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xing zheng wu

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Associate Professor, College of Civil Engineering and Architecture, Hebei University, ChinaPrevious Affiliations: [1] Department of Applied Mathematics, School of Applied Science, University of Science and Technology Beijing, China [2] University of Newcastle, UK [3] University of Dundee, UK [4] Chinese Institute of Water Resources and Hydropower Research, ChinaResearch interests focus on the general area of risk analysis, uncertainty modelling of Earth Systems Engineering. A cross-disciplinary endeavour aims to build an interface between the theoretical basic research, application oriented developments and numerical implementations. http://xingzhengwu.com/index.html

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Papers by xing zheng wu

Multivariate reliability method using the environment contour model based on C-vine copulas

Ocean Engineering, 2024

This study aims to develop a new technique for deriving the reliability index in original physica... more This study aims to develop a new technique for deriving the reliability index in original physical space using C-vine copula models integrated with the environmental contour method. In the proposed approach, the minimum of the reliability index is considered an objective function, whereas the coordinates on the environmental contour satisfying the limit state equation are treated as the constraint. These components are formulated as a non-linear constrained optimisation problem. The validity of this method is illustrated through four examples of its application, and the computed reliability index is compared with the results yielded by the existing sampling method. The findings confirm the superiority of the proposed technique. Compared with sampling simulation techniques, it is easier to obtain the critical environmental contour and the design point. Due to a pair-copula construction technique being applied to obtain high-dimensional vine copula models, the multivariate environmental contours are determined, which overcomes the limitation of existing contour line methods that are only used for solving low-dimensional problems. Unlike the first-order reliability method constructed in standard normal space, this new algorithm solves in the original physical space of random variables, making the calculation principles more transparent and easier to understand.

Solving the geometric reliability index for a case involving multivariate random variables in the original physical space

Quality and Reliability Engineering International, 2023

An effective interpretation of the complex reliability theory will promote its application in eng... more An effective interpretation of the complex reliability theory will promote its application in engineering practice. Based on an algorithm for solving the geometric reliability index for cases involving two or three random variables in the original physical space, an advanced algorithm for multivariate random variables is further developed by using a hypersphere model and inverse Nataf transformation. The main components of the algorithm include the definition of limit state surface, the construction of a hypersphere model, the determination of probability density iso-contour points, the identification of limit state of these points, and the definition of an actual reliability index. Several typical practical problems associated with two to five variables are analysed, and the results are compared with those obtained using the first-order reliability method (FORM) and the copulabased sampling method (CBSM), thus validating the accuracy of the proposed algorithm. With the help of the powerful and open-source R language platform, the visualisation of a reliability index in low dimensions and the solvability in high dimensions are implemented, which makes the interpretation of reliability theory more intuitive.

Development of environmental contours from site-specific regression parameters of load−settlement curves for piles -the global database

by xing zheng wu and He Liu

International Journal of Geomechanics, 2023

A database is compiled to establish the site-specific measured load−displacement responses of pil... more A database is compiled to establish the site-specific measured load−displacement responses of piles from different sites across the globe, which contains 188 curves from 12 sites. A curve-fitting technique is applied to each measured curve by using a two-parametric power law regression modelling. For an entire construction site, the scatter in the load−displacement behaviour of multiple curves is incorporated in the uncertainty of the regression pairs. Consequently, environmental contours for regression pairs are developed by using the Nataf transformation or a normal copula. Such a contour is generated to quantify the uncertainty in the regression pairs in which the target reliability index of piles is specified, as demonstrated in various cases with two, three, and four variables. Most importantly, the environmental contour bounded by the serviceability limit state of piles at each site is presented and the reliability index is solved through an evolution of the environmental contours. The computed reliability index of the bearing resistance for piles is compared with the values yielded by the geometric reliability analysis and the first-order reliability method (FORM). Since the reliability index is solved in the original physical variable space, it makes the reliability theory easy to understand. Such a site-specific reliability assessment of piles can be helpful for a practical project to implement the quality control during the construction of pile foundations.

Development of environmental contours from rainfall intensity and duration data for slopes

by xing zheng wu, Chenzhe Ma, and Rui-Kai Wang

Natural Hazards, 2023

Rainfall is an important environmental factor affecting the stability of natural slopes. In this ... more Rainfall is an important environmental factor affecting the stability of natural slopes. In this study, an environmental contour method is developed by incorporating the infiltration model proposed by Green and Ampt to evaluate the stability of slopes exposed to rainfall. Several environmental variables-such as rainfall intensity and rainfall duration, cohesion and internal friction angle of soil parameters-are treated as random variables. The influence of these variables, especially their joint characteristics, on the reliability of slopes is discussed. The environmental contour is determined by specifying a target reliability index or a return period. In addition, the reliability index is derived by using the environmental contour method on the basis of the limit state function. Based on the dataset pertaining to long-term rainfall observations for the Peloritani Mountains regions of Italy, the illustrative reliability analysis is performed for the hill slopes. The environmental contour in the case of multiple variables is demonstrated, and the reliability index of slope stability against sliding is obtained. The extreme slope responses are derived for the points on the environmental contour, which can be helpful for probabilistic slope stability assessment and design.

