---

In this paper, a procedure has been presented to develop fragility curves of structures equipped with optimal variable damping or stiffness semi-active tuned mass dampers (SATMDs). To determine proper variable damping or stiffness of semi-active devices in each time step, instantaneous optimal control algorithm with clipped control concept has been used. Optimal SATMDs have been designed based on minimization of maximum inter-story drift of nonlinear structure which genetic algorithm(GA) has been used to solve the optimization problem. For numerical analysis, a nonlinear eight-story shear building with bilinear hysteresis material behavior has been used. Fragility curves for the structure equipped with optimal variable damping and stiffness SATMDs have been developed for different performance levels and compared with that of uncontrolled structure as well as structure controlled using passive TMD. Numerical analysis has shown that for most range of intensity measure optimal SATMDs have been effective in enhancement of the seismic fragility of the nonlinear structures which the improvement has been more than passive TMDs. Also, it has been found that, although variable stiffness SATMD shows to be more reliable in lower mass ratios, however in higher mass ratios variable stiffness and damping SATMDs performs similarly to improve reliability of the structure.

---

Active tuned mass damper (ATMD) systems have attracted the considerable attention of researchers for protecting buildings subjected to earthquake loading. This paper presentes the development of an optimal sliding mode control (OSMC) system for a building equipped with ATMD. In the OSMC technique, a linear sliding surface is used with the slope of this surface designed such that a given (or desired) cost function is minimized. The design is obtained by transforming the system into the regular form. In the regular form, the system is divided into two subsystems inclding: a control term explicitly appears, and other control terms do not appear. In order to demonstrate the capability of the OSMC system, an 11–story realistic building with a TMD installed on the top story of the structure is considered. For achieving this purpose, four well–known earthquake records are selected to evaluate the performance of the OSMC system. Results show that the OSMC technique performs better than other control techniques in the reduction of seismic responses of the structure.

---

In this paper, a procedure has been introduced to the multi-objective optimal design of semi-active tuned mass dampers (SATMDs) with variable stiffness for nonlinear structures considering soil-structure interaction under multiple earthquakes. Three bi-objective optimization problems have been defined by considering the mean of maximum inter-story drift as safety criterion of structural components, absolute acceleration as the criterion of occupants’ convenience, and safety of non-structural acceleration sensitive components, as well as SATMD relative displacement as the cost criterion of the control device. The parameters of the weighting matrices of the instantaneous optimal control algorithm and the maximum and minimum level of variable stiffness of the semi-active device have been considered as design variables. An improved version of the non-dominated sorting genetic algorithm (NSGA-II), has been employed to solve the optimization problems and figure out the set of Pareto optimal solutions. SATMDs with different mass ratios have been designed for an eight-story shear type building with bilinear elasto-plastic stiffness model where the soil-structure interaction has been incorporated by Cone model with three degrees of freedom for the soil. Results show the capability and simplicity of the proposed procedure to design SATMDs considering multiple performance criteria. It is observed that this procedure can offer a wide range of optimal solutions throughout the Pareto front which can be chosen by the designer based on desired performance and application of the structure.