Reliability assessment of granular filters in embankment dams

Empirical criteria have been used successfully to design filters of most embankment large dam projects throughout the world. However, these empirical rules are only applicable to a particular range of soils tested in laboratory and do not take into account the variability of the base material and filter particle sizes. In addition, it is widely accepted that the safety of fill dams is mainly dependent on the reliability of their filter performance. The work herein presented consists in a new general method for assessing the probability of fulfilling any empirical filter design criteria accounting for base and filter heterogeneity by means of first‐order reliability methods (FORM), so that reliability indexes and probabilities of fulfilling any particular criteria are obtained. This method will allow engineers to estimate the safety of existing filters in terms of probability of fulfilling their design criteria and might also be used as a decision tool on sampling needs and material size tolerances during construction. In addition, sensitivity analysis makes possible to analyse how reliabilities are influenced by different sources of input data. Finally, in case of a portfolio risk assessment, this method will allow engineers to compare the safety of several existing dams in order to prioritize safety investments and it is expected to be a very useful tool to evaluate probabilities of failure due to internal erosion. Copyright © 2006 John Wiley & Sons, Ltd.     

Effect of 2D spatial variability on slope reliability: A simplified FORM analysis

To meet the high demand for reliability based design of slopes, we present in this paper a simplified HLRF(Hasofere Linde Rackwitze Fiessler) iterative algorithm for first-order reliability method(FORM). It is simply formulated in x-space and requires neither transformation of correlated random variables nor optimization tools. The solution can be easily improved by iteratively adjusting the step length. The algorithm is particularly useful to practicing engineers for geotechnical reliability analysis where standalone(deterministic) numerical packages are used. Based on the proposed algorithm and through direct perturbation analysis of random variables, we conducted a case study of earth slope reliability with complete consideration of soil uncertainty and spatial variability.     

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.     

Risk Analysis of Breakwater Caisson Under Wave Attack Using Load Surface Approximation

A new load surface based approach to the reliability analysis of caisson-type breakwater is proposed. Uncertainties of the horizontal and vertical wave loads acting on breakwater are considered by using the so-called load surfaces, which can be estimated as functions of wave height, water level, and so on. Then, the first-order reliability method（FORM） can be applied to determine the probability of failure under the wave action. In this way, the reliability analysis of breakwaters with uncertainties both in wave height and in water level is possible. Moreover, the uncertainty in wave breaking can be taken into account by considering a random variable for wave height ratio which relates the significant wave height to the maximum wave height. The proposed approach is applied numerically to the reliability analysis of caisson breakwater under wave attack that may undergo partial or full wave breaking.     

可靠度分析中梯度求解方法的研究

在岩土工程的可靠度分析中，功能函数的形式非常复杂，甚至是隐式的。而对于常用的可靠度分析方法，如一阶可靠性方法（FORM）、二阶可靠性方法（SORM）等，一个重要的问题是求功能函数对基本随机变量的梯度。因此，对于隐式的或复杂的显式功能函数，必须采用数值微分方法来求解功能函数的梯度。对于可靠度分析中常用的有限差分法（FDM）及有理多项式法（RPT）这两种数值微分方法，本文详细研究了二者问的异同及其求导结果与步长的关系，指出了对于相同的步长控制系数及取样方式，FDM是RPT在线性情况下的特例；对于连续的线性功能函数，可直接用3点FDM求导：对于连续的非线性功能函数，可用5点RPT求导；对于非连续的功能函数，应采用RPT求导。建议取步长控制系数等于1。     

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.     

Effect of surge uncertainty on probabilistically computed dune erosion

To assess the flood protection capacity of dunes in The Netherlands, a semi-probabilistic dune-erosion prediction method is currently in use in which uncertainties in input parameters of an empirical dune erosion model were taken into account, with the exception of the uncertainty in the extreme surge distribution. Previous research has shown that the surge is by far the most influential parameter affecting erosion in the currently used erosion model, which is due both to the influence of the surge level itself and to the conditional dependence of the wave height and period on the surge level in the probabilistic model used for the assessment. Furthermore, the distribution of extreme surge levels has been shown to contain large statistical uncertainty. The inclusion of uncertainty in input variables into probabilistic models results in more extreme events (in this case erosion) for the same exceedance probability, largely due to the incorporation of higher values of the input variables. The goal of the research described in this paper was to determine the impact of the inclusion of uncertainty in the extreme surge distribution on the estimate of critical erosion (erosion associated with an exceedance frequency of 10^− 5 per year). The uncertainty in the surge distributions was estimated and parameterized, and was incorporated into the probabilistic model. A reduction in uncertainty was subsequently imposed to estimate what value a reduction in uncertainty can offer, in terms of the impact on critical erosion. The probabilistic technique first-order reliability method (FORM) was applied to determine the relative contribution of the uncertainty in the surge distribution (as well as the remaining stochastic variables) to the critical erosion. The impact of the inclusion of uncertainty in the surge distribution on the critical retreat distance was found to be substantial with increases ranging from 34% to 93% of the original estimate at five locations along the Dutch coast. The reduced uncertainty showed a more subtle impact, with increases in critical retreat distance ranging from 10% to 26% of the original estimate. The relative importance analysis showed that the uncertainty in the surge distribution has a strong influence, with the relative importance ranging from 10% to 23% for an exceedance frequency of 10^− 5 per year.     

An approximation to the reliability of series geotechnical systems using a linearization approach

A method based on the linearization of the limit state functions (LSFs) is applied to evaluate the reliability of series geotechnical systems. The approach only needs information provided by first order reliability method (FORM) results: the vector of reliability indices, β, of the LSFs composing the system; and their correlation matrix, R. Two common geotechnical problems—the stability of a slope in layered soil and a circular tunnel in rock—are employed to demonstrate the simplicity, accuracy and efficiency of the suggested procedure, and advantages of the linearization approach with respect to alternative computational tools are discussed. It is also found that, if necessary, the second order reliability method (SORM)—that approximates the true LSF better than FORM—can be employed to compute better estimations of the system’s reliability.     

Probabilistic reliability analysis of multiple slopes with genetic algorithms

Uncertainty estimation and consideration in engineering is an important practice to design reliable structures especially in geotechnics since the level of control with regards to the material parameters is relatively low. The definition of reliability indices to approximate the probability of failure allows for a better assessment of stability with fewer computations than using alternative methods. Nonetheless, yet an optimisation problem needs to be solved. In this work, a genetic algorithm is developed to solve this optimisation problem considering the limit equilibrium method to search for multiple critical failures. Study cases are presented to illustrate the capabilities of the method.     

Environmental impact assessment using FORM and groundwater system reliability concept: case study Jining, China

In this paper, fist-order reliability method (FORM) is used to evaluate the impacts of uncertainties posed by traditional deterministic models on the environment in Jining, China. Because of groundwater contamination in shallow aquifer, and an increase in water demand, the new wells target the confined aquifer with constant pumping rate of 5,000 m³/d. Using Theis equation, the groundwater drawdown is analyzed to determine whether the confined aquifer will be contaminated. Although the piezometric level is higher than the phreatic level by 11.0 m, the risk of drawdown is still 19.49% when the pumping rate of 5,000 m³/d is maintained for 2 years. The deterministic model indicates a drawdown of 8.94 m which is lower than the maximum tolerance drawdown of 11.0 m. The sensitivity and uncertainty analysis reveal that the model result is more sensitive to transmissivity than specific yield, while the reliability analysis offers significant information for the decision makers. This approach exposes and minimizes the risk of undesirable consequences such as groundwater contamination.