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Article(id=1243226198482993394, tenantId=1146029695717560320, journalId=1242798230522609684, issueId=1243226190786441246, articleNumber=null, orderNo=null, doi=10.7511/jslx20240813001, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1723478400000, receivedDateStr=2024-08-13, revisedDate=1733760000000, revisedDateStr=2024-12-10, acceptedDate=null, acceptedDateStr=null, onlineDate=1774337823743, onlineDateStr=2026-03-24, pubDate=1761580800000, pubDateStr=2025-10-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774337823743, onlineIssueDateStr=2026-03-24, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774337823743, creator=13701087609, updateTime=1774337823743, 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=877, endPage=882, ext={EN=ArticleExt(id=1243226200026497278, articleId=1243226198482993394, tenantId=1146029695717560320, journalId=1242798230522609684, language=EN, title=Analytical algorithm for determining the reasonable completion state of suspension bridges based on multi-objective parameter optimization, columnId=1243226196507471936, journalTitle=Chinese Journal of Computational Mechanics, columnName=Research Notes, runingTitle=null, highlight=null, articleAbstract=

The computation mechanism of the calculation method for the completed state of existing suspension bridges is unclear, and the target state is unreasonable. A reasonable numerical analysis algorithm is proposed for bridge formation state. The cable theory consisting of the initial end angle and horizontal cable force is validated based on the relationship between the initial end angle and cable force in the theory of catenary equations. A system of bridge state analytical equations are constructed based on the optimization principle of the target parameters of each component of the suspension bridge. The calculation equation for the main cable configuration based on the geometric closure conditions of the suspension bridge's main cable. The mechanical equilibrium equations are constructed for each component based on the mechanical equilibrium conditions of the suspension cables and stiffening beams. Based on the principle of minimizing the bending moment of the stiffening beam components and the principle of uniform cable force of the suspension cable components in the completed state of the suspension bridge, a calculation equation system for the stiffening beam and suspension cable is established. The intelligent algorithm GRG is used to optimize the numerical solution of the objective function of the completed state of a suspension bridge. A case study of a kilometer-long level suspension bridge project. The derived analytical algorithm is compared with the calculation results of the finite element model and rigid supported continuous beam algorithm. The results show that the difference between the analytical algorithm and the finite element model calculation is relatively small in terms of force of main cable, shape-finding of main cable, and the bending moment of the stiffening beam. Compared with the rigid support continuous beam algorithm, the analytical algorithm has computational advantages in ensuring the uniformity of cable forces in bridge suspension cables and the extreme bending moment of stiffening beams.

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针对既有悬索桥成桥状态计算方法的计算机理不清晰、目标状态不合理等问题，提出合理成桥状态数值解析算法。根据悬链线方程理论中初始端缆力夹角与缆力间关系，推导由初始端夹角及水平缆力构成的缆索理论；由悬索桥各构件的目标参数满足最优化原则，构建成桥状态解析方程组；根据悬索桥主缆几何闭合条件，构建主缆位形计算方程；根据吊索及加劲梁力学平衡条件，构建各构件的力学平衡方程；基于悬索桥成桥状态加劲梁构件弯矩最小原则及吊索构件索力均匀性原则，建立加劲梁及吊索计算方程组；利用智能算法GRG对悬索桥成桥状态目标函数进行最优化数值求解。设置千米级悬索桥工程案例，将推导的解析算法与有限元模型、刚性支承连续梁算法的计算结果进行对比，结果表明，在成桥吊索力、主缆线形及加劲梁弯矩数值方面，解析算法与有限元模型计算差值较小；与刚性支承连续梁算法对比，解析算法在成桥吊索索力均匀性、加劲梁弯矩极值具有计算优势。

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黄练*（1990-），女，博士，讲师（E-mail：）.

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黄练*（1990-），女，博士，讲师（E-mail：）.

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黄练*（1990-），女，博士，讲师（E-mail：）.

