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Article(id=1243226201016353036, tenantId=1146029695717560320, journalId=1242798230522609684, issueId=1243226190786441246, articleNumber=null, orderNo=null, doi=10.7511/jslx20240614001, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1718294400000, receivedDateStr=2024-06-14, revisedDate=1719849600000, revisedDateStr=2024-07-02, acceptedDate=null, acceptedDateStr=null, onlineDate=1774337824348, onlineDateStr=2026-03-24, pubDate=1761580800000, pubDateStr=2025-10-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774337824348, onlineIssueDateStr=2026-03-24, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774337824348, creator=13701087609, updateTime=1774337824348, updator=13701087609, issue=Issue{id=1243226190786441246, tenantId=1146029695717560320, journalId=1242798230522609684, year='2025', volume='42', issue='5', pageStart='699', pageEnd='888', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=1, specialIssue=null, createTime=1774337821909, creator=13701087609, updateTime=1774338282025, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1243228120724128564, tenantId=1146029695717560320, journalId=1242798230522609684, issueId=1243226190786441246, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1243228120724128565, tenantId=1146029695717560320, journalId=1242798230522609684, issueId=1243226190786441246, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=819, endPage=824, ext={EN=ArticleExt(id=1243226201892962591, articleId=1243226201016353036, tenantId=1146029695717560320, journalId=1242798230522609684, language=EN, title=Interface stiffness transfer method for elastic multi-layered structures, columnId=1243226193193971746, journalTitle=Chinese Journal of Computational Mechanics, columnName=Research Papers, runingTitle=null, highlight=null, articleAbstract=

This paper presents an analytical method, namely interface stiffness transfer method, for evaluating the responses of multilayered elastic structures. Based on the Love function and general solutions, the stiffness matrix relationship of the displacement-stress state vectors is introduced to obtain the interface stiffness transfer matrix equation between adjacent layers, which satisfies an algebraic Riccati matrix equation. When the elastic layer is a half-space, an explicit solution is obtained directly for the interface stiffness matrix. The interface stiffness transfer matrix method starts from the bottom layer with a known stiffness, and then deals with one layer at a time until the uppermost layer is reached, obtaining the interface stiffness of the multilayered structure. Finally, by solving the symmetric equilibrium equations of the boundary conditions, the displacement-stress state vector of an arbitrary layer is obtained. This method keeps the advantages of the classical transfer matrix method, but naturally excludes its exponential growth terms. In particular, the proposed method is a powerful candidate for efficiently solving the algebraic Riccati equation for the optimal control problems. Numerical examples show the properties of the interface stiffness transfer method.

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针对弹性层状结构问题，本文提出一种界面刚度传递求解新方法。基于弹性层的Love通解，引入状态向量间刚度矩阵，得到弹性层界面刚度传递矩阵方程，即一种Riccati矩阵方程，以及半无限弹性层解析的界面刚度矩阵。界面刚度传递矩阵法通过分解传递矩阵将半无限底层界面刚度自下而上传递至顶层，再根据顶层表面边界条件建立与求解边界面对称的刚度平衡方程。本文方法保留了经典传递矩阵方法的优点，自然克服了指数增长计算问题，特别还为最优控制问题的Riccati方程提供了一种新解法。数值算例验证了界面刚度传递法。

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刘天云*（1968-），男，博士，研究员（E-mail：）.

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刘天云*（1968-），男，博士，研究员（E-mail：）.

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刘天云*（1968-），男，博士，研究员（E-mail：）.

