任伟新,靳启文
(合肥工业大学 土木与水利工程学院,合肥 230009)
摘要:
为进一步完善结构的鲁棒性、冗余度和易损性相关分析方法,更加准确地获取土木结构的抗力与外力关系,进而优化补强其薄弱易损部位,以减少结构因抗力不足而引发的损伤甚至是安全事故及其所造成的损失.通过综述结构的鲁棒性、冗余度和易损性分析研究进展,类比不同评价指标的表达形式,从而指出结构的鲁棒性、冗余度和易损性分析的本质含义,提出结构的鲁棒性、冗余度和易损性分析应变能评价指标,并据此对结构的鲁棒性、冗余度和易损性分析之间的关系以及基本影响因素展开研究.结果表明:结构的鲁棒性、冗余度和易损性源于不同时期、不同领域,已有文献大多是基于对其含义的不同程度理解,其评价指标也大多具有主观性、相互关联且计算较为复杂,也尚未提出可广泛接受的分析方法;结构的鲁棒性、冗余度和易损性本质均是反映结构特别是损伤结构抵抗外力作用的能力,其中结构鲁棒性、冗余度侧重结构在外力作用下安全储备或剩余容许能力,结构易损性则更直观反映了结构在外力作用下的破坏程度;结构的鲁棒性、冗余度评价指标具有相似的表达形式,并与易损性呈反比例变化;可通过提高结构构件的材料强度及截面尺寸、增加杆端约束、优化荷载路径等措施增强和提高结构的鲁棒性、冗余度,即降低结构的易损性,进而减少结构损伤甚至是安全事故及其所造成的损失.
关键词: 结构安全 鲁棒性 冗余度 易损性 应变能
DOI:10.11918/j.issn.0367-6234.201803128
分类号:TU31
文献标识码:A
基金项目:国家重点研发计划(2016YFC0701703)
Structural robustness, redundancy, and vulnerability
REN Weixin,JIN Qiwen
(School of Civil Engineering, Hefei University of Technology, Hefei 230009, China)
Abstract:
To further improve the analysis methods of structural robustness, redundancy, and vulnerability, and to obtain the relationship between structural resistance and external load effect, optimization and reinforcement measures can be implemented for the weak or vulnerable parts of a structure to reduce the loss caused by damage or even safety accident due to the lack of structural resistance. Firstly, by reviewing the research progress of structural robustness, redundancy, and vulnerability and comparing different expressions of the current evaluation indexes, the essence of structural robustness, redundancy, and vulnerability were pointed out, and a new evaluation index based on strain energy was proposed. A further study on the relationship between structural robustness, redundancy, and vulnerability, as well as the basic influencing factors was also performed. Results have shown that structural robustness, redundancy, and vulnerability are from different fields and periods, which can reflect the resistance capacity of a structure, especially that of a damaged structure. The evaluation indexes are mostly subjectively defined by a complicated calculation process based on different levels of understanding. A widely accepted analysis method has not been established so far. Specifically, structural robustness and redundancy share similar evaluation index expression, both of which can represent the safety capacity or the remaining tolerance capacity of a structure. Structural vulnerability can directly reflect the structural failure extent, and it is inversely proportional to structural robustness and redundancy in the physical sense. Moreover, the increase in the material strength, the section size, and the structural restraint, together with the optimization of the load path, can increase structural robustness and redundancy and reduce structural vulnerability. The optimization and reinforcement measures can be implemented for the weak or vulnerable parts, thus structural damage, safety accident, as well as the loss cause by them, can be further reduced.
Key words: structural safety robustness redundancy vulnerability strain energy
任伟新, 靳启文. 结构的鲁棒性、冗余度和易损性[J]. 哈尔滨工业大学学报, 2018, 50(9): 1-10. DOI: 10.11918/j.issn.0367-6234.201803128.
![复制到剪切板](http://hit.alljournals.cn/html/hitxb_cn/2018/9/images/copy.gif)
REN Weixin, JIN Qiwen. Structural robustness, redundancy, and vulnerability REN[J]. Journal of Harbin Institute of Technology, 2018, 50(9): 1-10. DOI: 10.11918/j.issn.0367-6234.201803128.
