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Article(id=1241793461054144895, tenantId=1146029695717560320, journalId=1241701559352995854, issueId=1241793456876618047, articleNumber=null, orderNo=null, doi=10.13197/j.eeed.2025.0414, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1709568000000, receivedDateStr=2024-03-05, revisedDate=1715184000000, revisedDateStr=2024-05-09, acceptedDate=null, acceptedDateStr=null, onlineDate=1773996232521, onlineDateStr=2026-03-20, pubDate=1755792000000, pubDateStr=2025-08-22, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773996232521, onlineIssueDateStr=2026-03-20, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773996232521, creator=13701087609, updateTime=1773996232521, updator=13701087609, issue=Issue{id=1241793456876618047, tenantId=1146029695717560320, journalId=1241701559352995854, year='2025', volume='45', issue='4', pageStart='1', pageEnd='222', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1773996231526, creator=13701087609, updateTime=1773997043565, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241796862877958695, tenantId=1146029695717560320, journalId=1241701559352995854, issueId=1241793456876618047, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241796862877958696, tenantId=1146029695717560320, journalId=1241701559352995854, issueId=1241793456876618047, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=141, endPage=150, ext={EN=ArticleExt(id=1241793461905588635, articleId=1241793461054144895, tenantId=1146029695717560320, journalId=1241701559352995854, language=EN, title=Research on laws of progressive collapse resistance of reinforced concrete planar frame structures based on strain energy of members, columnId=null, journalTitle=Earthquake Engineering and Engineering Dynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Progressive collapse is a nonlinear dynamic behavior of structural systems. In order to study the progressive collapse resistance laws of reinforced concrete planar frame structures after failure of important members, an important member identification method based on strain energy of members was proposed. The effectiveness of the method was verified by taking a four-bay and eight-story reinforced concrete frame structure as the target. It was found that the method identifies important members with high effectiveness. On this basis, eight reinforced concrete frame structures with different beam spans or total number of floors were designed in accordance with code. Firstly, important members were identified based on the above method and the adverse structural member removal scenarios were formulated. Then, impacts of different total number of floors and structural span on progressive collapse resistance capacity of reinforced concrete planar frame structures were analyzed by adopting the nonlinear dynamic alternate path method. Finally, the approximate functional curve of reinforced concrete planar frame structures between proportions of columns removed from ground floor and degrees of structural collapse was fitted. The results indicate that after removal of center column at ground floor, structures are mainly carrying vertical loads by ground floor frame beams, and frame beams of remaining floors cooperate with ground floor frame beams to support loads. The decrease in total number of floors and the decrease in span can increase redundant load carrying capacity of structures, which has a positive effect on resistance capacity of frame structures to progressive collapse. Frame beam span has a significant effect on degree of structural collapse and total number of floors has a limited effect on degree of structural collapse. The relationship between proportions of columns removed from ground floor and degrees of structural collapse of reinforced concrete plane frame structures can be approximated by a logistic function.

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连续倒塌属于结构系统的非线性动力行为，为研究平面RC框架结构在重要构件失效后结构的抗连续倒塌规律，提出一种基于构件应变能的结构重要构件评价方法，以一栋四跨八层的钢筋混凝土（reinforced concrete，RC）框架结构为目标验证该方法的有效性，发现该方法评价结构重要构件有效性高。在此基础上，按照规范设计8个不同框架梁跨度或总楼层数的典型规则RC框架，首先基于上述方法评价结构重要构件，制定结构不利拆除构件方案；然后采用非线性动力拆除构件法，分析框架梁跨度和总楼层数对平面RC框架结构抗连续倒塌能力的影响；最后拟合平面RC框架结构底层拆柱比例与倒塌程度近似函数曲线。结果表明：底层中柱拆除后，结构主要由底层框架梁承担竖向荷载，其余各层框架梁协同受力；总楼层数减少和跨度减少均能提高结构的冗余承载力，对结构抗连续倒塌起到积极作用；框架梁跨度对结构倒塌程度影响较大，总楼层数对结构倒塌程度影响有限，平面RC框架结构底层拆柱比例与倒塌程度近似可用logistic函数拟合。

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王贞（1983—），男，研究员，博士，主要从事结构抗震试验方法研究。E-mail：
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包宇（1989—），男，副教授，博士，主要从事结构减震隔震方面研究。E-mail：

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包宇（1989—），男，副教授，博士，主要从事结构减震隔震方面研究。E-mail：

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包宇（1989—），男，副教授，博士，主要从事结构减震隔震方面研究。E-mail：

