Probabilistic settlement analysis of rectangular footings

By modeling soil as a three-dimensional spatially random material, the reliability of shallow rectangular footings against serviceability limit state failure in the form of excessive settlement can be estimated. The methodology to be used is the 3-d random finite element method (RFEM) which combines finite element analysis with random field theory, where the mean, the standard deviation and the spatial correlation length of the modulus of the underlying soil can be controlled through the input data. The study starts by comparing settlements of rectangular footings against classical solutions for homogeneous deposits and then goes on to investigate the influence of footing aspect ratio on probabilistic settlements. Of particular interest is the relationship between the rectangular footing dimensions and the spatial correlation length of the underlying soil. The results of the studies are presented in probabilistic form, where for given statistics of soil input parameters, the probability of footing settlement exceeding a design criterion is assessed. Earlier studies on adjacent square footings indicated that distributions of settlements and differential settlements can be predicted using the geometric average of the underlying elastic soil modulus field. The current work on rectangular footings represents a further step towards developing a general probabilistic design framework for assessing settlements of shallow footings. RÉSUMÉ En modelant le sol comme un matériel spatialement fait au hasard à trois dimensions, la fiabilité de fondations rectangulaires peu profonds contre la limite d'aptitude à l'usage échec de l'état sous forme de règlement excessif peut être estimé. La méthodologie être utilisée est la méthode d'élément finie, faite au hasard et de 3 d (RFEM) qui combine l'analyse d'élément finie avec la théorie de champ faite au hasard, où les moyens, la déviation standard et la longueur de corrélation spatiale de la compressibilité (ou modulus) du dépôt fondamental peut être contrôlé par les données d'entrée. L'étude commence en comparant des règlements de fondations rectangulaires contre les solutions classiques pour les dépôts homogènes et va alors sur examiner l'influence de fondation de proportion d'aspect sur les règlements de probabilistic. D'intérêt particulier est la relation entre les dimensions de fondation rectangulaires et la longueur de corrélation spatiale du sol fondamental. Les résultats des études sont présentés dans la forme de probabilistic, où pour la statistique donnée de sol paramètres d'entrée, la probabilité de fondation de règlement dépassant un critère de conception est évalué. Plus tôt les études sur les fondations carrés adjacents indiqués que les distributions de règlements et de règlements différentiels peuvent être prédites utilisant la moyenne géométrique du champ de modulus de sol élastique fondamental. Une approche similaire sera utilisée dans le courant traite des fondations rectangulaires, comme une plus ample étape vers le développement d'un cadre de conception de probabilistic général pour évaluer de règlements de fondations peu profonds. 1 SETTLEMENT OF RECTANGULAR FOOTINGS The settlement of structures founded on soil is a subject of considerable interest to practicing engineers since excessive settlements can lead to serviceability or even failure states in the structural elements above. Settlements are typically predicted using elastic theory where soil properties are generally expressed in terms of “compressibilities” (essentially the reciprocal of “moduli” values preferred by structural engineers). Due to the common problem of limited site investigation data, engineers typically use a rather conservative approach to settlement prediction based on experience, combined with pessimistic estimates of compressibility and loading conditions. In view of the variable nature of soil properties, a probabilistic approach to this problem has attractions because it enables the analyst to estimate the probability of settlement limits being exceeded (see e.g. Baecher and Ingra 1981, Righetti and Harrop-Williams 1981, Paice et al 1996). The approach also facilitates an understanding of the sensitivity of settlement prediction to various input parameters describing the soil variability. In this paper, a probabilistic investigation of settlement of a single rectangular footing is described. Only the influence of uncertainties in the soil properties are considered here, with uncertainties that arise due to the model and loading conditions left for future study. In addition, the soil is assumed to be isotropic, that is, the correlation structure is assumed to be the same in both the horizontal and the vertical directions. Although soils generally exhibit a stronger correlation structure in the horizontal direction, and the analysis tools used in this study can model anisotropy, this site specific refinement is not considered here. Our priority in this work is to establish the probabilistic behavior of the settlement of rectangular footings as a function of the various statistics of the underlying soil. 2 THE RANDOM FINITE ELEMENT METHOD (RFEM) The method involves combining finite element analysis (e.g. Smith and Griffiths 2004) with random field theory (Vanmarcke 1984, Fenton and Vanmarcke 1990), where the latter is used to generate the material properties based on an underlying mean standard deviation and spatial correlation length. The analyses are then repeated using a Monte-Carlo approach until the output statistics of interest have stabilized. The authors have applied the method to a number of classical geotechnical problems (e.g. Griffiths and Fenton 2001,2004) and the interested reader is referred to those references for further details of the method. In this work the finite element simulations are simple elastic analyses, and the random field method is used to provide the