Variability response functions for stochastic plate bending problems

In this paper, the weighted integral method and the concept of variability response function are successfully extended to plate bending problems where the elastic modulus of the structure is considered to be a two-dimensional, homogeneous stochastic field, overcoming earlier computational problems associated with the large number of terms in the expression for the variability response function. The concept of the variability response function is used to compute spectral-distribution-free upper bounds of the response variability. Such bounds are of paramount importance for the majority of real-life problems where only first and second moments of the stochastic material properties can be estimated with reasonable accuracy. Under the assumption of a prespecified power spectral density function of the stochastic field describing the elastic modulus, it is also possible to compute the response variability (in terms of second moments of response quantities) and the reliability (in terms of the safety index) of the stochastic plate. The use of a variability response function based on the local averaging method reduces the computational effort associated with the weighted integral method, with only a small loss of accuracy in most cases. Numerical examples are provided to demonstrate all of the above capabilities. One of the conclusions is that the coefficient of variation of certain response quantities can become larger than the coefficient of variation of the elastic modulus (the input quantity).     

Effect of spatial variability on stability and failure mechanisms of 3D slope using random limit equilibrium method

It is well known that soils are prone to spatial non-uniformity, which affects evaluations of slope stability and failure mechanisms. This paper presents a probabilistic slope stability evaluation, considering the 3D spatial variation in the soil properties, by the random limit equilibrium method (RLEM). Specifically, 3D random fields of cohesion c, friction angle ?, and soil unit weight γ are generated using a fast Fourier transform. The RLEM is applied to evaluate the effects of the 3D spatial variability of the soil properties on slope stability and failure mechanisms. A Monte Carlo simulation is used to interpret the slope reliability and variation in slope failure dimension. Based on the critical slip surface passing different portions of a slope (slope base, inclined face, and crest), four main failure mechanisms (two base failures and two face failures), and one additional failure mechanism (toe failure), are identified for spatially variable slopes, and the corresponding distributions of the stability number (N_s) and sliding volume (V) are investigated in detail. The results show that the large variation in the soil properties induces changes in the failure mechanisms, and a threshold of c-? values is found for a shift from base failure to toe failure. Lastly, associated sensitivity studies are performed to explore the effects of the uncertainties of the input parameters on the uncertainty of the output. The results estimated by partial Spearman correlation coefficients show that cohesion has the greatest influence on the stability number, and that a positive influence of the unit weight, contributing to slope stability, is found for a base failure mechanism.     

Reliability analysis of slopes considering spatial variability of soil properties based on efficiently identified representative slip surfaces

Slope reliability analysis considering inherent spatial variability (ISV) of soil properties is time-consuming when response surface method (RSM) is used, because of the “curse of dimensionality”. This paper proposes an effective method for identification of representative slip surfaces (RSSs) of slopes with spatially varied soils within the framework of limit equilibrium method (LEM), which utilizes an adaptive K-means clustering approach. Then, an improved slope reliability analysis based on the RSSs and RSM considering soil spatial variability, in perspective of computation efficiency, is established. The detailed implementation procedure of the proposed method is well documented, and the ability of the method in identifying RSSs and estimating reliability is investigated via three slope examples. Results show that the proposed method can automatically identify the RSSs of slope with only one evaluation of the conventional deterministic slope stability model. The RSSs are invariant with the statistics of soil properties, which allows parametric studies that are often required in slope reliability analysis to be efficiently achieved with ease. It is also found that the proposed method provides comparable values of factor of safety (FS) and probability of failure (P_f) of slopes with those obtained from direct analysis and literature.     

