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Article(id=1228654099797836520, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228654089437901468, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.12.002, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1671120000000, receivedDateStr=2022-12-16, revisedDate=1675699200000, revisedDateStr=2023-02-07, acceptedDate=null, acceptedDateStr=null, onlineDate=1770863564681, onlineDateStr=2026-02-12, pubDate=1735315200000, pubDateStr=2024-12-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770863564681, onlineIssueDateStr=2026-02-12, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770863564681, creator=13701087609, updateTime=1770863564681, updator=13701087609, issue=Issue{id=1228654089437901468, tenantId=1146029695717560320, journalId=1225147924628267009, year='2024', volume='37', issue='12', pageStart='1993', pageEnd='2167', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770863562211, creator=13701087609, updateTime=1770863940325, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1228655675413299456, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228654089437901468, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1228655675413299457, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228654089437901468, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=2003, endPage=2011, ext={EN=ArticleExt(id=1228654101051933437, articleId=1228654099797836520, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Analytical solution for horizontal seismic response of shaft structure primary lining and secondary lining, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Based on the elastic foundation beam theory，the mechanical model for the seismic response of the vertical shaft is established. The shaft primary lining and secondary lining are simplified as Euler Bernoulli beams and the normal interaction between the primary lining and secondary lining is simulated by using uniformly distributed springs. The differential governing equations of the shaft under the horizontal seismic excitation are derived. The rapid solution of the horizontal seismic response of the primary lining and secondary lining is achieved through the distributed transfer function method and the correctness of the analytical solution is verified by comparison with the finite element numerical simulation method. The peak seismic response at the top of the shaft is parametrically analyzed from the perspectives of elastic foundation stiffness，secondary lining stiffness，shaft outer diameter，and stiffness of the elastic connection layer between the primary and secondary linings，respectively. The results show that the peak seismic response at the top of the secondary lining is greater than that of the primary lining. With the increase of the elastic foundation stiffness，the peak response of the top of the primary lining and the secondary lining decreases. The increase of secondary lining stiffness leads to the obvious increase of peak response at the top of secondary lining. With the increase of the shaft outer diameter，the peak response of both the primary and secondary linings increase significantly. When the elastic connection layer stiffness increases，the peak response of the primary lining increases slightly，but the peak response of the secondary lining decreases significantly.

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基于弹性地基梁理论建立了考虑竖井初衬与二次衬砌相互作用的地震响应力学模型，将竖井初衬和二次衬砌简化为欧拉-伯努利梁，采用均布弹簧模拟竖井初衬与二次衬砌的法向相互作用，推导了水平地震作用下的竖井振动微分控制方程。通过分布传递函数法实现了竖井初衬和二次衬砌水平地震响应的快速求解，并通过与有限元模拟数值解进行对比，验证了本文解析解的正确性。应用该解析解表达式分别从地基刚度、二次衬砌刚度、竖井外径和初衬与二次衬砌之间连接层刚度的角度出发，对竖井顶部峰值地震响应开展参数化分析，结果发现二次衬砌顶部的峰值地震响应大于初衬顶部，随着地基刚度的增大，初衬与二次衬砌顶部峰值响应均减小；二次衬砌刚度增加会导致二次衬砌顶部峰值响应明显增大；随着竖井外径的增大，初衬和二次衬砌顶部峰值响应均会明显增大；当弹性连接层刚度增大时，初衬顶部峰值响应会略微增大，而二次衬砌顶部峰值响应会显著减小。

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杜修力（1962―），男，博士，教授。电话： （010）67391652； E-mail： 。
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张卜（1990—），男，博士，助理研究员。E-mail： 。

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张卜（1990—），男，博士，助理研究员。E-mail： 。

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张卜（1990—），男，博士，助理研究员。E-mail： 。

