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Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2005, Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2005, Bu tez çalışmasında günümüzde gelişmekte olan ve uygulamada yaygınlaşmaya başlamış doğrusal olmayan analiz yöntemlerinden olan statik-itme (pushover) analizi anlatılmaya çalışılmış, konunun anlaşılması içinde mevcut bir yapı ve bu yapının güçlendirilmiş durumunun performans sonuçları değerlendirilmiştir. Birinci ve ikinci bölümde yöntemin genel açıklaması ve ek bilgiler verilmeye çalışılmıştır. Üçüncü, dördüncü ve beşinci bölümde yöntemin hesap prosedürleri anlatılmaya çalışılmış, yedinci bölümde ise yöntemin anlaşılması için örnek bir okul binası mevcut ve güçlendirilmiş durum olmak üzere SAP2000 Programı yardımıyla ATC40 Yöntemi kullanarak tahkik edilmiş ve sonuçlar incelenmiştir. Statik itme analizi genel anlamıyla tanımlanan malzeme ve kesit özelliklerine sahip yapının belirli bir yük dağılımıyla adım adım deplasman yapmaya zorlanması ve bu zorlama sürecinde yapının genel ve eleman bazında uygulanan yük dağılımına ne şekilde cevap verdiğinin incelenmesidir. Statik-İtme analizinde ilk aşama olarak yapının hangi yer hareketi için hangi performans seviyesinde cevap vermesi gerektiği öngörülür ve analiz sonucunda bu öngörünün gerçekleşmesi beklenir. İkinci aşamada ise uygulanan yük dağılımı sonucu elde edilen kapasite eğrisi yani yapının depreme verebileceği cevap hesaplanır. Kapasite eğrisi, deprem spektrumuyla yani deprem hareketinin yapıdan karşılamasını istediği eğri ile aynı eksen takımında kesiştirilir ve sonuçlar bu kesişim noktası için incelenir. Bu tez çalışmasında doğrusal olmayan analiz yöntemlerinden olan statik-itme analizi genel olarak ATC40 Dökümanında belirtilen yöntemlere dayanılarak anlatılmaya çalışılmış bir bölüm olarak ta FEMA356 Dökümanında belirtilen yöntem incelenmiştir., In this thesis work, the non-linear process of pushover analysis, which is progressing and has started to be used ever more widely has been tried to be explained, and an examplary structure with its retrofitted form has been analyzed for its performance results. In the first and the second sections, a general explanation of the method has been given. In the third, fourth and the fifth sections the calculation procedures of the method has been tried to be explained; and in the seventh section in order to give a clearer understanding of the method, an examplary school building has ben investigated in its original and strenghtened states with help of SAP2000 program and ATC40 method. The pushover analysis in general terms consists of; identification of a building in terms of its materials and section. The structure is then forced to step by step displacement with a given weight distribution and then the structure is investigated during this period of forcing. The first step in the pushover analysis is to determine against which magnitute of ground motion will the structure answer to and at which performance level. By the end of the analysis this determined level of performance is expected to be performed. In the second step, the capacity curve, the capacity of the structure to withstand earthquake, resulting from the performed weight distribution is computed. The capacity curve, is intersected with the desired level of resistance, i.e. the earthquage spectrum, on the same coordianate plane. The result of this intersection point is analyzed., Yüksek Lisans, M.Sc.

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2005, Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2005, Bu çalışmada, yeni dizayn edilen deprem direnim binalarında ve varolan yapıların deprem performanslarının iyileştirilmesinde, pasif enerji sönüm sistemlerinin kullanımı ile ilişkili bilgiler özetlenmektedir. Öncelikle, farklı tip girdi hareketleri için, durağan hal ve itici girdiler için yapısal sistemin temel dinamik tepkisiyle başlanılmak üzere, uzun süreli ve yakın bölge deprem yer hareketleri dikkate alınarak, bir yapıya enerji sönüm sistemlerinin ilave edilmesinin fiziksel sonuçları tartışılmıştır. Sonrasında enerji sönüm sistemlerinin karakteristikleri tartışılır ve seçilen dizayn kriterlerine karşı gelen belirli bir cihazın fayda ve zararları özetlenir. Sismik dizayn limitleri, seçilen enerji sönüm sistemleri için sınıflandırılabilir. Bu yüzden birincil yapısal eleman hem elastik kalacak hem de inelastiki cevap verecektir. Bu iki limit için dizayn yaklaşımları tanımlanmıştır. Bu çalışmanın amacı ilave enerji sönümü teknolojisinin temel kavramlarını sunmaktır. Bu nedenle bu teknolojinin pratikte getirebileceği artı ve eksileri daha iyi görmek için SAP2000 bilgisayar programıyla sayısal uygulamalar yapılmıştır. Bu uygulamalarla enerji sönüm cihazlarının seçimi, dizaynı ve cihaz yerleşiminin yapı tepkisi üzerindeki etkisi hakkında karşılaştırmalı sonuçlar elde edilmiştir., This study summarizes information on the use of energy dissipation devices in designing new earthquake-resistant buildings and upgrading the seismic performance of existing buildings. First we consider the physical consequences of adding energy dissipation systems to structure for various types of input motion, starting with the basic dynamic response of structural systems to steady-state and impulsive inputs and concluding with the response to long-duration and near-field earthquake ground motions. Generic energy dissipation device characteristics are then disscused, and pros and cons of specific device characteristics of meeting selected design objectives are summarized. The seismic design limits can be catagorized for selecting energy dissipation systems, so that the primary structuralg members either remain nearly elastic for the design earthquake or respond inelastically. The design approaches for these two limits are described. The purpose of this study is to impart basic concepts of the supplemental energy dissipation technology. Because of that the numerical examples were solved with SAP2000 computer program to see pros and cons of this tecnology more clearly. Thanks to these examples, the comparable results were provided about the effects of the selection of energy dissipation devices, design and positions of devices on building responce., Yüksek Lisans, M.Sc.

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2005, Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2005, Bu çalışmada viskoelastik Timoshenko kirişlerini çözmek için bir karışık sonlu elemanlar yöntemi geliştirilmiştir. Öncelikle Timoshenko kirişine ait denklemlerden ilki olan, kuvvet dengesinden yazılan bünye denklemine Laplace dönüşümü uygulanmış, daha sonra karışık sonlu elaman yöntemiyle birinci fonksiyonel elde edilmiştir. İkinci olarak moment dengesinden elde edilen bünye denklemi çeşitli bellekli integral ifadelerinin yardımıyla yazıldıktan sonra Laplace dönüşümü uygulanarak, karışık sonlu elaman yöntemine geçilmiştir. Buradan, ikinci fonksiyonel elde edilmiş ve fonksiyoneller toplanarak tek bir denklem altında birleştirilmiştir. Fonksiyonel elde edildikten sonra yaklaşım fonksiyonları kübik olacak şekilde seçilmiş ve Mathematica programı kullanılarak eleman matrisi elde edilmiştir. Bu aşamadan sonraki işlemler Mathematica programı kullanılarak gerçekleştirilmiştir. Bu program yardımıyla, farklı problemler için sistem matrisinin elde edilmesi ve sınır koşullarının uygulanmasıyla problemin çözümü mümkün hale gelmiştir., In this study, a mixed finite element method has been developed for solving viscoelastic Timoshenko beams. Firstly, Laplace transformation has been applied to the constitutive equation written from the force balance which is the first of the equations belonging to Timoshenko beam, then the first functional has been obtained via mixed finite element method. Secondly, after writing the constitutive equation obtained from the moment balance by the help of various hereditary integral expressions, it has been passed to the mixed finite element method by application of Laplace transformation. From this point, the second functional has been determined and combined under one equation by collection of all functionals. After obtaining the functional, approaching functions have been selected cubic and via Mathematica program, the element matrix has been determined. Processes after this step have been performed by Mathematica program. By the help of this program, obtaining system matrixes for different problems and applying boundary conditions has been made available., Yüksek Lisans, M.Sc.

