Dissertations / Theses on the topic 'Thin-walled open-section'

PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
PROGRAMA DE EXCELENCIA ACADEMICA
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.
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.

In this thesis, hybrid stress finite element is formulated for the analysis of the isotropic, thin walled, open section beams with variable cross sections. The beam element has two nodes each having seven degrees of freedom. Assumption of stress field is sufficient to determine the element stiffness matrix. Axial, flexural and torsional effects are taken into account in the analysis. The methodology can be applied both to the tapered and the uniform beams. Throughout this study, firstly element cross-sectional properties are computed using the flow analogy of the inter-connected elements which may have different thicknesses. Then another computer program calculates the displacements and stresses at the nodes along the beam. The results obtained are compared to the results taken from literature and commercial FEM program Nastran.

The current interest in collapse characteristics brought about by crashworthiness requirements ýas shown the need for a better understanding and predictive capability for the thin walled open section structures. In general three possible modes exist in which a loaded thin walled open section column can buckle: 1) they can bend in the plane of one of the principal axes; 2) they can twist about the shear. centre; 3) or they can bend and twist simultaneously. The following study was undertaken to investigate the general failure of thin walled open section structures. A literature survey was conducted and it prevailed that a basic fundamental theoretical study was vital in describing the behaviour of thin walled structural members. The following stages of theoretical study have been completed: 1) Formulation of the stiffness matrix to predict the generalised force-displacement relationships assuming the small displacement theory in the linear elastic range. 2) Formulation of the geometric stiffness matrix to predict the buckling criteria under generalised loading and end constraints in the linear elastic range. 3) Formulation of the compound coordinate transformation matrix to relate local and global displacements or forces. 4) Preparation of the associated finite element computer program to solve general thin walled open sections structural problems.

Les poutres à parois minces à sections ouvertes sont des éléments de base des ouvrages courants en génie civil, de l'automobile et de l'aéronautique. En raison de leur élancement et la forme des sections, elles sont très sensibles à la torsion et aux instabilités aussi bien en statique qu’en dynamique. En dynamique, les modes de vibration en torsion sont plus dominants par rapport au modes de flexion classiques. Pour ces raisons, les défaillances planaires de telles structures sont connues pour être une exception plutôt qu'une règle. Dans ce travail de thèse, on s’intéresse au comportement dynamique de poutres à parois minces et à section ouvertes arbitraires. En se basant sur le modèle de Vlasov qui prend en compte de la torsion et du gauchissement, les équations de mouvement 3D sont dérivées à partir du principe d’Hamilton. Des solutions analytiques originales pour différentes conditions aux limites sont dérivées pour des modes supérieurs en vibrations libres. Dans ces solutions, les effets des termes de rotation inertiels en flexion et torsion sont pris en compte. Pour des cas généraux, un modèle élément fini de poutre 3D est décrit et implémenté. Dans le modèle, un degré de liberté (ddl) est affecté au gauchissement. Toutes les matrices de rigidité masse de base sont calculées par intégration numérique (intégration de Gauss). Dans le modèle, les calculs en vibrations libres et forcées sont possibles. Le modèle est validé par comparaison aux solutions numériques et expérimentaux de la littérature. Une comparaison aux simulations des codes commerciaux est aussi suivie. Afin de valider le modèle théorique et numérique utilisé, une campagne d’essais a été suivie au LEM3 à Metz. Des essais de vibration libre et forcée sont effectués sur des poutres à parois minces avec différentes conditions aux limites. Les solutions analytiques, numériques et les mesures expérimentales sont comparées et validées. Un bon accord entre les différentes solutions est constaté. Le modèle est étendu aux poutres 3D retenues latéralement par des entretoises. Des ressorts élastiques et visqueux 3D sont ajoutés dans le modèle numérique. L'effet des entretoises est étudié dans le but d’améliorer le comportement des poutres à parois minces vis-à-vis des modes indésirables de type flexion latérale et torsion
Thin-walled beams with open section constitute main elements in engineering applications fields as in civil engineering, automotive and aerospace construction. Due to slenderness and cross section shapes, these elements are very sensitive to torsion and instabilities in both statics and dynamics. In dynamics, the torsional and flexural-torsional modes of vibration are often lower frequencies compared to the classical plane pure bending modes. Thus, planar failures of such structures are known to be an exception rather than a rule. In torsion, warping is important and governs the behavior. In this thesis work, we are interested with the dynamic behavior of thin-walled beams with arbitrary open cross sections. Based on the Vlasov’s model accounting for warping, the 3D motion equations are derived from the Hamilton’s principle. Original analytical solutions for different boundary conditions are derived for higher free vibration modes. In these solutions, the effects of the inertial rotation terms in bending and torsion are taken into consideration. For more general cases, a 3D beam finite element model is described and implemented. Compared to conventional 3D beams, warping is considered as an additional Degree Of Freedom (DOF). The mass and stiffness matrices are obtained by numerical integration (Gauss method). In the model, free and forced vibration analyses are possible. The model is validated by comparison with benchmark solutions available in the literature and other numerical results obtained from simulation on commercial codes. In order to validate the present model, laboratory test campaign is undertaken at the LEM3 laboratory in Metz. Tests are carried out on thin-walled beams with different boundary conditions. Free and forced vibration tests are performed using impact hammer and shaker machine. In the presence of arbitrary sections, flexural-torsional vibration modes are observed. The analytical, the numerical and the experimental solutions are compared and validated. Moreover, the numerical and experimental dynamic response spectra are compared. A good agreement between the various solutions is remarked. The model is extended to 3D beams in presence of lateral braces. 3D elastic and viscous springs are added in the finite element model. The effect of the springs is studied in order to improve the behavior of thin-walled beams against undesirable lateral bending and torsion modes