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(in Chinese), articleTitle=Collapse performance of RC frame structure, refAbstract=null), Reference(id=1241802935554868100, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, doi=null, pmid=null, pmcid=null, year=2017, volume=38, issue=2, pageStart=83, pageEnd=89, url=null, language=null, rfNumber=[21], rfOrder=34, authorNames=于晓辉, 钱凯, 吕大刚, journalName=建筑结构学报, refType=null, unstructuredReference=于晓辉，钱凯，吕大刚. 基于随机Pushdown方法的钢筋混凝土框架结构抗连续倒塌能力概率评估[J]. 建筑结构学报, 2017, 38(2): 83-89., articleTitle=基于随机Pushdown方法的钢筋混凝土框架结构抗连续倒塌能力概率评估, refAbstract=null), Reference(id=1241802935676502927, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, doi=null, pmid=null, pmcid=null, year=2017, volume=38, issue=2, pageStart=83, pageEnd=89, url=null, language=null, rfNumber=[21], rfOrder=35, authorNames=YU Xiaohui, QIAN Kai, LÜ Dagang, journalName=Journal of Building Structures, refType=null, unstructuredReference=YU Xiaohui, QIAN Kai, LÜ Dagang. Probabilistic assessment of structural resistance of RC frame structures against progressive collapse using random Pushdown analysis[J]. Journal of Building Structures, 2017, 38(2): 83-89. (in Chinese), articleTitle=Probabilistic assessment of structural resistance of RC frame structures against progressive collapse using random Pushdown analysis, refAbstract=null), Reference(id=1241802935777166236, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, doi=null, pmid=null, pmcid=null, year=2011, volume=null, issue=null, pageStart=null, pageEnd=null, url=null, language=null, rfNumber=[22], rfOrder=36, authorNames=null, journalName=null, refType=null, unstructuredReference=GB 50010—2010混凝土结构设计规范[S]. 北京：中国建筑工业出版社，2011., articleTitle=null, refAbstract=null), Reference(id=1241802935894606763, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, doi=null, pmid=null, pmcid=null, year=2011, volume=null, issue=null, pageStart=null, pageEnd=null, url=null, language=null, rfNumber=[22], rfOrder=37, authorNames=null, journalName=null, refType=null, unstructuredReference=GB 50010—2010 Code for design of concrete structures[S]. Beijing: China Architecture & Building Press, 2011. (in Chinese), articleTitle=null, refAbstract=null)], funds=[Fund(id=1241802927577301425, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, awardId=2022JCJQZD03803, language=CN, fundingSource=基础加强计划项目子课题(2022JCJQZD03803), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1241802914356855313, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, xref=null, ext=[AuthorCompanyExt(id=1241802914365243921, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, companyId=1241802914356855313, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=Schoolof Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, China), AuthorCompanyExt(id=1241802914373632530, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, companyId=1241802914356855313, 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figureFileBig=9Q6WNaTW9pGYDPmQhEGMqA==, tableContent=null), ArticleFig(id=1241802924431573263, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, language=CN, label=图10, caption=结构倒塌程度曲线, figureFileSmall=ZNRxtgaXIEsZqlJ3WiLkuQ==, figureFileBig=9Q6WNaTW9pGYDPmQhEGMqA==, tableContent=null), ArticleFig(id=1241802926025408796, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, language=EN, label=Table 1, caption=

Degrees of structural collapse under single column and double columns removal schemes

, figureFileSmall=null, figureFileBig=null, tableContent=
拆柱编号构件应变能/J倒塌程度/%拆柱编号构件应变能/J倒塌程度/%
310900.070（未塌）3、235230.523（未塌）
210250.067（未塌）3、727770.352（未塌）
87450.063（未塌）3、1222700.251（未塌）
76910.060（未塌）3、1718860.194（未塌）
), ArticleFig(id=1241802926159626532, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, language=CN, label=表1, caption=

单柱、双柱拆除方案下结构倒塌程度

, figureFileSmall=null, figureFileBig=null, tableContent=
拆柱编号构件应变能/J倒塌程度/%拆柱编号构件应变能/J倒塌程度/%
310900.070（未塌）3、235230.523（未塌）
210250.067（未塌）3、727770.352（未塌）
87450.063（未塌）3、1222700.251（未塌）
76910.060（未塌）3、1718860.194（未塌）
), ArticleFig(id=1241802926314815794, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, language=EN, label=Table 2, caption=

Degrees of structural collapse under three columns and four columns removal schemes

, figureFileSmall=null, figureFileBig=null, tableContent=
拆柱编号构件应变能/J倒塌程度/%拆柱编号构件应变能/J倒塌程度/%
3、2、4802075.122（塌）3、2、4、11319487.823（塌）
3、2、9685474.213（塌）3、2、4、61234487.064（塌）
3、2、1633562.815（塌）3、2、4、111160486.212（塌）
3、2、14595672.437（塌）3、2、4、71076675.121（塌）
), ArticleFig(id=1241802926503559493, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, language=CN, label=表2, caption=

