Simpliﬁed R-Factor Relationships forStrong Ground Motions

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Simplified R-Factor Relationships forStrong Ground Motions

Isabel Cuesta,a) M.EERI, Mark A. Aschheim,b) M.EERI, and Peter Fajfar,c)

M.EERI

Recent studies have demonstrated the need to consider the ground motionfrequency content in the development and use of RT relationships. Re-sults from two different approaches to determining these relationships areunified in the present paper. Two bilinear RT/Tg relationships are rec-ommended for most strong ground motions and structural systems. One ismore accurate, while the other, more conservative relationship is used in

FEMA 273, ATC-32, and the simple version of the N2 method. Both relation-ships are indexed by the characteristic period of the ground motion, Tg .Simple methods to determine Tg from smoothed design spectra and recorded

ground motions are provided. Neither recommended relationships are appli-cable to the nearly harmonic ground motions that may be generated at sitescontaining soft lakebed deposits. An example illustrates the application ofthese relationships to a code design spectrum in both the acceleration-displacement and yield point spectra formats. [DOI: 10.1193/1.1540997]

INTRODUCTION

Many research investigations conducted since the 1960s have determined strengthreduction (R) factors for limiting the peak ductility responses of simple single-degree-of-freedom (SDOF) systems. In general, these investigations have been carried out usingone of the following two approaches: (1) estimating R factors based on the computedresponses of a large number of SDOF oscillators to a number of ground motions, or (2)

using pulse waveforms to establish R-factor relations to be applied to elastic spectracomputed for recorded earthquake ground motions. Examples of the first approach in-clude Lai and Biggs (1980), Riddell et al. (1989), Hidalgo and Arias (1990), Nassar andKrawinkler (1991), Miranda (1993), Vidic et al. (1994), and Ordaz and Perez-Rocha(1998). Examples of the second approach include Newmark and Hall (1973, 1982), whoformulated the equal displacement, equal energy, and preservation of force rules basedon the response of elastoplastic systems to pulses; as well as Veletsos and Newmark(1964), Veletsos et al. (1965), Veletsos (1969), Veletsos and Vann (1971), and Cuesta andAschheim (2000, 2001ac).

These studies considered bilinear and stiffness-degrading SDOF systems with simi-lar values of parameters (post-yield stiffness and damping) subjected to a wide variety ofground motions, and generally determined similar R-factor relationships (e.g., as sum-

a) Los Alamos National Laboratory, P.O. Box 1663, MS C926, Los Alamos, NM 87545b) University of Illinois at Urbana-Champaign, 205 North Mathews, Urbana, IL 61801c) Faculty of Civil and Geodetic Engineering, University of Ljubljana, Jamova 2, SI-1000 Ljubljana, Slovenia

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Earthquake Spectra, Volume 19, No. 1, pages 2545, February 2003; 2003, Earthquake Engineering Research Institute


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marized by Miranda and Bertero, 1994). However, some distinctions have come forwardrecently. Miranda (1993), for example, finds that soil conditions affect the RT re-lationship, and proposes different R factors to use in firm, alluvium, and soft soil siteconditions. For soft soil conditions, the RT relationship is made dependent on a

parameter termed the predominant period of the ground motion.

Another study (Vidic 1993, Vidic et al. 1994) also proposed RT relationships,applicable to motions recorded on varied soil conditions that depend on a period, T1 ,

which is dependent on certain characteristics of the ground motion.Recognizing the similarity of RT relationships for ground motions and for

simple pulse accelerograms, Cuesta and Aschheim (2000; 2001a, c) applied R factorsdetermined for pulse excitations to the elastic spectrum of the ground motion to obtainan estimate of the inelastic response spectrum. The optimal pulses had characteristic pe-riods, Tp , which coincided approximately with a characteristic period of the ground mo-tion, Tg , for the set of fifteen ground motions that were investigated. These motions wereselected to have a range of frequency content for different classes of duration and dis-tance from the fault and included some near-fault records. A simple formula for esti-mating the characteristic period based on the elastic response spectrum was identified.

Cuesta and Aschheim (2001a, c) concluded that the Vidic et al. relationship wasnearly as accurate as the pulse R factors, and both were more accurate than other well-

known and commonly used models. This is illustrated in Figure 1, which compares themean of the unsigned errors in the estimated strengths for the seven R-factor models ofTable 1, the fifteen ground motion records of Table 2, and different bilinear and stiffness-degrading load-deformation models. Only the pulse and Vidic et al. relationships adjustto the frequency content of the ground motion. The pulse R factors, however, are un-wieldy for use in design contexts because they are not expressed by an explicit formula,

Figure 1. Mean of the error E2 rmodel for 6 SDOF systems and for the 7 models pre-sented in Table 1 (all ground motions in Table 2 were used).

