Sliding isolator Bearings

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2008; 37:163–183Published online 20 August 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.751

Spherical sliding isolation bearings with adaptive behavior: Theory

Daniel M. Fenz∗,†,‡ and Michael C. Constantinou§

Department of Civil, Structural and Environmental Engineering, University at Buffalo, The State Universityof New York, 212 Ketter Hall, Buffalo, NY 14260, U.S.A.

SUMMARY

The principles of operation and force–displacement relationships of three novel spherical sliding isolationbearings are developed in this paper. These bearings are completely passive devices, yet exhibit adaptivestiffness and adaptive damping. That is, the stiffness and damping change to predictable values at calculableand controllable displacement amplitudes. The primary benefit of adaptive behavior is that a given isolationsystem can be separately optimized for multiple performance objectives and/or multiple levels of groundshaking. With the devices presented here, this is accomplished using technology that is inherently no morecomplex than what is currently used by the civil engineering profession. The internal construction consistsof various concave surfaces and behavior is dictated by the different combinations of surfaces upon whichsliding can occur over the course of motion. As the surfaces upon which sliding occurs change, the stiffnessand effective friction change accordingly. A methodology is presented for determining which surfaces areactive at any given time based on the effective radius of curvature, coefficient of friction and displacementcapacity of each sliding surface. The force–displacement relationships and relevant parameters of interestare subsequently derived based on the first principles. Copyright q 2007 John Wiley & Sons, Ltd.

Received 16 March 2007; Revised 27 June 2007; Accepted 2 July 2007

KEY WORDS: adaptive seismic isolation system; triple Friction Pendulum bearing; seismic isolation

1. INTRODUCTION

An adaptive (or smart) seismic isolation system is capable of changing stiffness and damping prop-erties during its course of motion. To date, such systems have consisted primarily of conventionalisolation bearings used in conjunction with active or semi-active devices having variable stiffness

∗Correspondence to: Daniel M. Fenz, Department of Civil, Structural and Environmental Engineering, University atBuffalo, The State University of New York, 212 Ketter Hall, Buffalo, NY 14260, U.S.A.

†E-mail: [email protected]‡Ph.D. Candidate.§Professor.

Contract/grant sponsor: Multidisciplinary Center for Earthquake Engineering Research

Copyright q 2007 John Wiley & Sons, Ltd.

164 D. M. FENZ AND M. C. CONSTANTINOU

or damping properties. Systems reported in the literature include various permutations of slidingand elastomeric bearings combined with variable stiffness devices, variable friction devices, vari-able orifice fluid dampers, electrorheological dampers and magnetorheological dampers [1–10, forexample]. Generally, these studies conclude that properly designed active and semi-active hybridsystems can offer improved performance over passive systems in a wider range of earthquakes, butobstacles related to implementation and questions regarding reliability still persist. A compositeisolator consisting of two elastomeric bearings of different stiffness stacked one on top of theother was proposed in 1993 by A. G. Tarics and is presented in Imbimbo and Kelly [11]. To theknowledge of the authors, this has been the only passive isolation system proposed in the literatureto date that employs displacement-dependent behavior as a means of structural control rather thanas just a failsafe.

Three different multi-spherical sliding bearings that expand the definition of an adaptive seismicisolation system are introduced in this paper. These bearings are fully passive devices that exhibitadaptive stiffness and adaptive damping in and of themselves. By adaptive behavior, it is meantthat the stiffness and effective friction change to predictable values at calculable and controllabledisplacement amplitudes. This behavior naturally results from the internal construction of thebearings, not an externally applied active force. Moreover, each variation of bearing is a derivativeof the conventional Friction Pendulum (FP) bearing, a mature and established seismic protectivetechnology. As such, these devices are based on well-known engineering principles of pendulummotion and are constructed of materials with demonstrated longevity [12, 13]. This makes practicalimplementation more feasible.

Adaptive behavior can be used by designers to achieve benefits in performance that are notpossible with other isolation systems. Current practice is to design the structural system to resistthe base shear transmitted in the design basis earthquake (DBE) and to design the isolation system tohave sufficient displacement capacity to meet the demands of the maximum considered earthquake(MCE). This is a ‘Catch-22’ situation for designers; the desire to reduce displacement demand inthe MCE with increased stiffness and damping results in less than optimum performance in theDBE and vice versa. This situation is exacerbated due to the substantial differences in the DBE andMCE demands (the DBE spectrum prescribed by code is 2

3 of the MCE spectrum [14]). Moreover,the performance of the isolation system in more frequent events of smaller magnitude is typicallynot considered in the design process. Although low-level shaking is not a design issue in terms ofstrength or displacement capacity, it can be a performance issue. Isolation systems designed withsufficient damping and flexibility for larger earthquakes may not activate in more minor events,which can adversely affect secondary system response. Even if the isolation system does activate,re-centering can be an issue.