Interpretation of site-specific reliability index for piles using multiple bidirectional tests

Journal of Testing and Evaluation, 2022

The Osterberg cell test is an effective tool to evaluate the bearing behaviour for piled foundati... more The Osterberg cell test is an effective tool to evaluate the bearing behaviour for piled foundations. Due to the significant intrinsic variabilities in material properties of piles, soil−pile intermediate, and soils, the responses of multiple piles under loading are uncertain. These uncertainties should be considered to evaluate the bearing characteristics of piles, in this work, a methodology for estimating the geometric reliability index of piles using the Osterberg cell testing is proposed. A dataset of load−displacement curves including 26 Osterberg cell-tested drilled shafts at the Los Angeles Memorial Coliseum site is used to illustrate the proposed method. Regression analyses with a two-parametric power law curve-fitting model are firstly used to construct the bestfit model for each load−displacement curve along an upward or downward direction individually. As a result, the regression parameters for this site are combined into a random vector. Statistical analyses are then performed on these regression variables, and the bivariate normal and non-normal models are developed. These bivariate models are used to predict the reliability index of these drilled shafts via the geometric reliability method. Such an interpretation of reliability index by analysing the scatter in load test data can be helpful to design on pile foundations.

Determination of geometric reliability index of piles at site-specific scale: Case studies

Proceedings of the Institution of Civil Engineers: geotechnical engineering, 2021

This paper presents an application of geometric reliability techniques on various pile types base... more This paper presents an application of geometric reliability techniques on various pile types based on collected 67 load-displacement curves obtained from the pertinent literature. These static loading tests were performed on local scales (even building-specific sites) with essentially identical geotechnical conditions. A power law function with two parameters is utilised to fit the measured load-settlement curves. For each site, the means and coefficients of variation for the power law parameters are obtained. Since the number of tests at each site is usually small, it is extremely difficult to identify a certain distribution type for these regression parameters. Thus, for simplicity, bivariate normal distribution is assumed to represent the set of regression pairs. This joint distribution is incorporated into a geometric reliability method, which offers an estimation of the bearing capacity of piles at the serviceability limit state. Examples are provided to illustrate the application of the proposed reliability method to interpret the visually determined reliability index of the bearing capacity of these piles. These case studies confirm that the method is a useful and comprehensive tool for capturing the load-displacement response and for evaluating the bearing capacity of piles by considering the surrounding uncertainties in their load-displacement behaviour.

Geometric reliability analysis of composite foundations comprising of cement-fly ash-gravel piles at site-specific scale

Journal of Testing and Evaluation, 2020

Evaluating the pressure−settlement response of a composite foundation at a specific site is impo... more Evaluating the pressure−settlement response of a composite foundation at a specific site is important for supporting the superstructures safely. The uncertainties in the power law regression parameters stem from a significant scattering of pressure−settlement curves when a series of static load tests are conducted for the composite foundations comprising of cement-fly ash-gravel (CFG) piles at a particular site. Thus, in this work, a probability density contour (PDC) of these regression parameters is constructed using a bivariate distribution, which offers great flexibility in fitting the marginal distributions and the dependency between the two regression parameters. Based on an intuitive evolutionary geometry of the PDCs in the original physical space of two dependent random variables, a reliability index is defined as a distance ratio of two PDCs, i.e., the dispersed PDC that just touches the limit state curve and the one standard deviation (1 SD) PDC. These bivariate PDCs are discretised with a number of segments that adapt to the different spreads of the asymmetric data. Reliability indices are calculated based on the geometric reliability analysis, which offers great transparency for engineers in evaluating the safety of the composite foundation at a site-specific level. Using this reliability technique, the effects of uncertain regression parameters are explicitly explored at six sites consisted of 280 static loading tests.

Probabilistic analysis of site-specific load-displacement behaviour of cement-fly ash-gravel piles

