结构—热耦合问题及结构疲劳的可靠性分析方法研究
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摘要
在实际工程结构的分析与设计中，尤其在航天、航空等领域，结构的热问题逐步成为研究的热点，比较突出的问题如结构在热载荷作用下的热变形及热应力，在交变热载荷作用下的热疲劳。随着工程精度要求得日益提高和研究问题的不断深入，结构的热耦合响应问题也渐渐被广泛关注，如热弹耦合问题和热-结构耦合问题，它们分别对应不同的工程实际。同时考虑实际工程结构中存在着大量的误差和不确定性，使得结构的物理参数、几何参数以及载荷等均具有不确定性，从而导致结构的响应也具有不确定性，使得结构在热载荷下的可靠性分析也成为热点问题，所以有必要考虑这些因素，分析结构在热载荷下考虑热耦合情况时的可靠性，这对于热问题突出的工程结构的分析与设计是十分必要的，具有重要的理论意义及工程价值。本学位论文主要针对结构在热载荷下考虑热耦合因素时的随机响应及可靠性等问题进行了系统研究，并对疲劳可靠性分析方法做了一定的研究，主要研究工作如下：
1.随机弹性杆在随机瞬态温度场下的热响应分析
研究弹性杆在其物理参数、温度场同时具有随机性时的热响应。利用隐式差分法求解随机瞬态温度场，结合Monte-Carlo模拟法得到各时间节点处结构温度场分布的数字特征。利用随机因子的有限元法推导出在各时间节点处结构位移和单元应力响应的均值和方差计算表达式，其中加入了随机机械载荷的影响。通过悬臂杆的算例，得到结构各节点位移响应及单元应力响应在瞬态变温阶段内均值和均方差的变化规律；并考察了在随机力载荷和随机热载荷共同作用下，各随机参数的分散性对响应分散性的影响。
2.随机参数梁在考虑热弹耦合下的动力响应分析
以随机参数梁为研究对象，分析其在温度载荷和力载荷共同作用并考虑热弹耦合关系时的动力响应。建立了热弹耦合动力学有限元模型，给出了在时间域内差分离散、相互交替迭代的耦合计算方法。利用随机因子法推导了结构温度场和动力响应的数字特征表达式，其中温度场的求解利用θ时间积分法，动力响应则利用Newmark-β积分法。在求出结构各时间步温度场和动力响应数字特征的基础上，应用耦合算法获得了整个时间域内的结构响应数字特征。通过悬臂梁算例分析了热弹耦合项对动力响应的影响，并考察了诸随机参数分散性对结构动力响应分散性的影响。
3.热弹耦合下微谐振器的概率可靠性分析
建立了微谐振器的热弹耦合有限元方程，导出了热弹耦合的广义特征方程，基于左、右特征向量和分块Lanczos算法分析了热弹耦合频率随机性，通过对温度场简化并从特征值问题出发，进一步分析了温度场和热应力的随机性，分别建立了以断裂力学为基础的强度可靠性和激振频率影响下的频率可靠性，并综合考虑谐振器在这两种失效模式下的可靠性问题。通过算例，与一维Zener解和Lifshitz解对比，验证模型计算的正确性，并计算了频率偏移、可靠度随温度的影响变化情况。
4.空间结构在考虑热-结构耦合关系下的动力响应随机性分析
以空间薄壁圆管为研究对象，分析其在持续热流作用下并同时考虑热-结构耦合关系及参数具有随机性时的动力响应问题。为了便于分析由截面温差引起的热振动，将温度场分解为平均温度和扰动温度两部分，在此基础上建立了热-结构耦合动力学有限元模型，并采用时间域内温度场和动力响应交替迭代进行的近似计算方法，其中温度场的求解利用时间差分法和牛顿迭代法，动力响应则利用Newmark积分法。从各响应的求解迭代格式出发，分别使用矩法和随机因子法推导了温度场及动力响应的数字特征表达式，并通过各时间步内对温度场和动力响应的交替迭计算可求得整个时间域内的温度和动力响应均值与方差。最后以哈勃太空望远镜为算例考察了热-结构耦合对其悬臂薄壁圆管梁的热颤振影响以及参数的随机性对响应分散性的影响，并利用Monte-Carlo法验证了文中方法的可行性。
5.薄壁圆管空间结构的热疲劳可靠性分析
针对薄壁圆管的空间结构，分析其在交变热载荷下的疲劳可靠性问题。为同时考虑由截面平均温度和截面温差造成的疲劳损伤，提出了综合利用剩余强度和疲劳累积损伤模型的分析方法。首先根据疲劳累积损伤相等原理，将截面温差造成的多级扰动应力载荷作用频次等效为平均温度下的常幅应力载荷作用次数，从而将两者产生的热应力载荷统一为一常幅载荷，再利用剩余强度模型基于动态应力-强度干涉理论对疲劳可靠度进行分析，得到了结构在综合考虑两种热疲劳状态下的动态可靠度。本方法可避免直接利用疲劳累积损伤理论临界损伤值难以确定的问题，且能体现金属疲劳损伤的真实情况。最后以哈勃望远镜太阳能帆板的主梁为例，分析了其随疲劳热载荷循环作用下的动态可靠度，得出了一些有意义的结论。
6.结构疲劳可靠性分析的支持向量回归方法
从结构疲劳断裂力学角度出发建立了疲劳寿命的可靠性功能函数，考虑到该功能函数具有隐式、非线性，且较难直接利用经典的可靠性分析方法的特点，基于支持向量回归方法(SVR)出色的小样本学习能力和良好的泛化性，将其应用于结构疲劳可靠性功能函数的重构。为增强SVR的推广能力，采用遗传算法对SVR的参数进行优化，并利用优化的参数训练出最佳的回归函数，再根据所得的回归函数利用一次二阶矩法完成了疲劳可靠性的预测。最后通过算例验证了本文方法的可行性和有效性。
In the process of analysis and design of practical structures, especially in the fieldsuch as aerospace and aviation, the thermal problem is becoming serious. Theprominent problems including thermal deformation and thermal stress generated bythermal loads, thermal fatigue under alternating thermal loads. With the increasingdemand of engineering precision and research issue, the thermal coupling has graduallybeen widespread concern, such as thermoelastic coupling and thermal-structuralcoupling, which correspond to different actual engineering, respectively. Taking intoaccount that there are a large