THE U N I V E R S I T Y OF M I C H I G A N COLLEGE OF ENGINEERING Department of CiVil Engineering Progress Report ENERGY CONSUMPTION BY STRUCTURES IN STRONG-MOTION EARTHQUAKES G~Te. Berg S. S. Thomaides UMRI Project 2881 under contract with: NATIONAL SCIENCE FOUNDATION NSF GRANT NO. 7316 WASHINGTON, D.C. administered by: THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR March 1960

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ENERGY CONSUMPTION BY STRUCTURES IN STRONG-MOTION EARTHQUAKES by G. V. Berg* and S. S. Thomaides** SYNOPSIS The effect of energy dissipation upon structural response to earthquake is considered. A spectral study of the energy relations for an elasto-plastic system responding to strong ground motion, made by high-speed digital computer, is described. The results indicate that yielding does not increase the total energy input to the system, and that maximum drift may not be adversely affected by yielding as long as the yield level is maintained above a reasonable threshold. INTRODUCTION Engineers have long recognized that energy dissipation is a key factor in explaining the observed behavior of structures in strong-motion earthquakes. The earthquake feeds energy into the structure; and to survive, the structure must, without undue damage, consume all the energy imparted to it. Part of the energy is stored temporarily in the structure as strain energy and kinetic energy. Ultimately all the energy must be dissipated by internal friction and inelastic deformation in both the structural and nonstructural elements. Linear response spectra show that if structures responded elastically to strong-motion earthquakes, they would, even though heavily damped, be required to withstand forces far above those stipulated in seismic building codes.1 The reason they are able to survive is that both the main frame and the nonstructural components remain active when strained far beyond their *Associate Professor of Civil Engineering, The University of Michigan, Ann Arbor. **Graduate Student in Civil Engineering, The University of Michigan, Ann Arbor. 1

elastic limits, continue to offer resistance to motion, and thus consume the energy of the earthquake. If a structure were capable of storing as strain energy the maximum energy input from an earthquake, it would certainly survive regardless of its dissipative properties. On the other hand if a structure were able to dissipate energy as fast as the earthquake delivered it, without showing severe damage, it would survive regardless of its strain-energy capacity. Neither of these conditions is reasonable as a basis for aseismic design; indeed, the latter is impossible to achieve. Instead, a reasonable balance must be sought so that the strain-energy capacity and dissipative capacity, working together, can accommodate the energy delivered by the earthquake. Taking this view of the problem, one can immediately make an observation about present building codes; and the significant observation is not what is present in the codes but what is lacking. The strain-energy capacity appears in the codes in the form of seismic coefficients, but the dissipative characteristics of the structure are not reflected at all. One cannot say that energy dissipation is ignored completely, for it appears tacitly in the form of reduced seismic coefficients. But many existing codes place the same strain-energy requirements on the structure whether it is damped to one percent or thirty percent of critical damping. Also, the ductility of the structure is not accounted for. The danger of brittle fracture is circumvented byi detailed specifications, but existing codes give no reward for a large reserve of plastic st-rength and they levy no penalty for a lack of it. The traditional building of a decade or two ago, with its steel frame and masonry walls and partitions, had a fairly large energy-absorbing capacity. In contemporary architecture, light-weight metal panels and glass are more the rule than the exception. It is quite apparent that structures of today may differ widely in their dissipative properties, and it seems only logical that seismic codes should somehow take this into account. The question how best to do so may not be completely answerable in the present state of the art. If one adheres to the assumption that energy is dissipated only by viscous damping, one can use response spectrum techniques1'2 to appraise the effect of damping upon the response of the structure. For other than linear viscous damped systems, the effects of damping and energy dissipation are more difficult to assess. If only the velocity terms in the equations of motion are nonlinear, one can sometimes bring an equivalent viscous damping into the picture.3', However, an equivalent viscous damping is not very meaningful when significant nonlinearities are present in the restoring terms. There does appear to be good reason to give attention to systems which are strongly nonlinear. A real structure subjected to an earthquake strong enough to challenge its integrity significanty will respond quite differently from the ideal linear oscillator. In the early part of the response, the nonstructural elements may provide the dominant part of the restoring force. As the amplitude increases, the nonstructural components yield or crack, throwing 2