![复制到剪切板](http://hit.alljournals.cn/html/hitxb_cn/2018/9/images/copy.gif)
基金项目 国家重点研发计划(2016YFC0701703) 作者简介 任伟新(1960—),男,****特聘教授,博士生导师 通信作者 靳启文,jqw.sinx123@163.com 文章历史 收稿日期: 2018-03-25
Contents -->Abstract Full text Figures/Tables PDF
结构的鲁棒性、冗余度和易损性
任伟新, 靳启文
![通讯作者](http://hit.alljournals.cn/html/hitxb_cn/2018/9/images/REcor.gif)
![jqw.sinx123@163.com](http://hit.alljournals.cn/html/hitxb_cn/2018/9/images/REemail.gif)
合肥工业大学 土木与水利工程学院,合肥 230009
收稿日期: 2018-03-25
基金项目: 国家重点研发计划(2016YFC0701703)
作者简介: 任伟新(1960—),男,****特聘教授,博士生导师
通信作者: 靳启文,jqw.sinx123@163.com
摘要: 为进一步完善结构的鲁棒性、冗余度和易损性相关分析方法,更加准确地获取土木结构的抗力与外力关系,进而优化补强其薄弱易损部位,以减少结构因抗力不足而引发的损伤甚至是安全事故及其所造成的损失.通过综述结构的鲁棒性、冗余度和易损性分析研究进展,类比不同评价指标的表达形式,从而指出结构的鲁棒性、冗余度和易损性分析的本质含义,提出结构的鲁棒性、冗余度和易损性分析应变能评价指标,并据此对结构的鲁棒性、冗余度和易损性分析之间的关系以及基本影响因素展开研究.结果表明:结构的鲁棒性、冗余度和易损性源于不同时期、不同领域,已有文献大多是基于对其含义的不同程度理解,其评价指标也大多具有主观性、相互关联且计算较为复杂,也尚未提出可广泛接受的分析方法;结构的鲁棒性、冗余度和易损性本质均是反映结构特别是损伤结构抵抗外力作用的能力,其中结构鲁棒性、冗余度侧重结构在外力作用下安全储备或剩余容许能力,结构易损性则更直观反映了结构在外力作用下的破坏程度;结构的鲁棒性、冗余度评价指标具有相似的表达形式,并与易损性呈反比例变化;可通过提高结构构件的材料强度及截面尺寸、增加杆端约束、优化荷载路径等措施增强和提高结构的鲁棒性、冗余度,即降低结构的易损性,进而减少结构损伤甚至是安全事故及其所造成的损失.
关键词: 结构安全 鲁棒性 冗余度 易损性 应变能
Structural robustness, redundancy, and vulnerability REN
REN Weixin, JIN Qiwen
![Corresponding author](http://hit.alljournals.cn/html/hitxb_cn/2018/9/images/REcor.gif)
![jqw.sinx123@163.com](http://hit.alljournals.cn/html/hitxb_cn/2018/9/images/REemail.gif)
School of Civil Engineering, Hefei University of Technology, Hefei 230009, China
Abstract: To further improve the analysis methods of structural robustness, redundancy, and vulnerability, and to obtain the relationship between structural resistance and external load effect, optimization and reinforcement measures can be implemented for the weak or vulnerable parts of a structure to reduce the loss caused by damage or even safety accident due to the lack of structural resistance. Firstly, by reviewing the research progress of structural robustness, redundancy, and vulnerability and comparing different expressions of the current evaluation indexes, the essence of structural robustness, redundancy, and vulnerability were pointed out, and a new evaluation index based on strain energy was proposed. A further study on the relationship between structural robustness, redundancy, and vulnerability, as well as the basic influencing factors was also performed. Results have shown that structural robustness, redundancy, and vulnerability are from different fields and periods, which can reflect the resistance capacity of a structure, especially that of a damaged structure. The evaluation indexes are mostly subjectively defined by a complicated calculation process based on different levels of understanding. A widely accepted analysis method has not been established so far. Specifically, structural robustness and redundancy share similar evaluation index expression, both of which can represent the safety capacity or the remaining tolerance capacity of a structure. Structural vulnerability can directly reflect the structural failure extent, and it is inversely proportional to structural robustness and redundancy in the physical sense. Moreover, the increase in the material strength, the section size, and the structural restraint, together with the optimization of the load path, can increase structural robustness and redundancy and reduce structural vulnerability. The optimization and reinforcement measures can be implemented for the weak or vulnerable parts, thus structural damage, safety accident, as well as the loss cause by them, can be further reduced.
Keywords: structural safety robustness redundancy vulnerability strain energy
土木工程结构在长期的自然环境、外部车辆、自然灾害以及人为破坏等荷载作用下,会因为抗力不足出现损伤,如结构的强度降低、刚度退化及稳定性不足等都会导致结构的承载能力下降,进而大大增加服役风险.一方面,传统的结构设计和分析方法需要进一步完善,如结构的强度、刚度和稳定性分析,以便更加准确获得结构的抗力与外力关系;另一方面,还需要加强结构出现损伤后的性能分析,如结构的鲁棒性、冗余度和易损性,以最大程度地减少结构损伤所造成的损失.
然而,目前国内外还没有形成普遍可以接受的结构鲁棒性、冗余度和易损性相关分析方法,如何增强结构的鲁棒性、提高结构的冗余度、降低结构的易损性也有待完善.已有文献[1-6]大多是基于对其含义的不同程度理解、进而给出所定义的评价指标而展开研究.此类评价指标大多具有主观性、计算较复杂,且难以用于工程实践,也尚未提出可广泛接受的分析方法.
本文旨在通过综述结构的鲁棒性、冗余度和易损性研究进展及其评价指标,探讨结构的鲁棒性、冗余度和易损性的本质含义;针对其现有的不同评价指标表达形式展开对比分析,进而提出基于应变能的评价指标,并进一步分析了结构的鲁棒性、冗余度和易损性的相互关系以及基本影响因素.可据此进一步分析结构尤其是损伤结构抵抗外力作用的能力,找出并补强结构抗力薄弱部位,进而降低结构损伤甚至安全事故所造成的损失.
