereksinim vardır. Bu amaçla geliştirilen algoritma; her adımda doğrusal hesap yapılması, problemin genel bilinmeyen sayısının az tutulması, basit bilgisayar programlarının etkin kullanımı, gerçek yapı davranışını daha iyi yansıtabilecek mekanik modellerin oluşturulması ve kullanılması gibi ana esaslara oturtulmuştur. Önerilen hesap yönteminde, üç boyutlu yapı sistemi birbirlerine göre dik olarak yerleşik yatay ve düşey konumlu düzlem alt sistemlere ayrılmakta; her tür yapısal düzensizlik, düzlem alt sistemleri oluşturan çubuk elemanlar düzeyinde dikkate alınabilmektedir. Sarılmış betona ait gerilme-şekildeğiştirme ilişkisinin, olayı etkileyen en önemli değişkenleri içeren bir modelle ifadesinin ardından bileşik eğilmeye maruz genel betonarme kesitte bünye bağıntısı oluşturulmaktadır. Kesit dönme sünekliği bünye bağıntısından elde edilen önemli bir büyüklüktür. Doğrusal olmayan şekildeğiştirmelerin, çubuklar üzerinde iç kuvvet dağılımıyla uyumlu olarak yayılması durumu gözönüne alınmaktadır. Bir alt sistem olarak değerlendirilen çubuk eleman yeter sayıda alt parçaya bölünüp, her parçadaki ortalama iç kuvvetten sanal rijitliğe geçilmektedir. Değişken rijitliklikli çubukta birim yerdeğiştirme ve yükleme sabitlerinin hesabı için konsol mesnetlenmiş elemandan yararlanılmakta ve ardışık başvuru bağıntıları kullanılmaktadır. Döşemelerinde boşluklar bulunan büyük yapı sistemlerinde serbest titreşim hesabının önerilen algoritmayla yapılması, genel programlara göre önemli bir üstünlük getirmektedir. Gerçekleştirilen üç grup deneysel çalışmadan birincisi, bileşik eğilme etkisindeki 1/1 ölçekli betonarme kolonları; ikincisi, bölme duvarlarının yapı davranışına olan etkilerini ortaya çıkarmak üzere değişik inşaat aşamalarındaki gerçek binalar üzerinde yapılan küçük titreşim ölçümlerini; üçüncüsü ise 1/2 ölçekli bölme duvarlı ve duvarsız betonarme düzlem çerçevelerin iki yönlü yatay yükler etkisindeki davranışım konu almaktadır. Üçüncü grup deneysel çalışmanın ana amacı bölme duvarı ile betonarme çerçeve arasında beton kayma kamaları meydana getirerek oluşturulan, özel bölme duvarlı çerçevenin yatay yükler etkisindeki davranışını incelemektir. Deneysel ve kuramsal sonuçlar karşılaştırılarak aralarındaki uyum gösterilmiştir.

A general algorithm for the analysis of 3D large structures with certain irregularities such as discontinuities in floor diaphragms and big openings in shear walls has been proposed to take into account all kind of member irregularities including geometrical and material nonlinearities. Although the algorithm has developed to analyze three dimensional structures made of steel or reinforced concrete, special attention has been exercised on the overall behavior of reinforced concrete structures. The computer programs developed for this purpose can be feeded by the data obtained experimentally. The algorithm proposed in this work has not been only justified by theoretical comparisons but also by the results achieved experimentally. A versatile tool has been obtained at the end which can be used not only for design purposes but to criticize some of the design requirements taking place in the earthquake codes. The computer program developed herein is applicable to small size computers and can easily be adopted to the parallel computation techniques. The general purpose program called DOC3B developed here has been mainly based on two level substructuring technique. In the first level, beam elements are dealt with. And in the second level of substructuring, planar two dimensional frames and/or floors are dealt with. The following paragraphs are devoted to the summaries of the three major chapters and their complementary parts given in appendices. The basic definitions of sectional and structural ductilities are given in the first chapter together with the constitutive model chosen for plain and confined concrete. Structural ductility which is the major parameter to define the earthquake loads reduction factor R is strongly pertinent to the sectional ductilities and is selected at the very beginning of the aseismic design. A nonlinear analysis is needed to check at the end of design whether the R value chosen is reached or not, the ductility demands are provided by the existing sectional dimensions and configurations of reinforcement bars and confinement or not. This is why the constitutive model of concrete which carries the amount of volumetric confinement into analysis has been chosen in the xvm computer program developed. The details of this model is given in chapter one together with a short review of other models available in the literature. The second chapter of this work which covers essentially the nonlinear analysis of two dimensional planar systems with all kind of irregularities, consist of three parts. In the first part of the second chapter a computer program called M-KAPA has been developed to take into account sectional disordernesses and material nonlinearity. For this purpose laminas with different characteristics are defined within the total height of the section. Namely the characteristics of concrete and reinforcement configuration can be considered in this program and