ON THE APPLICATION OF STOCHASTIC DECOMPOSITION IN THE ANALYSIS OF WIND EFFECTS

Transcription

1 ON THE APPLICATION OF STOCHASTIC DECOMPOSITION IN THE ANALYSIS OF WIND EFFECTS Xinzhong Chen and Ahan Kareem Department of Civil Engineering and Geological Science, Univerity of Notre Dame, Notre Dame, IN , USA ABSTRACT Thi paper preent the formulation and application of tochatic decompoition of covariance and cropower pectral denity matrice in the analyi of wind load effect on tructure. Thi technique not only enhance our undertanding of the underlying phyic of wind load and their effect on tructure, but alo provide efficient mean to etimate tructural repone under multi-correlated excitation. Detailed dicuion on the truncation of higher wind loading mode in the modeling and imulation of random wind load procee and aociated analyi of wind effect are dicued uing an example high-rie building. KEYWORDS Proper orthogonal decompoition, Stochatic decompoition, Simulation, Wind effect, Equivalent tatic load, Building, Random vibration, Turbulence. INTRODUCTION The tochatic decompoition cheme involve decompoition of the covariance and/or the cro-pectral denity (XPSD) matrice of a multi-variate random proce. Thi decompoition technique i theoretically baed on the Karhunen-Loeve expanion which i alo known a proper orthogonal decompoition (POD) or principal component analyi (PCA). Armit (1968) pioneered the application of POD of the covariance matrix to wind-related problem, and it wa later ued by many reearcher in decribing preure fluctuation on building and tructure and a hot of wind-related problem (e.g., Lee 1975; Kareem 1978; Kareem and Cermak 1984; Holme 1992; Davenport 1995; Caraale et al. 1998; Tamura et al. 1999; and Kareem 1999). The tochatic decompoition of fluctuating preure field on a tructure provide ueful phyical inight to the patio-temporal nature of wind load utilizing underlying eiegnvalue and eigenvector. It alo provide a ueful mean of relating the preure field with the attendant load effect on tructure. The concept of tochatic decompoition of XPSD matrix a applied to probabilitic dynamic and digital imulation of multivariate random procee wa introduced in Li and Kareem (1993 and 1995). It i very effective in determining the repone of neted-cacade multiple input/output ytem that require

2 J # - - perturbation or iterative analyi technique. It can alo be ueful in the dynamic analyi of large cale tructure expoed to wind. Other application can be found in Lin (1992), Di Paolo (1998), Caraale et al. (1998), Benfratello and Mucolino (1999), Kareem (1999) and Kareem and Mei (2). In thi paper, the theoretical background of the tochatic decompoition technique i preented. Emphai i placed on it application to the analyi of wind effect on tructure. A detailed dicuion on the truncation of higher wind loading mode in the modeling and imulation of random wind load procee and analyi of wind effect i provided uing an example high-rie building. STOCHASTIC DECOMPOSITION Covariance Matrix Following Mercier theorem and the Karhunen-Loeve expanion, a dicrete random field,! "# can be expanded in term of patial optimal orthogonal matrix $ a: % $'& ( ) $ ( / & % $ # (1) The orthogonal matrix $ i the eigenvector matrix of the covariance matrix 21 with diagonal eigenvalue matrix 3. Thee are given by: 546$ 37$ (2) It i noted that the tochatic decompoition of the covariance matrix decompoe a et of correlated random procee into a number of component random ub-procee & (. Any two random procee -98 and -;: ( are tatitically non-correlated at zero time lag: <5= :?>@ BA 8C: <D= 8 ( E>@ FA 8G:H 8 (3) It can be hown that the integral of the mean quare value i equal to the um of the eigenvalue, i.e., <D= #>@ <5= & & >@ In above, A 8C: i the Kroneker delta; and upercript I Spectral Matrix <D= + E>@ H + (4) repreent the matrix tranpoe operator. The XPSD matrix of random procee can be decompoed a: J 4 KL NMOKL MQPKL (5) where R repreent tranpoe and conjugate operator. Thi decompoition can be obtained by the Choleky or Schur (modal) decompoition. When the decompoition i baed on the eigenvector of XPSD matrix, MOKL TSUKL V WDKL, where SUKL and W5KL are eigenvector and eigenvalue matrice, and given by: 4 KL SUKX BW5KL SUKL (6) The random procee can be viewed a the output of a ytem with a tranfer matrix SUKL and a vector-valued input Y with XPSD matrix of W5KL, i.e. KL TSUKX V WDKL ZS[KL Y KL S + KX \ + KL (7)