考虑参数空间变异性的失稳边坡参数概率反分析

概率反分析能够有效地考虑岩土体参数不确定性并融合现场监测数据和观测信息等更新岩土体参数统计特征,进而使得边坡稳定性评价更为符合客观工程实际,然而目前参数概率反分析几乎没有考虑参数固有空间变异性的影响。结合多重响应面和子集模拟提出了考虑岩土体参数空间变异性的边坡参数概率反分析方法,并以芝加哥国会街切坡为例,融合边坡失稳和滑动面入滑点与出滑点的大致位置这两个现场观测信息,概率反分析得到边坡不排水抗剪强度参数的后验统计特征。结果表明:本文提出方法可以有效地解决考虑参数空间变异性的低概率水平边坡参数概率反分析问题,具有较高的计算效率。子集模拟中每层随机样本数目对计算结果具有重要的影响,常用的500组样本点难以获得满意的计算结果。此外,土体参数空间变异性对概率反分析计算结果具有重要的影响,考虑参数空间变异性边坡参数由平稳随机场更新为非平稳随机场,与工程实际相符,然而忽略参数空间变异性更新后的参数仍服从平稳分布。     

基于高斯过程回归的响应面法及边坡可靠度分析

提出了一种新型高斯过程响应面法(GPRSM),通过高斯过程回归算法构建随机变量与功能函数响应值之间的关系。该方法相较多项式响应面法对于功能函数为高维和高度非线性的可靠度问题,具有更高的精度和计算效率。此方法可以通过新增加训练点以动态更新响应面函数。与此同时,为了模拟岩土参数的空间变异性,通过KL展开构建随机场,并与极限平衡法结合进行边坡稳定性分析。使用提出的GPRSM构建替代模型并用于蒙特卡洛模拟求解边坡失稳概率,在保障计算精度的同时减少了对边坡稳定性分析程序的调用。最后将所提出的方法分别应用于功能函数为显式和隐式两个案例,并与其他论文中的方法对比,证明了该方法的有效性和适用性。     

Soil spatial variability impact on the behavior of a reinforced earth wall

This article presents the soil spatial variability effect on the performance of a reinforced earth wall. The serviceability limit state is considered in the analysis. Both cases of isotropic and anisotropic non-normal random fields are implemented for the soil properties. The Karhunen-Loève expansion method is used for the discretization of the random field. Numerical finite difference models are considered as deterministic models. The Monte Carlo simulation technique is used to obtain the deformation response variability of the reinforced soil retaining wall. The influences of the spatial variability response of the geotechnical system in terms of horizontal facing displacement is presented and discussed. The results obtained show that the spatial variability has an important influence on the facing horizontal displacement as well as on the failure probability.     

考虑参数空间变异性的边坡可靠度分析非侵入式 随机有限元法

提出了考虑土体参数空间变异性的边坡可靠度分析的非侵入式随机有限元法。采用Karhunen-Loeve级数展开方法表征土体抗剪强度参数空间变异性,其中通过wavelet-Galerkin技术求解Fredholm积分方程得到相关函数的特征解。基于有限元滑面应力法计算边坡安全系数,采用随机多项式展开将隐式函数表达的安全系数替换为显式函数表达的安全系数,并编写了计算程序NISFEM。研究了所提方法在考虑土体参数空间变异性的边坡稳定可靠度分析中的应用。结果表明：提出的非侵入式随机有限元法极大地提高了考虑土体参数空间变异性的边坡可靠度分析的计算效率,为解决复杂边坡稳定可靠度问题提供了一条有效的途径。考虑抗剪强度参数空间变异性的边坡可靠度分析存在临界变异系数,其随边坡安全系数的增加而增大。当抗剪强度参数的变异系数小于临界变异系数时,忽略土体参数空间变异性将会高估边坡失效概率。当边坡安全系数小于1时,边坡失效概率并不总是随着抗剪强度变异系数的增加而增大。此外,土体黏聚力和内摩擦角随机场间相关性对边坡失效概率具有十分明显的影响。     

A fast and efficient response surface approach for structural reliability problems

