RELIABILITY OF SPATIALLY CORRELATED COMPONENTS UNDER NATURAL HAZARDS: EARTHQUAKE CASE

1. RELIABILITY OF SPATIALLY CORRELATED COMPONENTS UNDER NATURAL HAZARDS: EARTHQUAKE CASE Sevtap Selcuk-Kestel Semih M. Yucemen Albert-Ludwigs University Freiburg Middle East Technical University April 24, 2009

2. INTRODUCTION Assessment of thr reliability of a lifeline 1. Probabilistic seismic risk analysis Seismic activity propagated from seismic source zones to the components of the network to estimate the potential seismic hazard that the components are subjected to. Seismic hazard analysis is the formal procedure in which the uncertainty and the randomness in the various physical parameters are propogated through a stochastic model leading to a probability distribution for a selected earthquake severity parameter at a specified location 2. Capacity determination analysis The probability distribution of the earthquake resistance capacity of the elements of the network is assessed. Deterministic and random capacity cases are incorparated through a threshold value for the mean resistance to earthquake excitation. 3. Network analysis The reliability of a network is assessed based on the reliability of the elements of network. This analysis relates component performance to the system performance according to an algoriihm selected. The algorithm depends on the path enumeration method where breadth-first search is employed by the standard Boolean technique (Yoo and Deo, 1988)

3. PROBABILISTIC SEISMIC HAZARD MODEL Steps are probabilistic modeling of earthquake occurrences in time and space, identification of potential earthquake sources Preparation of a seismic data base using catalogs which list information on earthquakes that have occurred in the past Distribution of past earthquake data to different seismic sources consistent with their epicentral location (recurrence relationship providing information on the relative frequency of the earthquake occurrences with different magnitudes is obtained and the seismicity parameters are assessed for each seismic source zone). Development or selection of attenuation model Preparation of a computational algorithm aggregating the seismic threat nucleating from different sources, yielding the probability distribution for the specified location. P(Y>y)= ?? P(Y>y|(m,r) fM(m) fR(r) dm dr where Y: earthquake intensity; M:magnitude and R:distance are the random variables. Probability Distribution of M gives a doubly-truncated exponential relationship, number of earthquakes in time domain Poisson with mean rate of ?, R is unifom along with the fault, Attenuation Relationship:

4. CAPACITY DETERMINATION Seismic Capacity Model Failure probability of a component Pf under seismic hazard depends on the variables capacity (C) and the demand (D) on the element as follows: Assuming C and D are independent and normally distributed, Pf becomes Assuming C and D are lognormally distributed with c.o.v.�s ?C=?C /?C and ?D= ?D/?D , Pf becomes

5. RELIABILITY MODELS For spatially extended components, three reliabillity models are proposed: Point-site model (Selcuk, Yucemen, 1999,2000) The probability distribution of seismicity parameter is computed at a number of points of the component Then each component is idealized as a �point-site� with a seismic hazard distribution equal to the maximum hazard obtained along the component. Assumption: Perfectly correlated capacity and demand along the full length of the component Multi-site model (Selcuk, Yucemen,1999, 2000) Each member of the lifeline is divided into segments, transforming the whole component into a set of components formed in a series system Then the reliability of an element consisting of m-connected segments is expressed in upper and lower bounds due to the unmeasurable correlation among the segments: or Lower bound under independence asumption and the upper bound corresponds to a perfect dependence among all segments

6. RELIABILITY MODELS Spatial Correlation Model The correlation among the elements also influences the failure of the system. The degree of spatial correlation between two elements will depend on the distance separating them and is expected to be decaying function of this distance where ? is the correlation function and ? represents the scale of fluctuation, i.e. in space the distance over which safety margin shows a relatively strong correlation. In a network with n links the determination of seismic capcity is as follows: Dividing Link i into ki and Link j into kj subintervals proportional to ?, correlation matrix ?(i,j) with entries ?(ip,jt ) is where ?(ip,jt ) denotes the distance between pth and tth subintervals of links i and j.

