DESIGN SPECTRUM-BASED SCALING OF STRENGTH REDUCTION FACTORS

13 th World Conference on Earthquake Engneerng Vancouver, B.C., Canada August 1-6, 2004 Paper No. 539 DESIGN SPECTRUM-BASED SCALING OF STRENGTH REDUCTION FACTORS Arndam CHAKRABORTI 1 and Vnay K. GUPTA 2 SUMMARY The nelastc (desgn) spectra characterzng sesmc hazard at a ste are generally obtaned by the scalngdown of the elastc (desgn) spectra va the use of response modfcaton factors. These factors depend sgnfcantly on strength reducton factors (SRFs), where SRF represents the rato of elastc strength demand to the nelastc strength demand of a sngle-degree-of-freedom oscllator, wth the nelastc deformatons lmted to a specfed ductlty demand rato. SRF spectrum gves the varaton of ths factor wth the ntal perod of the oscllator. Ths study consders the scalng of SRF spectrum n case of an elasto-plastc oscllator undergong strength and stffness degradatons. A new model s proposed n terms of the pseudo-spectral acceleraton (PSA) values, when normalzed to unt peak ground acceleraton (PGA), and ductlty demand rato and a ductlty supply-related parameter. Least-square estmates of the coeffcents are obtaned through lnear regresson analyses of the data for 956 recorded accelerograms n western USA. Parametrc studes carred out wth the help of the proposed model show that hgher earthquake magntude and/or alluvum ste geology may result n hgher SRFs for medum- to long-perod structures. INTRODUCTION It s common to characterze sesmc hazard at a ste and to estmate the desgn forces or dsplacements of a lnearly behavng sngle-degree-of-freedom (SDOF) structure, wth specfed perod and dampng, through elastc desgn spectrum. From economc pont of vew, however, structures need to be desgned so as to permt dsspaton of nput energy by means of large nelastc deformatons durng severe ground shakng. Therefore, t s consdered convenent to obtan nelastc desgn spectra as scaled-down forms of elastc desgn spectra. The scalng-down s acheved by the use of response modfcaton factors, where a response modfcaton factor s a product of () strength reducton factor (SRF), () structural overstrength factor, and () redundancy factor (ATC [1]). SRFs account for the non-lnear characterstcs of the structure, and thus play the most mportant role n the determnaton of response modfcaton factors and n ther parametrc dependence on varous structural and ground moton characterstcs. The study of SRF was ntated by Newmark [2]. They appled the equal-dsplacement, equal-energy, and equal-acceleraton prncples to estmate analytcally the SRFs for long-perod, short-perod and zero- 1 Formerly Graduate Student, IIT Kanpur, Inda 2 Professor, IIT Kanpur, Inda

perod structures, respectvely, as functons of ductlty (demand) rato. Ths was followed by several studes based on actual computatons of SRFs for elasto-plastc or more refned oscllators subjected to artfcal or recorded ground motons (e.g., see Elghdams [3], Krawnkler [4], Mranda [5]). However, most of ths research consdered the effects of only one or two governng parameters smultaneously on SRFs, and thus, suffered from the lmtaton of the data-set not beng large enough. For a data-set practcally avalable for any study at present, varous source and ste parameters related to a ground moton have to be consdered smultaneously along wth the structural characterstcs. Twar [6] proposed a comprehensve model n terms of earthquake magntude, strong moton duraton, predomnant perod of ground moton, geologcal ste condton, and ductlty demand rato. However, some of these parameters may not always be convenently avalable to the desgner. Ordaz [7] ncorporated the effects of varous governng parameters by expressng SRFs as a functon of elastc spectral dsplacements and peak ground dsplacement. Use of peak ground dsplacement as an nput parameter s however nconvenent snce ts value s determned by twce-ntegraton of recorded accelerogram and s thus senstve to the ntegraton algorthm and mssng out of ground moton n the begnnng (due to delay n trggerng of the accelerograph). In fact, peak ground dsplacement has never been consdered mportant for sesmc hazard characterzaton. Snce desgn spectra (normalzed wth respect to peak ground acceleraton) form a more convenent nput, ths paper proposes an alternatve scalng model n terms of the normalzed pseudo-spectral acceleraton spectrum. A stffness-degradng and strength-degradng oscllator proposed by Gupta [8] has been used to model the non-lnear behavour of structures. Unlke earler hysteretc models, ths oscllator models the structural behavour at global level, not at the elemental level. Ths consders the ntal yeld dsplacement level and ductlty supply-related parameter as two addtonal nput parameters. Regresson coeffcents have been obtaned n case of the proposed model for several sets of ductlty (demand) rato and ductlty (supply) rato parameters at several tme perods. The regresson analyses have been carred out for a database of 956 horzontal moton accelerograms correspondng to 106 earthquakes n western Unted States, between 1931 Long Beach earthquake, Calforna and 1984 Morgan Hll earthquake, Calforna (wth detals as n Lee [9]). The error estmates for dfferent levels of confdence are presented along wth the smoothed regresson coeffcents. The proposed model has been used to carry out a parametrc study, and to see whether the dependence of SRFs on earthquake magntude and ste condton, as shown by the proposed model, s n agreement wth the trends shown by earler studes. CALCULATION OF RAW R µ DATA SRF for a non-lnear SDOF oscllator s defned as the rato of elastc strength demand to nelastc strength demand such that the dsplacement ductlty rato s lmted to a maxmum value of µ, where ductlty rato s the rato of the maxmum nelastc dsplacement of the oscllator to ts yeld dsplacement. Thus, for a target ductlty rato, µ, R µ µ = µ s defned as Fy, 1 Rµ µ = µ = (1) F where, F y, 1 s the mnmum strength for no yeldng n the oscllator (.e., when µ = 1), and F y, µ s the mnmum strength at frst yeld for nelastc deformatons lmted to the ductlty rato of µ. It s well known that Rµ ( T) 1, as T 0, and Rµ ( T) µ as T. For a gven dampng rato of the SDOF oscllator, ductlty rato and earthquake ground moton, Rµ ( T) s functon of the type of non-lnearty n the oscllator. Ths study uses a modfed Clough-Johnston y, µ