1 结构的鲁棒性、冗余度和易损性研究及其评价指标 1.1 结构的鲁棒性研究及其评价指标鲁棒性(robustness)也叫稳健性、强韧性等,早期主要用于经济、统计等领域,如鲁棒性估计、回归分析[7-8]. Davison等[9]较早将鲁棒性引入系统控制领域,用来研究系统在微小摄动作用下的稳定性.此后,鲁棒性逐渐用于结构分析和优化设计[10-11].
早在20世纪80年代,Morton[12]就针对砖石结构及随机系统鲁棒性展开了研究,并指出不能孤立地开展结构破坏分析,需考虑结构的鲁棒性和稳定性. Templeton等[13]先后针对民居建筑的鲁棒性展开系列研究,如墙体的意外损伤,并提出了相关完善措施. López-Almansa等[14]对建筑结构施加主动缆索控制可行性和鲁棒性展开了理论分析和实验验证. Beeby[15]指出结构失效多是由于缺乏鲁棒性,并提出了鲁棒性量化措施.
近年来,国际结构安全度联合委员会(JCSS)及欧盟科技合作项目(COST)先后就结构的鲁棒性展开了系列研究,Jen[16]指出鲁棒性是系统战略选择的适应性,高于稳定性;Faber[17]指出鲁棒性是决策分析的理论依据,并指出鲁棒性的概率分析十分必要;Baker等[18]指出鲁棒性反映了结构对外荷载作用的容许能力,指出系统鲁棒性是非直接风险占总风险的结果;S?rensen等[19]指出结构破坏时的严重后果及设计执行时的高效性使得结构鲁棒性至关重要. Fang等[1]将鲁棒性用来表征结构在发生破坏情况时维持其整体安全的能力, 提出了将结构吸收的能量合计无量纲形式作为鲁棒性评价指标,并研究了结构失效后的能量输入和转化. Biondini和Ghosn等[20-21]研究了结构鲁棒性、冗余度和超静定性,提出了可用于提高结构冗余度与鲁棒性的相关措施及设计方法. Starossek[22]指出鲁棒性是阻止结构破坏的一种方式,并提出了结构抵抗非比例破坏的总体设计方法. Rominu等[23]指出结构在服役期内应具有足够的抗力、耐久性和鲁棒性. Starossek等[2]将鲁棒性定义为结构对局部失效的敏感性,并提出了基于刚度、损伤或能量方法的简化公式.
目前,结构的鲁棒性已进入定量研究和结构性能分析评定阶段. Ghosn和Biondini等[24-27]先后研究了劣化混凝土结构的时变鲁棒性,提出了表征结构剩余承载能力的鲁棒性评价标准,研究了公路及劣化混凝土桥梁的冗余度和鲁棒性,后逐渐扩展到混凝土桥墩腐蚀后的全寿命鲁棒性及老龄结构的时变鲁棒性.吕大刚等[3]经总结指出,目前结构鲁棒性的定量研究主要是沿着结构属性和结构性能两方面展开,其中后者又包含了基于确定性结构性能的测度(如考虑冗余度的鲁棒性评价指标)、基于可靠度的概率测度以及基于风险的测度三大类指标.结构鲁棒性分析的部分评价指标表达式如下所述.
Fang和吕大刚等[1, 3]基于结构的能量变化,令Eu、Er、Ed分别为结构临界破坏状态、结构极限状态破坏时的能量,以及结构达到屈服或设计目标临界状态时的能量,表征结构鲁棒性的能量指标为
$\left\{ \begin{array}{l}{I_{{\rm{rob1}}}} = \frac{{{E_u}}}{{{E_d}}},\\{I_{{\rm{rob2}}}} = \frac{{{E_r}}}{{{E_d}}},\\{I_{{\rm{rob3}}}} = \frac{{{E_r}}}{{{E_u}}},\\{I_{{\rm{rob4}}}} = \frac{{{E_u}}}{{{E_u} - {E_r}}}.\end{array} \right.$ (1)
Starossek等[2]令Es, k、Ei, j分别为结构构件破坏的能量限值及构件破坏产生的能量,表征结构的鲁棒性的能量指标为
${R_e} = 1 - \max \frac{{{E_{i,j}}}}{{{E_{s,k}}}}.$ (2)
叶列平等[28]令k0、ki分别为完好结构和构件破坏后的刚度矩阵,考虑结构荷载途径,表征结构鲁棒性的刚度矩阵指标为
${R_I} = \frac{{{k_0} - {k_i}}}{{{k_0}}}.$ (3)
高扬[29]令λ0、λi分别为完好结构和破坏发生后的承载能力系数,考虑结构构件的重要性系数,表征结构鲁棒性的承载能力系数指标为
${R_n} = \frac{{{\lambda _0} - {\lambda _i}}}{{{\lambda _0}}}.$ (4)
1.2 结构的冗余度研究与评价指标结构的冗余度(redundancy)早期常用于数学、社科领域,用来表征信源信息率的多余程度[30-31]. Wilderd和Foulkes[32-33]较早研究了含有冗余杆件的桁架结构以及基于结构自重作用的框架结构优化设计问题.
