## 1 Introduction  
The fundamental principle of seismic base isolation is to modify the structural response by introducing a \"soft layer\" made of isolation bearings able to decouple the soil motion from that of the structure above, which almost behaves rigidly above the isolation system [1, 2, 3]. Consequently, the behavior of isolation devices plays a fundamental role in the seismic response of base-isolated structures, and the variability of their properties during the service life significantly influences the seismic response of both the isolation system itself and the superstructure. Thus, this variability cannot be neglected in the analysis and design of base isolated structures and must be properly accounted for to increase the seismic reliability of base-isolated buildings, especially the strategic ones, such as schools and hospitals, whose diffusion is more and more increasing worldwide [4; 5; 6]. Different sources of variability can be identified, from environmental conditions (such as temperature, ageing, etc.) to behavioral effects specific of each device typology. For example, for high damping rubber bearings (HDRBs), which are the focus of this paper, the behavior significantly depends on the strain amplitude, the strain rate and the load history (i.e., on the maximum strain amplitude attained and the number of experienced cycles) [7; 8]. This last effect results in a change in the device properties with cycles, and it is known as the Mullins effect [9] or \"scragging\" [7; 8; 10; 11]. For other kinds of bearings, such as lead rubber bearings (LRBs) or concave curved sliders (CSSs), the behavior significantly depends on the temperature increment during the load history, i.e., the heating effect [12; 13]. Other critical aspects influencing the performance of sliding isolation devices are, for example, the vertical pressure, the sliding velocity affecting the friction coefficient [12] and the breakaway effect [14]. Different papers were recently published on the importance of these effects on the actual behavior of base-isolated buildings [15; 16], as well as for other supplemental damping systems, e.g., [17]. The international seismic codes all agree in prescribing that the design process cannot be performed by only considering nominal properties of isolation bearings. However, in some cases, such codes provide only some general information or prescribe simplified approaches that are not always complete or clear, making difficult, de facto, the application of such prescriptions. In more detail, focusing on the European context and on HDRBs, Eurocode 8 for buildings [18] only specifies that the values of physical and mechanical properties of the isolation system should be the most unfavorable ones that can be attained during the lifetime of the structure. Both environmental and behavioral effects should be considered, where relevant. Eurocode 8 for bridges [19] gives more operative indications by proposing a simplified approach, described in Annex J, based on upper and lower bound coefficients to obtain upper bound design properties (UBDPs) and lower bound design properties (LBDPs) from nominal design properties (NDPs). Consequently, two analyses shall be performed: one using the UBDPs (generally leading to maximum forces and accelerations acting on the superstructure) and another using the LBDPs (generally leading to maximum displacements of the isolation system). This code also specifies that in absence of relevant test results, the upper and lower bound coefficients (also called \(\lambda\)-factors) given in Annex J may be assumed. The same approach is proposed by the European standard on anti-seismic devices (EN 15129) [20], which is a product standard mainly focused on qualification and control tests that also includes some design indications.