Crosswind response of tall building under wind load in time and frequency domains

Porto, Portugal, 30 June - 2 July 2014 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 2311-9020; ISBN: 978-972-752-165-4 Crosswind response of tall building under wind load in time and frequency domains A. Bakhshi_1 1, E. Nakhostin_2 1 1 Civil Eng. Dept., Sharif University of Technology, P.O. Box 11155-9313, Tehran 1458889694, Iran Email: bakhshi@sharif.edu, elhamnakhostin87@gmail.com ABSTRACT: Due to the progress in the construction technology, the height of buildings has risen and the weight of them has declined; therefore, affected structures increase against the wind load as fundamental periods of structures increase. Responses of tall buildings are important in crosswind under the turbulent component of wind loads, which will be more important when the side ratio is larger than one. Three tall buildings have been simulated for effects of the wind load in the time and frequency domain. Wind field in time domain has been simulated with the probabilistic approach. Regarding, the frequency domain, wind forces obtained from the wind tunnel at the university of Notre Dame have been used. Story displacement and story shear are critical parameters to evaluate the structural behavior; thus, the results of wind characteristics were compared in the form of them. Moreover, the base moment coefficient is another parameter for comparing wind loads obtained from the wind tunnel and the probabilistic approach. The results reveal that transverse displacements of models under the wind load are larger than longitudinal and the result of the story shear is reversed. Crosswind displacements under the wind load are compared for loading in two domains of time and frequency. There is a good agreement for these two methods for the gust load in transverse direction. KEY WORDS: Tall buildings; Side ratio; Wind loads; Wind tunnel, Crosswind. 1 INTRODUCTION Wind load is the important design load for structures like long span bridges, tall buildings, high towers or mast structures. Under the wind load, tall buildings fluctuate in the alongwind, crosswind and torsional directions; although, codes pay more attention to the alongwind responses, high side ratio of tall buildings may worsen the transverse and torsional responses. Most codes and standards provide procedures for loading in the alongwind direction and there is little guidance for crosswind and torsional loadings. To this end, the wind tunnel is one of the experimental methods that investigates the behavior of tall buildings under the wind load in the alongwind, crosswind, and torsional directions. High- Frequency Base Balance (HFBB) model appears more economical and time-efficient method to investigate the response of the model under the wind load [1]. Zhou, Kijewski and Kareem have investigated aerodynamic loads on tall buildings. They have measured base moments of rigid models tested in the wind tunnel [2]. Wind loads and base moment coefficient of tall building models that have been tested in wind tunnel with HFBB method are available on http://www.nd.edu/~nathaz, this work is in frequency domain [3]. Wind is composed of mean and turbulent components; forasmuch as, the property of fluctuating load in the time domain is because of turbulent component; in addition to empirical methods, there are theoretical methods for the simulation of turbulent component of wind in time domain. Shinozyka and Deodatis have simulated wind load in onedimensional with stationary Gaussian stochastic processes [4]. Solari and Piccardo have modeled three dimension turbulent components of wind with the probabilistic approach [5]. Carassale and Solari have simulated wind velocity field with Monte Carlo method [6]. Bakhshi and Nikbakht have investigated the loading pattern and spatial distribution of dynamic wind load along the height of tall building in alongwind direction [7]. According to the discussion, this study aims to investigate the dynamic and static effects of wind loads on tall buildings in longitudinal and transverse directions while ignoring the torsional response. To this end, three models with equal heights and different side ratios have been simulated with OpenSees software, lateral system of models is the bracedtube system. For wind loading, two base speeds from ASCE7-10 have been selected in open terrain [8]. For loading in the time domain, the wind field has been simulated in 3-D with the probabilistic approach. Concerning the frequency domain, the wind tunnel data from modeling tested in university of Notre Dame has been used [9], the maximum response of models under this loading is significant. To investigate the building behavior under the loads, displacement and story shear are suitable; thus, this study is concerned with two prominent structural demands such as the story displacement and story shear to describe the results. Moreover, to compare the response under the wind load in time and frequency domains, base moment coefficient results from wind tunnel and simulated record have been compared and crosswind displacements have been presented for two of the models. 2 WIND LOAD IN THE TIME DOMAIN In this study, the wind load effects have been examined in the time and frequency domains. For the time domain, 3-D turbulence model of wind field has been simulated with the 3821

