SUBSTRUCTURE SYNTHESIS ANALYSIS AND HYBRIDCONTROL DESIGN FOR BUILDINGS UNDER SEISMIC

EXCITATION

César A. Morales Velasco

Dissertation submitted to the faculty of the Virginia Polytechnic Institute and StateUniversity in partial fulfillment of the requirements for the degree of

Doctor of Philosophyin

Engineering Mechanics

L. Meirovitch, ChairR. A. HellerM. P. Singh

S. ThangjithamH. VanLandingham

April 18, 1997Blacksburg, Virginia

Keywords: Substructure Synthesis, Base Isolation, Structural Control, Hybrid Control,Earthquake Engineering

SUBSTRUCTURE SYNTHESIS ANALYSIS AND HYBRID CONTROLDESIGN FOR BUILDINGS UNDER SEISMIC EXCITATION

César A. Morales Velasco

(ABSTRACT)

We extend the application of the substructure synthesis method to morecomplex structures, and establish a design methodology for base isolation and activecontrol in a distributed model of a building under seismic excitation. Our objective is toshow that passive and active control complement each other in such an advantageousmanner for the case at hand, that simple devices for both types of control are sufficientto achieve excellent response characteristics with very low control forces.

The Rayleigh-Ritz based substructure synthesis method proved to be highlysuccessful in analyzing a structure more complex than the ones previously analyzedwith it. Comparing the responses of the hybridly controlled building and theconventional fixed building under El Centro excitation, we conclude that the stresses arereduced by 99.6 %, the base displacement is reduced by 91.7 % and the required controlforce to achieve this is 1.1 % of the building weight.

iii

ACKNOWLEDGMENTS

The author would like to express immense gratitude to his advisor, Dr. LeonardMeirovitch, for his guidance and support during the past three years and a half. Sincereappreciation is also expressed to the members of his advisory committee, Dr. Robert A.Heller, Dr. Mahendra P. Singh, Dr. Surot Thangjitham and Dr. Hugh VanLandingham,for overseeing this research project. Dr. Singh and Dr. Thangjitham reviewed severaldrafts and their suggestions made this work better.

iv

TABLE OF CONTENTS

ABSTRACT iiACKOWLEDGMENTS iiiLIST OF FIGURES vLIST OF TABLES vi1. INTRODUCTION 1

1.1. Model of the building 11.2. Literature survey 21.3. Organization 4

2. SUBSTRUCTURE SYNTHESIS ANALYSIS 52.1. The substructure synthesis method 52.2. Superstructure disjoint kinetic and potential energies 62.3. Constraint equations and eigenvalue problem 102.4. Quasi-comparison function selection 112.5. Eigenvalue problem solution 13

3. HYBRID CONTROL AND EQUATIONS OF MOTION 143.1. Base isolation 143.2. Active structural control 153.3. Equations of motion 15

4. CONTROL DESIGN AND SEISMIC RESPONSE ANALYSIS 194.1. Seismic response goals 194.2. Fixed building response 204.3. Frequency domain analysis 204.4. Passive control design 244.5. Isolated building response 254.6. Active control design 314.7. Isolated/controlled building response 34

5. CONCLUSIONS 39REFERENCES 40APPENDIX A- Disjoint mass and stiffness matrices 42APPENDIX B- Constraint matrix 44APPENDIX C- Integrals 45APPENDIX D- Displacement, acceleration and stress at critical points 47VITA 52

v

LIST OF FIGURES

1.1 Model of the building in disturbed configuration 32.1 Typical story in the building 74.1 Imperial Valley earthquake records at El Centro (S00E) 214.2 Generalized coordinates of the fixed building 224.3 Generalized accelerations of the fixed building 234.4 Power spectral density function of California earthquakes displacement 264.5 Function F(Mo,c,k) versus k with c [KN s/m] as a parameter 274.6 Base displacement and acceleration of the isolated building 284.7 Generalized coordinates of the isolated building 294.8 Generalized accelerations of the isolated building 304.9 Function Fc(g1,g2) versus g1 with g2 [KN s/m] as a parameter 324.10 Function Fc(400000,g2) evaluated at ω=2.5 r/s 334.11 Base displacement and acceleration of the isolated/controlled building 354.12 Generalized coordinates of the isolated/controlled building 364.13 Generalized accelerations of the isolated/controlled building 374.14 Control force and power 38D.1 Relative displacement and acceleration of the top of building (fixed) 48D.2 Relative displacement and acceleration of the top of building (isolated) 49D.3 Relative displacement and acceleration of the top of building (isol./cont.) 50D.4 Critical stress 51

vi

LIST OF TABLES

2.1 Parameters of the superstructure 132.2 Natural frequencies convergence 134.1 Peak values of the response variables 34

1

CHAPTER 1 INTRODUCTION

The objectives of earthquake engineering are 1) to prevent injury to theoccupants and damage to the contents of the building and 2) to protect the integrity ofthe structure. These objectives can be achieved through base isolation and activecontrol.

The subject of this dissertation is the dynamic modeling and control of structuressubjected to earthquakes. First, we extend the application of the substructure synthesismethod to more complex structures. Second, we establish a design methodology forbase isolation and active control (hybrid control) in a distributed model of a buildingunder seismic excitation.

The structural dynamic modeling part of this dissertation includes the mostadvanced application of the Rayleigh-Ritz based substructure synthesis method to date.

The structural control part of this dissertation includes the study of hybridcontrol in structures subjected to seismic excitation with a model much more realisticand reliable than the classical lumped-parameter model, and a design procedure for theisolation system parameters and the control gain.

Our objective is to show that, for the case at hand, passive and active controlcomplement each other in such an advantageous manner that simple devices for bothtypes of control are sufficient to achieve excellent response characteristics with verysmall control forces.

1.1. Model of the buildingCivil structures generally have distributed parameters and are of complex

geometries making them difficult to model and analyze. Spatial discretization methodsfor the analysis of distributed structures can be divided into two groups. The first groupincludes lumping procedures. In research on the response of buildings to earthquakesthe analysis is carried out mainly using lumped-parameter models in which the mass isconcentrated in the floors and the stiffness in the columns (Ref. 1). Commonly,damping is modeled as viscous, although this is not very realistic. Lumped-parametermethods lack mathematical rigor. The other group consists of series discretizationmethods. The finite element method (Ref. 2) is included in this group; it can be regardedas a version of the Rayleigh-Ritz method. The method has proved to be highlysuccessful in the dynamic analysis of very complex and irregular structures. Perhaps, itsonly disadvantage is the large number of degrees of freedom required for an accuratemodel. Also in the second group, the classical Rayleigh-Ritz method (Ref. 3) is limited tovery simple systems, because it is very hard to come up with suitable shape functionsfor complex structures.

2

Lately, two developments have renewed the interest in the classical Raleigh-Ritzmethod for the dynamic analysis of structures. First, Meirovitch and Hale (Refs. 4, 5 and6) showed that Hurty’s component-mode synthesis method (Refs. 7 and 8) and all itsvariants are basically extensions of the Rayleigh-Ritz method to complex structures.This has permitted an extension of the mathematical foundation of the classicalRayleigh-Ritz method to the component-mode synthesis method which was developedin a heuristic manner. This new approach to the method has come to be known as thesubstructure synthesis method. Second, Meirovitch and Kwak (Ref. 9) enhanced theclassical Rayleigh-Ritz method by conceiving a new class of approximating functions,namely, the class of quasi-comparison functions which are linear combinations ofadmissible functions capable of satisfying all the system boundary conditions. The useof this new class of functions improves dramatically the convergence characteristics ofthe approximation. In fact, it is shown in Ref. 9 that the new approach produces betterresults than the finite element method for systems with simple geometry.

We propose to model an N-story building as a two-dimensional assemblage ofportal frames. Each frame represents one story, with its beam and two columns beingmodeled as uniform Euler-Bernoulli beams. The compressive axial load carried by thecolumns due to weight above is also considered. The model, as shown in Fig. 1.1,includes the main characteristics of a building, i.e., the ones that contribute primarily toits dynamic behavior. It is the most advanced one found in the literature for the seismicanalysis of buildings, except for models derived by the finite element method. Internaldamping is ignored rather than modeling it as viscous, which is a conservativeassumption.

In the case in which base isolation is used, the superstructure is assumed to berigidly attached to the base, where the latter is mounted on the isolation system. Weidealize the isolation system as a linear spring and a linear dashpot. This is an excellentmodel in the case of laminated-rubber bearing/hydraulic damper type of isolationdevice (Ref. 10).

1.2. Literature surveyThe Rayleigh-Ritz based substructure synthesis method has not been applied to

complex structures. In Ref. (11) it was used to analyze a single frame mounted onsprings. In this case, a kinematical procedure taking into account all the compatibilityconditions could be followed. This is not the case in general. For instance, to analyze amultistory clamped frame we must combine a suitable kinematical procedure with aconstraining process.

On the other hand, the combination of base isolation and active control has beenstudied by Kelly, Leitman and Soldatos (Ref. 12) who used base isolation in conjunctionwith robust control. Pu and Kelly (Ref. 13) used optimal control instead, with emphasison the influence of time delay. Tadjbakhsh and Rofooei (Ref. 14) worked withinstantaneous optimal control. Meirovitch and Stemple (Ref. 15) used on-off controlwith a dead zone. Except for Ref. 15, only lumped-parameter models were used. Thereis not an adequate design procedure for the isolation system in the literature.

3

mc,EIc,L

mb,EIb,l

ck M

X x

ub

s

Y

Fig. 1.1 Model of the building in disturbed configuration

y

Fc

4

1.3. OrganizationThis dissertation is divided in three parts. In the first, we derive and solve the

eigenvalue problem of the framed structure or superstructure. In the second, we derivethe equations of motion of the base isolated and actively controlled structure. Finally, inthe third, we develop a procedure for the design of passive and active control systemparameters so as to enhance the seismic response of the structure, and obtain andanalyze the response of the controlled structure subjected to an actual earthquakeexcitation.

The superstructure eigenvalue problem is derived by means of a Rayleigh-Ritzbased substructure synthesis method. The analysis includes the use of quasi-comparisonfunctions (Ref. 3).

The equations of motion for the controlled building are derived by means ofLagrange’s equations. Direct output feedback is implemented as the control method.The control is carried out by means of a velocity/displacement sensor and an actuatorat the base.