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tableContent=null), ArticleFig(id=1228654125492142805, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228654099797836520, language=CN, label=图9, caption=衬砌连接层刚度对竖井顶部峰值响应的影响, figureFileSmall=b1UGzJWRCCFMEyJ5mCIXlA==, figureFileBig=T6YzTyrGDT4+x9rZwIPIcw==, tableContent=null), ArticleFig(id=1228654125584417496, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228654099797836520, language=EN, label=Tab.1, caption=

Maximum value comparison between analytical solution and numerical solution

, figureFileSmall=null, figureFileBig=null, tableContent=
地震响应位置解析解最大值数值解最大值误差/%
位移响应/m初衬0.0799550.0799550
二次衬砌0.0664680.0664470.03
转角响应/rad初衬0.0014350.0014320.16
二次衬砌0.0020460.0020390.34
弯矩响应/(kN∙m)初衬172243.7173022.20.45
二次衬砌112907.3114261.41.19
), ArticleFig(id=1228654125668303579, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228654099797836520, language=CN, label=表1, caption=

解析解和数值解最大值对比

, figureFileSmall=null, figureFileBig=null, tableContent=
地震响应位置解析解最大值数值解最大值误差/%
位移响应/m初衬0.0799550.0799550
二次衬砌0.0664680.0664470.03
转角响应/rad初衬0.0014350.0014320.16
二次衬砌0.0020460.0020390.34
弯矩响应/(kN∙m)初衬172243.7173022.20.45
二次衬砌112907.3114261.41.19
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竖井结构初衬及二次衬砌水平地震响应解析解
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张卜 1, 2 , 卢立东 1 , 钟紫蓝 1, 2 , 姬若愚 1 , 杜修力 1, 2
振动工程学报 | 2024,37(12): 2003-2011
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振动工程学报 | 2024, 37(12): 2003-2011
竖井结构初衬及二次衬砌水平地震响应解析解
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张卜1, 2 , 卢立东1, 钟紫蓝1, 2, 姬若愚1, 杜修力1, 2
作者信息
  • 1北京工业大学城市建设学部，北京　100124
  • 2北京工业大学城市与工程安全减灾教育部重点实验室，北京　100124

    • 张卜（1990—），男，博士，助理研究员。E-mail： 。

通讯作者:

杜修力（1962―），男，博士，教授。电话： （010）67391652； E-mail： 。
Analytical solution for horizontal seismic response of shaft structure primary lining and secondary lining
Bu ZAHNG1, 2 , Li-dong LU1, Zi-lan ZHONG1, 2, Ruo-yu JI1, Xiu-li DU1, 2
Affiliations
  • 1Faculty of Architecture，Civil and Transportation Engineering，Beijing University of Technology，Beijing 100124，China
  • 2Key Laboratory of Urban Security and Disaster Engineering，Ministry of Education，Beijing University of Technology，Beijing 100124，China
出版时间: 2024-12-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.12.002
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摘要
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基于弹性地基梁理论建立了考虑竖井初衬与二次衬砌相互作用的地震响应力学模型，将竖井初衬和二次衬砌简化为欧拉-伯努利梁，采用均布弹簧模拟竖井初衬与二次衬砌的法向相互作用，推导了水平地震作用下的竖井振动微分控制方程。通过分布传递函数法实现了竖井初衬和二次衬砌水平地震响应的快速求解，并通过与有限元模拟数值解进行对比，验证了本文解析解的正确性。应用该解析解表达式分别从地基刚度、二次衬砌刚度、竖井外径和初衬与二次衬砌之间连接层刚度的角度出发，对竖井顶部峰值地震响应开展参数化分析，结果发现二次衬砌顶部的峰值地震响应大于初衬顶部，随着地基刚度的增大，初衬与二次衬砌顶部峰值响应均减小；二次衬砌刚度增加会导致二次衬砌顶部峰值响应明显增大；随着竖井外径的增大，初衬和二次衬砌顶部峰值响应均会明显增大；当弹性连接层刚度增大时，初衬顶部峰值响应会略微增大，而二次衬砌顶部峰值响应会显著减小。

