A Monte Carlo Based Method for the Estimation of the Dynamic Response of High-Rise Buildings Induced by Turbulent Winds

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1 A Monte Carlo Based Method for the Estimation of the Dynamic Response of High-Rise Buildings Induced by Turbulent Winds Marra A. Smith 1, Luca Caracoglia 2 1 Graduate Student, Department of Civil Engineering, Northeastern University, Boston, MA, USA, marrasmith@coe.neu.edu 2 Assistant Professor of Civil Engineering, Northeastern University, Boston, MA, USA, lucac@coe.neu.edu ABSTRACT This paper presents a procedure for the calculation of the dynamic response of a tall building using Monte Carlo (MC) integration methods. The proposed MC numerical procedure was used to compute the power spectral density of the buffeting loads, and to derive statistical information about the dynamic response in the presence of uncertainty in the wind loading. The CAARC prototype building was utilized to validate the proposed algorithm. This study is used as a preliminary investigation for the fragility analysis of tall buildings under extreme winds. Future studies and analyses will be directed towards an application to performance-based engineering. INTRODUCTION Comprehensive research activities in the recent past have been undertaken in the area of riskbased assessment of structural integrity with a specific focus on earthquake engineering (performance-based engineering, PBE). Less emphasis has been devoted to the problem of PBE in the presence of wind hazards, especially for sensitive systems, such as high-rise buildings [1-3], where the presence of dynamic amplification of the vibration is of relevance. Even though risk-based methods for seismic design are readily available to practicing engineers, a direct technology transfer to wind hazards is not possible due to the uniqueness of the excitation mechanism. Current research activity is directed towards the development of a performance-based framework for wind engineering analyses with an application to tall buildings. The foreseen outcome of the studies will be the implementation of a set of numerical and analytical computer methods for the quantification of the conditional probability associated with the loss of an acceptable performance level against wind hazards. As part of these investigations we present a numerical method for estimating the dynamic response of high-rise structures due to buffeting loads. The significance of this approach is related to the possibility of replicating the closed-form analytical methods (based on random vibration techniques in the frequency domain), through equivalent numerical Monte Carlo (MC) algorithms. The advantage, from the computational standpoint, is that it further enables the subsequent implementation of numerical statistical analyses in the presence of uncertain wind loading scenarios. Capitalizing on the computational advantages of this proposed procedure, we have developed a two-step MC procedure, allowing for the estimation of the dynamic response due to the uncertainty of wind loading. The proposed approach aims to enable further implementation as a performance-based simulation technique, by potentially providing designers with a useful tool for practical application.

2 PROPOSED METHODOLOGY The proposed methodology utilizes standard frequency-domain random-vibration analysis for the estimation of wind-induced loading and dynamic response on a high-rise building [4-6]. The procedure, which was developed in this paper, assumes that the structure under consideration was linear-elastic. This assumption was valid since (at this stage) investigations were primarily concentrated on serviceability limit states for high-rise buildings. Moreover, few fundamental modes were required to capture the main response features (i.e., the magnitude of the dynamic response) [5, 7]. Therefore, since this study was primarily interested in displacements as opposed to acceleration, modal expansion of the dynamic response was limited to the fundamental modes. The simplified building that was considered has a prismatic floor plan and a main wind force resisting system composed of lateral frames, which are symmetric with respect to the primary axes of bending. Due to the exploratory nature of this study, it was assumed that the torsional effects are less significant for this building type. The proposed approach was employed to analyze the dynamic response at very large