Isolation devices are designed to uncouple the movement of the structure from the underlying ground, and this attribute is granted by their low horizontal stiffness. The most commonly used isolation devices are elastomeric and sliding bearings, such as curved surface sliders (CSSs) (Zayas et al., 1987), which are being increasingly used in the seismic isolation systems of buildings and bridges.

Double concave curved surface slider (DCCSS) isolation bearings consist of two facing concave stainless-steel surfaces separated by an inner rigid slider characterized by Φ s diameter and h s height (as shown in Figure 1A). The forces acting on the slider (see Figure 1B) are the vertical load W , the resultant forces of normal pressure acting on the sliding interfaces N , the lateral force F , and the friction force S , acting along the curved surfaces (Mostaghel and Davis, 1997). The radius of curvature and the friction coefficient are the main key parameters. Compared with traditional CSSs, the main advantage of DCCSS bearings is their lower cost due to their more compact size.

Previous studies have proposed a variety of numerical modelling and experimental tests for slider isolators (Constantinou et al., 1990; Almazán and De La LLera 1998, 2011; Fenz and Constantinou 2006; Fenz and Constantinou 2008a; 2008b; Becker and Mahin 2012a; 2012b; Lomiento et al., 2013; Sarlis and Constantinou 2016; Ponzo et al., 2017, 2019, 2020, 2021; De Domenico et al., 2018, 2019; Pavese et al., 2018; Di Cesare et al., 2019, 2021; Pigouni et al., 2019; Quaglini et al., 2012, 2019; Furinghetti et al., 2020) that also involve geometrical compatibility and multibody kinematics formulations (Belfiore et al., 2000; Shabana 2001; Tsai et al., 2005; Popov 2010; Mazza et al., 2017; Nikravesh 2018; Bianco et al., 2020, 2021). Additionally, the behaviour of single and multiple concave surface sliding bearings has been analytically characterized by Sarlis Constantinou (2013).

The analytical description of the DCCSS bearings characterized by the same radii of curvature R 1 = R 2 = R and same coefficients of friction μ 1 = μ 2 = μ is shown below. Angles ϑ i are formed by the lines connecting the centre of curvature of the concave plates (“1” for the bottom concave plate [BCP] and “2” for the top concave plate [TCP]) and the central point of the contact area between slider and plate, while angles ϑ s , i form between the latter of these points and the points of application of the resultant forces at the surface of the slider. Angles are considered to be small so that the normal components of force are equal to the applied load W (see Figure 1B). ϑ 1 = d R 1 − h s (1)

Imposing the constraint of TCP being horizontal during motion, the angles can be represented as: ϑ s 1 = ϑ s 2 = ϑ s (2) ϑ 1 = ϑ 2 = ϑ (3)

The resultant force N must be eccentric to satisfy moment equilibrium; accordingly, the pressure distribution on the sliding interface is not uniform. The equilibrium of moments for the slider around A (Eq. 4a) and (3horizontal) equilibrium of forces for the TCP (Eq. 4b) are as follows: W ϑ s R + ϑ s R − W ϑ s h s − S h s = 0   a W ϑ + ϑ s + S = F   b (4)

From Eq. 4 and (3the) value of angle ϑ s can be expressed as follows: ϑ s = μ h s R e f f (5)

Substituting Eq. 5 into Eq. 4 results in the following: F = μ W 1 + μ h s R e f f + W ϑ (6)

Considering that the displacement of the TCP is given by d = R − h s ϑ + R ϑ and that R e f f = R 1 + R 2 − h s = 2 R − h s , when substituting this into Eq. 6, the force-displacement relationship of the bearing is: F = W 2 μ R R e f f + W d R e f f (7)

Further details about this equation can be found in Sarlis and Constantinou (2013, Sarlis and Constantinou (2016).

During ground motions with intensities higher than those in simulations, the rigid slider in DCCSSs with flat rims and lacking elements can shift beyond the geometrical capacity of the housing plates in the so-called over-stroke regime (see Figure 2). Over-stroke displacement capacity is a crucial element that can reduce the annual frequency of the collapse displacement being exceed and improve the seismic resilience of a structure that is isolated with these bearings (Di Cesare et al., 2021). However, the mechanical description of this phenomenon has yet to be explored. This paper focuses on the development of an analytical model, based on fundamental mechanical principles, to describe the over-stroke behaviour of DCCSS devices and define their actual limit displacement.

