COLLAPSE OF A MODEL FOR DUCTILE REINFORCED CONCRETE FRAMES UNDER EXTREME EARTHQUAKE MOTIONS

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 8, 117-144 (1980)

COLLAPSE OF A MODEL FOR DUCTILE REINFORCED CONCRETE FRAMES UNDER EXTREME EARTHQUAKE

MOTIONS

HARUO TAKIZAWA~

Faculty of Engineering, Hokkaido University, Sapporo, Hokkaido, Japan

AND

PAUL c. JENNINGS~

California Institute of Technology, Pasadena, California, U.S.A

SUMMARY A mathematical formulation is presented for modelling the dynamic process of failure of a class of ductile, moment- resisting, reinforced concrete (R/C) frame buildings when subjected to intense earthquake motion. The formulation includes the geometrically non-linear term that accounts for the destabilizing action of gravity. In many cases of practical interest, in which the structures have strong columns and weak girders, the employed method ofsynthesizing the restoring force properties can provide a satisfactory description of the structural deformation at large deflections. By modelling approximately the effects of gravity, cracking, yielding and degradation of stiffness, the study is intended to aid the understanding of the process of failure in this type of ductile R/C structure, and to relate the mechanics of collapse to characteristics of the excitation. The collapse of R/C frames having strong girders and weak columns, which can develop sway mechanisms at a single storey, is not considered in this study.

Special examination is made of the capacity to resist short-duration motions consisting of a few pulses versus the capacity to resist motions of longer duration. For the class of structures modelled, the results indicate an extremely low destructive capability associated with short-duration motions, even when they have very high accelerations. The application in research of a two-parameter characterization of the severity of ground motion in terms of intensity and duration is also examined.

INTRODUCTION

When a structure is shaken hard enough by strong ground motion to produce significant yielding and damage, the important question is not how far into the yielding range the response has developed, but how close is the structure to collapse. The reliable description of the damage and ultimate failure of reinforced concrete (R/C) buildings under the combined actions of earthquake motion and gravity is, however, a very difficult task. One approach to modelling the inelastic and hysteretic behaviour begins with the direct use of empirical results derived from laboratory tests on structural elements. This approach can be developed into a generalized method of member-by-member analysis of the total structure. Important features in the non-linear response may also be described by use of a much simpler model, provided that the mechanism of overall structural deformation is adequately reflected in the modelling. This requires specifying the mode of storey drifts and incorporating the element properties into the simplified model in a manner consistent with the kinematics of yielding (e.g. Reference 18). Depending upon the associated degrees-of-freedom (DOF), this approach results in an ‘equivalent’ 1-DOF or multi-DOF model of the large deflection response of the total system.

t Associate Professor of Structural Engineering; formerly, Research Fellow in Earthquake Engineering, California Institute of Technology. $ Professor of Civil Engineering and Applied Mechanics.

0098-8847/80/0208-0117$01.OO @ 1980 by John Wiley & Sons, Ltd.

117

Received 13 April 1978 Revised 27 September 1979

118 H. TAKIZAWA AND P. C. JENNINGS

Frames of practical interest often have collapse mechanisms controlled by hinges in the girders or have a hybrid collapse mechanism controlled by hinges in both girders and columns. For multi-storey buildings of these types which are examined herein, modelling the structure as a shear beam can lead, in the yielding region of response, to a seriously unreliable evaluation of response l9 Another common type ofductile R/C frame is composed of strong girders and weak columns. This second type of structure, not considered in this

0 : y i e l d hinges

0 : ro ta t iona l s

structure ( b )

( C 1 Figure 1. Equivalent single-degree-of-freedom formulation of a structural system. The forcesi and pi are the lateral and vertical loads,

respectively, at the ith storey

study, tends to develop shear mechanisms at one or more levels in the structure and can be modelled more appropriately by a shear beam.

The major features of the static restoring force relationships of R/C frames are the marked softening due to cracking at relatively low stress levels and the trend toward degrading stiffness during cyclic loading at larger amplitudes. Thus, a modelling of the restoring force which adequately accounts for the effects of cracking, yielding and stiffness degradation is the minimum requisite in idealizing the behaviour of R/C structures at

DUCTILE REINFORCED CONCRETE FRAMES 119

large deflections. The 'degrading trilinear' formulation' is among the simplest models meeting this requirement.

Usually the earthquake response of R/C buildings is examined by disregarding the destabilizing action of gravity, which implicitly supposes limited deflections. This approach will be satisfactory for most practical purposes, since even seriously damaging response will be restricted to relatively small lateral deflections. However, as the response of a sufficiently ductile or damaged system progresses far into the yielded region, the influence of gravity eventually becomes dominant. What then occurs is a structural destabilization due to vertical compressive loads acting through the lateral drifts; this has been termed the P-A effect.

The literature on the effects of gravity on structural response dates back to early work by Ruge and Jacobsen, beginning in the 1930's. Recent work includes an examination of the effect of gravity upon hysteretic structures, reported by Jennings and Husid," and by Husid.', By electronic analogue simulation, Adu' examined the collapse of 1-DOF and 2-DOF elastoplastic systems under stationary random excitation, while the case of a sliptype hysteresis modelling of braced structures can be found in the study by Sun,I6 and by Sun et a/." References 28 and 23 are also along the lines of these investigations. Additional studies of the current problem have included the effects of gravity or non-linear geometry in member-by-member structural

The present study examines the ultimate capacity of a class of R/C frames under the combined action of intense ground shaking and gravity, by means of an approximate model of the collapse mechanism. The structural model employed is the equivalent 1-DOF system characterized by the degrading trilinear restoring force. Associated with a 1-DOF yield hinge mechanism such as shown in Figure l(a), the equivalent 1-DOF idealization is relevant to the class of R/C buildings which have the common property of strong columns and weak girders. Even though the model is necessarily somewhat restricted and should be applied with caution to practical problems, it is sufficiently general to illustrate important trends in the response of this type of structure.

' 9 22

EQUIVALENT 1-DOF STRUCTURAL SYSTEM

The first concern is the inclusion of the effect of gravity into the overall structural behaviour described by an 'equivalent' 1-DOF system. The 1-DOF idealization is equivalent to a structural model as illustrated in Figure l(b), in which the building frame is linked to an auxiliary rigid system that prevents the storey-drifts from deviating from a specified pattern. When including the gravity-effect term in the equation of motion, an additional assumption is used to avoid treating the full complexity of non-linear geometry in the deformed configuration:leach component of the structure is assumed to consist of a rigid bar with two rotational springs at its ends adjacent to the nodal points, as shown in the Figure l(b). Location of the rotational springs at critical sections would provide a better idealization of the deformations, but this results in rather unwieldy kinematic relations among the rotations, due to the effects of the so-called rigid zones.