三柱、四柱拆除方案下结构倒塌程度

, figureFileSmall=null, figureFileBig=null, tableContent=
拆柱编号构件应变能/J倒塌程度/%拆柱编号构件应变能/J倒塌程度/%
3、2、4802075.122（塌）3、2、4、11319487.823（塌）
3、2、9685474.213（塌）3、2、4、61234487.064（塌）
3、2、1633562.815（塌）3、2、4、111160486.212（塌）
3、2、14595672.437（塌）3、2、4、71076675.121（塌）
), ArticleFig(id=1241802926621000020, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, language=EN, label=Table 3, caption=

Reinforcement details for structures of different beam spans

, figureFileSmall=null, figureFileBig=null, tableContent=
梁跨度柱尺寸梁尺寸梁纵筋柱纵筋梁箍筋柱箍筋
4000400×400300×500顶216，底21681681008100
5000500×500300×500顶216，底316121681008100
6000600×600300×600顶218，底416121881008100
7000600×600300×600顶225，底420121881008100
8000800×800400×720顶420，底420162081008100
), ArticleFig(id=1241802926784577889, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, language=CN, label=表3, caption=

不同梁跨度结构配筋信息

, figureFileSmall=null, figureFileBig=null, tableContent=
梁跨度柱尺寸梁尺寸梁纵筋柱纵筋梁箍筋柱箍筋
4000400×400300×500顶216，底21681681008100
5000500×500300×500顶216，底316121681008100
6000600×600300×600顶218，底416121881008100
7000600×600300×600顶225，底420121881008100
8000800×800400×720顶420，底420162081008100
), ArticleFig(id=1241802926906212716, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, language=EN, label=Table 4, caption=

Strain energy of members in different columns removal schemes

, figureFileSmall=null, figureFileBig=null, tableContent=
拆柱方案梁跨度/mm层数
400050006000700080004层8层12层
方案3142268460116112781154601024
方案3、方案245387215033859421240615033194
方案3、方案2、方案41030198134298914968399734297065
方案3、方案2、方案4、方案116843240565614924163041850565611628
), ArticleFig(id=1241802927073984898, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, language=CN, label=表4, caption=

不同拆柱方案下构件应变能

, figureFileSmall=null, figureFileBig=null, tableContent=
拆柱方案梁跨度/mm层数
400050006000700080004层8层12层
方案3142268460116112781154601024
方案3、方案245387215033859421240615033194
方案3、方案2、方案41030198134298914968399734297065
方案3、方案2、方案4、方案116843240565614924163041850565611628
), ArticleFig(id=1241802927199814029, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, language=EN, label=Table 5, caption=

Degree of structural collapse with different columns removal schemes

, figureFileSmall=null, figureFileBig=null, tableContent=
拆柱方案底层拆柱比例梁跨度/mm层数
400050006000700080004层8层12层
3200.07325.04125.05425.07525.09325.03125.05425.074
3、24050.02150.03350.05250.06450.08750.03250.05250.095
3、2、46075.04275.05375.07475.08175.08675.04475.07475.102
3、2、4、18087.55187.57487.58587.60387.62187.55287.58587.623
), ArticleFig(id=1241802927304671636, tenantId=1146029695717560320, journalId=1241701559352995854, articleId=1241793461054144895, language=CN, label=表5, caption=

不同拆柱方案下结构倒塌程度

, figureFileSmall=null, figureFileBig=null, tableContent=
拆柱方案底层拆柱比例梁跨度/mm层数
400050006000700080004层8层12层
3200.07325.04125.05425.07525.09325.03125.05425.074
3、24050.02150.03350.05250.06450.08750.03250.05250.095
3、2、46075.04275.05375.07475.08175.08675.04475.07475.102
3、2、4、18087.55187.57487.58587.60387.62187.55287.58587.623
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基于构件应变能的平面RC框架结构抗连续倒塌规律研究
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包宇 , 徐必胜 , 王贞 , 吴斌 , 叶泽宇
地震工程与工程振动 | 2025,45(4): 141-150
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地震工程与工程振动 | 2025, 45(4): 141-150
基于构件应变能的平面RC框架结构抗连续倒塌规律研究
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包宇 , 徐必胜, 王贞 , 吴斌, 叶泽宇
作者信息
  • 武汉理工大学 土木工程与建筑学院，湖北 武汉 430070

    • 包宇（1989—），男，副教授，博士，主要从事结构减震隔震方面研究。E-mail：

通讯作者:

王贞（1983—），男，研究员，博士，主要从事结构抗震试验方法研究。E-mail：
Research on laws of progressive collapse resistance of reinforced concrete planar frame structures based on strain energy of members
Yu BAO , Bisheng XU, Zhen WANG , Bin WU, Zeyu YE
Affiliations
  • Schoolof Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, China
出版时间: 2025-08-22 doi: 10.13197/j.eeed.2025.0414
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摘要
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连续倒塌属于结构系统的非线性动力行为，为研究平面RC框架结构在重要构件失效后结构的抗连续倒塌规律，提出一种基于构件应变能的结构重要构件评价方法，以一栋四跨八层的钢筋混凝土（reinforced concrete，RC）框架结构为目标验证该方法的有效性，发现该方法评价结构重要构件有效性高。在此基础上，按照规范设计8个不同框架梁跨度或总楼层数的典型规则RC框架，首先基于上述方法评价结构重要构件，制定结构不利拆除构件方案；然后采用非线性动力拆除构件法，分析框架梁跨度和总楼层数对平面RC框架结构抗连续倒塌能力的影响；最后拟合平面RC框架结构底层拆柱比例与倒塌程度近似函数曲线。结果表明：底层中柱拆除后，结构主要由底层框架梁承担竖向荷载，其余各层框架梁协同受力；总楼层数减少和跨度减少均能提高结构的冗余承载力，对结构抗连续倒塌起到积极作用；框架梁跨度对结构倒塌程度影响较大，总楼层数对结构倒塌程度影响有限，平面RC框架结构底层拆柱比例与倒塌程度近似可用logistic函数拟合。

平面RC框架结构  /  构件应变能  /  非线性动力拆除构件法  /  连续倒塌  /  倒塌程度

Progressive collapse is a nonlinear dynamic behavior of structural systems. In order to study the progressive collapse resistance laws of reinforced concrete planar frame structures after failure of important members, an important member identification method based on strain energy of members was proposed. The effectiveness of the method was verified by taking a four-bay and eight-story reinforced concrete frame structure as the target. It was found that the method identifies important members with high effectiveness. On this basis, eight reinforced concrete frame structures with different beam spans or total number of floors were designed in accordance with code. Firstly, important members were identified based on the above method and the adverse structural member removal scenarios were formulated. Then, impacts of different total number of floors and structural span on progressive collapse resistance capacity of reinforced concrete planar frame structures were analyzed by adopting the nonlinear dynamic alternate path method. Finally, the approximate functional curve of reinforced concrete planar frame structures between proportions of columns removed from ground floor and degrees of structural collapse was fitted. The results indicate that after removal of center column at ground floor, structures are mainly carrying vertical loads by ground floor frame beams, and frame beams of remaining floors cooperate with ground floor frame beams to support loads. The decrease in total number of floors and the decrease in span can increase redundant load carrying capacity of structures, which has a positive effect on resistance capacity of frame structures to progressive collapse. Frame beam span has a significant effect on degree of structural collapse and total number of floors has a limited effect on degree of structural collapse. The relationship between proportions of columns removed from ground floor and degrees of structural collapse of reinforced concrete plane frame structures can be approximated by a logistic function.