早在20世纪80年代,Frangopol和Moses等[4, 34]先后针对土木结构工程的冗余度和优化设计展开了大量研究,并与Biondini等[20]先后指出结构鲁棒性和冗余度有本质上的内在联系,后者与结构的材料、荷载和几何拓扑密切相关. Frangopol等[35-37]还先后对桥梁冗余度展开了系列研究,如冗余结构的可靠性灵敏度、冗余对结构可靠性的影响、桥梁冗余度评估等,并指出结构在荷载和意外状况下的安全性使结构的冗余度相关研究显得十分必要. Shea[38]指出冗余度表征了结构具有多道抗力路径的特性. Connor等[39]根据荷载再分配能力不同的表现形式,指出冗余度包括杆件构造冗余、与外部约束相关的结构冗余及考虑荷载途径的结构体系冗余.于刚[40]指出冗余度包含了结构构件与结构体系的冗余度,前者是构件在特定荷载下的冗余安全储备,后者是构件失效后剩余结构对失效构件所释放荷载的再分配能力.
近年来,结构的冗余度经历了快速发展和不断完善,一些相关规范及设计指南先后出台. Ghosn等[41-43]先后研究了桥梁上部、下部结构的冗余度,提出了考虑冗余度的桥梁上部、下部结构设计和荷载容许能力的分析方法,并编制了相关规范.陈以一等[44]指出要求结构抵抗所有外部荷载是不经济的且很难实现,还讨论了冗余度的物理含义、结构材料和几何形式的影响,并给出了定性和定量分析方法及相关措施. Mertz[45]指出结构冗余度包括荷载路径冗余、结构及杆件构造冗余,研究了钢桥结构杆件冗余度和杆件失效,并提出了相关完善措施.结构冗余度分析的部分评价指标表达式如下所述.
Frangopol, Moses和Pandey等[4, 34, 46]及Faber等[47],令Lintact为完好结构的承载力、Ldesign为结构的设计承载力、Ldamage为受损结构的承载力,表征结构冗余度的承载力指标为
$\left\{ \begin{array}{l}{R_{{\rm{r1}}}} = \frac{{{L_{{\rm{intact}}}}}}{{{L_{{\rm{design}}}}}},\\{R_{{\rm{r2}}}} = \frac{{{L_{{\rm{damage}}}}}}{{{L_{{\rm{intact}}}}}},\\{R_{{\rm{r3}}}} = \frac{{{L_{{\rm{damage}}}}}}{{{L_{{\rm{design}}}}}},\\{R_{{\rm{r4}}}} = \frac{{{L_{{\rm{intact}}}}}}{{{L_{{\rm{intact}}}} - {L_{{\rm{damage}}}}}}.\end{array} \right.$ (5)
式中:Rr1为结构储备冗余系数(储备强度比、超强系数),Rr2、Rr3为结构剩余冗余系数(剩余影响系数、剩余或损伤强度比),Rr4为结构强度冗余系数.
于刚[40]令Ui、Uimax分别为结构荷载作用下构件i所存储应变能和所能存储最大应变能,令Usys0、Usysi分别为结构体系完好时的应变能和结构体系在杆件i失效后的应变能,表征结构冗余度的应变能指标为
$\left\{ \begin{array}{l}r_i^c = \frac{{U_i^{\max } - {U_i}}}{{U_i^{\max }}},\\r_i^s = \frac{{U_{{\rm{sys}}}^0}}{{U_{{\rm{sys}}}^i}}.\end{array} \right.$ (6)
式中:ric为构件层面的冗余度,ris为结构体系的冗余度(结构体系应变能是各个构件应变能之和).
Frangopol等[4]令Pf, d、Pr, 0分别为损伤结构和完好结构的失效概率,考虑结构体系的失效概率变化,表征结构冗余度的失效概率指标为
$RI = \frac{{{P_{f,d}} - {P_{f,0}}}}{{{P_{f,0}}}}.$ (7)
令β0、βd分别为损伤和完好结构的可靠指标,即采用可靠指标描述结构的可靠性,表征结构冗余度的可靠度指标为
${\beta _R} = \frac{{{\beta _0}}}{{{\beta _0} - {\beta _d}}}.$ (8)
1.3 结构的易损性研究与评价指标易损性(vulnerability)早期常用于生物、医学等领域[48],用来表征系统的脆弱性.此后易损性逐渐用于军事领域,如航空器颤振及弹片损伤影响[49-50],用来分析结构的易损薄弱部件以及损伤性能评价.
早在20世纪70年代,Greenspon[51]就指出结构的易损性取决于其保持正常服役所能承受的最大变形,即易损性表征结构的塑性变形和破坏. Shah[52]指出不同领域的易损性有不同含义,易损性意味着敏感性. Tassios[53]将结构易损性分析视为快速适用的整体评价技术,并给出了某结构的地震承载能力评估方法.