By use of the storey-height vector, {h} = Lh,, h,, ..., h,]', in which n is the number of storeys, and by referring to Figure l(c), the displacements of storeys are specified in the following forms.

- - -

- (u} = Lul,u2 ,..., u,]' = sinR{h)

-

{v} = Lo', ~ 2 , ..., u,JT = (COS R - 1) {h}

in which {u} and {v} are the vectors of horizontal and vertical storey-displacements, respectively, measured relative to the support. Figure l(c) also defines the horizontal and vertical storey force vectors, (f} and (p};

{f} = Lfi,fi,...,f"J', {P} = IPi,Pz,...,PJJT. Application of the principle of virtual work to the current system, using a virtual rotation 6R, leads to the equilibrium requirement :

cos RLf] {h} -sin RLp] {h} = M .


The generalized force M is defined by associating it with the generalized deformation R in the virtual internal- work relation of the total system, 6E = M 6R. Furthermore, since the horizontal and vertical forces are the inertial and gravity forces acting on the structure, these are represented in terms of the translational angle R as follows.

{f} = - [m] ({u} + u,{e}) = - [m] (cos RR-sin RR2 {h} + u,{e})

{p} = -CmI({v}+g+vo{e}) = - [ m l ( - s i n ~ R - ~ o s ~ R 2 { h } + g + i i , {el).

In this equation [m] is the inertia matrix of the structure; {e} equals [l, 1, ..., 1IT; ii, is the horizontal acceleration at the support of the building; uo denotes the vertical acceleration at the same point; andg is the gravity constant. The equation of motion then takes the form

L ~ J [m] {h) R + L ~ J [m] {el (ti, cos R - g + ii, sin R) + M = 0.

By noting certain equivalent relations, the equation can be cast into a more convenient form. The equation becomes

A + A = -uocos(A/H)+(g+u0)sin(A/H)

with the definitions

where A is the equivalent lateral drift along the circular arc;

- MIH - M A = - LhJ [m] {e} (P’/P”)(cW/g)

in which A equals the associated restoring force, expressed as an acceleration (c Wequals the total weight of the structure);

H = h,/P with H representing the equivalent total height of the structure; and

The betas are modification factors which depend upon the relative distribution of storey-mass and storey- height. [$’ = 3n/(2n + 1) = 1.0 - 1.5 and B” = 2n/(n + 1) = 1.0 - 2.0, for a uniform distribution of storey-mass and interstorey-height.]

This equation of motion has the same form as that used in describing simple 1-DOF systems.” In terms of the equivalent 1-DOF lateral drift, A, measured along the circular arc, the drift at each storey is specified by

d i = hi R = $’ A/(h,/hi) [ N $’(A/n)] - - -

with hi = hi - hi- I being the interstorey height at the ith storey As the mode of storey-drifts is specified, the dynamic characteristics of the overall structure can be

condensed into a single equivalent force. The synthesis of the restoring force is made by considering the non- linear hysteretic properties of elements by empirical means. 15, 19* 2 5 This approach yields the parameters (yield and crack moments, reduced stiffness at yield point, etc.) at critical sections for antisymmetric deformation. After having extrapolated the yield moments to the nodal points, the incremental internal energy of the total system during yielding (neglecting damping) can be expressed by

dE = M,dR


The notation M y stands for a partial sum of the yield moments, the summation being performed over the plastic hinges participating in the 1-DOF kinematic hinge-mechanism. An alternative expression at yielding is

in which A , is the yield capacity of the equivalent system. The synthesis of the restoring force for static loading generally results in a curvilinear relation on a diagram

of A versus A. In most cases of practical interest, however, the skeleton curve may be approximated by use of a piecewise linear function, which may then be associated with a model for degrading hysteresis.’g One of the simplest relations is the trilinear curve [Figure 2(a)] which requires two parameters : A , (equivalent crack

I A S ( = A y a S ) c

skeleton force curvebin proper t ies r e s t o r i n g / grav i t y - e t t e c t to rce . A’ I

! Q i = 4a2/T: I

= g s i n (A/H) 2 (g/H)A

Figure 2. Skeleton curves and gravity terms for hysteretic equivalent systems : (a) trilinear skeleton curve; and (b) quadrilinear skeleton curve

capacity) and u, (secant stiffness reduction factor, defined at the yield point). A more involved quadrilinear function permits idealization of the additional effects of crushing and spalling [Figure 2(b)]. Several procedures for evaluating the necessary parameters can be found in the literat~re’~. *’ and are not repeated here.

When applying the degrading trilinear hysteresis model [Figure 2(a)], the dynamic properties are characterized by five independent parameters; the system parameters usually adopted are T,( = 2n/Q,), A,, A,(a, = A,/A,), a, and re. (re is the equivalent damping factor.) Important parameters for this study can be then identified by introducing the dimensionless expressions :

A4 = A ( W Y 9 4t) = A(t)/A,


and by defining the horizontal excitation to have a peak value of unity :

4t) = &(t)/Amax

in which A,,, is the peak acceleration of the ground motion uo(t). Also, the dimensionless parameter

a* = (g/H)/!x is introduced which is the ratio of the initial gradient of the force of gravity to the elastic stiffness Of [see Figure qa)]. Then, neglecting vertical ground motion, the normalized drift p is seen to be a function of the following mutually independent factors :

P = At; T,, re, a,, ay, I,e(t), a,, Ay/g).

In this expression, the notation I = Amax/Ay represents the relative strength of the excitation. Consideration of gravity requires the last two factors which can be reduced to the single parameter a, if the first-order approximation to the geometrically non-linear terms is used.

The measure of the gravity effect, a,, may be related to fundamental parameters of the structure. With the average interstorey height h = h,/n taken as 375 cm,

T, = (W Jd Jh Jag JWP) = 3-89 Jag JWP)

K,/C w = (p/p")h;l a;' = 2-67(p/p")/(ag x 103)

and

again with hl = 375 cm. The first relation indicates the relation to the elastic period T, of an n-storey system, while the second factor is a measure of the relative flexibility of the total structure (K, is the elastic stiffness at the base storey). In addition, a, relates the yield deformability, expressed by the yield translational angle R, = Ay/H, to the yield capacity Ay/g ;

Ry = (a,by)(-4,/d

where the parameter uy does not vary significantly for most practical structures. Another useful parameter in studying the effects of gravity is

P, = = ay/ag

in which A, denotes the lateral limit of static stability of the structure [Figure 2(a)3. In terms of this parameter and the ductility factor p, the relative magnitude of the gravity force, A*, compared to the yield strength, is given by

A,/A, = sin (A/H)/(A,/g) = P/P,

In order to gain insight into the influence of gravity upon the response, it is important to know the range of the parameter ug in R/C building frames of practical interest. According to Umemura and Aoyama,26 the parameter K , E W in purely moment-resisting frames takes values around 0.5, with a possible minimum of 0.3, while frames with rigid spandrel walls, in particular with strong shear walls, are much stiffer, having K , E Wlarger than 08. In an extremely rigid case, this parameter may reach 3-0. The distribution of a, can therefore be estimated as

ug = (P/P")h; ' / ( K l E W ) = (9.0-0.7)~

for h, = 375 cm and P/p" in the range 1-0.75. In this study, representative values of 6 x structures) and 1 x low3 (for stiff structures) will be used for ug

In a similar vein, the system parameters Q, and u, are mostly distributed in the ranges

(for flexible

Q, = 3-4, U, = 0.10-0*30

Tabl

e I.

