Limit analysis is a very powerful tool to find accurate solutions of several geotechnical stability problems. This analysis is based on the theory of the plasticity and it provides two limiting solutions within lower and upper bounds. With the advancement of the finite elements and different robust optimization techniques, the numerical limit analysis approach in association with finite elements is becoming quite popular to assess the stability of various complicated structures. The present thesis deals with the formulations and the implementation of the finite element limit analysis to obtain the solutions of different geotechnical axisymmetric stability problems. The objectives of the present thesis are twofold: (a) developing limit analysis formulations in conjunction with linear and nonlinear optimizations for solving axisymmetric stability problems related with soil and rock mechanics, and then (b) implementing these axisymmetric formulations for solving various important axisymmetric stability problems in geomechanics. Three noded linear triangular elements have been used throughout the thesis. In order to solve the different problems, the associated computer programs have been written in MATLAB. With reference to the first objective of the thesis, the existing finite element lower bound axisymmetric formulation with linear programming has been presented. A new technique has also been proposed for solving an axisymmetric geomechanics stability problem by employing an upper bound limit analysis in combination with finite elements and linear programming. The method is based on the application of the von-Karman hypothesis to fix the constraints associated with the magnitude of the circumferential stress (), and finally the method involves only the nodal velocities as the basic unknown variables. The required computational effort becomes only marginally greater than that needed for an equivalent plane strain problem. The proposed methodology has been found to be computationally quite efficient. A new lower bound axisymmetric limit analysis formulation, by using two dimensional finite elements, the three dimensional Mohr-Coulomb (MC) yield criterion, and nonlinear optimization has also been presented for solving different axisymmetric stability problems in geomechanics. The nonlinear optimization was carried out by employing an interior point method based on the logarithmic barrier function. The yield surface was smoothened (i) by removing the tip singularity at the apex of the pyramid in the meridian plane, and (ii) by eliminating the stress discontinuities at the corners of the yield hexagon in the plane. No inherent assumption concerning with the hoop stress needs to be made in this formulation. The Drucker-Prager (DP) yield criterion was also used for computing the lower bound axisymmetric collapse load. The advantage of using the DP yield criterion is that it does not exhibit any singularity in the plane. A new proposal has also been given to simulate the DP yield cone with the MC hexagonal yield pyramid. The generalized Hoek-Brown (HB) yield criterion has also been used. This criterion has been smoothened both in the meridian and  planes and a new formulation is prescribed for obtaining the lower bound axisymmetric problems in rock media in combination with finite elements and nonlinear optimization. With reference to the second objective, a few important axisymmetric stability problems in soil mechanics associated with footings and excavations have been solved in the present thesis. In all these problems, except that of a flat circular footing lying over either homogeneous soil or rock media, it is assumed that the medium is governed by the MC failure criterion and it follows an associated flow rule. For determining the collapse loads for a circular footing over homogenous soil and rock media, the problem has been solved with the usage of Drucker-Prager, Mohr-Coulomb and Hoek-Brown criteria. The bearing capacity of a circular footing lying over fully cohesive strata, with an inclusion of a sand layer is evaluated. The effects of the thickness and internal friction angle of the sand layer () on the bearing capacity have been examined for different combinations of cu/(b) and q; where (i) cu defines the undrained shear strength, (ii)  is the unit weight of sand, (iii) b corresponds to the footing radius, and (iv) q is the surcharge pressure. The results have been presented in the form of a ratio () of the bearing capacities with an insertion of the sand layer to that for a footing lying directly over clayey strata. It is noted that an introduction of a layer of medium dense to dense sand over soft clay improves considerably the bearing capacity of the foundation. The improvement in the bearing capacity increases continuously (i) with decreases in cu/(b), and (ii) increases in  and q/(b). The bearing capacity factors, Nc, Nq and N, for a conical footing are obtained in a bound form for a wide range of the values of cone apex angle () and with  = 0, 0.5 and . The bearing capacity factors for a perfectly rough ( = conical footing generally increase with a decrease in . On contrary for  = 0, the factors Nc and Nq reduce gradually with a decrease in . For  = 0, the factor N for  ≥ 35o becomes minimum for  approximately equal to 90o. For  = 0, the factor N for  ≤ 30o, like in the case of  = , generally reduces with an increase in . It has also been intended to compute the bearing capacity factors Nc, Nq and N, for smooth and rough ring footing for different combinations of ri/ro and ; where ri and ro refer to inner and outer radii of the ring, respectively. It is observed that for a smooth footing, with a given value of ro, the magnitude of the collapse load decreases continuously with an increase in ri. On the other hand, for a rough base, for a given value of ro, hardly any reduction occurs in the magnitude of collapse load up to ri/ro ≈ 0.2, whereas beyond this ri/ro, the magnitude of the collapse load, similar to that of a smooth footing, decreases continuously with an increase in ri/ro. An attempt has also been made to determine the ultimate bearing capacity of a circular footing, placed over a soil mass which is reinforced with horizontal layers of circular reinforcement sheets. For performing the analysis, three different soil media have been separately considered, namely, (i) fully granular, (ii) cohesive frictional, and (iii) fully cohesive with an additional provision to account for an increase of cohesion with depth. The reinforcement sheets are assumed to be structurally strong to resist axial tension but without having any resistance to bending; such an approximation usually holds good for geogrid sheets. The shear failure between the reinforcement sheet and adjoining soil mass has been considered. The increase in the magnitudes of the bearing capacity factors (Nc and N) with an inclusion of the reinforcement has been computed in terms of the efficiency factors c and . The critical positions and corresponding optimum diameter of the reinforcement sheets, for achieving the maximum bearing capacity, have also been established. The increase in the bearing capacity with an employment of the reinforcement increases continuously with an increase in . The improvement in the bearing capacity becomes quite extensive for two layers of the reinforcements as compared to the single layer of the reinforcement. The stability of an unsupported vertical cylindrical excavation has been assessed. For the purpose of design, stability numbers (Sn) have been generated for both (i) cohesive frictional soils, and (ii) pure cohesive soils with an additional provision to account for linearly increasing cohesion with depth by using a non-dimensional factor m. The variation of Sn with H/b has been established for different values of m and ; where H and b refer to height and radius of the cylindrical excavation. A number of useful observations have been drawn about the variation of the stability number and nodal velocity patterns with changes in H/b,  and m. In the last, by using the smoothened generalized HB yield criterion, the ultimate bearing capacity of a circular footing placed over a rock mass is evaluated in a non-dimensional form for different values of GSI, mi, ci/(b) and q/ci. For validating the results, computations were exclusively performed for a strip footing as well. For the various problems selected in the present thesis, the failure and nodal velocity patterns have been examined. The results obtained from the analysis have been thoroughly compared with that reported from literature. It is expected that the various design charts presented here will be useful for the practicing engineers. The formulations given in the thesis can also be further used for solving various axisymmetric stability problems in geomechanics.

