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An Improved Quadrilateral Flat Element With Drilling Degrees of Freedom for Shell Structural Analysis

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This paper reports the development of a simple and efficient 4-node flat shell el ement with six degrees of freedom per node for the analysis of arbitrary shell structures
  1 Animprovedquadrilateralflatelementwithdrillingdegreesoffreedomforshellstruc-tural analysis H. Nguyen-Van, CESRC, Faculty of Engineering & Surveying, USQ, AustraliaN. Mai-Duy, CESRC, Faculty of Engineering & Surveying, USQ, AustraliaT. Tran-Cong, CESRC, Faculty of Engineering & Surveying, USQ, Australia. Abstract:  This paper reports the development of asimple and efficient 4-node flat shell el-ement with six degrees of freedom per node for the analysis of arbitrary shell structures. Theelement is developed by incorporating a strain smoothing technique into a flat shell finite el-ement approach. The membrane part is formulated by applying the smoothing operation ona quadrilateral membrane element using Allman-type interpolation functions with drillingDOFs. The plate-bending component is established by a combination of the smoothed cur-vature and the substitute shear strain fields. As a result, the bending and a part of membranestiffness matrices arecomputed on the boundaries ofsmoothing cells which leads to very ac-curate solutions, even with distorted meshes, and possible reduction in computational cost.The performance of the proposed element is validated and demonstrated through severalnumerical benchmark problems. Convergence studies and comparison with other existingsolutions in the literature suggest that the present element is efficient, accurate and free of lockings. Keyword  flat shell, strain smoothing method, shear-locking free, first-order shear deforma-tion theory, drilling degrees of freedom. 1 Introduction The wide application of shell structures in engineering practice has caught the interests of many researchers in the fields of analysis and design of such structures. A great body of research work has been proposed over several decades towards the development of simpleand efficient shell finite elements through three major approaches: (1) the curved shell  2elements based on classical shell theory with curvilinear coordinates; (2) the degeneratedshell elements derived from three-dimensional solid elements and (3) the flat shell elementsobtained by the combination of the membrane and bending behaviour of plate elements.In general, it is difficult to identify which shell element is the most advantageous. Amongthese approaches, the flat shell elements are regarded to be the most attractive as they canbe readily built by combining existing plate and membrane elements. They have been usedextensively because of the simplicity in their formulation, the effectiveness in performingcomputation and the flexibility in applications to both shell and folded plate structures. Inaddition, the performance of the flat shell elements for thick to thin structures also signifi-cantly improved with the aid of Reissner-Mindlin kinematics, the incorporation of drillingdegrees of freedom (Iura and Atluri, 1992) and the variational principles governing rotations(Atluri, 1980; Atluri and Cazzani, 1994; Atluri, 1984; Suetake, Iura, and Atluri, 2003).Although triangular flat elements are most efficient for discretizing arbitrary shell geome-tries, quadrilateral elements are usually used owing to their better performance with respectto convergence rates than that of triangular elements (Lee and Bathe, 2004). The difficultyin the development of the four-node shell element is that such elements are too stiff andsuffer from locking phenomenon. This phenomenon srcinates from the shortcoming in theinterpolation of the displacement. Two well-known locking types that may occur in four-node flat elements in analysis of shell structures are (1) the transverse shears locking whicharises as the ratio of the thickness-to-characteristic length of a shell becomes small (e.g. t  /  L  ≤ 1 / 100), and (2) the membrane locking which occurs when coarse or distorted meshesare used, especially in bending dominated problems.With the development of shell elements, many methods have been proposed to circumventthese disadvantages. For a summary, the readers are referred to (Yang, Saigal, Masud,and Kapania, 2000). Techniques to handle shear-locking commonly adopted are the re-duced/selective integration (Hughes, Cohen, and Haroun, 1978; Zienkiewicz, Taylor, andToo, 1971; Stolarski and Belytschko, 1983). However, it may lead to the possible mani-festation of hourglass