probabilistic approach; under these loads in time domain, the nonlinear time history analysis has been conducted. The results of this analysis are provided in the 7 th part of this study. The simulated wind speed is formed from mean and turbulent components. The time history of wind speed in 3-D coordinates with horizontal axis, x and y, and vertical axis z can be written as: Where and denote the mean and turbulent component of wind speed respectively. The mean component of wind speed is the function of height; whereas, the turbulent component in addition to the height depends on the coordinates of the points in the wind field. The accurate mathematical expression is used by the logarithmic law to simulate the mean wind speed profile; hence, the mean component of wind field has been simulated by the logarithmic law [5], [10]. Where is known as the friction velocity, is the roughness length and k is known as von Karman's constant. To determining the friction velocity, equation (3) and (4) have been applied, where K is the surface drag coefficient. Where is known as the basic wind speed, the relation between von Karman's constant and surface drag coefficient can be obtained from equations (2) and (3) for z=10 m. By using equations (2), (3) and (4), the mean wind speed is simulated [10]. Turbulent components in both alongwind and crosswind directions have been produced and the torsion are regardless. The turbulent component of the wind field is simulated by a stationary Gaussian multi-dimensional and multivariate random process with the average value of zero. The crossspectral density functions of the wind turbulence components are defined by Solari and Piccardo [5]: Where i and j denote the points of coordinates and l is the frequency; is the power spectral density function of the components of wind velocity turbulent, is the coherence function between the wind velocity turbulent components. Solari and Piccardo have defined the power spectral density matrix: (1) (2) (3) (4) (5) (6) Where and ; is known as the integral length scale of turbulent component, is the frequency and is the variance of : Where is a non-dimensional coefficient that is known as the turbulence intensity factor and and [10]. Turbulent components of wind are simulated by the Proper Orthogonal Decomposition method and fast Fourier transform. For alongwind, turbulent component was gathered by mean component but for crosswind, mean component of wind speed is zero. 3 WIND LOAD IN THE FREQUENCY DOMAIN To investigate the response of tall building in the frequency domain, determining the wind load in wind tunnel on the models is required. With respect to the frequency domain, the static forces obtained from the wind tunnel have been used. Modal analysis of tall building is used to determine the response of the simulated buildings in OpenSees software under the wind tunnel loads. Therefore, two required debates are briefly reviewed. 3.1 Modal analysis of tall building Dynamic equilibrium equation for tall buildings in the classic modal analysis can be written as follows: For the shear building with 3 degrees of freedom in each story,, and are mass, damping and stiffness matrix respectively. is known as the displacement vector in Lagrangian coordinates that is obtained from equation (9) and is known as the applied force vector at the mass center of each story. In the modal domain is determined by equation (10): (7) (8) (10) Where is known as the modal matrix, dynamic balance equation is written by equation (11): (11) Where, and denote modal mass, modal damping and modal stiffness matrices respectively and is known as the jth generalized force that obtained from equation (12): Where is known as jth mode shape vector [11]. (12) (9) 3822

3.2 High Frequency Base Balance method In wind engineering for estimating the generalized force acting on tall buildings, the HFBB technique is most popular. This method is based on rigid model on the wind tunnel. In this technique, base moments are measured with springs under it [1]. If it is assumed that the translation components of the first three modes vary linearly along the z-axis and torsional component constant, such as: Hardening Modulus have been assumed,, and respectively. Figure Figure3 depicts the Hardening material in OpenSees software [12]. (13) Where and are the values of the mode shapes at the top of buildings and is a constant value. The j is equal 1, 2 and 3 for the first three modes. The relationship between the story forces and base moments is shown in equation (14): (14) With equation (12), (13) and (14) following equation is simply obtained [11]: (15) Figure 1. Structural plans for model1, model2, and model3 Wind static forces obtain for each story by measuring the base moments in wind tunnel. 