to the main frame the burden of providing the restoring forces. Still, the nonstructural components continue to resist motion and dissipate energy, actting perhaps somewhat in the nature of Coulomb frictional forces. Finally, if the amplitude gets great enough, even the main frame yields and draws on its reserve plastic strength.5 ENERGY STUDIES Energy relations provide a common basis for appraising the dynamic response of systems of widely differing nonlinear characteristics. To explore some of the basic questions concerning energy components for structures responding to earthquakes, a research project was initiated at The University of Michigan about a year ago under the sponsorship of the U. S. National Science Foundation. The objective of the research was to study the epergy relations for elastic and elasto-plastic systems subjected to earthquake forces to learn how the energy components are influenced by variations in the structural parameters. The elasto-plastic system was chosen for study because of its simplicity. To avoid an excess of detail, the studies were restricted to an ideal single-degree-of-freedom elasto-plastic system with velocity damping, as shown in Fig. 1. The same system also serves as the linear viscous damped system when the yield strength is set high enough. The problem lends itself well to a spectral type of analysis using a high-speed digital computer. The restoring element in the system of Fig. 1 is an elasto-plastic spri, obeying the relations Q = kx if jQJ< y or Qx < 0 Q = O if IQ I Qy and Q x > 0 in which Q is the spring force, Qy is the yield strength of the spring, and k is the spring constant in the elastic range. These are the usual elasto-plastic relations, with hysteresis, as shown in F 2. The equation of motion for the system is m(R + y) + c x + Q = 0 in which c is a viscous damping coefficient, and the ground acceleration ~ i a known function of time. To reduce this to a unit mass basis, let q = Q/m,

qy = Qy/m = yield level Lco - = undamped natural frequency in the elastic range, and cm -B = - = fraction of critical damping in the elastic range. 2nmw The equation of motion then becomes x + 2x + q = -(t) whe re q = 0 if Iql = qy and q x > 0 q= 2 if il < or q x <_ The yield level gy is the lateral acceleration which would just bring the restoring force to the yield strength. The force per unit mass exerted on the structure by the foundation is - (q + 2powk), and the rate at which energy is being delivered to the system is E = - (q + 2cx) y The force per unit mass exerted on the damper is 2fWx, and the rate at which energy is being dissipated by the damper is L = 2f3x2 The kinetic energy per unit mass is T = l(x.)2 2 The energy of deformation may be resolved. into elastic strain energy, which is recovered when the system returns to the equilibrium position, and energy dissipated in plastic deformation. The recoverable strain energy per unit mass is q2 U, and the rate of energy dissipation in plastic deformation is D -= qx if 1 ql = qy and qx >'0 5 = O if jqj < qy o: qx < O

The equation of motion and the equations for E and L were rewritten for computer solution as the following system of first-order differential equations: z ='y(t) (z = ground velocity) v = - ty(t) + q + 2pwv] (v = system velocity relative to ground) x = v = - (q + 2 v) z L = 2:v2 The equations were solved by a Runge-Kutta third-order procedure.6 The plastic energy dissipation, D, and the restoring force, q, were obtained directly from difference relations rather than through differential equations. The reason is that both i and D are discontinuous whenever the system enters the plastic range. In a difference formulation the discontinuities cause no difficulty. Let xn, qn, and Dn denote the values of x, q, and D at the beginning of a small interval of time; let Xn+l, qn+l, and Dn+l be the values at the end of the interval; and let Ax, Aq, and AD be the changes during the interval. Assuming that x does not change sign during the interval, the equations for q and D lead to the following relations: If (qn + q2Ax) > qy, then qn+l = qy and AD = qn+ 8x - A ) or If (qn + 2Ax) <qy, then n+l = - y and AD = q+l x -) or If Iqn + c2AxI <_ qy, then qn+l q= + c2Ax and AD = 0 The assumption of monotonic x during the interval requires the interval to be small compared with the natural period of the system. This is required in any event, to insure accuracy in the numerical solution of the differential equations, so the use of a difference formulation for Aq and AD imposes no additional restrictions. In solving the system of equations, the computer evaluates all the variables at small intervals of time and records the maximum magnitude of each variable.

