In this paper, we first review the impact of the powerful finite element method (FEM) in structural engineering, and then address the shortcomings of FEM as a tool for risk-based decision making and incomplete-data-based failure analysis. To illustrate the main shortcoming of FEM, i.e., the computational results are point-estimates based on deterministic models with equations containing mean values of material properties and prescribed loadings , we present the FEM solutions of two classical problems as reference benchmarks: (RB-101) The bending of a thin elastic cantilever beam due to a point load at its free end, and (RB-301) the bending of a uniformly loaded square, thin, and elastic plate resting on a grillage consisting of 44 columns of ultimate strengths estimated from 5 tests. Using known solutions of those two classical problems in the literature, we first estimate the absolute errors of the results of four commercially-available FEM codes (ABAQUS, ANSYS, LSDYNA, and MPAVE) by comparing the known with the FEM results of two specific parameters, namely, (a) the maximum displacement and (b) the peak stress in a coarse-meshed geometry. We then vary mesh size and element type for each code to obtain grid convergence and to answer two questions on FEM and failure analysis in general: (Q-1) Given the results of 2 or more FEM solutions, how do we express uncertainty for each solution and the combined? (Q-2) Given a complex structure with a small number of tests on material properties, how do we simulate a failure scenario and predict time-to-collapse with confidence bounds? To answer the first question, we propose an easy-to-implement metrology-based approach, where each FEM simulation in a grid-convergence sequence is considered a numerical experiment, and a quantitative uncertainty is calculated for each sequence of grid convergence. To answer the second, we propose a collapse mechanism based on Weibulls weakest link theory. We conclude that in todays computing environment and with a pre-computational design of numerical experiments, it is feasible to quantify uncertainty in FEM modeling and progressive failure analysis based on a stochastic formulation and a multi-code approach.
Citation: ASME Transactions Journal of Pressure Vessel Technology
Pub Type: Journals
Applied mechanics, engineering statistics, failure analysis, finite element method, mathematical modeling, metrological science, stochastic modeling, structural engineering, uncertainty analysis