Professor Andrei Lyamin

The limit analysis and the finite element method are powerful tools for analysing the bearing capacity of foundations. Previous research mainly focused on the foundations in uniform soils. In realistic conditions, soil properties are always varying spatially due to complex physical, chemical, and biological process in earth evolution. This paper investigates the bearing capacity and failure mechanism of footings buried at various depths in clays with spatially variable distribution of undrained shear strength using the lower bound limit analysis, the upper bound limit analysis, and the finite element analysis. Results show that the bearing capacity increases with increasing buried depths in spatially random soils, which is the same as in the uniform soils. The bearing capacity factors calculated using the finite element method, the lower bound limit analysis, and the upper bound limit analysis for a footing in spatially varied soils are all smaller than the corresponding values in uniform soils. The majority of the bearing capacity factors obtained from the finite element method is bounded by those obtained from the lower bound and the upper bound limit analysis. The shear planes show a clearly unsymmetrical manner in spatially varied soils using the three methods, which is different from the symmetrical shear plane in uniform soils.

The problem of shakedown analysis is considered. The mathematical programming formulations of limit and incremental elastoplastic analysis are first briefly reviewed and a convenient standard form for shakedown analysis is then suggested. This standard form can formally be viewed as a problem of limit analysis. In this way, two different solution approaches are applicable: either the problem can be solved directly using general optimization methods or the problem can be converted into an equivalent fictitious incremental elastoplastic problem and solved as such. We further show that this result holds in general for arbitrary convex mathematical programs. Thus, all the methods and techniques developed for elastoplasticity are in principle applicable to general convex programming. For the solution of shakedown problems we employ a version of the well-known implicit solution procedure in combination with a number of equally well-known techniques from general optimization theory. The resulting algorithm enables an efficient and robust treatment of cone-shaped yield constraints of the Drucker¿Prager type.

In practical geotechnical engineering the factor of safety is still determined by means of simple limit equilibrium analysis in many cases. However, because displacement finite-element analysis is routinely applied for assessing displacements and stresses for working load conditions, this technique is increasingly being used to calculate ultimate limit states and, consequently, factors of safety, usually by means of the so-called strength reduction technique, and results which are comparable to those obtained with limit equilibrium methods have been reported in the literature. However, owing to the inherent assumptions of limit equilibrium analyses, they do not always provide unique factors of safety. The purpose of this paper is on the one hand to compare the strength reduction method with rigorous limit analyses which are based on collapse theorems of plasticity, and on the other hand to investigate if a shortcoming of the strength reduction method, namely possible numerical instabilities for non-associated plasticity, can be overcome. Two examples are considered, namely slope stability and tunnel face stability. Finally an important note on the definition of the factor of safety for effective and total stress analysis under undrained conditions is provided.

A new failure criterion, the generalised Tresca criterion, for undrained total stress analysis is presented. The criterion is consistent with an underlying effective stress Mohr-Coulomb model. It involves two parameters: the undrained shear strengths in triaxial compression and extension. As such, the model predicts different strengths in compression and extension without introducing any physical anisotropy due to layering, direction of deposition and so on. The model is applied to a number of boundary value problems, where the effects of an extension/compression shear strength ratio less than unity generally turns out to be relatively moderate.

A procedure for strength reduction analysis using finite-element limit analysis is presented. The scheme is completely general and does not require decision making regarding the loads needed to drive the system to failure. Rather, the scheme is based on the ability of modern interior-point methods to detect infeasibility in a controlled and reliable manner. The new scheme is illustrated by an example involving a strip footing on top of a slope.

In this paper, finite element limit analysis (FELA) and semi-analytical rigid block techniques are used to investigate the influence of tunnel spacing on the undrained stability of two unlined square tunnels constructed side by side. The tunnels, which are assumed to be straight and infinitely long, are modelled under conditions of plane strain. Upper and lower bounds on the stability of the tunnels are obtained using FELA; the numerical formulation of which is based upon the bounds theorems of classical plasticity. These bounds, which bracket the true collapse load from above and below, are found to be in good agreement with one another. Rigid block methods also provided an upper bound estimate on tunnel stability which was generally higher than, but still in good agreement with, the FELA upper bound. Failure mechanisms associated with the collapse of the dual tunnels were investigated, and for deeper tunnels, it was found that mechanisms extend much deeper below the tunnels than the collapse mechanisms associated with a single tunnel. Results from this study are summarised in dimensionless stability charts for use by practitioners.