oscllator, proposed by Gupta [8]. Ths oscllator s elasto-plastc n nature, and undergoes stffness and strength degradatons. Here, the strength degradaton s consdered to be a functon of ntal yeld dsplacement level, a ductlty supply-related parameter, n, and accumulated plastc deformatons. n s a 031 measure of ductlty supply rato of the oscllator (estmated as 344n.. by Gupta [8]). The stffness deteroraton s characterzed by the nstantaneous values of yeld dsplacement level and accumulated plastc deformatons. The dampng s assumed to be F-dampng (.e., wth no effect of non-lnear behavour) wth value equal to 5% of crtcal dampng. To compute the raw R µ data for a gven ground moton and ntal tme perod, T, the oscllator has been subjected to the ground moton, and the lateral yeld strength, F y, µ, has been terated untl the calculated dsplacement ductlty (demand) rato s wthn 1% of µ = µ. The non-lnear tme hstory analyss of the oscllator has been performed by usng the fourth order Runge-Kutta method wth an adaptve step sze control scheme and wth step sze taken as (1/ t) ground moton duraton, where t = 001. T. Durng the teratons, f more than one values of F y, µ are obtaned for the same ductlty rato, the largest value has been consdered for obtanng the mnmum value of R µ. The computaton of R µ for the gven ground moton record has been repeated for 56 ntal tme perods from T = 0.1 to 4.0 s, n case of n = 6, 10 and µ = 2, 4, 6. The complete database for the raw R µ data has been created by consderng 956 horzontal accelerograms whch were recorded durng the 106 earthquake events n the western U.S.A. regon from 1931 to 1984 (see Lee [9] for further detals). SCALING RELATIONSHIP AND REGRESSION ANALYSIS It may be observed that normalzed response spectrum, PSA( T )/ PGA, for a ground moton shows qute smlar trends as those shown by the SRF spectrum for the same ground moton at short and ntermedate tme perods. Both spectra approach unty as T 0. Ths smlarty s however lost as T and normalzed spectrum approaches zero value aganst SRF spectrum approachng the value of µ. Assumng that SRF spectrum attans the lmtng value of µ at T = 10 s, Rµ ( T) may be descrbed by the followng functonal form: α ( T ) PSA( T ) T Rµ ( T) β ( T) = + µ (2) PGA 10 Ths form s obtaned by supermposng a modfed form of PSA( T )/ PGA curve over a lne of µ/ 10 slope and passng through orgn. Based on Eq. (2), the scalng equaton for Rµ ( T) has been consdered to be T PSA( T) log 10 Rµ ( T) µ = b1 ( T)log 10 + b2 ( T) (3) 10 PGA where, b ( T ) and 1 b ( ) 2 T are (perod-dependent) regresson coeffcents for a set of ductlty (demand) rato, µ, and ductlty supply-related parameter, n. For convenence, the functonal, ( Rµ ( T) µ T/ 10), wll be referred to as Xµ ( T) hereafter. Lnear regresson analyses based on Eq. (3) have been carred out for 6 combnatons of µ = 2, 4, 6, and n = 6, 10 at 56 perods. In order to remove bas on the values of the regresson coeffcents, whch the uneven dstrbuton of data among magntude ranges, 3.0-3.9, 4.0-4.9, 5.0-5.9, 6.0-6.9, 7.0-7.9, may result n, data screenng has been carred out, as n Twar [6]. Thus, for each regresson analyss (for a set of T,