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Applied Mathematical Modelling, 2019, 67: 22-37., articleTitle=Assessing the mechanical responses for anisotropic multi-layered medium under harmonic moving load by Spectral Element Method (SEM), refAbstract=null), Reference(id=1243226254573420736, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, doi=null, pmid=null, pmcid=null, year=2023, volume=40, issue=6, pageStart=1000, pageEnd=1007, url=null, language=null, rfNumber=[17], rfOrder=22, authorNames=杨博, 汤跃文, 夏江海, journalName=计算力学学报, refType=null, unstructuredReference=杨博, 汤跃文, 夏江海, 等. 基于谱元法考虑层间接触状态的Rayleigh波频散特征正演计算[J]. 计算力学学报, 2023, 40(6): 1000-1007., articleTitle=基于谱元法考虑层间接触状态的Rayleigh波频散特征正演计算, refAbstract=null), Reference(id=1243226254690861250, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, doi=null, pmid=null, pmcid=null, year=2023, volume=40, issue=6, pageStart=1000, pageEnd=1007, url=null, language=null, rfNumber=[17], rfOrder=23, authorNames=YANG Bo, TANG Yuewen, XIA Jianghai, journalName=Chinese Journal of Computational Mechanics, refType=null, unstructuredReference=(YANG Bo, TANG Yuewen, XIA Jianghai, et al. Forward calculation of Rayleigh waves dispersion characteristics considering interlayer contact state based on spectral element method[J]. Chinese Journal of Computational Mechanics, 2023, 40(6): 1000-1007. (in Chinese)), articleTitle=Forward calculation of Rayleigh waves dispersion characteristics considering interlayer contact state based on spectral element method, refAbstract=null), Reference(id=1243226254829273282, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, doi=null, pmid=null, pmcid=null, year=2015, volume=70, issue=null, pageStart=48, pageEnd=59, url=null, language=null, rfNumber=[18], rfOrder=24, authorNames=Lin G, Han Z J, Li J B, journalName=Soil Dynamics and Earthquake Engineering, refType=null, unstructuredReference=Lin G, Han Z J, Li J B. General formulation and solution procedure for harmonic response of rigid foundation on isotropic as well as anisotropic multilayered half-space[J]. 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Computational Structural Mechanics and Optimal Control[M]. Dalian: Dalian University of Technology Press, 1993. (in Chinese)), articleTitle=null, refAbstract=null), Reference(id=1243226255257092300, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, doi=null, pmid=null, pmcid=null, year=1996, volume=124, issue=3, pageStart=869, pageEnd=887, url=null, language=null, rfNumber=[21], rfOrder=28, authorNames=Buchen P W, BenHador R, journalName=Geophysical Journal International, refType=null, unstructuredReference=Buchen P W, BenHador R. Free-mode surface-wave computations[J]. Geophysical Journal International, 1996, 124(3): 869-887., articleTitle=Free-mode surface-wave computations, refAbstract=null), Reference(id=1243226255382921419, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, doi=null, pmid=null, pmcid=null, year=2021, volume=null, issue=null, pageStart=null, pageEnd=null, url=null, language=null, rfNumber=[22], rfOrder=29, authorNames=黄晓明, 马涛, journalName=路面设计原理与方法, refType=null, unstructuredReference=黄晓明, 马涛. 《路面设计原理与方法》[M]. 北京: 人民交通出版社, 2021., articleTitle=null, refAbstract=null), Reference(id=1243226255504556237, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, doi=null, pmid=null, pmcid=null, year=2021, volume=null, issue=null, pageStart=null, pageEnd=null, url=null, language=null, rfNumber=[22], rfOrder=30, authorNames=HUANG Xiaoming, MA Tao, journalName=Principles and Methods of Pavement Design, refType=null, unstructuredReference=(HUANG Xiaoming, MA Tao. Principles and Methods of Pavement Design[M]. Beijing: People's Transportation Press, 2021. (in Chinese)), articleTitle=null, refAbstract=null)], funds=[Fund(id=1243226247170474091, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, awardId=52378350, language=CN, fundingSource=国家自然科学基金(52378350), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1243226235980071859, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, xref=null, ext=[AuthorCompanyExt(id=1243226235984266164, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, companyId=1243226235980071859, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China), AuthorCompanyExt(id=1243226235992654773, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, companyId=1243226235980071859, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=清华大学　水利水电工程系，北京　100084)])], figs=[ArticleFig(id=1243226242506407967, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, language=EN, label=Fig. 1, caption=Pavement structure and coordinate system situation, figureFileSmall=PmWnb0A0vsRbTlNT/UndzA==, figureFileBig=U7xHuUOT2+ubrqqTB4lJ8Q==, tableContent=null), ArticleFig(id=1243226242669985826, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, language=CN, label=图1, caption=弹性层及圆柱坐标系, figureFileSmall=PmWnb0A0vsRbTlNT/UndzA==, figureFileBig=U7xHuUOT2+ubrqqTB4lJ8Q==, tableContent=null), ArticleFig(id=1243226243219439663, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, language=EN, label=Fig. 2, caption=Variation of displacement and stress with depth, figureFileSmall=MkD99up70i0WKoyt2g5tAQ==, figureFileBig=A+SYcEEFevFmKrsolhqNzw==, tableContent=null), ArticleFig(id=1243226243508846645, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, language=CN, label=图2, caption=位移和应力随深度的变化, figureFileSmall=MkD99up70i0WKoyt2g5tAQ==, figureFileBig=A+SYcEEFevFmKrsolhqNzw==, tableContent=null), ArticleFig(id=1243226243760504890, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, language=EN, label=Fig. 3, caption=Variation of displacement and stress with radial distance, figureFileSmall=kpdmboft8335Gqf7exXocQ==, figureFileBig=QyhH58W7hDr7Pqg68rcDUQ==, tableContent=null), ArticleFig(id=1243226244163158079, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, language=CN, label=图3, caption=位移和应力随径向距离的变化, figureFileSmall=kpdmboft8335Gqf7exXocQ==, figureFileBig=QyhH58W7hDr7Pqg68rcDUQ==, tableContent=null), ArticleFig(id=1243226244603560006, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, language=EN, label=Tab. 1, caption=

