STOCHASTIC RESPONSE OF SINGLE DEGREE OF FREEDOM HYSTERETICi OSCILLATORS

byGustavo Omar Maldonado

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfrllment of the requirements for the degree of

Master of Science

in

Engineering Mechanics

APPROVEDÄ

Mahendra P. Singh

Robert A. Heller Daniel Frederick

April 1987

Blacksburg, Virginia

STOCIIASTIC RESPONSE OF SINGLE DEGREE OF FREEDOM HYSTERETIC IOSCILLATORS I

.I by II Gustavo Omar Maldonado I

. Mahendra P.

SinghEngineeringMechanics I

(ABSTRACT)During

strong ground shaking structures often become inelastic and respond hysteretically.Therefore, in this study some hysteretic models commonly used in seismic structural analysis are

studied. In particular the charactexistics of a popular endochronic model proposed by Bouc and

Wen are examined in detail. In addition, analytical expressions have also been developed for most

commonly used bilinear model as well as another model, herein called as the hyperbolic model. I

IAs stochastic response analysis with such models commonly use the stochastic linearization ap-

‘

proach which is necessarily iterative, here the convergence characteristics of such methods,whenapplied

to calculate the response of single degree of freedom oscillators, are studied in detail. Several I

oscillators with different pararneters are considered in the study. The ground motion is modeled I

by a stationary random process with Kanai·Tajimi spectral density function. It is noted that

someadjustmentsin the equivalent linear pararneters are necessary to achieve convergence. Also the rate Iof convergence to the final results is slower for the oscillators with low yield levels. IThe numerical results obtained by the equivalent linear approach are also compared with the · I

results obtained for an ensemble of ground motion time histories and the possible causes of the I

discrepancy between the two results are discussed. I

1

Acknowledgements

The author wishes to sincerely thank Dr. Mahendra P. Sir1gh, his advisor, for the assistance and

counsel throughout the course of this study. In addition, appreciation is extended to Dr. L.Faravelli and Dr. F. Casciati for the help they provided during their stay in Blacksburg, Virginia,in the summer of 1985.

The author would also like to express his gratitude to his Professors C. A. Barto, L. A. Godoyand C. A. Prato in Argentina for their encouragement and guidence.

This work was funded by the National Science Foundation Grant No. CEE-8412830. Thissupport is gratefully acknowledged.

Finally, the author gives his deepest thar1ks and love to his parents, wife and friends for their faith

3IlCl encouragement.

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Acknowledgements iii

I

Table of Contents

CHAPTER I: INTRODUCTION .......................................... 1

CHAPTER II: HYSTERETIC MODELS .................................... 42.1 INTRODUCTION .................................................... 4

2.2 EQUATION OF MOTION AND RESTORING FORCE ..................... 4

2.3 BOUC·WEN MODEL ................................................. 6

2.3.1 GAMMA PARAMETER ............................................ 8

2.3.2 BETA PARAMETER .............................................. 9

2.3.2.1 Softcning model ..........................................I...... 9

2.3.2.2 Hardcning model ............................................... 10

2.3.3 EXPONENT PARAMETER ........................................ 112.3.4 PATH EQUATIONS IN THE X-Z PLANE ............................. 12

2.3.5 ALTERNATIVE FORM OF THE MODEL ............................ 14

2.4 HYPERBOLIC HYSTERETIC MODEL .................................. 15

2.5 BILINEAR HYSTERETIC MODEL ..................................... 17

2.6 OSCILLATOR RESPONSE AND HYSTERETIC BEHAVIOR ................ 18

Table er Contents iv

i11

_ 11l

CHAPTER III: EQUIVALENT LINEARIZATION ............................. 20

3.1 INTRODUCTION ................................................... 203.2 EQUIVALENT LINEARIZATION TECHNIQUE ........................... 21

3.3 BOUC—WEN MODEL ................................................ 233.4 BILINEAR MODEL ................................................. 243.5 EQUIVALENT LINEAR STATIONARY RESPONSE ...................... 25

CHAPTER IV: NUMERICAL RESULTS ................................... 29

4.1 INTRODUCTION ................................................... 29

4.2 PROBLEM PARAMETERS ........................................... 30

4.2.1 OSCILLATOR PARAMETERS ..................................... 30

4.2.2 PARAMETERS OF THE HYSTERESIS MODEL ....................... 30

4.2.3 INPUT MOTION PARAMETERS ................................... 32

4.3 RATE OF CONVERGENCE RESULTS .................................. 33

4.4 COMPARISON OF RESPONSE ........................................ 35

CHAPTER V: SUMMARY AND CONCLUSIONS ............................ 39

APPENDIX ........................................................... 42

Derivation of Eq. 3.09 .................................................... 42

Expressions of C and K for Bouc·Wen’s model with n= 1 ......................... 44

FIGURES ............................................................ 47

TABLES ............................................................. 74

REFERENCES ........................................................ 84

Table of Contents v

I 1II

VITA ................................................................ 86

Table ofContentsvi

II

CHAPTER I: INTRODUCTION

The behavior of engineering structures which are affected by dynamic loads can not always bedescribed by simple linear-elastic models, Fig. 1. In many cases, the structures have nonlinear andinelastic stiffness characteristics. Often such stiffness characteristics also have, what we call, mem-ory; that is, the response of the system at a given time not only depends on the input at that timebut also on the input and response at an earlier time. If the applied load is cyclic, the structuralresponse may also be cyclic with dissipation of energy through hysteresis cycles. Structures sub-jected to strong ground shaking exibit such inelastic characteristics, and it is necessary to considerthese in the calculation of seismic response.

In earthquake structural engineering, the constitutive law of inelastic structures has often beenapproxirnated by the popular elasto·perfectly-plastic model, Fig. 2, or by the bilinear model, Fig.3. Tests on some structural elements have, however, demonstrated the need of using more complexmultilinear models, like the Takeda’s model shown in Fig. 4 and used for reinforced concrete. Se-

veral such models are being utilized now in dynamic analysis of structures subjected to earthquakeinduced ground motions. In such analyses, usually a step-by-step time history analysis is per-formed. Computer codes like DRAIN-2D [l] and ANSR-l [2] have been developed to carry outsuch analyses. It has, however, been found difficult to utilize these discrete multilinear models in

the stochastic response analysis of structures.

CHAPTER l: INTRODUCTION I

l

In addition to these discrete multilinear models, continuous models have also been developedto characterize the hysteretic force deforrnation characteristic of structural elements and materials.The earliest example of such a model is, probably, the Rarnberg-Osgood model [3]. This modelhas been utilized to characterize the hysteretic behavior of soils in several soil dynamics studies [4,5] in earthquake engineering. The utilization of this model for metal and concrete structures is,however, uncormnon. For such structures, the continous model proposed by Iwan [6] and anendochronic model proposed by Bouc [7] have recently been utilized. Bouc’s model has beenmodified and improved further to include the time dependent degradation and pinching effectscommonly observed in structures subjected to earthquake induced ground motions. Another modelsimilar to Bouc’s has also been developed by Ozdemir [8] which was utilized by Bhatti and Pister[9] in optirnization studies in seismic design of structures.

Because of the analytical sirnplicity, Bouc’s model has been of special interest lately. lt has beenextensively used by Wen [10], Baber and Wen [ll], Casciati and Faravelli [12], Casciati [13] andmany others ir1 their studies of stochastic response of structures subjected to stochastic loads. Inthese studies, the nonlinear differential equations associated with the constitutive law are linearizedthrough the concept of stochastic lineaxization [14]. The coeflicients of the linearized equation arefunctions of the response statistics, which are not known a priori. Some initial estimates of thesecoefficients are obtained which are then used in the calculation of the response from the linearizedequations. The calculated response is then used to modify or improve the earlier estimates of thecoefficients. This process is repeated till a convergence in the calculated response and coefficientvalues is acheived. Such approaches are thus iterative ir1 nature.