DIGITAL ACCELEROGRAMS The input functions "(t) consisted of punched card accelerograms7 of the two horizontal components of four strong-motion earthquakes: El Centro, California December 30, 1934 El Centro, California May 18, 1940 Olympia, Washington April 13, 1949 Taft, California July 21, 1952 The functions y(t) were piecewise linear approximations of the accelerograms. Because a strong-motion accelerogram is very nearly a broken line diagram, especially during the strongest part of the ground motion, the error in the piecewise linear approximation is probably not large compared with the error in the accelerogram itself. The time-acceleration coordinates of the intersection points of successive line segments were put on punched cards. An arbitrary zero axis was chosen for the accelerogram, and the coordinates were scaled from the record using an arbitrary scale of measurement. The resulting data were then processed by machine to introduce appropriate scale factors for time and acceleration. The initial ground velocity, initial ground displacement, and the adjusted location of the zero axis were chosen to minimize the mean square displacement; i.e., to make ( y2dt a minimum, where y is the ground displacement found by integration, and L is the length of the punched card accelerogram. This method of preparing the input data turned out to be quite satisfactory. A punched card accelerogram is compact, consisting of about 100 to 150 cards, each containing the coordinates of four points. Identification numbers and sequence numbers on the cards were checked by the machine during the computation to guarantee that no cards were missing or out of sequence. RESULTS The first computational phase of the project was the evaluation of linear response spectra. This was considered to be essential because it provided a basis for comparing the digitally computed data with previously published results obtained by electrical analog techniques." The digital response spectra compared quite favorably with the analog spectra, differences being on the order of ten or fifteen percent. Also, the ground-velocity and ground-displace6

ment curves obtained by integrating the punched card accelerograms were found to be in reasonable agreement with previously published results. This confirmed that the digital accelerograms are reasonably faithful representations of the records from which they were taken. Elasto-plastic analyses have thus far been made for only the N-S component of the El Centro, California, earthquake of May 18, 1940, and the N69~W component of the Taft, California, earthquake of July 21, 1952. Analyses of other components are planned, but have not yet been accomplished. The most significant results are the energy input curves and the maximum displacement curves shown in the figures.- - Figures 3 and 4 show spectral plots of energy input per unit mass versus period for damping ranging from zero to twenty percent of critical and for yield levels ranging from fully elastic (infinite yield level) down to one percent of gravity. The following observations are pertinent: The elastic undamped energy curve is a very jagged curve for which small changes in period may be accompanied by large changes in total energy input. The presence of dissipative forces in the system, either in the form of viscous damping or inelastic deformation, has a marked smoothing effect on the energy curves. In the elastic case an increase in the fraction of critical damping does not cause any decided change in the average value of the energy curve. However, at low values of yield level an increase in damping is accompanied by an increase in total energy input. On the average a decrease in yield level is accompanied by a decrease in total energy input at all levels of damping. An exception to this occurs at the low end of the period scale. In this region, for periods of about 0,4 second or less, the energy curves drop off as the period approaches zero; and the curves for high yield- level drop. more rapidly than the curves.for the low yield leve 1. Figures 5 and 6 show maximum displacement plotted against natural period. for various fractions of critical damping and various yield levels. The significant observation here is that a decrease in yield level is not accompanied by an increase in maximum displacement until the yield level gets below.06go Even with the yield level as low as.03g, the maximum displacement is not severely affected. However, when the yield level drops to.Olg, there is a marked increase in the maximum displacement. 7,

ACCURACY In this, as in any problem involving the response of a nonlinear dynamic system, one might justifiably question whether inaccuracies have crept into the analysis to destroy the validity of the results, or whether the problem is so ill-conditioned that small differences in the input data-might become greatly magnified to produce large differences in the response. The errors involved in the digital computation are small. Preliminary checks, using elementary input functions for which exact results were known, disclosed accuracy to four or more significant decimal digits. Each time an earthquake analysis was run, the energy balance, i.e., the difference between the energy input and the sum of the energy components, was recorded at the end of the solution. The balance should, of course, be zero. Its deviation from zero served as a check on the errors generated in the solution. It was found to be possible to shrink this energy balance nearly to zero simply by decreasing the size of the time interval used in the numerical integration process. This is not altogether desirable because it increases the time required to obtain the solution. For the most part the computations were controlled to keep this energy balance below five percent of the maximum energy input. In one or two cases it ran as high as seven percent, which was considered to be well within the range of tolerable error. Concerning the second point, the nonlinearities of the elasto-plastic system could conceivably cause some magnification of the input errors. If this magnification were extreme, it would destroy the worth of the results. Intuitively, one feels that the nearly random nature of the input would preclude any extreme magnification. As a rough check, several solutions were run twice, first, using the accelerogram without modification, and then using the accelerogram with all acceleration readings magnified ten percent. This magnification in acceleration would cause a 21* increase in the energy input to a linear system. For the elasto-plastic system the increase in energy input ranged from 17o to 22%. As a second check, several cases were run with the accelerogram modified so that the odd-numbered ordinates were increased.Olg and the even-numbered ordinates were decreased.01g. The effect of this was to superimpose a small noise on the earthquake signal. The change in output caused by this disturbance was almost immeasurably small. In view of these results, it seems that the problem is not ill-conditioned. CONCLUSIONS It would be presumptuous to draw any broad conclusions or design: recommendations from the data presented in this paper. The elasto-plastic analyses were made for only two earthquake components. The authors will hazard

