Thin walled beams warp under torsional and longitudinal loads. Warping restraint produces high longitudinal stresses. This is an analysis of the stress distribution in the side members of commercial vehicle chassis frames under the effects of the previously little studied longitudinal loads which may act on a truck chassis through spring hanger brackets. The structure analysed is a model chassis frame consisting of channel section side members and four cross members with different joint connections. The developed theories are incorporated into a special purpose finite element program which may be used in the preliminary stages of chassis frame design. Although the program is only used for the longitudinal load case in this thesis, it is generally applicable for other chassis load cases, including torsion, bending, etc. and combination of these. The theoretical results obtained from the program and the finite element analysis on complete chassis frame models are validated against experiments performed on a strain-gauged chassis frame model constructed with the same dimensions and constructional details as the finite element models with the appropriate loading and boundary conditions. Suggestions for the optimum design and attachment positions for components such as spring hanger bracket which may apply longitudinal loads to the side members of the chassis frame are discussed from the point of view of longitudinal loadings.

The considerable progress in the research and development of thin-walled beam structures responds to their growing use in engineering construction and to their increased need for efficiency in strength and cost. The result is a structure that exhibits large shear strains and important non uniform warping under different loadings, such as non uniform torsion, shear bending and distortion... A unified approach is formulated in this thesis for 3D thin walled beam structures with arbitrary profile geometries, loading cases and boundary conditions. A single warping function, defined by a linear combination of longitudinal displacements at cross sectional nodes (derived from Prokic work), is enhanced and adapted in order to qualitatively and quantitatively reflect and capture the nature of a widest possible range of behaviors. Constraints are prescribed at the kinematics level in order to enable the study of arbitrary cross sections for general loading. This approach, differing from most published theories, has the advantage of enabling the study of arbitrary cross sections (closed/opened or mixed) without any restrictions or distinctions related to the geometry of the profile. It generates automatic data and characteristic computations from a kinematical discretization prescribed by the profile geometry. The amount of shear bending, torsional and distortional warping and the magnitude of the shear correction factor is computed for arbitrary profile geometries with this single formulation. The proposed formulation is compared to existing theories with respect to the main assumptions and restrictions. The variation of the location of the torsional center, distortional centers and distortional rotational ratio of a profile is discussed in terms of their dependency on the loading cases and on the boundary conditions. A 3D beam finite element model is developed and validated with several numerical applications. The displacements, rotations, amount of warping, normal and shear stresses are compared with reference solutions for general loading cases involving stretching, bending, torsion and/or distortion. Some examples concern the case of beam assemblies with different shaped profiles where the connection type determines the nature of the warping transmission. Other analyses –for which the straightness assumption of Timoshenko theory is relaxed– investigate shear deformation effects on the deflection of short and thin beams by varying the aspect ratio of the beam. Further applications identify the cross sectional distortion and highlight the importance of the distortion on the stresses when compared to bending and torsion even in simple loading cases. Finally, a non linear finite element based on the updated lagrangian formulation is developed by including torsional warping degrees of freedom. An incremental iterative method using the arc length and the Newton-Raphson methods is used to solve the non linear problem. Examples are given to study the flexural, torsional, flexural torsional and lateral torsional buckling problems for which a coupling between the variables describing the flexural and the torsional degrees of freedom occurs. The finite element results are compared to analytical solutions based on different warping functions and commonly used in linear stability for elastic structures having insufficient lateral or torsional stiffnesses that cause an out of plane buckling.

