REPORT NO.

UCB/EERC - 80115

JUNE 1980

'1'l)Pl-122301

EARTHQUAKE ENGINEERING RESEARCH CENTER

ARESPONSE SPECTRUM METHODFOR RANDOM VIBRATIONS

by

ARMEN DER KIUREGHIAN

Report to

National Science Foundation

COLLEGE OF ENGINEERING

UNIVERSITY OF CALIFORNIA . Berkeley, California

50272-101

REPORT DOCUMENTATION 11. REPORT NO.PAGE NSF/RA-80020l I~

3. Recipient's Accession No.

< ,a81 12 23.0 14. Title and Subtitle

7. Author(s)

A Response Spectrum Method For Random Vibrations

Armen Der Kiureghian

5. Report Date

June 1980

a. Performlna 0'llanizatlon Rept. No.

UCB/EERC-80/l59. Performing O'llanization Name and Address

Earthquake Engineering Research CenterUniversity of California, Richmond Field Station47th and Hoffman Blvd.Richmond, California 94804

12. Sponsoring Organization Name and Address

National Science Foundation1800 G Street, N.W.Washington, D. C. 20550

15. Supplementary Notes

10. Project/Task/Work Unit No.

11. Contract(C) or Grant(G) No.

(C)

(G) ENG-7905906

13. Type of Report & Period Covered

14.

16. Abstract (limit: 200 words)A response spectrum method for stationary random vibration analysis of linear

structures is developed. The method is based on the assumption that the input excitation ~a wide-band, stationary Gaussian process and the response is also stationary. However, itcan also be used as a good approximation for the response to a transient stationary Gaussianinput with a duration several times longer than the fundamental period of the structure.Various response quantities, including the mean-squares of the response and its timederivative, the response mean frequency, and the cumulative distribution and the mean andvariance of the peak response are obtained in terms of the ordinates of the mean responsespectrum of the input excitation and the modal properties of the structure. The formulationincludes the cross-correlation between modal responses, which are shown to be significant for,modes with closely spaced natural frequencies.

The proposed procedure is demonstrated for an example structure that is subjected toearthquake induced base excitations. Computed results based on the response spectrum methodare in close agreement with simulation results obtained from time-history dynamic analysis.The significance of closely spaced modes and the error associated with a conventional methodthat neglects the modal correlations are also demonstrated through this example.

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OPTIONAL FORM 272 (4-7n(Formerly NTlS-35)Department of Commerce

, I

/1

A RESPONSE SPECTRUM METHOD FOR RANDOM VIBRATIONS

By

Armen Der KiureghianAssistant Professor of Civil Engineering

University of California, Berkeley

A report on research sponsored bythe National Science Foundation

Report No. UCB/EERC-80/15Earthquake Engineering Research Center

College of EngineeringUniversity of California, Berkeley

June 1980

• I

ill

TABLE OF CONTENTS

TABLE OF CONTENTS

SUMMARY

INTRODUCTION

RESPONSE OF MDF SYSTEMS TO STATIONARY EXCITATION

STATISTICS OF PEAK RESPONSE FOR GAUSSIAN EXCITATION

DEVELOPMENT OF THE RESPONSE SPECTRUM METHOD

APPLICATION TO EARTHQUAKE LOADING

EXAMPLE APPLICATION

CONCLUSIONS

ACKNOWLEDGEMENT

REFERENCES

NOTATION

TABLE 1 - SUMMARY OF RESULTS FOR EXAMPLE STRUCTURE

FIGURES

Page

i

1

2

4

7

s

13

15

17

13

19

20

21

22

-:L-

SUMMARY

A response spectrum method for stationary random vibration analysis of linear structures

is developed. The method is based on the assumption that the input excitation is a wide-band,

stationary Gaussian process and the response is also stationary. However, it can also be used as

a good approximation for the response to a transient stationary Gaussian input with a duration

several times longer than the fundamental period of the structure. Various response quantities,

including the mean-squares of the response and its time derivative, the response mean fre­

quency, and the cumulative distribution and the mean and variance of the peak response are

obtained in terms of the ordinates of the mean response spectrum of the input excitation and

the modal properties of the structure. The formulation includes the cross-correlation between

modal responses, which are shown to be significant for modes with closely spaced natural fre­

quencies.

The proposed procedure is demonstrated for an example structure that is subjected to

earthquake induced base excitations. Computed results based on the response spectrum

method are in close agreement with simulation results obtained from time-history dynamic

analysis. The significance of closely spaced modes and the error associated with a conventional

method that neglects the modal correlations are also demonstrated through this example.

- 2 -

INTRODUCTION

In random vibration analysis of structures subjected to stationary excitatiQns, a description

of the input excitation in terms of a power spectral density function is commonly used. For

linear systems, it is well known that the power spectral density of the stationary response is the

product of the system transmittancy function and the power spectral density of the input pro­

cess (8). It is also known that most response quantities of engineering interest can be obtained

in terms of the first few moments of the response power spectral density taken about the fre­

quency origin (16). For example, the zeroth moment, i.e. the area under the power spectral

density function, is the mean-square response, whereas the second moment, Le. its moment of

inertia, is the mean-square of the response rate. In the special case when the response process

is Gaussian, it has been shown that the first three spectral moments, Le., the zeroth, the first,

and the second, are also sufficient to determine the cumulative distribution and the mean and

variance of the peak response over a specified duration (I6,5). These response quantities are

fundamental to and, for the most part, are adequate for safety assessment of structural systems

subjected to random excitations.

In certain applications, such as in earthquake engineering, the specification of the input

excitation in terms of a power spectral density function is not the most convenient method. A

description that is often found to be more expedient is in terms of the mean response spectrum.

This is a function describing the mean of the peak response of an oscillator of varying fre­

quency and damping to a given input excitation. In structural engineering practice, a response­

spectrum description of an input is preferred because of a variety of reasons. Chief among

these is, perhaps, tradition; many existing structural codes and specifications are based on the

response-spectrum method, and most structural engineers are accustomed with this idea.

Another reason is convenience in generating design response spectra rather than power spectral

densities from existing data. Finally, as shown in this paper, for certain critical response quanti­

ties, such as the mean of the peak response, a formulation based on the response spectrum

method is computationally simpler than that in terms of the power spectral density function.

- 3 -

It is shown in this paper that, under a general set of conditions, the first three moments

of the response power spectral density for a linear structure can approximately be obtained in

terms of the mean response spectrum of the input excitation and the modal properties of the

structure. These conditions are that: (a) the structure have classical modes, (b) the input exci­

tation be a stationary Gaussian process, and (c) the input be a wide-band process, i.e. have a

smoothly varying power spectral density over a range of frequencies covering the significant

modes of vibration of the structure. The resulting expressions for the three spectral moments

lead to a set of modal combination rules whereby each statistical quantity of the response,

including the mean-squares of the response and its time derivative, the response mean fre­

quency and the mean and variance of the peak response, is expressed as a combination of the

mean values of maximum modal responses, where each maximum modal response is obtained

in terms of the ordinate of the mean response spectrum, associated with the corresponding

modal frequency and damping, and the modal properties of the the structure. This analysis also

yields the cumulative distribution of the peak response in terms of the input response spec­

trum. An important feature of this formulation is that it accounts for the cross-correlation

between modal responses. It is shown that this correlation can be highly significant for struc­

tures with closely spaced natural frequencies.

In practice, the restrictions under which the method is applicable can be considerably

relaxed. Specifically, the method may be used for the transient response of structures to wide­

band Gaussian inputs, provided the peak response occurs during a stationary phase of the input

with a duration several times longer than the fundamental period of the structure. Within this

context, the method can be particularly useful in earthquake engineering in determining the

response of structures to transient base inputs. Such an application is demonstrated for an

example structure with closely spaced frequencies subjected to an ensemble of artificially gen­

erated earthquake motions. Computed results based on the proposed response spectrum

method are shown to be in close agreement with Monte Carlo results obtained through time­

history analyses of individual motions. The example also serves to demonstrate the significance

- 4 -

of the cross-correlation between responses in modes with closely spaced frequencies.

RESPONSE OF MOF SYSTEMS TO STATIONARY EXCITATION

Consider an n-degree-of-freedom, viscously damped, linear structure. Assume that the

structure has classical modes with Wi' ~i' i=l, 2, ... ,n, denoting its modal frequencies and damp-

ing coefficients, respectively. It is well known (2) that, using a mode-superposition procedure,

any response R (I) of such a system can be expressed in terms of its modal responses as

(1)

where Ri{t) ='I'iSi{t) is the response in mode i, in which '1'; is the effective participation factor

for mode i and is a constant in terms of the i-th modal vector and the mass matrix (it is in gen-

eral the product of the conventional participation factor and a linear combination of the ele-

ments of the i-th modal vector (2», and Si{t) is the i-th normal coordinate, representing the

response of an oscillator of frequency Wi and damping coefficient ~i to the given input. Con-

sider the stationary response of the system to a stationary input F (t), described through a one-

sided power spectral density Gr(w). With no loss of generality, let F(t} be a zero-mean pro-

cess. Then, the response is also a zero-mean process. Its one-sided power spectral density is

given by

(2)i j

where Hi(w) = 1/(wl-w2+2i~iw;w) is the complex frequency-response function (for displace-

ment response) of mode i and the asterisk denotes a complex conjugate. Using Eq. 2,

moments of the response power spectral density about the frequency origin are obtained as

00

Alii = f wlllGR (w)dw = LL'I'i'l'jAIII.Uo ; .i

where

(3)

AIII.U = Re[~WIIIGr(W)Hi(W)H;'(W)dW] (4)

are cross-spectral moments of the normal coordinates associated with modes i and j (5). It is

noted that because of symmetry GR (w) is always real-valued; therefore, only the real parts of

- 5 -

the cross-spectral moments are of interest. Introducing coefficients Pm,u=Am,u/.JAm,iiAmJj, Eq,

3 can be written in terms of the spectral moments of the individual normal coordinates as

Am = r,r,'I';'I'jPm,u.JAm,i;Am,.i.i; j

(5)

It is well known that AO=(j~ and A2=(j~ are the mean squares of the response, R (t), and its

time derivative, R(t), respectively, whereas AO,i; and A2,ii are the mean squares of the i-th nor­

mal coordinate S;(t) and its time derivative, S;(t), respectively, Also, from the preceding

definition of Pm,U' it should be clear that PO,ij and P2,U are cross-correlation coefficients between

Si(t) and S/t) and between their time derivatives, Si(r) and Sj(t), respectively, The moments

A1 and Al,ii are related to the envelope of the response process, as described in Ref. 16. The

corresponding coefficients, P1.ij, have no obvious physical interpretation, However, their

behavior is also similar to a correlation coefficient, as shown in Ref. 5.

Closed-form solutions for Am,U and Pm,U for m = 0, 1,2, i.e., for the first three spectral

moments, are given in Ref. 5 for responses to white-noise and filtered white-noise inputs. For

these classes of inputs, the power spectral densities are of the form

(6)

and

4+ 41' 2 2 2W" ~owoWGF(w) = " "., G (7)

(w,3-w2)2+4~iw,3w2 0

respectively, where Go is a scale factor and wg and ~g are the filter frequency and damping

coefficient. By proper selection of these parameters, a variety of input power spectral density

shapes can be studied. In particular, the filter frequency, wg , determines the dominant range of

input frequencies, whereas ~g determines the smoothness of the power spectral density shape,

see Fig, 1, Observe that spectral amplitudes for frequencies greater than W g rapidly diminish

with increasing frequency. As a result of this, responses in modes with frequencies much

greater than wI( will generaily be small and insignificant in comparison to responses in modes

with frequencies within the dominant range. This fact plays an important role in the subse-

quent development, since results obtained in the response spectrum method for modes within

- 6 -

the dominant range of frequencies will be more accurate leading to accurate estimation of the

response. Also observe in Fig. 1 that the spectra are smoother for larger values of ~g' For the

purpose of this study, a filtered white-noise input with ~g ~ 0.6 will be considered as a wide-

band input. It is noted in passing that a filtered white noise with w.~ = 57T and ~g = 0.6 is com-

manly used in earthquake engineering to model the ground acceleration process (2,7).

