【作者】 张钰；

【导师】 徐军；

【作者基本信息】 湖南大学 ， 土木工程， 2021， 硕士

【摘要】 在实际的土木工程问题中,物理参数、外荷载和边界条件具有不确定性,这些不确定性会极大地影响到诸如位移、剪力和弯矩的系统输出。因此,为实现结构设计的可靠性和保证结构安全,考虑这些不确定性进行结构可靠度分析是十分必要的。近些年来研究人员发展了不同的数值方法用于可靠度分析,例如基于抽样的方法,基于矩的方法和代理模型等。随着计算机技术的发展,越来越精细的数值模型将应用于实际工程,探究高效的可靠性分析方法是一个关键问题。在本文中,首先利用一种称之为贡献度分析的高效灵敏性分析方法确定随机变量间的关系,随后基于贡献度分析的结果,本文提出了几种高效的数值方法用于可靠度分析:(1)通过引入贡献度分析,将输入系统的原始随机变量分为几个子向量:其中一个包含若干相互联系紧密的随机变量,其余子向量各包含一个与其他变量联系较弱的随机变量。根据贡献度分析结果和一种精确降维模型,本文建立了一种新的自适应降维模型。将求解统计矩的高维高斯权重积分分解为一个低维高斯权重积分和若干个一维高斯权重积分,该策略仅需要较少的计算量便可求解系统响应的统计矩。然后通过结合一种5-7阶的容积公式,提出了一种混合求矩方法。一旦得到功能函数的统计矩,可利用一种灵活的参数化概率密度函数模型(Pearson系统)来重构结构功能函数潜在的概率密度函数,而失效概率和可靠指标便可以很容易地通过该概率密度函数计算得到。(2)本文提出两种新的多项式混沌展开方法。根据基于贡献度分析的降维模型,原始结构功能函数可分解为一个低维子函数和若干一维子函数的和的形式。然后可利用完全的多项式混沌展开方法和稀疏的多项式混沌展开方法替代每个子函数。由于避免了直接重构复杂的原始功能函数,通过构造结合基于贡献度分析的降维模型的多项式混沌展开极大地降低了计算压力,同时精度也可得到保证。一旦建立了功能函数的多项式混沌展开代理模型,便可利用蒙特卡洛模拟方法进行结构可靠度分析。(3)本文提出一种改进的基于分数阶矩的最大熵方法。首先利用Laplace变换将分数阶矩转换为Laplace分数阶矩。通过结合基于贡献度分析的混合求矩方法和5阶容积公式求解Laplace分数阶矩。然后利用两种有效策略减少分数阶矩最大熵方法的分数阶阶数和拉格朗日乘子初值个数,从而使得改进的最大熵方法仅需要确定一个初值条件。因此功能函数的概率密度函数可以通过改进的最大熵方法建立,并且结果的鲁棒性得到了改善。(4)考虑到输入随机变量间的相关性,本文提出一种直接变换的方法用于结构可靠度分析。估计统计矩时,利用基于贡献度分析的混合求矩方法和协方差矩阵生成考虑变量间相关性的积分点。一旦求解得到功能函数的统计矩,便可利用改进的最大熵方法重构功能函数的概率密度函数。更多还原

【Abstract】 In practical civil engineering problems,uncertainties widely exist in physical parameters,external loads and boundary conditions,etc.,and significantly effect system outputs,such as displacements,shear forces and moments.Hence,to achieve a reliable structural design and ensure the safety of structures,it is necessary to take all these uncertainties into account and implement the reliability analysis.In recent years,various methods are developed for this purpose,such as sampling-based methods,moment-based methods and surrogated models,however,exploring the efficient numerical methods for reliability analysis is still a key issue especially the application of more and more elaborate numerical models.In this thesis,an efficient sensitivity analysis method called contribution-degree analysis(CDA)is used for determining relationships of the input random variables before implementing reliability analysis.Based on the results of the contribution-degree analysis,several efficient numerical methods for structural reliability analysis are proposed as followings:(1)The original input random variables are classified into several sub-vectors:one contains variables which have obvious interactions with each other and each of the remaining sub-vectors only contains one random variable that has less interaction with other variables via the contribution-degree analysis.According to the results of CDA and an exact dimension-reduction model,a novel adaptive dimension-reduction model is established.The Gaussian-weighted integration(GWI)for calculating the statistical moments of system outputs is divided into a lower-dimensional GWI and several one-dimensional GWIs,which means that fewer model evaluations are employed for approximating the statistical moments,and by combing with a 5-7degree cubature formula,a hybrid moments estimation approach is developed.Once the statistical moments are obtained,a flexible parametric probability density function(PDF)called Pearson system is used for recovering the underlying PDF of the performance function of the system output,then the failure probability and reliability index can be easily obtained.(2)Two novel surrogate models are established for structural reliability analysis.According to the CDA-based dimension-reduction model,the original performance function of the system output is decomposed as a summation of one lower-dimensional component function and several one-dimensional component functions.Then the full polynomial chaos expansion(PCE)and sparse PCE can be employed to reconstruct the component functions.Due to avoiding building the original complicated performance function,the computational burden for building the novel PCE is eased via the CDA-based dimension-reduction model and the accuracy can be also ensured.Once the PCE surrogate model is built,the Monte Carlo simulation(MCS)can be easily implemented by using the surrogate model for structural reliability analysis.(3)An improved fractional moments-based maximum entropy(FM-MEM)method is proposed for efficient reliability analysis.First,the Laplace transform is employed to transform the fractional moments into Laplace fractional moments.By combining the CDA-based hybrid moments estimation method with a 5th-degree cubature formula,the Laplace fractional moments can be approximated efficiently.Then two useful strategies are employed to reduce the number of the initial conditions of FM-MEM,i.e.,the order of fractional moments and Lagrangian multipliers,and thus only one initial condition is required to be specified.The PDF of the performance function can be easily established and the robustness of the result is improved.(4)Considering the correlation of input random variables,a straightforward transformation method is proposed for structural reliability analysis.For moments estimation,the integration points considering the correlation between random variables are generated by combing the hybrid moments estimation method and covariance matrix.Once the statistical moments are obtained,the improved FM-MEM method can be employed for constructing the unknown PDF of performance function.更多还原