Expanded reliability-based design of piles in spatially variable soil using efficient Monte Carlo simulations

Inherent spatial variability is considered as a major source of uncertainties in soil properties, and it affects significantly the performance of geotechnical structures. However, research that considers, directly and explicitly, the inherent spatial variability in reliability-based design (RBD) of geotechnical structures is limited. This paper develops a RBD approach that integrates a Monte Carlo Simulation (MCS)-based RBD approach, namely the expanded RBD approach, with random field theory to model, both directly and explicitly, the inherent spatial variability of soil properties in RBD of drilled shafts. The proposed approach is implemented in a commonly-available spreadsheet environment to effectively remove the hurdle of reliability computational algorithms and to provide a user-friendly graphical user interface to practicing engineers. To improve the efficiency and resolution of MCS at small probability levels, the expanded RBD approach is enhanced with an advanced MCS method called “Subset Simulation”. Equations are derived for the integration of the expanded RBD approach and Subset Simulation. The proposed approach is illustrated through a drilled shaft design example, and is applied to explore the effects of inherent spatial variability (including the scale of fluctuation and correlation structure) and to evaluate systematically the equivalent variance technique that is commonly used to indirectly model inherent spatial variability in current RBD approaches. It is found that inherent spatial variability significantly affects the RBD of drilled shafts, and its effects are considered in RBD using the proposed approach in a direct and explicit manner. In addition, the results show that the indirect modeling of inherent spatial variability using the equivalent variance technique with the simplified form of variance reduction function in RBD might lead to relatively conservative designs in design practice.     

考虑地层变异性和土体参数空间变异性的边坡可靠度全概率设计方法

目前边坡稳定设计研究中大多数考虑了土体参数的空间变异性,但忽略了地层变异性的影响。为此,提出了一种同时考虑这两类变异性的边坡可靠度全概率设计方法。在全概率设计框架内,将广义耦合马尔可夫链模型与随机场模型相耦合用于同时表征地层变异性和土体参数空间变异性,给出了所提方法的计算流程。利用澳大利亚珀斯市钻孔资料,以某边坡为例进行可靠度设计,为说明在边坡可靠度设计中同时考虑地层变异性和土体参数空间变异性的重要性,分析了仅考虑土体参数空间变异的情况,进一步分析了同时考虑两类变异性的情况,并对二者进行了比较。结果表明:所提出的边坡可靠度设计方法能够有效地考虑边坡中存在的地层变异性和土体参数空间变异性。当仅考虑土体参数空间变异性时,边坡可靠度设计结果很大程度上取决于所采用地层的分布情况,特别是地层分布中抗剪强度较强土体类型占比高于真实情况时,将导致得到的最优设计方案偏于危险。反之,若地层分布中抗剪强度较弱的土体类型占比高于真实情况,得到的最优设计方案偏于保守。因此,为准确地得到最优设计方案,在边坡可靠度设计中应同时考虑地层变异性和土体参数空间变异性的影响。     

单桩承载力计算模式的不确定性分析

岩土工程分析模型大都存在不确定性,可靠性分析需要考虑模型不确定性的影响, 本文提出了真实反映模型不确定性的计算方法。文中还根据实测资料分析了单桩极限承载力计算模式的不确定性, 通过比较分析表明用本文方法表示模型的不确定性是适当的。     

Reliability analysis of earth dams using direct coupling

Numerical methods are helpful for understanding the behaviors of geotechnical installations.However,the computational cost sometimes may become prohibitive when structural reliability analysis is performed,due to repetitive calls to the deterministic solver.In this paper,we show how accurate and efficient reliability analyses of geotechnical installations can be performed by directly coupling geotechnical software with a reliability solver.An earth dam is used as the study object under different operating conditions.The limit equilibrium method of Morgenstern-Price is used to calculate factors of safety and find the critical slip surface.The conmercial software packages Seep/W and Slope/W are coupled with StRAnD structural reliability software.Reliability indices of critical probabilistic surfaces are evaluated by the first-and second-order structural reliability methods(FORM and SORM),as well as by importance sampling Monte Carlo(ISMC)simulation.By means of sensitivity analysis,the effective friction angle(φ′)is found to be the most relevant uncertain geotechnical parameter for dam equilibrium.The correlations between different geotechnical properties are shown to be relevant in terms of equilibrium reliability indices.Finally,it is shown herein that a critical slip surface,identified in terms of the minimum factor of safety(FS),is not the critical surface in terms of the reliability index.     