26 I. CUESTA, M. A. ASCHHEIM, AND P. FAJFAR


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Table 1. Partial history of R-factor relations1

MODEL YEAR

NUMBEROF

RECORDS(PULSES)

SYSTEMS

R-FACTOR

DEPENDENCE, % , %

Newmark andHall

1973 (3) 0(b)

20 10 T, ,, Ta(ag,max , vg,max , dg,max ,ea , ev , ed)

Riddell, Hidalgo,and Cruz

1989 4 sets 0(b)

5 10 T,

Nassar andKrawinkler

1991 15 0, 2, 10(b, sd)

5 8 T, ,

Miranda 1993 124 3(b)

5 6 T, ,soil, TG

Vidic, Fajfar,and Fischinger

1994 40 10(b, sd)

5 10 T, ,Ta(ag,max , vg,max , ea , ev)

Ordaz and PerezRocha

1998 445 0(b)

5 8 T, ,dg,max , D(T)

Cuesta andAschheim

2000 15(24)

0, 2, 10(b, sd)

2, 5, 10 8 T, ,Tg

1bbilinear; sdstiffness degrading.

Table 2. Recorded ground motions1

IDENTIFIER EARTHQUAKE DATE ag,max /g MAGNITUDE Tg1 , s

SHORT DURATION

WN87MWLN.090 Whittier Narrows 8/1/87 0.175 ML5.9 0.20

BB92CIVC.360 Big Bear 6/28/92 0.544 ML6.5 0.40

SP88GUKA.360 Spitak 12/7/88 0.207 Mw6.8 0.55

LP89CORR.090 Loma Prieta 10/17/89 0.478 Mw6.9 0.85

NR94CENT.360 Northridge 1/17/94 0.221 Mw6.7 1.00

LONG DURATION

CH85LLEO.010 Central Chile 3/3/85 0.711 ML7.8 0.30

CH85VALP.070 Central Chile 3/3/85 0.176 ML7.8 0.55

IV40ELCN.180 Imperial Valley 5/18/40 0.348 ML6.3 0.65

LN92JOSH.360 Landers 6/28/92 0.274 Mw7.4 1.30

MX85SCT1.270 Michoacan 9/19/85 0.171 Mw8.0 to 8.1 2.00

FORWARD DIRECTIVE

LN92LUCN.250 Landers 6/28/92 0.733 Mw7.4 0.20LP89SARA.360 Loma Prieta 10/17/89 0.504 Mw6.9 0.40

NR94NWHL.360 Northridge 1/17/94 0.589 Mw6.7 0.80

NR94SYLH.090 Northridge 1/17/94 0.604 Mw6.7 0.90

KO95TTRI.360 Hyogo-ken Nanbu 1/17/95 0.617 Mw6.9 1.40

1 MwMoment magnitude, MLRichter magnitude, and ag,maxpeak ground acceleration.

SIMPLIFIED R-FACTOR RELATIONSHIPS FOR STRONG GROUND MOTIONS 27


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and while the basic trend represented by the pulse R factors is relevant, the exact signa-ture of the pulse RTp relationship may be overly precise for use with future un-known ground motions. The bilinear RT/Ta relationship proposed by Vidic et al.

(1994) is a useful approximation of this relationship. The present paper modifies theVidic RT/Ta relationship to be a function of the characteristic period estimate used

by Cuesta and Aschheim and investigates simplifications associated with the use ofmodified coefficients. The resulting recommendation is simpler to use than the pulse Rfactor in practice and provides results that are more accurate than many accepted mod-els. The formulation in terms of Tg is more useful in design contexts than the originalformulation in terms of Ta .

DEFINITIONS

To facilitate a subsequent discussion of the results obtained from several researchinvestigations, the strength parameter and the strength reduction factor are defined asfollows:

The dimensionless strength parameter, y , of a SDOF system having mass, m, pe-riod, T, and ductility demand, , subjected to a ground motion having peak ground ac-celeration, ag,max , is defined as the ratio

y ,T Vy ,T

mag,max, (1)

where Vy is the yield strength of the system. The strength parameter is also related to theyield coefficient, Cy :

CyVy

Wy

ag,max

g(2)

where g is the acceleration of gravity and Wmg is the weight of the system.

The dimensionless strength reduction factor, R, of a SDOF system is defined as theratio of the strength required for elastic response y(1, T) to the strength associatedwith a peak ductility demandy(,T):

R ,T y 1,T

y ,T. (3)

Note that the reduction factor of Equation 3 considers only the ductility of the sys-tem. It is not equivalent to the constant reduction factors used in building codes, e.g.,

FEMA 302 (BSSC 1998) andInternational Building Code (ICBO 2000). Code reductionfactors also account for both energy dissipation and overstrength of the structure.

PRIOR RESEARCH RESULTS

Seven of the many R-factor models developed in recent years are shown in chrono-logical order in Table 1. For each R-factor model, the year of publication is presented aswell as the number of recorded ground motions or pulses used, the types of systemsstudied, and the main parameters on which the R factors were found to depend on.



