# Static and Seismic Responses of Eco-Friendly Buried Concrete Pipes with Various Dosages of Fly Ash
## 2 Finite Element Modeling
### Determination of Basic Equations for the Stress-Strain Curve of GPC
\begin{table}
\begin{tabular}{c c} \hline \hline
**ANSYS Parameters** & **Values** \\ \hline The shear transfer coefficient for open cracks (\(\beta t\)) & 0.3 \\ \hline The shear transfer coefficient for closed cracks (\(\beta C\)) & 1 \\ \hline Uniaxial cracking stress (\(\beta t\)) & 46.23 MPa \\ \hline Uniaxial crushing stress (\(f_{c}\)) & 38 MPa \\ \hline Biaxial compressive strength (\(\beta c\)) & 45.40 MPa \\ \hline Ambient hydrostatic stress (\(\beta t\)) & 65.46 MPa \\ \hline Hydrostatic biaxial crush stress (\(\beta\)) & 54.86 MPa \\ \hline Hydrostatic uniaxial crash stress (\(\beta\)) & 65.08 MPa \\ \hline Tensile Crack Factor & 0.6 \\ \hline Elasticity modulus (E) & 22,000 MPa \\ \hline Poisson Ratio & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Concrete failure criteria constants in ANSYS (William–Warnke). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Concrete Cube** & **Ultimate Load (KN)** &
\begin{tabular}{c} **Compressive** \\ **Strength (MPa)** \\ \end{tabular} & **Strain at Peak Stress** \\ \hline GPC & & & \\ \hline Modified Popovics model & & & \\ ([PERSON] et al. [22]) & 851.4 & 37.84 & 0.002086 \\ \hline ANSYS (Present Study) & 858.15 & 38.14 & 0.002262 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of numerical results in the present study and the modified Popovics model. The following constants are desired for defining concrete properties in the ANSYS platform (Table 2):  
- Shear transfer coefficients for an open crack (\(\beta t\));  
- Shear transfer coefficients for a closed crack (\(\beta c\));  
- Uniaxial tensile cracking stress (\(\beta t\));  
- Uniaxial crushing stress (positive) (\(\ell cu\));  
- Biaxial crushing stress (positive) (\(\ell cb\));  
- Ambient hydrostatic stress state (\(\alpha h\));  
- Biaxial crushing stress (positive) (\(f1\));  
- Uniaxial crushing stress (positive) (\(f2\));  
- Stiffness multiplier for cracked tensile condition (\(T\!c\)). Four parameters, namely the shear transfer coefficients for an open crack (\(\beta t\)), the shear transfer coefficients for a closed crack (\(\beta c\)), the uniaxial tensile cracking stress (\(\beta t\)), and the uniaxial crushing stress (\(\ell cu\)), are more important compared to other concrete properties. The shear transfer coefficients, (\(\beta t\)) and (\(\beta c\)), are used to consider the retention of shear stiffness in cracked concrete; these parameters range from 0 to 1.0, with 0 representing a smooth crack (complete loss of shear transfer) and 1.0 representing a rough crack (no loss of shear transfer) (ANSYS [29]). For this study, the shear transfer coefficient for an open  
Figure 5: (**a**) 2D and (**b**) 3D pipe models in ANSYS 18.2 (current study). crack (\(\beta t\)) is set to 0.3 to avoid convergence problems and the shear transfer coefficients for a closed crack (\(\beta\)c) is chosen as 1.0 [3, 30, 31, 32, 33, 34]. Further details related to soil and concrete properties used in this validation are presented in Table 5. It is observed that the element types used in the current study are pipe 188 and solid 185 for concrete and soil, respectively, whereas the element type applied was solid 65 for both concrete and soil in the previous study. As can be seen in Figure 6, the values of the stress-strain curve have a relatively high approximation, so that the maximum values of stress for both numerical models have a minimum difference of 5%, which means that the numerical model used in this study is reliable.