2. Mathematical solution for representing the viscoelasticity

An increase in strain over time under constant stress is a viscoelastic phenomenon. Mathematically, viscoelasticity can be represented by a time-dependent function associated with rheological models capable of describing the phenomenon [6]. The slow deformation for concrete parts is a phenomenon that is related to loads and deformations but is partially reversible [7]. It is a phenomenon that is directly related to the movement of moisture inside concrete. When a sample of concrete is loaded for 90 days and then unloaded, the immediate or elastic recovery is approximately the same magnitude as the elastic deformation when the first load is applied [8].

It is conceptually convenient to consider classic viscoelastic models in which only two types of parameters, relating to elasticity and viscosity, appear [9]. Classic viscoelastic models are obtained by arranging springs and dampers, or dashpots, in different configurations. Springs are characterized by elastic moduli and dashpots by viscosity coefficients. The best known of these mechanical models are the Maxwell model, containing a spring in series with a dashpot, and the Kelvin-Voigt model, containing a spring and dashpot in parallel. One model used to represent the viscoelasticity of solids is the three-parameter model, in which the elastic parameter E0 is connected to the viscoelastic Kelvin-Voigt model with parameters E1 and η1, which is a simplification of the Group I Burgers model, as shown in Figure 1.

The three-parameter model sufficiently describes the viscoelastic nature of many solids and is often used to study the phenomenon in various scientific fields. The total deformations of the

Figure 1. Viscoelastic model of three parameters.

Kelvin-Voigt model are given by ε = ε<sup>e</sup> + ε<sup>v</sup> , where ε<sup>e</sup> is the deformation of the elastic model, and ε<sup>v</sup> is the deformation of the Kelvin-Voigt model. When differentiated with respect to time, the total deformation is obtained as

$$
\varepsilon = \dot{\varepsilon}^e + \dot{\varepsilon}^v \tag{1}
$$

The previous solution for consideration of the viscoelastic behavior of materials was used by [10] to evaluate the stability of a slender wooden column, for example. However, it is of interest, at this moment, to make clear that the present work is a numerical approximation, which takes into account the viscoelastic behavior of the concrete by assuming a viscoelastic

Analytical and Mathematical Analysis of the Vibration of Structural Systems Considering Geometric Stiffness…

http://dx.doi.org/10.5772/intechopen.75615

353

It is important to note that the viscoelastic behavior of the considered material is completely represented by the temporal modulus of elasticity. Therefore, the solid such behavior is wished to study should be according that adopted model. Any material can be represented by it, being, however, its usage conditioned by performing of experimental studies in order to confirm if it is correct or not. Keeping this in mind, the concrete viscoelastic behavior is assumed to be represented by the solid standard model, as an approximation of the reality. However, criteria from regulatory codes can be used in substituting of that model or even any

A piece can be considered as prestressed reinforced concrete when it is subjected to the action of the so-called prestressing forces and of permanent and variable loads, so that the concrete is not subjected to tension or it occurs below the limit of its resistance. As an example, take the normal stresses beam diagrams of the prestressed beam of Figure 2, where P is the prestressing force, MP the bending moment due to eccentricity of the load P, Mp is the bending moment due to uniformly distributed load p and R is the resultant, each one of these with their corresponding normal stresses. Under the conditions presented, the lower fibers of the beam, under positive bending moment, will have the tension stresses overturned by the superposi-

Prestressed concrete was developed scientifically from the beginning of the last century. Prestressing can be defined as the artifice of introducing, in a structure or a part, a previous state of stresses, in order to improve its resistance or its behavior in service, under the action of

tion of those produced by the normal stress of the applied stress eccentrically.

rheological model of three parameters or as also is known of the solid standard.

other rheological models can be adopted.

3. Beam as a basis of supporting

Figure 2. Normal stresses in a prestressed beam.

3.1. Basic considerations prestressing in reinforced concrete

which is the constitutive equation of the elastic and Kelvin-Voigt models, respectively. Considering E1 = E0 as the modulus of elasticity for both parts of the rheological model,

$$
\sigma = E\_0 \varepsilon^{\varepsilon} \text{ and } \dot{\sigma} + \frac{E\_0 + E\_0}{\eta\_1} \sigma = E\_0 \dot{\varepsilon} + \frac{E\_0 E\_0}{\eta\_1} \varepsilon \tag{2}
$$

are found. From the previous equations, one derives the following differential equation:

$$
\sigma = E\_0 \varepsilon^v + \eta\_1 \dot{\varepsilon}^v \tag{3}
$$

where σ = 0 for t < 0 and σ = σ<sup>0</sup> for t > 0, with t representing the time and t = 0 the instant of loading application. As the stress remains constant, the stress derivate with respect to time is zero. Applying the previous stress condition, the following ordinary differential equation is found:

$$E\_0 \dot{\varepsilon} + \frac{E\_0 E\_0}{\eta\_1} \varepsilon = \sigma\_0 \tag{4}$$

for which the general solution for t > 0, taking the initial condition ε(0) = σ0/E0, is

$$\varepsilon(t) = \sigma\_0 \left[ \frac{1}{E\_0} + \frac{1}{E\_0} \left( 1 - e^{-\frac{E\_0}{\eta\_1}t} \right) \right] \tag{5}$$

Obviously, if the stress level remains constant, the modulus of elasticity should decrease concurrently with increasing strain:

$$E(t) = \frac{1}{\frac{1}{\frac{1}{E\_0} + \frac{1}{E\_0}} \left(1 - e^{-\frac{E\_0}{\eta\_1}t} \right)}\tag{6}$$

The previous solution for consideration of the viscoelastic behavior of materials was used by [10] to evaluate the stability of a slender wooden column, for example. However, it is of interest, at this moment, to make clear that the present work is a numerical approximation, which takes into account the viscoelastic behavior of the concrete by assuming a viscoelastic rheological model of three parameters or as also is known of the solid standard.

It is important to note that the viscoelastic behavior of the considered material is completely represented by the temporal modulus of elasticity. Therefore, the solid such behavior is wished to study should be according that adopted model. Any material can be represented by it, being, however, its usage conditioned by performing of experimental studies in order to confirm if it is correct or not. Keeping this in mind, the concrete viscoelastic behavior is assumed to be represented by the solid standard model, as an approximation of the reality. However, criteria from regulatory codes can be used in substituting of that model or even any other rheological models can be adopted.
