## 1 Introduction  
In modern society, it is preferred that lifelines and tunnels are built underground due to the need for space for construction in urban areas. Damage to these underground structures can lead to a great loss of lives and economic impacts. Therefore, the evaluation of the stability and safety of underground structure systems is essential. Earthquake hazards are the main cause of damage to underground structures. The liquefaction hazard is one of the hazards associated with earthquakes. The buoyancy force and the movement of liquefied soil from soil liquefaction initiate the uplift of underground structures. This uplift deformation could damage the connections between lifelines, causing leaks of hazardous liquids or gases. Moreover, damage to the connections in tunnels could endanger passengers in vehicles or trains. Uplift mechanisms and factors affecting the magnitude of uplift have been studied via model tests and numerical simulations in the past. Several studies [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11] have observed that the magnitude of uplift is closely related to vertical and horizontal input motions. [PERSON] and [PERSON] [8] used a numerical approach (the FLAC program with the UBECSAND model) to explore the effects of horizontal input motion on the magnitude of uplift. Their results showed that the magnitude of uplift and the Arias intensity (the energy of an earthquake) of input motion are positively correlated. Regarding vertical motion, in practice, the effects of vertical motion on uplift are not typically included in uplift predictions [1; 12]. However, several studies have investigated the effects of vertical motion on uplift via numerical simulations. [PERSON] and [PERSON] [2] used a strong-motion record from the Kobe earthquake and sinusoidal motion to study the effects of vertical motion on uplift. The numerical simulation results obtained by [PERSON] and [PERSON] [2] showed that vertical motions cause an oscillation in excess pore pressure compared to cases without vertical motions, but the effects on uplift results are different. The vertical motion from the Kobe earthquake had a minimal effect on uplift magnitude, but the vertical motion of the sinusoidal motion significantly increased the uplift magnitude. [PERSON] and [PERSON] [2] concluded that the effects of vertical motion on uplift depend on the overall characteristics of the motions. [PERSON] et al. [6] used another strong motion record from the Kobe earthquake to study the effects of vertical motion on uplift magnitude. Their numerical simulation results showed that the magnitude of the uplift increases with the amplitude of the vertical motion. The effects of vertical input motions from the Kobe earthquake observed by [PERSON] and [PERSON] [2] differed significantly from those observed by [PERSON] et al. [6]. [PERSON] [9] used the FLAC program with the UBCSAND model to study the effects of the amplitude, frequency, and phase angle of vertical motion on uplift using sinusoidal waves. The simulation results showed that the effects of these characteristics of vertical motion do not present obvious trends with uplift magnitude. The above studies on vertical motion provided insights into the effects of single characteristics (amplitude, frequency, or phase angle) of vertical motion on uplift. However, during the shaking caused by an earthquake, these characteristics occur randomly and interact with each other. Additionally, these studies provide only a few simulation results, which are insufficient for thoroughly exploring and determining the effects of vertical motion on uplift. Therefore, a comprehensive study to explore the overall effects of these characteristics on uplift is essential. In this study, the FLAC program with the PM4 Sand model [13; 14; 15; 16; 17] was selected as the numerical model to study the effects of vertical motion on uplift. This numerical model was first validated and calibrated using the centrifuge test results from [PERSON] and [PERSON] [1].