The reliability analysis of large and complex structural requires approximate techniques in order to reduce computational efforts to an acceptable level. Since it is, from an engineering point of view, desirable to make approximative assumptions at the level of the mechanical rather than the probabilistic modeling, simplifications should be carried out in the space of physically meaningful system- or loading variables.Within the context of this paper, a new adaptive interpolation scheme is suggested which enables fast and accurate representation of the system behavior by a response surface (RS). This response surface approach utilizes elementary statistical information on the basic variables (mean values and standard deviations) to increase the efficiency and accuracy. Thus the RS obtained is independent of the type of distribution or correlations among the basic variables which enables sensitivity studies with respect to these parameters without much computational effort.Subsequently, the response surface is utilized in conjunction with advanced Monte Carlo simulation techniques (importance sampling) to obtain the desired reliability estimates.Numerical examples are carried out in order to show the applicability of the suggested approach to structural systems reliability problems. The proposed method is shown to be superior both in efficiency and accuracy to existing approximate methods, i.e., the first order reliability methods.     

On the stochastic nature of compact debris flight

The stochastic nature of debris flight is investigated through a series of Monte Carlo simulations based on the debris flight equations for compact debris presented by Holmes (2004). Any given debris flight situation presents a number of uncertainties such as the size of the piece of debris and the time-varying turbulent wind flow. Current debris flight models are largely deterministic and do not account for such uncertainty in input parameters. The simulations presented model the flight of a single spherical particle whose diameter is given by a probability distribution function, driven by a turbulent wind with velocity fluctuations appropriate to the atmospheric boundary layer. The model predicts the mean and standard deviation of the particle flight distance and impact kinetic energy. Results show that introducing uncertainty in particle diameter, horizontal turbulence intensity, or vertical turbulence intensity leads to larger mean values for flight distance and impact kinetic energy, compared to the condition where there is no variability in input parameters. Introducing input parameter variability also leads to variability in flight distance and impact kinetic energy that is quantified in this study. While the simulations presented do not realistically characterize the complex flow within an urban canopy, the results provide significant physical insight into the influence of particle size variability and turbulence on the mean and standard deviation of the flight distance and impact kinetic energy.     

High-order limit state functions in the response surface method for structural reliability analysis

The stochastic response surface method (SRSM) is a technique for the reliability analysis of complex systems with low failure probabilities, for which Monte Carlo simulation (MCS) is too computationally intensive and for which approximate methods are inaccurate. Typically, the SRSM approximates a limit state function with a multi-dimensional quadratic polynomial by fitting the polynomial to a number of sampling points from the limit state function. This method can give biased approximations of the failure probability for cases in which the quadratic response surface can not conform to the true limit state function’s nonlinearities. In contrast to recently proposed algorithms which focus on the positions of sample points to improve the accuracy of the quadratic SRSM, this paper describes the use of higher order polynomials in order to approximate the true limit state more accurately. The use of higher order polynomials has received relatively little attention to date because of problems associated with ill-conditioned systems of equations and an approximated limit state which is very inaccurate outside the domain of the sample points. To address these problems, an algorithm using orthogonal polynomials is proposed to determine the necessary polynomial orders. Four numerical examples compare the proposed algorithm with the conventional quadratic polynomial SRSM and a detailed MCS.     

非平稳随机场下饱和渗透系数空间变异性的无限长边坡稳定概率分析

土体饱和渗透系数表现为天然的空间强变异性,其均值通常沿土层深度方向呈递减趋势,但目前鲜有文献考虑这一特性。为此,以FLAC为平台,利用fish语言基于局部平均法建立了表征土体饱和渗透系数均值随深度递减的一维非平稳随机场模型,采用FLAC两相流模块模拟雨水的入渗过程,以蒙特卡罗法为框架,结合考虑正孔隙水压力的广义有效应力无限边坡稳定模型,探讨了饱和渗透系数空间变异性在不同降雨时刻下对边坡最危险滑裂面分布规律以及相应的边坡破坏概率的影响。结果表明：忽略饱和渗透系数的空间变异性的确定分析方法将不能真实反映边坡的安全性;随着降雨持时的增加,边坡最危险滑裂面发生在坡底基岩处的概率逐渐降低;随着饱和渗透系数竖向相关距离的增加,边坡最危险滑裂面发生在坡底的概率逐渐增加,而相应的边坡破坏概率却随竖向相关距离的增加而逐渐降低。     

