Poměrně velké množství soukromých firem v České republice vyrábí lehká sportovní letadla. Značná část těchto letadel využívá kompozitní pružiny ve svých přistávacích zařízení. Tyto pružiny jsou buďto menší díly, absorbující energii (na příďové noze či ostruze), anebo jde celé pružnice hlavního podvozku. Všechny tyto pružiny sdílí základní charakteristiky: jsou vyrobeny převážně z jednosměrného kompozitu s významnou tloušťkou, hlavním druhem zatížení je ohybový moment a jsou očekávány velké deformace. Podobnou charakteristiku můžeme použít i při popisu hlavního nosníku křídla. Jak vypadá návrh a analýza takovýchto dílů? V zásadě jsou dvě možnosti. První z nich je poměrně jednoduchá analytická analýza, případně naprogramovaná v tabulkovém výpočetním prostředí. Nevýhody tohoto řešení jsou limitované možnosti výpočtu a jeho nízká flexibilita. Druhou možností je využít komerční konečno-prvkový systém pro analýzu, případně i pro optimalizaci. Pochopitelnou nevýhodou této možnosti je cena programu a obsluhy. Cílem této disertační práce je vytvořit program, jež nabídne třetí možnost, která umožní provádět zevrubnou analýzu řešených produktů bez nutnosti pořizovat nákladný software. Tento program zjednoduší a urychlí návrh a pevnostní kontrolu. Umožní uživateli rychle analyzovat více návrhových variant. Program dále bude zohledňovat specifika analyzovaných produktů (například velké deformace a lokální koncentrace napětí kolmo na vlákno). Z pohledu uživatele by program měl být jednoduchý na ovládání. Minimum množství vstupních dat a přehledné grafické rozhraní zajistí komfortní používání. Samostatně spustitelný program (bez instalace a bez podpůrného softwaru) zlepšuje rozšiřitelnost programu.

Piston rings are installed in the piston and cylinder wall, which does not have a perfect round shape due to machining tolerances or external loads e.g. head bolts tightening. If the ring cannot follow these deformations, a localized lack of contact will occur and consequently an increase in the engine blow-by and lubricant oil consumption. Current 2D computational methods can not implement such effects – more complex model is necessary. The presented master’s thesis is focused on the developement of a flexible 3D piston ring model able to capture local deformations. It is based on the Timoshenko beam theory in cooperation with MBS software Adams. Model is then compared with FEM using software ANSYS. The validated piston ring model is assembled into the piston/cylinder liner and very basic simulations are run. Finally, future improvements are suggested.

A computational framework for linear static aeroelastic analyses is presented. The overall aeroelasticity model is applicable to conceptual aircraft design studies and other low-fidelity aero-structural analyses. A partitioned approach is used, i. e. separate solvers for aerodynamics and structure analyses are coupled in a suitable way, together forming a model for aeroelastic simulations. Aerodynamics are modelled using the vortexlattice method (VLM), a simple computational fluid dynamics (CFD) model based on potential flow. The structure is represented by a three-dimensional (3D) Euler-Bernoulli beam model in a finite element method (FEM) formulation. A particular focus was put on the modularity and loose coupling of aforementioned models. The core of the aeroelastic framework was abstracted, such that it does not depend on any specific details of the underlying aerodynamics and structure modules. The final aeroelasticity model constitutes independent software tools for the VLM and the beam FEM, as well as a framework enabling the aeroelastic coupling. These different tools have been developed as part of this thesis work. A wind tunnel experiment with a simple wing model is presented as a validation test case. An aero-structural analysis of a fully elastic unmanned aerial vehicle (UAV) (OptiMale) is described and results are compared with an existing higherfidelity study. / Rapporten beskriver en beräkningsmodell för linjära, statisk aeroelastiska analyser. Modellen kan användas för konceptuella designstudier av flygplan. En partitionerad metod används, d v s separata lösare för aerodynamik- och strukturanalyser kopplas på ett lämpligt sätt, och bildar tillsammans en modell för aeroelastiska simulationer. Aerodynamik modelleras med hjälp av en så kallad vortex-lattice method (VLM), en enkel modell för beräkningsströmningsdynamik (CFD) som är baserad på friktionsfri strömning. Strukturen representeras av en tredimensionell (3D) Euler-Bernoulli-balkmodell implementerad med hjälp av en finita elementmetod (FEM). Ovannämnda modeller har utvecklats med fokus på modularitet och lös koppling. Kärnan i den aeroelastiska modellen har abstraherats så att den inte beror på specifika detaljer i de underliggande aerodynamik- och strukturmodulerna. Aeroelasticitetsmodellen i sin helhet består av separata mjukvaruprogram för VLM och balk-FEM, såväl som ett ramverk som möjliggör den aeroelastiska kopplingen. Dessa olika program har utvecklats som en del av examensarbetet. Ett vindtunnelförsök med en enkel vingmodell presenteras som ett valideringstest. Dessutom beskrivs en analys av ett elastiskt obemannad flygplan (OptiMale) och resultaten jämförs med en befintlig studie som har genomförts med modeller av högre trovärdighet.