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Engineering Structures, 2018, 171: 202-213., articleTitle=Methods to correct unstrained hanger lengths and cable clamps' installation positions in suspension bridges, refAbstract=null)], funds=[Fund(id=1243226235891991474, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226198482993394, awardId=24B0734, language=CN, fundingSource=湖南省教育厅科学研究项目优秀青年科学项目(24B0734), fundOrder=null, country=null), Fund(id=1243226236286256056, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226198482993394, awardId=2023KJQD48, language=CN, fundingSource=广西民族大学科研基金（引进人才科研启动项目）(2023KJQD48), fundOrder=null, country=null), Fund(id=1243226236743435199, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226198482993394, awardId=301780209, language=CN, fundingSource=广西壮族自治区第一批青苗人才资助科研项目(301780209), fundOrder=null, country=null), Fund(id=1243226237154476999, tenantId=1146029695717560320, journalId=1242798230522609684, 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基于多目标参数优化的悬索桥合理成桥状态确定解析算法
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朱伟华 1 , 黄练 2 , 黄国平 1 , 杨山波 1 , 姜磊 1
计算力学学报 | 研究简报 2025,42(5): 877-882
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计算力学学报 | 研究简报 2025, 42(5): 877-882
基于多目标参数优化的悬索桥合理成桥状态确定解析算法
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朱伟华1, 黄练2 , 黄国平1, 杨山波1, 姜磊1
作者信息
  • 1.湖南城市学院　土木工程学院，益阳　413000
  • 2.广西民族大学　建筑工程学院　南宁　530006

    • 黄练*（1990-），女，博士，讲师（E-mail：）.

Analytical algorithm for determining the reasonable completion state of suspension bridges based on multi-objective parameter optimization
Weihua ZHU1, Lian HUANG2 , Guoping HUANG1, Shanbo YANG1, Lei JIANG1
Affiliations
  • 1.School of Civil Engineering, Hunan City University, Yiyang 413000, China
  • 2.School of Architecture Engineering, Guangxi Minzu Uiniversity, Nanning 530006, China
出版时间: 2025-10-28 doi: 10.7511/jslx20240813001
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摘要
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针对既有悬索桥成桥状态计算方法的计算机理不清晰、目标状态不合理等问题，提出合理成桥状态数值解析算法。根据悬链线方程理论中初始端缆力夹角与缆力间关系，推导由初始端夹角及水平缆力构成的缆索理论；由悬索桥各构件的目标参数满足最优化原则，构建成桥状态解析方程组；根据悬索桥主缆几何闭合条件，构建主缆位形计算方程；根据吊索及加劲梁力学平衡条件，构建各构件的力学平衡方程；基于悬索桥成桥状态加劲梁构件弯矩最小原则及吊索构件索力均匀性原则，建立加劲梁及吊索计算方程组；利用智能算法GRG对悬索桥成桥状态目标函数进行最优化数值求解。设置千米级悬索桥工程案例，将推导的解析算法与有限元模型、刚性支承连续梁算法的计算结果进行对比，结果表明，在成桥吊索力、主缆线形及加劲梁弯矩数值方面，解析算法与有限元模型计算差值较小；与刚性支承连续梁算法对比，解析算法在成桥吊索索力均匀性、加劲梁弯矩极值具有计算优势。

悬索桥  /  合理成桥状态  /  悬链线理论  /  优化原则  /  解析算法

The computation mechanism of the calculation method for the completed state of existing suspension bridges is unclear, and the target state is unreasonable. A reasonable numerical analysis algorithm is proposed for bridge formation state. The cable theory consisting of the initial end angle and horizontal cable force is validated based on the relationship between the initial end angle and cable force in the theory of catenary equations. A system of bridge state analytical equations are constructed based on the optimization principle of the target parameters of each component of the suspension bridge. The calculation equation for the main cable configuration based on the geometric closure conditions of the suspension bridge's main cable. The mechanical equilibrium equations are constructed for each component based on the mechanical equilibrium conditions of the suspension cables and stiffening beams. Based on the principle of minimizing the bending moment of the stiffening beam components and the principle of uniform cable force of the suspension cable components in the completed state of the suspension bridge, a calculation equation system for the stiffening beam and suspension cable is established. The intelligent algorithm GRG is used to optimize the numerical solution of the objective function of the completed state of a suspension bridge. A case study of a kilometer-long level suspension bridge project. The derived analytical algorithm is compared with the calculation results of the finite element model and rigid supported continuous beam algorithm. The results show that the difference between the analytical algorithm and the finite element model calculation is relatively small in terms of force of main cable, shape-finding of main cable, and the bending moment of the stiffening beam. Compared with the rigid support continuous beam algorithm, the analytical algorithm has computational advantages in ensuring the uniformity of cable forces in bridge suspension cables and the extreme bending moment of stiffening beams.

