System Reliability Assessment for a Rock Tunnel with Multiple Failure Modes

This paper presents a practical procedure for assessing the system reliability of a rock tunnel. Three failure modes, namely, inadequate support capacity, excessive tunnel convergence, and insufficient rockbolt length, are considered and investigated using a deterministic model of ground-support interaction analysis based on the convergence–confinement method (CCM). The failure probability of each failure mode is evaluated from the first-order reliability method (FORM) and the response surface method (RSM) via an iterative procedure. The system failure probability bounds are estimated using the bimodal bounds approach suggested by Ditlevsen (1979), based on the reliability index and design point inferred from the FORM. The proposed approach is illustrated with an example of a circular rock tunnel. The computed system failure probability bounds compare favorably with those generated from Monte Carlo simulations. The results show that the relative importance of different failure modes to the system reliability of the tunnel mainly depends on the timing of support installation relative to the advancing tunnel face. It is also shown that reliability indices based on the second-order reliability method (SORM) can be used to achieve more accurate bounds on the system failure probability for nonlinear limit state surfaces. The system reliability-based design for shotcrete thickness is also demonstrated.     

Probabilistic analysis of underground rock excavations using response surface method and SORM

Probabilistic analysis of underground rock excavations is performed using response surface method and SORM, in which the quadratic polynomial with cross terms is used to approximate the implicit limit state surface at the design point. The response surface is found using an iterative algorithm and the probability of failure is evaluated using the first-order and the second-order reliability method (FORM/SORM). Independent standard normal variables in U-space are chosen as basic random variables and transformed into correlated non-normal variables in the original space of random variables for constructing the response surface. The proposed method is first illustrated for a circular tunnel with analytical solutions considering Mohr–Coulomb (M–C) and Hoek–Brown (H–B) yield criteria separately. The failure probability with respect to the plastic zone criterion and the tunnel convergence criterion are estimated from FORM/SORM and compared to those obtained from Monte Carlo Simulations. The results show that the support pressure has great influence on the failure probability of the two failure modes. For the M–C model, the hypothesis of uncorrelated friction angle and cohesion will generate higher non-performance probability in comparison to the case of negatively correlated shear strength parameters. Reliability analyses involving non-normal distributions are also investigated. Finally, an example of a horseshoe-shaped highway tunnel is presented to illustrate the feasibility and validity of the proposed method for practical applications where numerical procedures are needed to calculate the performance function values.     

Reliability-based design and its complementary role to Eurocode 7 design approach

Reliability-based design (RBD) can play a useful complementary role to overcome some limitations in the Eurocode 7 (EC7) design approach, for example in situations with parameters not covered in EC7, different parametric sensitivities across different problems, cross-correlated or spatially correlated parameters, design aiming at a target reliability or failure probability, or when uncertainty in unit weight of soil is modeled. The complementary role played by RBD under these circumstances is illustrated and discussed for a shallow foundation, a reinforced rock slope, a Norwegian clay slope with spatial variability, a laterally loaded pile requiring implicit numerical analysis, and an anchored sheet pile wall. A pragmatic RBD approach involving first-order reliability method (FORM) only and a more rigorous RBD approach involving both first-order and second-order reliability method (SORM) are offered. Both approaches are implementable using either spreadsheet-based FORM and SORM procedures, or using various commercially available FORM/SORM packages.     

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.     

Efficient Probabilistic Algorithm Illustrated for a Rock Slope

Summary  A new spreadsheet-based algorithm for the first-order reliability method (FORM) is illustrated for a two-dimensional rock slope of Hong Kong. The new algorithm combines inverse distribution functions and a refined Newton method with the automatic constrained-optimization search of the design point in the original space of the random variables; it obviates the need for computations of equivalent normal means and equivalent normal standard deviations. In the rock slope analysis, the versatile 4-parameter beta distribution is used in lieu of a truncated normal distribution. Probabilities of failure inferred from reliability indices are compared with those from Monte Carlo simulations. The effects of parametric correlations on the required reinforcing force for a target reliability index value are studied. The intuitive perspective of an expanding equivalent dispersion ellipsoid in the original space of the basic random variables is also described as it is the basis from which the new approach evolved. Author’s address: Dr. Bak Kong Low, School of Civil and Environmental Engineering, Nanyang Technological University, Block N1 #1b-40, 50 Nanyang Avenue, 639798 Singapore     

