Analysis of earthquake wave propagation in buildings

Transcription

1 Earthquake Resstant Engneerng Structures V 33 Analyss of earthquake wave propagaton n buldngs H. Kawaka, M. Oyuncheg & E. A. J. Tngatnga 3 Geosphere Research Insttute, Sataa Unversty, Japan Graduate School of Scence and Engneerng, Sataa Unversty, Japan 3 Departent of Engneerng Scences, Unversty of the Phlppnes, Phlppnes Abstract Ths paper presents a wave propagaton odellng analyss of strong oton records for actual buldngs n coparson wth an analyss of sple buldng odels. In prevous studes on earthquake response of buldngs, vbraton approaches were used ore often than wave propagaton approaches; thus, wave propagaton through a buldng has not been nvestgated enough. We use the noralzed nput output nzaton (NIOM) ethod, whch can odel wave propagaton n ultple lnear systes by consderng the statstcal correlaton aong the strong otons at dfferent observaton locatons. Output wave odels splfed by ths ethod show two clear peaks that correspond to ncdent and reflected waves propagatng through the buldng n the vertcal drecton. The obtaned wave propagaton propertes, such as wave travel te and wave apltude rato, at each story of the buldng are found to reflect the structural propertes lke rgdty and dapng rato at that story. Further, a new ethod for wave-propagaton analyss called evolutonary noralzed nput-output nzaton s developed. It odels te-varant wave propagaton by consderng the te-varant statstcal correlaton between strong otons recorded at dfferent levels n the buldng. In the case of daaged buldngs, the travel te ncreased durng the earthquake; however, n the cases of undaaged buldngs and the elastc odel, t reaned alost constant durng the earthquake. Keywords: strong oton records, buldngs, wave propagaton, wave travel te, wave apltude, fundaental perod, dapng, daage. WIT Transactons on The Bult Envronent, Vol 8, ISSN (on-lne) 5 WIT Press

2 34 Earthquake Resstant Engneerng Structures V Introducton Analyzng the behavour of a structure durng an earthquake nvolves two probles, vbraton and wave propagaton, because the vbraton of a structure results fro sesc-wave propagaton n t. Vbraton ethods are well known and have been developed anly for structural engneerng. In contrast, wavepropagaton approaches have been anly used to nvestgate ground otons due to earthquakes, and researchers use several ethods (such as pulse response and correlaton functons) to splfy and clarfy the wave propagaton n sol layers and to deterne sol propertes. One of the wave-propagaton odellng ethods-the noralzed nput-output nzaton (NIOM) ethod [, ] s used n the present work. The NIOM ethod odels wave propagaton by consderng the statstcal correlaton of the earthquake otons at dfferent observaton locatons. Analyss results of sple buldng odel and actual buldng records show that wave propagaton propertes at each story of the buldng reflect the structural propertes at that story. We developed a so-called evolutonary NIOM ethod for exanng the te-varant propertes of buldngs. Ths ethod can splfy the relatonshps between waves observed at several levels n the buldng at dfferent tes fro the begnnng to the end of the earthquake, and t can deterne the change n travel te of the wave propagatng through the buldng [3]. The travel te was found to be a very good ndcator of daage, and ts change was related to the change n the degree of daage,.e., the change n the structural propertes of the buldng. Methodology. NIOM ethod The noralzed nput-output nzaton (NIOM) ethod can odel wave propagaton by consderng the statstcal correlaton between earthquake otons observed at dfferent ponts. Ths ethod can splfy the relatonshp between the observed waves and gve arrval tes, as well as ther relatve apltudes, of ncdent and reflected waves. Snce the wave-propagaton velocty depends uch on the characterstcs of the aterals through whch the wave propagates, such a ethod s applcable not only to ground-oton analyss but also to buldng-record (.e., of earthquakes) analyss. A bref revew of the NIOM ethod s gven below [, ]... NIOM ethod for a lnear syste When a te-nvarant lnear syste s subjected to an earthquake oton, the nput and output of the syste n the frequency doan can be related by eans of the transfer functon H ( ω). In the case of dgtzed earthquake otons, the output at each frequency s gven by WIT Transactons on The Bult Envronent, Vol 8, ISSN (on-lne) 5 WIT Press

