Three-dimensional finite element for analysing thin plate/shell structures

A three-dimensional (3-D) hexahedron finite element is presented for the analysis of thin plate/shell structures. The element employs an explicit algebraic definition of six uniform (continuum) strains, six rigid body modes and classical Lagrange-Germain-Kirchhoff thin plate bending modes. Nine additional stiffness factors are used to control higher-order hourglass modes. The element may be used for plate/shell analyses where the flat plate assumptions are appropriate. Also it can easily be adapted to form transition elements to lower order 2-D elements, or to higher-order 3-D continuum elements. The stiffness matrix satisfies the geometric isotropy requirement, passes the patch test, and gives essentially identical response to either applied transverse corner forces or to twisting moments applied on the corner, a requirement of Kirchhoff's corner conditions for a classical thin plate. Several examples are presented to demonstrate the performance of this finite element.     

Machine locking of degenerated thin shell elements

The degenerated shell element is one of the most efficient elements for analysing shell structures. However, it is known to result in rather stiff models when used in thin element applications. The phenomena associated with this behaviour are known as locking phenomena. This paper analyses the machine locking mechanism developed in thin to very thin Lagrangian and serendipity elements. The machine related locking phenomenon is distinguished from the shear and membrane locking phenomena. A remedy for the pure machine locking problem is developed for the two elements. The proposed remedy is based on the technique of the modified transverse shear modulus. It is also extended to control shear locking. The proposed technique is shown to completely eliminate machine locking. Also, it is shown to effectively alleviate stiffening effects due to the presence of spurious shear strain.     

J-integral analyses of plate and shell structures with through-wall cracks using thick shell elements

A new evaluation method using thick shell elements is proposed to calculate the distribution of the J-integral values along crack fronts of through-wall cracks in plate and shell structures. Dividing tentatively the thick shell elements into several layers in the thickness direction, integral paths are defined at each layer to obtain the thickness distributions of the J-integral values of through-wall cracks in plate and shell structures. The three-dimensional definition of the J-integral is used to calculate the J-integral value at each layer. The results of several stress intensity factor analyses are presented to show the effectiveness of the present evaluation method. The distribution of the J-integral value along a crack front is also compared between the methods using the thick shell elements and the three-dimensional solid elements.     

Sensitivity analysis with plate and shell finite elements

This paper addresses the problem of calculating sensitivity data by direct methods for isoparametric plate or shell elements. Sensitivity parameters of interest include intrinsic properties such as material modulus and plate thickness, as well as geometry variables which influence the size and shape of a structure. The sensitivity calculation therefore must consider the parametric mapping within an element, as well as the influence of geometric variables on the orientation of an element in space. The methods presented specialize directly to continuum elements, in which the co-ordinate transformation is omitted, or to simple structural members situated arbitrarily in space. Numerical examples are presented which illustrate the accuracy of the proposed techniques, and the effect of discretization error on computed sensitivities.     

Approximation of general shell problems by flat plate elements

The approximation of general shell problems by flat plate elements is very popular among engineering people. These methods have as common features a nonconforming approximation of the geometry of the considered shell using facet elements and a pseudo-conforming approximation of the components of the displacement, i.e., an approximation using conforming plate elements over every flat element. In this work, we analyze the compatibility conditions which have to be satisfied by the degrees of freedom at every node of the triangulation. Next, we obtain several interesting results valid for general shells and we prove the “pseudo-convergence” of the method for a class of shallow shells. Then, this careful study allows to introduce a perturbation of the bending terms upon each facet; the corresponding new method is convergent for arbitrary thin shells.     

Approximation of general shell problems by flat plate elements

In this paper, we analyse an approximation of general thin shell problems where the middle surface is approached by flat triangular facets, whereas the displacement field is approximated by triangles of type (1) for the membrane components and by reduced H.C.T. triangles for the bending component. In this second part of the paper, we define a sixth degree of freedom: the rotation around the normal. This introduces a small perturbation but has the advantage to make the implementation easier: indeed, the connection between two adjacent facets is simply realized by imposing the continuity of the displacement and rotation vectors at the vertices of the triangulation. We prove the pseudo-convergence of this method for sufficiently shallow shells; then we propose a new expression of the bending terms upon each facet for which the approximation method is unconditionally convergent, for arbitrary thin shells.     

