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Em engenharia, a quantidade de problemas geométricos complexos que precisam ser resolvidos empregando teorias de placas ou cascas é notável. Esta é a razão por que há tantas teorias que buscam simplificar os problemas tridimensionais em outros menos custosos computacionalmente. Além disso, o aumento atual do uso de estruturas sanduíche requer que as formulações bidimensionais sejam mais precisas. Esta tese, num primeiro momento, compila a maioria das teorias de placa, comentando as principais diferenças, vantagens e desvantagens de cada uma. As formulações bidimensionais de placas laminadas são classificadas principalmente de acordo com o tratamento da coordenada na direção normal a superfície da mesma: Camada Única Equivalente (ESL), ESL refinada (teorias Zig-Zag) e Teorias Discretas ou de Camada (LW). Cada uma destas teorias é revista juntamente com as hipóteses de placas que são feitas para cada uma das camadas ou para o laminado como um todo. Para resolver tais problemas estruturais em engenharia, métodos numéricos são normalmente utilizados. Portanto, num segundo momento, alguns métodos de solução são citados e revisados, mas o foco é dado ao Método dos Elementos Finitos (MEF). A contribuição deste trabalho consiste na implementação de um novo método de solução de compósitos laminados e estruturas sanduíche com base em um sistema unificado de Formulação Generalizada (GUF) via MEF. Um elemento quadrilátero de 4 nós foi desenvolvido e avaliado com um código de Elementos Finitos desenvolvido pelo presente autor. Os requisitos para continuidade do tipo C-1 são respeitados para a variável de deflexão da placa. Esse método é nomeado de Formulação Generalizada do Caliri (CGF). Resultados para placas isotrópicas, placas de laminado compósito e estruturas sanduíche consideradas finas ou espessas são comparados com dados da literatura e soluções via Abaqus. Os resultados obtidos ao longo da espessura reforçam a necessidade de soluções de placa não-lineares para placas espessas (laminadas ou não). Mostrou-se que as soluções estáticas e dinâmicas empregando o método proposto fornecem resultados coerentes quando comparados com outros métodos de solução. Dentre os diversos estudos de caso investigados, verificou-se que é possível se obter resultados com alta concordância. Para uma estrutura sanduíche com núcleo macio, o resultado de deslocamento previsto para um carregamento estático chega a 99.8% de concordância e o resultado de uma análise modal da mesma estrutura mostra uma concordância de 99.5% com os resultados de um modelo feito com elementos 3D em um programa comercial de elementos finitos. / In engineering, the amount of complex geometrical problems, which need to be solved by using plates and shells theories, is remarkable. This is the reason why there are so many plate and shell theories which attempt to simplify three dimensional problems into ones with low computational cost. Additionally, the current increasing use of sandwich structures requires that the two dimensional formulations be accurate enough. First, this thesis compiles most of the plate theories from the literature and quotes the main differences, advantages and weaknesses of each one. The bi-dimensional laminated plate formulations are mainly classified according to the treatment of the variable in the normal direction of the plate surface: Equivalent Single Layer (ESL), Refined ESL (Zig-Zag theories) and Layer-Wise (LW) theories. Each one of these theories is reviewed along with the plate hypotheses which are made for each ply and/or laminate. To solve such complex structural engineering problems, numerical methods are normally used. Second, few solution methods are reviewed and quoted, but focus is given to the Finite Element Method (FEM). The contribution of this work is the implementation of a new solution method for laminated composites and sandwich structures based on a Generalized Unified Formulation (GUF) via FEM. A quadrilateral 4-node element was developed and evaluated using in-house Finite Element program. The C-1 continuity requirements is fulfilled for the transversal displacement field variable. This method is tagged as Caliri\'s Generalized Formulation (CGF). Results for isotropic plates, laminated composite plates and sandwich structures for thin and thick laminates are compared with literature data and solutions via Abaqus. The through-the-thickness profile results reinforce the need for non-linear plate (laminated or not) solutions. It was shown that the static and dynamic solutions employing the proposed solution method yield coherent results when compared with other solution methods. Among the different case studies investigated, it was verified that it is possible to obtain results with high agreement. For a soft-core sandwich structure, the displacement result for a static loading is reported as high as 99.8% and the result of a modal analysis of the same structure shows an accuracy of 99.5%, comparing to the results from a 3D finite element model built with a commercial software.

