Üstyapı-zemin ortak sisteminin deprem hesabı

Abstract

ÖZET Bu çalışmada, üstyapı ile zeminden oluşan ortak siste min deprem hesabı için matematik modeller geliştir ilmi ş,bu modeller çerçevesinde üstyapı ile zemin arasındaki dinamik karşılıklı etki olayı incelenmiştir. Üstyapı-zemin ortak sisteminin deprem etkisi altındaki dinamik hesabı en genel şekli ile ele alınmış; geliştirilen modellerin sağladığı olanaklardan yararlanılarak» bir ve birden fazla üstyapıyı kapsayan ortak sistemlerin titreşim özellikleri araştırıl mıştır. Çalışma, oniki bölümden oluşmaktadır, Birinci bölümde konumun tanımı yapılmakta, dinamik karşılıklı etki olayı ile ilgili olarak daha önce. yapılan çalışmalar iki grup halinde incelenerek özetlenmektedir. Çalışmanın ikinci bölümü, üstyapı-zemin ortak sistemin de matematik model seçimi problemine ayrılmıştır. Bu bölümdes ortak sistemin idealieştirilmesi için daha önce kullanılan modeller eleştirilmiş, bu çalışmada geliştirilen modellerin özellikleri ve problemin çözümünde sağladığı olanaklar ay rıntılı olarak açıklanmıştır. İkinci bölümün son kısmında ise, geliştirilen her iki modelde de yer alan tek tabakalı sonsuz ortamın ayrıklaştırma prensibi açıklanmıştır. Üçüncü bölümde, dinamik rij itlik matrisi kavramı üze rinde durulmuş ve tek tabakalı sonsuz ortam için yaklaşık bir dinamik rij itlik matrisi Önerilmiştir. Yarı sonsuz or tam özel durumu için elde edilen kesin çözümün sonuçlarm- 'dan yararlanılarak, tek tabakalı ortamda frekansa bağlı bir yayılı kütle matrisinin tanımlanabileceği gösterilmiş ve önerilen dinamik -rij itlik matrisinin yaklaşıklığı irdelen miştir, Ayrık bir sistem. olarak idealleştiriien tek tabaka lı ortamda, birim durumların tanımladığı sınır şartlarına göre yapılan elastisite çözümü sonucunda,, ortamın birim dep lasman sabitleri ile birim ivme sabitleri elde edilmiştir.-3-1 Dördüncü bölümde, üstyapı ve zemin ile ilgili idealleş tirmelerden bağımsız olarak, deprem etkisi altında üstyapı- zemin ortak sisteminin hareket denklemleri çıkarılmıştır.Ze minin lineer elastik olup olmamasına bağlı olarak, üstyapı ve deprem koşulları ile ilgili çeşitli durumlar için hare ket denklemlerinin özel şekilleri elde edilmiştir. Çalışmanın beşinci bölümünde, tek tabakalı sonsuz or tam sınırında yer alan sonsuz rijit temel plakları için, or tamın rij itlik ve kütle matrisleri elde edilmiştir.Bu amaç la, karışık sınır değer probleminin ayrık çözümü için bir yöntem geliştirilmiştir. Bu yöntem, ortam sınırında birden fazla temel bulunması durumunda da uygulanabilmektedir. Bu çalışmada, tek temel plağı ile birlikte, yanyana aynı fazda titreşen iki eş temel plağı için ortamın rij itlik ve kütle matrisleri elde edilmiştir. Altıncı ve yedinci bölümler, üstyapı-zemin ortak sis temlerinin serbest titreşim hesaplarına ayrılmıştır. Bu he saplar için kullanılan ortak sistem modellerinde zemin orta mının rij itlik ve eylemsizliği, beşinci bölümde elde edilen küçük boyutlu rij itlik ve kütle matrisleri aracılığı ile ba sit bir şekilde gözönüne alınabilmektedir. Altıncı bölümde, tek üstyapı ile zeminden oluşan ortak sistemin serbest tit reşim denklemi ayrıntılı olarak yazılmış, birinci titreşim frekansının hesabı ve transfer fonksiyonlarının elde edilme si için izlenen yöntemler açıklanmıştır. Bu konudaki litera türde pek az rastlanan birden fazla üstyapı durumuna bir örnek olmak üzere, yedinci bölümde iki eş üstyapı ile zemin den oluşan ortak sistemin serbest titreşimi incelenmiştir. Çalışmanın sekizinci bölümünde, üstyapı-zemin ortak sisteminin deprem hesabı için uygulanan yöntem, ayrıntılı olarak açıklanmıştır. Deprem hesabı için bu çalışmada ge liştirilen ortak sistem modeli çerçevesinde hareket denkle minin kuruluşu incelenmiş, çözüm için kullanılabilecek sa yısal yöntemlerin özellikleri açıklanmıştır. Dokuzuncu bölümde, sayısal sonuçların elde edilmesi için hazırlanan Elektronik Hesap Makinası programlarının ay rıntıları açıklanmıştır.Ill Onuncu bölüm, sayısal örneklere ayrılmıştır, tik iki örnekte, tek ve yanyana iki eş üstyapıyı kapsayan ortak sistemlerin serbest titreşimi ile ilgili sayısal sonuçlar elde edilmiştir. Üçüncü örnekte ise, sekizinci bölümde açıklanan yönteme göre üstyapı-zemin ortak sistemi gerçek bir deprem kaydı için hesaplanmış, sayısal sonuçların dep rem süresince değişimi elde edilmiştir. Sayısal sonuçlar; üstyapı, temel ve zemin parametrelerine bağlı olarak, zemi nin varlığının üstyapının davranışına önemli derecede et ki edebileceğini göstermiştir. Onbirinci bölümde, bu çalışmada elde edilen genel so nuçlar açıklanmıştır. Çalışmanın ekleri, onikinci bölümde toplanmıştır.Tek tabakalı ortam sınırında birim durumların tanımladığı sı nır şartlarına göre yapılan elastisite çözümü ve çözüm so nucunda elde edilen gerilme ve deplasman alanları, bu bölü mün başında yer almaktadır. Daha sonra, statik birim dep lasman sabitleri ve birim ivme sabitlerinin kapalı sonuç ları integral ifadeler halinde verilmekte ve sayısal integ- rasyonda izlenen yöntemler açıklanmaktadır.

