Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1994. / Vita. Abstract. Includes bibliographical references (leaves 241-254). Also available via the Internet.

Multistable shells are not yet completely understood. Even under the widespread Uniform Curvature (UC) assumption, the limits of this behaviour have not been established and the influence of individual material and geometric parameters has not been described conclusively; this research explores these open questions. In addition, this project was motivated by the need for practical design guidelines and the pursuit of alternative construction and actuation methods for multistable shells. Our analysis is based on an expression for the strain energy of a shell under a set of simplifying assumptions— primarily the aforementioned UC assumption. We extend this concept beyond the work of previous authors by admitting a more diverse range of anisotropic materials. Furthermore, we take advantage of some aspects of the mathematical field of Catastrophe Theory (CT) to maximise the generality of available results. When appropriate, we examine aspects of our predictions by constructing relevant shell structures, with particular focus on material considerations. A commercial Finite Element Analysis package provides additional means of analysis and comparison. On the theoretical front, the influence of certain control parameters on the availability of multistability is described in closed-form while a unique graphical overview of the limits of this behaviour is provided. In the lab, a novel tristable shell is constructed from a laminate and the use of specialized materials is scrutinised. In a subsequent project, a bistable spherical cap made from a customized material is actuated by a magnetic field—the ensuing snap-through event is recorded with a high-speed camera, leading to valuable insights on the transition geometry. Furthermore, we confirm the possibility of bistability for developable, non-prestressed shells, composed of a single material, using grid shells and thin honeycomb shells.

A general analysis for a simply supported prismatic folded plate structure is presented. A stiffness matrix method is used with the individual plates taken as structural elements. When loads are applied to the structure the solution yields joint deflections and internal plate forces. The exact elasticity theory is used for the in plane or membrane solution and the classical thin plate theory is used for the out of plane or bending solution. By modifying a computer program used for stiffness matrix frame analysis, it is possible to solve a folded plate structure with more than two plates connected at the same joint. This is demonstrated by the solution of an I beam torsion problem, which also provides a check of existing torsion theories. Cylindrical barrel vaults of circular and other sections are analysed as folded plate structures. A study is made of the effect of the shape of the cross section on the shell stress distribution for long, short, thick and thin shells. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

Analysis of a general shell of revolution with arbitrary loading and boundary conditions using the Finite Element approach, well-suited for use with the electronic computer, is presented. The shell is approximated by an assemblage of flat, equilateral trapezoids and isosceles triangles connected to each other at the corners. The assumptions involved in transforming a piece of plate into a finite element are defined. Uncoupled plane stress and flexure stiffness matrices for the above-mentioned shapes of the finite elements are derived from considerations of (i) statics, and (ii) virtual work (energy). Statics matrices are asymmetric with the exception of the triangle plane stress stiffness matrix. However, it is important to note that irrespective of the size of the trapezoid element, in conditions of uniform stress the nodal forces satisfy Betti's reciprocal theorem. When a trapezoid reduces to a rectangle, the asymmetry of plane stress and flexure stiffness matrices disappears. Asymmetry of the Statics matrix is removed by averaging the matrix and its transpose. This process corresponds to introducing self-equilibrating nodal forces which disappear in conditions of uniform stress. Suitable direction cosine matrices are derived to transform the displacements and forces from the element coordinate system to the shell coordinate system. The accuracy of the formulation is demonstrated in several examples by comparing the finite element solution with the elasticity solution. The comparison suggests convergence of the results to the correct solution on reduction of the element size. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