modes and stabilization matrices are required to remove these spu-  3rious modes (Belytschko and Tsay, 1983; Belytschko, Lin, and Tsay, 1984). An alterna-tive scheme for dealing with the shear-locking problem is the hybrid/mixed formulation inwhich separate interpolations are used for the stresses and displacements (Lee and Pian,1978; Noor and Peters, 1981). In another approach to alleviate shear locking, the assumednatural strain method (ANS) first proposed in (MacNeal, 1978, 1982), is generally reportedto be an efficient method utilizing complete numerical integration rules. In this approach,the transverse shear strains areinterpolated from the displacement-dependent strains definedat the mid-side of element edges to reduce transverse shear locking. Based on this concept,some efficient models werepresented, including the mixedinterpolated tensorial component(MITC) family proposed by Bathe’s group (Dvorkin and Bathe, 1984; Bathe and Dvorkin,1985) and the discrete strain gap (DSG) elements proposed by Bischoff’s group (Bischoff and Bletzinger, 2001; Koschnick, Bischoff, Camprubi, andBletzinger, 2005). Another inter-esting scheme arising from mixed variational formulations is the enhanced assumed strain(EAS) method first presented by (Simo and Rifai, 1990) and further developed in the linearelastic range (Andelfinger and Ramm, 1993; Cardoso, Yoon, and Valente, 2006) and non-linear aspects (Bischoff and Ramm, 1997; Eckstein and Basar, 2000; Cardoso, Yoon, andValente, 2007). The key point of this method lies in the use of a strain field composed of a compatible strain field and an enhanced strain field based on the Hu-Washizu variationalprinciple to reduce shear locking.Some of these approaches mentioned above are also used to remedy membrane locking,especially the selective reduced integration (SRI) technique and the EAS method. Howeversome of them deteriorate significantly when mesh is distorted (Cardoso, Yoon, and Valente,2006). More works on the problems related to the membrane locking of flat shell elementscan be found in the references of (Cook, 1994), (Groenwold and Slander, 1995), (Choi andLee, 2003) and (Cui, Liu, Li, Zhao, Nguyen, and Sun, 2008).A large number of four-node shell element formulations have been presented to date, show-ing good performance, however, there is still room to improve the behaviour of flat shellelements, in order to enhance the efficiency, accuracy and stability even when meshes are  4coarse or elements are badly-shaped. The objective of this study is to propose an improvedformulation of a locking-free quadrilateral flat shell element with six degrees of freedomper node that is able to reduce the mesh distortion sensitivity and enhance the coarse meshaccuracy. The present flat element is obtained by applying the strain smoothing method(SSM) to a quadrilateral flat shell element with the combined characteristics of a membraneAllman-type element with drilling DOFs and the assumed strain plate-bending element of (Bathe and Dvorkin, 1985). The SSM was srcinally proposed by (Chen, Wu, and You,2001) as a normalization for nodal integration of mesh-free Galerkin weak form. Based onthis concept, (Liu, Dai, and Nguyen, 2007) first presented the application of the SSM tothe 2D elasticity finite element method as a new smoothed finite element method (SFEM).Further application of SSM for laminated composite plates/shells and piezoelectric solidswas presented by (Nguyen-Van, Mai-Duy, and Tran-Cong, 2007, 2008a,b).In this study, the membrane part of the proposed shell element is enhanced by applyingthe SSM instead of the use of hierarchical bubble interpolation mode. The SSM is alsoapplied to the curvature of the plate-bending part to improve the flexural behaviour in thedistorted as well as coarse meshes in particular. With the aids of the SSM, the evaluation of bending and membrane stiffness matrices are carried out by integration along the boundaryof smoothing elements which can give more accurate numerical integration even with badly-shaped elements or coarse meshes and also reduce computational time when comparedwith the evaluation of domain integration. Moreover, the incorporation of the SSM alsofacilitates relatively simple implementation procedure which makes coding much easier.Inthe following sections, abrief review ofthe four-node flatshell finiteelement withdrillingDOFs is first introduced. This is followed by the strain smoothing approach for the flat shellelement. Numerical benchmarks are then conducted to investigate and assess the perfor-mance of the proposed 4-node flat shell element before drawing the final conclusions.
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