Structural strength and stiffness were previously investigated to sufficiently improve the lateral load resistance against major events. Many buildings require appropriate design to effectively withstand the lateral seismic loads and reduce the corresponding damages. Design methodologies and structural elements were recently introduced to improve the energy dissipation capability and limit the high force demands under seismic loadings. The new systems are designed to protect the structural integrity and concentrate the inelasticity in a specific area, while the remaining parts are kept undamaged and intact. This study introduces a new structural system with dampers having strategic cutouts, leaving butterfly-shaped shear dampers for dominating the yielding mechanism over other brittle limit states. The new system is designed for re-establishing the conventional eccentrically braced frame system with simple linking beams. The system with strategic cutouts is subsequently used and compared with the eccentrically braced frames (EBF) system for seismic performance investigation and incremental dynamic analysis (IDA), using the OpenSees program, which is used to indicate the collapse behavior under forty-four selected ground motions. Results show that the butterfly-shaped multi-story buildings, compared to the corresponding conventional systems, are capable of enhancing the system resistance against lateral seismic loads by postponing the collapse state to the larger drift ratio values.