3 G( ω ) = H ( ω ) F( ω ) () π =,..., N and ω = ), N t ( where t s the saplng rate n the te doan; N s the nuber of saples; and F ω ) and G ω ) are the Fourer transfors of the dgtzed earthquake ( ( nput and output otons, respectvely. Transfer functons depend only on the physcal propertes of the syste. Therefore, the sae transfer functon as the F ω and output one that defnes the relatonshp between the observed nput ( ) G( ω ), eqn (), should satsfy the relatonshp between the splfed nput odel X ( ω ) and the splfed output odel Y( ω ). ( ) H X Y ω = () Mnzng the suaton of the squared values of Fourer apltude spectra of the nput and output when a constrant s n exstence [, ] would result n a splfed nput odel of X ( ω ) and a splfed output odel of Y ( ω ). The procedure that gves the splfed nput and output odels s shown scheatcally n fg.. In the analyss of a feedback syste, t should be noted that nput cannot be separated fro output and that the nput oton n ths paper s not the ncdent-wave oton. The oton observed at one arbtrary locaton (always the buldng s roof n ths paper) s called the nput oton n the followng nuercal analyses... NIOM ethod for ultple lnear syste Controllng the contrbuton of hgh- or low-frequency coponents n the procedure and generalzng to ultple lnear systes gves the followng equatons: X Y l = N t N M k + ω c + c H c = n= = N n= Earthquake Resstant Engneerng Structures V 35 M k + ωn c + c c H ω = ( ) M k + ω c + c H c = N t k + ωn c + c l M c = H H n n (3) (4) WIT Transactons on The Bult Envronent, Vol 8, ISSN (on-lne) 5 WIT Press

4 36 Earthquake Resstant Engneerng Structures V where l=,,m; M s the nuber of outputs; c to c M are weghtng constants of the squared Fourer apltude spectra of the nput and outputs, and k s weghtng constant of te dervatve of the squared Fourer apltude spectru of the nput. Effects of these weghtng constants on the results are shown n the referred papers [, ]. The nverse Fourer transfors of eqns (3) and (4) gve splfed nput and output odels n the te doan. These splfed nput and output odels llustrate the statstcal correlaton between the observed otons, and the procedure for obtanng these odels s called the NIOM ethod. It should be noted that the Fourer transfor defned n eqn (3) s real and syetrc wth respect to the foldng frequency. Ths eans that the nput odel s real and syetrc wth respect to te zero as n the case of auto-correlaton functons. f ( t ) F( ω ) X ( ω ) x( t ) Observaton Gl ( ω ) = H l ( ω ) NIOM Splfed odel F( ω ) g t ) G Y ω ) y ( t ) l ( l l ( l Fgure : Scheatc procedure of the NIOM ethod. 3 NIOM analyss of sple buldng odel The earthquake responses of a sple -story buldng odel were coputed. Floor weght and story stffness dstrbutons for ths odel are shown n fg. (a), and t s the sae odel as that n the book edted by M. Paz [4]. The acceleraton te hstory recorded n the baseent of Transaerca buldng durng the Loa Preta earthquake of October 8, 989 [5] (NS coponent) was used as a oton at the ground surface (fg. (b)). The responses of the assued buldng were coputed by odal analyss, n whch a Raylegh dapng rato was assued n the frst two odes. The ground oton and the coputed response acceleraton te hstores at the ffth floor and roof (dapng rato s 5%) are shown n fg. (b). We appled the NIOM ethod to the coputed response acceleratons of the assued buldng, such as shown n fg. (b), and obtaned the splfed nput and outputs as shown n fg. (c). Here, the response acceleraton at the roof was consdered as the nput, and the acceleraton responses at the other nne floors were consdered as outputs. Fgure (c) shows analyss results when the saplng rate of the te seres was.5 s and weghtng constants were k =., c = and c = = c 9 =. As shown n fg. (c), the nput was odelled such that ts apltude at te t = was unty and the apltudes at the other tes and output approached zero unless correlaton exsted between the. One ay understand sply that the apltude at the top story s assued to be., and relatve apltude copared to the top story s obtaned for the other stores. One can see two clear peaks n the splfed output odels correspondng to the WIT Transactons on The Bult Envronent, Vol 8, ISSN (on-lne) 5 WIT Press

