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CHAPTER 1
INTRODUCTION 1.1 Background
Over the last couple of decades, the world of structural engineering is going through tremendous revolution in terms of materials for structural applications. During the pre- historic civilization periods, different man-made structures were mainly built with the materials available naturally, like woods, stones etc. The early historic period of civilization saw the use of materials obtained by simple processing of natural materials, like bricks, a few binding materials etc. A revolution, in terms of much superior structural materials, occurred with the advent of steel and steel-reinforced concrete, which are being widely used for heavy structural components, mechanical equipments etc, till date. The unprecedented technological developments seen in the twentieth century, gave rise to the requirement of special lightweight structures for aerospace and marine vehicles, besides their strength requirements. This special demand gave birth to a completely new generation of structural materials, broadly referred to as composite materials. Amongst the different composite materials used today, the fiber reinforced plastic (FRP) and other laminated composites are definitely at the forefront in such lightweight structural applications, because of a number of advantages. However, the ever increasing scientific endeavors in recent times generated the requirement for another class of structural materials suitable for applications in high temperature environments. This very requirement, supported with intense research in the field of materials science in recent times, gave birth to the concept of functionally graded materials (FGM), introduced first time by a Japanese group of materials scientists (see, e.g., Koizumi, 1993).
The aforementioned progress made in the development of advanced laminated composites in last couple of decades has driven their use from secondary structural components to primary load bearing structural elements in recent times. Mainly it is the high strength-to-weight ratio of such composites that has popularized their use for
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Chapter 1 important weight-critical structural applications over conventional structural materials.
Other advantageous features like high fatigue strength, ease of maintenance/repairs, ease of tailoring their strength in required directions by considering fibre orientation, stacking sequence etc as design variables and ease of casting them into complex geometrical shapes have also played their role in popularizing such laminated composites. Among the vast range of applications of such laminated composites, to name a few, are the high-speed boat hulls, underwater vehicles, space crafts, components in automobile industry, some civil engineering structures and biomedical appliances etc. However, the main drawbacks with these materials come from the possibility of interface failure between consecutive layers and between fibers and matrix, due to sudden mismatch of mechanical properties at these interfaces. Possibility of such failure increases manifold in case of high-temperature applications.
On the contrary, functionally graded materials (FGM) provide means to circumvent the possibility of such interface failure (at the macroscopic level) and are greatly applicable in high temperature environments. The generic term FGM implies to those materials in which the volume fractions of two (or more) materials are varied continuously as a function of position along preferred direction(s) of the structure, to achieve required properties. FGMs are, therefore, composite materials with a microscopically anisotropic character, but considered isotropic and inhomogeneous at the macroscopic level. Hence, continuous/gradual changes in their macrostructural properties distinguish FGMs from other composite materials. Among the different kinds of FGMs developed so far, the ones obtained by using heat-resistant ceramic on the high-temperature side and tough metals on the other side, with a gradual compositional variation from ceramic to metal, stands high in popularity today. FGMs are nowadays being increasingly used as thermal barrier materials in space and other applications, nuclear fusion chambers etc (see e.g., Hirai and Chen, 1999). In the present work, structural elements with such FGM are considered, besides laminated FRP composites.
Structural elements in the form of plate or panel (shell) are the most indispensable load- bearing components for almost all engineering applications; and hence are of immense importance to structural engineers. Be it a bridge deck or the deck of a ship, the roof of a building or the body of a marine/space/automobile vehicle, examples of such structural components are abundant. Plates are definitely the simplest of such kind of structural elements. Moreover, these structural elements are frequently required to be strengthened with beam-like stiffener components, especially for high span to thickness ratio, to enhance
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Introduction their performance and to achieve overall weight and cost saving. The stiffeners/ribs, apart from serving the purpose of enhancing the strength at required positions, add to the aesthetics, at times. Hence, with the development of structures of aforementioned types of advanced materials, the possibility of the use of such stiffened structures follows naturally.
Such stiffened structural components of advanced materials, for their satisfactory performance in terms of carrying relevant loads during their life-cycle, require proper design of their geometric and material configurations. Consequently, the design requirement demands availability of accurate, efficient and robust analysis capability for such structures of any general configuration. This is because experimental verification in every case of such structures of advanced materials is difficult due to financial and other constraints. In this direction, some typical features like high orthotropy, complex interlaminar behaviour etc in case of laminated composites and the inhomogeneity of thermo-mechanical properties in case of FGMs, pose considerable difficulty to meet these demands. Addition of ribs/stiffeners to these structures further complicates the problem. However, to meet these challenges, the engineers in recent times, are equipped with powerful electronic computing facilities to employ different numerical- computational techniques, such as the finite element method (FEM). These developments have substantially driven the research community in this field in developing robust and efficient theoretical-numerical-computational tool for accurate analysis of complex structural elements with such advanced material. The FEM, being already recognized as highly robust and customizable to particular needs, has definitely been at the vanguard of these researches.