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2005, Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2005, Bu çalışmada, iki parametreli Vlasov zemin modelinde katmanlaşma, zemin şekli ile dağılım fonksiyonlarının etkisi incelenmiştir. Lineer dağılım fonksiyonlarının farklı katman sayısı durumları ile bulunan sonuçlar, hiperbolik dağılım fonksiyonları ile bulunan sonuçlarla karşılaştırılmıştır. Zemin şeklinin kademeli olması hali, kademenin başlangıcı ve bitiminden alınan tek ve iki katmanlı zemin modelleri ile birlikte örneklendirilmiştir. Tek katmanlı kirişsiz zemin elemanının diferansiyel denkleminden elde edilen tam şekil fonksiyonlarının kullanılması ile sonlu eleman rijitlik matrisi oluşturulmuştur. Yapılan uygulamalar sonucunda sonlu elemanlar yönteminin analitik çözüme yakın değerler aldığı görülmüştür., In this study, the effects of layering, soil shape and distribution functions on two-parameter Vlasov soil model were analized. The results obtained under the assumptions of linear distribution function with various layer situations and hyperbolic distribution functions were compared. Relations of stepped soil shape with one and two-layer soil which were taken from the beginning and end of step were exemplified. Stiffness matrix of a soil element was obtained from one-layer beamless soil differantial equation by using exact shape functions. Reasonable closer values were obtained from finite element method with analytical results in examples., Yüksek Lisans, M.Sc.

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2004, Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2004, Bu çalışmada, değişken derinlikli(eğik tabanlı) elastik zemin üzerine oturan kirişler, sonlu elemanlar yöntemi ile incelenmiştir. Zeminde oluşan reaksiyona ait matematiksel denklem, bir iki parametreli elastik zemin modeli olan Vlasov modelinde türetildiği şekilde alınmıştır. Vlasov modelinde değişken derinlik için türetilen zemin reaksiyon denklemi, bir Bernoulli-Euler kirişinin yönetici diferansiyel denklemi ile birleştirilerek, değişken derinlikli Vlasov zemini üzerine oturan bir kirişin yönetici denklemi elde edilmiştir. Ardından, beraberce dikkate alındıklarında, elde edilen bu denkleme karşılık gelen iki denge denklemi kullanılarak, kiriş için bir karışık sonlu eleman formülasyonu yapılmıştır. Yer değiştirme ve moment birincil, kesme kuvveti ve dönme ise ikincil değişkenler olarak alınmıştır. O halde; elde edilen sonlu elemanlar dört serbestlik derecelidir. Lineer yaklaşım fonksiyonları ile, momentin yer değiştirme üzerindeki etkisinin de hesaba katılmasını sağlayan kübik yaklaşım fonksiyonları kullanılmıştır. İkincil sınır koşulları, dikkate alınan elastik zemin modelinde türetilen kesme kuvveti denklemi yardımıyla tanımlanmıştır. Söz konusu elastik zemin modelinin kurucuları tarafından tanımlanan ve kiriş-zemin sisteminin elastik davranışını karakterize eden boyutsuz bir parametre de göz önüne alınarak uygulamalar yapılmıştır. Ayrıca, üzerinde kiriş bulunmayan serbest zemin için de bir uygulama yapılıp, aynı kişiler tarafından verilen analitik çözümle karşılaştırılmış ve yeterli yaklaşıklığın sağlandığı görülmüştür., In this study, the beams resting on elastic foundation with variable depth(inclined base), have been analysed. The mathematical equation corresponding to the reaction occuring in the foundation, has been assumed as the one derived in Vlasov model, which is a two-parameter elastic foundation model. The governing equation of a beam resting on Vlasov foundation with variable depth, has been obtained by combining the equation of foundation reaction derived for variable depth in Vlasov model and the governing equation of a Bernoulli-Euler beam. Then, a mixed finite element formulation has been done for the beam by using the two equilibrium equations which correspond to the equation obtained, when they are considered together. Displacement by bending moment have been taken as primary variables and shear force by rotation have been taken as secondary variables. So, each finite element obtained has four nodal degrees of freedom. Linear interpolation functions and cubic interpolation functions providing the consideration of the effect of bending moments upon displacements, have been used. Secondary boundary conditions have been defined with the aid of the equation of shear forces derived in the model considered. Applications have been done with the consideration of a non-dimensional parameter which characterizes the elastic properties of beam-foundation system. This parameter was defined by the builders of the model considered. Additionally, an application about a free foundation has been done and compared with the analitic solution given by same builders. This comparison have shown that, an enough approximation has been provided., Yüksek Lisans, M.Sc.

ÖZET Günümüzde deniz ticaretinin gelişmesi ve dışa açılımın sonucu olarak liman yapılan önem kazanmış ve son yıllarda Erdemir Liman Tesisleri ve Ambarlı Liman Tesisleri gibi büyük liman inşaatları yapılmıştır. Limanlarda dalga kıran olarak mendirek, yanaşma yeri olarak ise bloklu rıhtım veya kazıklı iskele ve rıhtımlar inşaa edilirler. Bu yapıların yanında açık denizde yanaşma yapısı olarak kazıklı dolfenler inşaa edilir. Bu tezde ise kazıklı bir yapıdaki, kazık-zemin etkileşimleri, kazıklı yapı çözümleri için modeller, kazık grupları ve kazıklara gelen dalga yükleri hakkında konular işlenmiştir. 1. bölümde kumlar ve killer için, kuvvet-deplasman eğrileri, maksimum zemin taşıma kapasiteleri, derinlerde ve yüzeyde göçme durumları kısaca açıklanmaya çalışılmıştır. Ayrıca bu bölümde kazıkların taşıma kapasiteleri konu edilmiş ve kazığın uç direnci ile yüzey sürtünme direncinin nasıl oluşacağı bu kuvvetlerin kıyaslanması ve kuvvet deplasman ilişkisi açıklanmıştır. 2. bölümde kazıklı yapıların eleman kuvvetlerinin bulunması için çeşitli modellemeler ve hesap yöntemleri gösterilmiştir. Sistem hesabın, yarı-statik analiz için lineerleştirme teknikleri kullanılarak eleman ve sistem rij itlik metrisleri kurulmuştur. Lineerleştirme tekniklerinde tanjant ve sekant rij itliklerinin bulunması ve buralarda sistem rij itliklerinin kurulumu adım adım gösterilerek,.elemanların `` kuvvet ve gerilmelerinin nasıl bulunacağı anlatılmıştır. 3. bölümde birbirlerine yakın çakılan kazıkların etkileşimleri ve grup taşıma kapasitesinin tek tek kazık taşıma kapasitelerinin toplamından farklı olduğu ve nasıl hesaplara dahil edileceği belirtilmiştir. 4. bölümde bu bölümde kazıklı yapıların hesaplarında önemli yeri olan dalga yükleri konu edilmiştir. Burada çoğunlukla silindirik kazıklara gelen kırılmamış dalgaların meydana getirdiği dinamik dalga kuvvetlerinin, statik kuvvetlere nasıl çevrileceği ve bu kuvvetlerin hesaplanmasında kullanılan formüllerdeki katsayılar anlatılmıştır. Bu konu dalga etkisine maruz kalan kazıkta, deniz yatağındaki kuvvetlerin bulunmasını anlatan bir örnekle bitirilmiştir. 5. bölümde silindirik elemanlar üzerinde wortex kopması etkisi göz önüne alınmıştır. Bu bölümde sabit akış hızlarında silindirlerin tepkisi ve sabit akımda yapı tepkisi harmonik modelle açıklanmaya çalışılmıştır. Bu konudada silindirik kazıkta vortex kopmalarından oluşan gerilmelerin nasıl bulunacağını anlatan bir örnekle sonlandırılmıştır. SUMMARY SOLUTION OF THE PILED WATER STUCTURES AND EFFECT OF WAVE AND VORTEX SHEDDING ON THE CYLINDIRICAL ELEMENTS 1. Bearing Capacity Of Single Piles : This section provides some background information on driven piles in clays and siliceous soils. Piles driven or grouted into predrilled holes or into calcareous soils are special cases. For pile foundations, the following simplifying assumptions regarding soil stiffness, damping and added mass are usually made: - The forces occurring between the soil and the pile any level on a pile are independent of the deflections at any other level. - The soil damping need not be calculated. - The effect of the mass of the soil acting with the piles has a negligible effect on the structural dynamics. When the piles are well spaced (spacing < 8 x diameter) it is not usually necessary to consider any interaction effects between the piles. However, for smaller spacing the deflection of one pile affects the other and the group or interaction effects also need to be taken into account. If a preliminary estimate of pile behaviour is required then the pile can be assumed fixed at a certain distance below the seabed. 