<p>U radu su izvedene diferencijalne jednačine tankozidnog kompozitnog štapa proizvoljnog poprečnog preseka, primenom principa virtualnih pomeranja, a polazeći od funkcije deplanacije koju je predložio A. Prokić, za tankozidne štapove homogenog poprečnog preseka. Ona omogućava jedinstvenu analizu tankozidnih štapova otvorenog i zatvorenog poprečnog preseka, pretpostavka o zanemarenju klizanja u srednjoj površi štapa nije<br />neophodna, pa se smičući naponi određuju direktno iz odgovarajućih deformacija. Raspodela normalnih napona nije više određena sektorskom koordinatom već parametrima pomeranja čvornih tačaka, i u opštem slučaju je promenljiva od preseka do preseka, što omogućuje registrovanje i analizu shear lag uticaja, koji se klasičnom teorijom tankozidnih nosača ne može opisati. Kao što je poznato, shear lag uticaj predstavlja neravnomernu<br />raspodelu normalnih napona u pojasevima, s tim što se maksimalna vrednost javlja na mestu spoja pojasa sa rebrima, i u opštem slučaju je veća od vrednosti napona koja se dobija klasičnom teorijom savijanja štapova zasnovanoj na Bernoullijevoj hipotezi. To je posebno izraženo kod štapova napregnutih na savijanje kod kojih dolazi do značajne deplanacije poprečnog preseka.<br />Dobijeni sistem diferencijalnih jednačina se ne može rešiti u zatvorenom obliku te se pristupilo numeričkoj metodi, odnosno primeni metode konačnih elemenata. Definisana su dva tipa elementa sa različitim polaznim pretpostavkama. Prvi tip elementa zasnovan je na teoriji Timoshenka, odnosno uticaj transverzalnih sila na deformaciju se uzima u obzir. Drugi tip elementa zanemaruje uticaj transverzalnih sila na deformaciju, odnosno usvaja se pretpostavka da poprečni preseci i nakon deformacije ostaju ravni i upravni<br />na srednju liniju štapa. Kao dokaz tačnosti prethodno izvedenih teorijskih razmatranja urađen je niz primera pomoću programa napisanog u programskom jeziku C.</p> / <p>Differential equations of thin-walled composite beams of arbitrary cross section were<br />derived, using the principle of virtual displacements and starting from function of deplanation suggested by A. Prokic, for thin-walled beams of homogeneous cross section. It enables unique analysis thin-walled beams of open and closed cross section, assumption of neglecting shear strain in the middle surface is not necessary and shear stresses can be calculated directly from the strains. Distribution of longitudinal stresses is not defined by warping function, but parameters of longitudinal displacement, and in general case it is variable of section to section, that enables registering and analysis of shear lag, which classical theory of thin-walled beams is unable to reflect. As it is known, shear lag effect presents a non-uniform distribution of normal stresses in the flanges, maximal values are on the connection of flange and web, in general case it is larger from the value of stress obtained by classical theory of beams based on the Bernoulli hypothesis. It is especially<br />expressed at beams subjected to bending where deplanation of cross section is significant. Derived system of differential equations can not be solved in closed form solution and it was accessed to numerical method, respectively on the finite element method. Two types of element with different starting settings were defined. First type of element is based on the theory of Timoshenko, apropos the influence of transversal forces on deformation were taken into account. The second type of element neglects influence of transversal forces on deformation, concerning assumption that cross section remain plane and orthogonal on the middle line is adopted. A number of numerical examples are calculated by a computer program written in program language C as a proof of accuracy of previously derived theoretical reviews.</p>

La presencia inadvertida de daños en elementos estructurales representa un aspecto crítico en la seguridad de los mismos. Tales fallas pueden causar irregularidades de funcionamiento e incluso conducir al colapso catastrófico. Por ese motivo, la detección de daños en forma temprana es de fundamental importancia. Un tipo de falla que puede pasar peligrosamente inadvertida es aquélla producida por fatiga, ya que resulta muy difícil de detectar a simple vista. Si bien existen técnicas adecuadas para la detección localizada, la aplicación de las mismas puede resultar impráctica en base a la dificultad en la revisión de estructuras complejas, que inclusive pueden pre-sentar sectores inaccesibles. En consecuencia, se han investi-gado otro tipo de procedimientos de carácter más global, entre los que ha adquirido gran importancia la identificación basada en la respuesta dinámica. Esencialmente, este método se basa en la comparación entre valores teóricos y experimenta-les de parámetros dinámicos de estructuras dañadas. Las incógnitas del mencionado problema corresponden a las magni-tudes asociadas al daño (localización, intensidad). Por lo tan-to, a los valores más próximos a los reales les corresponderá una mínima desviación entre los valores teóricos y experimen-tales. Este tipo de técnica inversa hace uso de un modelo teó-rico de la estructura dañada. En esta tesis se presenta un nuevo modelo unidimensional que permite reproducir el comportamiento