The objectives of earthquake engineering, identified before, determine theresponse variables of interest. The passive control parameters (spring stiffness,damping coefficient and base mass) and the active control gain vector are chosen usinga frequency domain design methodology enhancing those response variables. Theresponse of three different seismic designs, fixed, isolated and isolated-controlled, arecompared.

5

CHAPTER 2 SUBSTRUCTURE SYNTHESIS ANALYSIS

The building structure to be analyzed is composed of the superstructure (thebuilding itself), the base and the isolation system. To obtain the equations of motion ofthe base isolated building, we must first solve the eigenvalue problem of the clampedsuperstructure. The eigenvalue problem is derived by means of the substructuresynthesis method.

2.1. The substructure synthesis methodThe substructure synthesis method is a model reduction technique whereby a

relatively complex structure is analyzed as an assemblage of substructures. The methodis an extension of the classical Rayleigh-Ritz method to complex structures. It has beendeveloped and improved by Meirovitch et al (Refs. 4, 5, 6 and 11).

In the classical Rayleigh-Ritz method the eigensolution can be obtained byconstructing a sequence of approximating solutions from the space of admissiblefunctions instead of the space of comparison functions (Ref. 3). Admissible functions aredefined as p times differentiable functions satisfying only the geometric boundaryconditions of the problem, whereas comparison functions are defined as 2p timesdifferentiable functions satisfying both the geometric and natural boundary conditions,where 2p is the order of the differential operator involved.

It has been shown (Ref. 9) that these version of the Rayleigh-Ritz method haspoor convergence characteristics, because a finite sequence of admissible functions fromthe same family is not able to satisfy the natural boundary conditions, in addition to notsatisfying the differential equation. To remedy this problem, a new class of functionswas conceived in Ref. 9, namely, the class of quasi-comparison functions, defined in Ref.3 as “linear combinations of admissible functions capable of approximating thedifferential equation and the natural boundary conditions to any degree of accuracy bymerely increasing the number n of terms in the approximating solution”. The namecomes from the fact that these sequences act like comparison functions. For this to betrue the admissible functions must be from different families.

The eigenvalue problem is defined by the mass and stiffness matrices, whichrequires the derivation of the structure’s kinetic and potential energies. In thebeginning, only the kinetic and potential energies of the disjoint structure are available.The disjoint substructures are later coupled by constraining them to work together as asingle structure by enforcing the compatibility conditions. The constraint equationsresulting from these conditions can be reduced, or even eliminated, by a suitablekinematical procedure which also simplifies the task of deriving the equations ofmotion.

Next, in the spirit of the Rayleigh-Ritz method, the elastic deformation of thesubstructures is assumed in the form of a finite series of space-dependent admissible

6

functions, from different families, multiplied by time-dependent generalizedcoordinates. These series for the individual substructures are introduced in the disjointkinetic and potential energy expressions, which enables us to obtain the disjoint massand stiffness matrices. The mass and stiffness matrices for the assembled structures areobtained by imposing the compatibility conditions, which in turn defines the eigenvalueproblem for the full structure (Ref. 3). The convergence of the eigensolution requires alimiting process in which the number of admissible functions in the quasi-comparisonfunction for each substructure is increased continuously.

Because the substructure synthesis method is a Rayleigh-Ritz method, most ofthe theory behind the classical Rayleigh-Ritz method applies. For instance, we can statethat the computed eigenvalues converge to the actual eigenvalues from above (Ref. 3).

The substructure synthesis method is perfectly suited for the problem at hand,because the structural members are uniform and the boundary conditions are simple.Furthermore, the number of degrees of freedom involved is much smaller than thenumber of degrees of freedom required by the finite element method.

2.2. Superstructure disjoint kinetic and potential energiesTo derive the velocity field of the clamped superstructure we refer to Fig. 2.1 and

assume that the origin of axes xn,1-yn,1 and xn,2-yn,2 coincides with points n,a and n,b,respectively, and the slope of the axes xn,1 and xn,2 are the same as those of the previousstory’s columns at these points; the origin of axes xn,3-yn,3 coincides with point n+1,a.Axes xn,i represent body axes and they are along the undeformed beams. We note thatthe beam does not perform any rigid body rotation; it only undergoes rigid-bodytranslation. We denote by yn,i the elastic deformation of the beams relative to thoseaxes, and by θn,1 the rotation angle of xn,1 relative to the x axis (Fig. 1.1).

Assuming that the displacements are small, the velocity vector of a typical pointin column n,1, vn,1, is

vn,1 = vn,a + ˙ θ n,1xn,1jn,1 +

∂yn,1

∂tjn,1 (1)

where vn,a is the velocity vector of point a,n, jn,1 is the yn,1 axis unit vector (Fig. 2.1) andthe dot indicates d/dt. If we assume that the angular displacements are also small, the ji,1

unit vectors coincide with each other and with J, the Y axis unit vector (Fig. 1.1); and thevelocity vector of point n,a can be written as

vn,a = ( ˙ θ i,1

i =1

n − 1

∑ L +∂ yi,1 L

∂t)J (2)

where

θ i,1 =∂y j,1

∂xj,1j = 1

i −1

∑L

(3)

is the rotation angle of xi,1 relative to the x axis (Fig. 1.1) and L is the length of thecolumn. Thus, we have by substituting

7

n-1

n

n+1

n,1 n,2

yn,1

θn,1

n,3

n,a n,b

xn,1

yn,3

xn,3

xn,2

yn,2

n+1,a

Fig. 2.1 Typical story in the building

8

vn,1 = ( L∂ 2y j ,1

∂x j,1∂tj =1

i − 1

∑i =1

n −1

∑L

+∂y i,1 L

∂t+ xn,1

∂2y j,1

∂x j,1∂tL

+∂yn,1

∂t)

j =1

n− 1

∑ J (4)

On the other hand, the velocity vector of a point in beam n,3 is

vn,3 = vn +1,a +

∂yn,3

∂tjn,3 (5)

where vn+1,a is the velocity vector of point a,n+1 and jn,3 is the yn,3 axis unit vector.Substituting the corresponding form of Eq. (2) for vn+1,a we obtain

vn,3 = ( L∂ 2y j ,1

∂x j,1∂tj =1

i − 1

∑i =1

n

∑L

+∂y i,1 L

∂t)J −

∂yn,3

∂tI (6)

where I is the X unit vector (Fig. 1.1) which coincides with the -ji,3 direction.

So far, beginning at point 1,a, we have followed a kinematical procedure thatenforces automatically the displacement compatibility conditions at the n,a points, andpermits a systematic derivation of velocities. Unfortunately, we cannot continue doingthe same at points n,b to obtain the velocities of points in the right columns withoutcomplicating the procedure and the subsequent constraining process. At this point, it isbetter to start a similar kinematical procedure at point 1,b, and then establishcompatibility conditions at points 2,b; 3,b; ...; N+1,b. Hence, following the samereasoning as for columns n,1, we have the velocity vector in column n,2,

vn,2 = ( L∂ 2y j ,2

∂x j ,2∂tj = 1

i −1

∑i =1

n− 1

∑L

+∂ yi ,2 L

∂t+ xn,2

∂ 2y j,2

∂x j,2 ∂tL

+∂yn,2

∂t)

j = 1

n −1

∑ J (7)

Consequently, the disjoint kinetic energy of the superstructure can be written as

Td =1

2mc

n =1

N

∑ (0

L

∫ L∂ 2y j,1

∂x j,1∂tj = 1

i −1

∑i =1

n− 1

∑L

+∂ yi ,1 L

∂t+ xn1

∂ 2y j,1

∂x j ,1∂tL

+∂yn,1

∂t)2

j = 1

n −1

∑ dxn,1

+ mc (0

L

∫ L∂2 yj ,2

∂x j,2∂tj =1

i −1

∑i= 1

n− 1

∑L

+∂y i ,2 L

∂t+ xn2

∂ 2y j,2

∂x j ,2∂tL

+∂yn,2

∂t)2

j =1

n−1

∑ dxn,2

+ mb ( ( L∂ 2y j,1

∂xj,1∂tj= 1

i −1

∑i =1

n

∑L

+∂yi,1 L

∂t)2 + (

∂yn,3

∂t)2 )

0

l

∫ dxn,3 (8)

where mc and mb are the masses per unit length of the columns and beams, respectively,and l is the beam length. On the other hand, the disjoint potential energy of the systemis

Vd = 12

EI cn =1

N

∑ (∂ 2yn,1

∂xn,12

0

L

∫ )2dxn,1 + EIc (∂2yn,2

∂xn,22

0

L

∫ )2dxn,2 + EI b (∂2yn,3

∂xn,32

0

l

∫ )2dxn,3

− pn (∂yn,1

∂xn,10

L

∫ )2dxn,1 − pn (∂yn,2

∂xn,20

L

∫ )2dxn,2 (9)

where EIc and EIb are the flexural rigidities of the columns and beams, respectively, andpn is the axial force in the n floor columns due to the weight of the stories above them.

