I I I L1 I I I I I I I I I I I I I I I ..' , ... /... STRUCTURAL SYSTEM RESEARCH PROJECT

Transcription

1 AA3186,

2 L1 '".... " " / :--:::-...., Report No. SSRP-23/8 May 23..'... STRUCTURAL SYSTEM RESEARCH PROJECT ee 3/8G DEVELOPMENT OF TE TNG PROTOCOL FOR SHORT LNKS N ECCENTRJCALL Y BRACED FRAMES by PAUL RCHARDS CHlA- ling UA 'G Final Report Submitted to the Amerian Tnstitute of Steel Constrution Department of Strutural Engineenng University of Cali fomi a, San Diego La Jolla, California

3 University of California, San Diego Department of Strutural Engineering Strutural Systems Researh Projet Report No. SSRP-23/8 Development of Testing Protool for Short Links in Eentrially Braed Frames by Paul Rihards Gradl/ale Researh Assislalll Chia-Ming Uang Professor of Sn/ll/ra/ Engineering Final Report Submitted to the Amerian nstitute of Steel Constrution Department of Strutural Engineering University of California, San Diego La Jolla, California May 23

4 , A KNo\ LEDGE. E 'T The Amerian nstitute of Steel Con trution provided funding for lhi re earh; Mr. Tom Shlafly was the projet manager. Mr. James Malley and the A C Solutions Center developed designs for the prototype strutures. Professor llmut Krawlinkler provided guidane and insight in the seletion of ground motions and the development of the loading protool. Professors Mihael Engelhardt and Subhash Goel also provided helpful input on the projet.

5 BSTRACT An analytial study was onduted to devlop a loading protool to be used for experimental testing of short links in eentrially braed frames (EBF). Past experimental studies have demonstrated that link rotation apaity is dependent on the loading protool that is used in testing. There has been a shear link loading protool speified in the ASC ei mi Provisions sine 1997; however, it is a modified ver ion of the SA moment frame loading protool without any study to justity it. Three eentrially braed frames with short shear links (el',mp < 1.6) were designed for the study. Models were developed for these frames and nonlinear time-history analysis was perfom1ed using Los Angeles ground motions saled to math the design spetral aeleration for eah frame. Critial links were identified from the models and analysis r ults from these links were used to establish parameters for a new loading protool. The analysis results indiated that the loading sequene in the ASC Seismi Provisions for testing links is too onservative for short shear links. t has 1.5 times the umulative rotation demand and a higher perentage of large yles than the analysis results indiate is neessary. A new protool was developed following the same methodology as used in reating the SA moment frame loading history. The proposed loading protool has more total yles than the urrent ASC protool; however, there are fewer large inela ti yle and the umulative rotation demand is signifiantly less.

6 < TABLE OF CONTENTS ACKNOWLEDGEMENTS... i ABSTRACT... ii TABLE OF CONTENTS iii UST OF TABLES iv UST OF FGURES... v rntroducto Statement of Problem... 2 MODEL DEVELOPMENT A D ANAL YSS Prototype Buildings Models Beam, Column, and Brae Elements Shear Link Elements Gravity Loads and Seismi Masses Damping Modal and Pushover Analysis Time-History Analysis Earthquake Reords Analysis and Data Redution LOADrNG PROTOCOL DEVELOPMENT Cumulative Damage Modl Response Parameters for Link Loading Protool Target Values for Demand Parameters Governing Priniples in Seleting Target Valus Detrmining Target Values PROPOSED LNK LOADNG PROTOCOL Proposed Testing Protool Comparing the Proposed and Current Loading Protools SUMMARY AND CONCLUS ONS REFERENCES APPENDX A- Results from Time-History Analyses Using Simplified Models APPENDX B- Results from Time-llistory Analyses Using SA LA 1/5 Reords APPENDX C- Rainflow Cyle Counting Proedure

7 L T OFTABLE Table 2.1 Summary of Link Design Values Table 2.2 Gravity Loads Used for Building Models... Table 2.3 Seismi Masses Used for Building Models Table 2.4 Modal Analysis Results Table 2.5 LMSR (PEER) Ground Motion Reords Table 3. 1 Summary of Demand Parameter Target Values Table 4.1 Proposed Short Link Loading Protool Table 4.2 Comparing Protool Demands with Target Values Table B. Comparing Protool Demands with Target Values Table C. Rainflow Counting Results... 7 iv

8 C( Ll T O F FG RE Figure 1.1 Failure Mode Observed in TA Shon Links... 2 Figure 2.1 Plan Views of Prototype Buildings Figure 2.2 Elevations of Eentrially Braed Frames in 3-Story Building Figure 2.3 Elevation of Frame Figure 2.4 P-M nteration Curve for Beam-Column Elements Figure 2.5 Shear Link Element Proposed by Rahmadan and Ghoborah (1994) Figure 2.6 Comparing Methods of Modeling Beam Flexural Stiffiless Figure 2.7 Shear Link Element Used in Study Figure 2.8 Representative Results from Correlation Study Figure 2.9 Push-Over Analysis Results for Frame 3L Figure 2.1 Push-Over Analysis Results for Frame 3T... 2 Figure Push-Over Analysis Resu lts for Frame Figure 2.12 Aeleration Time Histories of Unsealed Ground Motions Figure 2.13 Design Spetra for 2 perent Damping Figure 2.14 Response Spetra and Saling Fators Figure 2.15 Representative Link Data (First Story Link, Frame 3T, PO-l Reord) Figure 3.1 Demand Parameter Value for Links in Frame 3L Figure 3.2 Demand Parameter Values for Links in Frame 3T Figure 3.3 Demand Parameter Values for Links in Frame Figure 3.4 Perentile Plaing of the Total umber of Cyles, N, Figure 3.5 Perentile Plaing of the Number of nelasti Cyles, N p Figure 3.6 Perentile Plaing of the Sum of the Rotation Rangs, L Y, Figure 3.7 Perentile Plaing of the Maximum Rotation Range, 6Ym" Figure 3.8 Perentile Plaing of the Maximum Rotation, YJ113X Figure 3.9 Perentile Plaing of the Seond Largest Rotation Range. y Figure 3.1 Perentile Plaing of the Third Largest Rotation Range, Y)... 4 Figure 3.11 Cumulative Distribution Funtion of Cyle Ranges, 6y, Figure 4.1 Proposed Shear Link Loading Protool v

9 Figure 4.2 Comparing Loading Protool CDFs with those from Frame 3T Links Figure 4.3 Comparing Cumulative Ranges for Proposed and 22 ASC Protool Figure A. Demand Parameter Values for Links in Frame 3L... 5 Figure A.2 Demand Parameter Values for Links in Frame 3T Figure A.3 Demand Parameter Values for Links in Frame Figure A.4 Perentile Plaing of the Total Number of Cyles. N, Figure A.5 Perentile Plaing of the Number of nelasti Cyles. Np Figure A.6 Perentile Plaing of the Sum of the Rotation Ranges. 1: y Figure A.7 Perentile Plaing of the Maximum Rotation Range. 6y,.", Figure A.8 Perentile Plaing of the Maximum Rotation. Y Figure A.9 Perentile Plaing octhe Ratio of 6Y2 to 6Y= Figure A.O Perentile Plaing of the Ratio of 6Y3 to 6Y""', Figure A. Cumulative Distribution Funtion of Cyle Ranges. Y, Figure B. SAC Los Angeles 115 Ground Motion Aeleration Time Histories Figure B.2 Response Spetra for SAC Los Angeles Ground Motions (2 perent damping) 59 Figure B.3 Demand Parameter Values for Links in Frame 3L Figure B.4 Demand Parameter Values for Links in Frame 3T Figure B.5 Demand Parameter Values for Links in Frame Figure 8.6 Perentile Plaing of the Total Number of Cyles. N, Figure B.7 Perentile Plaing of the umber of nelasti Cyles. Np Figure B.8 Perentile Plaing of the Sum of the Rotation Ranges, 1:6Y, Figure B.9 Perentile Plaing of the Maximum Rotation Range, 6y.,,, Figure B. Perentile Plaing of the Maximum Rotation. Y""', Figure B. Perentile Plaing of the Ratio of 6Y2 to 6yma., Figure 8.12 Perentile Plaing of the Ratio of 6Y3 to 6y..., Figure B.13 Cumulative Distribution Funtion of Cyle Range, y, Figure C. Example Rotation Time History Figure C.2 Reordering Rotation Time History Figure C.3 dentifiation of Peaks and Valleys Figure C.4 Counting Cyles in Reord V

10 Figure C.5 Counting Final yle Figure.6 Resulting Symmetri yles from Rainnow Counting... 7 V

11 'TRODU TlO 1.1 tatmnt of Problem Steel eentrially braed frames, EBFs, have been a popular alternative to moment frames and onentrially braed frames sine their int rodution to pratie in the early 19 O's. EBFs provide high initial elasti stiffness as well a dutile re ponse under extreme loading (Roeder and Popov 1977). The suessful performane of EBFs under sei mi loading depends on stable inelasti rotation of ative links while other frame omponent remain es entiallyelasti. Current design provisions (ASC 22) for ative link are ba ed primarily on a series of experimental studies onduted at the University of California, Berkeley (UCB) in the 198's with links of A36 teel (Hjelmstad and Popov 19 3; Malley and Popo 1983; Kasai and Popov 1986; Riles and Popov 1987; Engelhardt and Popov 1989). n , ASC spon ored a study to inve tigate the effets of higher strength steel (F)' = 5 ksi) on the perforn13ne of rolled setion links in eentrially braed frames. Experimental work was onduted at the University of Tex.as, Austin (UTA) and analytial work was performed at the University of California, San Diego (UCSD). During the experimental testing, most of the short links (ev,mp < 1.6) failed to reah.8 rad inelasti rotation, whih is the design value pernlitted in the ASC Seismi Provisions based on UCB studie. The most ommon failure mode of the UTA short link was one not previously observed in link testing, where horizontal fratures propagated out from the termination of stiffener-to-web fillet welds as shown in Figure 1.1 (Are 22). An analytial investigation of the failure found that lose stiffener spaing may explain this mode of failure, while the loading protool may have been responsible for links failing to reah.8 rad inela ti rotation prior to failure (Rihards and Uang 22). The UCB links from the 198's were tested under a variety of loading protools. The UTA links (Are 22) were tested with a protool, first speified in the 1997 A C Seismi Provisions, whih requires more yles and more umulative rotation to reah. rad inelasti rotation tban did any of the earlier UCB protools.