3.5D - 4.5D stiff clays 7D - 8.5D very soft silts - 6D general calculations p - y curves : The methods of estimating p-y curves for clays and sand are semi-empirical. The construction of the curves in both cases requires the ultimate lateral pressure pu to be determined. A non-linear curve is then fitted to pu value. XI0.1 0.1 p-y curve for sand Compression capacity of pile : The compression capacity of pile comprises : - the skin friction or shaft resistance acting along the pile - the end bearing at the tip Skin friction End bearing End bearing and skin friction Q = QP + Qs Q = compression capacity Qp = end bearing Qs = skin friction Tension capacity : The high moment applied to an offshore structure can result in tension forces in some of the piles. To calculate the ultimate tensile capacity, the end bearing component is neglected. The weight of the pile and soil plug may be included. XIIScour and cavities : These effects reduce the axial capacity of the pile. Both effects prevent the generation of skin friction at the top of the pile. Shaft resistance : The shaft resistance is essentially frictional and for sands the frictional stress can be estimated as: x = Ko CTvo tan 8 but x X u, du dt fi dz fD dz D /////// // // ////////// Wave force on the vertical cylindrical pile 7tD2 du 1,, f = f i + f D = CMP- -^ +CD-puufj = inertial force per unit length of pile, XIVfo= drag force per unit length of pile, p = density of pile, D = diameter of pile, u = horizontal water particle velocity at the axis of the pile, (calculated as if the pile were not there) du/dt = total horizontal water particle acceleration at the axis of the pile, (calculated as if the pile were not there) Cd = hydrodynamic force coefficient, the `Drag` coefficient, Cm = hydrodynamic force coefficient, the `Inertia` or `Mass` coefficient, 5. Vortex-induced vibrations : Vortex which form and shed into the flow past a bluff body produce a dynamic loading upon the body. The principal danger from this type of loading arises from the possibility of resonance between a natural frequency of the structure and the frequency of excitation. Large and damaging amplitudes of oscillation can be caused by a complicated and pernicious mechanism of resonance which occurs over a considerable range of conditions, when interactions between the flow and the structures motions causes the frequency of the excitation to be controlled by and to lock in to that of the response. Vortex shedding excitation is caused by pressure fluctuations associated with the motion of vortices in the wake of a body. XV 105

Sıvı dolu tankların titreşimleri ile ilgili problemler son 45 yıldan beridir tartışılmakta ve çözümler üretilmeye çalışılmaktadır. Konuyla ilgili ilk kaynaklar 1950'lerde, uçak yakıt tankları ve sıvı yakıtlı füzelerle ilgili olanlar 1960'larda görülmeye başlamıştır. ilk çalışmalarda sıvı depolayan tank rijit olarak gözönüne alınmış ve tankın içindeki sıvının dinamik davranışı incelenmiştir. 1964 Alaska depreminde sıvı dolu tankların büyük ölçüde hasara uğramasından sonra elastik tankların dinamik karakteristiklerinin araştırılması konusunda önemli bir dikkat sarfedilmiştir. Bu yıllarda sayısal tekniklerle birlikte bilgisayarlardaki gelişmeler çözümlerin sayısını ve hassasiyetini artırmıştır. Problemin matematik formulasyonunun oldukça karmaşık olması nedeniyle genel olarak; 1- Viskoz olmayan sıvı, 2- Sıkıştırılamayan sıvı, 3- Küçük yerdeğiştirmeler, hızlar ve eğimler, 4- Çevrintisiz akım alanı 5- Homogen sıvı varsayımları altında çözüm aranır. Islak yüzeyli yapılarda sıvı-yapı etkileşimi problemlerinde; 1- Sıvı hacmi üç boyutlu elemanlara ayrılarak sonlu elemanlar yöntemiyle, 2- Tüm sıvı yüzeyi iki boyutlu sınır elemanları yardımıyla direkt, indirekt veya varyasyonel sınır eleman yöntemleriyle, 3- Sıvının davranışını genelleştirilmiş fonksiyonlar yardımıyla ifade ederek çözümler üretmek mümkündür. Sınır üzerinde potansiyel ve akım ile bölgede potansiyel gibi üç bağımsız değişken içeren genelleştirilmiş varyasyonele dayalı yeni sınır eleman modelleri mevcuttur. Yaklaşım, bölge içindeki potansiyeli klasik temel çözümleri global interpolasyon fonksiyonu olarak kullanır ve bölge integrallerini sınır integraline dönüştürür. O nedenle sonuç denklemler sadece sınır bilinmiyenlerini içerir ve model simetrik matrisler üretir. Bu çalışmada, sıvı-yapı etkileşimi problemleri için mod süperpozisyonu yöntemi kullanıldı. Deprem altında sıvı-yapı sisteminin cosO tipi serbest titreşim modları kullanıldı. Sıvı analitik olarak incelenirken etkileşimde bulunduğu yapı elastik kabul edilip mod süperpozisyonu yöntemi kullanılarak çözüme gidildi. Elde edilen denklem takımındaki bilinmeyenler, yapıda düğüm noktalarının yerdeğiştirmeleri, sıvıda yerdeğiştirme potansiyeli ve hidrodinamik basınç ifadeleridir. SUMMARY THE SOLUTION OF FLUID-STRUCTURE INTERACTION PROBLEMS IN THE SEISMIC LOADED CYLINDRICAL TANKS BY NORMAL MODE SUPERPOSITION METHOD An extensive literature has developed during the past 45 years on the subject of waves caused by vibrating tanks of heavy fluid. Part of this literature appeared in the 1950s and 1960s in connection with the sloshing problem for aircraft fuel tanks. Several examples of this work, which was concerned primarily with calculating natural frequencies and modes rather than time-history responses to specified tank accelerations, are given by Graham and Rodrigez [1] and Silverman and Abramson [2]. The second part of this literature originated in 1949 with a paper by Jacobsen [3] and has continued to the present time. The performance of ground-based liquid storage tanks such as petroleum, LNG, LPG, nuclear containment vessels and so forth during recent earthquakes demonstrates the need for a reliable technique to assess their seismic safety. Early developments of seismic response theories of liquid storage tanks considered the container to be rigid and focused attention on the dynamic response of the contained liquid. A common seismic design procedure is based on the mechanical model derived by Housner [4] for tanks with rigid walls. In this approach, a mathematical model of the liquid-rigid tank system was used and the hydrodynamic affects were evaluated approximately as the sum of two components, viz. an impulsive part which represents the portion of the liquid which moves in unison with the tank and a convective component which represent the portion of the liquid sloshing in the tank. Epstein [5] improved Housner's work and presented design curves for estimating the bending and overturning moment induced by the hydrodynamic pressure, for cylindrical as well as rectangular rigid tanks. The following studies can be indicated as a sample on the dynamic characteristics of the liquid with rigid walls [6-12]. The 1964 Alaska earthquake caused the first large scale damage to tanks of modern design and initiated many investigations into the dynamic characteristics of flexible containers. In addition, the evolution of both the digital computer and various associated numerical techniques has significantly enhanced solution capability. Several studies were carried out to investigate the dynamic interaction between the deformable walls of the tank and the liquid [13-57], The exact mathematical procedure for describing fluid oscillations in a moving container is extremely complex. Therefore, the following simplifying assumptions are generally employed: 1- nonviscous fluid, 2- incompressible fluid, 3- smalldisplacements, velocities and slopes, 4- irrotational flow field and 5- homogeneous fluid. The assumption of irrotational flow ensures the existence of a fluid velocity potential,, which must satisfy the Laplace equation vV=o The mathematical boundary conditions for the solution are : 1- At the rigid tank walls, r=R, the normal component of the tank wall must equal to zero, therefore Here, the comma followed by a subscript designates a partial derivative with respect to radial direction. 