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The models proposed by Newmark and Hall, Riddell et al., and Ordaz and Perez-Rocha were determined for elasto-plastic systems, while Miranda considered bilinearsystems and Nassar and Krawinkler and Vidic et al. considered both bilinear and

stiffness-degrading systems. Cuesta and Aschheim proposed an R-factor model derivedfrom simple pulses that can be applied to different load-deformation models.

Two parameters appear in all the R-factor models: the initial period of the system, T,and the ductility of the system, . Some models are a function of additional parameterssuch as the post-yield stiffness ratio, , damping, , or the elastic response displace-ment, D(T). In other models, the soil conditions or parameters related to the ground mo-tions are required, such as the peak ground displacement, dg,max , peak ground velocity,vg,max , peak ground acceleration, ag,max , or the predominant or characteristic period(TG,Tg , or Ta) of the ground motion.

Veletsos and Newmark (1960), Veletsos et al. (1965), and later, Newmark and Hall(1973) established that (1) the R factors for elasto-plastic systems subjected to groundmotions are constrained to R1 for very short period systems, (2) R can be established

based on the equal energy rule for short-period systems, and (3) R for medium-and long-period systems. This latter relationship is known as the equal displacementrule.

Riddell, Hidalgo, and Cruz (1989) proposed a bilinear expression for the RTrelationship in which, for systems with large ductility demands (5) and periods T0.4 s, the R factors are less than the corresponding ductility values, departing from theequal displacement rule.

Miranda (1993) identified different R-factor relationships for different soil types. Forsoft soil sites, the R factor is a function of the parameter TG , termed the predominant

period of the ground motion. Miranda defined this period as the period at which themaximum relative velocity is reached in a 5% damped elastic spectrum.

BILINEAR R-FACTOR MODEL

Vidic (1993) developed strength reduction factor relations for bilinear and stiffness-degrading systems responding to ground motions recorded in California, Chile, Italy,Mexico City, Montenegro, and the former Yugoslavia. In the study, the post-yield stiff-ness was 10% of the initial stiffness, 10, T2.5 s, and viscous damping was either

proportional to mass or the instantaneous stiffness. The proposed bilinear RT/Tarelationships depend on a period, Ta , which depends on the ductility demand, , and a

period T1 , which is intended to represent the period at the intersection of the constantacceleration and constant velocity portions of the spectrum. The period T1 is calculated

based on the peak ground velocity, peak ground acceleration, and velocity and accelera-tion amplification factors. The relationship recommended for stiffness-degrading sys-

tems (Q-model proposed by Saiidi and Sozen, 1981) satisfies the equal displacementrule (R) for long-period systems, unlike the relationship recommended for bilinearsystems.

The R-factor relations determined for bilinear load-deformation models havingmass-proportional viscous damping equal to 5% of critical damping are given by thefollowing bilinear curve:



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R

c1 1

crT

Ta1

T

Ta1

c1 1 cr1 TTa1

, (4)

where c11.35, cr0.95,

Ta0.750.2T1 , (5)

T12evvg,max

eaag,max, (6)

and ag,max andvg,max are the peak ground acceleration and velocity, respectively. In thepreceding, the acceleration amplification factor ea2.5 and the velocity amplificationfactorev2.0, 1.8, 2.6, and 2.8 for Standard, U.S.A., Chilean, and soft-soil Mexico

City records, respectively. Different coefficients were proposed for Equation 4 for stiff-ness degrading systems and for damping proportional to instantaneous stiffness. Differ-ent coefficients were also proposed for Equation 5 for stiffness-degrading systems.

The above relations were used for determining the target displacement in the non-linear method for seismic performance evaluation known as the N2 method (Fajfar2000). In the most recent simplified version of the method, inelastic response spectra areestimated using c11, cr1, and TaT1 .

PULSE R-FACTOR MODEL

Cuesta and Aschheim (2000; 2001ac) investigated pulse R factors for the fifteenrecorded ground motions of Table 2. The fifteen motions comprise five motions selectedto have a range of frequency content in each of three categories: short duration (SD)

motions, long duration (LD) motions, and records with near-fault forward directivity ef-fects (FD). The study applied the RT relations determined for 24 simple accelera-tion pulses to the elastic response spectra of the ground motions to identify the pulsesthat resulted in the best estimates of the inelastic response spectra of the ground mo-tions. The study identified that good estimates of inelastic response spectra could be ob-tained using the qua(2) pulse (shown in Figure 2) for all motions except the nearly har-monic motion recorded on the soft lakebed deposits of Mexico City, for which asinusoidal pulse was needed. To obtain good results, the characteristic period of the

pulse, Tp , must be set equal to a characteristic period of the ground motion, Tg , whereTg is defined as the period at the transition between the constant acceleration andconstant velocity portions of a 5% damped elastic spectrum. For smoothed elastic de-sign spectra, Tg is equal to the period at the intersection of the constant acceleration and

constant velocity portions of the spectrum, and corresponds to the period Ts used in theNEHRP Recommended Provisions for Seismic Regulations for New Buildings and OtherStructures (FEMA 302) (1998) and in FEMA 356 (2000), andTc in the N2 method (Fa-

jfar 2000). For both smoothed design spectra and the irregular, jagged spectra computedfor real ground motions, Tg, may be estimated by



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Tg2

Sv max

Sa max(7)

where Sv

andSa are the elastic pseudo-velocity and pseudo-acceleration spectra, respec-tively, for linear elastic systems having 5%.

