Limit analysis based upon the theory of plasticity is one of the very useful numerical techniques to determine the failure loads of different civil and mechanical engineering structures for a material following an associated flow rule. The limiting values of the collapse loads, namely, lower and upper bounds, can be bracketed quite accurately with the application of the lower and upper bound theorems of the limit analysis. With the advancement of the finite elements and different robust optimization techniques, the numerical limit analysis approach in association with finite elements is becoming very popular to assess the stability of various complicated structures. Although two different optimization methods, namely, linear programming and nonlinear programming, have been both successfully implemented by various researchers for solving different stability problems in geomechanics, the linear programming method is employed in the present thesis due to its inherent advantage in implementation and ease in achieving the convergence. The objectives of the present thesis are (i) to improve upon the existing lower bound limit analysis method, in combination with finite elements and linear programming, with an intention of reducing the computational time and the associated memory requirement, and (ii) to apply the existing lower bound finite element limit analysis to various important planar stability problems in geotechnical engineering. With reference to the first objective of the thesis, two new methods have been introduced in this thesis to improve upon the existing computational procedure while solving the geomechanics stability problem with the usage of the limit analysis, finite elements and linear programming. In the first method, namely, the method-I, the order of the yield polygon within the chosen domain is varied, based on the proximity of the stress state to the yield, such that a higher order polygon needs not to be used everywhere in the problem domain. In the second method, the method-II, it has been intended to use only a few selected sides, but not all, of the higher order yield polygon which are being used to linearize the Mohr-Coulomb yield function. The proposed two methods have been applied to compute the ultimate bearing capacity of smooth as well as rough strip footings for various soil frictional angles. It has been noticed that both the proposed new methods reduce the CPU time and the total number of inequality constraints required as compared to the existing lower bound linear programming method used in literature. With reference to the second objective, a few important planar stability problems in geomechanics associated with interference of footings and vertical anchors have been solved in the present thesis. Footings are essentially used to transfer the compressive loads of the super structures to underlying soil media. On the other hand, vertical anchors are used for generating passive supports to retaining walls, sheet piles and bulkheads. A large number of research investigations have been reported in literature to compute the collapse load for a single isolated strip footing and a single vertical anchor. It is a common practice to estimate the bearing capacity of footings or pullout capacity of anchors without considering the effect of interference. There are, however, clear evidences from the available literature that (i) the ultimate bearing capacity of footings, and (ii) the ultimate pullout capacity of anchors, are significantly affected by their interference effect. Based on different available methods, the interference of footings, in a group of two footings as well as an infinite number of multiple footings, has been examined by different researchers in order to compute the ultimate bearing capacity considering the group effect. However, there is no research study to find the ultimate bearing capacity of interfering footings with the usage of the lower bound limit analysis. In the present thesis, the ultimate bearing capacity of two and an infinite number of multiple strip footings placed on sandy soil with horizontal ground surface, has been determined. The analysis has been performed for smooth as well as rough footings. The failure loads for interfering footings are found to be always greater than the single isolated footing. The effect of the footings' interference is expressed in terms of an efficiency factor ( ξγ); where, ξγ is defined as the ratio of the magnitude of failure load for a footing of width B in presence of the other footing to the magnitude of failure load of an isolated strip footing having the same width. The effect of the interference on the failure load (i) for rough footings becomes always greater than smooth footings, (ii) increases with an increase in soil frictional angle φ, and (iii) becomes almost negligible beyond the spacing, S > 3B. It is observed that the failure load for a footing in a group of an infinite number of multiple strip footings becomes always greater than that for two interfering footings. Attempts have been made in this thesis to investigate the group effect of two vertical anchors on their horizontal pullout resistance (PuT). The anchors are considered to be embedded at a certain clear spacing (S) along the same vertical plane. The group effect has been studied separately for anchors embedded in (i) sandy soil, and (ii) undrained clay, respectively. For anchors embedded in clays, an increase of soil cohesion with depth, in a linear fashion, has also been taken into consideration. The magnitude of PuT has been obtained in terms of a group efficiency factor, ηγ for sand and ηc for clay, with respect to the failure load for a single isolated vertical plate with the same H/B. The pullout capacity of a group of two anchors either in sand or in undrained clay becomes quite extensive as compared to a single isolated anchor. The magnitudes of ηγ and ηc become maximum corresponding to a certain critical value of S/B, which has been found to lie generally between 0.5 and 1. The value of ηγ for a given S/B has been found to become larger for greater values of H/B, φ, and δ. For greater values of H/B, the group effect becomes more significant in contributing the pullout resistance. The horizontal pullout capacity of a single isolated vertical anchor embedded in sand in the presence of pseudo static horizontal earthquake body forces has also been determined by using the lower bound finite element limit analysis. The variation of the pullout factor Fγ with changes in the embedment ratio of the smooth and rough anchor plates for different values of horizontal earthquake acceleration coefficient ( αh) has been investigated. The analysis clearly reveals that the pullout resistance decreases quite significantly with an increase in the magnitude of the earthquake acceleration coefficient. For the various problems selected in the present thesis, the failure patterns have also been exclusively drawn in order to understand the development of the plastic zones within the chosen domain for solving a given problem. The results obtained from the analysis, for the various problems taken up in this thesis, have been thoroughly compared with those reported in literature.