基坑体系可靠度的条件概率计算方法

基坑存在多种失效模式,考虑失效模式之间的相关性,双界限法计算体系失效概率存在计算结果区间范围较大的弊端。利用均匀试验和非参数回归方法建立响应面,在响应面的基础上,对Monte Carlo模拟生成的随机参数进行插值,得到各个失效模式指标,结合Pearson相关系数检验两两失效模式之间的相关性,用条件概率方法计算基坑体系失效概率,提出了基于条件概率考虑多失效模式相关的基坑体系可靠度分析方法。在此基础上,通过1个典型算例进行对比分析,计算结果表明,该方法不仅计算简便,而且结果可靠,其结果可为基坑体系可靠度分析理论提供一条新的途径。     

边坡可靠度分析的支持向量机法

提出了基于支持向量机（SVM）的边坡可靠度分析新算法。该方法采用均匀设计确定样本点,通过一定数量的确定性计算来训练SVM,拟合边坡的功能函数;采用一阶可靠度方法（FORM）和迭代算法优化SVM模型,获得可靠度指标和验算点信息;在SVM模型基础上进一步通过二阶可靠度方法（SORM）和蒙特卡罗模拟（MCS）计算边坡的失稳概率。以两个典型边坡为例,通过与其他方法比较,证明了该方法的准确性和高效性。结果表明：提出的在标准正态空间（U空间）中取样并构建SVM,在原始空间（X空间）中计算功能函数的算法,有效地解决了具有相关非正态分布变量的可靠度分析问题,并且可很容易扩展到SORM的计算。算例结果证明,该方法的精度高于FORM;而效率优于MCS。分析过程中,边坡安全系数计算和可靠度分析相互独立。因此,该方法既适用于具有显式功能函数的简单问题,也适用于需要软件计算安全系数的实际边坡问题。     

Reliability analysis of the induced damage for single-hole rock blasting

Rock explosion has always been a complex problem because neither rock characteristics nor explosion waves could be accurately estimated. As such, this imposes a high uncertainty on deterministic methodologies available for damage prediction. In this paper, by defining two damage zones around the blast hole, including crushed and cracked zones, a first-order reliability analysis (FORM) was adopted to address this issue. For this purpose, FORM was used in a double-loop algorithm, where the inner loop was responsible for converging the FORM, and the outer loop was assigned to feed the inner loop with new cases. Using such nested-loop algorithm, the probability of exceedance was calculated for any desired damage zone radius. The maximum effect of the involved parameters on the failure probability induced around the blast hole was additionally studied using a parametric reliability analysis. The results showed that the radii for crushed and cracked zones are limited to 0.5 and 4.2?m, respectively, so that the probability of going beyond these limits is less than 1%. Moreover, the analyses of decoupled explosions showed that increasing the gap between the explosion charge and wall of the borehole could severely reduce the failure probability; however, the maximum effect of decoupling ratio occurs in the small range of radii between 0.3?mm and 2.35?m.     

Reliability assessment of excavation systems considering both stability and serviceability performance

Excavation projects related to urban redevelopment and infrastructure improvement are often governed by serviceability-based design, rather than failure prevention criteria. Deformation tolerance specifications are often prescribed based on minimizing potential damage to adjacent structures. A risk-based approach to serviceability performance that systematically incorporates design parameter uncertainty will allow engineers to address soil uncertainty in performance-based design. This paper demonstrates the use of various kinds of reliability methods, such as response surface method (RSM), first-order reliability method (FORM), second-order reliability method (SORM), adaptive importance sampling (AIS), Monte Carlo simulation (MCS) and system reliability, to assess the risk of stability and/or serviceability failure of an entire excavation support system throughout the entire construction process. By considering multiple failure modes (including serviceability criteria) of an excavation, the component and system reliability indices for each excavation step are assessed during the entire excavation process. Sensitivity analyses are conducted for the system reliability calculations, which demonstrate that the adjacent structure damage potential limit state function is the dominant factor for determining excavation system reliability. An example is presented to show how the serviceability performance for braced excavation problems can be assessed based on the system reliability index.     

