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Seismic Fragility Evaluation with Incomplete Structural Appraisal Data An Iterative Statistical Approach Vincent Z Wang1 Michael Mallett2 and Andrew Priory3 Abstract This paper presents an iterative statistical approach to evaluating seismic structural safety using incomplete appraisal data Despite the continuous improvement to traditional structural assessment procedures and the recent progress in structural health monitoring method ologies practically acquired structural appraisal data may often be incomplete The occurrence of the appraisal data missingness could be ascribed to the malfunction of data acquisition systems the abnormality during data transfer and the inaccessibility of critical quantities among other reasons The study begins with a quantitative investigation into the sensitivity of the seismic fragility evaluation with respect to the structural appraisal data missingness through the defined additional information loss and probability of noninformativeness Sub sequently a remedy for the missingness of the structural appraisal data instead of a precaution against it is formulated by employing the expectation maximization EM algorithm With synthetic or real seismic ground accelerations involved the efficacy of the EM algorithm embedded remedy is demonstrated by examples of typical linear or nonlinear hysteretic systems in the framework of statistical hypothesis testing Resorting to the bootstrap technique the influence of the related correlations and missingness probability is also examined DOI 10 1061 ASCE ST 1943 541X 0000804 2013 American Society of Civil Engineers Author keywords Seismic fragility Statistics Incomplete data Missing data Uncertainty quantification EM algorithm Bootstrap Linear or hysteretic systems Analysis and computation Introduction Seismic activities have long been identified as one of the major natural hazards against which civil structures need to be designed Among various measures of the safety performance of a structure pertaining to seismic activities the fragility is to quantify the likelihood that a predefined event occurs This event is typically undesirable and can usually be formally described as such Corresponding to a given intensity of seismic ground motions the response of interest of the structure goes beyond a prescribed bound Practically there exist scores of options for the response of interest in accordance with the vast variety in engineering struc tures As a purpose related issue the final choice is contingent upon the structure being investigated and the specific structural aspects with which a study is concerned For instance for a given multistory reinforced concrete frame the bending moments at the beam column joints and the story drifts can respectively serve as the response of interest when its load carrying capacity and deformation performance are of primary concern Examples of the response of interest in the literature include displacement Sasani and Der Kiureghian 2001 Kafali and Grigoriu 2007 drift Kim and Shinozuka 2004 Lupoi et al 2006 Choe et al 2008 Park et al 2009 Celik and Ellingwood 2010 stress resultants Lupoi et al 2006 Casciati et al 2008 Choe et al 2008 and curvature Lupoi et al 2006 along with other derived quantities Oller and Barbat 2006 and combinations of multiple quantities Cimellaro et al 2009 Cimellaro and Reinhorn 2011 Once the response of interest has been selected it is also indis pensable to gather enough information to evaluate its value The information required comprises the structural condition loading configurations modeling considerations etc Of particular interest is the structural condition For structures to be constructed or recently constructed it may be obtained from relevant regulatory guidelines on structural design such as the code requirements for concrete buildings American Concrete Institute ACI 2011 British Standards Institution BSI 2004 Standards Australia 2009 and steel structures American Institute of Steel Construction AISC 2010 BSI 2005 Standards Australia 1998 Quality control procedures could also shed some light on the condition of a newly constructed structure When ready mixed concrete or precast prefabricated structural components and subassemblies are in volved it is often the case that standard quality control practice e g as per ASTM E122 ASTM 2009 and various in house quality control measures may provide some information useful for the evaluation of the structural condition By contrast if the condition of a structure having been in service for a considerable amount of time needs to be assessed those data directly from the aforementioned structural design requirements and quality control procedures may not be accurate enough for sub sequent fragility analysis as they generally do not allow for possible damage or deterioration to which the structure may have been subjected Thus structural appraisal techniques with the capability of inferring the up to date condition of the structure come into play ASTM C805 C805M ASTM 2008 and BS EN 13791 BSI 2007 are among the standards that direct the industry practice in this regard Diverse structural appraisal techniques are often applied simultaneously for improved accuracy Sullivan 1991 evaluated the in situ strengths of the concrete in an office building using nondestructive tests and core extraction Hassan et al 1995 investigated the modulus of elasticity of concrete in in service 1Lecturer in Civil Engineering School of Engineering and Physical Sciences James Cook Univ Townsville Queensland 4811 Australia corresponding author E mail vincent wang jcu edu au 2Former Honours Project Student School of Engineering and Physical Sciences James Cook Univ Townsville Queensland 4811 Australia 3Former Honours Project Student School of Engineering and Physical Sciences James Cook Univ Townsville Queensland 4811 Australia Note This manuscript was submitted on March 24 2012 approved on January 14 2013 published