New Multidimensional Visualization Technique for Limit-State Surfaces in Nonlinear Finite-Element Reliability Analysis
Publication: Journal of Engineering Mechanics
Volume 136, Issue 11
Abstract
Structural reliability problems involving the use of advanced finite-element models of real-world structures are usually defined by limit-states expressed as functions (referred to as limit-state functions) of basic random variables used to characterize the pertinent sources of uncertainty. These limit-state functions define hyper-surfaces (referred to as limit-state surfaces) in the high-dimensional spaces of the basic random variables. The hyper-surface topology is of paramount interest, particularly in the failure domain regions with highest probability density. In fact, classical asymptotic reliability methods, such as the first- and second-order reliability method (FORM and SORM), are based on geometric approximations of the limit-state surfaces near the so-called design point(s) (DP). This paper presents a new efficient tool, the multidimensional visualization in the principal planes (MVPP) method, to study the topology of high-dimensional nonlinear limit-state surfaces (LSSs) near their DPs. The MVPP method allows the visualization, in particularly meaningful two-dimensional subspaces denoted as principal planes, of actual high-dimensional nonlinear limit-state surfaces that arise in both time-invariant and time-variant (mean out-crossing rate computation) structural reliability problems. The MVPP method provides, at a computational cost comparable with SORM, valuable insight into the suitability of FORM/SORM approximations of the failure probability for various reliability problems. Several application examples are presented to illustrate the developed MVPP methodology and the value of the information provided by visualization of the LSS.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The writers gratefully acknowledge support of this research by (1) the National Science Foundation under Grant No. NSFCMS-0010112, (2) the Pacific Earthquake Engineering Research Center (PEER) Center’s Transportation Systems Research Program under under Award No. UNSPECIFIED00006493, and (3) the Louisiana Board of Regents through the Pilot Funding for New Research Program of the National Science Foundation Experimental Program to Stimulate Competitive Research under Award No. UNSPECIFIEDNSF(2008)-PFUND-86. Any opinions, findings, conclusions or recommendations expressed in this publication are those of the writers and do not necessarily reflect the views of the sponsors.
References
Ang, G. L., Ang, A. H.-S., and Tang, W. H. (1992). “Optimal importance sampling density estimator.” J. Eng. Mech., 118(6), 1146–1163.
Au, S. K., and Beck, J. L. (2001a). “Estimation of small failure probabilities in high dimensions by subset simulation.” Probab. Eng. Mech., 16(4), 263–277.
Au, S. K., and Beck, J. L. (2001b). “First excursion probabilities for linear systems by very efficient importance sampling.” Probab. Eng. Mech., 16(3), 193–207.
Au, S. K., and Beck, J. L. (2003). “Subset simulation and its application to seismic risk based on dynamic analysis.” J. Eng. Mech., 129(8), 901–917.
Au, S. K., Papadimitriou, C., and Beck, J. L. (1999). “Reliability of uncertain dynamical systems with multiple design points.” Struct. Safety, 21(2), 113–133.
Barbato, M. (2007). “Finite element response sensitivity, probabilistic response and reliability analyses of structural systems with applications to earthquake engineering.” Ph.D. thesis, Univ. of California at San Diego, La Jolla, Calif.
Barbato, M., and Conte, J. P. (2005). “Finite element response sensitivity analysis: A comparison between force-based and displacement-based frame element models.” Comput. Methods Appl. Mech. Eng., 194, (12–16), 1479–1512.
Barbato, M., and Conte, J. P. (2006). “Finite element structural response sensitivity and reliability analyses using smooth versus non-smooth material constitutive models.” Int. J. Reliab., 1(1/2), 3–39.
Barbato, M., Zona, A., and Conte, J. P. (2007). “Finite element response sensitivity analysis using three-field mixed formulation: General theory and application to frame structures.” Int. J. Numer. Methods Eng., 69(1), 114–161.
Breitung, K. (1984). “Asymptotic approximations for multinormal integrals.” J. Engrg. Mech. Div., 110(3), 357–366.
Bucher, C. G. (1988). “Adaptive importance sampling—An iterative fast Monte Carlo procedure.” Struct. Safety, 5(2), 119–126.
Conte, J. P., Barbato, M., and Gu, Q. (2008). “Finite element response sensitivity, probabilistic response and reliability analyses.” Chap. 2, Computational structural dynamics and earthquake engineering, Vol. 2, M. Papadrakakis, D. C. Champis, Y. Tsompanakis, N. D. Lagaros, eds., Structures & Infrastructures Series, Taylor & Francis, London.
Conte, J. P., Barbato, M., and Spacone, E. (2004). “Finite element response sensitivity analysis using force-based frame models.” Int. J. Numer. Methods Eng., 59(13), 1781–1820.
Conte, J. P., Vijalapura, P. K., and Meghella, M. (2003). “Consistent finite-element response sensitivity analysis.” J. Eng. Mech., 129, 1380–1393.
Der Kiureghian, A. (2000). “The geometry of random vibrations and solutions by FORM and SORM.” Probab. Eng. Mech., 15(1), 81–90.
Der Kiureghian, A., and De Stefano, M. (1991). “Efficient algorithms for second-order reliability analysis.” J. Eng. Mech., 117(12), 2904–2923.
Der Kiureghian, A., and Ke, J. -B. (1988). “The stochastic finite element method in structural reliability.” Probab. Eng. Mech., 3(2), 83–91.
Der Kiureghian, A., Lin, H. -Z., and Hwang, S. -J. (1987). “Second-order reliability approximations.” J. Engrg. Mech. Div., 113(EM8), 1208–1225.
Ditlevsen, O., and Madsen, H. O. (1996). Structural reliability methods, Wiley, New York.
Drenick, R. F. (1970). “Model-free design of aseismic structures.” J. Engrg. Mech. Div., 96(EM4), 483–493.
Filippou, F. C., and Constantinides, M. (2004). “FEDEASLab getting started guide and simulation examples.” Rep. No. NEESgrid-2004-22, NEESgrid SI, Davis, Calif., ⟨http://fedeaslab. berkeley.edu/⟩ (Aug. 2010).
Gill, P. E., Murray, W., and Saunders, M. A. (2002). “SNOPT: An SQP algorithm for large-scale constrained optimization.” SIAM J. Optim., 12, 979–1006.
Gill, P. E., Murray, W., and Wright, M. H. (1981). Practical optimization, Academic, New York.
Gu, Q. (2008). “Finite element response sensitivity and reliability analysis of soil-foundation-structure-interaction (SFSI) systems.” Ph.D. thesis, Univ. of California at San Diego, La Jolla, Calif.
Gu, Q., Barbato, M., and Conte, J. P. (2009a). “Handling of constraints in finite element response sensitivity analysis.” J. Eng. Mech., 135(12), 1427–1438.
Gu, Q., Conte, J. P., Elgamal, A., and Yang, Z. (2009b). “Finite element response sensitivity analysis of multi-yield-surface plasticity model by direct differentiation method.” Comput. Methods Appl. Mech. Eng., 198(30–32), 2272–2285.
Hagen, O., and Tvedt, L. (1991). “Vector process out-crossing as parallel system sensitivity measure.” J. Eng. Mech., 117(10), 2201–2220.
Haukaas, T. (2001). “FERUM (finite element reliability using Matlab).” User’s Guide, ⟨http://www.ce.berkeley.edu/projects/ferum⟩ (Aug. 2010).
Haukaas, T., and Der Kiureghian, A. (2004). “Finite element reliability and sensitivity methods for performance-based engineering.” Pacific Earthquake Engineering Research Center Rep. No. PEER 2003/14, Univ. of California, Berkeley, Calif.
Hohenbichler, M., and Rackwitz, R. (1986). “Sensitivity and importance measures in structural reliability.” Civ. Eng. Syst., 3(4), 203–209.