The bearings presented in this paper help to overcome these challenges, since adaptive behaviorpermits the isolation system to be separately optimized for low intensity, design level and maximumearthquake shaking. The work in Kelly [15] and Hall [16] indicates that to control displacementsin large earthquakes while still maintaining good performance in low-to-moderate earthquakesrequires designing an isolation system that is (a) very stiff with low damping at low-level shaking,(b) softens with increasing damping in the DBE, (c) further softens and increases damping in theMCE and (d) stiffens beyond the MCE. This desirable behavior can be achieved with properlydesigned multi-spherical sliding bearings.

Three variations of adaptive spherical sliding bearing are shown in Figure 1. The focus of thispaper is the formulation of the force–displacement relationships for these devices, with emphasisplaced on the triple FP bearing shown in Figure 1(a). Two additional variations that exhibit

Copyright q 2007 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2008; 37:163–183DOI: 10.1002/eqe

SPHERICAL SLIDING ISOLATION BEARINGS WITH ADAPTIVE BEHAVIOR 165

(a)

(b)

(c)

Figure 1. Variations of multi-spherical sliding bearings with adaptive behavior: (a) triple FP; (b) modifiedsingle FP; and (c) double FP with sliding surfaces of different displacement capacities.

adaptive behavior are the modified single FP bearing shown in Figure 1(b) and the double FPbearing with concave surfaces of different displacement capacities, shown in Figure 1(c). Thesevariations, together with the more familiar single and double FP bearings, offer a range of



adaptive behaviors from none in the single FP bearing to the most versatile in the triple FPbearing. It may be recognized that increases in adaptability are accompanied by increased com-plexity and, therefore, each of these variations may be of interest to a designer. A companionpaper [17] describes a program of characterization testing used to validate the theoretical workpresented here.

2. PRINCIPLES OF OPERATION OF THE TRIPLE FP BEARING

The triple FP bearing consists of two facing concave stainless steel surfaces separated by aninternal nested slider assembly. Referring to Figure 1(a), the outer concave plates have effectiveradii Reff1 = R1 − h1 and Reff4 = R4 − h4, where Ri is the radius of curvature of the i th sphericalsurface and hi is the radial distance between the i th spherical surface and the pivot point of thearticulated slider. The articulated slider assembly consists of two concave slide plates separatedby a rigid slider. The surface of the slide plates where they mate with the outer concave plates iscoated with a non-metallic sliding material. The coefficients of friction of these interfaces are �1and �4. The inner surfaces of the two slide plates have spherical concave recesses with effectiveradii Reff2 = R2 − h2 and Reff3 = R3 − h3. Both outer surfaces of the rigid slider are also coatedwith a non-metallic sliding material characterized by coefficients of friction �2 and �3. This permitssliding on the inner stainless steel surfaces of the slide plates.

The nominal displacement capacities of the sliders on surfaces 1–4 are denoted as d1–d4 (dueto the effects of slider height and slider rotation, the actual displacement capacities are slightlydifferent than as drawn). The unique behavior of the triple FP and the other adaptive bearingsrelies in part on the various sliders achieving the full horizontal displacement capacity of theirrespective sliding surfaces during the course of motion. Therefore, the displacement capacitiesd1–d4 are design parameters that significantly influence the global behavior, not just limits ofoverall capacity.

The adaptive behavior of the triple FP bearing results from the different combinations of slidingthat can occur on its multiple concave surfaces. The motion is organized into several slidingregimes, each corresponding to a distinct combination of surfaces upon which sliding occurs. Thestiffness of the bearing is inversely proportional to the sum of the radii of curvature of the surfaceson which sliding occurs. The effective coefficient of friction is also related to the coefficients offriction of the surfaces on which sliding is occurring.

Sequencing of the sliding regimes is determined by each surface’s coefficient of friction and itsratio of displacement capacity to radius of curvature. Starting from rest, sliding initiates on thei th surface when the horizontal force transmitted through the bearing, F , exceeds that surface’sfriction force, Ffi = �iW , where W is the vertical load on the bearing. Sliding is stopped by thedisplacement restrainer on the i th surface when the relative displacement of the slider on thissurface, ui , becomes equal to the displacement capacity, di . The lateral force at the instant theslider starts to bear upon this surface’s displacement restrainer is

Fdri = W

Reffidi + Ffi (1)

The sequence of activation and deactivation of sliding on various surfaces is determined bycomparing the relative values of Ffi and Fdri .