The purpose of this investigation was to develop an eighteen parameter flat triangular finite element for analyzing plate and shell structures. The development of the element was accomplished by combining a plate bending element with a new plane stress element. The well known nine parameter triangle using the normal displacement and two slopes at each vertex was used for the plate bending element. This element contains an incomplete cubic for the normal displacement. For the in-plane element, complete cubics were used initially for the displacements and then various constraints were imposed to reduce the number of generalized co-ordinates to nine, namely the two in-plane displacements and an in-plane rotation at each vertex. One of the constraints, namely that the included angle at each vertex was invariant, destroyed the completeness of the element. However, the element was compatible in the plane. A patch-type test of the in-plane element showed that it could not represent all constant strain states exactly. However, the errors were small. The complete element was then tested on a plane stress cantilever beam, a square plate subjected to membrane stresses only, a cylindrical shell, a spherical shell and a non-prismatic folded plate structure. In all cases, reasonable engineering accuracy was achieved with modest grids of elements. Thus it was concluded that the incompleteness of the in-plane element was not too important. Finally, a compatible beam element was formulated and tested to supplement the triangular element. The beam element formulation included unsymmetric crosssections. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

La transmission acoustique de plaques et de coques est un problème présent dans de nombreuses industries, de l'automobile à l'aéronautique en passant par le génie civil. Il s'agit alors, en présence d'une source externe, de limiter le niveau de bruit à l'intérieur d'une cavité enclose par une structure vibrante, qu'il s'agisse d'une voiture, d'un avion ou d'un immeuble.Les matériaux composites sont de plus en plus largement utilisés par toutes les industries du transport, du fait de leur faible masse rapportée à leur grande rigidité. Ces caractéristiques, utiles pour réduire la consommation énergétique des véhicules, sont cependant un handicap lorsqu'il s'agit de limiter le niveau de bruit intérieur. Il est alors nécessaire d'ajouter des protections acoustiques. On s'intéresse dans ce travail à la transmission d'un bruit produit à l'extérieur de la cavité, et à des solutions antibruit passives, utilisant des matériaux poreux absorbants. Ceux-ci, mousses ou laines minérales, présentent en général de bonnes performances acoustiques pour des fréquences élevées, mais nettement mois intéressantes en basses fréquences. Ce travail s'inscrit dans le cadre d'un projet international ayant pour objectif d'étudier la réduction de bruit à travers des structures composites incluant des protections acoustiques passives. Deux axes y sont privilégiés, la modélisation et l'analyse de sensibilité. En effet, les matériaux poreux et composites présentent souvent de grandes incertitudes, dues au procédé de fabrication, à la difficulté de mesure, ou encore simplement à des choix de conception à faire. On cherche donc à s'assurer de la robustesse des solutions étudiées.Cette thèse est organisée en trois parties. La première traite de la modélisation des structures composites et des traitements poroélastiques. On s'intéresse surtout à deux types de structures, les plaques et les cylindres, pouvant dans les deux cas inclure un grand nombre de couches et de matériaux différents. La deuxième partie traite des méthodes d'analyse de sensibilité, et d'applications dans le cas de la transmission acoustique à travers des structures composites. On s'intéressera enfin dans la troisième partie à l'effet du traitement poroélastique, à travers des études numériques et expérimentales. / Acoustic transmission through plates and shells is a problem that appear in many applications, for example in the automotive or aerospace industries, or in civil engineering. The idea is to reduce the noise level inside a caivty enclosed by a vibrating structure, which may be a car, a plane or a building. Composite materials are widely used in the transportation industries due to their light weight and high mechanic resistance, but these features tend to increase the acoustic transparency of the structures, thus making it necessary to add acoustic protections for reducing noise in the enclosed cavity. The most frequently used acoustic protections take the form of layers of poroelastic materials, which are very efficient noise absorbers in high frequencies, but perform less efficiently in lower frequency ranges.This thesis is part of an international project aimed at improving the noise reduction performance of composite structures through passive sound packages. This implies the development of reduced models of acoustic transmission through such structures, and optimization studies on these models. Besides, the robustness of a solution with respect to uncertainties in the model have to be ensured. This thesis is organised in three parts, dealing with numerical modelling, sensitivity analysis for acoustic transmission and the effect of a porous treatment on sound transmission.

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