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EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2002; 31:305332 (DOI: 10.1002/eqe.110)

Analytical model of structures with frictional

pendulum isolators

Jose L. Almazan and Juan C. De la Llera

Department of Structural Engineering; Ponticia Universidad Catolica de Chile; Casilla 306;Correo 22; Santiago; Chile

SUMMARY

This investigation centres on the development of a mathematically formal description of the dynamicresponse of structures isolated with the frictional pendulum system (FPS). It is shown that a theoreticallyexact model can be formulated to account for large deformation kinematics and the associated Peects in the isolators. The problem is of importance in light of the large deformations observed duringimpulsive ground motions like those that occurred during the Northridge, Kobe, Turkey, and Taiwanearthquakes. Besides, the model developed may be easily extended to other devices with kinematicconstraints other than the spherical one corresponding to the FPS. Results of the model are presented fortwo building examples. The rst one deals with the seismic response of a rigid superstructure supportedon two FPS isolators and is intended to provide a numerical example of the equations developed in thetext. The second example presents the three-dimensional earthquake response of a nominally symmetricstructure supported on four FPS isolators and subjected to dierent ground motions. Both examples pointout that small deformation kinematics may lead, in the case of impulsive motions, to discrepancies inglobal response quantities, relative to the actual response, as large as 30 per cent. These discrepanciesincrease up to 50 per cent for local response quantities such as normal isolator forces. Copyright ?

2001 John Wiley & Sons, Ltd.

KEY WORDS: frictional pendulum system (FPS); analytical model

INTRODUCTION

The frictional pendulum system (FPS) has become a widely accepted device for seismicisolation of new buildings, bridges, and industrial facilities, as well as for the retrot ofexisting structures [1]. The appeal of this device rests on the simplicity of the principlesthat govern its behaviour and the built-in self-centring action due to the concavity of thesliding surface (Figure 1(a)). During ground shaking, the slider moves on the spherical surface

Correspondence to: Juan C. De la Llera, Department of Structural Engineering, Ponticia Universidad Catolica deChile, Casilla 306, Correo 22, Santiago, Chile.

Contract=grant sponsor: Chilean National Fund for Research and Technology, Fondecyt: Contract=grant number:1000514, 2990069.Contract=grant sponsor: Funds for Foment and Technology, FONDEF; Contract=grant number: D96I1008.

Received 30 September 2000Copyright ? 2001 John Wiley & Sons, Ltd. Accepted 4 May 2001


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306 J. L. ALMAZ AN AND J. C. DE LA LLERA

Figure 1. (a) Components of a typical FPS: (1) spherical surface, (2) slider, and (3) stud. (b) Typicalstructural model and co-ordinate systems considered.

Copyright ? 2001 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2002; 31:305332


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STRUCTURES WITH FRICTIONAL PENDULUM ISOLATORS 307

lifting the structure and dissipating energy by friction between the spherical surface and theslider. Usually, the slider is locked on a vertical stud having a spherical hollowed end whichallows free rotation of the slider and a perfect contact with the sliding surface at all times(Figure 1(a)). To keep frictional forces relatively low, say a friction coecient = 510

per cent, the stainless steel slider is usually covered by a resistant teon layer.Most of the theoretical [24] and experimental research [2; 5] developed so far with the FPS

has been based on small-deformation constitutive laws of the devices. However, motivatedby the large deformations observed during the recent earthquakes of Northridge (1994), Kobe(1995), Turkey (1999), and Taiwan (1999), the large-deformation and the associated Peects may become an issue in the design of the isolators. As a result, coupling between thelateral and vertical motions, which is not considered in the small-deformation theory and thecurrently available structural analysis software [6; 7], needs further evaluation.

Experimental and analytical results suggest that the small-deformation hypothesis is accurateenough for estimating global building response quantities, such as storey and isolator defor-mations, or storey shears and torques. However, a recent study [8] showed that one of themost important aspects in modelling structures with FPS isolators is the correct evaluation ofthe normal force N, generated by the kinematic constraint imposed by the spherical surface.In order to evaluate correctly the normal force N, the vertical acceleration of the ground andthe true coupled lateral-vertical motion of the structure need to be considered. Because thelatter implies large deformations in the isolators, the FPS models available in the literature sofar need to be extended to account for these eects.

Consequently, it is the objective of this research to develop an analytical model for theanalysis of structures supported on FPS isolators considering large deformations and Peects. In this model each isolator is treated as a kinematic constraint, which can be arbitrary.The model used is an application of the approach known in rigid body dynamics as augmentedformulation. Besides being a nice tool for the analysis of structures with FPS isolators,the exact model presented herein helps in understanding the dynamic behaviour of these

structures; in particular, the natural separation between the pendular and frictional componentsof the isolator restoring force. Analytical and numerical examples of the earthquake responseof isolated structures are developed and studied in detail in order to evaluate the discrepanciesoccurring as a result of the hypothesis of small versus large-deformations. Moreover, theexact formulation constitutes a benchmark procedure that can be used to validate otherapproximate models for dynamic analysis of buildings isolated with FPS isolators.

FORMULATION OF THE PROBLEM

As opposed to other isolation devices such as rubber bearings, the FPS denes an isola-tion interface which is not planar. When the earthquake ground motion is such that sliding

of the structure occurs, each isolator is forced to move according to a nonlinear kinematicconstraint (spherical surface). Otherwise, the slider is xed to the spherical surface and theconstraint becomes a 3D spherical joint that allows the superstructure to rotate freely about that

point.There exist two approaches that can be followed to establish the equations of motion of

a structure with constrained support motions: (i) the embedding technique, and (ii) the aug-mented formulation. The embedding technique [9] works with geometric co-ordinates that are



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all independent, i.e. a base for representing the motions of the structure. This requires thatthe constrained motions be expressed as a function (usually non-linear) of the independentco-ordinates. On the other hand, the augmented formulation [9] allows to express the con-strained motions in terms of co-ordinates that need not be independent. For that purpose an

extended set of co-ordinates is dened, some of them interrelated by the constrained motionsof the structure. As with any constraint, the corresponding reactions are unknowns of the

problem and, hence, the size of the problem using the augmented technique is increased intwice the number of constraints. Although either formulation may be used to solve the prob-lem, the latter has been preferred in this case despite the larger dimension because it leads toa less coupled set of equations of motion, facilitating the numerical integration.