number of errors and uncertainties in the practicalstructures, which will cause the physical parameters, geometrical parameters and loadstaking on uncertainties, the analysis of reliability of the structures under thermal load ismust be considered. So it is necessary to consider these factors, analyzing the reliabilityconsidering thermal coupling under thermal load. This is very important for the analysisand design of engineering structures, and it has important engineering significance andtheoretical significance. This article mainly study the random responses and thermalreliability considering thermal coupling, and do some research for fatigue reliabilityanalysis methods, the main research works can be described as follows.
1. Thermal response analysis of stochastic pole structures under random transienttemperature field
The thermal response of stochastic elastic pole structure under random temperaturefield is researched in this paper. The transient temperature field is solved by using themethod of implicit difference and the numerical characteristics of node temperaturefield in every time step is obtained with Monte-Carlo simulation. The computationexpressions of mean and variance of displacement and stress in every time step arederived from finite element method based on random factor with consideration ofmechanical loads. An example of cantilever pole analysis shows the change to the meanvalue and mean-square deviation of displacement and stress of the random variables intransient duration, and the influence of the response dispersion due to the dispersion ofthe random parameters under both random static load and random thermal load.
2. Dynamic response analysis of stochastic beam structures under thermoelasticcoupling
To research the beam with random parameters, its dynamic response has been analyzed under both thermal load and force load when considering the effect ofthermoelastic coupling. The dynamic model considering thermoelastic coupling is setup using the finite element method, then a coupling calculate method is proposed whichmake use of the finite difference method in the time domain and alternative iteration ineach time step. The computational expressions for the numerical characteristics of thetemperature field and dynamic response are derived by using the random factor method.Temperature field is worked out by applying the time integral method, while dynamicresponse can be found through Newmark-β integral method. Base on the numericalcharacteristics expressions of temperature field and dynamic response in each time step,the expressions in whole time domain are obtained by using coupling algorithm. Finally,a cantilever beam is taken as an example and the influence of dynamic response due tothermoelastic coupling and randomness of the parameters is showed.