9

Following the discussion in Section 2.1, the substructures elastic deformations areexpressed in terms of quasi-comparison functions as follows:

yn,1(xn,1 ,t) = f T (xn,1 )qn,1(t)

yn,2(xn,2 ,t) = f T (xn,2 )qn,2 (t)

yn,3(xn,3,t) = gT (xn,3 )qn,3(t)

(10)

where f and g are vectors of admissible functions from different families, qn,i are vectorsof generalized coordinates of the substructures and T stands for transpose. SubstitutingEqs. (10) into Eq. (8), the disjoint kinetic energy can be written as

Td =1

2mc

n=1

N

∑ (0

L

∫ L fT ′

j = 1

i −1

∑i =1

n− 1

∑ (L) ˙ q j ,1+f T(L) ˙ q i,1 + xn,1 f T ′(L) ˙ q j,1 + fT ˙ q n,1 ) 2

j =1

n− 1

∑ dxn,1

+ mc (0

L

∫ L fT ′j =1

i −1

∑i= 1

n − 1

∑ (L) ˙ q j ,2+f T(L) ˙ q i,2 + xn, 2 fT ′(L) ˙ q j ,2 + fT ˙ q n,2 )2

j = 1

n −1

∑ dxn, 2

+ mb ( ( L f T ′j=1

i −1

∑i =1

n

∑ (L) ˙ q j,1+f T(L) ˙ q i,1 )2 + ( gT˙ q n,3 )2 )0

l

∫ dxn,3 (11)

where fT ′

(L) indicates derivative of f with respect to xn,1(2) evaluated at L. The firstintegrand in the previous equation has the expanded form

L2 fT ′(L)˙ q j,1

s =1

r − 1

∑j = 1

i− 1

∑r = 1

n− 1

∑i= 1

n − 1

∑ fT ′(L) ˙ q s,1+L f T ′

(L)˙ q j,1j = 1

i− 1

∑ f T(L)˙ q r,1+L fT(L)˙ q i,1fT′(L)˙ q s,1

s = 1

r − 1

∑ +f T(L)˙ q i,1fT(L)˙ q r,1 +

xn,12 f T′

(L)˙ q j,1fT ′

(L)˙ q s,1s= 1

n −1

∑j =1

n −1

∑ + 2xn,1 f T′(L)˙ q j,1f

T ˙ q n,1j= 1

n −1

∑ + (fT ˙ q n,1)2 +

2xn,1 L f T′(L) ˙ q k,1f

T ′(L)˙ q j,1

k =1

i− 1

∑j= 1

n −1

∑i = 1

n −1

∑ + fT(L)˙ q i,1fT ′(L)˙ q j,1

j = 1

n −1

∑ + 2 L fT′(L) ˙ q j,1fT ˙ q n,1

j = 1

i −1

∑i= 1

n− 1

∑ +f T(L)˙ q i,1fT ˙ q n,1(12)

Next, let us write the various quadratic forms in matrix notation and simplify to obtain

L2 ˙ q dT Nan

˙ q d + L ˙ q dT N bn

˙ q d + L ˙ q dT N cn

˙ q d + ˙ q dT Ndn

˙ q d + xn,12 ˙ q d

T N en˙ q d + xn,1

˙ q dT N fn

˙ q d

+ ˙ q dTN gn

˙ q d + Lxn,1˙ q d

TNhn˙ q d + xn,1

˙ q dTN in

˙ q d + L ˙ q dT Njn

˙ q d + ˙ q dTN kn

˙ q d (13)

where the N matrices are shown in Appendix A and qd is the disjoint configurationvector defined as

qd = q1,1T q1,2

T q1,3T q2,1

T q2,2T q2,3

T L qN,1T qN,2

T qN,3T( )T

(14)

In a similar manner, we expand the second and third integrands. For the secondintegrand, we obtain an expression identical to (12), but with the subscript 2 replacingthe subscript 1, and an expression identical to (13), although the matrices are slightlydifferent. For the third integrand we obtain

L2 fT ′(L)˙ q j,1

s =1

r − 1

∑j = 1

i− 1

∑r = 1

n

∑i= 1

n

∑ fT ′(L)˙ q s,1+L f T′

(L) ˙ q j,1j =1

i− 1

∑ f T(L)˙ q r,1+L fT(L)˙ q i,1fT ′

(L) ˙ q s,1s= 1

r− 1

∑ +f T(L) ˙ q i,1fT(L)˙ q r,1

+(gT ˙ q n,3)2 (15)

or

10

L2 ˙ q d

T Nln˙ q d + L˙ q d

TNmn˙ q d + L˙ q d

T Nnn˙ q d + ˙ q d

T N on˙ q d + ˙ q d

T Npn˙ q d (16)

where the N matrices are defined in Appendix A. Finally, integrating and summing, thedisjoint kinetic energy can be expressed as

Td =

1

2˙ q d

T Md˙ q d (17)

where Md is the disjoint mass matrix shown in Appendix A.

On the other hand, the potential energy can be written as

Vd = 12

EI cn =1

N

∑ (f T″qn,1

0

L

∫ )2dxn,1 + EI c (f T″qn,2

0

L

∫ )2dxn,2 + EIb (0

l

∫ gT″qn,3 )2dxn,3

− pn (f T′qn,1

0

L

∫ )2 dxn,1 − pn (f T ′qn,2

0

L

∫ )2 dxn,2 (18)

where the double prime indicates second derivative with respect to xn,i.

Vd = 12

EI cn =1

N

∑ qdTSanqd

0

L

∫ dxn,1 + EI c qdTSbnqd

0

L

∫ dxn,2 + EIb qdTScnqd

0

l

∫ dxn,3

− pn qdTSdnqd

0

L

∫ dxn,1 − pn qdTSenqd

0

L

∫ dxn,2 (19)

where the S matrices are shown in Appendix A. The disjoint potential energy can beexpressed as

Vd =

1

2qd

T Kdqd (20)

where Kd is the disjoint stiffness matrix also shown in Appendix A.

2.3. Constraint equations and eigenvalue problemAs pointed out earlier, geometric compatibility conditions not satisfied by the

kinematical procedure must be enforced by other means. We recall that the kinematicalprocedure took into account the displacement compatibility at points n,a, but not theslope compatibility, where the latter implies that the beams remain at right angles withrespect to each other. The slope compatibility between columns and beams can beexpressed as the constraint equations (Fig. 2.1)

θn,1 +

∂yn,1

∂xn,1 L

=∂yn,3

∂xn,3 0

(21)

The slope compatibility between columns will be satisfied automatically by a suitablechoice of admissible functions.

On the other hand, neither displacement nor slope compatibility were satisfied atpoints n,b. Slope compatibility between beams and columns is ensured by the constraintequations

θn,2 +

∂yn,2

∂xn,2 L

=∂yn,3

∂xn,3 l

(22)

11

The displacement compatibility conditions express that the displacement vectors of thebeams at l and of the columns at L must be the same. These conditions lead to theconstraint equations

Lθn,1 + yn,1 L= Lθn,2 + yn,2 L

(23a)

yn,3 l= 0 (23b)

Equations (23b) can be satisfied automatically by a suitable choice of admissiblefunctions, so that they are not considered in the following. Next, Eqs. (10) and (3) areinserted into Eqs. (21, 22 and 23a) to obtain

f T ′(L)q j,1

j =1

n

∑ = gT ′(0)qn,3 (24a)

f T ′(L)q j,2

j =1

n

∑ = gT ′(l)qn,3 (24b)

L f T ′(L)qj,1

j =1

n− 1

∑ + fT (L)qn,1 = L f T ′(L)q j,2

j =1

n− 1

∑ + f T(L)qn,2 (24c)

which can be written in the compact form qd = Cq (25)

in which C is the constraint matrix (Appendix B) and q is the vector of independentgeneralized coordinates. The mass and stiffness matrices for the assembled structure, Mand K, can then be expressed as (Ref. 5)

M = CT MdC

K = CTKdC (26a,b)where Md and Kd are the disjoint mass and stiffness matrices (Appendix A).

As pointed out earlier, the mass and stiffness matrices for the assembledstructure define the eigenvalue problem, which can be written as

Kq = λMq (27)

2.4. Quasi-comparison function selectionBefore the matrices M and K can be determined it is necessary to choose the

quasi-comparison functions entering into the series expansions (10). Certainly, series ofEuler-Bernoulli beam eigenfunctions are the best choice because they are the mostclosely related to the problem at hand. However, this can cause some problems,because M and K involve integrals of products of these trigonometric/hyperbolicfunctions. In fact, this difficulty is typical of the Rayleigh-Ritz method compared to thefinite element method. Fortunately, there is a technique for rendering the exactevaluation of these integrals easier (Ref. 12).

We recall that admissible functions must satisfy only the geometric boundaryconditions of the substructures; the differentiability requirement is seldom an obstacle.However, when these admissible functions are combined into a quasi-comparisonfunction, the linear combination must be able to approximate the natural boundaryconditions to any degree of accuracy. We begin with the selection of admissible

12

functions for the columns 1,1 and 1,2. First, we note that in the previous section all thecompatibility conditions at the internal boundaries were considered, but geometricconditions at the external boundaries were not mentioned. In order to satisfy theselatter conditions and be consistent with the kinematical procedure used, the admissiblefunctions for these two columns must have zero displacement and slope at x1,1(2)=0.Therefore, the only choice are beam functions clamped at that end. At the other endthere is no particular geometric condition to be satisfied. Now, we need at least twofamilies of beam functions, and a combination of the families must allow for nonzeroshearing force and bending moment at that end. Additionally, even though we do notneed to satisfy any additional geometric condition, the quasi-comparison functionsmust be able to approximate the displacement of the substructure reasonably well. Wenote that the displacement and slope at that end are nonzero. Consequently, we chooseclamped-free and clamped-clamped beam functions as the two families of admissiblefunctions.

For beam 1,3, we must first satisfy the geometric condition expressed by Eq.(23b). This implies zero displacement at x1,3=l. Obviously, to be consistent with thekinematical procedure, the displacement at x1,3=0 must be zero also. As before, toconstruct a genuine quasi-comparison function, the choice must allow for nonzeroshearing force and bending moment at both ends. In addition, we should try to permita reasonably good approximation of the displacement of the beam. To this end, a goodchoice is a combination of clamped-pinned and pinned-clamped shape functions. Webase this assertion on results for a single frame mounted on springs (Ref. 11) whereclamped-free and free-clamped proved to be a better choice than clamped-clamped andfree-free. In this regard, we remark that variety is what makes quasi-comparisonfunctions much more powerful than admissible functions from a single family.

Continuing with the column 2,1, we note that the displacements and boundaryconditions relative to the x2,1 axis are similar to the ones for column 1,1 relative to axisx1,1. We recall that we must satisfy the slope compatibility between columns. All thisapplies to column 2,2 also. Furthermore, the kinematics and boundary conditions of thebeam 2,3 are analogous to the ones of beam 1,3. The same reasoning applies to the restof the superstructure. As a consequence, we choose the first floor set of quasi-comparison functions for each frame of the superstructure.

The clamped-free, clamped-clamped, clamped-pinned and pinned-clamped Eulerbeam eigenfunctions, Ycf, Ycc, Ycp and Ypc, respectively, have the expressions

Ycfj = c1 j(coshβ 1j x − cos β1 j x − σ 1j(sinh β1j x − sinβ1j x)) (28a)

Ycc j = c 2j (coshβ 2jx − cosβ2 j x − σ 2j(sinh β2 jx − sinβ 2j x)) (28b)

Ycp j = c3 j(cosh β3j x − cosβ 3jx − σ 3j(sinhβ 3jx − sinβ 3j x)) (28c)

Ypc j = c4 j(cschβ 3jl sinhβ 3j x − cscβ 3j l sinβ 3j x) (28d)

The parameters β and σ along with the solution technique of the integrals involved inthe mass and stiffness matrix expressions are shown in Appendix C. The vectors ofadmissible functions in the quasi-comparison function expressions are given by

13

f = Ycf1 Ycc1 Ycf2 Ycc 2 L Ycfr Yccr( )T(29a)

g = Ycp1 Ypc 1 Ycp2 Ypc2 L Ycp r Ypcr( )T(29b)

2.5. Eigenvalue problem solutionAt this point, we must define the parameters of the superstructure. We consider

a 4-story concrete frame with parameters as defined in Table 2.1.

