Base Isolation of Buildings with Curved Surface Sliders: Basic Design Criteria and Critical Issues

Curved surface sliders are being used more and more in the seismic isolation of buildings. They are preferred not only because of their lower cost with respect to elastomeric isolators but also of their technical characteristics, such as the fact that the value of the period of vibration is independent of the mass and the automatic coincidence between the gravity mass center of the superstructure and the stiffness center of the isolation system. In this paper, these features are analysed with reference to simple structures, pointing out the possibility of rotations of the superstructure and the loss of contact in some devices. Finally, the importance of the static friction is also emphasized showing the experimental seismic response of an isolation system under a low-energy earthquake. For all these reasons, the use of nonlinear analysis, revised and detailed in this paper, is advisable for the isolation system made of curved surface sliders.


Introduction
In 1868, Stevenson developed an isolation system to protect the lighting system in Japan (Figure 1).It was called the "aseismatic joint" and consisted in spherical rollers in niches.ese were likely the first isolation devices of modern conception and were similar to the modern curved surface sliders.Stevenson understood the basic concepts of seismic isolation very well: "It is evident that any sudden lateral motion of the earth, on which the building rests, must be communicated to the foundation of the structure, and thence through all the rigid and unyielding materials of which it is composed to its very summit, where the violence of the shock will be aggravated by the greater elevation of the highest point of the building above the source of motion.On fully considering this action of earthquakes, it occurred to me that what was required to neutralize their shocks was a break in the continuity of the rigid parts forming the structure, so as to prevent the propagation of the shock, with increasing violence from its foundation, to its summit.e idea being that, in some horizontal plane, the building should be cut through and separated, so that the sudden motion of the lower portion should not be directly communicated to the superincumbent building" [1].
A few years later, in 1970, Touaillon proposed a very similar isolation system that used spherical rollers in niches between superstructure and foundations; the return to the initial position was guaranteed by the elliptical geometry of the housing system.
ese devices contained the basic concepts of the friction pendulum system (FPS), which was first developed in the USA in the early 90s of the last century.It made use of sliding elements covered with a special fabric.Later, the seismic isolation pendulum (SIP) was developed in Germany with polyethylene sliding materials and used in some applications also in Greece, Turkey, etc., and then with similar and other sliding materials, also in Italy.Nowadays, curved surface sliders (CSS) are widely used for buildings and bridges (Figure 2).e success of the CSSs is essentially related to two reasons: (i) e possibility of designing the isolation system independently of the e ective mass of the superstructure (ii) e automatic coincidence between the projection of center of the masses and the sti ness center, at least from a theoretical point of view Furthermore, the cost of a CSS is usually lower than the cost of a high damped rubber bearing (HDRB).
In this paper, after a short introduction on seismic isolation and on the main features of the curved surface sliders, the behavior of buildings seismically isolated by means of single CSSs is analysed.e attention is focused on the mathematical modelling of friction devices.Indeed, the original formulation [2] accounts for the variability of the friction coe cient with the sliding velocity, whereas other relevant aspects [3], such as the dependence on the contact pressure and the number of cycles, are usually neglected.In this paper, the e ect of di erent friction coe cients for di erent isolators in the same building is examined, which corresponds to eccentricity of the sti ness center of the isolation layer with respect to the center of masses.e results of a numerical investigation are shown.
is was carried out with reference to two simple framed structures, in which di erent distributions of the characteristics of the devices were considered.e analysis was performed using the acceleration time history recorded at L'Aquila during the 2009 earthquake.e study pointed out also the possibility of rotations of the superstructure and the loss of contact in some devices.
Finally, the importance of the static friction is emphasized showing the experimental seismic response of an isolation system under a low-energy earthquake.For all these reasons, the use of nonlinear analysis, revised and discussed in this paper, is advisable for the isolation system made of curved surface sliders.

Seismic Isolation and Curved Surface Sliders
It is well known that seismic isolation is based on the increasing of the fundamental period of vibration of the building, which corresponds to a terri c reduction of the seismic actions that a ect the structure.anks to the ltering due to the seismic isolation system the superstructure is loaded by low seismic action and can support it in the elastic range, preserving the structure and its content [4].
e number of applications of seismic isolation increased rapidly all over the world [5].As a matter of fact, several thousands of structures in the world are protected by new antiseismic techniques, located in more than thirty countries.e use of this new antiseismic system includes not only strategic structures (civil protection centers, army and police stations, etc.) and structures of relevant importance (schools, hospitals, hotels, theaters, airports, etc.) but also residential buildings and even small private houses.It also regards structures of all the types of material: reinforced concrete, steel, masonry, and even timber both for new constructions and the retro t of existing buildings [6].