It is shown in Ref. 5 that whereas A./lI,ij are sensitive to the shape of input power spectral

density (as described by parameters wg and ~g), the coefficients PIII.U remain relatively

indifferent for wide-band inputs, i.e. for ~g ~ 0.6. As an example, Fig. 2 shows a comparison

of Po.u for responses to white-noise and filtered white-noise inputs with.{~ = 0.6. Similar results

are given in Ref. 5 for PI.U and P2.U' Observe in this figure that the correlation coefficient

rapidly diminishes as the two modal frequencies Wi and Wj move further apart. This is especially

true at small damping values that are typical of structures. Thus, for wide-band inputs cross

terms in Eq. 5 are only significant for modes with closely spaced frequencies. Also note that

the coefficients for responses to the two types of inputs are nearly the same as long as the

modal frequencies are not far beyond the dominant range of input frequencies, as it happens at

the right end of the lower graph in Fig. 2 where both Wi and Wj are much greater than Wgo

Since the latter case is not critical, it follows that cross-correlation coefficients based on

response to a white-noise input are good approximations to the corresponding coefficients for

responses to wide-band inputs. Exact solutions of PIII.U' m = 0, 1, 2, for response to white-noise

input are given in Ref. 5. A set of approximate expressions obtained from these results are

(8)

(9)

00)

Pl.U =

2~[(Wi+w)2(~i+~) + (w?-w}) (~i-~)]Po.u =

4(Wi-W)2+ (Wi+w)2(~i+~)2

2~[(w,+wY(~;H)- 4(w;-w)2/7T ]

4(Wi-W)2+ (Wi+w)2(~i+~)2

2~[(Wi+w)2(~i+~)- (w?-w}) (~i-~)]P2,U =

4(Wi-Wj)2+ (Wi+WjV(~,+~)2

These expressions, which are simpler but good approximations of the exact results, are plotted

in Fig. 3 for selected values of damping. On the basis of the preceding discussion, these

(10

- 7 -

expressions will be used here for responses to wide-band inputs with arbitrary power spectral

density shapes. With these expressions given, to evaluate Eq. 5 it is only necessary to compute

the spectral moments for individual normal coordinates given by

00

Am.i; = f wmCF(w) IH;(w) 12dfl)

oThese moments for m = 0, 1, and 2 are subsequently obtained in terms of the ordinates of the

mean response spectrum.

STATISTICS OF PEAK RESPONSE FOR GAUSSIAN EXCITATION

It is well known that the response of a linear structure to a zero-mean Gaussian input is

also zero-mean and Gaussian (8). Vanmarcke (17) has derived a distribution for the first-

crossing time of a symmetric barrier for a zero-mean, stationary Gaussian process in terms of

its first three spectral moments. Using his formulation, the cumulative distribution of the peak

absolute response over a duration T, defined as

can be expressed as

, [ ] [ l-eXp(-.J'lT/28 eS)]FRr(r) = l-exp(-s2/2) exp -VT exp(s2/2)-I ,r >0

in which s = r/ IJ R = r/.J>:o is a normalized barrier level,

IJi? 1 r;;v = 'lTIJ R = -:;;" ~

is the mean zero-crossing rate of the process, and 8e = 81,2, where

(12)

(13)

(14)

(15).j:ff28 = 1__1_

AO"-2

is a shape factor for the response power spectral density with a value between zero and unity.

(A small value for 8 denotes a narrow-band process whereas a value near unity denotes a wide-

band process (17).) The mean and standard deviation of R T may in general be obtained as

RT=pIJR and IJRT=qIJR, respectively, where p and q are peak factors given in terms of the

three spectral moments and the duration To For 10 ~ VT ~ 1000 and 0.11 ~ 8 ~ 1, which are of

- S -

interest in earthquake engineering, approximate expressions for P and q from Ref. 5 are

~l+ 0.5772P = "\I LlnV e7

.J2lnv e7

1.2 5.4q= -

.J2lnve7 13+(2Inve7)3.2

where

v = 1(1.6380.4S- 0.3S )V, 8 < 0.69

e v, 8 ~ 0.69

(16)

(17)

(1S)

(19)

(20)

is an equivalent rate of statistically independent zero crossings. Fig. 4 shows plots of P and q

versus V7 for selected values of 8. Shown in this figure is also the ratio q / P which is the

coefficient of variation of the peak response.

For large values of V7 (say, V7 ~ 5000), which may be of interest in some applications

such as in wind and ocean engineering, asymptotic expressions of the peak factors given by

Davenport (3) may be used

~+ 0.5772P = "Lillv7 .J21nv7

1T 1q=-

.J6 .J21nv7

These expressions, which are independent of 8, disregard the dependence between the crossings

of the process. For this reason they can only be used for large V7 for which the influence of

such dependence on the mean and variance is negligible (17).

Results similar to the above also apply to each normal coordinate Si (t). It suffices to

replace R by Si and Am by Am./i in each of Eqs. 12-15. For notational purposes, the parameters

v, 8, P, and q for the i-th normal coordinate will be denoted in the subsequent analysis by Vi'

8 i , Pi' and qi' respectively.

DEVELOPMENT OF THE RESPONSE SPECTRUM METHOD

Let ST(W,~) represent the mean value of the maximum absolute response of an oscillator

of frequency wand damping ~ to a stationary input excitation, F(r), over a duration 7. The

function ST(W, {) for variable W and ~ is defined herein as the mean response spectrum associ-

ated with the input F (t) and the duration 7. It is the objective in this section to develop a

- 9 -

procedure for evaluating the response of a multi-degree-of-freedom structure when the input is

a zero-mean, wide-band, stationary Gaussian process specified through its mean response spec-

trum.

From solutions in Ref. 5, it can be shown that the mean zero-crossing rate,

V,=.jA2,,)Ao,Jrr, and the shape factor, 8,=.J1-Af.JJ/AO,JJA2,jj, associated with the response of

an oscillator of frequency Wi and damping coefficient L, are not too sensitive to the shape of

the input power spectral density, provided that the input is wide-band and that the oscillator

frequency is not far beyond the significant range of input frequencies. As an example, Fig. 5

shows the ratios of these quantities for response to a filtered white-noise input with (g = 0.6 to

those for response to a white-noise input. Observe that the ratio of the mean zero-crossing

rates is always very near unity. Also, the ratio of shape factors is near unity for values of the

oscillator frequency that are within the dominant range of input frequencies. From this argu-

ment, and the fact that the response is not overly sensitive to small variations in the mean

zero-crossing rate and the shape factor (as it is evident in Fig. 4 for the mean and variance of

the peak response), it follows that values of v, and 8, that are based on a white-noise input are

good approximations for the corresponding values for response to a wide-band input with arbi-

trary power spectral density shape. Thus, using results for a white-noise input (5,17),

W,v·=­I

7T

and

[ [ 121'h [ I'h8, = 1- 1 l-1.. tan-l (, ~2 ..h

.J1-(? 7T .J1-(? 7T

(21)

(22)

where the approximation is valid for small damping. These expressions are used in Eqs. 16-18

(or 19-20) to compute the peak factors P, and qi for each normal coordinate in terms of the

corresponding modal frequency and damping coefficient.

From the definition of the mean response spectrum, it is clear that ST(W;,~) is the mean

of the absolute maximum of the i-th normal coordinate, S; (t). Thus, using the relation

- 10 -

1 -2AO,ii = -2ST (Wi' U (23)

Pi

Furthermore, using the relations 1J j = .JA2,iJAo,;;/'rr and fj i = .J1 - Af./JAO.iiA2,ii together with the

above expression and Eqs. 21-22, the first and second spectral moments for the i-th normal

coordinate are obtained as

(24)

(25)

and

2Wi -2 ( )

A2.ii = -2ST Wi' ~iPi

respectively. Eqs. 23-25 give the first three spectral moments of the i-th normal coordinate in

terms of the corresponding modal frequency and damping coefficient and the ordinate of the

input response spectrum. These results are, of course, only valid for responses to wide-band

inputs and for modes whose frequencies are within the dominant range of the input frequen-

cies.

Using Eqs. 23-25 together with Eqs. 8-10 in Eq. 5, the moments AO, AJ, and A2 of the

response power spectral density are computed in terms of the response spectrum ordinates.

These moments can be used to evaluate the cumulative distribution of the peak response or the

various statistical quantities of the response as described in the previous section (Eqs. 13-20).

In particular, denoting RiT='J!iST(W,,~) as the mean of the maximum response in mode i, this

analysis yields the following modal combination rules for the response quantities:

Root-mean-square of response:

Root-mean-square of response rate:

[ I~

WiWi --(j" R = 1212--'P2.U RiT RjT

i j PiPi

Mean of peak response:

(26)

(27)

(28)

(29)

- 11 -

Standard deviation of peak response:

[2 I'/'q - -

U'RT

= qU'R = 1:1:-.-.PO,URjTRjTi j P,PI

where p and q in the last two equations are the peak factors for the response and are obtained

from Eqs. 16-18 (or 19-20) in terms of the mean crossing rate (Eq, 14) and the shape factor

(Eq. 15) of the response process. Another quantity that is of practical interest is the response

mean frequency, denoted by W, which is given by

,/,

U'RW = 1TV = -- =

U'R(0)

This frequency determines the average number of response cycles over a unit duration and is

useful in certain studies such as for fatigue related failures. Observe that, since Po,u and P2.U

are nearly the same (see Fig. 3), the mean response frequency is a weighted root-mean-square

of the modal frequencies of the structure.

It is important to note in the preceding expressions that since ST(Wj,~) is by definition

positive, the sign of liiT is the same as that of the corresponding effective participation factor,

'l'i' The sign of this factor depends on the modal characteristics of the structure and on the

direction of input. Since Po,u and P2,U are always positive, it follows that the cross terms in

Eqs. 26-30 are negative when 'l'i and '1') assume opposite signs. It is easy to show, however,

that the double summations inside parentheses in these expressions are always positive.

In many practical applications, the mean of the peak response is all that is required.

Therefore, a close examination and possible simplification of Eq. 28 is of special interest. It is

first noted from substituting Eq. 21 in Eq. 30 that the mean zero-crossing rate of the response

process, v, is a weighted root-mean-square of the mean zero-crossing rates, Vi' of the normal

coordinates. On the other hand, the shape factor, 0, of the response process would usually tend

to be greater than 0 i, since the response being contributed by all modes is usually a broader-

band process than each of the normal coordinates. This, however, may not be true when the

response is mainly contributed by one mode or two closely spaced modes, in which case 0 can

- 12 -

be equal to or smaller than f)i' In any case, since the peak factor is only slightly dependent on

the shape factor (see Fig. 4), it is easy to see that, because of the relation of v to Vi' P would

generally tend to be some sort of an average of Pi' It follows, then, that the ratios piPi are

around unity and, since the peak factor monotonically increases with the frequency, they tend

to decrease with increasing mode number. The role of these ratios in Eq. 28 for the mean

response, therefore, is to enhance the contributions from the lower modes and to reduce those

from the higher modes of the structure. Since the peak factor has slow variation with the fre-

quency, as is evident from its logarithmic relation to the mean-crossing rate (see Fig. 4), for

most structures the ratios piPi would tend to be near unity. Thus, these ratios in the expres-

sion for the mean response can be discarded without much loss of accuracy. With this

simplification, Eq. 28 reduces to

ii, ~ [~~Oijii"iilf (31)

The advantage gained from this simplification is that the mean response is now given directly in

terms of the maximum modal responses and the coefficients Po.u; Le., there is no need to com-

pute the spectral moments from Eq. 5. Also note that this expression for the mean response is

independent of the duration (except that which is implied through the input response spec-

trum).

A similar simplification of the expression for the response mean frequency, W, is also pos-

sible, which after multiplying the numerator and denominator in Eq. 30 by P and neglecting the

ratios piPi> reduces to

';'

LLW iW jP2,URiTRiT

W = I_i----::.

j__--==-=_-

LLPo.URiTR;Ti j

(32)

Observe that, with this simplification, wbecomes the root-mean-square of the modal frequen-

des as weighted by the maximum modal responses. Because of this, this frequency is a good

indicator of the significance of contributions from various modes of a structure to a particular

response, i.e. a larger wwould indicate larger contributions from higher modes.

- 13 -

For structures with well separated frequencies the coefficients Pm.U vanish; see Fig. 3. As

a result, all cross terms in the response expressions, i.e., Eqs. 5 and 26-32, can be neglected for

such structures. (As a simple rule, such terms can be dropped when w;;W j is less than

0.2/ ({ ;+{), which approximately corresponds to P m.U less than 0.1 (5).) In particular, Eq. 31 in

this case reduces to

(33)

This is the well known square-root-of-sum-of-squares (SRSS) rule for modal combination. It is

clear from this derivation that the SRSS rule for the peak response is only adequate for struc-

tures with well spaced frequencies. The error associated with the SRSS method for neglecting

the modal cross-correlations can be significant, as illustrated in the subsequent example.

