Kule türü yapıların deprem davranışının zemin-yapı etkileşimi gözönüne alınarak incelenmesi
On vibrations of towerlike structures under earthquake excitation considering soil-structure interaction
Yazar: KADİR GÜLER
Danışman: PROF. DR. HALİT DEMİR
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü
Derece: Doktora, 1992
Konu: İnşaat Mühendisliği = Civil Engineering
Dizin: Deprem = Earthquake ; Levhalar = Plates ; Titreşim = Vibration ; Yapı-zemin etkileşimi = Soil-structure interaction

Özet:
Bu tez çalışmasında çekme almayan Winkler zemini ne oturan dairesel plak ve bu plağa merkezinde mesnetlenen kolonun oluşturduğu plak-kolon sisteminin dinamik etkiler altındaki davranışı incelenmiştir. Kolon yayılı kütleli olup, ayrıca yüksekliği boyunca çeşitli seviyelerde toplu kütleler bulunmaktadır. Yüksekliği boyunca kolonun kütlesi ve eğilme rijitliği değişken olup toplu kütlelerin dönme ataletlerinin katkısı da hareket denklemine eklenmiştir. Kolonun mesnetlendiği dairesel plağın hem rijit hem de elastik olması durumları gözönüne alınmıştır. Kolon ve plak hareket denklemleri ayrı ayrı yazılmıştır. Kolon ve plağın elastik olması durumunda titreşim denklemleri diferansiyel denklem türünde olduğundan sistem hareket denklemini elde ederken denklemlerin birleştirilmesinde yaklaşık çözüm izlenmiştir. Kolon için kolonun geometrik sınır şartlarını sağlayan koordinat fonksiyonları kullanılmış, plak için ise plağın elastik yerdeğiştirmelerinin ilgili basit eğilme titreşim modlarının bir toplamı ve Bessel fonksiyonları formunda olduğu kabul edilerek hareket denklemlerine Galerkin yöntemi uygulanmıştır. Geliştirilen bir bilgisayar programı yardımıyla sistemin davranışını gösteren para metrelerin değişimleri elde edilmiş ve grafiklerle gösterilmiştir. Sisteme temelde verilen yatay bir yer hareketi veya kolona yatay olarak uygulanan bir kuvvetle sistem yatay titreşimler yanında düşey olarak ta titreşmekte, sistemin hareketi bu iki titreşimin süperpozisyonu olarak ortaya çıkmaktadır. Bu çalışmadaki analizin amacı yapı -zemin arasındaki ayrılmaların bir plak-kolon sisteminin serbest ve zorlanmış titreşimlerine olan etkisini araştırmaktır. Sistemin serbest titreşimlerine ek olarak harmonik yer hareketi ve El Centro 1940 depremi kuzey-güney bileşeni için davranış spektrum eğrileri elde edilmiştir. Elde edilen sayısal sonuçların incelenmesinden çekme almayan zemin durumunda, sistemin titreşim periyodunun büyüdüğü belirlenmiştir. Sistem serbest titreşime başladığında salınımlar oldukça karmaşık bir yapı göstermektedir. Periyodik kuvvetle zorlanmış titreşim halinde dış kuvvetin frekansının sistemin doğal titreşim frekansından yüksek olması durumunda her iki zemin (çekme alan ve almayan) için davranış spektrum eğrileri birbirine çok yakın olarak ortaya çıkmaktadır.