3 b l v or it i written in the time domain a: % + ( / + % ^]`_ a6b[c + edgfh i\ + fj k.f (8) where c + i the invere Fourier tranform of S + KL. It can be een that any two element procee from 8 are tatitically fully coherent, while any two procee from different 8 and : are noncoherent. Hence, each random proce i viewed a a ummation of mutually non-coherent component ub-procee. The relationhip between the eigenvector of the covariance and frequency dependent eigenvector of the XPSD matrice i: $237$ # ] ANALYSIS OF WIND-INDUCED RESPONSE S[KL WDKL msnpkl ikek (9) Wind effect on tructure can be conveniently eparated into background and reonant component. Typically, the dynamic repone analyi i conducted in modal pace for computational efficiency. In general, while only a few number of mode need to be included for the reonant repone, the background component require larger number of mode. Therefore, direct calculation of the background repone uing quai-tatic analyi enhance computational efficiency over the modal analyi approach. The background component of a pecific repone o l (i.e. diplacement, hear force, moment) and it mean quare value can be given a: oqp l Br5 % Fr $'& ( (1) hv teu Fr D4 r # Fr $`37$ # r #.w + H + (11) where w + yx :, {z :} ~:i+ ; r i the R influence vector; i the ƒr external wind load vector; & i the ' [R7 expanion coefficient vector of POD; $ i the R' eigenvector matrix of the covariance matrix D4 of ( ; and and Uˆ are the number of the dicrete load and the eigenvalue (wind load mode) conidered, repectively. Baed on the load-repone-correlation approach propoed by Kaperki (1992), the background component of the equivalent tatic load ditribution for the peak repone opš Œ E NŽ teu (where Ž i the peak factor) i expreed a: p Ž 1 r #X t u Ž $237$ # r #@ t u w +.H + $ + t u (12) It i noted that the background repone and the aociated equivalent tatic load can be expreed a a um of the contribution from wind loading mode. The contribution of each mode depend on the influence function, loading mode hape and eigenvalue. The reonant component can be calculated by uing modal analyi technique. Conidering the firt mode of the tructural ytem, the equation of motion are reduced to uncoupled equation in term of the modal coordinate. The XPSD matrix of i given by: J@ KL B KL m # J 4 KX 5 P KL FM KL im PKL (13) M KX B gkl # MšKL (14)