IV SUMMARY This work deals with the analysis of earthquake response of structures including soil-structure interaction. In recent years, many important structures such as tall buildings, dams, nuclear power plants etc., have been constructed in seismic zones, even on weak or moderate soil conditions. It is evident that, in the earthquake response analysis of such structures, besides the dynamic characteristics of the superstructure, those of the soil medium have to be considered. It is known that the soil conditions affect the seismic waves transmitted through the soil medium to the structure. Soil effect, which is independent of existing structure, causes in response an opposite effect, due to transmission of the stress waves from the vibrating structure into the soil. This dynamic interactive behaviour of soil and structure constitutes the subject of `soil-structure interaction`. In the first chapter of this study the problem is defined and a detailed review of the previous work is given. The previous investigations can be seperated mainly into two groups regarding different idealization procedures used for the soil medium. In the first group of investigations, the structure is assumed to be founded on a half space or a stratum through an infinetely rigid foundation plate. This approach utilizes the solution of steady state vibration of rigid circular, rectangular or strip plates bonded on a half space or on. a stratum. The resulting force-displacement relationships of the rigid plate constitute the complex compliance matrix of the soil medium. The imaginary part of the compliance matrix corresponds to the energy loss due to the infinite character of the medium. In the analysis of soil-structure systems, the soil medium can thus be considered as a simple subsystem, by the use of compliance matrices. But this model has some serious restrictions concerning the real dynamic behaviour. of the soil and the solution procedure of the soil- structure system : a) Homogeneous and linearly elastic soil assumption seems to be an. unrealistic approximation under actual soil conditions b) Since the elements of complex compliance matrices are frequency dependent, earthquake analysis of the soil-structure systems can only be done in frequency domain using Fourier Transform Technique, In the second model used in the previous work, the soil medium is assumed to be finite and composed of some discrete elements. The Finite Element Method is generally used for the discretization. In the solution of soil-structure systems, Finite Element Method has many advantages, such as; a) Geometrical» mechanical and constitutive complexities and irregularities of the soil medium can be taken into account properly. b) It is not necessary that the foundation plate be rigid as in the half space model and embedment of foundation can easily be considered. c) Earthquake analysis of soil-structure systems can be made directly in time domain, thus nonlinear soil and/or structural response can be taken into account using step by step integration algorithms. Although the discrete model is efficient and provides some flexibilities, it has also some drawbacks for the solution of the problem :. a) Large computational effort and computer storage are needed for sufficient; idealization. Since transmission of waves through the soil medium is a wave propagation problem, the discrete model has to contain enough number of nodal points for proper definition of waves.VI b) Since the model is finite, radiative energy loss can not be taken into consideration due to reflection of waves from the boundaries of the model. The second chapter of this investigation is devoted to the choice of appropriate models for the analysis of