Probabilistic assessment of tunnel convergence considering spatial variability in rock mass properties using interpolated autocorrelation and response surface method

This study aims at the probabilistic assessment of tunnel convergence considering the spatial variability in rock mass properties. The method of interpolated autocorrelation combined with finite difference analysis is adopted to model the spatial variability of rock mass properties. An iterative procedure using the first-order reliability method(FORM) and response surface method(RSM) is employed to compute the reliability index and its corresponding design point. The results indicate that the spatial variability considerably affects the computed reliability index. The probability of failure could be noticeably overestimated in the case where the spatial variability is neglected. The vertical scale of fluctuation has a much higher effect on the probabilistic result with respect to the tunnel convergence than the horizontal scale of fluctuation. And the influence of different spacing of control points on the computational accuracy is investigated.     

System probabilistic model of rock slope stability considering correlated failure modes

In this paper, a system reliability analysis of rock slope stability with considering all input parameters as stochastic parameter is presented. To perform reliability analysis a cut-set system has been used. For this purpose, Sequential Compounding Method (SCM) as a powerful method for reducing the computational time and accurate evaluation is employed to determine the reliability indices with considering correlations between failure modes which are calculated by defining equivalent linear safety margin for each failure mode. Furthermore, the 3-D system probability of failure surface is presented and the probabilistic model is developed to evaluate the rock slope probability of failure.     

A system reliability approach for evaluating stability of rock wedges with correlated failure modes

This paper proposes a system reliability approach for evaluating the stabilities of rock wedges considering multiple correlated failure modes. A probabilistic fault tree is employed to model the system aspects of the problem. The system reliability analysis is performed using an N-dimensional equivalent method taking into account correlations between different failure modes. Reliability sensitivity analyses at three different levels, namely, single limit state function level, single failure mode level, and system reliability level, were carried out to study the effect of changes in variables on the stability of the wedge. An example case was analysed to illustrate the proposed approach. The stability of the wedge can be evaluated efficiently using the proposed system reliability approach in a more systematic and quantitative way. The probabilities of failure of the wedge from the N-dimensional equivalent method are fairly consistent with those from the Monte Carlo simulation method. The results demonstrate that the probability of failure will be overestimated if the correlations between different failure modes of the wedge are not taken into account. They also demonstrate that the relative importance of different failure modes to the system reliability of the wedge can differ considerably and be treated systematically and quantitatively by the proposed approach. The sensitivity results are highly dependent on the selected sensitivity analysis level.     

Moving least squares method for reliability assessment of rock tunnel excavation considering ground-support interaction

A practical approach is proposed in this paper for the reliability assessment of rock tunnel excavations using the moving least squares method (MLSM) and the uniform design. The failure probability is computed by the first-order and the second-order reliability method (FORM/SORM), which is based on the generated MLSM response surface (MLSM-RS) via an iterative algorithm. The proposed approach is first implemented in the analysis of a circular tunnel that consists of three limit state functions to illustrate the efficiency and accuracy of the approach. Then, the method is applied to a non-circular tunnel to demonstrate the feasibility and validity of the method for practical problems, in which numerical procedures are commonly employed to solve the implicit limit state functions.     

Rock Wedge Stability Analysis Using System Reliability Methods

Summary We present a system reliability approach to rock wedge stability analysis. Different failure modes are considered, and a disjoint cut-set formulation is employed – with each cut-set corresponding to a different failure mode – to explore the system aspects of the problem, so that the reliability of the system is assessed by computing the probability of failure of the slope under each failure mode. An example case is used to demonstrate different approaches to compute the reliability of the slope design. Our results show that an approximation to the “exact” probability of failure – given by Monte Carlo simulation results – may be obtained using a first order approximation to the failure domain, and that linear programming techniques may be used to obtain bounds of the probability of failure. Furthermore, we identify the most likely failure mode, and we explore the sensitivity of the computed probabilities to changes in the random variables considered. The results indicate that the reliability results are quite sensitive to the geometry of the wedge. Changes in water conditions are also found to have a significant impact on the computed probabilities, while changes in unit weight of the rock have a considerably smaller effect on the reliability.     

Reliability analysis of circular tunnel under hydrostatic stress field

The first-order reliability method (FORM) is used to calculate the reliability index of a circular tunnel subjected to a hydrostatic stress field. The random variables are first assumed to follow the normal distribution. Comparison between analysis using negatively correlated and uncorrelated friction angle and cohesion indicates that the results of reliability analysis are conservative if negative correlation among strength parameters is not modeled. The reliability analysis involving correlated non-normal distributions is also investigated. The probability density functions are obtained from reliability indices and compared to those generated from Monte Carlo simulations. Reliability-based design of tunnel support pressure is also illustrated.     

