Application Of Variational Limiting Equilibrium Method To The Extreme Value Problem In Soil Mechanics

Many of the problems encountered in soil mechanics are of the extreme-value type. Such problems can be generally stated as: given all parameters defining the problem, it is required to find the extreme value of some unknown parameter. To make above problems statically determined, conventional limiting equilibrium approach commonly introduces several direct or indirect assumptions (such as assumption concerning interslice forces in general method of slices), which, in essence, is not rigorously founded. Based on the framework of limiting equilibrium approach, a variational proceduce is introduced to deal with various extreme value problems in soil mechanics. This procedure, without any a prior static or kinematic assumptions, only employs overall force and moment equilibrium equations for a test body and treats slip surface and normal stress distribution acting along it as unknows. It should be noted that, in the context of limiting analysis of plasticity, variational limiting equilibrium method is equivalent with upper bound analysis. The main work accomplished in this thesis is as follows:(1) Introducing some theoretical backgrounds and development of variational limiting equilibrium in t??iterature based on the author's recollection.(2) Highlighting the basic theory adopted to conduct variational analysis. Utilizing two Euler's equations the forms of potential slip surface and normal stress distribution are obtained. Variational analysis shows that the extreme value of an objective extremization parameter is independent of the normal stress distribution along the critical slip line. Also the test body possesses two modes of failure, that is rotational mode of failure corresponding to a log-spiral slip line and translational mode of failure corresponding to a straight line.(3) Proving the equivalence between variational limiting equilibrium method and upper bound method, providing an alternative explanation to the variational procedure and thus avoiding controversy concerning the existence of an extreme.(4) Applying variational limit equilibrium approach to some extreme-value problems in soil mechanics. For the problem of the critical height of a simple slope in homogeneous and isotropic soil, Numerical results show that the variational solution is almost identical with upper bound solution. Taking into account tension crack and its effect on the stability of slope, it is seen that in some limiting case, the depth of tension crack may extend to a considerable value up to 25 percent of the total height of slope. Therefore it deserves careful attention in engineering practice. Two stability charts in revelant with respect to stability factor are also given for the sake of convenience. For the problem of ultimate bearing capacity of reinforced foundation, only one sheet of geomembrane and the case of frictionless soil is considered. Results show that as tensile strength of geomembrane and its elevation increase, bearing capacity increases. However, influence of elevation of geomembrane is limited. Meanwhile it is feasible to assume that tension of geomembrane is parallal to horizontal direction since its inclination with respect to horizon executes little effect on bearing capacity. Comparison between different methods on the same problem reveals that the variational solution is higher than result obtained by finite element analysis. Hence much care should be taken when applying current method to the calculation of reinforced foundation.