Modal analiz yöntemi ile kat kuvvetleri hesabına kolonlarındaki normal kuvvetin etkisi
Başlık çevirisi mevcut değil.
- Tez No: 56039
- Danışmanlar: PROF.DR. NAHİT KUMBASAR
- Tez Türü: Yüksek Lisans
- Konular: İnşaat Mühendisliği, Civil Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1996
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 100
Özet
ÖZET Bir yapı dinamiği problemi statik probleminden iki önemli noktada ayrılır. Birinci fark yapı dinamiği probleminin zamana bağlı olmasıdır. Çünkü yük ve yapının bu yük altındaki davranışı zamanla değişir. I/K') lp 1 vsmtv/M atalet kuvvetleri ? Şekil A: Statik yükleme Şekil B: Dinamik yükleme Statik ve dinamik problemi arasındaki ikinci önemli fark şekil A ve şekil B'de gösterilmiştir. Eğer bir basit kiriş şekil A'da gösterildiği gibi statik bir p yüküne maruzsa, bu kirişin kesit tesirleri ve deformasyonlan direk olarak verilen p yüküne bağlıdırlar. Diğer yandan eğer p(t) yükü şekil B'de gösterildiği gibi dinamik olarak etkitilirse kirişin nihai şekil değiştirmesi ivmelere bağlı olur ki, bu ivmelerde kirişte kendilerine direnen atalet kuvvetleri doğururlar. Böylece şekil B'deki kirişte moment ve kesme kuvveti şeklindeki iç kuvvetler sadece dış yükleri değil ivmelerden doğan atalet kuvvetlerini de dengelemek zorundadırlar. Şekil B'deki dinamik sistemin analizi oldukça karmaşıktır. Çünkü yapıda atalet kuvvetleri doğuran yerdeğiştirmeler aynı zamanda atalet kuvvetlerinin büyüklüğüne bağlıdır. Atalet kuvvetleri ile yerdeğiştirmeler arasındaki bu karşılıklı bağımlılık gözönüne alındığında, problemin sadece diferansiyel denklemlerle ifade edilebileceği anlaşılır. Ayrıca kirişin kütlesi tüm uzunluğu boyunca düzgün olarak dağıldığından, eğer atalet kuvvetleri tam olarak belirlenecekse, kiriş ekseni boyunca yerdeğiştirmeler ve ivmeler tanımlanmalıdır. Bu durumda kiriş üzerindeki noktaların yeri ve zaman birbirinden bağımsız değişkenler olarak gözönüne alınacağından, olay kısmi diferansiyel denklemlerle formüle edilebilir. | Çok katlı binalardaki deprem etkilerinin hesabında çerçeveleri kayma çerçevesine dönüştürüp, hesaplan basitleştirmek için çoğu kere kolonlardaki normal kuvvetler ihmal edilir. Bu çalışmada kat kuvvetlerini mod süperpozisyonu yöntemi ile hesplayan bir bilgisayar programı hazırlanmış ve daha sonra bir açıklıklı bir çerçeve gözönüne alınarak kat sayısı 2'den 40'a kadar artırılıp çerçevelerdeki kat kuvvetleri kolonlardaki normal kuvvetlerin ihmal edilmesi ve edilmemesi halleri için ayrı ayrı hesap edilmiştir. Bu şekilde bulunan sonuçlar değerlendirilerek kolonlardaki normal kuvvetleri ihmal etmenin doğurduğu relatif hata miktarları hesaplanmıştır. Bu hata miktarlarının kimi durumlarda % 20 mertebelerine kadar vardığı gözlenmiştir. viii
Özet (Çeviri)
THE INFLUNCE OF NORMAL FORCES OF THE COLUMNS, TO THE CALCULATION OF STOREY FORCES USING MODE SUPERPOSITION METHOD SUMMARY An important fundamental for engineers to understand is how earthquake- induced forces are translated into the building. When any rupture occurs in an earthquake fault zone it produces a multitude of vibrations or seismic waves mat emanate in all directions. Although the focus of earthquake is identified as the initial point of energy release and source of rupture, the actual surface faulting that follows may spread in a predominant direction for many miles from this central spot. In any event, the result is a random pattern of ground-shaking motions that is felt throughout the entire impacted area. The effects of this ground shaking may be very severe on a building in terms of the motions amplitude, velocity, acceleration, displacement, and duration, depending of the earthquake. Essential, this ground shaking is caused by the seismic body and surface waves that are propagated in all directions (Figure A). These motions are translated into dynamic loads that cause the ground, and consequently any buildings located on that shaking ground, to vibrate in any complex manner. The result is that the structure is subjected to a combinations of horizontal and vertical loads introduced through the foundation system fixed to the ground. Of the two loads involved, the horizontal one is greater by far, although the vertical forces are currently under considerable study. LOVEWAVE Figure A: An earthhquake produces a set of primary seismic waves that are propagated in all directions and produce a random pattern of ground-shaking motions. IXAnother way of looking at the manner in which the seismic forces are translated to the structure is to realize that whereas the building foundation is fixed to the ground, thereby having a general tendency to be displaced along with the site, the upper portion of the building, or superstructure, is not,it is therefore free to move horizontally or oscillate at its own natural period. In tall building, depending on which mode of vibration the building enters, some floors may be moving in one direction while others may move in