Load cases

, figureFileSmall=null, figureFileBig=null, tableContent=
荷载单位压力/MPa荷载半径/cm位置r/cm位置z/cm
10.700010.65000.00000.0000
20.700010.650031.95000.0000
), ArticleFig(id=1243226245010407497, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, language=CN, label=表1, caption=

荷载工况

, figureFileSmall=null, figureFileBig=null, tableContent=
荷载单位压力/MPa荷载半径/cm位置r/cm位置z/cm
10.700010.65000.00000.0000
20.700010.650031.95000.0000
), ArticleFig(id=1243226245358534737, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, language=EN, label=Tab. 2, caption=

Surface displacements and stresses

, figureFileSmall=null, figureFileBig=null, tableContent=
 径向距离r/cm剪应力τrz/MPa竖向应力σr/MPa轴向应力σr/MPa切向应力σθ/MPa横向位移u/m竖向位移w/cm
10.000.0000－0.7000－0.6313－0.6746－1.9642e-40.01372
  （0.0000）（－0.7000）（－0.6308）（－0.6741）——（0.01401）
250.000.00000.0000－0.0083－0.1662－4.3887E-40.01246
  （0.0000）（0.0000）（－0.0078）（－0.1657）——（0.01276）
3100.000.00000.0000－0.0453－0.0780－5.4790E-40.01171
  （0.0000）（0.0000）（－0.0453）（－0.0780）——（0.01171）
4200.000.00000.0000－0.0095－0.0382－6.5503E-40.00984
  （0.0000）（0.0000）（－0.0095）（－0.0382）——（0.00984）
5300.000.00000.00000.0031－0.0206－6.0524E-40.00806
  （0.0000）（0.0000）（0.0031）（－0.0206）——（0.00806）
), ArticleFig(id=1243226245719244887, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, language=CN, label=表2, caption=

表面位移和应力

, figureFileSmall=null, figureFileBig=null, tableContent=
 径向距离r/cm剪应力τrz/MPa竖向应力σr/MPa轴向应力σr/MPa切向应力σθ/MPa横向位移u/m竖向位移w/cm
10.000.0000－0.7000－0.6313－0.6746－1.9642e-40.01372
  （0.0000）（－0.7000）（－0.6308）（－0.6741）——（0.01401）
250.000.00000.0000－0.0083－0.1662－4.3887E-40.01246
  （0.0000）（0.0000）（－0.0078）（－0.1657）——（0.01276）
3100.000.00000.0000－0.0453－0.0780－5.4790E-40.01171
  （0.0000）（0.0000）（－0.0453）（－0.0780）——（0.01171）
4200.000.00000.0000－0.0095－0.0382－6.5503E-40.00984
  （0.0000）（0.0000）（－0.0095）（－0.0382）——（0.00984）
5300.000.00000.00000.0031－0.0206－6.0524E-40.00806
  （0.0000）（0.0000）（0.0031）（－0.0206）——（0.00806）
), ArticleFig(id=1243226246306447456, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, language=EN, label=Tab. 3, caption=