3. Probabilistic reliability of micro-resonators with thermoelastic coupling in shockenvironments
Analyzing the probabilistic reliability of micro-resonators is essential for thedesign of Microelectromechanical Systems (MEMS). In order to investigate morecomplex structures and take into account the random variation of the design parameters,the numerical method is used, an efficient thermoelastic finite element formulation isderived from Hamilton’s variational principle, by the formulation, and a generalizedeigenvalue equation can be obtained. The effect of thermoelastic coupling and therandomness of parameters on the natural frequency are studied by using anon-symmetric block Lanczos algorithm combine with the right and left eigenproblem.Considering the influence of thermoelastic coupling on temperature is little, thethermoelastic coupling term in heat conduction equation can be negligible, then the heatconduction equation is simplified as the linearization. Depend on the eigenproblem fortemperature field and thermal stress in micro-resonators, further analyzing therandomness of temperature and stress in micro-resonators. For reliability analysis, amethod which both considering strength and frequency is proposed, the strength failurecriteria is set based on fracture mechanics in shock environments, and frequencyreliability is taken into account the probabilistic of resonators when the naturalfrequency is close to the excitation frequency. Take a simply supported model ofmicro-resonators as an example, the numerical solutions of quality factor have shownlittle difference with the analytical solutions proposed by Zener and Lifshitz, verifingthe correctness of the model calculations, finally, the natural frequency shift and theprobabilistic reliability with temperature are calculated.
4. Randomness analysis of space structures considering thermo-structural coupling
The dynamic response for space structures consisting of thin walled pipe which aresubjected to solar heat flux has been analyzed, and then this problem has beendeveloped taking into account the thermo-structural coupling and the randomness ofparameters. For analyzing thermal vibration due to temperature difference in the crosssection conveniently, temperature field is divided into two parts including averagetemperature and disturbance temperature, then the dynamic model consideringthermo-structural coupling is set up using the finite element method. The model can besolved by the approximate calculation method through alternating iterative betweentemperature field and dynamic response in the time domain. Temperature field is solvedby using the method of time integral and Newton-iteration, while dynamic response canbe found through Newmark integral method. Starting from the solving iterative formatof each response, the computational expressions for the numerical characteristics oftemperature field and dynamic response are derived, respectively, by the method ofmoments and random factor. Through the alternate iterative algorithm in each time step,the mean value and variance of response in whole time domain are obtained. Finally,Hubble Space Telescope is taken as an example. Thermal flutter for thin-walledcantilever beam due to thermo-structural coupling is showed, and the influence ofdynamic response due to the randomness of parameters also be found, throughcomparing with the Monte Carlo method, validating the feasibility of the method in thispaper.
5. Thermal fatigue reliability analysis for space structures composed of thin-walledtube
The fatigue reliability of space structures composed of thin-walled tube elementhas been analyzed under alternating thermal load. In order to take into account thefatigue damage due to both average temperature and temperature difference in crosssection, an analytical method which uses the models of residual strength and fatiguecumulative damage is proposed. First of all, according to the principle of fatiguecumulative damage equal, the frequency of multi-level disturbance stress load caused bytemperature difference in cross section is equivalent to the times of constant amplitudestress load action under average temperature, thus all thermal stress loads will be unifiedfor a constant amplitude load, then the fatigue reliability can be analyzed by using theresidual strength model, finally, the dynamic reliability of structures in thecomprehensive consideration of two kinds of thermal fatigue state is obtained. Thismethod avoids the problem that critical damage value is difficult to determine when directly using the theory of cumulated fatigue damage, even it can reflect the realsituation for fatigue damage of metal. In the end, take Hubble Space Telescope as anexample, the dynamic reliability of the main beam with the role of cycle fatigue thermalload is analyzed, some meaningful conclusions are drawed.