Soils and Foundations -Tokyo-, 2019

Cement-fly ash-gravel (CFG) pile is one of the novel ground improvement types. It is attracting c... more Cement-fly ash-gravel (CFG) pile is one of the novel ground improvement types. It is attracting considerable attention, as its beneficial role in accessible soft foundations is being increasingly exploited by the surging urban industrial development in China. Yet, despite its recognised significance, the load-settlement behaviour of CFG pile is poorly understood, particularly in the context of probabilistic assessments. This study focuses on six full-scale CFG pile load test datasets (consisting of 245 samples collected at six sites) under static axial compression loading from the Beijing region. As a part of this investigation, a regression curve is applied to the load-settlement data for each load test in the database using a two-parametric hyperbolic or power law curve-fitting equation. Moreover, an assessment of the multiple load-displacement curves based on the full set of pile load measurements conducted at a particular test site reveals that scatter observed in regression parameter values is caused mainly by inherent soil variability.
Thus, a bivariate copula-based mixed distribution is chosen to represent the dependence between these regression parameters. A simple copula-based simulation model is used to estimate the reliability index at any specific allowable settlement for the serviceability limit state (SLS) design. The correlation coefficients in copula-based distributions of regression parameters are proven to have an impact on the reliability index of this pile foundation. The scatter analysis of the load-displacement behaviour provides insight into the probabilistic design of site-specific CFG pile foundations.
Keywords: pile; CFG; correlation; copulas; regression; reliability index

Quantifying the non-normality of shear strength of geomaterials

European Journal of Environmental and Civil Engineering, 2018

Geotechnical shear strength variables generally include the cohesion and angle of internal fricti... more 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 (GCL, 2 types), with sample sizes ranging from 14 to 95. 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 0.05, 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.

Discussion of "Quantifying the cross-correlation between effective cohesion and friction angle of soil from limited site-specific data" by Wang and Akeju (2016)

Soils and Foundations -Tokyo-, 2017

This discussion pertains to the shear strength data of alluvial fine-grained soils in Central Ita... more

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

Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2017

Reliability analysis and multivariate statistical fitting are valuable techniques that enhance th... more 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.

Geometric reliability analysis applied to wave overtopping of sea defences

Ocean Engineering, 2015

A probability density contour-based expanding method (CBEM) is developed to provide an intuitive ... more A probability density contour-based expanding method (CBEM) is developed to provide an intuitive interpretation of calculating the reliability index of an engineered system in the original physical space of random variables. In this technique, a distance ratio is defined to quantify the reliability index in conjunction with a dispersive probability density contour (that just reaches the failure surface) and a unit contour (which is equivalent to the one-standard-deviation domain). The visualisation of the relative position, between the expanding probability density contour and limit state function, can be provided for bivariate and trivariate cases. The determination of the limit state function of the engineered system is discussed, which builds on an outer fence of the dispersive probability density contour. A spatial discretised approximation to both the joint probability density contour and the critical state surface is facilitated through taking advantage of the new multivariate visualisation techniques in R. Such an approximation allows the curvature of the limit state surface to be very complex, and the probability distribution contour of random variables to be any form, even non-parametric one. Several numerical examples are presented to illustrate the validity and applicability of the proposed method. A comparison between the proposed approach and the conventional reliability methods is provided.

Modelling dependence structures of soil shear strength data with bivariate copulas and applications to geotechnical reliability analysis

Soils and Foundations, 2015

Accurate estimates of the dependence of soil shear strength parameters (including cohesion and fr... more Accurate estimates of the dependence of soil shear strength parameters (including cohesion and friction angle) play a crucial role in decision making by civil engineers about geotechnical engineering safety. Increased site-specific information comes the need for joint soil strength models to account for correlation characteristics between shear strength properties. In this study, firstly, using 16 sets of soil shear strength observations (consists of 391 samples) as examples, the suitability of the dependence structure for these experimental observations is identified by a goodness-of-fit test based on the Bayesian Information Criterion (BIC) with the normal, Student's t, Clayton, Frank, Gumbel, and Plackett copulas. The dependence structure between shear strength components is found to be asymmetric in most cases. Secondly, a set of paired samples of shear strengths simulated from the different bivariate copulas, which contributed to various dependencies, is implemented as an input to two typical geotechnical probabilistic analyses, e.g., an infinite slope stability against a single sliding plane and a bearing capacity of shallow foundation. The impact of different choices of these dependence structures on the resulting reliability index is discussed. In both illustrative examples, the normal copula leads to an overestimation of the reliability index, whereas the Gumbel copula achieves the lowest reliability index. The conservative reliability indices are obtained when the joint behaviour of soil shear strengths follows a bivariate normal distribution.

Development of fragility functions for slope instability analysis

Landslides (Impact Factor: 2.81). 10/2014; 11(6). DOI: 10.1007/s10346-014-0536-3

This paper presents a methodology for constructing fragility functions to characterise slope stab... more 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.