Advances in the design and manufacturing technologies have greatly enhanced the utility of fiber reinforced composite materials in aircraft, helicopter and space- craft structural components. The special characteristics of composites such as high strength and stiffness, light-weight corrosion resistance make them suitable sub- stitute for metals/metallic alloys. However, composites are very sensitive to the anomalies induced during their fabrication and service life. Also, they are suscepti- ble to the impact and high frequency loading conditions because the epoxy matrix is at-least an order of magnitude weaker than the embedded reinforced carbon fibers. On the other hand, the carbon based matrix posses high electrical conductivity which is often undesirable. Subsequently, the metal matrix produces high brittleness. Var- ious forms of damage in composite laminates can be identified as indentation, fiber breakage, matrix cracking, fiber-matrix debonding and interply disbonding (delam- ination). Among all the damage modes mentioned above, delamination has been found to be serious for all cases of loading. They are caused by excessive interlaminar shear and normal stresses. The interlaminar stresses that arise in the case of composite materials due to the mismatch in the elastic constants across the plies. Delamination in composites reduce it’s tensile and compressive strengths by consid- erable margins. Hence the knowledge of these stresses is the most important aspect to be looked into. Basic theories like the Euler-Bernoulli’s theory and Timoshenko beam theory are based on many assumptions which poses limitation to determine these stresses accurately. Hence the determination of these interlaminar stresses accurately requires higher order theories to be considered. Most of the conventional methods of determination of the stresses are through the solutions, involving the trigonometric series, which are available only to small and simple problems. The most common method of solution is by Finite Element (FE) Method. There are only few elements existing in the literature and very few in the commercially available finite element software to determine the interlaminar stresses accurately in the composite laminates. Accuracy of finite element solution depends on the choice of functions to be used as interpolating polynomials for the field variable. In-appropriate choice will manifest in the form of delayed convergence. This delayed convergence and accuracy in predicting these stresses necessiates a formulation of elements with a completely new concept. The delayed convergence is sometimes attributed to the shear locking phenomena, which exist in most finite element formulation based on shear deformation theories. The present work aims in developing finite elements based on higher order theories, that alleviates the slow convergence and achieves the solutions at a faster rate without compromising on the accuracy. The accuracy primarily depends on the theory used to model the problem. Thus the basic theories (such as Elementary Beam theory and Timoshenko Beam theory) does not suffice the condition to accuratley determine the interlaminar stresses through the thickness, which is the primary cause for delamination in composites. Two different elements developed on the principle of super-convergence has been presented in this work. These elements are subjected to several numerical experiments and their performance is assessed by comparing the solutions with those available in literature. Spacecraft and aircraft structures are light in weight and are also lightly damped because of low internal damping of the material of construction. This increased exibility may allow large amplitude vibration, which might cause structural instability. In addition, they are susceptible to impact loads of very short duration, which excites many structural modes. Hence, structural dynamics and wave propagation study becomes a necessity. The wave based techniques have found appreciation in many real world problems such as in Structural Health Monitoring (SHM). Wave propagation problems are characterized by high frequency loads, that sets up stress waves to propagate through the medium. At high frequency, the wave lengths are small and from the finite element point of view, the element sizes should be of the same order as the wave lengths to prevent free edges of the element to act as a free boundary and start reflecting the stress waves. Also longer element size makes the mass distribution approximate. Hence for wave propagation problems, very large finite element mesh is an absolute necessity. However, the finite element problems size can be drastically reduced if we characterize the stiffness of the structure accurately. This can accelerate the convergence of the dynamic solution significantly. This can be acheived by the super-convergent formulation. Numerical results are presented to illustrate the efficiency of the new approach in both the cases of dynamic studies viz., the free vibration study and the wave propagation study. The thesis is organised into five chapters. A brief organization of the thesis is presented below, Chapter-1 gives the introduction on composite material and its constitutive law. The details of shear locking phenomena and the interlaminar stress distribution across the thickness is brought out and the present methods to avoid shear locking has been presented. Chapter-2 presents the different displacement based higher order shear deformation theories existing in the literature their advantages and limitations. Chapter-3 presents the formulation of a super-convergent finite element formulation, where the effect of lateral contraction is neglected. For this element static and free vibration studies are performed and the results are validated with the solution available in the open literature. Chapter-4 presents yet another super-convergent finite element formulation, wherein the higher order effects due to lateral contraction is included in the model. In addition to static and free vibration studies, wave propagation problems are solved to demonstrate its effectiveness. In all numerical examples, the super-convergent property is emphasized. Chapter-5 gives a brief summary of the total research work performed and presents further scope of research based on the current research.

