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Due to modernisation of main railway tracks in Poland there was a need to change the existing overhead electrical transmission lines together with their supports and foundations of the supports. One of possible way to construct a new foundation of such support is the direct connection of the support with a single pre-cast concrete pile embedded in soil. The piles used are usually short, then in a certain soil conditions have to be treated as rigid piles and the ultimate soil lateral resistance has to be considered. Computations of rigid piles by Brinch Hansen method demonstrates high sensitivity of ultimate lateral loading to precise determination of the rotation centre of the pile under consideration. The position of the centre is affected by some random factors, for example random variability of soil properties and loading applied. due to very complex nature of solution of equilibrium equations in the case of Brinch Hansen method being in use any evaluation of safety measure constitutes rather difficult problem. Two alternative approaches are suggested. one is an algorithm supported by some symbolic computations combined with some power series expansions. The second one bases on the response surface method and can be applied both for cohesive and non cohesive soils. Within the presentation some important numerical aspects will be discussed. Additionally computational examples allow the study a relationship between "classical" safety factor versus reliability index as well as an effect of spatial averaging. [ABSTRACT FROM AUTHOR]

The chapter deals with bearing capacity of spread foundations in the context of reliability computations. In the first part evaluations base on the recommendations given by Polish Standard (1981). Consequently some most important ideas concerning bearing capacity suggested by this standard are presented and compared with analogical statements of Eurocode EC7 (1997). Next some reliability computations are carried out under an assumption that each individual soil property is modelled by a single random variable throughout the earth body considered. But such approach seems to be too simple. In order to evaluate credible reliability indices when bearing capacity of a shallow foundation is considered it is reasonable to describe soil strength properties in terms of random field's theory. As a next step the selected random field can be spatially averaged by means of a procedure introduced by (1977). Earlier experiences have proved that, without applying spatial averaging procedure, reliability computations carried out in the context of foundation's bearing capacity had given significantly small values of reliability indices (large values of failure's probability) even for foundations which were considered as relatively safe. On the other hand the volume of the area under averaging strongly affects results of reliability computations. Hence the selection of the averaged area constitutes a vital problem and has to be dependent on the failure mechanism under consideration. In the present study local averages associated with kinematically admissible mechanism of failure proposed by (1920) are considered. Soil strength parameters are assumed to constitute anisotropic random fields with different values of vertical and horizontal fluctuation scales. These fields are subjected to averaging along potential slip lines within the mechanism under consideration. Next examples of equations for variances of random variables resulting from averaging procedure are shown. By numerical examples it is demonstrated that for reasonable proportions (from practical viewpoint) between horizontal and vertical fluctuation scales the reliability indices resulting in two-dimensional case only slightly differs from resulting that obtained in one-dimensional. This means that the simpler one-dimensional approach can be usually utilised when reliability measures of shallow strip foundation are carried out. [ABSTRACT FROM AUTHOR]

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 elastoplasticity combined with random field theory. The RFEM method is shown to offer 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. [ABSTRACT FROM AUTHOR]

The response surface method can be applied to numerous fields of knowledge. In general, this method consists in approximation an unknown function by known function chosen appropriately. It can be successively utilised in reliability measures computations when the failure criterion does not depend on random variables explicitly. Within this study an application of the response surface method in dealing with random settlement in geotechnical engineering is discussed. In practical applications, an explicit closed form of settlement function U(X) is only rarely known. Most often, if program of the finite element method is available, we are able to determine the function values for assumed material properties, loads and geometrical constants of the FEM computation model. In order to obtain the limit state function in the form appropriate for reliability computations, one can model a closed form of U(X) by means of non-linear regression. To simplify the reliability computations, rather simple functions, e.g. polynomials of the second degree are in use. The approximation in the vicinity of a design point (known from preliminary FORM computations) is especially convenient in computing the probability of failure, because the neighbourhood of this point affects most strongly the value of a failure probability. In order to make computational algorithm more efficient the application of neural network with a hyperbolical activation function is suggested. If U(X) is a continuous function then the three-layered neural network with one hidden layered containing necessary number of neurones could give a satisfactory result. Neural networks allow approximation in cases of large variability intervals of independent variables preserving sufficient accuracy. Furthermore they make possible to overcome some numerical drawbacks of second order approximation known as "false branches problem". [ABSTRACT FROM AUTHOR]

In this chapter, soils with spatially varying shear strengths are modeled using random field theory and elasto-plastic finite dement analysis to evaluate the extent to which spatial variability and cross-correlation in soil properties (c and ϕ) affect bearing capacity. The analysis is two dimensional, corresponding to a strip footing with infinite correlation length in the out-of-plane direction, and the soil is assumed to be weightless with footing placed on the soil surface. Theoretical predictions of the mean and standard deviation of bearing capacity, for the case where c and ϕ are independent, are derived using a geometric averaging model and then verified via Monte Carlo simulation. The standard deviation prediction is found to be quite accurate, while the mean prediction is found to require some additional semi-empirical adjustment to give accurate results for ‘worst case' correlation lengths. Combined, the theory can be used to estimate the probability of bearing capacity failure, but also sheds light on the stochastic behaviour of foundation bearing failure. [ABSTRACT FROM AUTHOR]

The majority of geotechnical analyses are deterministic, in that the inherent variability of the materials is not modeled directly, rather some "factor of safety" is applied to results computed using "average" properties. In the present study, the influence of spatially varying strength is assessed via numerical experiments involving the compressive strength and stability of pillars typically used in underground construction and mining operations. The model combines random field theory with an elasto-plastic finite element algorithm in a Monte-Carlo framework. It is found that the average strength of the rock is not a good indicator of the overall strength of the pillar. The results of this study enable traditional approaches involving factors of safety to be re-interpreted as a "probability of failure" in the context of reliability based design. [ABSTRACT FROM AUTHOR]