estructural de vigas de pared delgada con presencia de fisuras debidas a fatiga. En dicho modelo, se admite la posibilidad de considerar el efecto no lineal de cerra-miento parcial de fisura o breathing. El modelo asocia un enfoque de daño estructural por fisura basado en conceptos de mecánica de fracturas, que se desarrolla íntegramente en este trabajo, con una teoría de vigas de pared delgada recien-temente desarrollada para el caso de vigas intactas. Esta teo-ría debe ser extendida para considerar la presencia de fisuras. La premisa global de la teoría es que la presencia de una fisu-ra genera una reducción localizada en la rigidez de la viga. El modelo unidimensional de viga de pared delgada dañada es aplicable a vigas construidas con materiales isótropos, ortótro-pos y compuestos con laminación cross-ply simétrica o espe-cialmente ortótropa. Considera además flexibilidad por corte debido a flexión y alabeo. Los parámetros asociados a las fisuras son identificados mediante la minimización de una fun-ción objetivo, que se define en términos de diferencias norma-lizadas entre valores de indicadores de fisura obtenidos experi-mentalmente y calculados con el modelo viga. Como indica-dores de daño estructural se consideran los desplazamientos producidos por excitación forzada y también las frecuencias de vibración natural. En orden de resolver el problema inverso, se emplea el algoritmo de optimización Evolución Diferencial. / The unnoticed presence of damage in structural elements represents a critical issue in their security. Such flaws may generate malfunctions and even leading to catastrophic collapse. Thus, early detection of damage represents a topic of fundamental importance. A kind of flaw that can be dange-rously unnoticed is that one produced by fatigue, in terms of the difficulty to detect it with the naked eye. Although there are adequate techniques for local detection, their application may result impractical based on the difficulty represented by complex structures, which can have inaccessible parts. Consequently, other procedures of more global character have been topic of research. Among them, identification ba-sed on dynamic response has acquired major importance. Essentially, this method is based on comparisons among theo-retical and experimental values of dynamic parameters related to damaged structures. The unknowns of the mentioned problem correspond to the magnitudes associated to damage (location, severity). Thus, a minimal deviation among theore-tical and experimental results will correspond to the values closer to real ones. This kind of inverse technique makes use of a theoretical model of the damaged structure. In this the-sis, a new one-dimensional model is presented, which allows replicating the structural behavior of thin-walled beams with the presence of cracks generated by fatigue. In this model, the possibility of considering the nonlinear effect of crack partial closure or breathing is allowed. The model associates a structural damage approach based on fracture mechanics concepts, developed entirely in this work, with a recently developed thin-walled beam theory for the case of intact beams. This theory must be extended in order to consider the presence of cracks. The global premise of the theory is that the presence of a crack generates a localized reduction on the beam stiffness. The one-dimensional thin-walled da-maged beam model is applicable to beams made of isotropic material beams, orthotropic materials and composite materials with symmetric cross-ply or specially orthotropic stacking sequences. In addition, shear flexibility debt to flexure and warping is considered. Damage parameters associated to cracks are identified by means of the minimization of a target function, defined in terms of normalized differences among values of damage indicators obtained experimentally and calculated with the beam model. As indicators of structural damage, displacements debt to forced excitation and also frequencies of natural vibration are considered. In order to solve the inverse problem, the optimization algorithm Differential Evolution is employed.

[pt] Estruturas com elementos de seção aberta e paredes delgadas são amplamente utilizados em estruturas metálicas. Estes elementos têm, em geral, baixa rigidez a torção. Para seções monosimétricas e assimétricas, quando o centro de cisalhamento não coincide com o centro de gravidade, pode ocorrer acoplamento entre flexão e torção. Devido à baixa rigidez à torção, podem ocorrer grandes rotações das seções transversais da viga. Assim, uma análise do comportamento de tais elementos estruturais, levando em consideração a não linearidade geométrica, é desejável. Com este objetivo, equações diferenciais parciais de movimento que descrevem o acoplamento flexão-flexão-torção são utilizadas, em conjunto com o método de Galerkin, para se obter um conjunto de equações discretizadas de movimentos, que é resolvido pelo método Runge-Kutta. A partir das equações linearizadas, obtêm-se as frequências naturais, cargas críticas axiais e a relação entre carga axial e frequência para vigas com diferentes condições de contorno. A seguir, estudam-se as oscilações não lineares e bifurcações de uma viga engastada-livre submetida a cargas laterais harmônicas. Uma análise