Sensitivity analysis of support safety factor for predicting the effects of contributing parameters on roof falls in underground coal mines

Roof falls accounted for 18.18% of all fatal accidents in Indian coal mines, contributing about 35.29% of all fatal accidents in below-ground operations in 2005. The support safety factor, always preferred in support planning and design of underground coal mines, may be an important predictor for roof falls. In this paper, geotechnical data were collected from 14 roof fall incident places in an underground coal mine, located in the Eastern India, which has bord and pillar method of workings. The mean value of probabilistic support safety factor for the case study mine was found to be 1.24. However, the probability, of the estimated support safety factor of less than or equal to one, was found to be 0.246. Sensitivity analysis was conducted to analyze the effects of the contributing parameters on support safety factor and the likelihood of the roof fall. The multi-variate regression analysis was carried out for the data generated by Monte Carlo method to correlate the contributing factors to support safety factor. It ranked gallery width as the first parameter to control the support safety factor.     

Probabilistic stability assessment using adaptive limit analysis and random fields

For deterministic scenarios, adaptive finite element limit analysis has been successfully employed to achieve tight bounds on the ultimate load of a geotechnical structure in a much more efficient manner than a dense uniform mesh. However, no probabilistic studies have so far considered finite element limit analysis with adaptive remeshing. Therefore, this research explores the benefits of combining adaptive mesh refinement with finite element limit analysis for probabilistic applications. The outcomes indicate that in order to achieve tight bounds on probabilistic results (such as the probability of failure), the ultimate load in each individual simulation (e.g. factor of safety or bearing capacity) has to be estimated with a very high level of accuracy and this can be achieved more economically using adaptive mesh refinement. The benefits, assessed here for undrained conditions, are expected to be much more pronounced in the case of frictional soils and complex geometries.     

附加应力的概率式解答

基于概率论的思想及几个基本概念,没有采用任何复杂的模型和假设,将岩土体内附加应力的传播扩散看做概率事件（相互独立且机会均等）,推导了二维空间点荷载下附加应力的概率式解答,并延伸推导了二维空间均匀分布荷载下附加应力的概率式解答,且将两者推广到三维空间,最后与布辛内斯克解进行了（举例）比较,几种情形下的结果都非常接近,论证了概率式解答的思路和方法的合理性和可靠性。     

Robust geotechnical design of shield-driven tunnels

This paper presents a fuzzy set-based robust geotechnical design (RGD) methodology for the design of shield-driven tunnels. Here, uncertain geotechnical parameters required for analysis of tunnel performance (referred to herein as the structure safety and serviceability performance of tunnel cross section) are represented as fuzzy sets. Given fuzzy input parameters, the performance of a shield-driven tunnel will be uncertain, which is expressed in this study as a fuzzy factor of safety, according to the analysis of vertex method. Then, the fuzzy factor of safety for a given design is used to evaluate the failure probability and design robustness, which are, in turn, employed in the proposed RGD framework. Note that a design is considered robust if the performance of the shield-driven tunnel is insensitive to the variation of its uncertain geotechnical parameters. Within the RGD framework, each candidate design in the design space is analyzed for its safety state (in terms of failure probability), design robustness, and cost. The goal of the RGD of a shield-driven tunnel is to bring the safety state to an acceptable level, while maximizing the robustness and cost efficiency simultaneously. To this end, a multi-objective optimization is performed and a Pareto front is obtained, which provides a trade-off that may be used to select the most preferred design. Through an illustrative case, the effectiveness and significance of this new robust design methodology is demonstrated.     

Probabilistic key block analysis of a mine ventilation shaft stability – a case study

In this study, the probabilistic key block analysis was applied to evaluate the stability of a mine ventilation shaft developed in a rock mass of granite. The key blocks were identified based on the block theory. The variations of discontinuity orientations were fitted with the Beta distribution and taken into consideration. The key block forming probabilities were analyzed. For simplification of calculations the first-order second-moment (FOSM) approximation was employed for probability estimation. With the considerations of the rock properties as random variables and applications of several statistical analysis tools, the key block failure probabilities, the probabilistic distribution of safety factors, and the probabilistic distribution of potential maximum key block volumes were analyzed. The analysis indicated that although the safety factor calculated based on the mean values of the variables was above 1.0 for the stability of the most critical key block, the block had a considerable probability of failure with a significant rock volume due to variations in discontinuity orientations and rock properties. Without promptly applying supports to the rock excavation, the shaft had a significant likelihood of failure.     

