Aerodynamic Admittance Function o Tall Buildings in hou a Ahsan Kareem b a alou Engineering Int l, Inc., 75 W. Campbell Rd, Richardson, T, USA b Nataz odeling Laboratory, Uniersity o Notre Dame, Notre Dame, USA ABSTRACT: The aerodynamic admittance unction AAF has been widely inoked to relate wind pressures on building suraces to the oncoming wind elocity. In current practice, strip and quasi-steady theories are generally employed in ormulating wind eects in the along-wind direction. These theories permit the representation o the wind pressures on building suraces in terms o the oncoming wind elocity ield. Synthesis o the wind elocity ield leads to a generalized wind load that employs the AAF. This paper reiews the deelopment o the current AAF in use. It is ollowed by a new deinition o the AAF, which is based on the base bending moment. It is shown that the new AAF is numerically equialent to the currently used AAF or buildings with linear mode shape and it can be deried experimentally ia high requency base balance. New AAFs or square and rectangular building models were obtained and compared with theoretically deried expressions. Some discrepancies between experimentally and theoretically deried AAFs in the high requency range were noted. KEWORDS: Aerodynamic admittance unction; Wind eects; Tall Buildings; Turbulence; Wind tunnel; Code and Standard. INTRODUCTION The aerodynamic admittance unction AAF has been widely inoked to relate wind pressures on building suraces to the oncoming wind elocity in the requency domain [6]. In current wind engineering practice, strip and quasi-steady theories are generally employed in ormulating analysis o wind eects in the along-wind direction ollowing the gust loading actor GLF approach [,,7]. The application o these assumptions permitted the representation o the wind pressure ield on the building surace completely by the oncoming wind elocity ield. Figure a illustrates the chain relationship among arious loading and response eatures [,8]. The eectieness o these assumptions has been examined by limited ield measured data [3, 5] as well as wind tunnel model pressure tests [4,7]. Synthesis o the wind elocity ield utilizing its coariance and auto-spectral descriptions leads to a generalized wind load GWL that employs the AAF or GWL-AAF in the GLF ormulation. This paper irst reiews the theoretical background o the deelopment o current GWL-AAF to identiy some inherent ariations and discrepancies. This is ollowed by a comparison o GWL-AAFs used in seeral major international codes and standards, which exhibit a considerable scatter. Since the GWL-AAF has an inherent shortcoming in its deinition, thereore, no experimental alidation o this unction is aailable as it depends both on the mode shape and requency Figure a. In this paper, a new deinition o AAF is presented, which is based on the base bending moment BB. It is noted in Figure b that the BB-AAF is independent o the mode shape in contrast with the GWL- AAF. The BB-AAF is numerically equialent to the current GWL-AAF or most code applications due to implied assumption o linear mode shape []. Furthermore, the BB-AAF can be deried using wind tunnel tests employing a high requency base balance.
Gust spectrum Aerodynamic admittance DGLF Generalized orce spectrum S u, Displacement transer unction S d Displacement response spectrum S Log. Frequency a Gust spectrum Aerodynamic admittance GLF Aero - BB spectrum BB transer unction BB response spectrum S u S S Log. requency Figure. Aerodynamic admittance unctions in gust loading a DGLF; b GLF. b. TEORETICALL-DERIVED AAF.. Wind pressures on buildings Using strip and quasi-steady theories, the structure o wind pressures on building surace is replaced with that o the oncoming wind Figure. In particular, the wind load per unit height at any height z is treated as proportional to the square o the elocity at that height ˆ t / Vˆ tcd B where ˆ t z + t peak externally applied wind load in which mean wind load and zero-mean luctuating component o wind load; t time; V ˆ t V z + t peak wind elocity in which V mean wind elocity and zero-mean luctuating wind elocity; air density; C drag orce coeicient; and d B width o building normal to the oncoming wind. The mean wind elocity is usually expressed in the orm o V z V z / where V mean wind elocity ealuated at the building height and exponent o mean wind elocity proile. When neglecting the contribution o the quadratic luctuating wind elocity term, the luctuating wind load can be written as z, t V z z, t C d B which is a combined wind orce including wind pressures in both windward and leeward aces or t B pw x, t + pl x, t dx as shown in Figure, in which p wind pressure; and
Qp w x, x, p w t z, x, z, t l z, t z, t Q p w t z, z, x, z, t w Q z, z, w z, t w Q z, z, l z, t l Q z, l w z z Figure. Structure o wind elocity and pressure on a tall building subscripts w and l indicate windward and leeward, respectiely. In the requency domain, the ollowing spectral relationship can be introduced S C S 3 where S normalized spectrum o luctuating wind load per unit height at height z; S normalized spectrum o wind elocity; AAF with regard to unit wind load; and C a coeicient. The correlation structure o unit wind loads at dierent heights can be expressed by S z, z, S z, S z, Q z, z, S Q where 4 cross-spectral density o wind loads per unit height between height z and z ; and Q p correlation o wind load which is replaced by that o the wind elocity or Q according to the strip and quasi-steady theories... Generalized Wind Loading Based AAF Utilizing the aboe wind pressure inormation, a GWL is usually computed in deriing the gust loading actor GLF or along-wind load eects S S z z z z dz dz,, 5 where GWL in the undamental mode; and the undamental structural mode shape o a building assumed to be in the orm z c z / in which mode shape exponent and c normalization actor. In most o current codes and standards, a linear mode shape is assumed or. A GWL-AAF can be ealuated as C S / S 6
