In engineering, a large number of structures may be modeled as cantilevers. Due to their intrinsic characteristics, some of these structures are sensitive to dynamic actions. Gusts of wind are dynamic excitations for which the fundamental frequency of vibration is an important factor when calculating the structural response. Modeling the effects of the axial force on the natural frequencies of a structure usually results in systems of differential equations that are not solvable from a practical engineering perspective. This article develops a simple mathematical expression for calculating the fundamental frequency of cantilevered structures, within small ranges, that considers the presence of an axial demand. This expression has been validated by dynamic laboratory testing.

This study focused on improving the design of slender structures with reinforced concrete (RC) telecommunication towers as the main application. Analytical procedure based on Rayleigh's method to compute the first natural vibration frequency and the critical buckling load was development. All the nonlinearities present in the system were considered, in addition to the soil-structure interaction and the variation of the geometric properties along the length of the structure. The geometric nonlinearity and imperfections of the tower structure were computed as functions of the axial load using a geometric stiffness matrix. Further, the material nonlinearity was accounted for by reducing the flexural stiffness. As concrete structures exhibit viscoelasticity, creep was calculated using the Eurocode 2 model. The soilstructure interaction was modeled as a set of distributed springs. To validate the proposed method, the first frequency and critical buckling load were compared with those yielded by FEM simulations. The frequency results were in good agreement with those of the FEM simulations, indicating that the proposed method is sufficiently accurate for use in engineering design applications and easy to implement. On the other hand, the buckling load results obtained using the proposed method and FEM differed significantly, motivating further investigation.

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The ultimate limit state of stability by equilibrium bifurcation, the limit states for stress and strain resulting from this condition were evaluated for an extremely slender real structure of reinforced concrete, with geometry varying along its length. The aspects related to nonlinearities of the material were considered through the recommendations on NBR 6118:2014, from the Brazilian Association of Technical Standards (ABNT). In the analytical solution, developed for stability analysis, all elements of the structural dynamics present in the system were taken into account, including the column self-weight. The critical buckling load was then dynamically defined to different instants of time. Reductions of 70% for the modulus of elasticity and 59% for the critical buckling force were found in analyses performed from zero and five thousand days. It was also possible to obtain the induced stresses on the homogenized cross-sections and those transferred to reinforcement steel bars. KeywordsLimit states, critical buckling load, analytical solution, NBR 6118:2014. Graphical AbstractEvaluation of limit state of stress and strain of free-fixed columns with variable geometry according to criteria from the Brazilian code for concrete structures Alexandre de Macêdo Wahrhaftig et al.

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Eigenvector analysis can be performed to determine the shapes of the undamped free vibration modes of a system. Eigenvector analysis involves solving the generalized eigenvalue problem, which considers the stiffness and mass matrix of a structure. For a geometric nonlinear study, both parts of the total stiffness matrix are required. As modal analysis depends on the stiffness, the effect of its reduction on the modal shape of vibration of the structure must be determined. Case studies were evaluated using the finite element method, considering and neglecting the geometric portion of the stiffness matrix. Mathematic functions were applied for comparison.

-Creep is a phenomenon that can occur in wooden structures since wood is a viscoelastic material. Creep may change the purely elastic parameters determined in wood characterization initial tests, as its behavior depends on the rheology of the material, even under a constant stress level. Mathematically, creep can be characterized by models in which the immediate elastic deformation is increased by a viscous deformation, resulting in a temporal function. For this reason, the calculation of the natural frequency of vibration and the stability verification of a slender column should include the reducing effects of stiffness both of axial force and creep. The first one can be considered through the geometrical portion and the second one by the introduction, in the conventional portion, of a variable elasticity modulus over time, obtained in relation to the adopted rheological model. A numerical simulation was performed to evaluate the aspects above, considering a bar compressed by a force at the free end equivalent to 10% of the Euler critical force, plus its own weight, adopting a rheological model with three parameters for the variation of the elasticity modulus. The results show differences of 60% and 50% for the frequency and elasticity modulus, besides defining the exact instant of column collapse in the case of its non-observance.Keywords: Three-parameter model; Vibration; Stability.A FLUÊNCIA NA FREQUÊNCIA FUNDAMENTAL E NA ESTABILIDADE DE UMA COLUNA ESBELTA DE MADEIRA DE SEÇÃO COMPOSTA RESUMO -A fluência é um fenômeno que pode ocorrer em estruturas de madeira por ser esse um material viscoelástico. A fluência pode alterar os parâmetros puramente elásticos determinados nos ensaios iniciais de caracterização da madeira, pois o seu comportamento depende da reologia do material, mesmo com um nível de tensão constante. Matematicamente, a fluência pode ser caracterizada por modelos onde a deformação elástica imediata é acrescida de uma deformação viscosa, resultando em uma função temporal. Por essa razão, o cálculo da frequência natural de vibração e a verificação da estabilidade de uma coluna esbelta devem incluir os efeitos redutores da rigidez tanto da força axial quanto da fluência. O primeiro pode ser considerado por meio da parcela geométrica e o segundo pela introdução, na parcela convencional, de um módulo de elasticidade variável com o tempo, obtido em relação ao modelo reológico adotado. Para avaliar os aspectos mencionados, foi realizada uma simulação numérica, considerando uma peça comprimida por uma força na extremidade livre equivalente a 10% da força crítica de Euler, mais o seu peso próprio, adotando-se um modelo reológico de três parâmetros para a variação do módulo de elasticidade. Os resultados indicaram diferenças de 60% e 50% para a frequência e para o módulo de elasticidade, além de definir o instante exato de colapso da coluna em caso de sua inobservância.Palavras-chave: Modelo de três parâmetros; Vibração; Estabilidade.

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