考虑堆石料空间变异性的土石坝地震反应随机有限元分析

探讨了一种将随机场模拟技术与有限元方法相结合的随机有限元法在土石坝地震反应分析中的应用途径。采用局部平均细分法模拟堆石料物理力学性质随机场,使得能够考虑堆石料的空间变异性。提出尽量利用建立材料物理性质与力学性质关系的计算模型,以降低获取材料力学性质统计特性信息的成本。以一实际工程为例,采用该随机有限元法计算了堆石料的干密度、孔隙比、不均匀系数、平均粒径、Duncan–Chang E–B模型模量系数以及初始摩擦角具有不确定性时土石坝的随机地震响应及永久变形。结果表明:筑坝堆石料的不确定性会使大坝响应及永久变形发生一定程度的离散;忽略材料的不确定性可能导致低估大坝的地震反应;表征大坝地震反应的各性能指标并不一定都符合正态或对数正态分布。     

砂井地基固结概率特性的近似分析方法

利用蒙特卡罗模拟方法计算砂井地基固结度概率特征值。通过正交试验研究不确定性参数变异性对砂井地基固结概率特性的影响。研究了砂井地基固结概率分析的近似方法 ,并探讨了近似方法的精度。推导了砂井地基固结度期望值和方差的计算公式。研究表明 ,砂井地基固结度的概率特性对水平固结系数的不确定性最为敏感。提出了只考虑水平固结系数不确定性的砂井地基固结概率特性的近似分析方法。误差研究进一步证明了近似方法的实用性。砂井地基固结概率设计方法用设计系数将概率设计与常规设计联系起来。     

Reliability analysis and inspection updating by stochastic response surface of fatigue cracks in mixed mode

The analysis of engineering structures under fatigue crack growth aims at ensuring an appropriate reliability level over the entire operational lifetime. This paper deals with a new approach, namely the Stochastic Response Surface, to couple finite element analysis and reliability methods. The stochastic collocation method provides an explicit expression of the limit state function related to fatigue failure. This expression is used in first and second order reliability methods in order to compute the failure probability at a given structural age. When inspection is carried out, the structural reliability can be easily updated in terms of the observed crack length. Two numerical applications dealing with fatigue crack growth are presented to illustrate the proposed method, showing its performance in terms of numerical efficiency and accuracy.     

A sequential response surface method and its application in the reliability analysis of aircraft structural systems

The reliability analysis of aircraft structural systems is difficult to evaluate due to the complexity of the g-function g(x), the probability density function ƒ(x) and the dimension m of the problem. In many cases, the g-function can be very complex and the number of evaluation of g(x) may dominate the computation cost. Response Surface Methods may alleviate the problem by giving a simple approximation, g′(x), to the true g-function, which can then be used instead of g(x) for the reliability analysis.In this paper, a new Sequential Response Surface Method together with Monte Carlo Importance Sampling (MCIS) is suggested. Based on the method, a reliability analysis program RSM for aircraft structural systems is developed.Several examples are presented to illustrate the method.     