Structural reliability analysis for implicit performance functions using artificial neural network

The Monte-Carlo simulation (MCS), the first-order reliability methods (FORM) and the second-order reliability methods (SORM), are three reliability analysis methods that are commonly used for structural safety evaluation. The MCS requires the calculations of hundreds and thousands of performance function values. The FORM and SORM demand the values and partial derivatives of the performance function with respect to the design random variables. Such calculations could be time-consuming or cumbersome when the performance functions are implicit. Such implicit performance functions are normally encountered when the structural systems are complicated and numerical analysis such as finite element methods has to be adopted for the prediction.To address this issue, this paper presents three artificial neural network (ANN)-based reliability analysis methods, i.e. ANN-based MCS, ANN-based FORM, and ANN-based SORM. These methods employ multi-layer feedforward ANN technique to approximate the implicit performance functions. The ANN technique uses a small set of the actual values of the implicit performance functions. Such a small set of actual data is obtained via normal numerical analysis such as finite element methods for the complicated structural system. They are used to develop a trained ANN generalization algorithm. Then a large number of the values and partial derivatives of the implicit performance functions can be obtained for conventional reliability analysis using MCS, FORM or SORM. Examples are given in the paper to illustrate why and how the proposed ANN-based structural reliability analysis can be carried out. The results have shown the proposed approach is applicable to structural reliability analysis involving implicit performance functions. The present results are compared well with those obtained by the conventional reliability methods such as the direct Monte-Carlo simulation, the response surface method and the FORM method 2.     

不排水强度的空间变异性及单桩承载力可靠性分析

对于土性随深度变化的情况,讨论了不排水剪切强度cu空间变异性的随机场概率统计模型.由于现有的土层随机场模型都是针对平稳随机场,即土性的随机分量的,所以首先要用回归的方法分离出趋势分量来,然后才能对随机分量进行模拟.认为在进行岩土工程可靠度分析时,不仅要考虑抽样所带来的误差,还要考虑土性本身的空间变异性.并用不排水剪强度cu的变异性来计算粘性土中单桩承载力的方差,旨在阐明随机场理论在岩土工程可靠性分析中的应用.最后用一个算例作了具体说明.     

液化地基中群桩基础地震响应分析

可液化地基中桩基础地震响应分析一直是岩土工程抗震研究的热点和难点。针对这一问题,采用砂土液化大变形统一本构模型来描述可液化地基土体的应力应变关系,建立了一个3×5的群桩三维计算模型,采用三维弹塑性有限元动力时程分析,将地基、群桩基础和上部结构作为一个系统研究群桩基础的地震动响应规律,重点关注桩与土的运动相互作用以及水平方向的弯矩在地震荷载作用下的分配情况。结果表明可液化地基中桩基础的弯矩受桩与土运动相互作用影响显著;不同桩的弯矩最大值不同,角桩最大,边桩次之,中心桩最小;弯矩最大值出现的位置不相同,角桩和边桩弯矩最大值出现在上部非液化层与液化层界面处,中心桩弯矩最大值出现在桩头处。     