Inte

nse

eart

hqua

ke g

roun

d m

otio

ns e

xam

ined

iden

tific

atio

n Ep

icen

tral

A,,,

(SI)2

CPe

R

S CJ

E(03

)I R

MS (p')

2-

Ear

thqu

ake

reco

rds

sym

bol

Even

t M

agni

tude

di

stan

ce

(gal

) (c

m)

(cm

P2

) (g

al)

(s)

Ense

mbl

e a-

1 El

Cen

tro, S

OOE

com

p.

Oly

mpi

a, S

86W

com

p.

Taft,

S69

E co

mp.

Hac

hino

he, N

S co

mp.

Hac

hino

he, E

W c

omp.

Paco

ima

Dam

, S16

E c

omp.

Hol

iday

Inn

, NOO

W c

omp.

Ense

mbl

e a-

2 H

oshi

na, E

W c

omp.

Cho

lam

e Sh

ando

n N

o. 2

, N

65E

com

p.

Roc

ca, N

S co

mp.

Mel

endy

Ran

ch, N

29W

com

p.

Ense

mbl

e b

Jenn

ings

-Hou

sner

-Tsa

i,

Jenn

ings

-Hou

sner

-Tsa

i, Ty

pe A

-1

Type

A-2

EL

C

OL

M

TA

F

HC

N-N

S

HC

N-E

W

PAC

HO

L

HO

S

CH

O

RO

C

ME

L

A-1

A-2

Impe

rial V

alle

y ea

rthq

uake

in

Cal

if.;

18 M

ay 1

940

Puge

t So

und

eart

hqua

ke in

W

ash.

; 13

Apr

il 19

49

eart

hqua

ke in

C

alif.

; 21

Jul

y 19

52

eart

hqua

ke in

Ja

pan;

16

May

196

8

eart

hqua

ke in

C

alif.

; 9

Feb.

197

1

Ker

n C

ount

y

Toka

chi-O

ki

San

Fern

ando

Mat

sush

iro

eart

hqua

ke in

Ja

pan;

5

Apr

il 19

66

Park

field

ea

rthq

uake

in

Calif

.; 27

June

196

6

quak

e in

Ita

ly;

14 Ju

ne 1

972

Ston

e C

anyo

n ea

rthq

uake

in

Cal

if.;

4 Se

pt. 1

972

Anc

ona

earth

-

7.1

(65,

63)

7.1

(7.0

)

7.7

(7.5

)

7.9

6.6

(6.5

)

5.1

(5.4

)

5.5

(5.6

)

4.9

(4.5

)

4.7

10 k

m

341.

7 93

.2

329

64.4

18

.3

(6 k

m f

rom

faul

t)

60.6

12

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21 (1

6) km

27

4.6

61.0

25

9

56 (4

0,R

O) k

m

175.

9 42

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13.9

46

.4

189

190

km

225.

0

182.

9

1148

.1

250.

0

8 km

(4 k

m f

rom

su

rfac

e fa

ultin

g)

20 k

m (8

km

fro

m

surf

ace

faul

ting)

4km

48

6.9

0.2

km

479.

6 (0

.08

km f

rom

fau

lt)

in

252.

9 70

4 25

6,3

5.29

3 0 E

79.0

8.

76

0 8

907

253

51.7

16

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92.8

27

9 V

8 12

8.7

1.33

2 m

167.

3 3 3

4 25

8.9

1.17

p

69.6

17

7

4 i5

113.

7 1.

59

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402.

5 32

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171

83

km

617.

7 55

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205

215.

7 0.64

377.

4 13

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5 32

9

441.

6 13

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3 34

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(vid

e R

efer

ence

11)

c.

Num

eral

s in

pare

nthe

ses i

ndic

ate

the

para

met

ers

from

diff

eren

t lite

ratu

re so

urce

s. !2


which can be derived by consulting empirical equations.'" '' Typical values are used in this study : a, = 0.5 and a,, = 0.2. The remaining parameters for the two cases then take the values

Flexible case Stif case

T, = 0.301 J(t~/p)

R, = 0*030A,./g

ag = 6 x ag = 1 10-3 T, = 0-129J(n/p)

R, = 0.005AY/g KIE W = 0.33-0-44 KIE W = 2.0-2,7

p e = 33.3 pg = 200 where a typical value of h = h , = 375 cm has been used. In addition, the yield capacity of the total structure, A,,, is taken as 0.259 for specificity; however, because of the normalization procedures used its particular value is of little significance unless it is comparable to the gravity constant g.

El Centro, SOOE

Olympia, S 8 9 W

Hoshina E W

r, NS Melendy Ranch, N 2 9 W

(b) Hachinohe Harbor, EW

0 20 30 40 M 60 70 80 I I I I I I I I

10

(SECONDS)

Pacoima Dam, S16E

oow

Figure 3. Ensemble of accelerograms used in studies of damage and collapse : (a) strong earthquake motions of intermediate duration; (b) short earthquake motions with high peak accelerations; (c) artificial motions of long duration


TWO-PARAMETER CHARACTERIZATION OF EARTHQUAKE MOTIONS

The difficulty in selecting a suitable measure of the severity of earthquake motion arises mostly from the fact that ground motions can differ markedly in frequency content and duration, as well as in amplitude. Two ground motions having the same intensity may have different damage potentials primarily because of different durations; this is particularly the case for the highly non-linear response near collapse. Therefore, in addition to characterizing an individual motion by a single measure of intensity such as the peak amplitude, RMS (root mean square), RS (root square) or SI (spectral intensity), at least one more parameter measuring the duration is needed.6 For purposes of the present work, three groups of excitations having significantly different properties

time in seconds time in seconds 0 5 10 15 20 25 30 35 0 20 40 60 80

I I I l , , , I I I I ,

......................................................... " l r w 7 5 /

El Centro S 00 E component May 1940

I

I ............................................................ 7

S 86 W component

,

I .........................................................

July 1952 0

I L ...................... I ...................................................

May 1968

I ........................................................