12 The objeting of this study was to investigate the rotation demands on short links in eentrially braed frames under design earthquake loading and develop a loading protool for experimental link testing that reflets those demands. Additional work remains to be onduted to determine rotation demands for longer links (e V Mp> 1.6). Figure 1.1 Failure Mode Observed in UTA hon Links (Speimen 4a, Are 22) 2

13 2 MODEL DEVELOPMENT AND ANALY S 2.1 Prototype Buildings Two prototype eentrially braed buildings, one 3 stories and one 1 stories, were designed by the ASC Steel Solution Center with diretion from Mr. James Malley. Figure 2.1 shows the plan views of the two buildings with the bay dimensions, olumn orientation, and EBF bays indiated. These buildings were similar in dimension and gravity loading to buildings used for moment frame analysis in the SAC projet (Gupta and Krawinkler 1999). The 3-story building had two different EBFs, one for longitudinal loading ("Frame 3L") and the other for transverse loading ("Frame 3T"). The elevations of these two frames are shown in Figure 2.2 with the member sizes and support onditions indiated. A few of the olumn setions in Frame 3L were modified from the prototype design for the analysis. The original olumn setions are indiated in parentheses (see Setion 2.3 for explanation). Column bases in both 3-story frames were onsidered as pinned. The 1-story building had the same EBF in both diretions, resulting in one prototype 1-story frame ("Frame 1"). The elevation of Frame 1 is shown in Figure 2.3. The olumn bases in the 1-story frame were onsidered as fixed. The links in the EBFs of eah frame were sized to have simi lar demand/apaity ratios under the design earthquake load to enourage distributed rather than onentrated link yielding (Popov et al. 1992). Table 2.1 summarizes the links sizes, non-dimensional lengths. and demand/apaity ratios. The nature of EBF design tends to result in low shear demand/apaity ratios. This is buase the link shear under the design seismi loading, ditates a required web area, but in order to provide a short link, a setion with a large M/Vp must be seleted. Many setions with large M/Vp are disqualified by the urrent flange width/thikness requirement in the provisions (ASC 22). The hallenge to find suitable setions an result in atual web areas somewhat greater than the required. Reent studies suggest the flange width/thikness requirement for links an be relaxed (Are 22; Rihards and Uang 22). 3

14 2.2 Models Models for the three frames (3L, 3T, and 1) were developed and analyzed with DRA -2DX (Prakash et al. 1993). Beam and olumn enleriines were used to define model geometry. Panel zone shears in EBFs are typially muh lower than those in moment resisting frames, therefore panel zones were not modeled expliitly and panel zone rotations negleted. Connetions in the eentrially braed bays were onsidered as rigid while onnetions in the other bays were modeled as simple (see Figures 2.2 and 2.3). Rigid zones were inluded at the ends of beams and links to aount for olumn depth Beam, Column, and Brae Elements Beams, beam segments outside of the shear lin ks, braes, and olumns were all modeled as beam-olumns. The beam-olumn element used onsists of an elasti beam, two rigid-plasti hinges, and rigid end zones. The axial fore-moment yield surfae used for the beam-olumns is illustrated in Figure 2.4. A yield stress of 55 ksi was used throughout in the model. Beam, olumn, and brae elements had post-yield stiffness equal to 5 perent of the elasti stiffness. Signifiant yielding in members other th an the links was not antiipated or reali zed during the time-history analyses Shear Link Elements Shear links were modeled using a tehnique simi lar to that proposed by Ramadan and Ghobarah (1995) but with some modifiations. Ramadan and Ghobarah's approah is based on theory developed by Riles and Popov (1994) and results in a si mplified version of the element reated by Riles and Popov, whih an be inorporated in general analysis programs. Modifiations had to be made to Ramadan and Ghobarah's method to orretly model the elasti stiffness of the links. Element Proposed by Ramadan and Ghobarah (1995) Figure 2.5(a) illustrates the link element proposed by Ramadan and Ghobarah (1995) whih onsists of an elasti beam with translational and rotational springs at either end. Two nodes at eah end of the link, referred to as the external and internal nodes, are defined to have the same oordinates. The elasti beam onnets the internal nodes on either end of the link. Moment and shear hinging in the link is modeled by rotational and translational springs that ouple the rotational and vertial translational degrees of freedom of the external and 4

15 l r CD,.1) L- <:. internal nodes. Three translational and three rotational springs operate in parallel at eah end in order to ahieve multi-linear fore deformation relationships using bilinear spring elements. The horizontal displaement of eah internal node is onstrained to equal that of the orresponding external node. Link rotation is alulated as the venial distane between the external nodes divided by the length of the link. ndividual spring properties are alulated suh that the ombined fore-defornlation relationships of the springs at either end orrespond to those indiated in Figure 2.5(b). The yield points and post-yield stiffnesses, whih define the shear-foree-deformation and bending-moment-rotation relationships shown in the figure, were alibrated using results from UCB tests on A36 links. Post-yield stiffness is modeled with kinemati hardening in the rotational springs, and both isotropi and kinemati hardening in the translational springs. Coneptually, the element proposed by Ramadan and Ghobarah is sound. However the elasti shear and flexural stiffness is aounted for twie in the model, one in the elasti beam and again in the elasti stiffness of the rotational and translational springs, resulting in inaurate elasti sti ffness. Modified Element Used in This Study Sine the elasti stiffness is aounted for twie in the Ramadan and Ghobarah's element, the element should be modified by removing elasti stiffness from either the beam or the springs. The elasti shear stiffness an easily be removed from the beam lement, by speifying no shear deformation. Some modifiation of the springs is still needed, though. Ramadan and Ghobarah (1995) gives an elasti stiffness for the translational springs of GAsh.a,!e [Figure 2.5(b)]. However, the springs on either end should have a stiffness, Kv/, of 2GAsh<a,le beause the translational springs on either end of the link operate in series with eah other, and the ombined elasti stiffness of the springs on both ends beomes GAshra,le. n ontrast to the elasti shear stiffness, the elasti flexural stiffness annot be removed from the beam and lumped in the rotational springs without introduing errors. Figure 2.6 ompares the stiffness matrix of a standard elasti beam element (negleting shear and axial deformations) with that of a rigid beam with flexural stiffness modeled using rotational springs on either end. The four terms that are different in the two ases are highlighted. Under shear loading or loading with equal end moments, the two elements give idential results. However, under unequal or opposite end moments (typial under gravity loads) 5

16 ... disrepanies arise and the rigid-beam-with-springs element does not have the orret stiffness. n light of diffiulties assoiated with removing the elasti stiffness from the beam, it is more reasonable to keep the beam ela ti in bending and remove the ia ti stiffness from the rotational springs. Sine the standard beam element already has rigid-plasti hinges built in, the rotational springs an be eliminated altogether. Multi-linear hardening for the moment hinges was not deemed neessary, beause this study is onerned only with short links whih experiene only minor flexural hinging, so a single standard beam element was suffiient. Figure 2.7(a) illustrates the link element used in this studied, whih is a modified ver ion of the onept proposed by Ramadan and Ghobarah (1995). The translational spring fore-defornlation behavior on either nd is illustrated in Figure 2.7(b). The yield points and post-yield stiffnesses were alibrated based on the reent UTA link tests with A992 steel. The flexural hinges were defined to yield at Mp of the link (alulated using the expeted yield strength, 55 ksi). The post-yieldj7exliral stiffness of the link was equal to 5 perent of the elasti flexural stiffness. To vrify this link modeling approah, the UTA hnk test set-up (Are 22) was modeled in DRArN2DX and a orrelation study was perfonned. Eah of the UTA tests was simulated, with the links modeled as desribed above. Figure 2.8 ompares the analytial and experimental results for some of the UTA links. Similar orrelation was observed for all of the links and validates the link modeling tehnique. The orrelation study indiated that link behavior an be reasonably modeled with only kinemati hardening in the translational springs (rather than ombined isotropi and kinemati), although some auray is lost in small initial yles. Simplified Modeling Tehnique Whittaker et al. (1987) used a muh simpler approah in modeling a six-story EBF frame with links in the hevron-braed onfiguration. n the model, shear links were standard beam elements with the moment hinges alibrated to yield when the nominal shear strength was reahed. This approah is theoretially sound f the end moments of the link are equal and the post-yield stiffness of the moment hinges is properly alibrated. 6

17 D The equal end moment assumption is reasonable for links in the hevron-braed onfiguration, but unequal end momnts tend to our in lin ks adjaent to olumns. Although the end moments were not initially equal for the frames in this study, they did tend to balane afier minor yielding. To investigate the effet of modeling links in this simplified way, additional models were developed using this approah (see Appendix A) and analyzed for omparison with the more refined modeling tehniques. Overall results using this simplified tehnique were very lose to those obtained with the more preise modeling Gravity Loads and Seismi Masses The gravity loads and seismi masses used in the models were the same as those used for the models in the SAC moment frame study (Gupta and Krawinkler 1999). Un-fatored loads and masses are given in Tables 2.2 and 2.3. A gravity load ombination of 1.2D+.5L (CC 2) was applied to the strutures during the stati nonlinear pushover, or time-history analyses (Setions 2.3 and 2.4). The gravity loads for the half of the struture assoiated with eah frame, but not ating diretly on it, were applied to a P-delta olumn Damping Riles and Popov (1994) demonstrated that visous damping in link elements may result in unrealistially high brae fores and suggested that nonproportional damping may be more appropriate for eentrially braed frames than traditional Rayleigh damping. Nonproportional visolls damping was used in the models, where all elements exept for the links had visous damping. Damping oeffiients were based on 2 perent damping in the first mode (see Setion 2.3) and at a period of.2 seonds for eah frame. 2.3 Modal and Pu hover Analyse Natural periods determined from modal analyses are indiated in Table 2.4. For referene, the period alulated using an empirial formula (Equation 16-39) in the nternational Building Code (CC 2) is also indiated. A pushover analysis was performed for eah of the three frames. The base shear was distributed as speified in the BC (CC 2). Part (a) in Figures , shows the base shear versus roof displaement for eah of the frames. Tbe formation of hinges is indiated on the load-displaement urves. An absissa on the top of eah plot indiates the drifi at the 7

18 ,, 1 <Z, w.l. W roof. An ordinate on the right indiates the base shear normalized by the weight of the building assoiated witb eah frame (one half the total building weight). Labels in Pan (b) of Figures indiate th e sequene of shear and flexural hinge formation. The initial pushover analyses of Frame 3L indiated some undesirable olumn yielding, prior to the development of signifiant link rotation, in the first and seond story olumns of the right EBF bays. Yielding was the result of high axial fores and bending moments in the olumns. t should be noted that during the pushover analysis all of the links eventually yield and reah ultimate strength, representing a worst-ase situation. Sine the purpose of thi s study was to investigate link rotations and it is generally presumed that the olumns will not experiene signifiant yielding, the olumn sizes were simply inreased in the models to prelude any premature olumn yielding. Although only two of the olumns had problems, all of the large olumns were inreased in size for onsisteny. The results shown in Figure 2.9 are for the model that has the modified olumns. For all of the buildings the base shear at yield was greater tban the design base shear, by a degree that might be expeted from the demand/apaity ratios of the links (see Table 2.1). Links began to reah.8 rad inelasti rotation at roof dri fts between 1. and 1.3 perent in the models. Frame 3L, with signifiant gravity loads as ompared to the others, had links reah design rotation at the lowest roof drift. 2.4 T ime-history Analysis Earthquake Reords The suite of ground motions used for the development of the loading protool onsisted of twenty LMSR (large magnitude, M, small di stane, R) Los Angeles reords that have been used in reent PEER projets (Krawinkler et al. 23; Medina 23). The motions are referred to as PO -P2 herein. Table 2.5 provides information on these time hi tories. Figure 2.12 shows the aeleration time histories of the un-saled reords. Eah reord was saled differently for eah frame. Sale fators were alulated to make the spetral aeleration of eah reord, with 2 perent damping, equal to the design spetral aeleration, with 2 perent damping, at the period of eah frame. The 2 perent damping design spetra was obtained by adjusting the 1997 UBC, Soil Type D spetra (lcbo 1997), whih is for 5 perent damping, using the saling proedure in FEMA 356 8