soil- structure systems. Two different models are developed : The first intended to be used in the earthquake response analysis and the second to obtain rather qualitative results about the dynamic behaviour of soil-structure systems. The second model is also convinient for the interaction analysis of multi-structural systems. In the first model, which is called as Model I, the soil medium is idealized as two subsystems (Fig. II. 2). In the vicinity of the structure, the soil is discretized by means of two dimensional finite elements whereas the rest of the medium is assumed to be a linear elastic and homogenous stratum overlying the bedrock. Since the major soil deformations caused by the vibrating structure occur in the vicinity of the foundation, the finite element approximation can be used efficiently in this finite zone of the soil medium. On the other hand, the soil deformations decrease in the far region of the structure and therefore can be expected to be in the elastic range. This part of the soil medium which is idealized as a stratum, can thus be assumed to be an elastic sub-region. The second model developed in this study is rather simple. In this model the whole soil medium is idealized as a linear elastic and homogeneous stratum which has been already used in the first model as one of the subsystems. In both models the soil stratum is discretized by means of nodal points which are defined on the surface of the medium (Fig. II. 1). The discretization procedure used herein is essentially the same as used by. Chopra and Perumalswami [l3J, for the discretization of half plane. The linear variation of surface displacements provides the geometrical continuity between the two subsystems of Model I and also an efficient variation for the discrete solution of the mixed-boundary value problem which will be discussed in Chapter V.VI 1 One of the most important problems which arises in the steady state vibration, of half space or stratum is the energy loss due to radiation. Energy loss coefficients are frequency dependent and appear in the solutions as pseudo- viscous damping which is called in the literature as `radiation damping` or `geometric damping`. From the point of modal behaviour of soil-structure systems, it is a known fact that the participation of the first natural mode in the total response of systems is of prime importance [40,60J. Since radiational damping of the soil is proportional to the excitation frequency, for the lowest frequency of vibration this type of damping takes its minimum value. On the other hand, there are many uncertainties concerning the real nature of material damping in the soil and the structure. Under these circumstances, radiational energy loss mechanism of the soil has not been taken into consideration in this study and a quasi-static approach is utilized to obtain the dynamic characteristics of the soil medium. In the third chapter of this study, an approximate dynamic stiffness matrix for the discretized stratum is obtained. Using the discrete model shown in Fig. II. 1, the soil medium can be treated as a single `finite element`, with theoretically infinite but practically finite number of nodal points. The dynamic influence coefficients of the medium could be determined as nodal forces related to harmonic displacements with unit amplitude. However in this study, these coefficients are evaluated in an approximate manner by introducing the `consistent mass matrix` concept. As known from the theory of Finite Element Method, this concept is based on the approximation that inertia forces in the medium are proportional to the quasi-static acceleration field. Thus, approximate dynamic stiffness matrix of the medium is defined as, Md=Ms-w2[m] (III. 3) where [k],, [k] and [m] represent the dynamic and static stiffness matrices and consistent mass matrix respectively; w denotes the frequency of steady state vibration. The elements of static stiffness matrix are obtained as nodal forces related to unit displacements, from the distributedvııı stresses on the surface which are determined by the plane strain solution. The elements of the mass matrix, i.e. inertia influence coefficients, are evaluated as nodal forces related to unit accelerations, through the static displacement field in the medium. It has been shown that Eq.