Hachinohe Harbor E W component

I

Holidav Inn

time in seconds 0 5 10 15 20 I I 1

April 1966 0

I

0

June 1972 II

Figure 4. Accumulated energy functions for strong ground motions

were chosen. As listed in Table I and shown in Figure 3, these are Ensemble a of real accelerograms of engineering importance, further classified into a-1 and a-2, and Ensemble b of artificial motions. The strong- motion records in Ensemble a-1 have been obtained during Magnitude 6-7 events and their duration ofstrong shaking ranges from 10 to 25 s. Ensemble a-2 consists of short-duration, pulse-like motions with high peak accelerations. These have a duration of 1-2 s and were recorded in the epicentral areas of Magnitude 4 5 6


earthquakes. The longer motions of Ensemble b are two samples of motion designed to model the shaking expected in a great (Magnitude 8 +) earthquake."

For describing the pattern of build-up and decay in the ground motion, it is p~ssible '~, 6 , 7, lo to use the accumulated energy :

E(t) = [{uo(tl)]z dt,

which monotonically increases toward the total energy E( 00). For the three ensembles of ground shaking, Figure 4 displays this function, normalized by the factor E(m). In this figure, differences in duration are clearly seen among the ensembles; however, variations in the individual accelerograms are more prominent in Ensemble a-1, while a more consistent pattern can be found for Ensemble a-2.

0

0 5 10 15 2 0 2 5 30 35 40 4 5 I I

time in seconds

E l Centro-1940, SOOE component.

- Olympia-1949. S86W component.

Toft -1952, S69E component. *k Hochinohe Horbor-1968. NS component. ' r A r A

0

Hachinohe Hofbof - 1968. EW component.

A .-

Pocoimo Dom-1971. SIBE component.

A

- A-

A Holldoy lno-1971, NOOW componenl.

n--,----

0 10 2 0 30 4 0 50 60 7 0 80 90 r I

lime in seconds

A- I

I A - 2

0 5 10 15 2 0 2 5 30 35 4 0 4 5 I I

time in seconds

Cholome Shondon No.2-1966. N65E Component.

Melendy Ranch-1972, N29W component.

A = E * l t l / E Z .

0 8.J-

Figure 5. Energy density functions for strong ground motions

Also, the associated energy density (or power) has been employed for the present purpose :

Using a common interval for the running average of one second, the functions E*(t) are presented in Figure 5 for the same accelerograms. In addition to strong tails with notable power in some cases, prominent peaks and troughs are superimposed on the overall trends for the records in Ensemble a-1. Upon comparison of Figures 4 and 5 , it is seen that the description by E(t) tends to suppress the fluctuations, but preserves the significant trends; thus for displaying overall patterns in the ensembles. E(t) appears preferable to the energy density.


It is well known that it is difficult to define the duration of strong shaking quantitatively in a generally satisfactory way. Two approaches are examined here. One is to use the energy function E( t ) by defining the duration T as the interval between two characteristic fractions, aE(co) and flE(00). The upper part of Figure 6(a) illustrates this approach and a way of defining the associated RMS intensity P', the square root of average power. (a = 0.05 and p = 0.95 have been used by Trifunac and brad^.^^) The other approach is provided by

(a) (b) Figure 6. Duration of strong ground motion : (a) two methods for defining duration Tand intensity P'; (b) dependence of duration Ton

energy limits a (lower limit) and j3 (upper limit)

the energy density function E*(t). Again, a certain threshold level YE:,, is prescribed on the time function [see the lower part of Figure 6(a)]. A reasonable practice using this approach was found to be more difficult to determine and the first method, using a and f l , was selected.

Since there is little to guide the selection of suitable values for the parameters a and b, some variations were examined. Figure qb) summarizes the results for the selected accelerograms using durations comprising the middle 90,80,70,60 and 50 per cent of the total energy. The 5-95 definition was judged unsatisfactory for these accelerograms. The durations in this case are generally longer than is ordinarily understood; in particular, the value assigned to CHO seems unreasonably long. On the other hand, the 25-75 definition results in a set of extremely short durations. The serious problems that arise in this case are the large difference between the two components of HCN, and a meaningless value calculated for MEL. Even though there still remain problems, it appears that the time interval during which the central 70 per cent (a = 15 per cent, f l = 85 per cent) of the total energy is transmitted provides a relatively reliable definition of duration for the accelerograms used. Accordingly, the durations, T, and the RMS intensities over the duration, P', evaluated on this basis are used as two-parameter characterizations of the motions. Their numerical values are listed in Table I.

DAMAGE AND COLLAPSE IN RESPONSE TO STRONG GROUND MOTION

Selection of a normalized time-history, e(t), from the three ensembles and use of the fixed system parameters as noted prqviously allow the maximum response pmax to be considered a function of only three variables : T, or n (including p), the two values of ag and the relative intensity I . Examples of pmax versus I for different values of n and the two values of ag are shown in Figure 7. An alternative representation appears in Figure 8 wherein the

20

0 -

100

- a

0

60

40

- 30

-

20 - - -

A

-I

E I C

entr

o- 1

940;

S

OO

E c

ompo

nent

Hos

hino

-1966;

EW

Com

pone

nt

----

st

abili

ty li

mit

for

n

=2

a

,=1

;10

-3

A n

=7

0.31 1.0

1.5 2

3 4 5

7

I0

I5

20

30

-I

= A~

~~

/A

~

- Fi

gure

7. S

yste

m

I-A,O

X/Ay

- -

resp

onse

, mea

sure

d by

the

duc

tility

p,,,,

as

a f

unct

ion

of i

ncre

asin

g st

reng

th o

f gr

ound

mot

ion

r 0

0.31 I

I

0.1

0.2 0.3

0.5 0.7

I.O(s

ec)

-Te-

ug.rx10-3

-"&

a = 6

X1

0-3

I 2 3

57

10

111111

111111

I 2

3 571

0 - n-+

El C

enfr

o-19

40;

SO

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One of the notable points in Figure 8 is the gradually developing break between the full line for clg = 6 x as the intensity level is increased. This reflects the fact that the dynamic response under the lower intensities is not appreciably affected by the displacement-dependent effect of gravity; i.e. ag is insignificant, the response pmax being determined by the remaining parameters T, and I . For developing a criterion for denoting when the effects of gravity become dominant during dynamic response, a more detailed examination was made, selecting excitations ELC, HCN- EW and HOS. The results may be summarized as follows. The gradual reduction of restoring capacity which accompanies the increasingly larger drift is not accompanied by a gradual increase of response due to the effect of gravity. Instead, there exists a critical limit, beyond which the response increases abruptly, leading the system promptly to collapse. The critical limit can be much lower than the static stability limit pg, and ranges from 0.2 to 0 . 7 ~ ~ for ELC and HCN-EW, and from 0.4 to 1 . 0 ~ ~ for HOS, the difference seemingly reflects the influence of duration. Below the limits the gravity effect was insignificant, and the associated increase in drift is smaller than a factor of about 1.5. In particular, the effects were very minor in the region for which p,,, 5 0.1 pg

Where the effect of gravity on the response is negligible, attempts have been made13* l4 to formulate approximate rules describing the hysteretic response of yielding systems. When the period (To) dependence of the displacement response spectrum (sd) is expressed by S, a (Toy, a positive number, it is found in the current instance that pmax %. 12/T:(’ - I ) or pmax a 12”2 -I)/TZ. (The constants in these formulas account for the effects of a,, ay and the equivalent damping.) The exponent r appears to lie in the range from zero to two, so the features seen in Figures 7 and 8 are qualitatively consistent with existing correlations.