Because Sa(T)mag,maxy(1,T) and Sv(T)TSa(T)/2, Equation 7 can also beexpressed in terms of the strength parameter and the period of the system:

Tg T 1,T max

1,T max. (8)

The graphical determination of the period Tg according to Equation 8 for theCH85VALP.070 record (identified in Table 2) is illustrated in the Appendix.

DISCUSSION OF VIDIC MODEL AND PULSE R FACTORS

Figure 3 shows an example in which R-factor response spectra proposed by Vidicand the qua(2) pulse R factor are compared for elasto-plastic systems having 2, 4,and 8 and5%, based on the frequency content of the 1940 NS El Centro record. In

Figure 2. Normalized acceleration, velocity, and displacement time histories of the qua(2)pulse.



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the short-period range, both models tend to R1. In the long-period range, the pulsetends to Rwhile Vidics model for bilinear systems results in R greater than .

Strength estimates made using the pulse R factors were compared with those ob-tained using the six other RT relations shown in Table 1, for bilinear and stiffness-degrading models having 2, 4, and 8, subjected to the fifteen ground motion recordslisted in Table 2. Cuesta and Aschheim (2001b, c) concluded that the Vidic et al. RT/Ta relationship was nearly as accurate as the pulse R factors, and both were moreaccurate than other well-known and commonly used models (e.g., Figure 1). The relativeaccuracy of the pulse and Vidic et al. models was attributed to the fact that these models

explicitly consider the frequency content of the ground motion.

Because both models had similar overall accuracy, the precise curve described by thepulse R factor is not of critical importance. The bilinear approximation employed byVidic et al. appears to be well suited to the uncertainties inherent in future ground mo-tions. The pulse R factors, however, because of their implicit definition, may be usefulfor systems with load-deformation responses that differ from those studied in previousinvestigations. The pulse R factors also satisfy R for long-period systems. The othermodels of Table 1 were less accurate and in some cases also require posterior knowledgeof ground motion characteristics.

The Vidic et al. relation requires the specification of T1 , which is based on estimatesof the pseudo-velocity and pseudo-acceleration derived from peak ground velocity and

peak ground acceleration. Both Tg andT1 are intended to describe similar characteristics(note that they vary together with the ground motions in Table 3) but are evaluated bydifferent procedures. The period of the ground motion Tg1 in Table 2 was determinedconsidering both the elastic pseudo-acceleration response spectrum and the equivalentvelocity spectrum, as described in FEMA 307 (1997).

Figure 3. Comparison of pulse and Vidic et al. R factors, for elasto-plastic systems having 5% subjected to IV40ELCN.180 record.



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DEVELOPMENT OF RECOMMENDED R FACTORS

The following section develops bilinear approximations to the pulse R factors thatuse the analytical form of Vidics model (Equation 4) with the characteristic period, Tg ,of Equation 7. The objective is to develop a simple expression that captures the main

variables affecting the R factors, in accordance with the actual data.

FRAMEWORK OF STUDY

Potential simplifications of the bilinear Vidic et al. model were investigated. Inelasticspectra were computed for the following parameters:

Six SDOF systems having ductility demands 1, 2, 4, and 8 were considered:elasto-plastic systems having damping, , equal to 2, 5, and 10% of critical; bi-

linear systems having 5% and post-yield stiffness, , equal to 2 and 10% of

the initial stiffness; and, stiffness-degrading systems having 5% and

2%. The stiffness-degrading model is the same as the one described by Mahinand Lin (1983), applicable to systems that do not exhibit substantial degradation.

Periods: Forty-five periods, T, varying from 0.04 to 3 s, and spaced at 0.02 s inthe range 0.04 to 0.2 s, 0.05 s in the range 0.2 to 1 s, and 0.1 s in the range 1 to3 s.

Ground Motions: Fourteen of the fifteen ground motions listed in Table 2 wereused. Although the nearly harmonic MX85SCT1.270 record (obtained on softlakebed deposits in Mexico City) was considered in the development of the

pulse and Vidic et al. R factors, none of the simplified candidate R-factor rela-tionships could be recommended for these soil conditions, and hence this recordwas not used to evaluate the candidate relationships.

