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# جلسه سوم: آموزش جامع روش های پیشرفته آنالیز دینامیکی غیر خطی (دکتر حشمدار)

AN EFFICIENT METHOD FOR NONLINEAR DYNAMIC ANALYSIS OF 3D SPACE STRUCTURES

Prof.Dr.Hashamdar

Problem Statement

In general, various types of cable roof arrangements including beams, nets and grids have been investigated by previous researchers using the conventional linear method. The linear method overestimates the displacement when a structure is stiffening and underestimates the displacement when it is softening. It has been observed that when using conventional methods the number of iterations increased with the increase of degree of freedom and that these methods need large computer storage for solution of the equation of motion. The computational time in conventional methods increase with the increasing degree of freedom in the structures.

Several methods are available for the dynamic response analysis of a linear structure. The mode-superposition method has been used to comparison result in present study. However, nonlinear systems do not have fixed sets of eigenvectors and eigenvalues; instead new sets of eigenvectors and eigenvalues must be revaluated for each time step. This makes the use of the mode-superposition method very time consuming and costly. Apart from the above mentioned, the dynamic response analysis of nonlinear systems in general is based on the evaluation of the response for a series of short time intervals using different types of time integration techniques.

All the currently available methods predict the response of nonlinear assemblies by forward integration in the time domain. The methods are either implicit or explicit. They provide a numerical solution to the equation of motion set up for one interval of time. In the case of nonlinear systems, most of the methods assume that the structural properties remain constant during the interval, but revaluate 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 usually permit continuous revaluation of stiffness and damping during the iterative process, which is necessary to establish dynamic equilibrium at the end of each time step. However, the revaluation process makes these methods more expensive to use. The difference between the explicit and implicit methods is notable.

The implicit methods 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 systems. The explicit methods, on the other hand, have relatively less computer storage and computation requirements than implicit methods, but they 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, whereas 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 on the accuracy of different methods. However, little attention has been paid to the loss of accuracy caused by updating the stiffness at the end of each time step.

In many cases, the effect of variation of damping has been investigated but has received less attention and the stability criteria have been discussed to the smallest extent possible. For highly nonlinear structures such as cable structures the effect of assuming that stiffness remains constant during each time step can lead to a considerable degree of inaccuracy even when the time steps are small. There may not necessarily be one method for the whole time span of response, since the step-by-step integration permits switching from one method to another. For some type of structures it may be advantageous to apply one method while dynamic load is applied and another method for the continuation of response after the excitation has been ceased.

As mentioned above, one cannot determine which if any of these methods is the best unless the type of structure to be analysed is specified. Hence, it is necessary to find a common method which will be the optimum method based upon the minimization of the total potential dynamic work in order to achieve the dynamic equilibrium at the end of each time step.

آموزش جامع روش های پیشرفته آنالیز دینامیکی غیر خطی سازه های فضائی سه بعدی

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