A recursive methodology to determine the mechanical Response of thin laminated plates in bending

The paper’s objective is to present the development of a recursive methodology which is based on Adomian Decompostion Method in order to evaluate the mechanical response of thin laminated plates in linear bending. By the equivalent layer concept, the linear relation between the equivalent stresses and the strains, namely ABD matrix, is established. When viewed by the Adomian Decompostion Method perspective, it generates an interesting idea: each layer influence on the plate’s response can be recursively inserted into a base solution by a rearrangement of the plate’s properties. This base solution is previously obtained and it is, in this paper, an isotropic plate response for the same loading and boundary conditions. This approach can significantly increase optimization and delamination studies, given the simplicity on the layers modification, both in fiber orientation and constitutive properties, as these are considered on the recursive procedure. The pb-2 Rayleigh-Ritz Method is used to approximate the solution space and to generate analytic response surfaces. The methodology is applied to symmetrical and unsymmetrical stacking cases for different boundary conditions sets and loading types and the obtained responses are compared to those found on the literature. A study of case complements the methodology analysis: a simplified landing gear door is modeled considering a set of loading conditions as well as different stacking configurations. Good correspondence was found in all studied cases

FEM updating for damage modeling of composite cylinders under radial compression considering the winding pattern

This work aims at developing a strategy to obtain damage evolution parameters of wound cylinders to verify the influence of the winding pattern on them. First, a detailed description of the pattern generation is presented. Then, a finite element (FE) model is developed, in which the cylinders are constructed with winding patterns (WP) of 1/1, 2/1, and 3/1 and subjected to radial compressive loading. Since the cylinder-to-plate contact is considered, the variation of radial stiffness with respect to the parallel plate position is also analyzed. In addition, a damage model is used to predict the progressive failure of those cylinders. A finite element model updating (FEMU) routine is then developed to find the damage input parameters that best simulate experimental force–displacement curves. Key results show that the FEMU algorithm is strongly dependent on the initial guesses producing, however, an excellent correlation with experimental data. The predicted force versus displacement curves for all winding patterns are within the experimental standard deviation, except for the cases in which the winding pattern is not taken into consideration. The computational framework proposed is validated both quantitatively and qualitatively through post-mortem analysis of the specimens. The winding pattern affects the failure and damage mechanisms of the cylinders and, consequently conventional FE models that disregard the pattern cannot capture these mechanisms

On the winding pattern influence for filament wound cylinders under axial compression, torsion, and internal pressure loads

An intrinsic characteristic of components manufactured by the filament winding process is a winding pattern formation during the processing. This paper aims at unlocking and understanding how the winding pattern influences the mechanical behaviour of filament wound cylinders under different boundary conditions. To realize this, a series of finite element models followed by an original geometric approach to generate the pattern are herein developed. Four different patterns and six different winding angles are modelled. These are also modelled by varying the number of layers towards understanding whether there is a correlation between the pattern and the number of layers or not. Three loading cases are considered: axial compression, pure torsion, and internal pressure. Key results reveal that the more layers are stacked to the cylinder, the less impactful is the winding pattern to all loading cases herein investigated

A methodology for the achievement of semi-analytical response for linear and nonlinear bending of thick plates