The standard configuration of triple FP bearing is large and equal effective radii for the outer con-cave plates and small and equal effective radii for the inner slide plates, Reff1 = Reff4�Reff2 = Reff3.The coefficients of friction are selected so that the bearing exhibits high stiffness and low fric-tion initially and subsequently decreases in stiffness and increases in effective friction as theamplitude of displacement increases. This is accomplished by using friction materials that give�2 = �3<�1<�4. The displacement capacities of each surface are selected so that there is gradualstiffening at large displacement. The slider should contact the displacement restrainer on surfaces1 and 4 prior to surfaces 2 and 3. Provided that motion initiates on surfaces 2 and 3 prior tosurfaces 1 and 4, this is guaranteed as long as Ff1<Fdr2 and Ff4<Fdr3. In terms of displacements,this condition is d2>(�1 − �2)Reff2 and d3>(�4 − �3)Reff3. Furthermore, sliding should initiateon the surface of highest friction prior to the onset of any stiffening, that is Ff4<Fdr1. This is toavoid a situation in which the bearing stiffens, then softens, then stiffens again, which would occurif Fdr1<Ff4. The bearing would stiffen upon contacting the displacement restrainer of surface 1,soften when sliding started on surface 4, and then stiffen again upon contacting the displacementrestrainer of surface 4.

3. FORCE–DISPLACEMENT RELATIONSHIP OF THE TRIPLE FP BEARING

This section demonstrates how the force–displacement relationship is derived for a triple FP bearingof standard configuration. It is assumed that (a) Reff2 = Reff3 � Reff1 = Reff4, (b) �2 = �3<�1<�4,(c) d2>(�1 − �2)Reff2 and d3>(�4 − �3)Reff3 so that Ff1<Fdr2 and Ff4<Fdr3 and (d) Ff4<Fdr1.

3.1. Sliding regime I

Sliding regime I consists of sliding on surfaces 2 and 3 and no sliding on surfaces 1 and 4. Startingfrom rest, motion will initiate when the horizontal force, F , exceeds the friction force on thesurface(s) of least friction. Therefore, sliding begins on surfaces 2 and 3 when F = Ff2 = Ff3. Thedisplaced shape and free body diagrams of the components of the bearing during this regime areshown in Figure 2. In the free body diagrams, Ffi is the resultant friction force acting along thei th sliding interface and Si is the resultant force of normal pressure acting along the i th slidinginterface.

Based on FBD III of Figure 2(b), the following relationships are obtained by consideringequilibrium in the vertical and horizontal directions, respectively:

S1 + Ff2 sin �2 − S2 cos �2 = 0 (2a)

Ff2 cos �2 + S2 sin �2 − Ff1 = 0 (2b)

Also, from FBD IV of Figure 2(b),

F = Ff1 (3a)

W = S1 (3b)



(a)

(b)

Figure 2. Displaced shape (a) and free body diagrams (b) of the triple FP bearing during sliding regime I.Sliding occurs on surfaces 2 and 3 only; motion has not yet been initiated on surfaces 1 and 4.

From geometry, the relative displacement of the slider on surface 2, u2, is

u2 = Reff2 sin �2 (4)



Combining Equations (2)–(4) and assuming that the relative displacement u2 is sufficiently smallcompared with the effective radius Reff2 so that cos �2 ≈ 1:

F = W

Reff2u2 + Ff2 (5)

Equations (2)–(5) are the equations of equilibrium for the conventional single FP bearing [12].Similar analysis of equilibrium of FBD I and FBD II gives the following equation for surface 3:

F = W

Reff3u3 + Ff3 (6)

The force–total displacement relationship for the bearing during sliding regime I is determined bycombining Equations (5) and (6) based on the fact that the total displacement u is the sum of thedisplacements u2 and u3, as u1 = u4 = 0, resulting in

F = W

Reff2 + Reff3u + Ff2Reff2 + Ff3Reff3

Reff2 + Reff3(7)

Upon reversal of motion, the bearing unloads by 2Ff2(= 2Ff3) and sliding initiates again onsurfaces 2 and 3. As shown in Figure 3, the hysteretic behavior is rigid linear with post-elasticstiffness equal to the sum of the effective radii of surfaces 2 and 3 and strength equal to the averagecoefficient of friction on these two surfaces. The behavior is identical to a double FP bearing withconcave surfaces of equal radii and equal friction [18–20].3.2. Sliding regime II

When F = Ff1, motion begins on surface 1, marking the start of sliding regime II. The transitionoccurs at displacement u∗ given by

u∗ = (�1 − �2)Reff2 + (�1 − �3)Reff3 (8)

Sliding Regime I: umax < u*

Total Displacement, u

Hor

izon

tal F

orce

WReff2+ Reff3

2Ff2( =2Ff3)Ff 2 (= Ff 3 )

Figure 3. Force–displacement relationship of the triple FP bearing during sliding regime I.



