Probabilistic design and optimization of foundations on cohesionless soil using surrogate models

Abstract

The design of foundations depends on the properties of soil, which are affected by various forms of uncertainties. These uncertainties affect the foundation performance and hence are considered in the deterministic design using safety factors. Such factors may not be able to cover all design scenarios involving different degrees of variation of soil properties and hence may lead to an inefficient design. These variabilities can be assessed and quantified systematically using a probabilistic framework to provide an optimized design with a consistent level of safety using analytical or numerical models. The use of numerical modelling techniques enhances the analytical capabilities by providing numerical solutions considering interactions of the different elements involved, employing appropriate constitutive models and input properties. It also facilitates the modelling of random processes and random fields by providing random properties as inputs to the model and can be used for analysis of geotechnical structures. Thus, the conjunction of probabilistic and numerical modelling techniques provides enormous possibilities for considering variabilities in foundation design and its optimization. But the computational cost associated with the use of probabilistic methods and uncertainty assessment has been a constant hindrance to their incorporation in the foundation design methods.

This thesis proposes a methodology utilizing probabilistic techniques with analytical and numerical models for the design and optimization of foundations. The probabilistic technique incorporates the variability of the soil properties such as cohesion, angle of friction, soil modulus, and unit weight for design and analysis. The issue of computational cost is overcome by using surrogate modelling techniques that replace the numerical models to improve computational efficiency. Thus, the variability of soil properties is considered in the numerical models using surrogate-based numerical modelling techniques for the probabilistic design of foundations. The optimization algorithm developed for different discrete values of design dimensions can be used to arrive at the optimum design curve from which the final design dimensions can be chosen based on the desired level of failure probability. Further, the possibility of improving the computational efficiency of the surrogate-based design and optimization methodology using advanced techniques are explored.

The first objective of the thesis deals with the design and analysis of a shallow foundation resting on cohesionless soil to study the influence of different forms of variabilities on foundation performance. The reliability-based design is performed by considering the mean value of the soil properties, their COV, and the variation of COV. The robust geotechnical design methodology is modified to incorporate system reliability along with the consideration of variability of soil properties to simplify the multi-objective design problem. Constrained optimization is performed to determine the variability of failure probability of the foundation and its cost and obtain the optimized design curve. The results from the robust design show the importance of consideration of variations of COV in highly variable soil deposits as it affects the final design. Further, spatial variability studies are performed to assess their contribution to the foundation performance by considering numerical methods in conjunction with a random field modelling technique known as Karhunen-Loeve expansion. The spatial variability study considering different COV values and correlation distances of the soil properties shows the importance of incorporating them in the design and analysis of foundations as it is observed to affect the bearing capacity of foundations.

The second objective deals with developing a reliability-based design methodology to optimize foundations using numerical modelling techniques by utilizing surrogate models. Kriging models are used as surrogate models due to their efficiency in capturing the stochastic nature of the problem and good predictive capabilities. Kriging models are developed for different foundation dimensions, and their performance is assessed using the statistics of bearing capacity of foundation or failure probability. The methodology developed is explained with the example of a shallow foundation. The methodology is extended to study the variation of different factors and their effects on foundation design. The variation of COV of the soil properties and correlation among them are examined to study their effect on the failure probability of the foundation. The effect of B/L ratio on the failure probability is also analyzed to understand its importance in foundation design. The reliability-based design is also extended to the estimation of partial factors by comparing the design and mean values of the soil properties obtained from the reliability analysis. This can help to understand the effect of variability on the partial factors as well. Finally, the optimization algorithm is used to arrive at the design curve from which the final design dimensions can be chosen, corresponding to the desired level of reliability.

As the third objective, the reliability-based design methodology using kriging surrogate models is developed and applied to geogrid reinforced soil, thereby providing an opportunity to consider the variability of soil and geogrid properties in the design of foundations. Geogrid reinforcement can be considered when the limit states are not satisfied due to constraints in foundation dimensions. Three-dimensional numerical modelling is utilized, facilitating the consideration of the foundation shape and extent of reinforcement to be provided. The optimized design curve is obtained for the discrete foundation dimensions using the optimization algorithm, from which the design that satisfies the desired failure probability can be obtained. The study shows the applicability of the design and optimization methodology for various problems by using surrogate-based numerical modelling techniques. Reliability analysis also enables the estimation of partial factors from the design values of soil properties obtained.

The fourth objective addresses the improvement in computational efficiency of the reliability-based design and optimization methodology using advanced techniques such as adaptive kriging methods. The adaptive kriging algorithm is modified to bring in the variability of foundation dimensions to obtain an optimized design of the foundation from the different possible values of foundation dimensions. It improves the applicability of surrogate models by reducing the number of numerical simulations required to develop the surrogate model. Quantile-based adaptive kriging methodology is used, which considers the variation of foundation dimensions and adds new simulation points at the area of interest, i.e., at the threshold value of the performance function. This methodology provides the possibility of arriving at economic design using quantile values. The effectiveness of the adaptive kriging method for the design and optimization of shallow foundations and the robustness of the kriging model are illustrated. The optimized design curve is obtained using adaptive kriging from which the optimum design can be selected that satisfies the design requirements.

The fifth and final objective deals with the quantile-based design methodology which provides the design properties based on quantile estimates using simple statistical calculations. The quantile-based design methodology proves to be computationally efficient, reducing the number of computations required to obtain a probabilistic design. The study also shows the possibility of arriving at optimised design using simple calculations.

Thus, this thesis proposes a design methodology that can be used for probabilistic design and optimization of foundations using analytical or numerical models, along with a detailed study on different types of variability that affect the foundation performance. The major issue of computational cost encountered while using probabilistic methods in conjunction with numerical models is sorted out by using kriging surrogates or adaptive kriging methodology. The quantile-based design methods can be used for probabilistic design using simple statistical calculations. The probabilistic framework developed in this thesis is applied to the design of shallow and pile foundations, to illustrate their applicability and efficiency in the design process.