Cette recherche est une contribution à l'étude de la torsion des poutres en voiles minces ouverts et des poutres à sections transversales non symétriques. En effet, dans le cas des poutres en voiles minces ouverts, on a développé la statique de la torsion suivant la théorie de Vlassov, c'est-à-dire en tenant compte du gauchissement qui accompagne en général la torsion, et ses effets sur les éléments de structure. Notre travail commence par la résolution de l'équation différentielle de la torsion gênée pour diverses conditions aux limites et différents chargements. Des exemples numériques ont concrétisé l'importance de la prise en compte du gauchissement dans le cas des éléments minces. L'étude a été complétée par la formulation de la poutre continue soumise à la torsion et par une étude numérique comparative avec la méthode des éléments finis en utilisant une modélisation en éléments coques et montrant la concordance entre les deux méthodes. D'autre part, dans le cas des poutres a profil dissymétrique, le centre de torsion n'est pas en général confondu avec le centre de gravité. Notre travail était de prendre en compte cette excentricité et d'introduire son effet dans la matrice de rigidité d'un élément de poutre droite, ainsi que ses effets sur les sollicitations et les déplacements des éléments de structure. L'étude a été complétée par plusieurs exemples schématisant l'intérêt de l'utilisation de la matrice de rigidité modifiée.

碩士
國立交通大學
機械工程系所
102
A consistent co-rotational total Lagrangian finite element formulation for the geometric nonlinear buckling and postbuckling analysis of bisymmetric thin-walled Timoshenko beams is presented. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the centroid of the end cross-sections of the beam element and the axis of centroid is chosen to be the reference axis. The deformations of the beam element are described in the current element coordinate system constructed at the current configuration of the beam element. The exact kinematics of the Timoshenko beam is considered. The element nodal forces are derived using the virtual work principle with the consideration of the shear correction factor. The virtual rigid body motion corresponding to the virtual nodal displacements is excluded in the derivation of the element nodal forces. A procedure is proposed to determine the virtual rigid body motion. A consistent second-order linearization of the element nodal forces is used here. Thus, all coupling among bending, shearing, twisting, and stretching deformations of the beam element is retained. In the derivation of the element tangent stiffness matrix, the change of element nodal forces induced by the element rigid body rotations should be considered for the present method. Thus, a stability matrix is included in the element tangent stiffness matrix. An incremental-iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A bisection method of the arc length is used to find the buckling load. Numerical examples are studied and compared with the results obtained by using Euler beam element to demonstrate the accuracy and efficiency of the proposed method and to investigate the effect of the shear deformation on the loading–deflection curves and buckling load of the bisymmetric thin-walled beams.