Hohenbichler, M., and Rackwitz, R. (1988). “Improvement of second-order reliability estimates by importance sampling.” J. Engrg. Mech. Div., 114(12), 2195–2199.
Kleiber, M., Antunez, H., Hien, T. D., and Kowalczyk, P. (1997). Parameter sensitivity in nonlinear mechanics: Theory and finite element computation, Wiley, New York.
Kwon, M., and Spacone, E. (2002). “Three-dimensional finite element analyses of reinforced concrete columns.” Comput. Struct., 80, 199–212.
Liu, P. -L., and Der Kiureghian, A. (1991). “Optimization algorithms for structural reliability.” Struct. Safety, 9(3), 161–177.
Lutes, L. D., and Sarkani, S. (2004). Random vibrations, Elsevier, Burlington, Mass.
MathWorks. (1997). “Matlab—High performance numeric computation and visualization software.” User’s guide, Natick, Mass.
Mazzoni, S., McKenna, F., Scott M. H., and Fenves, G. L. (2007). OpenSees command language manual, Pacific Earthquake Engineering Center, Univ. of California, Berkeley, Berkeley, Calif., ⟨http://opensees.berkeley.edu/OpenSees/manuals/usermanual/OpenSeesCommandLanguageManual.pdf⟩ (Aug. 2010).
Melchers, R. E. (1989). “Importance sampling in structural systems.” Struct. Safety, 6(1), 3–10.
Menegotto, M., and Pinto, P. E. (1973). “Method for analysis of cyclically loaded reinforced concrete plane frames including changes in geometry and non-elastic behavior of elements under combined normal force and bending.” Proc., IABSE Symp. on “Resistance and Ultimate Deformability of Structures Acted on by Well-Defined Repeated Loads”, IABSE, Zurich, Switzerland.
Mirza, S. A., and MacGregor, J. G. (1979). “Variability of mechanical properties of reinforcing bars.” J. Struct. Div., 105(5), 921–937.
Mirza, S. A., MacGregor, J. G., and Hatzinikolas, M. (1979). “Statistical descriptions of strength of concrete.” J. Struct. Div., 105(6), 1021–1037.
Schueller, G. I. (2008). “Computational stochastic dynamics—Some lessons learned.” Computational structural dynamics and earthquake engineering, M. Papadrakakis, D. C. Charmpis, D. Nikos, N. D. Lagaros, and Y. Tsompanakis, eds., Taylor & Francis, London, 3–20.
Schueller, G. I., Pradlwarter, H. J., and Koutsourelakis, P. S. (2004). “A critical appraisal of reliability estimation procedures for high dimensions.” Probab. Eng. Mech., 19(4), 463–474.
Stoer, J., and Bulirsch, R. (2002). Introduction to numerical analysis, 3rd Ed., Springer, Berlin.
Zhang, Y., and Der Kiureghian, A. (1993). “Dynamic response sensitivity of inelastic structures.” Comput. Methods Appl. Mech. Eng., 108(1–2), 23–36.
Zona, A., Barbato, M., and Conte, J. P. (2004). “Finite element response sensitivity analysis of steel-concrete composite structures.” Rep. No. SSRP-04/02, Dept. of Structural Engineering, Univ. of California at San Diego, La Jolla, Calif.
Zona, A., Barbato, M., and Conte, J. P. (2005). “Finite element response sensitivity analysis of steel-concrete composite beams with deformable shear connection.” J. Eng. Mech., 131(11), 1126–1139.
Zona, A., Barbato, M., and Conte, J. P. (2006). “Finite element response sensitivity analysis of continuous steel-concrete composite girders.” Steel Compos. Struct., 6(3), 183–202.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 136 • Issue 11 • November 2010
Pages: 1390 - 1400
Copyright
© 2010 ASCE.
History
Received: Nov 24, 2009
Accepted: Apr 19, 2010
Published online: Apr 21, 2010
Published in print: Nov 2010
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.