Comparison of Force-Based and Displacement-Based seismic design of dissipative post-tensioned rocking timber wall systems

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1 Comparson of Force-Based and Dsplacement-Based sesmc desgn of dsspatve post-tensoned rockng tmber wall systems F. Sart, A. Palermo & S. Pampann Cvl and Natural Resource Engneerng Department, Unversty of Canterbury, Chrstchurch, New Zealand 215 NZSEE Conference ABSTRACT: Based on numercal studes performed by the authors, the paper presents to Force-Based (FBD) and Dsplacement-Based (DBD) sesmc Desgn provsons for dsspatve post-tensoned rockng tmber wall (Pres-Lam) systems. At frst, a dscusson on the sesmc desgn requrements of post-tensoned rockng tmber wall systems s provded, ncludng suggested materal lmt states to be consdered n the desgn phase (Ultmate Lmt State) also amed to prevent collapse n case of more severe events. The requred amendments to a Force-Based Desgn provson n accordance to NZS117.5 are then dscussed focussng on the development and sgnfcance of the nelastc spectrum scalng factor (k μ ). Smlarly, the key modfcatons to desgn equatons requred wthn a Dsplacement Based Desgn Provsons are proposed, based on extensve numercal analyses and focusng on the as the desgn nelastc dsplacement shape and the equvalent vscous dampng. In the fnal part of the paper a mult-storey buldng s desgned usng both FBD and DBD and the two dfferent approaches are dscussed. 1 INTRODUCTION Dsspatve post-tensoned rockng tmber (Pres-Lam) wall systems are low-damage sesmc-resstant elements whch provde a combnaton of re-centerng and dsspaton. Un-bonded post-tensonng renforcement (ether bars or strands) re-centers the system, whle several devces can be used to provde energy dsspaton. In walls systems, a frst layout opton s to concentrate the dsspatve renforcement at the base connecton of the wall. Generally ths confguraton has ether nternal or external (replaceable) tenson-compresson yeld mld steel dsspaters. Fgure 1. Qualtatve moment-rotaton behavour of post-tensoned rockng connectons. (a) re-centerng and dsspatve contrbutons to form a flag-shape and (b) nfluence of re-centerng rato. Paper Number O-31

2 The qualtatve behavour of a post-tensoned connecton can be represented by the combnaton of a mult-lnear elastc and a b-lnear hysteress as shown n Fgure 1a. The combnaton of the two hysteretc rules generates the flag-shape hysteress. The shape of ths relatonshp s governed by the re-centerng rato, β, defned as the rato of the post-tensonng and total moment contrbutons (β = M pt /M tot ) Fgure 1 qualtatvely shows the nfluence of ths parameter on the hysterc behavour of the connecton. For the unt value of β, the connecton behaves followng a mult-lnear elastc (posttensoned only) relatonshp, wth no hysteretc dsspaton; nstead, for β = the connecton s a mld steel only opton wth plenty of hysteretc dsspaton and sgnfcant resdual dsplacement. A mnmum value of.6 s suggested, ensurng acceptable levels of dsspaton and neglgble resdual dsplacements. The structural system was orgnally developed and tested for precast concrete applcaton durng the PREstress Sesmc Structural Systems (PRESSS) program n the 199s (Prestley 1991; Prestley et al. 1999) and more recently extended to structural steel (Chrstopoulos et al. 22) and engneered wood elements (Palermo et al. 25; Smth et al. 27; Newcombe et al. 28; Newcombe et al. 21b; Newcombe et al. 21a; STIC 213). The paper frst dscusses the performance-based desgn approach of dsspatve post-tensoned tmber walls wth detaled dscusson on materal lmt state to be adopted n the secton desgn also amed to prevent collapse under more severe events. The key modfcatons requred to Force-Based and Dsplacement-Based Desgn approaches are summarzed, and based on extensve numercal analyses n accordance to the FEMA P-695 procedure (ATC, 29) used n the last part of the paper whch provde a comparson and dscusson on the results of the two desgn approaches. 2 PERFORMANCE-BASED DESIGN PHILOSOPHY The general performance-based desgn phlosophy of post-tensoned tmber systems can be qualtatvely summarzed by Fgure 2. The general2herarchy of strength and sequence of events recommended for these systems (STIC, 213) would be: () Yeldng of the non-prestressed renforcement (or dsspaters); () Yeldng of the tmber at the rockng nterface ether n the column or the beam; and fnally () Yeldng of the post-tensonng renforcement. Such desgn phlosophy allows the system to () wthstand small and frequent events wth no sgnfcant damage; () provde the requred hysteretc dampng under a desgn level earthquake; () prevent structural collapse when subjected to a Maxmum Consdered Event (MCE). Fgure 2. Qualtatve push-over curve and performance lmts. 281