wind velocities, for which dynamic buffeting response is the main concern. The aerodynamic loading was based on equivalent quasi-steady formulation of wind forces [4-6] since the main objective of the simulations was the validation of the proposed numerical procedure. Mean wind direction was considered orthogonal to one of the vertical faces of the building, and the static moment coefficient was neglected (dynamic response principally in the along-wind direction). Load induced by vortex shedding was also neglected, as vortex shedding may be primarily considered of importance at lower wind velocities than those analyzed in this study [4, 8]. Figure 1 is a schematic depicting building and axis orientation, wind loading, F d (z,t), per unit height z, and bending displacements as a result of first-mode dynamic response. The dynamic displacements are denoted as x(z,t) and y(z,t) for the along-wind and across-wind dynamic response, respectively. The static response (not shown in the figure) is denoted as X(z). The normalized mode shape, which is not necessarily linear, is denoted by Φ g,d (z), with d=x when solving for the along-wind response and d=y when solving for the across-wind response. As an example, the total dynamic response in the along-wind direction was derived by modal superposition as x(z,t) = Φ g,x (z)ξ g,x(t), with ξ g,x (t) being the generalized coordinate of mode g. z x(z,t) or y(z,t) g (z) Φ g,d (z) F d F(z,t) ( ) h Figure 1: Schematic of building orientation, loading, and displacements

3 The main equations for the evaluation of the dynamic response in the along-wind direction are summarized below. Equation 1 corresponds to the modal force spectra as a function of frequency, n, with C D being the drag coefficient normalized with respect to the B dimension of the building. The mean wind velocity is denoted by U(z); S uu (n,z 1,z 2 ) is the crossspectrum of the along-wind turbulence, u, at z 1 and z 2. Equation 2 shows the total response autospectra without modal cross-coupling (with an exclusive contribution from g=1 in the analyses), * from which the RMS response (Equation 3) and mean response are found. In Equation 2, ζ stg, x is the structural damping, ζ aerog, x is the aerodynamic damping, and M gg,x is the modal mass. h h S Qg, x Q g, x (n) = ρ 2 B 2 C D Φ g,x (z 1 )Φ g,x ( z 2 )U ( z 1 )U ( z 2 )S uu (n, z 1, z 2 )dz 1 dz 2 (1) 0 0 Qg, xqg, x xx (, 1, 2 ) Φ g, x ( 1 ) Φ, ( 2) (2) g x 2 2 g n n * M, ( 2 π gg x n0 ) ( ) g, x 2 stg, x aerog, x n0 n ζ + ζ g, x 0g, x S n z z z z S ( n) σ x = S ξξ (n, z 1, z 2 )dn (3) Effects of uncertainty in the characterization of aerodynamic loading and their influence on the dynamic response were simulated through MC algorithms. The proposed methodology utilizes MC methods to accomplish the following: 1) compute the power spectral density of the modal buffeting force (Equation 1) as a function of frequency, for each mode of the structure, by numerical integration; and 2) derive statistical information about the dynamic response of a highrise building due to the uncertainty in the characterization of dynamic loading. By combining Steps 1 and 2, we are using what we have termed a Two-Step Monte Carlo Algorithm to estimate the dynamic response of high-rise buildings due to turbulence-induced loading. The MC approximation of a generic surface integral, f (z 1,z 2 ), was found by utilizing a set of randomly generated and uniformly distributed numbers (integration points) to numerically estimate the statistical expectation of the function [9]. In the case of a double integral (as in Equation 1), there are two randomly generated samples of size N MC defined on the intervals [Z 0,Z A ] and [Z 0,Z B ], where the variables Z 0, Z A, and Z B are the integration limits along the height of the structure. Equation 4 gives the general form of the equation used to approximate the double integral in Equation 1. Z A Z B f (z 1, z 2 )dz 1 dz 2 E f (z 1, z 2 ) Z A Z 0 Z 0 Z 0 [ ]( )( Z B Z 0 ) (4) DESCRIPTION OF SIMULATED BUILDING (CAARC) The CAARC building was analyzed as a prototype structure in this research. This prototype had been selected by the Commonwealth Advisory Aeronautical Research Council of Australia in the early Eighties as a benchmark high-rise structure to be employed for studying its dynamic response affected by wind loading [10]. Figure 2 shows a schematic view of the CAARC building with an indication of the main dimensions (depth D, width B, height h).