4 SIMULATION OF STRUCTURAL MODELS IN THIS STUDY In this study three models of tall buildings have been simulated with OpenSees software. To investigate the effect of side ratio in alongwind and crosswind, height of models are equal but the side ratio of first and second models are difference; also side ratio and aspect ratio of second and third models are equal but the fundamental periods of them are different for comparing wind tunnel results with probabilistic approach. The results of analysis for wind loads have been compared under loading in both the frequency and the time domains, for loading in time domain, time histories of wind loads have been simulated with probabilistic approach and for loading in frequency domain, have been used static wind loads that obtained from wind tunnel in the university of Notre Dame. For a batter comparison of the results of both static and time history analysis side ratio and aspect ratio of simulated models and wind tunnel models considered to have the same. The side ratios (D/B) of first, second and third models are 2, 3 and 3 respectively and the aspect ratios of them are 4.71, 5.77 and 5.77. Dimension of the structural plan for first model is 21m 42m and for second and third one is 14m 42m, height of three models is 140 meters. Lateral resistance system is the braced-tube system. The plans and 3- D views of models are shown in figures Figure 1 and Figure 2 respectively. In summary the first ten modes of three models are tabled in Table 1. The Hardening material has been used in the tube frame members in OpenSees software to model the nonlinear analysis. Hence, the values of Tangent Stiffness (, Yield Stress (, Isotropic Hardening Modulus and Kinematic Mode Figure 2. 3-D views of model1, model2, and model3 Table 1. Modal analysis results Periods of Model1 Periods of Model2 Periods of Model3 1 3.7213 4.3854 5.01647 2 2.704 2.3898 2.56042 3 1.7091 1.8385 1.86745 4 1.0816 1.1613 1.34135 5 0.8617 0.7432 0.83241 6 0.5594 0.5771 0.65928 7 0.5568 0.5611 0.61633 8 0.4893 0.4169 0.46156 9 0.3852 0.3688 0.413052 10 0.3674 0.3240 0.38172 3823

Figures Figure4 and Figure5 show a sample of wind load, simulated with MATLAB, at 140 meters up to ground in crosswind on alongwind direction in open terrain with speed 76 m/s respectively. Figure 3. Hardening material of OpenSees command language manual [13] 5 ASSUMPTIONS FOR SIMULATION OF THE WIND FIELD IN THE TIME AND FREQUENCY DOMAINS Two basic wind speeds selected from ASCE7-10 to simulate the wind field correspond to a 3-second gust speed at 10 meters above the ground. The exposure type in the wind field was assumed to be open terrain. The time step of data is 0.1. The cut off frequency of the fluctuating wind component is taken as 31.41 rad/s and the sampling interval of the spectrum is 0.05. The roughness length is taken as 0.01 m and Von Karman constant is 0.4. In the simulation of wind power, the drag force coefficient and air density are taken as 1.3 and 1.25 respectively. The angle between wind directions on the positive direction of the x-axis is assumed to be zero. The input information is presented in table Table2. In the probabilistic approach, the simulation of the mean wind speed is conducted by logarithmic law that is a deterministic process; whereas, the turbulent component of wind field is simulated by a stationary Gaussian multi-dimensional and multivariate random process with the average value of zero [8]. Figure 4. Alongwind time history in open terrain with speed 76 m/s Figure 5. Crosswind time history in open terrain with speed 76 m/s Table 2. Time history wind design parameters Wind Parameters value Basic wind speed (m/s) 47, 76 Exposure type open terrain Time step of data 0.1 Cut off frequency ( ) 31.41 Sampling interval of the spectrum 0.05 Roughness length ( ) 0.01 Von Karman constant (k) 0.4 Drag force coefficient ( ) 1.3 Air density ( ) 1.25 Figure 6. Wind Tunnel static load on model1 in open terrain with speed 76 m/s [9] 3824