In this work, the utilization of the endochronic model proposed by Bouc in stochastic seismic

response evaluation of a simple oscillator subjected to random earthquake inputs is further studied.In Chapter Il, the charactenstics of various parameters of Bouc’s model are studied. The 1imita~tions and appropriateness of this model in earthquake structural engineering are examined. Twonew endochronic laws are also proposed. One of them uses hyperbolic trigonometric functions tomodel a softening system and the other models bilinear hysteresis loops. In Chapter lll the ana-

CHAPTER I: INTRODUCTION 2

lytical details of the equivalent linearization process used for calculating the response of an oscillatorwith nonlinear hysteretic spring, and subjected to stationary iiltered white noise excitation at thebase are provided. In Chapter IV the convergence characteristics of the iterative procedure used inthe calculation of response are examined in detail. The comparison of the results obtained by thestochastic linearization with those obtained by the time history analysis for an ensemble ofaccelerograms is also given. The summary and concluding remarks are given in Chapter V.

CHAPTER I: INTRODUCTION 3

i11I 1I

2.1 INTRODUCTION

In this chapter we will discuss the characteristics of some commonly used hysteretic force mod-

els. In particular the model proposed by Bouc and subsequently modified by Wen, Baber, etc., is

discussed in detail. Based on the study of this model, analytical forms of two other models --the

hyperbolic and the bilinear-· are proposed.

2.2 EQUA TION OF MOTION AND RESTORING

FORCE

In general the equation of motion of a sing1e·degree-of-freedom (SDF) system can be writtenas follows:

CHAPTER ll: HYSTERETIC MODELS 4

l11

I

m.5% + Q’(x, x, t) = — m.5%g (2.01)

Where Q'(x,.5%, t) is the general nonlinear hysteretic force; x is the relative displacement of theoscillator with respect to its moving base; x and .5% are the relative velocity and acceleration respec-tively; m is the mass of the oscillator and 5%, is the ground acceleration. ln the present study wewill consider that the damping dependent restoring force is linear. Then, the equation of motionreduces to:

m.5% + cx + Q(x,:%, t) = — m.5%g (2.02)

Now Q represents the undamped nonlinear hysteretic term, which is a function of the displacementand velocity; c is the damping constant.

ln standard form, this equation can also be written as:

5é + (2 C., Q(x- 5%- ¢) = — 5%, (2-03)

where co, is the preyielding natural frequency and Q, is the viscous damping ratio of the system de-

fined as Q, = —€—.Zmco,

If the material is linear and elastic, Q(x, x, t) reduces to Q(x) = kx , where k is the elastic con-

stant of the spring supporting the mass, called the stiffness coefficient. To include the hystereticbehavior along with linear behavior, Wen [15] has proposed the following expression to model therestoring force

Q(x, x, t) = k[ax + (1 — ct)z] (2.04)

where z is an auxiliary variable which has hysteretic characteristics. Thus the second part in theabove equation represents a nonlinear element in parallel with a linear element represented by thefirst part. The constant 0. is a weighting constant representing the relative participations of the

CHAPTER II: HYSTERETIC MODELS 5

Iul

linear and nonlinear terms. We will discuss the meaning of this parameter for three differentmodels considered in this study.

The tangent stiffness for the nonlinear restoring force is defined as:

k=—c§—=k[a+(l—a)£-] (2.05)' dx dx

The stiffness when z = 0 is called as the initial stiffness. If we choose the auxiliary variable z suchthat = I at z = 0, we obtain the following for the initial stiffness:

k;=k[a+(l—

iII

z=Ax—yIxIzIz|"“‘—ßxIzI" (2.09)

This model is a generalization of the model proposed by Bouc [7] which has been extensively used

and modified by Wen [10, 15], Baber and Wen [ll], Casciati [12] and Casciati and Faravelli [13] in

their work.

This model has four parameters A, Y, ß and rz. The effect of these parameter values on the shape

of the hysteretic curve has been discussed earlier by Baber and Wen [11]. Here we will re-analyze

this model comprehensively to understand better the influence of its various parameters on the

characteristics of the model.

For this we devide Eq. 2.09 by ir, to obtain:

—-‘!¢Z—=A— IzI"|$ß+ly] (2.10)dx Ixzl

We now define the ultimate value of z as the value at which ä = 0 . By setting gxä- = 0 in Eq.2.10 we obtain:

A L*1 (2-11)xzuB + ‘*+ Y|x2„|

Since the positive value of z„ corresponds to the positive x and negative to the negative x, the termÄis equal to l. this defines z„ as:Ixz„I

A LI I = " 2.12A ß + Y ( 1

CHAPTER 11: HYSTERETIC MODELS 7——————— —

1II

I

I

It is seen from Eq. 2.10 that the slope dz/dx is equal to A at 2 = 0. By properly adjusting the I

value of k to define initial stiffness, we can choose A = 1. This will also permit us to define (1 as

the ratio of the final stiffness to the initial stiffness.

2.3.1 GAMMA PARAMETER

Various branches of a hysteretic loop, such as these shown in Fig. 5, can be devided into out-

ward ar1d inward paths. The paths starting at 2 = 0 and going towards :1: 2,, will be called as theoutward paths. For such paths the product x2 is always positive, and thus Eq. 2.10 for xz > 0 canbe written as:

iä=A—(ß+7)|zI" (2.13)dx

Likewise, the paths starting from the extreme position of 2 toward 2 = 0 will be called as the inwardpaths. For such paths, the product x2 is always negative, and thus Eq. 2.10 for xz < 0 becomes:

(2.14)dx

lt is noted that for 7 = 0, Eq. 2.13 and 2.14 are the same. That is, the outward and inward paths

coalesce into a single path. This irnplies that the relationship between 2 and x is not hysteretic,

although for rz > 1 it is still nonlinear.

Let us now consider a case with B > 0, and examine the effect of varying 7. If we choose

7 > 0, then the slope in the x·z plane defmed by Eq. 2.14 will always be greater than the slope ydefined by Eq. 2.13. Thus, the shape of the hysteresis loop will be like those in Fig. 6 or Fig.

7,withthe loop being traced in a clockwise direction. On the other hand, if we choose 7 < 0, then II

in the x-2 plane, the slope defined by Eq. 2.14 will always be smaller than the slope defined by (

Eq. 2.13. The hysteresis loop in this case will be as shown in Fig. 8 or Fig. 9, with the loop nowI

CHAPTER ll: HYSTERETIC MODELS 8 II

being traced in a counterclockwise direction. The counterclockwise loop is, however, not phys-ically possible as it implies a negative dissipation of the energy. This means that the parameter Yshould be positive, at least when B is positive.

A similar observation can also be made for the negative B values. We observe that when

Y > 0, the loops will again be traced like in Fig. 06 or in Fig. 07. Whereas if Y < 0, the loops willbe traced like in Fig. 08 or Fig. 09. Thus we can conclude that the parameter Y should always be

positive, independently of the value taken by B

2.3.2 BETA PARAMETER

We now observe that the softening and hardening characteristics can be created by proper choice

of B and Y values.

2.3.2.1 Softening model

For a softening model, the slope of outward paths must decrease with lzl. For this it is nec-

essary that:

(ß+v)>0 ¤r ß> —v (2-15)

Thus for softening stiffness characteristics, the parameter B could possibly take a negative value

since Y is always positive.

Now, to obtain a softening model with inward-path slopes decreasing as z —+ 0 , we need that:

(B — Y) < 0 or B < Y (2.1661)

CHAPTER Il: HYSTERETIC MODELS 9

11111

Thus, to satisfy the conditions expressed by Eq. 2.15 and Eq. 2.16a it is necessary that

-7 < B < 7. In such a case, the hysteresis loop will be as presented in Fig. 10.a.