The Vidic et al. R-factor model depends on the constant parameters c1 , cr, and theperiod Ta . Eight candidate RT relations are considered in the following: Vidicsoriginal proposal, the pulse R factors, and six simplified variations of the Vidic et al. R

factor using different values for c1 , cr, andTa , as summarized in Table 4. Model 6 is thebasis of the expression for C1 of the Nonlinear Static Procedure of FEMA 356 (2000),and is used in ATC-32 (1996) for bridge structures and the N2 method for buildings pro-

posed by Fajfar (2000).

The eight candidate R-factor models of Table 4 were applied to the elastic spectra of

Table 3. Periods (in s)

GROUNDMOTION Tg T1

GROUNDMOTION Tg T1

GROUNDMOTION Tg T1

WN87MWLN.090 0.17 0.11 CH85LLEO.010 0.41 0.38 LN92LUCN.250 0.41 0.89

BB92CIVC.360 0.30 0.29 CH85VALP.070 0.51 0.55 LP89SARA.360 0.59 0.38

SP88GUKA.360 0.40 0.42 IV40ELCN.180 0.56 0.43 NR94NWHL.360 0.69 0.74

LP89CORR.090 0.77 0.46 LN92JOSH.360 0.86 0.46 NR94SYLH.090 0.75 0.59

NR94CENT.360 0.73 0.52 MX85SCT1.270 2.00 2.53 KO95TTRI.360 1.25 1.00



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the 14 ground motions to determine estimates of the inelastic spectra. The computer pro-

gram PCNSPEC, a modified version of NONSPEC (Mahin and Lin 1983), was used todetermine iteratively the strengths required for each oscillator to achieve the specifiedductilities. For a given period, the strength-ductility relation is not necessarily monotonicsince the same ductility may result for different strengths; in such cases, the largeststrength required to achieve the specified ductility was retained.

The accuracy of a candidate RT relationship was evaluated by comparing theestimated inelastic strength response spectra (model) with the exact spectra computed foreach ground motion (r). The difference between the strength computed for a givenground motion and the estimated strength, for a given ductility demand and period, wascalculated as:

EijkE1rmodelyy 1

R(9)

and

EijkE2 rmodel (10)

where E1 may assume positive or negative values and E2 is always positive. The mean ofeither error measure, computed over all ground motions, ductilities, and periods is:

E1

nrndnp i1

nr

j1

nd

k1

np

Eijk, (11)

and the standard deviation, , is given by

E 1nrndnp1 i1nr

j1

nd

k1

np

EijkE2 (12)

where, nrnumber of recorded ground motions (14), ndnumber of ductility values(3, corresponding to 2, 4, and 8), and npnumber of periods considered (45,

between 0.04 and 3 s).

Table 4. Candidate R-factor models

Model

Number

Basic

Model

Characteristics

Period c1 cr

1 Vidic Original 1.35 0.95

2 Vidic Tg instead ofT1 1.35 0.95

3 Vidic Tg instead ofTa 1.35 0.95

4 Vidic Tg instead ofTa 1 0.95

5 Vidic Tg instead ofTa 1.35 1

6 Vidic Tg instead ofTa 1 1

7 Vidic Tg instead ofTa 1.3 1

8 Pulse qua(2) pulse, with TpTg



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RESULTS

Figures 4a and 4b show the results of the mean and standard deviation of the errorsE1 (Equation 9) and E2 (Equation 10), respectively, for the six SDOF systems, and for

Figure 4. Mean and standard deviation for 6 SDOF systems of the errors (a) E1rmodel ,(b) E2 rmodel , (c) E3(rmodel)/r, and (d) E4RrRmodel , for the eight models ofTable 4.



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each of the eight candidate R-factor models of Table 4. The mean of E1 often is negative,indicating overestimation of the strength parameter, a safe state of affairs for design. Themean ofE2 is always positive, because the absolute values of the errors are summed. The

largest mean and standard deviation correspond to the systems with smallest dampingratio, 2%, while the least dispersion is for systems with the largest damping ratio,10%.

Effect of the Period Ta

The periods Ta and T1 are approximately equal for 4. Comparison of the mean

errors for Models 1, 2, and 3 in Figure 4a shows that the errors E1 are very similar.Figures 4a and 4b show that the use of Tg in Model 2 in place of T1 (Model 1) reducesthe mean of errors E1 andE2 , and Figure 4a shows the standard deviation is smallest forModel 2. While the further simplification of using Tg in place ofTa in Model 3 causes aslight increase in the mean of error E2 (indicating support for the use of Ta in Model 2),the additional complexity of Model 2 may not be justified. The mean of errors E1 andE2

for Model 3 are still less than those for Model 1 (Figures 4a and 4b), and the standarddeviations for Model 3 are less than those for Model 1, for most of the load deformationmodels. Thus, the use of Tg in place of Ta , while not as accurate as using Tg in place ofT1 , results in a minor improvement in accuracy and a simpler formulation.

