Earthquake Distance Variable Effects on Response of Structures

Transcription

1 Australia Joural of Basic ad Applied Scieces, 5(8): , 211 ISSN Earthquake Distace Variable Effects o Respose of Structures Zaiar Tokmechi Departmet of Civil Egieerig, Sciece ad Research Brach, Islamic Azad Uiversity, Kordesta, Ira. Abstract: The formulatio of the respose i the frequecy domai is very efficiet i the computatio ad evaluatio of the stochastic respose of structures. The effect of the variability i earthquake variables o the respose is sufficietly importat to require careful cosideratio i desig. The respose reliability is depedet o the variability of the excitatio, groud motio. I this paper, the effects of earthquake magitude o the structure respose has bee studied. Aalytical results show that ucertaity i earthquake distace variable causes ucertaity i the respose of models ad this ucertaity i ear earthquakes is importat. It is also clear from the results that the icrease of the earthquake distace causes decrease i the coefficiet of variatio of respose. Key words: earthquake distace, modal aalysis, frequecy domai, seismological Fourier spectrum INTRODUCTION Amog all sources of ucertaity stemmig from the material properties, the desig assumptios, ad the earthquake-iduced groud motio, the later seems to be the most upredictable (Kappos, 22) ad it has a sigificat effect o the variability of structural respose (Padgett ad Desroches, 27). The problems of dyamic respose aalysis ad reliability assessmet of structures with ucertai system ad excitatio parameters have bee the subject of extesive research durig the last few years (Papadimitriou et al., 1995; Sakurai et al., 21; Chaudhuri ad Chakraborty, 26; Yazdai ad Komachi, 29; Yazdai ad Takada, 29). The eed for stochastic dyamic aalysis of egieerig systems stems from the fact that a importat class of structural loads, which evolves with time, exhibits strog variability i both amplitude ad frequecy cotet (Maolis ad Koliopoulos, 21). The simplified statistical structure model ca be efficietly used i aalytical radom vibratio studies ad for mathematically studyig the importace of the temporal o-statioary i both the amplitude ad frequecy cotet of groud motio o the respose of structures. For liear structures ad for Gaussia excitatio, the respose is also Gaussia (Soles, 1997). Therefore, oly the mea ad the covariace respose are eeded to completely determie the joit probability desity fuctio of the respose (Nigam, 1983). Excitatio processes display a wider power spectrum desity fuctio compared with the correspodig respose fuctio (Li ad Cai, 24). The sigificace of this result is that it verifies the assumptio of wide bad-limited oise (or eve white oise), which is ofte made for the iput of structural dyamic systems. As the iput process is filtered through the oscillatig system, oly a arrow bad of frequecies aroud the oscillator s atural frequecies are trasmitted, ad thus the output displays a arrow bad power spectrum. This is why the followig study is restricted to the peak of the arrow bad stochastic process. I radom vibratio theory, the probability desity for the respose of a system uder Gaussia white oise or badlimited excitatio ca be calculated based o the frequecy iformatio of excitatio (Su, 26). This iformatio oly requires the computatio of the Fourier amplitude spectrum of excitatio, which ca be calculated based o the seismological method. For regios where recorded groud motio data are scarce, it becomes imperative to use physical models to represet the groud motio geeratio ad propagatio. The formulatio of the respose i the frequecy domai is very efficiet i the computatio ad evaluatio of the stochastic respose spectrum, which the frequecy domai approach is appropriate for probabilistic aalyses. I the preset work a attempt has bee made to study the effect of the variability i earthquake distace variable o the stochastic respose of differet models. The preseted frequecy domai formulatio for the modal respose variability is the utilizatio of suitable explicit relatioships betwee the modal characteristics ad the ucertai earthquake groud motio variables. Correspodig Author: Zaiar Tokmechi, Departmet of Civil Egieerig, Sciece ad Research Brach, Islamic Azad Uiversity, Kordesta, Ira. TEL: , FAX: , Z.TOKMECHI@GMAIL.COM 574