Table 2.1 Parameters of the superstructureColumns Beams

Depth x width [m x m] 1/2 x 1/2 1 x 1/2Length [m] 3.5 7.0

Linear density [Kg/m] 580 1160Flexural rigidity [N m2] 1.3021 108 1.0417 109

The eigensolution consists of the natural frequencies and the eigenfunctions forthe full structure. The convergence of the natural frequencies to six digits is shown inTable 2.2, where q is the number of admissible functions in the quasi-comparisonexpression for each substructure (Eqs. (10) and (29)). We note that convergence is stillexcellent for ω10 , for which the difference in the sixth digit for q=7,8 is one unit.

Table 2.2 Natural frequencies convergence [r/s]q 4 5 6 7 8ω1 23.9314 23.9079 23.9076 23.9075 23.9075ω2 72.0684 71.9897 71.9888 71.9886 71.9886ω3 118.600 118.449 118.448 118.448 118.448ω4 155.534 155.312 155.311 155.310 155.310

To obtain the eigenfunctions we first insert the eigenvectors into Eq. (25), and theresulting vectors are inserted, in conjunction with the vectors of admissible functions(Eqs. (29)) into Eqs. (10). In our particular case, we normalized these vectors such thattheir norm is 1.

14

CHAPTER 3 HYBRID CONTROL AND EQUATIONS OF MOTION

Current earthquake engineering practice is to build very strong, well supportedstructures capable of resisting moderate earthquakes through plastic deformations.Failure of some structural elements is sometimes tolerated. However, such designs tendto be expensive and disastrous in major earthquakes. Therefore, new approaches areimperative.

Our goal is to show that, in the protection of structures against earthquakes,passive and active control complement each other in such an advantageous mannerthat simple devices for both types of control are sufficient to achieve excellent responsecharacteristics with very low control forces.

In this chapter we describe base isolation and active structural control. Afterdefining the passive and active control systems that we analyze in this work, theequation of motion of the hybridly controlled structure are derived.

3.1. Base isolationBase isolation is a technique for improving the overall performance of structures

during earthquakes. The simple idea behind this technology is to isolate the structurefrom the moving ground through flexible mountings so as to reduce the transmissionof forces. A review and bibliography was presented by Kelly (Ref. 17). Earlyinvestigators used mainly one degree of freedom models. Moreover, even when moreelaborate models are used, for the most part the displacement is measured relative tothe moving ground, which leads to the paradox that the structure is stabilized relativeto the moving ground, rather than relative to the inertial space. Only lately, themechanical models have been improved (Refs. 18,19 and 20), and the importance ofusing an inertial reference frame has been recognized (Refs. 18 and 12).

Several types of base isolation have been proposed and a complete descriptioncan be found in Ref. 10. In this dissertation, we perform the analysis for a linearisolation system, such as the laminated-rubber bearings which are the most used inbase isolated buildings. Because of the added flexibility, the base undergoes largedisplacements. Damping at the base is important in reducing this displacement (Ref. 21).Because damping provided by the rubber pads is very low, it is necessary to add otherdamping mechanisms. Our isolation system includes hydraulic dampers which are alsoused in base isolation practice.

In Ref. 21 base isolation was evaluated on a hollow Euler-Bernoulli beam modelfor the superstructure subjected to a stationary earthquake excitation. The resultsshowed that an isolated structure undergoes lower levels of displacement, accelerationand stress than the nonisolated counterpart, and that energy dissipation in the isolationsystem is important in controlling the displacement of the base. It was found that thesefavorable results are a consequence of the fact that in an isolated structure, the range of

15

the dominant excitation frequencies is reduced and is moved toward the lowerfrequencies (with respect to the nonisolated case), away from the resonant frequenciesrelated to the superstructure. Favorable theoretical results can also be found in Ref. 19for a shear beam model of the superstructure, and in Ref. 1 for lumped-parametermodels, although the ground-based reference is used. Similar experimental results canbe found in Ref. 22.

3.2. Active structural controlAnother appealing approach is the use of active control whereby structures are

made to behave responsively when subjected to earthquake excitations. Particularly,excellent performance characteristics can be attained by means of feedback control.

Active structural control research based on formal control theory began in theseventies. The feasibility of applying optimal control theory to reduce the vibrations ofcivil structures under stochastic loads was investigated by Yang (Ref. 23). Theapplication of pole placement methods to the control of structures was investigated byMartin and Soong (Ref. 24) and Abdel-Rohman and Leipholz (Ref. 25). Other class ofmodal control was developed by Meirovitch and coworkers (Refs. 26 and 27) wherebythe control design is carried out for each system mode independently. The method wasapplied to a structure subjected to earthquake excitation (Ref. 28) where the structurewas modeled as a 3-story frame and it was analyzed by the finite element method. In allthe other references lumped models were used.

As it is the case with base isolation, results show that active control is effective inreducing the response of structures subjected to earthquake excitations. Therefore,simpler, cheaper and more effective designs may be obtained by combining baseisolation and active control.

Base isolation is highly effective in reducing the superstructure displacementsrelative to the base (stresses) and the absolute accelerations. However, this is achievedat the expense of large base displacements. In order to reduce the base displacementbelow the ground displacement levels, large amounts of damping were required in thework reported in Ref. 21. On the other hand, the application of active control alonerequires, in general, a complex and expensive control system distributed over thestructure; not only that, the control forces required under earthquake excitation are solarge that practical application is doubtful.

In this dissertation, the base isolation system proposed in Section 3.1 iscomplemented with direct output feedback control. The control system consists simplyof a state sensor and an actuator collocated at the base because, after base isolation hasbeen implemented, we are interested in controlling the motion of the base. The controllaw is of the proportional type. We intend to show that excellent results can be obtainedwith this simple hybrid control design.

3.3. Equations of motionWe derive the equations of motion of the base isolated and actively controlled

building by means of the Lagrange’s equations. To this end, we refer to Fig. 1.1 where

16

X is an axis fixed to the inertial space and x-y is a system of axes fixed to the base. Wedenote by s the ground displacement and by ub the absolute displacement of the basefrom its original position. The velocity vector of a point in the superstructure can bewritten as

v(x,y,t) = ˙ u b(t)J + vr(x,y,t) (30)where vr is the velocity vector of the point relative to x-y, in this case equal to the timederivative of the elastic displacement vector relative to x-y. We express the elasticdisplacement vector of a point in the frame relative to x-y in the form

ue (x,y,t) = er

r = 1

m

∑ (x,y)qr(t) (31)

where er are 2-dimensional admissible functions of the superstructure representing thelowest eigenfunctions of the clamped superstructure obtained by substructuresynthesis, qr are the superstructure generalized coordinates of the isolated building andm is the number of clamped superstructure eigenfunctions considered in the expansion.The eigenfunctions of the clamped superstructure can be expressed as

e r(x,y) =Fr(x) j

Fr (x) j

−Gnr (y)i + Fr (nL) j

at

at

at

(x,0)

(x,l)

(nL,y)

(32)

where Fr are the modes of the superstructure left columns as a whole, Gnr are themodes of the n story beam, and i and j are the unit vectors of the x-y axes (Fig. 1.1).Note that Eq. (32) represents unsymmetric clamped superstructure eigenfunctionsonly.; the generalized coordinates corresponding to symmetric eigenfunctions are notexcited by horizontal ground excitations. Equation (31) can be simplified to

ue = Eq (33)where E is a 2xm matrix whose columns are the vectors er, and q is the m-dimensionalvector of superstructure generalized coordinates. Then, equation (30) can be written as

v = ˙ u bJ + E˙ q (34)

The kinetic energy of the system can be written as

T =

1

2M ˙ u b

2 +1

2mvT v

σ∫ dσ (35)

where M is the mass of the base and m(x,y) is the mass per unit length of thesuperstructure, which is equal to mc for the columns and mb for the beams. Afterexpanding we get

T =

1

2(M + M s) ˙ u b

2 + ˙ u b( meyTdσ

σ∫ ) ˙ q +

1

2˙ q T( mETE

σ∫ dσ )˙ q (36)

where Ms is the superstructure mass and ey is the second row of E. After performing theintegrals we obtain

17

T = 12(M+M s)˙ u b

2 + ˙ u b 2mc F1dx+mbl F1(nL)n =1

N

∑0

4L

∫ L 2mc Fmdx+mbl Fm (nL)n= 1

N

∑0

4L

∫

q +

1

2˙ q T

2mc F12dx

0

4L

∫ +mb Gn12dy+lF1

2(nL)0

l

∫n =1

N

∑ L 0

M O M

0 L 2mc Fm2dx

0

4L

∫ +mb Gnm2 dy+lFm

2(nL)0

l

∫n= 1

N

∑

˙ q (37)

or

T =

1

2(M + M s) ˙ u b

2 + ˙ u bmT ˙ q +

1

2˙ q TM s

˙ q (38)

where m is the mass vector in the second term of the right hand side of Eq. (37). Thefact that the matrix Ms is diagonal is a result of the orthogonality of the clampedsuperstructure eigenfunctions, which in turn is a consequence of the clampedsuperstructure being self-adjoint.