µ, and n ), there are a maxmum of 19 data ponts taken from each magntude range. Let b $ 1 ( T ) and b $ ( T ) denote the smoothed least-square estmates of the coeffcents, b 2 1 ( T ) and b 2 ( T ), respectvely. Those lead to the least-square estmate of Xµ ( T) as PSA( T ) log 10 X ( T) = b$ ( T) log 1 10 b ( T) µ + $ (4) 2 PGA The dfferences between the actual and the above estmates of log 10 Xµ ( T) gve the resduals whch have been used to obtan mean, mt ( ), and standard devaton, σ ( T ), for all 6 combnatons of n and µ. mt ( ) and σ ( T ) have been then smoothed along T. By assumng the normal dstrbuton to descrbe the dstrbuton of the calculated resduals, the error estmates at specfed levels of confdence can be calculated from the smoothed values of mt ( ) and σ ( T ). Those estmates may then be added to the calculated value of log 10 X ( T ) to obtan the value of R ( T) µ µ at the desred level of confdence. Goodness of ft tests have also been performed to check the valdty of normal dstrbuton assumpton, and t has been found that except for very short tme perods, the normal dstrbuton s a reasonable dstrbuton for the resduals. RESULTS AND DISCUSSION The smoothed least-square estmates of regresson coeffcents, b $ 1 ( T ) and b $ 2 ( T ), along wth the smoothed mt ( ) and σ ( T ) values, are shown n Tables 1 and 2 for n = 6, 10, and µ = 4. The probablstc estmates of SRF spectra obtaned from these estmates for p = 0.1, 0.5 and 0.9 have been obtaned and compared wth the actual spectra, n case of n = 6, µ = 4, for three recorded accelerograms. These accelerograms are (a) N75W component recorded at Coyote Lake dam durng the 1984 Morgan Hll earthquake, (b) east component recorded at Stone Corral, Parkfeld durng the 1983 Coalnga earthquake, and (c) S65E component recorded at 6074 Park Drve (ground level), Wrghtwood durng the 1971 San Fernando earthquake. Fgs. 1 3 show these comparsons for the Morgan Hll earthquake, Coalnga earthquake, and San Fernando earthquake motons, respectvely. It s observed that the proposed model ncely reflects the trends n the actual R µ spectrum; t captures the peaks n the low- to ntermedateperod range farly well. If we use a desgn spectrum wth less fluctuatons as PSA( T ), more smooth SRF spectra would be obtaned.

Table 1 Least Square Estmates of Regresson Coeffcents and Resdual Parameters for µ = 4 and n = 6 Perod, Least Square Estmates Resdual Parameters T (s) 10b 1 ( T ) 10b 2 ( T ) 100 mt ( ) 10 σ ( T ) 0.10 8.119 0.784-5.503 0.775 0.15 4.898 2.377-6.661 1.286 0.20 2.775 3.469-6.752 1.571 0.30 0.904 4.611-5.644 1.733 0.40 0.361 5.183-4.723 1.750 0.50 0.414 5.445-4.331 1.743 0.60 0.571 5.604-4.184 1.744 0.70 0.724 5.741-4.067 1.748 0.80 0.864 5.859-3.960 1.755 0.90 0.986 5.956-3.850 1.762 1.00 1.087 6.033-3.736 1.770 1.50 1.473 6.300-3.149 1.816 2.00 1.741 6.476-2.508 1.865 3.00 2.129 6.756-0.937 1.948 4.00 2.445 7.009-0.900 2.010 Table 2 Least Square Estmates of Regresson Coeffcents and Resdual Parameters for µ = 4 and n = 10 Perod, Least Square Estmates Resdual Parameters T (s) 10b 1 ( T ) 10b 2 ( T ) 100 mt ( ) 10 σ ( T ) 0.10 8.018 0.904-5.575 0.755 0.15 4.929 2.471-6.617 1.268 0.20 2.872 3.552-6.668 1.556 0.30 1.018 4.695-5.596 1.727 0.40 0.444 5.267-4.736 1.752 0.50 0.457 5.523-4.394 1.750 0.60 0.583 5.670-4.290 1.754 0.70 0.708 5.795-4.213 1.761 0.80 0.824 5.898-4.146 1.769 0.90 0.925 5.981-4.080 1.779 1.00 1.008 6.041-4.014 1.790 1.50 1.318 6.215-3.730 1.855 2.00 1.532 6.292-3.513 1.942 3.00 1.878 6.401-3.021 2.147 4.00 2.213 6.530-2.400 2.360

Fgure 1 - Comparson of the Actual and Estmated SRF Spectra for Morgan Hll Earthquake Case wth µ = 4 and n = 6. Fgure 2 - Comparson of the Actual and Estmated SRF Spectra for Coalnga Earthquake Case wth µ = 4 and n = 6. Fgure 3 - Comparson of the Actual and Estmated SRF Spectra for San Fernando Earthquake Case wth µ = 4 and n = 6.