19 (FEMA 2). Figure 2.13 illustrates the original and saled design spetra. Figure 2.14 shows the re ponse spetra of the un-saled reord with the 2 perent damping design spetra and the alulated values of the saling fators for eah frame. A separate group of analyses were also perfonned with the twenty Los Angeles ground motions used in the SAC projet, representing events with 1 perent probability of exeedane in SO years (Somerville et al. 1997); this set of reords ontained some nearsoure ground motions. These ground motions and results from those analyses are presented in Appendix B. Results were similar to those obtained using the PEER reords Analysi and Data Redution Eah of the models was analyzed using eah of the twenty speially saled ground motion reord. Link rotations were alulated at eah step of the time-history analysis u ing the oordinates of the link ends. Rotation time histories were generated for the links in all three frames under all 2 ground motions. Figure 2. S(a) shows a typial link rotation time history from a first story link of Frame 3T under the P4 ground motion. Figure 2.S(b) shows the link shear versus rotation hysteresis for the same link. The link rotation hi tories were the only neessary data for the development of the loading protool. The link rotation time histories needed to be onverted into series of yles, before they ould be used for loading protool development. The simplified minflow yle ounting method, used by Krawinkler et al. (2) in the SAC study, was u ed. This tehnique i desribed in Appendix C. When mean effets are not onsidered, the rain flow yle ounting proes results in a number of symmetri yles defined by their range (hange in deformation from peak to peak). Figure 2. S() shows yles alulated from the rotation time history in Figure 2.S(a). All of the link rotation time history data was redued in this way, so that for eah link in eah frame under eah seismi event there was an assoiated sequene of symmetri yles ordered with dereasing rotation range. 9

20 V1 Frame 3L 3T Table 2.1 Summary of Link Design Values Story Setion Non-Dimensional Length (e V,Mp) $,v, " 18x "" 14x N Ox " 16x nd 14x ,d Ox " 14x nd 18x l ,d 16x 'h 14x 'h 14x 'h 14x 'h 14x 'h Ox 'h 12x " Ox

21 Table 2.2 Gravity Loads Used for Building Models Load Type Load (psg Floor dead load for weight alulations 96 Floor dead load for mass alulations 86 Roof dead load 83 Redued live load per floor and for roof 2 Table 2.3 Seismi Masses Used for Bui lding Models Seismi Mass' Loation (kip-se 2 /ft) 3-story Struture: 1 st and 2 nd Stories story Struture: 3'd Story story Struture: 1 st Story lo-story Struture: 2 nd -9'" Sto_ry story Struture: 1tn Story 73.1 Values for entre buldng Table 2.4 Modal Analysis Results Frame T\ (se) T2(se) TJ (se) T4 (se) Ts(se) 3L.514 (.468)' T.617 (.468) ( ) 'Values in p arnlheses alulated usin g Ta-(.3)",.

22 w J'>. J Name PO P2 P3 P4 P5 P6 P7 P8 P9 PO Pil P2 P3 P4 P5 P6 P17 P8 P9 P2 Table 2.5 LMSR (PEER) Ground Motion Reords Event Year Station R (km) Lorna Prieta 1989 Agnws State Hospital 28.2 Lorna Prieta 1989 Capitola 14.5 Lorna Prieta 1989 Gilroy Array # Lorna Prieta 1989 Gi lroy Array # Lorna Prieta 1989 Gilroy Array # Lorna Prieta 1989 Holl ister City Hall 28.2 Lorna Prieta 1989 Hollister Differential Array 25.8 Lorna Prieta 1994 Sunnyvale-Colton Ave Northridge 1994 Canoga Park - Topanga Can orthridge 1994 LA - N Faring Rd 23.9 Northridge 1994 LA - Flether Dr Northridge 1994 Glendale - Las Palmas 25.4 Northridge 1994 LA - Hollywood Store FF 25.5 Northridge 1994 La Cresenta - New York 22.3 Northridge 1994 Northridge Satioy St San Fernando 1971 LA - Hollywood Store Lot 21.2 Superstition Hills 1987 Brawley 18.2 Superstition Hills 1987 E Cntro mp. Co. Cent 13.9 Superstition Hills 1987 Plaster City 21. Superstition Hills 1987 Wstmoreland Fire Station PGA Duration (g) (se)

23 '.D h C 6 BAYS@ 3' = 18',., Frame 3L. -. -:... EBF Bay,. :.., ', _.. ' Frame 3T (a) 3-Story Building 5 BAYS@ 3' = SO' , : : Frame 1, _ ' - t - (b) 1-Story Building Figure 2. 1 Plan Views of Prototype Buildings 13

24 Vb r Jlt 68 Wl 6a,.. 4 r. WO-S8 \.\11"&8 \"'C '.,,HSS2 '2 '. HSS ') 8..., ';' /' '".,.,., "- =' ;: ;: ;: ;: " 'N" S2 W14"82 W14 S2 W141<B2 ;: \'\114'182 N '" a. HSS14"'14"'S/8 HSS14<"5J N :n g M ;;; M " "',,, M " " :! ;: rg) ;: ;: '-\/1811'86 Vlla"S5 W81186 W8 8G M W18,Sj N a: ':,114 1 Ci J.;.. t:-6,/ N a M 'l',, ;! " ;: ;; ;: ;; "...,..., "'" (a) Frame 3L ".;, :11, "fl., \",/ Of ;: W8"6(' 1 ;: 1 1..''''. ;:, }1., ;: Slor,. 1;1 Cjl,".,..,. 4'- 4 Typ H Wl,45 Wl... S Wlx45 W.. 45 g HSS12.12xll2 ;;; ;;; ;;; HSS12x12.1, W,4x74 W b7. W4. 74 M '" g ;;; ; ; t, x x x x e.. M W16xn W18 11n W151<n g ;;; ; ;, Wl h74 g x.... 3rt! Story g 2nd Sto<y x g lat Story i (b) Frame 3T Figure 2.2 Elevations of Eentrially Braed Frames in 3- tory Building

25 uj V1,f'()"Typ W,O,,45 H Wl.45 W1Q.4S '\'1,,45 Wl.45 HSSt2"t2... \(2 M M M 1th Story.. W1245 W2.4$ W12.45 W12.45 M M M ; SS414"'12 HSS,4.. 1.,,' M ;,!.. 9th Stoty, Wl.. 68 WlM68 WO.68 'A 1.68 W,O.68 55,414,,112 8th Siory.. ill ill W4,,68 W,4.68 Wl,.. 68 W,4.68 SS4.'412 HSS4x"""! W W W ' W'4,,74 N P SS'.-1".S8 W14.82 W W"h82 "",4,,82 W.h82 N 5514<14.5/8 HSS4M14. N?!!'! S 7th Story 6th Story.., '"' SOr( W1482 W"'-82 W W,4,,82 W4 52 M ill 55'4.'4.518 HSS4",4.51 :;l.. -;.. 4.h SlOt)' W6,,77 W6.17 W,6,,77 W'6 77 = M HSS :1! W,8.1(l6 W8"'6 W8.:16 W W1816 :,.. ltd SOrY HSS 16.'6,,518 HSS1Sl'6.518 N N, 7, ': 7 ': 2nd Slaty W' W, W14.74 W W ;;; HSS\ t4. ': ': lit StorY Figure 2.3 Elevation of Frame 1 15

26 1.12 v,.6v. 1. V, V External Node P -Nt.2P.. A A1 Figure 2.4 P-M nteration Curve for Beam-Column Elements Shear Hinge L.LJSp"ngs Elasti Beam Element Zero Lenglh Link Lenglh. 1 1 (a) Shemati Translational Springs K,,} KVJ '" _ GAS1H'i'1" 1\."1 - e K", =.3K" K" =.15K" K" =.2K,.6M,.3M,.OOM,. Nt Shear Hinge Springs Moment Hinge Springs Zero Lenglh 1 1 Rolational Springs External Node K = 6 f "" e K", =.3K. " K"J =.15K." K", =.2K,,, :...: :.: e sym. sym. (b) Combined Spring Ation at Eah End Figure 2.5 Shear Link Element Proposed by Rahmadan and Ghoborah (1995) 16

27 r ' Experiment 1 Vi' a. 5., o tj -5 h-h-h-h-f-f-,fh'-f-'-f--h-h-i L- --' --'-..J Rotation (rad) Rotation (rad) , Experiment 1 Vi' 5 a. ;g, :. f--.h-h-h-+-h-(f-tf i., tj Vi' 5 a. E., o tj -5 (a) UTA Speimen C Analytial Modell..,:l , Analytial Model 1 Vi' a. 5., o tj -5 f h-h-/-f,fh-h-h-h---l '-----'----'------' -15 L-_--' --'-..J Rotation (rad) Rotation (rad) Vi' 5 a. ;g, Of--++++ffi4+f+++--i o '" tj -5 (b) UTA Speimen 2 15r , 1 o tj '" -5-1 L:::r:::,,--_--.J '-----'----'------' Rotation (rad) Rotation (rad) () UTA Speimen 4C Figure 2.8 Representative Results from Correlation Slldy 18

28 . ;g, "' o Q).r. en Q) Cl o lC.OO!l... Roof Drift.1.15.: 1 st Story links Begin. to Exeed Design :/ nelasti Rotation : (.8 rad) '--_-'- L..-_...J... L..-_...J... L..-_...J..._-' Roof Displaement (in) (a) Ba e Shear ver us Roof Displaement... _ -...., 2 (b) Sequene of Plasti Hinge Fonnation Figure 2.9 Pu hoover Analysis Results for Frame 3L ,g o tr o Q).r. en Q) Cl o

29 " U> a. :;2 '" (1).r: C) (1) /) CD '"..5 Roof Drift r ,----r-----, !e.?m :j.6 1 st Story links Begin to Exeed Design nelast Rotaton (.8 rad) L-_--'-_--'. -'--_-'-- L-_-"-_----'-_---' Roof Displaement (in) (a) Base Shear versus Roof Displaement."'-- _"", (b) Sequene of Plasti Hinge Formation Figure 2.1 Push-Over Analysis Results for Frame 3T 2 7 8, Q iii a: '" (1).r: C) (1) /) CD '"