6. Support vector regression for structure fatigue reliability analysis
Reliability performance function based on fatigue life is established from fatiguefracture mechanics, considering the performance function has the characteristics ofimplicit and non-linear, so it is difficult to use classic reliability calculation method. Inview of SVR has excellent learning capacity and generalization capability with a smallamount of samples, it can be used to reconstruct the fatigue reliability performancefunction. In order to enhance generalization ability, the genetic algorithm is introducedto optimize the SVR parameters, then the best regression function is trained by using theoptimal parameters, according to the regression function, using FOSM method, fatiguereliability analysis could be completed. Finally, take an example to verify the feasibilityand effectiveness of the proposed method.
1.随机弹性杆在随机瞬态温度场下的热响应分析
研究弹性杆在其物理参数、温度场同时具有随机性时的热响应。利用隐式差分法求解随机瞬态温度场，结合Monte-Carlo模拟法得到各时间节点处结构温度场分布的数字特征。利用随机因子的有限元法推导出在各时间节点处结构位移和单元应力响应的均值和方差计算表达式，其中加入了随机机械载荷的影响。通过悬臂杆的算例，得到结构各节点位移响应及单元应力响应在瞬态变温阶段内均值和均方差的变化规律；并考察了在随机力载荷和随机热载荷共同作用下，各随机参数的分散性对响应分散性的影响。
2.随机参数梁在考虑热弹耦合下的动力响应分析
以随机参数梁为研究对象，分析其在温度载荷和力载荷共同作用并考虑热弹耦合关系时的动力响应。建立了热弹耦合动力学有限元模型，给出了在时间域内差分离散、相互交替迭代的耦合计算方法。利用随机因子法推导了结构温度场和动力响应的数字特征表达式，其中温度场的求解利用θ时间积分法，动力响应则利用Newmark-β积分法。在求出结构各时间步温度场和动力响应数字特征的基础上，应用耦合算法获得了整个时间域内的结构响应数字特征。通过悬臂梁算例分析了热弹耦合项对动力响应的影响，并考察了诸随机参数分散性对结构动力响应分散性的影响。
3.热弹耦合下微谐振器的概率可靠性分析
建立了微谐振器的热弹耦合有限元方程，导出了热弹耦合的广义特征方程，基于左、右特征向量和分块Lanczos算法分析了热弹耦合频率随机性，通过对温度场简化并从特征值问题出发，进一步分析了温度场和热应力的随机性，分别建立了以断裂力学为基础的强度可靠性和激振频率影响下的频率可靠性，并综合考虑谐振器在这两种失效模式下的可靠性问题。通过算例，与一维Zener解和Lifshitz解对比，验证模型计算的正确性，并计算了频率偏移、可靠度随温度的影响变化情况。
4.空间结构在考虑热-结构耦合关系下的动力响应随机性分析
以空间薄壁圆管为研究对象，分析其在持续热流作用下并同时考虑热-结构耦合关系及参数具有随机性时的动力响应问题。为了便于分析由截面温差引起的热振动，将温度场分解为平均温度和扰动温度两部分，在此基础上建立了热-结构耦合动力学有限元模型，并采用时间域内温度场和动力响应交替迭代进行的近似计算方法，其中温度场的求解利用时间差分法和牛顿迭代法，动力响应则利用Newmark积分法。从各响应的求解迭代格式出发，分别使用矩法和随机因子法推导了温度场及动力响应的数字特征表达式，并通过各时间步内对温度场和动力响应的交替迭计算可求得整个时间域内的温度和动力响应均值与方差。最后以哈勃太空望远镜为算例考察了热-结构耦合对其悬臂薄壁圆管梁的热颤振影响以及参数的随机性对响应分散性的影响，并利用Monte-Carlo法验证了文中方法的可行性。
5.薄壁圆管空间结构的热疲劳可靠性分析
针对薄壁圆管的空间结构，分析其在交变热载荷下的疲劳可靠性问题。为同时考虑由截面平均温度和截面温差造成的疲劳损伤，提出了综合利用剩余强度和疲劳累积损伤模型的分析方法。首先根据疲劳累积损伤相等原理，将截面温差造成的多级扰动应力载荷作用频次等效为平均温度下的常幅应力载荷作用次数，从而将两者产生的热应力载荷统一为一常幅载荷，再利用剩余强度模型基于动态应力-强度干涉理论对疲劳可靠度进行分析，得到了结构在综合考虑两种热疲劳状态下的动态可靠度。本方法可避免直接利用疲劳累积损伤理论临界损伤值难以确定的问题，且能体现金属疲劳损伤的真实情况。最后以哈勃望远镜太阳能帆板的主梁为例，分析了其随疲劳热载荷循环作用下的动态可靠度，得出了一些有意义的结论。
6.结构疲劳可靠性分析的支持向量回归方法
从结构疲劳断裂力学角度出发建立了疲劳寿命的可靠性功能函数，考虑到该功能函数具有隐式、非线性，且较难直接利用经典的可靠性分析方法的特点，基于支持向量回归方法(SVR)出色的小样本学习能力和良好的泛化性，将其应用于结构疲劳可靠性功能函数的重构。为增强SVR的推广能力，采用遗传算法对SVR的参数进行优化，并利用优化的参数训练出最佳的回归函数，再根据所得的回归函数利用一次二阶矩法完成了疲劳可靠性的预测。最后通过算例验证了本文方法的可行性和有效性。