3 The proposed system desgn requrements nclude suggested values of re-centerng rato, stran lmts n the dsspaters at Desgn Base Event (DBE) and Maxmum Consdered Event (MCE) level and ntal tmber stress: Intal tmber stress should not exceed 1% of the tmber compressve strength. Ths stress lmt s set to allow the secton to develop the gap openng and neutral axs depth necessary to the effcent actvaton of the dsspaters. The maxmum stran n the tenson-compresson yeld dsspaters shall be n the range of 3-4% at DBE level and 6% at MCE level (Sart et al. 213). The stran range suggested allows the system to develop the necessary dampng whle provdng the dsspater wth some falure margn. Values of re-centerng rato (β) at desgn level should not be less than.6 (6% re-centerng contrbuton and 4% dsspatve contrbuton). 3 ANALYSIS METHODOLOGY The numercal analyses carred out to support the calbraton of the sesmc desgn provsons n accordance to ether a Force-Based was carred out followng the FEMA P-695 procedure. The procedure utlzes nonlnear analyss technques, and explctly consders uncertantes n ground moton, modellng, desgn, and test data. The techncal approach s a combnaton of tradtonal code concepts, advanced nonlnear dynamc analyses, and rsk-based assessment technques (ATC 29). The fnal am of the procedure s to determne sgnfcant FBD desgn parameters n accordance to ASCE/SEI 7-1 (ASCE/SEI, 21) such as the sesmc modfcaton factor (R), over-strength factor (Ω ) and deflecton amplfcaton factor (C d ). For brevty, the most sgnfcant aspects of the numercal methodology are dscussed n the paper and further nformaton can be found n ATC (29) and Sart et al. (214b). The numercal methodology wth mnor modfcatons was also used to determne sgnfcant Dsplacement-Based Desgn procedure parameters. 3.1 Case study buldngs The basc plan vew confguraton of the archetype buldngs s shown n Fgure 3. The buldng has an approxmate plan of 32m (4 bays) n the longtudnal drecton and 18m (3 bays) n the transverse drecton wth a floor area of approxmately 6m 2 per floor. The standard buldng confguraton has post-tensoned tmber walls provdng lateral load resstance n both the longtudnal and the transversal drecton (STIC 213). Fve three-bay frames spannng 6m n the transverse drecton support gravty loads. 282

4 Fgure 3.Case study buldngs parameters. Table 1. Case study buldngs summary. ID No. of Wall Floor Storey H storeys 1 H depth system mass (m) (m) (m) (t) TCC TCC TCC TCC TCC TCC TCC TCC CLT CLT CLT CLT CLT CLT CLT CLT 222 NOTE: TCC = Tmber-Concrete Composte floor system, CLT = sold cross-lamnated tmber floor system. H 1, H = frst and upper levels nter-storey heght. Several parameters affectng the structural behavour of the case study buldng were accounted for. Those ncluded the buldng and nter-storey heght, wall depth, gravty load magntude as summarzed n Fgure 3 and Table 1. For each case study buldng a dfferent sesmc resstng wall system was also consdered, namely sngle walls, Column-wall-column and coupled wall systems. 3.2 Model development The case study buldngs were modelled usng the model shown n Fgure 4. The model uses a multsprng element at the wall-foundaton nterface to smulate the gap openng. The modellng approach was frst ntroduced by Speth et al. (24) for modellng post-tensoned rockng concrete systems, but the approach can be extended to tmber systems as well (Newcombe, 212; Smth, 214). To ncorporate the P-Delta effects n OpenSEES (McKenna 211), an addtonal column (.e. P-delta column) was used. The P-Delta column was defned as an elastc beam-column element, wth hgh axal stffness and near-zero flexural stffness. The storey nodes were subjected to a constant vertcal force equal to the storey sesmc weght. Fgure 4. Mult-sprng model.(a) Sngle wall model and materal detals. (b) Column-wall-column model. (c) Coupled walls model. 283

5 Drft [%] Drft [%] Drft [%] (a) Top Dsplacement [mm] (b) Top Dsplacement [mm] (c) Top Dsplacement [mm] Base Shear [kn] Fgure 5. Numercal-expermental results comparson. (a) Sngle wall. (b) Pres-lamWEC system; (c) Couple walls. The numercal model were calbrated and compared to avalable expermental data. In partcular, expermental data by Sart et al. (214a) was used to calbrate the models for sngle walls and Pres- LamWEC system as shown n Fgure 5a,b. Quas-statc cyclc tests by Iqbal et al. (27) provded the calbraton data for coupled walls systems (Fgure 5c). As the results n Fgure 5 show, the numercal model s capable of closely match the observed expermental results. 4 CALIBRATION OF FORCE-BASED DESIGN PARAMETERS 4.1 Reducton Factors As part of the determnaton of the sesmc performance factors an extensve numercal study was carred out by the authors (Sart et al. 214b) n accordance to the FEMA P-695 methodology (ATC, 29). The objectve of such methodology was the determnaton of the sesmc performance factors n term of the sesmc modfcaton factor (R) and over-strength factors (Ω ). Whle more detals on the numercal work can be found n Sart et al. (214b), the paper focuses on a bref dscusson on the extenson of those numercal results to be appled to dfferent standards. Fgure 6. Defnton of the sesmc reducton factors n accordance to (a) ASCE 7-1 and (b) EC8 or NZS The man dfference n the sesmc desgn methodology s n the defnton of the desgn pont. As shown n Fgure 6a sesmc desgn n accordance to ASCE 7-1 (ASCE/SEI, 21) requres that the ductle sesmc resstng system s desgned elastcally for the reduced forces at yeldng pont. On the other hand, the desgn pont of the lateral resstng system n accordance to NZS117.5 (24) or EC8 (European Commttee for Standardzaton, 24) s desgned to fulfl some ductlty and the equvalent elastc-perfectly plastc approxmaton s secant to the ULS demand rather than the yeld pont. Therefore, the sesmc reducton factor, R = 7, determned n the numercal study by Sart et al. (214b) can be modfed to account for ths dfferent desgn approach gvng ether the suggested behavour factor (EC8), q, or the nelastc spectrum scalng factor (NZS 117.5), k µ, as defned by Equaton (1): 284