reinforced concrete planar frame structures  /  strain energy of members  /  nonlinear dynamic alternate path method  /  progressive collapse  /  degree of collapse
包宇, 徐必胜, 王贞, 吴斌, 叶泽宇. 基于构件应变能的平面RC框架结构抗连续倒塌规律研究. 地震工程与工程振动, 2025 , 45 (4) : 141 -150 . DOI: 10.13197/j.eeed.2025.0414
Yu BAO, Bisheng XU, Zhen WANG, Bin WU, Zeyu YE. Research on laws of progressive collapse resistance of reinforced concrete planar frame structures based on strain energy of members[J]. Earthquake Engineering and Engineering Dynamics, 2025 , 45 (4) : 141 -150 . DOI: 10.13197/j.eeed.2025.0414
正文
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0　引言
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连续倒塌是指偶然荷载作用下，结构局部构件出现损伤，随后损伤沿构件传递，引起其他位置处构件破坏，进而发生与初始损伤不匹配的大规模倒塌。其中，致使结构发生初始损伤的偶然事件包括：飞机撞击、煤气爆炸等。虽然结构发生连续倒塌属于小概率事件，但容易造成较为严重的人员伤亡，因此近年来国内外学者针对结构抗连续倒塌开展了许多相关研究。
针对结构抗连续倒塌研究主要采用拆除构件法，钱凯等[1]基于试验和数值模拟，研究钢筋混凝土（reinforced concrete，RC）梁-板子结构在边柱失效工况下的抗连续倒塌能力，对连续倒塌过程中荷载传递机制和抗力机制展开讨论。易伟建等[2]采用Pushdown分析方法，研究平面RC框架结构抗连续倒塌能力，提出一种RC框架结构可靠度计算方法，并总结楼层层数对结构抗连续倒塌能力的影响。肖宇哲等[3]针对钢筋混凝土梁柱子结构开展静力和动力试验，对子结构的连续倒塌机理和动力效应进行探讨。周云等[4]通过对装配式子结构开展试验，揭示中柱失效工况下，该类型结构的受力特征和荷载传递机理。ADAM等[5]对两层两跨RC板柱结构开展底层角柱失效工况下的结构动力试验，分析楼板抗弯机制和空腹机制对结构抗连续倒塌能力的影响。程小卫等[6]基于LS-DYNA采用拆除构件法以及撞击全过程法分析框架结构在撞击作用下结构的动力响应，对2种方法在模拟撞击作用下框架结构动力响应的精度和效率展开评估。KIM等[7]采用有限元分析软件OpenSees，对平面钢框架连续倒塌机理展开研究，对比线性动力分析方法以及非线性动力分析方法在模拟结构连续倒塌时的精度，总结建筑结构抗连续倒塌设计中的薄弱环节。于晓辉等[8]采用OpenSees建立RC框架模型，基于拆除中柱和角柱2种方案，研究悬链线机制对RC框架结构抗连续倒塌能力的影响。
目前，基于拆除构件法评估RC框架结构抗连续倒塌能力在面临构件数目较多的情况时，存在耗时长、拆柱模式多、计算量大等问题，需要优先确定结构不利拆除构件方案。结构系统的抗连续倒塌能力主要与冗余承载力[9-10]相关，而结构系统的冗余承载力与框架梁跨度、总楼层数等因素紧密相连，因此总楼层数或框架梁跨度对RC框架结构抗连续倒塌规律可能起到关键影响。
本文在已有研究的基础上，提出基于构件应变能的结构重要构件评价方法，假设构件应变能越大，该构件越重要，则拆除该构件对结构倒塌程度贡献越大，因此可依据构件应变能的大小近似确定拆除构件的顺序和方案。同时，利用OpenSees建立四跨八层平面RC框架结构模型，采用非线性动力拆除构件法验证上述评价方法的有效性，发现该方法有效性。在此基础上，考虑到在满足一般建筑使用功能条件下，常规框架结构布置形式比较固定，按照不同框架梁跨度、总楼层数设计共8个典型规则框架结构，采用非线性动力拆除构件法，分析框架梁跨度和总楼层数对平面RC框架结构抗连续倒塌能力的影响，拟合平面RC框架结构底层拆柱比例与倒塌程度的函数曲线，发现总楼层数减少和框架梁跨度减少均能提高结构抗连续倒塌能力，平面RC框架结构底层拆柱比例与倒塌程度的关系近似可用logistic函数拟合，框架梁跨度对结构倒塌程度影响较大，总楼层数对结构倒塌程度影响比较有限。
1　分析方法及数值模型
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1.1　基于构件应变能的结构重要构件评价方法