近年来,英国学者先后就结构的易损性展开了系列研究,如Wu、Lu和Agarwal等[54-56]将结构视为不同层级的环和簇,将其是否良好作为易损性度量方法,以识别结构最易损的部分,并指出如果由于结构的任何损伤导致/造成结构产生与损伤不成比例的后果,那么结构即是易损的;Agarwal等[56-57]指出易损性可用于大多数的结构分析,其目的是找出结构的薄弱连接,而非结构荷载作用下的响应理论;Hashimoto[58]较早提出了结构易损性分析的基本方法,并基于损伤场景展开了易损性量化研究;Yu[59]编制了结构易损性分析的工程应用计算程序;Lu等[55, 60]提出了结构环概念来识别结构的基本组成,并对某三维结构展开了易损性分析;Pinto[61]研究了不同损伤场景作用下的易损性量化.此外,Farrell等[62]还提出了目标易损性和防护预测的不确定性量化方法;Lind[63]提出了易损性和损伤容许限值的概率定义,并将损伤系统的易损性定义为失效概率与无损系统失效概率的比值. Blockley等[64]研究了结构的易损性、可靠性和风险性的相互关系.
目前,国内外学者正研究将结构易损性分析用于结构健康监测及性能评估中.于刚[40]指出结构易损性是对现有设计方法的补充,指出光靠安全储备及较高的可靠度不足以保证结构安全,还要使其具有较低的易损性. Sun等[65]指出易损性是结构性能对其局部损伤的敏感性,并提出了确定结构重点监测部件的易损性方法. Yan等[66]采用塑性极限分析实现了某斜拉桥的易损性评估. Torbol等[67]采用结构振动性能长期监测数据形成的脆弱性曲线表征桥梁的易损性,以修正服役期内桥梁结构参数变化. Cao等[68]基于某大跨桥梁动力响应的数值模拟,实现了该桥在风荷载作用下的易损性评估. Lonetti等[69]分析了斜拉悬索桥发生损伤后的易损性和失效可能.
Lind[5]令P(rd, s)、P(r0, s)分别为结构在荷载S作用下的失效概率和完好结构的失效概率,将结构易损性定义为承受能力的倒数,表征结构易损性的失效概率指标为
$V = \frac{{P\left( {{r_d},s} \right)}}{{P\left( {{r_0},s} \right)}}.$ (9)
Pinto等[6]令γ和Dγ分别为结构失效产生的后果和损伤相对需求,考虑完好结构和损伤结构的完整性以及结构的损伤需求,将结构易损性定义为结构失效产生后果与结构损伤相对需求的比值,表征结构易损性的失效和损伤相对需求指标为
${I_\varphi } = \frac{\gamma }{{{D_r}}}.$ (10)
式中:
Liu等[70]令C为结构破坏产生的后果、Dr为结构破坏的相对需求,考虑结构破坏路径,以应变能为评价指标,其本质还是结构的能力与需求的比值,表征结构易损性的破坏和需求指标为
$V = \frac{C}{{{D_r}}}.$ (11)
由1.1节~1.3节可以看出,在结构工程领域,结构的鲁棒性、冗余度和易损性研究是源于不同时期、不同的研究领域,均可用来反映结构尤其是损伤结构对外力作用的抵抗能力;部分学者也采用结构的冗余度来分析结构的鲁棒性.此外,现有的结构的鲁棒性、冗余度和易损性评价指标也大多采用类似的表达形式.因此,结构的鲁棒性、冗余度和易损性既有不同特点,又相互关联、具有相通之处.
2 结构的鲁棒性、冗余度和易损性本质含义、应变能评价指标及相互关系 2.1 结构的鲁棒性、冗余度和易损性本质含义由式(1)~(11)可知,结构的鲁棒性、冗余度和易损性分析多采用相似的表达式,可记为
$\left\{ \begin{array}{l}{R_1} = \frac{{{R_0}}}{{{R_p}}},\\{R_2} = \frac{{{R_0} - {R_p}}}{{{R_0}}},\\{V_1} = \frac{{{V_p}}}{{{V_0}}}.\end{array} \right.$ (12)
式中:R1、R2分别为结构的鲁棒性和冗余度(取相同英文打头字母),V1为结构的易损性. R0、V0和Rp、Vp分别为结构性能的极限状态和荷载响应,表征结构性能的参数可以是刚度、应变能、承载力、失效概率等.吕大刚等[3]指出R1本质是结构整体安全储备系数.
由式(12)可知,结构鲁棒性、冗余度侧重表征结构在外荷载作用下的安全储备或剩余容许能力,而结构易损性通过结构外荷载作用与其极限状态的比例,更直观反映了结构在外荷载作用下损伤程度.具体含义如下所述.
外荷载作用下,相对结构的极限状态,结构外荷载作用越大,结构的安全储备或抵抗能力剩余就越小,所占极限状态的比例就越大,此时结构的鲁棒性、冗余度就越小,而结构的易损性就越大,即结构容易出现损伤.当外荷载作用与结构的极限状态基本接近时,结构的安全储备或剩余抵抗能力最小,此时结构的鲁棒性、冗余度最小,而结构的易损性最大,即结构最容易出现损伤.
2.2 结构的鲁棒性、冗余度和易损性应变能评价指标考虑到能量原理具有普遍适用性,本节以结构在外荷载作用下的应变能变化为例,令UPs为结构及其构件在荷载作用下产生的应变能,UM为结构及其构件的极限弯曲应变能(UF为桁架杆件的极限拉压应变能),对结构的鲁棒性、冗余度和易损性评价指标展开进一步分析.
