0
Journal of Pressure Vessel Technology | Volume 128 | Issue 1 | Research Paper
RESEARCH PAPERS

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
Article
References
Figures
Citing Articles

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.
FIGURES IN THIS ARTICLE
<>
Copyright © 2006 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

1
Ross, B., 1984, “What is a Design Defect?” "Proceedings of the 1st International Conference on Structural Failure, Product Liability and Technical Insurance", Vienna, Austria, 26–29 September 1983, edited by H.P.Rossmanith, Elsevier Science Publishers B.V., North-Holland, pp. 23–71.
2
Greenman vs. Yuba Power Products, Inc., 1963, 59 Cal 2nd 57.
3
Rossmanith, H. P., 1984, “FPE - Failure Prevention Engineering: Fracture Mechanics Betwixt Designer and Failure Analyst,” in "Proceedings of the 1st International Conference on Structural Failure, Product Liability and Technical Insurance", Vienna, Austria, 26–29 September 1983, edited by H.P.Rossmanith, Elsevier Science Publishers B.V., North-Holland, pp. 11–21.
4
Google Scholar, 2005, http://www.scholar.google.com
5
Ciampoli, M., 1999, “A Probabilistic Methodology to Assess the Reliability of Deteriorating Structural Elements,” Comput. Methods Appl. Mech. Eng.  [CrossRef], 168 , pp. 207–220.
6
Haldar, A., and Mahadevan, S., 2000, "Reliability Assessment Using Stochastic Finite Element Analysis", Wiley.
7
Joshi, Y., Baelmans, M., Copeland, D., Lasance, C. J. M., Parry, J., and Rantala, J., 2001, “Challenges in Thermal Modeling of Electronics at the System Level: Summary of Panel Held at the Therminic 2000,” IEEE Trans. Compon. Packag. Technol.  [CrossRef], 24 , pp. 611–613.
8
Iwasaki, A., Sugiya, T., and Todoroki, A., 2002, “Unmanned Structural Health Monitoring via Internet with Unsupervised Statistical Diagnosis (Application for Damage Detection of Jet Fan),”"Proceedings of the 1st European Workshop on Structural Health Monitoring (SHM 2002)", Paris—ENS-Cachan—10–12 July 2002, Paper 2002-076, http://www.onera.fr/shm2002/shm2k2-41-80.pdf.
9
Gauss, C. F., 1871, see Carl Friedrich Gauss Werks, VII, Gottingen.
10
Galerkin, B. G., 1915, “Series Solution of Some Problems of Elastic Equilibrium of Rods and Plates,” Vestn. Inzh. Tekh.., 19 , pp. 897–908 (in Russian).
11
Biezeno, C. B., and Koch, J. J., 1923, “Over een Nieuwe Methode ter Berekening van Vlokke Platen,” De Ingenieur (in Dutch), 38 , pp. 25–36.
12
Strutt, J. W. (Lord Rayleigh), 1870, “On the Theory of Resonance,” Philos. Trans. R. Soc. London, 161 , pp. 77–118.
13
Ritz, W., 1909, “Uber Eine Neue Methode zur Losung Gewissen Variations—Probleme der Mathematischen Physik,” J. Reine Angew. Math., 135 , pp. 1–61.
14
Courant, R., 1943, “Variational Methods for the Solution of Problems of Equilibrium and Vibration,” Bull. Am. Math. Soc., 49 , pp. 1–23.
15
Prager, W., and Synge, J. L., 1947, “Approximation in Elasticity Based on the Concept of Function Space,” Q. Appl. Math., 5 , pp. 241–269.
16
Zienkiewicz, O. C., and Cheung, Y. K., 1964, “The Finite Element Method for Analysis of Elastic Isotropic and Orthotropic Slabs,” Proc.-Inst. Civ. Eng., 28 , pp. 471–488.
17
Hrenikoff, A., 1941, “Solutions of Problems in Elasticity by the Framework Method,” J. Appl. Mech., A8 , pp. 169–175.
18
McHenry, D., 1943, “A Lattice Analogy for the Solution of Plane Stress Problem,” J. Inst. Civ. Eng., 21 , pp. 59–82.
19