APPLICATION TO EARTHQUAKE LOADING

The proposed modal combination procedure should be particularly useful in earthquake

engineering, where a response spectrum description of the ground motion is widely used. How-

ever, for such an application, it is important to examine the validity of the basic assumptions of

the method relative to earthquake excitations. Specifically, assumptions to be examined are: (a)

that the ground motion is a stationary Gaussian process with a wide-band power spectral den-

sity, and (b) that the response of the (linear) structure is a stationary process. Whereas

earthquake-induced ground motions are inherently nonstationary, the strong phase of such

motions is usually nearly stationary. Since the peak response generally occurs during this

phase, it is reasonable, at least for the purpose of developing a response spectrum method, to

assume it to be a stationary process. This assumption would clearly become less accurate for

short-duration, impulsive earthquakes. The assumption of Gaussian excitation is acceptable on

the basis of the central limit theorem, since the earthquake ground motion is the accumulation

of a large number of randomly arriving pulses (2). The wide-band assumption for the earth-

quake motion has been verified based on recorded motions and is generally accepted (2,7).

Finally, for the assumption of stationary response, it is well known (e.g., Ref. 8) that the

- 14 -

response of a not-too-lightly damped oscillator to a wide-band input reaches stationarity in just a

few cycles. Thus, this assumption should be acceptable for structures whose fundamental

periods are several times shorter than the strong-phase duration of the ground motion. These

considerations also suggest that the strong-phase duration of the ground motion is the appropri­

ate value for the parameter T to be used in the response spectrum method.

It is clear from the above discussion that the response spectrum method for earthquake

loading will be most accurate for earthquakes with long, stationary phases Of strong shaking and

for not-too-lightly damped structures whose fundamental periods are several times shorter than

the duration of earthquake. Through a number of example studies, it has been found that the

procedure is quite accurate for typical structures and earthquakes (see the example below). It

has also been found that Eq. 31 for the mean response closely approximates the maximum

response for a deterministic ground motion with a non-smooth response spectrum. For this

reason, it has been proposed as a replacement for the SRSS method in deterministic analysis

(18). Based on the coefficient of variation of the peak response, i.e. the ratio q/p in Fig. 4,

maximum errors in such applications are expected to range within 10 to 30 percent, depending

on the response frequency.

Several formulations for the mean of the peak response to earthquake excitations have

previously been proposed (10,11,14). These are similar to Eq. 31 of the present formulation,

except for the difference in expressions given for the cross-correlation coefficient. The best

known among these methods is that of Rosenblueth et al. (10). In their formulation, which

has somewhat heuristic bases, the cross-correlation coefficients are given in terms of the modal

frequencies and damping coefficients and the duration of motion. For earthquake-type excita­

tions, this method appears to give good results if the duration is known (see Refs. 4 and 13 for

comparisons of the method with exact solutions). However, since the duration is often unk­

nown, Eq. 31, which is independent of duration, provides a better method to be used in such

applications. It must be pointed out that none of the existing response spectrum methods pro­

vide any means for computing the variance or the cumulative distribution of the peak response.

- 15 -

This aspect of the present formulation, therefore, is unique and a furtherance of the state of

the art.

EXAMPLE APPLICATION

As an example application of the proposed response spectrum method, the responses of a

5-story building structure to a set of 20 simulated ground motions are studied. The building is

assumed to have rigid floors with uniform mass and stiffness along the height. A typical floor

plan and the properties are shown in Fig. 6. The structure is subjected to ground motions in

the x direction only. However, because of asymmetry about the x axis, the center of mass at

each floor has a rotational as well as a translational degree of freedom. This results in modes

with closely spaced frequencies, as described in the table in Fig. 6.

The ground motions used in this study were obtained using a simulation method by Ruiz

and Penzien (12). In this method, ground acceleration records are generated as samples of a

filtered Gaussian white-noise process as modulated by an intensity function. The filter parame­

ters for the power spectral density were selected to be w.~ = 51T and ~g = 0.6. An intensity func­

tion similar to that of a type-B earthquake, as defined by Jennings et al. (6), was used for this

purpose. It includes a stationary strong-motion phase of 11 seconds (between 4 and 15

seconds), yielding T = 11 for the response spectrum analysis. The power spectral density scale

factor, Go (see Eq. 7), was selected such as to produce a mean peak ground acceleration of 0.5g

over the ensemble of records. A sample of the simulated ground motions is illustrated in Fig.

7.

Pseudo-velocity response spectra associated with each individual ground motion for 0, 2,

5, 10, and 20 percent damping were computed. Means, standard deviations, and coefficients of

variation of these spectra are shown in Fig. 8. For comparison, the coefficient of variation, q / p,

based on Eqs. 16-18 and 21-22, are also shown in Fig. 8(c) for the spectra with non-zero damp­

ing (smooth curves). Observe that for such spectra the analytical estimates of the coefficient of

variation closely agree with simulation results. It is interesting to note that the coefficient of

variation is relatively insensitive to damping. Also, note that it increa~es with increasing period

- 16 -

ranging from about 0.1 at short periods (0.05 seconds) to about 0.3 to 0.4 at long periods (5

seconds). This implies that the peak response of low-frequency oscillators is more sensitive to

details of the ground motion than that of high-frequency oscillators. This is in agreement with

Trifunac's results in a study of digitization noise for recorded accelerograms (15). It is also

interesting to note in the simulation results of Fig. 8(c) that for an undamped oscillator the

coefficient of variation in the peak response is much larger than that for a damped oscillator,

especially for short periodloscillators. (Note that for an undamped oscillator the response does

not reach stationarity and the analytical expressions in Eqs. 16-18 do not hold.) A somewhat

surprising result in Fig. 8(c) is the small magnitude of the coefficient of variation for damped

spectra. Studies based on recorded accelerograms, such as that by Newmark (9), generally indi­

cate much larger variability. This can be explained by the fact that acceleration records used in

such studies are for different earthquakes and site conditions and do not represent members of

a single stochastic process. The larger coefficient of variation is a consequence of this added

variability.

Using numerical integration, peak responses of the example structure to each of the 20

ground motions were computed. These samples are subsequently compared with response esti­

mates based on the proposed method using the mean response spectrum in Fig. 8(a). Table 1

shows a comparison of the means and standard deviations of peak responses. (The word "simu­

lation" in this table denotes results based on numerical integration). Included in this table are

estimates of the mean peak response based on Eq. 28, the simplified method of Eq. 31, and the

SRSS method of Eq. 33, and estimates of the standard deviation of peak response based on Eq.

29. Estimates of the root-mean-square response based on Eq. 26 are also included in the last

column of the table. It can be observed in this table that Eqs. 28 and 29 for the mean and

standard deviation of the peak response are in close agreement with the simulation results. The

simplified expression for the mean response, Eq. 31, also appears to give good results. How­

ever, the SRSS method, Eq. 33, is in gross error reflecting the significance of the correlation

between modal responses which is neglected in this approach. Note that this method of modal

- 17 -

combination underestimates the translational responses by as much as 27 percent and overesti­

mates the rotational responses by a factor greater than 2.

Fig. 9 shows a plot of the mean frequency, iii, for various responses orthe structure. As

can be observed, this frequency has a small value (close to the frequencies of the first two

modes) for floor displacements and rotations and for lower story shears and torques, whereas it

has a larger value for upper story shears and torques and for floor pseudo-accelerations. Since

iii is the root-mean-square of modal frequencies as weighted by the corresponding modal

responses, it follows from Fig. 9 that, as expected, floor displacements and rotations and lower

story shears and torques are mainly contributed by the first two modes, whereas upper-story

shears and torques and floor pseudo-accelerations have significant contributions from higher

modes.

Finally, Fig. 10 shows comparisons between the cumulative histograms of the simulated

samples and the cumulative distribution of Eq. 13, based on the response spectrum method, for

selected response quantities. In all cases, the theoretical distribution is acceptable based on the

Kolmogorov-Smirnov test (1) at all significance levels.

CONCLUSIONS

The principal results and conclusions of this study can be summarized as follows:

(1) A response spectrum method for stationary random vibration analysis of linear structures

subjected to wide-band, stationary Gaussian excitations is developed. Modal combination

rules are derived for the mean-squares of the response and its time derivative, the mean

and variance of the peak response and the mean frequency of the response. The cumula­

tive distribution of the peak response is also obtained in terms of the input response spec­

trum. The analysis properly accounts for the cross-correlation between modal responses.

(2) The cross-correlation between modal responses is significant for modes with closely spaced

frequencies. The conventional SRSS method of modal combination, which neglects this

correlation, can lead to gross errors in estimating the peak response when the structure

- 18 -

frequencies are closely spaced.

(3) The proposed response spectrum method can be used for structures subjected to transient

Gaussian wide-band inputs, such as earthquake induced base excitations.' In such applica­

tions, the method would be more accurate when the excitation has a long, stationary

phase of strong motion, and the structure is not too lightly damped and has a fundamental

period which is several times shorter than the duration of excitation.

(4) In an example application, results based on the proposed response spectrum method are in

close agreement with simulation results based on time-history computations. It is shown

that the coefficient of variation in the peak response of an oscillator increases with

decreasing oscillator frequency and damping. This coefficient is found to range between

0.1 to 0.4 for oscillators of period 0.05 to 5 seconds. It is also shown that the response

mean frequency, which is given as the root-mean-square of modal frequencies as weighted

by the corresponding modal responses, is a good indicator of the significance of higher

modes to a particular response.

ACKNOWLEDGEMENT

This research was supported by the U. S. National Science Foundation under Grant No.

ENG-7905906. The support is gratefully acknowledged.

- 19 -

REFERENCES

[I] Ang, A. H-S., and Tang, W.H., Probability Concepts in Engineering Planning and Design,John Wiley, New York, N.Y., 1975.

[2] Clough, R.W., and Penzien, J., Dynamics of Structures, McGraw-Hill, New York, N.Y.,1975.

[3] Davenport, A.G., "Note on the Distribution of the Largest Value of a Random Functionwith Application to Gust Loading," Proceedings, Institution of Civil Engineers, London,Vol. 28, 1964, pp. 187-196.

[4] Der Kiureghian, A., "Probabilistic Response Spectral Analysis," Proceedings, ThirdEngineering Mechanics Specialty Conference, Austin, Texas, September, 1079, pp. 339­343.

[5] Der Kiureghian, A., "On Response of Structures to Stationary Excitation," Report No.UCB/EERC 79-32, Earthquake Engineering Research Center, University of California,Berkeley, CA., December, 1979.

[6] Jennings, P.C., Housner, G.W., and Tsai, N.C., "Simulated Earthquake Motions," Earth­quake Engineering Research Laboratory, California Institute of Technology, Pasadena,CA, April, 1968.

[7] Kanai, K., "Semi-Empirical Formula for Seismic Characterization of the Ground," Bulletinof Earthquake Research Institute, University of Tokyo, Japan, Vol. 35, June 1967.

[8] Lin, Y.K., Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York, N.Y.,1967.

[9] Newmark, N.M., "A Study of Vertical and Horizontal Earthquake Spectra," WASH-1255,United States Atomic Energy Commission, Washington, D.C., April, 1973.

[10] Rosenblueth, E. and Elorduy, J., "Responses of Linear Systems to Certain Transient Dis­turbances," Proceedings, Fourth World Conference on Earthquake Engineering, Vol. I,Santiago, Chile, 1969, pp. 185-196.

[II] Ruiz, P., "On the Maximum Response of Structures Subjected to Earthquake Excitations,"Proceedings, Fourth Symposium on Earthquake Engineering, Roorkee, India, 1970, pp.272-277.

[12] Ruiz, P. and Penzien, 1., "PSEQGN - Artificial Generation of Earthquake Accelerograms,"Report No. UCB/EERC 69-3, Earthquake Engineering Research Center, University of Cali­fornia, Berkeley, CA, March, 1968.

[13] Singh, A.K., Chu, S.L. and Singh, S., "Influence of Closely Spaced Modes in ResponseSpectrum Method of Analysis," Report No. SAD-126, Sargent & Lundy Engineers, Chi­cago, Ill., December, 1973.

[14] Singh, M.P., and Chu, S.L., "Stochastic Considerations in Seismic Analysis of Structures,"Earthquake Engineering and Structural Dynamics, Vol. 4, No.3, March, 1976, pp. 295-307.

[15] Trifunac, M.D., "Response Spectra of Earthquake Ground Motion," Journal of theEngineering Mechanics Division, ASCE, Vol. 104, No. EM5, Proc. Paper 14051, October,1978, pp. 1081, 1097.

[16] Vanmarcke, E.H., "Properties of Spectral Moments with Application to Random Vibra­tion," Journal of the Engineering Mechanics Division, ASCE, Vol. 98, No. EM2, Proc. Paper8822, April 1972, pp. 425-446.

[17] Vanmarcke, E.H., "On the Distribution of the First-Passage Time for Normal StationaryRandom Processes," Journal ofApplied Mechanics, Vol. 42, March 1975, pp. 215 -220.

