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Research Papers

Nonlinear Finite Element Analysis of Frames Under Interval Material and Load Uncertainty

[+] Author and Article Information
Rafi L. Muhanna

School of Civil and Environmental Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: rafi.muhanna@gatech.edu

Robert L. Mullen

Department of Civil and Environmental Engineering,
University of South Carolina,
Columbia, SC 29208
e-mail: rlm@cec.sc.edu

M. V. Rama Rao

Department of Civil Engineering,
Vasavi College of Engineering,
Hyderabad 500 031, India
e-mail: mv.ramarao@staff.vce.ac.in

Manuscript received October 14, 2014; final manuscript received May 6, 2015; published online October 2, 2015. Assoc. Editor: Alba Sofi.

ASME J. Risk Uncertainty Part B 1(4), 041003 (Oct 02, 2015) (8 pages) Paper No: RISK-14-1071; doi: 10.1115/1.4030609 History: Received October 14, 2014; Accepted May 08, 2015; Online October 02, 2015

The present study focuses on the development of nonlinear interval finite elements (NIFEM) for beam and frame problems. Three constitutive models have been used in the present study, viz., bilinear, Ramberg–Osgood, and cubic models, to illustrate the development of NIFEM. An interval finite element method (IFEM) has been developed to handle load, material, and geometric uncertainties that are introduced in a form of interval numbers defined by their lower and upper bounds. However, the scope of the previous methods was limited to linear problems. The present work introduces an IFEM formulation for problems involving material nonlinearity under interval material parameters and loads. The algorithm is based on the previously developed high-accuracy interval solutions. An iterative method that generates successive approximations to the secant stiffness is introduced. Examples are presented to illustrate the behavior of the formulation. It is shown that bounding the response of nonlinear structures for a large number of load combinations under uncertain yield stress can be computed at a reasonable computational cost.