Following a report by Bao et al. (2017), the envisioned mode of behaviour in the over-stroke regime and the corresponding multi-body kinematic were calibrated in ad hoc experimental tests conducted on devices pushing the horizontal displacement beyond geometrical capability, revealing how the displacement limit better preserves the ability to support vertical loads (Furinghetti et al., 2021a; Di Cesare et al., 2021). The objective of this was to propose an easy tool for predicting limit displacement and the corresponding shear force, starting from the geometrical and mechanical characteristics of the devices.

The proposed formulation was applied in a case study of a six-storey RC-frame-isolated building. A multi-degree of freedom (MDoF) model was implemented for non-linear dynamic analyses that considered three sets of 20 horizontal earthquakes characterized by intensity values around the collapse limit state (CLS) design spectrum. The comparison between the non-linear numerical results accounting for the friction dependencies from variability in the dynamic condition of velocity and axial load, and the proposed algebraic solution defined in the static condition, accurately predicted of over-stroke force and displacement.

In the case of DCCSS bearings with a flat rim without restraining elements, the limit displacement d lim was able to correspond to the slider overturning around its centre of rotation ( d o t displacement) beyond the geometrical capacity d C = 2 u if failure for maximum contact pressure was avoided ( d p displacement) (see Figure 3, Figure 4B).

The assumptions of this study were that the bottom and top concave plates are characterized by the same radii of curvature and the same coefficients of friction ( R 1 = R 2 and μ 1 = μ 2 ), and the outcome is that the sliding occurs simultaneously on both surfaces. Two main sliding regimes were considered for horizontal displacement d , which were as follows:

-within the geometric capacity (Regime I): 0 < d < d c , see Eq. 7

The maximum shear force developed by the device and transferred from the superstructure to the foundation was defined as force F lim = F d lim at the limit displacement d lim (see Eq. 8).

Controlled displacement tests on DCCSS bearings were performed to investigate the actual response of the device when the sliding displacement exceeds the displacement capacity and runs over the sliding surfaces (Di Cesare et al., 2019; Ponzo et al., 2020). The main characteristics of the DCCSS specimen and the experimental testing results are summarized in Table 1 (Di Cesare et al., 2021). The testing protocol consisted of one cycle at constant velocity v = 2.5   m m / sec and constant vertical load W (Table 1), performed with a ‘triangular-shaped’ controlled displacement. The maximum displacement reached during the test d exp was more than two times the geometric capacity displacement d c ( d exp / d c ≈ 2.1 ).

Experimental results showed that when the rigid slider overcomes the concave sliding surface edge (position 1, Figure 3A), it can move up to half of its diameter (position 3, Figure 3A), resulting in a slight increase in force (black line in Figure 3B). In the loading and unloading phases of the first cycle, DCCSS over-stroke behaviour was characterized by a ‘sloping dog bone’ shape, preserving re-centring capability and the ability to support the gravity load. A sudden increase in horizontal force and local equivalent stiffness occured as the sliding pad travelled beyond the housing plate edge (Figure 3C). From a mechanics point of view, this behaviour may have occurred due to an increase in the friction coefficient ∆ μ = 0.015 as the sliding pad came into contact with the small edge of the housing plate sliding material (counterbore gap material). The test was stopped at displacement d p when the maximum contact pressure of the sliding material was reached (Figure 3C); polytetrafluoroethylene (PTFE)-based sliding material with an allowable contact pressure of σ a = 60   MPa was assumed based on experimental observations. Only a small amount of degradation of the inner slider PTFE surfaces, and zero isolator damage, were observed after one cycle of sliding above the edge of the device.

In the over-stroke stage of motion (Regime II), the previously exposed Eq. 7 could be properly modified to take an increased friction coefficient μ 2 = μ + ∆ μ into account, as follows: F = W 2 μ 2 R R e f f + W d R e f f (8)

Theoretically, in the over-stroke regime, the failure mechanism linked to the bearing kinematics occurs when overturning displacement d o t is reached (position 3 in Figure 3A, and black dashed line in Figure 3B). Referring to Figure 4 the d o t is identified as the displacement corresponding to the alignment of the TCP and the BCP opposite housing plate edges with the centre of rotation of the slider, and with the centres of curvature of both plates (Figure 4B), defined by the overturning angle ϑ o t and expressed as follows: ϑ o t = arcsin u + Φ s 2 / R (9) d o t = 2 u + Φ s − h s ∙ sin ⁡   ϑ o t (10)