Two specific intensities of ground motion, I d and I , , were introduced by determining the value of the relative intensity I associated with serious damage and ultimate collapse of the equivalent structures. The damage intensity I d is arbitrarily defined by pma, = 3, while the collapse intensity I , is the lowest intensity causing ultimate failure, within the precision of the discrete values used in the calculations. The intensities can be interpreted, therefore, as factors by which the peak acceleration must exceed the yield level before damage or collapse occurs, respectively. It should be realized that the collapse intensity is approximate and that complicated response is possible when gravity is considered. The system with n = 1 and ag = 1 x when subjected to HOS, for example, fails at the relative intensity of I = 20, yet is found to resist an intensity of I = 50 without failure.

Using the specific intensities, the results of the calculations are summarized in Figure 9(aHd). The factor I,& shown in these figures is a measure of the margin-of-safety against collapse, in comparison with motions that are damaging. Prior to discussing the important features in Figure 9, the expected trends of I d and I , are noted. First, the qualitative relation among pmax, T, and I can be written in the form Id a T;-I. The corresponding correlation for I , may be found by applying the empirical equation derived by Jennings and Husid.” Their result relates the average time to failure, to, of an ideal elasto-plastic system, to the ratio, 8, of the RMS strength (unit strength = 0697 ft/s2) of a stationary random excitation relative to the yielding strength A&, and to the height, I (in feet), of the simple system. The correlation derived was to = 20001/82; no appreciable influence of T, was indicated in their study. Converting to metric units and applying the notation used here, this empirical relation becomes (,/E)col,apsse = 0176A, J H , with E = (RMS)2f,, or I , = 0.176 ,/H/(,/(E)/A,,,). The notation E stands for the total energy of a truncated stationary excitation, which is equivalent to the accumulated total energy E( co)~ for this class of motions. Then, substitution for H leads to

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These equations imply that the collapse intensity I , , basically dependent upon a set of two factors T, (or n/P) and am, may be specified by the single parameter T;/a, (or nip) and may be proportional to the square root of nip. Another important suggestion is the possible use of the total energy E(m) as a single measure of the severity of excitation in this application. This can be seen more clearly from an alternative expression : A , = 0.294 ,/(E)/,/(njP), in which A , is interpreted as the average of the minimum yield strengths needed for resisting the excitations without collapse.

In Figure 9 the trend toward continuity in the two curves for I d derived for the two values of ug reflects the previously noted fact that the response at low levels is not noticeably influenced by the effects of gravity. The intensity I , is thus specified by a single factor, T,, and the relation is a monotonically increasing curve, convex downward.

Another important feature seen in Figure 9 is that the absolute magnitude of I d is clearly correlated with the type of excitation. The damage capabilities of ROC and MEL are extraordinarily low, particularly in the longer period range ( I , I: 3 at the lowest and I , N 70-100 at the highest). A low capability of HOS ( Id ‘v 2 at the lowest, and I , N 15 at the highest) follows these two cases, and then PAC ( I , N 2 at the lowest, and I d N 7 at the highest). For other excitations, I , is 1.0-1.5 in the shorter period range, and 3 4 (Ensemble b, HCN, HOL and CHO) or about 7 (ELC, OLM and TAF) at the highest. The high damage potential of CHO, comparable to Ensemble b and the destructive group in Ensemble a-1, is worthy of special notice.

More difficulties are found in describing properties of the collapse intensities. For instance, the differences in I, were found to be much larger than those for I , for the two motions of Ensemble b, which are samples of the same statistical process. Nevertheless, the collapse zones I , for the two cases of ug = 6 x have a strong tendency to overlap when n is employed as the common abscissa, and the combined zone appears to be a monotonically increasing function of n. Consistent with the empirical equation of Reference 12, these features suggest the relative unimportance of the factor T,; more significant factors governing the extremely non-linear response are A , and the parameter describing the absolute magnitude of the gravity effect, H = (n/jY)h.

The dependence of I , upon the types of excitation is seen to be far stronger than is the case for Id. Extremely high values occur for ROC and MEL ( I , N 100 on average), and HCN-EW is shown to be very destructive for low-storeyed structures ( I , 1: 2). The destructive group of accelerograms (Ensemble b, HCN and HOL) yields I, ‘v 2 4 for n = 1 and I , N 5-15 for n = 10, while I , N 3-7 for n = 1 and I , N 15-20 for n = 10 in the group of moderate excitations (ELC, OLM and TAF). Again, CHO is comparable to these records which are followed by PAC and HOS in decreasing order of destructive capability.

The margin of safety of collapse to damage, represented by ZJI,, follows the line of the preceding comments. The margin for urn = 6 x has a general tendency to be constant, independent of n or T,. In the destructive group, ‘v 1.5-2.5 with the exception of HCN-EW. The moderate group and PAC yield I,/Id I: 2-4, while I c j I d 21 3-5 in Ensemble a-2. On the other hand, the margin of safety depends noticeably upon n (or T,) in most cases for clg = 1 x and becomes higher than it is for soft structures with increasingly large n. An extremely high margin of 15-30 occurs for ROC and MEL.

If a bilinear regression is applied to the I , - T, correlation and a linear regression to the I , - n / p correlation, with all the variables on a logarithmic scale, some simple statistical relations can be derived. These approximations are motivated by the preceding findings, and the statistical relations, I , versus n, I d versus T, and I , /& versus [T, or n, ag], appear in Figure 10. This representation is intended to permit more direct comparisons and to display the characteristics of the three ensembles, under the same amplitude of peak acceleration. A low destructive capability and a high margin of safety are clearly seen for most excitations in Ensemble a-2.

and lag +

When the necessary parameters are reduced to a bare minimum, the cruder correlation forms :

in which k,, k, and k i are correlation constants, are obtained. The former is of the same type noted in Reference 12. These relations are not very accurate, however; for instance, the expression for I , leads to a relatively poor result for HCN-EW.

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H. TAKIZAWA AND P. C. JENNINGS

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Returning to the definition of the relative intensity I and substituting the recorded peak acceleration for A,,,, the following alternative expressions can be derived.

