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Conventional finite element method (FEM) relies on deterministic values to represent structural and load parameters and it does not have the capability of considering random parameters in response evaluation of the structures. In the context of stochastic computational mechanics, uncertainty modeling using random finite element method (RFEM) has by far received the major attention in recent years. The objective of the paper is to develop a computer program for stochastic analysis of reinforced retaining walls considering serviceability limit-state. The paper presents the formulation of the RFEM for random field modeling of the reinforced retaining walls. The developed program is validated using 1D bar and 2D plane-stress problems that involve random material properties. The random field is simulated by Cholesky decomposition of the covariance matrix. The Monte Carlo simulation (MCS) technique has been used in the finite element solution of the deformation response variability of the reinforced soil retaining wall. Results of the random finite element analysis are useful for serviceability limit-state design of the reinforced retaining walls.

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... Previous studies only considered random variables. However, Dodagoudar et al. [35] attempted to consider the influence of soil spatial variability on reinforced earth walls. A linear elastic behavior was assumed for the soil and the soil Young modulus was considered as a 2D isotropic random field and supposed to follow a normal distribution. ...

This article presents the soil spatial variability effect on the performance of a reinforced earth wall. The serviceability limit state (SLS) is considered in the analysis. Both cases of isotropic and anisotropic non-normal random fields are implemented for the soil properties. The Karhunen-Loeve (KL) expansion method is used for the discretization of the random field. Numerical finite difference models are considered as deterministic models. The Monte Carlo simulation (MCS) technique is used to obtain the deformation response variability of the reinforced soil retaining wall. The influences of the spatial variability response of the geotechnical system in terms of horizontal facing displacement is presented and discussed. The results obtained show that the spatial variability has an important influence on the facing horizontal displacement as well as on the failure probability.

... So far, few works have been performed on the application of these methods to determine the reliability of supported excavations. Fenton et al. [16] and Dodagoudar et al. [17] applied the random finite element method to problems of classical retaining walls and reinforced retaining wall design, respectively. The cantilever sheet pile wall for the Mohr-Coulomb model and independent anisotropic fields of unit weight γ, Young's modulus E and friction angle φ' has been considered by Papaioannou and Straub [18]. ...

The paper deals with reliability analysis of cantilever sheet pile wall located in non-cohesive soil with random properties. Spatial variability of friction angle has been described using random fields theory. The influence of both vertical as well as horizontal scale of fluctuation on the mechanical response of sheet pile wall is investigated. Deflection of wall top as well as maximum bending moment in the sheet pile wall are tested. The point distribution of soil friction and its vertical fluctuation scale is estimated using quasi-continuous results of cone penetrometer tests (CPTu). The boundary value problem is solved using finite difference code FLAC. The Fourier series method (FSM) allowing for non-uniform meshes is used to generate random fields for individual realizations. By utilizing Monte Carlo Simulation (MCS) technique the probability distributions of the results for different values of vertical and horizontal scales of fluctuation are obtained and used for reliability analysis. The results of analysis show that in case of cantilever sheet pile wall it is very important to properly estimate value of vertical fluctuation scale for the reliability analysis. It is also illustrated that in the considered problem horizontal scale of fluctuation significantly influences probability of failure.

... Various spatial random field generation techniques have been introduced in literature, such as the matrix decomposition method (Muller et al., 2009;Dodagoudar et al., 2015), the local average method (Fenton and Vanmarcke, 1990;Fenton and Griffiths, 2008), the turning bands method (Gui et al., 2000;Song et al., 2011), the spectral method (Sudret and Kiureghian, 2002;Farah et al., 2014), the Fourier transform method (Ching et al., 2011;Jha and Ching, 2013) and the KarhuneneLoève (KeL) series expansion method . However, these methods incorporated with finite element analysis (FEA) might result in large numbers of spatially correlated random variables that required to assign to elements of the numerical model, the MCS therefore is usually adopted to compute the failure probability. ...

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. © 2017 China University of Geosciences (Beijing) and Peking University.