碩士
國立交通大學
機械工程系所
103
A co-rotational finite element formulation for the nonlinear dynamic analysis of bisymmetric thin-walled Timoshenko beams is presented. The element deformation nodal force and tangent stiffness matrix are derived by consistent co-rotational formulation. The element inertia nodal force and inertia matrix are derived by co-rotational total Lagrangian formulation. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the centroid of the end cross-sections of the beam element and the axis of centroid is chosen to be the reference axis. A rotation vector is used to represent the finite rotation of coordinate systems rigidly tied to each node of the discretized structure. The incremental nodal displacement vectors and rotation vectors in global coordinates are used to update the node locations and orientation of the element. The deformations of the beam element are described in the current element coordinate system constructed at the current configuration of the beam element. Three rotation parameters are defined to describe the relative orientation between the element cross section coordinate system rigidly tied to the unwrapped cross section and the current element coordinate system. The exact kinematics of the Timoshenko beam is considered. The element deformation nodal forces and inertia nodal forces are derived using the nonlinear beam theory, d’Alembert principle, virtual work principle, and consistent second degree linearization at the current coordinate of the beam element. The terms up to the second order of spatial derivatives of deformation parameters are retained in the element deformation nodal forces, and the terms up to the second order of time derivatives of deformation parameters are retained in the element inertia nodal forces. However, the coupling between deformation parameters and their time derivatives are not considered in the element inertia nodal forces. The element tangent stiffness matrix is derived using the relations between the variation of the element nodal displacement vectors and rotation vectors and the corresponding variation of element nodal forces. The element inertia matrices may be obtained by differentiating the element inertia nodal forces with respect to the first and second time derivatives of the element nodal parameters. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method are employed here for the solution of the nonlinear equations of motion. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

碩士
國立交通大學
機械工程系所
104
A corotational total Lagrangian (CRTL) finite element formulation for the geometrically nonlinear dynamic analysis of asymmetric thin-walled beam with large rotations but small strain is presented. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear center of the end cross sections of the beam element and the shear center axis is chosen to be the reference axis. For the purpose of treating arbitrarily large rotation of node in space, the orientation of the node is described by a base coordinate system rigidly tied to each node of the discretized structure, and a nodal rotation vector is used to represent the finite rotation of the base coordinate system. The values of nodal rotation vectors are reset to zero at current configuration, thus, the values of the first and second time derivative of the nodal rotation vector are equal the values of the spatial nodal angular velocity and acceleration. The kinematics of the beam element is described in the current element coordinate system constructed at the current configuration of the element. The current element coordinate system is regarded as an inertial local coordinate system, not a moving coordinate system. Thus, the first and the second time derivative of the position vector defined in the element coordinates are the absolute velocity and absolute acceleration. Three rotation parameters referred to the current element coordinates are defined to determine the orientation of element cross section. The deformation of the beam element is determined by the displacements of the shear center axis and the rotations of element cross section. The element deformation nodal forces and inertia nodal forces are systematically derived by the d'Alembert principle, the virtual work principle and consistent second order linearization in the current element coordinates. The element stiffness matrix may be obtained by differentiating the element deformation nodal forces with respect to the element nodal parameters, and the element inertia matrices may be obtained by differentiating the element inertia nodal forces with respect to the element nodal parameters, and their first and second time derivatives. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the nonlinear equations of motion. The standard Newmark method is applied to the incremental displacement and rotational vectors, and their time derivatives. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

