# Reliability Analysis of Response-Controlled Buildings Using Fragility Curves
## 1 Introduction  
Earthquakes pose serious threats to life and infrastructure. The experience and knowledge gained through these events have improved our understanding of how to manage, mitigate and work towards the prevention of similar catastrophes. To reduce the impact of earthquakes on people and property, response-controlled systems are an advanced practice in managing the consequences of such disasters. In Japan, seismic response-controlled systems have been applied to almost all of the high-rise buildings constructed in the last several years in order to improve their structural safety and decrease damage sustained during seismic excitations [1]. Furthermore, these systems have been applied to rehabilitate the seismic resistance of existing structures. Different response-controlled techniques are presented, such as seismic isolation, dampers, and so on [2]. Moreover, researchers have combined different types of dampers to improve the seismic performance of structures. It is necessary to justify the adoption of a response-controlled system to building officials, owners, and users in terms of seismic performance during higher-intensity earthquakes. There have been several studies on the performance of dampers and structures equipped with dampers. Hysteresis dampers, oil dampers, viscous dampers, and viscoelastic dampers are four major types of energy dissipation device [3]. The Japan Society of Seismic Isolation (JSSI) guidebook [4] for the design, fabrication, testing, quality control, and analytical modeling of various passive control systems was issued to cover the mainconcerns at all stages of design, manufacture, and construction of the above four major passive dampers. To improve the damping, stiffness, and strength characteressistics of the buildings, passive energy dissipation systems make use of a variety of materials and technologies. The dissipation may be achieved either by transferring energy into vibrating modes or by the conversion of kinetic energy into heat. The first mechanism consists of devices involving deformation of viscoelastic solids or fluids and those employing fluid orificing, while the latter group includes devices that operate on principles such as yielding of metals and frictional sliding. A third classification consists of re-centering devices that use either a preload generated by fluid pressurization or internal springs, or a phase transformation to produce a modified force-displacement response that includes a natural re-centering component [5]. [PERSON] et al. [6] proposed a combination of various alternative energy dissipation elements (hysteretic, viscous, or visco-elastoplastic) in series and/or in parallel to self-centering elements and called it an advanced flag-shaped (AFS) system. They compared the seismic performance of AFS systems with that of the conventional system using a set of four single-degree of freedom systems under a suite of near-fault and far-field ground motions by performing nonlinear dynamic analyses. [PERSON] et al. [7] compared the seismic performance of an X-shaped metallic damper (XMD) and a fluid viscous damper (FVD) by analyzing five-, eight-, and ten-story reinforced concrete buildings without dampers, with XMD, and with FVD under eight different earthquake ground motions. They also discussed how the locations of the dampers affected the seismic response of the structure. To examine the performance of a structure with passive dampers, seismic fragility analysis is commonly used, where the probability of reaching or exceeding a specified limit state of damage measure (DM) is calculated as a function of a specified intensity measure (IM) in a structure. For a specific limit state of damage, several IMs can be achieved under multiple earthquake excitations, and collectively they are referred to as the multi-recorded IM cluster. The IM cluster is a random function, and the average and standard deviation can be obtained by calculating the mathematical characteristic values of multiple IM values.