... Stochastic considerations in geotechnical design have been gaining consideration in the past few years (Christian 2004). There are many geotechnical studies available in the literature, which incorporated random field modeling to various applications of geotechnical engineering like lateral thrust for a retaining wall (Fenton et al. 2005), bearing capacity of shallow footings (Soubra et al. 2008, Vessia et al. 2009and Li et al. 2016, determination of allowable load on a laterally loaded pile foundation in undrained clay (Haldar and Babu 2008), reinforced retaining walls (Dodagoudar et al. 2014). In random field theory, spatial variability is characterized by main parameters viz. ...

Present study investigates the influence of spatial variability of soil properties on
the lateral thrust on a rigid retaining wall in active condition. Friction angle of the soil
is modelled as a lognormal anisotropic random field in two dimensions, using the
Cholesky decomposition technique to study the effect of horizontal and vertical scale
of fluctuation on the lateral thrust. The Monte Carlo simulation approach is used in
association with FLAC3D to highlight the worst-case spatial variability configuration.
The study revealed that deterministic analysis employing mean friction angle
underestimates the lateral thrust on the wall, as compared to that obtained using a
spatially variable soil. For satisfactory performance of a wall, a scaling factor of
around 1.5 is required to be used, if lateral active thrust obtained from deterministic
analysis is employed in the design calculations. It is concluded from the reliability
analysis that between the two practically possible scenarios, high horizontal and
vertical scales of fluctuation scenario exhibit higher lateral thrust.

... Babu, Reddy, and Srivastava (2014) considered cohesion as an isotropic non-normal random field and observed its effects on a slope stability of MSW landfill. Dodagoudar, Sayed, and Rajagopal (2015) conducted studies on reinforced retaining walls taking elastic modulus of the soil as a normal random field and evaluated its effect on the settlement of the soil near the wall. Li, Tian, and Cassidy (2015) also conducted a study on footings buried in soil, where the effect of anisotropic variability of undrained shear strength was seen on the bearing capacity as well as the failure mechanism at various depths. ...

The present study investigates the influence of spatial variability of soil properties on the lateral thrust and failure surface of a 6 m high frictionless rigid earth retaining wall in active condition. The drained friction angle of the soil is modelled as a log-normal anisotropic random field in two dimensions, using the Cholesky decomposition technique. The effect of horizontal and vertical scale of fluctuation is observed on the lateral thrust. The failure surface in the backfill is found to be more or less similar for different combinations of spatial variability in vertical and horizontal directions. Monte-Carlo simulation technique is used to compute the probability of failure and to obtain the worst-case spatial variability configuration.

The main goal of this article is to perform a probabilistic analysis on a cantilever diaphragm wall constructed in sandy subsoil characterized by spatially variable strength and stiffness parameters. Three quantitates are considered, namely, the deflection of the wall, the settlements behind the wall and the maximum bending moment along the wall. To enable high accuracy of the evaluation of these quantities, a hardening soil model with a small strain overlay is used. To account for soil spatial variability, the friction angle of the sand is modelled using an anisotropic random field with specified vertical and horizontal scales of fluctuation (SOFs). All computations of the considered boundary value problem are carried out by means of the random finite element method (RFEM). The influence of both the vertical and horizontal SOFs on the probabilistic distributions of all the considered quantities is investigated. Reliability analyses are carried out based on the obtained distributions. Additionally, the effect of the number of realizations in the simulation process is analysed. It is shown that in the case of a diaphragm wall, the value of the horizontal SOF can substantially affect the probability of both excessive deformations as well as high bending moment values. It is also shown that additional modelling of soil stiffness moduli by a random field may induce small but noticeable changes in the probability of exceeding the conditions of the serviceability limit state.

