Xing Zheng Wu

Probabilistic slope stability analysis by a copula-based sampling method

In probabilistic slope stability analysis, the influence of cross correlation of the soil strength parameters, cohesion and internal friction angle, on the reliability index has not been investigated fully. In this paper, an expedient technique is presented for probabilistic slope stability analysis that involves sampling a series of combinations of soil strength parameters through a copula as input to an existing conventional deterministic slope stability program. The approach organises the individual marginal probability density distributions of componential shear strength as a bivariate joint distribution by the copula function to characterise the dependence between shear strengths. The technique can be used to generate an arbitrarily large sample of soil strength parameters. Examples are provided to illustrate the use of the copula-based sampling method to estimate the reliability index of given slopes, and the computed results are compared with the first-order reliability method, considering the correlated random variables. A sensitivity study was conducted to assess the influence of correlational measurements on the reliability index. The approach is simple and can be applied in practice with little effort beyond what is necessary in a conventional analysis.

Development of fragility functions for slope instability analysis

This paper presents a methodology for constructing fragility functions to characterise slope stability under a range of catastrophic earthquakes and rainfalls. The procedures for creating fragility functions, including the first-order reliability method (FORM) and the copula-based sampling method (CBSM), are demonstrated using a selection of typical slopes. The most common failure modes are included, such as the shallow sliding of an infinite slope, circular slip surface of a homogeneous slope, and tetrahedral wedge failure in a rock slope. Owing to the proposed approach, the fragility function can be applied to quantify the failure probabilities over a range of loading conditions with ease, as these are attributed to a function, rather than a design point. The advantage of these definitions is that the uncertainties of correlated soil shear strengths can be incorporated into the reliability models. The established procedure can provide a basis for describing vulnerable behaviour of a slope under various loading conditions and geometries.

Application of a stochastic differential equation to the prediction of shoreline evolution

Shoreline evolution due to longshore sediment transport is one of the most important problems in coastal engineering and management. This paper describes a method to predict the probability distributions of long-term shoreline positions in which the evolution process is based on the standard one-line model recast into a stochastic differential equation. The time-dependent and spatially varying probability density function of the shoreline position leads to a Fokker–Planck equation model. The behaviour of the model is evaluated by applying it to two simple shoreline configurations: a single long jetty perpendicular to a straight shoreline and a rectangular beach nourishment case. The sensitivity of the model predictions to variations in the wave climate parameters is shown. The results indicate that the proposed model is robust and computationally efficient compared with the conventional Monte Carlo simulations.

Probabilistic solution of floodplain inundation equation

Uncertainty in bed roughness is a dominant factor in providing a sufficiently accurate simulation of floodplain flows. This study describes a method to compute the transition probability density distribution of time-varying water elevations where the evolutionary process is based on a conventional one-dimensional storage cell model with governing stochastic differential equation. By including the random inputs (or noise terms) of bed roughness and initial water depth, time-dependent and spatially varying probability density function of the water surface leads to a Fokker–Planck equation. The model’s performance is evaluated by applying it to shallow water flow with a horizontal bed. Sensitivity of model predictions to variations in the bed friction parameters is shown. By comparing the result of the proposed method with that of conventional Monte Carlo simulation, the advantage of the former as a method for density function prediction is confirmed.

Analysing Flood and Erosion Risks and Coastal Management Strategies on the Norfolk Coast

Coastal systems are characterised by variability and interdependencies at a range of scales. On the Norfolk coast from Weybourne to Winterton in SMP6, variations in sediment supply, from cliff erosion and beach nourishment, have a profound influence upon the probability of failure of the flood defences that protect the large area of coastal lowlands, including the Norfolk Broads. The risk of flooding is therefore influenced by large-scale and long-term changes in sediment supply as well as by short-term fluctuations which are dominated by the arrival of extreme storms. The reliability of the flood defences also plays a crucial mediating role on the probability of flooding. When flooding does occur, the extent and severity of damage is influenced by patterns of inundation and the human and economic vulnerability of the communities that are flooded.

Implementing statistical fitting and reliability analysis for geotechnical engineering problems in R

ABSTRACT Reliability analysis and multivariate statistical fitting are valuable techniques that enhance the scientific basis of regulatory decisions in geotechnical problems. This study introduces the use of several R packages specifically developed to assist risk assessors in their geotechnical projects. Firstly, the fitting of parameterised models either to the distribution of observed samples or to characterise the dependence structures among variables, or both is presented. Secondly, the most popular reliability analysis methods, such as the first- and second-order reliability methods and the random sampling simulation method, are implemented in R. The efficiency of implementing these classical approximation methods is demonstrated through two example problems.

Liouville equation-based stochastic model for shoreline evolution

Long-term shoreline evolution due to longshore sediment transport is one of the key processes that need to be addressed in coastal engineering design and management. To adequately represent the inherent stochastic nature of the evolution processes, a probability density evolution model based on a Liouville-type equation is proposed for predicting the shoreline changes. In this model, the standard one-line beach evolution model that is widely used in coastal engineering design is reformulated in terms of the probability density function of shoreline responses. A computational algorithm involving a total variation diminishing scheme is employed to solve the resulting equation. To check the accuracy and robustness of the model, the predictions of the model are evaluated by comparing them with those from Monte Carlo simulations for two idealised shoreline configurations involving a single long jetty perpendicular to a straight shoreline and a rectangular beach nourishment case. The pertinent features of the predicted probabilistic shoreline responses are identified and discussed. The influence of the density distributions of the input parameters on the computed results is investigated.

Quantifying the non-normality of shear strength of geomaterials

Abstract Geotechnical shear strength variables generally include the cohesion and angle of internal friction based upon the Mohr-Coulomb failure criterion. In this study, the non-normality of a univariate probability density function (PDF) and the bivariate probability density contour (PDC) of observed shear strength pairs is examined for 33 geomaterial types, comprising of soils (24 types), rocks (7 types), and geosynthetic clay liners (2 types), with sample sizes ranging from 14 to 97. After a detailed analysis on the graphic features of probability density and box-whisker plots of shear strength parameters, normality testing is further quantified by skewness, kurtosis, and energy statistic tests. In most cases (23/31), distributions of cohesion are positively skewed, as are distributions of friction angle (17/29), while the distribution of these parameters is mostly platykurtic (characterised by negative excess kurtosis). Bivariate energy statistic analyses of shear strength pairs indicate that, in ten cases, p-values are below .05, demonstrating that the joint distribution differs from the bivariate normal distribution, and these results are largely consistent with those achieved by the Shapiro-Wilk test. Moreover, a slope stability analysis with different joint distributions is used to assess the impact of marginal PDFs on the failure probability.

Broadscale Coastal Inundation Modelling

To understand the implications of changes to marine climates and the impact on people, analysis is required of the spatial extent and depths of flooding in urban and rural areas. Over extended timescales, the uncertainties related to possible sea-level rise and changes in storminess increase significantly. As such, there is no certainty regarding when and if a major storm or breach will occur. The increasing availability of some details of changing wave climate (described in Chaps. 2 and 3) and full range of beach evolutionary behaviours (given in Chap. 7) offer an opportunity to improve the implications of these uncertainties on flood extents.