suspension bridge  /  reasonable bridge construction status  /  chain theory  /  optimization principle  /  analysis algorithm
朱伟华, 黄练, 黄国平, 杨山波, 姜磊. 基于多目标参数优化的悬索桥合理成桥状态确定解析算法. 计算力学学报, 2025 , 42 (5) : 877 -882 . DOI: 10.7511/jslx20240813001
Weihua ZHU, Lian HUANG, Guoping HUANG, Shanbo YANG, Lei JIANG. Analytical algorithm for determining the reasonable completion state of suspension bridges based on multi-objective parameter optimization[J]. Chinese Journal of Computational Mechanics, 2025 , 42 (5) : 877 -882 . DOI: 10.7511/jslx20240813001
正文
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1　引言
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悬索桥的设计使其能够承载较大的负荷，同时使用较少的支承点-索塔，这使其成为应对复杂地形的理想桥型[1-3]。悬索桥成桥平衡状态的确定为设计阶段首先需要计算的部分[4]，不少学者关于该部分的计算理论开展了大量的研究[5,6]。主缆是悬索桥主要受力构件，其具有强烈的几何非线性效应[7-9]，悬索桥成桥状态的计算方法主要包含两个部分，第一部分为计算吊索索力数值，第二部分为悬索桥主缆找形迭代计算[10-12]。
主缆找形的计算方法主要有分段悬链线方法[13,14]、解析算法[12]及有限元软件建模法[15,16]。悬索桥成桥状态吊索力确定的方法主要有刚性支承连续梁法，但是刚性支承连续梁法存在吊索力均匀性及极值较大等问题，只能作为悬索桥设计校核方法，一种各个构件力学状况合理的悬索桥合理成桥计算方法，值得桥梁工程研究人员探索。
主缆找形为悬索桥成桥状态设计计算内容的重要部分，为了解决解析算法主缆找形中特殊的理论问题，学者们提出创新性缆索找形计算理论与方法[17,18]。Wei等[19]为了准确获得主缆的无应力构形建立非线性方程，构建IPSO智能算法来求解力学模型的方程组。Song等[20]建立了一种外载作用悬索桥结构的解析力学模型，使用弹簧单元模拟悬索桥主要受力构件。Zhang等[21]构建了考虑施工期参数误差的无应力构形分析方法，利用智能算法（GRG）求解该力学模型。
悬索桥计算理论及成桥状态计算方法推动了悬索桥发展及应用[17]。现有研究理论中仍然存在问题，如刚性支承连续法中的成桥状态吊索索力均匀性差，塔区吊索索力及加劲梁弯矩极值较大，有限元软件不易开展参数化设计。在悬索桥成桥状态设计阶段，吊索索力均匀且极值较小、加劲梁线形平顺、加劲梁弯矩较小且极值较小共同组成悬索桥成桥控制目标参数，成桥目标参数最优化的计算过程为成桥无应力平衡态搜寻历程。因此，本文创建加劲梁受载显式计算方程，基于悬索桥各受力构件力学状态满足合理目标状态，构建悬索桥合理成桥矛盾方程组，通过智能算法GRG求解方程组最优化问题。
2　主缆非线性理论
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恒载下主缆力学简图如图1所示，以缆索为研究对象，缆索始端作为坐标原点O，缆索索段构形方程见式（1）。
式中　自重荷载集度为q，高度为c，索段跨度为l，O端水平缆力为H，初始段竖向缆力与水平缆力间夹角为μ。
构建笛卡尔坐标系，坐标原点为缆索始端，自重条件下缆索索段平衡态非线性方程为
根据缆索索段计算方程，在缆索索段初始端点处求导；根据计算方程中索端处导数的表达式、索端处竖向力与水平力计算公式一致性原则，始端水平缆力、始端缆力夹角及竖向高度推理计算关系式为
式中O端竖向缆力为H。
针对悬索桥成桥状态主缆线形典型计算工况，各吊点间跨度信息为已知量，则计算参数l为常数，式（4）进一步推导，可得到自重条件下缆索平衡态计算方程（5），式（5）表示计算高度表达式与初始端水平缆力、初始端缆索夹角有关。
缆索索段无应力长度s0见式（6）可知表示计算高度表达式与初始端水平缆力、初始端缆索夹角有关。
式中　弹性模量为E，A为截面面积。
