Dynamic Analysis of Gap Closing and Contact in the Mixed Lagrangian Framework: Toward Progressive Collapse Prediction
Publication: Journal of Engineering Mechanics
Volume 136, Issue 8
Abstract
Previous research has shown many advantages of the mixed Lagrangian formulation (MLF) for the solution of dynamic problems. In particular, it has shown a very stable and robust behavior with respect to the time step size required for convergence, even in cases where plasticity and fracture were considered. This paper presents another step toward enabling the prediction of progressive collapse of structures using MLF. A new gap element is added to the framework by formulating an additional component in the Lagrangian function. It is shown that by carefully formulating the new component, the optimization problem to be solved in each time step of the MLF algorithm retains its form which is quadratic in the cases considered. Hence, a unified formulation is attained for all stages of the analysis whether contact forces are present or not. After presenting details of the formulation, the proposed method is used for the solution of two examples. These examples illustrate that relatively large time steps can be considered even for contact problems. Furthermore, the reasons for this capability of the algorithm are discussed in the paper.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
Support for the work described in this paper was provided in part by the Multidisciplinary Center for Earthquake Engineering Research under a cooperative agreement from the National Science Foundation (Grant No. NSFEEC-9701471) as well as by the Kajima Corporation through the CUREE-Kajima joint program Phases VI and VII. The writer gratefully acknowledges this support.
References
Alart, P., and Curnier, A. (1991). “A mixed finite-element formulation for frictional contact problems prone to Newton-like solution methods.” Comput. Methods Appl. Mech. Eng., 92(3), 353–375.
Bazaraa, M. S., Sherali, H. D., and Shetty, C. M. (2006). Nonlinear programming: Theory and applications, Wiley, New York.
Consolazio, G. R., and Cowan, D. R. (2005). “Numerically efficient dynamic analysis of barge collisions with bridge piers.” J. Struct. Eng., 131(8), 1256–1266.
Cuomo, M., and Ventura, G. (2000). “A complementary energy formulation of no tension masonry-like solids.” Comput. Methods Appl. Mech. Eng., 189, 313–339.
Duvaut, G., and Lions, J. L. (1976). Inequalities in mechanics and physics, Springer, Berlin.
Filiatrault, A., Wagner, P., and Cherry, S. (1995). “Analytical prediction of experimental building pounding.” Earthquake Eng. Struct. Dyn., 24(8), 1131–1154.
Glowinski, R., Vidrascu, M., and LeTallec, P. (1985). “Augmented Lagrangian techniques for solving frictionless contact problems in finite elasticity.” Proc., U.S.-Europe Symp. on Finite-Element Methods for Non-Linear Problems, The Norwegian Institute of Technology, Trondheim, Norway, 745–758.
Johnson, K. L. (1985). Contact mechanics, Cambridge University Press, Cambridge, U.K.
Kikuchi, N., and Oden, J. T. (1988). Contact problems in elasticity: A study of variational inequalities and finite-element methods, SIAM Publications, Philadelphia.
Komodromos, P., Polycarpou, P. C., Papaloizou, L., and Phocas, M. C. (2007). “Response of seismically isolated buildings considering poundings.” Earthquake Eng. Struct. Dyn., 36(12), 1605–1622.
Laursen, T. A., and Simo, J. C. (1993). “A continuum-based finite-element formulation for the implicit solution of multibody, large-deformation frictional contact problems.” Int. J. Numer. Methods Eng., 36(20), 3451–3485.
Lavan, O., Sivaselvan, M. V., Reinhorn, A. M., and Dargush, G. F. (2009). “Progressive collapse analysis through strength degradation and fracture in the mixed Lagrangian framework.” Earthquake Eng. Struct. Dyn., 38(13), 1483–1504.
Papadrakakis, M., Mouzakis, H., Plevris, N., and Bitzarakis, S. (1991). “A Lagrange multiplier solution method for pounding of buildings during earthquakes.” Earthquake Eng. Struct. Dyn., 20(11), 981–998.
Reinhorn, A. M., Sivaselvan, M. V., Dargush, G. F., and Lavan, O. (2008). “Chapter 21: Mixed Lagrangian formulation in analysis of collapse of structures.” Progress in computational structural dynamics and earthquake engineering, M. Papadrakakis, D. C. Charmpis, N. D. Lagaros, and Y. Tsompanakis, Taylor and Francis, Leiden, The Netherlands, 325–337.
Simo, J. C., and Laursen, T. A. (1992). “An augmented Lagrangian treatment of contact problems involving friction.” Comput. Struc., 42(1), 97–116.
Simo, J. C., Wriggers, P., and Taylor, R. L. (1985). “A perturbed Lagrangian formulation for the finite-element solution of contact problems.” Comput. Methods Appl. Mech. Eng., 50(2), 163–180.
Simulia. (2007). ABAQUS theory manual, Providence, R.I.
Sivaselvan, M. V., et al. (2009). “Numerical collapse simulation of large-scale structural systems using an optimization-based algorithm.” Earthquake Eng. Struct. Dyn., 38(5), 655–677.
Sivaselvan, M. V., and Reinhorn, A. M. (2006). “Lagrangian approach to structural collapse simulation.” J. Struct. Eng., 132(8), 795–805.
Tsai, H. C. (1997). “Dynamic analysis of base-isolated shear beams bumping against stops.” Earthquake Eng. Struct. Dyn., 26(5), 515–528.
Villaverde, R. (1997). “Seismic design of secondary structures: State of the art.” J. Struct. Eng., 123(8), 1011–1019.
Zhu, P., Abe, M., and Fujino, Y. (2002). “Modelling three-dimensional non-linear seismic performance of elevated bridges with emphasis on pounding of girders.” Earthquake Eng. Struct. Dyn., 31(11), 1891–1913.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 136 • Issue 8 • August 2010
Pages: 979 - 986
Copyright
© 2010 ASCE.
History
Received: Feb 5, 2009
Accepted: Feb 4, 2010
Published online: Jul 15, 2010
Published in print: Aug 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.