4 Wind D-h face h D B Figure 2: Schematic view of the CAARC building The building is a prismatic structure with a rectangular floor plan of dimensions B=30.5m and D=45.7m. The reference wind direction, corresponding to a zero angle of incidence, coincides with the direction orthogonal to the vertical face designated as D-h in Figure 2. Tables 1(a) and 1(b) summarize the CAARC building full-scale model parameters that were employed in the simulations. Dynamic response was derived from wind tunnel-based aeroelastic model experiments conducted in a series of laboratories [10]. First mode response in the alongwind (Equation 2) and across-wind directions were simulated in this paper by considering the information available in the literature [10]. Building mass per unit length, m(z), structural frequencies, and modal damping were calibrated to represent the full scale response of the prototype. Mode shapes (Table 1(a)) were approximated as linear. Boundary layer wind loading was simulated by analyzing the approaching wind characteristics selected as reference in [10] for the tests. Turbulence intensities (Table 1(b)) in the direction parallel to the along-wind B direction (u) and parallel to the across-wind D direction (v) were also derived by calibration of the values employed in wind tunnel [10] to model fullscale behavior. A reduction of the turbulence intensity, I v, in comparison with I u was postulated [11]. The longitudinal velocity spectra employed to represent the stationary time-dependent dynamic loading were derived by fitting experimentally observed data in the wind tunnel [10], as indicated in Equation 5a, with f being a reduced frequency (Equation 5b): ns u (n) U (h) 2 I = 0.6 f 2 u (2 + f 2 ) (5a) 5 6 f = 1600n U (h) (5b) Only static coefficients of lift and drag and their first derivatives were considered in the analyses, while also neglecting any eccentricity in the wind loading, which is plausible for mean wind direction orthogonal to the D-h face. Reference values (Table 1(b)) of C D, C L, etc. per unit length, were derived by integration of mean surface pressure coefficients measured in the wind tunnel [10] around the building along a reference strip of unit height taken at approximately two-thirds of the total height. These values were held constant along the z direction for the

5 purpose of the simulations, even though the algorithm readily allows for the parameters to locally vary with z. Table 1: CAARC Building - (a) Numerical Model Parameters; (b) Wind Loading Parameters [10] Variable Value Assigned Variable Value Assigned h [m] 183 dz [m] (Equation 1) 1 m(z) [kg/m] 220,800 * ζ stg, x * = ζ stg, y 0.01 n 0g, x = n 0g, y = n 0 [Hz] 0.2 Φ g,x (z) = Φ g,y (z) (z/h) γ ; γ = 1 Z 0 [m] 0 Z A = Z B h (a) ρ [kg/m 3 ] 1.25 Roughness, z 0 [m] 0.5 U(z) [m/s] U(z)(z/h) / ; β = 0.25 I u (z) [a] (z/h) I v (z) 0.75I u (z) C z,u 10 C z,v 0.667C z,u C D 1 C L -0.1 dc D / dα -1.1 dc L / dα -2.2 (b) Note [a]: Derived from wind tunnel measurements; Figure 2 in [10] NUMERICAL VALIDATION The purpose of this set of initial simulations was two-fold: 1) verification and updating of the model and wind loading parameters in Table 1 to be compatible with the wind tunnel response in [10]; 2) validation of the first step of the proposed MC algorithm, and optimal selection of the number of MC integration points (i.e., sample size). To accomplish the first objective, the closed-form (CF) numerical solution was compared to wind tunnel based reference values. The CF solution employs a discrete integration scheme for the derivation of the modal force. The modal force was subsequently utilized to determine the dynamic RMS response of the building at the top floor, z=h, by integration (frequency by frequency) of the power spectral density of the response. An example of the results is shown in Figure 3, in which the numerically simulated RMS response in the along-wind direction is compared to wind tunnel measurements, as a function of the reduced velocity, U(h)/(n 0 B). The building response was normalized with respect to the dimension B. Excellent correspondence was found, as evident from the analysis of this figure, with an absolute error no more than two percent between the two curves. Similar results, not shown for brevity, were observed in the case of the mean wind response, while some discrepancy was seen when the comparison was extended to the across-wind direction (most likely attributed to effects of vortex shedding, not simulated by the current wind loading model). This result also justified the use of the CF numerical simulation results as exact target values in the performance analysis of the MC algorithm.