Calculation of displacements and stresses at different depths under load centers

, figureFileSmall=null, figureFileBig=null, tableContent=
 深度z/cm剪应力τrz/MPa竖向应力σr/MPa轴向应力σr/MPa切向应力σθ/MPa横向位移u/m竖向位移w/cm
14.00－0.0031－0.6656－0.3028－0.3203－1.4909e-40.01354
  （－0.0031）（－0.6656）（－0.3024）（－0.3199）——（0.01383）
218.00－0.0242－0.2634－0.0700－0.0556－3.9728e-50.01290
  （－0.0242）（－0.2634）（－0.0698）（－0.0555）——（0.01292）
337.00－0.0246－0.08210.00270.01433.1856e-50.01171
  （－0.0246）（－0.0821）（0.0027）（0.0143）——（0.01171）
456.00－0.0130－0.02260.06830.07579.2697e-50.01079
  （－0.0130）（－0.0226）（0.0680）（0.0754）——（0.01079）
576.00－0.0002－0.00240.09990.10661.7620e-40.00984
  （－0.0002）（－0.0024）（0.0995）（0.1062）——（0.00984）
), ArticleFig(id=1243226246759432293, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226201016353036, language=CN, label=表3, caption=

荷载中心下不同深度位移和应力计算

, figureFileSmall=null, figureFileBig=null, tableContent=
 深度z/cm剪应力τrz/MPa竖向应力σr/MPa轴向应力σr/MPa切向应力σθ/MPa横向位移u/m竖向位移w/cm
14.00－0.0031－0.6656－0.3028－0.3203－1.4909e-40.01354
  （－0.0031）（－0.6656）（－0.3024）（－0.3199）——（0.01383）
218.00－0.0242－0.2634－0.0700－0.0556－3.9728e-50.01290
  （－0.0242）（－0.2634）（－0.0698）（－0.0555）——（0.01292）
337.00－0.0246－0.08210.00270.01433.1856e-50.01171
  （－0.0246）（－0.0821）（0.0027）（0.0143）——（0.01171）
456.00－0.0130－0.02260.06830.07579.2697e-50.01079
  （－0.0130）（－0.0226）（0.0680）（0.0754）——（0.01079）
576.00－0.0002－0.00240.09990.10661.7620e-40.00984
  （－0.0002）（－0.0024）（0.0995）（0.1062）——（0.00984）
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弹性层界面刚度传递法
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王超 , 刘天云
计算力学学报 | 研究论文 2025,42(5): 819-824
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计算力学学报 | 研究论文 2025, 42(5): 819-824
弹性层界面刚度传递法
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王超, 刘天云
作者信息
  • 清华大学　水利水电工程系，北京　100084

    • 刘天云*（1968-），男，博士，研究员（E-mail：）.

Interface stiffness transfer method for elastic multi-layered structures
Chao WANG, Tianyun LIU
Affiliations
  • Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China
出版时间: 2025-10-28 doi: 10.7511/jslx20240614001
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摘要
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针对弹性层状结构问题，本文提出一种界面刚度传递求解新方法。基于弹性层的Love通解，引入状态向量间刚度矩阵，得到弹性层界面刚度传递矩阵方程，即一种Riccati矩阵方程，以及半无限弹性层解析的界面刚度矩阵。界面刚度传递矩阵法通过分解传递矩阵将半无限底层界面刚度自下而上传递至顶层，再根据顶层表面边界条件建立与求解边界面对称的刚度平衡方程。本文方法保留了经典传递矩阵方法的优点，自然克服了指数增长计算问题，特别还为最优控制问题的Riccati方程提供了一种新解法。数值算例验证了界面刚度传递法。