The efficiency of parallel preconditioned conjugate gradient (PCG) algorithm for solving large sparse linear systems arising from application of interior point methods to conic optimisation problems in the context of nonlinear finite element limit analysis (FELA) for computational geomechanics is studied. For large 3D problems, the use of direct solvers in general becomes prohibitively expensive owing to exponentially growing memory requirements and computational time. And the so-called saddle-point systems resulting from use of optimisation framework is not an exemption. On the other hand, although preconditioned iterative methods have moderate storage requirements and therefore can be applied to much larger problems than direct methods, they usually exhibit high number of iterations to reach convergence. In the present paper, we show that this problem can be effectively tackled using efficient variants of sparse approximate inverse preconditioners along with an elaborate parallel implementation on multicore CPUs and significant improvements can be achieved by parallel implementation on graphic processing unit (GPU). Furthermore, the efficiency of our proposed implementation is verified by the presented numerical results.

This paper uses the finite element upper and lower bound limit analysis methods to investigate the three-dimensional (3D) slope stability of two-layered undrained clay slopes. The solutions obtained from the slope stability analyses are bracketed to within ±10% or better. For comparison purposes, results from two-dimensional (2D) analyses based on the numerical limit analysis methods and the conventional limit equilibrium method (LEM) are also discussed. This study shows that 3D boundary of a slope can have significant effects on the slope stability. In addition, the results are presented in the form of stability charts which can be convenient tools for practicing engineers.

Rainfall induced landslides vary in depth and the deeper the landslide, the greater the damage it causes. This paper investigates, quantitatively, the risk of rainfall induced landslides by assessing the consequence of each failure. The influence of the spatial variability of the saturated hydraulic conductivity and the nature of triggering mechanisms on the risk of rainfall-induced landslides (for an infinite slope) are studied. It is shown that a critical spatial correlation length exists at which the risk is a maximum and the risk is higher when the failure occurs due to a generation of positive pore water pressure. © 2014 Elsevier B.V.

In the study of landslides, it is generally assumed that an impermeable boundary exists at a certain depth and failure occurs at this boundary. In reality this is not always the case and failures can occur at any depth. This paper aims to study the effect of boundary conditions on landslides, using a series of seepage and stability analyses performed over a range of rainfall intensities, and for different failure mechanisms, by studying the failure time and depths corresponding to fully drained, partially drained, and impermeable boundaries. It is shown that these conditions can significantly affect the occurrence and depth of rainfall-induced landslides. © 2014 Elsevier Ltd.

The problem of finite element simulation of incompressible fluid flow in porous medium is considered. The porous medium is characterized by the X-ray microtomography technique in three dimensions. The finite calculus-based stabilization technique is reviewed to implement the equal order finite element interpolation functions for both velocity and pressure. A noble preconditioner, the nodal block diagonal preconditioner, is considered whose performance is thoroughly investigated. Combining this preconditioner with a standard iterative solver during the computational homogenization procedure, it is possible to carry out the large-scale fluid flow simulation for estimating permeability of the porous medium with reasonable accuracy and reliability. © 2013 John Wiley & Sons, Ltd.

In this paper, numerical limit analysis and semianalytical rigid block techniques are used to investigate the effect of the tunnel spacing on the stability of two circular tunnels excavated side by side. The tunnels are modeled under plane-strain conditions, which implies that they are assumed to be infinitely long. Bounds on the stability of the tunnels are obtained using finite-element limit analysis, the numerical formulation of which is based on the upper and lower bounds theorems of classical plasticity. Solutions are obtained using advanced conic programming schemes to solve the resulting optimization problems, and upper and lower bound estimates on the stability of the tunnels are obtained for a range of geometries. These bounds, which bracket the true collapse load from above and below, are found to differ by at most 5% for the cases where the solution does not approach zero. Results from this study are summarized in stability charts for use by practitioners. © 2014 American Society of Civil Engineers.

Trench stability is a conventional geotechnical problem; however, current evaluations are often based entirely on empiricism. This paper uses numerical finite-element upper and lower bound limit analysis to produce stability charts for two-dimensional and three-dimensional homogeneous and inhomogeneous undrained diaphragm wall trenches. Using the limit theorems cannot only provide a simple and useful way of analyzing the stability of the trench, but also avoid the shortcomings and arbitrary assumptions underpinning the limit equilibrium method. By considering the effects from the bentonite slurry pressures, the collapse load in this study has been bracketed to within ±8.5 or better by the numerical upper and lower bound limit analyses. The chart solutions can be used to predict either the critical depth or the safety factor of the trench and provide a convenient tool for preliminary designs by practicing engineers. © 2014 American Society of Civil Engineers.

This paper presents a hybrid preconditioning technique for Conjugate Gradient method and discusses its parallel implementation on Graphic Processing Unit (GPU) for solving large sparse linear systems arising from application of interior point methods to conic optimization problems in the context of nonlinear Finite Element Limit Analysis (FELA) for computational Geomechanics. For large 3D problems, the use of direct solvers in general becomes prohibitively expensive due to exponentially growing memory requirements and computational time. Besides, the so-called saddle-point systems resulting from use of optimization framework is not an exemption. On the other hand, although preconditioned iterative methods have moderate storage requirements and therefore can be applied to much larger problems than direct methods, they usually exhibit high number of iterations to reach convergence. In present paper, we show that this problem can be effectively tackled using the proposed hybrid preconditioner along with an elaborate implementation on GPU. Furthermore, numerical results verify the robustness and efficiency of the proposed technique.

