SEISMIC FRAGILITY ANALYSIS

Transcription

1 9 th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability PMC24 SEISMIC FRAGILITY ANALYSIS C. Kafali, Student M. ASCE Cornell University, Ithaca, NY 483 M. Grigoriu, F. ASCE Cornell University, Ithaca, NY 483 Abstract Seisic fragility is the probability that a geotechnical, structural, and/or nonstructural syste violates at least a liit state when subjected to a seisic event of specified intensity. Current ethods for fragility analysis use peak ground acceleration (P GA), pseudo spectral acceleration (P S a ), velocity (P S v ), or spectral displaceent ( ) to characterize seisic intensity. While these descriptions of seisic intensity are attractive for applications, they cannot capture the essential properties of the ground otion, since the probability law of a stochastic process cannot be specified by, for exaple its axiu over a tie interval. The paper presents a ethod for calculating syste fragility as a function of oent agnitude and source-to-site distance r, referred to as fragility surface. The seisic ground otion intensity is characterized by seisic activity atrix, i.e., the relative frequency of the earthquakes with various and r. According to the specific barrier odel the probability law of the ground otion process is copletely characterized by, r, underlying soil at the site and other paraeters. A structural/nonstructural syste located in New York City is used to deonstrate the ethodology. Fragility surfaces for different liit states are obtained for the syste and its coponents. Introduction Fragility curves show the probability of a syste reaching a liit state as a function of soe easure of seisic intensity such as peak ground acceleration P GA (Hwang and Huo, 994; Hwang and Jaw, 99), pseudo spectral acceleration P S a (Singhal and Kireidjian, 99), or oent agnitude and source-to-site distance r of the seisic event (Seidel et al., 989). A single paraeter such as P GA or P S a cannot represent copletely an entire function of tie. P GA is an inadequate paraeter for characterizing ground otion (Swell, 989) and correlates weakly with both observed and theoretically coputed structural daages. This paper presents an alternative ethod for indexing fragility. The ethod is based on seisic activity atrix and a ground acceleration odel called specific barrier odel (Papageorgiou and Aki, 983a; Papageorgiou and Aki, 983b). Kafali and Grigoriu

2 Fragility Analysis Coon ground otion intensity characterization Suppose that the ground acceleration process W (t) can be odeled as a Gaussian white noise (GWN) excitation and consider a single degree of freedo (SDOF) syste representing a structure. The initial frequency and the daping ratio of the syste are ω = rad/sec and ζ = %, respectively and the paraeters of the Bouc-Wen odel (Wen, 9) representing a hysteretic daper are A =, α =, β =, n = 2 and the rigidity ratio ρ. The intensity of the GWN is g =. and the Nyquist frequency is 2 rad/sec. Figure reflects the lack of correlation between P GA and Correlation between PGA and (linear). Correlation between PGA and (nonlinear). Correlation between PS a and (nonlinear) PGA 4 PGA Figure. Lack of correlation PS a the spectral displaceent ( ) for ρ = (linear syste) and for ρ = (nonlinear syste), and between P S a calculated using the initial linear stiffness and for the nonlinear syste using saples of W (t). Proposed ground otion intensity characterization Exaple-: Unlike P GA, the intensity g copletely defines the GWN excitation. Consider the sae paraeters for the GWN input and the sae SDOF syste used previously with ρ =, so that the syste is linear. The correlation between g and the axiu response of the syste is estiated using (i) Monte Carlo siulation and (ii) crossing theory of stochastic processes. For the siulation approach saples of W (t) are generated for 2 different values of g covering the range fro. to 2. and corresponding s are calculated. Figure 2 shows spectral density of the GWN excitation and the correlation between g and. Probability of syste failure given a liit state, referred as the fragility, is also estiated using Monte Carlo siulation and crossing theory. For the siulation approach the fragility at a given intensity level g is approxiated by the ratio of the nuber of saples of exceeding a given liit state d to the total nuber of saples. The fragility of the syste can also be approxiated using the crossing theory. Accordingly, the probability that the (stationary) syste response process R(t) leaves the safe set D = ( d, d) in a tie interval of length t w is P [Syste failure] P [R() D] P [N D (t w ) = ], in which N D (t w ) is the average nuber of D-outcrossings of R(t) in t w and P [N D (t w ) = ] exp ( ν D t w ) is the reliability of the syste, the probability that R(t) stays in the safe set D during t w (Veneziano et al., 9). It is assued that P [R() D] for the selected safe Kafali and Grigoriu 2

