{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T07:20:39Z","timestamp":1781248839055,"version":"3.54.1"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,7,4]],"date-time":"2020-07-04T00:00:00Z","timestamp":1593820800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,7,4]],"date-time":"2020-07-04T00:00:00Z","timestamp":1593820800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"Natural Science Foundation of China","doi-asserted-by":"crossref","award":["51675428"],"award-info":[{"award-number":["51675428"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2022,4]]},"DOI":"10.1007\/s00366-020-01100-0","type":"journal-article","created":{"date-parts":[[2020,7,4]],"date-time":"2020-07-04T05:04:01Z","timestamp":1593839041000},"page":"1053-1070","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Reliability evaluation and importance analysis of structural systems considering dependence of multiple failure modes"],"prefix":"10.1007","volume":"38","author":[{"given":"Linjie","family":"Shen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5358-9283","authenticated-orcid":false,"given":"Yugang","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bifeng","family":"Song","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kunling","family":"Song","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,7,4]]},"reference":[{"key":"1100_CR1","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1007\/s00158-015-1239-7","volume":"52","author":"C Park","year":"2015","unstructured":"Park C, Kim NH, Haftka RT (2015) The effect of ignoring dependence between failure modes on evaluating system reliability. Struct Multidiscipl Optim 52:251\u2013268. https:\/\/doi.org\/10.1007\/s00158-015-1239-7","journal-title":"Struct Multidiscipl Optim"},{"key":"1100_CR2","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1016\/j.ress.2018.03.013","volume":"175","author":"P Wei","year":"2018","unstructured":"Wei P, Liu F, Tang C (2018) Reliability and reliability-based importance analysis of structural systems using multiple response Gaussian process model. Reliab Eng Syst Saf 175:183\u2013195. https:\/\/doi.org\/10.1016\/j.ress.2018.03.013","journal-title":"Reliab Eng Syst Saf"},{"key":"1100_CR3","doi-asserted-by":"publisher","first-page":"935","DOI":"10.1007\/s00366-017-0507-0","volume":"33","author":"A Kaintura","year":"2017","unstructured":"Kaintura A, Spina D, Couckuyt I et al (2017) A Kriging and Stochastic Collocation ensemble for uncertainty quantification in engineering applications. Eng Comput 33:935\u2013949. https:\/\/doi.org\/10.1007\/s00366-017-0507-0","journal-title":"Eng Comput"},{"key":"1100_CR4","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1007\/s00366-010-0190-x","volume":"27","author":"M Khandelwal","year":"2011","unstructured":"Khandelwal M (2011) Blast-induced ground vibration prediction using support vector machine. Eng Comput 27:193\u2013200. https:\/\/doi.org\/10.1007\/s00366-010-0190-x","journal-title":"Eng Comput"},{"key":"1100_CR5","doi-asserted-by":"publisher","first-page":"112","DOI":"10.1016\/j.strusafe.2006.03.002","volume":"29","author":"J Li","year":"2007","unstructured":"Li J, Chen J, Fan W (2007) The equivalent extreme-value event and evaluation of the structural system reliability. Struct Saf 29:112\u2013131. https:\/\/doi.org\/10.1016\/j.strusafe.2006.03.002","journal-title":"Struct Saf"},{"key":"1100_CR6","doi-asserted-by":"publisher","first-page":"9126","DOI":"10.1080\/03610926.2016.1205620","volume":"46","author":"X Jia","year":"2017","unstructured":"Jia X, Shen J, Wang L, Li Z (2017) Vine copula constructions of higher-dimensional dependent reliability systems. Commun Stat Theory Methods 46:9126\u20139136. https:\/\/doi.org\/10.1080\/03610926.2016.1205620","journal-title":"Commun Stat Theory Methods"},{"key":"1100_CR7","doi-asserted-by":"publisher","first-page":"1386","DOI":"10.1016\/j.ress.2011.05.008","volume":"96","author":"BJ Bichon","year":"2011","unstructured":"Bichon BJ, McFarland JM, Mahadevan S (2011) Efficient surrogate models for reliability analysis of systems with multiple failure modes. Reliab Eng Syst Saf 96:1386\u20131395. https:\/\/doi.org\/10.1016\/j.ress.2011.05.008","journal-title":"Reliab Eng Syst Saf"},{"key":"1100_CR8","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1016\/j.ress.2018.07.015","volume":"180","author":"M Sadoughi","year":"2018","unstructured":"Sadoughi M, Li M, Hu C (2018) Multivariate system reliability analysis considering highly nonlinear and dependent safety events. Reliab Eng Syst Saf 180:189\u2013200. https:\/\/doi.org\/10.1016\/j.ress.2018.07.015","journal-title":"Reliab Eng Syst Saf"},{"key":"1100_CR9","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1016\/j.compgeo.2015.04.002","volume":"68","author":"DQ Li","year":"2015","unstructured":"Li DQ, Zhang L, Tang XS et al (2015) Bivariate distribution of shear strength parameters using copulas and its impact on geotechnical system reliability. Comput Geotech 68:184\u2013195. https:\/\/doi.org\/10.1016\/j.compgeo.2015.04.002","journal-title":"Comput Geotech"},{"key":"1100_CR10","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1109\/TR.2017.2694459","volume":"66","author":"J Zhang","year":"2017","unstructured":"Zhang J, Ma X, Zhao Y (2017) A stress-strength time-varying correlation interference model for structural reliability analysis using copulas. IEEE Trans Reliab 66:351\u2013365. https:\/\/doi.org\/10.1109\/TR.2017.2694459","journal-title":"IEEE Trans Reliab"},{"key":"1100_CR11","doi-asserted-by":"publisher","first-page":"1316","DOI":"10.1080\/00949655.2012.658805","volume":"83","author":"Z Pan","year":"2013","unstructured":"Pan Z, Balakrishnan N, Sun Q, Zhou J (2013) Bivariate degradation analysis of products based on Wiener processes and copulas. J Stat Comput Simul 83:1316\u20131329. https:\/\/doi.org\/10.1080\/00949655.2012.658805","journal-title":"J Stat Comput Simul"},{"key":"1100_CR12","doi-asserted-by":"publisher","first-page":"624","DOI":"10.1109\/TR.2015.2513038","volume":"65","author":"W Peng","year":"2016","unstructured":"Peng W, Li Y-F, Yang Y-J et al (2016) Bivariate analysis of incomplete degradation observations based on inverse gaussian processes and copulas. IEEE Trans Reliab 65:624\u2013639. https:\/\/doi.org\/10.1109\/TR.2015.2513038","journal-title":"IEEE Trans Reliab"},{"key":"1100_CR13","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1016\/j.strusafe.2014.05.004","volume":"51","author":"S Eryilmaz","year":"2014","unstructured":"Eryilmaz S (2014) Multivariate copula based dynamic reliability modeling with application to weighted-k-out-of-n systems of dependent components. Struct Saf 51:23\u201328. https:\/\/doi.org\/10.1016\/j.strusafe.2014.05.004","journal-title":"Struct Saf"},{"key":"1100_CR14","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1023\/A:1016725902970","volume":"32","author":"T Bedford","year":"2001","unstructured":"Bedford T, Cooke RM (2001) Probability density decomposition for conditionally dependent random variables modeled by vines. Ann Math Artif Intell 32:245\u2013268. https:\/\/doi.org\/10.1023\/A:1016725902970","journal-title":"Ann Math Artif Intell"},{"key":"1100_CR15","doi-asserted-by":"publisher","first-page":"1031","DOI":"10.1214\/aos\/1031689016","volume":"30","author":"T Bedford","year":"2002","unstructured":"Bedford T, Cooke RM (2002) Vines\u2013a new graphical model for dependent random variables. Ann Stat 30:1031\u20131068. https:\/\/doi.org\/10.1214\/aos\/1031689016","journal-title":"Ann Stat"},{"key":"1100_CR16","doi-asserted-by":"publisher","DOI":"10.1002\/0470863072","volume-title":"Uncertainty analysis with high dimensional dependence modelling","author":"D Kurowicka","year":"2006","unstructured":"Kurowicka