igid nd non-leky two-comptment building. Yu et l [8] developed non-line govening equtions by consideing the effect of bckgound lekge. Howeve, thee e n

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1 The Seventh Intentionl Colloquium on Bluff Body Aeodynmics nd Applictions (BBAA7) Shnghi, Chin; Septembe -, Coupled vibtion between wind-induced intenl pessues nd lge spn oof fo two-comptment building with openings Yu Xinfeng, Qun Yong, Gu Ming Stte Key Lbotoy of Disste Reduction in Civil Engineeing, Tongji Univesity, Shnghi, Chin ABSTRACT: The equtions govening wind-induced intenl pessue esponses fo twocomptment building e deived bsed on some esonble ssumptions when the oof esponds in dynmic nd qusi-sttic mnnes, espectively. The gin functions, powe spect, nd RMS vlues of intenl pessue coefficients in both comptments e numeiclly nlyzed. The esults show tht the dynmic model must be used to clculte the esponses of intenl pessue when the oof is flexible, while the simplified qusi-sttic model cn be used fo igid oof. With incesed oof ntul fequency, ll the thee esonnce fequencies of the intenl pessues - oof coupled system incese; the pek intenl pessue esponse lso inceses t the fist two fequencies, but deceses t the thid esonnce fequency; the RMS intenl pessue coefficients lso incese, but thei diffeence deceses. KEYWORDS: building with intenl ptitioning nd openings, wind-induced intenl pessue, oof flexibility, govening equtions, gin functions, coupled system. INTRODUCTION The dominnt openings might be ceted due to eithe impct of flying debis o filue of windows nd doos in sevee wind stoms. In the lmin unifom wind, n initil intenl pessue oveshoot which is highe thn the men extenl pessue t the opening will be poduced when sudden opening is ceted, then the esponse of intenl pessue will exhibit dmped oscilltion, nd it finlly will keep constnt with the extenl pessue. Howeve, the oveshoot phenomenon existed in smooth flow is lost midst fluctuting tubulent wind in el life []. The stedy-stte intenl pessue esponse fo building with dominnt opening cn lso be descibed with the Helmholtz coustic esonto model. The esonnce phenomenon will be occued if thee is sufficient enegy t the Helmholtz fequency fo the extenl pessue t the opening. So it will hve pofound effect on building sfety. The effect of stuctue flexibility on intenl pessue hs been studied lot fo the single-cell building with dominnt opening. Vickey [] teted the building deflections s qusi-sttic nd suggested n effective volume method to conside the effect of building flexibility. Shm nd Richds [] deived n eqution govening the intenl pessue fo building with dominnt opening nd flexible oof, but the effect of extenl pessue on the oof movement filed to be consideed. Yu et l [] futhe consideed the joint ction of extenl nd intenl pessues on oof, nd pesented line intenl pessue model. Shm [] deived n eqution govening the esponse of intenl pessue in ny flexible building. Futhemoe, non-line intenl pessue govening eqution ws pesented when the oof stuctue esponds in qusi-sttic mnne, nd the oof is unde the joint ction of extenl nd intenl pessues. Guh [] deived n eqution govening the intenl pessue fo building with dominnt opening nd flexible oof, howeve only consideed the effect of intenl pessue on oof. The studies of windinduced intenl pessues fo two-comptment building e much less comped with the single-cell building. Shm [7] pesented lineized equtions govening the intenl pessues fo 8