(AYlC = "'$/&/P'), (Ayld = min PA;, '473 This substitution is made to identify the absolute severity of ground motions. In these relations, the notation (Ay)c or (Ay),, is interpreted as the minimum yielding capacity required for resisting the motion without


- PAC- -

CHO- -

A-2, - A- I

sustaining ultimate collapse or serious damage, respectively. Thus, the higher these values are, the more severe is the excitation. The scaled collapse and damage accelerations, 'A; ( = k , A,,,), 'A: ( = k, A,,,) and 'A; ( = kd A,,,), are independent of structural parameters, and can be used to represent the absolute severity of ground motions from these points ofview. For comparison, these intensity scales are summarized in Figure 11, which also includes the conventional parameters of A,,,, (SI)20% and RS [ = ,/E(oo)]. The two collapse accelerations 'A; and "A,', also shown in the figure, are those evaluated from the cases of n = 1 and n = 10, without use of the simplified correlation.

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Figure 11 illustrates the range of severities resulting from different points of view. In addition to having a peak acceleration far superior to others, PAC is shown to be most destructive by the measures (SI)20%, A; and A;; however, its capability as measured by the collapse acceleration A; is immediately followed by the two artificial accelerograms of Ensemble b. The artificial excitations, characterized by moderate peak accelerations and by the largest total energy, rank high on the basis of potential for collapse but are not so severe for elastic structures or for damaging yielding structures. Also it is seen that PAC loses its highest destructive rating in the case of failure of longer-period structures. On the other hand, a very low severity measure in all the scales is obvious for HOS, and particularly for ROC and MEL, in spite of their high peak accelerations. Also noteworthy in Figure 11 is that the HOL and HCN records are relatively destructive in comparison with their moderate peak accelerations.

The relatively high seventies for CHO according to these scales is inconsistent with the minor earthquake damage associated with the Parkfield earthquake. The lack of the damage observed in the sparsely settled area of strong shaking in this event has been attributed to the short duration of the intense ground shaking. However, it appears that its significant energy at longer periods compensates in most of the measures for the shortness of duration, even though the damage potential decreases considerably for longer-period structures as indicated by ''A;. It seems possible that the shaking recorded at Cholame might have been much more damaging had the earthquake occurred in a more heavily built-up area.


The preceding discussion has suggested that a single-parameter measure of the accelerograms cannot provide an adequate scaling of their destructive potential, as can be seen in Table I1 and Figure 11. An elementary two-parameter characterization is then examined which uses the power P' and the duration T. The three diagrams in Figure 12 contain the relations Tversus A;/P', T versus A ; / P and Tversus (SI)zoy4/P', respectively. Even though fluctuations are notable in these relations, the results indicate the importnnce of the duration of motion. Variations in the frequency content of the records are, in part, responsible for the ranges of values.

Table 11. Single-parameter characterizations of A;, A; and (SI)zoz

A-1 A-2

ELC OLM TAF HCN-NS HCN-EW PAC HOL

HOS CHO ROC MEL

151 145

55.4 44.2 299 62.6 55.2

95.0

26.8

155

106 5.47 5.95

156 159

65.3 46.7 37.3 65.9

106 175 103

32.7 153

7.08 8.71

147 134

47.8 42.4 24.5 59.8 30.9

88.4

22.4 75.9

140

4.40 4.28

0.40 0.33

0.16 0.16 0.17 0.28 0.30 0.14 0.38

0.055 0.22 0.014 00096

1.3 1.3

0.86 0.73 0.7 1 1.4 1.1 0.61 1.2

0.2 1 0.4 1 0.048 0028

0.19 0.18

0.17 0.17 0.16 0.26 0.22 0.22 0.34

0.15 0.32 0.032 0.029

332 329

264 170 144 156 165 557 192

262 489

135 82.8

0.88 2.8 0.4 1 0.75 2.9 0.42

0.77 0.62 0.82 0.70 0.90 0.49 0.77

0.54 1 .o 0.21 0.22

4.1 0.80 2.8 0.66 3.4 0.76 3.5 0.64 3.2 0.65 2.2 0.79 2.4 0.69

2.0 1.5 1.9 1.5 0.73 0.48 0.63 0.66

80.4 80.8

51.0 294 25.5 38.8 54.6

47.8

38.4

140

100 6.53 7.58

A-l A-2

ELC OLM TAF HCN-NS HCN-EW PAC HOL

HOS CHO ROC MEL

0.21 0.18

015 0.11 0.15 017 0.30 0.12 0.19

0-079 021 0.0 1 6 0012

0.69 0.71

0.79 0.49 0.60 0.8 7 1.1 0.55 0.60

0.30 0.39 0.058 0.035

0.10 0.10

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0.22 0.30 0.038 0.037

0.36 0.31

0.27 0.22 0.26 032 0.50 0.22 0.37

0-14 0.35 0.08 1 0.090

1.2 1.2

1.4 1 .o 1.1 1.6 1.8 0.99 1.2

0.54 0.65 0.29 0.26

0.17 0.17

0.28 0.24 0.25 0.29 0.36 0.36 0.33

0.39 0.50 0.19 0.27

An elementary regression analysis was done on the data in Figure 12 for aACy, 'A;, ''A;,, 'A:, 'A; and (SI)zo70. Using the correlation constants n1 and nz, the regression form used was n1 P'Tn2. This type of correlation was adopted mainly because the parameter n2 can clarify the effect of duration T with relation to the total energy and the RMS. A value of nz between 0 and f suggests a suitable single-parameter measure lying between P' and JE(co), while nz larger than f indicates an insufficient characterization of the effect of duration by the use of E(co). The correlations over the entire class of accelerograms are summarized in Table 111; the numbers in parentheses are those evaluated by excluding Ensemble a-2. (The corresponding regression lines appear in Figure 12.) Comparison of these results highlights the fact that the duration of ground shaking can influence


3.OC

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Table 111. Correlation constants in n, PT"*

"A; 0.10 (0.37) 0.80 (0.35) 'A; 0.14 (0.45) 0.75 (0.34) "A; 0.079 (0.31) 0.84 (0.36) sA; 1.1 (1.8) 0.34 (0.17) LA; 0.12 (04) 0.61 (0.19)

( W 2 W 0.41 (0.87) 0.39 (0.13)

DISCUSSION

By employing an equivalent 1-DOF system characterized by the degrading trilinear yielding model, the present study has investigated damage and collapse of a class of R/C frames subjected to intense ground motions. In this concluding discussion, some additional factors influencing the structural failure are first noted briefly.", 2 1

Since the total vertical forces on a structure undergo appreciable time-dependent fluctuations due to the action of vertical shaking, the vertical component of ground motion might be expected to have notable influence upon ultimate structural failure. However, calculations performed by the authors showed this effect to be very minor in almost all cases of practical interest, which is in agreement with one of the conclusions of a previous study.12