2- At the rigid bottom of the tank, z=0, the axial component of the liquid velocity must equal zero, therefore 4- At the liquid free surface, z=/7, imposing the condition that the fluid particles must stay on the surface, it follows that where g is the acceleration of gravity. There are two major cases of vibration associated with the system under consideration, for which the circumferential variation of the response is described by cosnO, (n is called the circumferential wave number and 0 is the circumferential coordinate angle). Case-/ corresponds to solutions with n-l and is called lateral sloshing. Case-// is named as breathing vibration, and it corresponds to all vibrations where n does not equal one. For a tall tank, the cos0-type modes can be denoted beam-type modes because the tank behaves like a vertical cantilever beam. This is not true for a broad tank because at 0=0 both the amplitude and the axial distribution of radial displacement are different from those of the circumferential displacement at 0=jt/2. This investigation is concerned for only case-/ which has lateral sloshing modes with n=l [28,46]. The liquid and the shell structure are two separate system that are coupled. Each system, acting alone, has an infinite number of modes of free vibration. If the coupled system is excited with some forcing frequency m, then the response will also have the same frequency a. The magnitude of the response will depend on the ratio of the forcing frequency a> and the natural frequencies of the coupled system. At least three different methods can be used in handling wetted structures:1- Dividing the whole fluid volume into finite elements which produces a three dimensional mesh, 2- Dividing the whole fluid surface into boundary elements, such as Direct, Indirect or Variational Boundary Element Method (BEM) which produces a two dimensional mesh, 3- Representation of fluid behaviour by a series of generalized functions. Using the Finite Element Method (FEM), one can obtain a linear interaction problem for both linear the structure and the fluid which can be solved by standard methods. A great simplification can be introduced by the assumption of incompressibility of the fluid. In this case, the coupled system can be reduced to the structural mass matrix plus a so-called added mass matrix system. The finite element procedure is used in conjunction with a simple source distribution on all surface of a liquid, which is contained in a tank with arbitrary geometry, to obtain the expression of the mass and stiffness (because of gravity) matrices of the liquid. Different methods can be used to extend the capabilities of finite element and finite difference programs to handle wet structures like surface source distribution or dipole distribution as a basis for the solution (thus, only the surface of the liquid need be considered and a two-dimensional grid results [14,22]), subdivision of the a three- dimensional mesh, representation of liquid behaviour by a series of generalized functions weighted by unspecified coefficients. In engineering problems involved fluid-structure interaction with sloshing, the fluid and sloshing behaviour is determined with a rigid wall assumption and than the structural response is obtained by imposing the dynamic pressure to the structural model. This approach generally yields conservative results since the rigid-wall forces are larger than the flexible wall forces. But an uncoupled analysis underestimates the structural response if the natural frequencies of the coupled system are close to the excitation frequencies, which is often the case in the seismic analysis of liquid-filled tanks and nuclear reactor systems [25]. Two basic approaches exist for the coupled analysis of Fluid-Structure Interaction (FSI) systems. In the first approach, the pressure or velocity potential formulation, the fluid is characterized by a single pressure or velocity potential variable at each node of the finite element mesh. The fluid and the structure are coupled through the fluid- structure interface. The coupled set of equations are, in general, unsymmetric. Therefore, a symmetrization of the resulting equations is needed and the implement of this formulation requires a special-purpose computer code. With this formulation, the sloshing effect cannot be corporated easily [29,40,47,57,59]. The second approach is the displacement formulation. The fluid motion is expressed in terms of the finite element nodal displacements. Hence, compatibility and equilibrium along the interface are automatically satisfied. The fluid is analyzed like an elastic solid but with a negligible shear modulus. This assumption leads to theappearance of non-physical circulation modes. The number of these modes tends to increase with mesh refinement. Different techniques are introduced to remove these non-physical models [48,63]. The main advantage of this approach is the similarity between the discretized forms of the fluid and the structure. Thus, easy implementation of the fluid equations into existing FEM codes is achieved. In addition, sloshing effects can easily be included in the formulation. Nevertheless, the existance of the non-physical circulation modes gives difficulties in numerical analysis of the problems, such as artificial viscosity and huge equation systems, among others. Another possibility to describe the fluid motion is to use both the displacement or velocity potential and the acoustic pressure together. The resulting finite element equations are symmetric, and circulation modes can be avoided without using penalty terms to restore the inviscid nature of the fluid, as it is the case for displacement formulation. Some variational formulation which incorporates fluid-structure interaction, sloshing, seismic and body forces have been considered in [25,40]. Coupled FEM and BEM are especially well-suited for dealing with problems that are defined over a combination of homogeneous regions free of body forces and regions within which body loads are present or where the material is inhomogeneous, even possibly nonlinear. Such methods have been used increasingly in engineering since the early 1970s; their mathematical analysis was initiated with the work of Brezzi, Johnson and Nedelec [69,70]. Many different variants of these methods have been developed over the years; for example, Hsiao [71] for a recent survey. In the most applications to date, while the domain finite element part of the formulation is generally derived from a variational principle, the boundary element part is usually obtained by direct collocation of the corresponding boundary integral equation. Only in isolated cases have variational principles been used to derive all the discretized equations. Hence, the resulting algebraic systems are often non-symmetric. Since symmetry is a desirable property in numerical computations, substantial effort has been devoted to developing symmetric formulations. A distinctive feature of fully variational procedures is that the discretized algebraic equations for the coupled problem are automatically symmetric since they are all derived from a single functional [39]. Recent papers on alternative finite or boundary element formulations have concentrated on the development of hybrid stress models. Schnack has used the boundary element method to generate a hybrid stress finite element model, which gives rapid convergence of the results and accurate solution for stress concentration problems. Other authors have developed variational formulations based on generalized principles of the type proposed by Hu-Washizu and Hellinger-Reissner [73] in which the unknowns on the boundary and in the domain are taken to be independent of each other. Dumont [74] has proposed a hybrid stress boundary element formulation based on boundary displacements and stresses inside the domain. PoHzzotto [75] has developed4wo^other variational formulations. The first based on the classical Washizu and Reissner principles [73] which are characterized by boundary functions independent of the field functions. The other is based on a boundary principle expressed in terms of the unknown boundary tractions and displacements and introduced by that author.The method described by De Figueiredo and Brebbia [55] uses a hybrid type functional with three kinds of independent variables, Le. potentials and fluxes on the boundary and potentials inside the domain. The approach applies classical fundamental solutions to interpolate inside the domain and thus allowing one to transfer the domain integrals to the boundary integrals. The resulting system of equations is written in terms of boundary unknowns. By rearranging them, one can obtain a final matrix equation system in which the potentials on the boundary elements are unknowns. This system has the main advantage of being symmetrical. The main objectives of this dissertation are to develop coupled My variational boundary element method-finite element method for fluid-structure interaction systems and to provide answers to the effect on natural frequency of the system of additional structural elements. Because the beam type (cos0) modes are important in the analysis of the vibrational behaviour of anchored storage tanks under seismic excitation, it is used cosö-type modes. The goal of chapter 2 is to develope the direct and variational boundary element formulation combined for liquid sloshing in rigid cylindrical tanks with a rigid baffle. In the next chapter, the governing equations for a thin cylindrical shell and plate are obtained by using Novozhilov's first approximate theory and Kirchoff thin plate theory. Chapter 4 is devoted to the fluid-structure with or without elastic baffle interaction problem by using coupled variational boundary element-finite element methods. The resulting system of equations is written in terms of boundary unknowns only. By rearranging them one can obtain a final matrix equation system in function of only boundary values of displacement of structure, potential of liquid and hydrodynamic pressure. The results are finally discussed. 62