Approximation of general shell problems by flat plate elements

An alternative to the approximation of general thin shell problems by flat plate elements (Parts 1 and 2) is proposed: the middle surface is now approached by curved triangular facets. It consists of a P ₂-Lagrange interpolation of the geometry of the shell, while the displacement field is approximated by triangles of type (1) for the membrane components and by reduced H.C.T. triangles for the bending component. To make the implementation easier, we also consider the addition of the drilling degree of freedom as a sixth independent degree of freedom by node (Part 2). We prove the convergence of the method for arbitrary thin shells.     

Shear correction for Mindlin type plate and shell elements

In this paper, two shear correction methods, which can guarantee good accuracy, suitable for different materials and easy to implement, are proposed for the first‐order theory based plate and shell elements. The general variational principles based on the thermodynamic potentials are first derived, to provide a uniform frame for the different material non‐linearity. Then, the two shear correction methods are derived from the obtained general variational principles using the assumed strain and assumed stress methods, respectively. The shear correction factors (SCF) for homogeneous elastic materials are evaluated using the two methods, and the same result of 0.8 is obtained. The proposed shear correction method is then applied to an elastic‐linear hardening plastic deep beam for verification, and the conformable result is obtained as compared with the plane strain result. Copyright © 2006 John Wiley & Sons, Ltd.     

A simple plate/shell triangle

A simple shear flexible triangular plate/shell element with three/five displacement degrees-of-freedom at the three corner nodes only, is identified. It does not lock or have zero-energy mechanisms. The specific geometrical configurations in which this is possible are derived through field consistency patch tests.     

Simple and efficient shear flexible two-node arch/beam and four-node cylindrical shell/plate finite elements

Several simple and accurate C° two-node arch/beam and four-node cylindrical shell/plate finite elements are presented in this paper. The formulation used here is based on the refined theory of thick cylindrical shells and the quasi-conforming element technique. Unlike most C° elements, the element stiffness matrix presented here is given explicitly. In spite of their simplicity, these C° finite elements posseses linear bending strains and are free from the deficiencies existing in curved C° elements such as shear and membrane locking, spurious kinematic modes and numerical ill-conditioning. These finite elements are valid not only for thick/thin beams and plates, but also for arches/straight beams and cylindrical shells/plates. Furthermore, these C° elements can automatically reduce to the corresponding C¹ beam and plate elements and give the C° beam element obtained by the reduced integration as a special case. Several numerical examples indicate that the simple two-node arch/beam and four-node cylindrical shell/plate elements given in this paper are superior to the existing C° elements with the same element degrees of freedom. Only the formulation of the rectangular cylindrical shell and plate element is presented in this paper. The formulation of an arbitrarily quadrilateral plate element will be presented in a follow-up paper³².     

Hellinger–Reissner principles for plate and shell finite elements

The Hellinger–Reissner generalized variational principle is used to deduce further principles directed especially towards formulation of high integrity finite elements for plates and curved shells in polynomial parametric representation. Assumed stresses in these principles are derived from displacements and supplemented with stresses derived from stress functions.     

A simple class of finite elements for plate and shell problems. I: Elements for beams and thin flat plates

This is the first paper of a pair which together discuss the development of a class of overlapping hinged bending finite elements which are suitable for the analysis of thin-shell, plate and beam structures. These elements rely on a simple physical analogy, involving overlapping hinged facets. They are based on quadratic overlapping assumed displacement functions. Only translational nodal degrees of freedom are necessary, which is a significant simplification over most other currently available beam, plate and shell finite elements which employ translational, rotational and higher-order nodal variables. In this paper the hinged bending element concept is introduced, and the hinged beam bending (HBB) and hinged plate bending (HPB) elements are formulated. In paper II these concepts are extended to develop a hinged shell bending (HSB) element. The HSB element can be readily combined with the constant strain triangular (CST) plane stress finite element for the modelling of thin-shell structures.     

Computational modeling of high performance cementitious thin shell elements under in-plane shear

High-performance cementitious materials are widely used in the construction of thin shell elements. This study investigates a simulation method based on composite layered shells for the nonlinear analysis of high-performance cementitious elements under in-plane shear. A tube torsion test is simulated and analyzed with MSC-MARC and its results are compared to an alternative calculation method, the Simplified Model for Combined Stress resultants (SMCS), as well as with experimental data. The simulation method is found to produce accurate results for fully under-reinforced elements with a range of strong to weak reinforcement ratios less than 2.     