The thesis describes the application of anchors in mining and tunneling and gives an up-to-date overview about anchor types, design principles and the interaction mechanisms between anchors and rockmass. A constitutive model was developed, implemented and tested for the 2- and 3-dimensional numerical codes FLAC and FLAC3D to simulate non-linear anchor behaviour including unloading and reloading. The interaction between rockbolts and rockmass was studied in detail via numerical simulations for 5 Vietnamese coal mines. An extended version of the so-called c-Φ reduction method and a new introduced reinforcement factor were applied to quantify the effect of bolting. Mine specific and generalised relations were deduced to quantify the influence of anchor length and distance between anchors on the effect of bolting.

The anisotropic behavior of a rock mass with persistent and planar joint sets is mainly governed by the geometrical and mechanical characteristics of the joints. The aim of the study is to develop a continuum-based approach for simulation of multi jointed geomaterials. There are two available numerical techniques for the strain-stress analysis of rock masses: continuum-based methods and discontinuum based methods. Joints are simulated explicitly in discontinuous methodology. This technique provides a more accurate description for the behavior of a rock mass. However, in some projects, the explicit definition becomes impractical, especially with increasing number of joints. Besides, the calculation efficiency will be significant reduced as the number of joints increases within the model. Considering the above mentioned shortcomings of the discontinuous method, the continuum-based approach is widely used in rock mechanics. Within the continuum methods, the discontinuities are regarded as smeared cracks in an implicit manner and all the joint parameters are incorporated into the equivalent constitutive equations. A new equivalent continuum model, called multi-joint model, is developed for jointed rock masses which may contain up to three arbitrary persistent joint sets. The Mohr-Coulomb yield criterion is used to check failure of the intact rock and the joints. The proposed model has solved the issue of multiple plasticity surfaces involved in this approach combined with multiple failure mechanisms. The multi-joint model is implemented into FLAC and is verified against the distinct element method (UDEC), analytical solutions, and experimental data. Uniaxial compression tests with artificial rock-like material (gypsum) are carried out in the laboratory in order to verify the developed constitutive model and to investigate the behavior of jointed specimen. Samples with two crossing joints covering more than 20 angle configurations and two different property sets were prepared and tested. Simulation results are in good agreement with experimental observations. The developed model is applied to two potential practical applications: the stability analysis of a slope and a tunnel under different stress conditions. Finally, the main achievements of the whole PhD study are summarized and future research work is proposed.