情况Ⅰ????用表达式
对于给定结构(即已知极限状态UM,下同),当UPs基本为零,即外荷载作用产生的应变能无穷小、基本为零时,R11趋近于无穷大而V11等于零,此时结构具有较高的鲁棒性、冗余度以及较低的易损性,结构不易出现损伤.此后,随着UPs逐渐增大,即外荷载作用产生的应变能逐渐增大,R11逐渐减小而V11逐渐增大,此时结构的鲁棒性、冗余度逐渐减小,而易损性逐渐增大,结构出现损伤的可能逐渐增加.当UPs趋近于UM,即外荷载作用产生的应变能趋近于结构的极限应变能时,R11及V11均趋近于1,此时结构具有较低的鲁棒性、冗余度和较高的易损性,结构容易出现损伤.
情况Ⅱ????用表达式
对于给定结构,当UPs基本为零,即外荷载产生的应变能无穷小、基本为零时,R12等于1而V12等于0,此时结构具有较高的鲁棒性、冗余度和较低的易损性,结构不易出现损伤.此后,随着UPs逐渐增大,即外荷载产生的应变能逐渐增大,R12逐渐减小而V12逐渐增大,此时结构的鲁棒性、冗余度逐渐减小,而易损性逐渐增大,结构出现损伤的可能逐渐增加.当UPs趋近于UM,即外荷载产生的应变能趋近于结构极限应变能时,R12趋近于0而V12趋近于1,此时结构具有较低的鲁棒性、冗余度和较高的易损性,结构易发生破坏.
2.3 结构的鲁棒性、冗余度和易损性相互关系情况Ⅰ????由结构的鲁棒性、冗余度评价指标R11和结构的易损性评价指标V11可知,
针对含有多个杆件的桁架结构,令i为结构杆件编号,则
情况Ⅱ????由结构的鲁棒性、冗余度评价指标R12和结构的易损性评价指标V12可知,R12=1-V12,即结构的鲁棒性、冗余度与结构的易损性呈现反比例变化.
针对含有多个杆件的桁架结构,令i为结构杆件编号,则R12i=1-V12i.当R12i>R12i+1时,有V12i < V12i+1;当R12i < R12i+1时,有V12i>V12i+1;当R12i=R12i+1时,有V12i=V12i+1.由此可知,结构鲁棒性、冗余度与易损性呈线性反向变化;不同的杆件因为可能因为外荷载产生相同的应变能而具有相同的结构鲁棒性、冗余度和易损性.
由2.1节~2.3节可知,外荷载作用下,结构鲁棒性、冗余度和易损性均呈反比例变化;不同杆件可能因为产生相同的应变能而具有相同的结构鲁棒性、冗余度和易损性.情况Ⅱ中的结构鲁棒性、冗余度和易损性表达式,因具有相同的取值区间而易于对比分析,可用来进一步分析或评价结构尤其是损伤结构抵抗外力作用的能力,据此找出并补强结构抗力薄弱部位,进而降低结构损伤甚至安全事故所造成的损失.
3 结构的鲁棒性、冗余度和易损性基本影响因素由力学知识可知,等截面曲梁微元段dx在力矩M(x)作用下的近似挠曲线方程及弯曲应变能,其表达式可记为
$M\left( x \right) = EI\omega ''\left( x \right),$ (13)
${\rm{d}}U = \frac{1}{2}M\left( x \right){\rm{d}}\theta = \frac{{{M^2}\left( x \right)}}{{2EI}}{\rm{d}}x.$ (14)
式中:M(x)、θ、E、I、ω(x)分别为微元段曲梁的力矩、转角、弹模、抗弯惯性矩及竖向位移.令l为梁长度,U为梁在力矩M(x)作用下产生的弯曲应变能,则通过积分可求梁长l的弯曲应变能,其表达式可记为
$U = \int_0^l {{\rm{d}}U} = \int_0^l {\frac{{EI{{\left[ {\omega ''\left( x \right)} \right]}^2}}}{2}{\rm{d}}x} .$ (15)
令R为结构的鲁棒性和冗余度(下同),由2.1节情况Ⅱ中结构的鲁棒性和冗余度表达式(R12=(R0-Rp)/R0)及式(13)~(15)可知,结构的鲁棒性和冗余度表达式可记为
$R = 1 - \frac{1}{{{U_M}}}\int_0^l {\frac{{EI{{\left[ {\omega ''\left( x \right)} \right]}^2}}}{2}{\rm{d}}x} .$ (16)
由力学分析求得结构在荷载作用下的位移函数,代入上式即可求得结构的鲁棒性、冗余度(可同理求得结构的易损性表达式).考虑到结构的鲁棒性、冗余度和易损性呈反比例变化,本节仅以单梁不同约束以及桁架梁为例,给出结构的鲁棒性、冗余度影响因素分析结果,也可据此反推对结构易损性的影响.
3.1 杆件强度此处以简单的矩形截面梁为例,令σs为材料的屈服强度、b为截面宽度、h为截面高度、Ms为梁屈服弯矩(Ms=σsbh2/6),由式(13)~式(16)可知,结构的鲁棒性和冗余度表达式可记为
$R = 1 - \frac{1}{{M_s^2l}}\int_0^l {{M^2}\left( x \right){\rm{d}}x} .$ (17)
将屈服弯矩Ms代入式(17),并令∫0lM2(x)dx等于数值C0可知,结构的鲁棒性和冗余度表达式可记为
$R = 1 - 36{C_0} \cdot \frac{1}{{{\sigma _s}^2}} \cdot \frac{1}{{{b^2}}} \cdot \frac{1}{{{h^4}}} \cdot \frac{1}{l}.$ (18)
由式(18)可知,增加材料屈服强度、截面的长度或宽度,尤其是高度均可以增大结构的鲁棒性/冗余度,即降低结构的易损性.
