THE UNIVERS ITY OF MI CHI GAN COLLEGE OF ENGINEERING Department of Civil Engineering THE RESPONSE OF STEEL FRAME STRUCTURES TO EARTHQUAKE FORCES PART I. SINGLE-DEGREE-OF-FREEDOM SYSTEMS M. J. Kaldjian Assistant Professor of Engineering Mechanics Wm. R. S. Fan Research Associate in Civil Engineering under contract with: AMERICAN IRON AND STEEL INSTITUTE NEW YORK, NEW YORK AISI Project No. 119 administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR June 1967

TABLE OF CONTENTS Page LIST OF FIGURES v NOMENCLATURE viii I. INTRODUCTION 1 A. The Earthquake Problem 1 B. Earthquake Characteristics 2 C. The Present Study 2 II. THE LINEAR OSCILLATOR 4 A. The Differential Equations of Motion 4 B. Steady-State Oscillation 6 C. Earthquake Response 6 III. THE ELASTO-PLASTIC SYSTEM 19 A. Load Displacement Relations 19 B. The Differential Equations of Motion For Earthquake Response 19 C. Energy Dissipation 22 D. Significant Response Parameters 23 E. Response Spectrum Concepts For Elasto-Plastic Systems 23 F. Elasto-Plastic Response Spectra For Three Strong-Motion Earthquakes 24 G. Programming Procedure 37 IV. THE RAMBERG-OSGOOD SYSTEM 40 A, Load Displacement Relations 40 B. The Differential Equation of Motion 40 C. Steady-State Oscillation 44 D. Energy Dissipation and Equivalent Viscous Damping 48 E. Ramberg-Osgood Response Spectra For Two Strong-Motion Earthquakes 56 V. EFFECT OF SHAPE OF THE FORCE-DISPLACEMENT CURVE UPON EARTHQUAKE RESPONSE 75 Ao Maximum Displacement 75 B. Maximum Energy 77 C. Yield Reversals and Energy Dissipation 77 iii

TABLE OF CONTENTS( Concluded) Page VI. INTENSITY AND TIME SCALE EFFECTS OF ACCELEROGRAM 86 VII. EFFECT OF AXIAL LOAD 92 VIII. SUMMARY AND CONCLUSIONS 102 ACKNOWLEGMENTS 105 REFERENCES 106 iv

LIST OF FIGURES Figure Page 1. Accelerograms, Taft, July 21, 1952. 3 2. Linear oscillator subjected to a sinusoidal force. 5 3. Frequency-amplitude response for steady sinusoidal load. 7 4. Response factors for a viscous-damped single degree-Of-freedom system for steady sinusoidal load. 8 5. Linear oscillator subjected to ground motion. 9 6. Velocity spectra, El Centro, May 18, 1940. 12 7. Displacement spectra for elastic systems, Taft, July 21, 1952. 14 8. Velocity spectra for elastic systems, Taft, July 21, 1952. 15 9. Acceleration spectra for elastic systems, Taft, July 21, 1952. 16 10. Response spectra for elastic systems, Taft, July 21, 1952. 18 11. Elasto-plastic force-displacement diagram. 20 12. Equivalent nonlinear system. 21 13. Response spectra for the elasto-plastic system, El Centro, May 18, 1940, S. 25 14. Response spectra for the elasto-plastic system, Taft, July 21, 1.952, S21W or Olympia, 1965, S86~W. 51 15. Reversals against period for the elasto-plastic system, El Centro, May 18, 1940, So 38 16. Ramberg-Osgood functions, 41 17. Experimental hysteresis loopso 42 18. Equivalent Ramberg-Osgood system. 43 19, Ramberg-Osgood load-displacement relations, 45 v

LIST OF FIGURES (Continued) Figure Page 20. Ramberg-Osgood hysteresis loops. 46 21. Steady-state response spectra for Ramberg-Osgood system. 49 22. Steady-state response spectra for elasto-plastic system, 53 23. Hysteresis loop for Ramberg-Osgood system. 54 24. Hysteresis loop for viscous damped linear system. 55 25. Equivalent viscous damping. 57 26. Displacement spectra for Ramberg-Osgood system, Taft, July 21, 1952, S21~W. 59 27. Displacement spectra for Ramberg-Osgood system, Olympia, April 29, 1965, S86~W. 67 28. Typical qy vs. i curves, El Centro, May 18, 1940, N-S componento 76 29. Typical qy vso. curves, El Centro, May 18, 1940, N-S component. 78 30. Yield reversal criterion and excursion ratio for Ramberg-Osgood systemso 80 31. Typical displacement-time curve for elasto-plastic system. 81 32. Typical displacement-time curve for Ramberg-Osgood system. 82 33. Typical force-displacement curves for elasto-plastic and RambergOsgood systems. 83 34. Yield level qy vs. number of yield reversals, El Centro, May 18, 1940, N-S component. 84 355 Intensity and time scale effects of accelerogram on response spectra. 89 36. Loading condition for cantilever beam with axial loado 93 37. Bending moment distribution for cantilever beam with. axial loado 95 380 Force-displacement relation for cantilever beam with axial load. 96 vi

LIST OF FIGURES (Concluded) Figure Page 39. Hysteresis-loops for cantilever beam with axial loado 99 if Ac -, - -. ---,_t,-..1._._.4i- _*__ __-,-1-. ---- -L.* r —~ I. -_-.- _ __

NOMENCLATURE c coefficient of viscous damping g acceleration of gravity k spring constant m mass qO reversal value of restoring force per unit mass = Qo/m p,q restoring force per unit mass = Q/m py, y yield or characteristic strength of spring per unit mass = Q/m r an exponent t time u x-displacement along cantilever beam relative to base v velocity xz relative displacement of mass to ground -0 extreme displacement of the restoring force XyZy yield or characteristic displacement of spring y ground displacement Cs seismic lateral load coefficient D dissipated energy per unit mass due to plastic deformation E input energy per unit mass Ec energy loss per cycle due to viscous damping Ed hysteresis energy dissipated per cycle EI energy input per cycle Es strain energy input viii

NOMENCIATURE (Continued) F0 maximum amplitude of the forcing function KIjKT a constant K.E. kinetic energy per unit mass L,L(t) energy loss per unit mass due to viscous damping M bending moment M yield or characteristic bending moment Qua restoring force Ql, yield or characteristic strength of spring Q reversal value of the restoring force Ra acceleration response factor Rd displacement response factor R', velocity response factor Sa approximate maximum absolute acceleration Sd approximate maximum relative displacement S approximate maximum relative velocity U maximum strain energy per unit mass (elastic) VB maximum base shear W weight of the system 3 fraction of critical damping C maximum strain energy input to recoverable strain energy at yield Ex excursion ratio ductility ratio T,T,T 2 a time parameter ix

NOMENCLATURE (Concluded) 10 curvature OY yield or characteristic curvature Cw frequency of input force l)2,co-a)n undamped natural frequency of a system 0~JD* damped natural frequency of a system NOTE: Differentiation with respect to time is denoted by dots. x

I. INTRODUCTION A. THE EARTHQUAKE PROBLEM It would be an ideal situation if the designer of an earthquake-resistant frame structure could know the response of the structure to the ground motion to which it would be subjected in its useful lifetimeo This response is not possible to obtain. The nature of the ground motion encountered in earthquakes and the type of structures the engineer has to work with are much too complicated for that. On the other hand, much can be learned about structural behavior in earthquakes by response spectrum analyses of past strong-motion earthquakes, Moreover, the response spectrum is a powerful tool to aid the designer of earthquake-resistant structures, A linear response spectruml 2' gives the maximum response of a single-degreeof-freedom damped linear oscillator to an earthquake as a function of the natural frequency and damping coefficient of the oscillator, The response may be expressed in terms of acceleration, velocity or displacement. The linear oscillator can be represented by a single mass, spring, and dashpot, In a multistory frame structure any one mode of vibration can be represented by an equivalent single-degree-of-freedom oscillator. To obtain the complete response of the structure, one must consider all modes of vibration simultaneously, using the method of superposition. Although this cannot be accomplished by response spectrum methods, since the maximums of the individual modal responses do not in general occur simultaneously, one can obtain the maximum response of each mode by spectrum methods and sum these to get an upper bound to the total response, A solution so obtained may be as useful as a precise solution, considering the uncertain characteristics of future earthquakes imperfect knowledge of the dynamic properties of a structure, nature of the soil which supports the structure, etc. Thus the utility of the response spectrum can be extended to multistory structures o Response spectrum analyses of strong-motion U. So earthquakes3 indicate seismic lateral forces to be much greater than the accepted code values currently in use in earthquake design, even when the structure is heavily damped, On the other hand, buildings designed in accordance with current seismic building codes have survived strong earthquakes without showing excessive structural damage. One possible explanation is that both the structural and nonstructural components remain active when strained beyond their elastic limits and the energy transmitted to the structure by the earthquake is dissipated by inelastic defor mationo, Dynamic response beyond the elastic range is therefore a topic worthy of further investigation, *Superscripts refer to reference numbers at the end of this report. 1

The elasto-plastic load-displacement relation has been used in a great majority of the studies of inelastic response to earthquake. The present study includes the elasto-plastic relation as a special case of a more general loaddisplacement relation called the Ramberg-Osgood relation,5 in which three parameters, a characteristic load, a characteristic displacement, and an exponent, characterize the behavior. Experimental work in progress at the University of California, Berkeley, on structural steel members and connections indicates that the Ramberg-Osgood relation can provide a good approximation of actual member behavior. A detailed discussion of the Ramberg-Osgood relation is presented later in this report. B- EARTHQUAKE CHARACTERISTICS U. S. Coast and Geodetic Survey records of strong-motion earthquakes show that during an earthquake the ground moves at random in all directions, and the ground acceleration is extremely irregular with respect to both frequency and amplitude. The three components of the ground motions of the earthquake recorded at Taft, California, on July 21, 1952, are shown in Fig. 1.6 This record is typical of "strong-motion" earthquake accelerograms. The maximum acceleration recorded on any U.S.C.G.S. strong-motion earthquake accelerogram to date is.50 g, recorded at Parkfield, California, on June 28, 1966. Prior to that time, the maximum recorded acceleration was the.33 g acceleration recorded at El Centro, California, on May 18, 1940. The duration of the intense portion of strong earthquakes ranges from 7 to 30 seconds.J C. THE PRESENT STUDY In this report the response of a single-degree-of-freedom structure subjected to strong motion earthquakes as well as to steady-state oscillations is studied. The principles and construction of the response spectra, which include the linear, the elasto-plastic and the Ramberg-Osgood systems, are discussed in detail. The effect of shape of the force-displacement curve on the earthquake response spectra is examined. The effect of changing the acceleration-intensity and the time scale of a known earthquake accelerogram on the response spectra is investigated. The influence of vertical dead load forces upon response of a structure is also examined briefly. 2