3 2 Spectral density of the input process. Correlation between g and Crossing theory Monte Carlo siulation.9 Failure probability vs intensity of the noise g...3 Crossing theory Monte Carlo siulation. 2 2 ω (rad/sec) g. 2 Figure 2. Gaussian white noise g. 2 set. The ean crossing rate of the syste ν D is the ean rate at which the response process R(t) leaves the safe set D and is given by ν D = σ exp ( ) d2 π σ 2σ assuing 2 R(t) N(, σ 2 ) and Ṙ(t) N(, σ2 ), (Veneziano et al., 9). Hence, the fragility of the syste can be approxiated by P [Syste failure] exp ( ν D t w ). () Figure 2 shows the fragility of the syste for d = against g estiated using Monte Carlo siulation and crossing theory. Exaple-2: Consider the sae linear SDOF syste used previously. The seisic ground acceleration at a site is odeled by a zero-ean, stationary Gaussian process F (t) with spectral density s F F given by the specific barrier odel (Papageorgiou and Aki, 983a; Papageorgiou and Aki, 983b). Accordingly, s F F (ω, r) = 2πt w a(ω, r) 2 (2) where t w is the duration of the strong ground otion (Halldorsson et al., 22), r denotes the distance fro the seisic source to site, a(f, r) is the Fourier aplitude spectru of the strong ground acceleration at the site and f = ω/2π is the frequency in Hertz. The spectral density function in Eq. 2 copletely defines the zero-ean, stationary Gaussian seisic ground acceleration process F (t) and is a function of the oent agnitude of the earthquake, source to site distance r, and the soil type at the site. Figure 3 shows the spectral density of ground acceleration for =.3 and r = k, the correlation between (,r) and, and the fragility of the syste for d = against (,r). Nuerical Exaple A siplified atheatical odel is developed for a hospital building constructed in 9 s and located in Southern California, referred to as the MCEER Deonstration Hospital Project, WC. It is assued that the structure (i) is linear elastic and does not fail, (ii) has a proportional daping, and that (iii) translation in the weak, x, and the strong, y, directions are decoupled, and (iv) cascade analysis applies, that is, the nonstructural syste does not affect the dynaics of the supporting Kafali and Grigoriu 3

4 .9 Spectral density of the input process Correlation between (,r) and Crossing theory Monte Carlo siulation Failure probability vs intensity g FF (ω) ω (rad/sec) Figure 3. Specific barrier odel structure. The odal properties corresponding to the first 2 odes are calculated using a three diensional odel of the structure. The direction of the seisic ground otion is assued to coincide with the weak direction of the structure. The non-zero odal participation factors in the x direction correspond to odes, 4, and. An illustration of the WC odel with a nonstructural syste (N S) consisting of two coponents C and C 2 attached to it, and the required odal properties of the structure are shown in Table. It is assued that daping ratio is 3% for all odes. joint k 24 C 9 joint l C 2 Mode ω i Γ i x Table. Illustration and odal properties of WC Structural syste Equation of otion of a ulti degree of freedo syste subjected to a seisic ground acceleration F (t) is given by Z(t) + cż(t) + kz(t) = F (t) (3) where Z(t) is the relative displaceent response in x direction, = [,..., ] T, and, c and k are the ass, stiffness and the daping atrices of the structural syste, respectively. The response Z(t) is a stationary Gaussian process since it is assued that the structural syste is linear and F (t) is a stationary Gaussian process. The spectral density between the absolute acceleration responses G k (t) and G l (s) at joints Kafali and Grigoriu 4

