# Attenuation Zones of Two-Dimensional Periodic Foundations Including the Effect of Vertical Loads
## 1 Introduction  
As a new kind of seismic isolation technique, periodic foundations possess good seismic isolation performance due to the unique dynamic characteristic of attenuation zones (AZs) [1; 2; 3; 4; 5]. Seismic waves with frequencies in the AZs will be attenuated effectively in the periodic foundations [6; 7]. Based on the generation mechanism of AZs, periodic foundations can be divided into Bragg scattering periodic foundations and locally-resonant periodic foundations [8; 9; 10]. The former enjoy AZs where the corresponding wavelength is of the same order of the geometric size of the unit cells of the periodic foundations. The latter possess AZs with the sizes of unit cells two-orders smaller than the relevant wavelength, providing a possibility to isolate seismic waves with dominant frequencies lower than 20 Hz [11; 12].  
The two-dimensional (2D) periodic foundation, which is composed of periodic arrays of heavy core cylinders coated with soft material in stiff matrix, is a typical locally-resonant periodic foundation. [PERSON] and [PERSON] [13] firstly proposed the application of 2D locally-resonant periodic foundations in seismic isolation and systematically the effect of geometric and material parameters on the AZs by using the finite element method. According to the numerical simulation of seismic responses of six-story reinforced concrete frame structures, with a 2D periodic foundation and aseismic rubber bearings, in [13], it is found that the 2D periodic foundation can reduce the seismic response more effectively than the aseismic rubber bearings. Using the finite element method, [PERSON] and [PERSON] [14] investigated theeffect of the reinforcing ratio in a concrete matrix, in which it is shown that the effect of the reinforcing ratio in a concrete matrix on the AZs is negligible. Thus, a reinforced concrete matrix can be simplified as plain concrete to determine AZs. More recently, [PERSON] et al. [15] conducted a free field test of a 2D periodic foundation that was made of periodic arrays of iron coated with rubber in concrete matrix, where it was found that the 2D periodic foundation can effectively mitigate the dynamic response of the upper structure for vibration excitations within its AZs.  
Periodic foundations, as a type of foundation, are required to support vertical loads due to heavy upper structures. According to the previous research in the aseismic rubber bearings, the seismic isolation performance of aseismic rubber bearings can be affected significantly by the vertical loads [16; 17]. However, the effect of vertical loads on AZs of 2D periodic foundations has not been studied, which will lead to the discrepancy between the designed and actual AZs of 2D periodic foundations.  
In this paper, the weak form quadrature element method is applied to study the AZs of 2D periodic foundations, including the effect of vertical loads. It should be pointed out that wave propagation in layered periodic foundations and 2D periodic foundations are different. Longitudinal waves (\(P\) waves) and shear waves (\(SH\) and \(SV\) waves) are decoupled in layered periodic foundations. However, \(P\) waves and \(SV\) waves are coupled in the plane of the cross section of 2D periodic foundations. The remaining sections are organized as follows: The computational methods are given in Section 2. In Section 3, a validation study of the present method is conducted by comparison with existing results in special cases, without the effect of vertical loads. In Section 4, the effects of vertical loads on the range of AZs and the attenuation coefficients are investigated. In addition, dynamic responses of a 2D periodic foundation with finite unit cells are analyzed. Finally, some conclusions are provided in Section 5.