## 2 Numerical Analysis of Creep Behavior
### Numerical Algorithm  
As already reported in the literature [17; 22; 29; 30], the finite element analysis of long term behavior with creep is much more efficient if a rate-type approach is used instead of an integral-type form. A Kelvin chain, as well as a Maxwell chain, arrangements of springs and dashpots can described the most general creep behavior [29; 30]. Since the material constitutive law of concrete is typically based on the assumption of the strain additivity, a Kelvin chain is more convenient [16; 31] than the Maxwell chain. Following the original idea of [PERSON] [32], the structural creep problem can be reduced to a sequence of elastic finite element analyses using an elastic stress-strain relation with inelastic strain, i.e, step-by-step linear elastic analysis for each time step. This means that in each time step the rheologic model can be considered non-aging and, consequently, its spring moduli and viscosity are constant and updated only at the beginning of the time step. As shown in the previous section, the incremental quasi-elastic stress-strain relation suitable for a general finite element program can be written as (Equation (8) or Equation (21))  
\[\Delta\sigma=E^{*}\left(\Delta\epsilon-\epsilon^{*}(t_{i})-\Delta\epsilon^{0} \right)\quad\text{with}\quad\epsilon^{*}(t_{i})=\frac{\sigma^{*}(t_{i})}{E^{*}} \tag{37}\]  
where \(E^{*}\) and \(\sigma^{*}(t_{i})\) have been derived in the previous section (see Equations (11) and (13) or Equation (22)) and \(\Delta\epsilon^{0}\) is the inelastic strain increment in each time step, such as the shrinkage strain or thermal strain [16]. This incremental stress-strain relation represent the quasi-elastic response of a non-aging kelvin chain that approximates a non-aging creep function \(J(t,t_{0})=C(t-t_{0})\). The formula of the compliance functions generally adopted in standards or codes can be put in the form  
\[J(t,t_{0})=d_{0}(t_{0})+C^{*}(t,t_{0})=d_{0}(t_{0})+c_{0}C_{1}(t_{0})C(t-t_{0}) \tag{38}\]where \(d_{0}(t_{0})\) indicates the instantaneous elastic strain caused by a unit applied stress at the time \(t_{0}\) and \(c_{0}C_{1}(t_{0})C(t-t_{0})\) represents the creep deformation, which is expressed as the product of a constant \(c_{0}\), an aging term, \(C_{1}(t_{0})\), and a non-aging term, \(C(t-t_{0})\). The compliance function in Equation (38) can be approximated by the following Dirichlet series  
\[J(t,t_{0})=d_{0}(t_{0})+c_{0}\sum_{j=1}^{N}\frac{C_{1}(t_{0})}{E_{j}}\left(1-e^ {-\frac{t-t_{0}}{\gamma_{j}}}\right) \tag{39}\]  
where the coefficients \(E_{j}\) are identified from the Dirichlet approximation of \(C(t-t_{0})\) using the expression in Equation (35). In this case, the general incremental quasi-elastic stress-strain relation is still given by Equation (37) with the following expression for \(E^{*}\) and \(\sigma^{*}(t_{i})\)  
\[(E^{*})^{-1}=d_{0}(\overline{t})+c_{0}C_{1}(\overline{t})\sum_{j=1}^{N}\frac{ 1}{E_{j}}\left[1-\left(1-e^{-\frac{\Delta t}{\gamma_{j}}}\right)\frac{\tau_{j }}{\Delta t}\right] \tag{40}\]  
\[ \begin{split}\epsilon^{*}(t_{i})=\frac{\sigma^{*}(t_{i})}{E^{*}}= &\sum_{j=1}^{N}\left(1-e^{-\frac{\Delta t}{\gamma_{j}}}\right) \epsilon_{j}^{*}(t_{i})\qquad\quad\text{with}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\epsilon_{j}^{*}(t_{i })=e^{-\frac{\Delta t}{\gamma_{j}}}\epsilon_{j}^{*}(t_{i-1})+c_{0}C_{1}( \overline{t})\frac{1}{E_{j}}\left(1-e^{-\frac{\Delta t}{\gamma_{j}}}\right) \frac{\tau_{j}}{\Delta t}\Delta\sigma\end{split}  \tag{41}\]  
where \(\overline{t}\) can be taken as the time at the middle of the time step, \(t_{i}+\Delta t_{i}/2\). When a specific creep formulation is considered, from the expression of \(C(t-t_{0})\) the coefficients \(E_{j}\) must be calculated first and then, using the specified expressions for \(d_{0}\), \(C_{1}\), and \(c_{0}\), one can calculate the value of \(E^{*}\) and \(\sigma^{*}(t_{i})\) to be utilized in the constitutive law of Equation (37).