The aim of this investigation was to review literature for the most suitable aeroelastic tailoring analysis tools for long slender composite structures, and integrate them into an aeroelastic tailoring process. The JS1C Revelation is a high performance sailplane made from modern composites, mostly carbon fibre. This has the advantage of being more rigid than traditional engineering materials, thereby reducing the effects of the twisting deflections on these long slender structures due to aerodynamic loads. The implementing of aeroelastic tailoring can create bend-twist couples for performance improvements. Composites enable the use of aeroelastic tailoring to improve gliding performance. Flaperon 3 of the JS1C 21 m was used as the design problem for aeroelastic tailoring. Aeroelastic tailoring was done by analysing the flaperon structure at the different layup angles to determine the correct design point to tailor the structure to improve aerodynamic performance at thermalling and cruise, but mostly cruise since it accounts for 70% of the flight time. The composite structure analysis tool has the objective to get results during concept design. This directed the line of research of analysis tools to a solution method of two dimensional cross-section mesh properties projected onto a one dimensional beam. The literature of Hodges had good verification and published data on the analysis tools. The analysis tools comprised of three programs that were not very user friendly. Thus the author compiled a Matlab program as a user interface tool to run the three programs together. The aeroelastic tailoring process systematically works through the known design variables and objectives, which are given as inputs to the analysis tool. The analysis tool plots the coupling data versus layup angle. From this the best layup angles for a sought-after bend-twist couple is used to aeroelastically tailor the wing. The composite structure analysis tool’s accuracy was verified by analysing cantilever beam deflections and comparing the results with hand calculations and SolidWorks Simulation FEM results. The analysis tool’s accuracy was further verified by comparing the aerodynamic torsional load’s twist deflections with thin walled tube theory. The analysis tool was validated by applying a torsional load at the tip of a JS1C production Flaperon 3 in an experimental setup and then comparing this result with the Flaperon 3 modelled in the analysis tool. These comparisons also ensured that the model’s composite material properties and the meshing of the flaperon cross-sectional properties were correct. This aeroelastic tailoring was validated with the advantage of then being used to improve the aerodynamic performance of the JS1C Revelation 21 m tip’s flaperon. This improvement could be made by making use of a tailored bend-twist couple to reduce the effect of the aerodynamic load’s twist deflections. A test sample of the JS1C 21 m flaperon 3 was used to validate aeroelastic tailoring. The test sample was designed to be 1 m in length and have all the specified tailoring coupling characteristics that could improve the aerodynamic performance of the JS1C 21 m flaperon 3. The test sample was manufactured according to Jonker Sailplanes manufacturing standards and experimentally set up with the same applied deflections as in the analysis tool. The calculated bend-twist values and the experimental setup results were similar with a negligible difference, assuming small displacements and an aspect ratio greater than 13; this confirmed that the PreVABS/VABS/GEBT composite structure analysis tool could be used in aeroelastic tailoring to predict and design the bend-twist couple needed to improve the aerodynamic performance of the JS1C 21 m. While the twist behaviour of Flaperon 3 was improved by the tailored bend-twist couple, it was still necessary to add pre-twist as well, to fully address the effects of twisting by aerodynamic forces. / MIng (Mechanical Engineering), North-West University, Potchefstroom Campus, 2014

The aim of this investigation was to review literature for the most suitable aeroelastic tailoring analysis tools for long slender composite structures, and integrate them into an aeroelastic tailoring process. The JS1C Revelation is a high performance sailplane made from modern composites, mostly carbon fibre. This has the advantage of being more rigid than traditional engineering materials, thereby reducing the effects of the twisting deflections on these long slender structures due to aerodynamic loads. The implementing of aeroelastic tailoring can create bend-twist couples for performance improvements. Composites enable the use of aeroelastic tailoring to improve gliding performance. Flaperon 3 of the JS1C 21 m was used as the design problem for aeroelastic tailoring. Aeroelastic tailoring was done by analysing the flaperon structure at the different layup angles to determine the correct design point to tailor the structure to improve aerodynamic performance at thermalling