paramétrica detalhada, usando várias ferramentas de dinâmica não linear, investiga o comportamento dinâmico não linear e não planar da viga nas três primeiras regiões de ressonância e a influência da não linearidade, posição do carregamento, restrições à torção e parâmetros de controle do carregamento na estabilidade dinâmica da estrutura. / [en] Structural elements with open and thin-walled section are widely used in metal structures. These elements have, in general, low torsional stiffness. For monosymmetric and asymmetric sections, when the shear center does not coincide with the center of gravity coupling between bending and torsion may occur. Due to the low torsional stiffness, large twist beam cross sections may arise. Thus, an analysis of the behavior of such structural elements, taking into account the geometric nonlinearity, is desirable. For this purpose, partial differential equations describing the flexural-flexural-torsional coupling are used in conjunction with the Galerkin method to obtain a set of discretized equations of motion, which is solved by the Runge-Kutta method. From the linearized equations, we obtain the natural frequencies, axial critical loads, and the axial load and frequency relationship for beams with different boundary conditions. Next, we study the nonlinear oscillations and bifurcations of a clamped-free beam subjected to harmonic lateral loads. A detailed parametric analysis, using various nonlinear dynamics tools, investigates the nonlinear dynamic behavior and nonplanar motions of the beam for the first three regions of resonance and the influence of the non-linearity, loading position, torsional restraints and load control parameters on the dynamic stability of the structure.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / A utilizaÃÃo de vigas laminadas de parede fina nas Engenharias AeronÃutica, Civil, MecÃnica e Naval tem crescido bastante nos Ãltimos anos, devido a suas elevadas relaÃÃes rigidez/peso e resistÃncia/peso. Esses elementos estruturais normalmente apresentam paredes finas devido à alta resistÃncia dos materiais compÃsitos. Outra caracterÃstica importante à que, mesmo sem apresentar grandes deformaÃÃes e sem que o limite elÃstico do material seja ultrapassado, estas vigas apresentam comportamento nÃo linear geomÃtrico devido à sua elevada esbeltez, acarretando em grandes deslocamentos e rotaÃÃes. Dependendo da laminaÃÃo utilizada, as vigas de materiais compÃsitos podem apresentar diversos acoplamentos entre esforÃos e deformaÃÃes, tornando a sua anÃlise bem mais complexa do que a anÃlise de vigas de material isotrÃpico. Neste trabalho, foram desenvolvidos dois elementos finitos de pÃrtico espacial para anÃlise nÃo linear geomÃtrica de vigas laminadas de parede fina. As propriedades seccionais da viga sÃo avaliadas atravÃs de teorias de vigas laminadas de parede fina apropriadas, em que desprezam-se os efeitos do empenamento e do cisalhamento transversal. Tais teorias de vigas laminadas conduzem a uma matriz constitutiva 4x4, onde diferentes nÃveis de acoplamento entre esforÃos e deformaÃÃes de viga sÃo considerados, por exemplo, pode-se obter uma matriz constitutiva diagonal ou cheia. A abordagem corrotacional independente do elemento à utilizada para o tratamento de grandes deslocamentos e rotaÃÃes de corpo rÃgido no espaÃo. No Ãmbito local, sÃo utilizados dois elementos, um baseado na teoria linear e outro na descriÃÃo Lagrangeana Total. O tratamento matemÃtico das grandes rotaÃÃes no espaÃo à realizado por meio do tensor das rotaÃÃes (fÃrmula de Rodrigues), juntamente com o conceito do pseudovetor. As implementaÃÃes dos elementos finitos propostos neste trabalho sÃo realizadas no software de cÃdigo aberto FAST. A metodologia de trabalho segue o roteiro clÃssico de mÃtodos computacionais, incluindo formulaÃÃo, implementaÃÃo, verificaÃÃo e validaÃÃo dos resultados. A verificaÃÃo à realizada atravÃs dos modelos tridimensionais de elementos finitos de casca e sÃlido desenvolvidos no software comercial ABAQUS. A validaÃÃo à realizada por meio da comparaÃÃo com resultados de ensaios experimentais encontrados na literatura. No que diz respeito à avaliaÃÃo das propriedades seccionais, pode-se verificar uma Ãtima concordÃncia entre as teorias de vigas laminadas adotadas neste trabalho e os resultados numÃricos e de ensaios experimentais, para todas as laminaÃÃes e carregamentos considerados. No caso dos elementos de pÃrtico espacial, verificou-se uma Ãtima concordÃncia entre os resultados dos elementos finitos propostos neste trabalho e os resultados analÃticos e computacionais disponÃveis na literatura. Observa-se tambÃm que o elemento baseado na descriÃÃo Lagrangeana à mais eficiente do que o elemento baseado na teoria linear no que tange à capacidade de apresentar uma resposta satisfatÃria com uma malha menos refinada. / The use of thin walled laminate beams in Aeronautical, Civil, Mechanical and Naval Enginee- ring is increasing in the last years. This is due to their high stiffness/weight and strength/weight ratios. Composite beams and other structural elements tend to have thin walls due to the elevated strength of the material. Other important aspect is that, even without reaching large strains and without overcoming the elastic limit of the material, such