弹性层状体系  /  对偶状态向量  /  界面刚度传递法  /  边界平衡方程  /  Riccati矩阵方程

This paper presents an analytical method, namely interface stiffness transfer method, for evaluating the responses of multilayered elastic structures. Based on the Love function and general solutions, the stiffness matrix relationship of the displacement-stress state vectors is introduced to obtain the interface stiffness transfer matrix equation between adjacent layers, which satisfies an algebraic Riccati matrix equation. When the elastic layer is a half-space, an explicit solution is obtained directly for the interface stiffness matrix. The interface stiffness transfer matrix method starts from the bottom layer with a known stiffness, and then deals with one layer at a time until the uppermost layer is reached, obtaining the interface stiffness of the multilayered structure. Finally, by solving the symmetric equilibrium equations of the boundary conditions, the displacement-stress state vector of an arbitrary layer is obtained. This method keeps the advantages of the classical transfer matrix method, but naturally excludes its exponential growth terms. In particular, the proposed method is a powerful candidate for efficiently solving the algebraic Riccati equation for the optimal control problems. Numerical examples show the properties of the interface stiffness transfer method.

elastic layered system  /  dual variable state vector  /  interface stiffness transfer method  /  boundary equilibrium equations  /  Riccati matrix equations
王超, 刘天云. 弹性层界面刚度传递法. 计算力学学报, 2025 , 42 (5) : 819 -824 . DOI: 10.7511/jslx20240614001
Chao WANG, Tianyun LIU. Interface stiffness transfer method for elastic multi-layered structures[J]. Chinese Journal of Computational Mechanics, 2025 , 42 (5) : 819 -824 . DOI: 10.7511/jslx20240614001
正文
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1　引言
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弹性层状模型可以合理模拟自然分层沉积的土、分层修建的人工建筑物等，在地震学、物探、岩土等领域获得非常广泛的应用，探索精确高效的求解方法一直是研究的重要内容[1-3]。
目前，弹性层状体系求解方法主要有传递矩阵法和整体刚度法两类。前者传递矩阵法有经典传递矩阵法[4,5]、改进传递矩阵法[6]，递推回代法[7]、反射/传递矩阵法（RT）[8]、精细积分法（PIM）[9]、对偶变量-位置法（DVP）[10,11]等。后者整体刚度法有解析刚度法[12-14]、薄层法[15]、谱元法（SEM）[16,17]、混合变量法[18]等。传递矩阵法计算过程简便，但初期传递矩阵中含有正指数项，在数值计算过程中容易出现指数溢出问题，导致计算失败。递推回代法计算效率高，避免了病态矩阵的问题，但传递变量与边界条件相关联，缺乏具体物理含义。随后的反射/传递法则完全解决了数值计算问题，尤其快速RT方法[19]、改进的传递矩阵法[6]既稳定又高效。精细积分法合并相邻层为等效层，实现层端物理量传递，架起了结构力学与最优控制交叉模拟研究的桥梁[20]。而对偶变量-位置法本质上为反射传递法[21]，但计算效率明显低于快速RT法[19]。整体刚度法是一种有限元法，层是自然的单元，薄层法将自然层再细分。层单元刚度矩阵可解析或数值建立。由于整体刚度法仅含负指数项，计算稳定，但静力问题高阶代数方程组或动力问题高阶矩阵特征值需要求解，计算效率较低[15]。
本文基于Love位移函数，引入弹性界面位移应力之间刚度矩阵，建立界面刚度传递方程即代数Riccati矩阵方程，提出一种界面刚度传递解析新方法，具有本质稳定性。数值算例验证了本文方法。
2　传递矩阵方程与Riccati方程
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对于一个N层弹性体，圆柱坐标系如图1所示，各层厚度为hi，弹性模量为Ei，泊松比为μi，1≤i≤N。层间为完全粘结，底层为半无限空间，表面作用轴对称应力。