Newmark, N. M., 1949, “Numerical Methods of Analysis in Bars, Plates and Elastic Bodies,” "Numerical Methods in Analysis in Engineering", edited by L.E.Grinter, Macmillan.
20
Argyris, J. H., 1960, "Energy Theorems and Structural Analysis", Butterworth (reprinted from vol. 26 , pp. 347–356 (1954) and vol. 27 , pp. 42–58 (1955)).
21
Turner, M. J., Clough, R. W., Martin, H. C., and Topp, L. J., 1956, “Stiffness and Deflection Analysis of Complex Structures,” J. Aero. Sci., 23 , pp. 805–823.
22
Bazeley, G. P., Cheung, Y. K., Irons, B. M., and Zienkiewicz, O. C., 1966, “Triangle Elements in Plate Bending: Conforming and Nonconforming Solutions,” "Proc. Conf. Matrix Meth. Struc. Mech.", Wright-Patterson AFB, Ohio.
23
Stummel, F., 1980, “The Limitations of the Patch Test,” Int. J. Numer. Methods Eng.  [CrossRef], 15 , pp. 177–188.
24
Irons, B., and Loikkanen, M., 1983, “An Engineers’ Defence of the Patch Test,” Int. J. Numer. Methods Eng.  [CrossRef], 19 , pp. 1391–1401.
25
Zienkiewicz, O. C., and Taylor, R. L., 2000, "The Finite Element Method", 5th ed., Vol. 1 , The Basis, Butterworth-Heinemann.
26
Hughes, T. J. R., 1987, "The Finite Element Method: Linear Static and Dynamic Finite Element Analysis", Prentice-Hall, and Dover reprint (2000).
27
Wiener, N., 1938, “The Homogeneous Chaos,” Am. J. Math., 60 , pp. 897–936.
28
Becus, G. A., and Cozzarelli, F. A., 1976, “The Random Steady State Diffusion Problem. I: Random Generalized Solutions to Laplace’s Equation,” SIAM J. Appl. Math.  [CrossRef], 31 , pp. 134–147.
29
Becus, G. A., and Cozzarelli, F. A., 1976, “The Random Steady State Diffusion Problem. II: Random Solutions to Nonlinear, Homogeneous, Steady State Diffusion Problems,” SIAM J. Appl. Math.  [CrossRef], 31 , pp. 148–158.
30
Becus, G. A., and Cozzarelli, F. A., 1976, “The Random Steady State Diffusion Problem. III: Solutions to Random Diffusion Problems by the Method of Random Successive Approximations,” SIAM J. Appl. Math.  [CrossRef], 31 , pp. 159–178.
31
Sobczyk, K., 1985, "Stochastic Wave Propagation", Elsevier (PWN-Polish Scientific Publishers, Warszawa).
32
Potapov, V. D., 1999, "Stability of Stochastic Elastic and Viscoelastic Systems", Wiley.
33
Xiu, D., and Karniadakis, G. E., 2002, “Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos,” Comput. Methods Appl. Mech. Eng.  [CrossRef], 191 , pp. 4927–4948.
34
Xiu, D., Lucor, D., Su, C. H., and Karniadakis, G. E., 2002, “Stochastic Modeling of Flow-Structure Interactions using Generalized Polynomial Chaos,” J. Fluids Eng.  [CrossRef], 124 , pp. 51–59.
35
Xiu, D., and Karniadakis, G. E., 2003, “Modeling Uncertainty in Flow Simulations Via Generalized Polynomial Chaos,” J. Comput. Phys.  [CrossRef], 187 , pp. 137–167.
36
Xiu, D., and Karniadakis, G. E., 2003, “A New Stochastic Approach to Transient Heat Conduction Modeling with Uncertainty,” Int. J. Heat Mass Transfer  [CrossRef], 46 , pp. 4681–4693.
37
Platen, E., 1999, “An Introduction to Numerical Methods for Stochastic Differential Equations,” Acta Numerica, 8 , pp. 197–246.
38
Weibull, W., 1939, “A Statistical Theory of the Strength of Materials,” published in English as Handbook No. 151 by the Royal Swedish Inst for Engrg. Research, Stockholm, Sweden.
39
Freudenthal, A. M., 1946, “The Statistical Aspect of Fatigue of Materials,” Proc. R. Soc. London, Ser. A, A-187 , pp. 416–429.
40
Freudenthal, A. M., 1950, “Statistical Concepts,” in "The Inelastic Behavior of Engineering Materials & Structures", Wiley, pp. 46–52.
