Keywords: nonlinear analysis, displacement method, large increment method (LIM), step-by-step,.

The purpose of this paper is to present the methodology and implementation of a flexibility-based large increment method (LIM) for solving nonlinear structural problems. In the traditional displacement method in order to represent the general forces in terms of general deformations, the constitutive relations have to be linearized and a step-by-step incremental procedure is used to reach a solution. The main advantage of the flexibility-based large increment method over the displacement method is that it avoids the linearization of the constitutive relations. The application of the LIM formulations is presented for 2D-beam elements controlled by an elastic-fully plastic material model. These formulations are demonstrated using a simple nonlinear problem and the results are compared to those obtained from the displacement method. The example highlights the accuracy and the computational efficiency of the flexibility based method.

In the nonlinear material problems, it is well known that the displacement method tends to use a step-by-step process to reach the solution, where the total load is divided into small steps and in each step it is assumed that the relation between s and e is linear. However, in order to get accurate results, the step size must be small enough and the element length should also be small, thereby increasing the computational time. In addition, we know that there are some errors due to the linearization of the constitutive model and this error will accumulate from one step to another. All of these problems are because the basic unknowns in the displacement method are the global displacement of the nodes. The displacement of these nodes at the end of a certain step is used as a seed to the next step, and of course the error up to this step will transfer also to the next step and accumulate with the error resulting from subsequent steps.

The large increment method (LIM) is a force based method first proposed by [3] and [1]. It is an iterative method which makes use of the three system equations: (1) the linear global equilibrium equations; (2) the linear global compatibility equations; (3) the constitutive relations or the physical equations. In this method the nonlinear behavior is manipulated in the local stage. Consequently, there would be no need to linearize the constitutive relations as in the displacement method. This aspect would overcome the weakness of the displacement method and make the LIM more suitable for solving nonlinear structural problems especially when dealing with complex structures with highly nonlinear material models.

The first step in the large increment method is to obtain an initial force vector that satisfies the equilibrium equations. The initial force vector resulting from this step is then used in the local stage to compute the element deformations which are controlled by the nonlinear constitutive material model. After that the compatibility equations are used to check whether the deformations are compatible or not. Next the error in the deformations resulting from the compatibility equations is employed to calculate a more accurate force vector in an iterative technique.

In this paper the formulation of the large increment method was applied to beam elements for the elastic-fully plastic case. The accuracy and the robustness of this method in dealing with this case have been presented in a numerical example for a beam structure. From the results of this example it was evident that the LIM was able to handle the elastic-fully beam problem with excellent accuracy and considerable potential saving in computational effort compared to the displacement method. This is especially true at high levels of loading, when the plastic hinges appear on the beam.

As a final conclusion for this paper, the large increment method appears to be suitable for solving nonlinear structural problems, and the capabilities of this method should be tested for various structural elements and different material models.

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

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

3

Zhang, C., Liu, X., "A Large Increment Method for Material Nonlinearity Problems", Advances in Structural Engineering, 1(2), 99-109, 1997.

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