The force-deflection curve of a 1-DOF model of a R/C skeleton cannot be truly elasto-plastic or bilinear beyond yielding even in the case of flexural failure. The actual load-carrying capacity decreases gradually as deterioration (e.g. crushing and spalling of concrete) progresses further. This was schematically shown in Figure 2(b). According to an exploratory study including moderate deterioration [as specified by a, = 1.05, p, = 3.0 and p = -0.01 in Figure 2(b)], collapse occurs at much lower intensities than without this deterioration, particularly for the stiffer frames. The marked redbction of the static stability limits implied by deterioration permits anticipation of these findings. The significance of this result is that the higher margin of safety for stiff structures (a, = 1 x low3) shown in the results of the present study can reduce, when deterioration of the force-deflection relation is considered, to values comparable to that for more flexible structures (ag = 6 x

In addition, the influences of biaxial shaking and response, and of localized structural failure, can also be of practical importance. Biaxial interaction of restoring forces occurs when the structure is acted upon by two- directional motion in the horizontal plane. In pilot calculations it was found that concurrently acting biaxial loading can speed the deterioration of R/C structures in comparison with one-dimensional response, and can lead to serious damage and collapse at lower intensities of motion. It appears that the margin-of-safety against collapse sometimes can become very small when the effects of biaxial response, deterioration and gravity are all combined.

The term 'localized failure' is used to describe types of failure different from the overall 1-DOF kinematic mechanism of structural collapse used in this study. In localized failure, inelastic deformations develop progressively in a comparatively soft and vulnerable portion of the structure. For example, in the cases of a weak base storey or penthouse the important features can be described approximately by means of an equivalent 2-DOF formulation. The gravity-effect parameter a,, associated with individual subelements of the equivalent structural system, generally takes a larger value, being influenced by the local distribution of rigidity. Moreover, the concentration of inelastic drift at the weaker element results in collapse at lower intensities of motion.

In view of these points and the limitations in the study reported here, considerable caution should be used in trying to apply the results to practical problems. This is particularly the case for the numerical values of A;, the collapse acceleration. The results can serve, however, to provide insight into the problem and as a reference case for more detailed studies. With this in mind, the main findings of the study are summarized in the following paragraphs.


For the purposes of characterizing the destructive potentials of the earthquake motions of differing durations used in this study, it was found that a fairly reliable result could be obtained by using as a duration the time for transmission of the middle 70 per cent of the total energy.

The results of the calculations are summarized in Figure 9. The most significant results seen in this figure are the low destructive capability of short-duration motions, even those with high peak acceleration, and the fact that for the conditions of the study the stiffer structures have a much larger margin of safety between damage and collapse than do the more flexible structures. The collapse and damage intensities and the margin of safety are also shown in Figure 10. This figure emphasizes the clearly differing severities of the three different ensembles under the same amplitude of peak acceleration. In particular, an extremely low destructive capability and a high margin of safety is obvious for the ensemble of short-duration motions.

Examination of the results showed that the damage intensity Id was specified by a single parameter'T,. The lack of influence of the factor ag reflects the fact that the response at this level (pmax = 3 ) was not noticeably affected by the deformation-dependent effect of gravity. On the other hand, examination of the collapse intensity I , showed it to be virtually independent of the factor T,. More significant factors in this instance are the yielding strength and the magnitude of the gravity effect; this allows I , to be related to the single factor n, the number of storeys of the equivalent structure.

To reduce the parameters in the I, ,., n and I d - T, correlations to a minimum and to eliminate the peak acceleration as a parameter, the collapse and damage accelerations A; and A; were introduced. These are correlation constants that indicate the minimum yielding capacity required for resisting the motion without sustaining ultimate collapse and serious damage, respectively. Being independent of structural parameters, these accelerations can be used to measure the absolute severity of ground motion. The ratings of the earthquake motions by different intensity scales, shown in Figure 11, demonstrate the unsatisfactory description of destructive potential by any conventional single-parameter measure. A two-parameter characterization by means of the RMS intensity and the duration has been examined. Even though fluctuations attributable mainly to variations in the frequency content were still present, the significance of the duration became clearer : the elastic response, serious damage and collapse were influenced in increasing order by the duration of motion. It was noted also that the yield acceleration of the structure required to resist collapse, A;, was not simply related to the total energy in the ground motion.

ACKNOWLEDGEMENTS

The material contained in this paper is part of the research conducted at the Earthquake Engineering Research Laboratory, California Institute of Technology, during the first author's visit from May 1975 to April 1976. Helpful suggestions and encouragement provided by Professor George W. Housner and the financial support given by the National Science Foundation under Grant ATA74-19135 during the course of study are gratefully appreciated.

APPENDIX

Notation The following symbols are used in this paper :

column vector, with T indicating the transpose row vector matrix equivalent 1-DOF restoring force of structure, associated with A and expressed as an acceleration crack capacity in A gravity force associated with A peak acceleration of a ground motion ti&) static representation of A yield capacity in A minimum A, of structure, without sustaining ultimate collapse


UY D', 8"

E i a x { e } = L1,1, ..., 1 JT

- hi

collapse acceleration of ground motion, with the discrimination of "A;, 'A; and ''A; minimum A , of structure, without sustaining serious damage damage acceleration of ground motion, with the discrimination of sA; and 'A;

dimensionless factors used in the restoring force model of Figure 2(b) threshold factors prescribed on E(t) or E*(t) in Figure qa ) gravity-effect parameter of structure (ratio of the initial gradient of gravity force to the elastic stiffness) secant stiffness reduction factor of structure, defined at yield point modification factors of equivalent 1-DOF structure, depending upon the relative distribution of storey-mass and storey-height incremental internal work of total structure incremental rotation of structure equivalent 1-DOF lateral drift of structure, measured along the circular arc static stability limit in A yield level in A lateral drift at ith storey, measured along the circular arc virtual internal work of total structure virtual rotation of structure total energy of truncated stationary excitation accumulated energy function of ground motion total energy of ground motion energy density function of ground motion peak energy density of ground motion

horizontal excitation with unit peak value horizontal force acting on ith storey horizontal storey-force vector of structure gravity constant equivalent 1-DOF height of structure average interstorey height of structure interstorey height at ith storey; in particular, h , is the interstorey height at base storey height at ith storey, measured from the ground level; in particular, h, is the total height of structure storey-height vector of structure strength of excitation relative to structure collapse intensity of ground motion measure of the margin-of-safety against collapse beyond damage damage intensity of ground motion elastic stiffness at base storey correlation constant in I , versus n/p correlation constants in Id versus T, generalized restoring force of equivalent 1-DOF structure, associated with R


4 ,

inertia matrix of structure normalized drift, or ductility factor, of structure static stability limit of structure, in terms of ductility factor peak value in p(t), or maximum ductility-factor response total number of storeys of structure elastic natural circular frequency of structure RMS intensity of ground motion, evaluated over T vertical force acting on ith storey vertical storey-force vector of structure correlation constants in X I P versus T [ X = A;, -4; or (SI)20z] translational angle of structure yield level in R exponent factor in S , cc (Tor displacement response spectrum of ground motion 20 per cent damped spectral intensity of ground motion partial sum of yield moments, contributing in 1-DOF kinematic hinge- mechanism of structure total weight of structure duration of ground motion elastic natural period of structure period of simple oscillators time time interval of running average for evaluating E*(t) horizontal displacement at ith storey, relative to ground horizontal storey-displacement vector of structure horizontal ground acceleration vertical displacement at ith storey, relative to ground vertical storey-displacement vector of structure vertical ground acceleration viscous damping factor of structure