(III.3) corresponds to the first two terms of the series solution of the steady state dynamic problem with the exception that zero lower limits of real Fourier inversion integrals are replaced by the nondimensional frequency aQ. IThe first static term [k(0)]s does not differ from [k(a0)]s for low frequencies, whereas inertia influence coefficients are determined as functions of a. In the frequency range of interest, in connection to the earthquakes, taking the first two terms of the series solution is shown to be sufficient. In the fourth chapter, the equations of motion of soil-structure systems are obtained. The problem is considered in a general manner, independent of idealization procedures used for the soil and the structure. In soil-structure systems, total displacements can be seperated into two parts as follows; [d] - [d]a ? [dj where [d] a represents earthquake free field displacement vector which_may have been recorded anywhere in the soil medium and [d] corresponds to interaction displacements due to vibrating structure. In this study, equations of motions are formulated in terms of interaction displacements. In the first section of Chapter IV, a general formulation of the problem, including the `traveling earthquake waves`, is given. In the following sections, the equations of motion are obtained for some special cases including rigid foundations. In the last section nonlinear soil-structure systems are formulated and nonlinearity of soils is discussed. The fifth chapter is devoted to the evaluation of stiffness and mass matrices for rigid strip plates bonded on an elastic soil stratum. In the analysis of rigidIX f oundations, an important problem arises in the application of mixed boundary conditions, especially in the case of multiple foundation systems. In this study, utilizing the discrete soil model, a procedure is developed for the solution of mixed boundary value problem. In the first step of the discrete solution procedure, the number of degrees of freedom of the contact surface is reduced to the rigid body degrees of freedom of the foundation. In the second step, the homogeneous equilibrium conditions are applied as a matrix condensation operation for the degrees of freedom of the free surface, i.e. the outer nodes of the foundation. This operation can be done by a standart condensation procedure, such as the Gauss-Jordan algorithm. The discrete procedure which is developed for the mixed boundary value problem can also be applied to multi- foundation systems. In this study, the stiffness and the mass matrices of the soil medium are obtained for two identical in-phase rigid foundations with a given spacing. According to the numerical results plotted in Fig. V.6, the rocking stiffness of each foundation increases and the swaying stiffness decreases with decreasing spacing of the two foundations. It is interesting to note that, in almost all previous studies, soil-structure system has been assumed to have a single superstructure. It is evident that interaction occurs not only between the soil and the single structure, but also between the neighbouring structures through the soil. The `cross interaction`, as called by Kobori and Minai [29], has a great importance in connection with the dense construction situations in big cities. In the sixth and the seventh chapters of this study, the vibrational characteristics of soil-structure systems are analyzed. In addition to the soil-single building system, a system which consists of two identical buildings is considered as an example for the cross interaction. Details of the analysis of the first natural frequencies and the displacement transfer functions of the soil-structure, systems are given.X In the eighth chapter, a solution procedure for the earthquake response analysis of the soil-structure system is presented. The elements of dynamic property matrices are described in detail and the numerical methods of the time- domain solutions are discussed. /The nineth chapter of the study is devoted to computer programs which are developed for the numerical analysis. The flow charts of the programs and the description of input data are given in detail. In the tenth chapter, some numerical examples are presented. In the first two examples, the first mode behaviour of soil-structure systems is analyzed. In both examples, a ten-story framed building /tfith a