Evaluation of the Instability Risk of the Dam Slopes Simulated with Monte Carlo method (Case Study: Alborz Dam)

Embankment dams are one of the most important geotechnical structures that their failures can lead to disastrous damages. One of the main causes of dam failure is its slope instability. Slope Stability analysis has traditionally been performed using the deterministic approaches. These approaches show the safety of slope only with factor of safety that this factor cannot take into account the uncertainty in soil parameters. Hence, to investigate the impact of uncertainties in soil parameters on slope stability, probabilistic analysis by Monte Carlo Simulation (MCS) method was used in this research. MCS method is a computational algorithm that uses random sampling to compute the results. This method studies the probability of slope failure using the distribution function of soil parameters. Stability analysis of upstream and downstream slopes of Alborz dam in all different design modes was done in both static and quasi-static condition. Probability of failure and reliability index were investigated for critical failure surfaces. Based on the reliability index obtained in different conditions, it can be said that the downstream and upstream slope of the Alborz dam is stable. The results show that although the factor of safety for upstream slope in the state of earthquake loading was enough, but the results derived from probabilistic analysis indicate that the factor of safety is not adequate. Also the upstream slope of the Alborz dam is unstable under high and uncontrolled explosions conditions in steady seepage from different levels under quasi-static terms.

     

β分布的参数确定及其在岩土工程中的应用

对岩土工程中随机变量的空间概率特征进行了统计分析,介绍了确定β分布各参数的迭代方法,提出了经验公式简化迭代过程,有效地处理了随机变量分布范围的估计问题,并阐明了β分布在可靠度领域中的适用性。作为对比,对同一样本采用了其他分布进行拟合。分析结果表明,β分布拟合精度高于其他分布。     

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。     

Stability analysis of rock wedges with multiple sliding surfaces

Summary Although wedge and plane sliding stability analyses are well established in the geotechnical literature, certain geologic environments produce blocks which cannot be adequately modelled as either wedges or plane slides. An example is blocks forming in cylindrically folded sedimentary rocks, where the surface of sliding is neither a single plane nor a double plane but is curved. This type of block may be idealized as a prismatic block with multiple sliding planes, all with parallel lines of intersection. If the sliding planes number three or more, the distribution of normal forces, and hence the factor of safety, is indeterminate. A new analytical model for sliding stability analysis is described in which the distribution of normal forces on the contact planes is chosen to minimize the potential energy of the system. The classic wedge and plane solutions are shown to be special cases of this more general model, which allows determination of the safety factor for any shape of prismatic contact surface. An example from Tennessee concerning a block with a curved sliding surface is described and the factor of safety compared with the standard wedge analysis. It is shown that with three or more contact planes, the safety factor may be significantly lower than that calculated from the wedge model, which provides an upper limit on stability.     

Probabilistic analysis of retaining walls

A methodology for the probabilistic analysis of reinforced concrete cantilever walls is developed and described in this paper. The wall’s external stability under static conditions is addressed and modeled as a series system with correlated failure modes. Computations of reliability are performed using Monte Carlo simulations for assumed probability distributions of the backfill and foundation material engineering properties. A case example is analyzed based on the described methodology. The results indicated that risk, measured by the system probability of failure, is not a linear function of safety ratios. All three safety ratios in question were positively correlated, with bearing capacity being subjected to higher degree of uncertainty. The degree of correlation was found to have an important effect on the system probability of failure. Considering the width of the base as varying design parameter, the study also showed that first-order reliability bounds, which are often applied in practice, may lead to a noteworthy over- or under-estimation of the design.     

Genetic algorithms in probabilistic finite element analysis of geotechnical problems

In application to numerical analysis of geotechnical problems, the limit-state surface is usually not known in any closed form. The probability of failure can be assessed via the so-called reliability index. A minimization problem can naturally be formed with an implicit equality constraint defined as the limit-state function and optimization methods can be used for such problems. In this paper, a genetic algorithm is proposed and incorporated into a displacement finite element method to find the Hasofer–Lind reliability index. The probabilistic finite element method is then used to analyse the reliability of classical geotechnical systems. The performance of the genetic algorithm (GA) is compared with simpler probability methods such as the first-order-second-moment Taylor series method. The comparison shows that the GA can produce the results fairly quickly and is applicable to evaluation of the failure performance of geotechnical problems involving a large number of decision variables.     