5 Earthquake Resstant Engneerng Structures V 37 ncdent and reflected waves, whch are ndcated by arrows () and () n the fgure, respectvely. The ncdent wave propagates fro the baseent to the roof, whereas the reflected wave propagates fro the roof to the baseent. Arrval tes of the ncdent and reflected waves were the sae:.4 s and.9 s at the ffth floor and baseent, respectvely. The wavefors of the splfed odels changed as we changed the value of k, whch deternes the contrbuton of hgh or low frequences [, ], but the obtaned wave arrval tes were slar. However, due to the lted space, only fgures showng NIOM analyss results for k =. are gven n ths paper. (a) Weght (t) Stffness 9 (tf/c) 8 (b) (c) A c c e l e r a t o n (c/s/s) 5th floor Baseent (Ground oton).5.5 () () () 5th floor () Baseent Fgure : Te (s) -4-4 Te (s) Assued -story buldng odel: (a) confguraton and propertes, (b) coputed acceleraton response, (c) NIOM ethod analyss result. Splfed odels obtaned fro ths analyss gve the arrval te and relatve apltude of ncdent and reflected waves at each level. Correspondng to the non-unfor buldng odel shown n fg. (a) (case n fg. 3), we assued a -story unfor buldng wth the sae stffness of 3 tf/c n all stores (case n fg. 3). Ths value of K = 3 tf/c was chosen by assung a seres sprng syste: = = K K, where K = stffness of each story n case (see fg. (a)) and K = story stffness (sae n all stores) n case. Fgure 3 (a) shows the wave arrval tes at the floors for three cases of story stffness dstrbuton for the assued buldng. In Case 3 the order of story stffness s reversed fro that n Case. Fro ths fgure one can see that how the wave travel te between floors changes when story stffness dstrbuton s changed. It s nterestng that the wave travel tes at baseent are equal to.9 s n all three cases. Ths eans that the arrval te obtaned fro the NIOM ethod s precsely the wave travel te fro the baseent to the roof. Fgure (c) shows that the reflected wave apltude was saller than the ncdent wave apltude n the output odels. The NIOM results for the WIT Transactons on The Bult Envronent, Vol 8, ISSN (on-lne) 5 WIT Press

6 38 Earthquake Resstant Engneerng Structures V assued buldng (Case ) were obtaned by changng the dapng rato fro % to 3%: cases of %, 5%, % and % dapng are shown n fg. 3 (b). One can see that the dfference between the ncdent and reflected wave apltudes ncreased wth ncreasng dapng rato, and that the arrval tes were slar for dfferent dapng ratos. 8 (a) Case Case Case 3 (b) Dapng rato: % 5% % % Floors th floor Wave arrval te (s).5 Baseent Te (sec) Fgure 3: (a) Wave arrval tes at the floors of the assued odel for three cases of story stffness dstrbuton, (b) NIOM results of assued -story buldng wth dfferent dapng ratos. 4 NIOM analyss of actual buldng records We have appled the NIOM ethod to a nuber of earthquake strong oton records observed n actual buldngs and fg. 4 shows soe results. Here, the record at the roof was consdered as nput and records at all other floors as outputs of the syste. 4.. Los Angeles, 5-story offce buldng Ths offce buldng located 3 k fro the epcentre of the January 7, 994 Northrdge earthquake and has 5 stores above and 5 levels below the ground. Its lateral force resstng syste conssts of concentrcally braced steel fraes at the core wth oent resstng connectons and outrgger oent fraes n both drectons [6]. Fgure 4 (a) shows NIOM analyss results of the buldng record n the NS drecton. Splfed wave odels obtaned at each story n the buldng show two clear peaks that correspond to ncdent and reflected waves propagatng through the buldng n the vertcal drecton. Arrval tes of the ncdent and reflected waves were ostly the sae, and reflected wave apltude was always saller than ncdent wave apltude. Lnes wth crcle n fg. 4(a) show wave arrval tes obtaned at the nstruented floors, and one can see a clear change n wave velocty between floors above and below ground. Quanttatvely, obtaned wave veloctes were about 8 /s and 43 /s above WIT Transactons on The Bult Envronent, Vol 8, ISSN (on-lne) 5 WIT Press