[I8] Wilson, E.L., Der Kiureghian, A. and Bayo, E., "A Replacement for the SRSS Method inSeismic Analysis," submitted for publication.

FR (r)T

GoGF(W)GR(W)H,(w)H,*(w)mpp,q

qirR(t)R(t)Ri(t)

/iT/iTR'TS

S,(t)Si(t)STeW, ~)to0"i'.ii~g

~,

AmAm.!iII

II"

II,

PIIl.!i(J'R(J't<

(J' R T

T

'1',w

Wg

WI

W

- 20 -

NOTATION

cumulative distribution of R T ;

power spectral density scale factor;power spectral density of F(t);power spectral density of R (t);frequency response function for mode i;complex conjugate of H,(w);a non-negative integer;peak factor for mean of peak response;peak factor for mean peak of i-th normal coordinate;peak factor for standard deviation of peak response;peak factor for standard deviation of peak of i-th normal coordinate;barrier level;random process describing response of structure;time derivative of R (t);random process describing response in mode i;peak of R (t) over T;mean of RT ;

mean of peak response in mode i;normalized barrier level;normal coordinate associated with i-th mode;time derivative of Sl(t);ordinate of mean response spectrum at frequency wand damping ~;

time;shape factor for power spectral density of R (t);an effective value of 0;shape factor for power spectral density of S,(!);filter damping coefficient;damping coefficient for mode i;m-th moment of response power spectral density;m-th cross-spectral moment of S,(t) and SiC!);mean zero-crossing rate of R (t);an equivalent mean zero-crossing rate;mean zero-crossing rate of S, (!);correlation coefficients associated with AmJi ;

root-mean-square of R (t);root-mean-square of (J'R;

standard deviation of peak response;

duration of excitation;effective participation factor for mode i ;circular frequency;filter circular frequency;natural circular frequency of mode i; andmean frequency of response.

-2b

Table 1. Summary of Results for Example Structure

Rr (J RT

(JR

Response LevelDescription

Simul.Eq. 28 Eq.31 Eq. 33

Simul.Eq. 29

Eq.26·Simul. Simul. Simul. Simul.Displacement, em. 5 8.05 0.99 0.97 0.74 1.59 0.92 2.96

4 7.38 0.98 0.98 0.74 1.43 0.90 2.683 6.10 0.99 0.99 0.75 1.19 0.88 2.232 4.36 1.00 0.99 0.75 0.85 0.88 1.591 2.29 1.02 0.99 0.75 0.43 0.87 0.82

Rotation, radx10-2. 5 0.463 1.06 1.14 2.32 0.088 1.01 0.2004 0.426 1.02 1.12 2.31 0.082 1.05 0.1783 0.355 1.00 1.11 2.30 0.070 1.07 0.1462 0.256 0.99 1.10 2.29 0.051 1.06 0.1041 0.134 1.01 1.10 2.28 0.026 1.03 0.054

Pseudo-Acceleration, g. 5 1.430 0.99 0.97 0.73 0.278 0.90 0.4864 1.270 1.02 0.98 0.74 0.237 0.87 0.4583 1.050 1.08 1.01 0.76 0.192 0.87 0.3842 0.843 1.08 0.99 0.74 0.114 1.09 0.2981 0.611 0.96 0.86 0.73 0.067 1.07 0.182

Angular Pseudo- 5 0.803 1.07 1.11 2.31 0.136 1.09 0.296Acceleration, rad/ S2. 4 0.716 1.10 1.12 2.31 0.140 0.93 0.278

3 0.624 1.16 1.09 2.25 0.131 0.81 0.2462 0.494 1.17 1.07 2.23 0.101 0.79 0.1921 0.296 1.21 1.10 2.30 0.051 0.89 0.112

Story Shear, kN. 5 1408 0.99 0.97 0.73 273 0.90 4784 2648 1.00 0.97 0.73 504 0.84 9453 3628 0.98 0.98 0.74 687 0.89 13052 4302 0.99 0.99 0.75 829 0.89 15691 4701 1.00 0.99 0.75 893 0.90 1681

Story Torque, kN.m. 5 2488 1.07 1.11 2.31 421 1.09 9144 4640 1.13 1.12 2.32 837 1.05 18973 6367 1.03 1.13 2.34 1246 1.08 26412 7686 1.01 1.12 2.33 1568 1.04 31661 8442 1.06 1.13 2.31 1727 0.94 3577

Note:R r = mean of peak response; (J R = standard deviation of peak response;(J R~ root-mean-T

square response.

-22-

3.0r-----------------~

4.03.0

FILTERED WHITE NOISE

2.0

W/Wg

1.0

1.0~-----~l<-----L..---------__1

2.0

Figure 1. Power Spectral Density Shapes for White-Noise andFiltered White-Noise Inputs.

-23-

---FILTERED WHITE NOISE INPUT. ~g =0.60----WHITE NOISE INPUT

1.5 2.0

...... --........ _-- ---

1.0 r------.,.---..,.-:=--;:-----------,

0.0 -- --0.5 0.6 0.7 0.8 0.9 1.0

Wi /Wj

0.2

Wj = OAwg Sj= S{O.20

0.80.10

0.6

0.4

0.2

0.0\.0

Wj =wg

0.8

0.6

PO,ij

0.4

0.2

0.01.0

0.8

0.6

0.4

Figure 2. Comparison of Correlation Coefficients for Responses toWhite-Noise and Filtered White-Noise Inputs.

-24-

~ i =0.1 , ~ j =0.05 -+-.1--_----1#>.

0.2

1.0 ,...-------,-----..------.-----.-----.-

0.8

0.6

~,ij0.4

Figure 3. Coefficients Pl11,ij for Response to White Noise.

-25-

4.0,-------------------,.------.

3.0

P (WHITE-NOISE INPUT)

2.0 8 =0.46 (~=O. 20)0.34 0.100.25 0.050.16 0.020.11 0.01

1.00.8

8= 0.11 (S=O.OI)0.16 0.020.25 0.05

0.6 0.34 0.100.46 0.20

q

0.4

500 100020050 100liT

200.1L-L_~-=~~~

10

0.20.4

8=0. II (~=0.01 )

0.3 0.16 0.020.25 0.05

q/p 0.34 0.100.46 0.20

0.2

Figure 4. Peak Factors for Response to Stationary Gaussian Input.

-26-

105.00.5 1.0 2.0wi /Wg

0.20.0

0.1

4.0

~g =0.60~=0.02

3.0--8FWN /8WN 0.05

0 --- 71FWN / 71WN

~2.0

0::

1.0

Figure 5. Ratios of Mean-Crossing Rates and Shape Factors for Responses toFiltered White-Noise (FWN) and White-Noise (WN) Inputs.

-27-

y

~k Y12

IkY/2~

ICENTER OF

MASS ~e~Ui -.LkiV/RlIL ZIM-.+

I· a a ·1

~Xg

11f,\V/A\\]II, /\v/;"\\Y/~

TYPICAL FLOOR PLAN ELEVATION

Properties of Typical Floor and Story:

Mass = 100,000 kg.Mass Radius of Gyration = 5.56 m.Stiffness: k x = k y = 205900 kN/m.e= 0.30 m. .a= 5.56 m.

Modal Properties

Mode Freq., cps Damp. Ratio

1 2.00 0.052 2.11 0.053 5.84 0.054 6.17 0.055 9.20 0.056 9.72 0.057 11.80 0.058 12.50 0.059 13.50 0.05

10 14.20 0.05

Figure 6. Properties of Example Structure.

0.6 iii I i a iii i i I I Iii iii I g I I Iii iii i i

Cl

IN00I

302520

r--- STATIONARY ..II I I I I I I I

5 10 15

TIME, SECONDS

-0.6 1 ' , I I I I

o

z 0.3otia:::w..J 00~nlW .U

~Cl

~ -0.3o0::t!)

Figure 7. Sample of Simulated Earthquake Ground Motion.

-29-

, , , , j'I 'l20 ~

0 10~Q)<II

5 r....-...lr

t:2 "u

:3w>00~

wen0..

(a) Mean Spectra! , !

0.05 0.1 0.2 0.5 2 5

PERIOD, SECONDS

2 V~/\!' ~ .~\I"'~-J v

/"1 /" --1\ r- /~r./

J\/ /'.\ I /----- ,-0 ._~//-~~/~Q)

~,In.... 0.5 • r'~;;: 'v-/;r

0.2 ~-----.jt:u Iff? \0...J 0.1 \w

\~0.00>60 0.05

\ 0.02~ "---- 0.05w(J) \-0.100..

0.02 0.20(b) Standard

0.01 Deviation Spectra0.05 0.1 0.2 0.5 2 5

PERIOD, SECONDS

0.5

~ 0.4

~cr~lJ..oI­Zw 0.2<3li:lJ..W

8 0.1

----1;=0.000.020.050.100.20

(c) Coefficient ofVariation Spectra

0.05 0.1 0.2 0.5

PERIOD, SECONDS

2 5

Figure 8. Pseudo-Velocity Spectra for 20 Generated Earthquakes.

-30-

FLOOR PSEUDO­ACCELERATIONS

STORY SHEAR ANDTORQUE

FLOOR DISPLACEMENTAND ROTATION

2

4

3

5~-----.--..,.----------------.

-'IJJ>IJJ-'

12 - MODES---3 4 56 78910U U U UUGROUND '-- --'- .....L.- ----L ---'

I 2 5 10 20

RESPONSE FREQUENCY ,W 127r,cps

Figure 9. Mean Frequency for Responses of Example Structure.

-31-

4000 6000 8000

BASE SHEAR, KN

o .....ooL---' ...I..-__......L.-'

1.0.---------::;;;;-.---,

0.6

0.8

0.2

0.4

2.52.01.51.0

ROOF PSEUDO-ACCELERATION, 9

O'--.....L--'-----'------'--......LJ

1.0 r---------::='",...---,

0.6

0.2

0.4

0.8

128o '-"''-'-__L-__--'__-'

4

0.2

0.4

1.0.-----------=...,...-..,

ROOF DISPLACEMENT, CM

>- 0.8I-::::iiiie::(

~ 0.60:::ll.

W

~l­e::(...J:::>~:::>u

1.0.-------r--:~--. 1.0 r-------.~..-....., 1.0 r-~-----.-::""""__,

o ""------'---'- -'-----__....L..J

140001000060000'-"0...---'.1.------'------'-----'

0.8

0.6

0.4

0.2

1.4100.6O......-:u-----'-----L.J

0.8

0.6

0.2

0.4

0.0080.0060.004

0.2

0.8

0.6

0.4

>­I-...J

CDe::(CDo0:::ll.

W>!;;:...J:::>~:::>u

ROOF ROTATION I RAD ROOF ANGULAR PSEUDO­ACCELERATION I RAD/ 52

BASE TORQUE I KN. m

Figure 10. Computed and Simulated Cumulative Distributions forSelected Peak Responses of Example Structure.

EERC-l

EARTHQUAKE ENGINEERING RESEARCH CENTER REPORTS

NOTE: Numbers in parenthesis are Accession Numbers assigned by the National Technical Information Service; these arefollowed by a price code. Copies of the reports may be ordered from the National Technical Information Service, 5285Port Royal Road, Springfield, Virginia, 22161. Accession Numbers should be quoted on orders for reports (PB------)and remittance must accompany each order. Reports without this information were not available at time of printing.Upon request, EERC will mail inquirers this information when it becomes available.

