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
Computer Methods in Applied Mechanics and Engineering (Impact Factor: 2.96). 10/2005; 37(39):4279-4300. DOI: 10.1016/j.cma.2004.11.005


A new quadrilateral shell element with 5/6 nodal degrees of freedom is presented. Assuming linear isotropic elasticity a Hellinger–Reissner functional with independent displacements, rotations and stress resultants is used. Within the mixed formulation the stress resultants are interpolated using five parameters for the membrane forces as well as for the bending moments and four parameters for the shear forces. The hybrid element stiffness matrix resulting from the stationary condition is integrated analytically. This leads to a part obtained by one point integration and a stabilization matrix. The element possesses the correct rank, is free of locking and is applicable within the whole range of thin and thick shells. The in-plane and bending patch tests are fulfilled and the computed numerical examples show that the convergence behaviour of the stress resultants is very good in comparison to comparable existing elements. The essential advantage is the fast stiffness computation due to the analytically integrated matrices.

  • Source
    • "There are several attempts to circumvent locking effects in the standard low-order displacement-based finite element models . These h-type shell elements are usually based on the various modifications of the principle of virtual work [9] [10] [11] [12] [13]. "
    [Show abstract] [Hide abstract]
    ABSTRACT: Dimensionally reduced cylindrical shell models using complementary energy-based variational formulations of a priori non-symmetric stresses are compared. One of them is based on the three-field dual-mixed Hellinger–Reissner variational principle, the fundamental variables of which are the stress tensor, the rotation and displacement vectors. The other one is derived from the two-field dual-mixed Fraeijs de Veubeke variational principle in terms of the self-equilibrated stress field and rotations. The most characteristic properties of the shell models are that the kinematical hypotheses used in the classical shell theories are not applied and the unmodified three-dimensional constitutive equations are employed. Our investigations are restricted to the axisymmetric case. The developed dual-mixed hp finite element models with C0 continuous tractions and with discontinuous rotations and displacements are presented for bending–shearing (including tension–compression) problems. The computational performance of the constructed shell elements is compared through two representative model problems. It is numerically proven that no significant differences can be experienced between the two well-performing shell elements in the convergence rates.
    Full-text · Article · Mar 2013 · Finite Elements in Analysis and Design
  • Source
    • "Thus the classical two-field dual-mixed variational principle of Hellinger–Reissner with a priori symmetric stresses can be obtained, where the displacement vector and the stress tensor are approximated as independent unknowns [27] [28] [30] [32]. Although this approach usually yields good results for stresses, the development of stable and efficient finite element models has proven to be much more difficult than that with not a priori symmetric ones [2] [7] [18] [48] [52]. "
    [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.
    Full-text · Article · Mar 2012 · ZAMM Journal of applied mathematics and mechanics: Zeitschrift für angewandte Mathematik und Mechanik

  • No preview · Article ·
Show more