Equivalent Static Wind Loads of an Asymmetric Building with 3D Coupled Modes

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1 Equivalent Static Wind Loads of an Asymmetric Building ith 3D Coupled Modes Yi Tang and Xin-yang Jin 2 Associate Professor,Wind Engineering Research Center, China Academy of Building Research, Beijing 3, China, yitang_hunan@63.com 2 Professor, Wind Engineering Research Center, China Academy of Building Research, Beijing 3, China, jinxinyang@cabrtech.com ABSTRACT Recent trends toards developing increasingly taller and complex buildings of irregular geometric shapes imply that these structures are potentially more responsive to ind excitation. Making accurate predictions of ind loads especially equivalent static ind loading is therefore a necessary step in structural design. In this paper, by introducing mode coupling coefficients, a modified SRSS method is proposed to calculate resonant components of the responses and the resonant static-equivalent ind loads. Three dimensional background static-equivalent ind loading is evaluated using Load-Response-Correlation (LRC) method. The background and resonant static-equivalent loads are combined ith eight factors. The consequential total equivalent ind load is verified through a 49-storey building ith three-dimensional mode shapes. KEYWORDS: COUPLED MODES; ESWL; MODIFIED SRSS METHOD; LRC METHOD. Introduction Modern high-rise structures continue to evolve ith unsymmetrical plan forms and irregular vertical arrangements. In addition, these buildings are being built in density developed suburban or urban environments. These trends have led to the effects of unsymmetrical loadings on high-rise structures, commonly referred to torsional effects. The unbalanced load effects can be further exacerbated by non-alignment of center of stiffness of the structural system and center of mass of the floor plates, hich result in three-dimensional (3D), coupled mode shapes and experience coupled responses hen exposed to ind loads. Design practice often requires dynamic ind loads on buildings to be modeled as equivalent static ind loads (ESWLs). Accordingly, a relatively simple static structural analysis procedure can be adopted for a more detailed structural analysis and design. The gust response factor (GRF) approach, introduced by Davenport(967) for along ind excited buildings, is most idely used ESWL evaluation method in building codes and standards. For the past decades, many researchers have developed ESWL calculation method ith regard to particular problems. But most methods are concerning one-dimensional ind excited response(holmes(22); Solari(99); Zhou(2)) (et al alongind, acrossind or torsional). Solari(993) introduced a global model for calculating 3-D ESWLs, but not considering modal coupling and cross correlation beteen forces along different directions. Chen(25) proposed a frameork to evaluate ESWLs of asymmetric tall buildings undergoing lateral-torsion coupled motion using ind loading measured by high frequency force balance technique.

2 In this paper, the ESWL of a super-all building of concrete construction ith coupled three-dimensional (3D) mode as calculated using pressure data measured in ind tunnel. In evaluating resonant static-equivalent load a mode coupling coefficient is introduced to considerate mode coupling effects and background static-equivalent ind load is calculated by Load-Response-Correlation (LRC) method. ESWL analysis based on measured pressure field Quasi-static ind loading The LRC method of Kasperski (992) has been using to determine the equivalent static load distributions of the background component. The LRC method can give an expected equivalent load distribution corresponding to a particular load effect or response. The theory is easily extended to to or three dimensions as presented by Chen(25). The equivalent ind loading is given by H g Cov ( h, h ) μ ( z; h ) dh B F l sl ˆ l= x, y, F B ( h ) =,( = x, y, ) () σ ( z) sb here g B is the peak factor for background response; μ ( zh ; ),( sl, xy,, ) sl = is the influence coefficient, hich represents response in s direction at z height ith an unit force acting at h height in l direction; Cov ( h, h ) is the covariant coefficient beteen to forces, one at h F l height in l direction and the other at h height in direction (, l = x, y, ); σ is background sb component of RMS response in s direction. Equivalent inertia force considering modal coupling effects The equivalent ind loading for the resonant response considering modal coupling effects is given by Fˆ ( h) = g R F ( h) + η,( = x, y, ) (2) Rm Rm m Rm m here g Rm is the peak factor for the mth resonant response; R is the RMS modal coordinate m neglecting modal coupling effects; F 2 ω φ, F ( h) = ω m ( h) φ ( h) 2 ( h) = m ( h) ( h) xrm m x mx yrm m y my, is the mth inertial force corresponding to unit modal coordinate; ω is m 2 F ( h) = ω m ( h) ( h) Rm m φm the mth modal circular frequency ; m ( h ), m ( h ) and m ( h) x y are the structural mass and polar moment of inertia per unit length; φ, φ mx my and φ m are mth modal shapes. η m is defined as