Geotechnical uncertainties are inevitable. The properties of the soil can disperse in a significant range on a domain. Thus, the safety factor is used in the deterministic approach that takes into account the uncertainty associated with soil properties. This does not take into account the sources and degree of uncertainty associated with the system. The limit state design of the geotechnical structure is difficult to estimate using deterministic methods. It is, therefore, reasonable to study the probability of failure of the systems. In this thesis, a reliability analysis was carried out for a reinforced earth wall, shallow circular tunnel, rigid inclusion ground reinforcement using the Finite Difference Method (MDF) and the bearing capacity of a shallow foundation under pseudo-static seismic loading using an analytical model in limit equilibrium. The deterministic models thus defined are then used in a probabilistic framework. The method of the response surface optimized by genetic algorithm (RSM-GA) has been presented, validated and perfected in order to make possible, for a low computation time, a parametric study of the input variables of the models used. The propagation of uncertainty through deterministic models has been evaluated and reliable dimensioning methods have been proposed. Lastly, the spatial variability of the soil was taken into account through random field theory and then applied to the reinforced earth wall model. This model developed to take into account this spatial variability for a low computing time, and subject to random draws in large numbers. The effect of the variability of soil properties has been evaluated and new phenomena have been brought to light.

Retaining wall design has long been carried out with the aid of either the Rankine or Coulomb theories of earth pressure. To obtain a closed-form solution, these traditional earth pressure theories assume that the soil is uniform. The fact that soils are actually spatially variable leads, however, to two potential problems in design: do sampled soil properties adequately reflect the effective properties of the entire retained soil mass, and does spatial variability of soil properties lead to active earth pressure effects that are significantly different from those predicted using traditional models? This paper combines non-linear finite element analysis with random field simulation to investigate these two questions and assess just how safe current design practice is. The specific case investigated is a two-dimensional frictionless wall retaining a cohesionless drained backfill. The wall is designed against sliding using Rankine's earth pressure theory. The design friction angle and unit weight values are obtained by sampling the simulated random soil field at one location, and these sampled soil properties are then used as the effective soil properties in the Rankine model. Failure is defined as occurring when the Rankine predicted force acting on the retaining wall, modified by an appropriate factor of safety, is less than that computed by the random finite element method employing the actual soil property (random) fields. Using Monte Carlo simulation, the probability of failure of the traditional design approach is assessed as a function of the factor of safety used and the spatial variability of the soil.

A parametric study was conducted using Monte Carlo simulation to assess how uncertainty in design parameters affects the probability of internal failure of mechanically stabilized earth (MSE) walls. Bishop's simplified method was used to conduct the internal stability analyses. The results of the analyses indicate that the mean and coefficient of variation of the backfill friction angle, mean and coefficient of variation of the tensile strength of reinforcement, mean unit weight of the backfill, mean surcharge, mean reinforcement vertical spacing, and mean reinforcement length have a significant effect on the probability of internal failure of MSE walls. Based on the results of the parametric study, a series of additional simulations were conducted where the significant parameters were varied over a broad range. The results of these simulations were used to develop a set of reliability-based design (RBD) charts for internal stability of MSE walls. A method to adapt these charts to address model bias and model uncertainty is also presented. A MSE wall was designed using the RBD method and two other deterministic design methods. The required tensile strength of the reinforcement obtained from the RBD method fell between the strengths determined from the deterministic methods.

With the aid of the finite element method, the present paper deals with the problem of structural response variability resulting from the spatial variability of material properties of structures, when they are subjected to static loads of a deterministic nature. The spatial variabilities are modeled as two-dimensional stochastic fields. The finite element discretization is performed in such a way that the size of each element is sufficiently small. Then, the present paper takes advantage of the Neumann expansion technique in deriving the finite element solution for the response variability within the framework of the Monte Carlo method. The Neumann expansion technique permits more detailed comparison between the perturbation and Monte Carlo solutions for accuracy, convergence, and computational efficiency. The result from such a Monte Carlo method is also compared with that based on the commonly used perturbation method. The comparison shows that the validity of the perturbation method is limited to the cases where the material property variation has a relatively small coefficient of variation, particularly when Young's modulus itself is assumed to form a stochastic field.