and cruise, but mostly cruise since it accounts for 70% of the flight time. The composite structure analysis tool has the objective to get results during concept design. This directed the line of research of analysis tools to a solution method of two dimensional cross-section mesh properties projected onto a one dimensional beam. The literature of Hodges had good verification and published data on the analysis tools. The analysis tools comprised of three programs that were not very user friendly. Thus the author compiled a Matlab program as a user interface tool to run the three programs together. The aeroelastic tailoring process systematically works through the known design variables and objectives, which are given as inputs to the analysis tool. The analysis tool plots the coupling data versus layup angle. From this the best layup angles for a sought-after bend-twist couple is used to aeroelastically tailor the wing. The composite structure analysis tool’s accuracy was verified by analysing cantilever beam deflections and comparing the results with hand calculations and SolidWorks Simulation FEM results. The analysis tool’s accuracy was further verified by comparing the aerodynamic torsional load’s twist deflections with thin walled tube theory. The analysis tool was validated by applying a torsional load at the tip of a JS1C production Flaperon 3 in an experimental setup and then comparing this result with the Flaperon 3 modelled in the analysis tool. These comparisons also ensured that the model’s composite material properties and the meshing of the flaperon cross-sectional properties were correct. This aeroelastic tailoring was validated with the advantage of then being used to improve the aerodynamic performance of the JS1C Revelation 21 m tip’s flaperon. This improvement could be made by making use of a tailored bend-twist couple to reduce the effect of the aerodynamic load’s twist deflections. A test sample of the JS1C 21 m flaperon 3 was used to validate aeroelastic tailoring. The test sample was designed to be 1 m in length and have all the specified tailoring coupling characteristics that could improve the aerodynamic performance of the JS1C 21 m flaperon 3. The test sample was manufactured according to Jonker Sailplanes manufacturing standards and experimentally set up with the same applied deflections as in the analysis tool. The calculated bend-twist values and the experimental setup results were similar with a negligible difference, assuming small displacements and an aspect ratio greater than 13; this confirmed that the PreVABS/VABS/GEBT composite structure analysis tool could be used in aeroelastic tailoring to predict and design the bend-twist couple needed to improve the aerodynamic performance of the JS1C 21 m. While the twist behaviour of Flaperon 3 was improved by the tailored bend-twist couple, it was still necessary to add pre-twist as well, to fully address the effects of twisting by aerodynamic forces. / MIng (Mechanical Engineering), North-West University, Potchefstroom Campus, 2014

Para o projeto de estruturas com perfis de aço formados a frio, é fundamental a compreensão dos fenômenos da instabilidade local e global, uma vez que estes apresentam alta esbeltez e baixa rigidez à torção. A determinação do carregamento crítico e a identificação do modo de instabilidade contribuem para o entendimento do comportamento dessas estruturas. Este trabalho avalia três metodologias para a análise linear de estabilidade de perfis de aço formados a frio isolados, com o objetivo de determinar os carregamentos críticos elásticos de bifurcação e os modos de instabilidade associados. Estritamente, analisa-se perfis de seção U enrijecido e Z enrijecido isolados, de diversos comprimentos e diferentes condições de vinculação e carregamento. Determinam-se os carregamentos críticos elásticos de bifurcação e os modos de instabilidade globais e locais por meio de: (i) análise com o Método das Faixas Finitas (MFF), através do uso do programa computacional CUFSM; (ii) análise com elementos finitos de barra baseados na Teoria Generalizada de Vigas (MEF-GBT), via uso do programa GBTUL; e (iii) análise com elementos finitos de casca (MEF-cascas) por meio do uso do programa ABAQUS. Algumas restrições e ressalvas com relação ao uso do MFF são apresentadas, assim como limitações da Teoria Generalizada de Viga e precauções a serem tomadas nos modelos de cascas. Analisa-se também a influência do grau de discretização da seção transversal. No entanto, não é feita avaliação em relação aos procedimentos normativos e tampouco análises não lineares, considerando as imperfeições geométricas iniciais, tensões residuais e o comportamento elastoplástico do material. / For the design of cold formed steel members, it is essential to understand the effects of local and global instability, since these members typically have a high slenderness and low torsion stiffness. The determination of critical loads and the associated buckling modes contribute to understand the behavior