Reinforced concrete slabs are a conventional type of foundation that is widely used in residential, commercial and industrial buildings. Most of the slab foundations are constructed directly on the ground without another structural medium in between and some of the slabs are therefore influenced by ground softening. A slab foundation is designed to transfer vertical loads and bridge imperfections in the ground, but excessive soil distortion may destroy the support conditions of the slab and through this influence the stability of the superstructures. The aim of this project is to study how ground softening and its further development influence a typical concrete slab. Furthermore, an approximate analytical method to evaluate the condition of a slab due to ground softening was studied. A practical case has been studied to capture the actual failure behaviours of a reinforced concrete slab, based on a previous project. For the case studied, possible future mining activities close to a high bay warehouse may cause damaging settlement in the underground. The fault that may appear softens the soils underneath the foundation, which may deteriorate and cause collapse of the concrete slab under the warehouse. Two types of failure scenarios were studied; subsoil softening and subsoil collapse. The Finite Element Method (FEM) was used to analyse the behaviour of the slab and the development of the failures in the subsoil. A commercial FEM software package, Abaqus, was used as the analytic tool, with a built-in Concrete Damaged Plasticity model (CDP) that for the concrete material model. Both linear and nonlinear material properties have been used in the analyses with same softening effect of the subsoil. A large number of models were analysed to simulate the development of the fault and capture the failure modes of the slab at different stages. Excessive tensile cracks and vertical deformations were found in both failure scenarios studied. A possible internal stability problem of the warehouse due to this was also found. The nonlinear behaviour of the concrete slab was captured through the failure scenarios before damage. It is shown that the serviceability of the reinforced concrete slab can be influenced by softening ground, and a more realistic description of the possible failures was here obtained based on the nonlinear model compared to previous elastic analyses. Keywords:     Reinforced concrete slab, Nonlinear, Ground softening, Finite Element Method (FEM), Abaqus, Concrete Damaged Plasticity (CDP). / Armerade betongplattor är en vanlig typ av fundament som ofta används för bostäder, kommersiella och industriella byggnader. De flesta av grundplattorna konstrueras direkt på marken utan andra strukturella medium emellan och vissa av plattorna påverkas därför av mjuknande undergrunder. En grundplatta är konstruerad för att överföra vertikala laster och överbrygga imperfektioner i marken, men stora sättningar i undergrunden kan förstöra plattans stödförhållanden och genom detta störa stabiliteten för överbyggnaderna. Syftet med projektet är att studera hur mjuknande undergrund och dess propagering påverkar en typisk betongplatta. Vidare används en approximativ analysmetod för att utvärdera tillståndet hos en platta på sådan mark. Ett praktiskt fall har studerats för att fånga det verkliga brottbeteendet hos en armerad betongplatta, vilket bygger på ett tidigare genomfört projekt. För det studerade fallet kan eventuell framtida gruvdrift nära ett höglager orsaka skadliga sättningar i undergrunden. Den förkastning som kan uppstå kan leda till uppmjukning av jordarna under fundamentet, vilket kan förvärras och orsaka kollaps av betongplattan under lagret. Två typer av brottscenarier studerades; uppmjuknande respektive full kollaps av undergrunden. Finita Element Metoden (FEM) användes för att analysera beteendet hos plattan och utvecklingen av fbrott och förskjutning i undergrunden. Ett kommersiellt FEM programpaket, Abaqus, användes som analytiskt verktyg, med en inbyggd plasticitets och skademodell (CDP) för att beskriva betongmaterialets egenskaper. Både linjärelastiska och icke-linjära materialegenskaper har använts i analyserna, för att simulera effekten av den mjukande undergrunnden. Ett stort antal modeller analyserades för att simulera skadepropageringen och fånga brottmoder i plattan genom olika skeden. Dragsprickor och vertikala deformationer uppstod i bägge de studerade scenariorna. Ett möjligt intern stabilitetsproblem hos lagret på grund av detta konstaterades också. Det olinjära beteendet hos betongplattan fångades fram tills allvarlig skada uppstod. Det visas att funktionen i bruksgränsstadiet hos den armerade betongplattan kan påverkas genom uppmjuknande undergrund, och att en mer realistisk beskrivning av möjliga brott här kunde erhållas genom den olinjära modellen, jämfört med tidigare elastiska analyser. Nyckelord:   Betongplatta , Icke-linjärt uppträdande, Finita Element Metoden (FEM), FE-modell, Abaqus, Plasticitets och skademodell för betong (CDP).