EERC 67-1

EERC 68-1

EERC 68-2

EERC 68-3

EERC 68-4

EERC 68-5

EERC 69-1

EERC 69-2

EERC 69-3

EERC 69-4

EERC 69-5

EERC 69-6

EERC 69-7

EERC 69-8

"Feasibility Study Large-Scale Earthquake Simulator Facility," by J. Penzien, J.G. Bouwkamp, R.W. Cloughand D. Rea - 1967 (PB 187 905)A07

Unassigned

"Inelastic Behavior of Beam-to-Column Subassemblages Under Repeated Loading," by V.V. Bertero - 1968(PB 184 888) A05

"A Graphical Method for Solving the Wave Reflection-Refraction Problem," by H.D. McNiven and Y. Mengi -1968(PS 187 943)A03

"Dynamic Properties of McKinley School Buildings," by D. Rea, J. G. Bouwkamp and R. W. Clough - 1968(PB 187 902)A07

"Characteristics of Rock Motions During Earthquakes," by H.B. Seed, LM. Idriss and F.W. Kiefer -1968(PS 188 338) A03

"Earthquake Engineering Research at Berkeley," - 1969 (PB 187 906)All

"Nonlinear Seismic Response of Earth Structures," by M. Dibaj and J. Penzien -1969 (PB 187 904)A08

"Probabilistic Study of the Behavior of Structures During Earthquakes," by R. Ruiz and J. Penzien - 1969(Ps 187 886) A06

"Numerical Solution of Boundary Value Problems in Structural Mechanics by Reduction to an Initial ValueFormulation," by N. Distefano and J. Schujman - 1969 (PB 187 942)A02

"Dynamic Programming and the Solution of the Biharmonic Equation," by N. Distefano - 1969 (PB 187 941) A03

"Stochastic Analysis of Offshore Tower Structures," by A.K. Malhotra and J. Penzien -1969 (PB 187 903)A09

"Rock Motion Accelerograms for High Magnitude Earthquakes," by H. B. Seed and I.M. Idriss - 1969 (FB 187 940) A02

"Structural Dynamics Testing Facilities at the University of California, Berkeley," by R.M. Stephen,J.G. Bouwkamp, R.W. Clough and J. Penzien -1969 (PB 189 11l)A04

EERC 69-9 "Seismic Response of Soil Deposits Underlain by Sloping Rock Boundaries," by H. Dezfulian and H.B. Seed1969 (PB 189 114)A03

EERC 69-10 "Dynamic Stress Analysis of Axisymmetric Structures Under Arbitrary Loading," by S. Ghosh and E.L. Wilson1969 (PB 189 026)A10

EERC 69-11 "Seismic Behavior of Multistory Frames Designed by Different Philosophies," by J .C. Anderson andV. V. Bertero - 1969 (PB 190 662)Al0

EERC 69-12 "Stiffness Degradation of Reinforcing Concrete Members Subjected to Cyclic Flexural Moments," byV.V. Bertero, B. Bresler and H. Ming Liao -1969 (PB 202 942)A07

EERC 69-13 "Response of Non-Uniform Soil Deposits to Travelling Seismic Waves," by H. Dezfulian and H.B. Seed -1969(PB 191 023) A03

EERC 69-14 "Damping Capacity of a Model Steel Structure," by D. Rea, R.W. Clough and J.G. Bouwkamp -1969 (PB 190 663)A06

EERC 69-15 "Influence of Local Soil Conditions on Building Damage Potential during Earthquakes," by H.B. Seed andI.M. Idriss - 1969 (PB 191 036)A03

EERC 69-16 "The Behavior of Sands Under Seismic Loading Conditions," by M.L. Silver and H.B. Seed -1969 (AD 714 982)A07

EERC 70-1 "Earthquake Response of Gravity Dams," by A.K. Chopra -1970 (AD 709 640)A03

EERC 70-2 "Relationships between Soil Conditions and Building Damage in the Caracas Earthquake of July 29, 1967," byH.B. Seed, I.M. Idriss and H. Dezfulian -1970 (PB 195 762)A05

EERC 70-3 "Cyclic Loading of Full Size Steel Connections," by E.P. Popov and R.M. Stephen -1970 (PB 213 545)A04

EERC 70-4 "Seismic Analysis of the C:larairna Building I Caraballeda, Venezuela," by Subcommittee of the SEAONC ResearchCommittee: V.V. Bertero, P.F. Fratessa, S.A. Mahin, J.H. Sexton, A.C. Scordelis, E.L. Wilson, L.A. Wyllie,H.B. Seed and J. Penzien, Chairman -1970 (PB 201 455)A06

EERC-2

EERC 70-5 "A Computer Program for Earthquake Analysis of Dams," by A.K. Chopra and P. Chakrabarti -1970 (AD 723 994)A05

EERC 70-6 "The Propagation of Love Waves Across Non-Horizontally Layered Structures," by J. Lysmer and L.A. Drake1970 (PB 197 896)A03

EERC 70-7 "Influence of Base Rock Characteristics on Ground Response," by J. Lysmer, H.B. Seed and P.B. Schnabel1970 (PB 197 897)A03

EERC 70-8 "Applicability of Laboratory Test Procedures for Measuring Soil Liquefaction Characteristics under CyclicLoading," by H.B. Seed and w.n. Peacock - 1970 (PE 198 016)A03

EERC 70-9 "A Simplified Procedure for Evaluating Soil Liquefaction Potential," by II.B. Seed and I.M. Idriss - 1970(PB 198 009)A03

EERC 70-10 "Soil Moduli and Damping Factors for Dynamic Response Analysis," by H.B. Seed and I.M. Idriss -1970(PB 197 S69)A03

E:S!~C 71-1 "Koyna Earthquake of December 11, 1967 and the Performance of Koyna Dam," by A.K. Chopra and P. Chakrabarti1971 (AD 731 49G)AOG

EERC 71-2 "Preliminary In-Situ Measurements of Anelastic Absorption in Soils Using a Prototype Earthquake SimUlator,"by R.D. Borcherdt and P.W. Rodgers - 1971 (PB 201 454)A03

EERC 71-3 "Static and Dvnamic Analysis of Inelastic Frame Structures," by F.L. Porter and G.H. Powell-1971(PB 210 135) A06

EERC 71-4 "Research Needs in Limit Design of Reinforced Concrete Structures," by V.V. Bertero -1971 (PB 202 943)A04

EERC 71-5 "Dynamic Behavior of a High-Rise Diagonally Braced Steel Building," by D. Rea, A.A. Shah and ,j.G. Bouwh\h\P1971 (PB 203 584)A06

EERC 71-6 "Dynamic Stress Analysis of Porous Elastic Solids Saturated with Compressible Fluids," by J. Ghaboussi andE. L. Wilson - 19"1 (PB 211 396)A06

EERC 71-7 "Inelastic Behavior of Steel Beam-to-Column Subassemblages," by H. Krawinkler, V.V. Bertero and E.P. Popov1971 (PB 211 335)A14

EERC 71-8 "Modification of Seismograph Records for Effects of Local Soil Conditions," by P. Schnabel, H.B. Seed andJ. Lysmer-1971 (PB 214 450)A03

EERC 72-1 "Static and Earthquake Analysis of Three Dimensional Frame and Shear Wall Buildings," by E.L. Wilson andH.H. Dovey -1972 {PB 212 904)A05

EERC 72-2 "Accelerations in Rock for Earthquakes in the Western United States," by P .B. Schnabel and H. B. Seed - 1972(PB 213 100)A03

EERC 72-3 "Elastic-Plastic Earthquake Response of Soil-Building Systems," by T. Minami -1972 {PB 214 868)A08

EERC 72-4 "Stochastic Inelastic RC8pons~ of Offshore TowE;rs to Strong ~10t:i.on Earth'luakes," by M.K. Kaul - 1972(PB 215 7l3)A05

EERC 72-5 "Cyclic Behavior of Three Reinforced Concrete Flexural Members with High Shear," by E.P. Popov, V. V. Berteroand H. Krawinkler - 1972 (PB 214 555)A05

EERC 72-6 "Earthquake Response of r;ravi ty Dams Including Reservoir Interaction Effects," by P. Chakrabarti andA.K. Chopra - 1972 (AD 762 330)A08

EERC 72-7 "Dynamic Properties of Pine Flat Dam," by D. Rea, C.Y. Liaw and A.K. Chopra-1972 (AD 763 928)A05

EERC 72-8 "Three Dimensional Analysis of Building Systems," by E.L. Wilson and H.H. Dovey -1972 (PB 222 438)A06

EERC 72-9 "Rate of Loading Effects on Uncracked and Repaired Reinforced Concrete Members," by S. Mahin, V. V. Bertero,

D. Rea and M. Atalay - 1972 (PB 224 520) AOS

EERC 72-10 "Computer Program for Static and Dynamic Analysis of Linear Structural Systems," by E.L. Wilson, K.-J. Bathe,J,E. Peterson and H.H.Dovey -1972 (PB 220 437)A04

EERC 72-11 "Literature Survey - Seismic Effects on Highway Bridges," by T. Iwasaki, J. Penzien and R.W. Clough -1972

(FB 215 613)A19

EERC 72-12 "SHAKE-A Computer Program for Earthquake Response Analysis of Horizontally Layered Sites," by P .B. Schnabel

and J. Lysmer - 1972 (PB 220 207) A06

EERC 73-1 "Optimal Seismic Design of Multistory Frames," by V.V. Bertero and H. Kamil-1973

EERC 73-2 "Analysis of the Slides in the San Fernando Dams During the Earthquake of February 9, 1971," by H. B. Seed,K.L. Lee, I.M. Idriss and F. Makdisi - 1973 (PB 223 402)A14

EERC-3

EERC 73-3 "Computer Aided Ultimate Load Design of Unbraced Multistory Steel Frames," by M.B. EI-Hafez and G.H. Powell1973 (PB 248 315)A09

EERC 73-4 IIExperimental Investigation into the Seismic Behavior of Critical Re0ions of Reinforced Concrete Componentsas Influenced by Moment and Shear," by M. Celebi and J. Penzien - 1973 (PB 215 884)A09

EERC 73-5 "Hysteretic Behavior of Epoxy-Repaired Reinforced Concrete Beams,1I by M. Celebi and :1. Penzien - 1973(PB 239 568)A03

EERC 73-6 "General Purpose Computer Program for Inelastic Dynamic Response of Plane Structures," by A. Kanaan andG.H. Powell-1973 (PB 221 260)A08

EERC 73-7 "A Computer Program for Earthquake Analysis of Gravity Dams Including Reservoir Interaction,lI byP. Chakrabarti and A.K. Chopra - 1973 (AD 766 271)A04

EERC 73-8 "Behavior of Reinforced Concrete Deep Beam-Column Subassemblages Under Cyclic Loads," by O. Kustu andJ.G. Bouwkamp - 1973 (PB 246 ll7)A12

EERC 73-9 "Earthquake Analysis of Structure-Foundation Systems," by A.K. Vaish and r•. K. Chopra -1973 (AD 766 272)A07

EERC 73-10 "Deconvolution of Seismic Response for Linear Systems," by R.B. Reimer -1973 (PB 227 179)A08

EERC 73-11 "SAP IV: A Structural Analysis Program for Static and Dynamic Response of Linear Systems,1I by K..-J. Bathe,E.L. Wilson and F.E. Peterson -1973 (PB 221 967)A09

EERC 73-12 "Analytical Investigations of the Seismic Response of Long, Multiple Span Hi<;lhway Bridges," by W.S. Tsengand J. Penzien - 1973 (PB 227 816) A10

EERC 73-13 "Earthquake Analysis of Multi-Story Buildings Including Foundation Interaction," by A.K. Chopra andJ.A. Guti",rrez -1973 (PB 222 970)A03

EERC 73-14 "ADAP: A Computer Program for Static and Dynamic Analysis of Arch Dams," by R.W. Clough, J.M. Raphael andS. Mojtahedi -1973 (PB 223 763)A09

EERC 73-15 "Cyclic Plastic Analysis of Structural Steel Joints," by R.B. Pinkney and R.W. Clough -1973 (PB 226 843)A08

EERC 73-16 "QttAD-4: A Computer Program for Evaluating the Seismic Response of Soil Structures by Variable DampingFinite Element Procedures," by I.M. Idriss, J. Lysmer, R. Hwang and H.B. Seed -1973 (PB 229 424)A05

EERC 73-17 "Dynamic ,,,'havior of a Multi-Story Pyramid Shaped Building," by R.M. Stephen, J.P. Hollings andJ. G. Bouwkamp - 1973 (PB 240 718) A06

EERC 73-18 "Effect of Different Types of Reinforcir:g On Seismic Behavior of Short Concrete Columns, II by V. V .. Bertero,J. Hollings, O. Kustu, R.M. Stephen and J.G. Bouwkamp -1973

EERC 73-19 "Olive View Medical Center Materials Studies, Phase I," by B. Bresler and V.V. Bertero -1973 (PE 235 986)A06

EERC 73-;'0 "Linear and Nonlinear Seismic Analysis Computer Programs for Long Multiple-Span Highway Bridges," byW.S. Tseng and J. Penzien -1973

EERC 73-21 "Constitutive Models for Cyclic Plastic Deformation of Engineering Materials," by J.M. Kelly and P.P. Gillis1973 (PB 226 024)A03

EERC 73-22 "DRAIN - 2D User's Guide," by G.H. Powell-1973 (PB 227 016)A05

EERC 73-23 "Earthquake Engineering at Berkeley - 1973, n {PB 226 033)All

EERC 73-24 Unassigned

EERC 73-25 "Earthquake Response ofAxi synunetric Tower Structures Surrounded by Water f' II by C .. Y .. Liaw and A .. K.. Chopra1973 (,\0 773 052)A09