Reliability analysis of wind response of flexibly supported tall structures

Statistics and probabilistic analyses and risk assessments can be very useful decision‐making tools when dealing with structural–geotechnical problems. Wind loads, dynamic properties of soil underneath the structure and material characteristics of the structure are important factors that affect the wind action on the structure and consequently the structural wind‐induced response. Uncertainties in the estimation of these factors as a result of human error or inherent variability are at the forefront for the use of reliability approaches to evaluate the risk of failure during the service period. In the present study, probabilistic base force analyses for tall structures are performed. The substructure approach in which the soil supporting the foundation is modeled by the foundation compliances as functions of soil shear wave velocity is used to account for the soil–structure interaction efficiently. A three‐variable probabilistic approach is used to account for the uncertainties in shear wave velocity of the soil underneath the foundation, the concrete strength and the design wind speed on the calculated response and the base forces. The second moment approximation using Taylor series expansion is used to perform the probabilistic analyses of the base cross‐section design and resistant forces of a free‐standing tower. The first‐order reliability method is used to examine the failure probability and the contribution to the total uncertainty. The results show that the dynamic response of the tower increases as soil shear wave velocity decreases. For the range of soil shear wave velocity encountered in practice, the base forces of the structure may increase by up to 20% as a result of the foundation flexibility. For the limit state considered in this study, it was found that the reliability index decreases by up to 15% and the probability of failure increases by up to one order of magnitude as a result of the soil–structure interaction effect. Copyright © 2003 John Wiley & Sons, Ltd.     

Kriging Numerical Models for Geotechnical Reliability Analysis

When the performance function is an implicit numerical model, geotechnical reliability analysis can be challenging due to the coupling between the deterministic numerical evaluation and reliability analysis. Previously, the kriging method was used in geotechnical engineering for modeling the spatial variability of soil properties. In this paper, we illustrate a first-order reliability analysis method based on a kriging approximation of the deterministic numerical model. The key idea in this method is to first calibrate a kriging model to approximate the deterministic numerical model, and then to evaluate the failure probability based on the kriging model. As any stand-alone software for deterministic geotechnical numerical analysis can be potentially used to generate samples for calibrating a kriging model, it can then be ultimately used for a reliability analysis. As such, this method provides a practical way for practitioners to perform reliability analysis based on existing deterministic geotechnical software. The effectiveness of the suggested method is illustrated through a pile foundation example, a shallow foundation example, and a slope example in which the performance functions do not have explicit forms. The kriging method is used here as a tool for interpolating and approximating deterministic numerical models. The present paper does not address any type of spatial variation of soil properties.     

一种新的边坡稳定性因素敏感性分析方法——可靠度分析方法

传统的边坡稳定性因素敏感性分析方法基于确定性计算,仅能得到边坡稳定性对各变量的敏感性大小,无法获得边坡稳定性对不同滑面位置上的同一岩土参数的敏感性。在可靠度分析基础上提出一种新的边坡稳定性因素敏感性分析方法——可靠度分析方法,该方法采用随机场模型来描述边坡滑面上岩土参数的空间变异性,通过验算点法优化求解实现,分析结果可得到边坡的最小可靠指标、概率临界滑面、参数敏感性因子在临界滑面上的分布曲线。研究结果表明,敏感性因子分布曲线随边坡尺度和岩土参数波动范围的相对大小变化而变化,根据边坡可靠度指标在临界滑面上的不同位置上的敏感性因子,可对边坡加固位置提出设计改进,将有限的加固措施加于敏感性因子较高的位置,能够更有效地提高边坡的稳定性,较之传统的边坡稳定性因素敏感性分析方法具有明显的优越性。     

Prediction of axial load bearing capacity of PHC nodular pile using Bayesian regularization artificial neural network

Pre-stressed precast high strength concrete (PHC) nodular piles with hyper-MEGA construction method are favorably used in medium to high-rise building foundations. In this study, a feed-forward neural network (FFNN) was adopted to investigate the ultimate axial load bearing capacity of the PHC nodular pile. The network receives the composite pile and geotechnical conditions with eight input neurons and outputs the nodular pile's ultimate axial load bearing capacity. Among numerous possible FFNN network architectures, the most accurate one is determined by optimizing the hidden layer. Network training is conducted with Bayesian regularization backpropagation (BRB); the training datasets consist of static pile load test and standard penetration test index of soil profile collected from various projects in Vietnam. The significance of each input parameter is quantified with importance-based sensitivity analysis. An explicit function has been constructed from weights and bias values at each neuron in the FFNN to estimate the axial load bearing capacity. The excellent agreement of all output values by the proposed FFNN with the measured values proved the model’s robustness and reliability. The predictive capacity of the proposed FFNN model has significantly outperformed all current empirical formulas. The outcome of this study can be directly put into engineering practice to furnish an economically optimal design of the composite nodular pile.     

