# Analysis of thick and thin shell structures by curved finite elements

01 Jul 1970-International Journal for Numerical Methods in Engineering (John Wiley & Sons, Ltd)-Vol. 2, Iss: 3, pp 419-451

TL;DR: In this article, a general formulation for the curved, arbitrary shape of thick shell finite elements is presented along with a simplified form for axisymmetric situations, which is suitable for thin to thick shell applications.

Abstract: A general formulation for the curved, arbitrary shape of thick shell finite elements is presented in this paper along with a simplified form for axisymmetric situations. A number of examples ranging from thin to thick shell applications are given, which include a cooling tower, water tanks, an idealized arch dam and an actual arch dam with deformable foundation.
A new process using curved, thick shell finite elements is developed overcoming the previous approximations to the geometry of the structure and the neglect of shear deformation.
A general formulation for a curved, arbitrary shape of shell is developed as well as a simplified form suitable for axisymmetric situations.
Several illustrated examples ranging from thin to thick shell applications are given to assess the accuracy of solution attainable. These examples include a cooling tower, tanks, and an idealized dam for which many alternative solutions were used.
The usefulness of the development in the context of arch dams, where a ‘thick shell’ situation exists, leads in practice to a fuller discussion of problems of foundation deformation, etc., so that practical application becomes possible and economical.

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17 Aug 2012

TL;DR: De Borst et al. as mentioned in this paper present a condensed version of the original book with a focus on non-linear finite element technology, including nonlinear solution strategies, computational plasticity, damage mechanics, time-dependent effects, hyperelasticity and large-strain elasto-plasticity.

Abstract: Built upon the two original books by Mike Crisfield and their own lecture notes, renowned scientist Rene de Borst and his team offer a thoroughly updated yet condensed edition that retains and builds upon the excellent reputation and appeal amongst students and engineers alike for which Crisfield's first edition is acclaimed. Together with numerous additions and updates, the new authors have retained the core content of the original publication, while bringing an improved focus on new developments and ideas. This edition offers the latest insights in non-linear finite element technology, including non-linear solution strategies, computational plasticity, damage mechanics, time-dependent effects, hyperelasticity and large-strain elasto-plasticity. The authors' integrated and consistent style and unrivalled engineering approach assures this book's unique position within the computational mechanics literature.

2,568 citations

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TL;DR: In this article, a method of non-destructively assessing the integrity of structures using measurements of the structural natural frequencies is described, where measurements made at a single point in the structure can be used to detect, locate and quantify damage.

Abstract: A method of non-destructively assessing the integrity of structures using measurements of the structural natural frequencies is described. It is shown how measurements made at a single point in the structure can be used to detect, locate and quantify damage. The scheme presented uses finite-element analysis, since this method may be used on any structure. The principle may, however, be used in conjunction with other mathematical techniques. Only one full analysis is required for each type of structure.Results are presented from tests on an aluminium plate and a cross-ply carbon-fibre-reinforced plastic plate. Excellent agreement is shown between the predicted and actual damage sites and a useful indication of the magnitude of the defect is obtained.

1,379 citations

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TL;DR: In this article, a simple extension is made which allows the element to be economically used in all situations by reducing the order of numerical integration applied to certain terms without sacrificing convergence properties.

Abstract: The solution of plate and shell problems by an independent specification of slopes and middle surface displacements is attractive due to its simplicity and ability of reproducing shear deformation. Unfortunately elements of this type are much too stiff when thickness is reduced.
In an earlier paper a derivation of such an element was presented1 which proved very successful in ‘thick’ situations. Here a very simple extension is made which allows the element to be economically used in all situations.
The improved flexibility is achieved simply by reducing the order of numerical integration applied to certain terms without sacrificing convergence properties. The process is of very wide applicability in improvement of element properties.

1,336 citations

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TL;DR: In this paper, a new decohesion element with the capability of dealing with crack propagation under mixed-mode loading is proposed and demonstrated, which is used at the interface between solid finite elements to model the initiation and non-self-similar growth of delaminations in composite materials.

Abstract: A new decohesion element with the capability of dealing with crack propagation under mixed-mode loading is proposed and demonstrated. The element is used at the interface between solid finite elements to model the initiation and non-self-similar growth of delaminations in composite materials. A single relative displacement-based damage parameter is applied in a softening law to track the damage state of the interface and to prevent the restoration of the cohesive state during unloading. The softening law is applied in the three-parameter Benzeggagh-Kenane mode interaction criterion to predict mixed-mode delamination propagation. To demonstrate the accuracy of the predictions, steady-state delamination growth is simulated for quasi-static loading of various single mode and mixed-mode delamination test specimens and the results are compared with experimental data.

