Article

Finite element corotational formulation for geometric nonlinear analysis of thin shells with large rotation and small strain

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Abstract

Based on the consistent symmetrizable equilibrated (CSE) corotational formulation, a linear triangular flat thin shell element with 3 nodes and 18° of freedom, constructed by combination of the optimal membrane element and discrete Kirchhoff triangle (DKT) bending plate element, was extended to the geometric nonlinear analysis of thin shells with large rotation and small strain. Through derivation of the consistent tangent stiffness matrix and internal force vector, the corotational nonlinear finite element equations were established. The nonlinear equations were solved by using the Newton-Raphson iteration algorithm combined with an automatic load controlled technology. Three typical case studies, i.e., the slit annular thin plate, top opened hemispherical shell and cylindrical shell, validated the accuracy of the formulation established in this paper.

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... Several methods are available for selection of the floating frame position. The corotational description is widely used for geometric nonlinear analysis of shells and beams with large displacements and rotations and small strain [8][9][10][11]. Use of an 'average' node position as the floating frame is described in [12,13]. The simplest method consists of connection of SCF with one of the SCi, i.e., the floating frame coincides with one of the node systems of coordinates. ...
... Using the normalization of the modes and skipping the modes related to zero frequencies, we obtain the desired transformation matrix (9) with the following properties: (10) where denotes a diagonal matrix. Consider the principle of virtual work for generation of equations of motion of the finite element [7], taking into account the work of inertia and elastic forces only (11) Here and a are the virtual displacement and acceleration of a point, respectively, whose position relative to SCF in Fig. 1 is determined by the vector r; dm and dV are the mass and volume of an infinitesimal part of the element; and are the elements of the stress and strain tensors. ...
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Equations of motion of a finite element in absolute coordinates including mass matrix, generalized inertia and internal forces are derived. A trapezoidal element for dynamic models of flexible shells in the shape of surface of revolution is considered. The element can be used for modeling dynamics of automotive tire and air spring bellows and some other flexible elements of transport systems undergoing large elastic deflections.
... However, these formulations appear to be intractable in a two-scale analysis method for composite plates [25], because the small strain framework is the premise for obtaining homogenized plate stiffnesses with NPT, and its extension to large deformations is far from trivial. To manage small strains and large translation/rotation simultaneously, we employ the so-called co-rotational (CR) formulation [31][32][33], in which the motion of each plate's FE is decomposed into two types: one for rigid body motion and the other for elastic deformation under the assumption of infinitesimal strain. In this way, in-plane homogenization [25] can be applied for the elastic deformation, and some minor adjustments are made to follow the rigid body motion. ...
... where k e T is the element tangent stiffness matrix whose derivation is detailed in the literature; see, for example, [31,33]. ...
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This study proposes a two-scale topology optimization method for a microstructure (an in-plane unit cell) that maximizes the macroscopic mechanical performance of composite plates. The proposed method is based on the in-plane homogenization method for a composite plate model in which the macrostructure is modeled using thick plate theory and the microstructures are three-dimensional solids. Macroscopic plate characteristics such as homogenized plate stiffnesses and generalized thermal strains are evaluated through the application of numerical plate tests (NPTs) applied to an in-plane unit cell. To handle large rotations of the composite plates, we employ a co-rotational formulation that facilitates working with the two-scale plate model formulated within a small strain framework. Two types of objective functions are tested in the presented optimization problems: one minimizes the macroscopic end compliance to maximize the macroscopic plate stiffness, whereas the other maximizes components of a macroscopic nodal displacement vector. Analytical sensitivities are derived based on in-plane homogenization formulae so that a gradient-based method can be employed to update the topology of in-plane unit cells. Several numerical examples are presented to demonstrate the proposed method's capability related to the design of optimal in-plane unit cells of composite plates. This article is protected by copyright. All rights reserved.
... Relatively recently, both the material and geometric stiffness matrices were found by Almeida and Awruch (2011) to be non-symmetric in a co-rotational 2D element shell formulation for nonlinear dynamic analysis of laminated composite shells. It was also found by Yang and Xia (2012) that the geometric stiffness matrix is asymmetric in the case of thin shells used for small-strain largedisplacement problems, similar to a simplified co-rotational method for quadrilateral shell elements developed by Tang et al. (2017). On the other hand, Izzuddin (Izzuddin, 2005;Izzuddin and Liang, 2015) showed that a symmetric tangent stiffness matrix can be realised in 2D shell elements with the adoption of a bisector or zero-macrospin local co-rotational system and the use of vectorial nodal rotational variables. ...
