Alexandre de Macêdo Wahrhaftig

By Alexandre de M. Wahrhaftig, Reyolando M. L. R. F. Brasil and Lázaro S. M. S. C. Nascimento

For a complete analysis of vibration, the stiffness of a structure must have two characteristics: one corresponding to conventional stiffness and the other to the geometric stiffness. Thus, the total stiffness takes form where the model to be used to represent any behavior of the material is introduced to the first part via the modulus of elasticity. The second is the geometric stiffness, through which it is possible to linearize a geometric nonlinear problem. To consider both aspects, a mathematical model based on the Rayleigh method has been elaborated. Two systems were numerically studied. First, the occurrence of resonance in the vibration of a prestressed reinforced concrete beam has been investigated. The results indicated resonant and non-resonant schemes between the natural frequency of the beam and the frequency of the engine. To the second system, the first natural frequency of a slender, 40-m-high concrete mobile phone mast, was calculated, and an evaluation of the structural collapse was performed. To the both systems, the cross section of reinforced concrete was treated by the theory for the homogenized section in order to consider the presence of the steel, and the viscoelasticity of the concrete was taken into account through a three-parameter rheological model.

Part of the book: Numerical Simulations in Engineering and Science

By Alexandre de Macêdo Wahrhaftig

A dynamic analysis of vibration for considering a three-parameter rheological model to fit the same results as predicted for creep by the Eurocode (EN 1992) criteria is performed based on the adjustment of its parameters. The use of a rheological model of three parameters as a valid alternative for real problems brings a huge facility for mathematical implementation and manipulation due the simplicity of the solution. For adjustment of the elastics and the viscous parameters, a numerical simulation to calculate the fundamental frequency of an actual reinforcement concrete pole is carried out in comparison with the standard Eurocode criteria. In this determination, the geometry variation, a concentrated force present at the free end of the structural element, and the self-weight of the structure are considered. The physical nonlinearity of the concrete due to the cracks is also considered by reducing of the flexural stiffness, and its viscoelastic behavior is included in the calculation through a temporal modulus of elasticity. In the analysis, the ground was modeled as a set of distributed springs along the foundation length. The frequency over time is then analytically calculated as the critical buckling load for different instants after the structure to be loaded.

Part of the book: Dynamical Systems Theory