Yüksek yapıların hesabında rüzgar spektrumu yöntemi

Abstract

Rüzgar yükleri, yüksek yapıların boyutlandınlmasında gözönünde tutulan belli başlı yüklerden birisidir. Bu yükün dinamik karekterinin, birçok halde statik olandan daha etkin olabildiği ispatlanmıştır. Bu nedenle de, rüzgarın dinamik etkisi yük standartlarrmn yapışma da girmiştir. Şu anda standartlarca önerilen ve pratikte kullanılan yöntem eşdeğer statik yük yöntemidir. Bu yöntemde rüzgarın statik ve dinamik etMerinin beraberce yapıda oluşturacağı deplasmanı tek basma oluşturabilecek bir eşdeğeryükbelirlemrve yapıya bu yüklenir. Bu şekilde yapının sadece deplasman cevabı elde edilebilir. Bu yöntemde, rüzgarın dinamik etkisi nedeniyle yapıda oluşan ivme değerleri belirlenemediğinden boyutlandırmada sınırlayıcı kriterler sadece deplasman ve gerilme olmaktadır. Bu nedenle özellikle yüksek yapıda çok önemli olan ivme ile ilgili konfor kriterlerinin dikkate alınması mümkün değildir. Bu çalışmada yüksek yapıların rüzgar etkisi altındaki dinamik cevabının (deplasman ve ivme) hesaplanabilmesi amacıyla bir cevap spektrumu yöntemi gehştirilmiştir. Bu teknik şu anda deprem yükleri için kullanılmakta olan cevap spektrumu yöntemine oldukça benzemektedir. Bu sayede, sadece yapıda oluşacak en büyük deplasmanlar hesaplanmakla kalmayıp, aynı zamanda yapının tepesinde yükün etki süresi içinde oluşacak en büyük ivme değeri de elde edilebilmektedir. Bu çalışmanın kapsamı içinde yapıya rüzgarın esiş yönünde etkiyen kuvvetler ve yapının da bu yöndeki cevabı dikkate alınmaktadır. Yöntem temel olarak üç aşamada uygulanmaktadır. İlk olarak, temel hizasından yere dönel bir yay ve titreşim söndürücü ile bağlanmış, tek serbestlik derecen, birim kütle dağılımına sahip ve dikdörtgen şeklindeki rijit bir blok olarak tanımlanan bir referans yapının, gerçek yapıyı etkileyen rüzgar alımda yaptığı en büyük tepe deplasman ve ivmeleri daha önceden hazırlanmış diyagramlardan yararlanarak belirlenir. Bu nedenle yapı parametrelerinin ve rüzgar şiddetinin değişik kombinezonları için 108 adet bu anlamda diyagram hazırlanmış ve çalışmanın içinde sunulmuştur. İkinci olarak gerçek yapıya ait mod şekil vektörleri bilinen standart yollardan herhangi birisiyle tespit edilir. Daha sonra bu gerçek mod şekillerini ve yapının gerçek parametrelerini dikkate alarak, bu çalışmada geliştirilmiş olan modal katkı veya düzeltme faktörleri hesaplanır. Son olarak bu katsayılar daha önceden belMenmiş olan, referans yapıya ait spektrum değerleriyle çarpılarak gerçek yapının rüzgarın dinamik etkisine cevabı hesaplanır. Modal katkı veya düzeltme faktörleri çıkartılırken iki sınır değer tespit edilmiştir. Gerçek sonuç bu iki değerin arasında çıkmaktadır. Bu sınır değerler birinci mod için birbirlerine çok yakın olduklarından, sadece birinci modun dikkate alınmasının yeterli olduğu deplasman hesabında direkt olarak kullanılabilirler. Bu çalışmada yüksek modlann etkisinin önemli olduğu ivme hesabmda da kullanılabilen ve daha doğru sonuç veren bir modal katkı veya düzeltme faktörü ifadesi de geliştirilmiştir. Bu yönteme göre çözülen örneklerde yapılan hatanın %5 in allında olduğu tespit edilmiştir.