D, Cooke R (2006) Uncertainty analysis with high dimensional dependence modelling. Wiley, Chichester, Hoboken"},{"key":"1100_CR17","doi-asserted-by":"publisher","first-page":"182","DOI":"10.1016\/j.insmatheco.2007.02.001","volume":"44","author":"K Aas","year":"2009","unstructured":"Aas K, Czado C, Frigessi A, Bakken H (2009) Pair-copula constructions of multiple dependence. Insurance Math Econ 44:182\u2013198. https:\/\/doi.org\/10.1016\/j.insmatheco.2007.02.001","journal-title":"Insurance Math Econ"},{"key":"1100_CR18","doi-asserted-by":"publisher","first-page":"803","DOI":"10.1002\/qre.2058","volume":"33","author":"D Xu","year":"2017","unstructured":"Xu D, Wei Q, Elsayed EA et al (2017) Multivariate degradation modeling of smart electricity meter with multiple performance characteristics via vine copulas: multivariate degradation modeling of SEM via vine copulas. Qual Reliab Eng Int 33:803\u2013821. https:\/\/doi.org\/10.1002\/qre.2058","journal-title":"Qual Reliab Eng Int"},{"key":"1100_CR19","doi-asserted-by":"publisher","first-page":"061405","DOI":"10.1115\/1.4030179","volume":"137","author":"C Jiang","year":"2015","unstructured":"Jiang C, Zhang W, Han X et al (2015) A vine-copula-based reliability analysis method for structures with multidimensional correlation. J Mech Des 137:061405. https:\/\/doi.org\/10.1115\/1.4030179","journal-title":"J Mech Des"},{"key":"1100_CR20","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-13785-4","volume-title":"Analyzing dependent data with vine copulas: a practical guide with R","author":"C Czado","year":"2019","unstructured":"Czado C (2019) Analyzing dependent data with vine copulas: a practical guide with R. Springer, Cham"},{"key":"1100_CR21","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1016\/j.ress.2015.04.016","volume":"142","author":"Y Dutuit","year":"2015","unstructured":"Dutuit Y, Rauzy A (2015) On the extension of importance measures to complex components. Reliab Eng Syst Saf 142:161\u2013168. https:\/\/doi.org\/10.1016\/j.ress.2015.04.016","journal-title":"Reliab Eng Syst Saf"},{"key":"1100_CR22","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1111\/risa.12110","volume":"34","author":"P Wei","year":"2014","unstructured":"Wei P, Lu Z, Song J (2014) Moment-independent sensitivity analysis using copula: moment-independent sensitivity analysis using copula. Risk Anal 34:210\u2013222. https:\/\/doi.org\/10.1111\/risa.12110","journal-title":"Risk Anal"},{"key":"1100_CR23","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1016\/j.ress.2015.05.018","volume":"142","author":"P Wei","year":"2015","unstructured":"Wei P, Lu Z, Song J (2015) Variable importance analysis: a comprehensive review. Reliab Eng Syst Saf 142:399\u2013432. https:\/\/doi.org\/10.1016\/j.ress.2015.05.018","journal-title":"Reliab Eng Syst Saf"},{"key":"1100_CR24","doi-asserted-by":"publisher","first-page":"04014084","DOI":"10.1061\/(ASCE)EM.1943-7889.0000798","volume":"140","author":"C Zhou","year":"2014","unstructured":"Zhou C, Lu Z, Ren B, Cheng B (2014) Failure-mode importance measures in system reliability analysis. J Eng Mech 140:04014084. https:\/\/doi.org\/10.1061\/(ASCE)EM.1943-7889.0000798","journal-title":"J Eng Mech"},{"key":"1100_CR25","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1016\/j.ress.2018.02.016","volume":"174","author":"L He","year":"2018","unstructured":"He L, Lu Z, Li Xinyao (2018) Failure-mode importance measures in structural system with multiple failure modes and its estimation using copula. Reliab Eng Syst Saf 174:53\u201359. https:\/\/doi.org\/10.1016\/j.ress.2018.02.016","journal-title":"Reliab Eng Syst Saf"},{"key":"1100_CR26","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1016\/0167-4730(82)90024-8","volume":"1","author":"M Hohenbichler","year":"1982","unstructured":"Hohenbichler M, Rackwitz R (1982) First-order concepts in system