### Abstract

Following previous work of [Dong, El-Gizawy, Juhany, Atluri (2014)], a simple locking-alleviated 3D 8-node mixed-collocation C^{0} finite element (denoted as CEH8) is developed in this study, for the modeling of functionally-graded or laminated thick-section composite plates and shells, without using higher-order or layer-wise zig-zag plate and shell theories which are widely popularized in the current literature. The present C^{0} element independently assumes an 18-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 18 pre-defined constraints at 18 pre-selected collocation points. The constraints are rationally defined to capture the basic kinematics of the 3D 8-node C^{0} element, and to accurately model each basic deformation mode of tension, bending, shear, and torsion. A 2×2×2 Gauss quadrature is sufficient for evaluating the stiffness matrix of CEH8 C^{0} 3D elements for homogeneous materials, but over-integration (with a higher-order Gauss Quadrature, a layer-wise Gauss Quadrature, or a simple Trapezoidal Rule in the thickness direction) is used for functionally-graded materials or thick-section laminated composite structures with an arbitrary number of laminae. Through several numerical examples, it is clearly shown that the present CEH8 3D C^{0} element can accurately capture the stress distribution of FG and thick laminated structures with an arbitrary number of laminae even when only one element is used in the thickness direction. In stark contrast to the higher-order or layer-wise zig-zag plate and shell theories, with assumptions for displacement or stress fields in the thickness direction, which may require complicated C^{1} finite element, the present C^{0} element can accurately compute the jumps in bending stresses at the interfaces of layers, while the out-of plane normal and shear stresses can be accurately recovered by exploring the equilibrium equations of 3D linear elasticity. By adding the contributing stiffness of z-pins into the stiffness matrix of CEH8, it is also demonstrated that the presently developed method can be used to study the effect of using z-pin reinforcements to reduce the inter-laminar stresses of composite structures, in a very simple and computationally-efficient manner.

Original language | English |
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Pages (from-to) | 163-192 |

Number of pages | 30 |

Journal | Computers, Materials and Continua |

Volume | 41 |

Issue number | 3 |

State | Published - 2014 |

### Keywords

- Collocation
- Functionally-graded material
- Mixed FEM
- Plate
- Shell
- Thick-section composite laminate
- Z-pins

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## Cite this

^{0}finite element with over-integration, for functionally-graded and laminated thick-section plates and shells, with & without z-pins.

*Computers, Materials and Continua*,

*41*(3), 163-192.