the opinion that other components will produce results agreeing with these in the over-all picture, although differing in the details. But this is yet unconfirmed. Also, one must remember that the ideal elasto-plastic system, while it is a convenient mathematical model and a step closer to reality than the linear viscous-damped system, still does not describe the behavior of a real structure very accurately. Despite these limitations, the results suggest the possibility of employing a design procedure that would take into account both the elastic and inelastic properties of a structure. This is essentially the same as a method first proposed by Housner8 and the foregoing results lend strength to Housner's views. The design philosophy is that a structure should be able to withstand an earthquake of moderate intensity with no more than superficial damage, and should also be able to survive a very intense earthquake without catastrophic damage. This dual objective can be attained by requiring that the structure remain in the elastic range in its response to a moderate earthquake, and also that it be able to consume the energy of a very intenge earthquake by calling on' its reserve strength beyond the yield deformation. For either the moderate or intense earthquake, the amount of energy input involved can be estimated by spectrum techniques. Because yielding tends to reduce the total energy input to the system, one can get a conservative estimate of the energy input from linear response spectrum curves or through the use of spectrum intensity concepts, rather than resorting to the more complex elasto-plastic analyses described above. Linear response spectrum curves and spectrum intensities have been published for most of the strong-motion earthquakes of record.l 8 A simple structure might be designed as follows: 1. Select velocity-response-spectrum values Svm to represent the moderate earthquake, and Svs to represent the strong earthquake. Because the Sv curves tend to be essentially horizontal lines, the natural period of the structure would not be critical. 2. Design the structure elastically to withstand the moderate earthquake without exceeding the design stress, i.e., so that 1 W Qd Xd > S2vm - 2g where W = effective weight of structure, Svm = velocity-response-spectrum value for moderate earthquake, taken at a fraction of critical damping appropriate to the type of structure, Qd = restoring force at design stress,, and Xd = lateral displacement at design stress. 9

3. To avoid excessive drift in the strong earthquake, impose the requirement that - > o05W whe re Qy = the restoring force at yield stress. 4. Check the design plastically for the intense earthquake, according to the criterion 1 W 2 Qy Xy+ 2 Qd Xd > 2g Svs whe' re Svs = the velocity response spectrum value for the strong earthquake, taken at the same fraction of critical damping as for Svm, and Xy = permissible total yield displacement. The procedure as stated here could be applied only to simple structures that have essentially one degree of freedom. Even so, there are questions that call for further study. The parameter Xy, for example, is the permissible total yield displacement, i.e., the sum, of the magnitudes of all displacements in the plastic range, in both directions. Yield could ordinarily be expected to occur in both directions. Present knowledge of structural materials is for the most part restricted to one-directional loading, and unloading, or to oscillatory loading at low stresses. Relatively few tests have been carried far beyond the yield displacement, and almost none has explored reversed loading cycles after exceeding the yield displacement. Attempts to assign numerical values to Xy would necessarily involve the use of unproved assumptions. The extension of these design concepts to multi-story structures is another topic for further investigation. On the basis of preliminary studies, the authors believe that the energy-input curves for elasto-plastic multistory structures will fall below the corresponding curves for elastic structures, much the same as in the single-degree-of-freedom systems. It then appears that a similar dual-purpose design might be appropriate for this case as well. The elastic portion of the design could be executed for a multistory structure with no difficulty, but the limit design portion is less clear. One cannot readily predict where inelastic deformation will occur. It might occur in many locations in the structure, or it might be concentrated at one or two locations. In exceptional circumstances it might even be desirable to control the location of yielding by purposely building a weak link in the structure-a structural fuse, so to speak. But this appears to have little merit as a general practice. 10

