Finite Element Limit Analysis Based On Non-linear Programming And Its Applications In Engineering

Finite element limit analysis is an international newly developed geotechnical stability analysis method. By finite element discretization of the upper and lower bound theorems into mathematical programming problems and using computers to search for the velocity field or stress field in limit state, this method can overcome the difficulty of constructing velocity field or stress field by hand in traditional limit analysis, thus, has a broad prospect in engineering practice. According to the differences in the constructed mathematical programming models, finite element limit analysis method can be divided into two groups, namely, the finite element limit analysis based on linear or non-linear programming. Due to the avoidance of linearization of yield functions, the latter is capable of dealing with geotechnical stability problems with a nonlinear failure criterion and is featured by its accuracy and high speed in computation and the economy in using computer memory. For these reasons, this paper mainly focuses on the research of finite element limit analysis based on nonlinear programming and give deep analysis and improvements for its theoretical research and engineering applications.Firstly, the theoretical fundamentals and basic hypothesizes of limit analysis are introduced in detail. On this basis, the concrete formulations of the mathematical variational principles of upper and lower bound theorems are given. To solve these mathematical variational problems as two mathematical programming problems, the stress field and velocity field are discretized using linear triangular finite elements, then, the statically admissible conditions and kinematically admissible conditions are converted into the constraints of nodal or elemental optimization variables, and the total external forces(lower bound analysis) or energy dissipation rates(upper bound analysis) are taken as the objective functions. By the above steps, the nonlinear programming models of upper and lower bound limit analysis are established.Secondly, in view of the shortcomings of existing algorithms, the feasible arc interior point algorithm and Wolfe’s inaccurate search technique are used to improve the optimization efficiency of the nonlinear upper and lower bound programming models. The deflection perturbation strategy of existing algorithms is replaced by feasible arc technique, which can overcome the problem of obtaining a too short search step when the iteration point reaches the nonlinear constraint boundary, thus, the number of iterations of the optimization solver decreases significantly. The Wolfe’s inaccurate search technique is used to replace the precise search technique used by existing algorithms, so that the efficiency of step-length searching is enhanced substantially and the time consuming of iteration is reduced greatly.Thirdly, by combining the bound gap error estimation theory and the advancing front mesh generation technique, an adaptive mesh refinement method of finite element limit analysis is proposed. With strictly mathematic deduction, a conclusion is demonstrated, that is, the overall bound gap between the upper and lower bound limit analysis is actually equated to the sum of elemental bound gaps in the mesh. Based on this principle, this paper implements a posterior error estimation method for finite element limit analysis by taking the overall bound gap as an indicator of the overall discretization error and the contribution of elemental bound gap to the overall bound gap as an estimator of local discretization error. Then, adaptive mesh refinement of finite element limit analysis is implemented successfull, by converting the distribution information of the local discretization error into the information of element sizes in the background mesh and calling the advacing front mesh generator to generate the calculation mesh.Next, the application of the finite element limit analysis method based on Hoek-Brown criterion in the stability analysis of jointed rock mass is discussed in detail. To this end, a brief review of the development history of Hoek-Brown criterion is reviewed. On this basis, the application conditions of the Hoek-Brown criterion are discussed. By strictly mathematic derivation, the formulae of the first and second order derivatives of Hoek-Brown yield function are given, and by compiling thsee formulae as computer codes and incorporating them into the finite element limit analysis programs, the stability analysis of rock mass based on the Hoek-Brown criterion is achieved successfully.Finally, using the finite element limit analysis programs developed in this paper, tunnel stabilities in jointed rock mass under gravity load are studied. Abstraction of the studied problem is carried out by proposing reasonable calculation hypotheses, and a simplified calculation model is proposed. Through a large number of numerical calculations, the tunnel stability analysis charts under undisturbed conditions are given and the influences of the Hoek-Brown parameters to tunnel stability in jointed rock mass are discussed. According to the energy dissipation and velocity distribution of the tunnel in limit state, the evolution laws of tunnel failure mode are studied. On this basis, the hypotheses on deep tunnel conditions proposed in the paper are demonstrated. By computing the stability numbers Nds under different disturbance degree, tunnel stability charts are constructed, and application examples are carried out to verify the practical application value of the proposed method.