Estimation of the ductility and hysteretic energy demands for soil–structure systems

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 single-degree-of-freedom oscillators with a period range of 0.1–3.0 s, as the representative of superstructures. The elastic-perfectly 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 7-10. A wide range of key parameters, including the strength reduction factors (2 ≤ R_μ ≤ 8), the foundation safety factor (3 ≤ SF ≤ 7), the foundation-to-structure height aspect ratio (1 ≤ h/b ≤ 5), and the foundation length-to-width 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 multi-story 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.

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 a₀, a₁, a₂, b₀, b₁, b₂ for an EPP model is obtained as:

  $$\left\{ \begin{aligned} a_{0} = \text{e}^{{\left( {0.9324\ln \left( {7.5} \right) - 0.0192} \right)}} = 6.42 \hfill \\ a_{1} = \text{e}^{{\left( {0.6101\left( {7.5} \right) - 4.3851} \right)}} = 1.21 \hfill \\ a_{2} = 2.037 \hfill \\ \end{aligned} \right.\,\,\,\quad \,\,\,\left\{ \begin{aligned} b_{0} = \text{e}^{{\left( {1.7064\ln \left( {7.5} \right) + 0.5631} \right)}} = 54.67 \hfill \\ b_{1} = \text{e}^{{\left( {2.0578\ln \left( {7.5} \right) - 2.9348} \right)}} = 3.36 \hfill \\ b_{2} = 2.6073 \hfill \\ \end{aligned} \right.$$

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• 2. Substituting the regression factors of a₀, a₁, a₂, b₀, b₁, b₂ into Eqs. (25) and (26):

  $$\mu_{{m\,\left( {fix} \right)}} = 6.42 + \frac{1.21}{{\left( {2.0} \right)^{2.037} }} = 6.71\quad E_{{N\,\left( {fix} \right)}} = 54.67 + \frac{3.36}{{\left( {2.0} \right)^{2.6073} }} = 55.22$$

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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 m₁, n₁, m₂, and n₂ by considering the P_ijs factors from Tables 4 and 5. For example, considering a non-sliding model of foundation, the regression factors of m₁, n₁, m₂, and n₂ are obtained as:

  $$\begin{aligned} m_{1} & = 1.41 - 0.3185\left( {2.56} \right) + 0.2808\left( 3 \right) - 0.01144\left( {2.56} \right)^{2} + 0.06539\left( {2.56} \right)\left( 3 \right) \hfill \\ & \quad - 0.03667\left( 3 \right)^{2} = 1.534 \hfill \\ n_{1} & = 1.831 + 0.1117\left( {2.56} \right) - 0.08769\left( 3 \right) + 0.003761\left( {2.56} \right)^{2} - 0.02298\left( {2.56} \right)\left( 3 \right) \hfill \\ \quad + 0.01183\left( 3 \right)^{2} = 1.8085 \hfill \\ \end{aligned}$$

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  $$\begin{aligned} m_{2} &= 1.012 - 0.1098\left( {2.56} \right) + 0.03543\left( 3 \right) - 0.0112\left( {2.56} \right)^{2} + 0.02847\left( {2.56} \right)\left( 3 \right) \hfill \\ & \quad - 0.00757\left( 3 \right)^{2} = 0.9143 \hfill \\ n_{2} & = 0.9676 + 0.04591\left( {2.56} \right) - 0.00337\left( 3 \right) + 0.009683\left( {2.56} \right)^{2} - 0.01489\left( {2.56} \right)\left( 3 \right) \hfill \\ & \quad + 0.002444\left( 3 \right)^{2} = 1.046 \hfill \\ \end{aligned}$$

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• 4. Substituting the regression factors of m₁, n₁, m₂, and n₂ 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:

  $$E_{{N\,\left( {SSI} \right)}} = 1.534\left( {7.5} \right)^{1.8085} = 58.66\,\quad \mu_{{m\,\left( {SSI} \right)}} = 0.9143\left( {7.5} \right)^{1.046} = 7.523$$

