Table 1 reports the variable geometric parameters considered in the numerical analyses, while Figure 1 gives a schematic of the generic rectangular-shaped U-FREI. Four and three different values of the base side in X ( 2 a ) and Y ( 2 b ) direction are considered, respectively; also, two total heights ( H ) and thicknesses of the elastomeric layers ( t e ) complete the variable geometric parameters. With the thickness of the elastomeric and reinforcements layers and the total heights defined in Table 1, four different values of the total number of rubber layers ( n = ( H + t f ) / ( t e + t f ) and of the total rubber height ( t r ) are obtained. Each bearing is loaded in five different horizontal directions, defined by the five angles ( ϑ ) listed in Table 1 and computed counterclockwise from the X axis (Figure 1B). Combination of the variable parameters leads to a total number of 240 Finite Element Models (FEMs), as part of the set presented in (Galano et al., 2021).

The mechanical parameters of each bearing are kept constant in the analyses, their numerical values shown in Table 2.

The primary shape factor ( S 1 = a b / t e ( a + b ) ), is included in the range 3.33–17.6 (Table 1). In this paper, the definitions of base side ( B ( ϑ ) ) and shear strain in the horizontal loading direction ( γ H ϑ = δ H ϑ / t r = δ H X ( ϑ ) + δ H Y ( ϑ ) / t r ) given in (Galano et al., 2021) are used; accordingly, the secondary shape factors in the horizontal loading direction ( S 2 ϑ = B ( ϑ ) / t r ) range from 1.02 to 12.5 (Table 1).

Each bearing was prior subjected to increasing vertical load up to a target vertical pressure (Table 2) and then displaced in the horizontal directions of Table 1 up to γ H ϑ = 100 % . Past this shear strain threshold, the overturning condition due to rollover deformation lead to increasing vertical and horizontal stiffnesses. Thus, no reduction of the bearing capacity of the U-FREIs would be obtained.

The numerical analyses are carried out using MSC Marc (MSC.Software Corporation, 2005), a general-purpose finite element software. The compressible Neo-Hookean hyperelastic material was used to model the elastomer. In this model, the strain energy density function is given by the following equation (MSC.Software Corporation, 2017a): W = C 1 ( I 1 − 3 − 2 ⁡ ln ⁡ J ) + C 2 ( J − 1 ) 2 , (1) Where C 1 and C 2 are material constants, I 1 = λ 1 2 + λ 2 2 + λ 3 2 is the first invariant of the right Cauchy-Green deformation tensor and J = λ 1 λ 2 λ 3 is the determinant of the deformation gradient. As for consistency with linear elasticity C 1 = G / 2 and C 2 = K / 2 ; thus, these constants can be set according to parameters shown in Table 2.

The fiber fabrics were modeled considering a bi-directional reinforcement mesh and using a linear elastic material model whose mechanical parameters are shown in Table 2 (last three columns).

The elastomer has been modeled using an eight node, isoparametric, arbitrary hexahedral element (element 7 in Marc (MSC.Software Corporation, 2017b), while the fiber layer has been modeled using a hollow, isoparametric 4-node membrane reinforced with rebars (element 147 in Marc (MSC.Software Corporation, 2017b). A “touch” type contact between the bearing and the upper and lower surfaces has been modeled, allowing the bearing to detach from the supports during roll-over to reproduce the unbonded condition. The supports are modeled as load-controlled rigid surfaces, while the bearings as deformable body. Based on the finite elements assigned to a contact body, the program will automatically set up the outer boundary of the deformable bodies. Also, the nonpenetration constraints are enforced using augmented Lagrangians.

Figure 2 shows a generic FEM used in the parametric FEA. Additional information on the mesh size can be found in (Galano et al., 2021).

Figure 3 shows the trends of the dimensionless ratio between vertical displacements and total height of the bearing ( δ v ( γ H ϑ ) / H ) with the primary (Figures 3A–C) and secondary shape factors (Figures 3D–F) at three significant levels of shear strain: 1) γ H ϑ = 0 % (i.e. pure compression, Figures 3A,D), 2) γ H ϑ = 50 % (Figures 3B,E) and 3) γ H ϑ = 100 % (Figures 3C,F). In Figure 4, the deformed configurations with contour plots of the vertical displacements, of a bearing with S 1 = 20 and S 2 ( 0 ° , … , 90 ° ) = ( 4.2,4.8,5.9,4.8,4.2 ) are illustrated for the shear strain thresholds of γ H ϑ = 50 % (Figures 4A,C,E,G,I) and γ H ϑ = 100 % (Figures 4B,D,F,H,J).

As expected, the vertical deformation decreases with increasing values of the primary shape factor (Figures 3A–C); an approximately exponential decreasing trends is found. The effect of the shear strain and of the horizontal loading direction can be seen comparing Figures 3A–C. The contact area between bearing and supports reduces with increasing shear strain, thus the vertical deformations increase accordingly. A higher increase is related to the smaller base side of the rectangular bearing ( ϑ ≤ 45 ° ), i.e. smaller primary shape factor; for larger values of the primary shape factor (i.e. S 1 ≥ 15 ) the effect of the horizontal loading direction is negligible.