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References

Koyluoglu, H. U., Cakmak, A. S., and Nielsen, S. R. K., 1995, “Interval Algebra to Deal With Pattern Loading and Structural Uncertainty,” J. Eng. Mech.-ASCE, 121(11), pp. 1149–1157. [CrossRef]
Muhanna, R. L., and Mullen, R. L., 1995, “Development of Interval Based Methods for Fuzziness in Continuum Mechanics,” Proceedings of ISUMA-NAFIPS’95, Sept. 17–20, IEEE Computer Society Press, Los Alamos, NM, pp. 145–150.
Nakagiri, S., and Yoshikawa, N., 1996, “Finite Element Interval Estimation by Convex Model,” Proceedings of 7th ASCE EMD/STD Joint Specialty, Conference on Probabilistic Mechanics and Structural Reliability, Aug. 7–9, WPI, MA.
Rao, S. S., and Sawyer, P., 1995, “Fuzzy Finite Element Approach for Analysis of Imprecisely Defined Systems,” AIAA J., 33(12), pp. 2364–2370. [CrossRef]
Rao, S. S., and Berke, L., 1997, “Analysis of Uncertain Structural Systems Using Interval Analysis,” AIAA J., 35(4), pp. 727–735. [CrossRef]
Rao, S. S., and Chen, Li., 1998, “Numerical Solution of Fuzzy Linear Equations in Engineering Analysis,” Int. J. Numer. Methods Eng., 43(3), pp. 391–408. [CrossRef]
Muhanna, R. L., and Mullen, R. L., 2001, “Uncertainty in Mechanics Problems-Interval-Based Approach,” J. Eng. Mech.-ASCE, 127(6), pp. 557–566. [CrossRef]
Neumaier, A., and Pownuk, A., 2007, “Linear Systems With Large Uncertainties, With Applications to Truss Structures,” Reliable Comput., 13(2), pp. 149–172. [CrossRef]
Impollonia, N., and Muscolino, G., 2011, “Interval Analysis of Structures With Uncertain-but-Bounded Axial Stiffness,” Comput. Methods Appl. Mech. Eng., 200(21), pp. 1945–1962. [CrossRef]
Guo S. X., and Lu Z. Z., 2001, “Interval Arithmetic and Static Interval Finite Element Method,” Appl. Math. Mech., 20(12), pp. 1390–1396. [CrossRef]
Qiu, Z. P., Wang, X. J., and Chen, J. Y., 2006, “Exact Bounds for the Static Response Set of Structures With Uncertain-but-Bounded Parameters,” Int. J. Solids Struct., 43, pp. 6574–6593. [CrossRef]
Verhaeghe, W., Munck, M. D., Desmet, W., Vandepitte, D., and Moens, D., 2010, “A Fuzzy Finite Element Analysis Technique for Structural Static Analysis Based on Interval Fields,” Proceedings of the 4th International Workshop on Reliable Engineering Computing, M. Beer, R. L. Muhanna, and R. L. Mullen, eds., Singapore, pp. 117–128.
Muhanna, R. L., Mullen, R. L., and Rama Rao, M. V., 2012, “Nonlinear Interval Finite Element for Structural Mechanics Problems,” Proceedings of the International Conference on Reliable Engineering Computing “Practical Applications and Practical Challenges,” Brno, Czech Republic, Jun. 13–15.
Mallet, R. H., and Maracal, P. V., 1968, “Finite Element Analysis of Nonlinear Structures,” Proc. ASCE, J. Struct. Div., 94(ST9), pp. 2081–2185.
Oden, J. T., 1967, “Numerical Formulation of Nonlinear Elasticity Problems,” J. Struct. Div., 93(3), pp. 235–356.
Oden, J. T., 1969, “Finite Element Applications in Non-Linear Structural Analysis,” Proceedings of the Conference on Finite Element Methods, Vanderbilt University, Tennessee.
Zienkiewicz, O. C., Valliapan, S., and King, I. P., 1969, “Elasto-Plastic Solutions of Engineering Problems. Initial Stress, Finite Element Approach,” Int. J. Num. Methods Eng., 1(1), pp. 75–100. [CrossRef]
Haisler, W. E., Stricklin, J. E., and Stebblins, F. J., 1971, “Development and Evaluation of Solution Procedures for Geometrically Non-Linear Structural Analysis by the Discrete Stiffness Method,” Proceedings of the AIAA-ASME 12th Structure, Structural Dynamics Conference, Anaheim, CA.
Sabir, A. B., and Lock, A. C., 1972, “The Application of Finite Elements to the Large-Deflection Geometrically Nonlinear Behavior of Cylindrical Shells,” Proceedings of International Conference on Variational Methods in Engineering, C. A. Brebbia, and H. Tottenham, eds., Southampton University Press, Southampton, UK, pp. 7–67.
Wright, E. W., and Gaylord, E. H., 1968, “Analysis of Unbraced Multistory Steel Rigid Frames,” Int. J. Struct. Div.-ASCE, 94(5), pp. 1143–1163.
Bergan, P. G., and Soreide, T. H., 1978, “Solution of Large Displacement and Instability Problems Using the Current Stiffness Parameter,” Proceedings of the Finite Element in Nonlinear Mechanics, P. G. Bergan, P. K. Larsen, H. Pettersson, A. Samuelsson, T. H. Søreide, and N. E. Wiberg, eds., Tapir Press, Trondheim, Norway, pp. 647–669.
Bergan, P.G., Horrigmoe, G., Krakeland, B., and Soreide, T. H., 1978, “Solution Techniques for Nonlinear Finite Element Problems,” Int. J. Numer. Methods Eng., 12(11), pp. 1677–1696. [CrossRef]
Fox, R. L., and Stanton, E. L., 1968, “Developments in Structural Analysis by Direct Energy Minimization,” AIAA J., 6(6), pp. I036–I042. [CrossRef]
Luenberger, D. G., 1984, Linear and Nonlinear Programming, 2nd ed., Addison-Wesley, Reading, MA.