基于弹性轴对称Love位移函数，Hankel变换域ξ中位移与应力通解为[7]
式中，，，Φ±为待定系数A、B、C、D；子矩阵Ti、Rj分别为
式中f=（1+μ）/E，μ为泊松比，E为弹性模量。
传递矩阵法建立任意层（如第j层）上下界面位移-应力状态向量传递方程
式中，hj为第j层的厚度，上标a表示该层上表面，b表示该层下表面。
假设第j层与j+1层界面刚度矩阵分别为Sj和Sj+1，利用层间连续条件，式（4）可改写为
而层单元刚度矩阵为
则式（5）本质上为一代数Riccati矩阵方程[20]
其中，，，。
3　界面刚度传递法
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引入层辅助界面向量、矩阵，则界面刚度传递方程式（5）可分解为
首先，计算式（8b）右侧
则式（8b）改写为
消去，得辅助界面刚度矩阵
由于式（11）不含正指数项，自然地克服了经典传递矩阵法数值计算问题。
其次，将式（11）代回式（8a），并消去辅助向量，得到第j层界面刚度矩阵
特别地，对于半无限层有，则式（1）位移-应力状态向量简化为
以及得到解析的界面对称刚度矩阵
由此可见，从式（14）界面刚度开始循环使用式（9、11、12）可获得顶层界面刚度S1。如已知第j+1层界面刚度矩阵Sj+1，将Sj+1代入式（9）可得和，继续代入式（11）得到，再将代入式（12）即递推得到第j层的界面刚度矩阵Sj。
4　边界平衡方程及其解
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假设弹性层表面作用轴对称垂直载荷
对其进行Hankel变换，得到应力边界条件
利用顶层界面刚度S1，建立边界应力平衡方程
求解获得顶层表面位移向量。
4.1　层界面位移应力
由式（8a）计算辅助变量
其中，第一行方程为
利用式（10），第j层底界面位移、应力分别为
4.2　层内位移应力
已知任意层上下界面的位移应力，式（18）第2方程
考虑式（19、21），则得到
以及式（1）变换为
4.3　半无限层内位移应力
利用式（1、13）得到半无限层任意位置位移与应力
最后，对上述变换的位移、应力进行逆Hankel变换得到原物理量。
5　算例分析与结果讨论
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5.1　算法验证
考虑六层弹性路面，各层弹性模量分别为10000 MPa、9000 MPa、8000 MPa、10000 MPa、7000 MPa、60 MPa，泊松比分别为0.25、0.25、0.25、0.25、0.25、0.35，厚度分别为4 cm、6 cm、8 cm、38 cm、20 cm，以及半无限。荷载工况列入表1。
利用载荷工况，进行位移、应力叠加，表2与表3分别为表面（深度z=0 cm）以及荷载中心下（径向距离r=0 cm）的位移和应力值。与文献[22]对比可知，数据完全一致（存在微小计算精度差异）。
5.2　位移和应力分布规律
图2与图3分别为表1中仅在荷载1的作用下位移和应力随深度、径向距离的变化结果。
图2竖向位移和竖向应力σz表现出来的规律为，竖向位移在荷载半径内部随深度增加逐渐减小，且越靠近荷载中心，最大值越大；在荷载半径外部由一非0值先增大至峰值，再逐渐减小至0，越远离荷载中心，该峰值越小。而竖向应力σz在荷载半径内部由最大值逐渐减小至0，且越靠近荷载中心最大值越大；在荷载半径外部由0先增大至峰值，再逐渐减小至0。
轴向附加应力σr和切向附加应力σθ在层间变化不连续，这是由不同层材料性质不同引起的。
此外，部分区域的土体受到了轴向拉应力和切向拉应力，可能导致裂缝和结构破坏。
由图3可见，浅层土中剪应力τrz在荷载边缘处取到峰值，且竖向附加应力σz、轴向附加应力σr和切向附加应力σθ在荷载半径内部的值远大于荷载半径外部的值。
由此可见，第一层和第二层路面的相互作用和剪切性能尤为关键，需保证第一层路面铺设质量。
6　结论
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本文针对弹性层结构提出了一种界面刚度传递解析新解法，在获得半无限层解析界面刚度后，逐层传递至顶层就可求解层中任意位置的位移与应力。新方法仅基于Love位移函数解决轴对称问题，也可推广采用其他位移函数或势函数求解非轴对称、波动等问题。特别地，基于结构力学与最优控制的模拟关系，新方法也是求解代数Riccati矩阵方程的可选方法。
基金
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  • 国家自然科学基金(52378350)
文献
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参考文献 引证文献
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2025年第42卷第5期
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doi: 10.7511/jslx20240614001
  • 接收时间：2024-06-14
  • 首发时间：2026-03-24
  • 出版时间：2025-10-28
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  • 收稿日期：2024-06-14
  • 修回日期：2024-07-02
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国家自然科学基金(52378350)
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    清华大学　水利水电工程系，北京　100084
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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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