应变能是指物体在受力条件下发生形变所储存的能量，RC框架结构在重力等荷载作用下产生初始形变，结构系统中各个构件具有一定的应变能，可通过构件应变能的大小来反映其在整体结构系统中的重要性程度。已有研究表明结构构件的重要性程度可通过重要性系数[11-12]来反映，在结构构件重要性评价指标方面，根据是否考虑荷载作用可将评价指标分为两类：基于结构自身属性和基于结构性能。单独以结构自身属性评价构件重要性程度，考虑不够全面，可能出现在整个结构荷载传递机制中贡献较小的构件，其重要性程度超过对荷载传递机制贡献较大的构件这一类现象。因此，结构构件重要性程度，不仅仅与结构自身属性有关，也与作用在结构上的外荷载分布和大小相联系。
基于构件应变能的结构重要构件评价方法考虑建筑结构外荷载分布，通过结构性能来评价构件重要性程度。在拆除构件之前，将整个结构看作为初始结构，假定构件的单元刚度矩阵为K（e），结构整体刚度矩阵为K，作用在整个结构上的外荷载向量为p，则整个建筑结构的位移向量v可表达为
由式（1）便可得到结构整体位移响应，同时也可得到各个构件的单元位移向量v（e），则构件应变能U（e）可表达为
式中：上标T为矩阵或向量的转置，假设作用在构件上的外荷载向量为p（e），构件在外荷载p（e）作用下的应变能可表达为
由式（3），可计算RC框架结构在外荷载p作用下所有梁、柱构件应变能U（e）。基于前面假定：构件应变能越大，则构件越重要（即对结构倒塌程度贡献越大），此时可根据构件应变能的大小，对所有柱构件进行重要性程度排序，制定单柱不利拆除方案。
同理，还可将拆除单柱后的剩余结构再次看作为初始结构，假设剩余结构的整体刚度矩阵为K’，同时认为拆除承力构件前后外荷载向量p不发生改变，采用上述公式计算剩余结构中各个构件的应变能并排序。此时，便可根据单柱拆除方案以及剩余结构柱构件应变能排序，制定双柱不利拆除方案。依此类推，还可制定三柱、四柱等不利拆除方案。本方法对于构件数目较多的结构，能快速评价结构重要构件，制定不利拆除构件方案，在评估结构抗连续倒塌能力时减少计算量，提高评估效率。
1.2　数值模型
本文以OpenSees作为分析平台，建立平面RC框架结构有限元模型，基于非线性动力拆除构件法，瞬时拆除柱构件模拟结构初始损伤，分析剩余结构的响应。数值模型通过共旋变换考虑框架梁大变形情况下附加重力二阶效应，采用基于位移的非线性梁柱单元模拟钢筋混凝土梁、柱构件，每根框架梁划分为6个位移单元，每个位移单元设置5个积分点，实现模拟框架梁大变形阶段中压拱效应和悬链线效应。采用节点连接单元考虑大变形过程中梁柱节点与地面的接触，通过梁柱节点单元模拟节点的力学性能。约束区与无约束区混凝土材料选用Concrete02本构模型，通过MANDER等[13]提出的理论模型考虑箍筋对约束区混凝土的强化效应，钢筋材料选用Steel02本构模型。
为考虑结构连续倒塌过程中构件可能出现的二次断裂和破坏失效，分别在梁、柱构件中定义失效准则，由于梁、柱构件均采用纤维截面进行建模，纤维截面建立在混凝土和钢筋2种材料的本构关系基础上，因此采用MinMax Material监测柱构件的损伤演化过程[14]。在设定梁构件失效准则时，考虑到梁构件变形能力的限制，将其失效准则设定为：拆柱上方节点竖向位移达到与节点连接最短梁跨度的18%[15]。
对于结构抗连续倒塌能力评价指标，参考其他宏观的整体结构抗连续倒塌能力评价指标[16-17]，采用结构倒塌程度来衡量拆除构件后结构抗连续倒塌能力。本文定义结构倒塌程度β为拆除构件后结构重心下降高度与结构原始重心高度的比值，假定拆除构件后结构倒塌程度越大，则结构抗连续倒塌能力越差，倒塌程度β的表达式为
式中：mi为各个节点质量；Δi为各个节点竖向位移；为节点总质量；H原为结构原始重心高度。
1.3　模型验证
为验证数值模型的准确性，选取初明进等[18]开展的RC单向梁-板子结构抗连续倒塌试验，何庆锋等[19]完成的单层两跨平面RC框架结构试验，易伟建等[20]开展的三层四跨平面RC框架结构抗倒塌试验，采用1.2节数值模型相同的建模方法进行模拟，验证数值模型的准确性。数值模拟结果与试验数据对比见图1。由图可知，数值模拟得到的荷载-位移曲线与试验数据吻合良好，表明该模型能较好模拟梁、柱构件在大变形下的力学行为，后续将采用该数值模型开展平面RC框架结构抗连续倒塌规律分析。
2　重要构件评价方法验证
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为验证基于构件应变能结构重要构件评价方法的有效性，选择于晓辉等[21]设计的四跨八层RC框架结构为目标，建立该结构有限元模型，单元编号见图2。采用1.1节构件应变能结构重要构件评价方法制定该结构单柱和多柱不利拆除方案，结果见表1和表2，表中仅展示前4种拆柱方案。
由上述不利拆柱方案，采用非线性动力拆除构件法，分析柱构件瞬时拆除后结构动力响应，计算不同拆柱方案下结构倒塌程度。值得注意的是，基于非线性动力拆除构件法分析多柱拆除方案倒塌程度时，同时拆除该方案中所有柱构件。