2 igid nd non-leky two-comptment building. Yu et l [8] developed non-line govening equtions by consideing the effect of bckgound lekge. Howeve, thee e no theoeticl nd expeimentl studies on the effect of oof flexibility. In this ppe, the equtions govening wind-induced intenl pessue esponses fo twocomptment building when the oof esponds in dynmic nd qusi-sttic mnnes is deived. Then, the gin functions, oof displcement, powe spect nd RMS vlues of intenl pessues e numeiclly nlyzed. GOVERNING EQUATIONS FOR A FLEXIBLE BUILDING. Bsic ssumptions The object is two-comptment building with flexible oof. Thee is n opening in the windwd wll nd n opening in the ptition wll, nd thee e no bckgound lekges. Besides, some esonble ssumptions e mde s follows: (i) The wll is igid, while the oof is flexible. (ii) The ptition wll only plys ole of wind shield, but it doesn t povide suppot to the oof. (iii) The oof behves linely. (iv) The oof will poduce tiny ottion ngle in theoy fo the fluctuting intenl pessue in the comptment without n extenl opening is little highe thn tht in the othe one. Howeve, it cn be ignoed fo the oof is lge nd the wll is igid. (v) Intenl pessues e completely elevnt in thei own comptments. This is the pecondition fo the theoy of the intenl pessue esponse nd hs ledy been vlidted by expeiments [9]. (vi) The dibtic lw is pplied fo i in both comptments. It mens the stte eqution gov ening the vition of the intenl i pessue nd density becomes: P i / i C () whee is the specific het tio fo dibtic i nd is genelly equl to., C is constnt, nd P i, i e the intenl i pessue nd density, espectively. Afte diffeentiting both sides of eqution () nd simplifying it, then: P dpi di () i whee P is the tmospheic pessue, which is the stedy-stte intenl pessue.. Govening equtions The simplified model of intenl pessues fo two-comptment building with flexible oof nd openings is shown in Fig.. P, W P i nd P i e the tnsient wind pessues of windwd wll, comptment nd comptment, espectively. L e nd L e e the effective lengths of i slugs in the opening nd. A o nd A o e the es of the opening nd. nd e the intenl volumes of comptment nd,. c nd c e the flow coefficients t the opening nd. C L nd C L e the loss coefficients t the opening nd. m, k nd e the mss, stiffness nd dmping tio of the oof, espectively. A nd A 9

3 The Seventh Intentionl Colloquium on Bluff Body Aeodynmics nd Applictions (BBAA7) Shnghi, Chin; Septembe -, e the oof es of comptment nd, A A A. x nd x e the displcements of i slugs in the opening nd, x is the displcement of flexible oof. Figue. Model of two-comptment building with linely flexible oof stuctue Accoding to the unstedy fom of the Benoulli eqution [], the eltionship between the iflow te nd the tnsient pessue dop fo opening is given by: LcAx e o cac o Lx x caq o CPW CPi () Similly, the eltionship between the iflow te nd the tnsient pessue dop fo opening is given by: LecA ox caocl x x caoqcpi CPi () whee CPi Pi / q nd CPi Pi / q e the tnsient intenl pessue coefficients of comptment nd comptment, CPW PW / q is the tnsient extenl pessue coefficient on the windwd wll. is the i density, q U is the efeence dynmic pessue of the ppoching flow. The chnges in volumes of both comptments e: A x c A x c A x () o o A x c A x () o Then, ccoding to the ssumption (vi): C dc P q q d pi pi Substtion of equtions () nd () into eqution (7), nd let x, thee e: H x H, (7) x, H

4 P C c A x c A x A H( ) Pi o o q (8) P C c A x A H( ) Pi o q whee H is the idge height of the building. Simultneously solving equtions (8) nd (9) esults in: x qc qc AH( ) Pi Pi () PcA o ca o x qc A H( ) Pi () Pc Ao cao Deivtion of both sides of equtions () nd () leds to: qc qc A H Pi Pi x () PcA o ca o qc A H Pi x () Pc Ao cao qc qc A H Pi Pi x () PcA o ca o qc A H Pi x () Pc Ao cao Substtion of equtions (~) into equtions (~) yields: L L L C C e e e Pi Pi PcA o PcA o ca oq CL qc PiqC Pi qc PiqC Pi C Pi C ( ca o) q P P L C L e e Pi Pc Ao caoq CL qc Pi qc Pi CPi CPi ( ca o) q P P Equtions (~7) e the genel equtions govening the intenl pessue esponses fo two-comptment building with flexible oof. PW (9) () (7)