The scalar elastic deformation of the beams and columns relative to their rigidbody axes, represented by zn,i, can be expressed as

zn,i (xn,i ,t) = hn,iT (xn,i )q(t) (39)

where hn,i are vectors of substructure admissible functions representing thesubstructure modes of the clamped superstructure obtained by substructure synthesis.The potential energy of the system can then be written as

V = 12

k(ub − s)2 + 12

EI cn= 1

N

∑ (hn,1T ″

0

L

∫ q)2dxn,1 + EI c (hn,2T ″

0

L

∫ q)2dxn,2 + EI b (hn,3T ″

0

l

∫ q)2dxn,3

− pn (hn,1T ′

0

L

∫ q)2dxn,1 − pn (hn,2T ′

0

L

∫ q)2dxn,2 (40)

where k is the spring stiffness and s is the ground displacement (Fig. 1.1). Becausehn,2=hn,1, we can write

V = 1

2k(ub−s)2 + 1

2qT

2EI c ′ ′ h n,11

2dxn,1+EIb ′ ′ h n,31

2dxn,3

0

l

∫0

L

∫ L 0

M O M

0 L 2EI c ′ ′ h n,1m

2 dxn,1+EI b ′ ′ h n,3m

2 dxn,3

0

l

∫0

L

∫

n= 1

N

∑ q

− qT

2pn ′ h n,11

2dxn,1

0

L

∫ L 0

M O M

0 L 2pn ′ h n,1m

2 dxn,1

0

L

∫

q (41)

This expression can be reduced to

18

V =

1

2k(ub − s)2 +

1

2qTK sq (42)

The virtual work of the nonconservative forces can be written as (Fig. 2.1) δW = Fcδub − c( ˙ u b − ˙ s )δub (43)

where c is the damping coefficient and Fc is the control force which can be written as Fc = −g1ub − g2

˙ u b (44)according to the discussion in Section 3.2; g1 and g2 are the displacement and velocitygains, respectively.

Applying Lagrange’s equations one obtains the equations of motion of the baseisolated and actively controlled building as

M˙ r + C˙ r + Kr = fc + fe (45)

where M, C and K are its mass, damping and stiffness matrices,

M =

M+Ms mT

m M s

C =

c zT

z Z

K =

k zT

z K s

(46)

where Z is the mxm zero matrix, and z is the mx1 zero vector. fc and fe are the controland earthquake induced force vectors, respectively, and r is the (m+1)x1 configurationvector,

r = ub qT( )T

(47)

We note that fe is not a simple disturbance but a strong external excitation; moreimportantly, we observe that it has the special form

fe = c˙ s +ks 0 L 0( )T (48)

furthermore, C and K are diagonal and M is diagonal except for the first row andcolumn (Eqs. 46). This implies that the disturbance only acts on the base (or baseequation), and that the superstructure generalized coordinates are excited only by thebase motion. Therefore, control force is necessary only at the base. Thus, the controldesign proposed in Section 4.2 is justified; this is,

fc = −g1ub−g2˙ u b 0 L 0( )T

(49)

We wish to recast the equations of motion in state form. To this end, we definethe 2(m+1)x1 state vector x = (r T ˙ r T )T . Then, the state equations have the familiar form

˙ x = Ax + Bfc + Bfe (50)

where

A =

0 I

−M−1K −M−1C

B =

0

M−1

(51)

The equations of motion of the fixed building can be directly obtained by makingub=s in the last m equations in (45). Because Ms and Ks (Eqs. (46)) are diagonal, the fixedbuilding equations of motion are uncoupled.

19

CHAPTER 4 CONTROL DESIGN AND SEISMIC RESPONSEANALYSIS

So far we have not dealt with the question of how to choose the parameters ofthe isolation system or the gain vector in the control law proposed. We develop adesign methodology in the frequency domain for the hybrid control proposed. Thisprocedure is based on enhancing the seismic response characteristics of the structure.

4.1. Seismic response goalsThe objectives of earthquake engineering are 1) to prevent injury to the

occupants, or even death, and damage to the contents of the building and 2) to protectthe integrity of the structure. Because we employ base isolation, we need to add a thirdobjective, which is to keep the motion amplitudes at reasonable levels. These objectivesdefine the response variables we need to analyze. Objectives 1) and 2), which are closelyrelated, point at keeping the stress levels below defined limits and maintaining theabsolute accelerations in the building at reasonable levels. Complying with the thirdobjective translates in maintaining the base displacement levels low.

We note that the analysis performed permits us to obtain the stress level at anypoint in the model structure. We recall that the maximum stress in a section isessentially defined by the bending moment M as

σm =

M ym

I (52)

where M is given by

M(xn,i ,t) = EI

∂ 2z(xn,i ,t)

∂xn,i2 (53)

Therefore, we have that

σm(xn,i ,t) = Eym hn,iT ″

(xn,i )q(t) (54)We conclude that the stress in a given section depends on the generalized coordinatesonly; the first one being the dominant.

On the other hand, the acceleration vector of a point in the superstructure can bewritten as

a = ˙ u b J + E˙ q (55)We conclude that the accelerations in the frame depend on the acceleration of the baseand the generalized accelerations. Consequently, the response variables desired to bereduced are actually: ub, q and their second time derivatives. These four variables will bereferred as the overall response.

We obtain, compare and analyze the overall response of three different designsof the four-story building model in the time domain. These different plans are:nonisolated building, base-isolated building and base-isolated/actively-controlled

20

building. We use an actual earthquake excitation in the analysis; specifically, the S00Ecomponent of the Imperial Valley earthquake at El Centro (California, May 1940)whose displacement and acceleration records are shown in Figs. 4.1. The records wereobtained on line from the Lamont Doherty/NCEER Strong-Motion Database(http://www.ldeo.columbia.edu). The records have a duration of 53.7 s. The samplingrate is 0.1, 0.04 and 0.02 s for the ground displacement, velocity and acceleration,respectively. We also obtain actual displacement, acceleration and stress levels atimportant points in the superstructure.

4.2. Fixed building responseThe overall response of the nonisolated building is shown in the following

figures. Figures. 4.2 show the generalized coordinates q, and Figs. 4.3 show thegeneralized accelerations. These plots of generalized coordinates and accelerations havephysical meaning only after defining how the clamped superstructure eigenfunctions(eigenvectors) are normalized, which was done in Section 2.5. Because, our study iscomparative, they suffice; nonetheless, the more physically meaningful top floor (point5,a) relative displacement and acceleration, and the critical stress in the column 1,1 (point1,a) are shown in Appendix D. In this dissertation the response of the systems isobtained by using the simulation command LSIM of the control system toolbox ofMATLAB (Ref. 29). LSIM uses the transition matrix approach. The accuracy of thesolution is guaranteed because this algorithm has been shown to be stable;furthermore, the order of the system is not high.

4.3. Frequency domain analysisThe frequency domain method is particularly suited for the analysis of stationary

random processes. Earthquakes are not stationary by any means but the followinganalysis will prove helpful in the design of hybrid control.

The power spectral density matrix of the state vector is expressed as (Ref. 30) Sxx = H∗Sff HT (56)

where H is the 2(m+1)x(m+1) frequency response matrix of the isolated structureexpressed as

H = iωI − A( )−1B (57)

and Sff(ω) is the (m+1)x(m+1) power spectral density matrix of the force vector fe. It isdefined as the inverse Fourier transform of the force vector correlation matrix Rff(τ),

Sff (ω) =

1

2πe−i ωτ Rff(τ )dτ

−∞

∞

∫ (58)

The force vector correlation matrix Rff(τ) is in turn defined by

R ff (τ) =E{f1(t)f1(t + τ )} L E{f1(t)fm+1(t + τ )}

M O ME{fm + 1(t)f1(t +τ )} L E{fm + 1(t)fm +1(t + τ)}

(59)

where E{ } is the expectation operator. Because only the first element of the forcevector is different from zero, the excitation power spectral density matrix has the form

21

0 5 10 15 20 25 30 35 40 45 50

-0.1

-0.05

0

0.05

0.1

0.15

Time [s]

Dis

pla

cem

en

t [m

]

Fig. 4.1a Displacement of the Imperial Valley earthquake at El Centro (S00E)

0 5 10 15 20 25 30 35 40 45 50-4

-2

0

2

4

Time [s]

Acc

ele

ratio

n [

m/s

^2]

Fig. 4.1b Acceleration of the Imperial Valley earthquake at El Centro (S00E)

22

0 5 10 15 20 25 30 35 40 45 50-0.05

0

0.05

Time [s]

Gen

. dis

plac

emen

t [m

]

Fig. 4.2a First generalized coordinate of the fixed building

0 5 10 15 20 25 30 35 40 45 50-0.05

0

0.05

Time [s]

Gen

. dis

plac

emen

t [m

]

Fig. 4.2b Second generalized coordinate of the fixed building

0 5 10 15 20 25 30 35 40 45 50-0.05

0

0.05

Time [s]

Gen

. dis

plac

emen

t [m

]

Fig. 4.2c Third generalized coordinate of the fixed building

23

0 5 10 15 20 25 30 35 40 45 50-30

-20

-10

0

10

20

30

Time [s]

Ge

n.

acc

ele

ratio

n [

m/s

^2]

Fig. 4.3a First generalized acceleration of the fixed building

0 5 10 15 20 25 30 35 40 45 50-30

-20

-10

0

10

20

30

Time [s]

Ge

n.

acc

ele

ratio

n [

m/s

^2]

Fig. 4.3b Second generalized acceleration of the fixed building

0 5 10 15 20 25 30 35 40 45 50-30

-20

-10

0

10

20

30

Time [s]

Ge

n.

acc

ele

ratio

n [

m/s

^2]

Fig. 4.3c Third generalized acceleration of the fixed building

24

Sff =

Sf1 f10 L 0

0 0 L 0

M M O M0 0 L 0

(60)

The force power spectral density function Sf1 f1 is the Fourier transform of the

force autocorrelation function which is expressed as R f1 f1

= E{( c˙ s (t) + ks(t))( c˙ s (t +τ ) + ks(t + τ ))} (61)Expanding Eq. (62) we have

R f1 f1= c2R˙ s s (τ ) + k2Rss(τ ) + ckR˙ s s(τ ) + ckRs ˙ s (τ ) (62)

so that Sf1 f1

= c2S ˙ s s (ω) + k 2Sss (ω ) + ck( S˙ s s(ω ) + Ss ˙ s (ω )) (63)Because the sum in the last term is zero (Ref. 30), we have finally

Sf1 f1= c2S ˙ s s + k2Sss (64)

4.4. Passive control designThe parameters of the isolation system are the base mass M, the damping

coefficient c and the stiffness constant k. The procedures found in the literature forselecting these parameters are either trial and error approaches (e.g. Ref. 31) ormethods based on the previous knowledge of the earthquake characteristics (Ref. 32).We propose to develop a design procedure for the selection of these parameters.

In general, any isolation system helps reduce the overall response except for thedisplacement of the base. Therefore, a passive control design procedure should bebased on minimizing the displacement of the base. Because we are concerned withrandom excitations, it is difficult to come up with a simple expression for thedisplacement of the base in the time domain. On the other hand, a frequency domainanalysis can provide such an expression because it is better suited for stochasticprocesses. Earthquake-induced ground motion is clearly nonstationary in nature.However, it can be considered stationary in the strong-motion portion, over which thepeak response occurs, in order to establish a design methodology.