reliability. Struct Saf 1:177\u2013188. https:\/\/doi.org\/10.1016\/0167-4730(82)90024-8","journal-title":"Struct Saf"},{"key":"1100_CR27","unstructured":"Nelsen RB (2006) An introduction to Copulas, 2. ed. 2006. Springer New York"},{"key":"1100_CR28","doi-asserted-by":"crossref","unstructured":"Joe H (1996) Families of m-variate distributions with given margins and m(m-1)\/2 bivariate dependence parameters. In: Lecture notes-monograph series. Institute of Mathematical Statistics, Hayward, pp 120\u2013141","DOI":"10.1214\/lnms\/1215452614"},{"key":"1100_CR29","doi-asserted-by":"publisher","first-page":"610","DOI":"10.1007\/978-1-4612-0919-5_38","volume-title":"Breakthroughs in statistics","author":"H Akaike","year":"1992","unstructured":"Akaike H (1992) Information theory and an extension of the maximum likelihood principle. In: Kotz S, Johnson NL (eds) Breakthroughs in statistics. Springer, New York, pp 610\u2013624"},{"key":"1100_CR30","doi-asserted-by":"publisher","first-page":"631","DOI":"10.1890\/13-1452.1","volume":"95","author":"K Aho","year":"2014","unstructured":"Aho K, Derryberry D, Peterson T (2014) Model selection for ecologists: the worldviews of AIC and BIC. Ecology 95:631\u2013636. https:\/\/doi.org\/10.1890\/13-1452.1","journal-title":"Ecology"},{"key":"1100_CR31","doi-asserted-by":"publisher","first-page":"899","DOI":"10.1016\/j.apm.2017.02.047","volume":"48","author":"B Keshtegar","year":"2017","unstructured":"Keshtegar B, Kisi O (2017) M5 model tree and Monte Carlo simulation for efficient structural reliability analysis. Appl Math Model 48:899\u2013910. https:\/\/doi.org\/10.1016\/j.apm.2017.02.047","journal-title":"Appl Math Model"},{"key":"1100_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/s00366-019-00793-2","author":"P Zeng","year":"2019","unstructured":"Zeng P, Li T, Chen Y et al (2019) New collocation method for stochastic response surface reliability analyses. Eng Comput. https:\/\/doi.org\/10.1007\/s00366-019-00793-2","journal-title":"Eng Comput"},{"key":"1100_CR33","doi-asserted-by":"publisher","first-page":"470","DOI":"10.1214\/aoms\/1177729394","volume":"23","author":"M Rosenblatt","year":"1952","unstructured":"Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Statist 23:470\u2013472. https:\/\/doi.org\/10.1214\/aoms\/1177729394","journal-title":"Ann Math Statist"},{"key":"1100_CR34","doi-asserted-by":"publisher","first-page":"172","DOI":"10.1016\/j.probengmech.2008.05.001","volume":"24","author":"R Lebrun","year":"2009","unstructured":"Lebrun R, Dutfoy A (2009) A generalization of the Nataf transformation to distributions with elliptical copula. Probab Eng Mech 24:172\u2013178. https:\/\/doi.org\/10.1016\/j.probengmech.2008.05.001","journal-title":"Probab Eng Mech"},{"key":"1100_CR35","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1016\/j.apm.2018.01.035","volume":"59","author":"F Wang","year":"2018","unstructured":"Wang F, Li H (2018) Distribution modeling for reliability analysis: impact of multiple dependences and probability model selection. Appl Math Model 59:483\u2013499. https:\/\/doi.org\/10.1016\/j.apm.2018.01.035","journal-title":"Appl Math Model"},{"key":"1100_CR36","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1016\/j.probengmech.2008.08.001","volume":"24","author":"R Lebrun","year":"2009","unstructured":"Lebrun R, Dutfoy A (2009) An innovating analysis of the Nataf transformation from the copula viewpoint. Probab Eng Mech 24:312\u2013320. https:\/\/doi.org\/10.1016\/j.probengmech.2008.08.001","journal-title":"Probab Eng Mech"},{"key":"1100_CR37","volume-title":"Optimal reliability modeling: principles and applications","author":"W Kuo","year":"2003","unstructured":"Kuo W, Zuo MJ (2003) Optimal reliability modeling: principles and applications. Wiley, Hoboken."