Using copulas to characterise the dependency of GCL shear strengths

Geosynthetics International (Impact Factor: 1.17). 10/2013; 20(5):344-357. DOI: 10.1680/gein.13.00021

The sensitivity of landfill design to shear strength variability is controlled by the measured i... more The sensitivity of landfill design to shear strength variability is controlled by the measured internal shear strength of the geosynthetic clay liner and the interface shear strength between the geosynthetic clay liner and the geomembrane. Based on the Mohr–Coulomb criterion, the measured shear strength values (including the internal and interface shear strength values) depend on the cohesion and friction angle. Here, a novel and efficient mathematical tool for analysing the properties of the bivariate and the correlated geosynthetic shear strength was constructed. The arbitrary marginal probability density distribution of the componential shear strength was organised as a bivariate joint distribution with a copula function to characterise the dependency between the different shear strengths. Generally, a copula-based method provides a simple and powerful framework for modelling interdependence among variables. Thus, the construction of copulas as an alternative to correlation coefficients is addressed. In addition, the shear strength marginal distributions were described by a non-normal distribution, which was identified by the Akaike information criterion. The dependency of these margins is implemented from several empirical copulas, which were chosen from the most frequently used Elliptical and Archimedean families. The simulated correlated material properties that were generated by the copula-based sampling method were used as input values to calculate the performance of a landfill lining problem. Moreover, the computed reliability indexes were compared with the reliability indexes that were obtained by the first-order second moment method. In addition, a sensitivity study with various correlation coefficients was conducted to determine the influence of the correlation measurements on the reliability index.

Assessing the correlated performance functions of an engineering system via probabilistic analysis

Structural Safety (Impact Factor: 2.39). 10/2014; 2015(52):10-19. DOI: 10.1016/j.strusafe.2014.07.004

An important issue regarding the use of probabilistic predictions for complex engineering systems... more An important issue regarding the use of probabilistic predictions for complex engineering systems is characterising the dependence structure among its correlated performance functions, which are driven by dependent or independent basic random variables. The interrelationship of these performance functions can be attributed to the same random variables and the cross correlation among the input parameters. An assessment of joint failure probability for an engineering system is proposed, which is associated with the correlated performance functions using a copula-based method by conveying the dependence structure of the performance functions. The method is demonstrated with four simple engineering problems, i.e., (a) bivariate distribution in which two predetermined performance functions are associated with each other; (b) pile bearing capacity in which the performance functions are related with the soil internal friction and the compressive strength of a concrete pile; (c) pipe flow in which the performance function of three pipes in a sewer system is assessed with six independent random variables; and (d) retaining wall in which the failure criteria for defining the performance functions include overturning failure about the toe point, sliding failure along the base, and bearing capacity instability considering uncertain soil properties. The computational efficiency is evaluated using the results based on the conventional bounding methods. The joint failure probability expressed by copulas provides a means to obtain the joint probabilities of multiple failure modes, which pave the way for an objective description of the overall failure probability of a practical engineering problem.

Trivariate analysis of soil ranking-correlated characteristics and its application to probabilistic stability assessments in geotechnical engineering problems

Soils and Foundations -Tokyo- (Impact Factor: 0.41). 08/2013; 53(4):540–556. DOI: 10.1016/j.sandf.2013.06.006

An important issue in the probabilistic prediction modelling of multivariate soil properties (usu... more An important issue in the probabilistic prediction modelling of multivariate soil properties (usually including cohesion, friction angle, and unit weight) is the measurement of dependence structure among these properties. The use of Pearson's correlation as a dependence measure has several pitfalls; therefore, it may not be appropriate to use probabilistic prediction models in geotechnical engineering problems based on this correlation. As an alternative, a copula-based methodology for prediction modelling and an algorithm to simulate multivariate soil data are proposed.

In this method, all different random variables are transformed to a rank/uniform domain in order to form a copula function by applying cumulative distribution function transformations. The technique of copulas, representing a promising alternative for solving multivariate problems to describe their dependence structure by a ranked correlation coefficient, is highlighted. Two existing observed soil data sets from river banks are used to fit a trivariate normal copula and a trivariate fully nested Frank copula. The ranking correlation coefficient Kendall's τ and the copula model parameters are estimated. The goodness-of-fit test to choose the best-fitting model is discussed.
A series of triplet samples (i.e., cohesion, friction angle, and unit weight) simulated from the trivariate normal copula with flexible marginal distributions are used as input parameters to evaluate the uncertainties of soil properties and to define their correlations.

The influence of the cross-correlation of these soil properties on reliability-based geotechnical design is demonstrated with two simple geotechnical problems: (a) the bearing capacity of a shallow foundation resting on a clayey soil and (b) the stability of a cohesive-frictional soil in a planar slope. The sensitivity analysis of their correlations of random variables on the influence of the reliability index provides a better insight into the role of the dependence structure in the reliability assessment of geotechnical engineering problems.

Probabilistic solution of floodplain inundation equation

Stochastic Environmental Research and Risk Assessment (Impact Factor: 2.67). 01/2015; 29(NA):NA. DOI: 10.1007/s00477-015-1025-5

Uncertainty in bed roughness is a dominant factor in providing a sufficiently accurate simulation... more 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 densityfunction 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.

Liouville Equation-Based Stochastic Model for Shoreline Evolution

Stochastic Environmental Research and Risk Assessment (Impact Factor: 2.67). 01/2015; 29(NA):NA.