30 <.D.." " (i) a. 1 6 '" 8 U) Q) /) o '" 1111 Shear Hinge o Moment Hange 6 4 Roof Drift _-,, 13-1 _, 1:_,"" 1-t2,-,,, 29}O.::. -:..:-,-;-'-.-; Be o "-----''----'_--'-_---'-_--'-_-'-_-'--_-'--_.L-_ '. o '" -- Some Links Begin to Exeed Design nelasti Rotation (.8 rad) Roof Displaement (in) (a) Base Shear versus Roof Displaement /.2.1 / /" 2. j /'32' 915 /, JLll /, "9 /" 2L' /. 17 2U /, 12 2 r / (b) Sequene of Plasti Hinge Formation Figure 2.11 Push-Over Analysis Results for Frame 1 21 ", g r '" '" Q).<::: U) Q) /) o '"

31 ..D " PO PGA =.1729 PGA= 443g : : Q.. " 8 u " " Time (se) Tmle (se) : :. j ' " 8 " 8 " i PGA =.3679 P4 PGA =.2129 :t rime (se) rime (se) P5 PGA =.2269 P6 PGA = 2479 : : g.2 e f!. " 8 " " Time (se) Time (se).5 P7 PGA =.279g P8 PGA =.27 : :.Q -" h "...'. '. A.... A i!.. "'',.1" -." V1l '''' '( 1M " ".5 J Time (se) Time (se) : P9 PGA".42g : PO PGA = 273g e ",.j! 2.., " 8 u 8 " " Time (se) rime (se) Figure Aeleration Time Histories of Unsealed Ground Motions 22

32 ... V1 5 5 Pl1 PGA::.24g P12 PGA.26g g f.!. e.,.,.5.(ls rime (se) Tm\e (se).5.5 P13 PGA' 231g P14 PGA' 1SQg. e,.",., eo O 1" 1 " "" " ".(l.s.(l.s rime (se) Time (se).5 P15 PGA 368g P16 PGA'.174g 8. " Time (se) " Tm18 (se).5.5 P17 PGA' 1569 P18 PGA'.3S8g.2.,L,L.2 e, OO,"'r', " ".(l.s S Tme (se) Tme (se) 5 P19 PGA'.186g P2 PGA =.172g.2,"...,J,,,h, g e.!.!. '.,., " " ''Y'" -OS Time (se) Time (se) Figure 2.12 Aeleration Time Histories of Unsealed Ground Motions (on't) 23 S

33 Sl :.2 ;; Q) a; u «iii U Q) a. C/l , USC (5% Damping) % Damping ",\... : , W.7. 1.m.. f: \ '...,: :..\......):).... : \ /O.SfT 1. O.64fT.i :-.(... 9 :..... : : : :: S2. L..: ----L...: L- ----ll- --L _.L Period (se) Figure 2.13 Elasti Design Spetra for 2 Perent Damping 24

34 ,., u.. -2r.J-- S pol 3l Sale Falor r =51 se) ;: 3T Sale Fator:: T=O 62 se) 1.5 _ 1 Sale Fator:: 4.75 =2.1 se).. 1 :'\,,, «-... O. 5 - U ot;-:...:::::::::::::::::::::= Vl,5 1 1, Penod (se) 3L Sale Fator:: =51 sec! 3T Sale Falor t; T=.62 se Sale Fator:: 1128 =21 se _ :'\ 1l1. " «"'- r5l_l.::::::::::::':==:::d Vl lo,5 1. 1, Penod (se) 2. rpo=5----';3::-l-;s;-a""''--.-;f""ac1-:-r-="'2"'=-31"1"(:;t-=""'5'-' - -"') 3T Sale Fator 1.83 ("1'.:.62 se) 1.5 _ 1 Sale Fator = (T=2.1 se) 11 :'\, (... O. 5 8.L- ========C=== Vl,,5 1, 1.5 2, 25 3 Period (se) S2. p7 3L Sale Fator:: (T=O 51 se! 3T Sale Fator = 1.31 (T=.62 se & 1.5.-_ 1 Sale Falor= 2664 {T=2.1se i 1 '\ 1j1. «O, i l ==== Vl,,,5 1 1,5 2 2,5 3. Period (se) 3l Sale Fator (T=O.S1 se) 3T Sale Fator = 57 (1= 62 se) '" 1!tT-, 1 Sale Fator::,76 (T=2.1 se) i!,5 i Vl, LO------:O-, 5---'-, ----:--'-, :-::'3 Period (se) 2. e 1.5.., 5 i (/) '%.5 3L Sale Fator= Fr=51 se) 3T Sale Fator =.823 =.62 se 1 Sale Falor = T=2 1 se Penod (se) 2r.---'3'L S7.7F7a,o=r="', ='9"8(T=='. <5'=",! 3T Sale Falor = (T=62 se iii Sale Falor = (T=-2.1 se «i: ::L_"':":_----:_=-==:::===:::J,, , 2,5 3 Pened (se) 2 p6 3L Sale Fator,; (T=.51 se)!; 3T Sale Fator = {T=O.62 se 1.5 _ 1 Sale Falor= 2.74 (T=2.1se.. 1 : '\ H 1. «r5-=-_::-===--':::==::j (/) O.OL,,5 1, 1.5 2, 2.5 3, Penod (se) 2. p8 3L Sale FaClor = (T=.51 se) :: 3T Sale FaClor = (T=.62 se) 1.5 _ 1 Sale Fator = 2.24 (T=2.1 se) 1,,1 :'\, < ' _ 1L-=---'::::::::::::::::==:J, , 2,5 3 Penod (se) a 2. p1 3L Sale FaClor =- 2.7 (T=.51 se) 3T Sale Falor = 1.26 (T=.62 se) _ 1 Sale Fator = (T=2.1 se) i / : '\ 1,, «'-... eo.s --_ 1 '% :-3 Pefloa (se) Figure 2,14 Response Spetra and Saling Fators 25 3

35 2 pt 3L Sale Fator:: 1.37 (T=O 51 se\ io _ 3T Sale Fator:: (T=O 62 se 1 Sale Fator:: (T=2.1 se) j1.1 : \'" PerOd (se), 2, pl3 3L Sale Fator:: 1.934JT::O.51 se\ 3T Sale Fator:: 2.17 T=.62 se iij 15 _ 1 Sale Fator = 4.41 ( =2.1se) 11 :\" O 5 '----- i '" Period (se) 2, Orp"""5-:----,;3'"L""S",a""""e "F""aC",o""rC-'- ' ''".6'''9''1"'''\''T-:''-.''"5'''"S'-e''C''\ 3T Sale Falor:: T=,62 se -15 1SaleFalor=.55(T=21se) j. \ 1 ", 'ii5 jool o Penod (se) a i :\ 1. ", : 2. p7 3lsaleFaClor::3729\T=o.51sel 3T Sale Fator:: T=62 se _ lqsalefator=11.14(t=2,1se i! : '" Penod (se) 2. pg 3L Sale Falor:: (T=.51 se) : 3T Sale Fator :: (T=62 sel _ 1SafeFatOl'=4641 (T=2.1se i :\ ", / , Penod (se) S2. p12 3LSaleFator:2656 CT:O.51seC\ 3T Sale Fator = 4929 {1=_62 se "; 15,- _ 1 Sale Fator:= (T=2 1 se) jl. 1 :\'" ii o.o.l_="-_--"":"'--_...j Penod (se) 2. p14 3l Sale Fator = 3.6 \T=.51 se) 3T Sale Fator = T=,62 se) e 1.5,-_ 1 Sale Faior = 14,767 (T=2 1 se) 1.1 :\" u C/), Period (se) 2rp6---'3"L'S'e'Fa'or'2'. (T'""'5"se"\' 3T Sale Fator = 6223 (1=.62 se "i! 1.5. \ 1 Sa'e FaC1or' (T 2.1 se) 1.. " :t ' v ol--:...::.:::::::::=====:j '" Pened (se) :S2. p18 3L Sale Fator = {!:=.51 se\ 3TSaleFator=1. 762r=O _ 62se 'l! 1,5 r-_" 1 Sale Fator = 1.88( =2.1se) "., 1. " '" ' '" i Period (se) S2- p2 3L Sale FaCO( = T=5' se! 3T Sale FaClor = T=_62 se l! 15 r-_ 1 Sale Falor= 2_54 =2 _1 se).., :\ i 1. " OS _!) ol2:...2::=::::=::::::::::: 3 '". 5 1, Period (se) Figure 2.14 Response Spetra and Saling fators (on't) 26

36 .4 r , v.n----; : «-.2 : g -.4 '" & -.6.>< ' -.1 L-_-'----_-'--_-'--_-'-_-'-_--'-_--'-_---'-_...J o Time (se) (a) Rotation Time History 3r ==-->l 2.>< : L--'---'-_'---'---'-_-'-----'----'-_-' L---'-_-' J Link Rotation (rad) (b) Shear versus Rotation Hysteresis Q) g>.2 «: a:: -.4 ::; -.6 L.-_-'-.l..-_--'- -'--_--'- -'-_--' -' o Cyle Number () Ordered Cyles from Rainnow Counting Proedure Figure 2.15 Representative Link Data (First Story Link, Frame n, P4 Reord) 27

37 (7' 3.1 umulative Damage Model 3 LO D:'"G PROTO OL DEVELOPME T The development of the shear link protool followed the arne methodology used in developing the basi SAC moment frame loading history (Krawinkler et a1. 2). The basi prmi es of that methodology will be re iewed here. Under yli loading. the damage to the link. D. is assumed to be de ribed by a umulative damage model of the type: where: 61i. N C e = = = deformation range of yle i. number of damaging exursion (yles). a strutural perfonnane parameter that may depend strongly on the type of omponelll and failure mode. and = a strutural performane parameter that i usually greater than. (3.1 ) From this model ome several important priniples of umulative damage that should be onsidered in loading protool devlopment. Among these are, first. the damage from inelasti exursions is umulative. omponent apaity is expeted to de rea e as the number of yles inreases. eond. large yles ause lillie;' more damage that small yles (e > ). For steel omponent experiening plasti deformations, is li kely between 1.5 and 2 (Krawinkler et a1. 22). Fina ll y. the model indiates that, for a simple loading history. the primary parameters should be the number of yles (N). the defomlation ranges of the yles (61i,). and the sum of the defonnation ranges (1: i,). Reall that the defonnation range for a yle is the hange in deformation from peak positive deformation to peak negative defonnation. 28