6 Normalized Along-wind RMS Response, σx/b Wind Tunnel Based Reference Values Numerical Simulation (CF) Reduced Velocity, U(h)/(n 0 B) Figure 3: Comparison of closed-form (CF) numerical simulation to the CAARC building response derived from wind tunnel tests [10] for wind orthogonal to the D-h face (Figure 2). The performance of the MC algorithm (second objective of the validation study) was assessed through repeated MC estimations of the integral in Equations 1 and 4, as a function of the sample size N MC. Figure 4 shows the main results of this study. The scatter plot corresponding to 100 consecutive MC simulations is shown in Figure 4(a). The normalized along-wind RMS response of the top floor of the CAARC building, computed by MC method at reduced velocity U(h)/(n 0 B)=8.2, is compared to the exact value (see Figure 3). The maximum absolute error is variable between 7% for N MC =1,000 and 2% for N MC =10,000, while it is almost negligible for larger sample sizes. The coefficient of variation (CoV), calculated for each sample population corresponding to N MC, was employed as a statistical indicator for the selection of the optimal N MC value (Figure 4(b)). Normalized Along-wind RMS Response, σx/b Target Value from CF Simul. at U(h)/(n 0 B)=8.2 (a) 1,000 5,000 10,000 50, ,000 N MC (No. of MC Integration Points) Coefficient of Variation 0.03 (b) ,000 5,000 10,000 50, ,000 N MC (No. of MC Integration Points) Figure 4: Normalized along-wind RMS response of the CAARC building at U(h)/(n 0 B)=8.2, computed by MC method (step 1) (a) scatter plot of 100 consecutive MC simulations as a function of the no. of MC integration points (N MC ); (b) Coefficient of Variation vs. N MC

7 As shown in Figure 4(b), the CoV was very small, generally less than 2% in all cases, and almost negligible for N MC beyond 10,000. Moreover, the decrement in the CoV was not proportional to the number of MC integration points. Practical implications associated with the selection of N MC were another important issue analyzed by this study. Therefore, the run time of the algorithm was considered as a second indicator of performance so that future implementations, such as a numerical tool for the practicing engineer, are plausible. Table 3 summarizes the effects of N MC on the run time of the MC algorithm. Values were normalized with respect to the time for N MC =100,000, and, for comparison purposes, the relative time corresponding to the CF simulation is indicated. The CF simulation took almost three times longer than the longest MC simulation, and, in particular, about 120 times longer than the MC simulation for N MC =10,000. Table 3: MC Validation Run Time vs. No. of Integration Points (N MC ) N MC Average Relative Run Time [b] 1, , , , , Closed Form Note [b]: Run time relative to N MC =100,000 By combining the results of Figure 4 with the average run time estimation, the optimal number of integration points was chosen as N MC =10,000. Even though run time is extremely small for N MC less than 10,000 points, significant error and variance are introduced (Figure 4), which do not justify their use in practical applications. FRAGILITY ANALYSIS The two-step MC algorithm is employed in this section as a numerical tool for the fragility analysis of the CAARC prototype building under the influence of high winds. Fragility analysis is a standardized methodology, utilized for performance-based structural design [1, 3, 12] against wind hazards. The objective of a fragility analysis is the computation of the (conditional) probability of exceedance of representative indicators, corresponding to a specific feature of the dynamic response. A set of thresholds is usually selected to represent different levels of structural performance based on such indicators. These indicators are either required or are prescribed by the designer. Examples of indicators include inter-story drift ratios, maximum lateral drift, and acceleration levels for occupant comfort. In this study, exceedance probabilities were represented as a function of the reference wind velocity and were computed by incorporating the effects of uncertainty in the description of the aerodynamic loading. Specifically, the effects of random aerodynamic static coefficients were analyzed. Emphasis was given to the drag coefficient, C D. The purpose of this study was related to the along-wind response estimation due to buffeting for wind direction perpendicular to the D-h face. Other static coefficients (C L, etc.) were not analyzed at this stage. Another objective of this study was the further analysis of the proposed numerical MC algorithm.