Long-term shoreline evolution due to longshore sediment transport is one of the key processes tha... more 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 the Liouville 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.

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

Stochastic Environmental Research and Risk Assessment (Impact Factor: 2.67). 12/2013; DOI: 10.1007/s00477-013-0715-0

Shoreline evolution due to longshore sediment transport is one of the most important problems in ... more 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.

Multivariate reliability method using the environment contour model based on C-vine copulas

Ocean Engineering, 2024

This study aims to develop a new technique for deriving the reliability index in original physica... more This study aims to develop a new technique for deriving the reliability index in original physical space using C-vine copula models integrated with the environmental contour method. In the proposed approach, the minimum of the reliability index is considered an objective function, whereas the coordinates on the environmental contour satisfying the limit state equation are treated as the constraint. These components are formulated as a non-linear constrained optimisation problem. The validity of this method is illustrated through four examples of its application, and the computed reliability index is compared with the results yielded by the existing sampling method. The findings confirm the superiority of the proposed technique. Compared with sampling simulation techniques, it is easier to obtain the critical environmental contour and the design point. Due to a pair-copula construction technique being applied to obtain high-dimensional vine copula models, the multivariate environmental contours are determined, which overcomes the limitation of existing contour line methods that are only used for solving low-dimensional problems. Unlike the first-order reliability method constructed in standard normal space, this new algorithm solves in the original physical space of random variables, making the calculation principles more transparent and easier to understand.

Solving the geometric reliability index for a case involving multivariate random variables in the original physical space

Quality and Reliability Engineering International, 2023

An effective interpretation of the complex reliability theory will promote its application in eng... more An effective interpretation of the complex reliability theory will promote its application in engineering practice. Based on an algorithm for solving the geometric reliability index for cases involving two or three random variables in the original physical space, an advanced algorithm for multivariate random variables is further developed by using a hypersphere model and inverse Nataf transformation. The main components of the algorithm include the definition of limit state surface, the construction of a hypersphere model, the determination of probability density iso-contour points, the identification of limit state of these points, and the definition of an actual reliability index. Several typical practical problems associated with two to five variables are analysed, and the results are compared with those obtained using the first-order reliability method (FORM) and the copulabased sampling method (CBSM), thus validating the accuracy of the proposed algorithm. With the help of the powerful and open-source R language platform, the visualisation of a reliability index in low dimensions and the solvability in high dimensions are implemented, which makes the interpretation of reliability theory more intuitive.

Development of environmental contours from site-specific regression parameters of load−settlement curves for piles -the global database

by xing zheng wu and He Liu

International Journal of Geomechanics, 2023

A database is compiled to establish the site-specific measured load−displacement responses of pil... more A database is compiled to establish the site-specific measured load−displacement responses of piles from different sites across the globe, which contains 188 curves from 12 sites. A curve-fitting technique is applied to each measured curve by using a two-parametric power law regression modelling. For an entire construction site, the scatter in the load−displacement behaviour of multiple curves is incorporated in the uncertainty of the regression pairs. Consequently, environmental contours for regression pairs are developed by using the Nataf transformation or a normal copula. Such a contour is generated to quantify the uncertainty in the regression pairs in which the target reliability index of piles is specified, as demonstrated in various cases with two, three, and four variables. Most importantly, the environmental contour bounded by the serviceability limit state of piles at each site is presented and the reliability index is solved through an evolution of the environmental contours. The computed reliability index of the bearing resistance for piles is compared with the values yielded by the geometric reliability analysis and the first-order reliability method (FORM). Since the reliability index is solved in the original physical variable space, it makes the reliability theory easy to understand. Such a site-specific reliability assessment of piles can be helpful for a practical project to implement the quality control during the construction of pile foundations.

Development of environmental contours from rainfall intensity and duration data for slopes

by xing zheng wu, Chenzhe Ma, and Rui-Kai Wang

Natural Hazards, 2023

Rainfall is an important environmental factor affecting the stability of natural slopes. In this ... more Rainfall is an important environmental factor affecting the stability of natural slopes. In this study, an environmental contour method is developed by incorporating the infiltration model proposed by Green and Ampt to evaluate the stability of slopes exposed to rainfall. Several environmental variables-such as rainfall intensity and rainfall duration, cohesion and internal friction angle of soil parameters-are treated as random variables. The influence of these variables, especially their joint characteristics, on the reliability of slopes is discussed. The environmental contour is determined by specifying a target reliability index or a return period. In addition, the reliability index is derived by using the environmental contour method on the basis of the limit state function. Based on the dataset pertaining to long-term rainfall observations for the Peloritani Mountains regions of Italy, the illustrative reliability analysis is performed for the hill slopes. The environmental contour in the case of multiple variables is demonstrated, and the reliability index of slope stability against sliding is obtained. The extreme slope responses are derived for the points on the environmental contour, which can be helpful for probabilistic slope stability assessment and design.

