Structural weakening and addition of damping is an approach previously proposed for the reduction of seismic forces and drifts in the retrofit of structures. It is also used in the design of new buildings with damping systems. While this approach is efficient, it does not significantly reduce and may even amplify inelastic excursions and permanent deformations of the structural system during a seismic event. This paper describes a negative stiffness device (NSD) that can emulate weakening of the structural system without inelastic excursions and permanent deformations. The NSD simulates yielding by engaging at a prescribed displacement and by applying a force at its installation level that opposes the structural restoring force. The NSD consists of (a) a self-contained highly compressed spring in a double negative stiffness magnification mechanism; and (b) a gap spring assembly (GSA) mechanism which delays the engagement of negative stiffness until the structural system undergoes a prescribed displacement. The NSD employs double chevron braces that self-contain the large vertical forces needed for the development of the horizontal negative stiffness without transferring these forces to the structure. This paper reports the development and operation of the NSD and presents analytical and computational tools that describe the behavior of the device. The principles of global control of structures using the NSD are presented in a companion paper. Additional papers present the results of testing of the device, and the results of analytical and experimental studies on the application of the device in a three-story conventional structure and a three-story seismically isolated structure. (C) 2013 American Society of Civil Engineers.