基于响应面法的边坡稳定二阶可靠度分析

提出了基于响应面法的边坡稳定二阶可靠度分析的实用算法。选择U空间中的随机变量,通过空间变换和相关矩阵分解,计算试验点的功能函数;通过迭代算法构造响应面、以确保通过最小的计算量获得最优精度,并在此基础上进行FORM/SORM计算。以一岩石边坡的平面滑动问题为例,通过与蒙特卡洛模拟、FORM及随机响应面法的比较,证明了该方法的准确性和高效性。分析了参数的相关性及试验点取值范围对计算结果的影响,讨论了可靠度分析结果中参数敏感性和物理属性问题。该方法可为实际边坡问题的可靠度分析提供参考,并可以用来进行基于可靠度分析的加固设计。     

Slope reliability analysis accounting for spatial variation

A practical and transparent procedure is described for implementing a generalized limit equilibrium method via cell-object-oriented constrained optimization in the spreadsheet platform. The formulation allows switching among the Spencer, Bishop simplified and wedge methods on the same template by specifying different side-force inclination and different constraints of optimization. Search for the critical circular or non-circular slip surface is possible. The deterministic procedure is extended probabilistically by implementing the first-order reliability method via constrained optimization of the equivalent dispersion ellipsoid in the original space of the random variables. This procedure is illustrated for an embankment on soft ground, and for a clay slope in southern Norway, both involving spatially correlated soil properties. The effects of autocorrelation distance on the results of reliability analysis are studied. Shear strength anisotropy is modelled via user-created simple function codes in the programming environment of the spreadsheet. The meaning of probability of failure is discussed.     

Probabilistic Stability Evaluation of Oppstadhornet Rock Slope,Norway

Probabilistic analyses provide rational means to treat the uncertainties associated with underlying parameters in a systematic manner. The stability of a 734-m-high jointed rock slope in the west of Norway, the Oppstadhornet rock slope, is investigated by using a probabilistic method. The first-order reliability method (FORM) is used for probabilistic modeling of the plane failure problem in the rock slope. The Barton–Bandis (BB) shear strength criterion is used for the limit state equation. The statistical distributions of the BB criterion parameters, for which comprehensive data were collected and statistically analyzed, are determined by using distribution fitting algorithms. The sensitivity of the FORM model for the BB criterion is also investigated. It is found that the model is most sensitive to the mean value of the residual friction angle (ϕ _r) and least sensitive to the mean value of the slope angle (β _f). It is also found that the standard deviation of joint compressive strength (JCS) causes the greatest difference in the reliability index, which has the least sensitivity to the change in the mean and standard deviation of joint roughness coefficient (JRC).     

基于模糊随机理论的广义可靠度在边坡稳定性分析中的应用

在常规的概率极限状态理论基础上 ,分析了边坡体安全状态的模糊性 ,在此基础上构造了基于MCS以及FORM算法的模糊随机可靠度算法 ,分别就边坡体的滑动失效、渗透破坏两种失效模式作了参数敏感性分析 ,并进一步对边坡体的系统失效模式作了探讨     

基于随机响应面法的边坡系统可靠度分析

提出了一套基于随机响应面法的边坡系统可靠度分析方法。该方法首先从大量潜在滑动面中筛选出代表性滑动面。针对每条代表性滑动面,采用Hermite多项式展开建立其安全系数与土体参数间的非线性显式函数关系（即随机响应面）。然后,采用直接蒙特卡洛模拟计算边坡系统失效概率。在蒙特卡罗模拟中,采用所有代表性滑动面的随机响应面计算每一组样本所对应的边坡最小安全系数。最后,以两个典型多层边坡系统可靠度问题为例验证了该方法的有效性。结果表明：文中提出的边坡系统可靠度分析方法能够有效地识别边坡代表性滑动面,具有较高的计算精度和效率,并且确定代表性滑动面时无需计算滑动面间的相关系数。同时该方法可以有效地计算低失效概率水平的边坡系统可靠度,为含相关非正态参数的边坡系统可靠度问题提供了一条有效的分析途径。此外,多层边坡可能同时存在多条潜在滑动面,基于单一滑动面（如临界确定性滑动面）或者部分代表性滑动面进行边坡系统可靠度分析均会低估边坡失效概率。