Yüksek yapıların hesabında rüzgar spektrumu yöntemi

Abstract

ÖZET 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. xıv

SUMMARY WIND RESPONSE TECHNIQUES FOR CALCULATION OF HIGH-RISE BUILDINGS INTRODUCTION 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 xvrandom 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 xviheight 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 (§)2<fc = f (4) One can write for the PSDF, Syr (zi&f), of the displacement responce of the referance system Sy&vnf) = fhbfrfa) /H(f) 1 2(pCDB)2V0(zi)Vo(z2) S$(zıS)S%(z2f)Coh(zıt2f) (5) The PSDF for the acceleration is xvnS*tei&J) = (&f)ASy&unf) (6) The RMS displacement Oyr{H), and the RMS acceleration Oar(H) at the top of the building are <Jyr(H) = f Syt{Hf)df and OaAH) = / Sat(W)df (7) For the peak displacement, max>v(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<fl (14) The frequency response function of the reference system, Hrj, is the same as that of the actual system for thej'.th mode, Hj(f), except the scaling factor (i.e., the mass term). The relationship can be written as Hrj(f) = ^'Hj(f) (15) Since the loading on the reference and actual systems are the same, and their frequency response functions are equal with a scaling factor, it is concluded that the spectral contents of the modal response and the response of the corresponding referance system are the same. Therefore the peak factors for each response can be assumed equal. Consequently, the ratio of the peak modal response to the peak response of the corresponding referance system is equal to the ratio of their RMS responses. If this ratio denoted by kj(k) for the responses at height z, one can write /oo Syj(zf)df fo-Tt-r (16) °*(z) }QS^{zf)df Because of the four-fold integration involved in the calculations of Si^(f) and SF$(f), a straightforward evaluation of fc/(z)is very complicated, and would not have much practical use. To simplify the calculations, one may consider two extreme cases regarding the spatial correlation of the pressures. Case-1 wul to the situation where the pressures are spatially uncorrelated, whereas Case-2 will refer to the situation where the pressures are fully correlated. In addition, it is assumed that the PSDF of velocity fluctuations is independent of the height as suggested by Davenport [1]. The simplified expressions for Jcj(z) can then be developed as follows: Case-1: Pressures are spatially uncorrelated If the pressures are uncorrelated it is assumed that Cdhip-yn&jCLf) = d(xı-X2)â(zı-z2) (17) where <5 denotes the Dirac's delta function. Using this expression, and also with the assumption that Sw(f) is independent of z, it can be shown that xixSlÇV) = (pBj)2Sw(f) J c]i(z)tf(z)VÎ(z)dz (18) It is assumed that Cp(z) can be taken out of the integral by using an averaged pressure coefficient, Co, calculated as One, therefore, can write for the PSDF of the/th modal response at the top 0 Similarly, for the referance system Syj(Hf)=tf(H)(pCDBj)2/Hj(f)/2Sw(f)f fij{z)V2o(z)dz (20) Sri(Hf)^fi}{H)ipCDBj)2/H,j{f)/2Sw(f)) ix2(z)V2o(z)dz (21) For Case-1, the ratio of the top-story RMS responses ky(H), from Eq.16 and also using Eq.15, becomes f tf(z)V2(z) dz /. rtwfa z)dz Vl (22) Case-2: Pressures are fully correlated If the pressures are fully correlated, the coherence function is unity, that is Coh(x/fi/piz74) = 1 (23) The PSDF of the generalized force then becomes n2 Sf/<S) = (pBj)2Sw(f) r Cp(z)fij(z)Vo(z)dz (24) Using the same approximation as in Case-1 for Cp(z), one can write for the PSDF of the/th modal response at the top SyjÇKf) = tf(H)(pCDBj)2/Hj(f)/2Sw(f) and similarly for the referance system J ` H 12 m(z)v0(z) dz (25) XXStfHf) = tı}(H)(pCDBj)2/Hq(f)/2Sw(f) The ratio of the RMS responses kqiH), becomes muff PfrWo&dz rH nAz)Vo(?)âz j Hriz)Vo{z)dz (26) k2jW MH)Mf fH (27) As stated earlier, since the peak factors for the modal response and the corresponding reference system response are equal, these ratios are also valid for the peak responses. Therefore, the peak value of the/th modal response at the top, vasx.yj{Hf), can be calculated in terms of the response ratio and the peak t response of the reference system as maxyjCHfy = kj(H) maxy^Hj) (28) t t Since we defined maxyAHj) as the spectral response (Eq.10), one can calculate t the peak modal response as maxyj(H^=kj(H)D(f0j) (29) t The total peak response can be approximated by combining peak modal responses. If SRSS (square-root-of-sum-of-squares) method is used for the combination, the total peak response becomes maxy(H,t) = t n 2 (maxyj{H,i)/ ;'=1 Vi 2 kf(H)D2(foj) V2 (30) The equations for accelerations are similar. Since the relationship between accelerations and displacements is the function of frequency only, the ratios ky andky calculated for displacements are also valid for the accelerations. Therefore, one can calculate the peak top-story acceleration for the/th mode, maxaj(H,t) in t terms of the spectral acceleration of the referance system as max aj(H J) =kj(H)A(f0j) (31) t The total peak acceleration is obtained by combining the peak modal accelerations as in Eq.30. For given building, the value of kj{z) is somewhere between ky(z) and k2j(z). There is no way of knowing the exact values without explicitly incorporating the correlation structure of the wind. Therefore, an approximation needs to be made regarding which value to use for kj(z). For the first mode, the two values XXIwould be very close since the first mode shape in most buildings is almost a straight line, same as that of the referance building. For higher modes, the ratio ky(z) would always be larger than the ratio fo/(z), because the value of the integral f fif(z)l%(z)dz is always larger than that of f fij(z)V0(z)dz (the negative portions of mode shapes become positive in the first integral due to square), whereas the values for the integrals / fi?(z)Vd(z)dz and / fir(z)Vo(z)dz are always close. Since, for the most of the multi-story buildings participation of higher modes other then first mode are not significant and can be ignored in dynamic displacement calculations, above given participation factors can suitable used for this purpose. However, for accelerations, participation of higher modes are significant and can not be ignored. Thus, the modal participation factors defined previously for two limit cases are not suitable for acceleration calculations. Because of this, in the final part of this study, employing a suitable simplification for the four-fold integration involved in the calculations ofSpffl and £f£(/), a simple and efficient expression for lq(z) has been obtained which works properly for the higher modes. xxu