EERC 73--26 "Investigation of the Failures of the Olive view Stairtowers During the San Fernando Earthquake and TheirImplications on Seismic Design," by V.V. Bertero and R.G. Collins -1973 (PB 235 l06)A13

EERC 73-27 "Further Studies on Seismic Behavior of Steel Beam-Column Subassernblages, n by v. V. Bertero, H. Krawinklerand E.P. Popov -1973 (PE 234 172)A06

EERC 74-1 "Seismic Risk Analysis," by C.S. Oliveira - 1974 (PB 235 920)A06

EERC 74-2 "Settlement and Liquefaction of Sands Under Multi-Directional Shaking," by R. Pyke, C.K. Chan and H.B. Seed1974

EERC 74-3 "Optimum Design of Earthquake Resistant Shear Buildings," by D. Ray, K.S. Pister and A.K. Chopra -1974(PB 231 172)A06

EERC 74-4 "LUSH - A Computer Program for Complex Response Analysis of Soil-Structure Systems," by J. Lysmer, T. Udaka,H.B. Seed and R. Hwang - 1974 (PB 236 796)A05

EERC 74-5

EERC-4

"Sensitivity Analysis for Hysteretic Dynamic Systems: Applications to Earthquake Engineering," by D. Ray1974 (PB 233 213)A06

EERC 74-6 "Soil Structure Interaction Analyses for Evaluating Seismic Response," by H.B. Seed, J. Lysmer and R. Hwang1974 (PB 236 519)A04

EERC 74-7 Unassigned

EERC 74-8 "Shaking Table Tests of a Steel Frame - A Progress Report," by R.W. Clough and D. Tang -1974 (PB 240 8f,9)A03

EERC 74-9 "Hysteretic Behavior of Reinforced Concrete Flexural Members with Special Web Reinforcement," byV.V. Bertero, E.P. Popov and T.Y. Wang - 1974 (PB 236 797)A07

EERC 74-10 "Applications of Reliability-Based, Global Cost Optimization to Design of Earthquake Resistant Structures,"by E. Vitiello and K.S. Pister -1974 (PB 237 231)A06

EERC 74-11 "Liquefaction of Gravelly Soils Under Cyclic Loading Conditions," by R.T. Wong, H.B. Seed and C.K. Chan1974 (PB 242 042)A03

EERC 74-12 "Site-Dependent Spectra for Earthquake-Resistant Design," by H.B. Seed, C. Ugas and J. Lysmer -1974(PB 240 953)A03

EERC 74-13 "Earthquake Simulator Study of a Reinforced Concrete Frame," by P. Hidalgo and R. W. Clough - 1974(PB 241 9 f 4)A13

EERC 74-14 "Nonlinear Earthquake Response of Concrete Gravity Dams," by N. Pal-1974 (AD!A 006 583)A06

EERC 74-15 "Modeling and Identification in Nonlinear Structural Dynamics - I. One Degree of Freedom Models," byN. Distefano and A. Rath - 1974 (PB 241 548)A06

EERC 75-1 "Determination of Seismic Design Criteria for the Dumbarton Bridge Replacement Structure, Vol. I: Description,Theory and Analytical Modeling of Bridge and Parameters," by F. Baron and S.-H. Pang -1975 (PB 259407)A15

EERC 75-2

EERC 75-3

EERC 75-4

EERC 75-5

EERC 75-6

"Determination of Seismic Design Criteria for the Dumbarton Bridge Replacement Structure, Vol. II: NumericalStudies and Establishment of Seismic Design Criteria," by F. Baron and S. -H. Pang - 1975 (PB 259 408) All(For set of EERC 75-1 and 75-2 (PB 259 406»)

"Seismic Risk Analysis for a Site and a Metropolitan Area," by C.S. Oliveira-1975 (PB 248 134)A09

"Analytical Investigations of Seismic Response of Short, Single or Multiple-Span Highway Bridges," byM.-C. Chen and J. Penzien -1975 (PB 241 454)A09

"An Evaluation of Some Methods for Predicting Seismic Behavior of Reinforced Concrete Buildings," by S.A.Mahin and V.V. Bertero -1975 (PB 246 306)A16

"Earthquake Simulator Study of a Steel Frame Structure, Vol. I: Experimental Results,1I by R.W. Clough andD.T. Tang-1975 (PB 243 98llA13

EERC 75-7 "Dynamic Properties of San Bernardino Intake Tower," by D. Rea, C.-Y. Liaw and A.K. Chopra-1975 (AD!A008406)A05

EERC 75-8 "Seismic Studies of the Articulation for the Dumbarton Bridge Replacement Structure, Vol. I: Description,Theory and Analytical Modeling of Bridge Components," by F. Baron and R.E. Hamati -1975 (PB 251 539)A07

EERC 75-9 "Seismic Studies of the Articulation for the Dumbarton Bridge Replacement Structure, Vol. 2: NumericalStudies of Steel and Concrete Girder Alternates," by F. Baron and R.E. Hamati -1975 (PB 251 540)AIO

EERC 75-10 "Static and Dynamic Analysis of Nonlinear Structures," by D.P. Mondkar and G.H. Powell - 1975 (PB 242 434) Aoa

EERC 75-11 "Hysteretic Behavior of Steel Columns," by E.P. Popov, V.V. Bertero and S. Chandramouli -1975 (PB 252 3(5)1\11

EERC 75-12 "Earthquake Engineering Research Center Library Printed Catalog," - 1975 (PB 243 711)A26

EERC 75-13 "Three Dimensional Analysis of Building Systems (Extended Version) ," by E.L. Wilson, J.P. Hollings andH.H. Dovey-1975 (PB 243 989)A07

EERC 75-14 "Determination of Soil Liquefaction Characteristics by Large-Scale Laboratory Tests," by P. De Alba,C.K. Chan and H.B. Seed -1975 (NUREG 0027)A08

EERC 75-15 "A Literature Survey - Compressive, Tensile, Bond and Shear Strength of Masonry," by R.L. Mayes and R.W.Clough - 1975 (PB 246 292)AIO

EERC 75-16 "Hysteretic Behavior of Ductile Moment Resisting Reinforced Concrete Frame components," by V.V. Bertero andE.P. Popov-1975 (PB 246 388)A05

EERC 75-17 "Relationships Between Maximum Acceleration, Maximum Velocity, Distance from Source, Local Site Conditionsfor Moderately Strong Earthquakes," by H.B. Seed, R. Murarka, J. Lysmer and I.M. Idriss-1975 (PB 248 172)A03

EERC 75-18 "The Effects of Method of Sample Preparation on the Cyclic Stress-Strain Behavior of Sands," by J. Mulilis,C.K. Chan and H.B. Seed-1975 (Summarized in EERC 75-28)

EERC-5

EERC 75-19 "The Seismic Behavior of Critical Regions of Reinforced Concrete Components as Influenced by Moment, Shearand Axial Force," by M.B. Atalay and J. Penzien -1975 (PB 258 842)All

EERC 75-20 "Dynamic Properties of an Eleven Story Masonry Building," by R.M. Stephen, J.P. Hollings, J .G. Bouwkamp andD. Jurukovski - 1975 (PB 246 945)A04

EERC 75-21 "State-of-the-Art in Seismic Strength of Masonry - An Evaluation and Review," by R. L. Mayes and R. W. Clough1975 (PB 249 040)A07

EERC 75-22 "Frequency Dependent Stiffness Matrices for Viscoelastic Half-Plane Foundations," by A.K. Chopra,P. Chakrabarti and G. Dasgupta - 1975 (PB 248 121)A07

EERC 75-23 "Hysteretic Behavior of Reinforced Concrete Framed Walls," by T.Y. Wong, V.V. Bertero and E.P. Popov-1975

EERC 75-24 "Testing Facility for Subassemblages of Frame-Wall Structural Systems," by V. V. Bertero, E.P. Popov andT. Endo - 1975

EERC 75-25 "Influence of Seismic History on the Liquefaction Characteristics of Sands,lI by H.B. Seed, K. Mori andC.K. Chan -1975 (SUImllarized in EERC 75-28)

EERC 75-26 "The Generation and Dissipation of Pore Water Pressures during Soil Liquefaction," by H.B. Seed, P.P. Martinand J. Lysmer - 1975 (PB 252 648) A03

EERC 75-27 "Identification of Research Needs for Improving Aseismic Design of Building 'Structures," by V.V. Bertero1975 (PB 248 136)A05

EERC 75-28 "Evaluation of Soil Liquefaction Potential during Earthquakes," by H.B. Seed, 1. Arango and C.K. Chan -1975(NUREG 0026)A13

EERC 75-29 "Representation of Irregular Stress Time Histories by Equivalent Uniform Stress Series in LiquefactionAnalyses," by H.B. Seed, 1.M. Idriss, F. Makdisi and N. Banerjee -1975 (PB 252 635)A03

EERC 75-30 "FLUSH - A Computer Program for Approximate 3-D Analysis of Soil-Structure Interaction Problems," byJ. Lysmer, T. Udaka, C.-F. Tsai and H.B, Seed - 1975 (PB 259 332)A07

EERC 75-31 "ALUSH - A Computer Program for Seismic Response Analysis of Axisymmetric Soil-Structure Systems," byE. Berger, J. Lysmer and H.B. Seed -1975

EERC 75-32 "TRIP and TRAVEL - Computer Programs for Soil-Structure Interaction Analysis with Horizontally TravellingWaves," by T. Udaka, ,T. INsmer and H,B. Seed-1975

EERC 75-33 "Predicting the Performance of Structures in Regions of High Seismicity,1l by J. Penzien -1975 (PB 24B 130)AO~~

EERC 75-34 "Efficient Finite Element Analysis of Seismic Structure - Soil - Direction," by J. Lysmer, H.B. Seed, T. Udaka,R.N. Hwang and C.-F. Tsai -1975 (PB 253 570)A03

EERC 75-35 "The Dynamic Behavior of a First Story Girder of a Three-Story Steel Frame Subjected to Earthquake Loading,"by R.W. Clough and L.-Y. Li -1975 (PB 248 841)A05

EERC 75-36 "Earthquake Simulator Study of a Steel Frame Structure, Volume II - Analytical Results," by D. T. Tang - 1975(PB 252 926)AIO

EERC 75-37 "ANSR-I General Purpose Computer Program for Analysis of Non-Linear Structural Response," by D.P. Mondkarand G.H. Powell - 1975 (PB 252 386)A08

EERC 75-38 "Nonlinear Response Spectra for Probabilistic Seismic Design and Damage Assessment of Reinforced ConcreteStructures," by M. Murakami and J. Penzien -1975 (PB 259 530)A05

EERC 75-39 "Study of a Method of Feasible Directions for Optimal Elastic Design of Frame Structures Subjected to J:~arth,·

quake Loading," by N.D. Walker and K.S. Pister -1975 (PB 257 781)A06

EERC 75-40 "An Alternative Representation of the Elastic-Viscoelastic Analogy," by G. Dasgupta and J.L. Sackman-1975(PB 252 173) A03

EERC 75-41 "Effect of Multi-Directional Shaking on Liquefaction of Sands," by H.B. Seed, R. Pyke and G.R. Martin -1975(PB 258 781)A03

EERC 76-1 "Strength and Ductility Evaluation of Existing Low-Rise Reinforced Concrete Buildings - Screening Method," byT. Okada and B. Bresler -1976 (PB 257 906)All

EERC 76-2 "Experimental and Analytical Studies on the Hysteretic Behavior of Reinforced Concrete Rectangular andT-Beams," by S.-Y.M. Ma, E.P. Popov and V.V. Bertero -1976 {PB 260 843)A12

EERC 76-3 "Dynamic Behavior of a Multistory Triangular-Shaped Building," by J. Petrovski, R.M. Stephen, E. Gartenbaumand J.G. Bouwkamp-1976 (PB 273 279)A07

EERC 76-4 "Earthquake Induced Deformations of Earth Dams," by N. Serff, H.B. Seed, F.I. Makdisi & C.-Y. Chang - 1976(PB 292 065)A08

EERC-6

EERC 76-5 "Analysis and Design of Tube-Type Tall Building Structures," by H. de Clercq and G.H. Powell-1976 (PB 252220)AIO

EERC 76-6 "Time and Frequency Domain Analysis of Three-Dimensional Ground Motions, San Fernando Earthquake," by T. Kuboand J. Penzien {PB 260 556)All

EERC 76-7 "Expected Performance of Uniform Building Code Design Masonry Structures," by R.L. Ma~es, Y. Ornote, S.W. Chenand R.W. Clough -1976 (PB 270 098)A05

EERC 76-8 "Cyclic Shear Tests of Maso'nry Piers, Volume 1 - Test Results," by R.L. Mayes, Y. Ornote, R.W.Clough - 1976 (PB 264 424) A06