-a:Fj:c -,_1 — c~~~. "-..I I -- -— " — -' I-" $~ --: c' Lf\ quo. _. -~lL ~~~~~~~~~~~~~~~~0 C>~~~~~~~ d~~~~~~~~~~~~~~~~oom ry\ r-4 ~~~~J~ —' 1:-c~ * ~~~~~~~MO - — j Cl) O a ~ ~ ~~~~- --- 0 - t dW ~ ~ ~ n ~ E-i d 0 43 _1, -L~r --- r- - U3 riQ) LO $i a) - ' ~ e ' ''.'~ — -- -.- I '" 0 C.) 0 a~~~- "H 1`~~~~~~~~~~~~~ f~~~~~~~ — -W- t0).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -i o~~~~~~\o ( No z (I) ~~~[,! -, o — 1-~. 6:I I I I I -:I I toOt n.;~.1 i. ~. "jj A0LiOfWlJ JO 1LNJ O d 'NOIV~I393:Dh3N

II. THE LINEAR OSCILLATOR A. THE DIFFERENTIAL EQUATIONS OF MOTION Consider a simple linear oscillator consisting of a single mass, spring, and dashpot subjected to a sinusoidal forcing function as-shown in Fig. 2. The differential equation of motion for this system is mx + ex + kx = F0 sin c t In terms of unit mass it becomes $2 2. F X + 2 Xnx + x = o sin C t (2.1) n n m where m = mass c = coefficient of viscous damping p= -/mc =fraction of critical damping k = spring constant (stiffness) x = relative displacement of mass to ground, (function of time) cun = = the undamped natural frequency = - forcing frequency t = time Fo = maximum amplitude of the forcing function (a constant) Differentiation with respect to time is denoted by dots. The steady-state solution of Eq. (2.1) can be written as Fo x - Rd sin (Ft - 2) (2.2) Rd = (2.3) 4

, k F sin wt m ---- x g. 2 Linear oscillator subjected to a sinusoidal force. Fig. 2. Linear oscillator subjected to a sinusoidal force. 5

B. STEADY-STATE OSCILLATION Curves showing Rd as a function of frequency ratio c/Cn for various values of P are plotted in Fig. 3. The velocity and acceleration responses, obtained by differentiating Eq. (2.2) with respect to time are X- aR= - Rd cos (CUt - ) = R cos (wDt - ) (2.4) and Fo/m K^n x_ =_ Rd sin (t -R sin (wt - )(25) The response factor Rd reaches its maximum value of l/(2p 1J - _2) at co/con = /1 - 232 R p has a maximum value of 1/25 at o/cn = 1 and Ra has a maximum of 1/(2 T1 - p2) at D/wn = / 1 21 2. A family of curves showing a four-way plot for response factors as functions of c/con for various P values is drawn in Fig. 4; the grid lines sloping upward to the right are for Rd, the horizontal grid lines are for Rv, and the lines sloping downward to the right are for Ra as indicated. The Rd curves of Fig. 4 are the same as those of Fig. 3 except for the logarithmic scales. C. EARTHQUAKE RESPONSE Consider next the linear oscillator of Fig. 2 subjected to ground motion only, as shown in Fig. 5. A simple one-story elastic frame with rigid foundations can be represented by a linear spring-mass-damper system. The motion considered here is translation in the direction of the spring and dashpot. The equation of motion of the system subjected to ground motion is: mx + ex + kx -= my (2.6) where y = ground displacement, a known function of time. Equation (2.6) can be rewritten as x+22onx+o4x = - (2.7) where ojn and l are the same as previously defined. 6

5.0.=, 4.0 -:=o.1I E 3.0 U-0 =0.2 R =0.5 1;0 Ii 0 0 1.0 2.0 3.0 Fig. 3. Frequency-amplitude response for steady sinusoidal load.18 7

VELOCITY RESPONSE FACTOR RV - _ ro 4N Ooo0 N _ io O bb ro O) 0o o o o oo00 op~~ ~~~~,;s~'\'ass - t II1, /IA,O )I ~ ~,,, / / / X /,N, 1x' 00, o\ 0 ^0 n \ 0 g^\N/ ' E r RESPONSE FACTOR RESPONSE FACTOR 0) 1q8 w Wtggltz<c os ^ c> 0 ^ (?6 8 ^ > / > C ^ X ACCELERATION ~ DISPLACEMENT RESPONSE FACTOR RESPONSE FACTOR Ra Rd Fig. 4. Response factors for a viscous-damped single degree-offreedom system for steady sinusoidal load. 8

R k x (Relative to Ground) - yL(t) Fig. 5. Linear oscillator subjected to ground motion. 9

The solution of Eq. (2.7) for zero initial conditions is: x = - 1 t y( T)e- & n(t ) sin wci (t-T) dT (2.8) CDn 0 where WA = nl - 2 = damped natural frequency t = time T = a time parameter of integration D. THE RESPONSE SPECTRUM Let X = ft y(T)e-Pn(t-T) sin w* (t - T)dT (2.9) 0 and I= rt y(T)e-on (t - T) co d (2.10) The displacement response of the oscillator, from Eq. (2.8), is then x= _ j (2.11) n Thef aand integrals can be differentiated under the integral sign, leading to x =- - (2.12) x +y (1 - 22) + 2n %Pwn2 (2.13) (i^~n Both - andCare oscillating functions of time, and for earthquake input they tend to oscillate at approximately the same amplitude and about 90~ out of phase. Hence when one of the integrals is at its maximum, the other is nearly zero, Define: Sv = U- max (2.14) The maximum of the absolute value of x, from Eq. (2.12), is 10

l2Imax I oP L-CImax (2m15) n Because P is small, the C term dominates and the right-hand side of this equation becomes approximately equal to I0lmax, which is nearly equal to [max, making Hlmax Sv (2016) Similar reasoning, and assuming small values of 3, leads to Sv Ixlmax Sd (2.17) Ix + Ylmax (- nSv = Sa (2.18) Note that Sd and Sv are approximately equal to the maximum relative displacement and velocity, respectively, and Sa is approximately equal to the maximum absolute acceleration. Combining Eqs. (2.17) and (2.18) results in the simplified relationship of (rSd a S= v Sa (2.19) c-n Sa is more directly applicable to design procedures because it is directly related to the force exerted on a structure. Sv finds more frequent application in theoretical developments. For a given ground motion, Sv can be calculated by numerical integration or by analog computer for different periods and fractions of critical damping. The plot of Sv against period T with damping ratio v as a parameter, is known as the velocity response spectrum. Different definitions of velocity response spectrum have been used by different authors. Some define it to be the maximum relative velocity, others as the maximum relative displacement times on, and some others as the maximum relative displacement times aj. Unless otherwise specified, the last is the definition used throughout this report. The real velocity spectrum is obtained by plotting the right-hand side of Eq. (2.15) against the period. Typical examples of this can be seen in Figs. 6a8 and 6b.8 Arithmetic plots of Sd, Sv, and Sa against the period T for various values of B are shown in Figs. 7, 8, and 9, respectively and are typical. When the fraction of the critical damping and the period for a single-degree-of-freedom elastic system are known, values of Sd, Sv, and Sa can be obtained readily from these plots. As an example let T = 0.5 sec and 5 = 0.0. The corresponding response 11

o 6o 0 \ (a) N-S component 50 40 P- - = '0 0 \ / B=~1 1 30 20 0 0.5 1.0 1.5 2.0 2.5 3.0 UNDAMPED NATURAL PERIOD, SECONDS Fig. 6. Velocity spectra, El Centro, May 18, 1940. 12

70 — (b) E-W component 060 -4o H50 ~ 20 =.20 0 0.5 1.0 1.5 2.0 2.5 3.0 UNDAMPED NATURAL PERIOD, SECONDS Fig. 6. Concluded. 13

N~ \\ C) C'JLO.\ roJ 000 C1 0 QQ I / /I0 co ooI I -, II U I ~O aL (U "U - r.. _ U -p C Y)C) '~. ^ \.% / o ~ / I0P0 (/) PS ^^ /I 1 1)

CD~ 0 Lo r rq I II a) I. H~~~~~~~~~~~~~~~~~~~~~~r:I,~~~~~~~~~~~I: -p U) *1 U)~~~~F Cy U) co EC) )) r 4 a) C) co: A 0 co tU) /I~~~~~~~~~~~~~~~~~~.'5~'

0 - 0) I I ):) -PArC\.J LL) 00 0 Hl In 1 CU It, C~j Cl) CJ -ON -p H I II r1i /i IP II da %CA~~~~~~~~~~~~~~~~~/~ 0 — 0 CP 0) 1 N 0 ^t '~~~~~~~~~~~~~~~" JC4 C 4J, Ca' LO ~ CI) -p Ca U" C-' CC 4l' co 0 4, 161 ON 07 (5) ~& 0 16~~~~~~~

spectra from Figs. 7 to 9 are Sd = 3.55 in., Sv = 44.4ino/sec, and Sa = 1.44 g. Figure 10 shows the same velocity spectra of Fig. 8 plotted in a log-log scale. Because of the relationship shown in Eq. (2.19), logarithmic diagonal scales can be constructed-for displacement sloping up to the right, and for acceleration sloping up to the left-and values of all response spectra (Sa, SV, and Sd) read directly from the same plot. To illustrate this, let us use the same example as before. To find the response spectra one follows, in Fig. 10, T = 0.5 sec vertically up until the curve 3 = 0.0 is reached. The intersection of a horizontal line through this point and Sv-axis (the vertical scale) gives Sv = 44.4 in./sec, the intersection of a sloping line parallel to Sd-axis through the same point and the Sa-axis gives Sa = 1.44 g. Similarly the intersection of a line parallel to Sa-axis through the same point and the S-axis give Sd = a dd 3.55 in. These are shown by the dotted lines in Fig. 10, and, as expected, are the same as the results obtained from Figs. 7 to 9o Thus the-:-maximum displacement, maximum velocity, and maximum acceleration, all three, can be obtained directly from a single log-log plot. The maximum base shear VB for a simple oscillator is related to the response spectrum as follows: V = k Ilmax (m n) ( = vm (2.20) Alternatively, one may write _a B = m x + Ymax - m Sa W (2.21) where W is the weight of the system and g is the acceleration of gravity. The quantity Sa/g corresponds to the seismic lateral load coefficient C in seismic building codes. Typical spectra for the latter are shown in Figs. 9 and 10. The values given by these graphs, which may be considered typical for strong motion earthquakes, are much greater than the code values currently in use in earthquake design. The maximum strain energy per unit mass U, developed in the system during the earthquake is U = 1 k x S2 (2.22) 2m max 2 v When a multi-degree-of-freedom system is elastic and its damping forces satisfy certain requirements, the structure possesses modes of vibration equal in number to the number of degrees of freedom. Each mode behaves as a single-degreeof-freedom system. Response spectrum techniques can be used to evaluate the maximum base shear in each mode, which in turn may be used to obtain an approximate value for maximum base shear in the structure. 17