The mean square value of the displacement of the base, αb, is defined by the 1,1element of the state vector power spectral density matrix (Ref. 30 and Eq. (57))

αb = H∗SffH

T( )11

dω−∞

∞

∫ (65)

or

αb = 2 H11

2S f1f1

dω0

∞

∫ (66)

We are interested in minimizing this integral. To this end, we express it as a function ofthe isolation system parameters, or

25

αb(M,c,k) = 2 H11(M,c,k)

2(c2ω 2 + k2 )Sss(ω )

0

∞

∫ dω (67)

We cannot carry out the integration because the ground displacement powerspectral density function Sss is unknown. However, we have noticed that Sss hasdominant components only in a very narrow frequency range. For earthquakes inCalifornia we find that the range is approximately 0.25-2.5 rad/s (0.04-0.40 Hz) (Figs.4.4). We propose to reduce the integrand over this frequency range. The integrand is aproduct of the unknown factor, Sss, and a known factor,

F(M,c,k) = H11(M,c,k)2(c2ω 2 + k2 ) (68)

We can only reduce this factor, and that is what we will do next.

We do not have much practical freedom in choosing the base mass M; thus, itsrange is limited. We plot F as a function of c and k for some reasonable values of M, andin the frequency range of interest. The base mass cannot be larger than the total massof the superstructure, for example; in particular, we constrain M to be less than themass of one floor. Because the effect of increasing the base mass is the lowering of thestructure first natural frequency away from the excitation dominant components, fromthese plots, not shown here for brevity, we conclude that the larger the mass is, thebetter designs may be obtained,. Therefore, we choose M=Mo=12000 Kg, one-floormass approximately. Next, the plot of F(Mo,c,k) in the frequency range of interest isshown in Fig. 4.5, where F is plotted against k (abscissa) for several values of c. If wereally mean “base isolation” we are interested in designing to the left of the intersectionof the curves for any frequency (in the range) because the curves reach their lowestvalues in that region; note that F=1 implies base displacements equal to the grounddisplacement (Eq. 67). However, this would imply a stiffness below 5000 N/m whichwould result in a very flexible isolation system, unable to support medium velocitywinds without deforming too much. In this regard, we propose to set a lower limit to k(ko) and choose c such that its corresponding curve at ω=0.25 r/s has the maximum at ko,which implies the exclusion of the resonant condition in the frequency range of interest.We choose ko=10000 N/m. From the plots, co=35000 N s/m.

4.5. Isolated building responseThe overall response of the base-isolated building is shown in the following

figures. Figure 4.6a shows the displacement of the base. This plot can be compared withthe one of the displacement of the ground (Fig. 4.1a) because in the unisolated case thebase moves with the ground. We note that the displacements of the base are a littlelower than the ground motion. This is a consequence of the design procedure usedbecause, in general, just any isolation system results in base displacements larger thanthe ground motion. The acceleration of the base is shown in Fig. 4.6b. It can be seen thatbase isolation reduces greatly the acceleration of the base from the levels shown in Fig.4.1b. The reductions are even much more impressive in the generalized coordinates andaccelerations, which are shown in Figs. 4.7 and 4.8. The reductions are above 200 and 50times for the maxima of the first and second generalized coordinates, respectively. Werecall that the stress amplitudes are defined directly by the generalized coordinates

26

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

Frequency [r/s]

PS

DF

x 1

0^3

[m^2

s]

Fig. 4.4a Power spectral density function of Imperial Valley earth. displacement at El Centro (S00E)

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

Frequency [r/s]

PS

DF

x 1

0^3

[m^2

s]

Fig. 4.4b Power spectral density function of Kern County earth. displacement at Taft (S69E)

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

Frequency [r/s]

PS

DF

x 1

0^3

[m^2

s]

Fig. 4.4c Power spectral density function of Loma Prieta earth. displacement at Corralitos (N90E)

27

Fig. 4.5 Function F(M0,c,k) versus k with c [KN s/m] as a parameter

28

0 10 20 30 40 50 60 70-0.1

-0.05

0

0.05

0.1

Time [s]

Dis

pla

cem

en

t [m

]

Fig. 4.6a Base displacement of the isolated building

0 10 20 30 40 50 60 70-0.2

-0.1

0

0.1

0.2

Time [s]

Acc

ele

ratio

n [

m/s

^2]

Fig. 4.6b Base acceleration of the isolated building

29

0 10 20 30 40 50 60 70

-2

-1

0

1

2

x 10-4

Time [s]

Gen. dis

pla

cem

ent [m

]Fig. 4.7a First generalized coordinate of the isolated building

0 10 20 30 40 50 60 70

-2

-1

0

1

2

x 10-4

Time [s]

Gen. dis

pla

cem

ent [m

]

Fig. 4.7b Second generalized coordinate of the isolated building

0 10 20 30 40 50 60 70

-2

-1

0

1

2

x 10-4

Time [s]

Gen. dis

pla

cem

ent [m

]

Fig. 4.7c Third generalized coordinate of the isolated building

30

0 10 20 30 40 50 60 70-0.2

-0.1

0

0.1

0.2

Time [s]

Gen

. ac

cele

ratio

n [m

/s^2

]

Fig. 4.8a First generalized acceleration of the isolated building

0 10 20 30 40 50 60 70-0.2

-0.1

0

0.1

0.2

Time [s]

Gen

. ac

cele

ratio

n [m

/s^2

]

Fig. 4.8b Second generalized acceleration of the isolated building

0 10 20 30 40 50 60 70-0.2

-0.1

0

0.1

0.2

Time [s]

Gen

. ac

cele

ratio

n [m

/s^2

]

Fig. 4.8c Third generalized acceleration of the isolated building

31

amplitudes (Eqs. (54)); therefore, Fig. 4.7 is one of the most important results in thisresearch project. The top floor (point 5,a) relative displacement and acceleration, and thecritical stress in the column 1,1 (point 1,a) are shown in Appendix D. We note that thecritical stress has been reduced around 200 times.

In short, base isolation reduces significantly all the variables in the overallresponse of the structure, except for the displacement of the base.

4.6. Active control designThe previous results show that, even after selecting the parameters of the

isolation system using the design procedure presented, the base displacement is close tothe level of the ground displacement itself. Consequently, the hybrid control designmethodology should continue concentrating on this variable. Of course, we cannotpermit the other response variables, that have been enhanced with passive control, toreturn to the nonisolated case levels, specially the generalized coordinates.

With the addition of active control the mean square value of the basedisplacement is

αb(g 1 ,g2) = 2 H11

c (g1 ,g 2)2(c o

2ω 2 + ko2 )Sss(ω)

0

∞

∫ dω (69)

where H11c is the component 1,1 of the frequency response matrix of the actively

controlled system, i.e.,

H c = iωI − A − b1g

TC( )−1B (70)

where C is the 2x2(m+1) measurement matrix,

C =

1 0 L 0 0 L L 0

0 L L 0 1 0 L 0

(71)

g is the gain vector

g = −

g1

g2

(72)

and b1 is the first column of B.

In this case we are interested in reducing the function

Fc(g1 ,g2) = H11

c (g1,g2 )2(c o

2ω 2 + ko2 ) (73)

We plot this function against g1 and g2 in the frequency range of interest; this is shownin Fig. 4.9, where Fc is plotted against g1 (abscissa) for several values of g2. We can seethat the interest would be in designing to the right of the resonant condition where thecurves reach their lowest values. We propose to choose g1 as the value for whichresonance occurs at 2.5 r/s, which ensures that resonance does not occur in thefrequency range. Thus, g1=400000 N/m. The role of g2 is to reduce the effect of the nearresonant condition at around 2.5 r/s. We choose g2 with the aid of Fig. 4.10 as the valuefor which the function Fc has reached 1 % of its value at g2=0 (no velocity feedback).Thus, g2=300000 N s/m. We note that no effort was made, neither here nor in Section

32

Fig. 4.9 Function Fc(g1,g2) versus g1 with g2 [KN s/m] as a parameter

33

Fig. 4.10 Function Fc(400000,g2) versus g2 evaluated at ω=2.5 r/s

34

4.4, to obtain the precise values of the parameters according to the design guidelinesproposed.

4.7. Isolated /controlled building responseThe overall response of the base-isolated and controlled building is shown in the

following figures. Figure. 4.11a shows the displacement of the base. It is another mostimportant result in this research project because the motion of the base has beenreduced 12 times compared to the original motion of the fixed base (Fig 4.1a). This is adirect result of active control and its design. Figure 4.11b shows the acceleration of thebase. We note that the amplitudes have increased compared to the isolated case (Fig4.6b), but they are still much lower than the original amplitude in the fixed case (Fig4.1b). Figures 4.12 show the generalized coordinates of the superstructure. We note thattheir levels are reduced a little more (1.5 and 1.8 times for the first and secondcoordinates) from the isolated case levels (Figs. 4.7), which is more important than theincrement in the previous figure. The generalized accelerations are shown in Figs. 4.13;they have increased compared to the ones in the isolated case, but they are still muchlower than the original ones in the fixed building case (Figs. 4.3).

The other most important result is shown in Fig. 4.14a. The control force is below7000 N. This is 1.09 % the weight of the structure (60720 Kg). The control power isshown in Fig 6.14b. There are no results comparable to these in the literature forstructures with similar hybrid control systems, except for Ref. 15 in which an additionalactuator is used at the top of the building. The peak values of the response variables ofinterest are shown in Table 4.1. The top floor relative displacement and acceleration,and the critical stress in the column 1,1 are shown in Appendix D.

Table 4.1 Peak values of the response variablesFixed Isolated Isol./Cont.

ub [m] 0.109 0.095 0.009

üb [m/s2] 3.417 0.196 0.479

q1 [m] 4.706 10-2 2.181 10-4 1.427 10-4

q2 [m] 2.826 10-3 4.873 10-5 2.748 10-5

q3 [m] 5.551 10-4 1.146 10-5 6.756 10-6

˙ q 1 [m/s2] 26.315 0.085 0.387

˙ q 2 [m/s2] 13.772 0.146 0.426

˙ q 3 [m/s2] 6.827 0.104 0.345

In short, active control reduces the displacement of the base with very lowcontrol forces and reduces even more the stress levels.