Interpretation of site-specific reliability index for piles using multiple bidirectional tests

Journal of Testing and Evaluation, 2022

The Osterberg cell test is an effective tool to evaluate the bearing behaviour for piled foundati... more The Osterberg cell test is an effective tool to evaluate the bearing behaviour for piled foundations. Due to the significant intrinsic variabilities in material properties of piles, soil−pile intermediate, and soils, the responses of multiple piles under loading are uncertain. These uncertainties should be considered to evaluate the bearing characteristics of piles, in this work, a methodology for estimating the geometric reliability index of piles using the Osterberg cell testing is proposed. A dataset of load−displacement curves including 26 Osterberg cell-tested drilled shafts at the Los Angeles Memorial Coliseum site is used to illustrate the proposed method. Regression analyses with a two-parametric power law curve-fitting model are firstly used to construct the bestfit model for each load−displacement curve along an upward or downward direction individually. As a result, the regression parameters for this site are combined into a random vector. Statistical analyses are then performed on these regression variables, and the bivariate normal and non-normal models are developed. These bivariate models are used to predict the reliability index of these drilled shafts via the geometric reliability method. Such an interpretation of reliability index by analysing the scatter in load test data can be helpful to design on pile foundations.

Determination of geometric reliability index of piles at site-specific scale: Case studies

Proceedings of the Institution of Civil Engineers: geotechnical engineering, 2021

This paper presents an application of geometric reliability techniques on various pile types base... more This paper presents an application of geometric reliability techniques on various pile types based on collected 67 load-displacement curves obtained from the pertinent literature. These static loading tests were performed on local scales (even building-specific sites) with essentially identical geotechnical conditions. A power law function with two parameters is utilised to fit the measured load-settlement curves. For each site, the means and coefficients of variation for the power law parameters are obtained. Since the number of tests at each site is usually small, it is extremely difficult to identify a certain distribution type for these regression parameters. Thus, for simplicity, bivariate normal distribution is assumed to represent the set of regression pairs. This joint distribution is incorporated into a geometric reliability method, which offers an estimation of the bearing capacity of piles at the serviceability limit state. Examples are provided to illustrate the application of the proposed reliability method to interpret the visually determined reliability index of the bearing capacity of these piles. These case studies confirm that the method is a useful and comprehensive tool for capturing the load-displacement response and for evaluating the bearing capacity of piles by considering the surrounding uncertainties in their load-displacement behaviour.

Geometric reliability analysis of composite foundations comprising of cement-fly ash-gravel piles at site-specific scale

Journal of Testing and Evaluation, 2020

Evaluating the pressure−settlement response of a composite foundation at a specific site is impo... more Evaluating the pressure−settlement response of a composite foundation at a specific site is important for supporting the superstructures safely. The uncertainties in the power law regression parameters stem from a significant scattering of pressure−settlement curves when a series of static load tests are conducted for the composite foundations comprising of cement-fly ash-gravel (CFG) piles at a particular site. Thus, in this work, a probability density contour (PDC) of these regression parameters is constructed using a bivariate distribution, which offers great flexibility in fitting the marginal distributions and the dependency between the two regression parameters. Based on an intuitive evolutionary geometry of the PDCs in the original physical space of two dependent random variables, a reliability index is defined as a distance ratio of two PDCs, i.e., the dispersed PDC that just touches the limit state curve and the one standard deviation (1 SD) PDC. These bivariate PDCs are discretised with a number of segments that adapt to the different spreads of the asymmetric data. Reliability indices are calculated based on the geometric reliability analysis, which offers great transparency for engineers in evaluating the safety of the composite foundation at a site-specific level. Using this reliability technique, the effects of uncertain regression parameters are explicitly explored at six sites consisted of 280 static loading tests.

Probabilistic analysis of site-specific load-displacement behaviour of cement-fly ash-gravel piles

Soils and Foundations -Tokyo-, 2019

Cement-fly ash-gravel (CFG) pile is one of the novel ground improvement types. It is attracting c... more Cement-fly ash-gravel (CFG) pile is one of the novel ground improvement types. It is attracting considerable attention, as its beneficial role in accessible soft foundations is being increasingly exploited by the surging urban industrial development in China. Yet, despite its recognised significance, the load-settlement behaviour of CFG pile is poorly understood, particularly in the context of probabilistic assessments. This study focuses on six full-scale CFG pile load test datasets (consisting of 245 samples collected at six sites) under static axial compression loading from the Beijing region. As a part of this investigation, a regression curve is applied to the load-settlement data for each load test in the database using a two-parametric hyperbolic or power law curve-fitting equation. Moreover, an assessment of the multiple load-displacement curves based on the full set of pile load measurements conducted at a particular test site reveals that scatter observed in regression parameter values is caused mainly by inherent soil variability.
Thus, a bivariate copula-based mixed distribution is chosen to represent the dependence between these regression parameters. A simple copula-based simulation model is used to estimate the reliability index at any specific allowable settlement for the serviceability limit state (SLS) design. The correlation coefficients in copula-based distributions of regression parameters are proven to have an impact on the reliability index of this pile foundation. The scatter analysis of the load-displacement behaviour provides insight into the probabilistic design of site-specific CFG pile foundations.
Keywords: pile; CFG; correlation; copulas; regression; reliability index