ÖZET Winkler tipi elastik zemine oturan, bazı bölgelerinde rijitliği azaltıcı kusurları olan değişik sınır koşullarında kirişler incelenmiştir. Bu incelemede sonlu elemanlar yöntemi kullanılarak kirişler 10 elemana bölünmüş elastik zemin sabiti kf nin değişik değerlerinde ( kr = 0, kf = 1000, kt- = 5000 t/m` ) çeşitli bozulma oranlarında doğal titreşim frekansları bulunmuştur. Ayrıca aynı bozulma oranlan için eksenel dış yük kirişin elemanlarına ayrı ayrı etki ederken burkulma yükleri bulunmuştur. Sonuçlar değerlendirilmiştir. Ek - A da kiriş 10 elemanlı düşünülmüş ve her elemanında rijitlikteki bozulma oranı ( 1/8,2/8, 3/8, 4/8, 5/8, 6/8, 7/8, 8/8 ) alınarak doğal titreşim frekansları bulunmuştur. Bulunan değerler grafik olarak gösterilmiştir. Bulunan tablolar birleştirilerek toplu halde verilmiştir. Ek - B de aynı bozulma oranlan ile eksenel dış yük sağ uçtan sol uca elemanlar boyunca kaydırılarak burkulma yükleri bulunmuştur. Yine bulunan değerler grafik olarak gösterilmiş ve her grup için tablolar birleştirilerek toplu halde verilmiştir. Ek - C ve Ek - D de basit kiriş için burkulma yükleri ve doğal titreşim frekanslarının bulunmasında standart tablo programlarından QUATTRO PRO da kullanılan şablon genelleştirilerek verilmiştir. SUMMARY DYNAMIC PROPORTIES and BUCKLING LOADS of CRACKED BEAMS A beam resting on an elastik foundation with spring constant of kt- in the length of L and uniform cros section and subjected to axial P enternal load and in the a-b region decreasing the stiffness is considered as seen in the figure ( 1 ). The beam is divided into ten elements. Except the boundary conditions every part of it have two unknovn objects which are v and S. H, is the ratio of deformed stiffness to the essential one. According to this, the potantial energy of the bar which has a changing stiffness between a-b is U= - o f.9V, 1 r,_`, `.. JY-S^ EIW^jdx + ^{EK(x,-EIW}^j dx - fk'GA[- -») dx + -Jk, v:dx-^J P /~ dx...(I) 2 i Vrx J 2J, 2 J, VdxJ Where E, Young modulus; I, inertia moment; K, inertia moment of the deformed part; k/ shear coefficiant; G, shear modulus; P, axial external load and A, area. Also the kinetic energy of the beam is -;M$)'fc4Mt)'* T'= 2İpAlrtJdX+lJPİlirJdX ?',:) Where p is the mass in unite volume of the beam material. For the displacement expression, the foolowing cubic displacement function is chosen on the point of a(>-a-, are constants: v = a().-? i// x ' - a>x t- a^x' (3)^srr- 1 - T -*-r Beam resting on an elastic foundation v = [ Nvi Nv2 Nv3 Nv4 ] Vj VJ »J (4) » = [N»i N»2 N»j NM] 3i »j Where Nvl=l-3^- + 2^ Nv2=[$-2Ç2 + !;3]// Nv3=3£2-2Ç3 N` = [-6^ + 6^]// Na2 = 1 - 4Ç + 3Ç2 Nw = [ 6Ç - 6£2 ] / / Nw = -2Ç + 3^2 Ç = x / / Bending displacement can be written as; (5)[BV] = A[NV]...(6) dx [Bb] = -£[N9]...(7) By the help of above equations, using displacement vectors, potential energy If and kinetic energy T can be written as; Ue = ^{q}eT[Kbr{q}e +^{qrT[Kfr{q}e -^{q}eT[Kg]e{q}e...(8) Te = {q'r-T[Mt]c{q'r.-(9) Where [Kb]- J[BJTEI[Bb]dx 0 = Bending stiffness matrix [Kf]`=J[Nv]Tkr[Nv]dx 0 = Foundation stiffness matrix [KJe= JP[Bv]T[Bv]dx 0 = Stability matrix [M,]e =J [Nv]TpA[Nv] (I = Mass Matrix XVIfi i V )'j 1- Şekil 2.2 Uç deplasmanlarının pozitif yönleri M. ~/MJ MT- v ! T,K Şekil 2.3 Uç kuvvetlerinin pozitif yönleri The clear expressions for the element matrix are given ; [Kb]c = EI r 12 6/ -12 6/ 4? -6/ 2/2 12 -6/ -/f [Kf]e = kf/ 13/35 11//210 9/70 -13//420 /2/105 13//420 -/2/140 L 13/35 -11//210 A2/ 105 XVII[Kg]e = P// 6/5 V10 -6/5 1/10 2/-/15 -110 /-/30 6/5 410 2f 15 [Mt]c = pA/ 13/35 11//210 9/70 -13//420 /2/105 13//420 -/2/140 13/35 -11//210 /7105 Equations The total energy or Lagrange function ^£ for the whole beam is given as the sum of energies of elements ; +£= I ( Ue - Te ) (10) According to Hamilton's princible ; +£ =0 If L is put in the Hamilton principle ;...(11) [K]{q}-[S]iqi + [M] jq }=0...(12).Willis obtained. Where ; = Displacement vector The above equation load us the solution of the two problems ; i. Free vibration : ([K]-co2[M]){q}=0...(17) h. Static buckling : ([K]-P[S]){q}=0...(18) Free vibration frequency (co) and buckling load (P) are solved by the above equations. Numerical values used in the solution is A = 0.25x0.25 = 0.0625 [m2] I = 0.254/ 12 = 0.000325521 [m4] E = 2850000 [t/m3] L = 5 [m] / = 0.5[m] p = 2.5 [t/m3] 159

ÖZET Bir çok mühendislik probleminin çözümünde, sayısal hesap yöntemlerinden olan sınır eleman yöntemi (BEM) kullanılmaktadır. Sınır eleman yönteminde sınırdaki ayrıklaştırma direkt veya indirekt olarak iki ayrı yaklaşımla yapılmaktadır. Direkt sınır eleman yönteminde sınırdaki bilinmeyenler doğrudan elde edilir. İndirekt yöntemde ise önce sınırda fiktif değerler elde edilir daha sonra bu fiktif değerler yardımıyla diğer bilinmeyenler hesaplanır. Şuurdaki bu fiktif değerler ise seçilen temel bilinmeyenler bakımından iki kısımda incelenebilir. Buna göre eğer şuurdaki bilinmeyenler fiktif gerilmeler ise fiktif değerler olarak fiktif gerilmeler alınır, eğer bilinmeyenler yer değiştirme süreksizlikleri ise fiktif değerler yerine, yer değiştirme süreksizlikleri alınır, Bu çalışmada, izotrop ve anizotrop ortamlar için, indirekt şuur eleman yöntemlerinden olan, fiktif gerilme yöntemi (FSM) ve yerdeğştirme süreksizliği yöntemi (DDM) kullanılmıştır. Her iki yöntemde de Kelvin ve Melan temel çözümleri kullanılarak, şuur integralleri sabit ve doğrusal olan elemanlar üzerinde kapalı olarak elde edilmiştir. Bu çalışmanın bir özelliğide, Kelvin temel çözümleri kullanılarak, sonsuz ve yan- sonsuz düzlemlerde izotrop-anizotrop ortamlar için tekil yüklerden dipoller oluşturulmuş ve bu dipoller yardımıylada DDMdeki temel çözümler elde edilmiştir. Ayrıca izotrop ortamdan farklı olarak, anizotrop ortamlarda, elastik sabitler doğrultusundaki eksenlerden her hangi bir açı yapacak şekilde bir eleman gözönüne alınmış olup, türevler ve integraller bu doğrultuda alınarak elemanlar daki bilinmeyen yer değiştirme süreksizlikleri için lineer denklem sistemi elde edilmiştir. Sayısal uygulama olarak sonsuz ve yan-sonsuz düzlemlerde, çeşitli boşluk ve çatlak problemleri incelenmiştir. Bunlardan, değişik şuur koşullan altında sonsuz ve yan- sonsuz düzlemlerde, dairesel kesitli bir boşluk için seçilen noktalarda meydana gelen gerilme ve yerdeğiştirme bileşenleri hesaplanmıştır, ikinci örnek olarak, iki malzemeli dairesel kesitli boşluk problemi DDM ile incelenmiştir. DDM için özel uygulama olması açısından, yan sonsuz düzlemde, yüzeyden itibaren derinlikle doğru orantılı değişen yükleme altında çatlak ve fay problemi ve ayrıca Mohr-Coulomb hipotezi kullanılarak, sonsuz düzlemde dairesel boşluk ve fay etkileşim problemi çözülmüştür. Son olarak anizotrop malzeme için, eksenel yük etkisi altında sonsuz düzlemde dairesel ve kare kesitli boşluk ile hidrostatik iç basınç ve eksenel yük etkisi altında yan sonsuz düzlemde dairesel boşluk problemleri çözülmüştür. Geliştirilen yöntemle çözülmüş olan tüm örnek problemlerde ankotropinin sonuçlan önemli ölçüde etkilediği ve gözönünde tutulması gerekli bir malzeme özelliği olduğu sonucuna varılmışta. XI SUMMARY DISPLACEMENT DISCONTINUITY METHOD IN ANISOTROPIC BODIES Realistic problems of engineering mechanics, usually formulated in terms of a set of governing differential equations together with appropriate boundary and initial conditions, are characterized by great complexities in geometry and material constitution. As a result of that, analytical solutions of these problem are practically impossible to obtain and resort has to be made to numerical methods for the determination of an approximate solution. Numerical solutions of very complicated problems are feasible today due to the availability of powerful digital computers. A numerical method is of the differential or the integral type depending on whether the numerical analysis precedes or follows the integration of the governing equations. The general problem two-dimensional elastostatics is that of determining the stresses, ay (i,j = 1,2), and displacements, u;, in a body of known shape subject to prescribed boundary conditions (traction, displacements). In a additions to these may be body forces which act throughout the interior of the body. The two-dimensional elastostatics problems considered here are those of plane strain and plane stress. The conditions to be satisfied by any solution for a particular problem geometry