3　主缆力学模型的构建
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3.1　中跨缆索系统解析方程组
悬索桥中跨主缆力学模型如图2所示，中跨主缆由n根主缆划分为n+1个索段，第i号吊索对应的吊索力为Pi。qs、Sdi分别为第i号吊索单位长度重量及吊索无应力长度。主缆计算控制点分别为IP2、IP3。IP2处的中跨水平缆力及缆力夹角分别为H1、μ1，第i号索段的高度与跨度分别为hi、li。
一跨含有n个索段的中跨缆索力学模型如图2所示。根据力学平衡条件，中跨主缆无应力构形递推公式
式中hi为主缆高程，H1为水平缆力，Vi为主缆第i#索段竖向缆力，第i#吊索无应力长度及吊索索力换算公式分别为
式中Pi为主缆吊点作用力，zi为加劲梁恒载作用下坐标数值，可由加劲梁设计坐标与恒载作用下位移数值表示，qd为吊索自重集度，EdAd为吊索抗拉刚度。
3.2　边跨缆索系统解析方程组
边跨主缆找形方程组可依次构建，以左边跨为例进行说明。悬索桥中跨主缆力学模型如图3所示。
中跨主缆与边跨主缆在主索鞍处的水平缆力满足平衡条件，Gi为边跨主缆第i号索夹重量，IP2处的边跨缆力夹角为，边跨主缆第i号索段的高度、跨度及无应力长度分别为、及s1，i，γ为边跨主缆计算高度。边跨主缆索段迭代计算递推公式为
由于IP2处水平缆力在中跨主缆迭代计算中为未知量，因此在边跨主缆解析方程中只需求解IP2缆力夹角。
4　基于多目标参数优化的成桥状态控制方程
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4.1　加劲梁弯矩控制方程
加劲梁吊点及各吊点间的中点作为弯矩控制节点，合理成桥的目标为恒载及吊索力作用下加劲梁控制节点弯矩数值最小。加劲梁恒载下弯矩力学模型如图4所示，根据理论力学相关知识，加劲梁控制节点为吊点及吊点间中点的弯矩计算公式分别为
4.2　吊索索力均匀性控制方程
悬索桥成桥吊索索力与斜拉桥成桥斜拉索索力均匀性一致，悬索桥成桥状态下各吊索索力尽量保持均匀性，同时吊索索力极值与各个吊索的差异性尽量偏小，这样可保证吊索索力的均匀性。为此，构建各吊索间索力均匀性计算函数，其计算格式为
4.3　多目标控制参数的成桥状态约束方程
根据文献[18]关于加劲梁挠度的解析计算公式，可建立加劲梁吊点位形为零的平衡方程组。根据IP间及矢高几何相容条件，可建立主缆无应力构形方程组。根据加劲梁吊点高程计算值与设计值应保持在一定容许范围内，若吊点高程值偏离设计值较大时，将引起桥面线性不平顺，影响桥梁结构静力与动力性能，本文算法加劲梁吊点计算差值控制在10 mm以内，加劲梁最终计算高程影响悬索桥成桥内力，同时为便于智能算法的迭代计算。
悬索桥成桥状态下各吊索力应满足均匀性要求，构建对应方程。根据加劲梁各吊点弯矩值最小原则，可求得加劲梁弯矩控制方程。同时加劲梁在自重恒载，吊索索力及支座反力作用下，其竖直方向应满足力学平衡条件，各荷载作用下，对于支座处进行求矩，满足力学平衡条件，依次构建平衡方程。
5　算例分析
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5.1　工程背景简介
拟定某悬索桥为研究算例，桥跨布置如图5所示，矢高为115.3 m，主缆的弹性模型为1.98×105 MPa。
5.2　三种算法成桥吊索索力对比分析
悬索桥成桥吊索索力对比如图6所示，可以发现，本文算法与有限元模型的吊索索力较为接近，两者间差值控制在1 kN以内，验证本文计算算法在吊索力计算过程中的可靠性。刚性支承连续梁算法的悬索桥成桥吊索索力极值为2863 kN，本文算法的悬索桥成桥吊索索力极值为2585 kN，本文算法的极值小于刚性支承连续梁算法。两种算法吊索力满足对称性要求，刚性支承连续梁算法第1#~3#及第61#~63#吊索的差异性较大，刚性支承连续梁算法的第1#~3#吊索索力依次为2863 kN、1583 kN、2010 kN，而本文推导的吊索索力对应依次为2585 kN、1913 kN、1913 kN，对比发现本文算法的吊索索力值更为均匀，同时极值较小；两者跨中区域吊索索力数值差异性较小。
5.3　三种算法成桥主缆线形对比分析
悬索桥成桥主缆线形对比如图7所示，三者间成桥主缆线形差值较小，中跨线形差值大于边跨线形差值，三者间边跨线形差值控制在2 mm以内。刚性支承连续梁法与本文算法在中跨线形差值计算过程中，中跨主缆塔区线性差值大于中跨跨中线形差值，考虑到两种算法在塔区吊索索力存在一定差异性，可作为主缆线形在塔区差值较大的解释，两者间差值呈现非线性形状，最大差值控制在18 mm以内，最大差值出现在2号吊点。有限元模型的中跨主缆线形高于本文算法的中跨主缆线形，两者间差值呈现非线性形状，最大差值控制在±18 mm以内。对于中跨跨度为1038米的悬索桥而言，本文算法与有限元算法线性差值相对较小，验证本文推导的缆索单元计算理论与合理成桥状态计算方法具有计算可靠性。