To decrease the area under the hysteresis loop, we can choose an inward path with increasing slope

as z —» 0. This can be achieved by choosing:

(B - 7) > 0 or B > 7 (2.16b)

In this case, satisfaction of Eq. 2.16b, i.e., 0 < 7 < B, will give us a loop traced in Fig. 10.b.

We can obtain a linear inward path by simply choosing B = 7.

2.3.2.2 Hardening model

Similarly, for hardening stiffness charactexistics, the slope of the outward path should increase

with Izl . This can be ensured by choosing

B+7

l

2.3.3 EXPONENT PARAMETER

We will now study the effect of changing the exponent parameter in the endochronic law. Iti

was mentioned by Wen [15] that as n —> 00 , the endochronic law approaches the commonly used

elasto-plastic model for the softening case. To show this analytically, we examine the softening

model. For a softening outward path (ß + Y) > 0. The expression for such path is:

äxä=A — IzI"(ß+y) (2.18)

Substituting for (ß + y) in terms of 2,, from Eq. 2.12, we obtain:

dz _ I z In—- — — A —— 2.19dx A Zu ( )

As noted before, the slope at 2 = 0 is A. Futhermore we note that when n approaches infinity, the

ratio (2/2,,) approaches zero as lzl < lz,,l . From Eq. 2.19 we note that in such a case the slope

at any point (x, 2) becomes equal to A, and the relationship between 2 and x is a straight line.

Similarly, we can obtain the slope on the inward path, defined by Eq. 2.14 as:

|iI" 220

This slope also approaches a constant value of A as rz becomes large.

Thus, for rt = ¤O, both the outward and inward path become straight lines, giving rise to an

elasto-plastic endochronic model in the x-2 plane. For a fmite value of cz in Eq. 2.4, the restoring

force is then defined by a bilinear hysteresis loop.

CHAPTER ll: HYSTERETIC MODELS ll

2.3.4 PATH EQUATIONS IN THE X-Z PLANE

Eq. 2.10 defines a set of differential equations for various paths of the endochronic law. De-pending upon the values of x and z, we will obtain different equations for different paths from Eq.2.10. Integration of these equations will provide the equations for these paths. For example, ifx > 0 and z > 0 , the differential equation for one of the two outward paths becomes:

ill = A — (ß + y)z" (2.21)dx

Integration of this will provide

fz $2% = (x — xo) (2.22)0 A "' (ß + Y)

Z,

Where x„ is the x-value when z = 0 The integration on the left hand side of Eq. 2.22 is difficultto evaluate for higher values of rz. However, we can for n= 1 obtain explicit expressions for z as afunction of x for various paths. Por this case, Eq. 2.10 for all paths can be written as:

d -E? - A + 6 z (2.23)

Where 5 takes on different values for different paths as:

ö=—-(ß+y) for x>0, z>0 (2.2461)

ö=—(ß—y) for x0 (2.24/2)

ö=+(ß+7) for x

z = —§ [6ö("‘ *0) — 1] (2.25)

Where at x = x„, z = 0. For the four paths defined by various values of 5 in Eq. 2.24, the following

expressions are obtained:

Outward path: x > 0 , z > 0

z = ——’l—— [1 — 6'“"’”"‘° *1)] (2.26a)(B + Y)

Inward path: x < 0, z > O

z (2.2612)(B -1)

Outward path: x < 0 , z < 0

z = ——l’i— [1 — 6‘ß+”"" *1*] (2.266)(B + v)

Inward path: x > 0, z < 0

z = ———’A [1 - €(ß—‘Y)(x— *0)] (2.2661)(B — Y)

For various possible values of B and y for the softening and hardening models, these paths are

schematically shown in Figs. ll and 12. The boundaries between the softening and hardening cases,

which correspond to the values of B + y = 0 and B — Y = 0 are also shown in these figures. The

~equation for all these boundaries can be simply writtenas:1

z = A (x — xo) (2.27) ‘

c11APT1:;R 11: HYSTERETIC MODELS 13

1

By a proper selection of ß and 7 values, as discussed earlier, a variety of hysteresis loops can be

traced. For -7 < ß < 7 , (I BI < 7), Eqs. 2.26a-d, will form the hysteretic loop shown in Fig. 13.

Similarly, if [3 = 7, the loop shown in Fig. 14 is obtained.

2.3.5 ALTERNATIVE FORM OF THE MODEL

For a comparison of different hysteretic models, it is sometimes convenient to use the same ul-

tirnate value of z. This value for the Bouc—Wen model is defined by Eq. 2.12. We will now rewrite

Eq. 2.09 with z„ as one of the model parameters.

We solve Eq. 2.12 for ß in terms of 2,, and 7 as:

ß = — 7 (2.28)|z„I

Introducing this in Eq. 2.09, we obtain:

é=AI:1—IéI”jI1i:+7(iIzI”-zIzI”_II>&I) (2.29)

For the special case of ß = 7, where the inward paths are linear, this expresion simplifies to:

· A n n - rr — 1 -Fi 2IzI — Izl x—zIzI Ixl (2.30)2 I zuln II

CHAPTER 11: 11YsTER1;T1c MODELS I4

I

2.4 HYPERBOLIC HYSTERETIC MODEL

Representation of Bouc-Wen model by Eqs. 2.26a-d for various branches of the loop suggeststhe possibility of developing the analytical expressions for other hysteretic models. For this wechoose appropriate expressions for the outward and inward paths. For a smooth transition betweenthese two paths it is necessary that the slopes of these paths be the same at z = 0. To demonstratethis, we choose the following expressions for the outward and inward paths:

Outward path:

z = a tanhlb (x — x0)] (2.3l)

Inward path:

z = a sinhlb (x — xO)] (2.32)

It is seen that both these functions are zero at x = x„. Also, the slopes of these two functions at

x = x„ or z = 0 are the same and equal to:

Tdi? = ab at z = 0 (2.33)

The differential equations of the two paths in the x-z plane are given by

Outward paths: (x > 0, z > 0) and (x < 0, z < 0)

cosh [b (x · x0)] cosh [tanh (2/61)]

Inward paths: (x < 0, z > 0) and (x > 0, z < 0)

CHAPTER ll: HYSTERETIC MODELS l5

= ab cosh[b (x — xO)] = ab cosh[ sinh— l(z/61)] (2.35)

The corresponding endochronic model is obtained as:

% = (2.36)

Which for the two paths can be written as:

Outward path: (iz) > 0

2 = abi sech2[ tanh—l (2.37)

lnward path: (X2) < 0

2 = a bx cosh'; sinh”l (2.38)

These two expressions can be combined into a single expression as:

(2.39)2 Izxl

where]; = sec/12[tanh"(z/a)] and]; = cosh [sinh"(z/a)] .

This model will trace a hysteretic loop as shown in Fig. I5. Eq. 2.39, however, can only producea softening system.

lt is also possible to have linear inward path in this case by simply adopting the following

equation in place of Eq. 2.38:

2 = abx, (iz) < 0 (2.40)

CHAPTER Il: HYSTERETIC MODELS I6

l

Combination of Eq. 2.37 and 2.40 into a single equation gives:

(2.41)2 lzxl

2.5 BILINEAR HYSTERETIC MODEL

Since the bilinear models have been very frequently used to define the restoring force, it is of

interest here also to develop an analytical expression for such models. lt is also necessary to have

an elasto-plastic law for z in Eq. 2.04 to obtain a post-yielding slope of 11. We will, therefore, dis-

cuss the formulation of an elasto-plastic model now.