Mu.ch additional research needs- to be done before these concepts can be applied effectively to multi-story or complex structures. The analyses presented herein are only a short step down a long road. ACKNOWLEDGMENT The research reported herein is a project administered by The University of Michigan Research Institute, under the direction of the Department of Civil Engineering and sponsored by the National Science Foundation. Electronic computation was performed in The University of Michigan Computing Center. The digital accelerograms were prepared from accelerogram records furnished by the U. S. Coast and Geodetic Survey. REFERENCES 1. J. L. Alford, G. W. Housner, and R. R. Martel, Spectrum Analyses of StrongMotion Earthquakes, Earthquake Research Laboratory, California Institute of Technology, August, 1951. 2. D. E. Hudson, "Response Spectrum Techniques in Engineering Seismology," Proceedings of the First World Conference on Earthquake Engineering, Earthquake Engineering Research Institute, June, 1956. 3. L. S. Jacobsen, "Steady Forced Vibration as Influenced by Damping," Transactions, American Society of Mechanical Engineers, 1930, Paper APM-54'-15. 4. L. S. Jacobsen, Frictional Effects in Composite Structures Subjected to Earthquake Vibrations, Stanford University, March, 1959. 5. J. A. Blume, "Structural Dynamics in Earthquake-Resistant Design," Journal of the Structural Division, American Society of Civil Engineers, July, 1958. 6. G. V. Berg and D. A. DaDeppo, "Dynamic Analysis of Elasto-Plastic Structures," Journal of the Engineering Mechanics Division, American Society of Civil Engineers. (In press.) 7. G. V. Berg and S. S. Thomaides, Punched Card Accelerograms of Strong-Motion Earthquakes, University of Michigan Research Institute, Report No. 2881-1-P, September, 1959. 8. G. W. Housner, "Behavior of Structures During Earthquakes," Journal of the Engineering Mechanics Division, American Society of Civil Engineers, October, 1959. 11

A w 0 Fig.. 1Elasto-plastc system. LL0 ID ISPL A CE MENT x 0 F-d Fig. 2. Force-displacement relations.

2400 T 2000t I Undormped ~~~~~~~~~~~~~~'V 1200/ qy = 2I' 1200[ 800- - ___=.03 400 O~~~~~~~~~~~~~~~=O u,o C)~~~~~~~~~~~~~q U U) eo. C 1600 cn cn.03 1200-,XI H M 066 z wI" 800 i —;~y/.....:.; — —._ _/ M ~ ~ ~ ~ 0. 0 (3 z 400- J. U~~~~~~~~~~~~~~~~ 1200 o~~~~~~~~~~~~~~~~~~~ I gI 80040 / 1200~~" A~ =.20q.06 8'. ".........0..+Z........................ q. o /g 400[ ~ 5 I 1,5 2 2.5 3 C ~~~~~~~~~~~~r% I Mei-PP%0 I I PERIOD, SECONDS Fig. 3. Energy input spectrum (N-S component, El Centro, California, earthquake of May 18, 1940).

600500-''''~'~-'~-_Undamped./2 40 \ 300 200- f,, I =.O/g 10,o........5.. W z ~N0 - 200 - w Z, - 300 - 1001 PERIOD, SECONDS Fig.. Energy input spectrum (N69W component, Taft, California, earthquake of July 21, 1952).

SaN093S' )01 3d ~' z gl I' 0 — 1 S a.,.,..... -01 90~ l o 0/.=',-. —-,.- 9 91z LV =,.......... pedw.u ~.. 0 X Oo........... r ~~~~ ~~ rr r~~~3 rr~~~~~',9 m(/3.......... eoe*~~~~~~~~ _~.. ~ -. _1 pedtopuR V r,~~~~~~~~~~~~~~~-i..".............

('FuThOJTTFe3'!JJPI'ql1uUuoduloo Io69N) iunaxql3ds;uaGuaeTdsFC[ 9 ATh, SaN003S'COl13d ~ ~1 2 E9 i I 00I I............. /0. _.|' I I I O ~0'' b "..-'' -— ~' oz.=l 110 -',/,,~ ~ -9 Z ~00 6/o'=r l. — ",'"''.,,,..... - -9'........ pedopn,/'~.,..... oO~4 4 Q~ ~ ~ ~~~~~~~~~~~~o,,o... Ii/ I/~~~~~~.... I~~~~~~~~~~~~o......0! i ~'~ -,1~ dl ~1''ll i

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