An improved constant membrane and bending stress shell element for explicit transient dynamics

Key, Samuel W.; Hoff, Claus

doi:10.1016/0045-7825(95)00785-y

Cited by 34 publications

(20 citation statements)

References 11 publications

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“…Reference [5]. Elements generated by this approach result in a de facto satisfaction of the first-order Irons patch test, provided a spatially-linear motion can be represented exactly by the displacement assumptions within the element domain.…”

Section: Gradient/divergence Operatormentioning

confidence: 99%

“…The four-node shell proved to be more difficult to develop than the solid finite elements. However, four-node quadrilateral shell finite elements are now available that perform well in transient dynamic simulations-for example, Belytschko et al [4] and Key and Hoff [5]. Striving for a constant membrane stress resultant, constant bending stress resultant formulation that has a minimum of arithmetic operations can lead to implementations that are not satisfactory approximations of the governing partial differential equations.…”

Section: Introductionmentioning

confidence: 98%

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A low‐order, hexahedral finite element for modelling shells

Key

Gullerud

Koteras

2004

Numerical Meth Engineering

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SUMMARYA thin, eight-node, tri-linear displacement, hexahedral finite element is the starting point for the derivation of a constant membrane stress resultant, constant bending stress resultant shell finite element. The derivation begins by introducing a Taylor series expansion for the stress distribution in the isoparametric co-ordinates of the element. The effect of the Taylor series expansion for the stress distribution is to explicitly identify those strain modes of the element that are conjugate to the mean or average stress and the linear variation in stress. The constant membrane stress resultants are identified with the mean stress components, and the constant bending stress resultants are identified with the linear variation in stress through the thickness along with in-plane linear variations of selected components of the transverse shear stress. Further, a plane-stress constitutive assumption is introduced, and an explicit treatment of the finite element's thickness is introduced. A number of elastic simulations show the useful results that can be obtained (tip-loaded twisted beam, point-loaded hemisphere, point-loaded sphere, tip-loaded Raasch hook, and a beam bent into a ring). All of the gradient/divergence operators are evaluated in closed form providing unequivocal evaluations of membrane and bending strain rates along with the appropriate divergence calculations involving the membrane stress and bending stress resultants. The fact that a hexahedral shell finite element has two distinct surfaces aids sliding interface algorithms when a shell folds back on itself when subjected to large deformations.

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Section: Gradient/divergence Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

A low‐order, hexahedral finite element for modelling shells

Key

Gullerud

Koteras

2004

Numerical Meth Engineering

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“…Thus, the volume of the eight-node Figure 2. A linear four-node tetrahedron (i; j; k; l = 1, 2, 3, 4) to which has been added four mid-face nodal points (m; n; o; p = 5, 6,7,8). A linear 'sub-tetrahedron' is associated with each mid-face nodal point so that-for example, ( j; k; l; m = 2, 3,4,5) enriched tetrahedron is given by the volume of the original or parent four-node tetrahedron V 0 and the volume changes introduced by movement of the mid-face nodes,…”

Section: Eight-node Tetrahedral ÿNite Elementmentioning

confidence: 99%

A suitable low-order, tetrahedral finite element for solids

Key

Heinstein

Stone

et al. 1999

Int. J. Numer. Meth. Engng.

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SUMMARYTo use the all-tetrahedral mesh generation capabilities existing today, we have explored the creation of a computationally e cient eight-node tetrahedral ÿnite element (a four-node tetrahedral ÿnite element enriched with four mid-face nodal points). The derivation of the element's gradient operator, studies in obtaining a suitable mass lumping and the element's performance in applications are presented. In particular, we examine the eight-node tetrahedral ÿnite element's behavior in longitudinal plane wave propagation, in transverse cylindrical wave propagation, and in simulating Taylor bar impacts. The element samples only constant strain states and, therefore, has 12 hourglass modes. In this regard, it bears similarities to the eight-node, mean-quadrature hexahedral ÿnite element. Comparisons with the results obtained from the mean-quadrature eight-node hexahedral ÿnite element and the four-node tetrahedral ÿnite element are included. Given automatic all-tetrahedral meshing, the eight-node, mean-quadrature tetrahedral ÿnite element is a suitable replacement for the eight-node, mean-quadrature hexahedral ÿnite element and meshes requiring an inordinate amount of user intervention and direction to generate.

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“…The KHQ4 Key-Hoff plane-stress 4-node quadrilateral finite element is documented in the publication by Key and Hoff [1995]. Using the current geometry of the finite element the vertex translational and rotational velocities are interpolated with bilinear shape functions.…”

Section: Khq4 Key-hoff Plane-stress Quadrilateralmentioning

confidence: 99%

Summary compilation of shell element performance versus formulation.

Heinstein¹,

Hales

Breivik³

et al. 2011

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This document compares the finite element shell formulations in the Sierra Solid Mechanics code. These are finite elements either currently in the Sierra simulation codes Presto and Adagio, or expected to be added to them in time. The list of elements are divided into traditional two-dimensional, plane stress shell finite elements, and three-dimensional solid finite elements that contain either modifications or additional terms designed to represent the bending stiffness expected to be found in shell formulations. These particular finite elements are formulated for finite deformation and inelastic material response, and, as such, are not based on some of the elegant formulations that can be found in an elastic, infinitesimal finite element setting. Each shell element is subjected to a series of 12 verification and validation test problems. The underlying purpose of the tests here is to identify the quality of both the spatially discrete finite element gradient operator and the spatially discrete finite element divergence operator. If the derivation of the finite element is proper, the discrete divergence operator is the transpose of the discrete gradient operator. An overall summary is provided from which one can rank, at least in an average sense, how well the individual formulations can be expected to perform in applications encountered year in and year out. A letter grade has been assigned albeit sometimes subjectively for each shell element and each test problem result. The number of A's, B's, C's, et cetera assigned have been totaled, and a grade point average (GPA) has been computed, based on a 4.0-system. These grades, combined with a comparison between the test problems and the application problem, can be used to guide an analyst to select the element with the best shell formulation.

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An improved constant membrane and bending stress shell element for explicit transient dynamics

Cited by 34 publications

References 11 publications

A low‐order, hexahedral finite element for modelling shells

A low‐order, hexahedral finite element for modelling shells

A suitable low-order, tetrahedral finite element for solids

Summary compilation of shell element performance versus formulation.

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