2 MATERIALS AND METHOD Computatio of Respose: The mathematic descriptio of excitatio respose characteristics of structural systems subjected to stochastic iputs, follows closely the stadard theory of mechaical vibratios with some chage i emphasis (Thomso, 1993). Physical systems whose behavior, characterized by a arbitrary respose quatity ca be represeted by the equatio 1: LXt [ ( )] Yt ( ) (1) where the differetial operator L[*] describes the mai properties of the system, that is, acts as a mathematical model for the system. X(t) is the respose vector of the system or the system output whe it is subjected to a excitatio vector or iput vector Y(t), stochastic or determiistic. The operatio performed o the iput vector sigal Y(t) is the iverse operatio L -1 [*], that is, the output vector sigal X(t) is obtaied by it. This iverse operatio is referred to as filterig of the sigal iput Y(t). The liear operator L[*] will be time ivariat ad act as a liear filter. Based o this format, the equatio of motio of proportioally damped liear elastic multi-degree-of-freedom (MDOF) subjected to stochastic ui-directioal horizotal groud acceleratio is obtaied as [ m]{ X} [ c]{ X } [ k]{ X} Y( t) where {X} is the vector relative displacemet (output) uder a stochastic excitatio vector (iput) ad [m], [k] ad [c] are mass, stiffess ad dampig matrix, respectively. Sice the iput vector (groud motio) is a radom process, the output vector {X} will also be a radom process (Elishakoff, 1999). It is coveiet that the frequecy respose method is applied for represetig the relatioship betwee iput ad output, i which both iput ad output processes are represeted by harmoic fuctios (Ordaz, et al., 23; Takewaki, 21). By takig the Fourier trasform of both sides, i frequecy domai the differetial equatio 1 is (2) { X( )} [ H( )]{ Y( )} (3) where [H(w)], as iverse operatio L -1 [*], is the structural trasfer fuctio matrix betwee the displacemets ad applied loads. Where the vector {Y} is the exteral excitatios of loadig ad {X}is the system displacemet vector i frequecy domai. Direct calculatio of [H(w)] requires a matrix iversio for each frequecy variatio. This is computatioal cumbersome, therefore, approximate aalysis ca be applied i practice. The modal aalysis techique is used for the respose calculatio of dyamically sesitive structures. It requires atural frequecies ad modal shapes of structural system, which are calculated from the eigevalue solutio. The, the structural trasfer fuctio matrix is stated as a summatio of the cotributio of each vibratio mode as N [ H( )] H ( ){}{} 1 T (4) i which N is the total umber of atural modes to be icluded, {f} is the eigevector of the mode, H (w) is the frequecy depedet modal participatio factor defied as 1 H ( ) k ( ) 2i (5) I this equatio, k is the geeralized stiffess, w is the atural frequecy ad ζ is the percetagedampig ratio for the mode. Equatio 3 idicates that the respose of damped liear elastic MDOF systems depeds explicitly o the dyamic properties of the systems ad smoothed versio of the amplitude spectrum of groud motio ad does ot deped o the phase iformatio of groud motio. Actually this fact is well kow based o previous studies (Lyo, 1975; Nakashima et al., 1996). 575

3 There is a vast amout of research aimed to predictig amplitude Fourier model, comig especially from the egieerig seismology field. Such models have usually bee developed i the cotext of the stochastic modelig approach ad radom vibratio theory (Boore, 23). The Fourier amplitude spectrum of motio, Y(M, r,w), expected at a average site at distace r from average earthquake of seismic momet M, as Y( M, r, ) E( M, ). P( r, ). S( ) (6) where ω is the agular frequecy. The fuctio o the right-had site represet the earthquake source radiatio E(M, w), propagatio path effects P(r,w), site respose S(w) (Lam et al., 2; Halldorsso ad Papageorgiou, 25) The acceleratio source spectrum of poit source is defied as 2 2 c EM (, ) CM (1 ( / ) ) (7) where C frequecy-idepedet scalig factor equal to FR fθ V/(4πρβ 3 ), where R fθ represets the averaged radiatio patter (.55 for S waves), F is the free surface amplificatio (a factor equal to 2), V represets the partitio oto two horizotal compoets (1/ 2), ρ ad β are the material desity ad shear-wave speed, respectively, i the source regio. I this equatio, w c is corer frequecy ad is related to the time take to rupture the fault causig a permaet slip. Followig Brue assumptio, the corer frequecy is give by the followig equatio (Brue, 197) (2 ) 4.91 ( M ) c 6 1/3 (8) where Δσ, i bar, is the stress drop, β i km/s, ad M i dye-cm. The seismic momet, M is ofte expressed i terms of the momet magitude M w which is defied as follows (Kaamori, 1977) Mw 2 log M The seismological model depeds o distace through the path atteuatio fuctio. The path fuctios represet the effects of geometric spreadig, o-elastic, the path idepedet loss of high-frequecy, ad scatterig atteuatio (Boore ad Atkiso, 1987). The theoretical geometric distace atteuatio is assumed to be 1/r, where the expoet,, depeds o r. I equatio 6, S(ω) is the upper crust amplificatio factor ad the quarter-wavelegth method is used to model the amplificatio factor of site soil (Boore ad Joyer, 1997). They have proposed the site-amplificatio factor S(ω), as a fuctio of the average shear wave velocities ( V S ), represetig the soil coditios i the upper 3 m. Accordig to above descriptio, the Fourier amplitude spectrum of motio ca be obtaied based o equatio 6. By substitutig this iput ito equatio 3, the displacemet of structures ca be calculated. RESULTS AND DISCUSSION Numerical Applicatio: Stochastic study is oe of the most importat studies i all field of the scieces (Tijai ad Aromolara, 29; Okereke ad Bassey, 21; Kadry et al., 27). I this paper, i order to study the stochastic respose of structural frame, the earthquake magitude M w, source-to-site distace r, ad amplificatio factors S(ω) were modeled as radom variables. Each radom variable is modeled as (9) Z (1 V ) Z Z Z (1) where μ z is the mea value, α z is a radom variable with a zero mea ad V z is the coefficiet of variatio of the radom variable. The overall variace i respose of the structural frame is affected by the variaces i 576