REFERENCES 1. R. A. Adu, ‘Response and failure of structures under stationary random excitation’, Earthq. Engng Res. lab. Rept., Calif. Inst. Tech.,

2. J. C. Anderson and V. V. Bertero, ‘Effects of gravity loads and vertical ground acceleration on the seismic response of multistory

3. H. Aoyama, ‘Simple non-linear models for the seismic response of reinforced concrete buildings’, Proc. Review Meeting of U.S.-Japan

4. A. Arias, ‘A measure of earthquake intensity’, in Seismic Designfor Nuclear Power Plants (Ed. R. J. Hansen), Mass. Inst. Tech. Press,

5. S. C Goel, ‘P-A and axial column deformation in aseismic frames’, J. Struct. Diu., A X E , 95, 1693-1711 (1969). 6. G. W. Housner, ‘Measures of severity ofearthquake ground shaking’, Proc. U.S. Nat. Con$ Earthq. Engng--1975, Ann Arbor, Mich.,

7. G. W. Housner and P. C. Jennings, ‘The capacity ofextreme earthquake motions to damage structures’, in Structural and Geotechnical Mechanics, a Volume Honoring Nathan M . Newmark (Ed. W. J . Hall), Prentice-Hall, Englewood Cliffs, N.J., 1977, pp. 102-116.

8. R. Husid, ‘Gravity effects on earthquake response of yielding structures’, Earthq. Engng Res. Lab. Rept., Calif. Inst. Tech., Pasadena, Calif. (1 967).

9. R. Husid, ‘The effect of gravity on the collapse of yielding structures with earthquake excitation’, Proc. 4th Wd Con$ Earthq. Engng, Santiago, Chile, 11, ( A 4 ) 3143 (1969).

10. R. Husid, Earthquakes: Spectral Analysis and Characteristics ofdccelerograms as a Basis of Earthquake-resistant Design, Editorial Andres Bello, Santiago, Chile, 1973, p. 447.

11. P. C. Jennings, G. W. Housner and N. C. Tsai, ‘Simulated earthquake motions’, Earthq. Engng Res. Lab. Rept., Calif. Inst. Tech., Pasadena, Calif. (1968).

12. P. C. Jennings and R. Husid, ‘Collapse of yielding structures under earthquakes’, J. Engng Mech. Din, A X E , 94, 104-1065 (1 968).

13. N. M. Newmark, ‘Current trends in the seismic analysis and design of high-rise structures’, in Earthquake Engineering (Ed. R. L. Wiegel), Prentice-Hall, Englewood Cliffs, N.J., 1970, chap. 16.

Pasadena, Calif. (1971).

frames’, Proc. 5th Wld Con$ Earthq. Engng, Rome, Italy, 2, 2914-2923 (1973).

Coop. Res. Program in Earthq. Engng, Honolulu, Hawaii, 291-309 (1975).

Cambridge, Mass., 1970, pp. 438483.

25-33 (1975).


14. A. Shibata and M. A. Sozen, ‘Substitute-structure method for seismic design in R/C‘, J . Struct. Div., ASCE, 102, 1-18 (1976). 15. S . Sugano and I. Koreishi, ‘An empirical evaluation of inelastic behavior of structural elements in reinforced concrete frames subjected

16. C.-K. Sun, ‘Gravity effect on the dynamic stability of inelastic systems’, Rept. Civil Engng Dept., Univ. Mich., Ann Arbor, Mich. (1971). 17. C.-K. Sun, G. V. Berg and R. D. Hanson, ‘Gravity effect on single-degree inelastic systems’, J . Engng Mech. Din, ASCE, 99, 183-200

18. H. Takizawa, ‘A simplified nonlinear structural model for estimating the response deflection of lowrise R/C buildings subjected to

19. H. Takizawa, ‘Non-linear models for simulating the dynamic damaging process of low-rise reinforced concrete buildings during severe

20. H. Takizawa and P. C. Jennings, ‘Ultimate capacity of lowrise R/C buildings subjected to intense earthquake motion’, Preprints, 6th

21. H. Takizawa, ‘Biaxial and gravity effects in modeling strong-motion response of R/C structures’, Preprints, 6th Wld Con$ Earthq.

22. R. Tanabashi, T. Nakamura and S. Ishida, ‘Gravity effect on the catastrophic dynamic response of strain-hardening multi-story

23. S . Tani and S . Soda, ‘Vertical load effect on structural dynamics’, Preprints, 6 th Wld Conf Earthq. Engng, New Delhi, India, 3,5540

24. M. D. Trifunac and A. G. Brady, ‘A study on the duration of strong earthquake ground motion’, Bull. Seism. SOC. Am., 65,581-626

25. H. Umemura and H. Aoyama, ‘Evaluation of seismic deflections of reinforced concrete frames based on the tests of members’, Proc.

26. H. Umemura and H. Aoyama, ‘From statics to dynamics; earthquake-resistant design of R/C lowrise buildings’ (in Japanese), Quart.

27. H. Umemura (Ed.), Dynamic Earthquake-resistant Design of Reinforced Concrete Buildings (in Japanese), Giho-do, Tokyo, Japan,

28. W. Y.-L. Wang, ‘Structural instability during earthquakes and accelerogram simplification’, Rept. Cioil Eng. Dept., Univ. Mich., Ann

to lateral forces’, Proc. 5 th Wld Con$ Earthq. Engng, Rome, Italy, 1, 841-844 (1973).

(1973).

severe ground shakings’ (in Japanese), Trans. Arch. Inst. Jap., Extra, 545-546 (1974).

earthquakes’, Earthqu. Engng Struct. Dyn. 4, 73-94 (1975).

Wld Con$ Earthq. Engng, New Delhi, India, 3, 79-84 (1977).

Engng, New Delhi, India, 3,49-54 (1977).

frames’, Proc. 5 th Wld Conf: Earthq. Engng, Rome, Italy, 2, 214G-2149 (1973).

( 1977).

(1975).

4th Wld Con$ Earthq. Engng, Santiago, Chile, I, (B-2) 91-108 (1969).

Column, Yahata Steel Corp., Tokyo, Japan, No. 30, 75-81 (1969).

1973, p. 442.

Arbor, Mich. (1975).

COLLAPSE OF A MODEL FOR DUCTILE REINFORCED CONCRETE FRAMES UNDER EXTREME EARTHQUAKE MOTIONS

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