Undrained bearing capacity of spatially random soil for rectangular footings

In this study, a new approach for the evaluation of the 3-D random bearing capacity in rectangular footings is described. The soil strength spatial variability is modelled by random fields. Those fields are discretized via Vanmarcke’s spatial averaging approach to single random variables, which correspond to particular slip surfaces or volumes (energy dissipation regions). A series of numerical analyses is performed for a variety of foundation shapes and isotropic and anisotropic correlation structures for soil strength parameters. The results show that the consideration of three-dimensional cases for relatively long foundations offers significantly lower bearing capacity coefficients of variation. Thus, its impact on reliability assessments is crucial. The worst case correlation length for the bearing capacity mean values is observed and discussed.     

Reliability-based settlement analysis of embankments over soft soils reinforced with T-shaped deep cement mixing piles

This paper presents a reliability-based settlement analysis of T-shaped deep cement mixing (TDM) pile-supported embankments over soft soils. The uncertainties of the mechanical properties of the in-situ soil, pile, and embankment, and the effect of the pile shape are considered simultaneously. The analyses are performed using Monte Carlo Simulations in combination with an adaptive Kriging (using adaptive sampling algorithm). Individual and system failure probabilities, in terms of the differential and maximum settlements (serviceability limit state (SLS) requirements), are considered. The reliability results for the embankments supported by TDM piles, with various shapes, are compared and discussed together with the results for conventional deep cement mixing pile-supported embankments with equivalent pile volumes. The influences of the inherent variabilities in the material properties (mean and coefficient of variation values) on the reliability of the piled embankments, are also investigated. This study shows that large TDM piles, particularly those with a shape factor of greater than 3, can enhance the reliability of the embankment in terms of SLS requirements, and even avoid unacceptable reliability levels caused by variability in the material properties.     

不同桩体刚度对复合地基的影响

结合两例不同桩体刚度的工程实例———石灰搅拌桩和CFG桩复合地基 ,进行了室内试验、现场载荷试验及非线性有限元分析 ,探讨了柔性桩复合地基和半刚性桩复合地基受力、变形性能的不同所在 ,指出桩体刚度的变化是影响复合地基承载特性的重要因素 ,进而提出了区分这两类复合地基的大致界限范围。同时 ,通过现场载荷试验 ,验证了数值分析方法的可靠性和地基土本构模型选取的合理性     

基于随机场理论的桩基沉降预测方法

针对土性指标普遍存在的变异性与自相关性 ,引入随机场理论分析方法 ,在研究桩基沉降概率分布特征及其特征值计算的基础上 ,提出了一种估算桩基沉降上限的方法 ,建立了一种新的桩基沉降预测方法     

预应力高强混凝土管桩防腐蚀问题的讨论

在地下土、水介质或其他环境、地质因素对管桩有侵蚀作用时,混凝土管桩能不能使用?要不要防腐处理?如何进行防腐处理?因目前无有关规范作出具体规定,本文分析了预应力混凝土管桩被腐蚀的原因,提出了防腐措施,可供桩基规范修订或设计与施工参考。     

Probabilistic analysis of site-specific load-displacement behaviour of cement-fly ash-gravel piles

Cement-fly ash-gravel (CFG) piles comprise one of the novel ground-improvement methods. They are attracting considerable attention, as their beneficial role in accessible soft foundations is being increasingly exploited by the surging urban industrial development in China. Despite their recognised significance, however, the load-settlement behaviour of CFG piles is poorly understood, particularly in the context of probabilistic assessments. This study focuses on six full-scale CFG pile load test datasets, consisting of 245 samples collected at six sites in the Beijing region, under static axial compression loading. As a part of this investigation, a regression curve is applied to the load-settlement data for each load test in the database using a two-parametric hyperbolic or power law curve-fitting equation. Moreover, an assessment of the multiple load-displacement curves, based on the full set of pile load measurements conducted at a particular test site, reveals that the scatter observed in the regression parameter values is mainly caused by the inherent soil variability.Thus, a bivariate copula-based mixed distribution is chosen to represent the dependence between these regression parameters. A simple copula-based simulation model is used to estimate the reliability index at any specific allowable settlement for the serviceability limit state (SLS) design. The correlation coefficients in the copula-based distributions of the regression parameters are proven to have an impact on the reliability index of this pile foundation. A scatter analysis of the load-displacement behaviour provides insight into the probabilistic design of site-specific CFG pile foundations.     

