<p>This study investigates the cyclic behavior of the X-Shape Damper (XSD). The parametric model was first calibrated using the experimental sample of XSD. Afterward, several parametric studies were carried out to investigate the effects of the width, height, and middle width of the XSD on various parameters, including effective and elastic stiffness, energy dissipation, and equivalent viscous damping ratio (EVDR). The purpose of these studies was to develop approximation formulas for computing these seismic damper parameters. In the second part of the study, the impact of axial force on the seismic parameters of XSD was also examined. Initially, the buckling load of the damper was calculated analytically, and an approximate equation was presented for easier estimation. The effect of axial force on XSD parameters such as elastic stiffness, ultimate strength, and ductility was then investigated. Parametric studies demonstrated that increasing the width and middle width while decreasing the height improved energy dissipation, effective stiffness, and EVDR. The approximate equations can estimate the seismic parameters of the damper with acceptable accuracy, making them useful for damper design. Furthermore, increasing axial force resulted in a decrease in elastic stiffness, yield strength, and ductility; the greatest effect of axial force was on ductility.</p>