1. Introduction

A column represents a continuous structural member whose vibrations are governed by nonlinear partial differential equations for which exact analytical solutions cannot be found, as pointed out by [1]. Columns constitute continuous systems, and their analysis can be reduced to an analogous system containing a single degree of freedom. The vibration mode is restricted to a configuration previously established by a mathematical function that describes the vibratory movement, and the properties of the system can be expressed as generalized coordinate functions [2]. In his study on the vibration of elastic systems applied this technique considering the function valid throughout the problem domain. However, for real cases, where the properties of the structural elements vary along their length, the formulation developed for calculating the stiffness and mass must be solved by

observing the intervals defined in the geometry. In these cases, the integrals obtained can be solved within the limits established for each interval, i.e., the generalized properties can be calculated for each discrete segment of the structure, as defined by its geometry within that segment. A variety of vibration problems using that mathematical concept were solved by [3] who mentioned a previous one [4] where the buckling load is calculated for stepped and tapered columns and where how laborious or even impossible it is to apply it for problems with variable geometry is registered. With the advent of digital computers, these problems passed to be solved by modeling that use discretization technics of the continuum [5].

To analytically define the fundamental frequency for the case modeled in this study, all the elastic stiffness components are considered in the calculation, including the conventional stiffness, which depends on the material behavior; the geometric stiffness, which depends on the normal force acting on the structure; and the soil parcel, which accounts for the soil-structure interaction. It is important to note that the soil-structure interaction cannot be ignored, particularly in the case of a monopile foundation, because it may significantly influence the dynamic behavior of the structure [6].

The structure selected for this study is a slender reinforced concrete (RC) having both full and hollow circular section with variable geometry, for which the natural frequency and the critical buckling load were calculated considering all nonlinearities present in the system. It is important to highlight that nonlinearities play an important role when calculating dynamic proprieties of a system, as well pointed by [7]. In this work, the geometric nonlinearity was taken in consideration by using the geometric stiffness parcel into the total stiffness of the system. The nonlinearity of the material was computed by reducing its flexural stiffness, as similarly done by [8], reflecting the development of cracking in the concrete when bended, which is dependent on the magnitude of the stress. Another kind of material nonlinearity is creep, which occurs due the viscoelastic behavior of the concrete, it being considered in two ways. The first one is the mathematical model for creep predicted by Eurocode 2 (European Standard EN 1992-1-1) [9]. The second one is a threeparameter viscoelastic model whose parameters are adjusted in order to meet the results obtained when using the Eurocode. In this sense, the use of the threeparameter viscoelastic model to represent the creep of concrete brings an enormous facility of employment for actual cases due the reduced number of variables which are manipulated. Indeed, just one of them is necessary because two of the three parameters can be expressed in terms of the modulus of elasticity of the concrete, a data easily calculated for any standard procedure or obtained in laboratory.