地下结构抗震  /  竖井地震响应  /  竖井衬砌  /  弹性地基梁

Based on the elastic foundation beam theory，the mechanical model for the seismic response of the vertical shaft is established. The shaft primary lining and secondary lining are simplified as Euler Bernoulli beams and the normal interaction between the primary lining and secondary lining is simulated by using uniformly distributed springs. The differential governing equations of the shaft under the horizontal seismic excitation are derived. The rapid solution of the horizontal seismic response of the primary lining and secondary lining is achieved through the distributed transfer function method and the correctness of the analytical solution is verified by comparison with the finite element numerical simulation method. The peak seismic response at the top of the shaft is parametrically analyzed from the perspectives of elastic foundation stiffness，secondary lining stiffness，shaft outer diameter，and stiffness of the elastic connection layer between the primary and secondary linings，respectively. The results show that the peak seismic response at the top of the secondary lining is greater than that of the primary lining. With the increase of the elastic foundation stiffness，the peak response of the top of the primary lining and the secondary lining decreases. The increase of secondary lining stiffness leads to the obvious increase of peak response at the top of secondary lining. With the increase of the shaft outer diameter，the peak response of both the primary and secondary linings increase significantly. When the elastic connection layer stiffness increases，the peak response of the primary lining increases slightly，but the peak response of the secondary lining decreases significantly.