Section of aseismatic tables
Steel spring  2

Advances in Civil Engineering
All the structures protected by seismic isolation exhibited an excellent behavior even under severe earthquakes.Most of the buildings showed no damage or very low damage, demonstrating that base isolation is a reliable technology for seismic protection of structures, ready for a wide application also in masonry and cultural heritage buildings [7,8].Furthermore, seismic monitoring systems are quite common also in base isolated buildings, providing several information on the actual seismic behavior of the base-isolated building under earthquakes of di erent energy [9,10].
It must be noted that in some countries seismic isolation is considered as an additional safety measure, so the use of seismic isolation represents an additional construction cost.In other countries, among these is Italy, the seismic codes allow for the lowering of the seismic actions to the superstructure when using base isolation [11,12].In any case, it is important noting that when entrusting the safety of a building to a limited number of isolators, the reliability of these devices is a fundamental issue.
e isolation devices have rst of all a bearing function.As a matter of fact, they have to support vertical loads always during their life, both in service conditions and under seismic actions.ey should have a low horizontal sti ness, in order to allow relative displacements under seismic actions but also a suitable sti ness to avoid displacements in presence of low horizontal actions, such as wind and tra cinduced vibrations, and to guarantee a lateral constraint for them.Furthermore, they should have a good dissipation capacity, to lower the displacements under seismic actions, and a recentering capacity, to bring back the structure to its natural con guration after the event.
Curved surface sliders (CSS, Figure 3) are sliding isolators with curved surfaces that have, without the contribution of other elements, the following characteristics: (i) e recentering capability, obviously due to curved surface, which allows the device to return to its initial position when the external horizontal action nishes (ii) e energy dissipation, due to the friction between contact surfaces [13] e behavior of a curved surface slider is similar to the behavior of a pendulum.An element sliding on a surface with a curvature radius R oscillates, for small amplitudes, with the period T 2π R/g , which is independent of the mass. is model represents well the behavior of a sliding pendulum only during the motion.It cannot account for what happens at the onset on motion and at the change of the velocity sign.In both cases, the behavior is governed by a static friction, which is usually much higher than the dynamic friction.
As shown in Figure 4, the motion of a curved surface slider is characterized by at least three phases, which corresponds to di erent friction factors: μ 0 is the static friction relative to the onset of motion; μ dyn μ is the dynamic friction during the sliding with velocity similar to the design one; μ 1 is the friction factor corresponding to the change of velocity sign.It is μ < μ 1 < μ 0 .For velocity lower than 50 mm/s, a further static friction μ st should be de ned; it is used in the laboratory experimental analyses.
e model usually adopted for CSSs refers to a bilinear behavior.It is composed by an initial rigid phase and a linear phase, whose slope is given by k mg/R (Figure 5).With reference to the generic displacement d, the secant equivalent sti ness to be considered in the linear modelling, the period of vibration, and the damping are, respectively, A fundamental parameter in the behavior of a CSS is its dynamic friction coe cient µ.
e experimental analyses demonstrated that µ [14] (i) decreases when the contact stress increases (ii) gets up when the velocity increases from 0 to 150 mm/s and keeps constant for velocities between  Advances in Civil Engineering 3 200 and 800 mm/s, which are reached during the quake (iii) decreases when the temperature increases In Figure 6, the sliding coe cient of friction is plotted versus velocity v and pressure p.
With reference to the recentering, the shape and the friction play di erent roles.At positions corresponding to very low displacements, the recentering force is lower than the friction force.As a result, if the slider reaches the con guration with d ≤ μR with velocity equal to zero, the device will not return to its initial con guration (d 0). e value d max μR is the maximum residual displacement.
Usually, the linear modelling is suitable only if some conditions are satis ed.ese relate to (i) the variability of the sti ness (ii) the in uence of the velocity and vertical load on friction (iii) the increment of the force for high values of the displacement e rst and the third conditions imply two upper limits to the curvature radius.ese are, respectively,