Reliability-based design concepts and their application to load and resistance factor design (LRFD or limit states design ( LSD) in Canada) are well known, and their adoption in geotechnical engineering design is now recommended for many soil-structure interaction problems. Two important challenges for acceptance of LRFD for the design of reinforced soil walls are (i) a proper understanding of the calibration methods used to arrive at load and resistance factors, and (ii) the proper interpretation of the data required to carry out this process. This paper presents LRFD calibration principles and traces the steps required to arrive at load and resistance factors using closed-form solutions for one typical limit state, namely pullout of steel reinforcement elements in the anchorage zone of a reinforced soil wall. A unique feature of this paper is that measured load and resistance values from a database of case histories are used to develop the statistical parameters in the examples. The paper also addresses issues related to the influence of outliers in the datasets and possible dependencies between variables that can have an important influence on the results of calibration.

In order to control serviceability problems arising from excessive settlement of shallow footings, geotechnical design codes generally include specifications regarding maximum settlement which often govern the footing design. Once the footing has been designed and constructed, the actual settlement it experiences on a real three-dimensional soil mass can be quite different than expected, due to the soil's spatial variability. Because of this generally large variability (compared to other engineering materials such as concrete and steel) and because this particular serviceability limit state often governs the design, it makes sense to consider a reliability-based approach to settlement design. This paper looks in some detail at a Load and Resistance Factor Design (LRFD) approach to limiting footing settlement. In particular, the resistance factors required to achieve a certain level of settlement reliability as a function of soil variability and site investigation intensity are determined analytically using random field theory. Simplified approximate relationships are proposed and tested using simulation via the Random Finite Element Method. It is found that the simplified relationships are validated both by theory and simulation and so can be used to augment the calibration of geotechnical LRFD code provisions with respect to shallow foundation settlement.

By modeling soil as a three-dimensional spatially random medium, the reliability of shallow foundations against serviceability limit state failure, in the form of excessive settlement and/or differential settlement, can be estimated. The soil's elastic modulus, , is represented as a lognormally distributed random field with isotropic correlation structure. The settlements of individual and pairs of square footings placed on the surface of the soil are computed using the finite element method. A probabilistic model for total and differential settlement is presented and compared to results obtained using Monte Carlo simulation. The distributions of total and differential settlement are found to be closely predicted using the distributions of geometric averages of the underlying soil elastic modulus field.

Stochastic finite element analysis is used to predict uncertainties in total and differential settlement under a large flexible footing. The results are compared with one-dimensional stochastic solutions already in the literature. Differences between the one- and two-dimensional analyses, particularly for differential settlement, are distinct. These differences seem primarily attributable to randomness in the stress field which cannot be included in one-dimensional models, and to mechanistic correlations by common dependence on the realizations of particular random variables. In principle, second-moment techniques can be extended to a broad range of analyses now performed using finite element and finite difference techniques.

A new stochastic analysis method based on the Bubnov-Galerkin approximation is proposed herein for estimating the response variability of systems with spatially varying flexural rigidity. Such a flexural rigidity is idealized as a multi-dimensional, statistically homogeneous, continuous Gaussian stochastic field. Two kinds of techniques for evaluating the response statistics are utilized: a first-order perturbation technique and the Monte Carlo simulation technique. Numerical examples are presented in this paper.

NEW PROBABILISTIC APPROACHES FOR REALISTIC RISK ASSESSMENT IN GEOTECHNICAL ENGINEERING. This text presents a thorough examination of the theories and methodologies available for risk assessment in geotechnical engineering, spanning the full range from established single-variable and "first order" methods to the most recent, advanced numerical developments. In response to the growing application of LRFD methodologies in geotechnical design, coupled with increased demand for risk assessments from clients ranging from regulatory agencies to insurance companies, authors Fenton and Griffiths have introduced an innovative reliability-based risk assessment method, the Random Finite Element Method (RFEM). The authors have spent more than fifteen years developing this statistically based method for modeling the real spatial variability of soils and rocks. As demonstrated in the book, RFEM performs better in real-world applications than traditional risk assessment tools that do not properly account for the spatial variability of geomaterials. This text is divided into two parts: Part One, Theory, explains the theory underlying risk assessment methods in geotechnical engineering. This part's seven chapters feature more than 100 worked examples, enabling you to develop a detailed understanding of the methods. Part Two, Practice, demonstrates how to use advanced probabilistic tools for several classical geotechnical engineering applications. Working with the RFEM, the authors show how to assess risk in problems familiar to all geotechnical engineers. All the programs used for the geotechnical applications discussed in Part Two may be downloaded from the authors' Web site at www.engmath.dal.ca/rfem/ at no charge, enabling you to duplicate the authors' results and experiment with your own data. In short, you get all the theory and practical guidance you need to apply the most advanced probabilistic approaches for managing uncertainty in geotechnical design.