100 80 60 = =-o. o 60, =0.02.-... --- =0.05 40 / I \ 5/. \, A \y/ /V n I0o 10 -/ ~ I / e / Component: S21~W '~Nr 1 *, / I 0.1.2.3.4.6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. 10. Response spectra for elastic systems, Taft, July 21, 1952. 18

III. THE ELASTO-PLASTIC SYSTEM A. LOAD DISPLACEMENT RELATIONS With the present trend toward lighter construction in buildings, due mainly to architectural and economic considerations, the structural frames may need to withstand strains beyond their elastic limit in order to survive a strong earthquake. Response spectrum analyses of strong motion earthquakes clearly show that elastic analysis cannot be reconciled with the observed behavior of actual structures in earthquakes. Invariably the elastic stresses indicated by the spectrum far exceed the yield strength of the structural mateterials. In the light of this knowledge, recent efforts have been made to extend spectrum concepts to inelastic systems. The force-displacement relationship of most structural members beyond the proportionality limit is difficult to characterize as a mathematical model. The model which has received the most attention in recent years, largely because of its simplicity, has been the elasto-plastic system, shown in Fig. 11. Ordinarily no attempt is made to account for softening or possible decaying of ultimate strength of the member due to load reversals. A single story frame with elasto-plastic members can be represented as an equivalent nonlinear damped oscillator consisting of a single mass, a spring, a dashpot, and a coulomb friction element of capacity Q as shown in Fig. 12. B. THE DIFFERENTIAL EQUATIONS OF MOTION FOR EARTHQUAKE RESPONSE The equation of motion of the system when subjected to ground motion is, m (x + y) + c x + Q = 0 (3-1) This can be rearranged and written on a unit mass basis as x + 2P x + q = - y (5.2) where Q q _ m and A and OIn are as defined earlier. The yield level of the system qy is defined as the amount of force per unit mass necessary to just bring the spring to its yield strength, i.e., qy = Qy/m. The force-deflection relation from Fig. 11 for the equivalent unit mass sys19

Xy O^ //- /x l=0 Qy u) k ih h /k ZF /} /1 /1 I DISPLACEMENT Z| IXmax= Xy a. / f/ QYQy = Yield Strength of Spring Qy 0 x = 0 y = Yield Displacement of Spring Fig. 11. Elasto-plastic force-displacement diagram. 20

m /^~~ c ^ O- y(t) K Fig. 12. Equivalent nonlinear system. 21

ter (Eq. (3.2)) obeys the relations: 4 = 0 if Iq = qy and q i > 0 (3.3) q = (o2 x if qal < qy or q x < O The equation of motion (3.2), a second-order differential equation, was rewritten for the computer solution as the following two simultaneous firstorder differential equations: x = v and v = x = - (y + 2P wn v + q) (3.4) A Runge-Kutta fourth-order procedurei' (with the aid of a digital computer) was used to solve Eqs. (3.4). C. ENERGY DISSIPATION The force per unit mass exerted on the structure by the foundation is - (q + 2p ca x), and the rate at which energy is being delivered to the system is, E = - (q + 2 cnx) y (3.5) The force on the damper per unit mass is 2p wni, and the rate of energy dissipation by the damper is, L. = 2B on2 (3.6) The kinetic energy per unit mass is, K.E. = (x 2(7) The strain energy is made of elastic (recoverable) energy and plastic (dissipated) energy. The elastic energy per unit mass is, u = _ (3.8) and the rate of energy dissipation in plastic deformation is8 22

D = qx if ql = q and qx > 0 (3 9) = o if Iql < q or q < 0 The sum of the various components of energy in the structure must, of course, equal the energy input E = L + K.E. + D + U (3.10) This relation provides a useful check on the accuracy of the computed response. D. SIGNIFICANT RESPONSE PARAMETERS Two response parameters, namely, the ductility ratio and the energy ratio, provide a meaningful characterization of the response of an elasto-plastic system to earthquake. The ductility ratio p is defined as the ratio of the maximum displacement to the yield displacement, p = Xmax/Xy. The energy ratio e is defined as the ratio of the maximum strain energy input per unit mass Es to the recoverable strain energy per unit mass at yield, e=:2(Es)max/qyxy. If all yielding occurred in the same direction, the two response parameters would have the relation C = 2pi -1. In any case, c > 2p. -1 (3.11) E. RESPONSE SPECTRUM CONCEPTS FOR ELASTO-PLASTIC SYSTEMS The maximum elastic spring force Qm can be expressed as Qm = CsW (5.12) Where the lateral load coefficient Cs corresponds to Ix + Y|max/g. For displacements in excess of the limiting elastic displacement xy, the maximum spring force is equal to the yield strength of spring. A plot of Ix + Ylmax against period T, for various values of ductility (or energy) ratio, and damping parameter P, results in elasto-plastic acceleration response spectra. When the ductility ratio is known the real maximum relative displacement can be obtained from the relation max = x (313) 25

In elasto-plastic systems, because the spring force is equal to Qy when displacements are in excess of xy, as previously noted, the absolute maximum displacement and the absolute maximum acceleration are related as follows, I x x IX + l max (3 14) n - I1. Wn This relation is exact when 3 = 0; otherwise it is approximate because it does not take the damping force into account. A plot of n Ix| ax/[i against period T on a log-log scale, for specified ductility (or energy) and damping ratios, results in a pseudo velocity response spectra.6,ll (Pseudo in the sense that nlxlmax/I is not equal to the absolute maximum velocity |limax, for there exists a discrepancy between the two and this discrepancy increases with increasing values of A.) From Eq. (3.14) and the pseudo-velocity concept, diagonal log scales can be constructed on the chart as was done in the elastic case, and the maximum displacement aXlmax/k, and the maximum acceleration |x + |max read from the diagonal scales directly. A typical example is shown in Fig. 13a. The maximum recoverable strain energy of the system per unit mass can be expressed by 1-~ 1xmax)2 2n ^2 F. ELASTO-PLASTIC RESPONSE SPECTRA FOR THREE STRONG-MOTION EARTHQUA.KES The input function y(t) consisted of punched card accelerograms12 of the following three strong-motion earthquakes: El Centro, California S May 18, 1940 Taft, California S21W July 21, 1952 Olympia, Washington S86W April 29, 1965 Figures 15a-f and 14a-f show pseudo-velocity response spectra plotted on a four-way logarithmic grid for various values of ductility ratio pi or energy ratio ~, and specified values of damping parameter P. This way of presenting the results, as previously mentioned, has the advantage that values of the maximum spectral acceleration and the maximum displacement can be read from the diagonal scales at the same time. To obtain the true maximum relative displacement, the values read from these plots must be multiplied by the corresponding ductility ratio go It was found that the maximum acceleration occurred inthe short period portion of the spectra, in contrast the displacement was maximum in the comparatively long period zone. The maximum pseudo-velocity in general was located between the periods ranging from 0.5 to 3.0 sec. The periods referred to above were computed 24

1 00 8 0.0 _ = 1.25 60 ----— / = 4 40. -1I.2 /.6. \ A.. / 6C y 5/ \ 2 A 'V (a),,~ / \^ l',.1.2.4.6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. 15a. Response spectra for the elasto-plastic system, El Centro, May l8, 1940 S. Constant ductility ratio " L..1~ ~~ ~ "2 '4 "'.. 24 6 8 'E10 ' " ~/ Fig.~_ X3a %epos Ipcr "%r ~h lsopasi y

100 80L -- -e 1.5._.c 3 60....-. 7 40 %, \' / Li\ 20 /.I.! I- / - /'*' ' " ' \ 0 I x J* E %.1.2.4.6.8. 2 4 6 8 10 V <j A s e ^ / /V '^ \ Vb PERIOD (SEC.) Fig. 11b. Response spectra for the elasto-plastic system, El Centro, May 18 1940, S. Constant energy ratio 'e." 26 Fig. 1~b. Response spectra for the elasto-plastic system,

100 80 - L ^.0 =.05 —. —/= 1.25 60~ > ---- = 2.0 60 --. _a40 = 4.0 40 20 Z \ ^/ \ o - / E, o,, * 6 -- O'/ /\ _ / /! ":D //~~~PERO (SEC.)' Fig c Response spectra for the elasto-plastic system,.1.2.4.6.8 1-.0 2 46 8 10 PERIOD (SEC.) Fig. 13c. Response spectra for the elasto-plastic system. El Centro, May 18, 1940, S. Constant ductility ratio " I." 27

100, E~ =1.0 80 tE =1.5 - =3.0 60 /- =.05 - e=7.0 40 o E\/ '/ ' \' \ 0 202 1090.1U...6.-8 1.0 2 4 6 8 10:' 8 /J* —,\28 c 6 28~~~~~~~~,ii

100 80.- = 1.0 - ~ —. —E= 1.25 60.-_= 2.0 40- ^=4.0 40 - 20 w U) Y.. /./'~..1.2.4.6.81.0 2 4 6 810 PERIOD (SEC.) Fig. l5e. Response spectra for the elasto-plastic system, El 8 ' M1- r-/ \ Centro May 19 S. Constant ductility ratio " 29 / / J. El Cen.ro, May l8, 1940, S. Constant ductility ratio "2." 29

100 80 = 1.0 _ =1.5 60 -- = 3.0 _._.... E = 7.0 / =.l1 40 20 s Y ^/ '*". //^. 10 o 8 E 10, 1.2.4 6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. 13f. Response spectra for the elasto-plastic system, El Centro, May 18 1940, S. Constant energy ratio "e. " * 7/ / " / "~$ 'S'~' ~r / / x <\ ^\ ~9.1 %/ ' /,....8 1. 2 4,6-8 10

100 80 -. — = 1.0 -- ~-.. —~~*-y~-P ~-/z~ = 1.25 60 = o ---- = 2.0 40_~ ~ ~ _ _ _-/ = 4.0 40 — 20 ~-~L~~'~~0 ~0 ~ XZ a <; \ * \ A 4\ **//I' ~ /2 6./8. 2 4 6 8 /,/ - /* I".,, / \,!''V \0 (a).1.2.4.6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. 14a. Response spectra for the elasto-plastic system, Taft, July 21, 1952, S210W. Constant ductility ratio "." 51