online on January 16 2013 Discussion period open until March 7 2014 separate discussions must be submitted for in dividual papers This paper is part of the Journal of Structural Engineer ing ASCE ISSN 0733 9445 04013048 13 25 00 ASCE04013048 1J Struct Eng J Struct Eng 2014 140 Downloaded from ascelibrary org by Changsha University of Science and Technology on 03 07 14 Copyright ASCE For personal use only all rights reserved bridges obtained by ultrasonic measurements and extracted cores and compared the results with those from load tests A series of nondestructive assessment methods was employed by Pascale et al 2003 to study the properties of high strength concrete It is also noted that statistical techniques are frequently invoked in a structural appraisal scenario to account for the uncertainty in volved Leshchinsky 1992 reviewed several statistical criteria relevant to the nondestructive assessment of concrete strength Geyskens et al 1998 proposed a Bayesian approach to relate the modulus of elasticity of concrete to the corresponding compressive strength Maes 2002 formulated an updating scheme in the empirical Bayes framework to assess the reliability of a structure subject to demanding environmental conditions Recently a multi disciplinary research field structural health monitoring SHM has been growing very fast Chang 2011 SHM aims to detect the pres ence of damage in a structure pinpoint the damage quantify its extent and prognosticate its potential development as well as its influence on the service life of the structure The advances in SHM have been complementing conventional structural appraisal techniques giving rise to enhanced capability for the acquisition of the data on the current condition of in service structures With continuing research effort put into the area of structural appraisal and health monitoring there is no doubt that more and more strategies will be constructed to further the state of the art of obtaining the latest data on the condition of the structure being investigated An accompanying issue arises that in practical situations the structural appraisal data collected can be incomplete The missingness of the structural appraisal data occurs when the data acquisition system does not function as it has been originally designed Nowadays it is not uncommon that a data acquisition sys tem consists of numerous sensors and such factors as installation errors mechanical actions electrical surges and power outage in any of them may lead to the data missingness to some extent For the sensors embedded into a structure and expected to work con stantly during the service life of the structure problems pertaining to the lifetimes of the sensors and the maintenance and replacement plans could result in the data missingness as well Another potential source that the appraisal data missingness can be ascribed to is the reliability robustness and working environment of the data transfer mechanism employed High levels of undesirable noise interfer ence and distortion may considerably affect the amount of the use ful information that can be extracted from the signals The appraisal data missingness happens if the disturbance cannot be allowed for by the data transfer mechanism More specifically this includes the occasions when the disturbance is of such a kind that the data transfer mechanism has not been designed and prepared to resist its influence or when some redundancy does exist in the data trans fer mechanism but is unfortunately not enough to compensate for the detrimental effect of the disturbance Besides the missingness of thestructuralappraisaldatacouldalsobeattributedtothefactthatthe quantities critical to the performance of a civil structure are not al ways readily convenient to be directly observed or measured For example in a typical frame building the nonstructural elements such as ceilings and finishes may prevent the frame joints from direct measurementaccess andthustherelevantappraisaldatacanbemiss ing Upon identifying representative scenarios in which the structural appraisal data may become incomplete appropriate measures that can be taken in these scenarios need to be sorted out Effort has rea sonably been focused on coming up with comprehensive precautions against the missingness of the appraisal data as exhibited by the rapid progress of a large number of structural appraisal and health monitoring techniques However the other side of the coin is that the likelihood of the occurrence of the appraisal data missingness is still far from negligible If a data missingness event does occur the consequence is that the structural appraisal data turn out to be incomplete Note that even when the probability of the data missing ness event is small the missingness probability associated with an individual appraisal data point can nevertheless be fairly high if the event occurs Therefore it would be interesting and worthwhile to formulate some remedies in contrast to precautions for the structural appraisal data missingness and examine their effectiveness in seismic safety performance evaluation In the next section a pilot study is presented in which the sensitivity of the seismic fragility evaluation with respect to the missingness of structural appraisal data is demonstrated Subsequently an iterative statistical algorithm is employed to formulate an efficacious approach to deal with incomplete structural appraisal data with application to both linear systems and nonlinear hysteretic systems The influence of some param eters involved is also discussed Sensitivity of the Seismic Fragility Evaluation to the Appraisal Data Missingness A Pilot Study Despite the relatively large number of extensively studied data missingness cases in other areas such as life sciences clinical re search etc to the authors knowledge few results on the seismic safety performance of civil structures are available Accordingly it becomes helpful to illustrate by some straightforward examples the structural appraisal