The effects of stochastic hydraulic conductivity on the slope stability of an embankment dam are investigated using a combination of random field simulation, seepage analysis, and slope stability analysis. The hydraulic conductivity distribution is treated as a spatially stationary random field following a lognormal distribution. The turning band method is used to generate the spatial variability of the saturated hydraulic conductivity Ks in the domain. Different standard deviations of log hydraulic conductivity σlnKs are investigated. For each value of σlnKs, various realizations of hydraulic conductivity were generated and combined with a numerical model to simulate water flow in an earth dam with variable Ks. The first-order second-moment reliability index β was employed to characterize the influence of the variability of Ks, and hence, pore-water pressures, on the stability of the downstream slope. A linear relationship between σlnKs and the standard deviation of the factor of safety σF was obtained from the simulation results. A relationship between β and σlnKs, in which every 0.1 increment of σlnKs results in a decrease of 1.0 in β, is deduced based on the simulation results. Results of a Shapiro-Wilk test for goodness of fit indicate that the factor of safety can be assumed to be normally or lognormally distributed when the saturated hydraulic conductivity follows a lognormal distribution and σlnKs is small (≤0.5). When σlnKs is large (>0.5), neither normal nor lognormal distributions provide a reasonable approximation of the factor of safety. Simulation results show that neither standard deviation nor coefficient of variation of the factor of safety is constant when only the variability of hydraulic conductivity is considered. While the results presented are directly applicable only to the particular earth dam geometry and boundary conditions studied, the methodology is general and may be extended to embankments with different boundary conditions.

A new approach to the limit equilibrium method is developed and used for the analysis of soil nailed walls. The basic procedure of the new approach is to compute the interslice forces by recursion and to fulfill the equilibrium requirement for interslice forces of the last boundary slice by iteration. Reliability analysis for soil nailed walls is carried out by considering the cohesion and internal friction angle of soil as random variables. The degree of mobilization of friction resistance between the nails and surrounding soil is taken as a third random variable. A parametric study is carried out to study the effect of soil behavior and the arrangement pattern of nails on the factor of safety and reliability index. Finally, an optimization technique is employed to obtain the minimum cost design of soil nailed walls with an object function expressed by the total nail length, which is considered an appropriate measure of the total cost of a soil nailed wall.

New approaches in the finite element method for stochastic structures are proposed. The FEM based on exact inverse of stiffness matrix is first proposed for bar extension problems with stochastic stiffness. The method is exemplified by the direct exact inverse of stiffness matrix for the deformation of the bar under extension. The second new FEM is based on the diagonalization of the element stiffness matrix and the inverse of the global stiffness matrix. The method is proposed for beam bending problems with stochastic stiffness. The third new FEM is based on the element-level flexibility and its idea is general applicable. The new methods avoid the error due to truncating the expansion series of random stiffness matrix, which appears in conventional finite element methods for stochastic structures based on either series expansion or perturbation technique. Examples of a stochastic bar under tension and stochastic beams under uniform pressure are analyzed. Comparison of the new finite element solution by new approaches and conventional finite element solution by the first-order perturbation is performed. Numerical results illustrates the superiority of the new proposed methods over the conventional FEM for stochastic structures.