碩士
國立交通大學
機械工程系所
105
The geometrical nonlinear static behavior and infinitesimal free vibration around the static equilibrium position are studied using total Lagrangian finite element method for three dimensional thin-walled beams with point-symmetric open section subjected to axial load with its resultant passing through the centroid of beam cross section. The bimoment induced by axial load is considered in this study. The element employed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear center of the end cross sections of the beam element and the shear center axis is chosen to be the reference axis. The kinematics of the beam element is described in the current element coordinate system constructed at the current configuration of the element. The current element coordinate system is regarded as an inertial local coordinate system. Thus, the first and the second time derivative of the position vector defined in the element coordinates are the absolute velocity and absolute acceleration. The element deformation nodal forces and inertia nodal forces are systematically derived by the d'Alembert principle, the virtual work principle and consistent second order linearization in the current element coordinates. The equilibrium equations may be obtained by dropping the terms of the inertia forces in the equation of motion. The governing equations for linear vibration around the static equilibrium position are obtained by the first order Taylor series expansion of the equation of motion at the static equilibrium position. An incremental-iterative method based on the Newton-Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of the nonlinear equilibrium equations. The subspace iterative method is used for the solution of natural frequencies and vibration modes for the free vibration. Numerical examples are studied to investigate the effects of the axial load and bimoment induced by axial load on the critical state, critical load and the natural frequencies of z cross section beams with different lengths and boundary conditions under axial loading. The objective of the paper is to analyze the influence of bimoment induced by constant axial loads on the free motion of thin-walled beams with point-symmetric open cross- section. For various boundary conditions, a closed-form solution for natural frequencies of free harmonic vibrations was derived by using a general solution of governing differential equations of motion based on Vlasov’s theory. In order to investigate the effect of the bimoment on natural frequencies, the numerical examples with symmetric Z cross-section are given. The obtained results, verified using an ANSYS finite element model, demonstrate that the influence of the bimoment is important in the assessment of torsional natural frequencies. A force with its resultant passing through the centroid of a particular section and being perpendicular to the plane of the section. A force in a direction parallel to the long axis of the structure

碩士
國立交通大學
機械工程系
90
A consistent procedure is proposed to derive the element internal nodal force and tangent stiffness matrix for asymmetric thin-walled beam element with open section using the virtual work principle combined with co-rotational total Lagrangian formulation. The effects of some higher order terms of element node forces and tangent stiffness matrix on the buckling load and postbuckling behavior of beam structures. In this thesis, a two-node element with seven degrees of freedom per node is developed. The element nodes are chosen to be the shear center of end section of the element. The shear center axis is employed as the reference axis of the beam element. The element deformation are referred to the undeformed geometry of the beam element and described in element coordinates which are constructed at the current configuration of the beam element. The element internal nodal forces are systematically derived by using virtual work principal and a consistent second-order linearization of the fully geometrically nonlinear beam theory. In this study, the terms up to the second order of rotation parameters and their spatial derivatives and the third-order term of twist rate are retained in the element nodal forces. When the virtual displacement method is used to derive the element nodal force, the external virtual work done by the element internal nodal force and the virtual nodal displacement are defined in the current element coordinates, which are regarded as fixed coordinates. However, the internal virtual work done by the element stress and the virtual strain corresponding to the virtual nodal displacement are defined in an element coordinates which are constructed at the disturbed configuration of the beam element corresponding to virtual nodal displacement. Note that the rigid body motion part in the virtual displacement is eliminated in the derivation of the internal virtual work. The tangent stiffness matrix is derived from the increment of the element nodal force corresponding to an infinitesimal incremental displacement. The increment of the element nodal force comprises the direction change of the element internal nodal force corresponding to the rigid body motion part in the infinitesimal incremental displacement and the increment of the element nodal force corresponding to the deformation part in the infinitesimal incremental displacement. An incremental-iterative method based on the Newton-Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. To verify the accuracy of present formulation, numerical examples are studied and compared with published experimental results and numerical results for geometrically nonlinear behavior and nonlinear buckling load. With the increase of element number, the length of beam element, the twist angle and slopes of the beam axis will approach to zero. Thus, The corresponding terms in element internal nodal forces and element stiffness matrices will approach to zero too. To investigate the effects of these terms on the convergent solution of buckling load and deflections for beam structures, convergence tests are carried out for different numerical examples.

