{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T04:42:13Z","timestamp":1774413733667,"version":"3.50.1"},"reference-count":24,"publisher":"Wiley","issue":"8","license":[{"start":{"date-parts":[[2025,8,18]],"date-time":"2025-08-18T00:00:00Z","timestamp":1755475200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"funder":[{"DOI":"10.13039\/501100018537","name":"National Science and Technology Major Project","doi-asserted-by":"publisher","award":["J2019\u2010IV\u20100001\u20100068"],"award-info":[{"award-number":["J2019\u2010IV\u20100001\u20100068"]}],"id":[{"id":"10.13039\/501100018537","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["52205143"],"award-info":[{"award-number":["52205143"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Quality &amp; Reliability Eng"],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>In recent years, time\u2010variant reliability issues have garnered widespread attention, particularly due to their consideration of the cumulative effects of material degradation and the time dependency of stochastic process loads. Methods based on the outcrossing rate are typically employed for time\u2010variant reliability analysis. However, they fail to effectively capture the correlations among multiple failure modes. To address this limitation, this paper proposes a time\u2010variant reliability analysis method that accounts for the correlation of multiple failure modes by integrating copula and PHI2. The method first utilizes Copula to analyze the correlations among various failure modes, followed by applying the PHI2 method for the time\u2010dependent reliability analysis of each individual failure mode. Ultimately, case studies confirm that the proposed method proficiently evaluates the time\u2010variant reliability of systems with multiple failure modes, yielding a maximum relative error below 5% compared to MCS and significant performance enhancements.<\/jats:p>","DOI":"10.1002\/qre.70046","type":"journal-article","created":{"date-parts":[[2025,8,19]],"date-time":"2025-08-19T05:45:42Z","timestamp":1755582342000},"page":"3396-3404","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Outcrossing Rate\u2010Based Time\u2010Variant Reliability Analysis Method Under Multiple Failure Modes"],"prefix":"10.1002","volume":"41","author":[{"given":"Yan\u2010Feng","family":"Li","sequence":"first","affiliation":[{"name":"School of Mechanical and Electrical Engineering University of Electronic Science and Technology of China  Chengdu Sichuan China"},{"name":"Center for System Reliability and Safety University of Electronic Science and Technology of China  Chengdu Sichuan China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Junfeng","family":"Feng","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering University of Electronic Science and Technology of China  Chengdu Sichuan China"},{"name":"Center for System Reliability and Safety University of Electronic Science and Technology of China  Chengdu Sichuan China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-2202-3772","authenticated-orcid":false,"given":"Hua\u2010Ming","family":"Qian","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering University of Electronic Science and Technology of China  Chengdu Sichuan China"},{"name":"Center for System Reliability and Safety University of Electronic Science and Technology of China  Chengdu Sichuan China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tudi","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering University of Electronic Science and Technology of China  Chengdu Sichuan China"},{"name":"Center for System Reliability and Safety University of Electronic Science and Technology of China  Chengdu Sichuan China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hong\u2010Zhong","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering University of Electronic Science and Technology of China  Chengdu Sichuan China"},{"name":"Center for System Reliability and Safety University of Electronic Science and Technology of China  Chengdu Sichuan China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,8,18]]},"reference":[{"key":"e_1_2_7_2_1","doi-asserted-by":"publisher","DOI":"10.1061\/AJRUA6.RUENG-962"},{"key":"e_1_2_7_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijfatigue.2023.107639"},{"key":"e_1_2_7_4_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.strusafe.2022.102204"},{"key":"e_1_2_7_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.113990"},{"key":"e_1_2_7_6_1","doi-asserted-by":"publisher","DOI":"10.1115\/1.4027865"},{"key":"e_1_2_7_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ymssp.2020.107159"},{"key":"e_1_2_7_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ymssp.2019.05.009"},{"key":"e_1_2_7_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ymssp.2020.107299"},{"key":"e_1_2_7_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.strusafe.2022.102286"},{"key":"e_1_2_7_11_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2019.106730"},{"key":"e_1_2_7_12_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2024.110102"},{"key":"e_1_2_7_13_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109614"},{"key":"e_1_2_7_14_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2022.108761"},{"key":"e_1_2_7_15_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2017.07.010"},{"key":"e_1_2_7_16_1","doi-asserted-by":"publisher","DOI":"10.1115\/1.4035792"},{"key":"e_1_2_7_17_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.strusafe.2017.02.004"},{"key":"e_1_2_7_18_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2022.108398"},{"key":"e_1_2_7_19_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109402"},{"issue":"1","key":"e_1_2_7_20_1","first-page":"1","article-title":"Copula Modeling: An Introduction for Practitioners","volume":"1","author":"Trivedi P. 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