Matthies, H. and Strang, G., 1979, “The Solution of Nonlinear Finite Element Equations,” Int. J. Numer. Methods Eng., 14(11), pp. 1613–1626. [CrossRef]
Crisfield, M. A., 1991, Non-linear Finite Element Analysis of Solids and Structures, Wiley, Chichester, UK.
Wriggers, P., 2008, Nonlinear Finite Element Methods, Springer-Verlag, Berlin, Heidelberg.
Riks, E., 1972, “The Application of Newton’s Method to the Problem of Elastic Stability,” ASME J. Appl. Mech., 39(4), pp. 1060–1065. 10.1115/1.3422829
Wempner, G. A., 1971, “Discrete Approximation Related to Nonlinear Theories of Solids,” Int. J. Solids Struct., 7, pp. 1581–1599. [CrossRef]
Keller, H. B., 1977, “Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems,” Application of Bifurcation Theory, P. Rabinowitz, ed., pp. 359–384, Academic Press, New York.
Ramm, E., 1981, “Strategies for Tracing the Nonlinear Response Near Limit Points,” Nonlinear Finite Element Analysis in Structural Mechanics, W. Wunderlich, E. Stein, and K. J. Bathe, eds., Springer, Berlin, Heidelberg, New York.
Crisfield, M. A., 1981, “A Fast Incremental/Iterative Solution Procedure that Handles Snap Through,” Comput. Struct., 13(1–3), pp. 55–62. [CrossRef]
Schweizerhof, K., and Wriggers, P., 1986, “Consistent Linearization for Path Following Methods in Nonlinear Fe-Analysis,” Comput. Methods Appl. Mech. Eng., 59(3), pp. 261–279. [CrossRef]
Wagner, W., 1991, “Zur Behandlung von Stabilit¨atsproblemen mit der Methode der Finiten Elemente,” , Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universität Hannover.
Riks, E., 1984, “Some Computational Aspects of Stability Analysis of Nonlinear Structures,” Comput. Methods Appl. Mech. Eng., 47(3), pp. 219–260. [CrossRef]
Wagner, W., and Wagner, P., 1988, “A Simple Method for the Calculation of Secondary Branches,” Eng. Comput., 5(2), pp. 103–109. [CrossRef]
Crisfield, M. A., and Shi, J., 1991, “A Review of Solution Procedures and Path-Following Techniques in Relation to the Non-Linear Finite Element Analysis of Structures,” Computational Methods in Nonlinear Mechanics, P. Wriggers, and W. Wagner, eds., Springer, Berlin.
Lin, Y., Gwaltney, C. R., and Stadtherr, M. A., 2006, “Reliable Modeling and Optimization for Chemical Engineering Applications: Interval Analysis Approach,” Reliable Comput., 12(6), pp. 427–450. [CrossRef]
Enszer, J. A., Lin, Y., Ferson, S., Corliss, G. F., and Stadtherr, M. A., 2011, “Probability Bounds Analysis for Nonlinear Dynamic Process Models,” AIChE J., 57(2), pp. 404–422. [CrossRef]
Rama Rao, M. V., Mullen, R. L., and Muhanna, R. L., 2011, “A New Interval Finite Element Formulation With the Same Accuracy in Primary and Derived Variables,” Int. J. Reliab. Saf., 5(3/4), pp. 336–357. [CrossRef]
Muhanna, R. L., Zhang, H., and Mullen, R. L., 2007, “Interval Finite Element as a Basis for Generalized Models of Uncertainty in Engineering Mechanics,” Reliable Comput., 13(2), pp. 173–194. [CrossRef]
Zhang, H., 2005, “Nondeterministic Linear Static Finite Element Analysis: An Interval Approach,” Ph.D. Dissertation, Georgia Institute of Technology, School of Civil and Environmental Engineering.
Moore, R. E., 1966, Interval Analysis, Prentice-Hall, Elingwood Cliffs, NJ.
Neumaier, A., 1990, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge.
Sun Microsystems, 2002, “Interval Arithmetic in High Performance Technical Computing,” Sun microsystems, Santa Clara (A WhitePaper).
Knüppel, O., 1994, “PROFIL/BIAS—A Fast Interval Library,” Computing, 53, pp. 277–287. [CrossRef]
Rump., S. M., 1999, “INTLAB—INTerval LABoratory,” Developments in Reliable Computing, T. Csendes, ed., Kluwer Academic Publishers, Dordrecht, pp. 77–104.
Dessombz, O., Thouverez, F., Laîné, J.-P., and Jézéquel, L., 2001, “Analysis of Mechanical Systems Using Interval Computations Applied to Finite Elements Methods,” J. Sound. Vib., 238(5), pp. 949–968 (Engineering Computations, 5, 103–110 (1988)). [CrossRef]
Bauchau, O. A., and Craig, J. I., 2009, Structural Analysis With Applications to Aerospace Structures, Springer, New York.
Mullen, R. L., and Muhanna, R. L., 1999, “Formulation of Fuzzy Finite Element Methods for Mechanics Problems,” Comput.-Aided Civ. Infrastruct. Eng., 14(2), pp. 107–117. [CrossRef]
Cook, R. D., Malkus, D. S., Plesha, M. E., and Witt, R. J., 2002, Concepts and Applications of Finite Element Analysis, 4th ed., Wiley & Sons, Singapore.

Figures

Grahic Jump Location
Fig. 1

Three nonlinear models: bilinear, cubic, and Ramberg–Osgood

Grahic Jump Location
Fig. 3

Cantilever beam-deflection of free end at various levels of load uncertainty

Grahic Jump Location
Fig. 4

Four-bay one-story frame

Grahic Jump Location
Fig. 5

Four-bay three-story frame

Grahic Jump Location
Fig. 6

Four-bay three-story frame loading pattern

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