moment-curvature relationship can be obtained automatically for the section. This nonlinear relationship has been linearized using the modified initial slope technique, Figure 1. Moment Mm Initial Stiffness M1"1 % Curvature Xmax Figure 1 Modified Initial Slope Method For Linearazition Using this technique one can easily obtain the effective flexural rigidity of the portion of the beam which is represented by the help of moment-curvature relation derived at the beginning of the analysis. In fact the algorithm can be modified so that only the necessary points which are corresponding to the observed value of bending moment can be calculated by the program M-KAPA and the effective flexural rigidity can be defined accordingly. Beam elements which are considered as the first level substructures are divided into smaller fictitious parts for which the flexural characteristics are calculated as it is explained in the previous paragraph. Since elements are divided to small fictitious parts all kind of irregularities can be taken into account at this preparatory stage of the analysis. It has to be kept in mind that this is done only when and where it is necessary. The irregularities which can be taken into account are as follows; /. Any kind of change in height and width of section including rigid ends, ii. Second order effects of axial force, Hi. Shear deformations together with flexural deformations. The deflected shape of the base element which has been chosen as a cantilever is obtained XIX and some fictitious forces are defined to account the above listed features. Two recurrence formulae which are suitable for quick hand calculations are given to obtain the necessary deflected shapes of base cantilever beams. And after having calculated the terms of flexibility matrix, the rigidity matrix is obtained by simple inversion of flexibility matrix. Loading terms and fixed end forces are all calculated using the stiffiiess matrix and the necessary edge displacement. The computer program called DOC2B which consist of two main sub programs has been developed for the nonlinear analysis of two dimensional structural systems which are going to be considered as the second level substructures in the preceding chapter three. The program named DOC2B-1 utilizes and composes the stiffness matrices which are prepared by the sub programs DOC2B-2 developed for this purpose. Composed global stiffness matrix is stored in one dimensional array taking into account the symmetry and half band width. And it can be organized so that the reduced stiffness matrices of two dimensional substructures can be reached. Several graphical features have been added to the program so that the internal forces, curvature distributions or curvature ratios with respect to certain curvature along the members can be displayed. There are several important features of the program DOC2B which can be feeded by ready moment-curvature relationships, force-displacement relationship which might be experimentally obtained and new moment-curvature relationship based on the effective axial force which is valid at a particular load increment, calculates the ultimate load corresponds to any local or global failure. Three type of executions can be identified for the program DOC2B which is able to combine the effects of vertical and horizontal loadings; i. Execution for a specified vertical and lateral single load case, /'/'. Execution for lateral load increments keeping the vertical loads constant. Lateral load-specific displacement relationship can be achieved at the end of this analysis. Hi. Execution for the critical load which corresponds to a certain curvature or structural ductility. The vertical loads are kept constant in this analysis even though it is not a restriction for the main logic of the program. One bay five story steel structure which had already been analyzed by other researchers has been used for the justification of the results produced by DOC2B. Lateral load factor-top displacement curves obtained by three different programs and different concepts are compared and very good agreement has been observed, see Page 53. Also, this comparative work has included a parametric study to find out the relative importance of some parameters on the top deflection and the number of successive approximations. The selected parameters are the amount of subdivisions on the elements, the number of points which are used to apply recurrence formulae on the elements and required stiffness approaching ratios, respectively. The parametric study shows that the first two parameters have to be increased and the third one be decreased after reaching the yielding level of the moment-curvature relationship. The theoretical analysis of a reinforced concrete cantilever which had been tested for constant axial and monothonicly increased lateral loads in the laboratory has been XX compared with experimentally found moment curvature and load-top displacement curves. And very good agreement has been observed, see Page 59. It means that the selected constitutive model, the assumptions made for the theoretical analysis are good enough to be used for predictions. The details