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 chapter 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. [ABSTRACT FROM AUTHOR]

The use of multi-dimensional random fields to model real soils is becoming ever more important, simply because soils are spatially random. Such random field models allow the rational quantification of the behaviour of spatially variable soils, which are inherently uncertain, and lead to reliability estimates, decision analysis, and. ultimately, optimal designs. Random models are commonly used either in analytical studies employing theoretical results or in Monte Carlo simulations. Since theoretical results do not exist for many problems of interest to geotechnical engineers, the Monte Carlo approach is often the practical choice. In that the accuracy of such models depends directly on the accuracy of the algorithm used to generate realizations of the representative random fields, there is a need to evaluate and compare various random field generators. To address this issue, three common random field generators are considered in this chapter: 1) the FFT method. 2) the Turning Bands Method (TBM). and 3) the Locai Average Subdivision (LAS) method. For each, an ensemble of realizations of a two-dimensional homogeneous Gauss-Markov process is generated and the field mean, variance, and covariance structures are checked for statistical accuracy. Concerns such as ease of use and efficiency are also considered. It is shown that all three methods have distinct advantages and disadvantages, and the choice of algorithm will depend on the particular application. A number of guidelines and suggestions are made to help avoid or minimize problems associated with each method. [ABSTRACT FROM AUTHOR]

Direct motivation for the present study was the reconstruction of the existing gantry crane foundation situated in one of the major factories in the Northern part of Poland. The problem has been analysed in connection with the excessive settlement of the part of the track founded on the approximately 9-meter thick layer of non-compact embankment. One of suggestions of repair works assumed the carrying out of the jet-grouting columns under the beam. It seemed that random fluctuations in flexibility or bearing capacity of piles had resulted in large random fluctuations of bending moments of the beam. A static scheme as well as random model has been proposed. Then a procedure of evaluating random fluctuations of bending moments, based on Monte Carlo method, has been elaborated. The procedure can serve the cases when piles work within the elastic range, as well as in the cases when some piles under the beam are over-loaded and the load has to be carried by other piles. The computations have provided evidence for a considerable influence of the fluctuation of the pile bearing capacities on the bending moments. The extent of that effect, of course, depends, first of all, on the bearing capacity fluctuation range, measured by the coefficient of variation. It can be demonstrated that in the situation when the limiting bearing capacities of the piles have been exceeded, the positive correlation causes a considerable increase of the absolute value of the bending moments. When accompanied by the high values of the reliability index, the values of bending moments obtained from the probabilistic analysis may considerably exceed the moments from the deterministic computations. [ABSTRACT FROM AUTHOR]

A procedure for combining a point estimate method (PEM) and deterministic finite element methods for geotechnical analysis is presented. A brief description of the theoretical background is given and an illustrative example analysing a sheet pile wall supporting an excavation is presented to demonstrate the general applicability of the proposed approach. Finally the application to numerical analyses of tunnel excavation is discussed in some detail. A NATM tunnel construction is considered and the results obtained from the suggested approach are compared to the ones obtained from conventional finite element analyses. The results clearly reveal the applicability of the proposed concept for solving practical problems and it is argued that the validity of finite element analyses is extended by introducing stochastic properties of input parameters. [ABSTRACT FROM AUTHOR]

This paper presents a possible framework for dealing with uncertainties in geotechnical engineering based on more recently introduced mathematical theories. Random sets are employed to describe the variability of material parameters and geometrical data. After a brief introduction of the basics of the proposed approach application to a boundary value problem, namely a deep excavation analysis, is presented. Comparison with in situ measurements is provided and it is shown that the calculated most likely range of displacements compares well with measurements. Furthermore an assessment of the probability of damage of a building, situated adjacent to the excavation, is in line with observed behaviour. [ABSTRACT FROM AUTHOR]

Working with uncertainty is an essential aspect of engineering—the larger the uncertainty and the closer to critical, the greater the need for evaluating its effect(s) on the results. To characterize the uncertainties in soil and/or rock properties, the engineer needs to combine, in addition to actual data, knowledge about the quality of the data, knowledge on the geology and, most importantly, engineering judgment. Once the uncertainty in input parameters and model(s) for solving a particular problem are quantified, the engineer has a variety of tools at his disposal to evaluate the uncertainty in the output. The most common practical tools are Monte Carlo simulation techniques, first-order, second moment (FOSM) approach, first- and second-order reliability methods (FORM and SORM), and event tree analysis. Each has its advantages and shortcomings. The more complicated methods often provide more useful information about the possible outcomes of a problem. These methods are described and their applications are demonstrated through example problems. Many geotechnical problems involve several possible failure modes, which may or may not be correlated. These problems should be treated as systems. Component reliability vs. system reliability are discussed, and example calculations are presented. [ABSTRACT FROM AUTHOR]

Regulatory bodies are increasingly asking geotechnical engineers to provide rational risk assessments to accompany their designs. In order to provide these assessments, practicing geotechnical engineers need a good understanding of both basic probability theory and the more sophisticated, but realistic, random field soil models. This chapter lays the groundwork for this understanding. Starting with the basics of probability, the reader is lead through the theory of random variables and random fields and how they can be used to realistically model spatially variable soils. [ABSTRACT FROM AUTHOR]