Fig. 16 shows the elasto·plastic behavior of z versus x . It is noted that on the outward path

A-B-C the time derivative of z can be simply written as:

2=-2-xIi...@;i.+1]; :&>0, z>0 (2.42)2 lz„ — z|

Where z„ is the maximum value. Similarly on the other outward path, D-E-F,

2=2-xl-i.‘2..+1;1é

I

z=)3 (2.466)Izxl

Where ß is defined by:

1·1= L LLL + L (2.466)2 Z | IxIIzl

2.6 OSCILLA TOR RESPONSE AND HYSTERETIC

In this section, we obtain the response of an oscillator subjected to some base motion time his-

tory. The three restoring force models discussed in previous sections have been used. The z(t) and

x(t) responses were obtained from the solution of the following equations:

£+2§Oc00+co§,[ax+(1—a)z]= -2Eg (2.4661)

2 = 2(ic, z) (2.46/2)

To solve these differential equations we used a step by step integrating procedure encoded in the

routine I1A2F of Abaci’s library written for personal computers. II

Fig. 17 shows the x-z response of the oscillator for the base motion time history shown in the I

iriset. The Bouc-\Ven model with ß = .5, y = 1.5, A = 1., rz = 1. and (1 = .25 has been used toIIICHAPTER 11: HYSTERETIC MODELS 18 I1

define the restoring force. The response after point A on the curve is the free vibration response,as the base motion ceases at this point.

Fig. 18 shows almost complete hysteresis loops obtained for the three restoring force modelsdiscussed in previous sections. For the Bouc-Wen model, Eq. 2.09, for the hyperbolic model, Eq.2.41, and for the bilinear, Eq. 2.45 were used. For comparison purposes the ultimate values z„ werekept the same in the three models. The base motion time history for this case is shown in the inset.

These complete loops fail to point out a serious drawback of the Bouc-Wen and hyperbolicmodels. The experiments on metals have shown that a partial loading-unloading path should belike the one shown in Fig. 19 and 20. After a partial unloading, the reloading should form a smallloop to reach the asymptotic yield level. However, this is not the case with the Bouc-Wen and thehyperbolic models. It can be clearly seen from Fig. 21, where the partial unloading paths A-B andreloading path B·C do not intersect. This softening behavior during a reloading followed by apartial unloading has not been observed in the experimental tests on metals. Probably there alsoexist more fundamental physical or thermodynamical reasons for the behavior shown in Fig. 20,but the author is not aware of this.

CHAPTER ll: HYSTERETIC MODELS 19

111

‘ 111111

3.1 INTRODUCTION

In Chapter II, we presented the equation of motion of a single degree of freedom oscillator witha nonlinear hysteretic constitutive law. That equation along with the endochronic law forms a

system of two differential equations. These equations can be rewritten in the following form:

5é+2§0600:i:+60(%[¤x+(l—a)z]=—5ég (3.0161)

2 + g(:é,z) = 0 (3.0lb)

Where the natural frequency 600 and the conventional damping ratio C0 are, respectively, defined as:

m 2 600 m

and k is the initial stiffness parameter, discussed in Chapter II.

CHAPTER Ill: EQUIVALENT LINEARIZATION 20

I

We also solved Eqs. 3.0la and 3.01b for various models of 2, for a prescribed base motion timehistory ix, by a step-by·step nonlinear equation solver. Here in this chapter, however, we are in-terested in obtaining the response of this system for ic: defined stochastically.

A general solution of these equations for an arbitrarily defined ix is almost impossible to obtain,e

even for the sirnplest of the endochronic models. Therefore, researchers have tried to obtain someapproximate solutions. For stochastically defined inputs, the stochastic linearization is cornmonlyused. In this chapter we will discuss the linearized solution of these equations for the Bouc-Wenand the bilinear hysteretic models.

The linearization techniques have been widely used in practice since the first proposal by Krylovand Bogoliubov [16]. For hysteretic models, Wen and others have used this technique extensivelyin the analysis of single and multiple degree of freedom structures subjected to random earthquakeloading. Here we will reiterate some important steps utilized in the linearization technique, not

because they are new but because of completeness of the topic discussed in this work.

For the solution of Eqs. 3.01, the nonlinear equation 3.0lb is replaced by a linear equation as

follows:

2+Cx+Kz=0 (3.03)

Where K and C are the linearization constants, yet to be detemiined. This linearization introduces

some error, 6

6 =g(2&,z)— C:&—Kz (3.04)

CHAPTER lll: EQUIVALENT LINEARIZATION 2l

The error 6 is a stochastic process. To obtain some estimates of K and C, it is customary to

minirnize the mean square value of this error. This minirnization requires that

ÖEKS2] : öE[62] :0 (305)öC Ö K °

Where E [ .] is the expected value of [ .] . Substitution of Eq. 3.04 into 3.05 and slight simp1ili·cation give the following two simultaneous equation for the coeflicients K and C

c EL?] + K Em] = E[Xg(X, 2)] (3.06)

C E[Xz] + K Eizz] = E[2g(X, 2)] (3.07)

These two equations can be solved for the coeflicients as:

C - E Ef 206, 2)]= [ci,] ‘ _ (3.08)K E [Z8(x.z)]

Where [C:] is the correlation function matrix for variables X and 2. For known joint probabilitydensity function of X and 2, the expected values on the right hand side of Eq. 3.08 can be calculated

in principle.

However, under some very general conditions fulfilled by the density function of X and 2 and

also the function g(X, 2) , it has been shown by Atalik and Utku [14} that Eq. 3.08 can also be

simplified to:

E= ix (3.09)“ E [ClOZThe conditions to be satisfied by the density function /(X, 2) and the function g(X, 2) are:

cuxprim ru: EQUIVALENT umzxnxzxrrow 22

1) /(:k, z) is Gaussian.

2) |g(x,z)| < A 6

where 0,,, 0, and 0, are the standard deviations of the x, x and 2 responses, respectively and ph isthe coxrelation coefiicient of x and 2 In both equations I", are the following Gamma functions:

_ + 1 _ — + 1 _ + 1¤ - de-1 » ¤ · de-6-1 » ¤ — 1%-1Futhermore r must be an even integer.

3.4 BILINEAR MODEL

We can also obtain the equivalent linear coefficients for a bilinear model using Eqs. 3.09. In this

case, the analytical expression for g(x, 2) is given by Eq. 2.45. lt is noted that for this case also, the

function g(x, 2) satisfies the conditions (2) and (3) given by Atalik and Utku. Again if we make the

assumption that the joint density function is Gaussian, we can obtain the coefficients by Eq. 3.09.

The function g(x, 2) in Eq. 2.45 can also be defined over the domain of x and 2 as follows:

—x for 0S2S2,, and x20

0 for 222,, and x20—x for 220 and xS0

g(x, 2) = I (3.14)—x for —2S2S0 and xs0

0—xfor zS0 and x20

1

In this domain the function is at least piecewise differentiable. The derivative required in Eq.

3.09 can be defined as:

CHAPTER 111: EQUIVALENT LINEARIZATION 24

-1 for Oszszu and X200 for z2z„ and X20y

'l forz20and-1 for -z„SzS0andXS0

0 for zS -2,, and Xs0-1 for zS0 and X20

Also, it is noted that the function is independent of z in these discrete domains. Thus, its de-

rivative with respect to z is zero. Substitution of these in Eq. 3.09 gives:

2,, ¤¤ ¤0 0C= -5 I fiz(X,z) dzdX -5 I fX2(X, 2) dzdX (3.16).. cg 0 “Zu — gp

K = 0 (3.17)

The integrals in Eq. 3.16 can be evaluated at least numerically and thus C can be defined. On

the other hand the coeflicient K is defined but its value being zero poses some problems ir1 sta-

tionary equivalent linear response analysis. This is discussed later.A

3.5 EQUIVALENT LINEAR STA TIONAR Y

RESPONSE

With the substitution of the equivalent Linear expression for g(X, z) in Eq. 3.01b we get a set

of coupled linear differential equations as follows:

X+2Q0coOX+coä[aX+(1—a)z]= —Xg (3.18)

CHAPTER 111: EQUIVALENT L1NEARlZAT1ON 25

I

2+Cx+Kz=0 (3.19)

These equations can be written as a system of first order differential equations for the state vectordefined as:

x,5 = z (3.20)

x

The differential equation is:

x 0 0 1 x 0ä Z = 0 —K —C Z + 0 (3.21). —¤¤>ä — (1 — ¤)

I

1I

x Q 0 1 0 0 x 0·? 0 —/< — c 0 0, ? 0coä(l — Q) —2C„

1111

It is, however, noted that the equivalent linear coeflicients C and K in this scheme are themselvesdefrned in terms of the response which is not known a priori. To start the solution therefore some

Vappropriate values are assumed for C and K, which are then used to define matrix G in Eq. 3.27.Solution of this equation provides the response covariances which are in tum used in the calculationof new values of C and K from Eqs. 3.10 and 3.11 or from 3.12 and 3.13 for the Bouc·Wen model.This iterative process is repeated till a convergence is achieved. The convergence characteristics ofthis procedure for the Bouc·Wen model are presented in the following chapter.

