A theoretically improved and easily implemented version of the Ahmad Thick Shell element

An implementation of the Ahmad Thick Shell element using vector manipulation, reduced integration and the incorporation of the missing term in the approximating polynomial is presented. The first two aspects are merely applications. The theory is presented elsewhere^6,7,11,18 and will not be repeated for the sake of conciseness. The third aspect was achieved by adding to the original nodal configuration a central node having only one degree-of-freedom. That node is then eliminated to preserve the original number of degrees-of-freedom. The Ahmad thick shell element is adequately presented in finite element literature.^1-3 The co-ordinate definition and the displacement field are retained and will not be dealt with henceforth. However, a different concept of the strain definition due to Irons is used and the bending terms are included in the modulus matrix. The element like all second generation isoparametric elements is easily implemented through a shape function subroutine suitable for quadrilaterals. It reproduces exactly rigid body motion for any combination of elements of any geometry even for elements withcurved sides and variable thickness. Although Ahmad's thick shell element passed the patch test for unequal parallelograms but not for quadrilaterals the version presented passes that test for quadrilaterals. In spite of its credentials, the new version denoted by A has a major setback. the presence of one too many spurious mechanism is reported.     

Improved numerical integration of thick shell finite elements

A quadratic thick shell element derived from a three-dimensional isoparametric element was first introduced by Ahmad and co-workers in 1968. This element was noted, however, to be relatively inefficient in representing bending deformations in thin shell or thin plate applications. The present paper outlines a selective integration scheme for evaluating the stiffness matrix of the element, in which each component of the strain energy is evaluated separately using a different Gaussian integration grid for each contribution. By this procedure, the excessive bending stiffness of the element, which results from the use of me quadratic interpolation functions, is avoided. The improved performance of this element, as compared with the original thick shell element, is demonstrated by analyses of a variety of thin and thick shell problems.

1 Editors' note: A similar development was outlined by O. C. Zienkiewicz and co-workers in lnt. J. num. Meth. Engng, 3 , 275–290 (1971). Some important details differ between the two papers which are thus complementary.

     

Automatic adaptive refinement for shell analysis using nine-node assumed strain element

An automatic adaptive refinement procedure for the analysis of shell structures using the nine-node degenerated solid shell element is suggested. The basic adaptive refinement principle and the effects of singularities and boundary layers on the convergence rate of the nine-node element used are discussed. A new stress recovery procedure based on the patch convective co-ordinate system concept is developed for the construction of a continuous smoothed stress field over the shell domains. The stress recovery procedure is easy to implement, requires a modest computational effort and needs only local patch information. It can be applied to shells with non-uniform thickness as well as to multi-layered shell structures. The smoothed recovered stress obtained is then used with the Zienkiewicz and Zhu error estimator for a posteriori error estimation during the adaptive refinement analysis. Numerical results which are in good agreement with theoretical predictions are obtained and they indicate that the current adaptive refinement procedure can eliminate the effect of singularities inside the problem domains so that a near-optimal convergence rate is achieved in all the numerical examples. This also indicates that the stress recovery procedure can produce an accurate stress field and as a result the error estimator can reflect the error distribution of the finite element solution. Even though in the current study only one type of element is used in the analysis, the whole adaptive refinement scheme can be readily applied to any other types of degenerated solid element. © 1997 John Wiley & Sons, Ltd.     

Finite deformation formulation of a shell element for problems of sheet metal forming

An elastic-plastic thin shell finite element suitable for problems of finite deformation in sheet metal forming is formulated. Hill's yield criterion for sheet materials of normal anisotropy is applied. A nonlinear shell theory in a form of an incremental variational principle and a quasi-conforming element technique are employed in the Lagrangian formulation. The shell element fulfills the inter-element C ¹ continuity condition in a variational sense and has a sufficient rank to allow finite stretching, rotation and bending of the shell element. The accuracy and efficiency of the finite element formulation are illustrated by numerical examples.     