This research endeavor develops Isogeometric approach for analysis of composite structures and take advantage of higher order continuity, smoothness and variation diminishing property of Nurbs basis for stress analysis of composite and sandwich beams and plates. This research also computes stress concentration factor in a composite plate with a hole. Isogeometric nonlinear/linear finite element code is developed for static and dynamic analysis of laminated composite plates. Nurbs linear, quadratic, higher-order and k-refined elements are constructed using various refinement procedures and validated with numerical testing. Nurbs post-processor for in-plane and interlaminar stress calculation in laminated composite and sandwich plates is developed. Nurbs post-processor is found to be superior than regular finite element and in good agreement with the literature. Nurbs Isgoemetric analysis is used for stress analysis of laminated composite plate with open-hole. Stress concentration factor is computed along the hole edge and good agreement is obtained with the literature. Nurbs Isogeometric finite element code for free-vibration and linear dynamics analysis of laminated composite plates also obtain good agreement with the literature. Main highlights of the research are newly developed 9 control point linear Nurbs element, k-refined and higher-order Nurbs elements in isogeometric framework. Nurbs elements remove shear-locking and hourglass problems in thin plates in context of first-order shear deformation theory without the additional step and compute better stresses than Lagrange finite element and higher order shear deformation theory for comparatively thick plates i.e. a/h = 4. Also, Nurbs Isogeometric analysis perform well for vibration and dynamic problems and for straight and curved edge problems. / Ph. D.

The steel industry has been producing flat plates through the process of hot rolling since the late 1600s. Hot rolling uses a series of rolls to progressively thin a strip of steel to a desired thickness. In deforming the strip, the rolling process causes variations in thickness across the width of the strip. These variations are commonly referred to as crown, which is specifically the difference in thickness between the center and edge of a strip. For most applications steel mill clients require flat products, or products with little variation in thickness. Therefore, variations represent wasted material which must be removed before the plate or sheet can be used in consumer products. Controlling the flatness of the metal strip is a high priority for the hot rolling business. The purpose of this work was to develop a 3-D dynamic model of the rolling process to simulate the behaviour of a strip while being rolled and predict its profile. To accomplish this task, much of the rolling process needed to be modeled. The profile of the strip is a product of the deformation of the rolls and frame within a mill stand. Therefore, not only did the geometry of these components need to be modeled, but the material properties and dynamic motion were required as well. The dynamic nature of the process necessitated the modeling of the rotation of the rolls and translation of the strip, aspects of rolling which are not typically simulated. Five models were developed during the project. The purpose of the first two models was to find the stiffnesses of the roll-stack and stand frame. The roll-stack refers to the rolls and their arrangement. The reference mill from which data was provided used a four-high roll-stack with two rolls above the strip and two below. The frame that holds the roll-stack, while massive, stretches when the strip is deformed between the rolls. This stretch changes the position of the rolls affecting the load and deformation of the strip. A lumped-mass model was created to simulate the dynamics of the roll-stack and frame. When the strip enters the gap between the rolls, there is a large impact force which causes the rolls to vibrate. The lumped-mass model was used to determine parameters to bring the system to steady state. The final two models simulated the entire rolling process with rotating rolls and moving strip. The 3-D dynamic rolling model was capable of predicting the strip profile due after exiting the rolls. Two calibrations were used to reduce model error before running a validation. The rolling causes thickness variation across the width of the metal strips; therefore, strips are intentionally rolled thick to meet a minimum thickness. In modern steel mills, specialized control systems are used to adjust parameters as the steel strip passes through each stand of rolls. Varying the parameters allows the thickness and profile of the strip to be controlled. Each stand may have several rolls in different configurations. These rolls are either work rolls, which directly contact the strip, or backup rolls, which contact the work rolls and stiffen the roll-stack. The stand frame holds the rolls and provides a means to position them. The validation results showed that the exit thickness, strip crown, and rolling load were less than 5% different from the values measured in the test data. The calibrated model was then used to derive strip crown sensitivities to gap, entry crown, work roll crown, and bending force. The 3-D dynamic model was able to predict the strip crown accurately when given calibrated information about the system. This model will be a useful tool for exploring the mechanics of hot rolling in ways that were not previously possible. / Master of Science

Yes / The design of reinforced concrete buildings must satisfy the serviceability stiffness criteria in terms of maximum lateral deflections and inter story drift in order to prevent both structural and non-structural damages. Consideration of plastic hinge formation is also important to obtain accurate failure mechanism and ultimate strength of reinforced concrete frames. In the present study, an iterative procedure has been developed for the analysis of reinforced concrete frames with cracked elements and consideration of plastic hinge formation. The ACI and probability-based effective stiffness models are used for the effective moment of inertia of cracked members. Shear deformation effect is also considered, and the variation of shear stiffness due to cracking is evaluated by reduced shear stiffness models available in the literature. The analytical procedure has been demonstrated through the application to three reinforced concrete frame examples available in the literature. It has been shown that the iterative analytical procedure can provide accurate and efficient predictions of deflections and ultimate strength of the frames studied under lateral and vertical loads. The proposed procedure is also efficient from the viewpoint of computational time and convergence rate. The developed technique was able to accurately predict the locations and sequential development of plastic hinges in frames. The results also show that shear deformation can contribute significantly to frame deflections.

