Keywords: beam structures, nonlinear analysis, displacement method, large increment method (LIM), bi-linear material model, generalized inverse.

The displacement-based finite element method is widely used in nonlinear analysis of structures, however, this method requires a step-by-step approach that depends on flow theory. Furthermore, significant mesh refinement is often required in plastic zones. These disadvantages motivate reconsidering force-based approaches as an alternative method of solution. One of these force algorithms that has been recently developed is the large increment method. The objective of this paper is to present an extension of the large increment method for the nonlinear analysis of beam structures controlled by a bi-linear material model (i.e. elastic with linear strain hardening). The efficiency and the accuracy of LIM to handle the nonlinear features of the problem are discussed especially for beams exhibiting many plastic regions. An illustrative example is presented, and the results are compared with those obtained from the displacement-based method.

In the nonlinear material problems, two major approaches were developed - namely, the displacement approach, and the force approach. The displacement approach uses a step-by-step solution procedure that depends on the decomposition of the total load into small increments. The history for the problem variables is known up to time and the problem is to solve for the state variables at . The Newton-Raphson method [3] is commonly used for solving the nonlinear system of equations at . Therefore, numerical errors are expected at the end of each step, which will accumulate to the subsequent steps. To improve the solution quality, one needs to reduce the load step size. Additionally, significant mesh refinement is needed in the plastic zone due to the complicated variation of the deformation field within that region.

The large increment method (LIM) is a force-based method first proposed by [1], and has recently been extended for new applications in [2]. The main advantage of this flexibility-based method over the displacement method is that it avoids the step-by-step solution procedure because it treats the nonlinear feature of the constitutive model in the element stage while the linear equilibrium and compatibility equations are used in global sense. Therefore, the load can be accommodated in one step or in few large steps. Furthermore, as with most force-based methods, there will be no need for any mesh refinement in the plastic zone. The problem is to find an accurate internal force vector that produces a compatible deformation vector through an iterative procedure. This algorithm is based on an optimization process to reduce the error in the deformation vector at the end of each iteration.

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 the present paper, the large increment algorithm is extended to analyze beam structures with a bi-linear material model. The accuracy and the robustness of the 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 beam problem with at least the same accuracy as the displacement finite element program ABAQUS. Moreover, LIM needs less computational effort compared to ABAQUS since it requires fewer elements, integration points and iterations.

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

Barham, W., Aref, A.J., Dargush, G.F., "Derivation and Implementation of a Flexibility-Based Large Increment Method for Solving Non-Linear Structural Problems." Proceedings of The Ninth International Conference on Civil and Structural Engineering Computing, Egmond-aan-Zee, The Netherlands, Sep. 2003, Civil-Comp Press, Stirling, UK. doi:10.4203/ccp.77.12

3

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

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