碩士
國立交通大學
機械工程系
89
A consistent procedure is proposed to derive the element internal nodal force and tangent stiffness matrix for doubly symmetric thin-walled beam element with open section using the virtual work principle combined with co-rotational total Lagrangian formulation. When the virtual displacement method is used to derive the element nodal force, the external virtual work done by the element internal nodal force and the virtual nodal displacement are defined in the current element coordinates, which are regarded as fixed coordinates. However, the internal virtual work done by the element stress and the virtual strain corresponding to the virtual nodal displacement are defined in an element coordinates which are constructed at the disturbed configuration of the beam element corresponding to virtual nodal displacement. Note that the rigid body motion part in the virtual displacement is eliminated in the derivation of the internal virtual work. The tangent stiffness matrix is derived from the increment of the element nodal force corresponding to an infinitesimal incremental displacement. The increment of the element nodal force comprises the direction change of the element internal nodal force corresponding to the rigid body motion part in the infinitesimal incremental displacement and the increment of the element nodal force corresponding to the deformation part in the infinitesimal incremental displacement. An incremental-iterative method based on the Newton-Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A bisection method of the arc length is used to find the buckling load. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed element and to investigate the buckling load of doubly symmetric thin-walled beams.

博士
國立交通大學
機械工程系
87
Studies on the geometrically nonlinear behavior and nonlinear buckling analysis of thin-walled open-section beams have been relatively rare. A two-node displacement-based thin-walled open-section beam element with seven degrees of freedom per node is developed by using corotational total Lagrangian formulation for the geometrically nonlinear, nonlinear buckling and postbuckling analysis of thin-walled open-section beams. In this thesis, element nodes are chosen to be the shear centers of end sections of the element. The shear center axis is employed as the reference axis of the beam element. The element deformations are referred to the initial undeformed geometry of the beam element and described in element coordinates which are constructed at the current configuration of the beam element. The internal nodal forces are systematically derived by using virtual work principal and a consistent second-order linearization of the fully geometrically nonlinear beam theory based on the exact kinematics of Euler beam to consider the coupling among bending, twisting and stretching deformations for the beam element. The third-order term of twist rate is the dominant term for the third-order terms and may be a very important term to reflect the nonlinear behavior of the beam subjected to a pure torque. Hence, the third-order term of twist rate is also considered in the element nodal forces. An incremental-iterative method based on the Newton-Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. For the buckling and postbuckling analysis of the structural system, only the nongyroscopic conservative system is considered in this thesis. A parabolic interpolation method of the arc length is proposed to find the nonlinear buckling load. An inverse power method for the solution of the generalized eigenvalue problem is used to find the corresponding buckling mode. In order to gain access to the secondary path from the primary path, at the bifurcation point a perturbation displacement proportional to the first buckling mode is added. To verify the accuracy of the present finite element formulation, numerical examples are studied and compared with published experimental results and numerical results obtained by nonlinear shell elements available in the literature. Comparisons between the present numerical results and those obtained by other beam elements or other methods available in the literature are also given in this thesis. Case studies are performed to investigate the effects of section geometry, slenderness ratio, warping boundary conditions, and location of loading point on the elastic buckling load and postbuckling behavior of the thin-walled beam structures.