Em engenharia, a quantidade de problemas geométricos complexos que precisam ser resolvidos empregando teorias de placas ou cascas é notável. Esta é a razão por que há tantas teorias que buscam simplificar os problemas tridimensionais em outros menos custosos computacionalmente. Além disso, o aumento atual do uso de estruturas sanduíche requer que as formulações bidimensionais sejam mais precisas. Esta tese, num primeiro momento, compila a maioria das teorias de placa, comentando as principais diferenças, vantagens e desvantagens de cada uma. As formulações bidimensionais de placas laminadas são classificadas principalmente de acordo com o tratamento da coordenada na direção normal a superfície da mesma: Camada Única Equivalente (ESL), ESL refinada (teorias Zig-Zag) e Teorias Discretas ou de Camada (LW). Cada uma destas teorias é revista juntamente com as hipóteses de placas que são feitas para cada uma das camadas ou para o laminado como um todo. Para resolver tais problemas estruturais em engenharia, métodos numéricos são normalmente utilizados. Portanto, num segundo momento, alguns métodos de solução são citados e revisados, mas o foco é dado ao Método dos Elementos Finitos (MEF). A contribuição deste trabalho consiste na implementação de um novo método de solução de compósitos laminados e estruturas sanduíche com base em um sistema unificado de Formulação Generalizada (GUF) via MEF. Um elemento quadrilátero de 4 nós foi desenvolvido e avaliado com um código de Elementos Finitos desenvolvido pelo presente autor. Os requisitos para continuidade do tipo C-1 são respeitados para a variável de deflexão da placa. Esse método é nomeado de Formulação Generalizada do Caliri (CGF). Resultados para placas isotrópicas, placas de laminado compósito e estruturas sanduíche consideradas finas ou espessas são comparados com dados da literatura e soluções via Abaqus. Os resultados obtidos ao longo da espessura reforçam a necessidade de soluções de placa não-lineares para placas espessas (laminadas ou não). Mostrou-se que as soluções estáticas e dinâmicas empregando o método proposto fornecem resultados coerentes quando comparados com outros métodos de solução. Dentre os diversos estudos de caso investigados, verificou-se que é possível se obter resultados com alta concordância. Para uma estrutura sanduíche com núcleo macio, o resultado de deslocamento previsto para um carregamento estático chega a 99.8% de concordância e o resultado de uma análise modal da mesma estrutura mostra uma concordância de 99.5% com os resultados de um modelo feito com elementos 3D em um programa comercial de elementos finitos. / In engineering, the amount of complex geometrical problems, which need to be solved by using plates and shells theories, is remarkable. This is the reason why there are so many plate and shell theories which attempt to simplify three dimensional problems into ones with low computational cost. Additionally, the current increasing use of sandwich structures requires that the two dimensional formulations be accurate enough. First, this thesis compiles most of the plate theories from the literature and quotes the main differences, advantages and weaknesses of each one. The bi-dimensional laminated plate formulations are mainly classified according to the treatment of the variable in the normal direction of the plate surface: Equivalent Single Layer (ESL), Refined ESL (Zig-Zag theories) and Layer-Wise (LW) theories. Each one of these theories is reviewed along with the plate hypotheses which are made for each ply and/or laminate. To solve such complex structural engineering problems, numerical methods are normally used. Second, few solution methods are reviewed and quoted, but focus is given to the Finite Element Method (FEM). The contribution of this work is the implementation of a new solution method for laminated composites and sandwich structures based on a Generalized Unified Formulation (GUF) via FEM. A quadrilateral 4-node element was developed and evaluated using in-house Finite Element program. The C-1 continuity requirements is fulfilled for the transversal displacement field variable. This method is tagged as Caliri\'s Generalized Formulation (CGF). Results for isotropic plates, laminated composite plates and sandwich structures for thin and thick laminates are compared with literature data and solutions via Abaqus. The through-the-thickness profile results reinforce the need for non-linear plate (laminated or not) solutions. It was shown that the static and dynamic solutions employing the proposed solution method yield coherent results when compared with other solution methods. Among the different case studies investigated, it was verified that it is possible to obtain results with high agreement. For a soft-core sandwich structure, the displacement result for a static loading is reported as high as 99.8% and the result of a modal analysis of the same structure shows an accuracy of 99.5%, comparing to the results from a 3D finite element model built with a commercial software.

