## 1 Introduction  
Previous studies conducted over the past two decades have proposed energy dissipation systems that use displacement-dependent dampers and/or velocity-dependent dampers [1]. Currently, a major concern is associated with the need to design a structure with dampers; therefore, many studies have been conducted on this subject [2]. An energy-based design procedure based on achieving a balance between the mean energy dissipated per cycle by a structure and that dissipated by dampers was developed for the seismic retrofitting of existing buildings [3]. By using parameters such as a maximum damper ductility value and an elastic stiffness ratio between the bracing-hysteretic device system and the moment frame system, a design procedure was investigated for frames equipped with a hysteretic energy dissipation device [4; 5]. By using a displacement-based design method, a design procedure was investigated for buildings with hysteretic dampers [6; 7; 8]. To determine the optimal damper volume related to seismic performance, extant studies have used a performance curve (called as performance spectra) for an elastic system [9; 10] and an inelastic system [2; 11; 12; 13] in the design process of buildings with dampers. However, these studies have not considered the turn from an elastic mode vector to an inelastic mode vector caused by the yielding of a main structure. This would lead to the loss of accuracy in a large drift range. On the one hand, nonlinear time history analysis (NTHA) using an elaborate analytical model is used to estimate the seismic performance of buildings with dampers. However, this is time-consuming and it is more convenient to perform a seismic performance assessment. In performance-based seismic design (PBSD), it is essential for the risk associated with the performance level of a structure to be clear and transparent, such that decision makers can understand the expected seismic performance of the structures [14]. Specifically, with respect to PBSD, NTHA using hundreds (or thousands) of ground motion records is required to perform a probabilistic seismic performance assessment of buildings. In recent years, for the seismic performance and probabilistic collapse resistance assessment of buildings with energy dissipation systems, many studies that use incremental dynamic analysis [15] and perform an NTHA using an elaborate analytical model have been conducted [16; 17; 18; 19; 20]. However, NTHA using an elaborate analytical model also requires intensive computations to estimate seismic demands [21]. Therefore, a simplified estimation method of inelastic seismic demands is useful, and thus several estimation methods have been proposed by previous studies [22; 23; 24; 25; 26]. These methodologies mostly focus on estimating the seismic demands of general buildings and involve the use of equivalent single-degree-of-freedom (SDOF) systems converted from a building structure. In most equivalent SDOF systems, a single skeleton curve is adopted to represent the characteristics of a building structure [24; 25; 26]. In order to consider the behavior of dampers, an equivalent SDOF system that includes two springs have been proposed for reinforced concrete buildings with hysteretic dampers by using the results of nonlinear static pushover analysis (NSPA) [27]. However, this model requires interpolation [27], and therefore, the model may necessitate time and effort to approximate the behavior of the springs for the dampers. Simplified Method to Estimate the Peak Inter-Story Drift Ratio of Steel Frames with Hysteretic Dampers  
In this section, a simplified method is proposed to estimate the inter-story drift ratio of a steel frame with steel hysteretic dampers (called steel metallic dampers), such as buckling-restrained brace dampers and added damping and stiffness dampers. The basic concept employed was presented in a previous study [28].