The analysis of soil–foundation–structure interaction can be carried out by different methods, depending on the part of the system that is examined. These methods can be classified into: (1) analytical, usually referring to simple foundation geometries lying on elastic half-space; (2) semi-analytical, combining analytical formulations for the half-space with numerical procedures; (3) numerical, usually FEM; (4) simplified discrete models, which allow fast calculation of the foundation–soil–structure system properties.
Discrete models for the analysis of soil–foundation–structure system have been developed by various researchers and are the most used for practice purposes. Focusing on pile foundations, according to these methods, the restraining action of soil is simulated by distributed springs and dashpots which substantially result in the evaluation of dynamic impedances for a single pile foundation.
Different formulations can be found in the literature to calculate the dynamic impedances, depending on various parameters among which the frequency of seismic excitation.
The general expression of a dynamic impedance along an arbitrary degree of freedom is given by: k + iΩc, where k and c represent the foundation stiffness and damping, while Ω is the circular excitation frequency.
Gazetas et al. (1993) for the dynamic impedances of a single pile in vertical and lateral directions proposed the following frequency-dependent expressions:
$$k_{z} = 0.6E_{\text{s}} \left( {1 + \frac{1}{2}\sqrt {a_{0} } } \right);$$
(5)
$$c_{z} = 2\xi_{\text{s}} \frac{{k_{z} }}{\varOmega } + \rho_{\text{s}} V_{\text{s}} R\left( {a_{0} } \right)^{ - 1/4} ;$$
(6)
$$k_{x} = 1.2E_{\text{s}} ;$$
(7)
$$c_{x} = 2\xi_{\text{s}} \frac{{k_{x} }}{\varOmega } + 6\rho_{\text{s}} V_{\text{s}} R\left( {a_{0} } \right)^{ - 1/4} ;$$
(8)
where Es is the Young’s modulus of elasticity of soil; ρs is the mass density of soil; ξs is the hysteretic damping coefficient of soil; Vs is the shear wave velocity of soil; R is the radius of the pile transversal section; a0 = Ω R/Vs is a dimensionless frequency parameter.
Gazetas (1984) also proposed approximate expressions not depending on the excitation frequency of soil.
Velestos and Tang (1990), for the calculation of lateral and rocking impedances of a single pile, furnished the following relations:
$$k_{x} = \frac{{8G_{\text{s}} R}}{{2 - \nu_{\text{s}} }}\alpha_{x} ;$$
(9)
$$c_{x} = \frac{{8G_{\text{s}} R}}{{2 - \nu_{\text{s}} }}\beta_{x} \frac{R}{{V_{\text{s}} }};$$
(10)
$$k_{\theta } = \frac{{8G_{\text{s}} R^{3} }}{{3\left( {1 - \nu_{\text{s}} } \right)}}\alpha_{\theta } ;$$
(11)
$$c_{\theta } = \frac{{8G_{\text{s}} R^{3} }}{{3\left( {1 - \nu_{\text{s}} } \right)}}\beta_{\theta } \frac{R}{{V_{\text{s}} }};$$
(12)
where Gs is the shear modulus of elasticity of soil and νs is the Poisson ratio of soil. For νs ~ 1/3, the coefficients α and β are given by:
$$\alpha_{x} = 1;\quad \beta_{x} = 0.65;$$
(13)
$$\alpha_{\theta } = 0.5\frac{{\left( {0.8a_{0} } \right)^{2} }}{{1 + \left( {0.8a_{0} } \right)^{2} }};\quad \beta_{\theta } = 0.4\frac{{\left( {0.8a_{0} } \right)^{2} }}{{1 + \left( {0.8a_{0} } \right)^{2} }}.$$
(14)
Finally, Maravas et al. (2014) proposed the following expressions, not depending from Ω:
$$k_{x} = \frac{{8G_{\text{s}} R}}{{2 - \nu_{\text{s}} }}\lambda_{x} ;$$
(15)
$$c_{x} = \frac{{8G_{\text{s}} R}}{{2 - \nu_{\text{s}} }}\chi_{x} \frac{R}{{V_{\text{s}} }};$$
(16)
$$k_{\theta } = \frac{{8G_{\text{s}} R}}{{2 - \nu_{\text{s}} }}\lambda_{\theta } ;$$
(17)
$$c_{\theta } = \frac{{8G_{\text{s}} R}}{{2 - \nu_{\text{s}} }}\chi_{\theta } \frac{R}{{V_{\text{s}} }}.$$
(18)
The coefficients λ and χ are given by:
$$\lambda_{x} = 1;\quad \chi_{x} = 0.575;$$
(19)
$$\lambda_{\theta } = 0.15;\quad \chi_{\theta } = 0.15.$$
(20)
As to the influence of pile group configuration, in current engineering practice, the dynamic impedances of pile groups are usually estimated using the impedances of a single pile and accounting for the group effect by means of interaction factors (static or dynamic). These group effects, due to the kinematic component of the pile-soil dynamic interaction, are quite small and can be neglected when, as in the case under examination, the ratio Ep/Es (L) between the pile and the soil (at the pile extremity) Young’s moduli of elasticity is \(\ll 1000\) (Gazetas et al. 1993).
Application to the case study
As above underlined, in the proposed study, it is possible to refer to single pile impedances, without accounting for the group effects. With reference to the analyzed spherical pressure vessel, the lateral impedance kx of a single pile was calculated by Eqs. (7), (9) and (15), so obtaining comparable values (~ 571,337.5 N/m). The total number of piles is 40. The period resulting from the simplified model described in Sect. "Fundamental period" is equal to 0.87 s. As a consequence, the consideration of soil–structure interaction would be beneficial in this case, leading to a higher value of natural period and to reduced seismic spectral forces. For this reason, this effect was neglected and the vessel was assumed perfectly constrained at the basis.