100 80 -- = 1.0 = 1.5 - — 3.0 60 -0 40 A\ w20 1 \^ / /^ \ r\ 111 \n ^y" /._ 03 r^/''f \- /.2 0 4. - '1. 2 6 0 / / \ ^ \ / \\ /E~ ~ ~~ i \ > ^/ \ ~\0 F. / R o o Tat,/ Jl 1' \ S /,,,' "" ' /~X '' /^ / 6.6 846 (b).2.4.6.8 1.0 2 4 6 8 l0 PERIOD (SEC.) Fig. 14b. Response spectra for the elasto-plastic system, Taft, July 21, 1952, S21~W. Constant energy ratio "e." 32

100 80 -- 1.0._ ---./:= 1.25 --- = 2.0 60 - -4.0 4.0 40 /9=.05 20 PERIOD (SEC.) U) 33(~ 10- \</\ 10; 8 C l<~: V^7 / 0-v~.1.2.4.6.8.0 2 4 6 8 10 PERIOD (SEC.) Fig. 14c. Response spectra for the elasto-plastic system, Taft, July 21, 1952) S21~W. Constant ductility ratio "T T" 33

100 80 _= 1.0 = 1.5 60 --- = 3.0. -.-.- e = 7.0 40 =.05 20 v), o s /...2.4'.26 8 8 " 6 -A ~. - d e, t.th \p s ~~~~4 J*\9^, ~3 0 ~ " N.,(", \' PERIOD (SEC.) Fig. l4d. Response spectra for the elasto-plastic system, Taft, July 21, 1952, S210W. Constant energy ratio "e." 34

100 80 =1.0 p.= 1.0 0 —. — = 1.25 60 - — a = 2.0 / A= o _.__ = 4.0 40 20 Cn o y \ d \ /' -/.! \ / o 3 E:a- 8 -C' 1~ / / '/< /,,' / \x ' \1! / \.~/~I,/ I~\ ~0\9 ~(e).1.2.4.6.8 10 2 4 6 8 10 PERIOD (SEC.) Fig. l4e. Response spectra for the elasto-plastic system, Olympia, July 21 1965, s860w. Constant ductility ratio 55

100 80 1.. = 1.0 = 1.5 60 - - =3.0 6P ~~ _ - 0= 7.0 40 20 - - I 1 i_ - ~. -/, "I ~1'P III1 \ " 1 ' \-. \\ ' l\\K%/ rci * \/ \ ^ \. / i O I "t8 -' 4/I I.1.2.4.6 8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. l4f. Response spectra for the elasto-plastic systems Olympia, July 21, 1965* S86~W. Constant energy ratio "e." 56

from the initial elastic behavior, i.e., the spring constant k. Ductility ratios of 1.0, 1.25, 2.0, and 4.0 were considered. With no reversal, the corresponding energy ratios would be, respectively, 1.0, 1.5 3.0, and 7-0. It is found that the spectral values for these energy ratios are much higher than those for the chosen ductility ratios. This was anticipated because the system is expected to experience yield reversals when subjected to an earthquake. Figures 15 a and b show the number of yield reversals plotted against the period of the system for a set of ductility ratios. A yield reversal is defined as a change of restoring force from yield in the positive sense to yield in the negative sense, or vice versa; strain reversals which do not involve going from yield to reverse yield are not counted. A lower bound for the maximum acceleration is obtained in terms of the energy ratio, by combining Eqs. (3.11) and (3.14). 2032 + Y n x (x.16) max - 1 + c Xmax G. PRGNG PROGRAMMING PROCEDURE As was described earlier, the response of a given system to an earthquake can be calculated by a numerical method. A fourth-order Runge-Kutta procedure was used for this purpose on the 7090 digital computer at The University of Michigan. The procedure used to obtain a response with a desired value of ductility ratio or energy ratio was as follows: The responses for a number of systems with arbitrarily assigned yield levels were first computed. Then a linear interpolation method was adopted to interpolate between the obtained responses in order to determine the approximate yield levels which would give the desired results. These yield levels were used as input data and the corresponding responses calculated. This procedure was repeated until the desired responses were obtained. In the elasto-plastic system the accuracy of the final results relative to the desired values was within half of one percent. 37

/ / / 0 ~\*. \3 -- p 0 c'J \ - %J. CH rj ~ * r ---........ i. -p.* C~O~~~~~~~~~~ ~ C( -rr-4 ^^^^^^*^^ / ^ ^~~~~~~~~~~~-r

LO ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ It 11'~~~~~~~~~~~~~~~~~~~~~~~~~~~ / j / ) ~ " oC Ic\ II t n / | / =t =i. / ~ -0 (C)C)~~~~~~~~~~~~~~~~~~~~~~~C C)~~~~~~~~~~~~~~~~~~~~~~~C I I p / ~~~~~~~/ \ o II' in I~~~~~~ I \ \ \ *~~~a \ / ~~~Q- III ^- ^. /^^' (~~~~ r-4i U) CY) I. I S1lVSaI3A3UI 59~~~~~~~~~~~~0

IV. THE RAMBERG-OSGOOD SYSTEM A. LOAD-DISPLACEMENT RELATIONS Actual structural members do not exhibit ideal elasto-plastic load-displacement relations, rather, the load-displacement curve has an elastic branch followed by a transition curve that leads to a plastic branch. Upon reversed displacement, the Bauschinger effect makes the transition more gradual. This behavior can be represented quite closely by a relatively simple mathematical model, the RambergOsgood function, shown in Fig. 16. Three parameters are employed, a characteristic or yield load Qy, a characteristic or yield displacement xy, and an exponent r. It is the exponent r that governs the sharpness of the break away from the elastic branch. The Ramberg-Osgood function includes as special limiting cases the elastic case, obtained by setting r = 1, and the elasto-plastic case, obtained as r tends to infinity. Some of the hysteresis loops obtained in recent tests of structural members 14 and connections at the University of California are shown in Fig. 17 along with a Ramberg-Osgood loop with the parameters Qy, xy, and r chosen to give the best fit in the sense of least squares. Fitting the curve to experimental data requires too much computation to be done by hand, but with the aid of a computer the task is relatively simple. Curves have been fitted to other experimental load-displacement data, and it is found that the closeness of fit shown in Fig. 17 is about typical. The Ramberg-Osgood representation of the load-displacement relation is considered realistic if the structure is capable of maintaining stable, nondeteriorating hysteresis loops.15,16 B. THE DIFFERENTIAL EQUATION OF MOTION A one-degree-of-freedom structure with Ramberg-Osgood characteristics can be represented by an equivalent oscillating system made of a single mass and a Ramberg-Osgood spring as shown in Fig. 18. The equation of motions for this system when subjected to ground motion is given by, m (x + y) + Q = 0 (4.1) which upon rearranging and expressing in terms of unit mass becomes, X +q = - (4.2) where < Q 40

LO -r C') 00 on~~~.~ ~ 0 10 -8 - 0 11 \ \ \ a *H0 \ \ I uY) ^s. c^ \ \ I x X C4l C1j C\ 4]I

16 -- Test Results --- Ramberg - Osgood 12 ~ 40 8WF20 4 fifi~ jfjI 1~ Top Fiber Strain -1% -.5% I 0l.5% 1%1 5% Fig. Eeri l -8 — 16 Fig. 17. Experimental hysteresis loops. 42

' Q(x) >. y(t) Fig. 18. Equivalent Ramberg-Osgood system. 45

The relation between the restoring force q and displacement x, see Fig. 19,is given by, x_ = q (1 + q-I Ir - xy qy qY where Qy = Q/m, the characteristics restoring force per unit mass. C. STEADY-STATE OSCILLATION The dynamic response of actual structures to steady-state sinusoidal excitation can be obtained experimentally. Thus a study of the resonant amplitude, as a function of frequency and maximum amplitude of the forcing function, equivalent viscous damping to express inelasticity, etc., would be useful in determining the Ramberg-Osgood parameters of real structures.l6 The steady-state oscillation of a single-degree-of-freedom system with hysteretic force-displacement relation of the Ramberg-Osgood type, shown in Fig. 20, has been studied both by the energy method5 as well as by the method of slowly varying parameters.5'17 The energy method is limited in scope, for it gives the response at resonance only. The results of the slowly varying parameters method are considered in this section. The latter approach gives the steady-state response for all values of /aw)n and can be used to plot amplitude against frequency curves. In the absence of viscous damping, the equation of motion for this system is, mx + Q(x) = F(t) = F cos cot (4.4) where Fo is the force amplitude and c is the frequency of excitation. The equation of motion, Eq. (4.4), in dimensionless form becomes, d2 x Q x F (_-) _- (_) = cos T (4.5) dT2 xy Q xy Qy where T = Co t n Q (x ) = Q(x)/Qy Qy x and on + = 4 my-, the undamped natural frequency 44

qy Xy qy y Xo q x+x1 q+q( I q+ql Xy ) 2xy 2xy 2xy Xy ~~/Y/~~~y r-I x-xo q-qo Q- q-qo ^^^ e^ 2Xy 2qy 2qy xi qi =- Xy qy Fig. 19. Ramberg-Osgood load-displacement relations. 45

!- I alaOa _ olco xol > > CY > CY P4 0 +Y\\O o\ xl \ \ \ \ \ u "r — 1 —. — 46 \ \ \ CY l \\ ta.r-I 0T xI. >^ 0 > 1 46

Let the solution for Eq. (4.5) be X Xo x = cos e (4.6) xy xy where e = (T + +), xo/Xy and B are slowly varying functions of T. Applying the method of slowly varying parameters5 Eqs. (4.5) and (4.6) result in the expressions, 2 CF) [ + Qo1r 2 16 r- 21o 2r (C) =- cx.(:1o)+ V y Qy (4.7) X"y [ + Qor-l 2 where -C o) 2 = - - ( ~ cos 9) cos 9 d e (4.8) xy t- I' o Qy xy Qo = the extreme value of the restoring force, and xo = the displacement corresponding to Q0. Equation (4.7) can be rewritten in the following form, )2 = c() + ( 2Fo 1 rl) 4 Qo o 2 (4.9) ~ ) = C(~) ~ () ~-()- -L) ~~- (~.9) where Xo For the elasto-plastic case, it can be shown that Eqs. (4.7) and (4.8) reduce to (I = c(4) + FO 1 4 -- (4.10) and C(4) = Fos (1 2) _ 2 (1-2) I] (4.11) 47