5 The Seventh Intentionl Colloquium on Bluff Body Aeodynmics nd Applictions (BBAA7) Shnghi, Chin; Septembe -, When the oof of the building is igid, i.e., nd, the equtions (~7) cn be simplified s follows: Le Le CLq C Pi C Pi C Pi C Pi C Pi C PiCPi CPW (8) PcA o PcA o ( PcA o) L C q ( ) (9) e L CPi C Pi CPi CPi CPi Pc Ao ( Pc Ao) If the dmping coefficients in equtions (8) nd (9) e lineized, they will become the intenl pessue govening equtions pesented by Shm [7]. Thus, the Shm s govening equtions [7] e specil cse of the equtions () nd (7) deived in this ppe... Responses of intenl pessue when the oof esponds in dynmic mnne In elity, mny lge spn industil building, thei component stuctues, in pticul the oof, exhibit gete flexibility thn the wll. So they will behve in dynmics mnne in stong wind. As fo two-comptment building shown in Fig., the esponse of the oof to the intenl pessues will be govened by: mx mx mx AqC A qc () Pi Pi The non-dimensionl fom of the eqution () is: Pi Pi mh q A C A C () Equtions () nd (~7) e the dynmic model of intenl pessues fo twocomptment building with flexible oof. The eqution () cn be solved simultneously with equtions (~7) to obtin the coupling esponses of intenl pessues nd oof fo specified pmete conditions... Responses of intenl pessue when the oof esponds in qusi-sttic mnne The dynmics model (Equtions () nd (~7)) is second-ode diffeentil equtions, nd it is hd to be solved fo its complexity. When the ntul fequency of the oof is lge thn the fist expected Helmholtz fequency, the intenl pessue esponses e dominted by vege nd bckgound components, while the esonnce component cn be ignoed fo its little contibution. Thus, the oof deflection is popotionl to the pplied lod t ll times [], i.e. qusi-sttic mnne []. q ( C C ) pi pi () kb whee kb ( f) m / A is the building bulk modulus defined s the tio of incese in pessue to volumetic stin. Substitution of the time deivtion of into equtions (~7), nd let b P / kb, yields:

6 L b L b C q C C e e L Pi Pi PcA o PcA o ( PcA o) b C b C b C b C C C Pi Pi Pi Pi Pi PW () Le( b) b C Pi C pi Pc A b o L C pi C pi C pi C pi CPi CPi ( Pc Ao) b b C q b b b Equtions (~) e the qusi-sttic model of intenl pessues fo two-comptment building with flexible oof. It is ctully n ppoximte model of the dynmic model when the ntul fequency of oof is considebly lge thn the lowe Helmholtz fequency. NUMERICAL EXAMPLE Tke two-comptment building s n exmple to nlyze the dynmic nd qusi-sttic esponses of intenl pessue. The two-comptment building is simil s the TTU test building, whose bsic pmete vlues e s follows: m, m, c., c., L.. kg / m, A m, f =Hz, Hz, 7Hz, Hz. A o m, A o m, Le Ao / (), L A /, C, CL., U m/ s,., P P, A m, H m, m kg,., f, 7 The time histoy of wind speed cn be simulted by the supeposition of the hmonic [] bsed on Dvenpot spectum, nd then the time histoy of extenl pessue coefficient t the windwd opening cn be futhe obtined ccoding to the qusi-stedy theoy. The esponses of intenl pessue cn be clculted by inputting the extenl pessue coefficient obtined in lst step into the govening equtions deived in the pesent ppe. The simulted wind speed nd its veifiction e shown in Fig.. e o b Dvenpot Spectum Wind Speed, m/s S(f) - 8 Time(t), s - - Simulted Spectum Fequency(f), Hz Figue. () Simulted time-histoy wind speed nd (b) its veifiction The gin functions of intenl pessues clculted fom the dynmic nd qusi-sttic models fo diffeent oof flexibilities e shown in Fig., in which Figs,b,c,d e the esults of comptment nd Figs e,f,g,h e the esults of comptment. The two-comptment building

