O principal objetivo deste trabalho é realizar a análise estrutural de edifícios de andares múltiplos que apresentam núcleos resistentes, considerando a deformação pelo esforço cortante nos mesmos, bem como nos pilares. Para atingir esse objetivo, será preciso que o comportamento à flexão dos elementos verticais de contraventamento passe a ser regido pela teoria de barras de Timoshenko e não mais pela de Euler-Bernoulli. Foram então desenvolvidos algoritmos que, utilizando o Método dos Elementos Finitos (MEF), permitem calcular os fatores de forma de quaisquer seções transversais abertas de paredes delgadas pertencentes a núcleos estruturais, bem como a distribuição da tensão de cisalhamento na seção transversa em função do esforço cortante atuante. As alterações acima descritas foram feitas em um programa de análise de edifícios denominado CEASO 01, de autoria de MATIAS JR (1997). Embora esse programa realize análise não-linear geométrica, a consideração da deformação por cortante só foi implementada na análise linear. Apresentam-se, ao final, exemplos numéricos que permitem avaliar a influência da deformação pelo esforço cortante sobre os deslocamentos e esforços de núcleos resistentes e pilares. / The main aim of this work is to perform structural analysis of multistory buildings with resistant cores, taking into account shear deformation in those elements, as well as in columns. To achieve this objective, the flexural behaviour of vertical elements must be governed by Timoshenko beam theory, rather than the Euler-Bernoulli theory. Procedures using the finite element method (FEM) were developped, which enable to evaluate shear correction factors of generic thin-walled open sections and shear stress distribution as a function of the shear resultant. Changes described above were made in a structural analysis program named CEASO 01, whose author is MATIAS JR (1997). Even though this program is able to perform nonlinear analysis, only in linear analysis the effect of shear deformation is taken into account. Numerical examples are provided, which enable to evaluate the influence of taking into account shear deformation on displacements and stress resultants of resistant cores and columns.

The conventional finite element models (FEM) of problems in structural mechanics are based on the principles of virtual work and the total potential energy. In these models, the secondary variables, such as the bending moment and shear force, are post-computed and do not yield good accuracy. In addition, in the case of the Timoshenko beam theory, the element with lower-order equal interpolation of the variables suffers from shear locking. In both Euler-Bernoulli and Timoshenko beam theories, the elements based on weak form Galerkin formulation also suffer from membrane locking when applied to geometrically nonlinear problems. In order to alleviate these types of locking, often reduced integration techniques are employed. However, this technique has other disadvantages, such as hour-glass modes or spurious rigid body modes. Hence, it is desirable to develop alternative finite element models that overcome the locking problems. Least-squares finite element models are considered to be better alternatives to the weak form Galerkin finite element models and, therefore, are in this study for investigation. The basic idea behind the least-squares finite element model is to compute the residuals due to the approximation of the variables of each equation being modeled, construct integral statement of the sum of the squares of the residuals (called least-squares functional), and minimize the integral with respect to the unknown parameters (i.e., nodal values) of the approximations. The least-squares formulation helps to retain the generalized displacements and forces (or stress resultants) as independent variables, and also allows the use of equal order interpolation functions for all variables. In this thesis comparison is made between the solution accuracy of finite element models of the Euler-Bernoulli and Timoshenko beam theories based on two different least-square models with the conventional weak form Galerkin finite element models. The developed models were applied to beam problems with different boundary conditions. The solutions obtained by the least-squares finite element models found to be very accurate for generalized displacements and forces when compared with the exact solutions, and they are more accurate in predicting the forces when compared to the conventional finite element models.

Adhesively bonded joints have been investigated for several decades. In most analytical studies, the Bernoulli-Euler beam theory is employed to describe the behaviour of adherends. In the current work, three analytical models are developed for adhesively bonded joints using the Timoshenko beam theory for elastic material and a Bernoulli-Euler beam model for viscoelastic materials. One model is for the peeling test of an adhesively bonded joint, which is described using a Timoshenko beam on an elastic foundation. The adherend is considered as a Timoshenko beam, while the adhesive is taken to be a linearly elastic foundation. Three cases are considered: (1) only the normal stress is acting (mode I); (2) only the transverse shear stress is present (mode II); and (3) the normal and shear stresses co-exist (mode III) in the adhesive. The governing equations are derived in terms of the displacement and rotational angle of the adherend in each case. Analytical solutions are obtained for the displacements, rotational angle, and stresses. Numerical results are presented to show the trends of the displacements and rotational angle changing with geometrical and loading conditions. In the second model, the peeling test of an adhesively bonded joint is represented using a viscoelastic Bernoulli-Euler beam on an elastic foundation. The adherend is considered as a viscoelastic Bernoulli-Euler beam, while the adhesive is taken to be a linearly elastic foundation. Two cases under different stress history are considered: (1) only the normal stress is acting (mode I); and (2) only the transverse shear stress is present (mode II). The governing equations are derived in terms of the displacements. Analytical solutions are obtained for the displacements. The numerical results show that the deflection increases as time and temperature increase. The third model is developed using a symmetric composite adhesively bonded joint. The constitutive and kinematic relations of the adherends are derived based on the Timoshenko beam theory, and the governing equations are obtained for the normal and shear stresses in the adhesive layer. The numerical results are presented to reveal the normal and shear stresses in the adhesive.