7 Earthquake Resstant Engneerng Structures V 39 and below ground floors, respectvely. Wave velocty dfference shows the dfference n story property. 6 (a) NS (center) (b) NS (center) 4 (c) NS (south wall) 5 49th floor 5 3 Floors th floor th floor 5 4th floor 8th floor nd floor Mall level 4th floor Level A Baseent - - Garage level B -5 Level E - - Te (s) - - Te (s) - - Te (s) Fgure 4: NIOM analyss results of actual buldng records: (a) 5-story offce buldng n Los Angeles, (b) Hollywood storage buldng n Los Angeles, (c) 3-story coercal buldng n Los Angeles. 4.. Los Angeles, Hollywood storage buldng Ths buldng stood n 3 k fro the epcentre of the January 7, 994 Northrdge earthquake and has 4 stores above and one level below ground. Its lateral load resstng syste conssts of renforced concrete fraes n both drectons [6]. Fgure 4 (b) shows NIOM analyss results n the NS drecton of the buldng, whch shows a slar trend wth the prevous buldng. There s also a dfference n wave velocty: about 9 /s between roof and 8th floor and about 5 /s between 8th floor and baseent Los Angeles, 3-story coercal buldng Ths buldng located n k fro the epcentre of the January 7, 994 Northrdge earthquake and t has three stores above and two parkng level below the ground. Its lateral load resstng syste conssts of steel braced fraes n the upper three stores and concrete shear walls n the parkng floors [6]. Fgure 4 (c) shows NIOM analyss results n the NS drecton of the buldng. There was also wave velocty dfference: about 95 /s above and 7 /s below ground floors. Here, one should notce that shear wave velocty for steel s about 3 /s, and even for rubber the wave speed s 98 /s, way above the values found. Such dfferences are because the obtaned values are not the shear wave veloctes n the dense aterals but those n the buldng frae. Clough and Penzen [7] have approxated the as v = EI /( h), where s the ass of each story, EI s the total rgdty of all coluns, and h s the heght of each story. WIT Transactons on The Bult Envronent, Vol 8, ISSN (on-lne) 5 WIT Press

8 3 Earthquake Resstant Engneerng Structures V 5 Daaged buldngs Strong otons recorded n a daaged buldng, whch has defored nto the plastc range, gve dfferent nforaton fro that expressed by the elastc characterstcs of the buldng. To reveal the daage process of such a buldng durng an earthquake, the change n buldng characterstcs wth te should be exaned fro the begnnng to the end of the earthquake. So, we tred to analyze the records segentally n te and copared the results. The observed strong otons were dvded nto short (but long enough to obtan relable results) portons and NIOM was appled to each porton. We naed ths ethod as the "evolutonary NIOM ethod". The shortest length of each porton was chosen to be fro τ to 4τ (τ s the wave travel te fro the baseent to the roof) so that a relable correlaton between otons at dfferent levels n the buldng could be obtaned. Wndows wth length of 5. to.4 s, shfted by or s fro the begnnng to the end of the record were used. Narrow wndows, strong oton records n the EW drecton for Van Nyus-7-story hotel buldng, and analyss results are shown n fg. 5. Ths 7-story hotel located 7 k fro the epcentre of the 994 Northrdge earthquake experenced serous structural and non-structural daage. It was found that ncdent-wave arrval te changed fro.4 s to.4 s (see fg. 5) at the ground floor n the EW drecton. As shown n fg. 5 the frst change n the ncdent-wave arrval te occurred at around 3-4 s, and an addtonal change occurred at around 7-8 s. These results ndcate how the daage developed n the buldng durng the earthquake. The possblty to locate the daage and dentfy the extent of daage depends on the avalablty of the strong oton records n the buldng. Arrval te (s).5 5. Acceleraton (c/s/s) th floor 3rd floor nd floor. Ground floor W- W-3W-5 W-7 W-9W- Wndow wdth and central te Wave arrval te W-3 W-5 W-5 W-6 W-7 W-7 W-8 W-9 W-9 W- W Te (s) Fgure 5: Narrow wndows used n evolutonary NIOM ethod and analyss results for Van Nyus-7-story hotel buldng. WIT Transactons on The Bult Envronent, Vol 8, ISSN (on-lne) 5 WIT Press