Equation (8) is obtained by solving Equation (7) for the displacement when F = Ff1. The displacedshape and free body diagrams for sliding regime II are shown in Figure 4. The rotation of thelower slide plate with respect to the lower concave plate is �1 and the rotation of the rigid sliderwith respect to the lower slide plate is �2. When the angles are defined in this way, the relativedisplacements u1 and u2 are

u1 = Reff1 sin �1 (9a)

u2 = Reff2 sin �2 (9b)

From FBD IV of Figure 4(b), the equilibrium equations for the conventional FP bearing areobtained, leading to the following relationship governing motion on surface 1:

F = W

Reff1u1 + Ff1 (10)

Although small in magnitude, rotation of the lower slide plate when sliding is occurring on surface1 has a significant impact on behavior. The angle that the rigid slider makes with respect to thevertical direction is now the sum of angles �1 and �2, as reflected in the equations of equilibriumfrom FBD III of Figure 4(b):

S1 cos �1 + Ff2 sin(�1 + �2) − S2 cos(�1 + �2) − Ff1 sin �1 = 0 (11a)

S2 sin(�1 + �2) + Ff2 cos(�1 + �2) − S1 sin �1 − Ff1 cos �1 = 0 (11b)

Using Equations (9)–(11) and the assumptions that the individual angles �1 and �2 are small sothat cos �1 ≈ cos �2 ≈ 1 and sin �1 × sin �2 ≈ 0, for surface 2 it is found that

F =W

(u1Reff1

+ u2Reff2

)+ Ff2 (12)

Substituting Equation (10) into Equation (12):

u2 = (�1 − �2)Reff2 (13)

Equation (13) reveals that the displacement on surface 2 is constant with magnitude equal to thevalue of u2 when motion transitions from sliding regime I to sliding regime II (solve Equation (5)for u2 with F = Ff1). This means that the instant motion starts on surface 1, it stops on surface 2.

Inspection of FBD I and FBD II of Figure 4(b) shows that there is no change from FBD I andFBD II of Figure 2(b), other than that the angle �3 is larger due to the increase of displacement u3.Therefore, there is no sliding on surface 4 and motion on surface 3 is still governed by Equation (6).The force–total displacement relationship for sliding regime II determined based on Equations (6),(10) and (12) is

F = W

Reff1 + Reff3u + Ff1(Reff1 − Reff2) + Ff2Reff2 + Ff3Reff3

Reff1 + Reff3(14)

This relationship is shown in Figure 5. Upon reversal of motion, the bearing unloads by2Ff2(= 2Ff3) and motion resumes on surfaces 2 and 3. Motion continues on surfaces 2 and 3for a distance of 2u∗ until the bearing has unloaded by 2Ff1, at which point sliding starts again



(a)

(b)

Figure 4. Displaced shape (a) and free body diagrams (b) of the triple FP bearing during slidingregime II. Sliding occurs on surfaces 1 and 3, motion has not yet been initiated on surface 4, and there is

constant displacement on surface 2.



Sliding Regime II:

u* < umax

< u**


Ho

rizo

nta

l F

orc

e

Sliding Regime II

Sliding Regime I

Ff 1

u*

WReff 1

+ Reff 3

WReff 2

+ Reff 3

2Ff1

2Ff 2

WReff 1

+ Reff 3

2u*

Figure 5. Force–displacement relationship of the triple FP bearing during sliding regime II shownin relation to sliding regime I.

on surface 1 and stops on surface 2. Sliding then continues on surfaces 1 and 3. In comparison tosliding regime I, transition to sliding regime II is accompanied by a reduction in stiffness and anincrease in effective friction.

3.3. Sliding regime III

Sliding initiates on surface 4 when F = Ff4, which occurs at displacement u∗∗ given by

u∗∗ = u∗ + (�4 − �1)(Reff1 + Reff3) (15)

Equation (15) is obtained by solving Equation (14) for the displacement when F = Ff4. Displace-ments u1 and u2 and angles �1 and �2 are defined as before; the rotation of the upper slide platewith respect to the upper concave plate is �4 and the rotation of the upper slide plate with respectto the rigid slider is �3. When the angles are defined in this way, the relative displacements u3 andu4 are

u3 = Reff3 sin �3 (16a)

u4 = Reff4 sin �4 (16b)

