THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING INTENSITY AND TIME SCALE EFFECTS OF ACCELEROGRAM M. J. KCaldjian WitiTam R, S. Fan April, 1968 IP-819

TABLE OF CONTENTS Page LIST OF FIGURES.................................................. ii INTRODUCTION....................................................... 1 LOAD-DISPLACEMENT RELATION......................................... 2 THE EQUATION OF MOTION............................................ 4 INTENSITY SCALE EFFECT............................................. 5 TIME SCALE EFFECT.................................................. 10 CONCLUSIONS........................................................ 13 ACKNOWLEDGMENTS.................................................... 15 NOMENCLATURE....................................................... 1i6 REFERENCES......................................................... 17 ii

LIST OF FIGURES Figure Page 1 Ramberg-Osgood Functions............................. 3 2 Response Spectra for the Elasto-Plastic System, Taft, July 21, 1952, S21~W. Constant Ductility Ratio "p.............................................. 8 3 Response Spectra for the Elasto-Plastic System, 3 x Taft, July 21, 1952, S210W. Constant Ductility Ratio "............................................ 4 Response Spectra for the Elasto-Plastic System Taft (0.5 x time scale), July 21, 1952, S21~W. Constant Ductility Ratio "p.".. 12 5 Intensity and Time Scale Effects of Accelerogram on Response Spectra.................................... 14 iii

INTRODUCTION The uncertainties involved in predicting the earthquake forces a structure is called to stand in its life time, often make it necessary for the designer to consider a stronger earthquake accelerogram in his analysis than the ones recorded to date. The same situation also arises when the spectral response of a system is to be compared for various recorded earthquakes. Since earthquake accelerograms do not have the same magnitudes, a modification of the accelerogram becomes necessary so that the spectrum intensity, defined as the area under the undamped velocity spectrum between the time (3, ) intervals t1 and t2, are the same for all.(3 It is also of interest to consider what effect modifying the time scale of an accelerogram has on the response spectra. In this study it is assumed that spectral response curves are available for a system to a given accelerogram. The new accelerogram is to be prepared by multiplying either the intensity of the acceleration or the time scale of the given accelerogram by a constant. The purpose of this brief discussion is to establish relations between response spectra of the modified accelerograms and spectra obtained from the original accelerogram. () Housner, G. W. "Behavior of Structures During Earthquakes," Journal of the Engineering Mechanics Division, ASCE, October 1959. (4) Goel, S. C. and Berg, G. V. "Inelastic Earthquake Response of Tall Steel Frames," ASCE Structural Engineering Conference, Seattle, Washington, May 1967. -1 -

LOAD-DISPLACEMENT RELATION The load-displacement curve for this study is a Ramberg-Osgood function( shown in Figure 1. This is a simple and a realistic model to represent actual structural member behavior.(6) Three parameters are employed, a characteristic or yield load qy, a characteristic or yield displacement xy, and an exponent r. The Ramberg-Osgood relation includes as special limiting cases the elastic case, obtained by setting r = 1, and the elasto-plastic case, obtained as r tends to infinity. Jennings, P. C. "Periodic Response of a General Yieldings Structure," Journal of Engineering Mechanics Division, ASCE, Volume 90, Number EM2, April 1964. (6) ) Kaldjian, M. J. "Moment-Curvature of Beams as Ramberg-Osgood Functions," Journal of the Structural Division, ASCE, Volume 93, Number ST5, October 1967. -2 -

-3 -I o?b0 00 + 0 >.~lo 2 ~~~ hD~~~O cV ~~~ O

THE EQUATION OF MOTION Consider the differential equation, x + 23w Ox + q(x) = - y(t) where x = relative displacement of mass to ground, a function of time y(t) = ground displacement, a function of time q(x) = restoring force per unit mass, a function of x P = fraction of critical damping Wo = the undamped natural frequency of small oscillations and differentiation with respect to time is denoted by dots. The numerical solution of this equation for various parameters, will result in the response spectra for the given accelerogram y(t) Tn dimensionless form Equation (1) becomes,(5) 2 + 2P d I + y(T/o) (2) dr2 xy dT xy qy ~ qy where qy = yield or characteristic strength of spring per unit mass Xy = yield or characteristic displacement of spring = qy/Xy T = 00t and q_ xy ' i qy y XY qy -4 -