Accordng to the studes of Trfunac [10, 11], both PSA( T ) and PGA may be estmated n terms of the parameters, earthquake magntude, M, geologcal ste condton, and epcentral dstance, R, for a gven level of confdence. Therefore, t may be nterestng to consder these models (say, by takng 0.5 confdence level and 100 km epcentral dstance) together wth the proposed model for Rµ ( T), and to study the varaton of SRF spectra due to parametrc varatons n M, µ, n and ste condton. The followng study has been carred out by consderng M = 5.5, n = 10, µ = 4, p = 0.5, and alluvum ste condtons, unless stated otherwse. Fg. 4 shows the varatons n Rµ ( T) for M = 4.5, 5.5, and 6.5 n case of alluvum ste condtons. It s observed that hgher magntude results n hgher SRFs for medum- to long-perod structures (T > 1.0 s), and n margnally lower SRFs for very stff structures (T < 0.3 s). Intermedate and hard rock stes also show these trends. These observatons contradct the fndngs of Mranda [5], who reported neglgble effect of magntude on SRF spectrum. Thus, t does not appear to be justfed to neglect the effect of magntude on SRFs. Further, as mpled by Fg. 4, t may be conservatve to estmate SRFs by usng the proposed model wth the desgn spectrum for a low-magntude earthquake. Fgure 4 - Estmated SRF Spectra for Dfferent Magntudes n Case of µ = 4, n = 10, p = 0.5, and Alluvum Ste Condtons. Fgure 5 - Estmated SRF Spectra for Dfferent Values of n n Case of M = 5.5, µ = 2, 4, 6, = 0.5, and Alluvum Ste Condtons. p

The effects of n and µ on SRFs have been shown n Fg. 5 for alluvum ste condtons. The curves wth dots are for n = 10, whle those wthout dots are for n = 6. It s observed, as expected, that hgher ductlty supply makes a dfference only n case of hgh ductlty demand ratos. Also, snce there s lesser strength degradaton n case of hgher values of n, those cases are assocated wth hgher SRFs. Fg. 5 also supports the b-lnear dealzaton of SRF spectrum, wth the selecton of proper slopes for both lnes. Fg. 6 shows the comparson of SRF spectra for dfferent ste categores. It s seen that except for stff oscllators, hard rock condtons are assocated wth smaller SRFs. Ths contradcts the observaton of Elghadams [3] that deamplfcaton of elastc response s slghtly more for a structure on rock than for a structure on alluvum ste for most tme-perods. Fg. 6 mples that t may be conservatve to estmate SRFs by usng the proposed model wth a desgn spectrum for hard rock ste geology. Fgure 6 - Estmated SRF Spectra for Dfferent Ste Condtons wth M = 5.5, µ = 4, n = 10, and p = 0.5. CONCLUSIONS A new model has been proposed n ths study for the scalng of SRF spectra, whle consderng pseudoacceleraton spectrum wth unt value of zero-perod-acceleraton as the nput data related to the ground moton. A recently developed elasto-plastc SDOF oscllator wth specfed strength and stffness degradaton characterstcs and 5% F-dampng has been consdered for calculatng SRFs from the recorded accelerograms. It has been found that the coeffcents obtaned from the lnear regresson analyses are able to predct the average trends of SRFs wth (ntal) tme perods of the oscllators farly well, and thus, the proposed model may be convenent when desgn spectrum s the only nput avalable to the desgner. A parametrc study carred out wth the help of the proposed model shows that hgher earthquake magntudes and alluvum ste condtons may be assocated wth greater SRFs, unless the oscllator s stff. REFERENCES 1. ATC. Structural response modfcaton factors. Report ATC-19, Appled Technology Councl, Redwood Cty, Calforna, U.S.A. 1995. 2. Newmark NM, Hall WJ. Procedures and crtera for earthquake resstant desgn. Buldng Practces for Dsaster Mtgaton. Buldng Scence Seres 46, Natonal Bureau of Standards, U.S. Dept. of Commerce, Washngton, D.C., U.S.A., 1973: 209 36.

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