Nonlinear Modelling of a CSS
e nonlinear modelling of the CSS behavior should account for (i) the variability of the friction coe cient due to the variations of the vertical load and the sliding velocity (ii) the rapid variations at the onset of motion and when the velocity sign changes e in uence of the sliding velocity v is described by the equation [15]: where f max is the friction coe cient at the high velocity, f min is the friction coe cient at the very low velocity, and 1/α is the characteristic sliding velocity.All these parameters are functions of the temperature.e parameter f max is also in uenced very much by the contact pressure p, while the in uence of the contact pressure on f min and α is lower and can be neglected.If f max 0 is the value for very low pressure and f max p the value for high pressure, the following expression is used: where ε is a constant.e friction force is Z is a nondimensional hysteresis variable that satis es the equation [16,17]: in which k represents the shear elastic sti ness of the sliding material in the absence of motion.During the motion, it is Z ±1, while |Z| < 1 during the elastic behavior (Figure 7). is model interprets well the actual behavior, in which displacements of about 0.13/0.5 mm were observed experimentally between steel and PTFE (Te on) during the rigid phase [2].
From the 1D model, the 2D model can be easily deduced.e forces-displacement relations are 4 Advances in Civil Engineering and the friction coe cients depend on the resultant sliding velocity v e hysteresis variables are given by the di erential equation system: where k x and k y are the elastic shear sti ness in the absence of sliding and α 2 and α 3 are equal to unity if v x Z x > 0 and v y Z y > 0, respectively, and equal to 0 elsewhere.e variables of hysteresis depend on the displacement, velocity, and initial sti ness k i xed by the yielding displacement.e interaction between the force components is governed by the condition |Z| ≤ 1.
e system is at the yield point when |Z| 1, otherwise it is in the elastic range.SAP2000 software was used for the numerical analyses.
e nite element friction-pendulum isolator was used, which accounts for the variation of the friction coe cient with the velocity but not of the contact pressure [18].

Vertical Displacement and Uplift
In the absence of uplift, a vertical component of the displacement is associated with a horizontal component.If ϑ is the arc length along the device, the horizontal and vertical components are, respectively, In Figure 8, the nondimensional vertical displacement δ v /R is plotted versus the nondimensional horizontal displacement d/R.e vertical load acting on each device is in uenced by the vertical seismic acceleration but also by the overturning moment due to the horizontal seismic actions.With reference to the rst aspect, consider a plane frame of width B and total weight W, composed of n spans of equal length, and seismically isolated by means of one curved surface slider under each column (Figure 9).If S e is the horizontal design acceleration, and the horizontal resultant force F S e • W is applied at the height H g from the isolation plane, the e ect on the rst isolator is where ΔW i is the load variations on the other isolators due to the seismic action.By substituting ΔW 1 0.1 • W 1 , we obtain the limit values for which the linear modelling is usually allowed according to the technical codes.If we suppose, for simplicity, W 1 W/2n the previous condition occurs when e curves relative to this case are plotted in Figure 10, for di erent values of n. e function f i (n, B) depends on the number of spans n and the width B. For n 3, the previous relation becomes H g /B (W/60 + ΔW 1 /3)/S e W.

Preliminary Design
e preliminary design of curved sliding devices is usually performed assuming constant values for the radius R, the vertical load W, and the friction μ.An iteration procedure can be used.is can be started by giving a rst value d 1 for the displacement and calculating the sti ness K e1 , the period T is1 , and the damping ξ is1 from Equation (1).On the spectrum relative to ξ is1 , a new value d 2 of the displacement, corresponding to T is1 , can be obtained.If d 2 ≠ d 1 , the procedure is continued.It will be stopped when d i ≈ d i-1 with a xed tolerance.

Isolation Systems with CSS Devices Having Different Frictions
It has already been said that one of the most interesting characteristics of the CSSs is the automatic coincidence between the projection of center of the masses of the superstructure and the sti ness center of the isolators.is is not exactly true.Actually, the friction force at each device depends on the vertical load acting on it.First of all, the static vertical loads are not the same on all the devices.Furthermore, they vary during the earthquake, being related to the seismic overturning moment.e friction coe cient is also in uenced by other phenomena.Among these, one of the most important phenomena is the cycling e ect, which consists in the reduction of the friction coe cient with temperature.erefore, some models have been proposed to address all these aspects, usually not totally considered in commercial codes.
In the following, the responses of some simple spatial frames are analyzed, in which the friction coe cients of the isolators are not the same for all devices.
Consider rst the simple spatial frame in Figure 11.It is composed of two oors, with four columns at the corners, connected by beams at each oor.e columns have the same squared cross section.e beams have all the same rectangular cross sections.erefore, the structure is symmetric around the two horizontal main axes.e structure is seismically isolated by means of four curved surface sliders placed under each column (C0 system, see Figure 12), characterized by a friction coe cient µ 0.035 for high speed and equal to 0.025 for low speed.e vertical load on each device is 1000 kN.A design displacement of 0.27 m was considered.
e previously described iteration procedure was used for the preliminary design of the sliding devices.
e following design parameters were obtained and kept constant for all cases: radius R � 4.0 m, vertical stiffness K v � 2.1•10 9 kN/m, and α � 43 s/m.e other characteristics are in the first column of Table 1