Crushing of compact elements in impact interaction with a thin plate

On the basis of investigations it was established that the limit speeds of crushing are affected by the scale, the ductility and other physicomechanical properties of materials. Crushing is the result of one or several mutually interacting effects (intense plastic deformation, wave processes, and fields of residual stresses).Translated from Problemy Prochnosti, No. 4, pp. 51–57, April, 1994.     

A simple triangular element for thick and thin plate and shell analysis

A new plate triangle based on Reissner–Mindlin plate theory is proposed. The element has a standard linear deflection field and an incompatible linear rotation field expressed in terms of the mid-side rotations. Locking is avoided by introducing an assumed linear shear strain field based on the tangential shear strains at the mid-sides. The element is free of spurious modes, satisfies the patch test and behaves correctly for thick and thin plate and shell situations. The element degenerates in an explicit manner to a simple discrete Kirchhoff form.     

A simple method for including shear deformations in thin plate elements

A method is described for including the effect of shear deformations in existing thin plate finite elements, and thereby extending their range of application to include moderately thick plates. The method does not add extra degrees of freedom to the final element, so the thick and thin plate elements can be used interchangeably, and the thick plate solution is not appreciably more expensive than the thin plate solution. It is assumed that the shear deformations are constant over the element and, to account for this, two extra internal shear strain variables are added to the element. Various methods for eliminating these internal variables are examined but it is shown to be impossible to simultaneously satisfy both the constant bending moment and constant shear patch tests, except for parallelograms. However, one method gives elements which pass the constant shear patch test and, although failing the constant bending moment patch test for arbitrary geometries, gives errors which are small enough to be neglected in most engineering applications. This method has been applied to a triangular plate element and it is shown that the results obtained with this element converge (for all practical purposes) to the correct thick plate results.     

A simple class of finite elements for plate and shell problems. II: An element for thin shells,with only translational degrees of freedom

This paper is the second of a pair which discuss the development of a class of overlapping hinged bending finite elements, which are suitable for the analysis of thin-shell, plate and beam structures. These elements rely on a simple physically appealing analogy, in which overlapping hinged facets are used to represent bending effects. They are based on quadratic overlapping assumed displacement functions, which results in constant stress/strain representation. Only translational nodal degrees of freedom are necessary, which is a significant advantage over most other currently available beam, plate and shell finite elements which employ translational, rotational and higher-order nodal variables. In paper I the hinged bending element concept has been introduced, and the hinged beam bending (HBB) and hinged plate bending (HPB) elements formulated. In the present paper these concepts are extended to develop a hinged shell bending (HSB) element. The HSB element can be readily combined with the constant strain triangular (CST) plane stress finite element for the modelling of thin-shell structures; and the combined HSB-CST element is tested against a number of 'standard' thin-shell problems. The present paper, like paper I, is conducted entirely in the context of small-displacement elastic behaviour.     

The mathematical shell model underlying general shell elements

We show that although no actual mathematical shell model is explicitly used in ‘general shell element’ formulations, we can identify an implicit shell model underlying these finite element procedures. This ‘underlying model’ compares well with classical shell models since it displays the same asymptotic behaviours—when the thickness of the shell becomes very small—as, for example, the Naghdi model. Moreover, we substantiate the connection between general shell element procedures and this underlying model by mathematically proving a convergence result from the finite element solution to the solution of the model. Copyright © 2000 John Wiley & Sons, Ltd.     

Variable-node plate and shell elements with assumed natural strain and smoothed integration methods for nonmatching meshes

In this paper, novel finite elements that include an arbitrary number of additional nodes on each edge of a quadrilateral element are proposed to achieve compatible connection of neighboring nonmatching meshes in plate and shell analyses. The elements, termed variable-node plate elements, are based on two-dimensional variable-node elements with point interpolation and on the Mindlin–Reissner plate theory. Subsequently the flat shell elements, termed variable-node shell elements, are formulated by further extending the plate elements. To eliminate a transverse shear locking phenomenon, the assumed natural strain method is used for plate and shell analyses. Since the variable-node plate and shell elements allow an arbitrary number of additional nodes and overcome locking problems, they make it possible to connect two nonmatching meshes and to provide accurate solutions in local mesh refinement. In addition, the curvature and strain smoothing methods through smoothed integration are adopted to improve the element performance. Several numerical examples are presented to demonstrate the effectiveness of the elements in terms of the accuracy and efficiency of the analyses.