Effect ofc1 and cr

Figures 4a and 4b indicate that the strengths are overestimated for all types of sys-tems when the coefficient c1 is equal to unity (Models 4 and 6), instead ofc11.35, eventhough c11.0 satisfies the equal displacement rule for long-period systems. Compari-son of Models 3 and 5 shows that setting the coefficient cr equal to unity (instead ofcr0.95) causes an insignificant increase in error at these ductility levels. Model 7 (c11.3 and cr1) tends to be slightly more conservative than Model 1 (c11.35 and cr

0.95) and is a simpler formulation. Figure 4a indicates that Model 6 (c11 and cr1), which satisfies the equal displacement rule and is used in several codes and meth-ods, is somewhat conservative in a mean sense, particularly in comparison with itssimple cousin, Model 7.

Effect of the Error Metric

The means of the errors E1 and E2 were used in the preceding to emphasize accu-rately estimating the strengths of short-period oscillators. This was intended, becausedifferences in the strengths of short-period oscillators can have a significant effect on the

peak response amplitudes of these systems, whereas differences in the strengths of long-period systems tend to have less effect on peak displacement response. Furthermore,drift limits often control the design of long-period systems, elevating the importance of

stiffness and reducing the importance of strength (and R factors) for such structures.Nevertheless, the conclusions made using the error measures E1 and E2 were re-examined using two additional error functions, as follows.

Figure 4c shows the mean and standard deviation of the normalized error



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EijkE3rmodel

r. (13)

This error metric examines relative differences in strength, and therefore, does not em-phasize the short-period range.

Past research on strength reduction factors often has focused on estimating R factors,and not their inverses, even though it is the inverse (1/R) that is applied to determinestrengths. Figure 4d presents data for the difference between the actual and estimated Rfactors, given by the error metric E4 :

EijkE4RrRmodel . (14)

Review of Figure 4 indicates that the conclusions made based on E1 andE2 also holdfor these error metrics. That is, Model 7 is a simple and reasonably accurate model forestimating the R factors of the six load-deformation models, and Model 6 providessomewhat conservative estimates of the R factors and associated strengths.

RECOMMENDED R FACTORS

Vidic et al. showed that there is a considerable difference between the R factors forbilinear systems and those for stiffness-degrading systems. According to Vidic et al., formass-proportional damping, stiffness-degrading systems require c11.0. This impliesthat Model 6 should provide good estimates for systems with substantial stiffness deg-radation. Model 6 also complies with the equal displacement rule, and was shown to

provide conservative estimates, on average, for the six load-deformation responses stud-ied (Figures 4a, 4b, and 4c). Given the record-to-record variability in RTrelations,some conservatism in the R-factor relation often is appropriate. Thus, the use of themodel in FEMA 273, ATC-32, and in the simplified form of the N2 method is supportedfor these conditions. Model 7 is recommended for systems with limited stiffness degra-

dation, where greater accuracy is desired, and where scatter associated with record-to-record variability is better tolerated. Thus, the analyses indicate that the original Vidicet al. R-factor model may be simplified to

R c1 1T

Tg1

T

Tg1

c1 1 1T

Tg1

, (15)

where Tg is given by Equations 7 or 8 and c11.3 for an accurate estimate of the Rfactors for systems with limited stiffness degradation or c11.0 for systems with sub-stantial stiffness degradation or for a somewhat conservative estimate for systems with

limited stiffness degradation. These recommendations are based on the response ofSDOF systems having 8, 210%, 010%, and T3 s to motions having arange of characteristic periods and durations. The motions include both far-fault motionsand a limited number of near-fault motions, recorded at various orientations relative tothe fault strike. The observation that a pulse R factor can be used with long-durationground motions may be more intriguing than the suggestion that a pulse R factor may be



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used with near-fault motions. Nevertheless, additional studies involving a larger numberof records may identify systematic differences between the R factors for near-fault mo-tions and far-fault motions that were not apparent in this study. Equation 15 is not rec-

ommended for use with soft soil sites that can generate nearly harmonic motions, suchas were observed in Mexico City in 1985.

EXAMPLE APPLICATION

Equation 15 with c11 is the basis for the determination of target displacements inthe nonlinear static procedures of FEMA 356 and the N2 method (Fajfar 2000). Thisrelationship has also been implemented in the draft Eurocode 8 standard (ECS 2002),with Tg set equal to the corner period, located at the intersection of the constant accel-eration and constant velocity portions of the design spectrum (Ts in FEMA 302 and

FEMA 356 and Tc in the N2 method). This section illustrates the application of this re-lationship with smoothed design spectra in order to visualize more easily the quantitiesrelevant to the seismic response of an idealized nonlinear SDOF system. Both the

acceleration-displacement (AD) and yield point spectra formats are shown. These rep-resentations are discussed in greater detail by Fajfar (2000) and Aschheim and Black(2000), respectively.

Both formats allow comparison of capacity curves with seismic demand curves toallow the peak displacement response of the system to be estimated. Seismic demandsare expressed in terms of accelerations and displacements, using spectral curves thatrepresent elastic or inelastic response. The capacity curve of the system is superimposedon the demand curves. The demand curves indicate the ductility demand expected forthe given hazard. The graphic representation of the demand and capacity curves enablesone to appreciate the influence of strength and stiffness on peak displacement and duc-tility demands. This information is useful for both the design of new structures and theevaluation of existing structures.