(μ).
e characteristics of the vibration modes, reported in Table 2, show that the first two modes of vibration have the same resonance frequency.ey consist in translations only along the two main directions, respectively, and involve almost the total mass.In the third mode, the structure just rotates around its main vertical axis.
In order to introduce accidental asymmetry in the structure, the friction coefficients of one or two devices have been changed, assuming alternatively (i) µ a � 1.5 µ, which corresponds to an equivalent viscous damping factor of 28%; the corresponding elastic and secant shear stiffness (in absence of sliding) were increased by about 20%, (ii) µ b � 2.5 µ, which corresponds to an equivalent viscous damping factor of 36% and therefore is out of the technical range; it allowed to obtain a stiffness of the devices able to modify significantly the dynamic behavior of the system.e characteristics of the isolation devices with these friction values are in the second and third columns of Table 1, respectively.Different cases were considered with reference to the isolator placements.e variation of the friction coefficient in one or more devices causes the reduction of the period of vibration but also the presence of important rotational component in the modal shapes.is is particularly evident in case C3 (Figure 12).
A nonlinear analysis (fast nonlinear analysis, FNA) was carried out by means of a finite element model with SAP2000, in which the inelastic effects were concentrated in a few points of the structural model.e structural damping was assumed to be equal to 5%.e three acceleration components recorded at the L'Aquila Park Station during the main shock of the 2009 earthquake were considered.e foundation of the models was first subjected to an acceleration component along the x axis.As one could expect, model C0 showed displacements along the x direction only, while model C3 presented also displacements along the y axis, which were higher in the case μ b (Figure 13).
en the model was subjected to the three acceleration components of the same seismic event, and the attention was focused on the vertical component of the displacement due to the curvature of the sliders.e software used does not account for this displacement component, which was evaluated separately.In the model with four columns, a maximum variation of the vertical load of about 60% was found. is allowed to state that no traction was present during the oscillation, and the vertical component of the displacement was well calculated.As shown in Figure 14, the vertical displacements were always the same in C0, while in model C3, it was verified that the maximum values were reached at the same time, and in any case, the contact points of the four sliders were on the same plane.e presence of traction in a device could determine the deformation of the superstructure needed to guarantee the contact; if the superstructure would not be able to absorb these deformations, the absence of contact would occur for a few instants.Now consider the spatial frame in Figure 15, which differs from the previous one for the presence of a column just in the center of the frame (C10, see Figure 16).e vertical load was equal to 1800 kN on the central isolator device and equal to 535 kN on the others.Also in this case, a design displacement equal to 0.27 m was considered, and the values of R � 4.0 m, vertical stiffness K v � 2.1•10 9 kN/m, and α � 43 s/m were assumed.e other characteristics of the sliding devices are in Tables 3 and 4, for the perimetral and the central isolators, respectively.As in the previous case, the first two modes of vibration have the same resonance frequency.ese involve translations only in the two main directions, and the mass participation factor is very close to unity along the relative direction (Table 5).
In order to introduce accidental asymmetry in the structure, the characteristics of some devices were changed as in the previous case.e corresponding characteristics of the devices are in Tables 3 and 4 (columns 2 and 3).Also in this case, different distributions of the devices were considered.e models with friction coefficient increased in one or more device present modal shapes with lower period of Advances in Civil Engineering  Advances in Civil Engineering vibration but also with an important rotational component.is is particularly apparent in the case C23 (Figure 16).e nonlinear analysis showed that in the presence of acceleration only along the x axis, model C10 showed displacements along x direction only, while all the other models presented also displacement along the y axis.
ese were higher in the model C23 in the case μ b (Figure 17).e maximum variation of the vertical load is about 60%; therefore, no uplift occurred, and the displacement vertical component was consistent with Equation (9).As shown in Figure 18, the vertical displacements were always the same in C10, while in model C23, it was veri ed that the maximum values were reached at the same time, and in any case, the contact points of the ve sliders were on a plane.