seismic resistance of underground structures  /  shaft seismic response  /  shaft primary lining  /  elastic foundation beam
张卜, 卢立东, 钟紫蓝, 姬若愚, 杜修力. 竖井结构初衬及二次衬砌水平地震响应解析解. 振动工程学报, 2024 , 37 (12) : 2003 -2011 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.12.002
Bu ZAHNG, Li-dong LU, Zi-lan ZHONG, Ruo-yu JI, Xiu-li DU. Analytical solution for horizontal seismic response of shaft structure primary lining and secondary lining[J]. Journal of Vibration Engineering, 2024 , 37 (12) : 2003 -2011 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.12.002
正文
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引言
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竖井结构作为铅直型地下结构，是连接地表和地下空间的重要“通道”，作为地下工程建设的首要建设对象和核心构筑物，被广泛应用于城市大深度地下空间开发、能源资源开采、隧道建设、水电站开发、大科学实验场和国防建设等领域［1-4］。竖井结构地震安全对保障能源资源开发、深层地下空间安全利用和深地工程等国家重大战略任务具有重要意义。
近年来，已有震害案例表明地下结构在地震中受到不同程度的损坏［5-9］，其中，唐山地震和日本阪神地震均有竖井结构地震破坏案例［10-11］。鉴于竖井结构和功能的重要性，建立竖井地震响应分析模型，实现竖井结构地震响应分析并进行初步地震响应计算，对于指导竖井结构抗震分析具有重要意义。
已有研究采用数值模拟方法开展了竖井与隧道结构地震响应研究［12-17］。由于数值计算方法建模工作量较大，且计算效率受计算平台限制，计算精度受参数取值等影响，相比于数值模拟方法，解析方法可以高效求解结构变形与内力，并从力学角度给出问题的直观解释，指导工程实践［18-20］。
近年来，诸多学者基于弹性地基梁理论或刚体理论开展了竖井结构地震响应解析方法研究，以快速求解和初步估计竖井结构地震动力响应。ZHANG等［21］基于文克勒弹性地基梁理论推导了竖井地震响应解析解，研究发现竖井地震动力响应主要由场地地基土层的变形控制。此外，在研究较深竖井的地震反应特性时，ZHANG等［22］基于拟静力法推导了名义柔度比公式，用于评估竖井的动力响应特征。在竖井与隧道的连接处，ZHANG等［23］提出横向荷载作用下竖井的动力响应解析解，分别针对地表横向荷载和横向地震激励两种情形与数值解进行了对比验证。此外，ZHANG等［24-25］推导了纵向地震激励和SH波激励下竖井-隧道连接处的动力响应解析解，分析发现在纵向激励下，隧道的约束作用使得竖井位移响应变小。志波由紀夫［26］将地下竖井结构简化为3D梁模型，并且推导出了简化抗震计算方法。
已有学者同样基于弹性地基梁理论开展了隧道结构地震响应分析研究。ST JOHN等［27］基于文克勒地基梁模型评估了地震作用下长隧道产生的内力，将自由场变形直接施加在隧道上，获得了隧道断面的最大轴力、剪力和弯矩的闭合表达式。为了考虑剪切变形效应对隧道响应的影响，SÁNCHEZ-MERINO等［28］将隧道衬砌简化为文克勒地基上的铁摩辛柯梁，推导了隧道任意截面的横向位移、弯矩和剪力表达式。YU等［29］使用弹性地基梁理论评估了突变刚度隧道的纵向地震响应。
上述研究大多将竖井或隧道简化为弹性地基上的单根地基梁以研究其地震响应，然而在工程实际中，许多结构由初衬和二次衬砌组成，往往将二次衬砌作为安全储备，忽略初衬与二次衬砌之间的动力相互作用可能会高估其地震响应。此外，竖井的二次衬砌作为重要结构部件，在地震中可能遭受破坏，其动力响应直接影响内部设施的正常运行，因此，推导建立考虑竖井初衬和二次衬砌相互作用的地震动力响应解析解十分必要。