41
Cornell, C. A., 1969, “A Probability-Based Structure Code,” J. Am. Concr. Inst., 66 , pp. 974–985.
42
Ang, A. H. S., 1973, “Structural Risk Analysis and Reliability-Based Design,” J. Struct. Div. ASCE, 99 , pp. 1891–1910.
43
Ang, A. H. S., and Tang, W. H., 1975, "Probability Concepts in Engineering Planning and Design, Vol. 1: Basic Principles", Wiley.
44
Fong, J. T., Rehm, R. G., and Graminski, E. L., 1977, “Weibull Statistics and a Microscopic Degradation Model of Paper,” Tappi J., 60 , pp. 156–159.
45
Fong, J. T., 1978, “Uncertainties in Fatigue Life Prediction and a Rational Definition of Safety Factors,” Nucl. Eng. Des.  [CrossRef], 51 , pp. 45–54.
46
Fong, J. T., 1979, “Statistical Aspects of Fatigue at Microscopic, Specimen, and Component Levels,” "Proc. ASTM-NBS-NSF International Symp. on Fatigue Mechanisms", Kansas City, MO, 22–24 May 1978, edited by J.T.Fong, ASTM STP 675, pp. 729–758.
47
Ang, A. H. S., and Shinozuka, M., 1979, in "Probabilistic Mechanics and Structural Reliability", Proc. ASCE Specialty Conf. in memory of Prof. A. M. Freudenthal (1906–1977), Tucson, Arizona, 10–12 January 1979, published by ASCE, New York, NY.
48
Sundararajan, C., 1995, in "Probabilistic Structural Mechanics Handbook: Theory and Industrial Applications", Chapman & Hall.
49
Hess, P. E., Bruchman, D., and Ayyub, B. M., 1998, “Uncertainties in Material and Geometric Strength Variables in Marine Structures,” Chap. 14 in "Uncertainty Modeling and Analysis in Civil Engineering", edited by B.M.Ayyub, CRC Press LLC, Boca Raton, Florida, pp. 245–288.
50
Thacker, B. H., Nicolella, D. P., Kumaresan, S., Yoganandan, N., and Pintar, F. A., 2001, “Probabilistic Finite Element Analysis of the Human Lower Cervical Spine,” Math. Models Meth. Appl. Sci., 13 , pp. 12–21.
51
Rahman, S., 2001, “Probabilistic Fracture Mechanics: J-Estimation and Finite Element Methods,” Eng. Fract. Mech.  [CrossRef], 68 , pp. 107–125.
52
Cambou, B., 1975, “Application of First-Order Uncertainty Analysis in the Finite Element Method in Linear Elasticity,” "Proc. 2nd Int. Conf. on Applications of Statistics and Probability in Soil and Structural Engrg.", Aachen, Germany, Deutsche Gesellschaft fur Grd-und Grundbau ev, Essen, FRC, pp. 67–87.
53
Handa, K., and Anderson, K., 1981, “Application of Finite Element Methods in the Statistical Analysis of Structures,” "Proc. 3rd Int. Conf. on Structural Safety and Reliability", Amsterdam, The Netherlands, Elsevier, Amsterdam, pp. 409–417.
54
Hisada, T., and Nakagiri, S., 1981, “Stochastic Finite Element Method Developed for Structural Safety and Reliability,” "Proc. 3rd International Conference on Structural Safety and Reliability", Amsterdam, Elsevier, Amsterdam, pp. 395–408.
55
Vanmarcke, E., and Grigoriu, M., 1983, “Stochastic Finite Element Analysis of Simple Beams,” J. Eng. Mech., 109 , pp. 1203–1214.
56
Der Kiureghian, A., and Ke, B. -J., 1985, “Finite-Element Based Reliability Analysis of Frame Structures,” "Proc. 4th Int. Conference on Structural Safety and Reliability", Kobe, Japan, Vol. I , published by Int. Soc. for Structural Safety and Reliability, New York, NY, pp. 395–404.
57
Liu, W. K., Belytschko, T., and Mani, A., 1986, “Probabilistic Finite Elements for Nonlinear Structural Dynamics,” Comput. Methods Appl. Mech. Eng.  [CrossRef], 56 , pp. 61–81.
58