Motion on surface 1 is still governed by Equation (10) and motion on surface 2 is still governedby Equation (12). From similar analysis of equilibrium as was carried out for FBD III and FBDIV of Figure 4(b), it follows that for surface 4,

F = W

Reff4+ Ff4 (17)



Sliding Regime III:

u** < umax

< udr1


Ho

rizo

nta

l F

orc

e

Sliding Regime III

Sliding Regimes I - II

Ff 4

u**

WReff1+ Reff 4

WReff1+ Reff4

WReff 2+ Reff 3

WReff 1

+ Reff 3

2Ff 2

2Ff 4

2Ff 1

2u**

Figure 6. Force–displacement relationship of the triple FP bearing during sliding regime III shown inrelation to sliding regimes I–II.

and for surface 3,

F = W

(u3Reff3

+ u4Reff4

)+ Ff3 (18)

u3 = (�4 − �3)Reff3 (19)

Equation (19) demonstrates that as soon as sliding starts on surface 4, it stops on surface 3. This canbe proven by solving Equation (6) for u3 with F = Ff4. The force–total displacement relationshipfor sliding regime III, determined by combining Equations (10), (12), (17) and (18), is

F = W

Reff1 + Reff4u + Ff1(Reff1 − Reff2) + Ff2Reff2 + Ff3Reff3 + Ff4(Reff4 − Reff3)

Reff1 + Reff4(20)

This relationship is shown in Figure 6. Compared with regimes I and II, transition to slidingregime III is accompanied by a reduction in stiffness and an increase in effective friction. Whenmotion reverses, the bearing unloads by 2Ff2(= 2Ff3) and sliding resumes on surfaces 2 and 3.Motion continues on surfaces 2 and 3 for a distance of 2u∗ until the bearing has unloaded by2Ff1, at which point sliding starts on surface 1 and stops on surface 2. From this point, motioncontinues on surfaces 1 and 3 for a distance of 2u∗∗ − 2u∗ until the bearing has unloaded by 2Ff4.At this point, motion resumes on surface 4 (and stops on surface 3) and sliding on surfaces 1 and4 occurs.

3.4. Sliding regime IV

Stiffening behavior of the triple FP bearing at large displacements is achieved by stopping motionon surfaces with large effective radius and forcing it to occur on surfaces with small effective



radius. Sliding regime IV begins when contact is made with the displacement restrainer on surface1 and sliding changes from surfaces 1 and 4 to surfaces 2 and 4. The displacement on surface 1is u1 = d1 and the horizontal force, Fdr1 is

Fdr1 = W

Reff1d1 + Ff1 (21)

The transition between sliding regimes occurs at a total displacement of udr1, given by

udr1 = u∗∗ + d1

(1 + Reff4

Reff1

)− (�4 − �1)(Reff1 + Reff4) (22)

Equation (22) is obtained by solving Equation (20) for u with F = Fdr1. The displaced shape andfree body diagrams for motion during regime IV are given in Figure 7. In FBD III and FBD IV ofFigure 7, it is shown that the effect of the displacement restrainer contacting the slider on surface 1is to introduce an additional force on the slider, Fr1. It is assumed that the displacement restraineris rigid, and therefore from FBD IV of Figure 7, the force–displacement relationship governingmotion on surface 1 is

F = W

Reff1d1 + Ff1 + Fr1 (23)

The force Fr1 accounts for the fact that the horizontal force F increases with no increase indisplacement on surface 1. In Figure 8, the horizontal force is plotted against the displacementon surface i , illustrating the meaning of the force Fri . The behavior shown in Figure 8 wasdemonstrated in the original experimental study of the FP bearing at UC Berkeley (see Figure 11of [12]).

Using FBD III and FBD IV of Figure 7(b), the force–displacement relationship governing motionon surface 2 is

F =W

(d1Reff1

+ u2Reff2

)+ Ff2 (24)

This demonstrates that sliding resumes on surface 2 when the displacement restrainer is contactedon surface 1. Equation (24) is simply Equation (12) with u1 = d1. Nothing has changed on the uppersurfaces so motion on surfaces 3 and 4 is still governed by Equations (18) and (17), respectively.Therefore, the force–total displacement relationship is

F = W

Reff2 + Reff4(u − udr1) + W

Reff1d1 + Ff1 (25)

This relationship is shown in Figure 9. Upon reversal of motion, the bearing unloads by2Ff2(= 2Ff3) and motion resumes on surfaces 2 and 3. As shown in Figure 8, after the slidercontacts the displacement restrainer on surface 1, motion will not start on this surface until thebearing has unloaded by Fr1 + 2Ff1 to Fdr1 − 2Ff1. Sliding resumes on surface 4 when the bear-ing has unloaded by 2Ff4. The order in which sliding resumes is determined by comparing thequantities Fr1 + 2Ff1 and 2Ff4. It can be shown that for sliding regime IV, if the magnitude of themaximum total displacement, umax, satisfies the following relation:

umax>udr1 + 2(�4 − �1)(Reff2 + Reff4) (26)



(a)

(b)

Figure 7. Displaced shape (a) and free body diagrams (b) of the triple FP bearing during sliding regimeIV. The slider is bearing on the displacement restrainer on surface 1, sliding occurs on surfaces 2 and 4,

and the displacement on surface 3 remains constant.