Combining which result in the simple expression of, ( )2 = [cos-1 (1 - ) - 2 (1- A) S ] + I( s)2 _ [ 1 2 (4.12) In order to check the accuracy of Eq. (4.7), a numerical analysis of Eq. (4.5) was performed on the digital computer. For given values of Fo/Qy and c/cn, values of xO/xy were found from Eq. (4.7) and used as initial starting points in Eq. (4.5). Then Eq. (4.5) was solved numerically by using a fourth order RungeKutta method. The error for each trial point was calculated from (xi + xf)2 + Xf2, where "i" and "f" stand for initial and final, respectively. With the help of a downhill climbing method an iteration procedure was established and the error minimized to the desired accuracy of less than 0.005. Figures 21 a-d and 22 show the results for various values of Fo/Qv and exponent r. D. ENERGY DISSIPATION AND EQUIVALENT VISCOUS DAMPING Past experience with viscous damped systems has given the engineer an intuitive feeling for the effect of viscous damping upon earthquake response. It would help him to visualize the effect of inelasticity if he could somehow express it in terms of an equivalent viscous damping coefficient. Consider the steady-state oscillation of a system with hysteretic force-displacement relations of the Ramberg-Osgood type. The force-displacement curve would follow a path such as that shown in Fig. 23 which is the same loop shown in Fig. 17 except for a change in scale. The area enclosed by the loop, Ed, is the energy dissipated in one cycle of oscillation. It can be shown that5 r - 1 Qo r+l Ed = 4 xy Q r (r0) (4.13) The force-displacement curve for a viscous damped linear system oscillating at the same restoring force and displacement amplitudes, QO and x0, respec — tively, is shown in Fig. 24. Here the width of the loop depends upon the fraction of critical damping B. Again, the area enclosed by the loop, Ec, is the energy dissipated per cycle of oscillation, and is given by the expression, 2T Ec = fnn c X2 dt (4.14) 48

-—..Varying Parameter Eq. (4.7).... Differential Equation Eq. (4.5) 16 r=3 QY!\ SXyOy 14 I F 1 \\:1.25x ~ Qy 12 Xy y 10 F F'1.00 Qy xjx~I~~~~~ j j \r \ ^~~~~~~~(a) ~~I F 6. /2. 75 Ir // Qy / / F _2~~~~~~ / Q~: so~~~~~25 0.2.4.6.8 1.0 1.2 1.4 FREQUENCY RATIO Fig. 21a. Steady-state response spectra for Ramberg-Osgood system. i9

45 -. — Varying Parameter Eq. (4.7) 45. Differential Equation Eq. (4.5) ~~I'~~~~~~~Xy' Q y Xy 35 - F 0 Q-' -= 2.00 ( -- Q 0Qy Kx Qy I/ \0 I F%(b) I I F.2.. 1.50 20 - - II.2.4.6.8 1.0 1.2 1.4 Fig. 21b. $teady-state response spectra for Ramberg-Osgood system. 50

Pcoro jet kE~.4.5) C, ry cdtio 31 \ '^^ r=~~~~~~~~5 00)^ z6~~~~~~~~~~o \ no 9t) \ %O 22 zo~~I 16 14 12 "\~~~~~~~~~~-" I~~~~~~~A 12~~..o ^a1 1 ^^e~^~~ ~,o Qy

9 \ ---- Varying Parameter Eq. (4.7) 8 Differential Equation Eq. (4.5) r \\ 10 6 ~ F.2.4.6 88 1.0 1.2 1.4 xk FREQUENCY RATIO /: I F (d) Fig. 21 Steady-state response spectra for aer-soo x^ 4 - —:52.75 3.50 0.2.4.6 88 1.0 1.2 1.4 FREQUENCY RATIO Fig. 21d. Steady-state response spectra for Ramberg-Osgood system. 52

II IIC D cs rlu~~~~~~~~~ — ' (]) LA t- C.//// o 4. o ~o ^ ^lo*/// - r- ^ _4 0. 1 0I '5, P a^o R/ sY.^ 1 2 P~ h. - L o - Q L / \ CC >~ / w I \ / 0~~~~'* * -p 'Ax

-Qo w 0 U: 0 0l/ ~~~/ i~Skeleton / ~/ | // Curve / / I // Xo x / /00o/0 Fig. 23. Hysteresis loop for Ramberg-Osgood system. 54

-Ix c. 00)~~~~~~~~~~~~~~~~~~~~() oc r o V oI 0 \ "oQ~~~~~~~~~~~~~~~~~~~~) \ \ 3a~~~~~C) wm~Q) ~~~~~~~~~~~~~~~~~QC) Cj

where 2:k Qo "In m m x On substituting Eq. (2.4) for x and 2Pcnm for c in Eq. (4.14) and integrating it, one obtains, E 2 * T P Q x (4. 15) An equivalent viscous damping coefficient Peq can now be obtained by equating Ec to Ed. The resulting equation for equivalent viscous damping is Peq - (4 ) (1 /QY) (4-16) where Qo/Q is the ratio of maximum force to yield force and 1i = xo/xy is the ratio of maximum displacement to yield displacement, popularly known as the ductility ratio. For the elasto-plastic system this reduces to Peq = (1 >) ir i >1 and (4.17) Q = 0 if i < eq The results are shown graphically in Fig. 25. Here the difficulty of trying to obtain an equivalent viscous damping coefficient becomes apparent. The damping coefficient depends upon the amplitude as well as upon the characteristics of the force-displacement relation. It should be observed that while equivalent viscous damping is a useful concept in giving the engineer an intuitive feeling for the behavior of an inelastic system, it has little merit toward producing meaningful quantitative results. E. RAMBERG-OSGOOD RESPONSE SPECTRA FOR TWO STRONG-MOTION EARTHQUAKES Equation (4.2) is evaluated, with the aid of the digital computer, for various values of qy, Xy and r, through the same procedure as used inthe elasto-plastic system. The maximum displacement, velocity, and acceleration are defined as before and the natural frequency wn is defined as fqy/Xy. The relation between o Ix lmax/k and Y + ima-x/ln established in the elastoplastic system is generally not valid for this system. Thus it is inadvisable to make a four way log-log plot. 56

0.6 0.5 - Q- %0 z 0.4 C.) / --- 0 3 cd 0.3 r0.1 0 0 1 2 3 4 5 6 7 =Xmox/XKy Fig. 25. Equivalent viscous damping. 57

The displacement and the acceleration spectra for Ramberg-Osgood systems are plotted separately on three way log-log plots in Figs. 26 a-h and 27 a-h. The values of r were chosen to be 5 and 10. To compare these parameters, ductility ratios of 1.0, 2.0, 4.0, and 6.0 were considered, and the corresponding spectra were computed directly by the digital computer. The spectra for energy ratios of 1.0, 3.0, 7.0 and 11.0 were obtained by interpolating the computed data above. The accelerograms used in this analysis were: Taft, California S21W July 21, 1952 Olympia, Washington S86W April 29, 1965 The response spectrum curves presented were constructed throughout on the basis of the maximum displacement or the maximum acceleration which occurred within the 30 sec duration of the earthquake. 58

100 80. = 1.0 60 -/ = 2.0 4 __.0. --- —.- =6.0 40 r= 5 ~~~~~~20 r-=n~~~~~~~~ ~(a) 20 0. / n 10,0 6 5 8,, X, \\ l Taft, July 21, 1952, S210W. Constant ductility ratio 'A " 59 C 4. -. ~ 1.2.4.6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. 26a. Displacement spectra for Ramberg-Osgood system, Taft, July 21, 1952, S21~W. Constant ductility ratio ".." 59

100 80, = 1.0 60. — = 3.0 _ ---= 7.0.... =11.0 40 r= 5 (b) 20 l(I 10.1 -..2.\ / \1. 6 — 6 4 '-~,, - '/", \ '.1.2.4.6.8 1.0.2.4.6.8 10 PERIOD (SEC.) Fig. 26b. Displacement spectra for Ramberg-Osgood system, Taft, July 21, 1952, S210W. Constant energy ratio "e." 60

100 80 -__ L:= 1.0 60 --- = 2.0 - _ ----,= 4. 0.-/.= 6.0 40 (c) r=5 20 w C,) 10 Q 8.1.2.4.6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. 26c. Displacement spectra for Ramberg-Osgood system, Taft, July 21, 1952, S21~W. Constant ductility ratio "t." 61.1.2.4.6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. 26c. Displacement spectra for Ramberg-Osgood system, Taft, July 21, 1952, S21~W. Constant ductility ratio "L." 6l

100 80 = 1.0 60 - ~ * = 3.0 ------ = 7.0..... - =11.0 40 r= 5 (d) 20 w Cl).1.2.4.6.8 1.0 2 4 6 8 10 0-0 Fig. 26d. Displacement spectra for Ramberg-Osgood system, Taft, July 21, 1952, S21~W. Constant energy ratio "e." 62 62

100 80 /: 1.0 60 - -- = 2.0 -— L 4.0......= /~:6.0 40 r= 10 (e) US. \.~ \ I \ i.. /20. L. 0' CU) < 10, — -,,,,," 44~~~~~~~~~~", 4 1~ PERIOD (SEC.) Fig. 26e. Displacement spectra for Ramberg-Osgood system, Taft^, July 21, 1952, S21~W. Constant ductility ratio?TI- 63 2 01-100'~~~~~~~~

100 80 = 1.0 60 -. e = 3.0 _ ---=- E = 7.0 r= 10. E =11.0 40 (f) 20 /J /%%\A. 6 10 / / 8:1 V/,-, \s.00. Tat July 21 1952,21W. Constant energy ratio. 6c I h~~~~~6.,e/\ PERIOD (SEC,) Fig. 26f. Displacement spectra for Ramberg-0sgood system, Taft, July 21, 1952, S21~W. Constant energy ratio "e." 64

100 80 I = 1.0 60 -- -~ = 2.0 _~._-:= 4.0. ----. = 6.0 -40 r= 10 (g) 20 U) 10 E 8:x 6- X ^-7^ \ \,65.1.2.4.6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. 26g. Displacenent spectra for Ramberg-Osgood. system, Taft, July 21, 1952, S210W. Constant ductility ratio "|i. "

100 80 - =.0 = 3.0 60 - = 7.0...... ~ =11.0 r =10 40- (h).2... PERIOD (SEC.) Fig. 26h. Displacement spectra for Ramberg-Osgood system, Taft, July 21, 1952~ S21~W. Constant energy ratio t"."T 66.1.2.4.6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. 26h. Displacement spectra for Ramberg-Osgood system, Taft, July 21. 1952, S21~W. Constant energy ratio "E." 66

100 80 - 1.0 60 *- = 2.0 - = 4.0..../ L= 6.0 40 r= 5 (a) 20 (/) co I. I z3 ^O ^/ 0 K 10 0 -..2.4.6.8 1.0 2 4 6 8 10 3 ~ " -'y6 \67.1.2.4.6.8 1.0 2 4 /6 8 10 April 29,, S860W. Constant ductility ratio

