Keywords: nonlinear dynamic analysis, displacement-based method, force-based method, large increment method (LIM), error accumulations, iterative procedures, elastic solution.

For structural dynamic analysis with nonlinear materials, the displacement-based approach is widely used. In this method, in order to quickly reach convergence, must be kept sufficiently small to control the accumulation of error and to avoid possible divergence of Newton-Raphson iterations. Thus, one can imagine the computational effort needed in each time step for each element, in each iteration, to reach a converged solution. The traditional displacement-based approach is obviously quite cumbersome. Thus the FE displacement approach may be associated with a lack of efficiency and computational speed. This paper presents a new force-based approach in an attempt to create robust and fast computational algorithms for nonlinear structural dynamics. The proposed algorithm is based on the developed flexibility-based large increment method (LIM) previously investigated by the present authors under quasistatic conditions. Unlike the displacement-based approach, LIM shows excellent computational benefits when used in nonlinear quasistatic analysis which can be summarized by large solution time steps, less number of iterations, and significantly fewer elements. Thus, this paper aims to present a framework to merge knowledge gained in the development of LIM for nonlinear structural analysis along with developments in mathematics ultimately to solve large and complex nonlinear structural dynamic problems in an efficient manner.

The displacement based approach relies on the flow theory to present the entire information of the loading path which is necessary for nonlinear analysis. Using this approach the solution process involves three basic operations; (1) incremental process, (2) time integration algorithm (implicit and explicit algorithms), and (3) an iterative solution procedure. All of those procedures involve many approximations that produce two serious problems - namely, error accumulation and computing time consumption. Adopting such methods for solving nonlinear structural dynamic problems results in substantial solution time and computational cost.

LIM is a force-based FE algorithm proposed originally in [1] and [3], and then developed further for quasistatic analysis by the present research team [2]. One main advantage of this method is that it can handle the complexity of the nonlinear problem without severe restrictions on step size. This is because the non-linearity of the problem is treated at the local stage (i.e. at the element level), while the linear equilibrium and compatibility equations are treated at the global stage. By separating these two stages, there is no need to linearize the constitutive model, and the load often can be handled in one large step (thus, the name LIM is designated as such) in the case of monotonic loading. LIM also shows similar robustness when used in our very recent work to solve cyclic elasto-plastic problems.

In our proposed algorithm, we aim to extend LIM to nonlinear structural dynamic analysis. The development includes formulation and application of new algorithms for a variety of nonlinear material dynamic problems. In static analysis, LIM demonstrated excellent performance in terms of accuracy, memory requirements and computational efforts. The idea of separating the nonlinear local stage and linear global stage must be taken into consideration in the final dynamics formulation because this separation is the main power of LIM over the displacement-based FE method. Although the use of LIM for large deformation analysis is quite viable, for simplicity, the discussion will focus on the small deformation case.

Various civil engineering applications would benefit from this development. For example, steel moment frames structures especially when plasticity starts to spread on the structure and plastic hinges start to form. This case is very problematic to be solved with the displacement method. Yet it was easily solved using LIM under static conditions.

In this paper, a framework for using the LIM for solving nonlinear dynamic problems is presented. To demonstrate the usage of the method, the new LIM formulation is applied to solve a quasistatic cyclic loading problem for a beam and a single degree of freedom elastodynamic problem. Afterwards, a discussion concerning the suitability of LIM for nonlinear dynamic analysis is presented.

1

Aref, A.J., Guo, Z., "Framework For Finite-Element-Based Large Increment Method For Nonlinear Structural Problems", Journal of Engineering Mechanics, 739-746,.July, 2001. doi:10.1061/(ASCE)0733-9399(2001)127:7(739)

2

W. Barham, G.F. Dargush and A.J. Aref, "On the Flexibility-based Solutions for Beam Elements with Bi-linear Material Model", in Proceedings of the Seventh International Conference on Computational Structures Technology, B.H.V. Topping and C.A. Mota Soares, (Editors), Civil-Comp Press, Stirling, United Kingdom, paper 139, 2004. doi:10.4203/ccp.79.139

3

Bathe, K., "Finite Element Procedures". Prentice-Hall International, Inc, 1996.

4

Zhang, C., Liu, Z., "A Large Increment Method for Material Nonlinearity Problems." Advances in Structural Engineering, 99-109, No. 2, 1997.

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