and loading condition are the differential equations of equilibrium, the constitutive equations, the equations of compatibility and the boundary conditions for the problem. The treatment of problems two-dimensional elastostatics simplifies somewhat when the body forces do not appear in differential equations. Since the equations for continuos, homogeneous form by finding one of the infinitely many particular integrals. The boundary element method has been used widely in geotechnical analysis and has performed well for elastic analysis of excavations in semi-infinite or infinite bodies. In this method, the problem is solved in terms of the conditions imposed at the surfaces of openings the problem domain. However, there are situations in geotechnical engineering for which it is difficult to use the boundary element method, especially those practical problems involving sequential excavation or construction, inhomogenius materials, material non-linearities, and the presence of joints. Natural rock masses are usually composed of blocks of intact material separated by joints or discontinuities. The behavior of these rock masses is complex as it is governed not only by the properties of the discontinuities. In general, the overall behavior of a jointed rock mass will be anisotropic. XUMost boundary element analyses carried out in geotechnical engineering assume that the rock mass can be modeled as an isotropic elastic material. There have been a few reported attempts to account for the structure of the rock mass in analyses, and examples of boundary element treatments of rock with distinct joints have been presented by Hocking and Brady et al however, when the spacing between the joint is small in comparison with the length scale of interest (such as tunnel width or foundation size), a simulation incorporating the jointing explicitly is very difficult and costly to implement. The problem of dealing with the structure of the rock mass explicitly can be avoided in cases where the jointing is closely spaced and regular, because it is convenient in such problems to idealize the rock mass and anisotropic elastic medium ; that is, the effects of the discontinuities is implicit in the choice of the stress-strain model adopted for the equivalent rock mass continuum. Some of the boundary element treatments of anisotropic materials relevant to jointed rock masses are reviewed briefly in this work. Increasing structural use is being made of materials with anisotropic elastic material properties. In recent years, various kinds of composite materials have been developed and used for structural components. The major reasons for their success are to be found in the fact that they can fulfill all the requirements for a given application. Orginally, structural composites were developed for the aerospace industry as they offered attractive properties of stiffness and strengthcompared to their weight. Further advantages such as high corrosion resistance and design flexibilty made composite materials the ideal replacement to the aluminum alloys previously used. Rizzo and Shippy and Crouch and Starfield have assumed that in some cases rock masses may be represented as transversely anisotropic materials, and they have implemented into boundary element formulations the fundamental solutions due to Green for a point force in an infinite sheet of transversely anisotropic elastic material. Brebbia has also suggested an iterative perturbation boundary element analysis for general anisotropic materials, and Dumir and Mehte have analyzed orthotropic half- plane problems using boundary element techniques using appropriate Airy stress functions. A boundary element formulation for general anisotropic materials has also been presented by Carter and Alehossein, using the fundamental solutions derived by Lekhnitskii. The most widely known and used numerical methods of solution are the Finite Difference Method (FDM) and the Finite Element Method (FEM), both of the differential type. The FDM replaces the differential equations by algebraic ones, valid at a set of nodes within the domain, through the approximation of derivatives by finite differences, while the FEM replaces the domain itself by a set of finite sub domains or elements connected through their nodes and approximately reproducing the behavior of the sub domain that represent. The FEM presents some very distinct advantages over the FDM such as better conformity to the domain geometry, much easier handling of the boundary conditions and easier construction of variable-size meshes. These advantages have made the FEM the most popular numerical method among scientists and engineers. Even the FEM, however, presents some disadvantages such as rather high costs for the preparation and input of data and time-consuming solutions of three-dimensional problems, especially those with distant boundaries. XlllThe Boundary Element Method (BEM), which is based on an integral formulation of the problem, has emerged during the last 15 years as a new powerful computational tool. This method usually requires only a surface discretization and not a discretization of both the interior and the surfaces of the domain of interest as it is the case with `domain-type` techniques, such as the FDM and the FEM. This fact makes the BEM more efficient than the FEM for quite a number of classes of problems. For some other problems the BEM might be equally good or inferior to the FEM, while in certain cases a combination of the two methods usually creates the optimum numerical scheme. The BEM has been successfully used to a great variety of problems in engineering science, such as potential theory, elastostatics, elastodynamics, elastoplasticity, viscoelasticity, viscoplasticity, fracture analysis, fluid mechanics, acoustics, heat conduction, electromagnetism, soil-structure interaction and fluid- structure interaction. There are basically two kinds of BEMs, the indirect and the direct ones. In the indirect approach the discretized integral equations are first solved for the density of the singular solutions over the boundary surface and then the remaining boundary quantities are computed in terms of these densities which have no physical significance. In the direct approach the discretized integral equations are formulated with the help of certain fundamental integral theorems and connect directly the unknown with the known boundary quantities. Even thought it has been shown by Brebbia and Butterfield in 1978 that the indirect and direct BEMs are formally equivalent, more emphasis is usually given to direct BEMs because they are more appealing to scientists and engineers. In addition, direct BEMs appear to be, at least so far, more easily amenable to improvements of the FEM type for the development of advanced BEMs than indirect ones. The term semi-direct BEM has also been used for some time in the past to indicate the treatment of some potential and elastostatic problems by a direct formulation with unknowns functions analogous to stream or stress function from which the physical quantities of the problem can be obtained by differentiation after the solution of the integral equations. This BEM category, however, is very limited and of a special character to be an independent one and for this it is usually considered as part of either the direct or, more frequently, the indirect BEM. In the direct boundary element formulation based on the point-force fundamental solution, considerable difficulties arise since the displacement discontinuities across the fracture surfaces are not explicitly accounted for. One approach to tackle this problem is partition the medium (which must be finite) into sub-regions However, this approach becomes inefficient when there are two or more cracks, or even for a single crack which propagates out-of-plane in mixed mode loading. In the indirect boundary element (or fictitious) formulation, difficulties similar to that of the direct boundary element method arise because continuous displacements are implicitly assumed. XIVThe indirect boundary element method (IBEM) makes use of `fictitious forces` distributed along the boundary of the region of interest. By means of the principle of superposition, an integral equation for these fictitious forces is derived. By representing the boundary by a set of elements and assuming some variation of fictitious forces over each element, a system of simultaneous equations approximating to the integral equation is obtained. Once these fictitious forces have been obtained by solving the equations, the stresses and displacements anywhere with in the region and on the boundary can be calculated by integration. For the sake of