Implementation of this equivalent linear procedure in the calculation of stationary response

covariance for a bilinear hysteretic model, however, has not been possible. This is because thecoeflicient K for this case is zero, which renders matrix G singular for this case. Thus, the numericalapproach used with Eq. 3.27 can not be utilized. The nonstationary solution can still be obtainecl

in principle. This will be attempted in future investigations.

CHAPTER III: EQUIVALENT LINEARIZATION 28

CHAPTER IV' NUMERICAL RESULTSO

4.1 INTRODUCTION

As mentioned in the previous chapter, the stochastic linearization approach is necessarily itera-

tive. In this chapter we will consider the convergence characteristics of the linearization approach.

The linearization coefiicients of the nonlinear equation depend upon the response, which in tum

is not known a priori. The process, thus, starts with some assumed values of the linearization co-

eflicients and then is obtained the response. The calculated response is then used to calculate the

new values of the coefiicients which are then used again to calculate a new response value. The

process is repeated till a desired convergence in the response quantities of interest is achieved.

The above procedure seems simple. Yet, some convergence problems were noted when no ad-

justments to the iterative process were made. It was also observed that the convergence also de-

pended upon the parameters of the problem. In some cases as high as 39 number of iterations are

required to achieve the desired level of convergence. In this Chapter, therefore, the experiencegained in achiveing convergence to the final results is discussed. Numerical results showing the

convergence rate for various cases are presented. Finally, the comparison of the response values

CHAPTER IV: NUMERICAL RESULTS 29

obtained by the stochastic linearization approach with the values obtained by the time history en-semble analysis are also presented and discussed.

4.2 PROBLEM PARAMETERS

The parameters considered in this study are the oscillator parameters, the parameters of thehysteresis model and the input motion parameters.

4.2.1 OSCILLATOR PARANIETERS

The oscillator parameters are: the initial frequency 60,,, the damping ratio C,,, and the yield levelIQ. The oscillator frequencies 60,, = 1, 1.5, 5, 6, 16 and 20 cps, and the damping ratios §,, = 0.05and 0.10 have been used. The yield displacement values of the stiifness element, y, were varied

between .003 to .800 inches.

4.2.2 PARAMETERS OF THE HYSTERESIS MODEL

For the Bouc-Wen model, the parameters are B, 7 and the exponent rz. In all the cases B = y

is assumed. The parameters B and y are related to the yield level as follows. For the ultimate valueof z = 2,,, we obtained Eq. 2.28, which for A = l gives:

ß + ~, = (4.01)

For an equivalent bilinear model, the yield force at the yield displacement x = y is given by

I

CHAPTER iv: NUMERICAL RESULTS 30

II

Ii},=aky+k(l—a)zy (4.02)

Where zy is the z·value at x = y. However, when rz —-> OO, 2, approaches z„, which in turn forA = l is equal to y. Thus in the limiting case of rz -» 00,

Fy=aky+k(l—

11

sponse can still be obtained, but it then depends upon the initial conditions. Analysis for such a

case is more complicated and is not considered in this work. We have therefore, assumed as low

values of a as possible without affecting the numerical accuracy of the results. For the elasto-plastic

case, a = 1/21 , has often been assumed in stochastic linearization approaches in the past. We have

used this value for obtaning the results for the elasto-plastic case, although some results with smaller

values of 11 we also reported. For the bilinear hysteretic case, the numerical results with a = .25

have been obtained,

4.2.3 INPUT MOTION PARAMETERS

The input to the oscillator is defined by a filtered white noise model. The most cornmonly used

seismic input of this type is defined by the Kanai-Tajirni spectral density function of the following

form:

4 2 2 2‘0g

u

3 4 + 4 2 4 2

!=1 (co, — co) + 4C, co, co

This is the multimodal form of the Kanai-Tajirni spectral density function defmed earlier. Theparameters of this model used in this study are given in Table 1. This spectral density function givesthe root mean square value (RMSV) of ground acceleration to be equal to 2.147 ß/ sec:. This withthe peak factor value of 3.00 for ground acceleration random process, gives a peak ground acceler-

ation of 6.44 f?/ secz or .2 g. The results obtained with this seismic input are compared with the

results obtained by time history analysis for an ensemble of ground motion time histories scaled to

.2g .

The results for the rate of convergence obtained for several oscillators with different frequencies,

damping ratios and yield level parameters are shown in Figs. 22 - 36. These curves are plotted for

rz = 1, 3 and 5. Each plot shows the logarithm (to the base 10) of the error versus the number of

iterations. The error is defined as:

= —l——— 4.10e Ri ( )

Where R, and R,+, are the mean square values of the response obtained at the i"* and (i + 1)"‘ iter-

ation. All three response quantities, x , x and z, were considered to obtain this error. Only the

maximum error obtained among all three response quantities is shown in the plots. The conver-

gence was assumed to have been achieved when this error was less than .01 for all response quan-

tities.

CHAPTER iv: NUMERICAL RESULTS 33

IBased in these results, it is felt that in a given problem the rate of convergence of the response

to the final response primarily depended on the choice of the linearization parameters C and K .

It has been suggested [22] that a value of C = 1.0 and K = .05 (ß + 7) provides a better conver-

gence. These initial values of C and K were observed to work very well with rz = 1 and rather well

with rz = 3 in almost all cases. However for rz = 5, some problems were observed. The experience

showed that if the initial value of K was increased to be of order l (between 1 and 10), the con-

vergence was assured in almost all cases.

Besides the choice of the initial values, it was also found to be important that the subsequent

values of C and K, for following iterative steps, be chosen carefully. It was observed that if the C

and K values, obtained according to Eqs. 3.10 and 3.11, were directly utilized then the convergence

was not always assured. It was found desirable to choose C and K values somewhere in between

the values obtained in the two consecutive steps. In fact, a simple average of the two values, if

chosen as the value for the next step worked very well. That is, for the rz"* step the initial values

C, and IQ calculated as:

C, = —&lläl& (4.11)

K, = —]$'¤§l

the stiffness element will yield. That is, for the oscillators with large inelastic response or ductilityratio the number of iterations required for convergence will be large.

In this section we compare the oscillator response results obtained by the equivalent linearizationapproach with those obtained for an ensemble of time histories.

The time history analysis results were obtained by Malushte [21]. For these results, the accel-eration time histories representing the ground motion were synthetically generated for a spectraldensity function, the parameters of which are given in Table 1. These time histories were synthe-sized from randomly phased harmonics, with amplitudes determined from the spectral densityfunction. The time histories were modified by a deterministic modulation function. These timehistories had a total duration of 15 seconds with the strong motion phase of 4 seconds. In all a totalof 75 such time histories were used. The maximum acceleration in each time history was normal-ized to a value of .2 g.

