Abstract
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.
1. 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. These 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. The 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.
These 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).
The success of the CSSs is essentially related to two reasons:(i)The possibility of designing the isolation system independently of the effective mass of the superstructure(ii)The automatic coincidence between the projection of center of the masses and the stiffness 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. The attention is focused on the mathematical modelling of friction devices. Indeed, the original formulation [2] accounts for the variability of the friction coefficient 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 effect of different friction coefficients for different isolators in the same building is examined, which corresponds to eccentricity of the stiffness center of the isolation layer with respect to the center of masses. The results of a numerical investigation are shown. This was carried out with reference to two simple framed structures, in which different distributions of the characteristics of the devices were considered. The analysis was performed using the acceleration time history recorded at L’Aquila during the 2009 earthquake. The 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.
2. 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 terrific reduction of the seismic actions that affect the structure. Thanks to the filtering 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].
The 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. The 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 retrofit of existing buildings [6].
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 different 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.
The isolation devices have first 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. They should have a low horizontal stiffness, in order to allow relative displacements under seismic actions but also a suitable stiffness to avoid displacements in presence of low horizontal actions, such as wind and traffic-induced 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 configuration 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)The recentering capability, obviously due to curved surface, which allows the device to return to its initial position when the external horizontal action finishes(ii)The energy dissipation, due to the friction between contact surfaces [13]
The 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 , which is independent of the mass. This 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 different friction factors: μ0 is the static friction relative to the onset of motion; μdyn = μ is the dynamic friction during the sliding with velocity similar to the design one; μ1 is the friction factor corresponding to the change of velocity sign. It is μ < μ1 < μ0. For velocity lower than 50 mm/s, a further static friction μst should be defined; it is used in the laboratory experimental analyses.
The 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 (Figure 5). With reference to the generic displacement d, the secant equivalent stiffness 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 coefficient µ. The 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 200 and 800 mm/s, which are reached during the quake(iii)decreases when the temperature increases
In Figure 6, the sliding coefficient of friction is plotted versus velocity and pressure p.
With reference to the recentering, the shape and the friction play different 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 configuration with d ≤ μR with velocity equal to zero, the device will not return to its initial configuration (d = 0). The value dmax = μR is the maximum residual displacement.
Usually, the linear modelling is suitable only if some conditions are satisfied. These relate to(i)the variability of the stiffness(ii)the influence of the velocity and vertical load on friction(iii)the increment of the force for high values of the displacement
The first and the third conditions imply two upper limits to the curvature radius. These are, respectively,
3. Nonlinear Modelling of a CSS
The nonlinear modelling of the CSS behavior should account for(i)the variability of the friction coefficient 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
The influence of the sliding velocity is described by the equation [15]:where is the friction coefficient at the high velocity, is the friction coefficient at the very low velocity, and 1/α is the characteristic sliding velocity. All these parameters are functions of the temperature. The parameter is also influenced very much by the contact pressure p, while the influence of the contact pressure on and α is lower and can be neglected. If is the value for very low pressure and the value for high pressure, the following expression is used:where is a constant. The friction force is is a nondimensional hysteresis variable that satisfies the equation [16, 17]:in which k represents the shear elastic stiffness of the sliding material in the absence of motion. During the motion, it is , while during the elastic behavior (Figure 7). This model interprets well the actual behavior, in which displacements of about 0.13/0.5 mm were observed experimentally between steel and PTFE (Teflon) during the rigid phase [2].
From the 1D model, the 2D model can be easily deduced. The forces-displacement relations areand the friction coefficients depend on the resultant sliding velocity :
The hysteresis variables are given by the differential equation system:where and are the elastic shear stiffness in the absence of sliding and α2 and α3 are equal to unity if and , respectively, and equal to 0 elsewhere. The variables of hysteresis depend on the displacement, velocity, and initial stiffness ki fixed by the yielding displacement. The interaction between the force components is governed by the condition . The system is at the yield point when , otherwise it is in the elastic range. SAP2000 software was used for the numerical analyses. The finite element friction-pendulum isolator was used, which accounts for the variation of the friction coefficient with the velocity but not of the contact pressure [18].
4. Vertical Displacement and Uplift
In the absence of uplift, a vertical component of the displacement is associated with a horizontal component. If is the arc length along the device, the horizontal and vertical components are, respectively,
In Figure 8, the nondimensional vertical displacement is plotted versus the nondimensional horizontal displacement .