In recent years, use of composites is increasing in most fields of engineering such as aerospace, automotive, civil construction, marine, prosthetics, etc., because of its light weight, very high specific strength and stiffness, corrosion resistance, high thermal resistance etc. It can be seen that the specific strength of fibers are many orders more compared to metals. Thus, laminated fiber reinforced plastics have emerged to be attractive materials for many engineering applications. Though the uses of composites are enormous, there is always an element of fuzziness in the design of composites. Composite structures are required to be designed to resist high stresses. For this, one requires a reliable failure criterion. The anisotropic behaviour of composites makes it very difficult to formulate failure criteria and experimentally verify it, which require one to perform necessary bi-axial tests and plot the failure envelopes. Failure criteria are usually based on certain assumption, which are some times questionable. This is because, the failure process in composites is quite complex. The failure in a composite is normally based on initiating failure mechanisms such as fiber breaks, fiber compressive failure, matrix cracks, matrix crushing, delamination, disbonds or a combination of these. The initiating failure mechanism is the one, which is/are responsible for initiating failure in a laminated composites. Initiating failure mechanisms are generally dependant on the type of loading, geometry, material properties, condition of manufacture, boundary conditions, weather conditions etc. Since, composite materials exhibit directional properties, their applications and failure conditions should be properly examined and in addition to this, robust computational tools have to be exploited for the design of structural components for efficient utilisation of these materials. Design of structural components requires reliable failure criteria for the safe design of the components. Several failure criteria are available for the design of composite laminates. None of the available anisotropic strength criteria represents observed results sufficiently accurate to be employed confidently by itself in design. Most of the failure criteria are validated based on the available uniaxial test data, whereas, in practical situations, laminates are subjected to at least biaxial states of stresses. Since, the generation of biaxial test data are very difficult and time consuming to obtain, it is indeed a necessity to develop computational tools for modelling the biaxial behavior of the composite laminates. Understanding of the initiating failure mechanisms and the development of reliable failure criteria is an essential prerequisite for effective utilization of composite materials. Most of the failure criteria, considers the uniaxial test data with constant shear stress to develop failure envelopes, but in reality, structures are subjected to biaxial normal stresses as well as shear stresses. Hence, one can develop different failure envelopes depending upon the percentage of the shear stress content. As mentioned earlier, safe design of the composite structural components require reliable failure criterion. Currently two broad approaches, namely, (1) Damage Tolerance Based Design and (2)Failure Criteria Based Design are in use for the design of laminated structures in aerospace industry. Both approaches have some limitations. The damage tolerance based design suffers from a lack of proper definition of damage and the inability of analytical tools to handle realistic damage. The failure criteria based design, although relatively, more attractive in view of the simplicity, it forces the designer to use unverified design points in stress space, resulting in unpredictable failure conditions. Generally, failure envelopes are constructed using 4 or 5 experimental constants. In this type of approach, small experimental errors in these constants lead to large shift in the failure boundaries raising doubts about the reliability of the boundary in some segments. Further, they contain segments which have no experimental support and so can lead to either conservative or nonconservative designs. Conservative design leads to extra weight, a situation not acceptable in aerospace industry. Whereas, a nonconservative design, is obviously prohibitive, as it implies failure. Hence, both the damage tolerance based design and failure criteria based design have limitations. A new method, which combines the advantages of both the approaches is desirable. This issue is also thoroughly debated in many international conference