The numerical results were obtained for the elasto-plastic case with a = 0 and for the bilinear

case with a = .25. Several yield levels were considered, as mentioned in the preceding section.

The time history response of each oscillator was obtained for each input motion time history.

Thus, for each oscillator the ensemble containing 75 time histories was available to obtain the re-

sponse statistics. From each response time history a maximum value was obtained which was then

used to detemiine the mean maximum response. Here this mean of the maximum response valueis compared with the mean of maxirna obtained by the stochastic linearization technique. It isnoted that individual maximum values for time histories did not occur at the same time.CHAPTER IV: NUMERICAL RESULTS 35

In the stochastic linearization approach the mean of the maxima was obtained by multiplyingthe root mean square response value by the peak factor. The root mean square response value wasobtained by a stationary random vibration analysis of the equivalent linear system.

The peak factor (PF) was calculated by the well-known Davenport fonnula [23] as:

PF = 2 1¤(i!§i) + (4.13)x/ " °>= td Gx

Where 6,, =standard deviation, 0; = standard deviation of the response derivative and td is theequivalent stationary duration of the response. 0, and 0; are directly available from the random

vibration analysis of the equivalent linear system. Since it is difficult to know the equivalent sta-

tionary duration for a nonstationary response, here two values of td = 4 sec. and td = 7 sec. wereused to obtain the peak factors. The numerical results for the peak factor calculated with td = 4sec. are given in Tables 2-5 and those for td = 7 sec. are given in Tables 6-9. The seismic input inthe equivalent linear approach is defined by a white noise. To ensure that the input used in the timehistory approach is somewhat comparable with the input in the stochastic approach, a filter wasadded between the oscillator and the base. Thus, the input motion at the base of the oscillator wasa frltered white noise of Kanai-Tajimi form. The numerical results for a single term Kanai-Tajimispectral density function (Eq. 4.08, in which the filter with parameters in Table l is used) and thethree-term Kanai·Tajimi spectral density function (Eq. 4.09, in which the three filters in parallel

with parameters in Table l are used) are presented. The intensity of the white noise at the base

of the filters was adjusted such that with a peak factor of 3, it gave a maximum ground accelerationof .2 g at the base of the oscillator.

As mentioned before, the nonlinear stiffness element in the stochastic approach is modeled by

the Bouc-Wen model with exponent rz = 1, 3 and 5. Since in a stationary response case we cannot

obtain results for a = 0, for a elasto-plastic model, we have used cz = 1/21 = .048 which has

commonly been used in the literature. For the bilinear model, since the time history response

CHAPTER IV: NUMERICAL RESULTS 36

I

I§1values are available for a = 1/4 = .25, the stochastic results have also been obtained with this

value.

Tables 2-9 show the results for various oscillators. Column 1 shows the oscillator identificationnumber and colurrms 2, 3 and 4 show the oscillator pararneters. The values in the remaining col-

urnrrs are for exponent n = 1, 3 and 5. For each case the following values are shown: the peak

factor calculated by Davenport’s forrnula, the maximum diplacement of the oscillator and the error

between the maximum displacement value obtained by the equivalent linear approach and the time

history ensemble results. This error is defined as:

Error = 100%.fi (4.14)rWhere x, and x, are the mean of the maximum values obtained by time history and stochastic

analysis approaches.

It is seen that the error is large in several cases and there is no particular trend, with respect to

the choice of exponent, and the equivalent time duration td used in the calculation of the peak

factors.

Probably a better comparison can be made by comparing the averages of the absolute errors in

these tables. These averages are given in the last row. A comparison of these averaged errors shows

that a smaller error is obtained for a = .25 than for rz = .048. It is noted that the results fora = .25 are for the same value of this parameter, both in the time history and stochastic approach.

Whereas, the results for a = .048 are for different values of this parameter in the time history and

stochastic approach. In the time history the results are for a = 0, and in the stochastic approach

the results for a = .048. It was not possible to obtain results for (1 values less than .048 because

in the stochastic approach the numerical error in the results started to increase when Cl values less

than .048 were used (see Fig. 38). Thus, it seems that a major share of the error in the results for

CHAPTER IV: NUMERICAL RESULTS 37

I

the elasto-plastic oscillators is due to our inability of modeling this case in the stationary stochastic Iapproach.

Comparison of the average errors for one term and three term spectral density function inputsshows that the error in the latter case is smaller. This is probably due to the fact that the input of3-term spectral density function is closer to the input used to define the acceleration time historyin the time history analysis.

In conclusion it is felt that errors are large, though they can be improved by improving theproper modeling of the system and consideririg the nonstationarity of the response properly in thecalculation of peak factors. To improve the accuracy of the stochastic linearization for the elasto-plastic case (a = 0), it seems to be necessary to include the nonstationarity of the motion and re-sponse.

CHAPTER IV: NUMERICAL RESULTS 38

Ä

ÄÄ

CHAPTER V: SUMMARY ANDCONCLUSIONS

The motivation for this work was to study the characteristics of some nonlinear hysteretic

models, commonly used in seismic structural analysis. In particular, here the model proposed by

Bouc and Wen has been studied comprehensively. The effect of various model parameters on the

characteristics of the model has been examined in detail in Chapter II. This study has led to the

development of the analytical forms for two other hysteretic models, viz, hyperbolic model and

bilinear model. It is expected that the expression for the bilinear model will be useful in future

studies. The nonlinear response of oscillators with these three models has also been obtained for

specified ground motion time histories. Some physical limitations of the commonly used Bouc-

Wen model have also been identified.

In Chapter III, the equivalent linearization of the equation of motion of an oscillator with

stiffness characterized by the Bouc-Wen model and by the bilinear model has been forrnulated. It

is observed that it is not possible to work with the equivalent linear equations of the bilinearoscillator for calculating the stationary response. That is, to obtain any meaningful response in this

case it is necessary to consider the initial conditions and the nonstationarity of the response. Such

CHAPTER V: SUMMARY AND CONCLUSIONS 39

I

Ia study is beyond the scope of this work. On the other hand, the Bouc-Wen model can still be usedfor stationary response calculations as long as the parameter a is nonzero.

In Chapter IV, numerical results have been obtained for oscillators with different characteristicsto study the convergence behavior of the equivalent linear iterative approach. It is observed thatthe number of iterations required for convergence can indeed change with the parameters of theoscillators, such as frequency, yield level and the exponent parameter of the Bouc-Wen model. Itis observed that the convergence is slowest for the exponent parameter value of n= 5. This con-vergence, however, can be improved by appropriately chosing the initial values of the linearizationcoefficients. It has been observed that better convergence is acheived when the initial values of thelinearization coefficients for an iteration are taken as the average of the two recently calculatedvalues. For oscillators with large inelastic response or ductility ratio the number of iterations re-quired for convergence will be large. The rate of convergence does not seem to be affected by theother parameters of Bouc·Wen’s model or the input characteristics.

The numexical results obtained by the equivalent linear approach are also compared with theresults obtained for an ensemble of ground motion time histories. This comparison is not observedto be good; that is, for some oscillators the response values calculated by the equivalent linear ap-proach differ significantly from the average values calculated by the time history ensemble analysies.The main reasons for this discrepancy are the inability of the equivalent linear approach (1) toprecisely model the elasto·plastic and bilinear hysteretic models in the analysis, (2) to consider thenonstationarity of the input and response. The input in the equivalent linear approach is definedstochastically and assumed to be stationary. Whereas the time history inputs used in the time his-tory calculations do not have stationary characteristics. In establishing the level of intensity of the

stochastic input so that it gives the same maximum ground acceleration as the inputs in the timehistory analysis, some assumptions are made about the ground acceleration peak factors. These

assumptions need to be verified. Also in calculating the peak response from the mean square re-sponse, the peak factors have been obtained from some simplified formulas which do not consider

the nonstationaxity of response. For a better corroboration of the two results it appears, therefore,

CHAPTER V: SUMMARY AND CONCLUSIONS 40

1

_ 1I

necessary that nonstationarity for the input and response be some how reflected in the calculationof response by the stochastic linearization approach.