beams present geometric nonlinear behavior due to high their slenderness, leading to large displacements and rotations. Depen- ding on the composite layup, the beams of composite materials can present several couplings between generalized stresses and strains, requiring a more complex analysis procedure when compared to isotropic beams. In this work, two three-dimensional space frame finite elements that can be used to analyze composite thin-walled beams subjected to geometric non-linearity were developed. The cross-section properties of the beams are evaluated through suitable thin walled beam theories, where the effects of the warping and transverse shear are neglected. Such theories yield a 4x4 constitutive matrix for the laminate and different levels of coupling between generalized stresses and strains can be considered. Depending of such couplings, the constitu- tive matrix can either be full or diagonal. The element independent corotational approach was used in order to consider large displacaments and rigid body rotations in space. In the local coordinate system, two elements are used, one based on the linear strain theory and the other on the Total Lagrangian formulation. The mathematical treatment of the large rotations in the space is performed by means of the rotation tensor (Rodriguesâs formula) in conjunction with the concept of the pseudovector. The computational implementations of the two finite elements proposed in this work were done in the open source software FAST ( Finite Element Analysis Tool ). The methodology used follows the classical steps used in computational methods, in- cluding formulation, implementation, verification and validation of results. Such verification is accomplished through shell and solid three-dimensional finite element models developed in the ABAQUS commercial software. The validation is performed by means of comparison with the experimental results found in literature. Regarding the evaluation of cross-sectional properties, one can observe a good agreement between the laminated beam theories adopted in this work and numerical and experimental results for all composite layups and load conditions conside- red. In the case of space frame elements, a good agreement is obtained between the results of finite elements proposed in this work and the analytical and computational results available in the literature. It is also observed that the element based on the Lagrangian formulation is more efficient than the element based on the linear theory regarding the ability to provide a satisfatory response with a less refined mesh

[pt] Há um interesse crescente na aplicação de vigas e colunas de paredes finas de materiais compostos em vários campos da engenharia. No entanto, pouco se sabe sobre seu comportamento não linear local e global sob cargas estáticas e dinâmicas. Aqui se apresenta a análise local e global de um perfil com seção canal de polímero reforçado com fibras. Na análise global, as equações não lineares de movimento da coluna de parede fina são derivadas em termos dos dois deslocamentos de flexão e do ângulo de torção, levando em consideração grandes deslocamentos, efeitos de empenamento e encurtamento. As equações de movimento não lineares governantes são discretizadas no espaço usando o método de Galerkin. Para a análise local, a seção do canal é discretizada em três placas, que são modeladas usando duas teorias não lineares de placas: a teoria clássica e a teoria de deformação por cisalhamento de primeira ordem. O sistema contínuo é discretizado usando o método de Ritz. Inicialmente são determinados analiticamente, através da resolução dos respectivos problemas de autovalor, a carga e modo crítico, as frequências naturais de vibração, bem como a relação carga-frequência do perfil em função da sua geometria e das propriedades do material. A seguir são obtidos, usando o método de Newton-Raphson e técnicas de continuação, os caminhos pós-críticos da estrutura perfeita e os caminhos não lineares de equilíbrio da estrutura imperfeita e investiga-se a sensibilidade a imperfeições, considerando diversos tipos de imperfeições geométricas. Finalmente, investigam-se as oscilações não lineares e a instabilidade paramétrica da coluna sob cargas axiais harmônicas. As equações de movimento não lineares são resolvidas numericamente pelo método de Runge-Kutta de quarta ordem. As regiões de instabilidade paramétrica são determinadas como uma função dos parâmetros do material ortotrópico, amortecimento e geometria da seção transversal. Os diagramas de bifurcação são obtidos empregando técnicas de continuação e o método da força bruta, e a estabilidade das soluções é posteriormente investigada usando a teoria de Floquet. A análise de bifurcação permite a identificação das bifurcações associadas às fronteiras de instabilidade paramétrica, bem como a existência de soluções coexistentes. Em seguida, a evolução das bacias de atração das soluções coexistentes em função da magnitude da excitação é investigada, a fim de avaliar a integridade dinâmica das soluções. Os resultados demonstram que a coluna pode perder estabilidade em níveis de carga bem abaixo da carga de flambagem estática e, portanto, o projetista deve ter cuidado ao lidar com essas estruturas sujeitas a cargas axiais