Keywords: approximate reanalysis, linear reanalysis, eigenproblem reanalysis, dynamic analysis, nonlinear analysis.

The cost of structural analysis and its practical feasibility depend to a considerable degree on the algorithms available for the solution of the resulting equations. The time required for solving the equilibrium equations can be a high percentage of the total solution time, particularly in nonlinear and dynamic analysis. In nonlinear analysis, a set of updated linear equations must be solved repeatedly during the solution process. Similarly, in dynamic analysis by mode superposition, it is necessary to solve the eigenproblem, which often involves repeated solution of a set of linear equations.

Reanalysis methods are intended to analyze efficiently structures that are modified due to changes in the design. The object is to evaluate the structural response for such changes without solving the complete set of modified simultaneous equations. The solution procedures usually use the original response of the structure. Multiple repeated analyses are needed in various problems, including structural optimization, structural damage analysis, and other applications related to probabilistic analysis, controlled structures, smart structures, adaptive structures, etc.

The object of this study is to show how an effective reanalysis approach, developed originally for linear reanalysis, can improve the solution efficiency of nonlinear and dynamic analysis problems. In such problems the analysis equations are modified due to the nonlinear behavior, the time dependent loads, or both. Nonlinear analysis is usually carried out in an iterative process. The solution process can be performed by different methods but, in general, a set of updated linear equations must be solved repeatedly. Similarly, many of the vibration (or eigenproblem) solution techniques are based on matrix iteration methods. To calculate the mode shapes it is necessary to solve repeatedly a set of updated linear equations.

Reanalysis methods are usually based on approximations. The common approximations can be divided into local, global and combined approximations. Local approximations are very efficient but they are effective only for small changes in the design variables. Global approximations are more accurate for large changes in the structure but they may require much computational effort. The Combined Approximations (CA) approach [1,2,3] developed recently is based on the integration of several concepts and methods, including series expansion, reduced basis, matrix factorization and Gram-Schmidt orthogonalization. The advantage is that efficient local approximations (series expansion) and accurate global approximations (the reduced basis method) are combined to achieve an effective solution procedure.

The main advantages of the CA approach, which have been studied in terms of several criteria, include:

a.

Generality. Various analysis models, different types of structures, and all types of changes in the design may be considered.

b.

Accuracy of the results. Accurate approximations are achieved for significant changes in the design of large structures.

c.

Efficiency. Similar to local approximations, the calculations are based on results of a single exact analysis. The number of algebraic operations and the total CPU effort are usually much smaller than those needed to carry out complete analysis.

d.

Flexibility. The efficiency of the calculations and the accuracy of the results can be controlled by the amount of information considered. Better accuracy can always be achieved at the expense of more computational effort.

e.

Ease of implementation. The solution steps are straightforward and the approach can be readily used with general finite element systems.

In the paper, formulation of linear reanalysis problems and solution by the CA approach are first described. Nonlinear analysis by the CA method, including both geometrical nonlinearity and material nonlinearity, are then presented. Dynamic analysis by the CA method, including linear and nonlinear behavior is discussed, and typical numerical examples are demonstrated. It is shown that the method is a powerful tool that can be used in various structural analysis problems.

1

Kirsch, U., "Design-oriented analysis of structures", Kluwer Academic, Dordrecht, 2002.

2

Kirsch, U., "Design-oriented analysis of Structures - A unified Approach", ASCE Journal of Engineering Mechanics, 129, 264-272, 2003. doi:10.1061/(ASCE)0733-9399(2003)129:3(264)

3

Kirsch, U., "A unified reanalysis approach for structural analysis, design and optimization", Structural and Multidisciplinary Optimization, 25, 67-85, 2003. doi:10.1007/s00158-002-0269-0

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £135 +P&P)