simplicity, represent a two-dimensional continuous, homogeneous, isotropic, linear and elastic region V bounded by the surface S/ acted upon by a surface traction t;'(x). We now want to find the stresses and displacements in the region V and on the boundary S/ However, it often is easier to find the solution to the relevant partial differential equations in an infinite region, because we do have the Kelvin solution which gives the displacements at the point x (referred to as the field point), U;(x), due to a concentrated unit force, Fj(£), acting at the point £ (referred to as the load point) in an elastic body of infinite extent. The displacement discontinuity method (DDM), as a means of solving boundary value problems in elasticity, has become a popular numerical method in the field of geomechanics. The most significant characteristic is its ability of handling rock discontinuities and fractures. The successful application of the DDM, however, relies on the derivation of the fundamental solution for a displacement discontinuity singularity, or mathematically a point dislocation in the infinite space. The displacement discontinuity method is particularly well suited to model fractures which have relative displacements across their surface (hence a displacement discontinuity results). In addition, it can also be used to model an ordinary boundary of a 2 or 3 dimensional body. However, as will be shown later strong stress singularities at the ands of the displacement elements make it undesirable to model boundaries with finite and smoothly distributed applied loads. The formulation of the displacement discontinuity method is identical to that of the fictitious stress method except that displacement discontinuities (or `fictitious cracks`) are used instead of fictitious stresses. (Naturally, the fundamental solution in also be different.) Hybridizing these two methods so that the advantages of both can be utilized seems to be very promising. In addition, the boundary element methods have further advantages over the finite element method when some special types of boundaries a present. For example, for a semi-infinite medium with a circular cavity. In this study investigated the indirect boundary element solution for isotrpic and anisotropic problems. The complete fundamental solutions due to unit loads within the infinite and semi-infinite plane are given. Expressions for stresses and displacements at internal points are also given. This formulation İs applied and compared to some classical problems. This solution procedure is highly accurate and computational more efficient than the boundary element formulation using the Kelvin fundamental solution for orthotopic problems. Throughout this work the so- called Cartesian tensor notation is used. This notation is not only a time-saver in XVwriting long expressions, but is also extremely useful in derivation and in the proof of theorems. Such notation makes use of subscript indices (1,2,3) to represent (x,y,z). The general problems two dimensional elastostatics are explained in chapter 2. This chapter is partly devoted to introducing some basic concepts of the theory of elasticity needed for developing boundary element models. The chapter starts by reviewing the small starin theory of elasticity. The complete Kelvin fundamental solutions are given in this chapter. The indirect boundary element method and numerical procedure are given in chapter 3. The fundamental solutions for a two-dimensional displacement discontinuity are also given in this chapter. The DDM is used to model joints and faults fillet with weak material in chapter 4. In this case the dispalcement discontinuity element surfaces are connected by springs which have shear stiffness K« and normal stiffness Kn. The other special application is to use DDM for inhomogeneous bodies in this chapter. In the last chapter anisotropic elasticity problems are investigated in the indirect boundary element method. Kelvin fundamental solutions for any direction are used and singular solutions found in the DDM. XVI 106

ÖZET Deprem esnasında elastik yapı ve bu yapıyı çevreleyen sıvı arasındaki etkileşim elastik yapının güvenliği açısından dikkate alınması gereken bir konudur. Şimdiye kadar bir çok makale ile birlikte baraj -rezervuar etkileşim problemi ele alınmıştır. Yapılan bu çalışmalarda kesin çözümler, sonlu eleman yöntemi,sınır eleman yöntemi ve hatta iki metodun bir arada kullanıldığı çözümler elde edilmiştir. Barajların tasarımında özellikle aktif sismik bölgeler için baraj yapısı ve rezervuar arasındaki etkileşim hesaplarda önemli bir faktördür. Sismik kuvvetlere maruz bırakılan barajlardaki hidrodinamik basınç dağılımları ilkin 1933 yılında Westergaard tarafından geliştirildi.Bu konudaki bir çok araştırma o zamandan beri yayınlandı.Zangar&Haefeli, Zienkiewicz & Nath, Chopra, Hanna & Humar hepsi bu dağılımın hesabında rijit bir cisim olarak barajı ele almışlardır. Yapılan çalışmalarda genelde sayısal teknikler kullanılırken kesin analitik çözümler genelde suda titreşen basit elastik yapılar için kullanılmıştır. Bu araştırmada incelenen çubuk bir elemanın sıvı ile etkileşim probleminde de SONLU ELEMAN YÖNTEMİ kullanılmıştır. Bu çalışmada baraja yakın kısımlarda değişken derinlikte belli bir uzaklıktan sonra sabit derinlikte olduğu rezervuarın baraj arkasında sonsuz uzunlukta olduğu eıw1 harmonik zorlama problemi ele alınmış daha sonra sonsuz uzun sabit derinliğe sahip rezervuarın harmonik zorlama haline ait problemler çözülmüştür.Problemin çözümünde baraj ekseni boyunca çubuk elemanlara ayrılarak sonlu elemanlar yöntemiyle incelenirken,rezervuara ait çözümlerde hız potansiyeli için rijit taban koşulu ve hidrodinamik basıncın sıfır olduğu serbest yüzey koşulunu sağlayan seri çözüm kullanılmıştır.Baraj-rezervuar etkileşimi problemi bir enerji prensibine dayalı varyasyonel yöntem ile çözülmüştür. Söz konusu yöntemde hız potansiyeli ile yapı üzerindeki düğüm noktalarının yer değiştirmeleri serbest değişken olarak seçilmiştir. Yapılan varsayımlar şöyledir: Sıvı, sıkıştırılabilir, homojen ve viskoz değildir. Çubuk elemandaki yer değiştirmeler küçük kabul edilecektir. VII SUMMARY SOLVING DAM - RESERVOIR INTERACTION PROBLEM BY USING VARIATIONAL HYBRID ELEMENT METHOD Exact analytical solutions are derived for simple elastic structures vibrating in water. Linear acoustic and beam theories are used to treat several cases some of which have been studied by more approximate methods before. A hybrid element method which is based on a localized variational principle is demonstrated numerically or a beam - dam; its theoretical bases is then generalized for arbitrary two - dimensional elastic structure. Foundation compliance is not included. In the presence of earthquakes the interaction between an elastic structure and the surrounding fluid can be important in considering the safety of the structure. The 1964 Alaska earthquake caused the first large scale damage to tanks of modern design and initiated many investigations into the dynamic characteristics of flexible containers. In addition, the evoluation of both the digital computer and various associated numerical techniques has significantly enhanced solution capability. Several studies were carried out to investigate the dynamic interaction between the deformable walls of the tank and the liquid. An extensive literature has developed during the past 45 years on the subject of waves caused by vibrating tanks of heavy fluid. Part of this literature appeared in the 1950s and 1960s in connection with the sloshing problem for aircraft fuel tanks. Several examples of this work, which was concerned primarily with calculating natural frequencies and modes rather than time - history responses to specified tank accelerations, are given by Graham and Rodriges and Silverman and Abramson. The second part of this literature originated in 1949 with a paper by Jacobsen and has continued to the present time. The performance of ground - based liquid storage tanks such as petroleum, LNG, LPG, nuclear containment vessels and so forth during recent earthquakes demonstrates the need for a reliable technique to assess their seismic safety. Early developments of seismic response theories of liquid storage tanks considered the container to be rigid and focused attention on the dynamic response of the contained liquid. A common seismic design procedure is based on the mechanical model derived by Housner for tanks with rigid walls. In this approach, a mathematical model of the liquid - rigid tank system was used and the hydrodynamic effects were evaluated approximately as the sum of two components, viz. an impulsive part which represents the portion of the liquid which moves in