In this thesis, a hybrid-stress finite element is developed for nonuniform Timoshenko beams. The element stiffness matrix is obtained by assuming a stress field only. Since element boundaries are simply the element nodes, a displacement assumption is not necessary. Geometric and mass stiffness matrices are obtained via equilibrium and kinematics of deformation equations which are derived in the beam arbitrary cross-section. Utilizing this method eliminates the displacement assumption for the geometric and mass stiffness matrices. The element has six degrees of freedom at each node. Axial, flexural and torsional effects are considered. The torsional and distortional warping effects are omitted. Deformations due to shear is also taken into account. Finally, some sample problems are solved with the element and results are compared with the solutions in the literature and commercial finite element programs (i.e. MSC/NASTRAN&reg / ).

Ce travail de recherche est dédié à la simulation de la réponse transitoire des assemblages de poutres soumis à des chocs. De tels chargements entraînent la propagation d’ondes haute fréquence dans l’ensemble de la structure. L’énergie qu’elles transportent peut être dommageable pour son fonctionnement ou celui des équipements embarqués. Dans des études précédentes, il a été observé sur des structures expérimentales qu’un régime vibratoire diffusif tend à s’installer pour des temps longs. Le but de cette étude est donc de développer un modèle robuste de la réponse transitoire des assemblages de poutres soumis à des chocs permettant de simuler, entre autres, cet état diffusif. Les champs de déplacement étant très oscillants et la densité modale élevée, la simulation numérique de la réponse transitoire à des chocs peut difficilement être menée par une méthode d’éléments finis classique. Une approche utilisant un estimateur de la densité d’énergie de chaque mode de propagation a donc été mise en œuvre. Elle permet d’accéder à des informations locales sur les états vibratoires, et de contourner certaines limitations intrinsèques aux longueurs d’onde courtes. Après avoir comparé plusieurs modèles de réduction cinématique de poutre à un modèle de Lamb de propagation dans un guide d’ondes circulaire, la cinématique de Timoshenko a été retenue afin de modéliser le comportement mécanique haute fréquence des poutres. En utilisant ce modèle dans le cadre de l’approche énergétique évoquée plus haut, deux groupes de modes de propagation de la densité d’énergie vibratoire dans une poutre ont été isolés : des modes longitudinaux regroupant un mode de compression et des modes de flexion, et des modes transversaux regroupant des modes de cisaillement et un mode de torsion. Il peut être également montré que l’´evolution en temps des densités d’énergie associées obéit à des lois de transport. Pour des assemblages de poutres, les phénomènes de réflexion/transmission aux jonctions ont du être pris en compte. Les opérateurs permettant de les décrire en termes de flux d’´energie ont été obtenus grâce aux équations de continuité des déplacements et des efforts aux jonctions. Quelques caractéristiques typiques d’un régime haute fréquence ont été mises en évidence, tel que le découplage entre les modes de rotation et les modes de translation. En revanche, les champs de densité d’énergie sont quant à eux discontinus aux jonctions. Une méthode d’éléments finis discontinus a donc été développée afin de les simuler numériquement comme solutions d’´equations de transport. Si l’on souhaite atteindre le régime diffusif aux temps longs, le schéma numérique doit être peu dissipatif et peu dispersif. La discrétisation spatiale a été faite avec des fonctions d’approximation de type spectrales, et l’intégration temporelle avec des schémas de Runge-Kutta d’ordre élevé du type ”strong stability preserving”. Les simulations numériques ont donné des résultats concluants car elles permettent d’exhiber le régime de diffusion. Il a été remarqué qu’il existait en fait deux limites diffusives différentes : (i) la diffusion spatiale de l’´energie sur l’ensemble de la structure, et (ii) l’équirépartition des densités d’énergie entre les différents modes de propagation. Enfin, une technique de renversement temporel a été développée. Elle pourra être utile dans de futurs travaux sur le contrôle non destructif des assemblages complexes et de grandes tailles. / This research is dedicated to the simulation of the transient response of beam trusses under impulse loads. The latter lead to the propagation of high-frequency waves in such built up structures. In the aerospace industry, that phenomenon may penalize the functioning of the structures or the equipments attached to them on account of the vibrational energy carried by the waves. It is also observed experimentally that high-frequency wave propagation evolves into a diffusive vibrational state at late times. The goal of this study is then to develop a robust model of high-frequency wave propagation within three-dimensional beam trusses in order to be able to recover, for example, this diffusion regime. On account of the small wavelengths and the high modal density, the modelling of high-frequency wave propagation is hardly feasible by classical finite elements or other methods describing the displacement fields directly. Thus, an approach dealing with the evolution of an estimator of the energy density of each propagating mode in a Timoshenko beam has been used. It provides information on the local behavior of the structures while avoiding some limitations related to the small wavelengths of high-frequency waves. After a comparison between some reduced-order beam kinematics and the Lamb model of wave propagation in a circular waveguide, the Timoshenko kinematics has been selected for the mechanical modelling of the beams. It may be shown that the energy densities of the propagating modes in a Timoshenko beam obey transport equations. Two groups of energy modes have been isolated: the longitudinal group that gathers the compressional and the bending energetic modes, and the transverse group that gathers the shear and torsional energetic modes. The reflection/transmission phenomena taking place at the junctions between beams have also been investigated. For this purpose, the power flow reflection/transmission operators have been derived from the continuity of the displacements and efforts at the junctions. Some characteristic features of a high-frequency behavior at beam junctions have been highlighted such as the decoupling between the rotational and translational motions. It is also observed that the energy densities are discontinuous at the junctions on account of the power flow reflection/transmission phenomena. Thus a discontinuous finite element method has been implemented, in order to solve the transport equations they satisfy. The numerical scheme has to be weakly dissipative and dispersive in order to exhibit the aforementioned diffusive regime arising at late times. That is the reason why spectral-like approximation functions for spatial discretization, and strong-stability preserving Runge-Kutta schemes for time integration have been used. Numerical simulations give satisfactory results because they indeed highlight the outbreak of such a diffusion state. The latter is characterized by the following: (i) the spatial spread of the energy over the truss, and (ii) the equipartition of the energy between the different modes. The last part of the thesis has been devoted to the development of a time reversal processing, that could be useful for future works on structural health monitoring of complex, multi-bay trusses.