on composites. Several pioneers in the composite industry indicated the need for further research work in the development of reliable failure criteria. Hence, this is motivated to carry out research work for the development of new failure criterion for the design of composite structures. Several experts meetings held world wide towards the assessment of existing failure theories and computer codes for the design of composite structures. One such meeting is the experts meeting held at United Kingdom in 1991.This meeting held at St. Albans(UK) on ’Failure of Polymeric Composites and Structures: Mechanisms and Criteria for the Prediction of Performance’, in 1991 by UK Science & Engineering Council and UK Institute of Mechanical Engineers. After thorough deliberations it was concluded that 1. There is no universal definition of failure of composites. 2. There is little or lack of faith in the failure criteria that are in current use and 3. There is a need to carry out World Wide Failure Exercise(WWFE) Based on the experts suggestions, Hinton and Soden initiated the WWFE in consultation with Prof.Bryan Harris (Editor, Journal of Composite Science and Tech-nology)to have a program to get comparative assessment of existing failure criteria and codes with following aims 1. Establish the current level of maturity of theories for predicting the failure response of fiber reinforced plastic(FRP)laminates. 2. Closing the knowledge gap between theoreticians and design practitioners in this field. 3. Stimulating the composites’ community into providing design engineers with more robust and accurate failure prediction methods, and the confidence to use them. The organisers invited pioneers in the composite industry for the program of WWFE. Among the pioneer in the composite industry Professor Hashin declined to participate in the program and had written a letter to the organisers saying that, My only work in this subject relates to failure criteria of unidirectional fiber composites, not to laminates. I do not believe that even the most complete information about failure of single plies is sufficient to predict the failure of a laminate, consisting of such plies. A laminate is a structure which undergoes a complex damage process (mostly of cracking) until it finally fails. The analysis of such a process is a prerequisite for failure analysis. ”While significant advances have been made in this direction we have not yet arrived at the practical goal of failure prediction”. Another important conference held in France in 1999, Composites for the next Millennium (Proceedingof Symposium in honor of S.W.Tsaion his 70th Birth Day Torus, France, July 2-3, 1999, pp.19.) also concludedon similar line to the meeting held at UK in 1991. Paul A Lagace and S. Mark Spearing, have pointed out that, by referring to the article on ’Predicting Failure in Composite Laminates: the background to the exercise’, by M.J.Hinton & P.D.Soden, Composites Science and Technology, Vol.58, No.7(1998), pp.1005. ”After Over thirty years of work ’The’ composite failure criterion is still an elusive entity”. Numerous researchers have produced dozens of approaches. Hundreds of papers, manuscripts and reports were written and presentations made to address the latest thoughts, add data to accumulated knowledge bases and continue the scholarly debate. Thus, the out come of these experts meeting, is that, there is a need to develop new failure theories and due to complexities associated with experimentation, especially getting bi-axial data, computational methods are the only viable alternative. Currently, biaxial data on composites is very limited as the biaxial testing of laminates is very difficult and standardization of biaxial data is yet to be done. All these experts comments and suggestions motivated us to carry out research work towards the development of new failure criterion called ’Failure Mechanism Based Failure Criterion’ based on initiating failure mechanisms. The objectives of the thesis are 1. Identification of the failure mechanism based failure criteria for the specific initiating failure mechanism and to assign the specific failure criteria for specific initiating failure mechanism, 2. Use of the ’failure mechanism based design’ method for composite pressurant tanks and to evaluate it, by comparing it with some of the standard ’failure criteria’ based designs from the point of view of overall weight of the pressurant tank, 3. Development of new failure criterion called ’Failure Mechanism Based Failure Criterion’ without shear stress content and the corresponding failure envelope, 4. Development of different failure envelopes with the effect of shear stress depending upon the percentage of shear stress content and 5. Design of composite laminates with the Failure Mechanism Based Failure Criterion using optimization techniques such as Genetic Algorithms(GA) and Vector Evaluated Particle Swarm