In the process of analysis and design of practical structures, especially in the fieldsuch as aerospace and aviation, the thermal problem is becoming serious. Theprominent problems including thermal deformation and thermal stress generated bythermal loads, thermal fatigue under alternating thermal loads. With the increasingdemand of engineering precision and research issue, the thermal coupling has graduallybeen widespread concern, such as thermoelastic coupling and thermal-structuralcoupling, which correspond to different actual engineering, respectively. Taking intoaccount that there are a large number of errors and uncertainties in the practicalstructures, which will cause the physical parameters, geometrical parameters and loadstaking on uncertainties, the analysis of reliability of the structures under thermal load ismust be considered. So it is necessary to consider these factors, analyzing the reliabilityconsidering thermal coupling under thermal load. This is very important for the analysisand design of engineering structures, and it has important engineering significance andtheoretical significance. This article mainly study the random responses and thermalreliability considering thermal coupling, and do some research for fatigue reliabilityanalysis methods, the main research works can be described as follows.
1. Thermal response analysis of stochastic pole structures under random transienttemperature field
The thermal response of stochastic elastic pole structure under random temperaturefield is researched in this paper. The transient temperature field is solved by using themethod of implicit difference and the numerical characteristics of node temperaturefield in every time step is obtained with Monte-Carlo simulation. The computationexpressions of mean and variance of displacement and stress in every time step arederived from finite element method based on random factor with consideration ofmechanical loads. An example of cantilever pole analysis shows the change to the meanvalue and mean-square deviation of displacement and stress of the random variables intransient duration, and the influence of the response dispersion due to the dispersion ofthe random parameters under both random static load and random thermal load.
2. Dynamic response analysis of stochastic beam structures under thermoelasticcoupling
To research the beam with random parameters, its dynamic response has been analyzed under both thermal load and force load when considering the effect ofthermoelastic coupling. The dynamic model considering thermoelastic coupling is setup using the finite element method, then a coupling calculate method is proposed whichmake use of the finite difference method in the time domain and alternative iteration ineach time step. The computational expressions for the numerical characteristics of thetemperature field and dynamic response are derived by using the random factor method.Temperature field is worked out by applying the time integral method, while dynamicresponse can be found through Newmark-β integral method. Base on the numericalcharacteristics expressions of temperature field and dynamic response in each time step,the expressions in whole time domain are obtained by using coupling algorithm. Finally,a cantilever beam is taken as an example and the influence of dynamic response due tothermoelastic coupling and randomness of the parameters is showed.