INTENSITY SCALE EFFECT Consider now another system, with damping property P1, undamped natural frequency wl, and force displacement relation Pl(x) It is desired to find the response of the new system when the latter is subjected to an earthquake having acceleration intensities KI times those of the original earthquake, the time characteristics remaining unchanged. KI is a constant. The differential equation for the new system is, z + 2plz + p1(z) KI y(t) (3) where z is the relative displacement of the mass with respect ot the ground. Introducing the dimensionless parameters; 2 1 =Ply/Zy 1= 1t and P1 z Pl(z) Ply lzy Ply in which ply and zy are yield or characteristic spring strength and spring displacement respectively, Equation (3) can be written as, d2"yi Pl z Y (Tl/Wl) 1 + 2 d + | KI (4) dT2 l 1 dT1 zy Ply Ply p ly zly Equations (4) and (2) can now be made identical by proper choice of parameters. This is done by letting -5 -

-6 -1 = 0o Pl! Z = _2 X( P~ lYz 1 qyl XY(5) ly Yy y and KI = Ply/y= y/xy Using the above values in Equation (4) one can show that at any instant the following relations are valid: d2z d_ d2x dz ~ -Y - K1 - dT2 Xy dv2 dT2 dz Zy dx dx dT1 xy dT dT Zy z = x = KI x Xy In terms of maximum values the above expressions become IZimax = KIjxmax Ilmax = KIIXmax (7) and z + KIy ax = KIIX + ymax Thus the ordinates of the new response spectra are KI times those obtained from the original accelerogram. Equation (7) is valid for linear as well as non-linear systems provided relations in Equation (5) are satisfied.

-7 -To illustrate the preceding, it is desired to use a given response spectra, Figure 2, in order to obtain the response spectra of an earthquake of acceleration intensity KI times the one used for the given spectra. This is done by multiplying the ordinates of the curves in the above figures by KI. On four-way-log plots this result is accomplished by shifting the curves vertically by log KI. Figure 3 shows the latter for KI = 3 applied to the curves of Figure 2.

-8 -100 80 -- M= 1.0 —.-= 1.25 ----—:= 2.0 60 60 —..... ~=:4.0 40 =.05 20 w 10 cor |:t \f / ^*-^ 10 0 6.14.6.8 IO 2 4 6 8 10 PERIOD (SEC.) Figure 2. Response Spectra for the Elasto-Plastic System, Taft, July 21, 1952, S21. Constant Ductility Ratio July 21, 1952, S21~W. Constant Ductility Ratio "\ -.

-9 -100 80..- -- = 1.0.-.- = 1.25 60 - KI =3,,- = 2.0.., -- 4.0 =.05 40 \W // / v'^"^\ \\ //-.-./ '.\. 2~~~~~~~~~~~~0~~~~ '' $,o-! " ' \ \ 4 4.. 2 4.1.2.4.6.8 10 2 46 8 10 PERIOD (SEC.) Figure 3. Response Spectra for the Elasto-Plastic System, 3 x Taft, July 21, 1952, S21W. Constant Ductility Ratio "p.". Ratio "~".

TIME SCALE EFFECT As a second case consider a system with damping property, undamped natural frequency w2 and a force-displacement relation P2(z) It is subjected to an accelerogram obtained by multiplying the time scale of y(t) in Equation (1) by KT, i.e., the duration of the modified accelerogram would be KT times the duration of the original accelerogram. The equation of motion for this can be written as, z+ 22 + 2 + 2(z) = - y(KTt) (8) In dimensionless form this becomes, dm2 (_L ) ~ ~ (k 4I2 T)/P2y (9) d | z g + 2P2 dT z P2Y z KT2 dT2 P2 Zy "2 where 2 2 = P2y/Zy T2 = 2t and P2 Z. P2(z P2y Zy 2y Again by proper choice of parameters, i.e., P2 = P P2y = qy P2 z (io) P2y Zy qy Xy/ and KT = c2/o = T2/T = (xy/zy)l/2 -10 -