IIII

III

11

1

CHAPTER V: SUMMARY AND CONCLUSIONS 4l ‘

I

APPENDIX

Derivation ofEq. 3.09

To prove Eq. 3.09, let us rename it as Eq. A.0l:

= _" (A.0l)OZEIÄI

We first evaluate:

EI-?] = H Ä; (x,z) dx dz (A.02)dx Z X dx”

Integration by parts of Eq. A.02 gives:

E Igßxzbä ZII — Ii gé- IAA!. ZI] d>%]dZ (403)

APPENDIX 42

1

K,

For the Bouc-Wen model and rz = 1, the function g(:2:, 2) is:

g(:&,z)= (ßlzl —A)x+ ylaklz (A.04)

The first term on the right hand side of Eq. A.03 is obviously zero at the limits. Thus, Eq. A.03becomes:

dx dz (A.06)öx 2 i 0:%"’

The joint distribution, if assumed Gaussian, can be written as:

expWhere(IT = (ak, 2). It is noted that because Izl is limited to a maximum value of z„, the assump-tion of the normality of the joint distribution is not correct. Nontheless, this assumption is com-monly made. The derivative of Eq. A.06 can be written as:

2)]SinceÖZIT/öx = (1, O), we can write:

ö - - 1 T — 1 xvr [,&,(x„ 2)] = · ß2(x„ 2) [Ce;} (A-08)Ox 0 z _

Thus, from Eq.

gf·(:t Z) 1T[e·}"X

dxdzZ X2 0 X2 Z

In a similar way we can obtairi E [ög/öz] as:

APPENDIX 43

50 de Zjf dk dz (AmZ Z X l ZCombining Eq. A.09 and A.10, we obtain:

6Elél-1Ö= [Cb] {Ji gßk2(x, z) dx dz (A.ll)Eli] ZözWhich means:

X -E ElgflE [QI E lg Z}öz

Substituting for the vector on the right hand side of Eq. 3.08 from Eq. A.12, we obtain Eq. 3.09.

Expressions of C and Kfor Bone- Wen’s model with n = I

To obtain the explicit expresions of C and K, we will use Eq. A.0l. For the particular case ofBouc’s model with exponent rz = l , the expression for g(x, z) is given in Eq. A.04. The derivatives

of g(x, z) are:

—A (A.13) rox ox Iög _ . . Ö lzl lEwlxl +ßx—E— (A.l4) 1

Aeeßwmx 44 1

I

Considexing Eq. A.0l, A.13 and A.14, we can write the lineanzation coeflicients as follows:

_ 6IxI _C—yE Z-? +ßE[|Z|] A (A.15)

K _ . . ölzl

For the jointly Gaussian density function of x and 2 , it is straightfoward to obtain the expectedvalues ir1 Eq. A.l5 and Eq. A.16 as:

E[IxI1=j_IxIg(x) dx (A.17)X

Where

.. · 2ß(x) = -2-.-- exp (A.18)\/ZT! Gx X

Integrating, from — 00 to 00, we get:

' .E[Ix|] \/E dx (A.19)

Inasimilar way we can obtainE[lzl]:_

_ T13[IZI] — X/Y 6, (A.20)

To determine E [2 (ölxl/öx)], we have to perform the following integrations:

E ölxl _ dlxl . .2+ -]‘ +j zj}z(x,z) dz dx (A.21)OX X. OX ZWhere

APPENDIX 45

f,,_,=——L—__—_: exp[—l-—J——[i—2gE-+—&H (A.22)2 (1-pZ) Ori O-xopz G;

After integrating from — GO to GO the variable z, we get:

exp dx (A.23)

Since the integrand in Eq. A.23 is the same as that in A.l7, we can write:

E[z%] = po, (A.24)

Where p = . Then,

Following a similar procedure, we find:

= (A.26)

Substituting Eqs. A.l9, A.20, A.25 and A.26 into A.15 and A.l6, we obtain:

C [yrääg- + pe,] — A (A.27)

K = [Y Gi + (A28)l

EAPPENDIX 46{

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FIGURES

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FIGURES 48

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TABLES

11III

TABLE 01 PARAMETERS FOR THE SPECTRAL DENSITY FUNCTIONS

Unimodal Kanai—Tajimi:°ä = 17.64 rad/secQ: .3535§r= .34125 (in/sec2)'2/(rad/sec)

Trimodal Kamai—Tajimi:44 = 13.50 rad/secQ: .39255/ = .21600 (in/sec2)2/(rad/Sec)

4@= 23.50 rad/secC2: .36005,: .07128 (in/secz)2/(md/Se;)Ҥ=

39.00 rad/secQ: .3350

.05400 (in/sec2)2/(rad/geg)

IIIII1

°I

75 ITABLES I· I

X

6 8 8 8 8 8 8 8 8 8ä

KEN f Elf} E?}%‘ E E8} ä E $8

Q ~ ~ ·· 6 E5 E8 6 6:" "‘ gi ETS 8 8 -1 -1 8 Q QLF “ E .7 Ei; Ei} 6 6 6 6 6Q xxLEO C\l xD E-a8 86 8 8 6 8 . 8 8N „-T E-; N N N m Nu

¤ 6N 8 8 8 8 8 8 8 8 8LE 7 6 E6 6 6 8 8 6 661 •—1 UW LVW

Ä C ;,„-Z N N 6 6 6 6 6

nä Q N 6 oo ä· 86 8 6 8 6 8 6 8 .__‘ Ln „-Ä E-4 E—« N N N N N86 8 8 8 8 8 8 8 8 8E ¤; LQ 6 6 6 6 8 6 92

m

I

3E 2 Rmäää Q:

co q 00

IIU

5 ::~22222 Ri

.8.

UW‘+

I I O I I I I

28228822 99.

2;,666666

Q CG I\r—•

¤"‘°„-«c\1c\1

11

6N 6 6 6 6 6 6 6 F: 6ä E 7 6 6 6 E6 6 6 6 66

·· 6 „ 6 6 6 6 6 6 6 6LF " E5 -3 63 63 0 0 0 0 0Q xxLiv-4 O (*7 LV7 P"*Q § . . 66FQ 4-4 4-4 N N N N NII

¤ 6 6 6 6 6 6 6 6 6 65,56 7 6 6 ,6 6 6 6* 66 ,61N (*7 ON I\ I\ GT 6 6 6 6 6 6 6 6 6

tä C:1-4 N N G G G G G

66 6 6 6 61 § 6 6 6· Qä 6

62 6 . . 6 6pq nz., 6] -4 -4 N N N N N66 6 6 6 6 6 6 6 6 66 6 6 6 6 6 6 6 6 6

q 4-4 LV76 6 6 6 6 6: 6 6 6 6J; .5 ,-3 63 61 0 0 0 0 0x/

L4Q ON N r\ N ä CN LFW F766 6 6 6 6 . 6 6 6:-1

>~§, . 6: C2 .$-4

6 ..6 6 6 6 6 6 6 6 6 Lg60

3G/R ,-4 -4 -4 LQ LQ m· 0 0

QX «-4 1-4 g)6 6

6 1, 6 61 6 6 6 6 6 6[E6 0 •—• N N 61 oo m 0I r-I r-I 1-·I v-*I

•—I

IIU 6 6% 6% 6 6 6 6 6 656 $2 6* : N SI 6 6 2.*

M r\ 0 r~ ä 0.. 36 6 6 6 6 6 . 6ä C "