where GWL-AAF hereater; S normalized spectrum o the GWL; and C a coeicient. The aboe deriation shows that the GWL-AAF depends mainly on the correlation structure o wind pressures and the structural mode shape..3. GWL-AAF in conentional GLF Adopting the aorementioned GWL, the conentional GLF or DGLF, which is the ratio between peak wind-induced displacement and the mean, can be obtained. This orm o the GLF has been widely used in almost all o the current wind loading codes and standards around the world ollowing its presentation by Daenport [,]. For quick reerence and comparison, a typical deriation o the DGLF is gien in the Appendix. Expressions o the DGLF similar to that in 6 hae been proided in most international codes and standards. Based on the deinition o DGLF in 7, the peak displacement response under the gusty wind is equal to the mean displacement multiplied by the DGLF. Similarly, reerring to, the peak GWL can also be computed by ˆ G + gr B + R 7 where ˆ peak GWL; G DGLF; mean GWL; g peak actor; r I z ; I z turbulence intensity at a reerence height o z / 3 ; B background actor; and R resonant actor which are all proided in the Appendix. For the background component the structural transer unction can be treated as to be unity. The RS alue o the background component is gien by r B 8 B where RS background GWL component; and RS aerodynamic or externally B applied GWL. Substituting 8 into results in K, S / S B 9 Comparing 9 to 6, the GWL-AAF can be computed by K, J J,, in which J and J,, joint acceptance unctions in horizontal and ertical directions, respectiely, as expressed in 4 and 5. A summary o GWL-AAFs in major international codes and standards is proided in Table [9]. Since all GWL-AAFs in current codes and standards are deeloped based on a similar procedure as outlined in the Appendix, it is not surprising to notice that all GWL-AAFs are expressed in terms o ery similar unctions inoling the size o buildings in the horizontal and ertical directions. oweer, since dierent codes and standards hae employed signiicantly dierent wind parameters, these details hae led to apparent scatter in the GWL-AAFs. 3. BASE BENDING OENT BASED AAF Unlike the conentional DGLF which deines an equialent static wind loading ollowing exactly the distribution o the mean wind orce, a new GLF has been introduced by the authors [8]. Among other adantages, this new GLF can proide more realistic equialent wind loading or dynamically-sensitie structures. The new GLF is deined as the ratio between the
Table. GWL-AAFs in major international codes and standards ASCE 7 AS7. NRCC RLB-AIJ Eurocode Expressions o GWL-AAF R RB.53+.47RD R l / / e, > ; R, R, 4.6 / V ; R B, 4.6 B / V ; R D, 5.4 D / V + 3.5 / V + 4 B / V + 8 /3V + B / V.84 +. / V +. B / R and l V R R B R are the same as those in ASCE7. B peak base bending moment response and the mean. Using procedure similar to the deriation o the DGLF, a GLF can be deried and expressed in similar orm as in 6. Applying the GLF to the BB response, the background BB component can be expressed as B r B where B RS background BB response; mean BB; and RS aerodynamic BB. eanwhile, the ollowing relationships releant to the BB-AAF can be obtained: K J J,, 3 B B S / S / I z where BB-AAF; and S SD o aerodynamic BB. Two important eatures related to the BB-AAF are noteworthy. Comparing 3 to, the BB-AAF is exactly the same as the GWL-AAF or a building with linear mode shape, an assumption oten implied in most current codes and standards. oweer, unlike the GWL-AAF that depends on some imaginary quantities such as the GWL, the BB-AAF is only related to realistic quantities through 4 and 5. eanwhile, the BB-AAF depends on the oerall wind pressure eects in terms o the BB, without the need to know the detailed wind pressure structure. The equialence between the GWL-AAF and the BB-AAF proides the basis or experimental inestigation o the GWL-AAF. 4 5
4. EERIENTAL RESULTS According to 4 and 5, the inormation needed or the deriation o BB-AAF includes the input wind spectrum and turbulence intensity, as well as the BB including both the mean and the luctuating components. The later components can be perectly measured with a strain type high requency base balance FBB. A 5-component FBB was employed in wind tunnel tests inoling two building models. odel is a square model with B D mm and odel with B mm and D 5 mm where D reers to the depth o the building model. Both models hae a height o 6 mm. Two oncoming boundary layers were simulated corresponding to an open country terrain BL and a city center BL, respectiely. The input wind elocity and the base bending moments were measured simultaneously. Figure 3 shows the test results or odel in BL wind boundary. It is noted that the wind 5/ 3 