Probabilistic analysis of strip footings resting on a spatially random soil using subset simulation approach

The failure probability of geotechnical structures with spatially varying soil properties is generally computed using Monte Carlo simulation (MCS) methodology. This approach is well known to be very time-consuming when dealing with small failure probabilities. One alternative to MCS is the subset simulation approach. This approach was mainly used in the literature in cases where the uncertain parameters are modelled by random variables. In this article, it is employed in the case where the uncertain parameters are modelled by random fields. This is illustrated through the probabilistic analysis at the serviceability limit state (SLS) of a strip footing resting on a soil with a spatially varying Young's modulus. The probabilistic numerical results have shown that the probability of exceeding a tolerable vertical displacement (P _e) calculated by subset simulation is very close to that computed by MCS methodology but with a significant reduction in the number of realisations. A parametric study to investigate the effect of the soil variability (coefficient of variation and the horizontal and vertical autocorrelation lengths of the Young's modulus) on P _e was presented and discussed. Finally, a reliability-based design of strip footings was presented. It allows one to obtain the probabilistic footing breadth for a given soil variability.     

Effect of spatial correlation structure and transformation model on the design of shallow strip footing

Directional variability of spatial correlation is observed in natural soils due to their depositional characteristics and it influences the response of structures founded on these deposits. Nonetheless, the results presented in most of the available literature are based on the assumption of either isotropic spatial correlation or perfect spatial correlation of soil properties in horizontal and vertical directions. It is also observed from past studies that the effect of transformation model on the total uncertainty is quite significant. Hence, an effort has been made in this paper to study the effect of anisotropy of autocorrelation characteristics of cone tip resistance (q_c) and the transformation model on the bearing capacity of a shallow strip footing, founding on the surface of a spatially varying soil mass. The statistics in the vertical direction of the soil mass are taken from 8 Cone Penetration Test (CPT) records and statistics in the horizontal direction are assumed. For the case considered, it is observed that the transformation model significantly influences the degree of variability of design parameter. The results also show that isotropic correlation structure based on the vertical autocorrelation distance underestimates the variability of design parameter. On the other hand, perfect correlation in horizontal or vertical, or both directions, overestimates the variability of design parameters, and produces conservative estimates of allowable bearing capacity.     

Probabilistic analyses of a strip footing on horizontally stratified sandy deposit using advanced constitutive model

An advanced hypoplastic constitutive model is used in probabilistic analyses of a typical geotechnical problem, strip footing. Spatial variability of soil parameters, rather than state variables, is considered in the study. The model, including horizontal and vertical correlation lengths, was calibrated using a comprehensive set of experimental data on sand from horizontally stratified deposit. Some parameters followed normal, whereas other followed lognormal distributions. Monte-Carlo simulations revealed that the foundation displacement u_y for a given load followed closely the lognormal distribution, even though some model parameters were distributed normally. Correlation length in the vertical direction θ_v was varied in the simulation. The case of infinite correlation length was used for evaluation of different approximate probabilistic methods (first order second moment method and several point estimate methods). In the random field Monte-Carlo analyses with finite θ_v, the vertical correlation length was found to have minor effect on the mean value of u_y, but significant effect on its standard deviation. As expected, it decreased with decreasing θ_v due to spatial averaging of soil properties.     

Reliability analysis for bearing capacity of surface strip footing using fuzzy finite element method

ABSTRACT

This paper presents the reliability analysis on the basis of the foundation failure against bearing capacity using the concept of fuzzy set theory. A surface strip footing is considered for the analysis and the bearing capacity is estimated using the conventional Finite Element Method (FEM). The spatial variability of the variables is taken into consideration to capture the physical randomness of the soil parameters for an isotropic field. A variation of the probability of failure (P_f) against a varying limiting applied pressure (q) is presented for different Coefficient of Variation (COV) of the variables and different scale of fluctuation (θ). The results reveal that the friction angle of soil (?) is the most influencing parameter among the other variables. Further, the influence of the scale of fluctuation (θ) on the probability of failure (P_f) is also examined. It is observed that for a particular COV of ?, higher value of θ predicts higher P_f whereas, P_f increases as COV of ? increases for a particular θ value. Later, a comparison study is accomplished to verify the viability of the present method and it can be noticed that the present method compares well with the other reliability method (First Order Reliability Method) to a reasonably good extent.     