... Relatively recently, both the material and geometric stiffness matrices were found by Almeida and Awruch (2011) to be nonsymmetric in a co-rotational 2D element shell formulation for nonlinear dynamic analysis of laminated composite shells. It was also found by Yang and Xia (2012) that the geometric stiffness matrix is asymmetric in the case of thin shells used for small-strain large-displacement problems, similar to a simplified co-rotational method for quadrilateral shell elements This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4445952 ...
... The number of elements is directly related to the number of design variables in an optimization simulation, so a higher-order element provides flexibility regarding the selection of the number of design variables as well as an effective solution. The plate and shell elements are selected and assembled in 18 DOF elements [31]. The plate element is the optimal triangular element (OPT) from Fellipa [30], and the shell element is the Discrete Kirchhoff triangular element (DKT) [32] as in figure 4. The assembled triangular element consists of 18 DOF elements (6 DOFs for each node). ...
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... Up to now, many researches have been conducted on the corotational method for geometrical nonlinear analysis of shells (Battini 2007;Caselli and Bisegna 2013;Li et al. 2008;Li et al. 2013;Yang and Xia 2012), plates (Izzuddin 2005), three-dimensional structures (Espath et al. 2014;Mostafa et al. 2013a;Norachan et al. 2012), and beams (Jonker and Meijaard 2013;Le et al. 2014;Li 2007;Urthaler and Reddy 2005). However, this method has been rarely utilized for analyzing plane problems. ...
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... Jetteur and Frey (1986) and Jaamei et al. (1989) developed and studied four-nodded flat shells with six dof based on the DKQ plate bending element in Marguerre's shallow shells theory for geometrically nonlinear analysis. Several other works dealing with the nonlinear analysis of flat shells with drilling rotational dof in the framework of the total Lagrangian formulation have been published (Simo et al., 1989;Fox and Simo, 1992;Samanta and Mukhopadhyay, 1999;Yang and Xia, 2012). Also the updated Lagrangian formulation has been adopted in many works (Lee and Yoo, 1988;Madenci and Barut, 1994;Kapania, 1997, 1998;Zhang and Kim, 2005;Kang et al., 2009;Boutagouga et al., 2010). ...
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The present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in [1]. It is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix T appertaining to a rotation about an arbitrary axis. Also analysed is the case of multiple rotations about fixed or follower axes. Particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations. This theme is discussed in some detail for a number of cases. Semitangential rotations—for which commutativity holds—first proposed in [2, 3]are also considered. Furthermore, an elementary geometrical analysis of large rotations is also given. Finally, we deduce in an appendix, using a judicious reformulation of quarternions, the compound pseudovector representing the combined effect of n rotations.In the author's opinion the present approach appears preferable to a pure vectorial scheme—and even more so to an indicial formulation— and is computationally more convenient.
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This paper investigates the formulation of co-rotational flat facet triangular elements for the numerical analysis of instability phenomena in shell structures. The elements have three nodes with six degrees of freedom at each node. The term ‘co-rotational’ relates here to the provision of a local system that continuously rotates and translates with the element. Following mainly Nour-Omid and Rankin [B. Nour-Omid and C.C. Rankin, Finite rotation analysis and consistent linearization using projectors, Comput. Methods Appl. Mech. Engrg. 93 (1991) 353–384], the definition of an element resorts to a change of variables from the local frame to the global one. This is done through the use of a projector matrix which relates the variations of the local displacements to the variations of the global ones, by extracting the rigid body modes from the latter. The main difference from the original formulation lies in the parameterization of 3D finite rotations. In contrast to the paper by Nour-Omid and Rankin, a parameterization based on the rotational vector is here adopted and thus, an additional change of variables has to be performed. As a result, the rotational variables become additive and the necessity of a special updating procedure is avoided. The main feature of the adopted formulation is its independence of the local assumptions used to derive the internal forces and tangent stiffness in local coordinates. For a certain class of elements (i.e. elements with the same number of nodes and degrees of freedom) the main co-rotational framework is the same. Using this property, three types of local formulations are considered. A set of carefully chosen test problems is used in order to assess the performances of the three element types.
Buckling and stability problems for thin shell structures using high-performance finite elements. Dissertation for Dotoral Degree
  • B Haugen
  • B. Haugen
A field consistency based co-rotational finite element procedure for 2D quadrilateral element (in Chinese)
  • S B Cai
  • P S Shen
  • B X Hu
  • S. B. Cai