of these members. This work performs a evaluation of three methods for linear stability analysis of isolated cold-formed steel members in order to determine the elastic critical loads and the corresponding buckling modes. Specifically, Ue and Ze shape members were studied with various length, different boundary conditions and loads. The elastic critical loads and buckling modes are determined by means of: (i) analysis with the Finite Strip Method (FSM), by the computer program CUFSM, (ii) beam finite element analysis based on the Generalized Beam Theory (FEM-GBT), by GBTUL program, and (iii) Finite Element Method with shell analysis using ABAQUS program. Some restrictions and warnings regarding the use of the FSM are presented, as well as limitations of the Generalized Beam Theory and precautions to be taken in the shell models. It is also analyzed the influence of the degree of discretization of the cross section. In the present study, no evaluation was made with respect to normative procedures neither nonlinear analyses considering the initial geometric imperfections, residual stresses and elastoplastic behavior of the material.

The aim of this work is to develop and validate a continuum model for the simulation of the thermomechanical response of a shape memory polymer (SMP). Rather than integral type viscoelastic model, the approach here is based on the idea of two inter-penetrating networks, one which is permanent and the other which is transient together with rate equations for the time evolution of the transient network. We find that the activation stress for network breakage and formation of the material controls the gross features of the response of the model, and exhibits a "thermal Bauschinger effect". The model developed here is similar to a thermoviscoelastic model, and is developed with an eye towards ease of numerical solutions to boundary value problems. The primary hypothesis of this model is that the hysteresis of temperature dependent activation-stress plays a lead role in controlling its main response features. Validation of this hypothesis is carried out for the uniaxial response from the experimental data available in the literature for two different SMP samples: shape memory polyurethane and Veriflex, to show the control of the evolution of the temperature sensitive activation stress on the response. We extend the validated 1D model to a three dimensional small strain continuum SMP model and carry out a systematic parameter optimization method for the identification of the activation stress coefficients, with different weights given to different features of the response to match the parameters with experimental data. A comprehensive parametric study is carried out, that varies each of the model material and loading parameters, and observes their effect on design-relevant response characteristics of the model undergoing a thermomechanical cycle. We develop "response charts" for the response characteristics: shape fixity, shape recovery and maximum stress rise during cooling, to give the designer an idea of how the simultaneous variation of two of the most influential material parameters changes a specific response parameter. To exemplify the efficacy of the model in practical applications, a thermoviscoelastic extension of a beam theory model will be developed. This SMP beam theory will account for activation stress governed inelastic response of a SMP beam. An example of a three point bend test is simulated using the beam theory model. The numerical solution is implemented by using an operator split technique that utilizes an elastic predictor and dissipative corrector. This algorithm is validated by using a three-point bending experiment for three different material cases: elastic, plastic and thermoplastic response. Time step convergence and mesh density convergence studies are carried out for the thermoviscoelastic FEM model. We implement and study this model for a SMP beam undergoing three-point bending strain recovery, stress recovery and cyclic thermomechanical loading. Finally we develop a thermodynamically consistent finite continuum model to simulate the thermomechanical response of SMPs. The SMP is modeled as an isotropic viscoplastic material where thermal changes govern the evolution of the activation stress of the material. The response of the SMP in a thermomechanical cycle is modeled as a combination of a rubbery and a glassy element in series. Using these assumptions, we propose a specific form for the Helmholtz potential and the rate of dissipation. We use the technique of upper triangular decomposition for developing the constitutive equations of the finite strain SMP model. The resulting model is implemented in an ODE solver in MATLAB, and solved for a simple shear problem. We study the response of the SMP model for shear deformation as well as cyclic shear deformation at different initial temperatures. Finally, we implement the thermomechanical cycle under shear deformations and study the behavior of the model.