An efficient isostatic mixed shell element for coarse mesh solution

A novel mixed shell finite element (FE) is presented. The element is obtained from the Hellinger–Reissner variational principle and it is based on an elastic solution of the generalized stress field, which is ruled by the minimum number of variables. As such, the new FE is isostatic because the number of stress parameters is equal to the number of kinematical parameters minus the number of rigid body motions. We name this new FE MISS-8. MISS-8 has generalized displacements and rotations interpolated along its contour and drilling rotation is also considered as degree of freedom. The element is integrated exactly on its contour, it does not suffer from rank defectiveness and it is locking-free. Furthermore, it is efficient for recovering both stress and displacement fields when coarse meshes are used. The numerical investigation on its performance confirms the suitability, accuracy, and efficiency to recover elastic solutions of thick- and thin-walled beam-like structures. Numerical results obtained with the proposed FE are also compared with those obtained with isogeometric high-performance solutions. Finally, numerical results show a rate of convergence between h² and h⁴ .     

Equivalent homogeneous finite element for composite materials via reissner principle. Part I: Finite element for plates

The stiffness matrix in the finite element method for multi-layered materials is generally computed by expressing the strain energy in each layer and adding them together. In order to lower the computing time, which may be prohibitive if the number of layers is high, and to get accurate information on the stresses, especially on transverse shear stresses, we present a new finite element using the Reissner principle. In the first part the case of plates will be detailed: extensions to shell problems will be presented in the second part. The efficiency of the method is tested on a special analytic solution, and some examples are given.     

Triangular finite element for analysis of thick laminated shells

The development of a general curved triangular element based on an assumed displacement potential energy approach is presented for the analysis of arbitrarily laminated thick shells. The associated laminated shell theory assumes transverse inextensibility and layerwise constant shear angle. The present element is a quadratic triangle of C⁰-type in the curvilinear co-ordinate plane, which is then mapped onto a curved surface. Convergence of transverse displacement, moments, stresses and the effect of two Gauss quadrature schemes also form a part of the investigation. Examples of two laminated shell problems demonstrate the accuracy and efficiency of the present element. Comparison of the present LCST (layerwise constant shear-angle theory) based solutions, with those based on the CST (constant shear-angle theory) clearly demonstrates the superiority of the former over the latter, especially in the prediction of the distribution of the surface-parallel displacements and stresses through the laminate thickness.     

Hybrid hexahedral element for solids,plates, shells and beams by selective scaling

A robust two-field hexahedral element capable of handling plate/shell, beam and nearly incompressible material analyses without locking are presented. Starting with the assumed stress element of Pian and Tong,⁷ parasitic strain components leading to locking in plate, shell and beam analyses are first identified. Locking can be alleviated by scaling down selectively the parasitic strain components in the leverage matrix. Unfortunately, the element then fails the patch test. However, patch test correction and reduction in computation can be achieved by the recently proposed admissible matrix formulation. The resulting element is lock-free and very efficient. All matrices involved in constructing the stiffness matrix can be derived explicitly. The accuracy of the element is tested by popular bench-mark problems.     

Treatment of material discontinuities in finite element computations

We consider finite element analysis of problems with discontinuous material coefficients. For applications in which the material interface crosses an element, we develop special elements with an embedded flux constraint at the interface. This new procedure is compared with the standard finite element method with interface coincident with the element boundary and with an existing method proposed by Steven.¹ Supporting numerical studies are conducted and rates of convergence for the solution and interface flux are examined. Some local superconvergence behaviour is observed.     

A degenerate solid element for linear fracture analysis of plate bending and general shells

A general curved element of arbitrary shape for both thick and thin shells is proposed for the linear fracture analysis of a through crack in a shell or a plate. The element is derived from a degenerate 20-noded solid isoparametric element using reduced integration technique. The 1/√(r) singularity of the strains is obtained by the same procedure proposed earlier for two- and three-dimensional problems,^1,2 viz. by placing the mid-side nodes near the crack at the quarter points. Several illustrated examples ranging from classical solutions to practical problems are given to assess the accuracy of solution attainable.     