Liu, W. K., Belytschko, T., and Mani, A., 1986, “Random Field Finite Elements,” Int. J. Numer. Methods Eng.  [CrossRef], 23 , pp. 1831–1845.
59
Benaroya, H., and Rehak, M., 1988, “Finite Element Methods in Probabilistic Structural Analysis: A Selected Review,” Appl. Mech. Rev., 41 , pp. 201–213.
60
Shinozuka, M., and Deodatis, G., 1988, “Response Variability of Stochastic Finite Element Systems,” J. Eng. Mech., 114 , pp. 499–519.
61
Shinozuka, M., and Yamazaki, F., 1988, “Stochastic Finite Element Analysis: An Introduction,” "Stochastic Structural Dynamics, Progress in Theory and Applications", edited by S.T.Ariaratnm, G.I.Schueller, and I.Eishakoff, Elsevier Appl. Sci., pp. 241–291.
62
Ghanem, R., and Dham, S., 1998, “Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media,” Transp. Porous Media  [CrossRef], 32 , pp. 239–262.
63
Matthies, H. G., and Bucher, C., 1999, “Finite Elements for Stochastic Media Problems,” Comput. Methods Appl. Mech. Eng.  [CrossRef], 168 , pp. 3–17.
64
Ghanem, R., 1999, “Ingredients for a General Purpose Stochastic Finite Elements Implementation,” Comput. Methods Appl. Mech. Eng.  [CrossRef], 168 , pp. 19–34.
65
Ostoja-Starzewski, M., and Wang, X., 1999, “Stochastic Finite Elements as a Bridge Between Random Material Microstructure and Global Response,” Comput. Methods Appl. Mech. Eng.  [CrossRef], 168 , pp. 35–49.
66
Eishakoff, I., and Ren, Y., 1999, “The Bird’s Eye View on Finite Element Method for Structures with Large Stochastic Variations,” Comput. Methods Appl. Mech. Eng.  [CrossRef], 168 , pp. 51–61.
67
Abdel-Tawab, K., and Noor, A. K., 1999, “Uncertainty Analysis of Welding Residual Stress Fields,” Comput. Methods Appl. Mech. Eng.  [CrossRef], 179 , pp. 327–344.
68
Anders, M., and Hori, M., 2001, “Three-Dimensional Stochastic Finite Element Method for Elasto-Plastic Bodies,” Int. J. Numer. Methods Eng.  [CrossRef], 51 , pp. 449–478.
69
Babuska, I., Tempone, R., and Zouraris, G. E., 2004, “Galerkin Finite Element Approximations for Stochastic Elliptic Partial Differential Equations,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.  [CrossRef], 42 , pp. 800–825.
70
Hlavacek, I., Chleboun, J., and Babuska, I., 2004, "Uncertain Input Data Problem and the Worst Scenario Method", Elsevier.
71
Yang, D., Oh, S. L., Huh, H., and Kim, Y. H., eds., 2002, “Numisheet 2002: Design Innovation Through Virtual Manufacturing,” "Proc. 5th Int. Conf. and Workshop on Numerical Simulation of 3D Sheet Forming Processes - Verification of Simulation with Experiment", 21–25 October 2002, Jeju Island, Korea, Vol. 2 , published by Korea Advanced Inst. of Science & Technology (KAIST), 373-1, Science Town, Taejon, 305–701, Korea.
72
Oberkampf, W. L., Trucano, T. G., and Hirsch, C., 2002, “Verification, Validation, and Predictive Capability in Computational Engineering and Physics,” "Proc. Workshop on Foundations for V & V in the 21st Century", 22–23 October 2002, John Hopkins Univ./Appl. Phys. Lab., Laurel, Maryland, edited by D.Pace and S.Stevenson, published by the Society for Modeling & Simulation International.
73
ISO, 1993, Guide to the Expression of Uncertainty in Measurement, prepared by ISO Technical Advisory Group 4 (TAG 4), Working Group 3 (WG 3), October 1993. ISO/TAG 4 has as its sponsors the BIPM (Bureau International des Poids et Mesures), IEC (International Electrotechnical Commission), IFCC (International Federation of Clinical Chemistry), ISO, IUPAC (Int. Union of Pure and Applied Chemistry), IUPAP (International Union of Pure and Applied Physics), and OIML (Int. Organization of Legal Metrology).