another at the upper stories. Interstory drift between floors may occur, leading to a foreshortened distortion of the building's stories, as illustrated in Figure B Figure B: Drift diagrams indicating lateral displacement and resulting foreshortening of a building system. In relatvely tall strucures, some floors tend to move in aone direction while floors above and below may move in opposite directions. Ordinarily, if dynamic lateral forces, such as those caused by earthquake or wind, were not taken into consideration. The structural design of a building system would be a relatively simple process, since only vertical gravity forces need be taken into account. In such instances, all forces produced by dead loads and live loads are calculated in terms of gravity (lg) and applied vertically. As these loads are considered static, there is no need for structural components, technically speaking, to resist any dynamic lateral forces (see Figure C-(l)). However, in the case of earthquake loads acting on a building, it is a totally different solution (see Figure C- (2)). [ (D (2) ? ? ? yi. (3) Figure C: (l):Building system subjected to vertical gravity loads/static forces only; (2): building system subjected to earthquake- induced ground motions and dynamic lateral and vertical forces; (3) combination of gravity/static, dynamic lateral, and dynamic vertical loads applied a building systemEarthquake loads are dynamic and are applied laterally (horizontally) to a building system. When forces induced by an earthquake are spread throughout the buildings as horizontal forces, their accumulations are applied by convention to the structural system at each floor line and roof level (see Figure C-(3)). The rationale assumptions behind this convention is the assumption that it is through the floor and roof systems that horizontal forces are distributed throughout the rest of the building eventually down through the vertical supports into the foundation. Once the forces are applied to the floor and roof levels and distributed to the other components of the basic structure, a vertical bracing system must then be designed to resist these forces and carry them throughout the rest of the building in a logical manner without breaking continuity in the integrity of the overall structural system. Basically then, we must design the vertical supporting system with the capacity to resist the lateral forces applied at the floor and roof levels throughout the building without excessive distortion and/or failure. An analogy to this is building a simple“house of cards”. As we discovered as children the house of cards was stable and remained intact as long as all the loads were predominantly vertical. However we soon learned that if someone pushed horizontally against the house of cards with a lateral force larger than the vertical pull of gravity holding these cards together, the house collapsed, much to everyone's dismay. To take this analogy one step closer in terms of earthquake loads, we also discovered that if someone gave the table on which the house of cards was assembled a hard, sharp shove laterally, a similar collapse of structure would take place. By visualizing the moving table as inducing motions into the house of cards, and noting its similarity to the dynamic ground shaking produced by an earthquake, we now have an excellent example of what type of action may occur to cause the collapse of a laterally unbraced structure impacted by a severe seismic event. In effect, we have produced a simple earthquake simulation model quite similar in basic principle to the engineering“shaking tables”used for experimental research in the laboratory testing of structural building systems and components. The conclusion to be drawn from this“house of cards”analogy is that some type of lateral bracing system must be introduced into the basic structural system to avoid collapse or failure. A stinctural-dynamic problem differs from its static-loading counterpart in two important respects. The first difference to be noted, by definition, is the time-varying nature of the dynamic problem. Because the load and the response vary with time, it is evident that a dynamic problem does not have a single solution, as a static problem does; instead the analyst must establish a succession of solutions corresponding to all times of interest in the response history. Thus a dynamic analysis is clearly more complex and time-consuming than a static analysis. p(0 1 ^fe*-iJ.iiiJ-J-'*^^ inertia forces Figure D : Static Loading Figure E: Dynamic Loading XIHowever, a more fundamental distinction between static and dynamic problems is illustrated in Figure D and Figure E. If a simple beam is subjected to a static load p, as shown in Figure D, its internal moments and shears and deflected shape depend directly upon the given load and can be computed from p by established principals of force equilibrium. On the other hand, if the load