1.2 Motivation for the Present Work
Laminated composite structures, carrying transverse loads, have been known to go through high transverse shear deformation in addition to their membrane and bending deformations, particularly when these are considerably thick. These transverse shear deformations may naturally become more important in beams / stiffeners, which are considerably thicker than the plates. Hence, to take this effect into account in the analysis models, several higher order plate and beam deformation theories (see, e.g., Lo et al., 1977; Kant and Pandya, 1988) have been put forward over the years; besides the elasticity (Pagano, 1970), layerwise (Reddy, 1987) and zigzag (Carrera, 1996) models.
However, none of the individual theories is found to be computationally cheap, to yield highly accurate results in all the cases and to be simple for implementation; all at the
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Chapter 1 same time. Further, the models for incorporating more accurate transverse shear
deformation effect have mostly been developed individually either for plate/panel or for beam components. Development of such model for stiffened plate / panel configurations is relatively scanty. Hence, it has been felt necessary to develop an accurate, economic and simple higher order shear deformation model for such stiffened plate configurations and has been pursued in the present work.
Further to the aforementioned observations, it is noticed that the analysis models presented for the stiffened plate configurations are mostly for isotropic materials and in a few cases for laminated composites. It seems to be very difficult to find some work on the analysis of stiffened plates of FGM, at least in open literature accessible to the present researcher. Geometrically, most of the stiffened plate configurations considered are of rectangular planform. Non-rectangular shapes, though may sometimes become essential in engineering applications, has also been given relatively less attention.
Hence, the development of accurate analysis tool for stiffened plates of non-rectangular planform, like the annular sector plates, is also felt to be of use.
Because of these reasons, the higher order shear deformation model, developed initially for the analysis of laminated composite stiffened rectangular plate configurations, has been extended for application to configurations with FGM and to annular sector stiffened plate configurations.
1.3 Objective of the Thesis
Based on the stated background and motivations, the objective of the present work has been the development of an accurate, efficient and simple higher order shear deformation theory for composite/sandwich laminates and extension of the developed formulations to the analysis of stiffened laminated composite rectangular and annular sector plates and stiffened plates of functionally graded materials. The finite element method is employed to achieve stable convergent results for all the cases.
1.4 Outline of the Thesis
The present thesis is organized in six different chapters followed by a list of references and an appendix. The contents of different chapters are briefly outlined below.
Chapter 1, i.e. the present one, provides a background of the present thesis work. It provides an overview, motivation and the objective of the present research.
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Introduction Chapter 2, presents a critical review of the significant works carried out and
presented in literature so far, in directions relevant to the present thesis. Broadly speaking, it first brings out the significant works done on development of different laminate deformation theories. Then it describes the works done on analysis of isotropic and laminated composite stiffened plate configurations mainly of rectangular plan- forms. Works on annular sector stiffened plate configurations are then discussed.
Analyses of rectangular and non-rectangular plates with functionally graded material are considered successively. It should be mentioned that the chapter on literature review includes only those papers, considered significant with respect to the current investigations, presented in this thesis. The direction of the present thesis is also set from the rational conclusions drawn from this chapter.
Chapter 3 starts with the presentation of the proposed semi refined higher-order
shear deformation theory (HSDT) and the corresponding finite element model for accurate analysis of laminated composite plates. It then, goes on developing a novel and efficient least square of error (LSE) method of transverse stress recovery at the post processing phase using the three-dimensional static equilibrium equations. Some numerical results in terms of displacements, through-the-thickness variation of different stress components and other quantities for some benchmark problems are then presented to validate the developments. Some important new results are also indicated for future references.
Extension of the semi refined HSDT, proposed in chapter 3, to stiffened plate configurations and the required stiffened-plate finite element model for its analysis, is presented in Chapter 4. Based on these developments, linear static and free vibration analyses of stiffened plates of isotropic and laminated composites are presented.
Chapter 5 is devoted to the applicability/extension of the proposed theory to other
geometric form such as annular sectorial form and plates with functionally graded material. The results that follow in this chapter are mostly new ones, since consideration of such cases have not been observed in the open literature accessible to the present author.
The important conclusions drawn from the present work are put forward in Chapter 6. The important contributions made in the present work are also mentioned.
Finally, the potential future scopes of the present work in different directions are also described and discussed briefly.
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Chapter 1 Apart from these, a list of references sought in different chapters is presented
thereafter. A short appendix, containing explicit forms of some matrices used earlier, is presented at the end, followed by a brief biography of the author of the thesis.
1.5 Summary
An overview of the whole thesis, starting from the historical background to the objective, promoted by the motivations generated from the literature review, is presented in this chapter. An outline of the contents of the different chapters is also presented. Hence, the reader is hoped to be provided with an indication of the content of the whole thesis, from a brief reading of this chapter.