74
Taylor, B. N., and Kuyatt, C. E., 1994, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, NIST Tech. Note 1297, Sep. 1994 edition (supercedes Jan. 1993 edition), prepared under the auspices of the NIST Ad Hoc Committee on Uncertainty Statements, U.S. Gov’t Printing Office, Washington, DC.
75
Liu, H. K., and Zhang, N. F., 2002, “Bayesian Approach to Combining Results from Multiple Methods,” Proc. Amer. Stat. Assoc., pp. 158–163.
76
Box, G. E. P., Hunter, W. G., and Hunter, J. S., 1978, "Statistics for Experimenters", Wiley, New York.
77
Nelson, P. R., Coffin, M., and Copeland, K. A. F., 2003, "Introductory Statistics for Engineering Experimentation". Elsevier Academic Press.
78
Fong, J. T., Filliben, J. J., Fields, R. J., deWit, R., and Bernstein, B., 2004, “A Reference Benchmark Approach to V & V of Computer Models of High-Consequence Engineering Systems,” "Proc. NIST-DOD Workshop", 8–9 November 2004, Gaithersburg, MD, pp. 77–94. A working document of NIST Mathematical & Computational Sciences Division, available upon request at fong@nist.gov.
79
Anon, 1998, Introduction to ABAQUS—Workshop 2 and Answer 2: Linear Statics, pp. W2.1-W2.5, WA2.1-WA2.2, and WR.2, part of a Workshop Course Notes distributed by ABAQUS, Inc., 1080 Main St., Pawtucket, RI 02860-4847.
80
Pilkey, W. D., 1994, "Formulas for Stress, Strain, and Structural Matrices", Wiley, pp. 520–521.
81
Timoshenko, S., and Young, D. H., 1955, "Vibration Problems in Engineering", 3rd ed., edited by D.Van Nostrand, p. 338.
82
Anon, 2005, ABAQUS User’s Manual, Version 6.5-5. ABAQUS, Inc., 1080 Main St., Pawtucket, RI 02860-4847.
83
Anon, 2005, ANSYS User’s Manual, Release 10.0. ANSYS, Inc., 275 Technology Dr., Cannonsburg, PA 15317.
84
Anon, 2003, LS-DYNA Keyword User’s Manual, Version 970, April 2003, Livermore Software Technology Corp., Livermore, CA.
85
Anon, 2001, TrueGrid User’s Manual, Version 2.1.0, Vols. 1 & 2, Dec. 26, 2001, XYZ Scientific Applications, Inc., 1324 Concannon Blvd, Livermore, CA.
86
Timoshenko, S., and Woinowsky-Krieger, S., 1959, "Theory of Plates and Shells", 2nd ed., McGraw-Hill, pp. 422–423.
87
Marcal, P. V., 2005, MPact, MPaveMesh and MPaveModel User’s Manual, Sep. 2005, MPave Corp., 1355 Summit Ave., Cardiff, CA 92007-2429.
88
Fletcher, F. B., 1990, “Carbon and Low-Alloy Steel Plate,” in "Metals Handbook", 10th ed., Vol. 1 , ASM International, Materials Park, OH 44073, pp. 226–239.
89
Fong, J. T., Filliben, J. J., Fields, R. J., and Bernstein, B., 2002, “A Stochastic Model of the Collapse of Two Simple Steel Grillages in a Fire: Part 1. Material Property Variability of Two Steels,” a Working Document of the NIST Mathematical & Computational Sciences Division, Gaithersburg, MD 20899, 25 August 2002, available upon request, fong@nist.gov.
90
Filliben, J. J., and Heckert, N. A., 2002, DATAPLOT: A Statistical Data Anal. Software System, a public domain software released by the U.S. National Inst of Standards & Tech (NIST), Gaithersburg, MD 20899, http://www.itl.nist.gov/div898/software/dataplot.html.
91
Gelman, A., and Meng, X. -L., 2004, in "Applied Bayesian Modeling and Casual Inference From Incomplete-Data Perspectives", Wiley.
92
Schafer, J. L., 1997, "Analysis of Incomplete Multivariate Data", Chapman & Hall/CRC.