35

0 10 20 30 40 50 60 70-0.01

-0.005

0

0.005

0.01

Time [s]

Dis

pla

cem

en

t [m

]

Fig. 4.11a Base displacement of the isolated/controlled building

0 10 20 30 40 50 60 70-0.5

0

0.5

Time [s]

Acc

ele

ratio

n [

m/s

^2]

Fig. 4.11b Base acceleration of the isolated/controlled building

36

0 10 20 30 40 50 60 70

-1

-0.5

0

0.5

1

x 10-4

Time [s]

Gen

. dis

plac

emen

t [m

]

Fig. 4.12a First generalized coordinate of the isolated/controlled building

0 10 20 30 40 50 60 70

-1

-0.5

0

0.5

1

x 10-4

Time [s]

Gen

. dis

plac

emen

t [m

]

Fig. 4.12b Second generalized coordinate of the isolated/controlled building

0 10 20 30 40 50 60 70

-1

-0.5

0

0.5

1

x 10-4

Time [s]

Gen

. dis

plac

emen

t [m

]

Fig. 4.12c Third generalized coordinate of the isolated/controlled building

37

0 10 20 30 40 50 60 70-0.5

0

0.5

Time [s]

Fig. 4.13a First generalized acceleration of the isolated/controlled building

Gen

. ac

cele

ratio

n [m

/s^2

]

0 10 20 30 40 50 60 70-0.5

0

0.5

Time [s]

Gen

. ac

cele

ratio

n [m

/s^2

]

Fig. 4.13b Second generalized acceleration of the isolated/controlled building

0 10 20 30 40 50 60 70-0.5

0

0.5

Time [s]

Gen

. ac

cele

ratio

n [m

/s^2

]

Fig. 4.13c Third generalized acceleration of the isolated/controlled building

38

0 10 20 30 40 50 60 70-50

0

50

100

150

Time [s]

Pow

er

[W]

Fig. 4.14b Control Power

0 10 20 30 40 50 60 70-1

-0.5

0

0.5

1x 104

Time [s]

Forc

e [N

]

Fig. 4.14a Control force

39

5. CONCLUSIONS

The Rayleigh-Ritz based substructure synthesis method proved to be highlysuccessful in analyzing a structure more complex than the ones previously analyzedwith it. The convergence of the eigenvalues is outstanding. The natural frequenciesconverge to 6 digits with 8 admissible functions per beam.

Passive and active control complemented each other exceptionally in protectingthe structure against earthquakes. The reason of this exceptional outcome is theapplication of the design methodology presented in this dissertation. The result is thatvery simple devices for both types of control are required. More importantly, thecontrol force is remarkably low.

Specifically, we found that for the structural model analyzed and the selectedparameters of the linear isolation system the critical stress is reduced 200 times with theimplementation of the base isolation. The accelerations in the structure are also greatlyreduced. However, the base displacements are not reduced much. Here is where activecontrol complements passive control. In the hybrid case the base displacement levelsare reduced 12 times from the levels in the fixed building case. The overall reduction inthe critical stress is 280 times in this case. The accelerations undergo an increase but stillthey are much lower than the ones in the fixed building case. Moreover, the controlforce is 1.1 % of the building weight.

40

REFERENCES

1. Chopra, A. K., Dynamics of Structures, Prentice Hall, U.S.A., 19952. Zienkiewicz, O. C. and Taylor, R. L., The Finite Element Method, 4 th Edition, McGraw-

Hill, England, 1989.3. Meirovitch, L., Principles and Techniques of Vibrations, Prentice Hall, U.S.A., 1997.4. Meirovitch, L. and Hale, A. L., “Synthesis and Dynamic Characteristics of Large

Structures with Rotating Substructures,” Proceedings of the IUTAM Symposium on theDynamics of Multibody Systems, Edited by L. Magnus, Springer-Verlag, Germany,1978, pp. 231-244.

5. Hale, A. L. and Meirovitch, L., “A general substructure synthesis method for thedynamic simulation of complex structures,” Journal of Sound and Vibration, Vol. 69,No 2, 1980, pp. 309-326.

6. Meirovitch, L. and Hale, A. L., “On the Substructure Synthesis Method,” AIAAJournal, Vol. 19, No 7, 1981, pp. 940-947.

7. Hurty, W. C., “Vibrations of structural Systems by Component-Mode Synthesis,”Journal of the Engineering Mechanics Division, Vol. 86, 1960, pp. 51-69.

8. Hurty, W. C., “Dynamic Analysis of Structural Systems Using ComponentsModes,” AIAA Journal, Vol. 3, No 4, 1965, pp. 678-685.

9. Meirovitch, L. and Kwak, M. K., “On the Convergence of the Classical Rayleigh-Ritz Method and the Finite Element Method,” AIAA Journal, Vol. 28, No 8, 1990, pp.1509-1516.

10. Skinner, R. I., Robinson, W. H. and McVerry, G. H., An Introduction to SeismicIsolation, John Wiley & Sons, England, 1993.

11. Meirovitch, L. and Kwak, M. K., “Rayleigh-Ritz Based Substructure Synthesis forFlexible Multibody Systems,” AIAA Journal, Vol. 29, No 10, 1991, pp. 1709-1719.

12. Kelly, J. M., Leitmann, G. and Soldatos, A. G., “Robust Control of Base IsolatedStructures Under Earthquake Excitation,” Journal of Optimization Theory andApplications, Vol. 53, No 2, 1987, pp. 159-180.

13. Pu, J-P. and Kelly, M., “Active Control and Seismic Isolation,” Journal of EngineeringMechanics, Vol. 117, No 10, 1991, pp. 2221-2236.

14. Tadjbakhsh, I. G. and Rofooei, F., “Optimal Hybrid Control of Structures UnderEarthquake Excitation,” Earthquake Engineering and Structural Dynamics, Vol. 21,1992, pp. 233-252.

15. Meirovitch, L. and Stemple, T. J., “Nonlinear Control of Structures in Earthquakes,”to appear in Journal of Engineering Mechanics.

16. Sharma, C. B., “Calculation of Integrals Involving Characteristic Beam Functions,”Journal of Sound and Vibration, Vol. 56, No 4, 1977, pp. 475-480.

17. Kelly, J. M., “Aseismic Base Isolation: Review and Bibliography,” Soil dynamics andEarthquake Engineering, Vol. 5, No 3, 1986, pp. 202-216.

18. Meirovitch, L. and Stemple, T. J., ”Base-Isolation Control of Structures inEarthquakes,” Proceedings of the 10th VPI&SU Structural Dynamics and ControlSymposium, 1995, pp. 263-274.

41

19. Su, L., Ahmadi, G. and Tadjbakhsh, I. G., “A Comparative Study of Performancesof Various Base Isolation Systems, Part I: Shear Beam Structures,” EarthquakeEngineering and Structural Dynamics, Vol. 18, 1989, pp. 11-32.

20. Shenton III, H. W. and Lin A. N., “Relative Performance of Fixed-Base and Base-Isolated Concrete Frames,” Journal of Structural Engineering, Vol. 119, No 10, 1993,pp. 2952-2968.

21. Morales-Velasco, C. A., Influence of Base Isolation on the Response of Structures toEarthquakes, M.S. Thesis, Virginia Polytechnic Institute and State University, 1995.

22. Paulson, T. J., Abrams, D. P. and Mayes, R. L., “Shaking-Table Study of Base-Isolation for Masonry Buildings,” Journal of Structural Engineering, Vol. 117, No 11,1991, pp. 3315-3336.

23. Yang, J. N., “Application of Optimal Control Theory to Civil EngineeringStructures,” Journal Engineering Mechanics, Vol. 101, No 6, 1975, pp. 819-838.

24. Martin, C. R. and Soong, T. T., “Modal Control of Multistory Buildings,” Journal ofEngineering Mechanics, Vol. 102, No 4, 1976, pp. 613-623.

25. Abdel-Rohman, M. and Leipholz, H. H., “Structural Control by Pole AssignmentMethod,” Journal of the Engineering Mechanics Division, Vol. 104, No 5, 1978, pp. 1159-1175.

26. Meirovitch, L. and Öz, H., “Active Control of Structures by Modal Synthesis,”Structural Control, Edited by H.H. Leipholz, North Holland Publishing, TheNetherlands, 1979, pp. 505-521.

27. Meirovitch, L. and Baruh, H., “Control of Self-Adjoint Distributed-ParameterSystems,” Journal of Guidance and Control, Vol. 5, No 1, 1982, pp. 60-66.

28. Meirovitch, L. and Silverberg, L. M., “Control of Structures Subjected to SeismicExcitation,” Journal of Engineering Mechanics, Vol. 109, No 2, 1983, pp. 604-618.

29. Grace, A., Laub, A. J., Little, J. N. and Thompson, C. M., Control System Toolbox, TheMathWorks, Inc., 1996.

30. Soong, T. T. and Grigoriu, M., Random Vibration of Mechanical and Structural Systems,Prentice Hall, U.S.A.,1993.

31. Adriono, T. and Carr, A. J., “A Simplified Earthquake Resistant Design Method forBase-Isolated Multistorey Structures,” Bulletin of the New Zealand National Society forEarthquake Engineering, Vol. 24, No 3, 1991, pp. 238-250.

32. Constantinou, M. C. and Tadjbakhsh, I. G., “Optimum Characteristics of IsolatedStructures,” Journal of Structural Engineering, Vol. 111, No 12, 1985, pp. 2733-2750.

33. Blevins, R. D., Formulas for Natural Frequency and Modes Shape, Van NostrandReinhold, U.S.A.,1979.

42

APPENDIX A Disjoint mass and stiffness matrices

Before defining the disjoint mass and stiffness matrices we need to establish anotation to work with these large matrices. First, any matrix in this appendix is a3qNx3qN supermatrix composed of N2 matrices which in turn have nine qxq submatrices.In order to define the disjoint mass and stiffness matrices we have to define some qxqsubmatrices, as follows

plpl = ′ f (L) f T ′(L) ff = ffT pppp = ′ ′ f f T″

lpl = f (L)f T ′(L) lf = f(L) fT gggg = ′ ′ g gT″

plf = ′ f (L)f T gg = ggT pp = ′ f fT ′ (a1)

Now, the notation

mm( )

a,b x,ya,b = 1,2,L,N; x,y = 1,2,3 (a2)

where mm can be any of the submatrices defined before, represents a supermatrix thathas the submatrix mm at the submatrix position x,y in the matrix a,b.