Optimization(VEPSO) and the comparison of design with other failure criteria such as Tsai-Wu and Maximum Stress failure criteria. The following paragraphs describe about the achievement of these objectives. In chapter 2, a rectangular panel subjected to boundary displacements is used as an example to illustrate the concept of failure mechanism based design. Composite Laminates are generally designed using a failure criteria, based on a set of standard experimental strength values. Failure of composite laminates involves different failure mechanisms depending upon the stress state and so different failure mechanisms become dominant at different points on the failure envelope. Use of a single failure criteria, as is normally done in designing laminates, is unlikely to be satisfactory for all combination of stresses. As an alternate use of a simple failure criteria to identify the dominant failure mechanism and the design of the laminate using appropriate failure mechanism based criteria is suggested in this thesis. A complete 3-D stress analysis has been carried out using a general purpose NISA Finite Element Software. Comparison of results using standard failure criteria such as Maximum Stress, Maximum Strain, Tsai-Wu, Yamada-Sun, Maximum Fiber Strain, Grumman, O’brien, & Lagace, indicate substantial differences in predicting the first ply failure. Results for Failure Load Factors, based on the failure mechanism based approach are included. Identification of the failure mechanism at highly stressed regions and the design of the component, to withstand an artificial defect, representative this failure mechanism, provides a realistic approach to achieve necessary strength without adding unnecessary weight to the structure. It is indicated that the failure mechanism based design approach offers a reliable way of assessing critically stressed regions to eliminate the uncertainties associated with the failure criteria. In chapter 3, the failure mechanism based design approach has been applied to a composite pressurant tanks of upper stages of launch vehicles and propulsion systems of space crafts. The problem is studied using the failure mechanism based design approach, by introducing an artificial matrix crack representative of the initiating failure mechanism in the highly stressed regions and the strain energy release rate (SERR) are calculated. The total SERR value is obtained as 3330.23 J/m2, which is very high compared to the Gc(135 J/m2) value, which means the crack will grow further. The failure load fraction at which the crack has a tendency to grow is estimated to be 0.04054.Results indicates that there are significant differences in the failure load fraction for different failure criteria.Comparison with Failure Mechanism Based Criterion (FMBC) clearly indicates matrix cracks occur at loads much below the design load yet fibers are able to carrythe design load. In chapter 4, a Failure Mechanism Based Failure Criterion(FMBFC)has been proposed for the development of failure envelope for unidirectional composite plies. A representative volume element of the laminate under local loading is micromechanically modelled to predict the experimentally determined strengths and this model is then used to predict points on the failure envelope in the neighborhood of the experimental points. The NISA finite element software has been used to determine the stresses in the representative volume element. From these micro-stresses, the strength of the lamina is predicted. A correction factor is used to match the prediction of the present model with the experimentally determined strength so that, the model can be expected to provide accurate prediction of the strength in the neighborhood of the experimental points. A procedure for the construction of the failure envelope in the stress space has been outlined and the results are compared with the some of the standard and widely used failure criteria in the composite industry. Comparison of results with the Tsai-Wu failure criterion shows that there are significant differences, particularly in the third quadrant, when the ply is under bi-axial compressive loading. Comparison with maximum stress criterion indicates better correlation. The present failure mechanism based failure criterion approach opens a new possibility of constructing reliable failure envelopes for bi-axial loading applications, using the standard uniaxialtest data. In chapter 5, the new failure criterion for laminated composites developed based on initiating failure mechanism as mentioned in chapter 4 for without shear stress condition is extended to obtain the failure envelopes with the shear stress condition. The approach is based on Micromechanical analysis of composites, wherein a representative volume consists of a fiber surrounded by matrix in appropriate volume fraction and modeled using 3-D finite elements to predict the strengths.In this