100 80 - = 1.0 60 = 3.0 — " — = 7.0 -.-... =11.0 40 r=5 (b) w 20 a" AAw z E 1 10 8 C - 6< - M^ <6\\.1.2.4.6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. 27b. Displacement spectra for Ramberg-Osgood system, Olympia, April 29j 19~65, S86W. Constant energy ratio "\." 68

100 80 -. 1.0 60 - -- = 2.0 = 4.0 - = 6.0 40 20 r= 5 (c) (I) I ^\ \I 10 2I I II. \, \E I, * -00 -.1.2.4.6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. 27c. Displacement spectra for Ramberg-Osgood system, Olympia, April 29, 1965, S861W. Constant ductility ratio "ft." 69

100 80 = 1.0 60 -- = 3.0 - -- = 7.0 _.. =1__ -1.0 40 r= 5 (d) 20 Cl) 0 6 0 ) ' ^ \0 L I I I I I I/I I I / _ 1 1 1 1 1\ sE,,\ *.. \.1.2.4.6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. 27d. Displacement spectra for Ramberg-Osgood system, Olympia, April 29, 1965, S86~W. Constant energy ratio "e." 70o

100 80 -- --- f = 1.0 60 _- s = 2.0 - ----- = 4.0..... M = 6.0 40 r 10 (e) 20 C,),,z~~~~~0 I' /\/ X \ ~/0 o 10.5 8 _ C.' ' ~, /:.' "\ /X -,"' - \ - - \% * /~#** \- ----- ^ \/^^ \ x-^/ ^ "N.1.2.4.6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. 27e~ Displacement spectra for Ramberg-Osgood system, Olympia, April 29, 1965, S86~W. Constant ductility ratio "1." 71

100 80 - = 1.0 60 -.~ = 3.0 - = 7.0 __. E __ =1 1.0 40 r= 10 (f) 20 C) 1o 0 43=t 8 i /\ A\ I NI I /1 2 *^/r% \\0 - 0.1.2.4.6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. 27f. Displacement spectra for Ramberg-Osgood system, Olympia, April 29, 1965, S86~W. Constant energy ratio "e." 72

100 80 _ - = 1.0 60 - = 2.0 — / —= 4.0....=,6.0 40 r= 10 (g) 20 C) 0 E 8 4 - ^^A /\ \ j::;>-/ \A^ \> ^ -.1.4 61.0 2 4 6 8 1.0 2 410 PERIOD (SEC.) Fig. 27g. Displacement spectra for Ramberg-Osgood system, Olympia, April 29, 1965, S86~W. Constant ductility ratio "p." 73

100 80 E = 1.0 60 = 3.0 _ - - -- 7.0..... ~ =11.0 40 r=10 (h) 20 U) 0I II 1.2.4.6 /\\.8 1.0 2 4 6\ E 8cl08 1 PERIOD (S\\EC.) /\\74 ~~~~~2 Nbloe~~~\ \ ~.1.2.4.6.8 1.0 2 4 6 8 10 PERIOD (SEC.) Fig. 27h. Displacement spectra for Ramberg'-Osgood. system, Olympia, April 29, 1965, s86~W. Constant energy ratio Ile.TT

V. EFFECT OF SHAPE OF THE FORCE-DISPLACEMENT CURVE UPON EARTHQUAKE RESPONSE The elasto-plastic system has been used as the basis for a great deal of earthquake research, and some basic concepts such as ductility ratio and energy absorption have been related to elasto-plastic system response. However, the actual structural members do not present ideal elasto-plastic load-displacement relation. It was found that the actual behavior of a member can be represented closely by Ramberg-Osgood function, thus the Ramberg-Osgood system has been applied to earthquake study to investigate the influence of the shape of the load-displacement curve upon the response. The results are compared for r values of 5.0, 10.0, and infinity (the elasto-plastic case). A. MAXIMUM DISPLACEMENT The question to be investigated here is how the yield level affects the maximum displacement of the system while all other properties of the system remain unchanged. Typical results are presented in Fig. 28, with the yield level qy, as a fraction of gravity, plotted against the ductility ratio ji, which is Xmax/xy/ These results are for the El Centro 1940 earthquake, N-S component. There are two sets of data with different periods as indicated to be 0.5 sec and 1.5' sec. The lines of constant maximum displacement which represent the maximum displacement for elastic systems of the periods indicated are straight lines with a 1:1 slope downward to the right, and can be determined as follows: Xmax _ 2nXmax -max on ~- ~ - -(5-1) Xy qy or Pqy n Xmax = onstant Hence log 4 + log q =contant It is observed that the computed maximum displacements for the inelastic systems are generally about the same magnitude as the maximum elastic displacement, often somewhat less at higher yield levels and sometimes greater at lower yield levels. The ratio of the maximum elastic to maximum inelastic displacement was found to be greatest between u = 2 and 6, and attained a maximum value of about 2. At very low yield levels the ductility ratios get extremely Large. 75

4-4)~~~ ~ ~ ~ ~ r0 X ci A a O / X / _r Q o /x ~y I 4) 00 U) II VI - t0 / X C) 0X / X o r- J n o.,. 1ox (a I = f ()C) 0 r- (/) / ^ ^ X ~ q ~ - 1 a) O4 X _.I 0 u 1 LL] O t Xe *O OOOco p — -- ~ Z~ 4ro X 0 I ii"I s " I I L) C)) C) (6) b13A 71 rU 0 0 0 0 0CD CD C COCD. CJ r — 0 0 0 CD (6) ~b '13A3'1 n]93IA

The data points for r = 5 and for r = 10 are not markedly different from those for the elasto-plastic system. It can be concluded that the maximum displacement is not adversely affected by yielding or by the shape of the force-displacement curve. B. MAXIMUM ENERGY INPUT The maximum energy input to the system is another response parameter of interest. This can be expressed as an energy ratio e defined as the ratio of the maximum energy input to the recoverable strain energy at yield. The results used in this analysis are the same as those used in the preceding case. Similarly, the maximum strain energy for the elastic system is a constant regardless of the changes of yield level. The response points are plotted in Fig. 29 to show how C is affected by varying the yield level qy alone. Lines of equal energy which represent the maximum strain energy for the elastic system of the periods indicated are straight lines with a 1:2 slope downward to the right. These lines can be determined as follows: = Emax = 2 max (.2) 2 Xyqy y or E q4 = 2 2n Em = constant then Log E + 2 Log qy = constant Figure 29 shows that the data points fall much closer here than in the Fig. 28, indicating that the maximum energy input for an inelastic system is approximately equal to the maximum strain energy for the elastic system. Thus we conclude that it is a good approximation to take the maximum energy input for inelastic systems to be the same as the maximum strain energy for the elastic system of the same period; moreover, it is independent of either yield level or the shape of the force-displacement curve. C. YIELD REVERSALS AND ENERGY DISSIPATION Yield reversal was well defined in the elasto-plastic system. However, the Ramberg-Osgood system has no single definition for yield reversal, and at least four definitions, for yield reversal are possible, namely, 77

/x a/) C /o 0 0 -I))CO >a o,,C o 0 1) ON / - -,8 o / C\~ J C:3-0O (::) 0 oLLn LLI 4-V/-. r- 4 LO co)(1o 4 (0(-) ()) 1 (,)3 CD —......0 0 0~~~~~~ 0,,,i 0 "' " '"~~~~~~V

(1) +Qy to -Q criterion, when a change from i in the positive sense to y in the negative sense is reached, or vice versa. (2) +x to -x criterion, when a change from xy in the positive sense to y y x in the negative sense is reached, or vice versa. (3) Sliding 2Qy criterion, when the absolute difference between Qi (current extreme point obtained by loading in one direction) and Qi+l (obtained by loading in opposite direction) is greater than twice the yield load Q, i.e., 12QI < Qi - Qi+l I (4) Sliding 2x criterion, when the absolute difference between xi (current extreme point obtained by loading in one direction) and xi+l (obtained by loading in opposite direction) is greater than twice the displacement xy,i.e.,. 2xy| < xi - xi+l| Unless otherwise specified, the last criterion is adopted in this report. A detailed description of this criterion as well as the definition of excursion ratio cx, are shown in Fig. 30. A practical way to explore the question of yield reversals is to examine the response of a particular system to an earthquake. Figure 31 shows the first 20 sec response to an elasto-plastic system to an earthquake, plotted as displacement against time. It can be noted that the yield level is 0.25 g, which is in excess of the seismic coefficients of the uniform building code, and the system tends to oscillate at its own natural period. Moreover, it is seen that the system yields not only once or twice, but twenty different times as shown. Figure 32 shows the response of a Ramberg-Osgood system (r = 10) of the same period and yield level subjected to the same earthquake. Although the maximum displacement is about the same the restoring force reaches yield only three times according to +Qy to -C criterion. The first few seconds of the response curve of Figs- 31 and 32 are shown in Fig. 335, plotted as force against displacement. It can be seen that the response follows the hysteresis loop and a complete circuit of the hysteresis loop involves two yield reversals. The energy dissipated by inelastic deformation is the sum of the areas enclosed by all the hysteresis loops. The effect of yield level upon the number of yield reversals is shown in Fig. 34, and this is for the El Centro 1940 N-S component again. The excursion ratio Ex for the Ramberg-Osgood function is defined as the sum of all deformation in the yield regions produced during the earthquake, to the yield deformation xy (see Fig. 30) The total energy dissipated by hysteresis in a Ramberg-Osgood system is related to the excursion ratio. However, unlike the elasto-plastic case there is no simple way of converting hysteresis energy to excursion ratio. If all yielding occurred in the same direction, the relation between the energy and ductility ratios can be given as 79

Q 2x_ (Q2,..I Qy //ff^ / // Qy// / /X Xy / / Fig. 30. Yield reversal criterion and excursion ratio for Ramberg-Osgood systems. / / /80 //! Qy o l/,/ / 0 (Q,X,) 4 Fig. 50. Yield. reversal criterion and. excursion ratio for Ramberg-Osgood. systems. 80

C\J co " Of E:- X) o r — C O a, LO c h,I: ~,, IC1 v 0 a4 co U. I LO (3 ) 4-~4 -r) o C) CY) C\J C) 0 - C\J CY ~~~I I a I~~a

o O co:: 0 O O 2>C 0 IS a ) 4 -cn o 0) U] C:L C\ b Q LO L Y)C; O m CCt a CY)C.rL( E. E IC) bO.r E4 I i I i I 82