38 CD (;) <. en 'w Response Parameters for Link Loading Protool For the SAC moment frame loading protool, the basi defonnation parameter was the interstory drift angle fl. For shear link loading protools, however, the link rotation angle y has typially been used as the deformation parameter. n terms ofy, the primary demand parameters disussed in Setion 3.1 are: N, = Total Number of Cyles (yles with rotation range >.75 rad) Rotations greater than hal f the yield rotation are onsidered damaging. The range of.75 rad was seleted beause a li nk rotation of.375 rad (half of that range) orresponds to an estimate of half the yield rotation. Cyles with range <.75 rad are not onsidered damaging. y, = Rotatioll Rallge of Cyle i Reall that yles are arranged in desending order so that yle 1 has the largest range, yle 2 as the seond largest, and so forth. ' y, = CUlllulative Deforlllatioll Rallge (yles with rotation range>.75 rad) This is the sum of all the yle ranges. n addition to these primary demand parameters, seondary parameters onsidered protool development ar: Np = NUlllber of ill elasti Cyles Link yield rotation varies somewhat depending on setion geometry and link length, but is generally lose to.75 rad for short shear links. nelasti yles an be roughly defined as those with Yi >.15 rad (2xO.75). y",,, = Yma\ = Y2 = YJ = Maximul Rotatioll Rallge Maximul Rotatioll Seolld Largest Rotatioll Rallge Third Largest Rotatioll Rallge These dmand parameters paralll those used by Krawinkler et al. (22). Values for thes primary and seondary demand parameters were alulatd for eah link in eah frame under eah earthquake using the yle data obtained from the rain flow ounts (Setion 2.4). 29

39 < 3.3 Target Values for Demand Parameters A loading protool an be haraterized by the primary and seondary demand parameters disussed in Setion 3.2. n developing a new protool, "target value" for eah of the demand paramtrs were seleted based on statistial analysis of the values alulated from the model link data Gonrning Priniple ill eleti ng Target Values Krawinkler et al. (22) outlined some guiding priniples in determining appropriate target values for the demand parameters. These priniples (re-written in terms ofy) are: 1. The loading hisloy shollid represenl a "reasonable and generally onsen'atil'e " demand on N " ay" alld r.ay, /1' the /1111 range / antiipated link rotalions (i.e.. /or links in EBFs / all periods, all stories in a stl'llclllre. all reasonable designs. all seismi regions, all types / ground motions. et.) 2. "Reasonable and generally onsermtil'e" implies that the total number / damaging yles, N " should be represented in Q erage. and that he umulatil'e de/ormation range. r.ay.. should be represented onservatively. Consideration should also be gil'en to the fat that small yles are muh lore frequelll to large ones. and that small elasti yles ontribute l'el)1 lillie to damage. 3. PrilllOlY onsideration should be given to the yles lvitll relatively large de/ormation ranges. 'hih \'i dominate damage aumul a t ions. 4. Additional onsideration should be given to onsermth'e representation a/the plasti de/ormation ranges. 5. El'en though it is desirable (see ilem 1). it will not be possible to separate tlte loading l,istoly /ully from tlte maximum de/ormation range. ay""" at 'hih aeptability is 1 be emluated. This annof be done beause r. y, depends strongly on aym" and r. y, is the most importanl parameter 1 be represented in the loading histol) Determining Target Values Figures 3. 1 shows values for three of the demand parameter (N" r.ay.. and ay"",) for Frame 3L. nfomlation is shown for links from eah story for all of the ground motions. For referene, eah plot also has the parameter value from the urrent A C link loading protool indiated by a dashed line. Results are shown in the same formant for Frames 3T and loin Figures 3.2 and 3.3. Note that the AS protool has a signifiantly higher umulative range 3

40 en., demand than that experiened by any of the inks under any of the earthquakes [plot (b) in Figures 3.1 through 3.3]. 1n the development of the SAC moment frame protool, a "ritial story" was identified from the strutures, and final statistial analysis and loading protool development was based on the data from the ritial story (Krawinkler et al. 2). A similar approah was used in this study. Comparing the results from the three frames, the 3-story frames have higher values than the 1-story frame. Of the two 3-story frames, Frame 3T has slightly higher overall values for the three parameters and represents the ritial frame. n Frame 3T the third story links have the highest umulative range, while the first story links have greatest maximum range. For the remaining analysis, data from the first and third story links from Frame 3T was used sine these represent the "ritial links". Statistis were used to haraterize the demand parameters for the ritial links. For eah parameter and link (e.g., 2:t.y, at third story) several probabilisti distributions were tested to desribe the variation of that parameter for the twenty earthquake reords. Lognormal distributions tended to provide the best fit for most of the parameters. With a distribution assigned, perentile values were omputed for eah parameter. Figures 3.4 through 3.1 show the perentile values for the parameters for the first and third story links in Frame 3T. These results were used to determine target demand parameter values for the proposd loading protool. A brief disussion about the seletion of the target values for eah parameter follows. Total NUll/bel' of Cyles, N, Figure 3.4 indiates that the number of damaging yles (t.y >.75 rad) was gratest for the third story links. Sine the total number of yles should be represented in average ( ee Setion 3.3.1), a target value of 36 was reasonable. NUll/bel' of nelasti Cyles, Np Figure 3.5 shows that the third story links experiened more inelasti yles than the first story links. A target of 18 inelasti yles, on'esponding to the 9'h perentile value, was seletd. t was deided to onservatively represent the number of inelasti yles, to aount for buildings shorter than 3 stories whih were not addressed speifially in the study. The two parameters, 36 total yles and 18 inelasti yles, provide a rough initial framework for the protool. 31

41 uj (!) "> Sum of ROlation Ranges, ".6y, This parameter is one of the most important as it represents the umulative rotation demand. From Figure 3.6, the third story links have higher values for ".6y, than the first story links. Sine this parameter should be represented onservatively, the 9th perentile value for the third story links, 1.1 rad, was used as the target value. Maximum Rotation Range, 6y",,,, The maximum rotation range in a protool indiates the point at whih the umulative rotation should be aomplished, and usually represents the aeptane riteria. Sine the urrent ASC Seismi Provisions speify a design inelasti rotation of.8 rad for short links (total rotation of about.9 rad assuming elasti rotation of about.75 rad), an appropriate maximum rotation range in terms of that value would be.18 rad (2 xo.9 rad). From Figure 3.7, the target value of.18 rad is higher than the 9th perentile value of the third story links, but a little lower than the 9th perentile value for the first story links. Maximum Rotation, Ym", Sine a simple protool onsists of symmetri yles, the target maximum rotation was onstrained by the target maximum rotation range to be.9 rad (6Ym,,/2). Figure 3.8 indiates this orresponds well with the 9th perentile value for the third story links. The 9'h perentile value for the first story links is somewhat greater, but this is not onsidered a problem for two reasons. First, damage is proportional to the range and not amplinlde, and the results in Figure 3.7 indiate that rotation ranges remain reasonable even though large one-sided exursions sometimes our. Seond, links that experiene a large maximum rotation generally have lower umulative rotations (reall lower sum of rotation ranges for the first story links in Figure 3.6). Testing of several links has demonstrated that large, onesided rotations are ahievable as long as umulative rotations are not high (Malley and Popov 1983; Kasai and Popov 1986). These results (Figure 3.8) suggest that links in EBFs, designed aording to the provisions (A SC 22), will generally not exeed the design rotation of.8 rad inelasti rotation under design earthquake loading. Those that do will likely have lower umulative rotations. Note that near fault reords were not inluded in the ground motion suite used to generate the data. Suh reords result in higher maximum rotations, but again, umulative range demands and maximum range values are still reasonable (see Appendix B). 32

42 G,.u i7> 'J Magnitlldes of D.Y2 and D.YJ From Figures 3.9 and 3.1, the target values of the seond and third largest ranges were seleted be.1 rad and.8 rad based on the 9th perentile values. Rotation Range of Cyle i. D.Yi This is one of the primary demand parameters and relates to the proper distribution of yle ranges within a protool. The model data for D.Yi is most useful in the foml of the umulative distribution funtion (CDF) for all the yles from all the reords, for a given link. Figure 3.11 shows the CDF for the first and third story links. The CDFs are similar and both indiate that the majority of yles have small ranges. Table 3.1 summarizes all the target demand parameter values obtained from the timehistory analysis data, as disussed above. 1 The umulative distribution funtion (CDF) should not be onfused with limulative rotation demand. Tle umulative distribution funtion indiates the perentage of yles ha ving a range less than some given range. The umulative rotation demand has units ofrad and is the sum of the yle ranges. 33

43 ;:;:) '. C') Table 3.1 Summary of Demand Parameter Target Values Demand Parameter Target Value N, 36 yles Np 18 yles LYi 1.1 rad Y",ax.18 rad Ymax.9 rad Y2.1 rad YJ.8 rad, Yi CDF (Figure 3.11) 34

44 () OJ U >- U - OJ.D E : z rn 2-2. " -; 1.5 '" : a: '" OJ 1. >.5 : u..3 ".!::;. '".25 OJ g'.2 a: '" E.15 : E.1 x.5.? j\ o -/-;'L'" / 1st Story A 2nd Story 3rd Story 3jo.!:!:; '"/ '"... e -"'-'" '" '" ", Eathquake Reord Number (a) Total Number of Cyles. JSC 1 st Story _.- A 2nd Story 3rd Story - )o- "' '8 8/... ",_",'8o '" /'" Q o "'-'" 8=8' fa /, "'-... _"'- -'" 6 A-o-A A A Eathquake Reord Number (b) Cumulative Range A 1st Story 2nd Story 3rd Story a f\scnrl_a a o / "'-"'\ '" _o-a-o/ - \ o\ o e- g-8=a/ 2_; : : Eathquake Reord Number () Maximum Range Figure 3.1 Demand Parameter Values for L inks in Frame 3L (Cyles with Range >.75 rad Considered)

45 Q) '" () 6 " ======== o 5 o Eathquake Reord Number (al Total Number of Cyles 2. JL._ -;; 1.5 OJ m 1.. o o o.. o 1st Story 2nd Story 3rd Story 1st Story 2nd Story 3rd Story ::J (). L- ----'- ----'- ----'- ----'----.J. 3 " 1l.25 Q) g'.2 m a: E.15 ::J E.1 'x.5 o Eathquake Reord Number (bl Cumulative Range o.. o 1st Story 2nd Story 3rd Story AtSC..Qen R<!!l9L. L- ----'-- --'-- -' '...J Eathquake Reord Number (l Maximum Range Figure 3.2 Demand Parameter Values for Links in Frame 3T (Cyles with Range >.75 rad Considered)

46 8 f/) C> U {S 6 C>. E ::J Z "" '" -=-.25 C> g>.2 :: '" E.15 ::J E.1 x.5 1st Story " 5th Story 1th Story ASC r ,,-ib\ e 8 - ;?9 ; 2 6 v a e= ' e:ae Eathquake Reord Number 5 o " 1 15 Eathquake Reord Number o A 1st Story 5th Story 1th Story o 1st Story "5th Story _ \- 1th Story. L- ---L.. --'- -'-- ---''--' Eathquake Reord Number Figure 3.3 Demand Parameter Values for Links in Frame 1 (Cyles with Range >.75 rad Considered)