Bayesian approach for calibrating transformation model from spatially varied CPT data to regular geotechnical parameter

Cone Penetration Test (CPT) is widely utilized to gain regular geotechnical parameters such as compression modulus, cohesion coefficient and internal friction angle by transformation model in the site investigation. However, it is challenging to obtain simultaneously the unknown coefficients and error of a transformation model, given the intrinsic uncertainty (i.e., spatial variability) of geomaterial and the epistemic uncertainty of geotechnical investigation. A Bayesian approach is therefore proposed calibrating the transformation model based on spatial random field theory. The approach consists of three key elements: (1) three-dimensional anisotropic spatial random field theory; (2) classifications of measurement and error, and the uncertainty propagation diagram of geotechnical investigation; and (3) the unknown coefficients and error calibration of the transformation model given Bayesian inverse modeling method. The massive penetration resistance data from CPT, which is denoted as a spatial random field variable to account for the spatial variability of soil, are classified as type A data. Meanwhile, a few laboratory test data such as the compression modulus are defined as type B data. Based on the above two types of data, the unknown coefficients and error of the transformation model are inversely calibrated with consideration of intrinsic uncertainty of geomaterial, epistemic uncertainties such as measurement errors, prior knowledge uncertainty of transformation model itself, and computing uncertainties of statistical parameters as well as Bayesian method. Baseline studying indicates the proposed approach is applicable to calibrate the transformation model between CPT data and regular geotechnical parameter within spatial random field theory. Next, the calibrated transformation model was compared with classical linear regression in cross-validation, and then it was implemented at three-dimensional site characterization of the background project.     

Risk-based characterisation of an urban building site

The evaluation of the underground soil stratigraphy is a key aspect in geotechnical site characterisation. However, these means of site exploration are only pinholing subsoil conditions and expert knowledge is needed to understand subsoil conditions in order to build a reliable geological-geotechnical model. This contribution employs a geostatistical simulation methodology for the simulation of random fields representing geological uncertainty. This combines borehole data and expert knowledge via a mathematical framework. Moreover a risk-based site characterisation scheme is developed for urban site characterisation. This novel characterisation scheme offers additional insight into the effects of large-scale, geological spatial variability by using fragility curves to quantify these effects.     

苏中腹地湖相沉积土层相关距离的统计分析

相关距离是用随机场理论建模土层剖面的一个非常重要的参数,也是利用随机场理论进行岩土工程可靠性分析的关键所在。基于苏中地区某建筑工程原位静力触探测试数据中的锥尖阻力指标,针对粉质黏土层,利用不同取样间距对相关距离进行了统计计算,分析了取样间距对相关距离计算结果的影响及原因,提出了实际应用中基于尺度匹配原则的取样间距确定方法;随后,基于江苏中部某高速公路工程地质勘察所提供的大量原位静力触探测试数据,结合相关距离计算的平均零跨距法、递推空间法和相关函数法,对该区湖相沉积土层土性参数的竖直向和水平向相关距离进行了系统地统计分析。研究成果不仅提供了土性参数相关距离计算过程中取样间距的确定原则,而且获得了相关距离的区域性代表值,为区域性土性参数随机场模型的建立打下坚实的基础,能对苏中地区岩土工程可靠性分析提供参考。     

苏中腹地湖相软土土性参数变异性统计描述

基于江苏中部某高速公路工程地质勘察所提供的大量室内试验测试数据,运用随机场理论,将土层剖面模拟为随机场而非传统意义上的随机变量,提出了基于随机场理论、考虑空间趋势分量的土性参数变异性统计方法,系统地统计分析了苏中腹地湖相软土土性参数的变异特性,对比了实验室物理性质指标参数、变形参数和强度参数变异性特征。统计结果表明,该地区软土实验室物理性质指标参数变异性较小,不存在明显的趋势分量,可用传统方法进行统计估计;实验室变形和强度参数存在明显的趋势分量,统计前需对数据进行回归分析和去趋势化处理,从而基于随机场理论进行统计分析。研究不仅提供了土性参数变异性统计的新方法,而且能对苏中地区工程建设提供必要的参考,为区域性软土土性参数随机场模型的建立打下了坚实的基础。     