unison with the tank and a convective component which represent the portion of the liquid sloshing in the tank. Epstien improved Housner's work and presented design curves for estimating the bending and overturning moment induced by the hydrodynamic pressure, for Vfflcylindrical as well as rectangular rigid tanks. The following studies can be indicated as a sample on the dynamic characteristics of the liquid with rigid walls. In engineering problems involved fluid - structure interaction with sloshing, the fluid and sloshing behaviour is determined with a rigid wall assumption and than the structural response is obtained by imposing the dynamic pressure to the structural model. This approach generally yields concervative results since the rigid - wall forces are larger than the flexible wall forces. But an uncoupled analysis underestimates the structural response if the natural frequencies of the coupled system are close to the excitation frequencies, which is often the case in the seismic analysis of liquid - filled tanks and nuclear reactor systems. The exact mathematical procedure for describing fluid oscillations in a moving container is extremely complex. Therefore, the following simplifying assumptions aregeneraly employed: 1. Nonviscous fluid, 2. Compressible fluid, 3. Small displacements, velocities and slopes, 4. İrrotational flow field, 5. Homogeneous fluid., ?'»'. The assumption of irrotational flow ensures the existence of a fluid velocity potential

ÖZET Sıvı dolu tankların titreşimleri ile ilgili problemler son 45 yıldan beridir tartışılmakta ve çözümler üretilmeye çalışılmaktadır. Konuyla ilgili ilk kaynaklar 1950'lerde, uçak yakıt tankları ve sıvı yakıtlı füzelerle ilgili olanlar 1960'larda görülmeye başlamıştır. İlk çalışmalarda sıvı depolayan tank rijit olarak gözönüne alınmış ve tankın içindeki sıvının dinamik davranışı incelenmiştir. 1964 Alaska depreminde sıvı dolu tankların büyük ölçüde hasara uğramasından sonra elastik tankların dinamik karakteristiklerinin araştırılması konusunda önemli bir dikkat sarfedilmiştir. Bu yıllarda sayısal tekniklerle birlikte bilgisayarlardaki gelişmeler çözümlerin sayısını ve hassasiyetini artırmıştır. Problemin matematik formulasyonunun oldukça karmaşık olması nedeniyle genel olarak; 1- Viskoz olmayan sıvı, 2- Sıkıştınlamayan sıvı, 3- küçük yerdeğiştirmeler, hızlar ve eğimler, 4- Çevrintisiz akım alam, 5- Homogen sıvı varsayımları altında çözüm aranır. Islak yüzeyli yapılarda sıvı-yapı etkileşimi problemlerinde; 1- Sıvı hacmi üç boyutlu elemanlara ayrılarak sonlu elemanlar yöntemiyle, 2- Tüm sıvı yüzeyi iki boyutlu sınır elemanları yardımıyla direkt, indirekt veya varyasyonel sınır eleman yöntemleriyle, 3- Sıvının davranışım geneUeştirilmiş fonksiyonlar yardımıyla ifade ederek çözümler üretmek mümkündür. Sınır üzerinde potansiyel ve akım ile bölgede potansiyel gibi üç bağımsız değişken içeren genelleştirilmiş varyasyonele dayalı yeni bir sınır eleman modelleri mevcuttur. Yaklaşım, bölge içindeki potansiyeli klasik temel çözümleri global interpolasyon fonksiyonu olarak kullanır ve bölge integrallerini sınır integraline dönüştürür. O nedenle sonuç denklemler sadece sınır bilinmiyenlerini içerir ve model simetrik matrisler üretir. Bu çalışmada, sıvı-yapı etkileşimi problemleri için birleşik varyasyonel sınır eleman- sonlu eleman yönteminin geliştirilmesi ve ek yapısal elemanların, serbest titreşim frekansına etkilerinin araştırılması üzerinde duruldu. Birleşik smır eleman-sonlu eleman yöntemi, 1970lerden beri kullanılmaktadır. Deprem altında sıvı-yapı sisteminin cosO tipi serbest titreşim modlan kullanıldı, önce direkt sınır eleman ve varyasyonel smır eleman formulasyonlan ile, rijit tankda sıvı titreşimi incelendi. Sonra, birinci mertebe Novozhilov yaklaşım teorisi ve Kirchoff ince plak teorisi kullanılarak elastik kabul ile sıvı-yapı etkileşimi probleminde, birleşik varyasyonel smır eleman-sonlu eleman yönteminin kullanılması ele alındı. Elde edilen denklem takımındaki bilinmeyenler yapıda düğüm noktalarının yerdeğiştirmeleri, sıvıda yerdeğiştirme potansiyeli ve hidrodinamik basmç ifadeleridir. SUMMARY FLUID-STRUCTURE INTERACTION IN THE CYLINDRICAL TANKS BY VARIATIONAL BOUNDARY ELEMENT-FINITE ELEMENT METHOD An extensive literature has developed during the past 45 years on the subject of waves caused by vibrating tanks of heavy fluid. Part of this literature appeared in the 1950s and 1960s in connection with the sloshing problem for aircraft fuel tanks. Several examples of this work, which was concerned primarily with calculating natural frequencies and modes rather than time-history responses to specified tank accelerations, are given by Graham and Rodrigez [1] and Silverman and Abramson [2]. The second part of this literature originated in 1949 with a paper by Jacobsen [3] and has continued to the present time. The performance of ground-based liquid storage tanks such as petroleum, LNG, LPG, nuclear containment vessels and so forth during recent earthquakes demonstrates the need for a reliable technique to assess their seismic safety. Early developments of seismic response theories of liquid storage tanks considered the container to be rigid and focused attention on the dynamic response of the contained liquid. A common seismic design procedure is based on the mechanical model derived by Housner [4] for tanks with rigid walls. In this approach, a mathematical model of the liquid-rigid tank system was used and the hydrodynamic allects were evaluated approximately as the sum of two components, viz. an impulsive part which represents the portion of the liquid which moves in unison with the tank and a convective component which represent the portion of the liquid sloshing in the tank. Epstein [5] improved Housner*s work and presented design curves for estimating the bending and overturning moment induced by the hydrodynamic pressure, for cylindrical as well as rectangular rigid tanks. The following studies can be indicated as a sample on the dynamic characteristics of the liquid with rigid walls [6-12]. The 1964 Alaska earthquake caused the first large scale damage to tanks of modern design and initiated many investigations into the dynamic characteristics of flexible containers. In addition, the evolution of both the digital computer and various associated numerical techniques has significantly enhanced solution capability. Several studies were carried out to investigate the dynamic interaction between the deformable walls of the tank and the liquid [13-57]. The exact mathematical procedure for describing fluid oscillations in a moving container is extremely complex. Therefore, the following simplifying assumptions are generally employed: 1- nonviscous fluid, 2- incompressible fluid, 3- small displacements, velocities and slopes, 4- irrotational flow field and 5- homogeneous fluid. The assumption of irrotational flow ensures the existence of a fluid velocity potential, , which must satisfy the Laplace equationvV=o The mathematical boundary conditions for the solution are : 1- At the rigid tank walls, r=R, the normal component of the tank wall must equal to zero, therefore Here, the comma followed by a subscript designates a partial derivative with respect to radial direction. 2- At the rigid bottom of the tank, z=0, the axial component of the liquid velocity must equal zero, therefore 4- At the liquid free surface, z=H, imposing the condition that the fluid particles must stay on the surface, it follows that where g is the acceleration of gravity. There are two major cases of vibration associated with the system under consideration, for which the circumferential variation of the response is described by cosnO, (n is called the circumferential wave number and 0 is the circumferential coordinate angle). Case-/ corresponds to solutions with n=l and is called lateral sloshing. Case-// is named as breathing vibration, and it corresponds to all vibrations where n does not equal one. For a tall tank, the cas#-type modes can be denoted beam-type modes because the tank behaves like a vertical cantilever beam. This is not true for a broad tank because at 0=0 both the amplitude and the axial distribution of radial displacement are different from those of the circumferential displacement at 0=7i/2. This investigation is concerned for only case-/ which has lateral sloshing modes with w=i [28,46]. The liquid and the shell structure are two separate system that are coupled. Each system, acting alone, has an infinite number of modes of free vibration. If the coupled system is excited with some forcing frequency a>, then the response will also have the same frequency to. The magnitude of the response will depend on the ratio of the forcing frequency