Displacement on Surface i,ui

Hor

izon

tal F

orce

Ffi

WReffi

Fdri

Fmax

2Ffi

Fri

di

Figure 8. Force–displacement relationship for a single concave surface when the slidercontacts the displacement restrainer.

Sliding Regime IV:udr1 < umax < udr4


Hor

izon

tal F

orce

Sliding Regime IVSliding Regimes I - III

Fdr1

Fdr1 - 2Ff 1

udr1

WReff 2 + Reff4

WReff1 + Reff4

WReff 2 + Reff 3

WReff 1 + Reff 3

2Ff 2

2Ff4

Figure 9. Force–displacement relationship of the triple FP bearing during sliding regime IV shown inrelation to sliding regimes I–III.

then upon unloading, motion will initiate on surface 4 prior to surface 1 (2Ff4<Fr1 + 2Ff1). IfEquation (26) is not satisfied, then motion will initiate on surface 1 prior to surface 4 (Fr1 +2Ff1<2Ff4). This demonstrates that it is possible to have different types of unloading behaviordepending on the maximum displacement achieved. However, based on Equation (28) that follows,one can show that for the typical configuration with d1 = d4 and Reff1 = Reff4, Equation (26) cannotbe satisfied prior to the start of sliding regime V. Therefore, motion will resume on surface 1 priorto surface 4 for the typical configuration of triple FP bearing.



3.5. Sliding regime V

Sliding regime V begins when contact is made with the displacement restrainer on surface 4. Motionchanges from sliding on surfaces 2 and 4 to sliding on surfaces 2 and 3, which is accompaniedby further stiffening. At the transition point, the relative displacement on surface 4 is u4 = d4 andthe horizontal force, Fdr4, is

Fdr4 = W

Reff4d4 + Ff4 (27)

The transition between sliding regimes occurs at a total displacement of udr4, given by

udr4 = udr1 +[(

d4Reff4

+ �4

)−

(d1Reff1

+ �1

)](Reff2 + Reff4) (28)

From similar analysis of equilibrium as was done for FBD III and FBD IV of Figure 7(b), itfollows that for surface 4,

F = W

Reff4d4 + Ff4 + Fr4 (29)

and for surface 3,

F =W

(d4Reff4

+ u3Reff3

)+ Ff3 (30)

Motion continues on surface 2 with the slider bearing on the displacement restrainer of surface 1.Combining the force–displacement relationships for surfaces 1–4 gives

F = W

Reff2 + Reff3(u − udr4) + W

Reff4d4 + Ff4 (31)

This is shown in Figure 10. When the motion reverses, the bearing will unload by 2Ff2(= 2Ff3)and sliding will occur initially on surfaces 2 and 3. Motion resumes on surface 1 when the bearingunloads to Fdr1 − 2Ff1 and motion resumes on surface 4 when the bearing unloads to Fdr4 − 2Ff4.Since the former is larger than the latter (assuming the standard configuration), sliding will initiateon surface 1 prior to surface 4 when unloading from umax>udr4.

3.6. Additional comments

(a) For the standard configuration of triple FP bearing that has been described in this section,forces Fdr2 and Fdr3 calculated using Equation (1) are not actually the forces at which the slidercontacts the displacement restrainer on surfaces 2 and 3. This is because motion on these surfacesis interrupted when sliding initiates on surfaces 1 and 4, respectively. Based on Equation (24), theactual force at which the slider contacts the displacement restrainer of surface 2 is

F =W

(d1Reff1

+ d2Reff2

)+ Ff2 (32)

A similar expression can be derived based on Equation (30) for surface 3. The forces Fdr2 andFdr3 are the forces at which the slider would contact the displacement restrainer if sliding did



Sliding RegimeV:umax > udr4


Hor

izon

tal F

orce

Sliding Regime V

Sliding Regimes I - IV

Fdr4

Fdr1 - 2Ff1

Fdr4 - 2Ff4

udr4

WReff2+Reff3

WReff1+ Reff3

WReff1+Reff4

2Ff2

Figure 10. Force–displacement relationship of the triple FP bearing during sliding regime V shown inrelation to sliding regimes I–IV.

not yet initiate on surfaces 1 and 4. Although not physically meaningful, they are used to checksequencing of regimes.