碩士
國立交通大學
機械工程系所
103
A consistent co-rotational (CCR) finite element formulation for geometrically nonlinear dynamic analysis of doubly symmetric thin-walled beam with large rotations but small strain is presented. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the centroids of the end cross sections of the beam element and the centroid axis is chosen to be the reference axis. A rotation vector is used to represent the finite rotation of coordinate systems rigidly tied to each node of the discretized structure. The incremental nodal displacement vectors and rotation vectors in global coordinates are used to update the node locations and orientation of the element. The deformations of the beam element are described in a current moving element coordinate system constructed at the current node locations and orientation of the beam element. Three rotation parameters are defined to describe the relative orientation between the element cross section coordinate system rigidly tied to the unwrapped cross section and the current element coordinate system. The element equations are derived in a fixed current element coordinates which are coincident with the current moving element coordinates. The perturbed displacements and spatial rotation, velocity and acceleration, angular velocity and angular acceleration of the current moving element coordinates, and the variation of the element nodal rotation parameters corresponding to the perturbation of element nodal displacement vectors and rotation vectors in the current fixed element coordinates are consistently determined and expressed in terms of the current element nodal displacements and rotation parameters, nodal velocities and accelerations, and nodal angular velocities and angular accelerations. The element deformation and inertia nodal forces are derived using the virtual work principle, the d’Alembert principle, and the consistent second order linearization of the fully geometrically nonlinear beam theory. In element deformation nodal forces, all coupling among bending, twisting, and stretching deformations of the beam element is considered. In the element inertia nodal forces, the terms up to the second order of time derivatives of deformation parameters are retained. However, the coupling between rotation parameters and their time derivatives are not considered in the element inertia nodal forces. In this study, the element inertia nodal forces are expressed in terms of element nodal velocities and accelerations, and nodal angular velocities and accelerations. Thus, the element inertia matrices may be obtained by differentiating the element inertia nodal forces with respect to the element nodal velocities and accelerations. There is a slight difference between the present element inertia nodal forces and that derived using corotational total Lagrangian (CRTL) formulation. However, the effect of the difference may be negligible with the decrease of element size. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed for the solution of nonlinear equations of motion. Numerical examples are presented to investigate the accuracy and efficiency of the proposed method. It is found that the difference between the dynamic responses obtained using CCR formulation and CRTL formulation is negligible for all examples studied.

碩士
國立交通大學
機械工程系所
94
A six node degenerate element is proposed based on the concept of Ref. [1] by using consistent co-rotational finite element formulation for the geometric nonlinear analysis of doubly symmetric thin-walled I beam with slow varying flange. Only the axial displacement and axial rotation are considered for the element developed here. The kinematics of the element is governed by two sectional nodes and four true element nodes. The deformations of the element are described in the current element coordinate system, which is constructed at the current configuration of the element. In element nodal forces, all coupling between twisting and stretching deformations of the element is considered by consistent second-order linearization of the large displacement theory. An incremental-iterative method based on the Newton-Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. Numerical examples are studied to verify the accuracy of the present method and investigate the torsional buckling load and post-buckling behavior of thin-walled I beams with different variable sections subjected to different axial loads. The geometric nonlinear behavior of thin-walled I beams subjected to axial load and axial torque simultaneously are also studied.

博士
國立交通大學
機械工程系所
96
A consistent co-rotational finite element formulation for the free vibration analysis and geometric nonlinear dynamic analysis of thin-walled beams with generic open section is presented. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear centers of the end cross sections of the beam element and the shear center axis is chosen to be the reference axis. The deformations of the beam element are described in a current moving element coordinate system constructed at the current configuration of the beam element. Three rotation parameters are used to describe the orientation of the beam cross section in the moving element coordinate system. However, the rotation vector is used to describe the element nodal rotations in fixed coordinates. The values of the nodal rotation vectors are reset to zero at current configuration. The element equations are derived in a fixed current element coordinates which are coincident with the current moving element coordinates. The perturbed moving element coordinates and the variation of the element nodal rotation parameters corresponding to the perturbation of element nodal displacements and rotations referred to the current fixed element coordinates is consistently determined using the first order linearization of the way used to determine the current element coordinates and element nodal rotation parameters corresponding to the incremental element nodal displacements and rotations referred to the global coordinates. The angular velocity and acceleration of the current moving element coordinates and the first and the second time derivative of the element nodal rotation parameters are consistently determined using the current element nodal displacements and rotations, nodal velocities and accelerations, and nodal angular velocities and accelerations. The element deformation and inertia nodal forces are derived using the virtual work principle, the d’Alembert principle, and the consistent second order linearization of the fully geometrically nonlinear beam theory. In element nodal forces, all coupling among bending, twisting, and stretching deformations of the beam element is considered. For convenience, in the derivation of the element deformation nodal force, the generalized nodal moments corresponding to the variation of the nodal rotation parameters are derived first, and then transformed to the conventional moments and forces using controgradient law. Because the element nodal displacements and rotations with value of zero are retained in the relationship between the variation of the element nodal rotation parameters and the variation of element nodal displacements and rotations, the element tangent stiffness matrix may be obtained by differentiating the element deformation nodal force with respect the element nodal parameters. An incremental-iterative method based on the Newton-Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The subspace iterative method is used for the solution of natural frequencies and vibration modes for the free vibration of beam structures. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed for the solution of nonlinear equations of motion. Numerical examples are presented to investigate the accuracy and efficiency of the proposed method. The effect of different cross sections, boundary conditions and different loads on the natural frequencies, vibration modes, and nonlinear dynamic behavior of three dimensional thin-wall beam structures are also investigated through numerical examples.