Combined micro- and macromechanical considerations of layered composite shells

This contribution deals with a computational treatment of the non-linear behaviour of composite sandwich shells with extremely thin face layers and a relatively thick orthotropic core, and multi-layered fiber reinforced polymer or metal matrix composite shells. Special emphasis is laid on algorithms which allow the consideration of local effects, like face layer wrinkling in sandwich shells and progressive damage or microplasticity in fibre composites, by semi-analytical approaches implemented into a finite element formulation. This leads to the advantage that the shell structure must be discretized only up to an element mesh density which is sufficient for describing the global deformation and stability behaviour.     

4-Node combined shell element with semi-EAS-ANS strain interpolations in 6-parameter shell theories with drilling degrees of freedom

A 4-node C ⁰ shell element with drilling degrees of freedom is presented in this paper. The element is developed within the nonlinear 6-field shell theory. Kinematics of the shell is described by two independent fields: the vector field for translations and the proper orthogonal tensor field for rotations. Within the theoretical formulation no restriction is applied on magnitudes of displacements and rotations. To avoid locking phenomena the proposed element combines two interpolation schemes: the assumed natural strain (ANS) for transverse shear strains and the enhanced assumed strain (EAS). The latter interpolation is used with asymmetric (in-plane) membrane strains. The performance of the element is evaluated by example of benchmark problems with special emphasis on shell structures containing orthogonal intersections.     

Elasto‐plastic finite element analysis of shells with damage due to microvoids

This paper presents a non‐linear finite element analysis for the elasto‐plastic behaviour of thick/thin shells and plates with large rotations and damage effects. The refined shell theory given by Voyiadjis and Woelke (Int. J. Solids Struct. 2004; 41 :3747–3769) provides a set of shell constitutive equations. Numerical implementation of the shell theory leading to the development of the C⁰ quadrilateral shell element (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted) is used here as an effective tool for a linear elastic analysis of shells. The large rotation elasto‐plastic model for shells presented by Voyiadjis and Woelke (General non‐linear finite element analysis of thick plates and shells. 2006, submitted) is enhanced here to account for the damage effects due to microvoids, formulated within the framework of a micromechanical damage model. The evolution equation of the scalar porosity parameter as given by Duszek‐Perzyna and Perzyna (Material Instabilities: Theory and Applications, ASME Congress, Chicago, AMD‐Vol. 183/MD‐50, 9–11 November 1994; 59–85) is reduced here to describe the most relevant damage effects for isotropic plates and shells, i.e. the growth of voids as a function of the plastic flow. The anisotropic damage effects, the influence of the microcracks and elastic damage are not considered in this paper. The damage modelled through the evolution of porosity is incorporated directly into the yield function, giving a generalized and convenient loading surface expressed in terms of stress resultants and stress couples. A plastic node method (Comput. Methods Appl. Mech. Eng. 1982; 34 :1089–1104) is used to derive the large rotation, elasto‐plastic‐damage tangent stiffness matrix. Some of the important features of this paper are that the elastic stiffness matrix is derived explicitly, with all the integrals calculated analytically (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted). In addition, a non‐layered model is adopted in which integration through the thickness is not necessary. Consequently, the elasto‐plastic‐damage stiffness matrix is also given explicitly and numerical integration is not performed. This makes this model consistent mathematically, accurate for a variety of applications and very inexpensive from the point of view of computer power and time. Copyright © 2006 John Wiley & Sons, Ltd.     