Este trabalho apresenta uma metodologia para obtenção de benchmarks e prontuários de soluções para flexão de placas semi-espessas sob grandes deslocamentos. O modelo de placa de Mindlin foi o empregado para considerar a deformação cisalhante, e soluções semi-analíticas foram obtidas através de uma variação do método de Rayleigh-Ritz. O método, denominado pb- 2, facilita a imposição das condições de contorno cinemáticas, ampliando consideravelmente a aplicabilidade do método de Rayleigh-Ritz convencional. A metodologia foi implementada utilizando programas de computação simbólica, o que permitiu a obtenção de soluções analíticas aproximadas para diversos problemas lineares. A geração de soluções similares no âmbito não linear não foi possível, e neste caso recorreu-se a ajuste de parâmetros para obtenção de superfícies de resposta representativas de um grande número de análises realizadas através do método dos elementos finitos. Esta abordagem permitiu incorporar às soluções as diversas variáveis que influenciam na resposta, particularmente a espessura, razão de aspecto da placa e a compressibilidade do material. Uma nova parametrização do carregamento transversal foi proposta a fim de minimizar a influência da compressibilidade nos resultados para deslocamento central da placa, levando assim a soluções parecidas com as já reportadas para placas finas. Curvas carga ´ deslocamento podem assim ser extraídas diretamente dos ajustes realizados. Resultados para diversos casos de geometria e condições de contorno foram comparados com soluções disponíveis na literatura, mostrando boa concordância.This work presents a methodology for generating of benchmark solutions and reference formulas for large displacement analysis of thick plates under bending. The Mindlin’s plate model was used to take into account the shear deformability, and semi-analytical solutions were obtained through a variation of the Rayleigh-Ritz method. The method, called pb-2, facilitates the imposition of kinematically admissible conditions, extending considerably the applicability of the conventional Rayleigh-Ritz method. The methodology was implemented in a symbolic computation program, and approximated analytical solutions were generated for linear cases. Similar solutions for non-linear problems were not possible, and in such cases response surfaces were obtained using data provided by finite element analysis. The approach allowed incorporating explicitly to the approximate solution the influence of parameter such as thickness of the plate, aspect ratio of the plate, and the compressibility of the material. A new nondimensional loading is proposed in order to minimize the influence of the compressibility on the response surfaces for central displacement, leading to displacement solutions similar to those reported for thin plates. Load ´ displacement curves can be extracted directly from the fitted response. Results for several cases of geometry and boundary conditions are compared with other available solutions, and good agreement was found

A methodology for the achievement of semi-analytical response for linear and nonlinear bending of thick plates

Este trabalho apresenta uma metodologia para obtenção de benchmarks e prontuários de soluções para flexão de placas semi-espessas sob grandes deslocamentos. O modelo de placa de Mindlin foi o empregado para considerar a deformação cisalhante, e soluções semi-analíticas foram obtidas através de uma variação do método de Rayleigh-Ritz. O método, denominado pb- 2, facilita a imposição das condições de contorno cinemáticas, ampliando consideravelmente a aplicabilidade do método de Rayleigh-Ritz convencional. A metodologia foi implementada utilizando programas de computação simbólica, o que permitiu a obtenção de soluções analíticas aproximadas para diversos problemas lineares. A geração de soluções similares no âmbito não linear não foi possível, e neste caso recorreu-se a ajuste de parâmetros para obtenção de superfícies de resposta representativas de um grande número de análises realizadas através do método dos elementos finitos. Esta abordagem permitiu incorporar às soluções as diversas variáveis que influenciam na resposta, particularmente a espessura, razão de aspecto da placa e a compressibilidade do material. Uma nova parametrização do carregamento transversal foi proposta a fim de minimizar a influência da compressibilidade nos resultados para deslocamento central da placa, levando assim a soluções parecidas com as já reportadas para placas finas. Curvas carga ´ deslocamento podem assim ser extraídas diretamente dos ajustes realizados. Resultados para diversos casos de geometria e condições de contorno foram comparados com soluções disponíveis na literatura, mostrando boa concordância.This work presents a methodology for generating of benchmark solutions and reference formulas for large displacement analysis of thick plates under bending. The Mindlin’s plate model was used to take into account the shear deformability, and semi-analytical solutions were obtained through a variation of the Rayleigh-Ritz method. The method, called pb-2, facilitates the imposition of kinematically admissible conditions, extending considerably the applicability of the conventional Rayleigh-Ritz method. The methodology was implemented in a symbolic computation program, and approximated analytical solutions were generated for linear cases. Similar solutions for non-linear problems were not possible, and in such cases response surfaces were obtained using data provided by finite element analysis. The approach allowed incorporating explicitly to the approximate solution the influence of parameter such as thickness of the plate, aspect ratio of the plate, and the compressibility of the material. A new nondimensional loading is proposed in order to minimize the influence of the compressibility on the response surfaces for central displacement, leading to displacement solutions similar to those reported for thin plates. Load ´ displacement curves can be extracted directly from the fitted response. Results for several cases of geometry and boundary conditions are compared with other available solutions, and good agreement was found