本文首先基于分布传递函数法基本理论［30］，采用广义位移法的基本思想［31］，即竖井的地震响应受到地基地层响应控制，将竖井-土动力模型简化为由初衬、二次衬砌和文克勒地基组成的双层梁振动系统，建立了竖井双层梁-弹簧模型，推导了水平向地震作用下初衬和二次衬砌的动力响应解析解。其次将解析解与有限元法数值解进行对比验证。最后从场地地基刚度、二次衬砌刚度、竖井外径和初衬与二次衬砌之间连接层刚度的角度出发，分别对初衬和二次衬砌顶部位移和转角峰值地震响应进行参数化分析。
1　竖井模型与假定
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图1所示为地震作用下竖井-场地相互作用模型示意图，场地深度为L，竖井底部嵌入刚性基岩，地震作用下的自由场变形使得初衬与二次衬砌产生水平地震响应。为快速求解竖井地震响应解析解，基于文克勒弹性地基梁理论，建立了如图2所示的竖井-场地相互作用梁-弹簧简化力学模型。通过将地震波垂直入射引起的自由场位移施加于地基弹簧远离结构的一侧，使得初衬与二次衬砌组成的双层梁系统产生地震响应。其中，场地土层简化为黏弹性地基弹簧和阻尼，地基刚度为Ke，地基阻尼系数为Cs；同时，将竖井初衬及二次衬砌简化为两根平行的欧拉-伯努利梁，并用沿竖井深度线性分布的水平向连接弹簧来替代初衬与二次衬砌相互作用，弹簧刚度为K，忽略其切向作用。初衬水平位移响应为w1（z，t），二次衬砌水平位移响应为w2（z，t）。
模型假定如下：①竖井初衬、二次衬砌和场地土层为各向同性且均质的线弹性体；②竖井初衬与二次衬砌顶部为自由端，底部嵌入基岩为固定端；③地震激励为垂直入射的S波，在地震作用下场地土层产生自由场响应，竖井初衬和二次衬砌在自由场变形作用下产生水平位移响应。
2　解析解推导
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2.1　运动控制方程
根据经典欧拉-伯努利梁理论，得到竖井初衬与二次衬砌耦合的微分控制方程：
式中　 为初衬弹性模量；为二次衬砌弹性模量；为初衬截面惯性矩；为二次衬砌截面惯性矩；和分别为初衬和二次衬砌等效线密度；为自由场位移。
将式（1）中自由场位移项移到等式右端，等效为施加在初衬上的荷载，变形后的控制方程为：
式中　 为施加于竖井初衬的等效荷载。
2.2　地震响应解析解
将式（2）进行初始条件为0的拉普拉斯变换，关于时间t的函数可以变换为关于复数参数s的函数，得到下式：
其中，变量上方的横线表示该变量关于时间t的拉普拉斯变换形式，将式（3）写为状态空间表达式：
式中　 为一个由控制方程参数组成的8×8矩阵。，和可以分别表示为：
根据竖井模型边界条件，初衬与二次衬砌顶部为自由端，底部为固定端，边界条件可以表示为：
将上述边界条件进行拉普拉斯变换得到矩阵形式：
其中，矩阵和分别为：
根据LIU等［30］的分布传递函数法公式，在s域内控制方程式（3）的解表示为：
其中，的形式为：
其中，
为状态转移矩阵，其形式为：
其中，
自由场位移可以表示为：
式中　 为场地自由场频域位移响应幅值，式（17）可以进一步表示为［32］：
式中　 为土层表面的简谐位移幅值；ω为圆频率；为竖井地基土层的剪切波速，可以根据下式求得：
式中　 ，，和分别为土层的剪切模量、泊松比、弹性模量和密度。
在考虑土层的阻尼比时，剪切波速可以表示为。将式（18）代入荷载向量中并把s替换为，通过式（12）可以得到解析解向量为：
其中，的表达式为：
求解竖井地震响应过程中，场地土层频域自由场位移响应可以通过将场地顶部地震动位移时程经快速傅里叶变换（FFT）后由式（18）求得，通过式（21）获得竖井频域解析解，经快速逆傅里叶变换（iFFT）即可求得竖井地震响应时域解，即初衬和二次衬砌水平位移、转角、弯矩和剪力，其中，弯矩和剪力的表达式为：
式中　 M为弯矩；Q为剪力；下标i=1，2分别表示初衬和二次衬砌。
3　数值验证
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本文使用大型通用有限元软件ABAQUS验证竖井地震响应解析解的准确性。通过建立二维欧拉-伯努利梁-弹簧模型模拟竖井与场地土层动力系统，采用直接稳态分析方法进行竖井地震响应数值解求解。
3.1　数值建模
验证工况参数如下：竖井深度60 m，初衬和二次衬砌的弹性模量，密度，竖井的外径d=8 m，厚度为0.3 m，二次衬砌的厚度为0.4 m。竖井地基土层密度，阻尼比，泊松比，弹性模量。根据MAKRIS等［32］的场地土层参数取值方法，地基刚度取值为，所以，地基阻尼系数取值为，其中。在实际工程中，初衬与二次衬砌之间的弹性连接层由防水层、隔震层等多部分构造组成，假设弹性连接层刚度取值为。