1,285 citations

01 Jun 2002

TL;DR: In this article, a decohesion element with mixed-mode capability is proposed and demonstrated at the interface between solid finite elements to model the initiation and non-self-similar growth of delaminations.

Abstract: A new decohesion element with mixed-mode capability is proposed and demonstrated. The element is used at the interface between solid finite elements to model the initiation and non-self-similar growth of delaminations. A single relative displacement-based damage parameter is applied in a softening law to track the damage state of the interface and to prevent the restoration of the cohesive state during unloading. The softening law for mixed-mode delamination propagation can be applied to any mode interaction criterion such as the two-parameter power law or the three-parameter Benzeggagh-Kenane criterion. To demonstrate the accuracy of the predictions and the irreversibility capability of the constitutive law, steady-state delamination growth is simulated for quasistatic loading-unloading cycles of various single mode and mixed-mode delamination test specimens.

909 citations

### Cites methods from "Analysis of thick and thin shell st..."

...The definition of the relative displacements for an element with a general orientation in space is obtained using a procedure based on the work of Ahmad [23] and Beer [24]....

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##### References

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01 Jan 1959

TL;DR: In this article, the authors describe the bending of long RECTANGULAR PLATES to a cycloidal surface, and the resulting deformation of shels without bending the plates.

Abstract: CONTENTS: BENDING OF LONG RECTANGULAR PLATES TO A CYLINDRICAL SURFACE PURE BENDING OF PLATES SYMMETRICAL BENDING OF CIRCULAR PLATES SMALL DEFLECTIONS OF LATERALLY LOADED PLATES SIMPLY SUPPORTED RECTANGULAR PLATES RECTANGULAR PLATES WITH VARIOUS EDGE CONDITIONS CONTINUOUS RECTANGULAR PLATES PLATES ON ELASTIC FOUNDATION PLATES OF VARIOUS SHAPES SPECIAL AND APPROXIMATE METHODS IN THEORY OF PLATES BENDING OF ANISTROPIC PLATES BENDING OF PLATES UNDER THE COMBINED ACTION OF LATERAL LOADS AND FORCES IN THE MIDDLE PLANE OF THE PLATE LARGE DEFLECTIONS OF PLATES DEFORMATION OF SHELLS WITHOUT BENDING GENERAL THEORY OF CYLINDRICAL SHELLS SHELLS HAVING THE FORM OF A SURFACE OF REVOLUTION AND LOADED SYMMETRICALLY WITH RESPECT TO THEIR AXIS.

10,200 citations

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2,971 citations

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TL;DR: The program given here assembles and solves symmetric positive–definite equations as met in finite element applications, more involved than the standard band–matrix algorithms, but more efficient in the important case when two-dimensional or three-dimensional elements have other than corner nodes.

Abstract: The program given here assembles and solves symmetric positive–definite equations as met in finite element applications. The technique is more involved than the standard band–matrix algorithms, but it is more efficient in the important case when two-dimensional or three-dimensional elements have other than corner nodes. Artifices are included to improve efficiency when there are many right hand sides, as in automated design. The organization of the program is described with reference to diagrams, full notation, specimen input data and supplementary comments on the ASA FORTRAN print-out.

884 citations

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TL;DR: In this article, the theory of a new family of isoparametric elements for use in two-dimensional situations is described, and examples illustrating the accuracy improvement are included in the paper.

Abstract: An increase of available parameters associated with an element usually leads to improved accuracy of solution for a given number of parameters representing the whole assembly. It is possible thus to use fewer elements for the solution. As such element have yet to be able to follow prescribed boundaries to a good degree of approximation curved shapes are desirable. The paper describes the theory of a new “family” of “isoparametric” elements for use in two-dimensional situations. Examples illustrating the accuracy improvement are included.

397 citations

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TL;DR: In this paper, a method for the structural analysis of shells of revolution, composed of materials with orthotropic properties, is discussed based on the direct stiffness method and a truncated cone element is introduced to take advantage of symmetry.

Abstract: A method for the structural analysis of shells of revolution, composed of materials with orthotropic properties, is discussed. The development is based on the direct stiffness method. A truncated cone element is introduced to take advantage of symmetry. Derivations of the stiffness and stress matrices for the truncated cone element are given. Several examples are solved on the digital computing machine using a program that is based on the truncated cone element. The results are compared to other theoretical results, and the correlation is excellent. Extension of the technique to handle linear unsymmetric deformation and nonlinear symmetric deformation is discussed.

140 citations