Methods on additive decompositions of elastic tridimensional and thin plates theory fundamental solutions

A presente tese tem por objetivo o desenvolvimento de metodologias para a decomposição de soluções fundamentais da elasticidade tridimensional e da teoria de placas finas por intermédio de soluções fundamentais do mesmo operador diferencial. A diferença entre as soluções fundamentais utilizadas na decomposição é a simetria do tensor constitutivo de cada uma dessas soluções, cujas propriedades elásticas são diretamente proporcionais aos coeficientes da(s) equação(ões) diferencial(ais). O objetivo dessa decomposição é evitar um efeito conhecido como degeneração matemática. Esse efeito cria singularidades não regulares nessas soluções quando considerado altas simetrias, tanto na teoria de placas finas quanto na elasticidade tridimensional. Dois tipos de decomposições – cinemática e constitutiva – são apresentados e discutidos. Devido a relações hiperdefinidas, comprova-se que a decomposição via critérios cinemáticos é impossível. Mostra-se a possibilidade da decomposição constitutiva, via duas diferentes abordagens. Suas propriedades e particularidades bem como outros resultados são discutidos no decorrer do texto e são comparados, quando possível, com a literatura disponível.The present thesis has as objective the development of a methodology for the decomposition of elastic tridimensional and thin plates fundamental solutions through fundamental solutions of the same differential operator. The difference between the fundamental solutions is the constitutive tensor symmetry of each solution, in which is directly proportional to the differential equation(s) coefficients. The objective of the decomposition is to avoid an effect known as mathematically degeneration. This effect creates non regular singularities in these solutions when high material symmetries are considered in the thin plate’s theory as well in the tridimensional elasticity. Two types of decompositions – kinematics and constitutive – are presented and discussed. Due to hyper-defined relationships, it is proved that the decomposition by kinematics criteria is impossible. It is also shown the possibility of the constitutive decomposition, by two different approaches. Their properties and particularities as well other results are discussed through the text and are compared, when possible, with the available literature

ON A RECURSIVE METHODOLOGY FOR SEMI-ANALYTICAL SOLUTIONS OF SYMMETRIC AND UNSYMMETRIC LAMINATED THIN PLATES

 The present paper has as objective the introduction and analysis of a new procedure in order to derive semi-analytical solutions of symmetric and unsymmetrical laminated rectangular thin plates in a recursive manner. The methodology is based on three main characteristics: (a) decomposition of the differential operator into two or more components, (b) an infinite expansion of the differential equation solution and, (c) determination of each superposed solution by a relationship between the divided operators and previous solutions. The first expanded term concerns to the plate’s isotropic solution and, in each step, orthorhombic laminae influence is inserted. In order to approximate the solutions, the pb-2 Rayleigh-Ritz Method is used. Obtained solutions are discussed and compared to those found in the literature

A recursive methodology to determine the mechanical Response of thin laminated plates in bending

The paper’s objective is to present the development of a recursive methodology which is based on Adomian Decompostion Method in order to evaluate the mechanical response of thin laminated plates in linear bending. By the equivalent layer concept, the linear relation between the equivalent stresses and the strains, namely ABD matrix, is established. When viewed by the Adomian Decompostion Method perspective, it generates an interesting idea: each layer influence on the plate’s response can be recursively inserted into a base solution by a rearrangement of the plate’s properties. This base solution is previously obtained and it is, in this paper, an isotropic plate response for the same loading and boundary conditions. This approach can significantly increase optimization and delamination studies, given the simplicity on the layers modification, both in fiber orientation and constitutive properties, as these are considered on the recursive procedure. The pb-2 Rayleigh-Ritz Method is used to approximate the solution space and to generate analytic response surfaces. The methodology is applied to symmetrical and unsymmetrical stacking cases for different boundary conditions sets and loading types and the obtained responses are compared to those found on the literature. A study of case complements the methodology analysis: a simplified landing gear door is modeled considering a set of loading conditions as well as different stacking configurations. Good correspondence was found in all studied cases