8 Figure 5 shows typical results of the analyses conducted on the CAARC prototype building. The two-step proposed MC algorithm is employed for numerically deriving the histogram, and the subsequent probability density function (PDF), of the along-wind response for a random C D coefficient. Results for reduced velocity U(h)/(n 0 B)=8.2, referenced at z=h, are shown. Figure 5(a) analyzes the effects of a random static drag coefficient per unit height, assumed as a constant throughout the height, and simulated as a gamma-type random variable with mean 1.0 and standard deviation 0.3. The mean and the standard deviation were derived through comparison of the wind tunnel tests conducted at various research institutions and described in [10]. Figure 5(b) shows the PDF of the RMS response, σ x, at the top of the building. The results of the MC algorithm (Figure 5(b)) confirm that, because of the linear dependence between C D and modal force (Equation 1), the response PDF also follows a gamma distribution. Dynamic structural response was investigated in the absence of other sources of uncertainty. Probability Density Function (a) (b) Probability Density Function Drag Coefficient, C D Along-wind RMS Response, σ x Figure 5: Results of the MC simulation algorithm (step 2) for gamma-type random distribution of drag coefficient (C D ) (a) C D PDF; (b) σ x PDF for U(h)/(n 0 B)=8.2 s The method of moments, based on empirical mean and standard deviation of the sample population generated by MC algorithm, was employed to calculate the scale and dispersion parameters of the input and output gamma distributions. The proposed MC algorithm can be accepted based on numerical validation depicted in Figure 5(b). An example of fragility curves for an uncertain distribution of C D as a function of reduced velocity, U(h)/(n 0 B), is shown in Figure 6. Figure 6(a) shows the along-wind RMS response at top of the building, which corresponds to Figure 5 when U(h)/(n 0 B)=8.2; Figure 6(b) shows the along-wind peak response derived from Figure 5(a) by application of Professor Davenport s peak factor standard equation [6, 13]. The interval of reduced velocities is variable between 4 and 10, which coincides with medium to severe wind velocities for the CAARC building between 25 m/s and 60 m/s, measured at the top floor. In both figures, dimensional thresholds limits (T1 to T5) were pre-selected as a function of the building width (B=30.5m, see Figure 2), variable between 0.01%B to 1.5%B. It must be noted that threshold T5 may be associated with the maximum lateral drift, usually acceptable in the context of dynamic full-scale analyses of building response under wind excitation [14]; the total drift of threshold T5 also corresponds to an inter-story drift ratio of 1/400 for the same building [14].