With the intention of clarifying the presentation, the augmented formulation presented nexthas been cast into a well known structural engineering analysis format. In spite of that, sincethe notation used in the equations may become cumbersome to the reader, an example has

been included in Appendix A to help in their interpretation.

EQUATIONS OF MOTION

The model considered for the superstructure is a conventional one with six degrees of freedomper node and rigid-in-plane oor diaphragms (Figure 1(b)). The degrees of freedom q of thestructure are measured with respect to frame 1, xed to the ground. On one hand, themodel assumes small deformations in the structural elements, i.e. linear kinematics; but onthe other, large deformation kinematics are considered for the FPS isolators. Notice that itonly makes sense to talk about large deformations at the FPS interface and not for thesuperstructure; otherwise, the whole concept of isolation in building design would be of littleuse. The link between the superstructure and the FPS system is through conventional nodesthat include six degrees of freedom, three translations and three rotations (Figure 1(b)). The

model developed includes both possibilities for the isolator placement, denoted hereafter asupward and downward (Figure 2). Although they are conceptually equivalent in terms of theirisolation eect, they have quite dierent implications for the design of the superstructure andfoundation system.

There exist two assumptions in the equations presented next that need to be stated clearly.First, the isolators are assumed to be always in a sliding phase, and, second, uplift of thestructure is impaired. Although sticking and uplift eects can be incorporated in the analyticalmodel presented by changing the sliding constraints by either hinges or releases, the increasein complexity of the model would opaque the neatness of this formulation. Moreover, for alarge number of cases the structural model developed next based on these assumptions willlead to earthquake responses that are in excellent agreement with the true response in thestructure. Consequently, the study of these two phenomena, their numerical implementationand interpretation is left for a sequel paper in which a physical model for the FPS with largedeformations will be introduced.

The most general equations of motion of a linear structure with n degrees of freedom andsupported on p FPS isolators may be written as

M q + Cq + Kq + Q(n) + Q(\) =MLww (1)



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Figure 2. FPs bearing in (a) downward and (b) upward position and denition of degrees of freedom.

where q{n1} = [q1; : : : ; qi; : : : ; qn]T

is the vector of augmented degrees of freedom and in-cludes the constrained motions of the sliders along the spherical surfaces; M, C and K are then n well known mass, damping, and stiness matrices, respectively; w = [ ugx ugy ugz + g]

T

is the ground motion excitation vector, where ugi is the ith component of ground acceleration,and g represents the gravity; Lw{n3} the excitation inuence vector; and Q

(n) and Q(\) arethe normal and tangential (frictional) components of the FPS forces, respectively, applied inthe degrees of freedom q of the structure.



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It is convenient to start by expressing the kinematic isolator constraints in terms of thedegrees of freedom q of the structure as

(q) = 0 (2a)

where (q){p1} = [G1(q); : : : ; G k(q); : : : ; G p(q)]T is the matrix of isolator constraints. By tak-ing the rst and second time derivative of Equation (2a), the following relationships areobtained between the degrees of freedom q, and their velocities and accelerations, q and q(Appendix B)

(q; q) = J(q)q = 0 (2b)

(q; q; q) = A(n)(q; q) + J(q) q = 0 (2c)

where and are the rst and second derivatives of the matrix of constraints (q) withrespect to time; J(q){pn} = @(q)=@q is the Jacobian matrix of the constraints with elements

J(k; i) = @Gk=@qi; and A(n)

{p1} = (@(Jq)=@q)q. Equations (2b) and (2c) imply that: (i) the veloc-

ities q must be orthogonal to the gradients of the constraints, and (ii) the relative accelerationsq projected in the normal direction to each constraint must equal A(n)(q; q). After somealgebra (Appendix B), it can be shown that the kth component of this vector has the form

A(n)(k) = qTHkq (3)

where

Hk(nn) = @2Gk=@q@q

T (4)

is the Hessian matrix of the kth constraint Gk with elements Hk(i; j) = @2Gk=@qi@qj. As it

will be shown later, each component A(n)(k) corresponds to the relative normal accelerationbetween the slider and the spherical surface.

The procedure used to integrate Equation (1) while satisfying constraints (2a) (2c) is

similar to that used in conventional structural analysis. Such procedure has the following steps:(i) dene a local system of co-ordinates for each isolator; (ii) compute the deformations anddeformation velocities of each isolator in terms of the degrees of freedom q of the structure byconsidering the non-linear constraints (2a)(2c) (kinematics step); (iii) compute the restoringforces generated by each isolator in local co-ordinates (actiondeformation step); (iv) apply aset of virtual displacements to compute the non-linear forces Q(n) and Q(\) acting along theglobal degrees of freedom q of the structure (equilibrium step); and (v) nd the accelerationsq satisfying Equation (1) and the constraints (2a)(2c). Next, this sequence of ve steps isdescribed in detail.

Local system of co-ordinates

Shown in Figure 2 is a detailed view of the FPS in the downward and upward positions.Consider a local system of co-ordinates 2 = {Ok: xk yk zk}, which is solidary to the sphericalsurface at the origin Ok. The convention is that the local axis zk always points toward thecentre of curvature Ck, forming with the two other unitary vectors a right-handed triplet,

It is assumed that the isolator constraints are the only non-linear constraints in the structure; other linear constraintsare considered in the assemblage of the structural matrices M, K and C.



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such that xk yk =zk. The deformation of the isolator Tk =OkSk = [xk yk zk]

T denotes theinstantaneous position Sk of the slider relative to the origin Ok.

The constraint imposed by the spherical surface on the deformation Tk of the slider can beeasily written in local co-ordinates at any instant. Indeed, Tk must satisfy the equation of a

sphere tangent to the origin at point Ok, i.e.

Gk(Tk) =1

2Rk[2xk +

2yk

+ (zk Rk)2 R2k] = 0 (5a)

and its time derivative

Gk(Tk; Tk) =1

Rk[xkxk + ykyk + zk(zk Rk)] = 0 (5b)

where Rk is the radius of the spherical surface of the isolator. The factor 1=(2Rk) has beenintroduced in Equation (5a) in order to make the gradient of the constraint a unitary vector,i.e., Gk= 1. Please notice that Equations (5a) and (5b) do not explicitly constrain the

degrees of freedom q and q as required by Equations (2a) (2c). Therefore, the kinematicsteps shown next are required in order to relate the isolator deformations Tk with the structuraldegrees of freedom q.