Then, if geometric capacity displacement d c = 2 u and Eq. 9 are substituted into Eq. 10, the final expression for the overturning displacement can be written as follows: d o t = R e f f 2 R d c + Φ s (11)

It should be noted that the overturning displacement is often not implemented as a failure condition for DCCSSs because the failure of maximum contact pressure on the reduced slider area occurs for a generally shorter displacement d p < d o t . Displacement d p , associated with the attainment of the limit contact σ a = W / A e f f on the sliding interface between the rigid slider and concave plate, is shown in the scheme of Figure 4C. A e f f is the effective contact area between the slider and housing plate, and is equal to the slider surface reduced by the area of the slider external to the sliding A e x t A e x t , as follows: A e f f = π Φ s 2 2 − A e x t (12)

The external area A e x t formulation, using simple geometric considerations, is: A e x t = 1 2 Φ s 2 2 β − c ∙ r b c 2 (13) where β is the angle described by the connection of the intersection point between the slider and the sliding pad circumferences, c is the circumference chord that connects these intersection points, and r b c is the distance between the sliding pad border and the slider centre (see Figure 4C), represented as follows: β = 2 ∗ arcsin c / 2 Φ s / 2 (14) c = 2 Φ s 2 2 − r b c 2 (15) r b c = d c + Φ s 2 − d p (16)

Substituting Eqs 12–15 into Eq. 16, the displacement d p = d σ = σ a associated with the limit contact pressure is expressed as: d p = 2   W c   σ a − 2 c Φ s 2 2 π − arcsin c Φ s +   Φ s 2 + d c (17)

The proposed analytical formulations for sliding in the over-stroke regime (Regime II) is reported below, together with the basic formulation for general sliding conditions (Regime I), as set out by (Sarlis and Constantinou., 2013; Sarlis and Constantinou., 2016).

The backbone curve of the force-displacement relationship is shown in Figure 5. Compared with another recent study (Furinghetti et al., 2021b), the analytical formulation of the over-stroke stage of motion and the definition of the domain of application (Eqs 8, 17) are novel to this study.

The proposed formulation has been compared with a few over-stroke tests described by Furinghetti et al. (2021b) in which a DCCSS specimen characterized by an effective radius of curvature of R e f f = 3080 m m , a slider diameter of Φ s = 260 m m , and a geometric capacity displacement of d c = ± 275 m m was tested under constant velocity v = 2.5   m m / sec and constant vertical load W , which were applied to the test apparatus, as shown in Figure 6. Tested DCCSSs were characterized by three different sliding materials: graded polytetrafluoroethylene (PTFE) filled with carbon fibres ( μ = 0.05 ) Figure 6A); ultra low-density polyethylene ( μ = 0.03 ) Figure 6B); and virgin PTFE ( μ = 0.01 ) Figure 6C). The sliding materials implemented into the adopted device have been tested at an average contact pressure of σ = 45   M P a (Furinghetti et al., 2021a). The ratio between the maximum displacement reached during the tests and the geometric capacity displacement was d exp / d c ≈ 1.5 . The experimental results showed that in the over-stroke regime, the friction coefficient increases by μ = 0.015 for all tests. Experimental force-displacement results have been replicated using the proposed formulation Figure, which shows that the analytical model is capable of representing DCCSS behaviour during tests with a significantly good approximation. For the experimental cases, the limit displacement can be calculated using the proposed formulation d lim = min d o t = 515   m m ;   d p = 415   m m . For the sake of safety, in these experimental tests, the isolators were excited at maximum displacements less than d lim , as predicted by analysis. It is worth noting that the experimental tests (Furinghetti et al., 2021a; Di Cesare et al., 2021) were performed assuming a low velocity and constant axial force, so the dependency of the extra-stroke parameters on higher sliding velocities and different contact pressure values still needs to be investigated.

A case study prototype structure was selected for the application of the proposed analytical model. The case study is representative of an existing building, designed according to the outdated İtalian seismic code (Decreto Ministreriale, 1986), with a low seismic design approach, retrofitted using the seismic isolation technique. The building, which is located in the city of L’Aquila (Iervolino et al., 2019), has a regular plan of approximately 240 m² square meters and is characterized by a six-storey reinforced concrete (RC) frame structure (see Figure 7,B). The ground level height is 3.4 m, whereas all the other stories are 3.05 m in height. A staircase designed with knee beams is included, and slab thickness is 25 cm for all stories. Infill panels were considered to be regularly distributed in plan and elevation, with different opening percentages.