9 1.0 (a) 1.0 Probability of Exceedance Probability of Exceedance T1=0.0010B T2=0.0025B T3=0.0050B T4=0.0100B T5=0.0150B Reduced Velocity, U(h)/(n 0 B) 0.0 (b) Reduced Velocity, U(h)/(n 0 B) Figure 6: Example of fragility curves for uncertain distribution of C D as a function of reduced velocity, U(h)/(n 0 B) (a) Along-wind RMS response; (b) Peak Response. NOTE: T1 to T5 are dimensional preselected exceedance thresholds, which are the same for both figures (B=30.5m, see Figure 2) From the figures it can be seen that if the RMS is employed as a measure of performance and T5 is selected as the reference threshold for design purposes, the probability of exceedance is very low, below 1% even during severe wind conditions for dynamic vibration. However, if the peak response is the criterion, which includes the effects of the mean drift, the probability of exceedance becomes large (about 90% at U(h)/(n 0 B)=8.2). This fact may possibly lead to unacceptable drift ratios, which are mainly associated with the mean wind pressure and force distribution in this example. Even though this example suggests a predominant effect by the static wind load on the probability of exceedance for the CAARC building, the contribution of the dynamic response and of σ x cannot be ruled out for tall flexible structures in general. Therefore, fragility curves, which also incorporate structural dynamic response, seem preferable in the context of design against high winds. CONCLUSIONS This study summarizes recent research activities aimed at the development of a performance-based methodology for the design of high-rise buildings against wind hazards. As part of these activities, the development of a numerical MC algorithm for the derivation of the dynamic response was implemented. The simulation algorithm was employed to analyze the effects of uncertain wind force estimation on the dynamic response of a simulated tall building, modeled after the CAARC benchmark building. Comprehensive validation of the numerical procedure was conducted by comparing the simulated results to the reference values extracted from wind tunnel tests. Moreover, this algorithm was successfully employed in the derivation of fragility curves associated with serviceability limit states. Along-wind response and displacement-based criteria were mainly utilized for the validation of the probability-based methodology and the development of the fragility curves.

10 Although neglected at this stage, torsional response, higher-mode dynamic influence (especially important for acceleration-based design criteria), and a more rigorous description of the wind loading in the across-wind direction will possibly be included in future studies. REFERENCES [1] Bashor, R., Kareem, A., 2007, Probabilistic performance evaluation of buildings: An occupant comfort perspective, in: Proceedings of the 12th International Conference on Wind Engineering (12- ICWE), Cairns, Australia, Vol. 2, pp [2] Jain, A., Srinivasan, M., Hart, G.C., Performance based design extreme wind loads on a tall building, The Structural Design of Tall Buildings, 10 (1), [3] Norton, T., Abdullah, M., Stephens, D., 2008, Proposed methodology for performance-based vulnerability assessment of wind-excited tall buildings, in: Proceedings of the Fourth International Conference on 'Advances in Wind and Structures (AWAS'08)', Jeju, South Korea, pp [4] Kareem, A., Model for predicting the acrosswind response of buildings, Engineering Structures, 6 (2), [5] Piccardo, G., Solari, G., Generalized equivalent spectrum technique, Wind and Structures, 1 (2), [6] Davenport, A.G., Response of six building shapes to turbulent wind, Phil. Transaction of the Royal Society London Ser. A, 269 (1199), [7] Humar, J.L., Dynamics of Structures, Taylor & Francis, London. [8] Kareem, A., Aerodynamic response of structures with parametric uncertainties, Structural Safety, 5 (3), [9] Robert, C.P., Casella, G., Monte Carlo statistical methods, Springer Science, New York. [10] Melbourne, W.H., Comparison of measurements on the CAARC standard tall building model in simulated model wind flows, Journal of Wind Engineering and Industrial Aerodynamics, 6 (1-2), [11] Dyrbye, C., Hansen, S.O., Wind Loads on Structures, John Wiley & Sons, Inc., New York. [12] Filliben, J.J., Gurlye, K., Pinelli, J.-P., Simiu, E., 2002, Fragility curves, damage matrices, and ind induced loss estimation, in: Proceedings of the Third International Conference on Computer Simulation in Risk Analysis and Hazard Mitigation, Sintra, Portugal, pp [13] Davenport, A.G., Note on the distribution of the largest value of a random function with application to gust loading, Proceedings of the Institute of Civil Engineering, 28, [14] Chan, C.M., Chui, J.K.L., Huang, M.F., Integrated aerodynamic load determination and stiffness design optimization of tall buildings, The Structural Design of Tall and Special Buildings, 18 (1),