# 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  
After reviewing the variety of constitutive models for GPC in compression state, Table 1 represents the six most widely accepted models which are relatively compatible for GPC in low-calcium classification (class F). The linear state of stress-strain relations in the ascending branch was kept unchanged for all these six models. However, the curvatures of the stress-strain relations after peak stress in the descending branch have a distinctive difference which varies by replacing diverse values for the parameter 'n' in the empirical formula experimentally obtained in previous investigations. As shown in Figure 1, 'n' is attributed to the peak stress value. Additionally, [PERSON] et al. [19] demonstrated that the parameter 'n' at different temperatures can play an effective role in the nonlinear behavior of concrete in the descending region at the collapse moment. It was proved that 'n' has a significant effect on the nonlinear behavior of concrete resulting in less plastic strain in the concrete specimens containing FA compared to those with conventional paste such as OPC. This occurrence can be ascribed due to the increase of the CSH chain caused by a chemical reaction between FA and PC. The increase of 'n' that rapidly occurred at an early stage is due to amplification of the reaction at temperatures between 9-200 \({}^{\circ}\)C as shown in Figure 1. On the contrary, the decrement of 'n' at a later stage might be caused by the appearance of the thermally induced microcracks in cement paste and decomposition of the CSH chain. In this study, the value of n = 4 was chosen for the Popovic relation proposed in 1971, given that the curing condition for the experimental schedule of the current study was kept under the temperature of 21 \({}^{\circ}\)C (the temperature at water bath). With respect to \(Ec\) as the significant parameter in the stress-strain curve, most early studies, as well as current investigations, centralize on \(Ec\) of GPC mixture at 28 days of age. For instance, [PERSON] and [PERSON] [20] presented a relationship (Equation (1)) based on their experimental observations to predict the elastic modulus of ambient cured blended low calcium FA-based GPC mixes as follows:  
\[\mathrm{E_{cj,a}}=3510\sqrt{f_{c}} \tag{1}\]  
where \(\mathrm{E_{cj,a}}\) is the elastic modulus of ambient cured FA-based GPC in MPa. In addition, [PERSON] et al. [21] have recently explored an empirical formula to predict the elasticity modulus of ambient cured fly ash and slag-based geopolymer concrete (FS-GPC) as given by Equation (2):  
\[E_{c}\ =\ 2.5(f_{c}^{\prime})^{\frac{3}{5}} \tag{2}\]where \(Ec\) is the elastic modulus of concrete (GPA) and fc is the average compressive strength measured on cubes (MPa). In the current study, the two above-mentioned relations for the value of \(Ec\) are selected for the modified Popovics relations proposed by [PERSON] et al. [22] and [PERSON] [23] (Table 1). It is essential to mention that only the amounts of strain corresponding to the peak-stress of these two relations are attributed to \(Ec\); whereas the value \(\varepsilon_{\text{c}}\) for other proposed constitutive models are in relation to \(f_{\text{c}}\) as shown in Table 1.  
Validation of the Developed Constitutive Model for GPC Concrete  
The comparison between stress-strain curves subjected to FE results of the GPC cube in the present study and those relations discussed in Section 2.1 are represented in Figures 2 and 3. It is clear from this comparative assessment that the numerical model of this study behaves excellently with the proposed relations. However, there is a relatively significant difference that can be seen from the relation presented by [PERSON] [25] and [PERSON] et al. [26] referring to the different values of 'n' as explained in Section 2.1.