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A Simple Locking-Alleviated 4-Node Mixed-Collocation Finite Element with Over-Integration, for Homogeneous or Functionally-Graded or Thick-Section Laminated Composite Beams

Leiting Dong1, Ahmed S. El-Gizawy2, Khalid A. Juhany2, Satya N. Atluri3
Department of Engineering Mechanics, Hohai University, China.
King Abdulaziz University, Jeddah, Saudi Arabia.
Center for Aerospace Research & Education, University of California, Irvine, Distinguished Adjunct Professor, KAU, Saudi Arabia.

Computers, Materials & Continua 2014, 40(1), 49-78.


In this study, a simple 4-node locking-alleviated mixed finite element (denoted as CEQ4) is developed, for the modeling of homogeneous or functionally graded or laminated thick-section composite beam structures, without using higher-order (in the thickness direction) or layer-wise zig-zag theories of composite laminates which are widely popularized in current literature. Following the work of [Dong and Atluri (2011)], the present element independently assumes a 5-parameter linearly-varying Cartesian strain field. The independently assumed Cartesian strains are related to the Cartesian strains derived from mesh-based Cartesian displacement interpolations, by exactly enforcing 5 pre-defined constraints at 5 pre-selected collocation points. The constraints are rationally defined to capture the basic kinematics of the 4-node element, and to accurately model each deformation mode of tension, bending, and shear. A 2 by 2 Gauss quadrature is used when each element is used to model a piece of a homogeneous material or structure, but over-integration (using a higher-order Gauss Quadrature, a layer-wise Gauss Quadrature, or a simple Trapezoidal Rule in the thickness direction) is necessary if functionally-graded materials or thick-section laminated composite structures are considered. Through several numerical examples, it is clearly shown that the present CEQ4 is much more accurate than the well-known Pian-Sumihara (1984) element as well as the primal four-node element, for the modeling of homogeneous beams. For functionally-graded materials, the presently-developed element can accurately capture the stress distribution even when very few elements are used; but the Pian-Sumihara element fails, because the assumption of linearly-varying stressfield is generally invalid unless a very fine mesh is used in the thickness direction. For thick-section laminated composite beams, reasonably accurate solutions (for axial as well as transverse stresses) are obtained even when only one CEQ4 element is used in the thickness direction. Without using higher-order theories or layer-wise zig-zag assumptions for displacement or stress fields in the thickness direction, for thick-section laminates, the present method can accurately compute the jumps in axial stresses at the interfaces of layers. Extension of the present CEQ4 concept to C0 elements of higher-order, for plates and shells as well as for multi-physics will be pursued in future studies.


mixed FEM, collocation, beam, functionally-graded material, thicksection composite laminates.

Cite This Article

L. . Dong, A. S. . El-Gizawy, K. A. . Juhany and S. N. . Atluri, "A simple locking-alleviated 4-node mixed-collocation finite element with over-integration, for homogeneous or functionally-graded or thick-section laminated composite beams," Computers, Materials & Continua, vol. 40, no.1, pp. 49–78, 2014.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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