The vertical load acting on each device is influenced by the vertical seismic acceleration but also by the overturning moment due to the horizontal seismic actions. With reference to the first 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 is the horizontal design acceleration, and the horizontal resultant force is applied at the height from the isolation plane, the effect on the first isolator iswhere is the load variations on the other isolators due to the seismic action. By substituting , we obtain the limit values for which the linear modelling is usually allowed according to the technical codes. If we suppose, for simplicity, the previous condition occurs when
The curves relative to this case are plotted in Figure 10, for different values of n. The function fi(n, B) depends on the number of spans n and the width B. For n = 3, the previous relation becomes .
5. Preliminary Design
The 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. This can be started by giving a first value d1 for the displacement and calculating the stiffness Ke1, the period Tis1, and the damping ξis1 from Equation (1). On the spectrum relative to ξis1, a new value d2 of the displacement, corresponding to Tis1, can be obtained. If d2 ≠ d1, the procedure is continued. It will be stopped when di ≈ di-1 with a fixed tolerance.
6. 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 stiffness center of the isolators. This 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. The friction coefficient is also influenced by other phenomena. Among these, one of the most important phenomena is the cycling effect, which consists in the reduction of the friction coefficient with temperature. Therefore, 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 coefficients of the isolators are not the same for all devices.
Consider first the simple spatial frame in Figure 11. It is composed of two floors, with four columns at the corners, connected by beams at each floor. The columns have the same squared cross section. The beams have all the same rectangular cross sections. Therefore, the structure is symmetric around the two horizontal main axes. The structure is seismically isolated by means of four curved surface sliders placed under each column (C0 system, see Figure 12), characterized by a friction coefficient µ = 0.035 for high speed and equal to 0.025 for low speed. The vertical load on each device is 1000 kN. A design displacement of 0.27 m was considered. The previously described iteration procedure was used for the preliminary design of the sliding devices. The following design parameters were obtained and kept constant for all cases: radius R = 4.0 m, vertical stiffness = 2.1·109 kN/m, and α = 43 s/m. The other characteristics are in the first column of Table 1 (μ).
The characteristics of the vibration modes, reported in Table 2, show that the first two modes of vibration have the same resonance frequency. They 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)µa = 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)µb = 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.
The 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. The 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. This 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. The structural damping was assumed to be equal to 5%. The three acceleration components recorded at the L’Aquila Park Station during the main shock of the 2009 earthquake were considered. The 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 μb (Figure 13).
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Then 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. The 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. This 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. The 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.
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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). The 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 = 2.1·109 kN/m, and α = 43 s/m were assumed. The 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. These 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. The 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. The models with friction coefficient increased in one or more device present modal shapes with lower period of vibration but also with an important rotational component. This is particularly apparent in the case C23 (Figure 16).
The 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. These were higher in the model C23 in the case μb (Figure 17).
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The 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 verified that the maximum values were reached at the same time, and in any case, the contact points of the five sliders were on a plane.
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7. The Influence of the Static Friction
The static friction is usually much higher than the dynamic friction. If μ0 is very high, then sliding could not occur, the system behaves as rigid, and there is no decoupling of 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.
The longitudinal components of the acceleration at the three levels are plotted in Figure 20. The amplification from the basement to the top is apparent.
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The absence of the decoupling of motion is confirmed by the Fourier spectra (Figure 21), which present amplifications 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 first floor and the foundation, and between the top and the first floor, respectively. It is apparent that the isolation system is non put in action properly, due to a very high static friction.
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8. 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)The dynamic behavior of a curved surface slider is governed by three friction coefficients, relative to the onset of motion, to the dynamic phase, and to the inversion of the velocity sign.(ii)The variability of the dynamic friction μ with the vertical load acting on it. These are different from one column to another and are also influenced by the overturning moment due to the horizontal seismic action. The effects 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. The rotations determine the loss of contact in some devices and also different displacements in the devices.(iii)The recentering of the isolation system, related to curved surface that allows the device to return to its initial position when the external horizontal action finishes, could not be guaranteed when velocity is very low; in detail, if the slider reaches the configuration with d ≤ μR with velocity equal to zero, the device will not return to its initial configuration.(iv)The static friction should not be very high. Actually, the isolation system should have an adequate stiffness against low horizontal actions, such as wind, traffic, or low-energy earthquakes. This 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. The 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. These should account for the static friction and the variability of friction especially with the vertical load, which makes the friction force different from one device to another.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The monitoring of the mentioned building in L’Aquila is part of the Seismic Observatory of Structures project, organized and managed by the Italian Department of the Civil Protection (DPC Osservatorio Sismico delle Strutture–OSS Download Service, http://www.mot1.it/ossdownload).