The severe social and economic impacts of recent earthquakes have inspired a growing interest in smart structural systems that offer immediate post-disaster occupancy. Post-tensioned rocking frames are emerging damage-avoiding seismic-resistant structures that employ rocking joints at member connections (to avoid major damage to primary structural elements) and unbonded post-tensioned strands (to provide self-centring capability). Nevertheless, currently available passive load-resisting systems to control the peak structural responses in steel rocking frames rely on sacrificial yielding components that accumulate damage during strong dynamic action. This results in a system with limited durability and a requirement for regular maintenance throughout the building's lifetime. By contrast, the recently proposed Buckling-Enabled Composite Bracing (BECB) elements can provide a thorough damage-avoidance solution by means of carefully controlled elastic buckling behaviour. In these systems, compression-only elements with circular-arc-shaped cross-sections are incorporated into steel rocking frames as lattice bracing in order to improve their dynamic performance. The proposed system has been shown to perform successfully under static loading and discrete sine-sweep ground motions for single-storey rocking frames. This further examines this innovative concept by performing numerical investigations on three-storey four-bay post-tensioned steel rocking buildings under real earthquake ground motions. The performances of conventional moment frames (MRFs) and their rocking frame counterparts (RFs) with and without BECB elements are compared through numerical simulations. Glass-fibre reinforced polymer (GFRP) is selected for the BECB elements. Static Pushover, Discrete Sine-sweep and Incremental Dynamic (IDA) analyses are performed to evaluate the buildings' performances. Damage measures investigated include maximum inter-storey drifts and floor accelerations. It is demonstrated that BECB members enhance the dynamic response of steel rocking frames by significantly reducing maximum storey drifts and accelerations.