构件应变能与结构倒塌程度关系见图3。由图可知，单柱和双柱拆除方案下，结构未发生大范围连续倒塌，构件应变能与结构倒塌程度呈现良好的相关性，拆柱方案对应的构件应变能越大，拆除后结构倒塌程度越高；在计算三柱和四柱拆除方案下结构倒塌程度时，结构均发生大范围连续倒塌，存在构件应变能更大的拆柱方案其倒塌程度更小的情况，由于应变能为线性指标，在结构发生大范围连续倒塌时，整体结构处在强非线性阶段，因此采用应变能线性指标制定结构多柱不利拆除方案时可能存在误差，但此时构件应变能与结构倒塌程度仍表现出一定的相关性，由本方法制定的三柱、四柱不利拆除方案的结构整体倒塌程度较为接近。整体上看，本方法在评价结构重要构件以及制定结构不利拆除构件方案上具有较高的有效性，对于构件数目较多的结构，能快速评价结构的重要构件，制定结构不利拆除构件方案。
3　抗连续倒塌规律分析
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3.1　典型框架结构设计
按照规范[22]设计典型规则RC框架结构，结构横向、纵向框架梁跨数均选择四跨，底层层高为4.5 m，其余层层高为3.3 m。RC框架结构抗震设防烈度为7度，场地类别为Ⅱ类，抗震设防类别为乙类建筑，设计分组第1组，地面粗糙度为C类，基本风压为0.4 kN/m2，梁、柱截面纵向钢筋采用HRB400，箍筋采用HPB300，混凝土材料选用C40，楼面、屋面恒荷载标准值按照钢筋混凝土容重及板厚计算得到，活荷载取2 kN/m2。
为研究结构总楼层数对平面RC框架结构抗连续倒塌规律的影响，不改变结构设计模型的梁、柱截面尺寸和配筋信息，框架梁跨度选择6 m，分别设计4层、8层和12层结构模型，梁、柱截面尺寸及配筋信息见图4。为研究框架梁跨度对平面RC框架结构抗连续倒塌规律的影响，结构总楼层数选择8层，改变框架梁跨度，分别设计4、5、6、7、8 m这5种框架梁跨度下的结构模型，梁、柱截面尺寸及配筋信息见表3。针对上述设计的不同框架梁跨度和总楼层数的典型规则RC框架结构模型，分别选取8个典型RC框架结构中一榀框架，见图5（a）。建立平面RC框架结构OpenSees有限元模型，结构平、立面布置及底层柱编号见图5。
3.2　非线性动力分析
依据上述不同总楼层数与框架梁跨度OpenSees有限元模型，采用1.1节构件应变能结构重要构件评价方法，制定8个平面RC框架结构不利拆除构件方案，结果见表4。加载方面，首先在原始结构施加静力荷载，静力荷载分析完成后，通过瞬时拆除柱构件，模拟结构初始局部破坏，分析剩余结构整体动力响应，采用结构倒塌程度β衡量结构抗连续倒塌能力。
为说明非线性动力拆除构件法的全过程，以8层（框架梁跨度6 m）平面RC框架结构为例，瞬间拆除结构底层中柱。中柱节点上方框架梁不同变形破坏阶段见图6。由图可知，框架梁先后经历压拱阶段和悬链线阶段，最后达到承载能力极限状态，发生破坏。
根据表4的方案对不同框架梁跨度和总楼层数结构开展拆柱分析，得到不同拆柱方案下结构响应。不同结构发生初始破坏时，结构冗余承载力有所差异，不同框架梁跨度结构底层中柱拆除后结构响应见图7。由图可知，在梁跨为4 m时，拆除底层中柱后，剩余结构具有足够的冗余度，失效柱所承受的重力荷载通过框架梁传递至相邻柱构件，底层框架梁轴力发展较为稳定，在内力重分布后结构未发生连续倒塌。在梁跨不小于5 m时，拆除底层中柱后梁端出现大位移，底层框架梁先后经历压拱阶段和悬链线阶段，由轴压力转变为轴拉力，剩余结构冗余承载力不足，未能找到有效的荷载传递路径，拆柱上方节点竖向位移随时间持续增加，梁跨越长，竖向位移增长越快，框架梁超出极限承载能力发生破坏，结构发生连续倒塌。不同梁跨度下底层框架梁在压拱阶段和悬链线阶段的峰值轴力也有所差异，梁跨度的增加导致作用在框架梁上线荷载增加，框架梁截面内力迅速发展，轴力峰值提高，压拱阶段和悬链线阶段承载能力提高，但拆除柱构件后梁端位移发展越快，框架梁更快进入到压拱阶段和悬链线阶段，结构倒塌程度越大。
梁跨度的不同，底层中柱拆除后，BC跨各层框架梁截面轴力发展也存在差异。连续倒塌过程中不同梁跨度结构BC跨各层框架梁截面轴力随时间发展曲线见图8。由图可知，底层中柱拆除后，结构主要由底层框架梁承担荷载，其余各层梁协同受力。梁跨度为6 m时，底层框架梁与第2层框架梁经历比较明显的轴压力转变为轴拉力的过程，顶层框架梁轴力始终保持为轴压力的受力状态，其余中间层框架梁轴力较小；梁跨度为8 m时，各层框架梁截面轴力均经历由轴压力转变为轴拉力的过程，最后发生失效，但轴力峰值较底层框架梁较小。由于结构在抵抗竖向荷载时各层框架梁存在协同受力机制，框架梁跨度的增加导致拆柱上方框架梁依次经历轴力转化过程。
不同总楼层数结构底层中柱拆除后结构响应见图9。由图可知，3种总楼层数下结构均发生连续倒塌，随楼层数增加，拆柱上方节点竖向位移发展越快，底层框架梁轴力增大，但增大幅度有限。由于平面RC框架结构拆除底层中柱后主要由底层框架梁抵抗竖向荷载，其余层框架梁协同受力，楼层数的增加表现为底层框架梁截面轴力增大，在相同梁柱截面尺寸和配筋条件下，框架梁端发展的位移越快，结构倒塌程度越大，不利于结构抵抗连续倒塌。