O principal objetivo deste trabalho é realizar a análise estrutural de edifícios de andares múltiplos que apresentam núcleos resistentes, considerando a deformação pelo esforço cortante nos mesmos, bem como nos pilares. Para atingir esse objetivo, será preciso que o comportamento à flexão dos elementos verticais de contraventamento passe a ser regido pela teoria de barras de Timoshenko e não mais pela de Euler-Bernoulli. Foram então desenvolvidos algoritmos que, utilizando o Método dos Elementos Finitos (MEF), permitem calcular os fatores de forma de quaisquer seções transversais abertas de paredes delgadas pertencentes a núcleos estruturais, bem como a distribuição da tensão de cisalhamento na seção transversa em função do esforço cortante atuante. As alterações acima descritas foram feitas em um programa de análise de edifícios denominado CEASO 01, de autoria de MATIAS JR (1997). Embora esse programa realize análise não-linear geométrica, a consideração da deformação por cortante só foi implementada na análise linear. Apresentam-se, ao final, exemplos numéricos que permitem avaliar a influência da deformação pelo esforço cortante sobre os deslocamentos e esforços de núcleos resistentes e pilares. / The main aim of this work is to perform structural analysis of multistory buildings with resistant cores, taking into account shear deformation in those elements, as well as in columns. To achieve this objective, the flexural behaviour of vertical elements must be governed by Timoshenko beam theory, rather than the Euler-Bernoulli theory. Procedures using the finite element method (FEM) were developped, which enable to evaluate shear correction factors of generic thin-walled open sections and shear stress distribution as a function of the shear resultant. Changes described above were made in a structural analysis program named CEASO 01, whose author is MATIAS JR (1997). Even though this program is able to perform nonlinear analysis, only in linear analysis the effect of shear deformation is taken into account. Numerical examples are provided, which enable to evaluate the influence of taking into account shear deformation on displacements and stress resultants of resistant cores and columns.