Traditionally, the reliability of retaining walls is achieved through the use of safety factors or margins and adopting conservative assumption in the process of design, that is, by ascertaining that a mini- mum supply condition will remain adequate under a maximum demand condition. However that is often de- fined on the basis of subjective judgments. Such a traditional design methods are difficult to quantify and lack the logical basis of describing uncertainty. Especially, reinforced walls consider not only soil properties but soil-reinforcement interaction uncertainties. There has been much emphasis recently the use of probabilistic method in the geotechnical engineering. The most effective applications of probabilistic methods are involv- ing relative probabilities of failure or illuminating the effects of uncertainties in the parameter. This thesis de- scribed how probabilistic description of soil parameters and soil-reinforcement interaction parameters were applied to the stability analysis. The first-order, second moment approach was explored and applied to the de- sign of reinforced retaining walls. An example illustrated the relative contribution of uncertainties about dif- ferent parameters to the reliability of the reinforced retaining walls. The results obtained from this study were follows; the reliability of the soil-reinforcement interface friction angle, δ was highly sensitive to the coeffi- cient of variation. However, when the reinforced fill unit weight γr, and the reinforcement length, L were lower than the limited values, the probabilities of failure were increased. The reliability of the retained back- fill soil unit weight, γf in the unreinforced area was lowly sensitive to the coefficient of variation.

In this paper, a stochastic finite element method is presented for the analysis of structures having statistical uncertainities in both material properties and externally applied loads, which are modelled as a homogeneous Gaussian stochastic process. The Neumann expansion technique has been used for the inversion of a stochastic stiffness matrix. The digital simulation technique was adopted to generate the deviatoric part of the stiffness matrix and load vector. A beam problem was taken up for comparison of results obtained by Neumann expansion and the direct Monte Carlo simulation technique. The comparison of the results shows that the values approach those obtained by direct Monte Carlo solutions as the order of expansion of the Neumann series is increased. An excellent agreement was achieved for cases when the coefficient of variation is comparatively small.

A method of stochastic finite element analysis is developed for solving a variety of engineering mechanics problems in which physical properties exhibit one-dimensional spatial random variation. The method is illustrated by evaluating the second-order statistics of the deflection of a beam whose rigidity varies randomly along its axis. A key component of the approach is a new treatment of the correlation structure of the random material property in terms of the variance function and its principal parameter, the scale of fluctuation. The methodology permits efficient evaluation of the matrix of covariances between local spatial averages associated with pairs of finite elements. Numerical results are presented for a cantilever beam, with deformation controlled by shear, subjected to a concentrated force at its free end or to a uniformly distributed load.

The paper investigates the probability of failure of a cohesive slope using both simple and more advanced probabilistic analysis tools. The influence of local averaging on the probability of failure of a test problem is thoroughly investigated. In the simple approach, classical slope stability analysis techniques are used, and the shear strength is treated as a single random variable. The advanced method, called the random finite element method (RFEM), uses elasto- plasticity combined with random field theory. The RFEM method is shown to oer many advantages over traditional probabilistic slope stability techniques, because it enables slope failure to develop naturally by "seeking out" the most critical mechanism. Of particular importance in this work, is the conclusion that simplified probabilistic analysis, in which spatial variability is ignored by assuming perfect correlation, can lead to unconservative estimates of the probability of failure. This contradicts the findings of other investigators using classical slope stability analysis tools.

The traditional factor of safety does not account for the variability and uncertainty of the different parameters considered in the analysis. On the other hand, the reliability approach considers the possible variations in the design parameters and gives more realistic estimates of the safety of the structure and the possible risk of failure. In the present investigation, reliability analysis was carried out to study the stability aspects of reinforced soil retaining walls under static and seismic conditions using three methods: first-order second-moment method (FOSM), point estimate method (PEM) and first-order reliability method (FORM). A parametric sensitivity analysis is presented to bring out the effect of uncertainties in the soil and reinforcement parameters on the value of the minimum reliability index for internal and external stability of reinforced soil walls under static and seismic loads. The reliability indices evaluated by the three methods of reliability analyses were compared for both the internal and external stability of the wall. Based on the results, it is concluded that the friction angle (φ) is the most sensitive random variable affecting the overall stability of the reinforced soil walls under static and seismic loads.

