## 1 Introduction  
Existing seismic damages have shown that soil sites significantly affect the seismic response of structures [1, 2, 3]. Since soil sites generally have layered characteristics, the one-dimensional (1D) equivalent linearization [4] is the most commonly used method for seismic response analysis [5, 6, 7, 8]. However, [PERSON] et al. [9] and [PERSON] [10] analyzed the actual seismic records and found that ground motions at ridges were much larger than at the mountain foot. [PERSON] et al. [11] compared 1D and two-dimensional (2D) seismic responses of three representative cross-sections of the Santiago Basin, Chile, and the results showed that 1D simulations cannot capture the basin edge effects in the fine-grained soils. Therefore, although the soil sites are layered, two-dimensional or three-dimensional (3D) seismic response analysis methods need to be adopted to analyze the seismic responses [12, 13, 14]. Due to the complexity of soil constitutive models, the equivalent linearization method is often used for seismic response analysis. The computational process for 2D and 1D equivalent linearization is the same, with linear calculations performed at each iteration [15, 16]. Sincethe finite element model of the 2D site is enormous, it is of great importance to improve the computational efficiency of the linear seismic response of each iteration. The frequency domain method and the time domain method are often used to analyze the linear seismic response analysis of sites. In the frequency domain method, it is necessary to solve the complex algebraic equation to obtain the response at each exciting frequency [17; 18]. Solving complex algebraic equations is relatively time-consuming. The time domain method includes direct integration methods and modal superposition methods. The direct integration method [19] is convenient for linear and nonlinear seismic response analysis of the site [20; 21]. [PERSON] and [PERSON] [22] used the FLAC-2D program to analyze the 2D nonlinear site response. The modal superposition method decouples the coupled motion equations into a series of single-degree-of-freedom motion equations. Because only a few low-mode dynamic responses are required, it has high computational efficiency and has become the most common method for dynamic response analysis of linear elastic systems. However, modal truncation [23] is an important issue in the modal superposition method for site seismic response. [PERSON] [24] proposed the static load participation ratio and dynamic load participation ratio to be the index of modal truncation. For seismic excitation, the dynamic load participation ratio index is the same as the modal participation mass ratio. [PERSON] [25] proposed the modal contribution factor to estimate the modal truncation error. [PERSON] et al. [26] compared structural response errors by the use of different modal truncation indexes. The calculation results showed that modal truncation based on the modal contribution factor can effectively control the error of displacement. However, two concerns are raised by the use of the modal contribution factor for the site seismic response, as follows: (1) The modal truncation based on the displacement response leads to a larger calculation error of the acceleration response because the higher modes have a greater impact on the structural acceleration response than on the displacement response; (2) In the iterative calculation process, displacement time history and its corresponding maximum strains are used to determine the convergence of the iteration, while the result generated is the acceleration response. In this paper, to establish a calculation method for site acceleration response under seismic excitations using the modal superposition method, the concept of modal contribution factor of acceleration is proposed for the modal truncation index in acceleration response analysis.