Kinematics

It is the objective of this section to establish a relationship between the kth isolator defor-mations Tk and the degrees of freedom q of the structure. Consider rst the intermediate set

of nodal displacements uk = [u(J)k ; u

(I)k ],

with components u(J)k = [u

(J)x u

(J)y u

(J)z r

(J)x r

(J)y r

(J)z ]T and

u(I)k = [u

(I)x u

(I)y u

(I)z r

(I)x r

(I)y r

(I)z ]

T, dening the motions of nodes J and I connected by the kth FPS(Figures 1 and 2).

Starting from these displacements, a full non-linear kinematic relationship between uk andTk may be established [10]. This relationship may be conveniently simplied by preservinghigher order terms for the displacements but simultaneously assuming small rotations in theisolator. The latter is motivated by the physical observation that nodal rotations tend to besmall for essentially all practical cases. Based on these kinematic assumptions, it is possibleto construct by simple geometry a quadratic non-linear relationship of the form [10]

Tk = Lk(uk)uk (6)

where Lk(uk) represents a rst order approximation for the fully non-linear kinematic trans-formation between Tk and q; and Lk(uk) can be expressed as [10]

Lk =

1 0 0 0 lJ 0 1 0 0 0 (uz lI) uy

0 1 0 lJ 0 0 0 1 0 uz lI 0 ux

0 0 1 0 0 0 0 0 1 uy ux 0

(7)

Throughout this paper the semicolon denotes dierent rows of a matrix.



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where ux = u(J)

x u(I)

x , uy = u(J)y u

(I)y , and uz = u

(J)z u

(I)z , are the relative displacements

between nodes J and I; and lI and lJ are the vertical distances in the undeformed congu-ration between the origin Ok and nodes I and J, respectively. The upper and lower signs inthis equation correspond to the downward and upward position of the isolator, respectively

(Figure 2). Furthermore, since the deformations of the superstructure are small, the nodaldisplacements uk of the kth isolator may be expressed as a linear function of the degrees offreedom q of the structure

uk = Pkq (8)

where Pk is the kinematic transformation matrix between the degrees of freedom of the struc-ture and the nodal isolator displacements. Hence, the resulting relationship Tk = Lk(uk)Pkq.

Similarly, the deformation velocity of the isolator Tk can be obtained by taking the time

derivative of Equation (6), i.e., Tk = Lkuk, where uk = Pkq are the nodal isolator velocities

and the transformation matrix Lk = @Tk=@uk between Tk and uk. As before, it can be shownthat Lk may be approximated by the following expression [10]:

Lk =

1 r(I)z r

(I)y 0 lJ 0 1 r

(I)z r

(I)y 0 (uz lI) uy

r(I)z 1 r(I)

x lJ 0 0 r(I)

z 1 r(I)

x uz lI 0 ux

r(I)y r(I)

x 1 0 0 0 r(I)

y r(I)

x 1 uy ux 0

(9)

By using this result, the nal relationship between the isolator deformation velocities and ve-

locities of the structural degrees of freedom becomes Tk = @Tk=@uk@uk=@q dq=dt= Lk(uk)Pkq =Lkq.

Equations (6)(9) enable us to express the constraint Equations (2a)(2c) in terms of thedegrees of freedom q of the structure. Indeed, each row of the Jacobian dened in Equa-tion (2a) can be expressed as (Appendix B)

J(k; 1 : n) =@Gk@q

=GTkLkPk (10)

where the gradient Gk = @Gk=@Tk of the kth constraint in local co-ordinates is, for the caseof a spherical constraint, equal to (Equation (5))

Gk =[xk=Rk yk=Rk (zk=Rk 1)]T (11)

Please notice that the gradient has unitary norm. Analogously, the Hessian matrices intro-duced in Equation (3) may be computed by the matrix product (Appendix B)

Hk = PT

k LTkHkLkPk (12)

where Hk = @2Gk=@Tk@T

Tk = 1=RkI is the Hessian matrix of the constraint Gk expressed in local

co-ordinates and I the rank 3 identity matrix. As it should, the Hessian Hk has a simplestructure in this case, since it is related to the inverse of the radius of curvature of thekth isolator sliding surface, which is constant. Substituting Equation (12) into the normal



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Figure 3. Action of the structure on the isolator in the downward position: (a) normaland frictional components in the local system of co-ordinates; (b) equivalent nodal forces

and torques in the 1 system of co-ordinates.

acceleration expression (Equation (3)), it can be proven that

A(n)(k) = qTHkq =

TTk

(LkPkq)T HkTk

(LkPkq) =Tk

2

Rk

(13)

Two comments on Equation (13). First, as one would expect, the normal accelerationcoincides with the centripetal acceleration of a particle moving on a spherical motion (pendularmotion). Second, although Equation (13) has been applied to the case of a spherical slidingsurface, it could be used as well for other functional forms of kinematic constraints. Next,the solution requires to state the actiondeformation relationship for the FPS.

Actiondeformation

Shown in Figure 3(a) is the resultant isolator force fk projected into the normal and frictional

components f(n)k and f

(\)k , respectively, generated as a result of the applied deformation Tk

and the corresponding deformation velocity Tk of the isolator. Since the motion in the normaldirection is known to be zero, the normal force in the isolator is unknown. However, sincethe direction of this normal force is known to be perpendicular to the spherical surface, onlyits magnitude is unknown. This can be expressed mathematically by the following constitutiverelationship:

f(n)k = kGk =Nknk (14)



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where k is the Lagrangian multiplier associated with the normal constraint imposed by Gkand equal in this case to the magnitude of the normal force Nk. Please notice that in thisdenition the unitary normal vector nk =Gk always points outward of the sliding surface(Figure 3(a)).

On the other hand, the frictional forces may be computed from the constitutive Coulombfriction relationship

f(\)k = kNktk (15)

where k represents the sliding coecient of friction which may or may not be assumeddependent on velocity and contact pressure [11]; and tk = Tk=Tk (Figure 3(a)), is the unitaryvector tangent to the trajectory of the isolator, i.e. the direction of the isolator velocity. Next,the nal step in the formulation is to state the equilibrium conditions.