},{"key":"1100_CR38","doi-asserted-by":"publisher","DOI":"10.1002\/9781118314593","volume-title":"Importance measures in reliability, risk, and optimization: principles and applications","author":"W Kuo","year":"2012","unstructured":"Kuo W, Zhu X (2012) Importance measures in reliability, risk, and optimization: principles and applications. Wiley, Chichester"},{"key":"1100_CR39","doi-asserted-by":"publisher","first-page":"428","DOI":"10.1016\/j.engfailanal.2016.10.003","volume":"70","author":"W Guo","year":"2016","unstructured":"Guo W, Cui W, Shi Y et al (2016) Function failure and failure boundary analysis for an aircraft lock mechanism. Eng Fail Anal 70:428\u2013442. https:\/\/doi.org\/10.1016\/j.engfailanal.2016.10.003","journal-title":"Eng Fail Anal"},{"key":"1100_CR40","doi-asserted-by":"publisher","first-page":"1076","DOI":"10.1016\/j.engfailanal.2019.06.035","volume":"104","author":"L Shen","year":"2019","unstructured":"Shen L, Zhang Y, Song K, Song B (2019) Failure analysis of a lock mechanism with multiple dependent components based on two-phase degradation model. Eng Fail Anal 104:1076\u20131093. https:\/\/doi.org\/10.1016\/j.engfailanal.2019.06.035","journal-title":"Eng Fail Anal"},{"key":"1100_CR41","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1016\/j.engfailanal.2015.11.032","volume":"60","author":"H Pang","year":"2016","unstructured":"Pang H, Yu T, Song B (2016) Failure mechanism analysis and reliability assessment of an aircraft slat. Eng Fail Anal 60:261\u2013279. https:\/\/doi.org\/10.1016\/j.engfailanal.2015.11.032","journal-title":"Eng Fail Anal"},{"key":"1100_CR42","doi-asserted-by":"publisher","first-page":"2066","DOI":"10.1109\/TPWRS.2011.2113380","volume":"26","author":"Zhen Shu","year":"2011","unstructured":"Shu Zhen, Jirutitijaroen P (2011) Latin hypercube sampling techniques for power systems reliability analysis with renewable energy sources. IEEE Trans Power Syst 26:2066\u20132073. https:\/\/doi.org\/10.1109\/TPWRS.2011.2113380","journal-title":"IEEE Trans Power Syst"},{"key":"1100_CR43","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198523963.001.0001","volume-title":"Applied smoothing techniques for data analysis: the kernel approach with S-Plus illustrations","author":"AW Bowman","year":"1997","unstructured":"Bowman AW, Azzalini A (1997) Applied smoothing techniques for data analysis: the kernel approach with S-Plus illustrations. Clarendon Press, Oxford University Press, New York, Oxford"},{"key":"1100_CR44","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1016\/j.csda.2012.08.010","volume":"59","author":"J Dissmann","year":"2013","unstructured":"Dissmann J, Brechmann EC, Czado C, Kurowicka D (2013) Selecting and estimating regular vine copulae and application to financial returns. Comput Stat Data Anal 59:52\u201369. https:\/\/doi.org\/10.1016\/j.csda.2012.08.010","journal-title":"Comput Stat Data Anal"},{"key":"1100_CR45","unstructured":"Nagler T, Schepsmeier U, Stoeber J et al (2019) VineCopula: statistical inference of vine copulas, R package version 2.2.0. https:\/\/CRAN.R-project.org\/package=VineCopula"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-020-01100-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-020-01100-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-020-01100-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,9]],"date-time":"2024-08-09T05:54:48Z","timestamp":1723182888000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-020-01100-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,4]]},"references-count":45,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,4]]}},"alternative-id":["1100"],"URL":"https:\/\/doi.org\/10.1007\/s00366-020-01100-0","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,4]]},"assertion":[{"value":"11 November 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 June 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 July 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}