L’hydrogène liquide est un ergol utilisé en complément de l’oxygène liquide pour la propulsion de nombreux lanceurs en particulier ceux de la famille Ariane. Cependant, sa dangerosité interdit la réalisation d’essais, en particulier vibratoires, sur des réservoirs remplis. Cette thèse explore une piste d’étude permettant de réaliser ces essais vibratoires sur le réservoir rempli d’un matériau de substitution : un ensemble de billes pré-contraintes. L’objectif est d’obtenir un comportement modal similaire en termes de modes et de fréquences propres à celui du réservoir rempli d’hydrogène liquide pour l’étude des premiers modes. Pour répondre à cet objectif, cette étude est développée en deux parties. Dans la première partie, une approche analytique basée sur une méthode par équivalences fréquentielles est détaillée. Après avoir présenté les grandes lignes de la méthodologie utilisée et l’ensemble des modèles développés, la méthodologie est appliquée au cas étudié expérimentalement de manière à mettre en évidence l’influence des différents paramètres et de proposer une première méthode pour choisir un jeu de billes adapté. La seconde partie, traite principalement des aspects expérimentaux et numériques. Après avoir détaillé la caractérisation des billes utilisées, le montage expérimental conçu et réalisé pour effectuer des essais vibratoires sur un réservoir rempli de billes pré-contraintes est présenté. Les différents résultats obtenus avec ce montage sont étudiés au regard de l’approche analytique, mais aussi de modèles numériques utilisant des éléments-discrets et des éléments-finis. / Liquid hydrogen is a propellant alternatively used with liquid oxygen for the propulsion of many launchers, especially those of the Ariane family. However, its dangerousness prohibits vibration tests on filled tank. The aim of this PHD is to explore the possibility to carry out such vibration tests by filling these tanks with a substitute material : a set of pre-stressed balls. As further argued and developed in the report, the objective is to obtain a modal behavior similar in term of mode shapes and natural frequencies to those of the tank filled with liquid hydrogen for the first modes. In the first part, an analytical approach based on a method using frequency equivalences is detailed. After presenting the outlines of the methodology used and all the models developed, the methodology is applied to the experimentally studied case in order to highlight the influence of the different parameters as well as then to propose a first method to choose a set of well adapted balls. The second part deals mainly with experimental and numerical aspects. Firstly, the balls used in the thesis are characterized. Then, the experimental set-up designed and realized to carry out vibration tests on a tank filled with pre-stressed balls is presented. The different results obtained are compared with both analytical results and numerical based on discrete-elements and finite-elements results.

Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2017-07-18T12:41:11Z No. of bitstreams: 1 arquivototal.pdf: 2094299 bytes, checksum: 783d6f9949086fb75e4a51fc3adbd48a (MD5) / Made available in DSpace on 2017-07-18T12:41:11Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 2094299 bytes, checksum: 783d6f9949086fb75e4a51fc3adbd48a (MD5) Previous issue date: 2016-04-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, new solutions based on the Boundary Element Method (BEM) are established for the linear analysis of bending and stability problems of Reddy-Bickford beams. All mathematical steps to write the BEM representation are properly presented: transformation of governing differential equations into equivalent integral equations, deduction of fundamental solutions, formation and solution of algebraic representation.In addition, elastic foundations (winkler and pasternak’s types) attached to Reddy-Bickford beams are solved by BEM as well. It is also addressed a convenient strategy for discontinuities in the area such as abrupt change in geometry of the cross section (stepped beam), intermediate axial load, intermediate supports (continuous beam). Numerical examples incorporating various types of discontinuities and boundary conditions in the field are presented to validate the solutions proposed BEM. / Neste trabalho, novas soluções, baseadas no Método dos Elementos de Contorno (MEC), são estabelecidas para a análise linear de problemas de flexão e estabilidade de barras de Bickford-Reddy. Todos os passos matemáticos para estabelecer a representação do MEC são apresentados: transformações das equações diferenciais governantes em equações integrais equivalentes, dedução das soluções fundamentais, obtenção e solução do sistema alébrico. Além disso, fundações elásticas (Winkler e Pasternak) em barras de Bickford-Reddy também são analisados pelo MEC. É também abordada uma conveniente estratégia para de discontinuidades no domínio tais como: mudança abrupta de geometria da seção transversal (viga escalonada), carga axial intermediária, apoios rígidos no domínio (viga contínua). Exemplos numéricos incorporando vários tipos de condições de contorno e discontinuidades no domínio são apresentadas para validar as soluções do MEC propostas.