47 '" 6 > () '" <1l 6 () u {'l 4 : 1 5t Story 3rd Story o 3 Figure Perentile Perentile Plaing of the Total umber of Cyles, N, (Cyles with Range>.75 rad) 15t Story 3rd Story o Q:; 2..!!Qe.t.= :.:.:...:.::.:..:..:.:..;...; J:l E :J Z " 2. '" Q) 1.5 o :: o Perentile Figure 3.5 Perentile Plaing of the Number of nlasti Cyles, Np (Cyles with Range>.15 rad) 1 5t Story 3rd Story --.. T g.e.= '.:.-" iii '.5 o E :J en. L... -'- -'- -' '-' 5 Figure Perentile Perentile Plaing of the Sum of the Rotation Ranges, 1:6.Yi (Cyls with Range >.75 rad) 38

48 --..J W " '" -'=- Q) Ol '".3.25 ::.2 g.15 i'l :: E : E.5 'x :::;: '" Perentile 1 5t Story 3rd Story Figure 3.7 Perentile Plaing of the Maximum Rotation Range, D.Ym3X.15.2 o o ::.1. _"_ a=9:e.1.o:9:. :d. ''.:. '... ':':";. ' ''= ' ''='..... :. :-=:.. =..... =... = E : E ---- 'x.5 :::;: '". --,- --,- -,- --,---.J Perentile :.2 Q) g'.15 :: '" Vl Q).1 -' '" -g.5 8 Q) 8 15t Story 3rd Story Figure 3.8 Perentile Plaing of the Maximum Rotation, Ym3X 1 5t Story 3rd Story..! 9.e.t.= _ Perentile Figure 3.9 Perentile Plaing of the Seond Largest Rotation Range, D.Y1 39

49 '.2 g 8,.15 OJ n: 'ii).1 '" e> 5.5 u. u..!..= st Story 3rd Story L...L -'-- --' --'- ---''--' Perentile 8 9 Figure 3.1 Perentile Plaing of the Third Largest Rotation Range, YJ 1..9 /' /'.8 - ' st Story 3rd Story ',,,, y, Figure 3.11 Cumulative Distribution Funtion of Cyle Range, 6y, (Cyle with Range>.75 rad, for Links in Frame 3T with all Ground Motion) 4

50 4 PROPOSED HORT LNK LO DNG PROTOCOL 4.1 Propo ed Testing Protool A proposed loading protool for short links was developed, based on the demand parameter values desribed in Setion 3.3. The protool onsist of several rotation amplitude steps, eah onsisting of a number of symmetri yles. The protool is summarized in Table 4.1 and illustrated in Figure 4.1 (a). The 22 ASC protool is shown in Figure 4. (b) for omparison. Table 4.2 ompares the demand parameter values of the proposed loading protool with the target values. The target va lues are onservatively represented in the new protool. The values of the 22 ASC protool are also shown in the table for omparison. Figure 4.2 ompares the disrete COF of the proposed protool wi th those of the links from Frame 3T (see Setion 3.3.2). The COF of the ASC protool is also shown for ompanson. ote that a protool COF "below" the data is onservative, indiating the protool has a greater perentage of large amplitude yles than th data. The proposed protool is reasonable in omparison with the data. 4.2 Comparison of the Proposed and ASC Loading Protools The proposed protool differs from the ASC link protool, and the ASC moment frame loading protool, in that the deformation inrement hanges for the latter stps and only one yle is applied at the latter steps rather than two. These harateristis were neessary to provide a distribution of yles onsistent with the data. From Table 4.2, the proposed protool has more total yles, the same number of inelasti yles, and lower umulative range demand as ompared with the 22 ASC protool. Figure 4.3 illustrates the umulative range demands of the two protool plotted against yle range. The ASC protool requires 48% more umulative rotation than the new protool. n addition, and perhaps more signifiant, 72% of the total umulative range demand in the ASC protool omes from yles with rallges greater th an. 1 rad. n omparison, only 37% of the total Ulllulati ve range demand omes from yles with ranges greater than.1 rad in the proposed protool. Reall from the damage model that large exursions ause 11l1ell more damage than small exursions (see Setion 3.1). This highr 41

51 perentage of large range yles in the ASC protool is also illustrated by the CDF (see Figure 4.2) Based on the demand parameters and the assumed damage model, the proposed protool is signifiantly less severe than the ASC protool. t is antiipated that a short shear link would reah a greater rotation level prior to failure with the proposed protool than it would with the ASC protool. 42

52 J J Table 4.1 Proposed Short Link Loading Protool Load Step Peak Link Rotation Angle, y Number of yles '.9 'Continue Wllh llcrelllnts 111 Y oro.2, and perrorm yle at ah step untli raliure Demand Parameter Table 4.2 Comparing Protool Demands with Target Values Target Value Proposed Protool ASC Protool N 36 yles 36 yles 24 yles N' p 18 yles 18 yls 18 yles LLl.y, 1.1 rad 1.14 rad 1.69 rad Ll.Yn,"..18 rad.18 rad.18 rad Ynl3,,(.9 rad.9 rad.9 rad nyl.1 rad.14 rad.16 rad nyj.8 rad.1 rad.16 rad ny, See Figure 4.2 See Figure 4.2 See Figure 4.2 'Assullling Yy '".75 rad 43

53 :> LO... 1.:;>.8.6 u '".4.B1 CJ).2 <{.,g (5 '" -.2 :: x -.4 :.::; U '".4,.B1 CJ).2 <{. :;:; (5 '" -.2 :: X -.4 C :.::; r 375rad - <y<;e. A ::':"A A A OO:5A'": A A vvvvvvyyyyyyvvvvvv t,; A 6 ydes, P yles. 6 yles. 25,,. A s: A o;,, A 3 oyo. OOS 4 OOJ 15 rad.2... yles Cyle Number vvvyyv 3 yle. 3 yles. V (a) Proposed ooe tad '(t.7 rad O.6 rad 1)5 raa 4 rad 3rad '2 red 2 eyelet 2 ylo. r 2 ylet 2 yle. 2 oyo. - 2 yde> 2 "",., Cyle Number (b) ASC Figure 4.1 Shear Link Loading Protools 44

54 . O. 1. F---==---==::::===;;,;;r= Data -..- J....9.; :-- / r_jproposed Protool.8 - /...J r.7 f- / ASC Protool r-'.6 t- / LL 8.5 f- rl.4,j ,,, 1 5t Story 3rd Story y, (rad) Figure 4.2 Comparing Loading Protool CDFs with those from Frame 3T Links :-.g,.: <l w ASC Protool J Proposed Protool J 1 1.6, ' r--'.4 r-' r...j.2 rj....,'---'-----"'-----"--" ' " ' ' ' ' ',.J y, rad) Fi gure 4.3 Comparing Cumulative Ranges for Proposed and 22 ASC Protools 45 r

55 5 i\ Ji\1 RY A ' QXCL 1 There has been a shear link loading protool speified in the Al ei mi Provi ions sine 1997; however, it is a modified version of the SAC moment frame loading protool without any study to justify it. The objetive of this study was to in\' tigate the rotation dmands on short links in entrially braed frames under design earthquakes and develop a loading protool for link testing. One 3-story building and one O-story building, with a total of three unique eentrially braed frames, were designed by the Al C Solution Center. The designs alled for lnk in onfigurations where one end of eah link was onneted to a olumn. Models were developed for eah of the frames, and nonlinear timehistory analysis was performed using a suite of Los Angeles earthquakes saled to math the 1997 UBC design spetra. The model results indiatd that the protool in the ASC eismi provisions is overly onservative in representing design earthquake demands. A new loading protool was developed following the same genral proedure as was used in devloping the SA moment frame loading protool. The proposed protool requires only 67% of the umulative rotation speified by the ASC protool in order to reah the link design inlasti rotation for short links (.8 rad). The proposed protool also requires fewer large inelasti yles as ompared to the ASC protool. Aording to the assumed damage model, the proposed protool is signifiantly less severe than the urrent ASC protool and link would ahieve higher maximum rotation when tsted with the proposed protool. t is reommended that the proposed protool be used for future te ting of short shear links. Additional work needs to be ompleted to address loading for longer links (ev,.mp > 1.6). 46

56 REFERENCE ASC. (22). Seismi Provisiolls for Strutural Steel Buildillgs, Amerian nstitute of Steel Constrution, Chiago, L. Are. G. (22). "mpat of Higher Strength Steels on Loal Bukling and Over trength of Links in Eentrially Braed Frames." Masters Thesis, University of Texa at Austin, Austin, TX (advisor: M.D. Engelhardt). Engelhardt, M. D., and Popov, E. P. ( 1989). "Behavior of Long Links in Eentrially Braed Frames." Report No. UBCEERC-8911, Earthquake Engineering Researh Center, University of California at Berkeley, Rihmond, CA. FEMA. (2). "Prestandard and Commentary for the Seismi Rehabilitation of Buildings." FEMA 356, Federal Emergeny Management Ageny, Washington, D.C. Gupta, A., and Krawinkler, H. (1999). "Predition of Seismi Demands for SMRFs with Dutile Connetions and Elements." Reporr No. SACBD SAC Joint Venture, Saramento, CA. Hjelmstad, K. D., and Popov, E. P. (1983). "Seismi Behavior of Ative Beam Links in Eentrially Braed Frames." Report No. UBCEERC-83115, Earthquake Engineering Researh Center, University of California at Berkeley, Rihmond, CA. CBO (1997). Ulliforlll Buildillg Code. nternational Conferene of Building Offiials, Whittier, CA. CC (2). /lerllatiollal Bllildillg Code. nternational Code Counil, n., Whittier, CA. Kasai, K., and Popov, E.P. (1986). "A Study of Seismially Resistant Eentrially Braed Steel Frame Systems." Report No. UCBEERC-8611, Earthquake Engineering Researh Center, University of California at Berkeley, Rihmond, CA. Krawinkler, H., Gupta, A., Medina, R., and Luo, N. (2). "Loading Histories for Seismi Perfornlane Testing of SMRF Component and Assemblies." Report No. SACBD- 1/. SAC Joint Venture, Saramento, CA. Krawinkler, H., Medina, R., and Alavi, B. (23). "Seismi Orin and Dutility Demands and Their Dependene on Ground Motions." Ellgilleerillg Strutures, 25(5),