The main purpose of this thesis is to develop a beam element in order to model the wing of a drone, made of composite materials. The proposed model consists of the framework for the structural design and analysis of long slender beam like structures, e.g., wings, wind turbine blades, and helicopter rotor blades, etc. The main feature consists of the addition of the coupling between axial and bending with torsional effects that may arise when using composite materials and the coupling stemming from the inhomogeneity in cross-sections of any arbitrary geometry. This type of modeling approach allows for an accurate yet computationally inexpensive representation of a general class of beam-like structures. The framework for beam analysis consists of main two parts, cross-sectional analysis of the beam sections and then using this section analysis to build up the finite element model. The cross-sectional analysis is performed in order to predict the structural properties for composite sections, which are used for the beam model. The thesis consists of the model to validate the convergence of the element size required for the cross-sectional analysis. This follows by the validation of the shell models of constant cross-section to assess the performance of the beam elements, including coupling terms. This framework also has the capability of calculating the strains and displacements at various points of the cross-section. Natural frequencies and mode shapes are compared for different cases of increasing complexity with those available in the papers. Then, the framework is used to analyze the wing of a drone and compare the results to a model developed in NASTRAN. / Master of Science

Electromagnetic launchers (railguns) are set to replace traditional large caliber ship mounted cannons in the near future. The success of the railgun depends heavily upon a comprehensive understanding of beam behavior during periods of heavy dynamic loading. It is hypothesized that the combination of velocity transition effects, electromagnetic loading, and other non-linear or design specific effects contribute to areas of high stresses/strains over the length of the rail/beam during launch. This paper outlines the use of the Timoshenko beam model, a model which builds upon the traditional Bernoulli-Euler beam theory with the addition of shear deformation and rotary inertia effects, a necessity for high wave velocities. Real-world experimental setups are simplified and approximated by a series of linear springs and dampers for model prediction and validation. The Timoshenko beam model is solved using finite difference (FD) methods for the approximation of spatial derivatives and MATLAB ordinary differential equation (ODE) solvers. The model shows good convergence and precision over a large range of system parameters including load velocities, foundation stiffness values, and beam dimensions. Comparison to experimental strain data has validated model accuracy to an acceptable level. Accuracy is further enhanced with the inclusion of damping and non-linear or piecewise effects used to mimic experimental observations. The MATLAB software package presents a valid preliminary analysis tool for railgun beam and foundation design while offering advantages in ease of use, computation time, and system requirements when compared to traditional FEA tools. / Master of Science

In this thesis, transverse vibration of a cracked beam on an elastic foundation and the effect of crack and foundation parameters on transverse vibration natural frequencies are studied. Analytical formulations are derived for a beam with rectangular cross section. The crack is an open type edge crack placed in the medium of the beam and it is uniform along the width of the beam. The cracked beam rests on an elastic foundation. The beam is modeled by two different beam theories, which are Euler-Bernoulli beam theory and Timoshenko beam theory. The effect of the crack is considered by representing the crack by rotational springs. The compliance of the spring that represents the crack is obtained by using fracture mechanics theories. Different foundation models are discussed / these models are Winkler Foundation, Pasternak Foundation, and generalized foundation. The equations of motion are derived by applying Newton&#039 / s 2nd law on an infinitesimal beam element. Non-dimensional parameters are introduced into equations of motion. The beam is separated into pieces at the crack location. By applying the compatibility conditions at the crack location and boundary conditions, characteristic equation whose roots give the non-dimensional natural frequencies is obtained. Numerical solutions are done for a beam with square cross sectional area. The effects of crack ratio, crack location and foundation parameters on transverse vibration natural frequencies are presented. It is observed that existence of crack reduces the natural frequencies. Also the elastic foundation increases the stiffness of the system thus the natural frequencies. The natural frequencies are also affected by the location of the crack.

abstract: In order to verify the dispersive nature of transverse displacement in a beam, a deep understanding of the governing partial differential equation is developed. Using the finite element method and Newmark’s method, along with Fourier transforms and other methods, the aim is to obtain consistent results across each numerical technique. An analytical solution is also analyzed for the Euler-Bernoulli beam in order to gain confidence in the numerical techniques when used for more advance beam theories that do not have a known analytical solution. Three different beam theories are analyzed in this report: The Euler-Bernoulli beam theory, Rayleigh beam theory and Timoshenko beam theory. A comparison of the results show the difference between each theory and the advantages of using a more advanced beam theory for higher frequency vibrations. / Dissertation/Thesis / Masters Thesis Civil Engineering 2016