Keywords: re-analysis, dynamic response, nonlinear systems.

In the framework of computational mechanics re-analysis techniques provide an efficient strategy in order to evaluate the static and dynamic response of modified structures while reducing the computational effort. Specifically, by using the response of a reference structure, the modified structure is analyzed without solving the pertinent set of implicit equations governing either the static or dynamic problem. Clearly, the time-saving is evident in particular when repeated analysis of slightly modified multi-degree-of-freedom systems have to be performed, for example optimization problems and Monte Carlo simulations.

The re-analysis techniques can be defined as topological or non-topological if the modifications involved produce a change or not in the number of the degrees of freedom of the structure. Also, re-analysis techniques refer as static or dynamic respectively if the static or dynamic response has to be reanalyzed. A review of static re-analysis techniques can be found in [1]. Regarding the dynamic re-analysis it is possible to distinguish three different levels that involve the evaluation of the eigenvalues, of the eigensystem or the time history of the modified response. In [2] most recent contributions in this field can be retrieved.

Recently, re-analysis techniques have been used also to cope with the analysis of nonlinear systems. Static re-analysis of nonlinear systems has been dealt with principally exploiting iteratively methods developed for the linear case [3,4,5]. Regarding the dynamic re-analysis, in reference [6] it has been shown that reduction methods along with static re-analysis techniques can be efficiently applied to both geometric and mechanical nonlinear problems. According to the modal analysis, equation of motion of nonlinear systems can be projected in a reduced subspace. It follows that for each single step, the pertinent eigenproblem has to be solved. In reference [7] the computational effort involved in the repeated evaluation of the eigenproperties are reduced exploiting the combined approximation method.

It is noted that that integration schemes based on the transition matrix afford more accurate results with respect to Newmark's method [8], used in references [6,7] to integrate the equations governing the nonlinear problem. In this paper the dynamic response of nonlinear systems is determined extending the method proposed in [9] to nonlinear dynamic problems. Specifically, the dynamic response of elasto-plastic systems is evaluated assuming the changes in mechanical properties of the system due to the nonlinearity as dynamic modifications. It follows that the modified transition matrix and the related operators are obtained at each step through the knowledge of the reference transition matrix. Remarkably, via the proposed approach the repeated solution of the modified eigenproblem is avoided, as shown in [10], for the case of geometric nonlinearities. Several numerical applications show the accuracy and the efficiency of the proposed procedure

1

Abu Kasim A.M., Topping B.H.V. "Static re-analysis: a review", J. Struct. Engng, ASCE, 1987; 113(6), 1029-1045.

2

Muscolino G. and Cacciola P., "Re-Analysis Techniques in Structural Dynamics", Progress in Computational Structures Technology, Ed. by B.H.V. Topping and C.A. Mota Soares, pp. 31-58. Saxe-Coburg Publications, Stirling, Scotland, 2004. doi:10.4203/csets.11.2

3

Makode P.V., Corotis R.B., Ramirez M.R., "Nonlinear analysis of frame structures by Pseudodistorsions", J. of Struct. Engrg. (ASCE), 125(11), 1309-1317, 1999 doi:10.1061/(ASCE)0733-9445(1999)125:11(1309)

4

Deng L.Z. and Ghosn M., "Pseudoforce method for nonlinear analysis and reanalysis of structural systems", J. of Struct. Engrg. (ASCE), 127(5), 570-578, 2001. doi:10.1061/(ASCE)0733-9445(2001)127:5(570)

5

Hurtado J.E., "Re-analysis of linear and nonlinear structures using iterated Shanks transformation", Comp. Meth. Appl. Mech. Engrg, 191, 4215-4229, 2002. doi:10.1016/S0045-7825(02)00373-0

6

Leu L.J. and Tsou C.H., "Applications of a reduction method for reanalysis to nonlinear dynamic analysis of framed structures", Computational Mechanics, 26, 497-505, 2000. doi:10.1007/s004660000200

7

Kirsch U., Bogomolni M., Sheinman I., "Nonlinear dynamic reanalysis of structures by combined approximations", Comp. Meth. Appl. Mech. Engrg. 195, 4420, 4432, 2006. doi:10.1016/j.cma.2005.09.013

8

Muscolino G.. "Dynamically Modified Linear Structures: Deterministic and Stochastic Response", J. of Eng. Mech. (ASCE) 1996; 122: 1044-1051. doi:10.1061/(ASCE)0733-9399(1996)122:11(1044)

9

Cacciola P., Impollonia N., Muscolino G., "A dynamic reanalysis technique for general structural modifications under deterministic or stochastic input", Computers and Structures, 83, 1076-1085, 2005. doi:10.1016/j.compstruc.2004.11.017

10

Cacciola P., Muscolino G., Sofi A., "Dynamic analysis of non-linear structures by modal superposition approach", IMAC-XVIII Conference on Strucutural Dynamic, San Antonio, Texas 7-10 February, 2000.

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