-11 -Equation (9) is made this time similar to Equation (2), and at any instant T and corresponding T2 it can be shown to result in the equalities, d2 (z \ 1 d2z 1 d2x d Iz 1 dz _ 1 dx (11) dT2 Z KT Zy dT x dT z x z x zy xy The desired relations can now be found from Equation (11) to be, IZImax= Xlmax/KT I Zmax = IXlmax/KT z Ymax Ix + max (12) max max Thus to obtain the new acceleration spectra, the period scale of the original acceleration spectrum curves are divided by KT. The new velocity spectra'are obtained from the original velocity spectrum curves by dividing the period and the velocity scales by KT. The new displacement spectra are obtained from the original displacement spectra by dividing the period scale by KT and the displacement scale by KI On four-way-log plots the modified response spectra described above are readily obtained by shifting the original curves horizontally as well as vertically by log(1/KT). Figure 4 shows the latter for KT = 0.5 applied to spectra of Figures 2.

-12 -100 80 80 =: ___ ^ 1.0 60 = --- —- 1.25 K =0.5 _ 2.0 T~: 4.0 40 / =.os 20 - /' '~'1 - \\ I,,1 O/ 0.2.4.6.8'1.0 2 4 6 8/10 Ductility Ratio "".' /. / \...0.2.4.6.8 - - 2 4 6 8 10 (0.5 x time scale),'""", l-192, S2lW. Constant

CONCLUSIONS It has been shown that when an accelerogram is modified by multiplying the acceleration scale of a given earthquake accelerogram by an arbitrary constant KI, the response spectra for the new accelerogram are KI times the spectral values obtained from the original accelerogram. On the other hand if the new accelerogram is obtained by multiplying the time scale by a factor KT, all quantities involving time must be changed appropriately. The new acceleration, velocity and displacement response spectra are obtained in this case by dividing the corresponding original spectral values by 1, KT, and K, respectively, and dividing the period values by KT When the intensity as well as the time scale of the accelerogram are modified simultaneously then the above two cases must be superimposed to obtain the desired response spectra. The above three cases are shown plotted side by side in Figure 5 for curve t = 2 of Figure 2. -13 -

-14 -100 F- -— ~~~~~~~~~ —~ Original Spectra 80 Modified for KI = 3 - -~.... Modified for KT = 0.5 60 =.05 --- Modified for KI = 3 & =_ /A= 2.0 KT= 0.5 40 2 L / I\/ l / I.7 >/ \ ^/ IPT\8PT ^ \/ /s3<<,O./II 2 - /~S IW I - a: / \ ^ ' - 4..2.4.6.8og 0 /.2 4 6 -/ // \ ' Log 2 / --- -,..Taft 7/21/52 0 0 N.1.2.4.6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Figure 5. Intensity and Time Scale Effects of Accelerogram on Response Spectra.

ACKNOWLEDGMENTS This research was carried out at the University of Michigan as part of AISI Project 119 on earthquake response studies. The writers are indebted to Glen V. Berg, who initiated the project and to Robert D. Hanson who reviewed this report. -15 -

NOMENCLATURE KI, KT a constant q, Pi' P2 restoring force per unit mass qy, Ply, P2y yield or characteristic strength of spring per unit mass r an exponent t time x, z relative displacement of mass to ground xy, zy yield or characteristic displacement of spring y ground displacement P, P1, P2 fraction of critical damping \I ~ ~ ductility ratio = Ixlmax/Xy T, T1 T2 a time parameter 0) 3W1, W2 undamped natural frequency of small oscillations Note: Differentiation with respect to time is denoted by dots. -16 -

REFERENCES 1. M. J. Kaldjian. Assistant Professor, Department of Engineering Mechanics, University of Michigan, Ann Arbor, Michigan. 2. William R. S. Fan. Research Associate, Department of Civil Engineering, University of Michigan, Ann Arbor, Michigan. 3. Housner, G. W. "Behavior of Structures During Earthquakes," Journal of the Engineering Mechanics Division, ASCE, October 1959. 4. Goel, S. C. and Berg, G. V. "Inelastic Earthquake Response of Tall Steel Frames," ASCE Structural Engineering Conference, Seattle, Washington, May 1967. 5. Jennings, P. C. "Periodic Response of a General Yielding Structure, Journal of Engineering Mechanics Division, ASCE, Volume 90, No. EM2, April 1964. 6. Kaldjian, M. J. "Moment-Curvature of Beams as Ramberg-Osgood Functions, Journal of the Structural Division, ASCE, Volume 93, No. ST5, October 1967. -17 -

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