Equilibrium

In this section the normal and frictional force components in each isolator are projected intothe global system of co-ordinates attached to the ground 1. By using the principle of virtualdisplacements, the projection of these components for the kth isolator are

Q(n)k =

@Tk@q

Tf

(n)k = L

Tkf

(n)k (16)

and

Q(\)k =

@Tk@q

Tf

(\)k = L

Tkf

(\)k (17)

where Lk = LkPk is the whole kinematic transformation matrix for the kth isolator. SubstitutingLk into Equations (16) and (17) and adding vectorially both force components, the total FPSforce fk projected in global co-ordinates is

Qk = PTk Lk

Tfk = PTk Fk (18)

where fk = [fxk fyk fzk]T = f

(n)k + f

(\)k ; and Fk{121} = Lk

Tfk = [F(J)

k ; F(I)

k ] is the force vector fornodes J and I expressed in the co-ordinate system 1. By using Equation (9) the force

Fk = LkTfk may be computed as

F(J)

k = [(fxk fykr(I)

z fzkr(I)

y ); (fyk fxkr(I)

z fzkr(I)

x ); (fzk fxkr(I)

y + fykr(I)

x );

; : : : (fyklJ); (fxklJ); 0]T (19)

F(I)k = [(fxk + fykr

(I)z fzkr

(I)y ); (fyk fxkr

(I)z fzkr

(I)x ); (fzk fxkr

(I)y fykr

(I)x )

; : : : (fyk(uz lI) fzkuy); (fxk(uz lI) fzkux); (fxkuy fykux)]T

(20)



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in which, as before, the upper and lower signs correspond to the upward and downwardposition of the isolator, respectively. Shown in Figure 3(b) are the force components inlocal and global co-ordinates for the downward position of the isolator. Notice that sincethe deformations between the slider and node J are small, the P eect (equilibrium in the

deformed position) is included only in the bending and torsional moments transmitted to nodeI (components 4 through 6 of F

(I)k ). Thus, in the downward position of the isolator, the P

eect is transmitted to the portion of the structure below the isolation system, which in thecase of a building usually coincides with the foundation of the structure. On the other hand,if the FPS isolator is positioned upward, the same P eect is transmitted to the resistingelements of the superstructure. Either solution may be possible, but the designer must beaware of the dierence between both cases, which is sometimes overlooked as a consequenceof the essentially identical isolation eects in both situations.

By adding the FPS forces for all isolators in the structure, the restoring force componentsin global co-ordinates Q(n) and Q(\) introduced earlier in Equation (1) end up being

Q(n) =p

k=1

Q(n)k =p

k=1

LTkf(n)k = L

TF(n) (21)

Q(\) =p

k=1

Q(\)k =

pk=1

LTkf(\)k = L

TF(\) (22)

where F(n) = [f(n)1 ; : : : ; f

(n)k ; : : : ; f

(n)p ] and F

(\) = [f(\)1 ; : : : ; f

(\)k ; : : : ; f

(\)p ] are the column vectors of

normal and frictional forces in the isolators; and L = [L1; : : : ; Lk; : : : ; Lp] is the compositekinematic transformation matrix for the whole system.

An interesting result can be obtained for the normal force component by means of introduc-

ing the expression for f(n)k (Equation (14)) and Lk into Equation (21), leading to (Appendix B)

Q(n)

= JT

(23)

where J is the Jacobian matrix of the constraints and = [N1; : : : ; N k; : : : ; N p]T is the column

vector of normal forces (Lagrangian multipliers). Although Equation (23) is a well knownresult in Lagrangian dynamics that could have been stated directly [9], it seemed useful torecast it by using a conventional structural analysis approach. The setup is now complete inorder to integrate the exact large deformation equations of motion of structures supportedon FPS isolators.

INTEGRATION OF EQUATIONS OF MOTION

This section describes some interesting aspects of the time integration of the dierential equa-tions of motion of a structure supported on FPS isolators (Equations (1) and (2)). Let usstart by recasting these equations into a single system of second order coupled dierentialequations, i.e.

M JT

J 0

q

=

Qe()

A(n)(q; q)

(24)



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where Qe() =MLww (Kq + Cq + Q(\)()) is the resultant of all external forces acting

on the structure with exception of the normal restoring force vector Q(n) =JT. Beforeattempting an integration algorithm, it is important to notice from Equation (24) that theaccelerations q and the normal forces may be computed once q and q are known. Since

the total force Qe is a function of , such computation corresponds to the solution at each stepof a system of n+p non-linear equations. This can be accomplished by using any algorithm tosolve non-linear equations, such as xed-point iteration or Newton Raphson. It turns out thatfor the integration procedure used, less than three xed-point iterations are required to achieveconvergence. The procedure assumes in the rst iteration that Q(\) = 0, and determines q and from Equation (24). Such is used to compute Q(\)() in the second iteration and soforth; the iteration converges swiftly.

The integration in time of Equation (24) may be performed more eciently by writing theequations of motion of the structure as a rst order system of dierential equations. In orderto do so, the state of the system z = [q q]T is dened. Consequently, the rst order systemof equations can be written as

z =q

q

= g(z; t) (25)

where g(z; t) is a non-linear function of q and q only. Equation (25) seems simple to compute,but it must be recognized that the last n equations are precisely those dened by the solutionof q from Equation (24). In this format, Equation (25) may be integrated by any of thewell known rst order integration strategies such as the explicit fourth order RungeKutta

procedure used in this investigation. To better understand the integration of Equation (25),let us start considering that z, q, and are known at instant t. To compute these variablesat instant t+ t, Equation (25) is written as

z(t+ t) = z(t) + t+t

t

g(z; ) d (26)

where the integral is evaluated in this investigation by using the RungeKutta algorithm.Such algorithm uses predictors of z at four intermediate instants in the interval (t; t + t)and requires evaluations of g(z; ) at these points. Since the last n equations of g(z; ) are q,each of these evaluations requires the solution of Equation (24) for q. Such solution is betterexplained in Figure 4 where a pseudo code version of the implemented procedure is presented.

By looking at Equation (24) it is apparent that the FPS constraints are satised in termsof the accelerations q as stated by Equation (2c). It is an important numerical aspect ofthe integration procedure developed to guarantee that the constraints as stated by Equations(2a) and (2b) are also satised [9]; otherwise the solution may drift away from the truesolution. Indeed, Equation (2c) is also satised by a perturbed solution q() of the form

q() = q() + b1 + b2, where q() is the true solution, and b1, b2 are constants that dependon the initial conditions ( = 0) of the integration interval (t; t+ t). A plausible numericalsolution to this problem is presented next.