Quantifying the non-normality of shear strength of geomaterials

European Journal of Environmental and Civil Engineering, 2018

Geotechnical shear strength variables generally include the cohesion and angle of internal fricti... more 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 (GCL, 2 types), with sample sizes ranging from 14 to 95. 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 0.05, 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.

Discussion of "Quantifying the cross-correlation between effective cohesion and friction angle of soil from limited site-specific data" by Wang and Akeju (2016)

Soils and Foundations -Tokyo-, 2017

This discussion pertains to the shear strength data of alluvial fine-grained soils in Central Ita... more

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

Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2017

Reliability analysis and multivariate statistical fitting are valuable techniques that enhance th... more 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.

Geometric reliability analysis applied to wave overtopping of sea defences

Ocean Engineering, 2015

A probability density contour-based expanding method (CBEM) is developed to provide an intuitive ... more A probability density contour-based expanding method (CBEM) is developed to provide an intuitive interpretation of calculating the reliability index of an engineered system in the original physical space of random variables. In this technique, a distance ratio is defined to quantify the reliability index in conjunction with a dispersive probability density contour (that just reaches the failure surface) and a unit contour (which is equivalent to the one-standard-deviation domain). The visualisation of the relative position, between the expanding probability density contour and limit state function, can be provided for bivariate and trivariate cases. The determination of the limit state function of the engineered system is discussed, which builds on an outer fence of the dispersive probability density contour. A spatial discretised approximation to both the joint probability density contour and the critical state surface is facilitated through taking advantage of the new multivariate visualisation techniques in R. Such an approximation allows the curvature of the limit state surface to be very complex, and the probability distribution contour of random variables to be any form, even non-parametric one. Several numerical examples are presented to illustrate the validity and applicability of the proposed method. A comparison between the proposed approach and the conventional reliability methods is provided.

Modelling dependence structures of soil shear strength data with bivariate copulas and applications to geotechnical reliability analysis

Soils and Foundations, 2015

Accurate estimates of the dependence of soil shear strength parameters (including cohesion and fr... more Accurate estimates of the dependence of soil shear strength parameters (including cohesion and friction angle) play a crucial role in decision making by civil engineers about geotechnical engineering safety. Increased site-specific information comes the need for joint soil strength models to account for correlation characteristics between shear strength properties. In this study, firstly, using 16 sets of soil shear strength observations (consists of 391 samples) as examples, the suitability of the dependence structure for these experimental observations is identified by a goodness-of-fit test based on the Bayesian Information Criterion (BIC) with the normal, Student's t, Clayton, Frank, Gumbel, and Plackett copulas. The dependence structure between shear strength components is found to be asymmetric in most cases. Secondly, a set of paired samples of shear strengths simulated from the different bivariate copulas, which contributed to various dependencies, is implemented as an input to two typical geotechnical probabilistic analyses, e.g., an infinite slope stability against a single sliding plane and a bearing capacity of shallow foundation. The impact of different choices of these dependence structures on the resulting reliability index is discussed. In both illustrative examples, the normal copula leads to an overestimation of the reliability index, whereas the Gumbel copula achieves the lowest reliability index. The conservative reliability indices are obtained when the joint behaviour of soil shear strengths follows a bivariate normal distribution.

Development of fragility functions for slope instability analysis

Landslides (Impact Factor: 2.81). 10/2014; 11(6). DOI: 10.1007/s10346-014-0536-3

This paper presents a methodology for constructing fragility functions to characterise slope stab... more 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.

Using copulas to characterise the dependency of GCL shear strengths

Geosynthetics International (Impact Factor: 1.17). 10/2013; 20(5):344-357. DOI: 10.1680/gein.13.00021

The sensitivity of landfill design to shear strength variability is controlled by the measured i... more The sensitivity of landfill design to shear strength variability is controlled by the measured internal shear strength of the geosynthetic clay liner and the interface shear strength between the geosynthetic clay liner and the geomembrane. Based on the Mohr–Coulomb criterion, the measured shear strength values (including the internal and interface shear strength values) depend on the cohesion and friction angle. Here, a novel and efficient mathematical tool for analysing the properties of the bivariate and the correlated geosynthetic shear strength was constructed. The arbitrary marginal probability density distribution of the componential shear strength was organised as a bivariate joint distribution with a copula function to characterise the dependency between the different shear strengths. Generally, a copula-based method provides a simple and powerful framework for modelling interdependence among variables. Thus, the construction of copulas as an alternative to correlation coefficients is addressed. In addition, the shear strength marginal distributions were described by a non-normal distribution, which was identified by the Akaike information criterion. The dependency of these margins is implemented from several empirical copulas, which were chosen from the most frequently used Elliptical and Archimedean families. The simulated correlated material properties that were generated by the copula-based sampling method were used as input values to calculate the performance of a landfill lining problem. Moreover, the computed reliability indexes were compared with the reliability indexes that were obtained by the first-order second moment method. In addition, a sensitivity study with various correlation coefficients was conducted to determine the influence of the correlation measurements on the reliability index.