边坡安全系数概率密度研究

边坡安全系数概率分布形态是边坡可靠指标、破坏概率的计算基础,目前对边坡安全系数的影响参数如摩擦系数、凝聚力、重度等的变异性及分布形态有一定的研究,但对边坡安全系数的变异性及概率分布形态缺乏研究。以三峡库区巴东沿江路云沱段（狮子包）边坡为例,基于岩土参数c、? 值服从正态分布,通过Monte-Carlo模拟计算,研究得到安全系数的概率分布为正态分布,并利用Q-Q图分析对概率分布形态进行了定量检验。由安全系数服从正态分布得到狮子包边坡安全系数的平均值和失稳风险。     

Uncertainty and Reliability Analysis Applied to Slope Stability: A Case Study From Sungun Copper Mine

Nowadays, there are many new methods for slope stability analysis; including probabilistic methods assessing geotechnical uncertainties to develop safety factors. In this paper, a reliability index analysis for the Sungun copper mine slope stability is evaluated based on three methods of uncertainties consisting Taylor series method, Rosenblueth point estimate method and Monte-Carlo simulation method. Sungun copper mine will be one of the Iran’s biggest mines with final pit’s height of 700 meters. For this study two of its main slopes were assessed, one dipping to the NE (030) and the other to the SE (140). Probability density function of cohesion and angle of friction for the slopes were developed using limit equilibrium methods. These shear strengths were then used to determine the probability density function of safety factor and reliability index using the probabilistic methods. Results of the probabilistic analysis indicate that with ascending values of the uncertainties the reliability index decreases. Furthermore, it was determined that with the Monte Carlo simulation the seed number used has little effect on the reliability index of the safety factor especially with seed numbers in excess of 1200. Variations in the overall reliability index of safety factor were observed between the two slopes and this difference is explained by the differences in complexities of the geology within the cross-section.     

节理分布空间变异的地下洞室稳定性概率分析

以岩土材料力学参数空间变异性的"点估计-有限元"分析方法为基础,结合节理分析时自身存在几何模型、网格划分等特性,扩展了该方法在节理分布空间变异性分析方面的适用性,明确了具体的研究步骤与方法。以某抽水蓄能水电站为例,通过分析节理空间变异性对围岩变形与塑性区的影响,验证了扩展后该方法的准确性和合理性。对工程案例开挖揭露的1400余条节理进行概率统计,建立了节理空间变异性的有限元分析模型;采用扩展后的概率分析方法,研究了节理分布对地下洞室群围岩开挖稳定性的影响。研究结果表明:(1)对比概率分析得到的围岩变形概率分布与现场监测结果,发现剔除变形异常点后监测变形量值大部分位于得到的位移概率分布范围内,说明节理的空间变异性是导致监测变形波动的主要影响因素;(2)围岩变形概率分布的标准差能有效识别出围岩开挖变形受节理空间变异性的影响程度,对于所给出的案例依次为:机窝>边墙>顶拱;(3)围岩塑性区的概率分区能合理判断地下洞室群开挖时受节理影响较大的区域和范围,为工程施工的支护设计提供依据。     

Probabilistic analysis of a one-dimensional soil consolidation problem

In this paper, a probabilistic study of a one-dimensional soil consolidation problem has been carried out. The Collocation-based Stochastic Response Surface Method (CSRSM) was employed for the probabilistic analysis. The Young modulus E, the Poisson ratio ν, the hydraulic conductivity k _h and the uniform surcharge loading q applied at the ground surface were considered as random variables. The probabilistic system responses considered in the analysis were the surface settlement and the consolidation time. Numerical simulations that make use of Biot theory were used for the computation of these system responses. A global sensitivity analysis based on Sobol indices was performed to identify the random variables that have the most contribution in the variability of the system responses. Also, a parametric study was undertaken to investigate the effect of the input geotechnical parameters and the statistical parameters of the random variables on the probability distribution functions of the system responses.     