Wind loading is one of the most important factors to be considered in designing high-rise buildings. The dynamic componenet of wind forces have long been recognized and incorporated in design codes. The current practice of design for wind is based on the equivelend static load, under which the static deflection of the building is equal to the dynamic deflection [4, 5]. This load, along with the static component of wind load, are applied to the building, and a staticanalysis is performed for design. This approach is known as the gust factor approch [1]. The design criterion for the gust factor method is to limit the stress and deflections, same as for any other static load, since the method is based on the equivelent static load concept. It is well known, however, that one of the major problems inhigh-rise buildings is the wind-induced discomfort of occupants. The occupant discomfort occurs due to excessive acceleration, rather than deflections. This observation suggests that the wind design criterion of high-rise buildings should be based on peak accelerations, as well as peak displacements. Current codes do not have any provision for wind-induced peak accelerations. Using existing theory, analytical expression can be developed for peak accelerations [3, 28]. However, the expressions are probably too complex to use for practicing design engineers. There is a need for a simple wind design methodology, that will not only incorporate peak displacements and peak accelerations, but also will be consistent with the current methods of analysis, so that design angineers can easily adopt it. One such method is the response spectrum method. Response spectrum method has been widely used for earthquake design, and is well known among engineers. It is very simple, and can incorporate peak accelerations, velocities, and displacements. It was first suggested Newmark [36], and later shown by Cevallos-Candau [37], that the earthquake and wind loads, and corresponding building responses have a lot of similarities. Therefore, similar methods of analysis, such as the response spectrum technique, can be used for both loads. An important advange of using the response spectrum technique for both wind and earthquake loads is that, when both loads need to be considered for design, the designer would know beforehand which load will dominate his design, without doing a separate analysis for each load. In this study, a response spectrum technique presented for predicting wind-induced response of high-rise buildings. The technique is similar to that used for earthquake loads, and incorporates not only peak displacements, but also peak accelerations of the building. Therefore, the method can be used for design for safety (i.e., considers peak displacements), and also for design for comfort (i.e., considers peak accelerations). Is this study current techniques used for wind and earthquake response analysis of structures will be first outlined. Then how the xv random vibration technique used for wind loads can be put into a responce spectrum form will be explained. Following this wind response spectra for different wind and terrain conditions will be presented, and a parametric analysis to investigate the effect of various parameters on spectra will be performed. It is show that existing computer programs that perform spectral analysis for earthquake loads can be easily modified to perform spectral analysis for wind loads as well as earthquake loads. WIND FORCES ON HIGH-RISE BUILDINGS Wind induced vibrations in high-rise buildings are due to indiwidual or combined effects of following dynamic force mechanisms in the wind: along-wind forces due to turbulance, across-wind forces due to vortex sheding, wake buffeting, and galloping. Along-wind forces are in the direction of main wind flow. They induced static and dynamic component, generated by the steady and fluctuating components of the wind, respectively, and are the most dominant force mechanism in a typical building. In general, along-wind forces are in the form of pressures on the frontal (i.e., windward) face, and suctions on the back (i.e., leeward) face of the building. Across-wind forces are generated by vortecies that develop at the sides of the building moving clockwise and counterclockwise, and shed in an alternating fashion in the direction perpendicular to the mean wind flow. Across-wind forces canbe critical for slender buildings, such as buildings with very large height to width ratios, smokestacks, and transmission towers. Wake buffeting occurs if one structure is located in the wake of another structure, and can cause large oscillations in the downstream structure if the two structures are similar in shape and size, and less than ten-diameter apart. Galloping is an oscillation induced by the forces which are generated by the motion itself. It corresponds to an unstable motion with negative damping, and can be seen in structures like transmission lines, or long slender towers with sharp edgedcrossections. More detail on wind force mechanism can be found in Simiu and Scanlan [3] and Safak and Foutch [28]. In this study, only the along-wind forces in the direction of main wind flow will be dealed. RESPONSE SPECTRA FOR WIND LOADS Development of response spectra for wind loads can be accomplished following a similar approach to that for earthquake loads. As earthquake spectra, wind responce spectra should also be defined for a given site, since the velocity and turbulance structure of the wind is strongly site dependent. Earthquake loads are inertia loads, therefore the spectral responce involves only the damping and the natural frequency of the structures, but no any other structural parameter. Wind loads, however, are strongly dependent on the outside geometry of the structure. They are the size and the shape of the wind exposure area that determine the total wind load on the building. Therefore, the wind response is dependent not only on the natural frequency and damping, but also on outside geometry of the structure. Since we are dealing with buildings with rectengular cross-section and normally incident wind, and also considering only along-wind vibrations at this phase of the study, we can define the outside geometry in terms of the height and frontal width of the building. Further simplification can be achived for very tall buildings by neglecting the variation of wind pressures in the horizontal direction and using pressure coefficients avaraged over the height of the building. In the formulation that follows both the height and the width of the building will be considered. Wind response spectra for a given site, and given structural damping, height, and height to width ratio will be considered. The dependence of wind response spectra on xvi height and height to width ratio is the major difference when compared to earthquake response spectra A reference building for wind spectra: In order to develop wind response spectra, we will consider a referance building as schematically shown in Fig.6-1 will be considered. The referance building can be visualied as a rigid block of specified width, height, and mass, connected to the base by a rotational spring-dashpot system. Therefore, the referance system is a SDOF system, and its single mode shape is a straight line. For the referance system for different damping ratios, wind velocities, heihgts, and height-to-width ratios a wind response spectra will be developed. It is assumed that the reference system has unit mass per unit height, and a location in the middle of the city. Using the coordinate system shown in Fig.6-1, It is possible to write for the response of the referance system as yr(zfy=Hr(z)qr(t) (1) where fife) denotes the single mode shape of the system. Since the building has only one degree of freedom, the rigid body rotation with respect to base, one can write, for the mode shape M*)=| (2) The equition for qAf), can be obtained as, qAt) + 2t,or{2jlfor)qr{t) + <&for?q*t) = ^jjp. (3) where itaor and/or are the damping ratio and natural frequency, and K and M? are the generalized load and mass aof the referance building, respectively. For unit mass per unit length, one can calculate the generalized mass of the referance system as M*r=fQtf(z)-l-dz = fQ (§)2v(fl^), and the peak acceleration maxarfHj), at t t the top, one can write mıatfHj) =y0AH) + gyr{H)Oyr(H) (8) t mzmr(H,t) = gaAH)(Jar(H) (9) t where yoAJB) is the static displacement due to static wind load, and gyr{H) and g sub ar (H) are the displacement and acceleration peak factors, respectively, at the top of the referance system. For the displacement response spectra, only the dynamic displacement will be considered, since the static dispalcement can easily be calculated using static analysis. It should be noted here that, if desired, the static diplacementcan also be included in the response spectrum by expanding it into static modal components. We will define the displacement response spectra as the plot of the peak dynamic displacement response at the top of the referance building against the natural frequency for a range of frequencies. Therefore, for the natural frequenciy/0; the displacement spectra, D(fof), is Difoj) = max^)]^^ = gyrWoyriH) (10) Similarly for the acceleration spectra, it is possible to write A(f0j) = m3xar(H,t) = ga^Hpa^H) (11) t Modal Participation Factors for Wind Spectra In order to calculate the wind response of a given building by using the response spectra of the referans building, one has to determine the modal participation factor first which will be defined as ratio of the peak modal response of a given system to that of the referance system that has the same modal frequency and damping. The PSDF, Syr(f), of they.th modal displacement of a given building is Syj(z/) = fif(z)\Hj(f)\2SF;(f) (12) where Hj(f), the frequency response function for the;.th mode, can be written as, xviu #/(/)= - 5 ' T (13) İW?[-(2?r/)2 + i(2nf)(%oj)(2n:foj) + (2jtfoj)2] The PSDF of the referance systen, Srj(f), corresponding to ;'. th mode (i.e., the reference system with frequency faj, damping itaoj, and same outside dimensions) is Srjizf) «*fc)|J5#)|2S*3