57 Malley, 1.., and Popov, E. P. (1983). "Design Considerations for Shear Links in Eentrially Braed Frames." Report No. UBCEERC-83124, Earthquake Engineering Researh Center, University of California at Berkeley, Rihmond, CA Medina, R. A. (23). "Seismi Demands for Nondeteriorating Frame Strutures and Their Dependene on Ground Motions." Ph.D Disertatioll, Department of Civil and Environmental Engineering, Stanford University, Palo Alto, CA. Popov, E.P., Riles, 1.M., and Kasia, K. (1992). "Methodology for Optimum EBF Link Design," Proeedillgs, Tenth World Conferene of Earthquake Engineering, Vol. 7, pp , Balkema, Rotterdam. Prakash, V., Powell, G.H., and Campbell, S., (1993). "DRAN-2DX: Base Program Desription and User Guide." Report No. UCBSEMM-9317, Department of Civil Engineering, University of California, Berkeley, CA. Ramadan, T., and Ghobarah, A. (1995). "Analytial Model for Shear-Link Behavior." Journal oj Strutural Ellgilleerillg, 121 (), Rihards, P., and Uang, C-M. (22). "Evaluation of Rotation Capaity and Overstrength of Links in Eentrially Braed Frames (Phase )." Report No. SSRP-22118, Department of Strutural Engineering, University of California at Sun Diego, La 11la, CA. Riles, J. M., and Popov, E. P. (1987). "Experiments on Eentrially Braed Frames with Composite Floors." Report No. UBCEERC -8716, Earthquake Engineering Researh Center, University of California at Berkeley, Rihmond, CA. Riles, 1.M., and Popov, E.P., (1994). "nelasti Link Element for EBF Seismi Analysis." Joul'/lal oj Stmtural Ellgilleerillg, 12(2), Roeder, C. W., and Popov, E. P. (1977). "nelasti Behavior of Eentrially Braed Steel Frames under Cyli Loadings." Report No. UCBEERC-77118, Earthquake Engineering Researh Center, University of California at Berkeley, Rihmond, CA. Somerville, P., Smith, N., Punyamurthula, S., and Sun, 1. (1997). "Development of Ground Motion Time Histories for Phase 2 of the FEMNSAC Steel Projel." Report No. SACBD SAC Joint Venture. Whittaker, A. S., Uang, C-M., and Bertero, V. V. (1987). " Earthquake Simulation Tests and Assoiated Studies of a.3-sale Model of a Six-story Eentrially Braed Steel Struture." UBCEERC Report No. 87-2, Earthquake Engineering Researh Center, University of California at Berkeley, Rihmond, CA. 48

58 w 1 APPE DX A-Rsults from Tim-History Analyses Using Simplified lodel Models were developed using the simplified tehnique that Whittaker et al. (1987) used. The same data redution tehniques were used and results are presented in the fol lowing pages in the same manner as in the body of the report. Re ults ar very similar, and the same loading protool would be developed based on the simplified models. 49

59 ) Q) u u > Q) D E ::> Z '" 2 1:) 2. ';' 1.5 OJ : '" 1.. iii.5 ::> U..3 1:).!::;..25 '" Q) g>.2 :: '" E.15 ::> E.1 'x.5. 5 o.. [J 1 15 Eathquake Reord Number (a) Total Number of Cyles _. A.!S.L.. " 1 st Story 2nd Story 3rd Story 1st Story 2nd Story 3rd Story OJ! o 6, 6'6 /-e,d7-6,?d-d- /, 6- -g 9/ 9 _H Eathquake Reord Number (b) Cumulative Range.. " 1st Story 2nd Story 3rd Story AnRL //6\ i. ' / ",- J: 6\6 6 D_g / D?G B=8D/ g-?;= Eathquake Reord Number () Maximum Range Figure A. Demand Parameter Values for Links in Frame 3L (Cyles with Range>.75 rad Considered) 5 2 2

60 V1 (/) Q) 8 u u '" 6 Q).n E ::> Z.l!l - " 4 2 1st Story.. 2nd Story [] 3rd Story o h- O iyi: - 'g;-:8- -1 / - _" Of / -- 8/ / 6 6 / 6 - /6- _ 6 - _ / '_ 6 - / ' '6-6 = - s:::: -&-----"::" Eathquake Reord Number (a) Total Number of Cyles 2. _.._ 1.5 Cl o.. [] 1st Story 2nd Story 3rd Story 2 _.- '; '" 1. '".5 ::> U. ' '- ----'- ----'- ---<--.J. 3 " '" -=-.25 Q) g'.2 :: '" E.15 ::> E.1 x Eathquake Reord Number (b) Cumulative Range o.. [] 1st Story 2nd Story 3rd Story. '-- --'-...L.- -' '---l Eathquake Reord Number () Maximum Range Figure A.2 Demand Parameter Values for Links in Frame 3T (Cyles with Range».75 rad onsidrd)

61 en 8 > 4> U 66-4>. E ::l Z 4 '" 2 (5 f-.3 ".!;:..25 '" 4> g'.2 :: '" E.15 ::l E.1 x.5. 1st Story.. 5th Story D 1th Story """ J " -----o_----o-.- o - Yo 6 =g-o - -aee Ealhquake Reord Number 2. r-----;;-;l: ;==o==1s=t=st=ory==::::; Ealhquake Reord Number.. 5th Story D 1th Story 1st Story.. 5th Story D 1th Story JL._ g e / o > j /o--gfo'e=q,/ _ 8- o - /,- O/oe/ '8 - -, Ealhquake Reord Number Figure A.3 Demand Parameler Values for Links in Frame 1 (Cyles with Range >.75 rad onsidered) 52 2

62 <> J en., 6 u -o 1 st Story 3rd Story 2 4..!Qel.=}...'"'-'_ "l"":i"". :":'7.-:-:-.-:-: E z " 2 (5 - en Q) 6 U.>! iii 4 -o " 2. en., o Perentile Figure A.4 Perentile Plaing of the Total Number of Cyles, Nt (Cyles with Range>.75 rad) 1 st Story 3rd Story o Perentite Figure A.5 Perentile Plaing of the Number oflnela ti Cyles, Np (Cyles with Range >.15 rad) 1 st Story 3rd Story OJ 1.5 ::.. J[!je!.=.l P." _.... Q 1. _ S o ::.5 -o E (/) ". L...>. --'- --'- ----L ----L-' Perentite Figure A.6 Perentile Plaing of the Sum of the Rotation Ranges, Lily; (Cyles with Range >.75 rad) 53

63 " o <l.3 Ol.25 o :: o (5 ::.1 E :::l E.5 'x o :2..2 ".15 a <5 ::.1 E ::J..5 o :2 iii Ol e> ",-...J 151 Story 3rd Story.. Trt. ':.. '?d = 7 Perentile Figure A.7 Perentile Plaing of the Maximum Rotation Range, 6Ymo< 15t Story 3rd Story == L... --' ' --'- L...J 1.5 -au::: 1. Co _ UOl Ol", (/)C O&..5 a :: Perentile Figure A.S Perentile Plaing of the Maximum Rolation, Yma., 8 1 st Story 3rd Story..!e..= L... -'- --'--.L '---' Perentile Figure A.9 Perentile Plaing of the Ratio of 6yz to!.ynm

64 iii > >_ <O '" J i: 1. 'Eo :; t-o> o iii 5.Q ::. iii :: LL.. t) 1st Story 3rd Story.. :r l..q? '_ ' L... "-- ---'- --'...L...J Perentile Figure A. Perentile Plaing of the Ratio of Y3 (.9.,-,,---.- /.8 - /.7 l- J.6 l-.5 l J 1 st Story 3rd Story 8 Ym,,,.1 -.,, />"y, Figure A. Cumulative Distribution Funtion of yle Ranges, y, (Cyles with Range >.75 rad, for Links in Frame 3T under all Ground Motions)

65 'Zl Q;),1) O Q;) APPENDX B-Rsults from Time-History Analyses Using SAC LA 1/5 Reords Analyses were also perfonned on the simplified models using the SAC Los Angeles ground motions with 1 perent probability of exeedane in 5 years. Figure B. shows these ground motions (already saled for the SAC projet) with name and PGA indiated in eah plot. Figure 8.2 shows the spetral aelerations with 2 perent damping. Also inluded in the plots is the 2 perent design spetra used the study (see Setion and Figure 2. 13). Analysis results for the key parameters are shown for the three frames in Figures AJ to A.5. Based on these plots, the " and 3rd story links in Frame 3T were the ritial links. Figures A.6 to A.12 show the perentile values for the protool parameters for the ritial links. Figure A.13 shows the CDFs for the ritial links. Table A. ompares the target values obtained from these results with those from the runs under the PEER reords (see Setion 3.3.2). The PEER reords resulted in a higher number of yles, both elasti and inelasti, but the umulative range was only slightly higher. The signifiant differene between the SAC reords and the PEER suite was the presene of near fault reords in the SAC suite. This r ulted in high 9'h perentile values for!!,y",,, and Ymax. These seondary protool parameters would end up being ditated by the design range of the links (.18 rad), so vry sim ilar protools would have developed from the SAC and PEER data. Table B. Comparing Protool Demands with Target Values Protool Demand Parameter PEER Reords SAC Reords N, 36 yles 32 yles N' p 18 yles 15 yles L!!,Y;.rad 1.2 flyma".18 rad.18 Ym3X.9 rad.9!!,y2.1 rad.14!!,yj.8 rad.1!!'y; CDF (Figure 3.11) CDF (Figure B.13) 56

66 'LJ u)... LAO' PGA: 1749 LA2 PGA-O '3511 g 1... _'" e... -,.,.- J! '" J!.,., Tme (se) Tme (se) LA3 PGA - 366g LAO. PGA - 662g.a.tt. L g E ""l... J!.,., 5 ' ' ' ' Time (se) Tme (se) LAOS PGA LAOS PGA 352g 2 E J! J! ".,., ' ' Time (se) Time (se) E LA7 PGA - 295g LA8 PGA - 389g g 'h","...,..,." '' '- i.,., ' ' TfTe (se) Time (se) LA9 PGA: 588g PGA. 569g 2. J! i.,., ' ' Time (se) Time (se) Figure B. SAC Los Angls 1/5 Ground Motion Aeleration Time Histories 57

67 <J) PGA LA'3 LA'5 LA17. LA'9 U) > e i e = 7529 PGA'.596g. ' 1l < <.,., ' ' Tme (se) Time (se) PGA=O.369g LA,4 PGA = 339 e Q. i 8 u <.,., Time (se) Time (se) e e g.,.., 8 u < e i < e Q PGA = 29g LA'6 PGA =.574g.,., '2 '4 ' ',2,4 '6 Tme (se) Time (se) PGA = 6989 LA'8 PGA = 67'g e.g.,., Time (se) Time (se) PGA = 5419 LA2 PGA =.384g g i <.,., Tml& (se) Time (se) Figure B.l SAC Los Angeles 1/5 Ground Motion Aeleration Time Histories (on't) 58 e

68 CJ CJ u) u) J) '" Qi " 2 1i " 2 u r-""'- u «1 :: '- «ii. 1M -- ii. : LAOl LA ti ti ---- ". ". en en Period (se) Period (se) Q : LA3 g : LA4 " " " Qi Qi 2 u u ««ii iii --- ti --- 1i ---- ". %. ". %. en en Period (se) Period (se) ; LAOS.Q : LA6. " Qi Qi u u " 2 " 2 ««ii --- ii ti ti ---.:... ". %..% % Period (se) en Period (se) Q : LA7 : LA8 " " u " 8 ««ii ii -. ti --- ti " ".. %. en en Period (se) Period (se) : LA9 g : LA1 '" Qi " 1i " 2 u u u ««ii --- ti ". %. ". % "' en en.5 1..,as Period (se) Peri (se) rigure B.2 Response Spetra for SAC Los Angeles Ground Motions (2 perent damping) 59

