## 1 Introduction  
Fiber-reinforced elastomeric isolators (FREIs) are made by bonding together layers of elastomer and fiber fabrics [1]. These devices were first used as bridge bearings [2], and then proposed as low-cost seismic devices as alternatives to the more common steel-reinforced elastomeric isolators (SREIs) [3, 4]. Several studies demonstrated the advantages of using fiber reinforcements in lieu of steel sheets as the fiber fabrics are lighter than steel [5] and allow for an affordable and rapid manufacturing process (i.e., cold vulcanization) [6]. Although FREIs could be used both in bonded (bonded fiber-reinforced elastomeric isolators, B-FREIs) and unbonded (unbonded fiber-reinforced elastomeric isolators, U-FREIs) applications [7], they are generally adopted in an unbonded configuration. When unbonded, fiber-reinforced bearings under shear loads experience a rollover deformation as the top and bottom surfaces of the devices detach from the supports [8]. The rollover continues until the initial vertical surfaces of the FREI gradually become horizontal and touch the supports, resulting in the full rollover condition [9]. Complex hysteretic behavior of FREIs under lateral loading has been recently addressed by different authors [10, 11, 12]. The rollover deformation reduces the horizontal stiffness of the device, increasing the efficiency of the isolation system [13]. However, the horizontal stiffness needs to be positive throughout the horizontal deformation in order for the bearing to be stable (stable unbonded fiber-reinforced elastomeric isolators, SU-FREIs); unstable bearings (unstable unbonded fiber-reinforced elastomeric isolators, UU-FREIs) show softening at large lateral deformation prior to full rollover [14; 15]. The stability of the U-FREIs depends mostly on the secondary shape factor \(S_{2}\), defined as the ratio between the base side in the direction of the horizontal displacement and the total rubber height [16]. Secondary shape factors greater than 2.5 were seen to ensure a stable response to U-FREIs under combined axial and both uni- and bi-directional shear loads [17; 18; 19; 20]. Full-scale FREIs have been recently tested under combined vertical and horizontal uni- and bi-directional motion with different levels of axial pressure [21; 22]. Recent studies also demonstrated FREIs being a valuable option for reduction of seismic risk in unreinforced masonry buildings [23; 24; 25]. In order to further reduce both the environmental impact and the cost of FREIs, reclaimed rubber compounds are under development to replace virgin compounds [26; 27]. During the horizontal deformation, the overlap area between upper and lower surfaces reduces, both in bonded and unbonded configurations [28; 29]. Thus, the vertical response of the bearing depends on the horizontal displacement as recently addressed by shaking table tests, hybrid simulations and finite element analyses [30; 31; 32]. Analytical solutions for the vertical stiffness or the effective compressive modulus were proposed for bearings under pure compression [33; 34; 35; 36; 37; 38; 39]. In this condition, the vertical responses of the bearings is greatly affected by the primary shape factor \(S_{1}\), defined as the ratio of the loaded area to the lateral free-to-bulge area of the single elastomeric layer. To the best of the authors' knowledge, analytical studies that consider the variation of the axial stiffness as a function of the imposed lateral deformation were developed solely for bearings with rigid reinforcements [40]. This paper studies the vertical stiffness of strip-shaped U-FREIs under combined axial and shear loads. A closed-form analytical solution for the vertical stiffness as a function of the horizontal displacement is proposed. The influence of the main geometric and mechanical parameters of the bearings, such as the primary and secondary shape factors, as well as the shear modulus of the rubber, are studied. Finally, a parametric finite element analysis is used to validate the proposed analytical solution.