Hyperbolic paraboloid shell analysis via mixed finite element formulation

An isoparametric rectangular mixed finite element is developed for the analysis of hypars. The theory of shallow thin hyperbolic paraboloid shells is based on Kirchhoff–Love's hypothesis and a new functional is obtained using the Gâteaux differential. This functional is written in operator form and is shown to be a potential. Proper dynamic and geometric boundary conditions are obtained. Applying variational methods to this functional, the HYP9 finite element matrix is obtained in an explicit form. Since only first-order derivatives occur in the functional, linear shape functions are used and a C⁰ conforming shell element is presented. Variation of the thickness is also included into the formulation without spoiling the simplicity. The formulation is applicable to any boundary and loading condition. The HYP9 element has four nodes with nine Degrees Of Freedom (DOF) per node—three displacements, three inplane forces and two bending, one torsional moment (4 × 9). The performance of this simple, and elegant shell element, is verified by applying it to some test problems existing in the literature. Since the element matrix is obtained explicitly, there is an important save of computer time.     

3D‐based hp‐adaptive first‐order shell finite element for modelling and analysis of complex structures—Part 1: The model and the approximation

The paper presents a 3D‐based adaptive first‐order shell finite element to be applied to hierarchical modelling and adaptive analysis of complex structures. The main feature of the element is that it is equipped with 3D degrees of freedom, while its mechanical model corresponds to classical first‐order shell theory. Other useful features of the element are its modelling and adaptive capabilities. The element is assigned to hierarchical modelling and hpq‐adaptive analysis of shell parts of complex structures consisting of solid, thick‐ and thin‐shell parts, as well as of transition zones, where h, p and q denote the mesh density parameter and the longitudinal and transverse orders of approximation, respectively. The proposed hp‐adaptive first‐order shell element can be joined with 3D‐based hpq‐adaptive hierarchical shell elements or 3D hpp‐adaptive solid elements by means of the family of 3D‐based hpq/hp‐ or hpp/hp‐adaptive transition elements. The main objective of the first part of our research, presented in this paper, is to provide non‐standard information on the original parts of the element algorithm. In order to do that, we present the definition of shape functions necessary for p‐adaptivity, as well as the procedure for imposing constraints corresponding to the lack of elongation of the straight lines perpendicular to the shell mid‐surface, which is the procedure necessary for q‐adaptivity. The 3D version of constrained approximation presented next is the basis for h‐adaptivity of the element. The second part of our research, devoted to methodology and results of the numerical research on application of the element to various plate and shell problems, are described in the second part of this paper. Copyright © 2006 John Wiley & Sons, Ltd.     

A geometrical non-linear brick element based on the EAS-method

This paper presents the development of a 3-D brick element with enhanced assumed strains for a geometrically non-linear theory. Some linear and non-linear examples show that this element can be used successfully in the whole range of solid structures. Thin 2-D- and 3-D-beam and shell structures are calculated with few 3-D elements and the results are the same as for shell or beam elements. © 1997 John Wiley & Sons, Ltd.     

An assessment of the semiloof shell element

The semiloof shell element is the final product of years of research by Professor Irons and his colleagues, during which period the three-dimensional quadratic displacement isoparametric elements were successively modified into Ahmad thick shells, then eventually, using discrete Kirchhoff hypotheses, into the thin semiloof shells. The retention of curved sides and midside nodes retains the familiar appearance of the two- and three-dimensional isoparametric families, which enables ease of use with existing pre- and post-processing facilities. The element has been widely adapted and used frequently for elastic analyses of wide-ranging structures. It has also been developed for nonlinear geometric and material behaviour. However, despite the unparalleled generality in performance of this element, some unresolved problem areas still exist, particularly with use of reduced integrating rules. Such problems are highlighted in the paper and rectified where possible. A variety of problems are described to illustrate the above points and other interesting features, such as the treatment of thermal loading, and to emphasize the importance of this contribution to finite element technology by Professor Irons.     

Anisotropic cylindrical shell element based on discrete Kirchhoff theory

A triangular cylindrical shell element based on discrete Kirchhoff theory is developed. It is a three-node, 27-degrees-of-freedom element using cubic polynomials for the tangential and normal displacement interpolations. The normal rotations are independently interpolated by quadratic polynomials. The formulation is capable of modelling general anisotropy representative of multi-layered, multi-directionally oriented composite construction. The numerical results indicate that the solution for displacements and stresses of cylindrical shells converge monotonically and rapidly to those based on deep shell theory.