Modeling of non-stationary random field of undrained shear strength of soil for slope reliability analysis

The spatial variability of soil properties is often assumed to be modeled as stationary or weakly stationary random fields in slope reliability analyses. However, abundant site-specific data have revealed that the mean and standard deviation of soil properties, such as the undrained shear strength of soil, change with depth. Thus, the non-stationary characteristics of soil properties need to be properly accounted for. The aim of this paper is to propose a non-stationary random field (RF) model for the characterization of the spatial variability and the depth-dependent nature of the undrained shear strength of soil. With the proposed model, the uncertainties of the trend and fluctuating components can be modeled individually. As an example, a clay slope under undrained conditions is investigated to illustrate the proposed model. A subset simulation is carried out to evaluate the slope reliability incorporating the non-stationary characteristics of soil properties. The advantages of the proposed model, relative to the existing non-stationary RF models and the commonly-used stationary RF model in the literature, are demonstrated through a series of sensitivity studies.     

Settlement predictions of footings on sands using probabilistic analysis

The design of footings on sands is often controlled by settlement rather than bearing capacity.Therefore,settlement predictions are essential in the design of shallow foundations.However,predicted settlements of footings are highly dependent on the chosen elastic modulus and the used method.This paper presents the use of probabilistic analysis to evaluate the variability of predicted settlements of footings on sands,focusing on the load curve(predicted settlements)characterization.Three methodologies,the first-and second-order second-moment(FOSM and SOSM),and Monte Carlo simulation(MCS),for calculating the mean and variance of the estimated settlements through Schmertmann(1970)'s equation,are presented and discussed.The soil beneath the footing is treated as an uncorrelated layered material,so the total settlement and variance are found by adding up the increments of the layers.The deformability modulus(ESi)is considered as the only independent random variable.As an example of application,a hypothetical case of a typical subsoil in the state of Espirito Santo,southeast of Brazil,is evaluated.The results indicate that there is a significant similarity between the SOSM and MCS methods,while the FOSM method underestimates the results due to the non-consideration of the high-order terms in Taylor's series.The contribution of the knowledge of the uncertainties in settlement prediction can provide a safer design.     

Proposed hybrid approach for three-dimensional subsurface simulation to improve boundary determination and design of optimum site investigation plan for pile foundations

Geological uncertainty refers to the changeability of a geomaterial category embedded in another. It arises from predicting a geomaterial category at unobserved locations using categorical data from a site investigation (SI). In the design of bridge foundations, geological uncertainty is often not considered because of the difficulties of assessing it using sparse borehole data, validating the quality of predictions, and incorporating such uncertainties into pile foundation design. To overcome these problems, this study utilizes sparse borehole data and proposes a hybrid approach of various spatial Markov Chain (spMC) models and Monte Carlo simulation to predict three-dimensional (3D) geomaterial categories and assess geological uncertainties. The 3D analysis gives realistic and comprehensive information about the site. Characteristics of the proposed hybrid approach include the estimation of transition rates, prediction of 3D geomaterial categories, and simulation of multiple realizations to propagate the uncertainties quantified by information entropy. This proposed hybrid approach leads to specific novelties that include the development of optimal SI plans to reduce geological uncertainty and the determination of geomaterial layer boundaries according to the quantified geological uncertainty. Reducing the geological uncertainties and accurately determining spatial geomaterial boundaries will improve the design reliability and safety of bridge foundations. The hybrid approach is applied to the Lodgepole Creek Bridge project site in Wyoming to demonstrate the application of the hybrid approach and the associated novelties. Outcomes are cross-validated to evaluate the geomaterial prediction accuracy of the hybrid approach.