In time domain, the responses of models obtain by nonlinear analysis under the time history wind loads. To investigate the response in frequency domain, static load of wind calculated in NatHaz modeling laboratory. Figure Figure6 shows a sample of wind load from ground to 140 meters height for the first model in the open terrain with 76m/s speed in the wind tunnel that is available in NatHaz modeling laboratory. In figure Figure6 A is mean component, B is alongwind background component, C is alongwind resonant component, D is crosswind background component, and E is crosswind resonant component [9]. 6 BASE MOMENT COEFFICIENT Base moment coefficient is appropriate parameter for comparing obtained wind load with probabilistic approach and wind tunnel test. Figures Figure7 and Figure8 present base moment coefficient of wind load for two models in alongwind and crosswind directions for basic wind speed 76 m/s for first and second models respectively that calculated with probabilistic approach. Longitudinal axis is frequency and vertical axis is non dimensional moment coefficient obtain as follow [2], [3]: Figure7. BMC of model1 with basic wind speed 76 m/s (16) Where is power spectral density of the external aerodynamic base moment, f is frequency in Hertz and here is known as root mean square of the fluctuating base moment which can be computed by: (17) Where is power spectral density of the fluctuating base moment response, the is computed by following equation, (18) Where is structural first mode transfer function which can be computed by equation (19). Figure8. BMC of model2 with basic wind speed 76 m/s (19) Where is natural frequency, is damping ratio in the first mode [7]. Due to the limitation of the models tested in the wind tunnel, two models are selected whose side and aspect ratios are 3 and 5.77, respectively [2]. Non-dimensional moment coefficients obtained from Notre Dame wind tunnel is plotted in figures Figure9 and Figure10 for base wind speed 76 m/s in open terrain for Model2 and Model3 respectively[9]. The base moment coefficient obtained from the probabilistic approach for the second and third models are plotted too for comparing these coefficients with measurements performed in the wind tunnel. Figure 9. Base moment coefficient from wind tunnel and probabilistic approach in crosswind direction with basic wind speed 76 m/s Model2 3825

larger than transverse. Due to the geometry of the structure, story shear force has reduced as the side ratio rises. Values of story shear in longitudinal and transverse are close to each other for base wind speed 47 m/s, with increasing the wind speed (speed 76 m/s), they have increased. Thus, the difference between the values for both directions for each model has considerably increased (comparing M2.L with M2.T and M1.L with M1.T for speed 76 m/s in figure Figure13). Comparing M1.T with M2.T and M1.L with M2.L with the same speed in figure Figure13, one notices that in the first model this response of structure is more than the second one. Hence, story shear on the two models are decreased by enhancing the side ratio. Figure 10. Base moment coefficient from wind tunnel and probabilistic approach in crosswind direction with basic wind speed 76 m/s Model3 7 ANALYSIS RESULTS This study addresses three prominent structural demands such as the story displacement, story drift, and story shear that were evaluated and also the results of the wind load. Maximum values of structural response under the wind are plotted. In the figures legend, M1, M2 and M3 mean model1, model2 and model3, L and T are longitudinal and transverse displacements of models under the wind load with speed 47 and 76 m/s that speed word have shown with S. Figure Figure11 demonstrates that the displacement of the two models (M1 and M2) under the time history wind load. The transverse displacements for both models are larger than longitudinal displacement; the difference between longitudinal and transverse for the second model with the side ratio of 3 is larger than the first model with the side ratio of 2 (comparing M1.T with M1.L and M2.T with M2.L in figure Figure11). It indicates that tall buildings with a large side ratio are more sensitive to wind load in crosswind direction; this is true for both speeds. Comparison of M1.T with M2.T and M1.L with M2.L in figure Figure11 reveals that the responses in alongwind for both speeds for the first model are larger than the second model. However, in crosswind the reverse is true; on the other hand, there is a big difference between the responses in transverse displacement. In comparison, the difference among the alongwind responses is relatively slight; thus, it can be concluded that for wind loading, changes in the side ratio of tall building are not very effective in the responses of the alongwind; special effects will be in the transverse displacement. To investigate the behavior of structures under the wind load in time domain, figure Figure12 shows maximum drifts of model1 and model2. This figure shows how the brace of structure affects the story displacement. Figure Figure13 shows maximum story shear of first and second models under the wind load simulated with probabilistic approach. According to Figure Figure13, wind loading on two models renders the longitudinal story shear Figure 11. Maximum displacements of model1 and model2 under the wind load Figure 12. Maximum drifts of model1 and model2 under the wind load 3826

