Neste trabalho analisam-se grelhas de concreto armado, adaptando-se modelos não-lineares baseados na teoria mecânica do dano contínuo, com a incorporação da distorção da seção transversal ao estado de deformações. Discutem-se os modelos do CEB-158 (1985), da NBR-6118, para o concreto armado e modelos elastoplásticos para as barras da armadura longitudinal. Estuda-se a teoria de vigas de Timoshenko, onde são incorporadas as distorções da seção transversal à matriz de rigidez do elemento finito de viga. Discute-se o modelo de dano de MAZARS(1984), mostrando a sua formulação e parâmetros. Dedica-se especial atenção também à análise não-linear de grelhas de concreto armado, enfatizando-se as técnicas de solução dos sistemas de equações não-lineares. Por fim, faz-se uma revisão sobre o modelo da treliça clássica de Mörsch, tomando-a como base para a formulação de um modelo de resistência de esforços cortantes em uma seção transversal de concreto armado. / In this work, Reinforced concrete grids are analysed, assuming non-linear models based on the damage theory, and also incorporating into shear deformations. The proposed CEB-158 and the NBR-6118 models for reinforced concrete members are discussed, as well as the elastoplastic model assumed for the reinforcement. The Timoshenko's beam theory is studied and adopted to modify the beam stiffness matrix to take into account the shear deformation. The MAZARS' damage model is discussed, presenting the main aspects and parameters. Particular attention has been given to the reinforced concrete grid, pointing out the usual non-linear system solution techniques. Finally, the Mörsch truss model is revised and adopted to formulated an algorithm to deal with reinforced concrete beam elements, dividing the shear forces into two parts: the concrete and the reinforcement resultants.

Two methods for overcoming locking phenomena in isogeometric beams are presented. The first method blends the rotation of a Timoshenko beam with the rotation of a Bernoulli beam to produce realistic displacements in straight beams. The second method uses mixed variational principles, specifically the Hu-Washizu Principle, to produce realistic displacements as well as realistic strains without post-processing.

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.

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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

Neste trabalho analisam-se grelhas de concreto armado, adaptando-se modelos não-lineares baseados na teoria mecânica do dano contínuo, com a incorporação da distorção da seção transversal ao estado de deformações. Discutem-se os modelos do CEB-158 (1985), da NBR-6118, para o concreto armado e modelos elastoplásticos para as barras da armadura longitudinal. Estuda-se a teoria de vigas de Timoshenko, onde são incorporadas as distorções da seção transversal à matriz de rigidez do elemento finito de viga. Discute-se o modelo de dano de MAZARS(1984), mostrando a sua formulação e parâmetros. Dedica-se especial atenção também à análise não-linear de grelhas de concreto armado, enfatizando-se as técnicas de solução dos sistemas de equações não-lineares. Por fim, faz-se uma revisão sobre o modelo da treliça clássica de Mörsch, tomando-a como base para a formulação de um modelo de resistência de esforços cortantes em uma seção transversal de concreto armado. / In this work, Reinforced concrete grids are analysed, assuming non-linear models based on the damage theory, and also incorporating into shear deformations. The proposed CEB-158 and the NBR-6118 models for reinforced concrete members are discussed, as well as the elastoplastic model assumed for the reinforcement. The Timoshenko's beam theory is studied and adopted to modify the beam stiffness matrix to take into account the shear deformation. The MAZARS' damage model is discussed, presenting the main aspects and parameters. Particular attention has been given to the reinforced concrete grid, pointing out the usual non-linear system solution techniques. Finally, the Mörsch truss model is revised and adopted to formulated an algorithm to deal with reinforced concrete beam elements, dividing the shear forces into two parts: the concrete and the reinforcement resultants.

