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Mode Pressure Coefficient Maps As an Alternative to Mean computation Article Mode Pressure Coefficient Maps as an Alternative to Mean Pressure Coefficient Maps for Non-Gaussian Processes: Hyperbolic Paraboloid Roofs as Cases of Study Alberto Viskovic Department of Engineering and Geology, G. D’Annunzio University, viale Pindaro 42, 65127 Pescara, Italy; [email protected] Received: 9 November 2018; Accepted: 10 December 2018; Published: 12 December 2018 Abstract: Wind tunnel experiments are necessary for geometries that are not investigated by codes or that are not generally and parametrically investigated by literature. One example is the hyperbolic parabolic shape mostly used for cable net roofs, for which codes do not provide pressure coefficients and literature only gives mean, maxima, and minima pressure coefficient maps. However, most of pressure series acquired in wind tunnels on the roof are not Gaussian processes and, for this reason, the mean values are not precisely representative of the process. The paper investigates the ratio between mean and mode of pressure coefficient series acquired in wind tunnels on buildings covered with hyperbolic paraboloid roofs with square plans. Mode pressure coefficient maps are given as an addition to traditional pressure coefficient maps. Keywords: wind tunnel tests; non-Gaussian processes; pressure coefficient; mode; hyperbolic paraboloid roofs 1. Introduction Wind tunnel tests are necessary for structure sensitivity to wind action regarding shapes not discussed in technical codes or literature. This is the case of the structures with hyperbolic paraboloid shapes mostly used for roofs made of cable nets and concrete shells, widely used, for example, in sport arenas, meeting rooms, or music halls [1–4]. The shortcoming relates the generalization and parametrization of aerodynamic coefficients to the widest number of geometries derived from the hyperbolic paraboloid. Codes of practice, such as those found in the literature [5–14], for example, do not give values of pressure coefficients for buildings covered with hyperbolic paraboloid roofs. The literature discusses a great number of examples of wind tunnel test campaigns [11,15–32]. However, all references regard particular cases that are not generalizable for different geometries. There are some exceptions [26,27] that have followed a parametric approach in order to give generalizable values. In addition, the authors of [33,34] have synthetized experimental results in order to give simplified pressure coefficient maps. The simplified pressure coefficient maps are used by designers to calculate the structure and evaluate the cost–benefit convenience in advance. All publications and codes give mean or maxima and minima pressure coefficient maps. The maxima and minima pressure coefficients are usually considered to estimate the local extreme values of wind action, whereas the mean values are used to investigate the global wind action distribution on the surface. However, some information about the processes that have been used to generate the maps should be given too. Computation 2018, 6, 64; doi:10.3390/computation6040064 www.mdpi.com/journal/computation Computation 2018, 6, 64 2 of 17 InComputation fact, if 2018 the, 6 processes, x FOR PEER are REVIEW non-Gaussian, the mean value is not precise enough to represent2 of 17 them. The non-Gaussianity generally depends on very big flow fluctuation because of, for example, However, some information about the processes that have been used to generate the maps flow separationshould be given near too. edges or vortex shedding caused by a complex geometry [27,35]. Recently, the authorsIn offact, [35 if, 36the] illustratedprocesses are the non-Gaussian, peak factor the distributions mean value on is anot hyperbolic precise enough paraboloid to represent roof and showedthem. that The many non-Gaussianity processes acquired generally by depends pressure on taps, very particularlybig flow fluctuation near the because roof borders of, for example, and corners, are non-Gaussianflow separation processes. near edges For or thesevortex reasons,shedding the caus meaned by valuesa complex of thegeometry pressure [27,35]. coefficient Recently, series the are representativeauthors of as [35,36] much illustrated as the mode, the forpeak example. factor distributions on a hyperbolic paraboloid roof and Inshowed cases wherethat many the difference processes between acquired mean by pressure and mode taps, values particularly is significant, near the if the roof mean borders is lower and than the modecorners, (i.e., are which non-Gaussian is the most processes. frequent For value), these the reas useons, of meanthe mean pressure values coefficients of the pressure can affectcoefficient structure series are representative as