Figure 13. Maximum story shear of model1 and model2 under the wind load The base moment coefficient in crosswind direction with base wind speed 76 m/s obtained from wind tunnel and the probabilistic approach have shown in figures Figure9 and Figure10 for the second and third models. These loads were applied to these models, the crosswind displacement was obtained from the static and dynamic analysis for the wind tunnel and probabilistic approach loads respectively. Figures Figure14 and Figure15 demonstrate that there is a satisfactory agreement for estimating drift and displacement responses between these two methods for the gust load in transverse direction. Figure 14. Crosswind displacements of model2 and model3 under wind tunnel and probabilistic approach loads with basic wind speed 76 m/s Figure 15. Crosswind drifts of model2 and model3 under wind tunnel and probabilistic approach loads with basic wind speed 76 m/s 8 CONCLUSION Increasing the basic wind speed results in increase of the difference between responses in orthogonal directions and also the increase in the side ratio will intensify this problem. The sensitivity of this type of buildings in the transverse direction is high and differences in both longitudinal and transverse displacement response for these models are impressive; whereas, the transverse displacement is dominant with respect to longitudinal. The buildings have a different approach in displacement and story shear responses; thus, transverse displacements always dominate longitudinal one on each model. However, the opposite applies to story shear where longitudinal shears are larger than transverse; since the increase of side ratio enhances displacement and reduces story shear it can be concluded that tall buildings tend to release the energy from the wind with displacement in high side ratio. The reduction of building surface in alongwind direction leads to a decrease in the displacement response in this direction. Nonetheless, the excitability of tall buildings under the turbulent component of wind strongly increases in crosswind. In this study, only the turbulent component of wind load in crosswind direction was considered for comparing the wind tunnel results with the probabilistic approaches regardless of the answer in longitudinal direction. The crosswind displacement for the gust load in open terrain reveals that there is a good congruity between these two experimental and analytical methods; as a result, the displacements results from the two methods are very close to each other. 3827

ABBREVIATIONS The abbreviation curtails figures description ac follow: Sigma Y: yield stress E: tangent stiffness H_iso: isotropic hardening modulus H_kin: kinematic hardening modulus M1: model 1 M2: model 2 M3: model 3 L: longitudinal displacement or story shear T: transverse displacement or story shear S.47: base wind speed 47 m/s S.76: base wind speed 76 m/s FD: frequency domain TD: time domain ACKNOWLEDGEMENTS Writers wish to acknowledge Sharif University of Technology, NatHaz Modeling Laboratory and Network for Earthquake Engineering Simulation (NEEShub) for their respective contribution. REFERENCES [1]. B. S. Taranath, Wind and earthquake resistant building: structural analysis and design, Marcel Dekker, Los Angeles, California, 2005. [2]. Y. Zhou, T. Kijewski and A. Kareem, Aerodynamic loads on tall buildings: interactive database, J. Struc. Eng., 129 (2003) 394-404. [3]. D.K. Kwon, T. Kijewski-Correa and A. Kareem, e-analysis of high-rise buildings subjected to wind loads, J. Struc. Eng., 134 (2008) 1139-1153 [4]. M. Shinozyka and G. Deodatis, Simulation of stochastic processes by spectral representation, Appl. Mech. 44(4) (1991) 191-204. [5]. G. Solari and G. Piccardo, Probabilistic 3-D turbulence modeling for gust buffeting of structures, Prob. Eng. Mech. 16 (2001) 73 86 [6]. L. Carassale and G. Solari, Monte Carlo simulation of wind velocity fields on complex structures, J. Wind Eng. Ind. Aerodyn., 94 (2006) 323 339. [7]. A. Bakhshi and H. Nikbakht, Loading pattern and special distribution of dynamic wind load and comparison of wind and earthquake effects along the height of tall buildings, 8th International Conference on Structural Dynamics, EURODYN 2011, 1607-1614 [8]. American Society of Civil Engineering (ASCE), ASCE 7-10, Minimum design loads for buildings and other structures, 2010. [9]. NatHaz Modeling Laboratory, University of Notre Dame, Aerodynamic load database, http://www.nd.edu/~nathaz. [10]. F. Petrini, A probabilistic approach to performance-base wind engineering, University of Rome (La Sapienza), Depart. Stru. Geo. Eng., Thesis, 2009. [11]. S. M. J. Spence, E. Bernardini and M. Gioffre, Influence of the wind load correlation on the estimation of the generalized forces for 3D coupled tall building, J. Wind Eng. Ind. Aerodyn. 99 (2011) 757 766 [12]. S. Mazzoni, F. McKenna, M. H. Scott and G. L. Fenves, OpenSees command language manual, 2007. [13]. Y. Zhou and A. Kareem, Gust loading factor: new model. J. Stru. Eng., 127(2) (2001) 168 175. 3828