Abstract
This study aims to consider the effect of soil–structure interaction (SSI) on the ductility and hysteretic energy demands of superstructures and propose empirical equations for demand prediction in soil–structure systems. To this end, the FEMA 440 procedure was considered to develop nonlinear singledegreeoffreedom oscillators with a period range of 0.1–3.0 s, as the representative of superstructures. The elasticperfectly plastic and a moderate pinching degrading hysteretic models were considered for the nonlinear response of the superstructure. The model of the nonlinear soil–foundation system was developed through the Winkler method. In this regard, the type of soil beneath the foundation was assumed as D category, according to the site classification in ASCE 710. A wide range of key parameters, including the strength reduction factors (2 ≤ R_{μ} ≤ 8), the foundation safety factor (3 ≤ SF ≤ 7), the foundationtostructure height aspect ratio (1 ≤ h/b ≤ 5), and the foundation lengthtowidth ratio (3 ≤ L_{f}/B_{f} ≤ 20) was introduced into the analytical models to conduct parametric studies. Results show the considerable effect of SSI on reducing the ductility and hysteretic energy demands in superstructures with short fundamental periods. More demand reduction can be achieved by providing the lateral sliding of the foundation on the soil surface, especially for systems with a small aspect ratio. The pinching–degrading hysteretic behavior of the superstructure remarkably modifies the level of demands. Moreover, predictive models were proposed for estimating the ductility and hysteretic energy demands in flexible base systems. These models modify demands in the rigid base structures based on their physical and mechanical properties. The developed models consider the effects of structural hysteretic behavior as well as foundation flexibility. The efficiency of the proposed model was assessed on a multistory frame. Finally, the required ductility capacity of the systems was determined through the Park–Ang damage index and by using the developed predictive models. Results show the efficiency of the empirical models to reasonably estimate the required ductility capacity.
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Acknowledgements
The author would like to acknowledge the financial support of the Institute of Science and High Technology and Environmental Sciences, Graduate University of Advanced Technology, Kerman, Iran, under grant number of 98.314.
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Appendix
Appendix
This appendix deals with details for computing the predicted values of E_{N} and μ_{m} from the proposed formulation. First, considering the studied structure in this research (see Sect. 6), in which, the fundamental natural vibration period of the structure is T_{fix} = 2.0 s.

1.
Based on Eqs. (27) and (28), the regression factors of a_{0}, a_{1}, a_{2}, b_{0}, b_{1}, b_{2} for an EPP model is obtained as:
$$\left\{ \begin{aligned} a_{0} = \text{e}^{{\left( {0.9324\ln \left( {7.5} \right)  0.0192} \right)}} = 6.42 \hfill \\ a_{1} = \text{e}^{{\left( {0.6101\left( {7.5} \right)  4.3851} \right)}} = 1.21 \hfill \\ a_{2} = 2.037 \hfill \\ \end{aligned} \right.\,\,\,\quad \,\,\,\left\{ \begin{aligned} b_{0} = \text{e}^{{\left( {1.7064\ln \left( {7.5} \right) + 0.5631} \right)}} = 54.67 \hfill \\ b_{1} = \text{e}^{{\left( {2.0578\ln \left( {7.5} \right)  2.9348} \right)}} = 3.36 \hfill \\ b_{2} = 2.6073 \hfill \\ \end{aligned} \right.$$(47) 
2.
Substituting the regression factors of a_{0}, a_{1}, a_{2}, b_{0}, b_{1}, b_{2} into Eqs. (25) and (26):
$$\mu_{{m\,\left( {fix} \right)}} = 6.42 + \frac{1.21}{{\left( {2.0} \right)^{2.037} }} = 6.71\quad E_{{N\,\left( {fix} \right)}} = 54.67 + \frac{3.36}{{\left( {2.0} \right)^{2.6073} }} = 55.22$$(48)
For the SSI model, since the fundamental natural vibration period of the structure is T_{fix} = 2.0 s, the 2nd part of Eqs. (30) and (31) were used for computing the predicted values of E_{N(SSI)} and μ_{m(SSI)}. In this regard:

3.
Equations (32) and (33) are essential for computing the regression constants of m_{1}, n_{1}, m_{2}, and n_{2} by considering the P_{ij}s factors from Tables 4 and 5. For example, considering a nonsliding model of foundation, the regression factors of m_{1}, n_{1}, m_{2}, and n_{2} are obtained as:
$$\begin{aligned} m_{1} & = 1.41  0.3185\left( {2.56} \right) + 0.2808\left( 3 \right)  0.01144\left( {2.56} \right)^{2} + 0.06539\left( {2.56} \right)\left( 3 \right) \hfill \\ & \quad  0.03667\left( 3 \right)^{2} = 1.534 \hfill \\ n_{1} & = 1.831 + 0.1117\left( {2.56} \right)  0.08769\left( 3 \right) + 0.003761\left( {2.56} \right)^{2}  0.02298\left( {2.56} \right)\left( 3 \right) \hfill \\ \quad + 0.01183\left( 3 \right)^{2} = 1.8085 \hfill \\ \end{aligned}$$(49)$$\begin{aligned} m_{2} &= 1.012  0.1098\left( {2.56} \right) + 0.03543\left( 3 \right)  0.0112\left( {2.56} \right)^{2} + 0.02847\left( {2.56} \right)\left( 3 \right) \hfill \\ & \quad  0.00757\left( 3 \right)^{2} = 0.9143 \hfill \\ n_{2} & = 0.9676 + 0.04591\left( {2.56} \right)  0.00337\left( 3 \right) + 0.009683\left( {2.56} \right)^{2}  0.01489\left( {2.56} \right)\left( 3 \right) \hfill \\ & \quad + 0.002444\left( 3 \right)^{2} = 1.046 \hfill \\ \end{aligned}$$(50) 
4.
Substituting the regression factors of m_{1}, n_{1}, m_{2}, and n_{2} into the 2^{nd} part of Eqs. (30) and (31), the predicted values of E_{N(SSI)} and μ_{m(SSI)} for the SSI model is obtained as:
$$E_{{N\,\left( {SSI} \right)}} = 1.534\left( {7.5} \right)^{1.8085} = 58.66\,\quad \mu_{{m\,\left( {SSI} \right)}} = 0.9143\left( {7.5} \right)^{1.046} = 7.523$$(51)
Now assuming another structure with a sliding foundation (SLD) and T_{fix} = 1.5 s, R_{µ} = 4, h/b = 2, SF = 3, and a PD model; the following steps are used for predicting E_{N} and μ_{m}:

1.
Computing the regression factors of a_{0}, a_{1}, a_{2}, b_{0}, b_{1}, b_{2} from Eqs. (27) and (28) for a PD model:
$$\left\{ \begin{aligned} a_{0} = \text{e}^{{\left( {0.976\ln \left( 4 \right)  0.0318} \right)}} = 3.748 \hfill \\ a_{1} = \text{e}^{{\left( {1.7336\ln \left( 4 \right)  3.8084} \right)}} = 0.245 \hfill \\ a_{2} = 2.146 \hfill \\ \end{aligned} \right.\,\quad \,\,\left\{ \begin{aligned} b_{0} = \text{e}^{{\left( {1.6411\ln \left( 4 \right) + 0.3537} \right)}} = 13.856 \hfill \\ b_{1} = \text{e}^{{\left( {2.0493\ln \left( 4 \right)  3.1421} \right)}} = 0.740 \hfill \\ b_{2} = 2.7761 \hfill \\ \end{aligned} \right.$$(52) 
2.
Computing E_{N(fix)} and μ_{m(fix)} for the rigidbased structure by substituting the regression factors of a_{0}, a_{1}, a_{2}, b_{0}, b_{1}, b_{2} into Eqs. (25) and (26):
$$\mu_{{m\,\left( {fix} \right)}} = 3.748 + \frac{0.245}{{\left( {1.5} \right)^{2.146} }} = 3.85\,\quad E_{{N\,\left( {fix} \right)}} = 13.856 + \frac{0.74}{{\left( {1.5} \right)^{2.7761} }} = 14.096\,$$(53)
For the SSI model, the 1st part of Eqs. (30) and (31) is used for computing the predicted values of E_{N(fix)} and μ_{m(fix)}. In this case:

3.
The regression constants e_{i}, f_{i}, g_{i}, and h_{i} are computed using the polynomial function of degree 3 in Eq. (38) through regression factors in Table 7 for E_{N} and Table 9 for μ_{m}, which yields: e_{1} = 0.1234, f_{1} = 1.5592, g_{1} = 0.1052, h_{1} = 2.5189, e_{2} = 0.17988, f_{2} = 1.1796, g_{2} = 0.1684, h_{2} = 2.1. The computation details are presented for one of the coefficients (say e_{1}):
$$\begin{aligned} e_{1} & =  0.4968 + 1.47\left( {\frac{2}{4}} \right) + 0.2119\left( 4 \right)  2.386\left( {\frac{2}{4}} \right)^{2}  0.1799\left( 4 \right)\left( {\frac{2}{4}} \right) \\ & \quad  0.03523\left( 4 \right)^{2} + 0.5958\left( {\frac{2}{4}} \right)^{3} \, + 0.1747\left( {\frac{2}{4}} \right)^{2} \left( 4 \right)  0.01014\left( {\frac{2}{4}} \right)\left( 4 \right)^{2} \\ & \quad + 0.00223\left( 4 \right)^{3} =  0.1234 \\ \end{aligned}$$(54) 
4.
Computing the regression factors of c_{i} and d_{i} through Eqs. (36) and (37):
$$\begin{aligned} c_{1} & = \exp \left[ {  0.1234\left( 3 \right) + 1.5592} \right] = 3.284 \\ d_{1} & =  \exp \left[ {  0.1052\left( 3 \right) + 2.5189} \right] =  9.055 \\ c_{2} & = \exp \left[ {  0.17988\left( 3 \right) + 1.1796} \right] = 1.8964 \\ d_{2} & =  \exp \left[ {  0.1684\left( 3 \right) + 2.10} \right] =  4.93 \\ \end{aligned}$$(55) 
5.
Now substituting these factors into the 1^{st} part of Eqs. (30) and (31) yields to:
$$\begin{aligned} c_{1} T_{fix} + d_{1} & = 3.284\left( {1.5} \right)  9.055 =  4.129 \\ c_{2} T_{fix} + d_{2} & = 1.8964\left( {1.5} \right)  4.93 =  2.085 \\ \end{aligned}$$(56)
On the other hand, from Fig. 15, the mean elastic acceleration response spectrum of the ground motion records at T_{fix} = 1.5 s is about 1.48. Thus, substituting into the left side of Eqs. (30) and (31):
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Homaei, F. Estimation of the ductility and hysteretic energy demands for soil–structure systems. Bull Earthquake Eng 19, 1365–1413 (2021). https://doi.org/10.1007/s10518020010282
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DOI: https://doi.org/10.1007/s10518020010282