Figures

Grahic Jump Location
+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)
View Large  |  Save Figure  |  Download Slide (.ppt)
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)
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)

Grahic Jump Location
+One of three benchmark test problems of NUMISHEET (71), an international interlaboratory validation exercise for simulating three-dimensional aluminum and steel sheet-forming processes
View Large  |  Save Figure  |  Download Slide (.ppt)
One of three benchmark test problems of NUMISHEET (71), an international interlaboratory validation exercise for simulating three-dimensional aluminum and steel sheet-forming processes
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

Grahic Jump Location
+(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.
View Large  |  Save Figure  |  Download Slide (.ppt)
(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.
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.

Grahic Jump Location
+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.
View Large  |  Save Figure  |  Download Slide (.ppt)
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.
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.

Grahic Jump Location
+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.
View Large  |  Save Figure  |  Download Slide (.ppt)
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.
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.

Grahic Jump Location
+A typical result of a nonlinear (large deformation) analysis of RB-301 using MPAVE (87)
View Large  |  Save Figure  |  Download Slide (.ppt)
A typical result of a nonlinear (large deformation) analysis of RB-301 using MPAVE (87)
Figure 6

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

Grahic Jump Location
+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).
View Large  |  Save Figure  |  Download Slide (.ppt)
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).
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).

Grahic Jump Location
+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)
View Large  |  Save Figure  |  Download Slide (.ppt)
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)
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)

Grahic Jump Location
+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.
View Large  |  Save Figure  |  Download Slide (.ppt)
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.
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.

Grahic Jump Location
+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).
View Large  |  Save Figure  |  Download Slide (.ppt)
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).
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).

Grahic Jump Location
+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
View Large  |  Save Figure  |  Download Slide (.ppt)
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
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

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Data Tabulations
Structural Shear Joints> Chapter 12
Utility Function Fundamentals
Decision Making in Engineering Design> Chapter 3
Subsection NB—Class 1 Components
Companion Guide to the ASME Boiler & Pressure Vessel Code, Volume 1, Second Edition> Chapter 6
Introduction and Definitions
Handbook on Stiffness & Damping in Mechanical Design> Chapter 1
Estimating Resilient Modulus Using Neural Network Models
Intelligent Engineering Systems Through Artificial Neural Networks, Volume 17
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
This site uses cookies. By continuing to use our website, you are agreeing to our privacy policy. | Accept
Sign In