EERC 76-9 "A Substructure Method for Earthquake Analysis of Structure - Soil Interaction," by J .A. Gutierrez andA.K. Chopra - 1976 {PB 257 783)A08

EERC 76-10 "Stabilization of Potentially Liquefiable Sand Deposits using Gravel Drain Systems," by H.B. Seed andJ.R. Booker - 1976 (PB 258 820)A04

EERC 76-11 "Influence of Design and Analysis Assumptions on Computed Inelastic Response of Moderately Tall Frames," byG.H. Powell and D.G. Row-1976 (PB 271 409)A06

EERC 76-12 "Sensitivity Analysis for Hysteretic Dynamic Systems: Theory and Applications," by D. Ray, K.S. Pister andE. Polak - 1976 (PB 262 859)A04

EERC 76-13 "Coupled Lateral Torsional Response of Buildings to Ground Shaking," by C.L. Kan and A.K. Chopra ­1976 (PB 257 907)A09

EERC 76-14 "Seismic Analyses of the Banco de America," by V.V. Bertero, S.A. Mahin and J.A. Hollings - 1976

EERC 76-15 "Reinforced Concrete Frame 2: Seismic Testing and Analytical Correlation," by R. W. Clough andJ. Gidwani - 1976 {PB 261 323)A08

EERC 76-16 "Cyclic Shear Tests of Masonry Piers, Volume 2 - Analysis of Test Results," by R.L. Mayes, Y. Ornoteand R.W. Clough - 1976

EERC 76-17 "Structural Steel Bracing Systems: Behavior Under cyclic Loading," by E.P. Popov, K. Takanashi andC.W. Roeder - 1976 (PB 260 715)A05

EERC 76-18 "Experimental Model Studies on Seismic Response of High Curved Overcrossings," by D. Williams andW.G. Godden - 1976 (PB 269 548)A08

EERC 76-19 "Effects of Non-Uniform Seismic Disturbances on the Durnbarton Bridge Replacement Structure," byF. Baron and R.E. Hamati - 1976 {PB 282 981)A16

EERC 76-20 "Investigation of the Inelastic Characteristics of a Single Story Steel Structure Using SystemIdentification and Shaking Table Experiments," by V.C. Matzen and H.D. McNiven - 1976 (PB 258 453)A07

EERC 76-21 "Capacity of Columns with Splice Imperfections," by E.P. Popov, R.M. Stephen and R. Philbrick - 1976(PB 260 378) A04

EERC 76-22 "Response of the Olive View Hospital Main Building during the San Fernando Earthquake," by S. A. Mahin,V.V. Bertero, A.K. Chopra and R. Collins - 1976 (PB 271 425)A14

EERC 76-23 "A Study on the Major Factors Influencing the Strength of Masonry Prisms," by N.M. Mostaghel,R.L. Mayes, R. W. Clough and S.W. Chen - 1976 (Not published)

EERC 76-24 "GADFLEA - A Computer Program for the Analysis of Pore Pressure Generation and Dissipation duringCyclic or Earthquake Loading," by J.R. Booker, M.S. Rahman and H.B. Seed - 1976 {PB 263 947)A04

EERC 76-25 "Seismic Safety Evaluation of a R!C School Building," by B. Bresler and J. Axley - 1976

EERC 76-26 "Correlative Investigations on Theoretical and Experimental Dynamic Behavior of a Model BridgeStructure," by K. Kawashima and J. Penzien - 1976 (PB 263 388)All

EERC 76-27 "Earthquake Response of Coupled Shear Wall Buildings," by T. Srichatrapimuk - 1976 (PB 265 157)A07

EERC 76-28 "Tensile Capacity of Partial Penetration Welds," by E.P. Popov and R.M. Stephen - 1976 (PB 262 899) A03

EERC 76-29 "Analysis and Design of Numerical Integration Methods in Structural Dynamics," by H.M. Hilber - 1976(PB 264 410)A06

EERC 76-30 "Contribution of a Floor System to the Dynamic Characteristics of Reinforced Concrete Buildings," byL.E. Malik and V.V. Bertero - 1976 (PB 272 247)A13

EERC 76-31 "The Effects of Seismic Disturbances on the Golden Gate Bridge," by F. Baron, M. Arikan and R.E. Harnati ­1976 (PB 272 279)A09

EERC 76-32 "Infilled Frames in Earthquake Resistant Construction," by R.E. Klingner and V.V. Bertero - 1976(PB 265 892)A13

EERC-7

UCB/EERC-77/01 "PLUSH - A Computer Program for Probabilistic Finite Element Analysis of Seismic Soil-Structure Inter­action," by M.P. Romo Organista, J. Lysmer and H.B. Seed - 1977

UCB/EERC-77/02 "Soil-Structure Interaction Effects at the Humboldt Bay Power Plant in the.Ferndale Earthquake of June7, 1975," by J.E. Valera, H.B. Seed, C.F. Tsai and J. Lysmer - 1977 (PB 265 795)A04

UCB/EERC-77/03 "Influence of Sample Disturbance on Sand Response to Cyclic Loading," by K. Mori, H.B. Seed and C.K.Chan - 1977 (PB 267 352)A04

UCB/EERC-77 /04 "Seismological Studies of Strong Motion Records," by J. Shoja-Taheri - 1977 (PB 269 655)AIO

UCB/EERC-77/05 "Testing Facility for Coupled-Shear Walls," by L. Li-Hyung, V.V. Bertero and E.P. Popov - 1977

UCB/EERC-77/06 "Developing Methodologies for Evaluating the Earthquake Safety of Existing Buildings," by No.1 _B. Bresler; No.2 - B. Bresler, T. Okada and D. Zisling; No.3 - T. Okada and B. Bresler; No.4 - V.V.Bertero and B. Bresler - 1977 (PB 267 354)A08

UCB/EERC-77/07 "A Literature Survey - Transverse Strength of Masonry Walls," by Y. Ornote, R.L. Mayes, S.W. Chen andR.W. Clough - 1977 (PB 277 933)A07

UCB/EERC-77/08 "DRAIN-TABS: A Computer Program for Inelastic Earthquake Response of Three Dimensional Buildings," byR. Guendelman-Israel and G.H. Powell - 1977 (PB 270 693)A07

UCB/EERC-77/09 "SUBWALL: A Special Purpose Finite Element Computer Program for Practical Elastic Analysis and Designof Structural Walls with Substructure Option," by D.Q. Le, H. Peterson and E.P. Popov - 1977(PB 270 567)A05

UCB/EERC-77/10 "Experimental Evaluation of Seismic Design Methods for Broad Cylindrical Tanks," by D. P. Clough(PB 272 280) A13

UCB/EERC-77/11 "Earthquake Engineering Research at Berkeley - 1976," - 1977 (PB 273 507)A09

UCB/EERC-77/12 "Automated Design of Earthquake Resistant Multistory Steel Building Frames," by N.D. Walker, Jr. - 1977(PB 276 526)A09

UCB/EERC-77/13 "Concrete Confined by Rectangular Hoops Subjected to Axial Loads," by J .. Vallenas, V.V. Bertero andE.P. Popov - 1977 (PB 275 165)A06

UCB/EERC-77/14 "Seismic Strain Induced in the Ground During Earthquakes," by Y. Sugimura - 1977 (PB 284 201)A04

UCB/EERC-77/15 "Bond Deterioration under Generalized Loading," by V.V. Bertero, E.P. Popov and S. Viwathanatepa - 1977

UCB/EERC-77/16 "Computer Aided Optimum Design of Ductile Reinforced Concrete Moment Resisting Frames," by S.W.Zagajeski and V.V. Bertero - 1977 (PB 280 137)A07

UCB/EERC-77/17 "Earthquake Simulation Testing of a Stepping Frame with Energy-Absorbing Devices," by J.M. Kelly andD.F. Tsztoo - 1977 (PB 273 506)A04

UCB/EERC-77/18 "Inelastic Behavior of Eccentrically Braced Steel Frames under Cyclic Loadings," by C.W. Roeder andE.P. Popov - 1977 (PB 275 526)A15

UCB/EERC-77/19 "A Simplified Procedure for Estimating Earthquake-Induced Deformations in Dams and Embankments," by F.I.Makdisi and H.B. Seed - 1977 (PB 276 820)A04

UCB/EERC-77/20 "The Performance of Earth Dams during Earthquakes," by H.B. Seed, F.I. Makdisi and P. de Alba - 1977(PB 276 821) A04

UCB/EERC-77/21 "Dynamic Plastic Analysis Using Stress Resultant Finite Element Formulation," by P. Lukkunapvasit andJ.M. Kelly - 1977 (PB 275 453)A04

UCB/EERC-77/22 "Preliminary Experimental Study of Seismic Uplift of a Steel Frame," by R.W. Clough and A.A. Huckelbridge1977 (PB 278 769)A08

UCB/EERC-77/23 "Earthquake Simulator Tests of a Nine-Story Steel Frame with Columns Allowed to Uplift," by A.A.Huckelbridge - 1977 (PB 277 944)A09

UCB/EERC-77/24 "Nonlinear Soil-Structure Interaction of Skew Highway Bridges," by M. -C. Chen and J .. Penzien - 1977(PB 276 176)A07

UCB/EERC-77/25 "Seismic Analysis of an Offshore Structure Supported on Pile Foundations," by D.D.-N. Liou and J. Penzien1977 (PB 283 180)A06

UCB/EERC-77/26 "Dynamic Stiffness Matrices for Homogeneous Viscoelastic Half-Planes," by G. Dasgupta and A.K. Chopra ­1977 (PB 279 654)A06

UCB/EERC-77/27 "A Practical Soft Story Earthquake Isolation System," by J.M. Kelly, J.M. Eidinger and C.J. Derham ­1977 (PB 276 814)A07

UCB/EERC-77/28 "Seismic Safety of Existing Buildings and Incentives for Hazard Mitigation in San Francisco: AnExploratory Study," by A.J. Meltsner - 1977 (PB 281 970)A05

UCB/EERC-77/29 "Dynamic Analysis of Electrohydraulic Shaking Tables," by D. Rea, S. Abedi-Hayati and Y. Takahashi1977 (PB 282 569)A04

UCB/EERC-77/30 "An Approach for Improving Seismic - Resistant Behavior of Reinforced Concrete Interior Joints," byB. Galunic, V.V. Bertero and E.P. Popov - 1977 (PB 290 870)A06

UCB/EERC-78/01

UCB/EERC-78/02

UCB/EERC-78/03

UCB/EERC-78/04

UCB/EERC-78/05

UCB/EERC-78/06

UCB/EERC-78/07

UCB/EERC-78/08

UCB/EERC-78/09

UCB/EERC-78/10

UCB/EERC-78/11

UCB/EERC-78/12

VCB/EERC-78/l3

VCB/EERC-78/14

UCB/EERC-78/15

UCB/EERC-78/16

EERC-8

liThe Development of Energy-Absorbing Devices for Aseismic Base Isolation Systems," by J.M. Kelly andD.F. Tsztoo - 1978 (PB 284 978)A04

"Effect of Tensile Prestrain on the Cyclic Response of Structural Steel Connections, by J.G. Bouwkampand A. Mukhopadhyay - 1978

"Experimental Results of an Earthquake Isolation System using Natural Rubber Bearings," by J.M.Eidinger and J.M. Kelly - 1978 {PB 281 686)A04

"Seismic Behavior of Tall Liquid Storage Tanks," by A. Niwa - 1978 (PB 284 017)Ai4

"Hysteretic Behavior of Reinforced Concrete Columns Subjected to High Axial and Cyclic Shear Forces,"by S.W. Zagajeski, V.V. Bertero and J.G. Bouwkamp - 1978 (PB 283 858)A13

"Inelastic Beam-Column Elements for the ANSR-I Program," by A. Riahi, D.G. Rowand G.H. Powell - 1978

"Studies of Structural Response to Earthquake Ground 1.lotion," by O.A. Lopez and A.K. Chopra - 1978(PB 282 790)A05

"A Laboratory Study of the Fluid-Structure Interaction of Submerged Tanks and Caissons in Ea,'thquakes,"by R.C. Byrd - 1978 (PB 284 957)A08

"Model for Evaluating Damageability of Structures," by I. Sakamoto and B. Bresler - 1978

lfSeisrnic Performance of Nonstructural and Secondary Structural Elements," by I. Sakamoto - 1978

"Mathematical ~·1odelling of Hysteresis Loops for Reinforced Concrete Columns," by S. Nakata, T. Sprouland J. Penzien - 1978