of the experimental work carried out are given in Appendix A. The measured and calculated structural ductilities are compared in a table, see Page 61. Another experimental work carried out on a 1/2 scale reinforced concrete frame subjected to displacement reversals has been used to verify the theoretical results obtained through the computer program DOC2B. The results are in very good agreement in the lower rate of loadings and higher level of displacements, see Page 67. However theoretical results do not coincide very much with the experimental results around the yielding level of reinforcement. This may be because of the lack of bond between the concrete and reinforcement at that stage, which has not been reflected into the theoretical models. Since the initial slopes, ultimate loads and displacements are in good agreement the proposed model can be used for predictions. Theoretically and experimentally found plastified regions are good agreement as well, see Page 68. The details of this experimental work can be found as a part of Appendix C. Another theoretical work carried out on a similar nonsymmetric frame, to have better understanding about the length of plastified zone. The results indicate that loading and configuration of the structure or in more general terms the distribution of internal forces are effective on the lengths of plastified zones. One of the strength upgrading technique of an existing structure is shear wall adding to the building. An existing wall can be modified using special techniques to resist higher lateral loads, instead. As another alternative a partitioning wall can be prepared so that it could have a chance to act with the peripheral reinforced concrete element and the integrated structure can carry higher lateral loads. A group of pilot tests which are summarized in Appendix C has been carried out in the laboratory to show the importance of the integration of partitioning wall to frame. The theoretical findings of this experimental work has been utilized in the last example taking place in chapter two of this work. Experimentally found load-deflection curves have been implemented into the program DOC2B by means of a fictitious flexural rigidity which contains the shear deformations. Integrated walls are represented in the computer program by means of another approach observing from the test results that the shear resistance between the wall and reinforced concrete elements are perfect. The results achieved by means of these two approaches are in good agreement, see Page 76. Taking this opportunity the shear wall orientation has been changed and the behavior of two planar structures strengthened by two different shear walls have been observed theoretically. And it has been indicated that in the case of shear wall-weak beam connection, shear strength of the structures are restricted by the existing flexural ductility of weak beam elements, see Page 77. It is very well known that the amount of unknowns are tremendously increased as soon as the inplane deformations of slabs are taken into consideration during the 3D analysis of a building system subjected to both vertical and lateral loading. Most of the computer programs which are available are not capable enough to analyze the XXI structural systems even in the elastic range unless their slabs are assumed as rigid diaphragms. However not only the big openings of slabs jeopardize this assumption done for the plane rigidity of slabs but the nonlinear behavior of slabs also. An algorithm proposed in Chapter 3 enables the nonlinear analysis of 3D structures with flexible slab diaphragms. The proposed algorithm which is based on two level of substructuring technique utilizes mainly the reduced lateral rigidity matrices of planar elements such as lateral load carrying vertical frames, shear walls or shear walls with openings and lateral load distributing slabs and slabs with openings. There exist too many simplified methods or algorithms to obtain the reduced lateral rigidity matrices of substructures. In the program called DOC3B, the computer program DOC2B which has already mentioned in chapter 2 is being used as a subprogram to derive the lateral rigidity matrices of slabs and lateral load resisting substructures. DOC3B combines effectively the matrices supplied by DOC2B and solve the equations at each level of loading. All the characteristics described for DOC2B are valid for DOC3B. Static equilibrium equations used for this algorithm can be modified to dynamic equilibrium equations simply by adding the inertia forces and damping forces. It has been shown in this chapter that these equations can be solved readily for free vibrational characteristics of the 3D building systems. Even the general formulation can be simplified omitting some of the unimportant interaction terms between the coupled rigidities. All the proposed algorithms have been verified and used to demonstrate the importance of flexible diaphragm action. The achieved accuracy and efficiency of the proposed algorithm has been proven by means of the first example enclosed to this chapter. A three storey, 3 by 8 bays structure has been analyzed both