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

[+] Author and Article Information
Jeffrey T. Fong

Mathematical & Computational Sciences Division, National Institute of Standards & Technology (NIST), Gaithersburg, MD 20899fong@nist.gov

James J. Filliben

Statistical Engineering Division, National Institute of Standards & Technology (NIST), Gaithersburg, MD 20899filliben@nist.gov

Roland deWit

Metallurgy Division, National Institute of Standards & Technology (NIST), Gaithersburg, MD 20899dewit@nist.gov

Richard J. Fields

Metallurgy Division, National Institute of Standards & Technology (NIST), Gaithersburg, MD 20899rjfields@nist.gov

Barry Bernstein

Departments of Mathematics and Chemical Engineering, Illinois Institute of Technology (IIT), Chicago, IL 60616bernsteinb@iit.edu

Pedro V. Marcal

 MPave Corp., 1355 Summit Avenue, Cardiff, CA 92007-2429marcalpv@cox.net

J. Pressure Vessel Technol 128(1), 140-147 (Oct 23, 2005) (8 pages) doi:10.1115/1.2150843 History: Received September 30, 2005; Revised October 23, 2005

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.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

The addition of two types of uncertainties in a representation of the interrelation of fracture mechanics, failure analysis, product liability, and technical insurance by Rossmanith (3)

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Figure 2

One of three benchmark test problems of NUMISHEET (71), an international interlaboratory validation exercise for simulating three-dimensional aluminum and steel sheet-forming processes

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Figure 3

(a) Force versus displacement results of five simulations and one experiment (BE-01, Korea) reported in 2002 NUMISHEET (71). The simulations resulted from five different FEM models, i.e., BS-07 (LS-DYNA3D, 384 elem.), BS-08 (ABAQUS, 4400 elem.), BS-09 (PAM-STAMP, 3600 elem.), BS-10 (DD3 IMP, 7380 elem.), and BS-12 (Indeed 7.3.1, 4000 elem.); (b) details of each team of investigators.

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Figure 4

A typical result of the free vibration solution of the reference benchmark problem, RB-101, using ANSYS (83) with 5120 solid elements (ST45). This run is part of a 96-run design.

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Figure 5

A typical result of estimating uncertainty of a natural frequency of RB-101 using a three-parameter exponential fit of a grid convergence run. Note the comparison with an exact solution.

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Figure 10

A histogram of the yield strength of 224 heats of ASTM 285 Grade C steel comparable to ASTM A36) from the purchase record of a single fabricator (88). This set of data was fitted by Fong (89) with a three-parameter Weibull (γ=1.8).

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Figure 11

A typical unsymmetrical pattern of failure of a thin square plate loaded at its center with a point load when the material is given a failure strain distribution of the Weibull type

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Figure 6

A typical result of a nonlinear (large deformation) analysis of RB-301 using MPAVE (87)

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Figure 7

Plot of a three-parameter Weibull distribution of the column ultimate strength based on 5 test data (3, 4, 5, 5, and 8). Note in small print the goodness of fit equal to 0.964 546 (90).

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Figure 8

A typical sample of a progressive failure analysis of the collapse of a 44-column grillage loaded at a constant rate with a 5-test ultimate strength distribution given in Fig. 7(90)

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Figure 9

A typical time-to-collapse distribution based on 100 simulations of progressive failure of a 44-column grillage with one of the sample simulations given in Fig. 8. Note the difference between the simulation mean and the deterministic mean.




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