much as the mode, for example. reliability. In particular, this variation can affect the structural reliability for flexible structures and in general In cases where the difference between mean and mode values is significant, if the mean is lower for cables structures. In fact, analyses of instability on tensile structures, in particular on suspended bridges than the mode (i.e., which is the most frequent value), the use of mean pressure coefficients can affect understructure flutter (i.e., reliability. [37–39]), In have particular, shown that this smaller variation variations can affect of wind the structural action can reliability also give globalfor flexible instability, contrarystructures to the case and ofin concretegeneral for bridges, cables which structures. are more In fa sensitivect, analyses to seismicof instability action on [40 tensile]. In addition, structures, the in cable net tensileparticular structures’ on suspended sensitivity bridges to wind under action flutter is well-known (i.e., [37–39]), from have many shown studies that smaller [1,2,23 ,variations41–52]. of Thiswind paper action investigatescan also give global this aspect instability, on fourcontra geometriesry to the case of of buildings concrete bridges, covered which with are hyperbolic more paraboloidsensitive roofs to seismic withsquare action [40]. plans. In addition, This shape the cabl wase chosennet tensile because structures’ it is sensitivity frequently to usedwind foraction tensile structuresis well-known that are veryfrom many sensitive studies towind [1,2,23,41–52]. action. For this reason, the paper proposes to optimize the structuralThis design paper of these investigates structures this usingaspect mode on four instead geometries of mean of valuesbuildings from covered wind tunnelwith hyperbolic experiments. paraboloid roofs with square plans. This shape was chosen because it is frequently used for tensile In total, two different heights and hyperbolic paraboloid surface curvatures are considered. The paper structures that are very sensitive to wind action. For this reason, the paper proposes to optimize the aims tostructural show the design differences of these between structures pressure using coefficient mode instead maps of obtained mean values by mean from and wind mode tunnel values in orderexperiments. to propose alsoIn total, taking two into different account heights this value and forhyperbolic non-Gaussian paraboloid processes. surface The curvatures paperdoes are not aim toconsidered. propose a The prediction paper aims method to show of pressurethe differences coefficient between distribution pressure coefficient on the roof maps and, obtained consequently, by peak factorsmean and of pressuremode values coefficients in order areto propose not investigated also taking in into this account phase. Thethis value differences for non-Gaussian between mean and modeprocesses. values The are paper shown does in not terms aim to of propose wind loads a prediction on the method roof too. of pressure coefficient distribution Sectionon the roof2 discusses and, consequently, the geometrical peak factors sample of pres andsure the coefficients experimental are not data investigated set given in by this [ 40 phase.], used to investigateThe differences the mean between and mode mean pressure and mode coefficient values are ratio, shown while in terms Section of wind3 discusses loads on the the main roof too. results of the research.Section It is 2 important discusses the to notegeometrical that in sample this study, and the the experimental author used data the sameset given experimental by [40], used data to set investigate the mean and mode pressure coefficient ratio, while Section 3 discusses the main results discussed in the work of [27], made available by authors. Finally, some maps contain asymmetries of the research. It is important to note that in this study, the author used the same experimental data becauseset experimentaldiscussed in the results work of are [27], affected made available by inaccuracies by authors. in Finally, the experimental some maps set-upcontain given asymmetries by models. because experimental results are affected by inaccuracies in the experimental set-up given by models. 2. Geometrical Sample and Experimental Setup Wind2. Geometrical tunnel tests Sample were carriedand Experimental out by [27] andSetup [34 ] on 12 plan shapes, 2 different heights, and 2 different curvaturesWind of buildings tunnel tests covered were with carried hyperbolic out by [27] paraboloid and [34] roofs.on 12 plan shapes, 2 different heights, and 2 Thisdifferent paper curvatures is focused of buildings on four of covere thesed sampleswith hyperbolic and, in paraboloid particular, roofs. on buildings that have a square plan, twoThis different paper
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