9 Earthquake Resstant Engneerng Structures V 3 Arrval te (s).5 5. Acceleraton (c/s/s) th floor 3rd floor nd floor. Ground floor Te (s) Fgure 6: buldng. Evolutonary NIOM ethod analyss results for 7-story elastc To test the ethod thoroughly, t was appled to an assued elastc sevendegrees-of-freedo shear-sprng odel, whch represents a seven-story buldng. In the odel, all floors have the sae ass of 45.3 t and all the stores have equal stffness of kn/c, and the fundaental perod by odal analyss s.43 s. Elastc response of the odel was coputed by the conventonal vbraton ethod. In ths calculaton, the ground oton was consdered as the strong oton (EW coponent) recorded at the ground floor of the seven-story hotel n Van Nuys durng the 994 Northrdge earthquake. Fgure 6 shows the coputed acceleraton responses when the dapng rato n the frst ode s assued to be % and wave arrval tes obtaned by the evolutonary NIOM ethod. Accordng to the elastc odel, there should be no change, theoretcally, n wave travel te durng the earthquake. Fgure 6 clearly proves the aboveentoned theory; naely, dfferences n the arrval tes obtaned between wndows were only ±. s, whch s the sae as the saplng rate. 6 Conclusons Splfed wave odels obtaned by the NIOM ethod at each story n the buldng show two clear peaks that correspond to ncdent and reflected waves propagatng through the buldng n the vertcal drecton. Arrval tes of the ncdent and reflected waves were ostly the sae, and reflected wave apltude was always saller than ncdent wave apltude. Wave travel tes obtaned n each story n actual buldngs were found to reflect the structural propertes n that story, such as rregularty n heght and dfference n structural types. Evolutonary ethod for noralsed nput-output nzaton (NIOM) enables te-varant wave propagaton odellng by takng nto account the te-varant statstcal correlaton aong the strong otons recorded at dfferent levels n the buldng. The NIOM results for actual daaged and undaaged buldngs, as well as those fro an assued elastc buldng odel, were WIT Transactons on The Bult Envronent, Vol 8, ISSN (on-lne) 5 WIT Press

10 3 Earthquake Resstant Engneerng Structures V copared. In the case of the daaged buldngs, the travel te ncreased durng the earthquake; however, n the case of the elastc odel, t reaned alost constant durng the earthquake. It was found that the change n the travel te s related to the change n the structural propertes and to the degree of daage to a buldng. These results show that evolutonary NIOM s an effectve new ethod for nvestgatng the change n structural propertes and the daage to buldngs. Ths ethod has an advantage of nvestgatng buldng behavour durng earthquakes wthout assung any structural propertes, because only strong oton te hstores recorded n the buldng are needed. References [] Kawaka, H. & Haddad, H.R., Modellng wave propagaton by usng noralzed nput output nzaton (NIOM). Sol Dynacs and Earthquake Engneerng, 7, pp. 7-6, 998. [] Haddad, H. R. & Kawaka, H., Modellng wave propagaton by usng noralzed nput-output nzaton (NIOM) ethod for ultple lnear systes. Journal of Structural Mechancs and Earthquake Engneerng, JSCE, 584/I-4, pp. 9-39, 998. [3] Oyuncheg, M., Sesc wave propagaton n actual buldngs. Doctoral thess, Sataa Unversty, Japan, 4. [4] Paz, M., (ed). Internatonal handbook of earthquake engneerng: codes, progras and exaples. Chapan & Hall: New York, 994. [5] Natonal Geophyscal Data Centre. Earthquake strong oton 3-volue CD-ROM collecton. Boulder, Colorado, 996. [6] Nae, F., Perforance of extensvely nstruented buldngs durng the January 7, 994 Northrdge earthquake. Los Angeles: John A. Martn & Assocates, JAMA report nuber , 997. [7] Clough R.W. & Penzen. J., Dynacs of structures. McGraw-Hll: New York, 975. WIT Transactons on The Bult Envronent, Vol 8, ISSN (on-lne) 5 WIT Press