[pt] No projeto de estruturas de engenharia há, freqüentemente, incertezas associadas µas propriedades dos materiais, nas propriedades geométricas e aos carregamentos. A maneira mais comum e tradicional para se levar em conta estas incertezas é através da definição dos valores de projeto como o resultado do produto do valor característico das variáveis aleatórias por um fator parcial de segurança. Esta solução, no entanto, falha ao não permitir a quantificação da confiabilidade do projeto ótimo uma vez que um fator grande de segurança pode não significar uma confiabilidade mais alta. Para se considerar a natureza probabilística de quantidades como propriedades dos materiais, carregamentos, etc., tem-se que identificar e definir estas quantidades como variáveis aleatórias no modelo de análise. Desta maneira, a probabilidade de falha (ou a confiabilidade) de uma estrutura sujeita a uma restrição de desempenho na forma de uma função de estado limite pode, então, ser calculada e formulada como uma restrição num problema de otimização. Neste trabalho, restrição probabilísticas são incorporadas ao esquema tradicional de otimização estrutural. A formulação e os métodos numéricos para este processo, comumente chamado de otimização baseada em confiabilidade, são descritos. O objetivo principal é apresentar um sistema computacional capaz de resolver problemas de otimização de forma e de dimensões de treliças espaciais baseado em confiabilidade. Podem ser consideradas como variáveis, determinísticas ou aleatórias, as seções transversais, as coordenadas nodais, as propriedades dos materiais (módulo de elasticidade e tensão de escoamento) e os carregamentos. De maneira a tratar os problemas de instabilidade global são considerados os efeitos da não-linearidade geométrica no comportamento da estrutura e uma restrição formulada para uma função de estado limite associada na carga de colapso é incluída. Funções de estado limite referentes aos deslocamentos e nas tensões também são consideradas. A flambagem global das barras é considerada por meio da carga crítica de Euler / [en] Uncertainties associated with random variables, such as, the material proprieties and loads, are inherent to the design of structures. These uncertainties are traditionally taken into account in the project before the design by defining design values for the random variables. The design values of the random variables are obtained from statistical properties of the random variables and from partial safety factors. Once these values are defined the variables are treated as deterministic variables in the design process. This approach has been followed in the conventional design optimization and in many design codes such as the Brazilian code for the design of steel and concrete structures. This simple approach, however, does not allow an estimate of the structural reliability of the resulting project which may have a low (unsafe structure) or a very high (expensive structure) reliability. To overcome this problem a reliability analysis must be incorporated into the traditional design optimization. Design optimization, incorporating reliability analyses, has been denoted Reliability-Based Design Optimization (RBDO). In RBDO, the constraints are defined in terms of the probabilities of failure associated with some prescribed failure functions and therefore, it requires, as in the reliability analysis, the definition of the random variables and information about their statistical properties. In this work, RBDO is applied to the shape and sizing optimization of spatial trusses considering geometric nonlinearities. The constraints considered in the RBDO problem are related to the following failure mechanisms: to the global collapse (limit load), to local buckling and yield stress and to serviceability conditions (displacement bounds). The algorithms used for solving the optimization problem and for performing the reliability analysis are described.