Tese no âmbito do Doutoramento em Engenharia Civil na Especialidade de Estruturas apresentada à Faculdade de Ciências e Tecnologia da Universidade de Coimbra
The main body of the thesis is divided into two largely self-contained parts. The first one is devoted to the development of a continuous one-dimensional linear model for the stretching, bending and twisting of tapered thin-walled bars with open cross-sections under general quasi-static loading conditions. These bars are treated as two-dimensional Kirchhoff-Love shells, exhibiting both membrane and flexural behaviours. To achieve the necessary dimensional reduction, the classical assumptions of Vlasov and Kirchhoff-Love are regarded systematically as internal constraints, that is, a priori restrictions, of a constitutive nature, on the possible deformations of the bars (alternatively, they may also be viewed as holonomic-scleronomic constraints). Moreover, the internal forces are decomposed additively into active and reactive parts and this is shown to lead to a dual one-dimensional description of kinematics and statics. Two examples illustrate the application of the developed one-dimensional model, shed light on its physical aspects and demonstrate the shortcomings of piecewise prismatic models, regardless of the number of prismatic segments used (indeed, even in the limit when the length of these segments tends to zero). The main original contributions in this first part of the thesis may be summarized as follows: (i) The second fundamental form of the middle surface of a bar and the change of curvature tensor are established in general form. (ii) The displacement field of a whole bar (not just of its middle surface) is completely characterized, thus including the so-called through-the-thickness (or secondary) warping deformation. (iii) In the characterization of the internal forces in the bar, the shell bending and twisting moments and the transverse shear forces are taken into account, in addition to the membrane forces. (iv) The Saint-Venant contribution to the strain energy and the corresponding component of the total torque are derived consistently. (v) A set of fundamental inequalities concerning the cross-sectional properties is established. The second part of the thesis is restricted to the important special case of depth-tapered singly symmetric I-section bars and deals with one-dimensional models of the Hencky bar-chain type, whose nature is intrinsically discrete. Indeed, a Hencky bar-chain model consists of a finite number of rigid units linked by elastic springs (or, more generally, by rheological elements) – it can be thought of not only as an idealization of a (continuous) member, but also as an actual mechanical structure in its own right, the inherent simplicity and transparency of which make its qualitative behaviour more easily grasped. Two types of problem are addressed in successive chapters: (i) the linear mechanical behaviour in three-dimensional space under general quasi-static loading conditions and (ii) the linearized flexural-torsional buckling behaviour under bending (in the plane of symmetry, which is also the plane of greatest flexural rigidity) and compression, including the so-called Wagner effect associated with the asymmetry of the flanges. Particular attention is paid to the calibration of the spring stiffnesses and to the appropriate definition of boundary conditions. It is shown that the bar-chain models are consistent with (but not subordinate to or in any away dependent on) previously developed Vlasov-type continuum models, in the sense that the local truncation errors tend to zero as the length of the rigid units approaches zero. Several illustrative examples, including prismatic and flangeless members (i.e., members with narrow rectangular cross-sections), are solved in order to verify the discrete Hencky bar-chain models and to assess their convergence rates.