An isolation system composed of DCCSS bearings was designed for the collapse limit state (CLS), following the Italian seismic code (NTC 2018). The equivalent parameters are summarized in Table 2, where T e q is the equivalent period of the isolated building, ξ e q is the equivalent damping, and d b d is the design displacement (Ponzo et al., 2021). Once the isolation system was designed, the geometrical and mechanical parameters of the isolator shown in Figure 9 were used in the proposed algebraic solution to define displacements d p and d o t , limit displacement d lim , and the shear force limit F lim . Results are shown in Table 2 and the force-displacement law is shown in Figure 9.

Non-linear time history analyses were carried out to consider 20 different earthquakes (EQ) per three intensity measure levels characterized by return periods T R of: 500 y corresponding to the life safety limit state (LLS); 1000 y corresponding to the CLS; and a highest return period of 2500 years from the current Italian seismic code (NTC 2018). Figure 8 shows the elastic spectra of the 20 selected earthquakes for each return period for the main directions (X and Y), and the mean spectrum. All records were selected using spectrum-compatibility criteria, with a fundamental period T = 3.0   sec . Detailed information about the ground motion selection is provided by (Iervolino et al., 2011, Iervolino et al., 2018).

The comparison between the mean spectra at the different return period and the design spectrum is shown in Figure 8B and Figure 8C in terms of pseudo-acceleration and pseudo-displacement. Figure 8 also shows, for each return period, the mean value of the spectral acceleration Sa(T) corresponding to a vibration period of T = 3.0   sec due to record-to-record variability. Figure 8 highlights how the selected earthquakes cover a wide range of spectral acceleration (Figure 8) and displacements (Figure 8) around the design spectrum.

Starting with the formulations of past theories, this paper presents an extension of the DCCSS bearing force-displacement relationship to describe over-stroke behaviour in the simplified case of a DCCSS with an equal radii of curvature and equal friction coefficients for both concave plates.

The proposed algebraic solution is capable of representing forces that act on the isolation device for displacements higher than the geometric housing plate capacity, and is capable of identifying the displacements that correspond with the attainment of real limit conditions.

The proposed formulation has been provided to describe the sloping dog bone shape constitutive law estimated through experimental tests of the over-stroke regime when the rigid slider of the DCCSS bearing runs on the edge of the housing plates, exceeding its geometric capacity displacement. The algebraic solution for two failure mechanisms, which accounts for the rigid slider overturning kinematic and maximum contact pressure on the sliding interface, has been provided in terms of algebraic equations for the force-displacement relationships. The solution is valid for DCCSS bearings with a rigid slider and flat rim, equal radii of curvature, and friction coefficients on both of the concave plates.

The utility of the proposed solution is in checking the validity of numerical solutions of more complex models, and in designing the actual limit displacement and maximum shear force of the device with high accuracy. Furthermore, the maximum shear force F lim algebraic expression provided is of crucial importance to the design of the connections between the isolation devices and structure.

To validate the application of the proposed solution, non-linear dynamic analyses were implemented in OpenSees software to represent the case study of a six-storey RC frame building seismically isolated at the ground level with DCCSS bearings with over-stroke capacity. A three-dimensional MDoF model of the superstructure was upgraded with base isolation, implementing a multivariable friction model for the over-stroke displacement of DCCSS bearings, which also accounts for the effects of velocity and axial load variabilities. Additionally, structural analyses at intensity levels around design one have been carried out.

The results obtained by non-linear analysis strongly concurred with the results from the direct implementation of the proposed algebraic equations derived from the static condition. In particular, the proposed algebraic solution very accurately represented the shear forces acting on the single DCCSS bearing in the over-stroke regime. Furthermore, the results showed how the high accelerogram variability significantly affects the structural response. The proposed algebraic solution accounting for the over-stroke behaviour of DCCSS isolators should provide a safer tool for designers.

Further research should be carried out on the topic, and additional experimental testing campaigns are needed to evaluate possible dependencies of the over-stroke effect of DCCSS devices. The findings could help to facilitate highly accurate estimations of the seismic risk of seismically isolated structures with slider bearings, and develop appropriate safety factors in future building codes.