4 Á ³ l b : v Å ] l b Ÿ gkl Bk9 - Ž =œ : KL E>/ œ : KL ž : id Ÿ e 8 : : Ÿ : Ÿ (15) where MšKL i the ' 'R decompoed matrix (Eq. 5,!ˆ ). In the cae of decompoition baed on the eigenvector of XSPD matrix, MOKL SUKL V WDKL ; i the R tructural natural modal hape matrix; ž : iÿ : K : are the ma and frequency in ª -th mode; : and : are the tructural and aerodynamic damping ratio in ª -th mode (ª ); and e T«d. The mean quare value of ª -th modal repone i expreed a (Kareem 1999): ] œ : K # : JL KL : kekq ²± œ : KL m³ :i+ KL Eµ + KL ikek (16) :i+ KL # : S + KX. The reonant repone component can be evaluated by auming that the where forcing function i placed by a white noie with a contant pectral denity at the tructural natural frequency (Kareem 1987): j ¹ h± The reonant component of o l i given by: where ¾ K ³ : :E+ K : iµ + K : º K : E» ( : : Ež o¼ l % :, q¾ : ': ¼ ( / t ¹ :, t ¹ :, q¾ : j ¹ (18) : ÀrnÁ l : K : i the participation coefficient of ª -th mode to the repone o l ; and l i the ma matrix in phyical coordinate. Accordingly, the ª -th mode reonant equivalent tatic load ditribution can be expreed in term of inertial force ditribution a: ¼ : ŽÂÁ l : K : : (17) j ¹ (19) It i noted that the reonant modal repone and equivalent tatic load can be expreed a a um of the component aociated with frequency dependent loading mode. The contribution of each mode depend on the tructural dynamic characteritic, loading mode hape and eigenvalue. Alternatively, the background repone can alo be given in term of modal repone. The mean quare value of the background repone in modal and phyical coordinate are given by (Kareem 1999): u jã XÄ where Å :i+ T # : $ +. b l ³ :E+ KL iµ + KL kek K :» ž :i+ H + K :» ž The total abolute peak value of repone o l i then given by (Chen and Kareem 2): o Œ E Ž t NŽ V teu t ¹ Ž)Æ teuç p where Ç p and ÇQ: ¼ are the weighting factor given by: :, : / t ¹ ÇQ: ¼iÈ teu BrÉÆ :, p Ç p t u :, 9¾ :, : u (2) Ê: ¼ ÇË: ¼È (21) Ç p teu t / ÇË: ¼ t ¹ t (22) Accordingly, the total equivalent tatic peak load ditribution including tatic load i given a:

5 â K w ÍÌ APPLICATION Ì p ÇQ: :, Š: ¼ ÇQ: ¼ (23) Ω n 15 x 14 1 A 76 tory 36 meter with 76 DOF i ued to dicu the formulation preented here. The firt natural frequency and damping ratio are.16 Hz and.1, repectively. The mean wind velocity at height 8 above ground i given by the power law Î 8 Î l 8 Ï lð», where Î l i the mean wind velocity at 1 m above the ground and i choen to be 15 m/ec. The oneided cro-pectrum of along-wind fluctuation Ñ 8 and Ñ : e ÒqÓ9 i given by: ºÙØ Ô@ÕÖ Õ KL l Î l»iú Û w}üý 8 : KX wüqý where = 8 : KL NÞß6àid w ÏqÏ K K 8 dá : Î l Î l, Ø l =.3 and w (24) (25) =7.7 Baed on the quai-teady and trip theory, the drag on the -th tory i given by: 8 ( ) ÏÂäã å z 8(æ ç Î 8 Ñ 8 Î 8 (26) where drag coefficient æ ç =1.2 and z 8 i the tributary area for the -th tory unit. Figure 1 and 2 how the eigenvalue of the covariance matrix and the contribution of each loading mode to the background diplacement at the top of the building. The total RMS repone i given by the quare root of the um of the quare of each component. The loading mode hape baed on the covariance and XPSD matrice are hown in Fig. 3. The equivalent tatic load ditribution (preure) for the diplacement at the top, the bae hear force and bae over-turning moment are hown in Fig. 6(b). Figure 6(a) alo how the mean tatic load (preure) given by = Ï6äãå æ ç Î 8 ". It i noted that the background repone and load are dominated by the firt wind load mode. It contributed about 95è, 95è and 99è (i.e. w H xqé ê + H + ) to the mean quare diplacement, bae hear and bae moment, repectively. RMS diplacement [m] Wind load mode number Fig. 1 Eigenvalue of covariance matrix Wind load mode number Fig. 2 Background diplacement Mode 1 Mode 2 Ω 2 =.51e+5 Ω 1 = 1.33E Wind load mode hape Λ 2 (f 1 ) Mode 1 Mode 2 =2.69e+4 Λ 1 (f 1 ) =3.28e Wind load mode hape Fig. 3 Wind load mode hape (a) from covariance matrix; (b) from XPSD matrix Λ(f) 6 x x Frequency (Hz) Fig. 4 Eigenvalue of XPSD matrix