69 s 3r ,.2 : LAll ".9! 2 i /) o ---. o Penod (se) 3r , : LA13., 2 13 lli /)..5 s : Period (se) 3r : -LA--,5----' ".9! 2 --" :: ol--=-----=:::::::==:;::=::::d /) Period (se) s.5 : LA Period (se). 3 r-----7t : -LA--,9----' ".9! 2 u --- lli OL ::::::::====:= /) Penod (se) s. 3 nr :-l-a-, , " " 2 1 u L-_-=== Penod (se).o.5.5 : LA Penod (se) 3r ,.g : LA16 f! " ---- lli /) Period (se) 3r--nr ,.2 : LA18 " " 2 i /) s. o Period (se) 3, ,.2 : LA2 " " 13 lli /).5 1. _'"5 2. Penoo (se) Figure B.2 Response Spetra for SAC Los Angeles Ground Motions (an't) 6

70 8 V ) U >- u 6 - ) J:J E :J Z - 1:) ; 1.5 Cl '" 1.. io.5 :J u..3 1:).!::..25 '" ) g>.2 &. E.15 :::> E.1 'x Eathquake Reord Number (a) Total Number of Cyles r st Story -- <> 2nd Story 3rd Story <> - Ej;;; Eathquake Reord Number (b) Cumulative Range 1st Story 2nd Story 3rd Story i\ / _ 6-98 / _ o =,8a6-6 /g_6 _,6 '_6/ _--AnRL---_.t\--o- " oo /'<tle 8 7 o LO r a/ a=8 g/ e _88ye,- 'a Eathquake Reord Number () Maximum Range Figure 8.3 Demand Parameter Values for Links in Frame 3L (Cyles with Range >.75 rad Considered)

71 :;: CD. U) V 8 () Q) u >- 6 u - Q). E :J Z (ij ( st Story.. 2nd Story 3rd Story "/ \"'" \ 1< /"'-"'... / 1:- AlO:::::O /-'" '" 1:1/ '" '" -9_ /"'_"'_'" -6-6_ Eathquake Reord Number (a) Total Number of Cyles 2 2. " -; 1.5 Ol 1..2 iii.5 :J u..3 "!:..25 Q) g>.2 a: E.15 :J E.1 'x.5. f- A.lli.C "'\ " 1st Story -_... 2nd Story 3rd Story / / o o Q /;\ / '" 8- O_\O/ O... /, /8/ / ",/ \ - '" - '" 1:1- "'_'".. C?a 6,'/ '6-6/ '6-6'_6_6_6/ 6_ -6_'/ Eathquake Reord Number (b) Cumulative Range 1st Story 2nd Story 3rd Story : j 8- /J8g... h! - - V - Al-:::-6-L _Q.:..a...".. t-- - -G f-/-anr-- a- : :_\ "8-\,(/:/ \til \' > o - '" OJ '" / '2 6_ / g "'-'" 6/ "'-'" "'-'" 78 1 /\ '" '" Eathquake Reord Number () Maximum Range Figure B.4 Demand Parameter Values for Links in Frame 3T (Cyles with Range >.75 rad Considered)

72 u),. J 8 V QJ u >- 6 u - QJ.D E :J Z rn (5 f ' ro 1.5 Ol ro 1. > :;:; ro.5 :J U. 2nd Story.. 6th Story 1th Story r Q_6\ _e=e-d o 6_6_ _- -O /6 _O_C D<Q - o _ Eathquake Reord Number (a) Total Number of Cyles f nd Story r- 6th Story 1th Story a 6_ e / ---'6-e-e O --8 D e?8---8:=8 D C= Eathquake Reord Number (b) Cumulative Range.3 '.25 QJ g>.2 ro :: E.15 :J E.1 'x.5. 2nd Story.. 6th Story 1th Story r AnRL /,!6 8 /6, H<6 Q, H /- 6 B =O8 e o-o D Eathquake Reord Number () Maximum Ranqe Figure B.5 Demand Parameter Values for Links in Frame 1 (Cyles with Range >.75 rad Considered) 63 2

73 U) C) Q) u 6 C) Q) U.!.! iii < Qi -o Q) J:l E z " 8 1st Story 3rd Story o.. r! : :.: :.::-",',r., ' :... -;.- ' Perentile Figure B.6 Perentile Plaing of the Total 8 9 umber of yles, Nt 8 1st Story 3rd Story 6 4 'U 2. C) Q) en 1.5 &. g.!!l o a:: -o E CJ " 2.. :r.. :: ',:: :.-. ''':... :.. ::. :": ".::': :.: :': '.: :.:.:: '.:: ':" ', :....:.,._.. ",'.: :-: ':: :-. ':... o Perentile 8 9 Figure B.7 Perentile Plaing of the umber of nela ti yes, Np 1 st Story 3rd Story 1... T!i.t."". 1 9!q :.:';::.::_""".5.- ",.. -.-, ".-,.. - ", " " -." - '.. --,--...J Perentile Figure B.8 Perentile Plaing of the Sum ofth Rotation Ranges, L Y, 64

74 ".3.25 C Cl ::.2 C g.15 o ::.1 E ::J E.5 'x Cl ::;;.2 " C.15 g o ::.1 E ::J..5 Cl ::;; in a> e> " ", - '" -,.. -., ,. _. - '''- Cl ". - ". _... _ ".." - '.- ' " 1 st Story 3rd Story. L...o. --'- ----' --'- o...j Perentile 8 9 Figure 8.9 Perentile Plaing of the Maximum Rotation Range, Ym,..,, - "' -.. -,., - 1st Story 3rd Story - ". - ".:.:": :.: :.. : :.; : :.:.; _.,.,,_ '",._ 1\',-: : ',' :-: :': :-: ' L.. ---'- --'- -'- ---'--l 1.5 -' "OiL 1. Co - Ua> a> > C/l &.5.Q iii :: Perentile Figure B. Perentile Plaing of the Maxi mum Rotation, Ym,,, 8 1 st Story 3rd Story Target = :....: " - ". - "' -,,.. L '- -'- -'-- ---'--l Perentile Figure B. Perentile Plaing of the Ratio of Y2 to y o u., 65

75 1.5 iii Q) 1_ CJ) j it 1. "Eo :-;; t-o '.5 oel: & u. u 1st Story 3rd Story.. T!.= L.J... -'- ---'- -'- ---'-.J Perentile Figure B.12 Perentile Plaing of the Ratio of fl:.yj to y..., ' st Story 3rd Story 8,,...,, /1y i Figure B.13 umulntive Distribution Funtion of yl Ranges, ',. (Cyles with Range >.75 rad, for Links in Frame 3T under all Ground Motion)

76 - APPE DX - Rainnow yl ounting Proedure Cyle ounting is needed to onven a link rotation time history into a serie of yles from whih a loading history an be developed. The rain now yle ounting proedure u ed 1 this study (Krawinkler et al. 2) is desribed below and illustrated in an example.. First the rotation time history i drawn so as to begin and end at the greate t rotation, to eliminate the ounting of half yles,. This is done by moving the ponion of the time history that follows the maximum amplitude point to the front of the history. The end of the original time history is then anifiially onneted to the beginning of the time hi tory [impliations disussed in Krawinkler et al. (2).) 2. All of the peaks and valleys are identified in the time history. 3. Cyle ounting stans at the beginning of the time history. One a yle is ounted and reorded, the peak and valley assoiated with the yle are not onsidered for further yle ounting purposes. A yle is ounted when the seond range in a pak-valleypeak or valley-peak-valley ombination is greater than the first range. The yle ounted is defined by the first peak-valley or valley-peak ombination (see example on next page). The range of the yle is the differene in rotation between the peak and valley. The mean value a soiated with the yle is the average of the rotation values at the peak and valley assoiated with the yles. Counting ontinue from left to right and stan again at the beginning when the end is reahed. Counting ontinues until the entire history is exhausted. 4. yles are arranged from grea test ran ge to least. Mean effets are not onsidered for basi loading history development, so symmetri yle result. 67

77 An example of the rainnow yle ounting tehnique follows. A short rotation time hi tory with some residual drift is shown below in Figure C.. Rotation o re..:lual rotallon t \-- '-- Time Figure C. Example Rotation Time History r n Step, the part of the history following the maximull1 rotation is identi fied and moved to the beginning (Figure C.2). The original end of the reord is artifiially onneted to the original beginning of the reord. RotatOn 4 < 1!\. o Tm" - O'.,.,lv\ , 'v RotatOn.,f\,/\., 1 /\. o --i-'hi Tm. u V -.5-6,7 Figure C.2 Reordering Rotation Time Hi tory n Step 2 peaks and valleys are identified (Figure C.3). 68

78 o o r t lisa) Figure C.3 dentifiation of Peaks and Valleys n Step 3, yles are ounted as the reord is read from left to right. The first peak-valleypeak or valley-peak-valley ombination to be read is A-B-C. ombination B-C is /lot greater than or equal to A-B, /1 ine the eond leg of the yle is ounted (Figure C.4). The next peak-valley-peak or valley-peak-valley to be onsidered is B- -D. ine seond leg D C is greater than the first leg B-,the eyle B-C is ounted, and points B and are not onsidered anymore. Similarly, ontinuing to the right, yle E-F is ounted sine F-G is greater than or equal to E-F; and yle G is ounted sine H- i greater than or equal to G H. RotatOn o.1 Figure o /--"---\ T... 1(1.) ounting Cyles in Reord A nr reading through the reord one, and removing the peak and valleys that were "ounted", the only remaining points are A, D. and (Figure.5). When the reord is read a seond time, yle A-D is ounted sine D- is greater than or equal to D-A. tne there are no peak-valley-peak or valley-peak-valley ombinations remaining. the reord is exhausted and the proess is finished. 69

79 ..- CD '" h RotatOn,4,3 2,1 T1me Ml 2 -, Figure C5 Counting Final Cyle The results of the yle ounting are tabulated, with yles arranged from greatest range to least as shown in Table C, The ranges define the resulting yles. For a basi loading history mean effets are not onsidered, so the set of yles are symmetri (mean value of zero). Figure C6 shows the resulting symmetri yles from the original time history, The maximum rotation (,5 rad) is equal to half of the maximum range (.11 rad), Cyle No, (A-D) 2 (B-C) 3 (E-F) 4 (G-H) Table C Rainflow Counting Results RotatOn Range (rad),1\ ,5,4 3,2 1 o-+--+n---,1 -,2 3,4-5,6 Mean (rad) -,15, Figure C6 Resulting Symmtri Cyles from RainflolV Counting 7

80 -V"