3. Creep consideration

The creep represents the increase of deformation under constant stress, which occurs in some materials due to its viscoelastic nature. It is essential to consider it in the analysis of slender structural elements, because the stiffness of these members is modified as a function of the rheology of the material. Usually, viscoelasticity is associated with creep of structural elements and can be characterized by models where the immediate elastic deformation is increased by viscous deformation, resulting in a temporal function for deformation. Consequently, the modulus of elasticity must also be provided as a temporal function that provides accurate results under normal levels of stress. Due to the viscous nature of the concrete, even at a constant stress level, the deformation of a structural element tends to increase over time. An increase in strain over time under constant stress is a viscoelastic phenomenon.

#### 3.1 Solution of the three-parameter rheological model

Mathematically, viscoelasticity can be represented by a time-dependent function associated with rheological models capable of describing the phenomenon. It is conceptually convenient to consider classic viscoelastic models in which there are only two types of parameters, relating to elasticity and viscosity. 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 [12]. One model used to represent the viscoelasticity of solids is the three-parameter model, in which the elastic parameter Ee is connected to the viscoelastic Kelvin-Voigt model with parameters Ev and η, which is a simplification of Group I of the Burgers model, as shown in Figure 2.

The three-parameter model is an appropriate model for describing the viscoelastic nature of many solids [13] and is often used to study the phenomenon in various scientific fields. Adaptation of the Burgers model in different fields of structural analysis can be found in [14–19]. The total deformations of the Kelvin-Voigt model are given by ε = ε <sup>e</sup> + ε v , where ε <sup>e</sup> is the deformation of the elastic model

Figure 2. Viscoelastic model of three parameters.

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

$$
\dot{\varepsilon} = \dot{\varepsilon}^{\varepsilon} + \dot{\varepsilon}^{v},
\tag{12}
$$

which includes the constitutive equations of the elastic and Kelvin-Voigt models, respectively. Considering the modulus of elasticity for both parts, elastic and viscous, the stress becomes

$$
\sigma = E\_{\epsilon} \epsilon^{\epsilon} \text{and } \sigma = E\_{\nu} \epsilon^{\nu} + \eta \stackrel{\cdot}{\epsilon^{\nu}}. \tag{13}
$$

From the previous equations, one derives the following differential equation:

$$
\dot{\sigma} + \frac{E\_{\epsilon} + E\_{v}}{\eta} \sigma = E\_{\epsilon} \,\dot{\varepsilon} + \frac{E\_{\epsilon} E\_{v}}{\eta} \,\varepsilon,\tag{14}
$$

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 derivative of the stress with respect to time is zero. Applying the previous stress condition, the following ordinary differential equation is found:

$$E\_e \left\| \varepsilon + \frac{E\_e E\_v}{\eta} \varepsilon = \sigma\_0. \tag{15}$$

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

$$
\sigma(t) = \sigma\_0 \left[ \frac{\mathbf{1}}{E\_\varepsilon} + \frac{\mathbf{1}}{E\_\nu} \left( \mathbf{1} - e^{-\frac{E\_\nu}{\eta}t} \right) \right]. \tag{16}
$$

From Eq. (16), it is possible to extract the temporal function for the modulus of elasticity of the three-parameter model:

$$E(t) = \frac{1}{\left[\frac{1}{E\_r} + \frac{1}{E\_v}\left(1 - e^{-\frac{E\_{\sigma\_I}}{\eta}}\right)\right]}\,. \tag{17}$$

Using Dynamic Analysis to Adjust the Rheological Model of Three Parameters… DOI: http://dx.doi.org/10.5772/intechopen.88665

Therefore, it is easily seen that for

$$t = 0 \Rightarrow e^{-\frac{E\_\theta}{\eta}} = 1 \; \text{.} \; \varepsilon(\mathbf{0}) = \frac{\sigma\_0}{E\_\epsilon};\tag{18}$$

$$t \to \infty \Rightarrow e^{-\frac{E\_v}{\eta}} = 0 \; \therefore \; \varepsilon(\infty) = \frac{\sigma\_0 (E\_\varepsilon + E\_v)}{E\_\varepsilon E\_v},\tag{19}$$

$$E(\infty) = \frac{E\_e E\_v}{E\_e + E\_v} \Rightarrow \varepsilon(\infty) = \frac{\sigma\_0}{E(\infty)}.\tag{20}$$

It is important to note that the viscoelastic behavior of the considered material is completely represented by the temporal modulus of elasticity and it can be used for static or dynamic applications. For instance, the previous solution was used in numerical simulations as can be seen in [20, 21].

It is possible to transform the parameters of the viscous part to being just a function of the modulus of elasticity of the elastic part, which can easily be calculated by any standard procedure or obtained in the simplest laboratory. Therefore, these parameters can be written as

$$E\_v = aE\_e; \eta = \chi E\_e. \tag{21}$$

where α is a real positive number and γ brings together a temporal unit.