The vertical deformation appears to be slightly affected by the secondary shape factor (Figures 3D–F) when γ H ϑ ≤ 50 % , as scattered values of vertical deformations in the range 0%–5% are obtained for the same values of S 2 ϑ . When γ H ϑ > 50 % the vertical deformation appears to decrease with an exponential trend, similar to the trend of δ v ( γ H ϑ ) / H with S 1 . This confirms how the secondary shape factor plays a key role on the horizontal rather than vertical deformation of the U-FREI. Larger values of S 2 ϑ ensue stable bearings, thus according to Figure 3F, the vertical deformation tends to increase solely when S 2 ≤ 2.5 . Such threshold value matches what previously found on the stability of U-FREIs both on the vertical and horizontal response (Galano et al., 2021b; Galano et al., 2021; Galano, 2022).

The vertical stiffness as a function of the horizontal deformation of each U-FREI is defined as: K v ( γ H ϑ ) = F v δ v ( γ H ϑ ) . (2)

The trends of K v ( γ H ϑ ) with the primary and secondary shape factors are plotted in Figure 5. Similar to vertical deformations, the vertical stiffness appears to be greatly affected by the primary shape factor (Figures 5A–C), while S 2 ϑ plays a minor role (Figures 5D–F).

For increasing values of shear strain, the vertical deformation increases accordingly (see Section 3.1) and from Eq. 2 the vertical stiffness reduces. However, the coupled horizontal response of both base sides due to bidirectional shear loads may lead to a slightly increase of the vertical stiffness when S 1 ≥ 15 (Figure 5C). Marked reductions of the vertical stiffness with the shear strain are obtained from the FEMs solely when S 1 < 10 (Figures 5B,C).

Starting from Eq. 2, the effective compressive modulus as a function of the horizontal deformation can be obtained as: E c ( γ H ϑ ) = K v ( γ H ϑ ) ⋅ t r A c = F v ⋅ t r δ v ( γ H ϑ ) ⋅ A c . (3)

Figure 6 shows the trends of E c ( γ H ϑ ) with S 1 and S 2 ϑ . The effective compressive modulus appears to depend on the primary shape factor with an increasing linear trend (Figure 6A). With increasing values of the shear strain E c ( γ H ϑ ) generally reduces (see Eq. 3). The horizontal loading directions appears to affect the effective compressive modulus as greater decrease is related to U-FREI loaded along the smaller base side (i.e. ϑ ≤ 30 ° , Figure 6C).

Here again, the secondary shape factor plays a minor role when γ H ϑ ≤ 50 % (Figures 6D,E). When γ H ϑ > 50 % , smaller values of S 2 ϑ (i.e. S 2 ϑ ≤ 2.5 ) correspond to larger reduction of E c ( γ H ϑ ) , while increasing values of the secondary shape factor leads to stable bearings with E c ( γ H ϑ ) almost independent on the horizontal deformation (Figure 6F).

The trends of vertical deformation, vertical stiffness and effective compressive modulus with both primary and secondary shape factors are illustrated in Figure 7. This figure shows how the shape factors affect the whole vertical response of the U-FREIs under combined axial and multi-directional shear loads. In each plot of Figure 7, data fitting with regression surfaces are also proposed.

The surface fitting on the vertical deformation shows the influence of the shape factors with the shear strain. Under pure compression the vertical deformation slightly depends on S 2 ϑ and is greatly affected by S 1 (Figure 7A), while at larger horizontal deformation the influence of S 2 ϑ is greater (Figures 7B,C).

Both the vertical stiffness and the effective compressive modulus assume negative values as S 1 tends to zero, as expected, almost independent on the corresponding values of S 2 ϑ (Figures 7D,G). These two parameters largely increase when S 1 range from 10 to 20. The influence of S 2 ϑ is relevant solely when γ H ϑ > 50 % (Figures 7F,I).

Tables 3, 4 report the percentage reductions of the vertical stiffness with increasing shear strains, in the different horizontal loading directions. In Table 3 the influence of the primary shape factor is highlighted, considering U-FREIs with increasing values of S 1 and an almost constant S 2 ϑ , while in Table 4 variable values of the secondary shape factors in the horizontal displacement directions are considered.

From Table 3 it can be seen how the percentage reductions of K v ( γ H ϑ ) in the generic horizontal loading direction are almost independent on the primary shape factor. Average reductions of the order of 17.4%, 16.9%, 14.9%, 10.2% and 5.12% for γ H = 50 % and of the order of 39.4%, 35.0%, 29.9%, 18.0% and 11.0% for γ H = 100 % are obtained for ϑ = 0 ° , ϑ = 30 ° , ϑ = 45 ° , ϑ = 60 ° and ϑ = 90 ° , respectively. These values prove how greater reductions of the vertical stiffness can be expected when the U-FREI is loaded along the smaller base side, regardless of the value of S 1 . In other words, the main influence on K v ( γ H ϑ ) is related to the secondary shape factor in the horizontal displacement direction. This concept is illustrated in Table 4 in numerical terms. This table shows how, when the bearing is loaded in a generic horizontal direction, the vertical stiffness always reduces with the shear strain according to the secondary shape factor in the same horizontal direction. It is worth mentioning how great percentage reductions of K v ( γ H ϑ ) are obtained when S 2 ϑ < 2.5 , while the little reductions of the same parameter are obtained for S 2 ϑ > 3 .