7 The Seventh Intentionl Colloquium on Bluff Body Aeodynmics nd Applictions (BBAA7) Shnghi, Chin; Septembe -, with flexible oof nd openings is simil to coupled dynmic system which contins thee mss-sping-dmping subsystems. It cn be seen fom Fig. tht thee esonnce peks e evident fom dynmic nlysis while only two esonnce peks e obtined fom qusi-sttic nlysis. Futhemoe, the fist two esonnce fequencies obtined fom dynmic nlysis is the sme s tht obtined fom qusi-sttic nlysis. Besides, the intenl pessue esponse in comptment is highe thn tht in comptment clculted fom both models. Howeve, the thid esonnce pek gdully disppes when the oof ntul fequency gdully becomes lge, nd the gin function obtined fom the dynmic model is gdully consistent with tht fom qusi-sttic model... Comptment f=hz Qusi-sttic nlysis.. e Comptment f=hz Qusi-sttic nlysis b Comptment. f=hz Qusi-sttic nlysis f Comptment. f=hz Qusi-sttic nlysis c Comptment f=7hz Qusi-sttic nlysis g Comptment f=7hz Qusi-sttic nlysis

8 7 d Comptment f=hz Qusi-sttic nlysis 7 h Comptment f=hz Dynmics nlysis Qusi-sttic nlysis Fig. Gin function of intenl pessue fo dynmic nd qusi-sttic nlysis The gin functions of intenl pessues in both comptments, the oof displcement ove extenl pessue t the windwd opening clculted fom dynmic model fo diffeent oof ntul fequencies e shown in Fig.. It is shown fom Figs,b tht thee esonnce fequencies of the intenl pessue - oof coupled system incese with incesed oof ntul fequency. The fist two esonnce peks of intenl pessue e lso incese with incesed oof ntul fequency, while the thid esonnce pek deceses nd finlly disppes. It is shown fom Fig c tht the estoing foce of the oof inceses with incesed oof ntul fequency t the fist two esonnce fequencies, nd the estoing foces t the fist esonnce fequency e much lge thn these t the lst two esonnce fequencies. 7 Comptment 7 b Comptment c Roof Hz 7Hz Hz Hz Hz Hz Hz Hz K.X / C pw.q.a Hz 7Hz Hz Hz Figue. Gin function of () intenl pessue fo comptment (b) intenl pessue fo comptment nd (c) oof displcement ove extenl pessue fo diffeent oof flexibilities The powe spect of intenl pessue coefficients in both comptments obtined fom the dynmic model fo diffeent oof flexibilities e shown in Fig.. It is shown tht the esonnce enegy of intenl pessue in comptment t the fist esonnce fequency is highe thn tht in comptment fo ll kinds of oof flexibilities. The esonnce enegies of intenl pessue in both comptments t the thid esonnce fequency decese with incesed oof ntul fequency. This is in geement with the nlyticl esults of Fig.. The RMS intenl pessue coefficients in both comptments nd RMS oof displcement fo diffeent oof flexibilities e list in Tble. It is shown tht the RMS intenl pessue coefficient in comptment is lge thn tht in comptment. The RMS intenl pessue coefficients obtined fom both models incese with incesed oof ntul fequency, but thei diffeence deceses. It is becuse with incesed oof ntul fequency, i.e. the oof flexibility gdully deceses, the dditionl dmping lso deceses. Moeove, the RMS vlue of the oof displcement deceses with incesed oof ntul fequency.