A version of the Particle Finite Element Method applicable to geomechanics applications is presented. A simple rigid-plastic material model is adopted and the governing equations are cast in terms of a variational principle which facilitates a straightforward solution via mathematical programming techniques. In addition, frictional contact between rigid and deformable solids is accounted for using an approach previously developed for discrete element simulations. The capabilities of the scheme is demonstrated on a range of quasi-static and dynamic problems involving very large deformations. © 2013 Elsevier Ltd.

The displacement finite element, lower and upper bound finite element limit analysis and analytical upper bound plasticity methods are employed to investigate the undrained limiting lateral resistance of piles in a pile row. Numerical analyses and analytical calculations are presented for various pile spacings and pile-soil adhesion factors. The numerical results are shown to be in excellent agreement with each other and also with the theoretical upper bounds produced by the analytical upper bound calculations. Based on the numerical and analytical results, an empirical equation is proposed for the calculation of the ultimate undrained lateral bearing capacity factor. This equation is subsequently used to calculate p-multipliers applicable to the lower part of piles in pile rows, which are compared to multipliers available in the literature (that are constant with depth). The comparison shows significant differences, indicating that the amount of reduction in lateral resistance due to group effects is not constant with depth as routinely assumed in practice. © 2013 Elsevier Ltd.

SUMMARY: This study extends the limit analysis techniques used for the computation of strict bounds of the load factors in solids to stability problems with interfaces, anchors and joints. The cases considered include the pull-out capacity of multi-belled anchors and the stability of retaining walls for multiple conditions at the anchor/soil and wall/soil interfaces. Three types of wall supports are examined: free standing wall, simply supported wall and anchored wall. The results obtained are compared against available experimental and numerical data. The conclusion drawn confirms the validity of numerical limit analysis for the computation of accurate bounds on limit loads and capturing failure modes of structures with multiple inclusions of complex interfaces and support conditions. © 2012 John Wiley & Sons, Ltd.

The efficiency of several preconditioned Conjugate Gradient (PCG) schemes for solving of large sparse linear systems arising from application of interior point methods to nonlinear Finite Element Limit Analysis (FELA) is studied. Direct solvers fail to solve these linear systems in large sizes, such as large 2D and 3D problems, due to their high storage and computational cost. This motivates using iterative methods. However, iterative solvers are not efficient for difficult problems without preconditioning techniques. In this paper, the effect of various preconditioning techniques on the convergence behavior of the preconditioned Conjugate Gradient (PCG) is investigated through a detailed comparative study. Furthermore, numerical results of applying PCG to several sample systems are presented and discussed thoroughly in a parametric study. Our results suggest that while incomplete Cholesky preconditioners are by far the most efficient techniques for sequential computations, significant gains may result from use of sparse approximate inverse methods in parallel environment in this field.

Bearing capacity and failure mechanism of a buried footing in uniformsoils have been simulated using limit analysis and finite element analysis in the past decades. In realistic conditions, soil properties always vary spatially. This dramatically affects the failure mechanism of a footing and, in turn, its bearing capacity. This paper illustrates an investigation into the failure mechanism and bearing capacity of a vertically and centrally loaded footing embedded in spatially variable clayey soils using random lower bound limit analysis, random upper bound limit analysis and random finite element analysis. The footing was embedded to 4 times its width. Monte Carlo simulation was performed for 400 realizations of random fields of undrained shear strength. The majority of the bearing capacity factors obtained from the finite element method is bounded by those obtained from the lower bound limit analysis and the upper bound limit analysis, but more close to the upper bound results. A full-flow failure mechanism is observed for the deeply embedded footing in spatially variable soil. The shear path of the footing shows an unsymmetrical pattern, which results from the spatially variable and unsymmetrical random field of soil shear strength. © 2015 Taylor & Francis Group, London.

Rainfall induced landslides can vary in depth and deeper the landslide, greater is the damage it causes. This paper investigates, quantitatively, the risk of rainfall induced landslides by assessing the consequence of each failure. The influence of the spatial variability of the saturated hydraulic conductivity on the risk of landslides is studied. It is shown that a critical spatial correlation length exists at which the risk is a maximum. © 2015 Taylor & Francis Group, London.

