## 1 Introduction  
Large concrete dams are crucial infrastructures generating backbone energy power and controlling water resource distribution. The earthquake safety of those structures is an important issue, since damaged large dams have an extremely negative impact on social and economic systems across a sizable region, and often those damages are inrecoverable. Especially, the possibility of sliding during an earthquake is important in the safety assessment of a concrete gravity dam, in that sliding can cause the total failure of a dam [1, 2].  
To quantify the chance of sliding as a whole dam structure, the concept of the Sliding Safety Factor (SSF) has been used widely [3, 4]. The SSF of a dam is generally defined as the ratio between the shear strength of the dam-foundation interface and the external shear force applied to the interface. The shear strength consists of the cohesive force, and the multiplication of the vertical force and the coefficient of the internal friction. When the external force exceeds the shear strength, in which the SSF is lower than unity, a dam possibly slides. Since this definition of SSF is straightforward, major dam agents including USACE and USBR have used them to evaluate the sliding stability of their dams [5, 6]. However, the conventional SSF formulation cannot explain the sliding stability along the dam-foundation interface, since it does not take into account the fractured area in the interface [7, 8]. A concrete gravity dam usually has a pre-existing fractured area due to the hydraulic pressure at the heel area and/or past earthquakes [9, 10]. In addition, when a dam is subjected to strong earthquakes, the fracture initiates and gradually propagates at the bottom of a dam [10; 11; 12]. These pre-existing and progressive fractures reduce the resistant shear strength against the external shear force [13; 14; 15; 16]. Hence, if the fractured area is not considered in the SSF evaluation, the shear strength can be exaggerated against sliding, and this would have an adverse impact on the precision of the sliding safety evaluation. The existing conventional SSF is not able to describe these important situations for the sliding stability of a dam.  
To overcome the shortcomings, an improved SSF has been proposed to consider the fractured area in the dam-foundation interface [17; 18], where the resistant shear forces at the undamaged area and the fractured area are determined separately. However, in the improved SSF formulation, the total vertical force is decomposed in proportion to the ratio of those areas for simplicity, which produces inaccurate results in dynamic analysis of a dam. In addition, the effect of fracture progression is not considered important in this improved SSF, as well as in the conventional SSF.  
In this study, a new SSF is suggested to take account of the progressive fractured area at the dam-foundation interface. In order to use the suggested SSF, a contact and sliding model for the dam-foundation system is required to compute the dynamically varying normal forces and sliding motions, which is presented in the context of the finite element method. To investigate the effect of the fractured area progression on the sliding safety evaluation, the conventional, improved, and newly suggested SSFs were computed using the dynamic earthquake analysis results of a concrete gravity dam subjected to two artificial earthquakes. After comparing the SSF values, there is a discussion about the possibility that the conventional SSF significantly overestimates the resistant capacity of a dam against sliding, and about the importance of an appropriate SSF formulation.