Effect of soil variability on the bearing capacity of clay and in slope stability problems

Site-specific geotechnical data are always random and variable in space. In the present study, a procedure for quantifying the variability in geotechnical characterization and design parameters is discussed using the site-specific cone tip resistance data (q_c) obtained from static cone penetration test (SCPT). The parameters for the spatial variability modeling of geotechnical parameters i.e. (i) existing trend function in the in situ q_c data; (ii) second moment statistics i.e. analysis of mean, variance, and auto-correlation structure of the soil strength and stiffness parameters; and (iii) inputs from the spatial correlation analysis, are utilized in the numerical modeling procedures using the finite difference numerical code FLAC 5.0. The influence of consideration of spatially variable soil parameters on the reliability-based geotechnical deign is studied for the two cases i.e. (a) bearing capacity analysis of a shallow foundation resting on a clayey soil, and (b) analysis of stability and deformation pattern of a cohesive-frictional soil slope. The study highlights the procedure for conducting a site-specific study using field test data such as SCPT in geotechnical analysis and demonstrates that a few additional computations involving soil variability provide a better insight into the role of variability in designs.     

Estimating horizontal scale of fluctuation with limited CPT soundings

The paper explores the possibility of estimating the horizontal scale of fluctuation(δ_h) with limited CPTs.The following conditions are desirable:(1) the CPT depth is large;(2) there are more than two CPTs;(3)the CPT separation distances are distinct and preferably less than 2×δ_h; and(4) the Whittle-Matern auto-correlation model is adopted.     

Effect of spatial variability of cross‐correlated soil properties on bearing capacity of strip footing

Geotechnical engineering problems are characterized by many sources of uncertainty. Some of these sources are connected to the uncertainties of soil properties involved in the analysis. In this paper, a numerical procedure for a probabilistic analysis that considers the spatial variability of cross‐correlated soil properties is presented and applied to study the bearing capacity of spatially random soil with different autocorrelation distances in the vertical and horizontal directions. The approach integrates a commercial finite difference method and random field theory into the framework of a probabilistic analysis. Two‐dimensional cross‐correlated non‐Gaussian random fields are generated based on a Karhunen–Loève expansion in a manner consistent with a specified marginal distribution function, an autocorrelation function, and cross‐correlation coefficients. A Monte Carlo simulation is then used to determine the statistical response based on the random fields. A series of analyses was performed to study the effects of uncertainty due to the spatial heterogeneity on the bearing capacity of a rough strip footing. The simulations provide insight into the application of uncertainty treatment to geotechnical problems and show the importance of the spatial variability of soil properties with regard to the outcome of a probabilistic assessment. Copyright © 2009 John Wiley & Sons, Ltd.     

Consolidation in spatially random unsaturated soils based on coupled flow‐deformation simulation

This paper integrates random field simulation of soil spatial variability with numerical modeling of coupled flow and deformation to investigate consolidation in spatially random unsaturated soil. The spatial variability of soil properties is simulated using the covariance matrix decomposition method. The random soil properties are imported into an interactive multiphysics software COMSOL to solve the governing partial differential equations. The effects of the spatial variability of Young's modulus and saturated permeability together with unsaturated hydraulic parameters on the dissipation of excess pore water pressure and settlement are investigated using an example of consolidation in a saturated‐unsaturated soil column because of loading. It is found that the surface settlement and the pore water pressure profile during the process of consolidation are significantly affected by the spatially varying Young's modulus. The mean value of the settlement of the spatially random soil is more than 100% greater than that of the deterministic case, and the surface settlement is subject to large uncertainty, which implies that consolidation settlement is difficult to predict accurately based on the conventional deterministic approach. The uncertainty of the settlement increases with the scale of fluctuation because of the averaging effect of spatial variability. The effects of spatial variability of saturated permeability k_sat and air entry parameters are much less significant than that of elastic modulus. The spatial variability of air entry value parameters affects the uncertainties of settlement and excess pore pressure mostly in the unsaturated zone. Copyright © 2016 John Wiley & Sons, Ltd.     