ABAQUS竖井建模分析步骤为：首先，分别建立二维欧拉-伯努利梁用于模拟竖井初衬和二次衬砌，使用梁单元B33进行模拟，网格尺寸为1 m，初衬与二次衬砌之间节点采用线性弹簧连接，实现初衬与二次衬砌的相互作用；其次，设置初衬与二次衬砌顶部为自由边界，底部为固定边界；再次，通过在初衬的每个单元节点上建立地基弹簧，将式（18）计算得到的场地自由场位移幅值施加在地基弹簧远离结构的一侧，采用直接稳态法进行频域地震响应求解。最后，将数值模拟结果经iFFT转换为时域解，再与解析解进行对比验证。
3.2　解析解与数值解对比
本文对z=30 m处初衬和二次衬砌的位移、转角和弯矩响应进行验证，分别绘制其在地震作用下的时程响应曲线。地震波选取Kobe波，其位移时程曲线及位移傅里叶幅值谱如图3所示。
在地震动作用下，z=30 m处初衬和二次衬砌的位移、转角和弯矩时程响应曲线对比验证如图4所示。对比本文解析解与数值解可以看出，分析结果较吻合。
为了进一步验证解析解的合理性，分别取初衬和二次衬砌的位移、转角和弯矩最大值进行对比，解析解和有限元法数值解的对比结果如表1所示，其中，误差定义为（解析解最大值/数值解最大值-1）×100%。结果表明，本文解析解与数值解的误差可以控制在2%以内，进一步更加验证了本文解析解的正确性。
4　参数化分析
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在验证了解析方法准确性的基础上，本文从地基刚度、二次衬砌刚度、竖井外径和初衬与二次衬砌连接层刚度四个角度开展了参数分析，以研究其对竖井初衬和二次衬砌地震响应的影响，分别选取Kobe波与ChiChi波作为输入地震激励，调整二者峰值加速度均为0.3447g，计算竖井顶部峰值位移响应Wp与峰值转角响应Rp，其中Kobe波和ChiChi波的位移时程曲线及傅里叶幅值谱分别如图3和5所示。
4.1　地基刚度影响
图6从改变地基刚度的角度研究了竖井顶部位移和转角峰值响应，分别取地基刚度为Ke=96，144，192，240和288 MPa，其余参数与验证工况保持一致。
从图6中可以看出，两种地震动下二次衬砌位移和转角峰值响应比初衬响应大，这是由于竖井初衬既受到场地土层弹簧作用，又受到二次衬砌作用，使得初衬地震响应较小，后文的参数分析部分可以得出相同的结论。此外，当地基刚度增大时，初衬与二次衬砌顶部位移和转角峰值响应均在减小，同时，初衬与二次衬砌的峰值位移差值在逐渐增大。由图6（a）可得，当地基刚度由96 MPa增大至288 MPa时，Kobe波激励下二次衬砌的位移峰值响应降低约4%，而初衬位移峰值响应降低约10%，初衬与二次衬砌位移差值近似扩大为之前的两倍，在ChiChi波中峰值位移差值也有增大趋势。竖井顶部位移和转角峰值响应均减小是由于场地地基刚度增大，使得场地自由场变形减小，进而导致竖井结构地震响应减小。初衬地震响应降低较明显是由于初衬直接受场地地基影响，相比较而言，地基刚度的变化对竖井初衬顶部峰值响应影响较大。
4.2　二次衬砌刚度影响
为了研究二次衬砌刚度对竖井顶部地震响应的影响，本文通过改变二次衬砌的弹性模量进行参数化分析，考虑二次衬砌选用C25，C30，C40，C50和C60五种等级混凝土的情况，分别取它们对应的弹性模量E2=28，30，32.5，34.5和36 GPa进行计算，其余参数与验证工况保持一致。
图7为竖井在不同二次衬砌刚度下初衬和二次衬砌的顶部位移和转角峰值响应。从图7中可以看出，在其他条件不变的情况下，两种地震动作用下初衬和二次衬砌的位移和转角峰值响应会随二次衬砌刚度的增大略微线性增大。其中初衬峰值响应几乎无明显变化，二次衬砌响应变化较为明显，特别是转角峰值响应变化较大。弹性模量由28 GPa增大至36 GPa时，ChiChi波作用下二次衬砌位移峰值响应大约增大2%，而转角峰值响应大约增大20%。这可能是由于竖井整体刚度变大，使得场地施加的土层荷载增加，引起转角增大。此现象表明，对于竖井顶部地震响应，当二次衬砌刚度较大时，应重点关注二次衬砌在地震作用下的峰值响应，使其满足抗震设计要求。
4.3　竖井外径影响
图8所示为竖井外径对初衬和二次衬砌顶部位移和转角峰值响应的影响，分别取竖井外径d=4，6，8，10和12 m进行计算，其余参数同验证工况保持一致。
从图8中可以得出，随着竖井外径的增大，两种地震激励下初衬和二次衬砌顶部位移和转角峰值响应均有增大趋势，其中二次衬砌增大速率较快。在ChiChi波地震作用下，当竖井外径由4 m增大至10 m时，初衬位移峰值响应增大约0.02 m，二次衬砌增大约0.06 m，约为初衬响应增大值的3倍，在Kobe波激励下呈现相似的规律。但是在10~12 m范围内二次衬砌的位移峰值响应略微下降，这是由于本文解析解基于欧拉-伯努利梁模型建立，忽略了竖井地震响应过程中的剪切变形。此外，本模型没有考虑初衬与场地土层之间的切向剪应力，使得竖井二次衬砌位移峰值响应出现下降的趋势。结果表明，竖井外径对于初衬与二次衬砌峰值响应均有较大影响，在工程实践中应予以关注。
