Uncertainty in Finite Element Modeling and Failure Analysis: A Metrology-Based Approach

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 the 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 two 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 question, we propose a progressively weakening model based on a small number (e.g., 5) of tests on ultimate strength such that the failure of the weakest column of the grillage causes a load redistribution and collapse occurs only when the load redistribution leads to instability. This model satisfies the requirement of a metrology-based approach, where the time to failure is given a quantitative expression of uncertainty. We conclude that in today’s computing environment and with a precomputational “design of numerical experiments,” it is feasible to “quantify” uncertainty in FEM modeling and progressive failure analysis.