4 each of the radom variables of excitatio. Table 1 shows the mea value set of earthquake groud motio variables for calculatio of the Fourier amplitude spectra of groud motio which is used for achievig horizotal displacemet of structures. Table 1: Set of parameters of earthquake groud motio Parameters (radom variables) mea value Earthquake magitude, M w 7 Focal distace, R (km) 2, 4, 6, 1 Desity, ρ (gr/cm^3) 2.8 Shear-wave velocity, β (km/s) 3.5 Stress drop, Δσ (bar) 1 Geometric Atteuatio 1/r Amplificatio factor, S(ω) NEHRP Class C Figure 1 shows two shear buildig models, deoted as case 1 ad 2, which are three stories, ad five stories i height. For the two models the floor masses, the story stiffess ad dampig have bee illustrated. I the stochastic calculatio of the respose resultig from the variability of the earthquake groud motio variables, the coefficiet of variatio, COV, is assumed to be equal to.2 for earthquake magitude ad.1 for distace variables, ad.2 for site amplificatio owig to the later variable is more ucertai tha earthquake magitude ad distace (Atkiso ad Silva, 1997;. Aderso et al., 1996; Fraceschia et al., 26). Fig. 1: Models of frames. The horizotal displacemet of the frame is evaluated based o the values of all structural ad earthquake groud motio variables. The trials are created a large umber of sets of radomly geerated values for the ucertai parameters ad compute the fuctio for each set. The method has the advatage of coceptual simplicity, but it ca require a large set of values of the fuctio to obtai adequate accuracy. It is possible to study the output statistically ad to obtai values of meas, stadard deviatio, ad other statistical parameters. The overall horizotal displacemet of the frame ad iter-story drift at each floor is first evaluated usig the values of all excitatio variables. As a example i case 2, the regular five-story, the values of the overall horizotal displacemet of the frame ad iter-story drift at odal poit are illustrated i Figures 2 ad 3 for differet earthquake distace. The traces of this figure demostrated the variatio of odal displacemets ad their dispersio whe earthquake distace of site chages. Table 2 compares the coefficiet of variatio i the maximum horizotal displacemet ad iter-story drift ratio by variatio of earthquake distace. The earthquake distace iflueces the respose of structure. The mea values ad the dispersio of iter-story drift are illustrated i Figures 2 to 3 for differet earthquake distaces. Table 2 idicated that the icrease of the earthquake distace causes decrease i the coefficiet of variatio of respose. 577

5 Table 2: Coefficiet of variatio of the stochastic respose i 5DOF i the case of M w 7., NEHRP class D, ad differet distaces R2 R4 R6 R1 Horizotal displacemet at top of frame Maximum iter-story drift ratio Fig. 2: Mea values of overall horizotal displacemet ad dispersio (M w 7, soil coditio of NEHRP class C ad differet distace). Fig. 2: Mea values of overall horizotal drift ad dispersio (M w 7, soil coditio of NEHRP class C ad differet distace). Coclusios: The formulatio i the frequecy domai is appropriate for computig the structural respose whe the Fourier acceleratio amplitude spectrum is available. This formulatio requires oly the Fourier amplitude spectrum of iput groud motio ad the real part of trasfer fuctio. Oe of the essetial characteristics of the seismological method is that it distills what is kow about the various factors affectig groud motios ito differet fuctioal forms. The preseted expressio i this study provides a importat basis for a wider use of seismological theory i the uderstadig of the relatio betwee seismological ad earthquake distace variable. Dispersio i the drift demad for a give itesity measure of excitatio is importat i calculatig the probability of exceedig a structural limit state. The results reveal the coefficiet of variatio ad dispersio i the maximum horizotal displacemet ad iter-story drift ratio decreases by icrease of earthquake distace. 578

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