rigid foundation is considered (Fig.X.l). The first example deals with the soil-single building system. It has been shown that soil and structural parameters and the height/width ratio of the building affect the first natural frequency of vibration. Numerical results are outlined in Table X.l. In addition, the amplitude transfer functions of the tenth story and the base displacements are obtained and plotted versus frequency (Fig.X.2 to Fig. X. 12). In the second example, a soil-structure system consisting of two identical buildings is analyzed (Fig. X. 13). In order to compare the results, the building used in the first example is considered as one of the two buildings. In addition to other parameters, the effect of the spacing of the two buildings is also studied. As it is shown in Table X.2, the first natural frequency of vibration increases with the decreasing spacing of the two buildings. The amplitude transfer functions of the tenth story and the base displacements are shown in Fig.X.13 to Fig. X. 19. It is observed in both examples that the amplitudes of relative displacements of the buildings decrease with decreasing frequencies whereas those of the total displacements increase. These results can be interpreted as follows : The decrease of the first natural frequency, mainly depends on the rotation of rigid foundation which produces the large quasi-static displacements in the building. As a consequence of the decreasing frequency, induced inertia forces and the relative displacement amplitudes decrease. Although it is not shown here, for very rigid and low buildings with small height/width ratios, the first natural frequency of two-building system may decrease with decrasing spacing of the buildings due to the decreasing variation ofXI swaying stiffness of the soil (See Fig.V.6). In the third example, a soil-structure system is subjected to a real earthquake motion and time-histories of the roof displacement and the base shear of the building are obtained. In the soil-building system shown in Fig. X. 20, the soil medium is assumed to be composed of three layers where the one overlying the bedrock is idealized as an elastic stratum. In order to compare the results with the case of rigid soil, the building system is analyzed seperately. The earthquake record of `Latino Americana Tower` of May 19, 1962 is taken as input data and the equations of motion are integrated step by step using the modified Linear Acceleration Method [7l]. The time-histories of the tenth story displacements and the base shear of the building are given in Fig. X. 21 to Fig. X. 25. The results show that, in the case of flexible soil, an appearent increase occur in the period of vibration and the amplitudes of responses. The conclusions of this study are presented in the eleventh chapter and the Appendices are given in the last chapter.BÖLOM I G İ R î Ş 1.1. KONU I. 1.1. Tanıra Yüksek binalar, barajlar, nükleer güç santral lan gibi, depreme karşı davranışlarının önemli olduğu bilinen yapı sistemleri, günümüzde zorunlu olarak aktif deprem bölgele rinde de yapılmaktadır. Bu tür yapıların gerekli bazı durum larda, çok değişik özellikler taşıyan zeminler üzerinde ku rulması zorunluluğu, üstyapı ile zemin arasındaki `dinamik karşılıklı etki` probleminin konusunu oluşturmaktadır. Yapa Mühendisliği açısından deprem, birtakım jeofizik nedenlerle oluşan yer hareketinin etkisi ile üstyapının tit reşimi olayıdır. Ancak üstyapının doğal olarak belirli bir zemin ortamı üzerine oturmuş bulunması, bu titreşim olayın da zeminin de üstyapı ile birlikte gözönüne alınmasını zo runlu kılmaktadır. Üstyapı ve zeminden oluşan `ortak sistem` çerçevesinde dinamik karşılıklı etki süreci kısaca şu şekil de tanımlanabilir : Deprem nedeni ile zemin ortamından üstyapıya aktarılan titreşimler burada, üstyapının dinamik özelliklerine bağlı birtakım etkiler meydana getirmektedir. Böylece bir titre şim kaynağı durumuna gelen üstyapı bu kez zeminde, deprem titreşimlerine ek olarak, zeminin dinamik özelliklerine bağ lı birtakım karşı etkiler oluşturmaktadır. Günümüzde deprem hesabı için uygulanmakta olan alışıla gelmiş yöntemler ve üstyapıya gelen dinamik etkileri belir leyen deprem yönetmelikleri, zemini genellikle üstyapının titreşimlerinden etkilenmeyen bir ortam olarak kabul etmek tedir. Zeminin sonsuz rijit olarak alınması bu kabulün doğal