基于K-L信息距离的多源信息融合法

为解决小子样条件下岩土参数概率分布推断的难题,并克服基于专家信息的融合法不可避免地带有主观随意性的弊端,引入信息论中K-L信息距离的概念,基于先验信息可信度,提出一种新的多源信息融合方法。利用K-L信息距离作为参数分布之间距离的度量,定义先验分布差异率,确定融合权重,进而根据Bayes原理得到后验分布,优化岩土参数分布概型。工程实例分析结果表明,该法计算简单,且克服了推断过程中的主观随意性。计算结果显示该法所得融合分布的方差比已有成果所得方差偏小,说明该法可实现统计意义上的参数概型优化,为岩土参数设计值的合理选取提供了参考。     

岩土工程风险分析及应用综述

系统分析了岩土工程不确定性的根源及其分析方法,总结了降低不确定性的途径。系统评述了岩土参数随机场估计、随机变量描述方法以及模型不确定性研究。对几类典型岩土工程问题的风险分析方法、研究现状及其工程应用进行了总结。对目前岩土工程风险理论及其应用的前沿问题进行了研究和展望。研究指出岩土工程风险分析待研问题主要有：系统的风险源辨识方法包括风险源辨识的可视化研究,不确定性因子的合理描述研究,极限状态包括失效模式的仿真研究,不确定性计算包括应用信息融合技术数据挖掘技术研究,目标可靠度研究,风险转嫁风险交换的量化及风险跟踪的实施研究,风险的反分析研究等。展望岩土工程风险分析与应用：风险分析反映了对工程设计的综合性要求,表现了安全与经济的统一;定量风险分析作为决策工具或传统设计的补充,可给决策者提供更多的辅助评价信息,提高了结果的置信程度;但现在能描述的还只是“随机性的确定性模式”,工程师们只有使用“保守的”选择,以适应和弥补不完全的认识与有限的资料的条件。     

Application of the dynamic bounds method in the safety assessment of flood defences,a case study: 17th Street flood wall,New Orleans

In this paper we provide a computational framework for evaluation of reliability and safety assessment of infrastructures. It is based on the combined application of the dynamic bounds (DB) method and a probabilistic finite element model (FEM). The DB improves the computational efficiency of the FEM when calculating time-dependent failure analyses of coastal and offshore structures, and can speed up the simulation process by several orders of magnitude.

Our approach is demonstrated here for an example problem, and shown to be the most efficient method in applications with a limited number of influential variables, which is true for geotechnical and coastal flood defence systems. It is applied to the 17th Street flood wall, a failing component of the flood defence system in New Orleans during Hurricane Katrina. The variation in soil parameters is a critical input in the reliability estimation of this structure, and the calculated probability of failure depends on these assumed values.     

Probabilistic analysis of rock slope stability and random properties of discontinuity parameters, Interstate Highway 40, Western North Carolina, USA

Probabilistic analysis has been used as an effective tool to evaluate uncertainty so prevalent in variables governing rock slope stability. In this study a probabilistic analysis procedure and related algorithms were developed by extending the Monte Carlo simulation. The approach was used to analyze rock slope stability for Interstate Highway 40 (I-40), North Carolina, USA. This probabilistic approach consists of two parts: analysis of available geotechnical data to obtain random properties of discontinuity parameters; and probabilistic analysis of slope stability based on parameters with random properties. Random geometric and strength parameters for discontinuities were derived from field measurements and analysis using the statistical inference method or obtained from experience and engineering judgment of parameters. Specifically, this study shows that a certain amount of experience and engineering judgment can be utilized to determine random properties of discontinuity parameters. Probabilistic stability analysis is accomplished using statistical parameters and probability density functions for each discontinuity parameter. Then, the two requisite conditions, kinematic and kinetic instability for evaluating rock slope stability, are determined and evaluated separately, and subsequently the two probabilities are combined to provide an overall stability measure. Following the probabilistic analysis to account for variation in parameters, results of the probabilistic analyses were compared to those of a deterministic analysis, illustrating deficiencies in the latter procedure. Two geometries for the cut slopes on I-40 were evaluated, the original 75° slope and the 50° slope which has developed over the past 40 years of weathering.     

Probabilistic assessment of soil liquefaction considering spatial variability of CPT measurements

The conventional liquefaction potential assessment methods (also known as simplified methods) profoundly rely on empirical correlations based on observations from case histories. A probabilistic framework is developed to incorporate uncertainties in the earthquake ground motion prediction, the cyclic resistance prediction, and the cyclic demand prediction. The results of a probabilistic seismic hazard assessment, site response analyses, and liquefaction potential analyses are convolved to derive a relationship for the annual probability and return period of liquefaction. The random field spatial model is employed to quantify the spatial uncertainty associated with the in-situ measurements of geotechnical material.