Demand curves in both the acceleration-displacement (AD) format, developed origi-nally by Freeman (1978), and the yield point spectra (YPS) format are shown togetheron one plot in Figure 5. Although accelerations and displacements are represented in

both formats, we continue the traditional use of AD to designate plots of ultimate dis-placement and use YPS to designate plots of yield displacement. Accelerations arenormalized by gravity; the normalized acceleration at yield is equivalent to the yieldstrength coefficient, Cy , where the yield strength VyCyW. The AD and YPS curves co-incide for elastic response (1), for which uyy .

Equation 15 with c11 can be used to obtain simple and instructive graphic repre-sentations of inelastic demands. The AD and YPS curves for higher ductilities are ob-tained by computing the R factors for a given ductility demand using Equation 15. Ac-celeration and displacement demands for a given ductility demand are calculated as C

ySa /(Rg) anduSd/R orySd/R, where Sa and Sd are the spectral accelerationand spectral displacement of an elastic system having the same period. The period isconstant along lines that radiate from the origin. The intersection of the radial line cor-responding to the elastic period T of the idealized bilinear system with the elastic de-



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mand spectrum identifies the normalized acceleration demand or strength required forelastic response, as well as the corresponding peak elastic displacement, Sd, given by ufor1.

The intersection of the bilinear capacity curve (with normalized yield accelerationCySa /R and associated yield displacements yCyg(T/2)

2) and demand spectrum(for the given ductility) defines the demand of the inelastic system. Figure 5 illustratesthat the demand can be determined equivalently using AD spectra and YPS spectra.When AD spectra are used, the displacement demand is identified by the intersection ofthe peak displacement of the capacity curve and the AD demand curves. When YPSspectra are used, the ductility demand is identified by the intersection of the yield dis-

placement of the capacity curve and the YPS demand curves; the peak displacement iscomputed as uy . Demand spectra in AD format can be transformed to YPS format

by dividing the displacements by the corresponding ductility, whereas the multiplicationof the yield displacements is needed for the opposite transformation.

Figure 5 illustrates the demand for a medium-period system. For medium- and long-period systems (TTs), Equation 15 with c11 expresses the equal displacement rule,which postulates that R, or the peak displacement demand(u) is equal to the peakdisplacement (Sd) of an elastic system having the same initial period. For short-periodsystems (TTs), Equation 15 indicates that R, resulting in peak displacements thatexceed the corresponding elastic displacements.

Figure 5 also illustrates that the displacements d obtained from elastic analysis us-ing reduced seismic forces, corresponding to the design strength coefficient C

d, have to

be multiplied by a displacement amplification factor given by the product of the over-strength factor, defined as Cy /Cd, and the ductility associated with the term R, where

RSa /Cyg(Sa /Cdg)/(Cy /Cd).

The graphical techniques used to estimate seismic demand can be used to visualizemore complex cases, in which different relations between elastic and inelastic quantities

Figure 5. AD and YPS curves for the case 3, determined by applying the recommendedR-factor relationship (c11) to a smooth code design spectrum.



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and different idealizations of capacity curves may be used. However, in such cases thesimplicity of relations, which is of paramount importance for practical design, is lost.

The plots of Figure 5 can be used for both traditional force-based design as well asfor deformation-controlled or displacement-based design. The usual force-based designtypically starts by assuming the stiffness or period of vibration, given the estimatedmass, and a strength reduction factor whose value is prescribed by the code for the in-tended structural system. The seismic forces (defining the strength) are then determined,and finally an estimate of displacement demand is determined. In a displacement-baseddesign, the starting points are typically displacement and/or ductility limits, and thequantities to be determined are the required stiffness and strength. In the displacement-

based evaluation of an existing structure (or an existing design), the strength and stiff-ness (or period) of the structure being analyzed are known, and the displacement andductility demands are to be determined. For the displacement-based approaches, thestrength corresponds to the actual strength and not to the design base shear given bycurrent seismic codes, which is less than the actual strength in all practical cases. All

approaches can be easily visualised using the AD or YPS formats of Figure 5.

SUMMARY AND CONCLUSIONS

Recent studies have shown that improved estimates of inelastic spectra can be ob-tained when the R-factor relations reflect the frequency content of the ground motion.Good estimates of inelastic spectra were obtained using the bilinear model, proposed byVidic et al., and the R factors derived from simple pulse waveforms, proposed by Cuestaand Aschheim. The present study considered possible simplifications to the precedingmodels, consisting of restating the Vidic et al. proposal in terms of the characteristic pe-riod identified by Cuesta and Aschheim and modifying the original coefficients.