(b) Although the coefficient of friction, �i , appears as a single-valued parameter in the expressionsthroughout this paper, in reality it varies as a function of several factors including sliding velocityand pressure [21]. The single-valued coefficient of friction is a simplification, not a limitation ofthe theory. When modeling of velocity dependence is required, the equations are used in their sameform with �i (u̇i ), a friction coefficient that is updated at each time step based on the instantaneoussliding velocity of surface i .

(c) The pivot point is the point about which the articulated slider rotates. In single and doubleFP bearings, there is only one articulation in the slider. Rotation is physically constrained to occurabout the center of the sphere defined by the ball and socket articulation. In the triple FP bearing,there is no such mechanical restraint defining the location of the pivot point. Instead, the pivotpoint corresponds to the instantaneous center of zero rotational velocity of the slider assembly,which is not a fixed point. In most cases, there is little error introduced by assuming the center ofzero rotational velocity is fixed at the mid-height of the articulated slider assembly.

(d) In previous work on the double FP bearing, it was observed that the slider becomes offsetwithin the bearing during displacement-controlled harmonic tests when friction is different on theupper and lower surfaces [20]. At zero total displacement, the individual displacements on eachsurface are equal and opposite rather than both zero. The same phenomenon occurs in the triple FPbearing. The offset occurs because the effective coefficient of friction (the normalized horizontalforce at zero total displacement) is different from the coefficients of friction on the individualsurfaces (the normalized horizontal force at zero relative displacement). Explicit expressions forthe individual offsets are not included here; however, they can be determined by tracking thehysteresis loops. The offsets are related to displacement-controlled testing and are different fromthe phenomenon of permanent total displacements at the end of earthquake excitation.



4. BEHAVIOR OF THE MODIFIED SINGLE FP BEARING

The modified single FP bearing is similar in construction to the conventional FP bearing but with anintermediate slide plate. Accordingly, its behavior follows directly from simplification of that of thetriple FP bearing and is only presented briefly herein. In the standard configuration of Reff1 � Reff2and �1>�2, the force–displacement relationship is composed of three sliding regimes: (i) initialsliding only on surface 2 with high stiffness and low friction; (ii) sliding only on surface 1 witha decrease in stiffness and an increase in friction; and (iii) stiffening as the slide plate contactsthe displacement restrainer on surface 1 and sliding on surface 2 resumes. This configuration willreadily activate and re-center well in minor events, provide sufficient flexibility and damping formore severe shaking and then stiffen substantially before the maximum displacement capacityof the bearing is achieved. However, since most of the sliding occurs on one surface, the plandimensions of modified single FP bearings will need to be much larger than those of the doubleor triple FP bearings.

Upon application of horizontal force, motion will initiate on surface 2 when F = Ff2. Slidingthen occurs on surface 2 only with the force–displacement relationship:

F = W

Reff2u + Ff2 (33)

If the direction of motion reverses prior to the initiation of motion on surface 1, the bearing willunload by 2Ff2 and sliding will continue only on surface 2. As shown in Figure 11(a), the hystereticbehavior is identical to that of the conventional FP bearing.

The friction force on surface 1 is overcome at displacement u∗, which is given by

u∗ = (�1 − �2)Reff2 (34)

At this point, motion starts on surface 1 and stops on surface 2. When sliding occurs on surface 1,the displacement on surface 2 remains u∗. The force–displacement relationship is

F = W

Reff1(u − u∗) + Ff1 (35)

Upon reversal of motion, the bearing will unload by 2Ff2 and motion will initiate on surface 2.Motion continues on surface 2 over a distance 2u∗ until the bearing has unloaded by 2Ff1, at whichpoint motion will resume on surface 1 and stop on surface 2. The force–displacement behaviorfor this sliding regime is shown in Figure 11(b).

Sliding occurs on surface 1 only until the slide plate starts to bear on the displacement restrainer,which occurs at total displacement udr1, given by

udr1 = d1 + u∗ (36)

For u>udr1, the force–displacement relationship is

F = W

Reff2(u − udr1) + W

Reff1d1 + Ff1 (37)

Upon reversal of motion, the bearing unloads by 2Ff2 and sliding occurs on surface 2. Slidingresumes on surface 1 and stops on surface 2 when the bearing unloads to Fdr1 − 2Ff1. This isshown in Figure 11(c).



