## 2 Rocking Foundations for Seismic Loading
### Experimental Results  
The experimental results utilized in this study are obtained from a total of nine series of centrifuge and shaking table experiments on rocking shallow foundations conducted at the University of California at Davis, the University of California at San Diego, and the National Technical University of Athens in Greece [30]. The details and major results of these individual test series are available in separate publications [18; 29; 31; 32; 33; 34; 35; 36]. A summary of these experimental results and the effects of key rocking system capacity parameters (e.g., \(A/A_{c}\) and \(C_{r}\)) and earthquake demand parameters (e.g., \(a_{max}\)) on the performance  
Figure 1: Simplified schematic of a rigid structure-foundation system rocking on soil and the major forces acting on it during earthquake loading. parameters of rocking foundations (e.g., AAR), derived from the data obtained from these experiments, are available in [PERSON] et al. (2021) [26]. Altogether, results obtained from 140 experiments on rocking foundations from the abovementioned series of experiments are utilized in this study. Figure 2 presents the variation of AAR with \(a_{max}\) (experimental results) for rocking structure-foundation systems with three different clusters of \(C_{r}\), of which two clusters are for sandy soil foundations and one cluster is for clayey soil foundations. The AAR of rocking foundations are smaller than 1.0 for more than 82% of the experiments considered (for 116 out of 140 experiments), indicating that rocking foundations reduce the accelerations transferred to the structures they support (de-amplifying effect). This effect increases as \(C_{r}\) decreases, indicating that the foundations that have more tendency to rock (smaller \(C_{r}\)) de-amplify the acceleration more. The reduced acceleration demand on the structure during foundation rocking is a consequence of mobilization of bearing capacity and yielding of soil beneath the foundation during rocking. This de-amplifying effect is more pronounced for large amplitude shaking events (greater \(a_{max}\)) than for small shaking events, as soil yielding is even more significant during large amplitude shaking (a beneficial consequence of nonlinear soil-foundation interaction). Note that, for the experiments considered in this study, there is no noticeable difference between the AAR values of sandy and clayey soil foundations as long as the \(a_{max}\) and the range of \(C_{r}\) remain the same. It should be noted that simplified procedures for estimating the peak acceleration demands on traditional fixed-base structures use a trapezoidal distribution, where peak acceleration at roof level could be about 3.0 to 4.0 times the peak ground acceleration of the earthquake [12]. For example, the American Society of Civil Engineers' document Minimum Design Loads for Buildings and Other Structures (ASCE 7-16) indicates that the floor acceleration amplification factor can be as high as 3.0 at roof level [37], while the National Earthquake Hazards Reductions Program's (NEHRP) Building Seismic Safety Council (BSSC) indicates that the floor acceleration amplification factor can be as high as 4.0 at roof level [38]. For the purpose of comparison, the abovementioned values correspond to an equivalent AAR of 3.0 to 4.0. This clearly shows that rocking foundations are much more efficient in de-amplifying the accelerations transferred to the structures during seismic loading (AAR < 1.5 for the vast majority of the experiments plotted in Figure 2). Figure 3 plots the experimental results of AAR against \(C_{r}/a_{max}\) for all 140 rocking foundation experiments considered in this study (the same data plotted in Figure 2). Also,  
Figure 2: Results obtained from 140 centrifuge and shaking table experiments: Variation of AAR with \(a_{max}\) for rocking structure–foundation system with three clusters of rocking coefficient (\(C_{r}\)).