p(t) is applied dynamically, as shown in Figure E, the resulting displacements of the beam are associated with accelerations which produce inertia forces resisting the accelerations. Thus the inertial moments and shears in the beam in Figure E must equilibrate not only the externally applied force but also the inertia forces resulting from the accelerations of beam. Inertia forces which resist accelerations of the structure in this way are the most important distinguishing characteristic of a structural dynamics problem. In general, if the inertia forces represent a significant portion of total load equilibrated by the inertial elastic forces of the structure, then the dynamic character of the problem must be accounted for in its solution. On the other hand, if die motions are so slow that the inertia forces are negligibly small, the analysis for any desired instant of time may be made by static structural-analysis procedures even though the load and response may be time-varying. In the dynamic system of Figure E, the analysis obviously is greatly complicated by the fact that the inertia forces result from structural displacements which in turn are influenced by magnitudes of inertia forces. This closed cycles of cause the effect can be attacked directly only by formulating the problem in terms of differential equations. Furthermore, because the mass of the beam is distributed continuously along its length, the displacements and accelerations must be defined for each point along the axis if the inertia forces are to be completely defined, in this case, the analysis must be formulated in terms of the partial differential equations because the position along the span as well as the time must be taken as independent variables. 'M/sM. ^fr y(agf \air WsW/s I''. i'', K'3 Figure F: Lumped-mass idealization of a simple beam. On the other hand, if the mass of the beam were concentrated in a series of discrete points or lumps, as shown in figure F. The analytical problem would be greatly simplified because inertia forces could be developed only at the mass points, in this case it is necessary to define the displacements and accelerations at these discrete points. The number of the displacements components which must be considered in order to represent the effects of all significant inertia forces of a structure may be xutermed the number of dynamic degrees of freedom of the structure. For example, if me system of Figure F were constrained so that the three mass points could move only in a vertical direction, tins would be called a three-degree of freedom (3 DOF) system. It is clear that a system with continuously distributed mass, as in Figure E, has an infinite number of degrees of freedom. A single degree of freedom (SDOF) system the dynamic response of which can be evaluated by a solution of a single differential equation of motion. If the physical system of the system are such that its motion can be described by a single coordinate and no other motion is, then it actually is a SDOF system and the solution of the equation provides the exact dynamic response. On the other hand, if the structure actually has more than one possible mode of displacement and it is reduced mathematically to a sdof approximation by assuming its deformed shape, the solution of the equation of motion is only an approximation of the true dynamic behavior. Dynamic response of a SDOF system can be evaluated by the solution of a single differential equation of motion. The displaced position of any arbitrary N degree of freedom system can be defined by the N components of the displacement vector {v}. However, for dynamic response analysis of linear systems, a much more useful representation of the displacements is provided by the free vibration mode shapes. These shapes constitute N independent displacement patterns, the amplitude of which may serve as generalized coordinates to express any form of displacement. The mode shapes thus serve the same purpose as the trigonometric functions in a Fourier series, and they are advantageous for the same reasons because of their ortogonality properties and because they describe the displacements efficiently so that good approximations can be made with few terms. In most cases, during the calculation of earthquake effects on the multi-storey buildings, to consider the frames as shear frames, normal forces in the columns are neglected. In this study, a computer program has been prepared to calculate storey forces using mode superposition method.A one spanned frame considered and number of storey is increased from 2 to 40. For all frames storey forces are calculated firstly considering normal forces and secondly neglecting it for the columns. Relative errors, which occurs because of neglecting normal forces for the columns, calculated. The calculations shows that, sometimes amount of the relative errors can reach 20%. xm
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