CO H o^ ^0 o1' - I I Id -0) C.) o c 0.-O (V) rcc LI P C1k C) II85

x c Lf Cq) l> -, d t~~~~~~~~~~~~~/~~~, -- I "> — 0. ~i~~~~0 I I H i / X - x LU o /I /(6 ) ^ b 13.A31 ~( 01I.x,/x~8 Ic OH z I/~~~~~~~~C C CC I r/ N A k -7 -'I A -7 -7 7 H \^ ^ ^ ^ 11 I X~<( rc\1 u x 0 I /,'"II II 0 O o o.o O __. 0 cO 1 (6)^b -3A31 0 03I0 84

/Q 2r t - (r-1) L (5.5) in any case /1 2r - (r-) (5.4) When r = oo, Eq. (5.4) results in the elasto-plastic case, i.e., Eqs. (5.4) and (3.11) become identical. The energy ratio e has been proposed as a more critical parameter in inelastic earthquake design than the ductility ratio Ai, because it is felt that the energy ratio, along with the number of times the system reverses from yield in the positive sense to yield in the negative sense during the earthquake will help provide indication of how a structure would perform in a strong-motion earthquake. 85

VI. INTENSITY AND TIME SCALE EFFECTS OF ACCELEROGRAM It is of interest to consider what effect modifying an earthquake accelerogram will have on the response spectra. 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 acceleration or the time scale of the given accelerogram by a constant. The purpose of this section is to establish relations between response spectra of the modified accelerograms and spectra obtained from the original accelerogram. Consider the differential equation, x + 2 P On x + q(x) - (t) (6.1) The numerical solution of this equation for various parameters, will result in the response spectra for the given accelerogram y(t). Equation (6.1) in dimensionless form becomes, d xx ( (6.2) ~ (i-) + 2P — (-)+ () - -_ (6.2) dTa Xy dT xy qy Xy qy where, 2 = Ult n = qy/xy and (x) - (x) qy qy Xy (a) Consider now another system, with damping property P1, undamped natural frequency 1w, and force displacement relation p(z). It is desired to find the response of the new system when the latter is subjected to an earthquake having accelerations KI times those of the original earthquake, the time characteristics remaining unchanged. The differential equation for the new system is, z + 2l3lz + p(z) = -KI y(t) (6.3) 86

where z = relative displacement of mass to ground. In dimensionless form this can be written as, d Z 2) + d ( ) + = KI. 1/y( (6.4) dT12 Zy dTl zy y y Py where T1 = 1 t C? = Py/Zy and P (z) - p(Z)/Py py Zy By proper choice of parameters, i.e., P1 = P w1l = Cn P (z) = ( (6.5) Py Zy qy Xy and KI = y/y = zy/y Equation (6.4) is made identical to Eq. (6.2). Thus at any instant the following relations must be valid. d2z Zy d2x d2x dT2 x dT2 dT2 1 y dz Zy dx dx d1- = = KI dT Xy dT dT Z z = - x = K x (6.6) x y where T = T1 In terms of maximum values the above expressions can be shown to become 87

Izlmax = KI lImax lIlmax = KI Ilmax (67) and z KI + Ylmax K + max The ordinates of the new response spectra are therefore KI times those obtained from the original accelerogram. Equation (6.7) is valid for linear as well as non-linear systems provided Eq. (6.5) relations are satisfied. To illustrate the preceding, it is desired to modify a given response spectra, say Fig. 14c, in order to obtain the response spectra for an earthquake of intensity KI times that used in the former. This is done by multiplying the ordinates of the curves of Fig. 14c by KI. On four-way-log plots this result is accomplished by shifting the curves vertically by log (KI). The dash-dot curve in Fig. 35 shows the latter for K- = 3 applied to curve p = 2 of Fig. 14c. (For clearity the remaining curves of Fig. 14c are not shown in Fig. 35). (b) As a second case consider a system with damping property P2' undamped natural frequency A2 and a force-displacement relation p(z). It is subjected to an accelerogram obtained by multiplying the time scale of y(t) in Eq. (6.1) by 1/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 + 252u z + p(z) = - y (KTt) (6.8) In normalized form this becomes, d dz) + -2-<- - y(2Y( 2).F' ~ (6.9) ^ (-_) + 2z -- ( ) + ( ) - - ) (6.9) dT2 Zy 2dT2 Zy Py Zy Py where T2 = %2t.2 = P 2 Zy and p (z-) (z) Py Zy Py Again by proper choice of parameters, i.e., 88

100 -__- —. Original Spectra 80 Modified for K1 = 3......~ Modified for KT = 0.5 60 -- f =.05 Modified for Kr = 3 & 60 / = 2.0 P KT 0.5 40 / \.Y ~ 4N0 \ '. 20 -- < / '/,, ' o>\ 4 Log ( 10 / KT),,i Fi 55. Ine t an t s e ef -~~/ I-/~ \Log 2 "' J S21W ~.~.2.4.6.8 1.0 2 4 6 8 10 Fig_ 355 Intensity and time scale effects of accelerogram on response spectra. 89

P2 = P py = qy p () z = @ (x) (6.10) py zy qy Xy and KT / W/ = xy/zy Equation (6.9) is made identical to Eq. (6.2), and can be shown to result in the equalities, d2 z 1 d2 1 d2x 2 y K y dT22 Kzy dT2 d z 1 dz 1 dx dT2 Zy KTZy d xy dT z _ x (6.11) zy xy where T = T2/KT, The desired relations can now be found from Eq. (6.11) to be, 1 I Zmax KT~ Ix Imax +z + Ylmax = + (6+12) Thus to obtain the new acceleration spectra, the period scale of the original acceleration spectrum curves are divided by KTo 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 KT2. On four-way-log plots the modified response spectra described above are readily obtained by shifting the original curves horizontally as well as vertically 90

by log (1/KT). The dotted line in Fig. 55 shows the latter for KT = 0.5 applied to curve u = 2 of Fig. 14c. 91

VII. EFFECT OF AXIAL LOAD It was shown that in the absence of axial load, the force-displacement relationship for structural steel members can be expressed closely by a RambergOsgood function. Likewise, the relationship between the moment, M, and the curvature, 0, of a structural member can be expressed in the same form, = <1 )+ M ) (7.1) where M = a yield or a characteristic moment and Oy = a yield or a characteristic curvature The parameters My, Oy, and r are chosen by means of a least square method to give the best fit to the experimental data. It is known that the plastic-hinge-moment capacity of a steel member reduces with increase in axial load acting on the member. However, the value of the plastic-hinge-moment remains practically unchanged when the axial load P is small in comparison with the ultimate or critical load Py of the member.2 As a result the moment-curvature relationship in this case does not change appreciably and Eq. (7.1) holds. For large values of P, the values of My, y, and r should be reestablished. Due to insufficient experimental data available in the literature, the latter is not possible to accomplish at this time. Hence the analysis that follows will be limited to structural members subjected to small axial loads only. Consider the cantilever column shown in Fig. 36, subjected to a constant vertical load P and a variable horizontal force Q. The force-displacement relationship for this column can be obtained from Eq. (7.1) as, d2u = y _M (1 + I M I-) (7.2) dy- 2 W My where u = transverse displacement relative to base. d2u = y = length along cantilever 92

y P X r I!I I I / I u I L y I I I I I 7r77 /S Fig. 36. Loading condition for cantilever beam with axial load. 93

M = P(x - u) + Q(L - y) (7.3) The numerical solution to Eqs. (7.2) and (7.3) is very time consuming, hence an approximation was made. The actual bending moment curve was replaced by a linear moment variation as shown in Fig. 37. As a result, Eqs. (7.2) and (7.3) reduce to d u MF(l ) - (+ Ml - YL l ( Oy - (1 4I L (7.4) dy2 (M M) where MF = QL + P x (7-5) This is the moment at the fixed end of the column. Integrating Eq. (7.4) twice and using the boundary conditions u'(0) = u(0) = 0, the following expression is obtained, x = yL MF (1 + 3 L Mir-l) (7.6) 5 r +2 The lateral load from Eq. (7.5) is, Q = (MF- P x) (7-7) Equations (7.6) and (7.7) can now be used to construct a force-displacement curve. A comparison of the approximate method results with the exact solution is presented in Figs. 38a and 38b. It can be seen that the deviation between the results of the two methods depends on the value of the three parameters employed, namely, ductility ratio, exponent r and axial load ratio P/Py. It is appropriate to investigate how axial load affects the shape of the force-displacement hysteresis loop. The skeleton curve for this is obtained from Eqs. (7.6) and (7.7), and the branch curves are derived (see Fig. 20), from the branch moment-curvature expression of - _ o _ M- Mo M- M+ r-l =- (2 My 1 + O ) (7.8) 20y 2 2 and from which, making the same approximation for moment distribution as before, and integrating twice over the proper limits, the branch equation for the lateral displacement is obtained, 94

P Q t u ___ M(L) = 0 Approximate L\ Exact y 0 M(O) = MF Fig. 37. Bending moment distribution for cantilever beam with axial load. 95

1.25 (a) r = 10 1.00 Exact and Approximate Results Coincide.75 P.50 - 0.0 p/ My.25 Y L o 1 2 3 4 5 1.25 - Approximate Results 1.00 Exact Results.75 0Q ~~~~p0.2 0Y.50 Py- / 2.50~~~/ ~~Qy=[My-.2Py xy (l+3/(r+2)]/L.25 0 1 2 3 4 5 1.25 - Approximate Results 1.00.75 - /Exact Results.50 / -=0.4.25 /Qy =[My-.4Py xy (I+3/(r+2))]/L 1 2 3 4 5 X Xy Fig. 38. Force displacement relation for cantilever beam with axial load. 96

1 50 1.25 1.25/ - (b) r = 5 1.00 Exact and Approximate Results.75 P =0.0.50 P/ y ~/ Q^~~Y L.25 0I I I 0 1 2 3 4 5 Approximate Results i.25 Exact Results 1.00 P. 2 Q / Qy.75 Py / Qy=[My-.2Pyxy (1+3/(r+2))]/L.50.25 / o.- I I I I 0 1 2 3 4 5 Approximate Results 1.25 1.00 Exact Results.75 / p -=0.4.50 r / Qy= [My -.4Py Xy (1+3/(r+2))] /L.25 0 / I 0 1 2 3 4 5 -y Fig. 38. Concluded. 97