data missingness and its effect on the fragility evaluation More details along this line can be found in Wang et al 2011 Consider for instance a three story shear frame modeled as a three degree of freedom DOF linear dynamic system Suppose that the story stiffness has a trivariate normal distribution Practi cally the parameters in this probability distribution i e the mean vector and the covariance matrix can be estimated from pertinent structural appraisal data which may be incomplete Throughout the study presented in this paper the missing completely at random MCAR assumption Heitjan and Basu 1996 is used to model the incomplete structural appraisal data MCAR refers to the situa tion in which the conditional probability of an observed missing ness pattern given the observed and missing data equals the probability of the observed missingness pattern That is it essen tially assumes that the observed missingness pattern is statistically independent of the observed and missing data Letting the mean and the coefficient of variation of each of the three random vari ables for the story stiffness be 1 5 104kN m and 0 3 respec tively and assuming that the correlation between any two of the three random variables is 0 5 Fig 1 illustrates the corresponding complete and incomplete structural appraisal data with a sample size m of 15 Notice that the missing data are indicated by blanks As one can see in Fig 1 b where the appraisal data for each story are subjected to a missingness probability p of 0 3 a total of six five and five data points are missing for the first second and third stories respectively To estimate the mean vector the incomplete appraisal data can be treated as three individual univariate samples For the estimation of the covariance matrix two immediate options exist i e pairwise manner and listwise manner In the pairwise manner the covariance between two random variables is estimated based on the missing ness pattern exhibited by the realizations of these two random variables only The ostensible advantage of the pairwise manner is that it appears to make the best of the situation of the data missingness while when the bigger picture of estimating the full covariance matrix for the story stiffness is considered it turns out to be dubious as the full covariance matrix thus obtained may not ASCE04013048 2J Struct Eng J Struct Eng 2014 140 Downloaded from ascelibrary org by Changsha University of Science and Technology on 03 07 14 Copyright ASCE For personal use only all rights reserved always be positive semidefinite Instead the listwise manner can be used where only the observed units without any missing data i e the five shaded units in Fig 1 b are employed to estimate the covariance matrix Obviously the listwise manner imposes a second round of information loss on top of the data missingness having occurred For the incomplete appraisal data illustrated in Fig 1 b besides the 16 missing data points 14 observed data points do not essentially contribute to the covariance matrix estimation if the listwise manner is applied This corresponds to a 48 3 additional information loss Here the additional information loss is quantified by the percentage of the number of noncontributory observed data points with respect to the total number of observed data points More generally Fig 2 a plots the average additional information loss for different values of the missingness probability p and the number of stories It is apparent that on average a considerable portion of the observed data points is noncontributory when the number of stories is large Another concern is that the probability of the event that each observed unit contains at least one missing data point which is defined as the probability of noninformativeness becomes higher and higher if either the missingness probability or the number of stories increases Several examples of the probability of noninformative ness corresponding to selected values of the sample size m and the missingness probability p are presented in Fig 2 b For those cases with notably high probabilities of noninformativeness very often the listwise estimation procedures cannot even be implemented not to mention the level of accuracy achieved With some examples of the seismic fragility evaluation in the listwise manner documented by Wang et al 2011 exploring alternative procedures to cope with the incomplete appraisal data would be preferable Fragility Evaluation of Linear Systems with Incomplete Appraisal Data The fragilities of typical linear systems under the circumstances of the appraisal data missingness are dealt with in this section Fig 1 An illustration of the structural appraisal data missingness a complete appraisal data and b incomplete appraisal data for the story stiffness of a three story shear frame a b 2468101214161820 0 10 20 30 40 50 60 70 80 90 100 Number of stories Additional information loss p 0 1 p 0 2 p 0 3 p 0 4 p 0 5 1012141618202224262830 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 Number of stories Probability of noninformativeness m 15 p 0 1 m 15 p 0 2 m 15 p 0 3 m 30 p 0 1 m 30 p 0 2 m 30 p 0 3 Fig 2 Concerns about the listwise estimation procedures a additional information loss and b probability of noninformativeness ASCE04013048 3J Struct Eng J Struct Eng 2014 140 Downloaded from ascelibrary org by Changsha University of Science and Technology on 03 07 14 Copyright ASCE For personal use only all rights reserved The equation of motion of a linear structural system subject to a seismic ground acceleration time history can be written as M U t C U t KU t MX t 1 where M C and K denote the mass damping and stiffness matri ces respectively t is the time U t is the displacement time history random vector of the structural system and X t is the seismic ground acceleration random vector Assume that onlythe horizontal vibrations of the system are looked at in Eq 1 as is the case for the vast majority of relevant structural engineering practice Accordingly U t o