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# A linear quadrilateral shell element with fast stiffness computation

Institut für Werkstoffe und Mechanik im Bauwesen, Technische Universität Darmstadt, Petersenstraße 12, Darmstadt D-64287, Germany; Institut für Baustatik, Universität Karlsruhe (TH), Kaiserstraße 12, Karlsruhe D-76131, Germany

Computer Methods in Applied Mechanics and Engineering (Impact Factor: 2.63). 10/2005; DOI: 10.1016/j.cma.2004.11.005 - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper describes an eight-node, assumed strain, solid-shell, corotational element for geometrically nonlinear structural analysis. The locally linear kinematics of the element is separated into in-plane (which is further decoupled into membrane and bending), thickness and transverse shear components. This separation allows using any type of membrane quadrilateral formulation for the in-plane response. Assumed strain fields for the three components are constructed using different approaches. The Assumed Natural Deviatoric Strain approach is used for the in-plane response, whereas the Assumed Natural Strain approach is used for the thickness and transverse shear components. A strain enhancement based on Enhanced Assumed Strain concepts is also used for the thickness component. The resulting element passes well-known shell element patch tests and exhibits good performance in a number of challenging benchmark tests. The formulation is extended to the geometric nonlinear regime using an element-independent corotational approach. Some key properties of the corotational kinematic description are discussed. The element is tested in several well-known shell benchmarks and compared with other thin-shell and solid-shell elements available in the literature, as well as with commercial nonlinear FEM codes. Copyright © 2013 John Wiley & Sons, Ltd.International Journal for Numerical Methods in Engineering 07/2013; 95(2):145-180. · 1.96 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**A dimensionally reduced cylindrical shell model using a three-field complementary energy-based Hellinger–Reissner's variational principle of non-symmetric stresses, rotations, and displacements is presented. An important property of the shell model is that the classical kinematical hypotheses regarding the deformation of the normal to the shell mid-surface are not applied. A dual-mixed hp finite element model with stable polynomial stress- and displacement interpolation and C0 continuous normal components of stresses is constructed and presented for the bending-shearing problem, using unmodified three-dimensional inverse stress-strain relations for linearly elastic materials. It is shown through an example that the convergences in the energy norm as well as in the maximum norm of stresses and displacements are rapid for both h-extension and p-approximation, not only for thin but also for moderately thick shells loaded axisymmetrically, even if the Poisson ratio is close to the incompressibility limit of 0.5.ZAMM Journal of applied mathematics and mechanics: Zeitschrift für angewandte Mathematik und Mechanik 03/2012; 92(3):236-252. · 1.01 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Since many piezoelectric components are thin rod-like structures, a piezoelectric finite beam element can be utilized to analyse a wide range of piezoelectric devices effectively. The mechanical strains and the electric field are coupled by the constitutive relations. Finite element formulations using lower order functions to interpolate mechanical and electrical fields lead to unbalances within the numerical approximation. As a consequence incorrect computational results occur, especially for bending dominated problems. The present contribution proposes a concept to avoid these errors. Therefore, a mixed multi-field variational approach is introduced. The element employs the Timoshenko beam theory and considers strains throughout the width and the thickness enabling to directly use 3D constitutive relations. By means of several numerical examples it is shown that the element formulation allows to analyse piezoelectric beam structures for all typical load cases without parasitically affected results.Computational Mechanics 12/2013; 52(6). · 2.04 Impact Factor

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