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Sismik yükler etkisindeki silindirik tanklarda mod süperpozisyonu yönetimi ile sıvı yapı etkileşimi problemlerinin çözümü

Başlık çevirisi mevcut değil.

  1. Tez No: 75444
  2. Yazar: MUSTAFA SAATÇI
  3. Danışmanlar: PROF. DR. M. ERTAÇ ERGÜVEN
  4. Tez Türü: Yüksek Lisans
  5. Konular: İnşaat Mühendisliği, Civil Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1998
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Yapı Mühendisliği Bilim Dalı
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 62

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. 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.

Özet (Çeviri)

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.

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