The Influence of the Static Friction
e static friction is usually much higher than the dynamic friction.If μ 0 is very high, then sliding could not occur, the system behaves as rigid, and there is no decoupling of   Advances in Civil Engineering motion between the superstructure and the soil.Furthermore, during the sliding, the stick-slip phenomenon could occur.In the following, the recordings obtained on a seismically CSS-isolated building (Figure 19) at L'Aquila, Italy, during the event of 30th October, 2016 (Mw 6.5), are analysed and discussed.e longitudinal components of the acceleration at the three levels are plotted in Figure 20.e ampli cation from the basement to the top is apparent.e absence of the decoupling of motion is con rmed by the Fourier spectra (Figure 21), which present ampli cations at frequency much higher than the design frequency of the isolated structure (equal to 0.25 Hz).In Figure 22, the time histories of the relative displacement along the longitudinal direction are plotted, between the rst oor and the foundation, and between the top and the rst oor, respectively.It is apparent that the isolation system is non put in action properly, due to a very high static friction.Advances in Civil Engineering 11

Conclusions
Curved surface sliders present interesting characteristics that make them very promising for a large application of base isolation in civil structures.Anyway, some features are to be studied in detail.Some of these have been analysed in this paper: (i) e dynamic behavior of a curved surface slider is governed by three friction coe cients, relative to the onset of motion, to the dynamic phase, and to the inversion of the velocity sign.(ii) e variability of the dynamic friction μ with the vertical load acting on it.ese are di erent from one column to another and are also in uenced by the overturning moment due to the horizontal seismic action.e e ects of this occurrence has been analysed with reference to two simple framed structures, and the possibility of rotations of the superstructure has been pointed out.e rotations determine the loss of contact in some devices and also di erent displacements in the devices.12 Advances in Civil Engineering (iii) e recentering of the isolation system, related to curved surface that allows the device to return to its initial position when the external horizontal action nishes, could not be guaranteed when velocity is very low; in detail, if the slider reaches the conguration with d ≤ μR with velocity equal to zero, the device will not return to its initial con guration.(iv) e static friction should not be very high.Actually, the isolation system should have an adequate sti ness against low horizontal actions, such as wind, tra c, or low-energy earthquakes.is can be obtained by means of a suitable static friction, but this should not be so high in order to guarantee that the devices are put in action under certain seismic loadings.e importance of this aspect has been emphasized showing the experimental seismic response of an isolation system under a low-energy earthquake at the site of the building.
It appears obvious that the design of isolation systems with curved surface sliders should be done using nonlinear modelling.ese should account for the static friction and the variability of friction especially with the vertical load, which makes the friction force di erent from one device to another.

Figure 3 :
Figure 3: Schematic view of a single curved surface slider (CSS).

Figure 4 :
Figure 4: E ective force-displacement diagram of a CSS. e three di erent phases with di erent friction factors are apparent.

Figure 6 :FFigure 5 :
Figure 6: Typical diagram of the dynamic friction coe cient versus velocity and pressure.

Figure 7 :
Figure 7: Z function in the case of seismic acceleration recorded during the 2009 L'Aquila earthquake.

Figure 10 :Figure 11 :Figure 12 :Figure 9 :Figure 8 :
Figure 10: H g /B ratio versus the design acceleration S e for di erent values of n.

Figure 13 :Figure 14 :
Figure 13: Four-column frame: displacements in the plane xy due to an acceleration input along x for (a) C0, (b) C3 with μ a , and (c) C3 with μ b .

Figure 16 :
Figure 16: Five columns frame: di erent deployments of the sliding devices: circle normal friction coe cient and square increased friction coe cient.

Figure 17 :
Figure 17: Five columns frame: displacements in the plane xy due to an acceleration input along x for (a) C10, (b) C23 with μ a , and (c) C23 with μ b .

Figure 19 :
Figure 19: e building at L'Aquila, seismically isolated by means of CSSs, deployed at the top of the circular columns, which raise up from the basement.

Figure 20 :Figure 21 :
Figure20: Accelerometric recordings obtained during the October 30, 2016, earthquake (M 6.5) in the longitudinal direction (a) at the basement, (b) at the rst oor just above the isolation system, and (c) at the top of the building, respectively.

Table 2 :
Four-column frame: periods and participant masses (translation X, translation Y, and rotation Z in the vibration modes).

Table 3 :
Five columns frame: characteristics of the four perimetral isolators.

Table 4 :
Five columns frame: characteristics of central isolator.

Table 5 :
Five columns frame: periods and participant masses (translation X, translation Y, and rotation Z in the vibration modes).