Let us assume rst that the response of the structure at instant t is known and that the valuesq(t) and q(t) at that time satisfy the kinematic constraints given by Equations (2a) and (2b).Because the values q(t+t) and q(t+t) are computed from a numerical integration of Equa-tion (25) for the interval (t; t+t); they will fail in general to satisfy Equations (2a) and (2b).



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Figure 4. Pseudo code algorithm used to solve Equation (24).

As said before, they do satisfy Equation (24), i.e. the dynamic equilibrium and the secondderivative of the constraints, but not the actual constraints. Therefore, as said before, any so-lution of Equation (1) diering from the true solution in a constant or linear term will equallysatisfy Equation (24)other sources of this compatibility error are the second order approxi-mation of the true non-linear kinematic transformation (Equation (6)), and the accuracy of the

integration procedure used. The error in satisfying Equations (2a) and (2b) for the kth isolatormay be dened as the dierence between the deformation in the zk-direction,

zk

, computedfrom Equation (6), and the deformation, zk, computed from the true equation of the constraint(5a). This compatibility error in the local vertical deformation, k, may be evaluated as

k =

zk zk = (uz r

(I)y ux r

(I)x uy) (Rk

R2k (

2xk

+ 2yk)) (27)



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and its velocity as

k =

zk

zk= (uz r

(I)y ux r

(I)y ux r

(I)x uy r

(I)x uy)

xkxk + ykyk

(Rk zk)

(28)

In order to avoid that these integration errors accumulate in the response of the structure,the vertical nodal displacements and velocities of the dierent isolators are corrected usingEquations (27) and (28), so as to guarantee that k and k are equal to zero at every time step.Then, for isolators in downward position (Figure 2(a)), node-J displacements are corrected

as u(J)

z corr = u(J)

z unc k and u(J)

z corr = u(J)

z unc k, while for the upward position (Figure 2(b)) thesame correction is imposed to node-I displacements. This correction will force the slider toremain in contact at all times with the sliding surface.

A nal aspect related to the integration of the Equation (24) in how to deal with the initialcondition of the system. Such condition correspond to the deformations of the structure dueto gravitational loads. One possible solution to the initial condition corresponds to q(0)= 0and q(0) evaluated as

q(0)= K10 P0 (29)

where P0 =MLw[0 0 g]T is the vector of gravitational loads; and K0 is the stiness matrix

of the structure with de FPS isolators modelled as 3D hinges. This assumption implies thatsliding of the structure does not occur as a result of the application of the gravitational loads,which is true in most cases. It is important to emphasize that the initial normal load vector(0) need not be computed since it depends on q(0) and q(0) as stated by Equation (24).Finally, if sliding occurs under gravitational loads, the initial condition may be computed byapplying gravitational loads w(t) = [ 0; 0; g r(t)]T through a ramp function r(t) dened as

r(t) =t=Tr if t6Tr

1 if tTr

(30)

where Tr is an arbitrary and suciently long time so as to achieve a stationary condition inthe system (q(Tr) 0).

EXAMPLES

A convenient vehicle to better understand the previous equations will be the two buildingexamples developed in this section. The results presented next are intended also to showthe expected discrepancies between the responses obtained from large (LDM) and small-deformation (SDM) models [6; 7]. The latter model is derived from the same exact solution

by replacing the spherical sliding surface by a planar surface and an elastic horizontal springwith variable stiness keq(t) =N(t)=R (Figure 5(b)). Moreover, for the SDM the kinematic

transformation matrices Lk (Equation (7)) and Lk (Equation (9)) are identical matrices sincede P eect is not included in such model.

Let us consider rst the planar motion of a rigid superstructure (Figure 5) supported on twoFPS isolators of radii R = 100cm and constant friction coecient . Because of the simplicityof the example, it is possible to construct an analytical solution that helps interpreting the



15/28


Figure 5. System considered for example 1: (a) large deformation model (LDM), and(b) small deformation model (SDM).

dierent steps and matrices that participate in the solution. The rigid body motion of thesuperstructure is dened by three degrees of freedom q = [qx qz q]

T, two translations andone rotation, and two kinematic constraints imposed by the isolators. In this problem, the

system is subjected to two excitations, a velocity impulse and a ground motion correspondingto the NS and vertical components of the Sylmar record (Northridge, 1994). It can be shownthat the mass matrix M of the system is diagonal with translational mass m and polar momentof inertia m2; the stiness K as well as the damping matrix C are both equal to zero. Therigid superstructure is dened with a height to base aspect ratio H=B.

Details of the closed-form analytical solution of this problem may be found in Appendix A; thesolution is actually a very interesting exercise since it clearly states the dierences betweenthe LDM and SDM. In order to state more clearly these discrepancies, shown in Table I arethe relevant matrices for both models, assuming = 0 (pendular action only) and weight Wfor the superstructure. In this example, Equation (24) may be solved explicitly for the normalforces Nk as a function of the degrees of freedom q. Notice that for the SDM the externalloads Qe include both, the horizontal force (W=R)qx and the torque H(W=R)qx caused by

the horizontal isolator springs. It is apparent from this table that the solution for the SDM maybe obtained by making R in the exact LDM. Moreover, Table I shows the proportionaland antisymmetric eect created by the aspect ratio of the superstructure H=B on the normalisolator forces as a result of the overturning of the structure. It is this antisymmetry in thenormal forces of the isolators the one that creates a plan asymmetry in the FPS isolationinterface, leading to accidental torsion in structures subjected to real earthquakes motions[12].



16/28


TableI.

Summaryofmatrixresultsforthe

large(LDM)andsmalldeformationmodel(SDM)forexample1(=

0).

