AN EFFICIENT METHOD FOR NONLINEAR DYNAMIC ANALYSIS OF 3D SPACE STRUCTURES
The Trujillo method
Trujillo presented an explicit algorithm for the dynamic response analysis of structural systems in 1977 . and tested the method for linear problems. For the linear undamped systems the method was shown to be unconditionally stable. An algorithm based upon Trujillo’s method has been developed for nonlinear systems by Raman & Kumar but it does not take into account the effect of damping
All the methods reviewed by the author predict the response of nonlinear assemblies by forward integration in the time domain. In general, the methods are either implicit or explicit provide numerical solutions to the equations of motion set up for one interval of time. They are mainly concerned with the two ends of a given interval and how to get from one end to the other, and for establishing starting values for the next time step. This is satisfactory for the methods developed for the analysis of linear systems.
In the case of nonlinear systems, most of the methods assume the structural properties to remain constant during the interval, but revaluates them at the end and in some cases also in the middle of the time step. For highly nonlinear assemblies this may not be sufficient and in such cases it is important to revaluate both the stiffness and the damping during the time step. The implicit methods do usual permit continuous revaluation of the stiffness and damping during the iterative process to establish dynamic equilibrium at the end of each time step. The revaluation process, however, makes the methods more expensive to use.
The implicit method offer unconditional stability at the expense of operating with relatively dense decomposed matrices when applied to linear structures, but lose the advantage of unconditional stability when applied to nonlinear system. The explicit methods, on the other hand, have relatively less computer storage and computation than the implicit methods, but are hampered by instability which limits the size of the time steps. The implicit methods when applied to nonlinear structures require the solution of a set of nonlinear equations whilst most explicit methods require the inversion of a non-diagonal matrix if consistent mass and non-diagonal damping matrices are used. A considerable amount of information is available concerning the effect of the size of the time intervals on the stability as well as the accuracy of the different methods.
The effect of variation of damping has attracted even less attention. In many cases the stability criteria has been discussed in the absence of damping even though for assemblies with high damping. For highly nonlinear structures such as cable and membrane structures, the stiffness is assumed to be constant during each time step. This assumption can lead to a considerable degree of inaccuracy even when the time steps are small. The degree of inaccuracy will increase as the prediction time increases. One cannot choose any of these methods as the best, unless the type of structure to be analysed is specified.
Once known problem is when the most suitable method is selected for the analysis, this may not necessarily be suitable method for the whole time span of response. Hence, the step by step integration that permits switching from one method to another method is useful to nonlinear analysis. Some types of structures it is advantageous to apply one method while dynamic loads such as sudden shocks or wind gusts are applied and another method for the continuation for response is used after the excitation has ceased.
The present method is based upon the minimization of the total dynamic work in order to achieve in accuracy result in during less time to compare conventional methods. In the following chapter, therefore, a review and a comparison are made of the relevant optimization methods in order to choose the most appropriate minimization algorithm.
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