6 ë ³ Figure 4 how the eigenvalue of the XPSD matrix at different frequencie. It i noted that at the lower frequency range the firt wind load eigenvalue dominate, wherea at higher frequency range all the eigenvalue are of the ame order. Figure 5(a) and 5(b) how :i+ K 8 [ 7# : S + K 8 ( =ª =1,2; ì ' which ugget relative ignificance of each loading mode to the firt and econd tructural mode repone. Thee alo how approximate orthogonality among loading mode and tructural mode. Since the firt loading mode hape i imilar to the firt tructural mode, the orthogonality among the higher loading mode and firt tructural mode approximately hold. However, it i not true in the cae of econd tructural mode, in which not only the firt and econd loading mode, but alo the higher loading mode have non-negligible contribution. In fact, the loading mode depend on the patial variation of the fluctuating wind field, thee do not necearily guarantee the orthogonality with the tructural mode which depend on the ma and tiffne ditribution of the tructure. In thi example, the RMS reonant repone i dominated by the repone in the firt mode (more than 97è ), which i contributed mainly by the firt loading mode (about 9è ). The ratio between mean quare background and reonant repone are.32,.4 and.33 for diplacement, bae hear and bae moment, repectively. The reonant equivalent tatic load for the firt and econd natural mode are hown in Fig. 6(c). The total peak equivalent tatic load i hown in Fig. 6(d). The peak factor Ž i aumed to be 3.5. The gut repone factor for the diplacement at the top of the building, bae hear force and bae over-turning moment are found to be , and 1.832, repectively. The tochatic decompoition technique ha been ued in the imulation of random procee uing pectral or hybrid pectral and time erie cheme (Kareem 1999). It ha alo been implemented in tate-pace modeling of multicorrelated wind field (Benfratello and Mucolino 1999; and Kareem and Mei 2). The decompoition in Eq. 5 uing modal cheme require more computational effort in comparion χ jn 2 (f1 ) χ jn 2 (f2 ) Wind load mode number (a) Firt natural mode Wind load mode number (b) Second natural mode Fig. 5 Significance of loading mode to tructural mode repone Preure F [kpa] (a) Mean tatic load Mode 1 Mode Preure F eri [kpa] (c) Reonant load Di. Shear force Moment Preure F eb [kpa] (b) Background load Di. Shear force Moment Preure F +F e [kpa] (d) Total load Fig. 6 Equivalent tatic load ditribution

7 with the Choleky decompoition. In both cheme, truncation of higher mode or reduction of the order of decompoition enhance computational í efficiency. In the tate-pace modeling, further approximation of the frequency dependent eigenvector of XPSD matrix i poible in pecial cae. The eigenvector can be evaluated at a fixed frequency if the eigenvector change very lowly with repect to the frequency (Kareem and Mei 2). Figure 7 how the influence of the truncation of higher loading mode on the PSD of the generalized force in the firt tructural mode and of the force on the 5th tory. The number of loading mode included i elected a 1, 2, 1 and 2. Thee recontructed PSD are compared to the original untruncated. Reult indicate that an accurate repreentation of the local element force, epecially at higher frequency range, uing the tochatic decompoition technique require large number of loading mode, while only a few loading mode can accurately repreent the global wind load and their effect. The PSD in the high frequency range i particularly important for the etimate of reonant repone of more flexible tructure uch a highrie building. PSD S F1 (f) PSD S P (5,5,f) = Frequency (Hz) Target =1 =2 (a) Generalized modal force (mode 1) =1 = frequency (Hz) (b) Local force on 5-th tory Target =2 =1 Fig. 7 Effect of higher mode truncation to PSD of rorce From the coherence function of the wind field that decreae with the increae in frequency and patial eparation, it can be eaily undertood that at higher frequency range or with large patial eparation the wind fluctuation will become tatitically non-coherent. Weakly coherent random field require relatively large number of mode to be conidered in the recontruction. In the extreme cae when the random procee are tatitically non-coherent, i.e. the XPSD matrix i diagonal, all the mode will be needed for an accurate repreentation of the original random procee. Therefore, the efficiency and accuracy of thee modeling and imulation technique depend on the propertie of the XPSD matrix and the frequency range of interet. The tochatic decompoition technique i epecially very effective and efficient in imulating and modeling well correlated random field uch a the preure field around low-rie building, roof and ide face of tall building. CONCLUSIONS Examining the wind load effect in light of the loading mode enhance our undertanding of the dynamic of wind effect on tructure. It alo provide efficient way to etimate the background wind load effect, and erve a a ueful tool for imulating and modeling multi-correlated wind field by truncating higher mode. Reult indicate that accurate repreentation of PSD at higher frequency range require incluion of larger number of mode, epecially for local wind load effect. ACKNOWLEDGMENTS The upport for thi work wa provided in part by NSF Grant CMS and CMS Thi