Matrices

Largedeformatio

n

Smalldeformation

model

mod

el

Nodalmatrices

P1

1

0

H

0

1

B

0

0

0

1

0

H

0

1

B

0

0

0

P2

1

0

H

0

1

B

0

0

0

10

H

01

B

00

0

Kinematictransf

ormationmatrices

L1=

L2

1

0

0

0

1

0

1

0

0

0

1

0

Constraints

G1

(1

)

12R

1(

2 x1+(z1R1)

2)=

0

z1=

0

G2

(2

)

12R

2(

2 x2+(z2R2)

2)=

0

z2=

0

Gradients

G

1(

1)

1 R1

[x1

(z1R1)]

T

[0

1]T

G

2(

2)

1 R2

[x2

(z2R2)]

T

[0

1]T

Jacobianmatrix

J

1 Rqx

|(qz

R)|

Hqx+

B(R

qz

)

qx

|(qz

R)|

Hqx

B(R

qz

)

0

1

B

0

1

B

Hessianmatrices

H1=

H2

1 RI{

33}

0{3

3}

Force

Qe

[0;

W;

0]T

W Rqx;

W;

HW Rqx

T

Normalacceleration

A(n)

=an

1

an

2

1 R[

1

2;

2

2]T

0{2

1}

Normalforceve

ctor

=

N

1N

2

W 2

cos()

H Bsin()

cos()+

H Bsin()

+man 2

1

H Btan()

1+

H Btan()

W 2

1 1

+H 2B

W Rqx

11

Totalnormalforce

NtW

cos()+an g

1



17/28


Figure 6. Impulse velocity response of the structure of example 1 for frictioncoecient = 0 (left) and 0.07 (right).

Shown in Figure 6 is a comparison between the LDM and SDM response of the structureof Figure 5 subjected to an initial velocity of the centre of mass (CM) of 200 cm =s in theX-direction. Two frictional coecients are considered in the analysis, =0 (left) and 0:07(right). The rst row of plots presents the displacement history of qx, the second row, thenormal force over the weight of the structure W, and the third row, the total normal force(Nt =N1 + N2) over the weight W. It is interesting to notice that the assumption of smalldeformations produces a slight shortening of the apparent nominal isolated vibration periodof the structure dened at 2 seconds, i.e. the SDM tends to overestimate the stiness ofthe isolation system. Further, as it has been shown previously [8], the small deformationassumption leads to a conservative displacement demand which is within 10 per cent relativeto the exact response. However, the normal force in isolator #1 is underestimated at severalinstants by the response of the SDM. This eect is also apparent in the resultant of thenormal forces of both isolators, which diers considerably from 1 as obtained from the SDM.Finally, results of Figure 6 show that although the frictional coecient aects considerablythe traces of the response, the maxima remains relatively the same.

The hysteresis loops for the initial velocity response of the structure and = 0:07 arepresented in the rst row of plots of Figure 7. From left to right, the plots show in columns the

normalized total pendular force (f(n)

1 + f(n)

2 )=W, the normalized total frictional force (f(\)

1 +

f(\)

2 )=W, and the normalized total force (f1 + f2)=W. As shown in the gure, the SDMoverestimates the FPS displacements as well as the peak total forces [8]. This is explained bythe counteraction of both terms of the normal force, i.e., Nt=W = cos() + an=g. While the rstgeometric term leads to a predominant softening of the isolator force as the angle increasesfor large deformations (Figure 5), the second term leads to a stiening of the isolator that



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Figure 7. Comparison of the forcedeformation hysteresis loop for the structure of example 1( = 0:07)subjected to: (a) velocity impulse of 200 cm=s, and (b) the horizontal (NS) and vertical components

of Sylmar record (Northridge, 1994).

predominates the response for small isolator deformations; their combined eect is presentedin Figure 7 and shows that smaller peak deformations and forces are obtained relative to the

SDM. Finally, the second row of plots of Figure 7 shows the earthquake response of the samerigid superstructure system subjected to the horizontal and vertical components of the Sylmarrecord (Northridge, 1994). Although, considerably more complex, the earthquake response ofthe structure shows the same trends as stated above for the initial velocity. As shown bythis example, the discrepancies between the LDM and SDM is bounded by, say 10 per centerror, which is rather small. Unfortunately, this is not always the case as shown by the nextexample.



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Figure 8. Structural system considered in example 2: (a) rst storey planview, (b) lateral view ofresisting plane 1, and (c) 3D structural model.

The second example considered is a three-dimensional nominally symmetric two-storey R=Cframe structure (Figure 8). Four isolators of radii R = 150cm are installed in upward positionon top of four identical columns in the rst storey. The fundamental apparent periods ofvibration of the isolated structure are 2.53, 2.52, and 1:51 s in the X, Y, and -directions,respectively; the periods of the structure before sliding occurs are 0:36 s, 0:35 s, and 0:21 s inthese directions, respectively.



20/28


Figure 9. Earthquake response of the structure of example 2 subjected to theNewhall record (Northridge, 1994).

The damping matrix of the system has been dened in block-diagonal form as

C =

C1 0

0 C2

(31)

where C1 and C2 are the damping matrices of the sub- and super-structure, respectively.Sub-matrix C1 corresponds to the classical damping matrix of the structure below the isolationlevel (nodes 18, Figure 8), working as an independent structure. In computing matrix C2 forthe superstructure, it has been assumed that nodes 913 (Figure 8) are completely restrained.Rigid body modes are then incorporated into the resulting damping matrix C2, so that there iszero damping associated with them [10]. Such denition of the damping matrix also ensuresthat there is no coupling between the two parts of the structure. In this example both matrices

were computed assuming a constant damping ratio of = 0:05.In this example the response of the structure subjected to the three components of increas-ingly stronger ground motions Newhall, Sylmar (Northridge, 1994), and TCU052 (Taiwan,1999) was studied. The earthquake response of the structure for the dierent ground motionsis compared in Figures 912. The rst three gures show three response quantities for the X-

and Y-direction of analysis; say for the X-direction, the roof displacement at the CM q(r)

x (t),

the deformation of the superstructure relative to the isolation level q(r=i)

x (t) = q(r)

x (t) q(i)

x (t),



21/28


Figure 10. Earthquake response of the structure of example 2 subjected to theSylmar record (Northridge, 1994).

and the normalized base shear Vx

=W. The trends of the response with increasing intensity ofthe ground motion are apparent from the gures. The peak responses and trends dier sub-stantially from earthquake to earthquake and discrepancies between the SDM and the LDMincrease with increasing ground motion intensity.

As before, roof displacements are overestimated by 1020 per cent by the SDM. Also,due to the P eect, the superstructure deformations predicted by the SDM underesti-mate the true deformations for the three ground motions in approximately 35, 40, and 48

per cent, respectively. Furthermore, base shears are usually overestimated by the SDM in lessthan 36 per cent. Moreover, the normal forces in the isolators are poorly predicted by theSDM (Figure 12), leading to underestimations in the three cases of 10, 19, and 32 per cent,respectively. In summary, the results show that although global structural responses such as

base shear and oor displacements are reasonably predicted by a SDM, local responses such

as the superstructure deformations and the normal isolator forces, need to be computed froma more accurate model.