This paper investigates the stability of fill slopes often found in embankment cases where frictional fill materials are placed on purely cohesive undrained clay with increasing strength. By using finite element upper and lower bound limit analysis for this investigation, the limit load can be truly bounded. It is known that two-dimensional analysis yields a more conservative result due to plain strain condition when compared to three-dimensional analysis. Therefore, this paper will focus on three-dimensional (3D) slope stability analysis and for comparison purposes two-dimensional analysis results will be employed. In fact, the final results are presented in the form of comprehensive chart solutions for the convenience of practicing engineers during preliminary slope design. The failure mechanism will also be discussed in order to further illustrate the situation during failure. It should be highlighted that the failure mechanisms are obtained through the numerical method itself and no prior assumptions are required, therefore, are more realistic and able to provide a better understanding for the slope failure surfaces.

For general stability analysis of rock slopes, rock mass strength and rock mass disturbance are definitely should be considered. In addition, the impact of earthquakes must be taken into account. In fact, the rock mass strength is very difficult to be assessed which causes the difficulty of analysing rock slope stability. Therefore, an empirical failure criterion, the Hoek-Brown failure criterion, has been proposed. It is one of the most widely accepted approaches to estimate rock mass strength. The rock mass disturbance is important and was found having significant influence on evaluating rock slope stability, especially for rock slope with poor quality rock mass. In the Hoek-Brown failure criterion, the disturbance factor can represent the level of the rock mass disturbance which would provide a reasonable basis for estimating rock mass strength. This research will not only discuss the slope factor of safety, but also consider the influence of the seismic force on rock slope stability assessment using pseudo-static method. In practice, only horizontal seismic coefficient is used. Various magnitudes of the disturbance factor and recommended blasting damage zone thickness are also taken into account. The blasting damage zone thickness considered ranges from 0.5 to 2.5 times of slope height. The research results have potential to be extended and then sets of comprehensive stability charts can be provided for the rock slope stability evaluations. They will be convenient tools for practising engineers. In this study, finite element upper bound and lower bound limit analysis methods are employed. Their applicability has been investigated in some previous studies. The differences between upper bound and lower bound solutions are less than ±10% which would provide reasonable and acceptable range for rock slope stability safety factor estimation.

This paper uses the finite element upper and lower bound limit analysis to assess the stability of slopes mostly found in embankment cases where frictional materials are filled on purely cohesive undrained clay. For comparison purposes, the commonly used stability assessment method, limit equilibrium method (LEM) is also employed. The final results for both methods are then presented in the form of comprehensive chart solutions for the convenience of practicing engineers during preliminary slope designs. The failure mechanism will also be discussed in this paper. Ultimately, it should be noted that finite element limit analysis method holds the upper hand as its prior assumptions are not required. Thus, the obtained failure mechanism from the slope stability analysis will be more realistic. Hence, it will provide a better understanding for the slope failure surface. Therefore, engineers should design more carefully when the LEM is applied to the slopes with frictional materials filled on purely cohesive undrained clay. © 2014 American Society of Civil Engineers.

The mechanical properties of natural materials such as rocks and soils vary spatially. This randomness is usually modelled by random field theory so that the material properties can be specified at each point in space. When these point-wise material properties are mapped onto a finite element mesh, discretization errors are inevitable. In this study, the discretization errors are studied and suggestions for element sizes in relation with spatial correlation lengths are given. © (2014) Trans Tech Publications, Switzerland.

Some common criteria for predicting the plastic failure of geomaterials, including the Mohr-Coulomb model, can be represented as conic constraints. Thus, by formulating finite element limit analysis (FELA) problems with such materials as second-order cone programs (SOCPs), solutions to small- and medium-scale problems are readily obtained using current state-of-the-art optimisation software that are based on polynomially-bounded interior-point methods (IPMs). Unfortunately, when progressing to large-scale 2D and 3D problems, these schemes often struggle with the increased computational burden of obtaining a search direction at each iteration of the IPM using conventional direct solvers that implement some form of Gaussian elimination. In this paper, current conic optimisation software is compared for some medium and large-sized FELA problems. Additionally, a scheme which avoids the factorisation of large linear systems, through the use of a Krylov subspace solver, is developed and compared to existing software. © (2014) Trans Tech Publications, Switzerland.

Computational limit analysis provides a fast and convenient means of evaluating the stability or bearing capacity of geostructures. It is based on numerical optimization techniques and the latest trend is to use robust conic programming algorithms. The shortcoming, however, is that the types of problems covered by conic programming are not very general. In practice, this means that only criteria containing linear and quadratic terms (such as Drucker-Prager) or those that involve linear terms in the principal stresses (such as Mohr-Coulomb) can be considered. In the present paper this shortcoming is addressed. The idea is to maintain an efficient and robust conic programming algorithm as the main solution engine. Nonlinear criteria are then handled by a dynamic linearization procedure that involves a sequence of standard conic programming solutions in an iterative scheme that turns out to converge relatively rapidly.