Ribbed plates comprising cross-laminated timber slab, CLT, with glulam beams allow for increased span length compared to pure CLT slabs. At the moment the Eurocodes do not provide any recommendations to design a CLT plate or a ribbed plate. There are handbooks for designing ribbed plates. But are their methods the best suited? Should one use Eurocode’s gamma-method, Timoshenko, shear analogy method or maybe an FE-model? This is difficult to answer, but nevertheless important questions for a structural engineer today. This master thesis investigates some of the different design methods used today in a numerical study. The study compares analytical methods and FE-models by investigating different parameters, like span lengths 6 m, 12 m, and 18 m, or uniform load and point load. Furthermore, this thesis provides guidance and help to apply design methods for ribbed plates. The outcome of the study shows a tendency for the gamma-method and the extended gamma-method giving the lowest deflections and normal stresses in many situations. Thus, there could be a risk of underestimating these properties by using the gamma-method. On the other hand there were consistent results with the gamma-methods compared to Timoshenko- and shear analogy-method which showed more deviations from the extended gamma-method as reference. The Timoshenko-method showed anomalies with thicker CLT. In addition, an Abaqus model with 3D elements and RFEM models with 2D and 1D elements were applied. The Abaqus models gave in general higher deflections and consistent results. Concluding from this thesis the Abaqus model together with the extended gamma-method would complement each other most efficiently in the design process of ribbed plates.

This thesis focuses on the development and convergence study of finite element methods for eigenvalue analysis of arbitrarily shaped domains with multi-material and material-void interfaces. Such configurations can be found in problems with evolving discontinuities and interfaces as in fluid-structure interaction or topology optimization problems. The differential equations considered in this thesis include the elliptic operator, Timoshenko beam and Mindlin plate. The compatibility conditions at the interface are weakly imposed using either Nitsche's method or Lagrange multipliers. The variational statements are derived for each case. The analysis results are benchmarked using Galerkin finite element discretization with bodyitted grids. Nitsche's method shows a direct dependence on a penalty term and for Lagrange multipliers method, additional degrees of freedom are added to the solution vector. The convergence rate of the discretized forms is computationally determined and is shown to be optimal for both Timoshenko beams and Mindlin plates.

La presente tesis trata sobre la novedosa aplicación de materiales funcionalmente graduados (-FGM- Functionally Graded Materials) en el diseño de estructuras tipo viga. Como es sabido, los estudios de rigidización dinámica, se llevan a cabo para evitar resonancias y/o reducir tensiones dinámicas. La forma tradicional que se emplea para optimizar la rigidez de la viga, es variando su sección transversal; ya sea reduciéndola de forma discontinua (multi-steps beams tal como se las conoce en inglés), ahusada (tapered beams tal como se las denomina en inglés) o bien empleando combinaciones de ambas. Surge la originalidad, desde el punto vista de la optimización dinámica, de adicionar una nueva solución al problema de la rigidización. La misma consiste en hacer variar las propiedades de estos modernos materiales en la dirección axial de las vigas (-AFGM-Axially Funtionally Graded Materials). El problema de vibraciones libres para las vigas AFG, se estudia implementando las teorías clásicas de Bernoulli-Euler y de Timoshenko, mediante tres enfoques aproximados; el método energético de Rayleigh-Ritz, el método de cuadratura diferencial generalizada (GDQM, por sus siglas en inglés) y el método de elementos finitos (FEM, por sus siglas en inglés). Los resultados obtenidos se comparan con valores disponibles en la literatura. La muy buena correlación de los mismos, permite dar certezas de que el método de Rayleigh-Ritz, GDQM y FEM, son herramientas de gran precisión y eficiencia para la resolución de esta clase de problemas. Los casos de vigas propuestos se modelan con condiciones de bordes clásicas y con extremos elásticamente restringidos. Para la modelación numérica de los problemas, se implementa el software Wolfram Mathematica 9.0. En el estudio se tratan leyes de distribución del tipo axial asimétrica y axial simétrica; para varias composiciones y distintos materiales AFG. En particular, se da especial énfasis al material compuesto por alúmina (Alum) y acero (Ac), ya que desde el punto de vista ingenieril presenta propiedades atractivas para la rigidización dinámica de vigas. Se analiza el efecto dinámico que tiene adosar una o varias masas a la viga. Evidenciando que su presencia disminuye, en todos los casos, a los coeficientes de frecuencias naturales de la viga sin masa adosada. Y además, constituye un factor importante, desde el punto de vista del diseño, en la elección de la composición del material AFG a implementar. Para las vigas AFG cantiléver con características asimétricas del material, la composición a) Ac-Alum rigidiza más que la composición b) Alum-Ac, cuando el exponente n (de las leyes de variación) es menor a 1, exista o no masa adosada en el extremo libre. En cambio, para 1n> la composición b) es quien optimiza la solución. El caso 1n= es particular, ya que la composición a) rigidiza levemente más que b) cuando la viga no tiene masa adosada. Pero, cuando se adosa la masa, se aprecia que el coeficiente de frecuencia fundamental aumenta más para la composición b). Para vigas AFG con características simétricas, se evidencia que la composición c) Ac-Alum- Ac optimiza la rigidización haya o no masa adosada, frente a la composición d) Alum-Ac- Alum. / This thesis deals with the novel application of functionally graduated materials (FGM) in the design of beams. It is known that dynamic stiffening studies are carried out to avoid resonances and/or reduce dynamic stresses. The traditional method to optimize the rigidity of the beam consists in variations of its cross section; either by reducing it in a discontinuous way (multi-steps beams), in a tapered way (tapered beams) or by using a combination of both. In this Thesis, from the point of view of dynamic optimization, an original solution of the problem of stiffening is proposed. This novel solution consists in varying the properties of these modern materials in the axial direction of the beams (AFGM). With this aim, the problem of free vibrations for the AFG beams is studied through the classical theories of Bernoulli-Euler and Timoshenko, using three approximation approaches: the Rayleigh-Ritz energy method, the generalized differential quadrature method (GDQM) and the finite element method (FEM). The obtained results are compared with those available in the related literature. The very good correlation of these, allows to give certainty that the Rayleigh-Ritz method, GDQM and FEM, are tools of great precision and efficiency for the resolution of this class of problems. The cases of proposed beams are modeled with classic edge conditions and with elastically restrained edge supports. For the numerical modeling of the problems, Wolfram Mathematica 9.0 software is implemented. The used distribution laws are the axial asymmetric and axial symmetric and were applied for various compositions and different AFG materials; with a particular emphasis on the material composed of alumina (Alum) and steel (St), since it has attractive properties for the dynamic stiffening of beams from the engineering point of view. The dynamic effect of attaching one or several masses to the beam is analyzed. It is shown that their presence decreases, in all cases, the natural frequency coefficients of the beam that do not have attached mass. In addition, it constitutes an important factor, from the point of view of the design, in the choice of the composition of the AFG material to be implemented. In cantilever AFG beams with asymmetric characteristics of the material, the composition a) St-Alum stiffens more than the composition b) Alum-St, when the exponent n (of the laws of variation) is less than 1, whether or not there is an attached mass at the free end. On the other hand, for the composition b) it is the one that optimizes the solution. The case 1n= is particular, since the composition a) stiffens slightly more than b) when the beam has not attached mass. But, when the mass is attached, the fundamental frequency coefficient increases more for the composition b). In AFG beams with symmetrical characteristics, it is evident that the composition c) St- Alum-St optimizes the stiffening whether or not there is an attached mass, compared to the composition d) Alum-St-Alum.