8 upport i gratefully acknowledged. REFERENCES Amitt, J. (1968). Eigenvector analyi of preure fluctuation on the wet burton intrumented cooling tower. Central Electricity Reearch Laboratorie, UK, Internal report RD/L/N 114/68. Benfratello, S. and Mucolino, G Filter approach to the tochatic analyi of MDOF wind-excited tructure. Prob. Engrg. Mech., 14, Bienkiewicz, B., Tamura, Y., Hann, H. J., Ueda, H. and Hibi, K Proper orthogonal decompoition and recontruction of multi-channel roof preure. J. of Wind Engrg. and Ind. Aerodyn., 54-55, Caraale, L., Piccardo, G. and Solari, G Wind repone of tructure by double modal tranformation. Proc. of the 2nd Eat European Conference on Wind Engineering, Prague, Chen, X., and Kareem, A. 2. Equivalent tatic wind load ditribution for coupled buffeting repone of bridge, Proc. of 8th ASCE Specialty on Prob. Mech. and Struct. Reliab., Notre Dame. Davenport, A. G How can we implify and generalize wind load? J. of Wind Engrg. and Ind. Aerodyn., 54-55, Di Paolo, M Digital imulation of wind field velocity. J. of Wind Engrg. and Ind. Aerodyn., 74-76, Holme, J. D Optimized peak load ditribution. J. of Wind Engrg. and Ind. Aerodyn., 41-44, Kareem, A Wind excited motion of building. Ph.D diertation, Department of Civil Engineering, Colorado State Univerity. Kareem, A Wind effect on tructure: a probabilitic viewpoint. Prob. Engrg. Mech., Vol.2, No.4, Kareem, A Analyi and modelling of wind effect: numerical technique. Proc. of 1th Int. Conf. on Wind Engineering, Copenhagen, Kareem, A. and Cermak, J. E Preure fluctuation on a quare building model in boundary-layer flow. J. of Wind Engrg. and Ind. Aerodyn., 16, Kareem, A., and Mei, G. 2. Stochatic decompoition for imulation and modal pace reduction in wind induced dynamic of tructure. Application of Statitic and Probability, Melcher î Stewart (ed), Balkema, Rotterdam, Kaperki, M Extreme wind load ditribution for linear and nonlinear deign. Engrg. Strut., 14, Lee, B. E The effect of turbulence on the urface preure field of a quare prim. J. of Fluid Mech., Vol.69, Li, Y. and Kareem, A Simulation of multivariate random procee: a hybrid DFT and digital filtering approach. J. of Engrg. Mech., ASCE, 119(5), Li, Y. and Kareem, A Stochatic decompoition and application to probabilitic dynamic. J. of Engrg. Mech., ASCE, 121(1), Lin, J. H A fat CQC algorithm of PSD matrice for random eimic repone. Computer and Structure, Vol.44, No.3, Tamura, Y., Suganuma, S., Kikuchi, H. and Hibi, K Proper orthogonal decompoition of random wind preure field. J. of Fluid and Struct., 13,