Application of the stochastic finite element method to a probabilistic characterization of the system based on a random field model appears to be a promising method of analyzing the reliability of structural and geotechnical complex systems. Following this method, it is possible to model a structure by a limited number of accessible stochastic data and subsequently to compute the characteristics of the displacement and stress random fields in the structure by use of the first-order second-moment approximation to the stochastic finite element equations. This paper presents an application of the methodology for a soil profile with a random distribution of the elastic modulus. A parametric study is carried out in order to evaluate the significance of stochastic modelling of such a random system by use of the adopted methodology.

As modern structures require more critical and complex designs, the need for accurate approaches to assess uncertainties in loads, geometry, material properties, manufacturing processes and operational environments has increased significantly. Reliability assessment techniques help to develop safe designs and identify where significant contributors of uncertainty occur in structural systems, or, where further research, testing and quality control could increase the safety and efficiency of the structure. Reliability-based Structural Design provides readers with an understanding of the fundamentals and applications of structural reliability, stochastic finite element method, reliability analysis via stochastic expansion, and optimization under uncertainty. Probability theory, statistic methods, and reliability analysis methods including Monte Carlo sampling, Latin hypercube sampling, first and second-order reliability methods, stochastic finite element method, and stochastic optimization are discussed. In addition, the use of stochastic expansions, including polynomial chaos expansion and Karhunen-Loeve expansion, for the reliability analysis of practical engineering problems is also examined. Detailed examples of practical engineering applications including an uninhabited joined-wing aircraft and a supercavitating torpedo are presented to illustrate the effectiveness of these methods. Reliability-based Structural Design will be a valuable reference for graduate and post graduate students studying structural reliability, probabilistic analysis and optimization under uncertainty; as well as engineers, researchers, and technical managers who are concerned with theoretical fundamentals, computational implementations and applications for probabilistic analysis and design.

In recent years, a considerable amount of effort has been made on the part of structural engineers to model the excitation to structures as a random process and to analyze the structural response so that Structural behaviors under nondeterministic disturbances can be predicted with certain probability statements. However, much less work has been accomplished dealing with structures with a random property. This appears to be due to the fact that the governing equations of motion for structures with spatial statistical variation of material property involve stochastic parameters and are usually extremely difficult to solve. This paper introduces a Monte Carlo approach to solve structural problems of this kind and deals, as an example, with the stress-wave propagation through a random structure under impact loading. Through this is example, total compatibility of the proposed approach with the infinite element analysis, which is capable of taking into consideration the effects of irregular boundaries, nonlinear material properfies, and finite displacements, is demon- strated. Such generality is beyond the reach of analytical mcthods, but is required for problems of engineering importance. The complexity of the structure is limited only by the computing facilities available, and the accuracy of the sample statistics by the computational time required.

A stochastic meshless method is presented for solving boundary-value problems in linear elasticity that involves random material properties. The material property was modelled as a homogeneous random field. A meshless formulation was developed to predict stochastic structural response. Unlike the finite element method, the meshless method requires no structured mesh, since only a scattered set of nodal points is required in the domain of interest. There is no need for fixed connectivities between nodes. In conjunction with the meshless equations, classical perturbation expansions were derived to predict second-moment characteristics of response. Numerical examples based on one- and two-dimensional problems are presented to examine the accuracy and convergence of the stochastic meshless method. A good agreement is obtained between the results of the proposed method and Monte Carlo simulation. Since mesh generation of complex structures can be a far more time-consuming and costly effort than the solution of a discrete set of equations, the meshless method provides an attractive alternative to finite element method for solving stochastic mechanics problems. Copyright © 2001 John Wiley & Sons, Ltd.

Generalised nth order stochastic perturbation technique that can be applied to solve some boundary value or boundary initial problems in computational physics and/or engineering with random coefficients is presented here. This technique is implemented in conjunction with the finite element method (FEM) to model 1D linear elastostatics problem with a single random variable. Main motivation of this work is to improve essentially the accuracy of the stochastic perturbation technique, which in its second order realization was ineffective for large variations of the input random fields. The nth order approach makes it possible to specify the accuracy of the computations a priori for the expected values and variances separately. The symbolic computer program is employed to perform computational studies on convergence of the first two probabilistic moments for simple unidirectional tension of the bar. These numerical studies verify the influence of coefficient of variation of the random input and, in the same time, of the perturbation parameter on the first four probabilistic moments of the final solution vector.