The dissertation thesis deals with the properties of hard rocks important for designing geotechnical structures, such as strength and deformation characteristics, as well as rock failure criteria. This work examines the possibility of using correlation relations to estimate the strength characteristics of rocks from index tests, which can make the rock testing process more efficient. The author described selected laboratory tests of rocks and identified several limitations of the tests procedures based on his own practical experience. The correlation analysis of an extensive data set and the derivation of regression relations for selected dependencies were performed. Furthermore, the rock strength estimation quality of the newly derived regressions was compared with the already published regressions. The analysis shows that the achieved degree of correlation is not sufficient to generalize the examined regressions. A significant reason of the low degree of correlation is the combination of the variability of rock properties and limitations of practical testing procedures. Thus, focus should be paid on calibration of the regression relationships within smaller areas in order to precisely estimate the rock properties as reliable input to the geotechnical design.

A lifetime prediction scheme is proposed based on the assumption that the lifetime (time to failure) of rocks under load is governed by the growth of microstructual defects (microcracks). The numerical approach is based on linear elastic fracture mechanics. The numerical calculation scheme is implemented as a cellular automat, where each cell contains a microcrack with length and orientation following certain distributions. The propagation of the microcrack is controlled by the Charles equation, based on subcritical crack growth. The zone inside the numerical model fails if the microcrack has reached the zone dimension or the stress intensity factor of the crack reached the fracture toughness. Macroscopic fractures are formed by these coalesced propagating microcracks, and finally lead to failure of the model. In the numerical approaches, elasto-plastic stress redistributions take place during the forming of the macroscopic fractures. Distinct microcrack propagation types have been programmed and applied to the proposed numerical models. These numerical models are studied under different loading conditions. Numerical results with excellent agreement with the analytical solutions are obtained with respective to predicted lifetime, important parameters for the microcracks, fracture pattern and damage evolution. Potential applications of the proposed numerical model schemes are investigated in some preliminary studies and simulation results are discussed. Finally, conclusions are drawn and possible improvements to the numerical approaches and extensions of the research work are given. / 本文认为微结构缺陷（微裂纹）的扩展决定了受力岩石的寿命（破坏时间）。基于此假设，提出了岩石寿命预测方法。利用线弹性断裂力学理论，通过FLAC进行了数值模拟。数值模型中每个单元定义一条初始裂纹，其长度与方向服从特定分布。基于亚临界裂纹扩展理论，由Charles方程决定微裂纹的扩展（速度）。如微裂纹发展至单元边界，或应力强度系数到达断裂韧度，则单元破坏。宏观裂纹由微裂纹所联合形成，并最终贯穿模型导致破坏。在形成宏观裂纹的过程中，发生弹塑性应力重分布。在数值模型中，编制了不同类型的微裂纹扩展方式，并在不同的受力条件下加以分析。数值模型的岩石寿命，裂纹形状，破坏方式以及一些重要的参数的数值模拟结果与解析解有较好的一致性。对本文所提出的数值模型的初步实际应用进行了分析，并讨论了计算结果。最后讨论了本文所提出的岩石寿命预测方法的可能改良与发展，并对进一步的研究工作给出建议。

The Garpenberg mine, owned by the Boliden Mining group, has established a trial area at Dammsjön orebody in order to examine the possibility of increasing the productivity of the mine. The mine uses the rill mining method with a current rill height of 15 m. In order to increase the productivity, the mine is examining the possibility of increasing the height of the rill. The trial area is located at 882 m depth surrounded by dolomite on the hangingwall and quartzitic rock on the footwall side. Rock support arches have been installed, in addition to the regular support pattern, to test their effectiveness on stabilizing the ground around the drifts. The arches have been installed in every 6 m and every 3 m in different parts of the test area. Rock samples from the trial area were brought to the university laboratory for testing. The data gathered from the laboratory tests along with the data from the monitoring of the trial area were used to develop a calibrated numerical model. A three-dimensional (3-D) model was therefore created, by using the FLAC3D numerical code. After the calibration of the model a parametric study was conducted for different rill heights and different arch spacing to investigate the performance of the arches. Specifically, the case of no arch installation along with the cases of an installed arch every 6 m and 3 m were tested, for the rill heights of 15 m, 20 m, 25 m and 30 m. The study concluded that the arches assisted in reducing the ground convergence in the production drift. The results also showed that the total height of the rill bench yields regardless of its height. After the yielding, the rockmass can no longer support itself and caves under its own weight. The larger the rill height, the larger the volume of loose rock that has to be supported and thus, higher the convergence. Furthermore, it was also observed that, significant amount of convergence in the production drift occurred during the drifting of the top drive and less during the stoping of the rill bench. This indicates that, the timely installation of the arches is an important criterion for their performance.