by DOC3B and very well known computer program SAP90, and the results are compared. The following conclusions are achieved; /'. The differences between the results are negligible, the assumption made in the proposed algorithm are acceptable, ii. SAP90 is able to analyze the structures with 10000 unknowns or less for static loadings and the efficiency drops down for free vibrational analysis. On the other hand DOC3B has practically no limits for both type of analysis, Hi. Inplane elastic deformations of slabs may become very effective on the overall structural behavior of 3D buildings, iv. The inplane flexibility of slabs can be controlled by means of peripheral beam elements. Another numerical example to show the importance of inelastic deformations of slabs has been prepared, which can not obviously be analyzed by SAP90. Once again it has been observed in this analysis that if the slab elements undergo to plastic deformations, the shear walls which are very important in elastic analysis becomes unimportant elements as for as the lateral loads are concerned. In the last numerical 3D example which has been designed according to the Turkish Earthquake Code of 1975 analyzed previously by another program which is based on plastic section assumption, DOC3B has been tested once again. The following results are achieved at the end of comparations, i. Load-deflection curves of pushover analyses by two programs are in good agreement, see Page 126, //. The actual behavior of the 3D structure without any deformation restrictions can be followed up easily by DOC3B. Doing that some higher displacement ductilities are reached. And even higher ductilities can be achieved introducing higher volumetric ratios for XXll confinements in critical regions. This means that the required overall ductility can be controlled in a certain extent. Appendix A has been devoted to the details of an experimental work for which theoretical predictions have been prepared by the help of computer program mentioned in Chapter 2, DOC2B. Doing that the test results are compared with theoretical results to justify the chosen constituve model for concrete and to test the program. Three columns with 1/1 scale has been prepared in Structural and Earthquake Engineering Laboratory of ITU and tested for monothonically increased displacements. The results shows that both the moment-curvatures and load- displacement curves are in good agreement, which means that the preferences and assumptions done at the beginning of programming are acceptable. Very brittle bricks are widely used in local practice to construct the partitioning walls of low rise reinforced concrete structures which have dual actions on the vibrational characteristics of the buildings. They are effective on the lateral rigidities of structures and on the mass of the structures. Even they have a certain amount of shear strength which becomes important if the concrete quality of structures reduce for any reason. Plastering on brick wall are also effective on the structural behavior in the same direction. In order to quantify all this effects an experimental preliminary study has been carried out in the field on three building with identical structural systems. Micro tremor measurements have been picked up and evaluated by means of a specially developed computer program called MIC I. Test results which have initiated the experimental program outlined in Appendix B, indicated that contribution of walls to the stiffness is larger than to the mass of structure, because the first free vibrational periods of structures are getting relatively smaller after the construction of either plane or plastered walls. The contribution of walls to the shear strength of integrated wall-frame system has been launched in the laboratory by 7 early experiments. The experimentally produced load-displacement diagrams which are already implemented into the computer program DOC2B, have been compared in Appendix C which contains all the details of testing program. The most spectacular part of this program is related to the specimen which has prepared referring to a local practice in which the brick walls are constructed first and then the peripheral reinforced concrete elements are casted. In the laboratory the brick wall constructed so that concrete and brick shear connectors obtained at the interfaces of concrete and brick masonary domains. It has been proven at the end of the tests that these ordinary shear connectors made of the same materials always used in practice are enough to integrate the partitioning wall to the reinforced concrete frame. The integrated system has not only better behavior against earthquake loads but the total shear strength of composite action is higher than the summation of individual shear capacities. For any easy upgrading and strengthening purposes wire meshes have been mounted on two sides of the tested infill frame and coated by simple plaster with higher cement/water ratio and tested. Approximately 30% of increment in stiffness has been achieved. The expected strength increment which is obvious will be tested after having improved the loading capacity of testing setup.