7. RELIABILITY MODELS Ultimate correlation for Link i associated with other links in the network is where denotes the average correlation of pth partition of Link i with respect to all possible subintervals of (n-1) remaining links in the network

8. RELIABILITY MODELS The correlation coefficient calculated for each subdivision will be incorparated into the determination of the capacity by modeling each link being partitioned into subintervals as a series system with unequal correlation attained for each interval. Failure probability for such series system having spatial correlation is (Thoft and Christensen, 1982): where F and f denote the distribution and density functions for standard normal random variable and �e is the common reliability index.

9. NETWORK RELIABILITY A lifeline is represented by a graph in which each node and each branch has a given probability of being in �operation�. Forms of network reliability problem are 1. probability that specified nodes can communicate in a network in which edge failures are statistically independent 2. networks in which nodes are also subject to failure 3. Networks with more than one source or sink 4. Networks having uni-or bi-directional communication links Terminal-Pair Reliability: two nodes function as alternatives to provide communication between the source and the sink given the probability of survival for each communication link in network. The algorithm based on Terminal-pair reliability (Yoo and Deo, 1988) generates collection of success, S, and failure, F, sets. The reliability of the network, therefore is

10. CASE STUDY Input Seismicity parameters Attenuation relationship Geographical locations of seismic sources Geographical locations of the components Assumptions: All links have the same seismic capacity Seismic demand varies depending on the location of components w.r.to the seismic sources Case 1: Deterministic link resistances: a constant resistance value assigned to each link Case 2: Link resistance follow Gaussian distribution with mean and standard deviation via a software coded for this purpose called LIFEPACK (Selcuk and Yucemen 1999, 2000) A lifeline as case study is considered and the results are compared with the ones in literature : Boston Highway system Output Network reliability of lifeline under seismic hazard

11. LIFEPACK

12. BOSTON HIGHWAY SYSTEM Studied by Panoussis (1974) and Taleb-Agha (1977) 18 nodes and 23 links. The objective is to reach from Node 1 to Node 5 8 Seismic sources (Cornell and Merz, 1975) Attenuation equation:Y is the peak ground accelaration Deterministic model: Seismic capacity of each link with a resistance of 0.075g Gaussian Model: Seismic capacity of each link ~N(0.100g, 0.002g)

13. BOSTON HIGHWAY SYSTEM

14. BOSTON HIGHWAY SYSTEM- Deterministic Model

15. BOSTON HIGHWAY SYSTEM Multi-site Model Each element is subdivided into segments of length ?=1km Only Gaussian random seismic capacity with �=0.100g and s= 0.002g Using upper bound reliability (Perfectly correlated case) bounds for annual reliability of the highway system is 0.999956 < RS < 0.999957 Using lower bound element reliability (independent segments) RS=0.998685 Combining those two bounds the system reliability for multi-site model is 0.998685 < RS < 0.999957 Reliability bounds for each links are presented as follows:

17. BOSTON HIGHWAY SYSTEM Spatial Correlation Model Scale of fluctuation varies with respect to the length of the link Correlation between two segments of each partition falling below 0.001 is neglected in calculation System reliability for Gaussian random seismic capacity with �=0.100g and s= 0.002g is 0.997360 < RS < 0.997953

18. REFERENCES Cornell C.A. and Merz H.A., (1969). A Seismic Risk Analysis of Boston, Journal of Stuctural Division of ASCE, 101, pp. 2027-2043. Panoussis, G. (1974). Seismic Reliabiliy of Lifeline Networks, SDDA Report No. 15, MIT, Dept. of Civil Eng., Rept R74-57, Cambridge USA. Selcuk, A.S. and Yucemen, M. S. (1999), Reliability of Lifeline Networks under Seismic Hazard, Reliability Engineering & System Safety, 65, pp. 213-227. Selcuk, A.S. and Yucemen, M, S. (2000). Reliability of Lifeline Networks with Multiple Sources under Seismic Hazard, Natural Hazards, 21, pp.1-18. Taleb-Agha, G. (1977). Seismic Risk Analysis of Lifeline Networks, Bull. Seism. Soc. Am., 67, pp.1625-1645. Thoft-Christensen, P., Sorensen, D.J. (1982). Reliability of Structural Systems with Correlated Elements, Applied Math. Modeling, 6, pp. 171-178. Yoo, Y.B. and Deo, N. (1988). A Comparison of Algorithms for Terminal Pair Reliability, IEEE, Transactions on Reliability, 37, pp. 210-215.