## 3 Modeling
### Nonlinear Structural Analysis  
After the moisture transport analysis, the nonlinear structural analysis using _Model-2_ is performed by applying the equivalent load (calculated as described in Sections 3.2 and 3.3) together with other permanent loads.  
In the nonlinear structural analysis, the creep of the concrete is described by the Kelvin chain model. The Kelvin chain is a chain of Kelvin-Voigt model which is composed of a parallel arrangement  
Figure 1: Curvature and strain profile of a beam.  
of a linear spring and a dashpot [14; 15]. By the Kelvin chain model, the creep compliance function can be expressed by a Dirichlet series.  
\[J(t)=\sum_{\alpha=0}^{n}\frac{1}{E_{\alpha}}\Big{(}1-e^{\frac{t-1}{\lambda_{ \alpha}}}\Big{)} \tag{15}\]  
Here, \(E_{\alpha}\) is the elastic modulus of each chain, \(\tau\) is the age of concrete at the time of load application, \(\lambda_{\alpha}\) is the retardation time of each chain (defined as \(\eta_{\alpha}/E_{\alpha}\)), and \(\eta_{\alpha}\) is the damping coefficient of each chain. The elastic modulus and retardation time of each chain can be determined by curve fitting to experimental data or by using the creep equation of the design code. In this study, the creep equation provided by MC2010 [13] was used as shown in Equation (16).  
\[J(t,\tau)=\frac{1}{\beta_{E}(\tau).E(t=28)}+\frac{\varphi_{0}(t=28)}{\beta_{ \varphi}(\tau).E(t=28)} \tag{16}\]  
Here \(\varphi_{0}(t=28)\) is the creep coefficient obtained for loading at the age of 28 days, \(E(t=28)\) is the modulus of elasticity at the age of 28 days. The scaling functions \(\beta_{E}\) and \(\beta_{\varphi}\) are aging factors to adjust the instantaneous and transient part of the creep respectively, for loads that are applied other than 28 days.  
\[\beta_{E}(t)=\sqrt{e^{-[s[1-(\frac{28}{\
u})^{0.5}]]}} \tag{17}\]  
where s varies from 0.2-0.38 depending on the type of cement. For normal hardening and normal weight cement with cubic strength less than or equal to 60 MPa, s = 0.25. \(\beta_{\varphi}\) is used with a value of unity.  
The uniform shrinkage strain was also calculated according to MC2010 [13], in which the autogenous and drying shrinkage are explicitly defined as functions of the concrete age, compressive strength of the concrete, type of cement used, atmospheric RH, and notional size of the member. Details can be found in the DIANA User's Manual [15; 16].  
For the crack analysis, the constitutive model of concrete is defined using normal to the crack (mode-I) parameters. In mode-I fracture definition, the important material parameters are the fracture energy (\(G_{F}\)) and the tensile softening function. In this study, a total strain crack formulation with the exponential tension-softening curve as shown in Figure 2 was adopted. For the tension-softening curve, the fracture energy was calculated using the following equation [13]:  
\[G_{F}=73\;f_{cm}{}^{0.18} \tag{18}\]  
In the above, \(f_{cm}\) is the mean compressive strength (MPa) and is defined as \(f_{cm}=f_{ck}+8\).  
Same with the moisture transport analysis, the prestressing steel and non-prestressing reinforcing bars were modeled as embedded bar elements and assumed to be in a perfect bond with the surrounding concrete. The prestress force after initial loss due to the combined effects of the elastic shortening,  
Figure 2: Exponential tension softening diagram.  
friction, and anchorage slip was applied as an imposed stress on the prestressing steel, so that the changes in stress over time could be calculated. Quadratic hexahedron elements were also used for the concrete.