Predicting standard penetration test N-value from cone penetration test data using artificial neural networks

Standard Penetration Test(SPT) and Cone Penetration Test(CPT) are the most frequently used field tests to estimate soil parameters for geotechnical analysis and design.Numerous soil parameters are related to the SPT N-value.In contrast,CPT is becoming more popular for site investigation and geotechnical design.Correlation of CPT data with SPT N-value is very beneficial since most of the field parameters are related to SPT N-values.A back-propagation artificial neural network(ANN) model was developed to predict the N6o-value from CPT data.Data used in this study consisted of 109 CPT-SPT pairs for sand,sandy silt,and silty sand soils.The ANN model input variables are:CPT tip resistance(q_c),effective vertical stress(σ'_v),and CPT sleeve friction(f_s).A different set of SPT-CPT data was used to check the reliability of the developed ANN model.It was shown that ANN model either under-predicted the N_(60)-value by 7-16%or over-predicted it by 7-20%.It is concluded that back-propagation neural networks is a good tool to predict N_(60)-value from CPT data with acceptable accuracy.     

基于地质统计学方法的孔压静力触探锥尖阻力固有空间变异性研究

地质统计学是用于模拟土体固有空间变异性的方法之一,以变差函数为工具,采用Kriging插值提供未采样点处土工参数值的最优线性无偏估计。将地质统计学方法应用于宿-新（宿迁至新沂）高速公路某试验段内孔压静力触探（piezocone penetration test,CPTU）锥尖阻力qt空间变异性研究中,采用回归分析移除数据中的趋势项,从而获得具有弱平稳性的残差数据。指数型理论变差函数能够准确描述试验段内土体的连续空间变异性特征。根据估计结果,试验段内锥尖阻力qt残差的变程具有显著各向异性,在水平方向和竖直方向分别为4.05 m和1.2 m。采用普通Kriging插值结合趋势分析,绘制了qt在试验段的空间分布图和平面投影图,用于指导工程实践。结果表明,普通Kriging插值的估计结果能够与试验段内实测资料形成较好的对比,仅仅在部分极值变化和远离采样点的位置处估计值可靠性会降低。     

Reliability analysis of soil nail walls

In India, soil nail walls are being extensively used for supporting vertical excavations below ground level to accommodate construction of one-or two-storied basements. Generally, the depth of excavations for basement construction ranges from 10 m to 15 m. For such large depth of excavation, variability of in-situ soil properties has significant influence on the stability of the soil nail walls. In the present study, using reliability analysis, an attempt is made to study the influence of variability of in-situ soil properties on the stability of soil nail walls. For better understanding, a case of 10 m high soil nail wall constructed to support a vertical cut is considered for the study and its stability is evaluated for various failure modes. Additionally, the influence of correlation among soil parameters on soil nail wall stability is assessed. In-situ soil friction angle and correlation between in-situ soil cohesion and angle of friction are found to influence soil nail wall stability significantly. In general, reliability analysis provided a better insight into the assessment of stability of soil nail wall.     

Bearing capacity of strip footings on spatially random soils using sparse polynomial chaos expansion