This dissertation is focused on understanding the performance of a particular fiber-reinforced polymeric composite structural beam, a 91.4 cm (36 inch) deep pultruded double-web beam (DWB) designed for bridge construction. Part 1 focuses on calculating the Timoshenko shear stiffness of the DWB and understanding what factors may introduce error in the experimental measurement of the quantity for this and other sections. Laminated beam theory and finite element analysis (FEA) were used to estimate the shear stiffness. Several references in the literature have hypothesized an increase in the effective measured shear stiffness due to warping. A third order laminated beam theory (TLBT) was derived to explore this concept, and the warping effect was found to be negligible. Furthermore, FEA results actually indicate a decrease in the effective shear stiffness at shorter spans for simple boundary conditions. This effect was attributed to transverse compression at the load points and supports. The higher order sandwich theory of Frostig shows promise for estimating the compression related error in the shear stiffness for thin-walled beams. Part 2 attempts to identify the failure mechanism(s) under quasi-static loading and to develop a strength prediction for the DWB. FEA was utilized to investigate two possible failure modes in the top flange: compression failure of the carbon fiber plies and delamination at the free edges or taper regions. The onset of delamination was predicted using a strength-based approach, and the stress analysis was accomplished using a successive sub-modeling approach in ANSYS. The results of the delamination analyses were inconclusive, but the predicted strengths based on the compression failure mode show excellent agreement with the experimental data. A fatigue life prediction, assuming compression failure, was also developed using the remaining strength and critical element concepts of Reifsnider et al. One DWB fatigued at about 30% of the ultimate capacity showed no signs of damage after 4.9 million cycles, although the predicted number of cycles to failure was 4.4 million. A test on a second beam at about 60% of the ultimate capacity was incomplete at the time of publication. Thus, the success of the fatigue life prediction was not confirmed. / Ph. D.