为得到具有一定普适性的平面RC框架结构底层拆柱比例与倒塌程度关系曲线，计算不同框架梁跨度和总楼层数结构在不同拆柱方案下的倒塌程度，结果见表5。由表可知，单柱拆除方案下，仅在框架梁跨度为4 m时，结构未发生连续倒塌；在多柱拆除方案下，8个平面框架均发生大范围连续倒塌，随着底层拆柱比例的增大，结构倒塌程度也呈现大幅度增加，框架梁跨度对结构倒塌程度影响较大，总楼层数对结构倒塌程度影响相对较小。根据表中数据，选用logistic函数，拟合底层拆柱比例与结构倒塌程度函数曲线，结果见图10。由图可知，底层拆柱比例与结构倒塌程度两者具有良好的相关性，两者近似呈logistic函数关系，底层拆柱比例n与倒塌程度β的函数关系如式（5）所示：
式中，A0、A1、A2和p为对应的函数参数。
从整体上看，平面RC框架结构在初始局部构件破坏时，结构的抗连续倒塌能力与结构冗余承载力相关，不同初始破坏位置、总楼层数和梁跨度下，结构的冗余承载力也各不相同，不同底层拆柱比例下，结构倒塌程度也存在较大的区别，且结构的总楼层数减少和跨度的减小都能增加结构的冗余承载力，最终对结构抗连续倒塌起到有利作用。
4　结论
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本文通过考虑外荷载分布对结构重要构件评价的影响，提出一种基于构件应变能的结构重要构件评价方法，以一栋四跨八层RC框架结构为研究对象，利用OpenSees有限元软件验证该方法的有效性。同时，设计8个典型规则RC框架结构，基于非线性动力拆除构件法，分析结构总楼层数、框架梁跨度对平面RC框架结构抗连续倒塌能力的影响，拟合平面RC框架结构底层拆柱比例与结构倒塌程度的近似函数曲线，得出如下结论：
1）基于构件应变能的结构重要构件评价方法在评价结构重要构件方面具有较高的有效性，能够针对结构制定单柱和多柱不利拆除构件方案，在评估结构抗连续倒塌能力时减少计算量，提高评估效率。
2）底层中柱拆除后，结构主要由底层框架梁抵抗竖向荷载，其余各层框架梁协同受力。随着框架梁跨度的增加，底层框架梁在压拱阶段和悬链线阶段轴力峰值提高，拆柱上方框架梁依次经历轴力转化过程，框架梁端位移发展加快；随着总楼层数的增加，底层框架梁轴力峰值有所增大，框架梁端位移发展加快，但增大幅度有限。
3）平面RC框架结构抗连续倒塌能力主要与结构冗余承载力相关，框架梁跨度的减小或总楼层数的减少均能提高结构冗余承载力，对结构抗连续倒塌起到积极作用。框架梁跨度对结构倒塌程度影响较大，总楼层数对结构倒塌程度影响有限，平面RC框架结构底层拆柱比例与倒塌程度近似可用logistic函数拟合。
基金
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  • 基础加强计划项目子课题(2022JCJQZD03803)
文献
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2025年第45卷第4期
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doi: 10.13197/j.eeed.2025.0414
  • 接收时间：2024-03-05
  • 首发时间：2026-03-20
  • 出版时间：2025-08-22
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  • 收稿日期：2024-03-05
  • 修回日期：2024-05-09
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基础加强计划项目子课题(2022JCJQZD03803)
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    武汉理工大学 土木工程与建筑学院，湖北 武汉 430070

通讯作者:

王贞（1983—），男，研究员，博士，主要从事结构抗震试验方法研究。E-mail：
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2种不同金属材料的力学参数
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Family		 属数
Number of
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Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
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鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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