## 1 Introduction  
Flat-plate (FP) structures consist of horizontal reinforced concrete (RC) slabs supported on isolated columns. One version of this type of system is the waffle-flat-plate (WFP) structure, which features RC lightened flat-plates supported on isolated columns. The lightening facilitates achievement of bay lengths of up to 10-12 m with plate depths of 40-45 cm. Further, the bidirectional action of FP or WFP systems under gravity loads provides flexibility in the architectural design of the building; that is, it permits a non-regular layout of columns. These characteristics have led to a widespread use of FP and WFP structures in many earthquake-prone countries, including Spain and Mexico. Nevertheless, FP and WFP structures exhibit some important drawbacks when they are used as the primary system under seismic loadings. Past earthquakes, such as Michoacan (Mexico, 1985) and Northridge (USA,1994), unveiled these drawbacks. As a consequence, worldwide standards have implemented new rules and recommendations for the use of FP and WFP structures. Spanish standard NCSE-02 (2018) establishes a low ductility class for FP and WFP buildings, and limits the ductility factor, \(\mu\), to \(\mu=2\). Eurocode 8 (2018) also proposes a low ductility class (DCL) for FP buildings with a behavior factor, \(q\), of 1.5 and limits their use for low seismicity areas. ACI 318M-14 (2018) indicates that two-way slabs without beam structures can be used only in areas with a moderate--level C--seismic category design (SCD). Despite these limitations, there are experiences for which FP structures exhibit good performance under heavy seismic actions (2018); Kalamata (Greece, 1986); Aegio (Greece, 1995), and Athens (Greece, 1999). Therefore, new research and studies are needed to improve the knowledge of the seismic behavior of this type of structures.  
Research involving WFP structures was limited until the Michoacan earthquake (1985). [PERSON] and [PERSON] (2018) tested a two-story WFP structure designed with seismic codes prior to the Michoacan quake and found it to show low energy dissipation and ductility capacities. [PERSON] et al. (2018, 2019) investigated the seismic behavior of WFP structures with a non-regular layout of columns under unidirectional and bidirectional seismic loadings. They corroborated the high flexibility of this type of system (the yield interstory drift being around 1%) and identified sources that limit the energy dissipation capacity (e.g. brittle torsional failure of the spandrel beams, weak column-strong plate mechanisms in interior columns).  
The use of energy dissipation devices (EDDs) as primary seismic elements has proven effective to control the response of a structure under seismic actions (2018, 2019, 2018, 201  
Past experimental and numerical studies on FP or WFP systems with EDDs entail application of a single horizontal component of the ground motion to the structure. To date, no published studies on the seismic behavior of FPs or WFPs with hysteretic dampers have focused on quantifying their ultimate energy dissipation capacity under bidirectional seismic loadings, in particular. The energy capacity of structural systems is a poorly known aspect of the earthquake engineering field [23]. This research effort is aimed to help fill this gap, testing a WFP system equipped with hysteretic dampers under loading conditions (bidirectional loadings). The different parameters related to the capacity of both the WFP system and the hysteretic dampers in terms of energy are presented and discussed. These findings are particularly useful for designers and researchers involved in the energy-based approach to seismic design.