3.2 杆端约束 3.2.1 单梁结构为分析同一荷载作用下不同梁端约束对结构鲁棒性、冗余度的影响,以单一的悬臂梁、两端固定梁及简支梁为例,分别计算不同梁在跨中截面集中力作用下的结构鲁棒性、冗余度变化,如图 1所示,R0=1表示结构初始时的鲁棒性、冗余度(荷载无穷小、基本为零).
Figure 1
![点击查看原图](http://hit.alljournals.cn/html/hitxb_cn/2018/9/PIC/hebgydxxb-50-9-1-1.jpg)
由图 1可知,单梁跨中截面在集中荷载作用时,结构的鲁棒性和冗余度沿着约束端至跨中方向逐渐减小;两端固定梁的鲁棒性和冗余度变化较小,简支梁居中,悬臂梁的鲁棒性和冗余度变化较大;两端固定梁和简支梁的鲁棒性和冗余度沿跨中截面对称变化,悬臂梁无荷载作用区域的鲁棒性和冗余度保持不变.
3.2.2 桁架结构以三角桁架(图 2)为例,令结构杆件分别取相同的长度l、横截面面积A以及弹性模梁E,外荷载为P,杆件极限拉压应变能和弯曲应变能分别为U1、U2,分析铰接和刚性连接对结构的鲁棒性和冗余度的影响.
Figure 2
![点击查看原图](http://hit.alljournals.cn/html/hitxb_cn/2018/9/PIC/hebgydxxb-50-9-1-2.jpg)
分别计算节点3铰接(图 2(a))和刚性连接(图 2(b))作用下的结构支反力及各杆件轴力和应变能,其中节点1水平反力均为零,节点1、2竖向反力均为
令RiⅠ与RiⅡ为不同约束时杆件i的鲁棒性和冗余度,将各杆件在外荷载作用下产生的应变能代入2.1节情况Ⅱ中结构的鲁棒性和冗余度表达式(R12),可知不同杆件的结构的鲁棒性和冗余度,其表达式可记为
$R_{13}^{\rm{I}} = R_{23}^{\rm{I}} = 1 - \frac{{{P^2}l}}{{6EA{U_1}}},$ (19)
$R_{12}^{\rm{I}} = R_{12}^{{\rm{II}}} = 1 - \frac{{{P^2}l}}{{24EA{U_1}}},$ (20)
$R_{13}^{{\rm{II}}} = R_{23}^{{\rm{II}}} = 1 - \left( {\frac{{{P^2}l}}{{6EA{U_1}}} + \frac{{{P^2}{l^3}}}{{96EI{U_2}}}} \right).$ (21)
由式(19)~(21)可知,同一荷载作用下,杆件12端部均为铰接约束,其结构的鲁棒性和冗余度相等;杆件13和23在端部铰接和一端铰接一端固定约束时,其结构鲁棒性和冗余度有较大差异,后者明显小于前者.
3.3 荷载路径以桁架梁为例(图 3),结构各杆件仍然取相同的长度l、横截面面积A、弹性模梁E,外荷载分别为P1、P2及P3且P1=P2=P3=P0,极限轴向力为F,极限应变能为U0,分析不同外荷载作用下结构的鲁棒性和冗余度变化.