O O CD O O

@6 @6 6* F2 6 g g 6cw :66 62 62 'Ü . . clLv-• N N N N cd N N

gw 6 6 6 6 6 6 6 6I I

6* ^ § E 6 6 6Q; EE 6 6 6 6 6 6 6E LaO LO O LO ·-·—• N6 6 6 66umU-· N N N N ow N NCD

6 6 § 6 6 6V . C2 Q 6ä Le6 6 6 6 6 6 6 6 I Q;LR 0LOQ6 66 ..1 6 S6 S6 6ä 6*, 66* ob 6 6 6 6TABLES 79

II

SS S S S S S S S.5 S S S S S S. S S SN

LVW .-4 CD FW C"W LVW*1, -1: S S S S S S S S SU C,. Q N N

22, Ü Ü Ü Ü Ü Ü Ü ÜQ7 E

-— 2 2 7 2 2 E8 LET 2 2 gg ä 2 g 2 2'U •5

O? • •'—:

•*4.: C: V 0 -« 0 0 0 0 0

$-1O (*7 U7 G) N I\2 22 2 2 2 2 2 2 2• ¤”°[_;·_, N N (*7 N (*7 ("7 (*7IIU 2N Ü Ü Ü Ü Ü Ü ÜJ5 N T RI EI 2 2 2

or') xx LF7 LV7 (I) I\¤¤ 88ä;; " 0 ~• 0 0 0 0 0

ä $-4° O O\ N :-1‘ 22 2 2 2 2 2 2 2”—°V-‘-· N N 07 N 02 07 07

2 Ü Ü 2 Ü Ü 51 Ü ÜDQN 92 -*7 2 -2 2 2 2 2II\Ti ,22 2 2 2 2 2 2 2

C " O O O CD O CD C)

$-1O (I3 I\ LO |\ (I)ä 2 2 2 2 2 2 2N N N N 07 0*7 0*7CD

/\ 5% 2 22 *2 2 2 2ä O O O CD O OCD‘~’”—: 2 —: ·—: 2: 0: —: LE„ /\. GJI\ 0 Lf'7 M7

OO

E

gw Q Q Qi Q Q Q Q Q1 Qä $3 Q 2 2 2 2 2 2 2 2'L, T} E- Q Q -1 Q1 Qi öl QL1 N N 0 0 0 0 0 0

$-••——« 0 N LO F7 0* N ··—-• (XDQ EQ Q Q Q Q Q Q Q QH

LL, N N v—4 CO N N 07 F7U

v—OQ2 Q Q Q Q Q Q Q Q Q2$3 2 Q Q 2 2 EQ 2 2 2gr) |\ |\ v-4 LO (X).. Q Q Q Q Q Q Q Q

ÄE-_ g, N N M 0 0 0 0

QQ 2 -· F2 2 - 2 S3 g4, 2% Q2 Q Q2 C2 Q Q Q .Q-• N N .—• M N N M M*6 Q Q Q Q Q Q Q Q QEQ Q Q Q Q Q Q Q Q Q

LÜ LO N Q xD7 3- Q Q Q Q Q Q Q QN N M 0 0 0 0 0E LeO N |\ |\ ONQ2 Q Q Q Q Q Q Q Q

Lu N N N N N N M MLu0 ^ ä R Q ä "ä >~§, . -1 Q . Q

S-eQ 2 Q Q Q Q Q Q Q Q LgC1)·· Q .2 2 QUW LOUWQQ Q

F ' C\l Q 0 r~ig; N 2 N 2 N N QTABLES 82

I

_ II

*61 8 8 8 8 8 8 8EIN „-I :1

F

1. A. E. Kanaan and G. H. Powell, "DRAIN-2D General Purpose Program for Dynamic Anal-ysis of Inelastic Plane Structures", Earthquake Engineering Research Center, Report No.EERC 73-6 and EERC 72-22, University of California, Berkeley, April 1973.

2. D. P. Monkar and G. H. Powell, "ANSR-1 General Purpose Program for Analysis of Non-linear Structural Response", Earthquake Engineering Research Center, Report No. EERC75-37, University of California, Berkeley, December 1975.

3. W. Ramberg and W. T. Osgood, "Description of Stress-strain curves by Three Parameters",Technical Note 902, National Advisory Committee for Aeronautics, 1943.

4. E. Faccioli and J. Ramirez, "Earthquake response of Non-linear hysteretic Soil Systems,Earthquake Engineering and Structural Dynamics, Vol 4, 261-276, 1976.

5. M. P. Singh and Ghafory-Ashtiany, ”Seismically Induced Stresses and Stability of Soil Media",Soil Dynamics and Earthquake Engineering, Vol. 1, No. 4, 167-177, 1982.6. W. D. Iwan, ”The Distributed Element Concept of Hysteretic Modeling and its Applicationto Transient Response Problems", Procedings of the Fourth World Conference on Earthquake

Engineering, Vol. II, A-4, 45-47, Santiago, Chile, 1969.

7. R. Bouc, "Forced Vibration of Mechanical Systems with Hysteresis", Abstract, Procedings ofthe Fourth Conference on Nonlinear Oscillation, Prague, Czechoslovakia, 1967.8. H. Ozdemir, "Nonlinear Transient Dynamic Analysis of Yielding Structures", Ph. D. Disser-

tation, Division of Structural Engineering and Structural Mechanics, Department of CivilEngineering, University of California, Berkeley, 1976.

9. M. A. Bhatti and K. S. Pister, "A Dual Criteria Approach for Optimal Design of Earthquake-resistant Structural Systems", Earthquake Engineering and Structural Dynamics, Vol. 9,557-572, 1981.

10. Y. K. Wen, "Method for Random Vibration of Hysteretic Systems". Engineering MechanicsDivision, Proceedings, ASCE, Vol. 102, No. EM2, 249-263, April 1976.

11. T. T. Baber and Y. K. Wen "Stochastic Equivalent Linearization for Hysteretic, Degrading,Multistory Structures", Report Submitted to the National Science Foundation, ResearchGrant ENV 77-09090, April 1980.

12. F. Casciati and L. Faravelli, "Non-linear Sthocastic Dynamics by Equivalent Linearization",Department of Structural Mechanics, University of Pavia, Italy, 1986.

13. F. Casciati, "Equivalent Linearization Technique in the Analysis of Seismic~excited Structures”,Procedings of the Euro-China Joint Seminar on Earthquake Engineering, Beijing, June 1986.

14. T. S. Atalik and S. Utku, "Stochastic Linearization of Multi-degree-of-freedom Non-linearSystems", Earthquake Engineering and Structural Dynamics, Vol. 4, 411-420, 1976.

15. Y. K. Wen, "Equivalent Linearization for Hysteretic Systems Under Random Excitation",Applied Mechanics, Vol 47, 150-154, March 1980.

16. N. Krylov and N. Bogoliubov, "lntroduction to Nonlinear Mechanics", Princeton, New Jersey,Translated from 1937 Russian Edition.

17. Y. K. Lin, "Probabilistic Theory of Structural Dynamics", Mc Graw-Hill, New York, 1967.18. R. H. Bartels and G. W. Steward, "Solution of the Matrix Equation AX + XB = C", Com-

munication of the ACM, Vol. 15, No. 9, 820-826.

19. U.S. Nuclear Regulatory Commission: "Design Response Spectra for Nuclear Power Plants",Nuclear Regulatory Guide, No. 1.60, Washington, D.C., 1975.

20. M. P. Singh and S. L. Chu, "Stochastic Considerations in Seismic Analysis of Structures",Earthquake Engineering Structural Dynamics, Vol. 4, 295-307, 1976.

21. S. R. Malushte, "Seismic Design Response of Simple Nonlinear Hysteretic Structures", M. S.Thesis, Virginia Polytechnic Institute and State University, September 1984.

22. F. Casciati, Personal communication.23. A. G. Davenport, "Note on the Distribution of the Largest Value of a Random Function with

Application to Gust Loading, Proc. Inst. Civil Engineering, Vol. 28, 187-196.

REFERENCES 85