CONCLUSIONS

An analytical model for the analysis of structures supported on FPS isolators experiencinglarge deformations has been developed and implemented. The model presented can be readily



22/28


Figure 11. Earthquake response of the structure of example 2 subjected to theTCU052 record (Taiwan, 1999).

implemented in a format similar to that present in currently available software packagessuch as SAP2000 [7] and, hence, become available to the engineering profession. Resultsobtained from the application of the model in the study of the earthquake response of structuressubjected to impulsive ground motions show that the SDM may lead to discrepancies up to20 per cent in global response quantities and over 50 per cent in local response quantitiessuch as the normal force in the isolators or the interstorey deformations. Although shownonly for two examples in this study, these results are representative of more general trends,also presented in an earlier publication [8].Therefore, it is concluded that a LDM, like the one

presented here, should be used in the design of structures prone to undergo large deformationsin the isolation system as a result of impulsive ground motions. Particularly important is thecorrect estimation of normal forces in the isolators that may lead to accidental torsion eects

not accounted for in the current design procedures.

ACKNOWLEDGEMENTS

This investigation has been supported by the Chilean National Fund for Research and Technology,FONDECYT under Grants # 1000514 and # 2990069 (doctoral dissertations). Part of the research was



23/28


Figure12

.Responsehistoryofthenormalf

orcesinthefourisolatorsofthestructureofexample2

,subjectedtotheNewhall,

Sylmar,

andTCU052records.



24/28


also supported by the Fund for Foment and Technology, FONDEF under Grant #D96I1008. The authorsare grateful for this support.

APPENDIX A

The structure considered in this example was introduced earlier in Figure 5 and consists ofa rigid superstructure of base dimension 2B and height 2H supported on two FPS isolatorsidentied hereafter by indices 1 and 2. Although the system considered is simple, its solution isillustrative and incorporates the most relevant aspects associated with the behavior of structuressupported on FPS isolators.

The motion of the superstructure is dened by three degrees of freedom q = [qx qz q]T.

Because the displacements of the superstructure are considered to be small, a linear trans-formation will exist between the degrees of freedom q and the displacement of any pointin the superstructure, in particular the nodes of the isolators. Thus, by Equation (8) the dis-

placements u1 and u2 of the isolator nodes are u1 = P1q and u2 = P2q, where the kinematictransformation matrices P1 and P2 are

P1 =

1 0 H

0 1 B

0 0 1

and P2 =

1 0 H

0 1 B

0 0 1

(A1)

On the other hand, since the problem is planar, only two deformations are dened for theisolators, namely the horizontal and vertical motion of each slider. By using Equation (9),

the deformations for isolator 1 are T1 = L1P1q = [x1 z1]T = [qx + Hq qzBq]

T; and for

isolator 2, T2 = L2P2q = [x2 z2]T

= [qx + Hq qz + Bq]T

, where

L1 = L2 =

1 0 0

0 1 0

(A2)

are the kinematic transformation matrices for the two isolators relating the nodal displacementsuk of the kth isolator and its deformations. Finally, the diagonal mass matrix for the systemis M = diag([m m m2]), where the mass m has been adopted as 1 and is the radius ofgyration of the superstructure relative to an axis perpendicular to the XZ plane passingthrough the CM. Notice also that by the symmetry of the problem q = 0.

The non-linear kinematic constraints (large deformations) imposed by isolators 1 and 2 may

be stated in this case as (Equation (5a))

G1(T1) =1

2R1[2x1 + (z1 R1)

2] = 0

G2(T2) =1

2R2[2x2 + (z2 R2)

2] = 0

(A3)



25/28


where R1 and R2 are the radii of curvature of isolators 1 and 2, respectivelyassumedequal to R from now on. By using Equation (11), the gradients of the constraints (A3)are G1(T1) = 1=R1[x1 (z1 R1)]

T and G2(T2) = 1=R2[x2 (z2 R2)]T. These gradients are

now used to compute the Jacobian matrix of the constraints using Equation (10), i.e.

J =

GT1 L1P1

GT2 L2P2

=

1

R

qx (qzR) Hqx + B(R qz)

qx (qzR) Hqx B(R qz)

(A4)

Similarly, the Hessians may be computed in the local system of coordinates as H1 = @2G1=

@T1@TT1 = (1=R)I and H2 = H1.

At this point it becomes interesting to compare these matrices with their small-deformationcounterparts. In the case of small deformations, the spherical constraints reduce to that of asliding joint (planar surface)

F1(T

1) =

z1= 0

F2(T2) = z2 = 0(A5)

having gradients F1(T1) = [ 0;1]T and F2(T2) = [ 0;1]

T. Analogously the Jacobian of theconstraints will look much simpler

J =

FT1 L1P1

FT2 L2P2

=

0 1 B

0 1 B

(A6)

and the Hessian matrices in local coordinates H1 = H2 = 0.Back to large deformations, the next step in the solution is the computation of the normal

forces in the isolators. The only external force acting on the system is the weight of thestructure, i.e. Qe = [0;W; 0]

T (Equation (24)). Replacing the Hessians in Equation (13), thenormal accelerations A(n) = [an1; an2]

T are

A(n) =

TT1 H1T1

TT2 H2T2

=

1

R

T1

2

T22

(A7)

Now, from Equation (24) it is possible to see that the normal forces in the isolators are: = HqQe + HA

(n), where H = (JM1JT)1 and Hq =(M

1JTH)T [9]. Expanding

this result it is possible to obtain the normal forces as

=

N1

N2

=

W

2

cos() H

Bsin()

cos() +H

Bsin()

+ man2

1 HB

tan()

1 +H

Btan()

(A8)

where sin() = x=R = qx=R (Figure 5) and Nt=W = (N1 + N2)=W = cos() + an=G.



26/28


The same analysis may be performed under the assumption of small deformations. For thatcase, the external load vector is redened as Qe = [Wqx=R; W; HWqx=R]

T and the normalacceleration vector is the zero vector, A(n) = 0. By using the general equation = HqQe +HA

(n) [9] it can be proven that

=

N1

N2

=

W

2

1

1

+

W

2

H

B

qxR

1

1

(A9)

where the total normal force Nt=W = 1.