Thus, the N matrices in expression (13) are

N an = plpl( )j ,s1,1

s= 1

r− 1

∑j =1

i −1

∑r = 1

n − 1

∑i =1

n− 1

∑ N gn = ff( )n,n 1,1

N bn = lpl( )r , j 1,1

j =1

i − 1

∑r = 1

n − 1

∑i= 1

n− 1

∑ Nhn = plpl( )k, j1,1

k = 1

i − 1

∑j =1

n− 1

∑i =1

n− 1

∑

Ncn = lpl( )i ,s1,1

s= 1

r −1

∑r =1

n− 1

∑i = 1

n−1

∑ N in = lpl( )i , j1,1

j= 1

n−1

∑i =1

n −1

∑

Ndn = ll( )i,r 1,1r =1

n −1

∑i = 1

n −1

∑ N jn = plf( )j,n 1,1

j = 1

i − 1

∑i = 1

n− 1

∑

N en = plpl( )i,s 1,1

s = 1

n− 1

∑j= 1

n −1

∑ Nkn = lf( )i,n 1 , 1

i = 1

n− 1

∑

N fn = plf( )j ,n1,1

j = 1

n − 1

∑

(a3)

It was said that an identical expression to (13) is obtained for the second integrand in Eq.(11). In this case, the N matrices are as shown before except for the sub-subindices;instead of 1,1 we have 2,2. The N matrices in Exp. (16) are

N ln = plpl( )j,s 1,1

s =1

r −1

∑j= 1

i − 1

∑r =1

n

∑i = 1

n

∑ Non = ll( )i,r 1,1

r = 1

n

∑i =1

n

∑

Nmn = lpl( )r,j 1 , 1j = 1

i −1

∑r= 1

n

∑i =1

n

∑ Npn = gg( )n,n 3,3

Nnn = lpl( )i ,s1,1s= 1

r −1

∑r =1

n

∑i = 1

n

∑

(a4)

43

If Exp. (13) is written as

˙ q dTN1n

˙ q d (a5)and similar expressions are obtained for the second and third integrands, we can writeEq. (11) as

T =

1

2mc

n =1

N

∑ ˙ q dT N1n

˙ q d0

L

∫ dxn,1 + mc˙ q d

TN 2n˙ q d

0

L

∫ dxn,2 + mb˙ q d

TN3n˙ q d

0

l

∫ dxn,3 (a6)

or

T =

1

2˙ q d

T M d ˙ q d (a7)

where Md is a nonsymmetric disjoint mass matrix,

M d = mc

n = 1

N

∑ N1n

0

L

∫ dxn,1 + mc N2n

0

L

∫ dxn,2 + mb N3n

0

l

∫ dxn,3 (a8)

Finally, the symmetric disjoint mass matrix Md can be written as

Md =

1

2(M d + M d

T ) (a9)

On the other hand, the S matrices in Eq. (19) are

San = pppp( )n,n 1,1

Sdn = pp( )n,n 1,1

Sbn = pppp( )n,n 2,2S en = pp( )n,n 2,2

Scn = gggg( )n,n 3,3

(a10)

and Kd can be written as

Kd = EI c

n= 1

N

∑ San

0

L

∫ dxn,1 + EI c Sbn

0

L

∫ dxn,2 + EIb S cn

0

l

∫ dxn,3 − pn Sdn

0

L

∫ dxn,1 − pn Sen

0

L

∫ dxn,2 (a11)

44

APPENDIX B Constraint matrix

In this appendix we assume that the vectors of admissible functions f and g arethe ones selected in Section 3.4 (Eqs. (29)). Furthermore, the functions are normalizedsuch that

Ycf′ (L) = 1 Ycp

′(l) = 1

Ycc′(L /4) = 1 Ypc

′(0) = 1 (b1)and here q is considered even. In this case, Eqs. (24) can be written as

qj ,1ii= 1,3

q − 1

∑j = 1

n

∑ = qn,3ii = 2,4

q

∑

qj,2 ii= 1,3

q − 1

∑j =1

n

∑ = qn,3 ii= 1,3

q − 1

∑

L qj ,1ii= 1,3

q −1

∑j = 1

n− 1

∑ + fT (L)qn,1 = L qj,2 ii =1,3

q− 1

∑j = 1

n −1

∑ + fT (L)qn,2 (b2)

Next, we select 3N dependent coordinates, and write them as

qn,3q= q j,1 i

i =1,3

q − 1

∑j =1

n

∑ − qn,3 ii = 2,4

q − 2

∑

qn,3 q − 1= qj ,2i

i =1,3

q− 1

∑j =1

n

∑ − qn,3 ii =1,3

q − 3

∑

qn,2q − 1= β q j,1 i

i= 1,3

q − 1

∑j =1

n− 1

∑ + αiqn,1 ii =1,3

q −1

∑ − β qj,2 ii =1,3

q −1

∑j = 1

n −1

∑ − α i qn,2 ii =1,3

q − 3

∑ (b3)

where

β =

L

fq− 1(L)α i =

fi(L)

fq− 1(L) (b4)

The constraint matrix C can be readily obtained by writing Eq. (25) in its expandedform.

45

APPENDIX C Integrals

The β’s in Eqs. (28) are defined by the nonlinear equations (Ref. 33),

cosβ 1j Lcosh β1j L = −1

cosβ2 j Lcosh β 2jL = 1

tanβ3 jl = tanh β3j l (c1)

β is related to the Euler beam natural frequencies, ω , as

β =

mω 2

EI4 (c2)

where m and EI are the mass per unit length and flexural rigidity of the beam,respectively. We obtained 17-digit roots which is necessary due to the nature of thehyperbolic functions. The σ’s are

σ1j =sinhβ1j L − sinβ 1j L

coshβ 1j L + cosβ 1j L

σ 2j =cosh β2j L − cosβ2 j L

sinhβ 2jL − sinβ 2j L

σ 3j = cot β3jl (c3)

The c’s are obtained trough normalization of the eigenfunctions. We have alreadydefined a normalization course in Appendix B.

We follow the technique presented in Ref. 12 to solve the integrals of products ofbeam eigenfunctions involved in the analysis. We show the method with the product ofclamped-clamped and clamped-free functions. First, the eigenfunctions satisfy thedifferential equations

Ycf'''' = β1

4Ycf

Ycc'''' = β2

4Ycc (c3)which permit us to write

Ycfr0

L

∫ Yccsdx =1

β 2s4 Ycfr

0

L

∫ Yccs''''dx (c4)

Next, using integration by parts repeatedly above we can write

1

(β2s4 − β1r

4 )(YcfrYccs

''' − Ycfr'Ycc s

'' + Ycfr''Ycc s

' − Ycfr'''Yccs) 0

L (c5)

which after using the boundary conditions can be written as

1

(β2s4 − β1r

4 )(Ycfr(L)Yccs

'''(L) − Ycfr'(L)Ycc s

''(L) ) (c6)

Finally, using Eqs. (28) and some trigonometric and hyperbolic identities we obtain

46

Ycfr0

L

∫ Yccsdx =4c1rc2sβ2s

2

β2 s4−β1r

4 sinhβ 1rLsinβ 1rLsinhβ2sLsinβ2sL(β 1r

(sinhβ1r L+sinβ 1r L)(coshβ2sL−cosβ2 sL)

−β 2s

(sinhβ2sL−sinβ 2sL)(coshβ1sL+cosβ1sL)) (c7)

47

APPENDIX D Displacement, acceleration and stress at critical points

The displacement and acceleration of the top of the building (point 5,a) relative tothe ground are shown in figures D.1, D.2 and D.3 for the fixed, isolated andisolated/controlled building, respectively. The critical stress in column 1,1 (point 1,a) isshown in Fig. D.4 for the three cases.

48

0 5 10 15 20 25 30 35 40 45 50-50

0

50

Time [s]

Acc

ele

ratio

n [

m/s

^2]

Fig. D.1b Absolute acceleration of the top of building (fixed)

0 5 10 15 20 25 30 35 40 45 50-0.1

-0.05

0

0.05

0.1

Time [s]

Dis

pla

cem

en

t [m

]

Fig. D.1a Displacement of the top of building relative to the base (fixed)

49

0 10 20 30 40 50 60 70

-0.2

-0.1

0

0.1

0.2

0.3

Time [s]

Acc

ele

ratio

n [

m/s

^2]

Fig. D.2b Absolute acceleration of the top of building (isolated)

0 10 20 30 40 50 60 70-4

-2

0

2

4x 10-4

Time [s]

Dis

pla

cem

en

t [m

]

Fig. D.2a Displac. of the top of building relative to the base (isolated)

50

0 10 20 30 40 50 60 70-4

-2

0

2

4x 10-4

Time [s]

Dis

pla

cem

en

t [m

]

Fig. D.3a Displac. of the top of building relative to the base (isol./cont.)

0 10 20 30 40 50 60 70

-0.2

-0.1

0

0.1

0.2

0.3

Time [s]

Acc

ele

ratio

n [

m/s

^2]

Fig. D.3b Absolute acceleration of the top of building (isol./cont.)

51

0 5 10 15 20 25 30 35 40 45 50-4

-2

0

2

4x 105

Time [s]

Str

ess

[Pa]

Fig. D.4b Critical stress (isolated)

0 5 10 15 20 25 30 35 40 45 50-4

-2

0

2

4x 105

Time [s]

Str

ess

[Pa]

Fig. D.4c Critical stress (isolated/controlled)

0 5 10 15 20 25 30 35 40 45 50-1

-0.5

0

0.5

1x 108

Time [s]

Str

ess

[Pa]

Fig. D.4a Critical stress (fixed)

52

VITA

César A. Morales Velasco

The author was born to Agustín Morales and Virginia Velasco on March 18, 1969in Puno, Perú, ashore the lake where the Incas’ empire was born. He moved toVenezuela, land of South America’s Liberator Simón Bolívar, in 1975 and is aVenezuelan citizen since 1982. He received a Mechanical Engineer degree, Cum Laude,from Universidad Simón Bolívar, Caracas, in October 1991. He immediately joined theMechanics Department in that institution, and was awarded a joint scholarship fromFundación Sivensa and his university in 1993 to pursue the Master of Science degree inEngineering Mechanics at Virginia Polytechnic Institute and State University, which wasconcluded in July 1995. He immediately started the doctoral program in the sameuniversity with the support of a joint scholarship from the Organization of AmericanStates and his university. He wants to go back, has to go back and must go back toAmérica Latina.