3 mth modal coupling factor given by η m = n= n m φ ( z) R n φ ( z) R m n m r mn (3) here r = ρ Re ( ) / ( ) ( ) mn mn S ω S ω S ω Qmn ; Qmm Qnn = morωn ω ω ρ mn 3/2 8 ζmζn ( βmnζm + ζn ) βmn ( βmn ) + 4 ζmζnβmn ( + βmn ) + 4( ζm + ζn ) βmn = ; Re S ( ω) Q mn is cross poer spectral density(psd) beteen the nth and mth generalized ind forces; S ( ) Q mm ω and ζ m are mth generalized ind force PSD and damping ratio, and SQ nn ( ω ) and ζ n are the nth ones; β = m mn n ω ω. It is noted that in Eq. (2) + η m is used to describe contributions of structural mode to the mth resonant response. If modal coupling factor η m is zero, hich means the mth resonant response is mainly contributed by the mth structural mode, Eq. (2) has the same form as the ordinary inertial force equation as presented by Chen(25). Equivalent static ind loading Combining LRC method and equivalent inertia force considering modal coupling effects, the total equivalent static ind loading for the peak load effect is ritten as Fˆ ( ) ( ) ˆ ( ) ˆ h = F h + BF B h + RmF Rm( h), = x, y, (4) m= here F ( h) is the mean ind loading; F ˆ ( h) B is the background ind loading determined by LRC method; F ˆ ( h) mr is the equivalent ind loading for the mth resonant response; B = gbσ B( h) 2 2 [ gbσ B( h)] + [ grmσ Rm( h)] m= and Rm = grmσ Rm ( h) 2 2 [ gbσ B( h)] + [ grmσ Rm( h)] m= are eighting factors; σ B is RMS of background response; σ is the RMS of resonant response in mode m. Rm Case study Wind tunnel test A 49-storey concrete building as used to test the proposed frameork for 3D ESWL analysis. The building has an overall height of 98m and a floor plan dimension of about 27m by 62m. The fluctuating ind pressures acting on building ere obtained from a ind tunnel test. The test as carried out in the Boundary Layer Wind Tunnel of Shantou University hose orking section is 3m ide and 2m high. Experiment model ith geometric scale of

4 :3 is made of organic glass as shon in Fig.. The typical boundary layer as simulated for suburban terrain category B in accordance ith the building s surroundings. The ind pressures at 43 measuring points ere simultaneously measured ith sample frequency 3Hz for 36 ind directions shon in Fig.2. In this paper, ind load case in the most unfavorable ind direction of 8-degree as considered. And selected results for the full-scale ind speed of 5 m/ s at m height are summarized belo. Equivalent static ind loading Structure FEM analysis results indicate that this building s mass center and stiffness center are not coincident as shon in Fig.3. The building has the first three fundamental frequencies of.2hz (saying primary in the Y-direction),.28Hz (saying in the X-direction) and.32hz (torsional mode). As expected, due to the asymmetric structural layout of the building, the mode shape of each of the three fundamental modes is three dimensional as shon in Fig.4. Fig.5 shos the peak modal inertial loading for the three fundamental modes, ith a peak factor of 3.5, hich include only resonant components and neglect modal coupling effects. The loads in x and y directions are expressed in terms of load per unit height, hereas the torque is given in terms of torque per unit height. The eighting factors of the peak modal inertial loads and quasi-static loads for selected response components are summarized in Table, hich are calculated based on the proposed frameork. The response components considered are the displacement in X, Y direction at the top of the building, hich are referred to as u, v respectively. Weighting factors of the torsional angle at the building top referred to as, base bending moment responses in X and Y direction M x and M y, and the base torque response M are also calculated. Using the eighting factors, the overall ESWL for peak response can be conveniently determined for immediate applications to building design. Corresponding to the six responses presented above, ESWLs including the quasi-static and the inertial loadings are shon in Fig.6. The peak torsional angle, displacements at the geometric center of the building top storey in X-direction and Y-direction are.82-3 rad,.24m and.4m respectively, hich are calculated by applying ESWLs through simple static structural analysis procedure. By comparison, the three peak responses are also calculated using CQC method hich are.84-3 rad,.22m and.6m. It indicates that responses evaluated using the 3D ESWLs agree ell ith the results calculated using CQC method.

5 Y Z X Y Z X Fig. : Photograph of experiment model Fig. 2: Definition of ind directions Fig. 3: Coordinates of centers Table : The eighting factors u v M M M x y R R R B Fig. 4: First three 3D Mode shapes

6 F xr 5 F yr2 5 F xr3 F yr3 5 F R /2m F yr 5 F R2 /2m F xr2 5 F R3 /2m F R (K/m) F R2 (K/m) F R3 (K/m) Fig. 5: Inertial forces of the first three mode(neglecting mode coupling effects) F xeswl (K/m) A: Base torque; B: Base bending moment in Y-direction; C: Base bending moment in X-direction; D: Torsionan angle at the building top; E: Y-displacement at the building top; F: X-displacement at the building top Fig. 6: Equivalent static ind loadings Conclusion The analysis of 3D ESWL for a given peak response of buildings ith 3D coupled mode shapes of vibration ere presented. In this context, the spatiotemporally varying dynamic ind loads may be derived through multiple point synchronous scanning of pressures on the building model surface. Both the cross correlation of ind loads acting in different directions and intermodal coupling among modal response components ere taken into consideration in the analysis of response and modeling of ESWL. Using the proposed method, 3D ESWLs of a full-scale 49-storey asymmetric building ere calculated. It is found that for a particular response, the ind loadings in all three directions contribute to the response. The peak responses by applying 3D ESWLs are very close ith results calculated by CQC methods, hich very the effectiveness of the ESWL evaluation method. References Chen, X.,A. Kareem (25). "Coupled dynamic analysis and equivalent static ind loads on buildings ith three-dimension models." Journal of Structural Engineering 3(7): 7-82.

7 Davenport (967). "Gust loading factors." Journal of the Structural Division( ASCE ) 93(ST3): -34. Holmes, J. D. (22). "Effective static load distributions in ind engineering." Journal of Wind Engineering and Industrial Aerodynamics(9): 9-9. Solari, G. (99). "A Generalized Definition of Gust Factor." Journal of Wind Engineering and Industrial Aerodynamics 36: Solari, G. (993). Toards a Global Model For Calculating 3-D Equivalent Static Wind Forces on Structures. Proceedings of Seventh United States ational ind Engineering Conference. Zhou, Y.,A. Kareem (2). "Equivalent Static Buffeting Loads on Structures." Journal of Structural Engineering 26(8):