Neste trabalho são discutidos os aspectos referentes à análise não-linear física de painéis de contraventamento em alvenaria estrutural. O objetivo foi a obtenção de coeficientes a serem aplicados ao produto de rigidez à flexão (EI) para a consideração da não-linearidade física de forma simplificada. As estruturas foram analisadas de forma paramétrica através de um modelo de elementos finitos lineares (pórtico plano), com a utilização de duas diferentes taxas de armaduras para uma combinação de ação horizontal e vertical aplicada. A ação horizontal limitou o deslocamento do topo do painel em H/3500, nas análises lineares. A ação vertical simulou a reação de uma laje com peso próprio e sobrecargas de utilização. A alvenaria foi considerada como um meio contínuo e homogêneo (macro-modelos). Foi apresentada uma metodologia para a obtenção do coeficiente de rigidez à flexão para ser aplicado nas análises lineares e simular o efeito da não-linearidade física. Os valores do coeficiente de redução do produto de rigidez à flexão apresentaram-se muito baixos, com uma variação de 0,1 a 0,4, sugerindo a verificação do modelo matemático bem como do programa utilizado nas análises. / This study deals with aspects related to the nonlinear analysis of physical shear walls in masonry. The objective was to obtain coefficients to be applied to the product of bendind stiffness (EI) for considering physically nonlinear behavior in a simplified analysis. The structures were analyzed usig a parametric finit element model using linear elements (plane frame), with two different reinforcement ratios for one combination of horizontal and vertical actions. The horizontal action limited de top displacement of the shear wall in H/3500. The vertical loads simulated the reaction of a slab with self-weight and burdens of use. The masonry was considered as a means continuous and homogeneous (Macro-models). Was presented a methodology for obtaining the coefficient of bending stiffness to be applied in linear analysis and simulate the effect of non-linearity. The coefficient of reduction product of bending stiffness were very low, with a variation of 0.1 to 0.4, suggesting the verification of mathematical model and the program used in the analysis.

A eletroencefalografia (EEG) é uma das evidências tomadas na avaliação de indicação cirúrgica, em casos de pacientes portadores de epilepsia refratária a medicamentos, que pode auxiliar na localização da área responsável pela origem das crises epilépticas. Ao longo das últimas décadas, além das bandas de frequências já tradicionalmente avaliadas (até cerca de 40Hz), a EEG tem despertado o interesse de pesquisadores também para bandas de frequências mais altas. Passaram a ser encontradas evidências de que oscilações de alta frequência, conhecidas por HFO (High Frequency Oscillations), podem ser usadas como biomarcadores de epilepsia. Diversos estudos têm sido realizados em busca de uma melhor compreensão sobre HFO, a fim de viabilizar sua utilização em aplicações clínicas. Entretanto, características não lineares e de complexidade, que podem contribuir na análise de sinais com origem em sistemas biológicos, não têm sido investigadas neste tipo de sinais. Este estudo propôs a investigação de características extraídas de sinais de EEG com presença de HFO, de pacientes portadores de epilepsia refratária, através de métodos considerados como de análise não linear. Análise de Dinâmica Simbólica, Análise de Flutuações Destendenciadas (DFA), Entropia Multiescala (MSE) e Análise qSDiff foram aplicadas em segmentos de sinais de EEG intracraniano, amostrados a 5kHz, de pacientes portadores de epilepsia refratária, e também em alguns sinais simulados de características conhecidas para fins de comparação. Os resultados dos diferentes métodos investigados apontaram características semelhantes entre os segmentos de EEG analisados e séries simuladas de ruído browniano, sugerindo que os sinais de EEG em geral têm perfil bastante suavizado, são não estacionários e exibem correlações de longo alcance. Foram também levantadas evidências de que tanto HFO quanto os segmentos de EEG onde estão inseridas têm padrões mais regulares de variação e são menos complexas que segmentos de EEG sem HFO, sugerindo a degradação da complexidade fisiológica desta região cerebral, que poderia estar relacionada com mecanismos fisiopatológicos da epilepsia. Todos os métodos investigados sugeriram que as características e propriedades não lineares, relacionadas a complexidade inerente dos sinais de EEG, podem ser úteis na análise de HFO, principalmente pelas evidências de que estas características se alteram nas HFO, quando comparadas ao restante do sinal onde elas se encontram e também a outros sinais sem sua presença. / Electroencephalography (EEG) is one of the evidences taken in the evaluation of surgical indication, in cases of patients with drug refractory epilepsy, which may help in locating the area responsible for the origin of epileptic seizures. Over the last few decades, in addition to the frequency bands that have traditionally been evaluated (up to about 40Hz), the EEG has attracted researchers also to higher frequency bands. Evidence has been found that high frequency oscillations (HFO), can be used as biomarkers of epilepsy. Many studies have been carried out in search of a better understanding about HFO, in order to make it feasible to use in clinical applications. However, nonlinear and complex features, which may contribute to the analysis of signals originating