Results of decades of research in earthquake engineering have led to the classification of earthquake resisting systems by materials as; steel, concrete, masonry, wood, etc., types of system as; moment frame, bearing wall, building frame, braced frame, cantilever column, dual, etc., and levels of seismic detailing as; ordinary, intermediate, special and nondetailed. Recent studies have resulted in more specific classifications for each material type, e.g., for steel structures as; ordinary concentric, special concentric, buckling restrained and eccentrically braced frames, and special frame/plate shear walls (FEMA 2009). The vast majority of earthquake resisting structures consist of fixed base Moment Frames (MF) combined with fixed base shear walls and/or braced frames. While these dual systems have served their functions rather well, they are not free from technical flaws and socio-economic drawbacks, yet they remain the most popular construction systems worldwide. To address some of these issues, the authors propose a new dual earthquake resisting system consisting of Grade Beam Supported MFs (GBSMF) in combination with Rigid Rocking Cores (RRC) that promises greater utility and futuristic options than its conventional counterparts. The idea that rocking motions may reduce damage to structures during earthquakes in not new and was first observed by Housner (1963), half a century ago. The pioneering work in this field is attributed to Aslam et al. (1980), Ajrab (2000), Ajrab et al. (2004), Panian et al. (2007), Deierlein et al. (2009), Seymour and Laflamme (2011), Wada et al. (2012), and Grigorian et al. (2017). The successful use of RCMFs for new and retrofit projects has been reported by Wada et al. (2009, 2012), Janhunen et al. (2012) and Takeuchi et al. (2015). MacRae et al. (2004) and Ji et al. (2009) have proposed similar paradigms in the prevention of large deformations due to drift concentrations in multistory frames. The theoretical basis of the proposed development together with a number of illustrative examples, have been compiled in Sects. 1–۴ of the current article. Sections 5 and 6 discuss the practical design aspects of the proposed solutions. A brief discussion of the proposed structural system is presented in Sect. 7. The loading and configuration of the example problems have been devised to be as general, informative and as tractable as possible. The results of all numerical examples have been verified through independent computer analysis. No specific code issue is discussed. All symbols are defined as they first appear in the text.