9 The Seventh Intentionl Colloquium on Bluff Body Aeodynmics nd Applictions (BBAA7) Shnghi, Chin; Septembe -, - - b - - SCpi(f) , f=hz Comptment Comptment SCpi(f) , f=hz Comptment Comptment c - d - - SCpi(f) , f=7hz Comptment Comptment - - SCpi(f) , f=hz Comptment Comptment - - Figue. Powe spect of intenl pessue coefficients in both comptments Tble. RMS intenl pessue coefficients in bot comptments nd RMS oof displcement fo diffeent oof flexibilities Roof fequency C pi () pi C (Qusi-sttic nlysis) x () Comptment Comptment Comptment Comptment Hz Hz Hz Hz CONCLUSIONS As fo two-comptment building with flexible oof nd openings, the equtions govening wind-induced intenl pessue esponses e deived bsed on some esonble ssumptions when the oof esponds in dynmic nd qusi-sttic mnnes. The unstedy fom of the Benoulli eqution, the lw of mss consevtion, nd dibtic eqution e used fo the deivtion. A two-comptment building simil to the TTU test building is tken s n exmple to nlyze the chcteistics of gin functions, powe spect nd RMS vlues of the intenl pessue coefficients nd oof displcement. Some min conclusions e dwn: () The govening equtions deived in the pesent ppe cn be used to nlyze the intenl pessue esponses fo two-comptment building with flexible oof nd openings. The dynmic model must be used to clculte the esponses of intenl pessue when the oof is flexible, othewise, the simplified qusi-sttic model cn be insted.

10 () The fist two esonnce fequencies of the intenl pessue - oof coupled system obtined fom the dynmic model is consistent with tht obtined fom the qusi-sttic model. The intenl pessue esponse in comptment is lge thn tht in comptment. When the oof ntul fequency gdully becomes lge, the gin functions obtined fom the dynmic model e gdully in ccod with these fom the qusi-sttic model. () With incesed oof ntul fequency, the thee esonnce fequencies incese, nd the fist two esonnce peks of intenl pessue e lso incese, while the thid esonnce pek deceses nd finlly disppes. Besides, the estoing foces of the oof t the fist two esonnce fequencies lso incese with incesed oof ntul fequency. () The RMS intenl pessue coefficient in comptment is lge thn tht in comptment. The RMS intenl pessue coefficients obtined fom both the dynmic nd qusi-sttic models incese with incesed oof ntul fequency, but thei diffeence deceses. Moeove, the RMS vlue of the oof displcement lso deceses with incesed oof ntul fequency. ACKNOWLEDGEMENTS We gtefully cknowledge the joint suppot of the Key Poject of the Mjo Resech Pogm of Chinese Ntionl Ntul Science Foundtion (Gnt No. 97) nd the Genel Pogm of Chinese Ntionl Ntul Science Foundtion (Gnt No. 8789). REFERENCES Stthopoulos, T., Luchin, H. D., 989. Tnsient wind-induced intenl pessues. Jounl of Engineeing Mechnics, -. Vickey, B. J., 98. Gust-fctos fo intenl-pessues in low ise buildings. Jounl of Wind Engineeing nd Industil Aeodynmics, 9-7. Shm, R. N., Richds, P. J., 997. The effect of oof flexibility on intenl pessue fluctutions. Jounl of Wind Engineeing nd Industil Aeodynmics 7, 7-8. Yu Shice. Wind-induced intenl pessue of stuctue with openings nd its coupuling effect with flexible oof.[phd Thesis]. Zhejing: Zhejing Univesity,.~. (in Chinese) Shm, R. N., 8. Intenl nd net envelope pessues in building hving qusi-sttic flexibility nd dominnt opening. Jounl of Wind Engineeing nd Industil Aeodynmics 9,7-8. Guh, T. K., Shm, R. N., Richds, P. J.,. On the intenl pessue dynmics of leky nd flexible low ise building with dominnt opening. in: Poceedings of the th Int. Conf on wind Engineeing, Amstedm, Nethelnds. 7 Shm, R. N.,. Intenl Pessue Dynmics with Intenl Ptitioning. in: Poceedings of the th Intentionl Confeence on Wind Engineeing, Lubbock Texs. 8 Yu Xinfeng, Gu Ming, Qun Yong, et l.,. Study on esponse of wind-induced intenl pessue fo stuctue with intenl ptitioning nd n opening when consideing bckgound poosity, Act Aeodynmic Sinic, -. (in Chinese) 9 Holmes, J. D., 979. Men nd fluctuting intenl pessues induced by wind. in: Poceedings of the th Int Conf on Wind Engineeing, Fot Collins,USA. Wng Zhihong, 99. Simultion of wind loding. Jounl of building stuctues, -. (in Chinese) 7