3. Probabilistic reliability of micro-resonators with thermoelastic coupling in shockenvironments
Analyzing the probabilistic reliability of micro-resonators is essential for thedesign of Microelectromechanical Systems (MEMS). In order to investigate morecomplex structures and take into account the random variation of the design parameters,the numerical method is used, an efficient thermoelastic finite element formulation isderived from Hamilton’s variational principle, by the formulation, and a generalizedeigenvalue equation can be obtained. The effect of thermoelastic coupling and therandomness of parameters on the natural frequency are studied by using anon-symmetric block Lanczos algorithm combine with the right and left eigenproblem.Considering the influence of thermoelastic coupling on temperature is little, thethermoelastic coupling term in heat conduction equation can be negligible, then the heatconduction equation is simplified as the linearization. Depend on the eigenproblem fortemperature field and thermal stress in micro-resonators, further analyzing therandomness of temperature and stress in micro-resonators. For reliability analysis, amethod which both considering strength and frequency is proposed, the strength failurecriteria is set based on fracture mechanics in shock environments, and frequencyreliability is taken into account the probabilistic of resonators when the naturalfrequency is close to the excitation frequency. Take a simply supported model ofmicro-resonators as an example, the numerical solutions of quality factor have shownlittle difference with the analytical solutions proposed by Zener and Lifshitz, verifingthe correctness of the model calculations, finally, the natural frequency shift and theprobabilistic reliability with temperature are calculated.
4. Randomness analysis of space structures considering thermo-structural coupling
The dynamic response for space structures consisting of thin walled pipe which aresubjected to solar heat flux has been analyzed, and then this problem has beendeveloped taking into account the thermo-structural coupling and the randomness ofparameters. For analyzing thermal vibration due to temperature difference in the crosssection conveniently, temperature field is divided into two parts including averagetemperature and disturbance temperature, then the dynamic model consideringthermo-structural coupling is set up using the finite element method. The model can besolved by the approximate calculation method through alternating iterative betweentemperature field and dynamic response in the time domain. Temperature field is solvedby using the method of time integral and Newton-iteration, while dynamic response canbe found through Newmark integral method. Starting from the solving iterative formatof each response, the computational expressions for the numerical characteristics oftemperature field and dynamic response are derived, respectively, by the method ofmoments and random factor. Through the alternate iterative algorithm in each time step,the mean value and variance of response in whole time domain are obtained. Finally,Hubble Space Telescope is taken as an example. Thermal flutter for thin-walledcantilever beam due to thermo-structural coupling is showed, and the influence ofdynamic response due to the randomness of parameters also be found, throughcomparing with the Monte Carlo method, validating the feasibility of the method in thispaper.
5. Thermal fatigue reliability analysis for space structures composed of thin-walledtube
The fatigue reliability of space structures composed of thin-walled tube elementhas been analyzed under alternating thermal load. In order to take into account thefatigue damage due to both average temperature and temperature difference in crosssection, an analytical method which uses the models of residual strength and fatiguecumulative damage is proposed. First of all, according to the principle of fatiguecumulative damage equal, the frequency of multi-level disturbance stress load caused bytemperature difference in cross section is equivalent to the times of constant amplitudestress load action under average temperature, thus all thermal stress loads will be unifiedfor a constant amplitude load, then the fatigue reliability can be analyzed by using theresidual strength model, finally, the dynamic reliability of structures in thecomprehensive consideration of two kinds of thermal fatigue state is obtained. Thismethod avoids the problem that critical damage value is difficult to determine when directly using the theory of cumulated fatigue damage, even it can reflect the realsituation for fatigue damage of metal. In the end, take Hubble Space Telescope as anexample, the dynamic reliability of the main beam with the role of cycle fatigue thermalload is analyzed, some meaningful conclusions are drawed.
6. Support vector regression for structure fatigue reliability analysis
Reliability performance function based on fatigue life is established from fatiguefracture mechanics, considering the performance function has the characteristics ofimplicit and non-linear, so it is difficult to use classic reliability calculation method. Inview of SVR has excellent learning capacity and generalization capability with a smallamount of samples, it can be used to reconstruct the fatigue reliability performancefunction. In order to enhance generalization ability, the genetic algorithm is introducedto optimize the SVR parameters, then the best regression function is trained by using theoptimal parameters, according to the regression function, using FOSM method, fatiguereliability analysis could be completed. Finally, take an example to verify the feasibilityand effectiveness of the proposed method.
引文
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