- = L2 ME - M (+ 3 IMF - rl) ( -) 2 y 5 2My r+2 2My The corresponding lateral load Q = L (M - P x) (7.10) In Eqso (7.9) and (7o10), x0, Moy and Qo are, respectively, the displacement, the end moment and the lateral load at the reversal point. The branch curves [Eq. (7.9)] are similar to the skeleton curve [Eq. (7.6) ] but twice as large. This is the same relationship that exists between Eqs. (7.8) and (7.1). Figures 59a and 39b show hysteresis loops computed from Eqso (7~9) and (7~6) for various values of P/Py and exponent r. Unlike ordinary Ramberg-Osgood functions, when axial load is present the maximum lateral force does not always occur on the skeleton curve. In steady-state vibration, for (Qmax)skeletnn greater than Q,, the maximum lateral force will occur on the loop, see Fig. 39a(iv) 9 and is given by the relation (Qmax)branch = 2(Qmax)skeleton - Qo (7.11) Figure 40 shows the response of a single degree of freedom system for the first few seconds of El Centro earthquake using Eqs. (7.9) and (7.6) as the restoring force Q in Eq (4.1). The response is plotted as lateral force against displacement for P = 0 (without axial load), and for P = 0.4 Py. In this case Qmax and xmax are found to be greater without axial load. Equations (7,9) and (7.6) give exact results only when the axial load is zero. Otherwise the results obtained from them are approximate and must be used for small values of P/Pyo The presence of axial load, see Figs 38a and 38b, seems to reduce the stiffness of the system. When axial load is present, it is no longer possible to express the forcedisplacement relation by ordinary Ramberg-Osgood functions as displacements become large. The brief discussion presented in this section seems to point out clearly that further investigation into the subject, especially in the experimental side, is needed. 98

Cy) d (l a. a\'V c co a) Q1) ci C\l CH N CM 01) 0 CY') CH Ia., a. I ii 4' \ U> ~ i \ _\ ~ \ alC _Coy I C\ \,, kI 99

II) g>w >bII I N I \ *id~r ^^a a. CC I rd -H C) CV 0 dV Q ojcu IT \ \010I II aC XI I N 1 II!IC m 100

P = 0.4 Py -------- P = 0 Shear Force (Kips) _6 5 0 -OVI,~ ~I I / I 1 I with axial load subjected to ground motion.(Inhes) -5 -4 -3/ -2 -1 0 12 101!l iL-1 I I,if /- -2 J i! 1 I / -3 / [__L^ ^ -4 I 0 —. Fig. 40. Shear-displacement response of cantilever beam with axial load subjected to ground motion. 101

VIII SUMMARY AND CONCLUSIONS In this report, the response of a single-degree-of-freedom structure to strong-motion earthquake was studied. The principles and the construction of the response spectra were discussed in detail for three different systems, namely, the linear system, the elasto-plastic system and the Ramberg-Osgood system. The discussion for each system generally included subjects such as force-displacement relation, equation of motion, energy dissipation, response spectrum concepts, steady-state oscillation, and response spectra for strong-motion earthquakes. Also, a family of spectral curves was presented for these systems, where the damping coefficient, the ductility and energy ratios were the main parameters considered. The effect of the shape of the force-displacement curve upon the maximum displacement and upon the maximum energy input of a system was examined. Furthermore, the influence of modifying the intensity and the time scale of an earthquake accelerogram as well as the behavior of a structure with axial loads were also investigated. From the results presented in this report the following conclusions can be drawn: (1) The Ramberg-Osgood representation of the force-displacement relation is considered realistic if the structure is capable of maintaining stable, nondeteriorating hysteresis loops. The Berkeley experiments have produced remarkably stable hysteresis loops at large cyclic strains17 which can be approximated closely by a Ramberg-Osgood functiono (2) The spectral relation Ixlmax lilmax /ax n I + YlImax/42 is exact for undamped linear systems, and is a good approximation for damped linear systems provided the damping is small. The response spectra for linear systems are represented by the top curves of Figs. 13 and 14. (The elasto-plastic system reduces to the elastic system when the ductility ratio t or the energy ratio c becomes equal to 1.) (3) In nonlinear systems the spectral relation above is not valid. For the elasto-plastic systems this relation takes the form, Ixlmax Ix + Y|max n The expression on the left above, a pseudo velocity, i's the quantity that was plotted to obtain the response spectra in this case. 102

(4) For Ramberg-Osgood systems there exists no simple spectral relation. It is noted however that in Ramberg-Osgood systems if Vd and Va are defined as follows, lX Imax_ X + Ylmax - nlmax V = and V Vd - a V '~n then Vd > Va for P- < 2 Vd = Va for i = 2 Vd < Va for 1:> 2 For values of ductility ratio less than two, the maximum difference between Vd and Va occurs when ui equals one. It is also observed that the difference between the displacement and the acceleration spectrum curves for constant ductility ratio is constant, and thus produces only a vertical shifting of the curves. This does not apply to curves of constant energy ratio because the ductility ratio is not a constant in this case. For this reason, displacement and acceleration spectra for Ramberg-Osgood systems are plotted separately. (5) It is noticed that in the Ramberg-Osgood system, the acceleration spectra is much more sensitive to changes in exponent r than is the displacement spectra. The spectral characteristics of the elasto-plastic system are quite different from those of the Ramberg-Osgood system. (6) Equivalent viscous damping is helpful to the intuition in comprehending response phenomena, but it appears ill suited to the earthquake problem for quantitative purposes. (7) The maximum displacement and the maximum energy input for RambergOsgood systems are comparable with those obtained.forelasto-plastic systems of the same period and yield level. (8) For the steady-state vibration response, slowly varying parameter results showed good agreement with those of "downhill-climbing-method." The discrepancy between the two increased as exponent r and the ratio of input force to yield level of the system F/Qy, became large. Neither result showed existence of unstable zone for the Ramberg-Osgood system. (9) When an accelerogram is modified by multiplying the acceleration readings of a given earthquake accelerogram by an arbitrary constant K-, the response spectra for the new accelerogram are KI times the spectral values obtained from the original accelerogram. 103

If, however, the new accelerogram is obtained by multiplying the time readings 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 2 values by 1, KT, and 1, respectively, and multiplying the period values by KT~ If the intensity of the accelerogram is also modified then the above two cases must be superimposed to obtain the desired response spectra, as shown by the dashed line in Figo 35~ (10) When axial load is present, for large displacements, the force-displacement relation of a cantilever column is no longer of the Ramberg-Osgood type. Nevertheless, a Ramberg-Osgood curve can still be used as a good approximation, when the axial load and the displacement are small. Figures 39a and 39b illustrate the effect of axial load upon the shape of the hysteresis loop. It can be seen that for small values of axial load the loop is narrowo As the axial load increases the loop broadens and the curves show an unloading region even when the ductility ratio is small. Figures 38a and 38b show that in the absence of axial load the approximate method yields the same results as the exact method. With axial force present the results of these two methods no longer coincide. The discrepancy is found to be proportional to the axial load in the column. A column without axial load is stiffer than the same column with axial load resulting in different response to the same earthquake. 104

ACKNOWLEDGMENTS This report was prepared for a research project "Response of Steel Frame Structures to Earthquake Motion," conducted at the University of Michigan under the sponsorships of the American Iron and Steel Institute. The writers are deeply indebted to Glen V. Berg, director of the project, for his valuable assistance and helpful suggestions. The assistance and advice of Robert D. Hanson who reviewed this report, is also gratefully acknowledged. 105

REFERENCES 1. Hudson, D.E., "Response Spectrum Techniques in Engineering Seismology," Proc. I. World Conf. on Earthquake Engrg., Berkeley, Calif., June, 1956. 2. Harris and Crede, Shock and Vibration Handbook, Vol. I, McGraw-Hill, 1961. 3. Alford, J. L., Housner, G. W., and Martel, R. R., "Spectrum Analyses of Strong-Motion Earthquakes," Earthquake Research Lab., Calif. Inst. of Tech., August, 1951. 4. Berg, G. V., and Thomaides, S. S., "Energy Consumption by Structures in Strong-Motion Earthquakes, Univ. of Mich. Research Inst., Report No. 2881-2-P, March, 1960. 5. Jennings, P. C., "Response of Simple Yielding Structures to Earthquake Excitation," Ph.D. Thesis, Calif. Inst. of Tech., June, 1963. 6. Blume, J. A., Newmark, N. M., and Corning, L. H., Design of Multistory Reinforced Concrete Buildings for Earthquake Motions, PCA, 1961. 7. Jacobsen, L. S., and Ayre, R. S., Engineering Vibrations, McGraw-Hill, 1958. 8. Thomaides, S. S., "Effect of Inelastic Action on the Behavior of Structures During Earthquakes," Ph.D. Thesis, Univ. of Mich., Dept. of Civil Engrg., 1961. 9. Biot, M. A., "Analytical and Experimental Methods in Engrg. Seismology," ASCE Trans., Vol. 108, Paper No. 2183, 19435 10. Berg, G. V., and DaDeppo, D. A., "Dynamic Analysis of Elasto-Plastic Structures," Jour., Engo Mech. Div., ASCE, April, 1960. 11 Newmark, N. M. and Veletsos, A. S., "Effects of Inelastic Behavior on the Response of Systems" Proc. II World Conf. on Earthquake Engrg., Tokyo, Japan, July, 1960. 12o Berg, G. V. and Thomaides, S. S., "Punched Card Accelerogram of StrongMotion Earthquakes," Univ. of Mich. Research Inst., Report No. 2881-1-P, September, 1959. 135 Berg, G. V., "A Study of Error in Response Spectrum Analyses," Proc. Primeras Jornadas Chilenas de Sismologia E Ingenieria Antisismica, Santiago, Chile, July, 1963. 106

REFERENCES (Concluded) 14. Popov, E. P., Proceedings of Structural Engineers Association of California, 1965. 15. Jacobsen, L. S., "Frictional Effects in Composite Structures Subjected to Earthquake Vibrations," Report of the Dept. of Mechanical Engrg., Standford University, March, 1959. 16. Hanson, R. D., "Post-Elastic Dynamic Response of Mild Steel Structures," Ph.D. Thesis, California Institute of Technology, June, 1965. 17. Caughey, T. K., "Sinusoidal Excitation of a System with Bilinear Hysteresis," Journal of Applied Mechanics, Vol. 27, No. 4, December, 1960. 18. Biggs, J. M., Structural Dynamics, McGraw-Hill Book Co., 1964. 19. Morrow, J., "Cyclic Plastic Strain Energy and Fatigue of Metals," ASTM Special Technical Publication No. 378, 1965. 20. Penzien, J., "Elasto-Plastic Response of Idealized Multistory Structures Subjected to a Strong-Motion Earthquake," Proc. II World Conf. on Earthquake Engrg., Tokyo, Japan, July, 1960. 21. Massonnet, C. E., and Save, M. A., Plastic Analysis and Design, Blaisdell, 1965. 107

UNIVERSITY OF MICHIGAN 3 9015 03024 4449