APPENDIX B

This appendix summarizes the basic denitions of the dierential operators used in demon-strating the most relevant results of Equations (2) through (26). Included are also the demon-strations of these equations.

Denition 1. The partial derivative of a scalar function F with respect to each componentof a column vector X = [x1; x2; : : : ; xn]

T of dimension n 1, is dened for convenience as the1 n row vector (transpose of the gradient):

F;X =@F

@X=

@F

@x1

@F

@x2

@F

@xn

(B1)

Denition 2. The partial derivatives of each component of a vector function F of dimensionm 1 with respect to each of the components of the column vector X of dimension n 1,dene the m n Jacobian matrix:

F;X =

@F

@X = @F1@X ; @F2@X ; : : : ; @Fm@X (B2)In the proofs that follow the matrix (q){p 1} = [G1q; : : : ; G k(q); : : : ; G p(q)]

T represents thematrix of constraints for the p isolators and q is the n 1 vector of augmented degrees offreedom of the structure which also includes the constrained motions of the sliders along thespherical surfaces.

By using Equations (B1) and (B2) all results stated previously may be proven. Let usconsider rst Equations (2b) and (2c)

(q; q) =d

dt=

@

@q

dq

dt= J(q) q (B3)

which coincides with Equation (2b) and

(q; q; q) =d (q; q)

dt=

d[J(q) q]

dt=

@[J(q) q]

@q

dq

dt+

J(q)

@[

J(q) q]

@ q

dq

dt

=@[J(q) q]

@qq + J(q) q = A(n)(q; q) + J(q) q (B4)



27/28


with Equation (2c). In these equations, J(q) represents the Jacobian of the vector of constraintswith respect to q. Furthermore, A(n)(q; q) = (@[J q]=@q) q represents the p1 dimension vectorof normal accelerations. According to Equation (B2), the matrix @[J q]=@q has dimensionpn, with rows of the form @[Jk q]=@q, where Jk = J(k; :) = @Gk=@q is the kth row component

of the Jacobian matrix. Thus, each element of the matrix A(n) can be obtained as

A(n)(k) =@[Jk q]

@qq =

@[ qTJTk]

@qq = qT

@JTk@q

q = qT@

@q

@Gk@q

Tq = qTHk q (B5)

which coincides with Equation (3) and where

Hk =@

@q

@Gk@q

T=

@2Gk@q@qT

(B6)

was dened earlier in Equation (4) as the Hessian matrix of the constraint Gk.These results may be obtained alternatively by relating the structural degrees of freedom q

with the FPS deformations Tk (Equations (6) and (8)). By using the chain rule, the following

relationships may be proven

Jk = J(k; :) =@Gk@q

=

1 3@Gk

@Tk

3 12@Tk

@uk

12 n@uk

@q

= 1 nGTk Lk Pk (B7)

which corresponds to Equation (12). Similarly, Equation (14) for the Hessian Hk may becomputed by

Hk =@

@q

n n@Gk

@q

T

=

@

@Tk

n 3@Gk

@q

T

3 12@Tk

@uk

12 n@uk

@q

=

n3

@JTk@Tk

3 12@Tk

@uk

12 n@uk@q

= : : :

=

n 12@uk

@q

T 12 3@Tk

@uk

T

3 3

@2Gk@Tk@T

Tk

3 12@Tk

@uk

12 n@uk

@q

= PTk LTk HkLkPk (B8)

Finally, the restoring force Q(n), resulting from the normal constraints and using Equations(14) and (18) is

Q(n) =p

k=1

Q(n)k =

pk=1

LTkf(n)k =

pk=1

LTk [PTk L

T

k]

f(n)

k GkNk = : : :

=p

k=1

[PTk LTkGk] JT

k

Nk = JT1N1 + J

T2N2 + + J

TpNp = J

T (B9)



28/28


which coincides with Equation (23) and indicates that the global normal restoring force maybe computed as the product of the transpose of the Jacobian of the constraint matrix andthe vector of Lagrangian multipliers corresponding in this case to the normal forces in theisolators.

REFERENCES

1. Mokha A, Amin N, Constantinou M, Zayas V. Seismic Isolation Retrot of Large Historic Buildings. Journalof Structural Engineering ASCE 1996; 122:298308.

2. Zayas V, Low S, Mahin S, The FPS Earthquake resisting system. Report UCB=EERC-87=01, EarthquakeEngineering Research Center, University of California at Berkeley.

3. Zayas V, Low S, Bozzo L, Mahin S, Feasibility and performance studies on improving the earthquakes resistanceof new and existing buildings using the frictional pendulum system. Report UCB=EERC-89=09, EarthquakeEngineering Research Center, University of California at Berkeley.

4. Zayas V. Low S, Mahin S. A simple pendulum technique for achieving seismic isolation. Earthquake Spectra1990; 6:317334.

5. Al-Hussaini T, Zayas V, Constantinou M. Seismic isolation of multi-storey frame structures using sphericalsliding isolation system. Report NCEER-94-0007. National Center for Earthquake Engineering Research, State

University of New York at Bualo.6. Tsopelas P, Constantinou M, Reinhorn A. 3D-BASIS-ME: computer program for nonlinear dynamic analysis of

seismically isolated single and multiple structures and liquid storage tanks. Report NCEER-94-0010. NationalCenter for Earthquake Engineering Research, State University of New York at Bualo.

7. SAP 2000. Computers and Structures Inc.: Berkeley, CA, 1999.8. Almazan J, De la Llera J, Inaudi J. Modeling aspects of structures isolated with the frictional pendulum system.

Earthquake Engineering and Structural Dynamics 1998; 27:845867.9. Shabana A. Computational Dynamics. Wiley: New York, 1994.

10. Almazan J. Accidental and natural torsion in structures isolated with frictional pendulum system. Doctoral thesisdissertation, May 2001. In Spanish.

11. Constantinou M, Mokha A, Reinhorn A, Teon bearings in base isolation, Part II: modeling. Journal ofStructural Engineering ASCE 1990; 116:455474.

12. Almazan J, De la Llera J. Lateral torsional coupling in structures isolated with the frictional pendulum system.Proceedings, of the 12a World Conference of Earthquake Engineering (12WCEE), Auckland, New Zealand,January 30February 4, 2000, paper 1534=6=A.


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