4.4　初衬与二次衬砌连接层刚度影响
图9所示为初衬与二次衬砌之间连接层刚度对竖井顶部峰值响应的影响情况。在这里分别取弹性连接层刚度为K=0.2×105，0.6×105，1.0×105，1.4×105和1.8×105 kN/m进行计算，其余参数保持不变。由图9可得，随着连接层刚度的增大，两种地震波激励下初衬顶部峰值位移和转角响应均呈现增大趋势，而二次衬砌顶部峰值位移和转角响应逐渐减小，原因可能是衬砌之间连接层刚度增大使得二次衬砌地震响应减小。特别值得注意的是，当衬砌连接层刚度较低时，二次衬砌的位移和转角峰值响应均较大，其中ChiChi波地震作用下位移峰值响应达到将近0.24 m，Kobe波地震作用下达到0.12 m，转角峰值响应分别达到0.004和0.002 rad，应被重点关注。因此，在实际工程中，当初衬与二次衬砌之间材料刚度较小时，应该重点关注二次衬砌峰值地震响应，防止其产生较大的位移和转角响应。
5　结论
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本文聚焦于考虑竖井结构初衬与二次衬砌相互作用的竖井地震响应分析，建立了场地-竖井结构相互作用的梁-弹簧模型，场地地基简化为文克勒地基，竖井结构初衬与二次衬砌简化为欧拉-伯努利梁，通过分布传递函数法推导了竖井初衬和二次衬砌水平地震响应解析解，并从地基刚度、二次衬砌刚度、竖井外径和初衬与二次衬砌连接层刚度的角度分别对初衬和二次衬砌顶部位移和转角峰值响应进行了参数化分析，主要结论如下：
（1）通过与有限元模拟数值解进行对比验证，结果表明解析解与有限元模拟数值解基本一致，验证了解析解的准确性和合理性。
（2）二次衬砌顶部的峰值响应一般大于初衬，地基刚度对初衬与二次衬砌顶部峰值响应均有明显影响，随着地基刚度的增大，初衬与二次衬砌峰值响应均会减小。
（3）随着二次衬砌刚度的增大，初衬和二次衬砌顶部峰值响应均会增大，其中二次衬砌增大较为显著。
（4）随着竖井外径的增大，初衬和二次衬砌顶部峰值响应均会明显增大，其中二次衬砌增长速率较快。
（5）当弹性连接层刚度增大时，初衬顶部峰值响应会略微增大，而二次衬砌顶部峰值响应会显著减小。
本文推导的竖井地震响应解析解和参数化分析结论可为竖井抗震设计提供一定理论参考。
基金
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  • 国家自然科学基金联合基金资助项目(U1839201)
  • 中国博士后科学基金资助项目(2021M700311)
  • 朝阳区博士后工作经费资助项目
  • 北京市博士后国际交流培养资助派出项目(2022-PC-03)
文献
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参考文献 引证文献
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2024年第37卷第12期
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doi: 10.16385/j.cnki.issn.1004-4523.2024.12.002
  • 接收时间：2022-12-16
  • 首发时间：2026-02-12
  • 出版时间：2024-12-28
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  • 收稿日期：2022-12-16
  • 修回日期：2023-02-07
基金
国家自然科学基金联合基金资助项目(U1839201)
中国博士后科学基金资助项目(2021M700311)
朝阳区博士后工作经费资助项目
北京市博士后国际交流培养资助派出项目(2022-PC-03)
作者信息
    1北京工业大学城市建设学部，北京　100124
    2北京工业大学城市与工程安全减灾教育部重点实验室，北京　100124

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杜修力（1962―），男，博士，教授。电话： （010）67391652； E-mail： 。
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
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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