Figure 3
![点击查看原图](http://hit.alljournals.cn/html/hitxb_cn/2018/9/PIC/hebgydxxb-50-9-1-3.jpg)
分别计算不同外荷载作用下结构的支反力及各杆件轴力、应变能,并参考第2.1节情况(Ⅱ)中结构的鲁棒性和冗余度表达式(R12),计算各杆件的结构鲁棒性和冗余度如下,其中
3.3.1 荷载P1单独作用下的结构鲁棒性和冗余度荷载P1单独作用下,节点1水平反力为零,节点1、3竖向反力分别为
将各杆件在荷载P1单独作用下产生的应变能代入结构的鲁棒性和冗余度表达式,可知不同杆件的结构鲁棒性和冗余度,其表达式可记为
$R_{12}^{{P_1}} = 1 - \frac{{3{P_1}^2l}}{{32EA{U_0}}} = 1 - \frac{3}{{16}}{\left( {\frac{{{P_1}}}{F}} \right)^2},$ (22)
$R_{14}^{{P_1}} = 1 - \frac{{3{P_1}^2l}}{{8EA{U_0}}} = 1 - \frac{3}{4}{\left( {\frac{{{P_1}}}{F}} \right)^2},$ (23)
$R_{23}^{{P_1}} = 1 - \frac{{{P_1}^2l}}{{96EA{U_0}}} = 1 - \frac{1}{{48}}{\left( {\frac{{{P_1}}}{F}} \right)^2},$ (24)
$\begin{array}{*{20}{c}}{R_{24}^{{P_1}} = R_{25}^{{P_1}} = R_{35}^{{P_1}} = R_{45}^{{P_1}} = 1 - \frac{{{P_1}^2l}}{{24EA{U_0}}} = }\\{1 - \frac{1}{{12}}{{\left( {\frac{{{P_1}}}{F}} \right)}^2}.}\end{array}$ (25)
3.3.2 荷载P2单独作用下的结构鲁棒性和冗余度荷载P2单独作用下,节点1的水平反力为零,节点1、3竖向反力为
将各杆件在荷载P2单独作用下产生的应变能代入结构的鲁棒性和冗余度表达式,可知不同杆件的结构鲁棒性和冗余度,其表达式可记为
$R_{12}^{{P_2}} = R_{23}^{{P_2}} = 1 - \frac{{{P_2}^2l}}{{24EA{U_0}}} = 1 - \frac{1}{{12}}{\left( {\frac{{{P_2}}}{F}} \right)^2},$ (26)
$\begin{array}{*{20}{c}}{R_{12}^{{P_2}} = R_{24}^{{P_2}} = R_{25}^{{P_2}} = R_{35}^{{P_2}} = R_{45}^{{P_2}} = 1 - \frac{{{P_2}^2l}}{{6EA{U_0}}} = }\\{1 - \frac{1}{3}{{\left( {\frac{{{P_2}}}{F}} \right)}^2}.}\end{array}$ (27)
3.3.3 荷载P3单独作用下的结构鲁棒性和冗余度荷载P3单独作用下,节点1的水平反力为零,节点1、3竖向反力分别为
将各杆件在荷载P3单独作用下产生的应变能代入结构的鲁棒性和冗余度表达式,可知不同杆件的结构鲁棒性和冗余度,其表达式可记为
$R_{12}^{{P_3}} = 1 - \frac{{{P_3}^2l}}{{96EA{U_0}}} = 1 - \frac{1}{{48}}{\left( {\frac{{{P_3}}}{F}} \right)^2},$ (28)
$R_{23}^{{P_3}} = 1 - \frac{{3{P_3}^2l}}{{32EA{U_0}}} = 1 - \frac{3}{{16}}{\left( {\frac{{{P_3}}}{F}} \right)^2},$ (29)
$R_{35}^{{P_3}} = 1 - \frac{{3{P_3}^2l}}{{8EA{U_0}}} = 1 - \frac{3}{4}{\left( {\frac{{{P_3}}}{F}} \right)^2},$ (30)
$\begin{array}{*{20}{c}}{R_{14}^{{P_3}} = R_{24}^{{P_3}} = R_{25}^{{P_3}} = R_{45}^{{P_3}} = 1 - \frac{{{P_3}^2l}}{{24EA{U_0}}} = }\\{1 - \frac{1}{{12}}{{\left( {\frac{{{P_3}}}{F}} \right)}^2}.}\end{array}$ (31)
由式(22)~式(31)可知,不同荷载作用下,各杆件的结构鲁棒性和冗余度均与其外荷载值和杆件轴向力极限值相关,并受外荷载值与杆件轴向力极限值之比的直接影响.又因为P1=P2=P3=P0,所以
Figure 4
![点击查看原图](http://hit.alljournals.cn/html/hitxb_cn/2018/9/PIC/hebgydxxb-50-9-1-4.jpg)
由图 4可知,结构不同杆件因为可能产生相同的应变能而具有相同的结构鲁棒性和冗余度.不同荷载位置作用下,杆件14、35的结构鲁棒性和冗余度较小,即杆件14、35对外荷载作用下的容许能力较小,为抗力较薄弱杆件.
由3.1节可知,结构的鲁棒性、冗余度和易损性受杆件强度、杆端约束以及荷载路径影响显著.结构设计及分析过程中,可通过提高结构构件的材料强度及截面尺寸、增加杆端约束、优化荷载分配等措施增强和提高结构的鲁棒性、冗余度,即降低结构的易损性;也可据此找出并补强结构的抗力薄弱杆件,进而降低结构损伤甚至安全事故所造成的损失.
4 结论1) 通过综述结构的鲁棒性、冗余度和易损性研究进展,探讨了结构的鲁棒性、冗余度和易损性本质含义,对比分析了已有各评价指标的表达形式,进而提出了基于应变能的评价指标,并进一步分析了结构的鲁棒性、冗余度和易损性之间的相互关系及其基本影响因素.
2) 结构的鲁棒性、冗余度和易损性分析源于不同时期、不同领域,其本质均是反映结构特别是损伤结构抵抗外力作用的能力,其中结构鲁棒性、冗余度侧重结构在外力作用下安全储备或剩余容许能力,而结构易损性则更直观反映了结构在外力作用下的破坏程度.
3) 结构鲁棒性、冗余度评价指标具有相似的表达形式,并与易损性呈反比例变化;可通过提高结构构件的材料强度及截面尺寸、增加杆端约束、优化荷载分配等措施增强和提高结构的鲁棒性、冗余度,即降低结构的易损性.
4) 基于所提出的应变能评价指标,可进一步分析结构尤其是损伤结构抵抗外力作用的能力,找出并补强结构抗力薄弱部位,进而降低结构损伤甚至安全事故所造成的损失.
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