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CivilComp Proceedings
ISSN 17593433 CCP: 79
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 139
On the Flexibilitybased Solutions for Beam Elements with Bilinear Material Model W. Barham, G.F. Dargush and A.J. Aref
Department of Civil, Structural and Environmental Engineering, University at Buffalo  State University of New York, United States of America W. Barham, G.F. Dargush, A.J. Aref, "On the Flexibilitybased Solutions for Beam Elements with Bilinear Material Model", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 139, 2004. doi:10.4203/ccp.79.139
Keywords: beam structures, nonlinear analysis, displacement method, large increment method (LIM), bilinear material model, generalized inverse.
Summary
The displacementbased finite element method is widely used in nonlinear
analysis of structures, however, this method requires a stepbystep approach that
depends on flow theory. Furthermore, significant mesh refinement is often required
in plastic zones. These disadvantages motivate reconsidering forcebased 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 bilinear 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 displacementbased method.
In the nonlinear material problems, two major approaches were developed  namely, the displacement approach, and the force approach. The displacement approach uses a stepbystep 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 NewtonRaphson 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 forcebased method first proposed by [1], and has recently been extended for new applications in [2]. The main advantage of this flexibilitybased method over the displacement method is that it avoids the stepbystep 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 forcebased 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 bilinear 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. References
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