Contributos para a Modelação Unidimensional do Comportamento Tridimensional de Barras Não Prismáticas com Secção de Parede Fina Aberta A tese encontra-se dividida em duas partes em larga medida independentes. A primeira é dedicada ao desenvolvimento de um modelo linear unidimensional contínuo para a flexão e torção de barras com secção aberta de paredes finas, continuamente variável, submetidas a carregamentos quase-estáticos genéricos. Estas barras são tratadas como cascas de Kirchhoff-Love (bidimensionais), considerando tanto o comportamento de membrana como o de flexão. Para levar a cabo a necessário redução dimensional, as hipóteses clássicas de Vlasov e Kirchhoff-Love são tratadas sistematicamente como constrangimentos internos, isto é, restrições de natureza constitutiva às possíveis deformações de uma barra (alternativamente, aquelas hipóteses podem também ser vistas como constrangimentos holonómicos-escleronómicos). Assim, as forças internas são decompostas em parcelas activa e reactiva, o que conduz a uma descrição dual (unidimensional) da cinemática e da estática. São apresentados dois exemplos que ilustram a aplicação do modelo unidimensional desenvolvido, esclarecem os seus aspectos físicos e atestam as limitações dos modelos seccionalmente prismáticos (ou “em escada”), independentemente do número de segmentos prismáticos utilizados (de facto, estas limitações mantêm-se mesmo no processo de passagem ao limite quando o comprimento dos segmentos tende a para zero). Os principais contributos originais nesta primeira parte da tese podem ser resumidos da seguinte forma: (i) Obtêm-se expressões gerais para a segunda forma fundamental da superfície média de uma barra e para o tensor de mudança de curvatura (ii) Generaliza-se a definição do campo de deslocamentos da superfície média para todo a barra, incluindo assim a caracterização do empenamento na espessura das paredes (também designado por empenamento secundário). (iii) Na caracterização dos esforços internos, são tidos em consideração não apenas os esforços de membrana, mas também os momentos flectores e de torção e as forças de corte transversais “de casca”. (iv) A contribuição de Saint-Venant para a energia de deformação e a componente correspondente do momento torsor total são obtidas de forma consistente. (v) Estabelece-se um conjunto de desigualdades fundamentais relativas às propriedades mecânicas das secções transversais. A segunda parte da tese, cujo âmbito se restringe ao importante caso particular de barras com secção em I monossimétricas e altura variável, trata de modelos unidimensionais do tipo Hencky, cuja natureza é intrinsecamente discreta. De facto, um modelo de Hencky consiste num número finito de unidades rígidas ligadas por molas elásticas (ou, mais geralmente, por elementos reológicos) e pode ser encarado não apenas como uma idealização de um elemento estrutural contínuo, mas também como uma estrutura real por direito próprio. A sua simplicidade e transparência faz com que o seu comportamento, de um ponto de vista qualitativo, seja mais facilmente apreendido. São abordados dois tipos de problema em capítulos sucessivos: (i) o comportamento linear no espaço tridimensional, sob acções quase-estáticas genéricas e (ii) a encurvadura por flexão-torção (linearizada) de vigas e colunas-viga solicitadas à flexão no seu plano de simetria (que é também o plano de maior rigidez à flexão), incluindo o chamado efeito Wagner associado à assimetria dos banzos. É dada uma especial atenção à calibração das rigidezes da mola e à definição apropriada das condições de fronteira. Mostra-se que os modelos de Hencky, se bem que desenvolvidos de forma totalmente independente, são consistentes com modelos contínuos do tipo Vlasov previamente desenvolvidos, na medida em que os erros de truncatura locais tendem para zero à medida que o comprimento das unidades rígidas também se aproxima de zero. Apresentam-se vários exemplos ilustrativos, que incluem elementos prismáticos e elementos de secção rectangular fina, de forma a verificar os modelos discretos de Hencky e avaliar as suas taxas de convergência.