考虑岩土参数空间变异性的盾构隧道地表沉降分析

现有随机场理论能够较好地描述单一岩土参数的天然随机性、空间结构性和局部奇异性。当前面临的挑战是如何考虑多个土层、多元及统计相关的岩土参数空间变异性对岩土工程的影响。考虑岩土参数如黏聚力、内摩擦角和压缩模量等具有明显的空间变异性,将随机场理论引入盾构隧道地表沉降可靠指标分析。主要研究工作包括:(1)采用局部均值方法将多个非平稳土层同一岩土参数转化为平稳各向异性随机场变量,并给出随机场的统计特征;(2)简化岩土参数的空间变异性为一般变异性,直接采用响应面方法计算盾构隧道地表沉降可靠指标;(3)提出协同序贯高斯离散算法,在空间网格上实现多元岩土参数随机场的同步离散化,并基于经典Monte-Carlo随机模拟原理和盾构隧道数值计算,直接统计地表沉降的可靠指标;(4)引入子集Monte-Carlo随机模拟加速算法。将以上方法应用于天津~#5、~#6地铁环湖西路站—宾馆西路站区间盾构隧道四线交叠工程,结果表明,采用经典随机理论描述岩土参数的变异性,所得到的盾构隧道地表沉降可靠指标小于基于随机场理论的对应值。该结论为复杂盾构隧道的施工控制与设计优化提供了理论依据。     

基于区间变量的响应面可靠性分析方法

通过对响应面方法、区间变量和地下结构不确定性特点的研究发现:(1)响应面方法在具有不确定性结构的静力响应分析中具有相当高的准确性;(2)地下岩体结构物理、力学参数及力学响应的不确定性特征非常适合通过区间数进行表达,但结构功能函数解析表达式往往很难直接求出。因此,对基于数理统计特征的响应面方法进行了改进,提出了基于区间变量的响应面函数形式选取,试验设计方法及函数解析式的拟合程序。在此基础上,根据函数中自变量的特点,研究了区间变量函数值域区间的求解方法。其中结合响应面函数形式,重点研究了同一自变量多次出现的函数形式的值域求解方法和具体计算程序。从而形成了完整的基于区间变量的响应面非概率可靠性分析方法,利用该方法分析了某一工程实例,展示了其有效性和可行性。     

An observational method for consolidation analysis of the PVD-improved subsoil

Surcharge preloading together with prefabricated vertical drains (PVDs) have been widely used to improve properties of thick clay deposits. To assess the performance of soil improvement works, the average degree of consolidation needs to be estimated. A curve fitting formula is proposed in this paper to simulate the degree of consolidation versus a non-dimensional time factor relationship. The proposed formula fits the theoretical consolidation solutions well with a regression coefficient R² larger than 0.9996 and an error of less than 1.2%. Based on the formula, a modified Asaoka's observational method is proposed to predict the ultimate settlement and calculate the coefficient of consolidation using field settlement monitoring data. The effectiveness of the proposed observational method has been verified using some well-documented case histories. Comparisons between the proposed method and the Asaoka's method indicate that the proposed method will give a less than 1.0% higher ultimate settlement than that by the Asaoka's method and the proposed method is able to predict the c_h value with the consideration of both vertical and horizontal flow through the ratio of time factor in horizontal and vertical direction ν_hv.     

Stochastic analysis of progressive failure of fractured rock masses containing non-persistent joint sets using key block analysis

A stochastic key block method is developed for the analysis of complex blocky rock masses containing non-persistent joint sets. A robust block generation program is developed to model the non-persistent discontinuities. Various uncertainties of geological and geometrical parameters of the discontinuities are considered and Monte Carlo simulations of key blocks are performed. Based on the present analysis, progressive failure of a rock mass can be evaluated in a stochastic manner and the statistics of the key blocks including the total number and volume, the maximum and mean volume, shape and failure mechanism, etc. can be assessed. This approach is applied to a hypothetical horseshoe shaped tunnel in a highly fractured rock mass. Three scenarios with varying mean discontinuity size are analyzed to consider size effect on the predicted blocks and key blocks. It is shown quantitatively that a persistent discontinuity network assumption causes over-fragmentation of predicted blocks, overestimation of key blocks, and underestimation of the largest key block volume compared with non-persistent ones. More realistic representation of the discontinuities by considering the non-persistence is important to give out more reliable failure estimation of fractured rock mass. In addition, a case study application to a slope at the right bank of the Jinping I hydropower station has been conducted. Key block statistics is also helpful in support design.