Mixed Shell Element Research Articles

A geometrically nonlinear Hellinger–Reissner shell element for the postbuckling analysis of variable stiffness composite laminate structures

Variable stiffness (VS) composite laminates provide larger freedom to design thin-walled structures than constant stiffness (CS) composite laminates. They showed to allow the redistributing of stresses, improving buckling and post-buckling performance and, therefore, reducing material weight and costs. This work extends a recently developed mixed shell element, MISS-4C, to the postbuckling analysis of VS composite laminate structures. MISS-4C has a linear elastic closed-form solution for the stress interpolation of symmetric composite materials. Its stress field interpolation is obtained by the minimum number of parameters, making it an isostatic element. Moreover, its kinematic is only assumed along its contour, leading to an efficient evaluation of all operators obtained through analytical integration along the element contour. MISS-4C uses a corotational approach within a fast multi-modal Koiter algorithm to efficiently obtain the initial post-buckling response of VS composite laminate structures.First, the element performance is investigated by analysing a carbon fibre VS composite laminate plate subjected to compressive stresses. Numerical results obtained with MISS-4C are compared with those obtained with the MISS-4 element, showing good accuracy and a high convergence rate. Subsequently, the structural response of a glass fibre VS composite laminate girder of a short-length bridge is optimised through a multi-objective optimisation that exploits the robustness of the MISS-4C element and the efficiency of the multi-modal Koiter algorithm.

Mixed shell element for static and buckling analysis of variable angle tow composite plates

A mixed quadrilateral 3D finite element, obtained from the Hellinger–Reissner functional, is presented for linear static and buckling analyses of variable-angle tow (VAT) composite plates. Variable-angle tows describe curvilinear fiber paths within composite laminae and are a promising technology for tailoring the buckling and post-buckling capability of plates. Due to the variable stiffness across the planform of the VAT plates, pre-buckling stresses can be tailored and redistributed towards supported edges, thereby greatly improving the buckling load. A linear mixed element called MISS-4 is used as starting point for this work. The element presents a self-equilibrated and isostatic state of stress. The kinematics lead to element compatibility matrix calculations based solely on the interpolation along element edges. The drilling rotations do not require penalty functions or non-symmetric formulations, thus avoiding spurious energy modes. The buckling analysis is reliably performed via a co-rotational formulation. In this work VAT plates with linear fiber angle variation in one direction, and constant stiffness properties in the orthogonal direction are studied. Numerical examples of VAT plates subjected to different loads and boundary conditions are investigated herein. The convergence of displacements, stress resultants and buckling loads are presented, and comparisons with numerical results, obtained using the S4R finite element of Abaqus and the pseudo-spectral Differential Quadrature Method, are shown.

Energy‐momentum conserving discretization of mixed shell elements for large deformation problems

AbstractIn the present paper we consider structure‐preserving integration methods in the context of mixed finite elements. The used low‐order mixed finite elements typically exhibit improved coarse mesh accuracy. On the other hand energy‐momentum (EM) consistent time‐stepping schemes have been developed in the realm of nonlinear structural dynamics to enhance the numerical stability properties. EM schemes typically exhibit superior robustness and thus offer the possibility to use large time steps while still producing physically meaningful results. Accordingly, combining mixed finite element discretizations in space with EM consistent discretizations in time shows great promise for the design of numerical methods with superior coarse mesh accuracy in space and time. Starting with a general Hu‐Washizu‐type variational formulation we develop a second‐order accurate structure‐preserving integration scheme. The present approach is applicable to a large number of mixed finite element formulations. As sample application we deal with a specific mixed shell element. Numerical examples dealing with large deformations will show the improved coarse mesh accuracy in space and time of the advocated approach. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A continuum-based mixed shell element for shakedown analysis

The objective of this paper is to present a new triangular shell element for shakedown analysis. The element is based on a mixed variational formulation with displacement/velocity and stress fields independently interpolated. We also employ the continuum based (CB) approach which allows the definition of the yield function at the continuum level without any approximation in terms of generalized stresses. The formulation of the proposed element is presented in details after a brief review of relevant classical shakedown theory concepts. The performance of the element is assessed by means of a set of selected representative examples. The numerical tests include: (i) analyses of thin and thick-walled straight pipes under combined loads, (ii) the shakedown analysis of a pipe bend under internal pressure and in-plane bending and (iii) limit analysis of a cylinder–cylinder intersection subjected to bending and internal pressure. • A new CB mixed finite element is devised for limit and shakedown analysis of shells. • The use of generalized stresses in shakedown formulation is avoided with the element. • We focus on the ability of the element to provide accurate critical load domains. • Numerical tests confirm the accuracy and convergence of the element.

Development of New Hybrid Shell Element Takinging the Normal Stress into Account

The calculation of the stress distribution in the forming stage of sheet metal forming is critical to the accuracy of springback simulation. In order to improve the accuracy of stress prediction, a new mixed shell element is presented. Two pseudo nodes are added in the thickness direction so that the normal strain can be calculated using the displacements of the nodes. Mixed formulation is used instead of that of pure displacement. As a result, the accuracy of stress prediction is improved. Examples are given to show the improvement.

Mixed shell element for seven-parameter formulation

SUMMARY A new mixed shell element is developed for a seven-parameter formulation in this paper. The mixed shell element is constructed by assuming stress field and displacement field together. Assumed stress field and assumed displacement field can be combined by stress–strain relationship with Hu-Washizu functional. The developed mixed shell element can provide more flexible stiffness than other commercial softwares. Additionally, seven-parameter shell formulation is used instead of Reissner/Mindlin formulation, since it can provide the thickness change. Even though some commercial engineering software are not proper for very thick shell structure, the developed mixed shell element for sevenparameter formulation can be used without distinction of thick shell and thin shell. An example of shell models with different thickness is provided with solid model. Static and modal analyses are also performed for verification. Copyright 2005 John Wiley & Sons, Ltd.

A 9-node mixed shell element based on the Hu-Washizu principle

This paper proposes a new 9-node degenerated shell element based on a nonlinear mixed formulation. To avoid locking phenomena, we present a mixed formulation based on a three-field Hu-Washizu principle in which displacements, the Green strain tensors, and the second Piola-Kirchhoff stress tensors are independently assumed. In approximating strain and stress fields, covariant components of the strains and stresses measured in the element curvilinear coordinate system are interpolated by the common polynomial functions over an element. Parameter vectors of stress and strain interpolants are elementwise eliminated so that we may obtain an element stiffness matrix similar to that of the displacement model. This formulation is mathematically clear in the variational context, and can include geometrical and material nonlinearities without spoiling such clearness. Numerical results based on our approach are illustrated with satisfactory behavior of the element observed.