A probabilistic model is presented to compute the probability density function (PDF) of the ultimate bearing capacity of a strip footing resting on a spatially varying soil. The soil cohesion and friction angle were considered as two anisotropic cross‐correlated non‐Gaussian random fields. The deterministic model was based on numerical simulations. An efficient uncertainty propagation methodology that makes use of a non‐intrusive approach to build up a sparse polynomial chaos expansion for the system response was employed. The probabilistic numerical results were presented in the case of a weightless soil. Sobol indices have shown that the variability of the ultimate bearing capacity is mainly due to the soil cohesion. An increase in the coefficient of variation of a soil parameter (c or φ) increases its Sobol index, this increase being more significant for the friction angle. The negative correlation between the soil shear strength parameters decreases the response variability. The variability of the ultimate bearing capacity increases with the increase in the coefficients of variation of the random fields, the increase being more significant for the cohesion parameter. The decrease in the autocorrelation distances may lead to a smaller variability of the ultimate bearing capacity. Finally, the probabilistic mean value of the ultimate bearing capacity presents a minimum. This minimum is obtained in the isotropic case when the autocorrelation distance is nearly equal to the footing breadth. However, for the anisotropic case, this minimum is obtained at a given value of the ratio between the horizontal and vertical autocorrelation distances. Copyright © 2012 John Wiley & Sons, Ltd.     

祁连山退化高寒草甸土壤水分空间变异特征分析

利用传统统计学方法和地统计学方法,对祁连山地区受到过度放牧影响而退化为以狼毒为优势种的高寒草甸的土壤水分垂直变异特征、水平空间异质性以及分布特征进行了系统分析. 结果表明：在垂直方向上,0~100 cm土壤水分含量随深度的增加而逐渐减少,土壤水分含量的变化速度随深度的增加也趋于减少;土壤水分分布的变异系数在浅层和深层土壤较大,在中层土壤较小. 在水平方向上,0~40 cm土壤水分具有中等空间变异性,其中10~20 cm土壤水分变异性主要受根系的影响,随机部分引起的变异性最大;而在其他土壤层,随着深度的增加土壤水分含量由随机部分引起的空间异质性程度减弱,由空间自相关部分引起的异质性程度增强. 整体上,土壤水分含量与微地形关系密切,与距离溪流的远近程度正相关,与高程分布负相关.     

Uncertainty reduction and sampling efficiency in slope designs using 3D conditional random fields

A method of combining 3D Kriging for geotechnical sampling schemes with an existing random field generator is presented and validated. Conditional random fields of soil heterogeneity are then linked with finite elements, within a Monte Carlo framework, to investigate optimum sampling locations and the cost-effective design of a slope. The results clearly demonstrate the potential of 3D conditional simulation in directing exploration programmes and designing cost-saving structures; that is, by reducing uncertainty and improving the confidence in a project’s success. Moreover, for the problems analysed, an optimal sampling distance of half the horizontal scale of fluctuation was identified.     

路堤边坡稳定可靠度计算中的模型不确定性分析

基于均质路堤边坡Monte Carlo法的稳定可靠度计算,分析了临界滑面搜索策略和稳定分析方法两类模型不确定性对边坡可靠度的影响特性,讨论了边坡失效概率随土工参数变异性的变化规律。研究表明,选用不同的临界滑面搜索策略所得可靠度结果差异不大,参数滑面法（overall slope）的失效概率略大于均值滑面法（global minimum）,但差别对边坡稳定性分析没有实质性影响;土性参数变异水平是影响边坡可靠度的最重要因素,边坡在相同设计参数安全系数下的可靠度指标随参数变异性增大而急剧降低;不同稳定性分析方法对应的安全系数概率密度函数曲线形态基本一致,但失效概率差异明显,因此目标可靠度指标取值应与稳定性分析方法相适应。提出的考虑土工参数变异水平的安全系数取值修正原则,对改进确定性设计的边坡稳定分析技术有积极意义。     

Uncertainty in spatial average undrained shear strength with a site-specific transformation model

ABSTRACT

Transformation models are used to infer geotechnical properties from indirect measurements. A site-specific transformation model can be calibrated with direct and indirect measurements from a site. When such a model is used, then spatial variability, measurement errors and statistical uncertainty propagate into the uncertainty of the spatial average, which is the variable of interest in most geotechnical analyses. This research shows how all components enter the total uncertainty of a transformation model for undrained shear strength from cone resistance. A method is proposed to estimate the uncertainty in the spatial average undrained shear strength, particularly focusing on the role of averaging of all spatially variable error components. The main finding is that if a considerable share of the measurement and transformation errors is random or spatially variable, the uncertainty estimates can be considerably lower compared to methods proposed earlier, and hence, characteristic values can be considerably higher.