Assessing the correlated performance functions of an engineering system via probabilistic analysis

Structural Safety (Impact Factor: 2.39). 10/2014; 2015(52):10-19. DOI: 10.1016/j.strusafe.2014.07.004

An important issue regarding the use of probabilistic predictions for complex engineering systems... more An important issue regarding the use of probabilistic predictions for complex engineering systems is characterising the dependence structure among its correlated performance functions, which are driven by dependent or independent basic random variables. The interrelationship of these performance functions can be attributed to the same random variables and the cross correlation among the input parameters. An assessment of joint failure probability for an engineering system is proposed, which is associated with the correlated performance functions using a copula-based method by conveying the dependence structure of the performance functions. The method is demonstrated with four simple engineering problems, i.e., (a) bivariate distribution in which two predetermined performance functions are associated with each other; (b) pile bearing capacity in which the performance functions are related with the soil internal friction and the compressive strength of a concrete pile; (c) pipe flow in which the performance function of three pipes in a sewer system is assessed with six independent random variables; and (d) retaining wall in which the failure criteria for defining the performance functions include overturning failure about the toe point, sliding failure along the base, and bearing capacity instability considering uncertain soil properties. The computational efficiency is evaluated using the results based on the conventional bounding methods. The joint failure probability expressed by copulas provides a means to obtain the joint probabilities of multiple failure modes, which pave the way for an objective description of the overall failure probability of a practical engineering problem.

Trivariate analysis of soil ranking-correlated characteristics and its application to probabilistic stability assessments in geotechnical engineering problems

Soils and Foundations -Tokyo- (Impact Factor: 0.41). 08/2013; 53(4):540–556. DOI: 10.1016/j.sandf.2013.06.006

An important issue in the probabilistic prediction modelling of multivariate soil properties (usu... more An important issue in the probabilistic prediction modelling of multivariate soil properties (usually including cohesion, friction angle, and unit weight) is the measurement of dependence structure among these properties. The use of Pearson's correlation as a dependence measure has several pitfalls; therefore, it may not be appropriate to use probabilistic prediction models in geotechnical engineering problems based on this correlation. As an alternative, a copula-based methodology for prediction modelling and an algorithm to simulate multivariate soil data are proposed.

In this method, all different random variables are transformed to a rank/uniform domain in order to form a copula function by applying cumulative distribution function transformations. The technique of copulas, representing a promising alternative for solving multivariate problems to describe their dependence structure by a ranked correlation coefficient, is highlighted. Two existing observed soil data sets from river banks are used to fit a trivariate normal copula and a trivariate fully nested Frank copula. The ranking correlation coefficient Kendall's τ and the copula model parameters are estimated. The goodness-of-fit test to choose the best-fitting model is discussed.
A series of triplet samples (i.e., cohesion, friction angle, and unit weight) simulated from the trivariate normal copula with flexible marginal distributions are used as input parameters to evaluate the uncertainties of soil properties and to define their correlations.

The influence of the cross-correlation of these soil properties on reliability-based geotechnical design is demonstrated with two simple geotechnical problems: (a) the bearing capacity of a shallow foundation resting on a clayey soil and (b) the stability of a cohesive-frictional soil in a planar slope. The sensitivity analysis of their correlations of random variables on the influence of the reliability index provides a better insight into the role of the dependence structure in the reliability assessment of geotechnical engineering problems.

Probabilistic solution of floodplain inundation equation

Stochastic Environmental Research and Risk Assessment (Impact Factor: 2.67). 01/2015; 29(NA):NA. DOI: 10.1007/s00477-015-1025-5

Uncertainty in bed roughness is a dominant factor in providing a sufficiently accurate simulation... more 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 densityfunction 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.

Liouville Equation-Based Stochastic Model for Shoreline Evolution

Stochastic Environmental Research and Risk Assessment (Impact Factor: 2.67). 01/2015; 29(NA):NA.

Long-term shoreline evolution due to longshore sediment transport is one of the key processes tha... more 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 the Liouville 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.

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

Stochastic Environmental Research and Risk Assessment (Impact Factor: 2.67). 12/2013; DOI: 10.1007/s00477-013-0715-0

Shoreline evolution due to longshore sediment transport is one of the most important problems in ... more 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.