from biological systems, have not been investigated in this type of signals. This study proposed the investigation of features extracted from EEG signals with HFO of patients with refractory epilepsy, using nonlinear analysis methods. Symbolic Dynamics Analysis, Detrended Fluctuation Analysis (DFA), Multiscale Entropy (MSE) and qSDiff Analysis were applied to segments of intracranial EEG signals, sampled at 5kHz, from patients with refractory epilepsy, as well as some features-known simulated signals for comparison purposes. Results of the different investigated methods pointed out similar features between the analyzed EEG segments and the simulated series of Brownian noise, suggesting that EEG signals, in general, have a very smoothed profile, are nonstationary and exhibit long- range correlations. Evidence has also been raised that both HFO and the EEG segments where they are inserted have more regular patterns of variation and are less complex than EEG segments without HFO, suggesting the degradation of the physiological complexity of this brain region, which could be related to pathophysiological mechanisms of epilepsy. All the investigated methods suggested that nonlinear features and properties, related to the inherent complexity of EEG signals, may be useful in HFO analysis, mainly because of the evidence that these features change in HFOs when compared to the rest of the signal where they are and other signals without their presence.

Neste trabalho desenvolveu-se um método para análise de estruturas planas em concreto armado, em que considerou-se os efeitos de segunda ordem e da fluência. Foi utilizado o método dos elementos finitos (MEF) com elemento de barra com seis graus de liberdade, formulado com as hipóteses da teoria de viga Bernoulli-Euler, com seção transversal subdividida em lamelas. A não linearidade geométrica é representada pela descrição corrotacional para grandes deslocamentos e rotações e pequenas deformações. O efeito viscoelástico para prever a fluência no concreto é obtido pelo modelo de Maxwell em camadas, calibrado a partir de funções de fluência fornecidas por normas. Uma análise envolvendo o modelo desenvolvido em plataforma MATLAB pode ser dividida em duas etapas: na primeira busca-se a resposta da estrutura para um carregamento instantâneo e na segunda o comportamento é dependente do tempo. Inicialmente fizeram-se comparações com alguns resultados disponíveis na literatura a fim de avaliar o comportamento do modelo implementado. Por fim, foi feita a análise de um modelo reduzido. Esse modelo consistiu em um pórtico moldado em microconcreto armado, que foi instrumentado com sensores elétricos para medir suas deformações e deslocamentos. O pórtico foi ensaiado na idade de trinta e três dias e monitorado durante cento e vinte e cinco dias. A partir dos resultados analisados por este aplicativo, concluiu-se que o mesmo é capaz de fazer análise não linear física e geométrica de estruturas planas em concreto armado, considerando a fluência. Os resultados de previsão de fluência fornecidos pelo programa para elementos de concreto armado submetidos à força axial são muito próximos dos resultados experimentais de Kataoka (2010). No entanto, para a análise do pilar do pórtico, que corresponde a um caso de flexo-compressão, houve uma diferença significativa, pois a análise numérica gerou curvaturas maiores do que as experimentais. Em relação aos resultados monitorados foi observado que a umidade tem grande influência nas deformações medidas. / In this work a methodology for the analysis of plane structures of reinforced concrete was developed. The second-order effect and creep were considered. The methodology is based in the finite element method (FEM). Beam elements with 6 degrees of freedom (d.o.f), which takes into account the Euler-Bernoulli beam theory with cross-section divided in layers, was used. The geometric nonlinearity is described by the co-rotational formulation for large displacements and rotations and small deformations. The viscoelastic effect of the creep in the concrete is obtained with the use of the Maxwell model in layers, calibrated with creep functions obtained in the standards. The model developed was implemented in MATLAB language, in which the analysis is sub-divided in two phases: in the first phase one seeks the response of the structure for an instantaneous load and in the second, the behaviour is time dependent. Firstly, some comparison with some available results in the literature were done with the aim to assess the accuracy of the model implemented. After that, the analysis of a prototype was done. The prototype consists of a frame made of reinforced micro-concrete. The frame was instrumented with electric sensors to measure the displacements and deformations. The frame was loaded after at the age of 33 days and monitored during 125 days. The results showed that the program is able to analysis plane structures of reinforced concrete with material and geometric nonlinearity, with the consideration of creep. The results obtained with the analysis program for an element of reinforced concrete under axial load are very close to the experimental results obtained by Kataoka (2010). However, in the analysis of framecolumns, corresponds to combined bending and compression loads, noticeable differences appears due to bigger curvatures in the numerical analysis than in the experimental one. With respect to the monitored results it was noted that the humidity has large influence in the deformations measured.