Sliding Regime III: u* < umax < udr1

Total Displacement, u(c)

Hor

izon

tal F

orce

Sliding Regime III

Sliding Regimes I - II

Fdr1

udr1

Fdr1 - 2Ff1

WReff2

WReff2

2Ff2

Total Dispalcement, u(a)

Hor

izon

tal F

orce Ff2

2Ff2

W Reff2

Sliding Regime I:umax < u* W

Reff2

Total Dispalcement, u(b)

Hor

izon

tal F

orce

Sliding Regime IISliding Regime I

Sliding Regime II:u* < umax < udr1

2Ff2

2Ff1

W Reff1

W Reff1

W Reff2

u*

2u*

Ff1

Figure 11. Force–displacement relationship of the modified single FP bearing: (a) for sliding regime I;(b) for sliding regime II; and (c) for sliding regime III.

5. BEHAVIOR OF THE DOUBLE FP BEARING WITH DIFFERENTDISPLACEMENT CAPACITIES

Adaptive behavior can also be achieved using the double FP bearing with sliding surfaces ofdifferent displacement capacities. For example, in a configuration with �1<�2, Reff1�Reff2 andd1�d2, the bearing will exhibit high stiffness and low friction initially, then decrease in stiffnessand increase in friction with increasing displacement and finally stiffen with even greater frictionprior to achieving the bearing’s total displacement capacity (provided that the slider impacts



the displacement restrainer on surface 1 prior to surface 2). Unlike the configurations presentedpreviously, prior to achieving the total displacement capacity of the bearing, it is possible to haveboth an increase in the stiffness and an increase in the effective friction.

Assume a configuration of double FP bearing to give the adaptive behavior just described, thatis �1<�2, Reff1�Reff2 and d1�d2. The behavior of the double FP with �1 �= �2 prior to contactingthe displacement restrainer has been described in [20]. Sliding will start on surface 1 when theapplied lateral force exceeds Ff1. Motion continues on surface 1 only for a distance of u∗, at whichpoint motion starts on surface 2. With increasing displacement, sliding continues on both surface 1and surface 2 until the slider begins to bear on one of the displacement restrainers. To ensure thatthe slider contacts the displacement restrainer of surface 1 prior to surface 2, it is necessary thatFdr1<Fdr2.At the instant the slider makes contact with the displacement restrainer on surface 1, u1 = d1

and the total displacement is

udr1 = d1

(1 + Reff2

Reff1

)− (�2 − �1)Reff2 (38)

When the total displacement exceeds udr1, sliding occurs only on surface 2 and the force–displacement relationship is given by

F = W

Reff2(u − d1) + Ff2 (39)

After the slider makes contact with the displacement restrainer on surface 1, upon reversal ofmotion different orders of unloading are possible. Sliding resumes on surface 1 when the bearinghas unloaded to Fdr1 − 2Ff1 and sliding resumes on surface 2 when the bearing has unloaded by2Ff2. Therefore, if

umax>udr1 + 2Reff2(�2 − �1) (40)

sliding resumes on surface 2 prior to surface 1. These different types of possible unloading behaviorare denoted as regimes III(a) and III(b) in Figure 12.


Hor

izon

tal F

orce

Sliding Regime III(a)Sliding Regimes I - II

Sliding Regime III(b):

u > u + 2R ( - )


Hor

izon

tal F

orce

Sliding Regime III(b)Sliding Regimes I - III(a)

Sliding Regime III(a):

u < u + 2R ( - )

F

u

WR

WR

F - 2F2F

WR + R

WR

2F

WR

FF - 2F

uW

R + R

(a) (b)

Figure 12. Force–displacement relationship of the double FP bearing: (a) for sliding regime III(a); and(b) for sliding regime III(b).



6. CONCLUSION

The force–displacement relationships of three novel variations of multi-spherical sliding bearinghave been presented in this paper. Furthermore, general principles of operation have been explainedso that alternate configurations can be analyzed. It has been shown that when properly configured,these bearings provide stiffness and damping that change desirably with increasing displacement.These changes in stiffness and damping result from the various combinations of surfaces uponwhich sliding can occur. The appeal and potential of the bearings presented in this paper are notonly the performance and economic benefits that result from the adaptive behavior they exhibit, butalso the fact that they are simple passive devices. They are based on technologies and principlesthat are universally accepted in practice, just applied in an innovative manner. The interested readeris referred to the companion paper [17] for experimental verification of the theory that has beenpresented in this paper.

ACKNOWLEDGEMENTS

Financial support for this project was provided by the Multidisciplinary Center for Earthquake EngineeringResearch (Thrust Area 2) and Earthquake Protection Systems Inc. This support is gratefully acknowledged.

REFERENCES

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