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.

O escopo do trabalho é estudar o problema da torção uniforme em barras de seção maciça e resolvê-lo analiticamente para obter o momento de inércia à torção da seção transversal e os deslocamentos ao longo de toda a barra. Este trabalho foi desenvolvido no contexto da Teoria da Elasticidade, utilizando o método semi-inverso para determinar as equações de Saint-Venant para a torção uniforme. As seções em forma de elipse e triângulo eqüilátero foram resolvidas utilizando a função de tensão de Prandtl, a função empenamento e a sua conjugada harmônica. A seção retangular foi resolvida utilizando as funções empenamento e de Prandtl desenvolvidas em séries infinitas. Foi desenvolvida uma formulação matricial utilizando o Método de Galerkin para resolver problemas que não possuem solução fechada. / The main purpose of this essay is to present the issue of the uniform torsion in solid section bars and to solve it analytically to achieve the moment of inertia to the torsion of the transversal section and the displacements throughout the whole bar. This essay was developed in the Elasticity Theory context, using the semi-inverse method to determine the Saint-Venant equations to the uniform torsion. The sections in ellipse and equilateral triangle were solved using the Prandtl stress function, the warping function and its harmonic conjugate. The rectangular section was solved using the warping and the Prandtl functions developed in infinite series. A formulation based on matrixes was developed using the Galerkin method to solve problems that do not have closed solution.

O escopo do trabalho é estudar o problema da torção uniforme em barras de seção maciça e resolvê-lo analiticamente para obter o momento de inércia à torção da seção transversal e os deslocamentos ao longo de toda a barra. Este trabalho foi desenvolvido no contexto da Teoria da Elasticidade, utilizando o método semi-inverso para determinar as equações de Saint-Venant para a torção uniforme. As seções em forma de elipse e triângulo eqüilátero foram resolvidas utilizando a função de tensão de Prandtl, a função empenamento e a sua conjugada harmônica. A seção retangular foi resolvida utilizando as funções empenamento e de Prandtl desenvolvidas em séries infinitas. Foi desenvolvida uma formulação matricial utilizando o Método de Galerkin para resolver problemas que não possuem solução fechada. / The main purpose of this essay is to present the issue of the uniform torsion in solid section bars and to solve it analytically to achieve the moment of inertia to the torsion of the transversal section and the displacements throughout the whole bar. This essay was developed in the Elasticity Theory context, using the semi-inverse method to determine the Saint-Venant equations to the uniform torsion. The sections in ellipse and equilateral triangle were solved using the Prandtl stress function, the warping function and its harmonic conjugate. The rectangular section was solved using the warping and the Prandtl functions developed in infinite series. A formulation based on matrixes was developed using the Galerkin method to solve problems that do not have closed solution.

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