5 k and l can be written as S Gk G l (ω) = n α k α l S F F (ω) S F F (ω) Γ i (α k φ i (l) H i (ω) + α l φ i (k) H i (ω)) (4) i= n + S F F (ω) φ i (k)φ j (l)γ i Γ j Hi (ω) H j (ω), () i,j= where α k = n j= Γ j φ j (k), φ j (k) is the k th coordinate of the odal shape j, H i (ω) = ( ζi 2 ωi 2 ωd,i 2 + 2ζ iω i ωd,i 2 ) ζ i ω i + hω H i (ω) 2ζ iω i ζ i ω i + hω with ω d,i = ω i ζ 2 i, h =, H i (ω) = / (ω 2 i ω 2 + 2hζ i ω i ω) and H i (ω) is the conjugate of Hi (ω). Mean and correlations, naely the second oent properties, define the response process copletely. Nonstructural syste The nonstructural syste consists of a water tank and a power generator located at the roof (joint-24) and at the first floor (joint-), respectively (Table ). It is assued that (i) the coponents are not interacting, (ii) C is drift sensitive and C 2 is velocity sensitive, and (iii) both coponents are linear SDOF oscillators with paraeters ω C = 8. rad/sec, ζ C =.2, ω C2 = 2. rad/sec, ζ C2 =.3. Equation of otions for C i is R i (t) + 2ζ Ci ω Ci Ṙ i (t) + ω 2 C i R i (t) = G k (t) () where R i (t) is the relative displaceent response of C i for i =, 2, k = 24 for i = and k = for i = 2. Fragility surfaces by crossing theory Mean crossing rates are used to obtain an upper bound for the probability of failure of C, C 2 and NS. It can be shown that the ean crossing rate ν NS of the nonstructural syste can be bounded by ν NS ν C + ν C2, where ν Ci is the ean crossing rate of the coponent C i, i =, 2 (Veneziano et al., 9). The liit states for the relative displaceent response of C and the relative velocity response of C 2 are d = c and v = 4 c/sec, respectively. Figure 4 shows the fragilities of the coponents and an upper bound for the syste fragility for the linear nonstructural syste defined previously. Conclusions Fragility is the probability of a syste reaching a liit state as a function of soe easures of seisic intensity and it needs to be plotted against paraeters of the probability law of ground acceleration rather than properties of its saples such as () Kafali and Grigoriu

6 Fragility of C Fragility of C 2 Fragility of NS Figure 4. Fragility surfaces of the nonstructural coponents and syste P GA or P S a. This paper presents an alternative ethod for indexing fragility. The ethod is based on a ground acceleration odel called specific barrier odel. Acknowledgeents The work in this study was supported by MCEER grant E-883. This support is appreciated. References Halldorsson, B., Dong, G., and Papageorgiou, A. S. 22. Earthquake Motion Input and its Disseination via the Internet, ( Journal of Engineering and Engineering Vibration, (), 2 2. Hwang, H. H. M., and Huo, J-R Generation of Hazard-Consistent Fragility Curves. Soil Dynaics and Earthquake Engineering, 3, Hwang, H. H. M., and Jaw, J-W. 99. Probabilistic Daage Analysis of Structures. Journal of Structural Engineering, (), Papageorgiou, A. S., and Aki, K. 983a. A Specific Barrier Model for the Quantitative Description of the Inhoogeneous Faulting and the Prediction of Strong Ground Motion. Part I. Description of the Model. Bulletin of the Seisological Society of Aerica, 3, Papageorgiou, A. S., and Aki, K. 983b. A Specific Barrier Model for the Quantitative Description of the Inhoogeneous Faulting and the Prediction of Strong Ground Motion. Part II. Applications of the Model. Bulletin of the Seisological Society of Aerica, 3, Seidel, M. J., Reinhorn, A. M., and Park, Y. J Seisic Daageability Assessent of R/C Buildings in Eastern U.S. Journal of Structural Engineering, (9), Singhal, A., and Kireidjian, A. S. 99. Method for Probabilistic Evaluation of Seisic Structural Daage. Journal of Structural Engineering, 22(2), Swell, R. T Daage Effectiveness of Earthquake Ground Motion: Characterizations Based on the Perforance of Structures and Equipent. Ph.D. thesis, Stanford University, Stanford, CA. Veneziano, D., Grigoriu, M., and Cornell, A. 9. Vector process odels for syste reliability. Journal of the Engineering Mechanics Division, June, Wen, Yi-Kwei. 9. Method for rando vibration of hysteretic systes. Journal of the Engineering Mechanics Division, April, Kafali and Grigoriu