Abstract:
This paper presents a finite element model based on mathematical non-linear programming to determine the upper bounds of collapse loads for a mechanical structure. The formulation uses a kinematical approach with two independent field approximations for velocity and strain rate, with an augmented Lagrangian used to ensure compatibility between them. The model uses only continuous velocity fields and applies Uzawa's minimization algorithm to find the optimal kinematic field that minimizes the difference between external and dissipated work rates. This approach bypasses the complexity of the problem's non-linear aspects by addressing non-linearity through a set of small, local subproblems for each finite element. This makes the model versatile and capable of solving a wide range of collapse problems, including 3D strut-and-tie structures, 2D plane strain/stress, and 3D solid problems. The finite element model's efficiency, accuracy, and robustness are demonstrated through several numerical examples and compared with analytical solutions and other numerical formulations. The study's results for problems like Prandtl's punch problem, a perforated square plate, a defective pipeline, and a circular footing punch problem show excellent agreement, with low error rates, even for non-smooth yield criteria. While the assumption of continuous velocity fields means the exact collapse load isn't guaranteed, the model still yields accurate upper bound estimates.