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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 a₀, a₁, a₂, b₀, b₁, b₂ from Eqs. (27) and (28) for a PD model:

  $$\left\{ \begin{aligned} a_{0} = \text{e}^{{\left( {0.976\ln \left( 4 \right) - 0.0318} \right)}} = 3.748 \hfill \\ a_{1} = \text{e}^{{\left( {1.7336\ln \left( 4 \right) - 3.8084} \right)}} = 0.245 \hfill \\ a_{2} = 2.146 \hfill \\ \end{aligned} \right.\,\quad \,\,\left\{ \begin{aligned} b_{0} = \text{e}^{{\left( {1.6411\ln \left( 4 \right) + 0.3537} \right)}} = 13.856 \hfill \\ b_{1} = \text{e}^{{\left( {2.0493\ln \left( 4 \right) - 3.1421} \right)}} = 0.740 \hfill \\ b_{2} = 2.7761 \hfill \\ \end{aligned} \right.$$

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• 2. Computing E_N(fix) and μ_m(fix) for the rigid-based structure by substituting the regression factors of a₀, a₁, a₂, b₀, b₁, b₂ into Eqs. (25) and (26):

  $$\mu_{{m\,\left( {fix} \right)}} = 3.748 + \frac{0.245}{{\left( {1.5} \right)^{2.146} }} = 3.85\,\quad E_{{N\,\left( {fix} \right)}} = 13.856 + \frac{0.74}{{\left( {1.5} \right)^{2.7761} }} = 14.096\,$$

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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: e₁ = -0.1234, f₁ = 1.5592, g₁ = -0.1052, h₁ = 2.5189, e₂ = -0.17988, f₂ = 1.1796, g₂ = 0.1684, h₂ = 2.1. The computation details are presented for one of the coefficients (say e₁):

  $$\begin{aligned} e_{1} & = - 0.4968 + 1.47\left( {\frac{2}{4}} \right) + 0.2119\left( 4 \right) - 2.386\left( {\frac{2}{4}} \right)^{2} - 0.1799\left( 4 \right)\left( {\frac{2}{4}} \right) \\ & \quad - 0.03523\left( 4 \right)^{2} + 0.5958\left( {\frac{2}{4}} \right)^{3} \, + 0.1747\left( {\frac{2}{4}} \right)^{2} \left( 4 \right) - 0.01014\left( {\frac{2}{4}} \right)\left( 4 \right)^{2} \\ & \quad + 0.00223\left( 4 \right)^{3} = - 0.1234 \\ \end{aligned}$$

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• 4. Computing the regression factors of c_i and d_i through Eqs. (36) and (37):

  $$\begin{aligned} c_{1} & = \exp \left[ { - 0.1234\left( 3 \right) + 1.5592} \right] = 3.284 \\ d_{1} & = - \exp \left[ { - 0.1052\left( 3 \right) + 2.5189} \right] = - 9.055 \\ c_{2} & = \exp \left[ { - 0.17988\left( 3 \right) + 1.1796} \right] = 1.8964 \\ d_{2} & = - \exp \left[ { - 0.1684\left( 3 \right) + 2.10} \right] = - 4.93 \\ \end{aligned}$$

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• 5. Now substituting these factors into the 1^st part of Eqs. (30) and  (31) yields to:

  $$\begin{aligned} c_{1} T_{fix} + d_{1} & = 3.284\left( {1.5} \right) - 9.055 = - 4.129 \\ c_{2} T_{fix} + d_{2} & = 1.8964\left( {1.5} \right) - 4.93 = - 2.085 \\ \end{aligned}$$

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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):

$$\begin{aligned} & Ln\left( {\frac{{E_{{N\,\left( {SSI} \right)}} }}{{\left( {14.096} \right)^{2} }}} \right)\left( {1.48} \right) = - 4.129 \Rightarrow E_{{N\,\left( {SSI} \right)}} = 12.21 \\ & Ln\left( {\frac{{\mu_{{m\,\left( {SSI} \right)}} }}{{\left( {3.85} \right)^{2} }}} \right)\left( {1.48} \right) = - 2.085 \Rightarrow \mu_{{m\,\left( {SSI} \right)}} = 3.63 \\ \end{aligned}$$

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