chapter, different failure envelopes are developed by varying shear stress say from 0% of shear strength to 100% of shear strength in steps of 25% of shear strength. Results obtained from this approach are compared with Tsai-Wu and Maximum stress failure criteria. The results show that the predicted strengths match more closely with maximum stress criterion. Hence, it can be concluded that influence of shear stress on the failure of the lamina is of little consequence as far as the prediction of strengths in laminates. In chapter 6, the failure mechanism based failure criterion, developed by the authors is used for the design optimization of the laminates and the percentage savings in total weight of the laminate is presented. The design optimization of composite laminates are performed using Genetic Algorithms. The genetic algorithm is one of the robust tools available for the optimum design of composite laminates. The genetic algorithms employ techniques originated from biology and dependon the application of Darwin’s principle of survival of the fittest. When a population of biological creatures is permitted to evolve over generations, individual characteristics that are beneficial for survival have a tendency to be passed on to future generations, since individuals carrying them get more chances to breed. In biological populations, these characteristics are stored in chromosomal strings. The mechanics of natural genetics is derived from operations that result in arranged yet randomized exchange of genetic information between the chromosomal strings of the reproducing parents and consists of reproduction, cross over, mutation, and inversion of the chromosomal strings. Here, optimization of the weight of the composite laminates for given loading and material properties is considered. The genetic algorithms have the capability of selecting choice of orientation, thickness of single ply, number of plies and stacking sequence of the layers. In this chapter, minimum weight design of composite laminates is presented using the Failure Mechanism Based(FMB), Maximum Stress and Tsai-Wu failure criteria. The objective is to demonstrate the effectiveness of the newly proposed FMB Failure Criterion(FMBFC) in composite design. The FMBFC considers different failure mechanisms such as fiber breaks, matrix cracks, fiber compressive failure, and matrix crushing which are relevant for different loadin gconditions. FMB and Maximum Stress failure criteria predicts byupto 43 percent savings in weight of the laminates compared to Tsai-Wu failure criterion in some quadrants of the failure envelope. The Tsai-Wu failure criterion over predicts the weight of the laminate by up to 86 percent in the third quadrant of the failure envelope compared to FMB and Maximum Stress failure criteria, when the laminate is subjected to biaxial compressive loading. It is found that the FMB and Maximum Stress failure criteria give comparable weight estimates. The FMBFC can be considered for use in the strength design of composite structures. In chapter 7, Particle swarm optimization is used for design optimization of composite laminates. Particle swarm optimization(PSO)is a novel meta-heuristic inspired by the flocking behaviour of birds. The application of PSO to composite design optimization problems has not yet been extensively explored. Composite laminate optimization typically consists in determining the number of layers, stacking sequence and thickness of ply that gives the desired properties. This chapter details the use of Vector Evaluated Particle Swarm Optimization(VEPSO) algorithm, a multi-objective variant of PSO for composite laminate design optimization. VEPSO is a modern coevolutionary algorithm which employs multiple swarms to handle the multiple objectives and the information migration between these swarms ensures that a global optimum solution is reached. The current problem has been formulated as a classical multi-objective optimization problem, with objectives of minimizing weight of the component for a required strength and minimizing the totalcost incurred, such that the component does not fail. In this chapter, an optimum configuration for a multi-layered unidirectional carbon/epoxy laminate is determined using VEPSO. The results are presented for various loading configurations of the composite structures. The VEPSO predicts the same minimum weight optimization and percentage savings in weight of the laminate when compared to GA for all loading conditions.There is small difference in results predicted by VEPSO and GA for some loading and stacking sequence configurations, which is mainly due to random selection of swarm particles and generation of populations re-spectively.The difference can be prevented by running the same programme repeatedly. The Thesis is concluded by highlighting the future scope of several potential applications based on the developments reported in the thesis.