Wojciech Puła

Reliability with respect to settlement limit-states of shallow foundations on linearly-deformable subsoil

Abstract A step by step procedure for applying the response surface and SORM methods in estimating the reliability index associated with exceeding a certain allowable settlement level by a shallow foundation is presented in this paper. Two random variables, the Young modulus and Poissons ratio, of lognormal and beta distribution respectively, in a single soil layer are taken into account. A linearly-deformable model of soil is assumed which is most frequently used in engineering practice when the serviceability limit state is considered. The main problem encountered in using the response surface methodology was the existence of false design points that prevented coordinate calculations of the real ones. Two procedures were employed. The first one consisted of widening the area covered by the response surface (polynomial of the second degree) with an additional “oedometric” term. Inserting the oedometric term improves the quality of the fitting and enables one to extend the range of approximation. The latter added a barrier to prevent the procedure from moving into the false design point region. Moreover, the paper presents the effect of random variation of the Young modulus E and Poissons ratio ν as well as their mutual correlation, on the reliability index associated with exceeding the assumed level of a shallow foundation settlement.

A probabilistic analysis of foundation settlements

Abstract This paper deals with a settlement analysis of shallow foundations resting on a layered subsoil. Three basic sources of randomness are discussed: random shape of the subsoil (location of an interface between two strata), random material parameters and random loads. The numerical analysis is based on the finite element method coupled with stochastic versions of the perturbation and the Neumann expansion methods. Both methods lead to a simple evaluation of the response variability in terms of the mean value and the covariance matrix for foundation settlements. More sophisticated analyses introduce the probability of failure as a measure of safety. In this context, failure means either exceeding the bearing capacity or exceeding allowable displacements. The probability of failure is usually evaluated by means of FEM-oriented gradient algorithms or some versions of the Monte Carlo simulation techniques. Numerical examples are presented for a raft foundation resting on an elastic subsoil. The obtained results compare the scale of random fluctuations of soil properties or loading with the scale of foundation settlement fluctuations.

INFLUENCE OF VARYING SOIL PROPERTIES ON EVALUATION OF PILE RELIABILITY UNDER LATERAL LOADS

Abstract A three dimensional probabilistic approach to analyzing laterally loaded piles is presented. Two typical subsurface models are used in the analyses: the first one consists of layered linear elastic soil where each layer has a random modulus of elasticity; while the second model takes the form of linear elastic soil with a random modulus of elasticity that increases with depth. Efficient step by step procedures for the reliability computation involving pile displacements are proposed. The solution is based on three-dimensional modeling by the finite element method. A series of results has been obtained for various values of elastic parameters of the soil. Next by a non-linear regression procedure a response surface is obtained. To get the final response surface allowing for a reliability analysis, an iterative algorithm based on the so-called design point concept is applied. The failure criterion is defined as the pile head displacement exceeding displacement threshold. The two cases of piles subj...

Probabilistic analysis of the stability of massive bridge abutments using simulation methods

Abstract This study provides a probabilistic procedure for analysing the stability of massive bridge abutments. The procedure is based on simulation techniques. To analyse the stability a model is applied which considers five cases of losing the equilibrium. To describe the random variability of soil properties, polygonal probability distributions are proposed, which are especially useful in sumulation calculations. Examples of description of some soil parameters by polygonal distributions together with verification by statistical testing are presented. The proposed method is examined in numerical examples and applied to the analysis of the influence of individual parameters on the probability of loss stability. The problem of the accuracy of calculations is also discussed. Numerical examples show the usefulness of the presented methodology. The randomness of the friction angle of the geotechnical layer directly under the base of an abutment is shown to have the greatest effect on the probabilistic characteristics of the stability. The probability distribution of the stability measures in the case of exceeding the bearing capacity may differ significantly from the normal distribution.

Reliability of Laterally Loaded Rigid Piles

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.

Application of the response surface method

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”.

On Some Aspects of Reliability Computations in Bearing Capacity of Shallow Foundations

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.

On Some Methods in Safety Evaluation in Geotechnics

Abstract The paper demonstrates how the reliability methods can be utilised in order to evaluate safety in geotechnics. Special attention is paid to the so-called reliability based design that can play a useful and complementary role to Eurocode 7. In the first part, a brief review of first- and second-order reliability methods is given. Next, two examples of reliability-based design are demonstrated. The first one is focussed on bearing capacity calculation and is dedicated to comparison with EC7 requirements. The second one analyses a rigid pile subjected to lateral load and is oriented towards working stress design method. In the second part, applications of random field to safety evaluations in geotechnics are addressed. After a short review of the theory a Random Finite Element algorithm to reliability based design of shallow strip foundation is given. Finally, two illustrative examples for cohesive and cohesionless soils are demonstrated.

Estimation of the probability distribution of the random bearing capacity of cohesionless soil using the random finite element method

Soil mathematical modelling is a complex task when considering random soil parameters. To consider soil spatial variability, one can assume that soil properties are random variables or random fields. This study considered the estimation of the probability distribution of the bearing capacity for shallow footings in cohesionless soil using the random finite element method. In the deterministic case, the FEM code was calibrated to receive capacity evaluations similar to the results obtained using Hills mechanism. Final capacity simulations were conducted under the assumption that the friction angle of the investigated subsoil constitutes a random field characterised by a bounded distribution. The random field was obtained by applying a hyperbolic tangent transformation to a Gaussian random field with an ellipsoidal (anisotropic) correlation function. The final outcome of the numerical analysis was an evaluation of the probability distribution that fits the bearing capacity for various foundation widths and depths. In all the cases considered, the empirical probability distribution of the bearing capacity closely resembled the Weibull distribution. Moreover, the estimated distribution of the bearing capacity was used as the basis for a foundation safety assessment.

Application of HDMR method to reliability assessment of a single pile subjected to lateral load

Abstract The paper presents an application of High Dimensional Model Representation (HDMR) to reliability assessment of a single pile subjected to lateral load. The purpose is to compare HDMR with some classical method based on response surface technique. First 3D numerical model of the problem for finite elements computations in the ABAQUS STANDARD program has been presented. The soil model is assumed to be linear elastic. However, contacts between the sidewall and the foundation of the pile and the soil are modelled as Coulomb one with friction and cohesion. Next the Response Surface Method is briefly reviewed in conjunction with reliability approach. Then the High Dimensional Model Representation approach is presented. In our approach the HDMR algorithm is based on polynomial of the second degree. Finally the numerical studies have been carried out. The first series of computations demonstrate the efficiency of HDMR in comparison to neural network approach. The second series allows comparison of reliability indices resulting from three different approaches, namely neural network response surface, first-order HDMR and second-order HDMR. It has been observed that for increasing values of the length of the pile reliability indices reach similar values regardless of the method response surface applied.