variáveis no tempo. / [en] The continuous system is discretized using the Ritz method. Initially, the load and critical mode of the profile, its natural frequencies, as well as the load-frequency relation are determined analytically as a function of the column geometry and material properties by solving the respective eigenvalue problems. Next, using the Newton-Raphson method and continuation techniques, the post-critical paths of the perfect structure and the non-linear equilibrium paths of the imperfect structure are obtained and the imperfection sensitivity is investigated, considering several types of geometric imperfections. Finally, the nonlinear oscillations and parametric instability of the column under harmonic axial loads are investigated. Non-linear equations of motion are solved numerically by the fourth-order Runge-Kutta method. The regions of parametric instability are determined as a function of the parameters of the orthotropic material, damping ratio and cross-sectional geometry. The bifurcation diagrams are obtained using continuation techniques and the brute force method, and the stability of the solutions is further investigated using Floquet s theory. The bifurcation analysis allows the identification of the bifurcations associated with the boundaries of parametric instability, as well as the existence of coexisting solutions. Then, the evolution of the basins of attraction of the coexisting solutions as a function of the forcing magnitude is investigated, in order to evaluate the dynamic integrity of the solutions. The results demonstrate that the column can lose stability at load levels well below the static buckling load and, therefore, the designer must be careful when dealing with these structures subject to time-varying axial loads.

The considerable progress in the research and development of thin-walled beam structures responds to their growing use in engineering construction and to their increased need for efficiency in strength and cost. The result is a structure that exhibits large shear strains and important non uniform warping under different loadings, such as non uniform torsion, shear bending and distortion.<p><p>A unified approach is formulated in this thesis for 3D thin walled beam structures with arbitrary profile geometries, loading cases and boundary conditions. A single warping function, defined by a linear combination of longitudinal displacements at cross sectional nodes (derived from Prokic work), is enhanced and adapted in order to qualitatively and quantitatively reflect and capture the nature of a widest possible range of behaviors. Constraints are prescribed at the kinematics level in order to enable the study of arbitrary cross sections for general loading. This approach, differing from most published theories, has the advantage of enabling the study of arbitrary cross sections (closed/opened or mixed) without any restrictions or distinctions related to the geometry of the profile. It generates automatic data and characteristic computations from a kinematical discretization prescribed by the profile geometry. The amount of shear bending, torsional and distortional warping and the magnitude of the shear correction factor is computed for arbitrary profile geometries with this single formulation.<p><p>The proposed formulation is compared to existing theories with respect to the main assumptions and restrictions. The variation of the location of the torsional center, distortional centers and distortional rotational ratio of a profile is discussed in terms of their dependency on the loading cases and on the boundary conditions.<p><p>A 3D beam finite element model is developed and validated with several numerical applications. The displacements, rotations, amount of warping, normal and shear stresses are compared with reference solutions for general loading cases involving stretching, bending, torsion and/or distortion. Some examples concern the case of beam assemblies with different shaped profiles where the connection type determines the nature of the warping transmission. Other analyses –for which the straightness assumption of Timoshenko theory is relaxed– investigate shear deformation effects on the deflection of short and thin beams by varying the aspect ratio of the beam. Further applications identify the cross sectional distortion and highlight the importance of the distortion on the stresses when compared to bending and torsion even in simple loading cases. <p><p>Finally, a non linear finite element based on the updated lagrangian formulation is developed by including torsional warping degrees of freedom. An incremental iterative method using the arc length and the Newton-Raphson methods is used to solve the non linear problem. Examples are given to study the flexural, torsional, flexural torsional and lateral torsional buckling problems for which a coupling between the variables describing the flexural and the torsional degrees of freedom occurs. The finite element results are compared to analytical solutions based on different warping functions and commonly used in linear stability for elastic structures having insufficient lateral or torsional stiffnesses that cause an out of plane buckling. <p> / Doctorat en sciences appliquées / info:eu-repo/semantics/nonPublished