Basic error estimates for elliptic problems

SUMMARY An assessment of flat triangular plate bending elements with displacement degrees-of-freedom at the three comer nodes only is presented, with the purpose of identifying the most effective for thin plate analysis. Based on a review of currently available elements, specific attention is given to the theoretical and numerical evaluation of three triangular 9 degrees-of-freedom elements; namely, a discrete Kirchhoff theory (DKT) element, a hybrid stress model (HSM) element and a selective reduced integration (SRI) element. New and efficient formulations of these elements are discussed in detail and the results of several example analyses are given. It is concluded that the most efficient and reliable three-node plate bending elements are the DKT and HSM elements.

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A study of three‐node triangular plate bending elements

1 Formulation of the equations of motion 2 Element energy functions 3 Introduction to the finite element displacement method 4 In-plane vibration of plates 5 Vibration of solids 6 Flexural vibration of plates 7 Vibration of stiffened plates and folded plate structures 8 Vibration of shells 9 Vibration of laminated plates and shells 10 Hierarchical finite element method 11 Analysis of free vibration 12 Forced response 13 Forced response II 14 Computer analysis technique

Introduction to finite element vibration analysis

A simple and efficient finite element is introduced for plate bending applications. Bilinear displacement and rotation functions are employed in conjunction with selective reduced integration. Numerical examples indicate that, despite its simplicity, the element is surprisingly accurate.

A simple and efficient finite element for plate bending

A simple quadrilateral shell element

The application of the high precision triangular plate bending element to problems with curved boundaries is considered. Appropriate edge conditions for nodal points on these boundaries are derived. The error inherent in representing the shape of a curved boundary by a series of straight segments is found to be the limiting factor on accuracy, while the effect of approximations in the actual boundary conditions is minor. To overcome the first type of error, the high precision element is modified to include one curved edge. Substantial improvements in accuracy are obtained, as demonstrated in example calculations for circular and elliptical plates.

Finite element analysis of plates with curved edges

: The stiffness and mass matrices for a relatively simple finite cylindrical shell element are presented. The element has 28 degrees of freedom corresponding to the seven generalized coordinates at each corner. All six rigid body modes for the element are adequately represented by this model. The element is used to predict the vibrations of a curved fan blade, and the results are verified experimentally. It is found that a 4 x 4 grid of the finite elements is sufficient to predict the first twelve vibration frequencies for the fan blade to within ten percent. The agreement between experimental and theoretical mode shapes is generally very good.

Vibration Analysis of Cantilevered Curved Plates Using a New Cylindrical Shell Finite Element

Annular and circular sector finite elements for plate bending

The convergence rates of eigenvalue solutions using two finite plate bending elements are studied. The elements considered are the well-known 12 degree of freedom, non-conforming rectangular element and the 16 degree of freedom, conforming rectangular element. Three problems are analysed, a square plate simply supported on two opposite sides with the other two sides clamped, simply supported, or free. Closed form, finite element solutions for these problems are obtained by using shifting E-operators.

With few exceptions, eigenvalue solutions found with the non-conforming element converge from below the exact answers at an asymptotic rate of n−2, where n is the number of elements on a side. However, since the array size needed for such convergence is very large, little can be said about the convergence rates for practical arrays. The conforming element solutions converge from above at an asymptotic rate of n−4. A comparison of the errors involved in using these two elements shows that the conforming element is far superior to the non-conforming element.

Convergence studies of eigenvalue solutions using two finite plate bending elements

element uses as generalized displacements the tangential displacements and their first derivatives, plus the normal displacement and its first and second derivatives at each vertex, a total of 36 in all. The transverse displacement function for the element contains a complete quartic polynomial plus some higher degree terms and allows a cubic variation of normal slope along each edge. The tangential displacement functions are complete cubic polynomials, and it is shown that this formulation leads to a consistent asymptotic strain energy convergence rate of n~6, where n is the number of elements per side of a shell. Results show that this element is exceedingly accurate and far outperforms early lower order elements in predicting stresses as well as displacements. This element may easily be converted into an efficient and useful cylindrical shell element simply by substituting cylindrical shell theory for the shallow shell theory. The purpose of this Note is to present the necessary derivations for doing this and to illustrate the element's usefulness on an example application. The problem of a cylindrical shell with a circular cut-out is analyzed, and the stress concentration results are compared with those from both an approximate analytic analysis2 and a new finite difference variational approach.3 The following presentation is necessarily brief, but more details are available in Ref . 4.

A high-precision triangular cylindrical shell finite element

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