The computational study compared the errors in the inelastic spectra obtained forseveral candidate bilinear models for a set of fourteen ground motions and for differentload-deformation relationships. The ground motions consisted of five motions in theShort Duration and Forward Directive categories, and four motions in the Long Durationcategory. Ground motions within a category were selected to represent different fre-quency contents. Bilinear and stiffness-degrading load deformation models were consid-ered, having varied amounts of viscous damping and different values of the ratio of post-yield stiffness to initial stiffness.

The simplified R-factor model (Model 7, Equation 15) is found to be a good ap-proximation for use with bilinear and stiffness degrading systems which have limitedstiffness degradation for all motions except the nearly harmonic motions that may begenerated at soft soil sites. The R-factor model used in FEMA 273, ATC-32, and in thesimple version of the N2 method (Model 6, Equation 15) is indicated where some con-servatism in the estimate of required strength is desired and for systems with more sub-stantial degradation of stiffness. Both models are simpler than the original formulation,with Model 7 being of comparable accuracy to the Vidic et al. and pulse R-factor rela-tionships, and therefore, an improvement over many conventional R-factor models. Ad-ditional study to confirm the finding that a single bilinear R-factor relationship is suit-able for both near- and far-fault motions would be useful.



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The increased accuracy of the strength estimates is attributed to the explicit consid-eration of the frequency content of the ground motions by means of a characteristic pe-riod, Tg . When the response spectrum is available, Tg can be obtained directly from the

spectrum as described in the Appendix, or by Equations 7 or 8. However, if only attenu-ation relations are available for peak ground velocity and peak ground acceleration,Equation 6 may be preferred.

The recommended relationships can be applied to a code design spectrum in theacceleration-displacement and yield point spectra formats, as illustrated in an example.

ACKNOWLEDGMENTS

The work was supported in part by the Earthquake Engineering Research CentersProgram of the National Science Foundation under Award Number EEC-9701785 and in

part by an NSF CAREER Award to the second author (Award Number CMS-9984830).

NOTATION

ag pulse acceleration

ag,max peak ground acceleration

cr, c1 coefficients used in Equation 4

C1 coefficient used in FEMA 273

Cd design strength coefficient

Cy yield strength coefficient

dg,max peak ground displacement

E error

E1 , E2 , E3 , E4 errors using Equation 9, 10, 13, and 14, respectivelyE mean of the error

FD forward directive records

g acceleration of gravity

LD long duration records

m mass of the system

nd number of ductility values

np number of periods

nr number of records

R strength reduction factor

Sa pseudo acceleration

Sd pseudo displacement

Sv pseudo velocity



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SD short duration records

SDOF single-degree-of-freedom

t timetd duration of the pulse

T natural period of the system

Ta period determined in Equation 5

Tg , Tg1 characteristic ground motion periods

Tp characteristic period of the pulse

T1 period determined in Equation 6

vg,max peak ground velocity

Vy yield strength

W weight of the SDOF system

post-yield stiffness

viscous damping ratio

d displacement determined using design forces

u ultimate displacement

y yield displacement

ea , ev acceleration and velocity amplification factors used in Equation 6

model estimated strength parameter

r strength parameter associated to a ground motion

y strength parameter (Vy /m ag,max)

ductility demand of the system

standard deviation using Equation 12

APPENDIX: DETERMINATION OF THE PERIOD Tg

An example of the graphical determination of the period Tg according to Equation 8is presented in Figure 6 for the CH85VALP.070 record. The curves (1,T) andT(1,T) are plotted as well as the maximum of both curves. The period T*1.5 sdefines the period at which the product of T and (1,T) is a maximum, and *1.173 is the elastic strength at this period. Since the maximum of (1,T) is 3.482,Equation 8 provides

Tg1.1731.5s

3.4820.505 s. (16)

Figure 6 also shows a graphical method for establishing Tg . The largest-valued curveproportional to 1/T that intersects (1,T) does so precisely at the location that



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T(1,T) is a maximum. This intersection defines the period T and the strength pa-rameter (1,T) at which the largest T(1,T) occurs. The values of T(1,T) max and (1,T) max so determined can be used in Equation 8 to determine Tg ,or the intersection of the corresponding constant strength and largest valued 1/T curvescan be determined graphically, with this intersection identifying Tg . This graphical in-tersection is seen to correspond exactly to the intersection of the constant acceleration

and constant velocity portions of a smoothed design spectrum that bounds the actualspectrum.

This technique is also applicable to harmonic motion. In this case, resonance causesthe peak values ofT(1,T) and(1,T) to be reached at T/Tp1; therefore, Equa-tion 8 gives TgTp resulting in a definition for Tg that is consistent with the character-istic period of the motion. Thus, the definition Tg by Equations 7 and 8 accommodates awide range of excitations.

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Figure 6. Graphical determination of the period Tg for the CH85VALP.070 record.



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(Received 7 September 2001; accepted 11 September 2002)


Documents

Simpliﬁed R-Factor Relationships forStrong Ground Motions