Varvasyonel hibrit eleman metodu ile baraj-rezervuar etkileşim probleminin çözümü
Solving dam-reservoir interaction problem by using variational hybrit element method
- Tez No: 66794
- Danışmanlar: PROF. DR. M. ERTAÇ ERGÜVEN
- Tez Türü: Yüksek Lisans
- Konular: İnşaat Mühendisliği, Civil Engineering
- Anahtar Kelimeler: Barajlar, Rezervuarlar, Sonlu elemanlar yöntemi, Dams, Reservoirs, Finite element method
- Yıl: 1997
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: İnşaat Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
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
Özet (Çeviri)
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, <P which must satisfy the Helmholtz equation c~ 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 o, then the response will also have the same frequency o. The magnitude of the response will depend on the ratio of the forcing frequency co 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. In recent years, the finite element method has become widely accepted by the engineering professions as an extremely valuable method of analysis. Its application has enabled satisfactory solutions to be obtained for many problems which had hitherto been regarded as insoluble, and the amount of research effort currently being devoted to the finite element method ensures a rapidly widening field of application. IXThe finite element method is extremely useful for modelling complicated structures and machines such as the aircraft and bridge. The Finite Element Method is a powerful numerical technique that uses variational and interpolation methods modeling and solving boundary value problems. The method is also extremely useful for complicated devices and structures with unusual geometric shapes (e.g., trusses, frames, machines parts). The finite element is very systematic and modular. Hence the finite element method may easily be implemented on a digital computer to solve a wide range of practical vibration problems simply by changing the input to a computer program. The finite element method approximates a structure in two distinct ways. The first approximation made in finite element modeling is to divide the structure up into a number of small simple parts. These small parts are called finite elements and the procedure of dividing up the structure is called discretization. Each element is usually very simple, such as a bar, beam, plate, which has an equation of motion that can easily be solved or approximated. Each element has endpoints called nodes, which connect it to the next element. The collection of finite elements and nodes is called a finite element mesh or finite element grid. 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. Another possibility is to use approximating functions that satisfy the governing equations in the domain but not the boundary conditions. These techniques are called boundary methods and for a number of reasons they have developed slowly up to the present. Boundary methods can be of as many different types as c domain ' methods, ranging from simple techniques such as the so called 'indirect ' method to the more versatile 'direct ' techniques. Both techniques will be explained in more detail in what follows but it is interesting to point out that they can be interpreted as different weighted residual formulations. The Boundary Element Method is a technique which offers important advantages over 'domain' type solutions, such as finite elements and finite differences. One of the most interesting features of the method is the much smaller systems of equations and considerable reduction in the data required to run a problem. In addition the numerical accuracy of boundary elements is generally greater than that of finite elements. These advantages are more marked in two and three - dimensional problems. The method is also well suited to problem solving with infinite domains such as those frequently occuring in soil mechanics, hydraulics, stress analysis etc. for which the classical domain methods are unsuitable.The term ' Boundary Elements ' originated within the Department of Civil Engineering at Southampton University. It is used to indicate the method whereby the external surface of a domain is divided into a series of elements over which the functions under consideration can vary in different ways, in much the same manner as in finite elements. This capability is important as, in the past, integral equation type formulations were generaly restricted to constant sources assumed to be concentrated at a series of points on the external surface of the body. 3. Representation of fluid behaviour by a series of generalized functions. Using the Finite Element Method, one can obtain a linear interaction problem for both linear the structure and the fluid whichcan be solved by standart 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 The problem of dam - reservoir interaction has received considerable attention. The similar problem of an ocean structure excited by earthquakes has also been studied before. Linear elasticity and linear acoustics in water are frequently assumed for the frequency range typical of earthquakes. İn additions, many existing studies assume, for simplicity, the ground to be rigid and its motion to be prescribed. İn more recent studies the important effect of ground compliance has also been included to account for radiation damping; the required analysis often involves simplifications which are either valid only for a limited range of frequencies or result in expensive computations. In cases where ground compliance is ignored (i.e. rigid ground) available solutions are also approximate in varying degrees. Sometimes only the lowest few modes are included; this is probably very good for some practical purposes when the earthquake spectrum is narrow banded and the peak frequency is low. For earthquakes of short duration, it is desirable to have accurate information of the frequency response for as wide a range as possible before Fourier synthesis. The method of Finn and Varoğlu which is exact in principle, is to express the beam and water motions in terms of their respective eigen modes. Suitable matching conditions lead to infinite by infinite matrix for the expansion coefficients. As the matrix elements need to be computed numerically, the theory, strictly speaking, is semi - analytical. The numerical method of Finn and Varoğlu applies only to a reservoir of constant depth. In this study, we first report some exact solutions in series form whose expansion coefficients are obtained analytically. Therefore the final results involve only numerical summation of infinite series. The analytical method is convenient only for the special case of a thin beam or column of uniform rigidity. We next describe a more general hybrid element method based on a localized variational principle. This method combines an analytical solution in most of the XIsurrounding water with finite elements only in the structure and its neighbouring water where the depth is constant. In this study, finite element method is used in examining beam element - liquid interaction problem. Problems about harmonic motion of the reservoir that has infinite constant deep is solved. In solution of problem, beam element is examined by using finite element method and the series solution that is employing rigid base condition for velocity potential and free surface condition where the hydrodynamic pressure is zero for solution about reservoir. The dam - reservoir interaction problem is solved by variational method that is based an energy principle. xn
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