Stochastic finite element-based reliability analysis is applied to structures with distributed parameters that can be modeled as random fields. In this method, reliability is estimated through analytical computation of the sensitivity of stochastic response to the basic random variables. The random fields are discretized into sets of correlated random variables using two methods of discretization. The sensitivity measures are further used to selectively consider only a few of the distributed parameters as random fields, to ensure computational efficiency. The issue of choosing the appropriate mesh for the discretization of the random field is addressed through mesh refinement studies. With the help of three numerical examples, the paper examines the effects of the correlation characteristics of the random field on discretization and reliability analysis, and develops guidelines for efficient application of stochastic finite element analysis to structures with distributed parameters.

This paper describes a stochastic finite element method using the first-order approximation at a failure point of a set of random variables. The method is extended to equivalent normal represtation of non-normal distributions and offers two advantages: (1) It gives a consistent measure of failure probability for the limit-states defined in terms of different but equivalent performance function formulations, (2). It can be applied to reliability analysis for non-normal variants. Results using this method are compared favorably with that of Monte Carlo simulation in a simple example. Furthermore, this method will be applied to earth slope stability analysis to give probability levels for local and global failures on a potential failure surface.

This paper presents a procedure which allows for a stochastic finite element (SFE)-based reliability analysis of large nonlinear structures under dynamic loading involving both structural and loading randomness with relatively little computational effort when compared to traditional Monte Carlo methods. The analysis is based on the identification of important random variables by means of a transformation of the vector of original random variables to the uncorrelated space and subsequent sensitivity analyses. Only few nonlinear computations using the most important identified random variables are performed to determine points on the limit state surface. Subsequently, the response surface method (RSM) is employed to estimate the reliability of the structure.

The main sources of uncertainties involved in the analysis of structures and solids are shown and the tools available to deal with them. While trying to cover the complete modeling process, ranging from the problem formulation via the mathematical model all the way to the numerical approximation, we have tried to expose areas in need of further research. The techniques and methods involved in stochastic modeling are explained in somewhat more detail, as they are newer and less known than those used for the deterministic modeling.

First-order reliability and finite element methods are used to develop a methodology for reliability analysis of structures with stochastically varying properties and subjected to random loads. Two methods for discretization of random fields are examined and the influence of the correlation length of random property or load fields on the reliability of example structures are investigated. It is found that the correlation length of load fields has significant influence on the reliability against displacement or stress limit states. The correlation length of property fields is significant for displacement limit states, but may not be significant for stress limit states. Examples studied include a fixed ended beam with stochastic rigidity and a plate with stochastic elasticity.

This state of the art paper identifies as the distinguishing feature of stochastic finite element analysis that it involves the discretization of the parameter space of a random field of material properties and / or loads. This discretization implies that the stochastic input consists of a vector of random variables whose covariance matrix depends on the finite element mesh. The paper provides an overview of basic concepts underlying random field theory, describes specific analytical tools to convey first- and second-order information about homogeneous random fields, and surveys available information on the space-time variation of random loads and material properties encountered in structural engineering. Stochastic finite element formulations covering a wide range of applications to both static and dynamic problems in structural engineering are examined, and a parallel approach to stochastic finite difference analysis is outlined.

User’s manual for geotechnical finite element modelling - GEOFEM

- K Rajagopal

Tolerable deformations

- H E Wahls

Reliability analysis for geotechnical problems via finite elements-a practical application

- Thurner R
- H F Schweiger

Towards reliable and effective site investigations

- Mb Jaksa
- Js Goldsworthy
- Ga Fenton
- Ws Kaggwa
- Dv Griffiths
- Yl Kuo
- Hg Poulos

Stochastic FEM in settlement predictions

- Gb Baecher
- Ts Ingra

Probabilistic slope stability by finite elements

- Dv Griffiths
- Ga Fenton

A study on reliability analysis for reinforced earth retaining walls

- B S Chun
- Kim Km

Mechanically stabilized earth walls and reinforced soil slopes design and construction guidelines

- Fhwa