5.4　三种算法成桥加劲梁弯矩对比分析
悬索桥成桥加劲梁弯矩对比如图8所示，可以发现，本文算法与有限元模型的加劲梁弯矩值较为接近，两者间差值控制在450 kN•m以内，验证本文算法在加劲梁弯矩计算过程中的可靠性。刚性支承连续梁算法的悬索桥成桥吊索索力极值为5914 kN•m，本文算法的悬索桥成桥吊索索力极值为5473 kN•m，本文算法的极值小于刚性支承连续梁算法。刚性支承连续梁算法与本文算法在塔区弯矩差值较大，最大数值达到2762 kN•m，由于加劲梁抗弯刚度较大，两种算法中跨主缆塔区弯矩差值大于中跨跨中弯矩差值，考虑到两种算法在塔区吊索索力存在一定差值，可作为塔区加劲梁弯矩存在较大差异性的解释。
5.5　成桥加劲梁吊点位移分析
成桥加劲梁吊点位移差值计算如图9所示，本文算法的加劲梁偏离设计值控制在6 mm以内，计算差值满足对称性要求。本文算法的加劲梁吊点偏离设计数值总体较小，塔区加劲梁吊点线形计算差值大于跨中侧计算差值，最大计算差值出现在4#吊点，而主缆线形差值出现在2#吊点，验证加劲梁线形差值呈现滞后性。
综上，本文算法与有限元算法在成桥吊索索力、主缆线形及弯矩数值计算差值较小，验证了本文推导的算法的计算可靠性；本文算法与刚性支承连续梁算法对比，在吊索索力极值与均匀性及加劲梁弯矩极值等方面具有优势，吊索极值与均匀性影响悬索桥设计阶段吊索索力设计及悬索桥运营期间的吊索索力增量，从而验证了本文算法在悬索桥设计与运营阶段具备一定优势。
6　结论
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基于缆索计算理论及各构件合理目标状态，构建悬索桥合理成桥状态解析算法，得出以下结论。
（1）本文根据悬链线方程理论中初始端缆力夹角与缆力间关系，由缆索微分方程推导了由初始端夹角及水平缆力构成的缆索单元新理论，通过算例验证其计算可靠性。新型缆索计算理论与加劲梁位形解析算法可作为大跨度悬索桥缆索单元与长跨主梁结构计算方法。
（2）基于多目标参数优化的解析算法可有效解决刚性支撑连续梁吊索索力极值大且分布不均匀及加劲梁塔区弯矩极值较大等问题。
（3）本文解析算法有效解决塔区吊索力及加劲梁弯矩数值极值较大等问题。为达到悬索桥合理的均匀成桥吊索力，可适当放宽主梁位形收敛容差限值。
基金
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  • 湖南省教育厅科学研究项目优秀青年科学项目(24B0734)
  • 广西民族大学科研基金（引进人才科研启动项目）(2023KJQD48)
  • 广西壮族自治区第一批青苗人才资助科研项目(301780209)
  • 广西青年科技人才托举工程(GXYESS2025176)
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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文章信息
doi: 10.7511/jslx20240813001
  • 接收时间：2024-08-13
  • 首发时间：2026-03-24
  • 出版时间：2025-10-28
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  • 收稿日期：2024-08-13
  • 修回日期：2024-12-10
基金
湖南省教育厅科学研究项目优秀青年科学项目(24B0734)
广西民族大学科研基金（引进人才科研启动项目）(2023KJQD48)
广西壮族自治区第一批青苗人才资助科研项目(301780209)
广西青年科技人才托举工程(GXYESS2025176)
作者信息
    1.湖南城市学院　土木工程学院，益阳　413000
    2.广西民族大学　建筑工程学院　南宁　530006
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2种不同金属材料的力学参数
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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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