"Damageability in Existing Buildings," by T. Blejwas and B. Bresler - 1978

"Dynamic Behavior of a Pedestal Base Multistory Building," by R.M. Stephen, E.L. Wilson, J.G. Bouwkampand M. Button - 1978 (PB 286 650)A08

IlSe ismic Response of Bridges - Case Studies," by R.. A. Imbsen, V. Nutt and J. Penzien - 1978(PB 286 503)A10

"A Substructure Technique for Nonlinear Static and Dynamic Analysis," by D.G. Rowand G.H. Powell ­1978 (PB 288 077)A1D

··Seismic Risk Studies for San Francisco and for the Greater San Francisco Bay Area, II by C.S. Oliveira ­1978

UCB/EERC-78/17 "Strength of Timber Roof Connections SUbjected to Cyclic Loads," by P. Gulkan, R.L. Mayes and R.W.Clough - 1978

UCB/EERC-78/18 "Response of K-Braced Steel Frame Models to Lateral Loads," by J.G. Bouwkamp, R.M. Stephen andE.P. Popov - 1978

UCB/EERC-78/19 "Rational Design Methods for Light Equipment in Structures Subjected to Ground Motion," byJ.L. Sackman and J.M. Kelly - 1978 {PB 292 357)A04

UCB/EERC-78/2D "Testing of a Wind Restraint for Aseismic Base Isolation," by J.M. Kelly and D.E. Chitty - 1978(PB 292 833) AD3

UCB/EERC-78/21 "APOLLO - A Computer Program for the Analysis of Pore Pressure Generation and Dissipation in HorizontalSand Layers During Cyclic or Earthquake Loading," by P.P. Martin and H.B. Seed - 1978 (PB 292 835)AD4

UCB/EERC-78/22 "Optimal Design of an Earthquake Isolation System," by M.A. Bhatti, K.S. Pister and E. Polak - 1978{PB 294 735)A06

UCB/EERC-78/23 "MASH - A Computer Program for the Non-Linear Analysis of Vertically Propagating Shear Waves inHorizontally Layered Deposits," by P.P. Martin and H.B. Seed - 1978 (PB 293 lDl)AD5

UCB/EERC-78/24 "Investigation of the Elastic Characteristics of a Three Story Steel Frame Using System Identification,"by I. Kaya and H.D. McNiven - 1978

UCB/EERC-78/25 "Investigation of the Nonlinear Characteristics of a Three-Story Steel Frame Using SystemIdentification," by I. Kaya and H. D. McNiven - 1978

UCB/EERC-78/26 "Studies of Strong Ground Motion in Taiwan," by Y.M. Hsiung, B.A. Bolt and J. Penzien - 1978

UCB/EERC-78/27 "Cyclic Loading Tests of Masonry Single Piers: Volume 1 - Height to width Ratio of 2," by P.A. Hidalgo,R.L. Mayes, H.D. McNiven and R.W. Clough - 1978

UCB/EERC-78/2B "Cyclic Loading Tests of Masonry Single Piers: Volume 2 - Height to Width Ratio of 1," by S.-W.J. Chen,P.A. Hidalgo, R.L. Mayes, R.W. Clough and H.D. McNiven - 1978

UCB/EERC-78/29 "Analytical Procedures in Soil Dynamics," by J. Lysmer - 1978

EERC-9

UCB/EERC-79/0l "Hysteretic Behavior of Lightweight Reinforced ConcreteBeam-Column Subassernblages," by B. Forzani, E.P. Popov,and V.V. Bertero - 1979

UCB/EERC-79/02 "The Development of a Mathematical Model to Predict theFlexural Response of Reinforced Concrete Beams to CyclicLoads, Using System Identification," by J.F. Stanton andH.D. McNiven - 1979

UCB/EERC-79/03 "Linear and Nonlinear Earthquake Response of SimpleTorsionally Coupled Systems," by C.L. Kan andA.K. Chopra - 1979

UCB/EERC-79/04 "A Mathematical Model of Masonry for Predicting ItsLinear Seismic Response Characteristics," by Y. Mengiand H.D. McNiven - 1979

UCB/EERC-79/05 "Mechanical Behavior of Lightweight. Concrete Confinedby Different Types of Lateral Reinforcement," byM.A. Manrique, V.V. Bertero and E.P. Popov - 1979

UCB/EERC-79/06 "Static Tilt Tests of a Tall Cylindrical Liquid StorageTank," by R.W. Clough and A. Niwa - 1979

UCB/EERC-79/07 "The Design of Steel Energy Absorbing Restrainers andTheir Incorporation Into Nuclear Power Plants forEnhanced Safety: Volume 1 - Summary Report," byP.N. Spencer, V.F. Zackay, and E.R. Parker - 1979

UCB/EERC-79/08 "The Design of Steel Energy Absorbing Restrainers andTheir Incorporation Into Nuclear Power Plants forEnhanced Safety: Volume 2 - The Development of Analysesfor Reactor System Piping," "Simple Systems" byM.C. Lee, J. Penzien, A.K. Chopra, and K. Suzuki"Complex Systems" by G.H. Powell, E.L. Wilson,R.W. Cloughand D.G. Row - 1979

UCB/EERC-79/09 "The Design of Steel Energy Absorbing Restrainers andTheir Incorporation Into Nuclear Power Plants forEnhanced Safety: Volume 3 - Evaluation of CommericalSteels," by W.S. Owen, R.H.N. Pelloux, R.O. Ritchie,M. Faral, T. Ohhashi, J. Toplosky, S.J. Hartman, V.F.Zackay, and E.R. Parker - 1979

UCB/EERC-79/l0 "The Design of Steel Epergy Absorbing Restrainers andTheir Incorporation Into Nuclear Power Plants forEnhanced Safety: Volume 4 - A Review of Energy-AbsorbingDevices," by J.M. Kelly and H.S. Skinner - 1979

UCB/EERC-79/ll "Conservatism In Summation Rules for Closely SpacedModes," by J.M. Kelly and J.L. Sackman - 1979

UCB/EERC-79/l2

UCB/EERC-79/13

UCB/EERC-79/l4

UCB/EERC-79/l5

UCB/EERC-79/l6

UCB/EERC-79/l7

UCB/EERC-79/1B

UCB/EERC-79/l9

UCB/EERC-79/20

UCB/EERC-79/2l

UCB/EERC-79/22

UCB/EERC-79/23

UCB/EERC-79/24

UCB/EERC-79/25

EERC-10

"Cyclic Loading Tests of Masonry Single Piers Volume3 - Height to Width Ratio of 0.5," by P.A. Hidalgo,R.L. Mayes, H.D. McNiven and R.W. Clough - 1979

"Cyclic Behavior of Dense Coarse-Grained Materials inRelation to the Seismic Stability of Dams," by N.G.Banerjee, H.B. Seed and C.K. Chan - 1979

"Seismic Behavior of Reinforced Concrete Interior Beam-ColumnSubassemblages," by S. Viwathanatepa, E.P. Popov andV.V. Bertero - 1979

"Optimal Design of Localized Nonlinear Systems withDual Performance Criteria Under Earthquake Excitations,"by M.A. Bhatti - 1979

"OPTDYN - A General Purpose Optimization Program forProblems with or without Dynamic Constraints," byM.A. Bhatti, E. Polak and K.S. Pister - 1979

"ANSR-II, Analysis of Nonlinear Structural Response,Users Manual," by D.P. Mondkar and G.H. Powell - 1979

"Soil Structure Interaction in Different SeismicEnvironments," A. Gomez-Masso, J. Lysmer, J.-C. Chenand H.B. Seed - 1979

"ARMA Models for Earthquake Ground Motions," by M.K.Chang, J.W. Kwiatkowski, R.F. Nau, R.M. Oliver andK.S. Pister - 1979

"Hysteretic Behavior of Reinforced Concrete StructuralWalls," by J.M. Vallenas, V.V. Bertero and E.P.Popov - 1979

"Studies on High-Frequency Vibrations of Buildings I:The Column Effects," by J. Lubliner - 1979

"Effects of Generalized Loadings on Bond Reinforcing BarsEmbedded in Confined Concrete Blocks," by S. Viwathanatepa,E.P. Popov and V.V. Bertero - 1979

"Shaking Table Study of Single-Story Masonry Houses,Volume 1: Test Structures 1 and 2," by P. GUlkan,R.L. Mayes and R.W. Clough - 1979

"Shaking Table Study of Single-Story Masonry Houses,Volume 2: Test Structures 3 and 4," by P. GUlkan,R.L. Mayes and R.W. Clough - 1979

"Shaking Table Study of Single-Story Masonry Houses,Volume 3: Summary, Conclusions and Recommendations,"by R. W. Clough, R. L. Mayes and P. Gulkan - 1979

UCB/EERC-79/26

UCB/EERC-79/27

UCB/EERC-79/28

UCB/EERC-79/29

UCB/EERC-79/30

UCB/EERC-79/3l

UCB/EERC-79/32

UCB/EERC-79/33

UCB/EERC-79/34

UCB/EERC-80/0l

UCB/EERC-80/02

UCB/EERC-80/03

UCB/EERC-80/04

UCB/EERC-80/0S

UCB/EERC-80/06

EERC-ll

"Recommendations for a U.S.-Japan Cooperative ResearchProgram Utilizing Large-Scale Testing Facilities," byU.S.-Japan Planning Group - 1979

"Earthquake-Induced Liquefaction Near Lake Amatitlan,Guatemala," by H.B. Seed, I. Arango, C.K. Chan,A. Gomez-Masso and R. Grant de Ascoli - 1979

"Infill Panels: Their Influence on Seismic Response ofBuildings," by J.W. Axley and V.V. Bertero - 1979

"3D Truss Bar Element (Type 1) for the ANSR-II Program,"by D.P. Mondkar and G.H. Powell - 1979

"2D Beam-Column Element (Type 5 - Parallel ElementTheory) for the ANSR-II Program," by D.G. Row, G.H. Powelland D. P. Mondkar

"3D Beam-Column Element (Type 2 - Parallel ElementTheory) for the ANSR-II Program," by A. Riahi, G.H. Powelland D.P. Mondkar - 1979

"On Response of Structures to Stationary Excitation," byA. Der Kiureghian - 1979

"Undisturbed Sampling and Cyclic Load Testing of Sands,"by S. Singh, H.B. Seed and C.K. Chan - 1979

"Interaction Effects of Simultaneous Torsional andCompressional Cyclic Loading of Sand," by P.M. Griffinand W.N. Houston - 1979

"Earthquake Response of Concrete Gravity Dams IncludingHydrodynamic and Foundation Interaction Effects," byA.K. Chopra, P. Chakrabarti and S. Gupta - 1980

"Rocking Response of Rigid Blocks to Earthquakes,"by C.S. Yim, A.K. Chopra and J. Penzien - 1980

"Optimum Inelastic Design of Seismic-Resistant ReinforcedConcrete Frame Structures," by S.W. Zagajeski and V.V.Bertero - 1980

"Effects of Amount and Arrangement of Wall-PanelReinforcement on Hysteretic Behavior of ReinforcedConcrete Walls," by R. Iliya and V.V. Bertero - 1980

"Shaking Table Research on Concrete Dam Models," byR.W. Clough and A. Niwa - 1980

"Piping With Energy Absorbing Restrainers: ParameterStudy on Small Systems," by G.H. Powell, C. Oughourlianand J. Simons - 1980

UCB/EERC-80/07

UCB/EERC-80/0B

UCB/EERC-80/09

UCB/EERC-80/10

UCB/EERC-80/11

UCB/EERC-80/12

UCB/EERC-80/13

UCB/EERC-80/14

UCB/EERC-80/15

EERC-l2

"Inelastic Torsional Response of Structures Subjectedto Earthquake Ground Motions," by Y. Yamazaki - 1980

"Study of X-Braced Steel Frame Structures UnderEarthquake Simulation," by Y. Ghanaat - 1980

"Hybrid Modelling of Soil-Structure Interaction," byS. Gupta, T.W. Lin, J. Penzien and C.S. Yeh - 1980

"General Applicability of a Nonlinear Model of a OneStory Steel Frame," by B.I. Sveinsson and H. r'1cNiven - 1980

"A Green-Function Method for Wave Interaction with aSubmerged Body," by W. Kioka - 19BO

"Hydrodynamic Pressure and Added Mass for AxisYmmetricBodies," by F. Nilrat - 1980

"Treatment of Non-Linear Drag Forces Acting on OffshorePlatforms," by B. V. Dao and J. Penzien - 1980

"2D Plane/AxiSYmmetric Solid Element (Type 3 - Elasticor Elastic-Perfectly Plastic) for the ANSR-II Program,"by D.P. Mondkar and G.H. Powell - 1980

"A Response Spectrum Method for Random Vibrations," byA. Der Kiureghian - 1980