Abstract:
In this thesis, dynamic response of a plate-column sHstem on a tensionless Winkler foundation is studied. Free and forced vibrations of the system are investigated in detail and numerical results are presented in figures. The system consists of a circular plate and a column. The plate of constant thickness is supported on a tensionless Winkler foundation and carries a uniformly distributed load. On the other hand, the column is supported by the plate and has a continuously distributed mass and rigidity along its height. In addition to the column's own mass, a concentrated tip mass having transversal and rotational inertias is considered when the governing equations of the motion are derived. Both, the cases of rigid plate and elastic circular plate are investigated. The governing equations of the system are obtained by considering the plate and the column separately. At first it is shown that, governing equation of the column can be written as LlU(X.t), e(t)l= IEI(X)U (x,t)l + C U(X,t) + c + tm + £ M.5(X-X. )1 V + £ J. 5 (X-X. ) V - C l. t. I t t I -P (t) 5(X-X. >-P (X,t> = 0 (1) c v d Where EI(X) denotes flexural rigidity, m (X) mass per unit length of the column, C damping coefficient, M. concentrated masses and J.'s their rotatory inertias. It is assumed that the masses are attached at points located at distances of X.'s from the support of the column. U denotes the structural displacement of the column. 5 Di rac delta function, P (t) and P,(X,t) are c dconcentrated and distributed loads which are applied to the column, respectively. V(X,t> denotes the total lateral displacement of the column and it consists of the horizontal ground motion, the displacement due to the rotation of the support and the elastic displacement of the column; V = U + © X + U , (2) 9 where U (t> denotes the horizontal displacement of the g ground motion, © , denotes i. o bending moment and EI flexural rigidity at the bottom of the column. On the other hand, in the case of elastic plate- column system, the dynamic governing equation of the plate can be written as LCR.e.tl- DAAW + Cf W H - 3P(t> 2M(t> - ph(g - W > - Q K 5 6(©> - - B - S<©> 5 denote polar co-ordinates, D flexural rigidity of the plate, W(R,©,t) displacement function of the plate, C and K damping coefficient and stiffness of the foundation, respectively, p is unit mass of the plate, XXXP(t) and M(t) are the force and moment at the center of the plate respectively. They are the normal force and bending moment of the column at the support» i.e., at the point where the column Joints the plate. H(R,e,t) denotes the contact function and it is defined as follows. H = 1 for W > 0 <6.a) H = 0 for W < 0 <6.b> It means that the function is unity for a point when the contact is occurs at that point, and it vanishes for a point when a lift-off takes place at that point. However, the full contact throughout the plate surface can be assumed when the contact function is assumed to be equal one checking if the displacements correspond to the contact. This case corresponds to the conventional elastic foundation of Winkler type. As explained above, the governing equations of the motion is written by considering the plate and column separately. These two systems of the equations are coupled through the equilibrium condition and the continuity of the deformation at the Joint between the plate and column. The vertical force and the moment equilibriums at the Joint between the plate and column can be written as follows P 1 c i v W M and M ; where as the suitable combination of equations <1>, (5>, <7) and <8> yields the governing equation of the elastic plate and elastic column. Although equations (1) and (B) are of continuous form i.e., they are differential equations with respect to X, whereas equations (3), (4), <7> and (8) are of discret form. For the discretization of the differential equation, the Galerkin method is adopted. For this purpose the lateral displacement is approximated as oo _ V = U + T T (t> U CX> <9> g r> n n=l where the co-ordinate functions U + (1-v) n2 l(n-l)J -X J 1 n r> n n+ i r> r> n+ 1 n X2 = 0 (15) F Thus, the governing equation of the elastic plate-column system, equation (7), (8), (10) and (15) can be combined in the_matrix form_as it is done with equation (11) where VT = £T,T, T,f, T,T 1 Although the matrix,v 12 O X 1 2 equation (11) represents the small amplitude motion of the plate-column system, it is actually highly non-linear due to the tensionless character of the foundation. Computer programs are developed for the rigid plate-column and for the elastic plate-column system in order to evaluate the behaviour of the systems. In order to cover a wide range of the parameter combinations the non-dimensional i zed quantities are used and the numerical results are illustrated in various figures comparatively for the conventional as well as for the tensionless foundations. As a first step the static behaviour of the circular plate on a tensionless Winkler foundation is obtained under an axial force and a moment at the centre of the plate. The relationship between the external moment and the rotation at the middle of the plate is obtained and represented in figures for various values of the axial force and those of the foundation rigidity. Displacements of the plate are determined at three points when a lift-off arises. Moreover, the behaviour of the rigid plate-column system subjected to an external harmonic horizontal force applied at the top of the column is obtained. The oscillations of the system is illustrated in figures. The inspection of these figures reveals that in case of an lift-off of the plate the system softens and vibration period of the system becomes higher. Horizontal displacements at the top of the column and at the centre of the plate experience larger values when the non-dimensional frequency of the applied force is lower. The full contact develops when the frequency of the external force is big. Furthermore, xxi ithe behaviour of the elastic plate-column system are obtained and illustrated in figures for free vibration. Besides the free vibrations of the system, the forced vibrations of the system are investigated for an assumed harmonic ground motion excitation. Additionally, the dynamic response of the system to the El Centro 1940 ground motion is presented. From various conclusions of the study the followings are of particular interest: The period of the system becomes larger when the foundation is of tensionless character. The oscillations of the system are quite complex when the system experiences free vibrations and when the lift-off is present. However, if the frequency of the external excitation is large, the response spectra of the system are almost the same for these two types of the foundation. In addition to these parametrical and numerical results, chimney and water tower problems available in the literature are also solved and their first mode periods are obtained. It is seen that the present model yields quite accurate results for the problems mentioned. As it is shown, the periods of the first modes agree well with the results given in the literature. The numerical calculation is carried out by considering elastic and rigid plate-column systems. The present study enables the analysis of the structures of tower type by considering the interaction between the plate-column system and the elastic foundation. XXlll