#### 3.2 Model predicted by Eurocode

The method specified in European Standard EN 1992-1-1 for incorporating creep into structural analysis considers the effects of the creep behavior and its variation with time. Eurocode 2 provides hypothetical and model limitations for creep calculation, wherein the creep coefficient φ is predicted as a function of the tangent modulus of elasticity Ec. The creep deformation of concrete is computed by multiplying the immediate deformation by the creep coefficient. The total concrete deformation at time t, under constant temperature, can be obtained as the sum of the terms that represent the immediate deformation and creep. All the factors related to the phenomenon, such as loading and environment humidity, are calculated under the assumption that they remain constant over the considered time interval, affording a specific result for the creep coefficient φ. This coefficient is then directly introduced into the slow deformation equation and used as input data for various procedures. The basic equations for determining the creep coefficient of concrete over time are based on the average compressive strength fcm (fcm = fck + 8, fck in MPa). The creep coefficient φ(t, t0), as defined in Eq. (22), is the product of two factors, namely, φ<sup>0</sup> and βc(t, t0), which, respectively, characterize the effects of the rheological properties of the concrete under environmental conditions and the evolution of creep with time after loading of the structure:

$$
\rho(\mathbf{t}, t\_0) = \rho\_0 \beta\_c(\mathbf{t}, t\_0). \tag{22}
$$

The first factor φ<sup>0</sup> defined in Eq. (23) consists of three other factors. The first of them, φRH (given by Eq. (24)), considers concrete compressive strengths >35 MPa (as in the case that will be seen) and accounts for the effects of the environmental relative humidity RH, the equivalent thickness h<sup>0</sup> of the member which is a function of the cross-sectional area Ac, and the external perimeter ue of the member in contact with the environment. The second one, β(fcm) (Eq. (26)), represents the direct effect of the resistance on φ0. The third, β(t0) (Eq. (27)), takes into account the age of the concrete at the beginning of loading, i.e., at t0.

Dynamical Systems Theory

$$
\varphi\_0 = \varphi\_{RH} \beta(f\_{cm}) \beta(t\_0),
\tag{23}
$$

$$\rho\_{RH} = \left[1 + \frac{1 - RH/100}{0.1\sqrt[3]{h\_0}\alpha\_1}\right] a\_2,\tag{24}$$

$$h\_0 = \frac{2A\_c}{u\_e},\tag{25}$$

$$
\beta \left( f\_{cm} \right) = \frac{16.8}{\sqrt{f\_{cm}}},
\tag{26}
$$

$$\beta(t\_0) = \frac{1}{(\mathbf{0}.\mathbf{1} + t\_0^{0.20})}.\tag{27}$$

The second factor, βc(t, t0) (Eq. (28)), is a function of the coefficient β<sup>H</sup> (given by Eq. (29) for average concrete compressive strengths upper than 35 MPa), and it is used to regulate the combined effects of the relative humidity and the equivalent member thickness. The percolation path of the adsorbed water in a robust section of concrete is so large that the effects of creep due to differential moisture are less important for slimmer sections.

$$\beta\_{\epsilon}(t, t\_{0}) = \left[ \frac{(t - t\_{0})}{\beta\_{H} + (t - t\_{0})} \right]^{0.3},\tag{28}$$

$$
\beta\_H = \mathbf{1.5} \left[ \mathbf{1} + \left( \mathbf{0.012RH} \right)^{18} \right] h\_0 + \mathbf{250} a\_3 \le \mathbf{1500},\tag{29}
$$

$$a\_1 = \left(\frac{\mathfrak{B}}{f\_{cm}}\right)^{0.7}, a\_2 = \left(\frac{\mathfrak{B}}{f\_{cm}}\right)^{0.2}, a\_3 = \left(\frac{\mathfrak{B}}{f\_{cm}}\right)^{0.5}.\tag{30}$$

Thus, the creep coefficient can be obtained using Eq. (22), and the temporal function that describes the deformation in accordance with EN 1992-1-1 can be expressed as

$$
\sigma(t, t\_0) = \sigma\_\varepsilon(t\_0) \left[ \frac{1}{E\_\varepsilon(t\_0)} + \frac{\rho(t, t\_0)}{E\_\varepsilon(t\_{28})} \right]. \tag{31}
$$

Based on the above equations, the modulus of elasticity with respect to time can be expressed as

$$E(t, t\_0) = \frac{1}{\frac{1}{E\_c(t\_0)} + \frac{q(t, t\_0)}{E\_c(t\_{2b})}},\tag{32}$$

where Ec(t0) is the modulus of elasticity at the beginning of loading and Ec(t28) is the modulus of elasticity 28 days after the commencement of loading.
