## 4 Finite Element Modeling
### Thermo-Mechanical Properties  
Following the Eurocode 2 EN 1992-1-1 model available in the DIANA FE program, the young hardening of concrete with total strain crack model is considered to simulate early-age crack induction in the CRCP segment under external varying temperature conditions [45]. The concrete class C40/50 is generally used in the construction of CRCP sections in Belgium [12]. The specific heat capacity and thermal conductivity of concrete are the essential parameters for predicting the temperature gradient in early-age concrete. The former represents the heat storage capacity and the latter determines the heat transfer through conduction [47]. The specific heat capacity is taken as 2.87 \(\times\) 106 J/m\({}^{3}\)\({}^{\circ}\)C in the FE analysis based on previous studies [48; 49; 50]. The value of thermal conductivity is considered to be 2.7 W/m \({}^{\circ}\)C [12; 41; 48]. Constant values have been considered in the present study because of the difficulties in determining accurate early-age concrete thermal properties [51].  
The coefficient of thermal expansion (CTE) is a parameter of paramount importance required for the thermal analysis of concrete as it provides a measure of volumetric changes at varying temperatures. As the concrete material is composed of aggregates up to 70 to 85% of the total solid volume, the type of aggregate and its mineral composition largely determine the values of CTE of concrete due to different thermal properties possessed by various types of aggregates [51; 52]. Experimental studies have indicated that the CTE of concrete remains constant after the final setting [12; 47; 50]. Therefore, in the present study, a constant value of CTE is assumed for the simulation of early-crack induction. The thermal and mechanical properties of concrete used in the FE analysis are demonstrated in Table 3.  
The early-age cracking of concrete is significantly influenced by the type of cement. The blast furnace slag cement (CEM III) is mostly preferred for the construction of CRCPs in Belgium owing to lesser heat discharge, low permeability, and good durability [53]. The heat of the hydration model valid for blast furnace slag cement is used to define the heat development in terms of the adiabatic heat curve [54]. The evolution of concrete mechanical properties (tensile strength, compressive strength, and elastic modulus) with respect to time is formulated based on the degree of reaction in accordance with the Eurocode 2 EN 1992-1-1 model code [45]. Input parameters for Eurocode 2 EN 1992-1-1 creep and shrinkage model are demonstrated in Table 4.  
The longitudinal and transverse reinforcing steel bars are modeled as embedded reinforcement. The Young's modulus and yield strength of steel are taken as 200 GPa and 500 MPa, respectively. The constitutive material behavior of reinforcing steel is defined as elastoplastic with no hardening. To ensure the smooth convergence of the FE analysis  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Input Parameters** & **Value** \\ \hline Ambient temperature (\({}^{\circ}\)C) & 20 \\ Notional size (mm) & 439 \\ Relative humidity (\%) & 80 \\ Curing age (Days) & 3 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Input parameter for Euro-code 2 EN 1992-1-1 creep and shrinkage model.  
\begin{table}
\begin{tabular}{c c} \hline \hline Concrete class & C40/50 \\ Aggregate type & Limestone \\ Coefficient of thermal expansion of concrete (1/\({}^{\circ}\)C) & \(10.0\times 10^{-6}\) \\ Concrete thermal conductivity (W/m \({}^{\circ}\)C) & 2.7 \\ Concrete volumetric heat specific capacity (J/m\({}^{3}\)\({}^{\circ}\)C) & \(2.87\times 10^{6}\) \\ Convection-radiation coefficient between concrete and air (W/m\({}^{2}\)\({}^{\circ}\)C) & 7.55 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Input parameters used in the finite element (FE) simulation.  
as well as to reduce the computational time and data storage space, full-bonding contact between the steel bars and concrete is assumed in the present study [15; 26; 27; 38; 41; 42].