elocity spectrum ollows the rule closely in the high requency region. Using the measured wind elocity and BB spectra, the BB-AAF is plotted against the theoretical GWL-AAF using the orm proided by the NBC code or Daenport []. Notable dierences between the tested BB-AAF and the code based GWL-AAF can be noted in the high requency region o interest to building analysts... odel ; BL 3 5 E-3 E-4 S / S / AAF-BB AAF in NBC E-5 O Engineering Interest E-6 E-3.. B/U Figure 3. Wind spectrum, BB spectrum and AAFs or a square building model 5. CONCLUDING REARKS The theoretical background in the deelopment o GWL-AAF, which is used in current GLF based codes and standards, is irst reiewed to identiy releant ariations and discrepancies. Seeral major international codes and standards are compared in terms o their representation o GWL-AAFs, which exhibited considerable scatter. Since the GWL-AAF has intrinsic shortcomings in its deinition and no experimental conirmation is aailable, this paper presented
a new concept o AAF that is based on the base bending moment BB. It was shown that the BB-AAF was numerically equialent to the current GWL-AAF or buildings with linear mode shapes, an assumption implied in most codes and standards. In addition, the BB-AAF can be deried using measured BB inormation. igh requency base balance wind tunnel tests were perormed or a square and a rectangular building model. The BB-AAF results obtained rom this wind tunnel test were compared to the corresponding GWL-AAFs rom codes and standards. Noteworthy discrepancies in the high requency region between the measured and the theoretical AAF were noted. This suggested that the scatter in the response prediction o dierent codes and standards may be attributed in part to the choice o aerodynamic admittance unction which did exhibit departure rom those based on the strip and quasi-steady theories. 6. AENDI: AN TICAL DERIVATION OF DGLF In the current GLF approach, the ESWL on tall buildings is deined as the mean wind load multiplied by a magniication actor [] ˆ z G z 6 where ˆ ESWL; G DGLF which can be computed by G ˆ z / z + g / 7 where ˆ and peak and mean displacement, respectiely; g peak actor which is usually about 34; and S d RS displacement in which S SD o displacement response. The DGLF takes into account the oerall wind-structure-interaction including both structural dynamics and gust luctuation. Usually, the mean structural displacement can be approximated by that in the irst mode z / k z 8 where z z dz, k m and m m z z dz are the GWL, stiness and mass o the irst mode, respectiely; natural requency o the irst mode; and m z mass per unit height. Using the luctuating GWL in 5 and ollowing the ramework in Fig. a, the SD o the luctuating displacement in the irst mode can be computed by S S / k z 9 where [ / ] + / irst mode structural transer unction; and critical damping ratio in the irst mode. Combining 5 and 8 results in S S The integration o is the luctuating portion o the DGLF. When assuming a ull-correlation between the wind pressures on windward and leeward suraces and using the detailed wind pressure description in -4, the SD o the generalized wind load in 5 can be written as
+ + B B z z S c C dv Q x, x, Q S dxdxdzdz C C where Q x, x, exp x x and Q exp z z horizontal V z V z and ertical correlation unctions, respectiely, in which C and C exponential decay coeicients; and z reerence height. Substituting and the mean GWL into leads to S r K, S where r I z in which I z + + / + + I, I / V turbulent intensity ealuated at height ; and z / 3 ; and K, J J,, 3 in which J and J,, joint acceptance unctions in horizontal and ertical directions, respectiely, that can be computed by B B J Q x, x, dxdx B 4 + + z + z + J,, Q dzdz 5 It has been reported that the joint acceptance unctions ulill the role o the aerodynamic admittance, but in addition, take into account the inluence o mode shape on the eectie excitation orce []. Substituting and into 7, and diiding the integration into background and resonant portions, the DGLF in codes and standards has been expressed by where G + g r B + R 6 S B K, S d background actor; and R SE / 4 resonant actor in which E gust energy actor and S K, size reduction actor. 7. REFERENCES:. A. G. Daenport, The application o statistical concepts to the wind loading o structures, roc. Inst. Ci. Engrs., 9 96449-47.. A. G. Daenport, Gust loading actors, J. Struct. Di., ASCE, 933 967-34. 3. J. D. olmes, ressure luctuations on a large building and along-wind structural loading. J. Indust. Aerodyn., 3 976. 4. A. Kareem, Synthesis o luctuating along-wind loads on tall buildings, J. Engrg. ech., ASCE, 986-5. 5.. Kawai, ressure luctuations on square prisms applicability o strip and quasi-steady theories, J. Wind Engrg. Indust. Aerodyn., 3983. 6.. W. Liepmann, On the application o statistical concepts to the bueting problem, J. Aeronautical Sciences, 9 95 793-8. 7. B. J. Vickery, Load luctuations in turbulent low, J. Engrg. ech. Di., ASCE, 94 968 3-46. 8.. hou, and A. Kareem, Gust loading actor: new model, J. Struct. Engrg., ASCE, 7 68-75. 9.. hou, T. Kijewski, and A. Kareem, Along-wind load eects on tall buildings: A comparatie study o major international codes and standards, J. Struct. Engrg., ASCE, 86 a 788-796... hou, A. Kareem, and Gu, ode shape corrections or wind eects on tall buildings, J. o Engineering echanics, ASCE, 8 b.