A voz tem sido objeto de estudos em diferentes áreas da ciência. Nas últimas duas décadas os pesquisadores demonstraram a presença do caos na produção de voz. O objetivo deste trabalho é estabelecer padrões visuais de sinais de voz humana através da técnica não linear de reconstrução de espaço de fase e associá-los com suas correspondentes análises auditivo perceptiva e acústica. Foram analisados sinais de voz humana de sujeitos de ambos os gêneros, na faixa etária de 19 a 39 anos. Foram gravadas as vogais sustentadas /a/, /e/ e /i/ do português do Brasil, com uma taxa de amostragem de 22.050 Hz. Os sinais foram analisados a fim de obter medidas acústicas (Jitter, Shimmer e Coeficiente de Excesso). Foi utilizada a técnica de reconstrução de espaço de fase para descrever as características de dinâmica não linear dos sinais de voz, e para análise perceptivo auditiva foi utilizada a escala RASAT. Os resultados demonstram que métodos de dinâmica não linear como a reconstrução do espaço de fase, parece ser uma técnica apropriada para análise de sinais de voz, incorporando os componentes caótico e determinístico da voz humana. É sugerido que análise da dinâmica não linear não substitui as técnicas existentes, embora possa aperfeiçoar e complementar os métodos de análise disponíveis para os profissionais da saúde, como fonoaudiólogos e otorrinolaringólogos / Human voice has been the focus of study for different areas of science. Researches in the last two decades have demonstrated the existence of chaos in human voice production. The purpose of this work is to use nonlinear dynamics techniques in the analysis of normal voices from healthy subjects and correlate them to traditional acoustic parameters as well as perceptual analysis. Human voice signals from healthy subjects, both male and female, ranging in age from 19 to 39 years old were analyzed. Sustained vowel sounds /a/, /e/ and /i/, from brazilian Portuguese were recorded at a sampling rate of 22,050 Hz and analyzed in order to obtain acoustic measures (Jitter, Shimmer and coefficient of excess – EX). The phase space reconstruction technique was used to describe the nonlinear dynamic characteristics of voice signal samples. The results show, that non-linear dynamical method as phase space reconstruction seems to be a suitable technique for voice signals analysis, due to the chaotic component of the human voice. It is suggested, that non-linear dynamic analysis does not replace existing techniques instead, it may improve and complement the recent voice analysis methods available for health professionals, speech therapist and clinician

Este trabalho utiliza uma teoria não linear tridimensional de barras de [1], que considera o empenamento da seção transversal como uma das variáveis do problema, para a análise de perfis de aço por meio do Método dos Elementos Finitos. A equação constitutiva apresentada em [2] é aqui expressa de forma completa, com os termos de ordem cúbica e superiores antes desprezados. A influência da inclusão desses termos sobre a estabilidade da barra pode assim ser estudada com maior profundidade. O modelo constitutivo obtido foi implementado em um programa computacional de elementos finitos, e diversos exemplos numéricos são apresentados para validar a teoria aqui desenvolvida. Foi possível obter a instabilidade por flexo-torção com a equação constitutiva completa aqui desenvolvida, onde antes não era possível em [2]. Melhorando o modelo constitutivo. / This work presents a nonlinear tridimensional beam theory of [1], which encompasses cross-sectional warping as one of the model variables, and its application to the analysis of steel profiles with the Finite Element Method. The constitutive equation presented in [2] is fully developed, including the third and higher order terms neglected therein. The influence of the inclusion of such terms on the beam stability could then be studied more deeply. The obtained constitutive model was implemented in a computational Finite Element program and several numerical examples are presented to validate the theory here developed. The complete constitutive equation made possible to determine instability in composite flexural torsional states, a clear contribution to the model developed in [2].