Numerical Methods For Plastic Upper Bound Limit Analysis

The collapse load and failure mechanism of solid or structure play an important role in engineering structure design and safety assessment in civil engineering. In general, the methods used to determine the collapse load and failure mechanism of a structure can be classified into two types: a) elastoplastic incremental approaches, and b) upper and lower bound approaches of limit analysis. The upper and lower bound approaches which based on limit theorem are more direct and rigorous method for designing engineering structures. In general, the numerical upper bound limit analysis is more popular in pracitcal engineering due to the facility of constructing the kinematically admissble velocity field. In the upper bound limit analysis, the linear or nonlinear mathematical programming problems will be solved ultimately. With the increasing of dimension of kinematically admissble velocity, these mathematical programming problems will become too difficult to be solved. Therefore, in this paper, the research focuses on the numerical theories of plastic upper bound approach and the application of new algorithm of optimization. The summary of this paper is illustrated as follows:If the safety factor is treated as the objective function in RFEM (rigid finite element method)-based upper bound approach, the mathematical programming will be a nonlinear programming problem with constraints. In genearl, the so-called sequential quadratic programming (SQP) method is used to solve the constrained nonlinear programming. However, with an arbitrary starting point, it is quite time consuming and difficult to search the optimum based on the SQP-type algorithms because the sub-QP problem is infeasible at every iteration step. Fortunately, a QP-free algorithm based on the so call pivoting operation and active-set strategy can be convergent toward the optimal points with arbitrary start point. The infeasibility of sub-QP problem can be avoided using poviting operation, and the QP can be transformed into three linear systems of equations with the same coefficient matrixs. For overcoming the constant derivatives of objective function, the objective function (safety factor) was reformulated as a function of the kinematically admissible velocity field based the virtual work rate equation. Two classical problems of slope stability are solved by this QP-free algorithm, the results show that QP-free method is more efficiency than the SQP method.In the above RFEM-based upper bound approach, the nonlinearity of mathematical programming is caused by the definition of the safety factor. For avoiding this nonlinearity, the limit load multiplier is used to evaluate the limit load of the structures. And according to the concept of critical acceleration, the safety factor of stablility of strucutres can be computed iteratively from the limit load multiplier using linear programming. In addtion, the rotational failure mechanism can be considered in constructing kinematically admissible velocity field by using enforce the associated flow rule at two points along the interface between two adjacent rigid elements. Therefore, the presented RFEM-based upper bound approach can be earlier applied into slope stability and bearing capacity analysis. In addition, the linear programming problem of RFEM-based upper bound approach can be solved by using simplex algorithm or primal-dual interior point method. The numerical results of several test problems show that the primal-dual interior point method is very suit for the large-scale linear programming problem with sparse coefficient matrix.For rock or soil materials, the non-associated flow rule should be satisfied when the kinematically admissible velocity field was constructed. Therefore, in RFEM-based upper bound approach, the non-associated flow rule was enforced at two end points along interface between two adjacent rigid elements. And then, the linear programming problem of upper bound limit analysis was formulated considering the non-associated flow rule. Therefore, the influence of non-associated flow rule on safety factor of slope stability can be solved based on the present method. Furthermore, the pull out resistance of soil nailing within the dilative soils can also be computed by using the present method.For overcoming the difficulty of adaptive mesh, the kinematically admissible velocity field can be constructed by using radial point interpolation method (RPIM). Then, the external work rate and internal energy dissipation rate are computed by using Cartesian transformation method (CTM). As a result, the nonlinear programming problem of upper bound approach is presented based on meshless method (MM). For solving the presented MM-based upper bound approach, a direct iterative algorithm based on the distinguishing rigid/plastic zone is adopted. Regarding the direct iterative algorithm for frictional materials, the iteration control parameter is identically vanishing for all the shear strength parameters. Therefore, the direct iterative algorithm can't be used to calculate the limit load of frictional materials which follow the Mohr-Coulomb or Drucker-Prager yield criterion, and the direct iterative algorithm can be only used to compute the limit load of non-frictional materials which follow the ellipsoid yield function. To verify and extend the presented MM-based upper bound approach, the limit loads of some classical problems including stability of slopes, plate and thick-walled cylinder are calculated by using the presented method.