6 R qk, µ = Ω (1) Although a system over-strength factor of 3.5 may be adopted for the system as defned above and n accordance to the FEMA P-695 procedure, a qute sgnfcant varaton n the over-strength factor for the dfferent systems was observed; therefore, the average value across the dfferent systems was adopted n the defnton of a sesmc reducton factor n accordance to other standards. As shown n Sart et al. (214b), the system over-strength factors were respectvely 2.5, 3. and 3.5 for sngle wall, Pres-LamWEC and coupled walls systems and the average value across the dfferent systems was approxmately 3.. As a result, sesmc reducton factors of Equaton (2) can be adopted: R 7. qk, µ = = = Ω Estmaton of natural perod Another key pont n the sesmc force-based desgn of a mult-storey buldng s the estmaton of the natural (or desgn) perod of vbraton. Usually the coeffcents of emprcal formulas are calbrated for tradtonal steel or concrete structures and the estmaton of the ntal perod was done by consderng other structures type of coeffcents. From a comparson of the numercal results ths assumpton was deemed over-conservatve and dfferent coeffcents were used. In partcular, among the coeffcents reported n the standards, those showng a better ft to the numercal results were the ones related to moment-resstng concrete frames. Fgure 7a shows that the moment-resstng concrete frames emprcal formulas provde a better estmaton of the perod of the structure up to 1.5s, whle for hgher perods the values were stll underestmated. The Raylegh method provded a better and more relable estmaton of the natural perod, yet a modfcaton shall be appled. The elastc stffness of a sesmc resstng system shall be defned as the secant-to-yeld stffness (representng the equvalent elastc-plastc system) rather than the ntal. Adoptng the ntal flexural stffness would agan result n an under-estmaton of the natural perod of vbraton of the structure. Unlke concrete structures, the yeld pont of a post-tensoned rockng system s not of easy evaluaton from a prelmnary desgn pont of vew; therefore, t s suggested to evaluate the perod consderng the stffness at decompresson pont and apply a magnfcaton factor of 1.4, proposed n ths secton based on numercal results. 1 n = 1 π n g = 1 T = Wd 2 Fd Where d s the horzontal dsplacement of the center of mass at level, F the dsplacng force actng at level, the level under consderaton of structure, n number of levels n a structure, W the sesmc weght at level The chart n Fgure 7b reports the comparson of the numercal values aganst the value evaluated from Equaton (3). (2) (3) 285

7 Estmated perod (a) Numercally determned perod (b) Numercally determned perod Fgure 7. Natural perod of vbraton estmaton methods. (a) Emprcal method (24); (b) modfed Raylegh method. 5 CALIBRATION OF DISPLACEMENT-BASED DESIGN PARAMETERS The calbraton of the Dsplacement-Based Desgn (Prestley et al., 27) parameters focussed on the determnaton of modfcaton factors for the area-based hysteretc vscous dampng and the nelastc dsplacement profle. The former, was determned through quas-statc numercal analyss based on expermental data on sngle wall systems by Sart et al. (214a) and dscussed n Secton 5.1 below. The model shown n Secton 3.2 was used. The calbraton of the desgn nelastc dsplacement profle was carred out through non-lnear tmehstory analyss adoptng the methodology descrbed above. 5.1 Area-Based hysteretc vscous dampng As hghlghted n sngle dsspatve post-tensoned tmber wall expermental tests (Sart et al. 214a), the area-based hysteretc dampng was strongly nfluenced by the ratchetng behavour of the dsspater as well as the dsspater connecton flexblty (see Fgure 1b). Those effects were mplemented n the mult-sprng numercal model and quas-statc analyses were carred out wth the varaton of several desgn parameters. The analyss consdered a set of case study walls wth dfferent cross-sectonal dmensons (.24m 2.4m and.36m 3.6m) and three elements heghts (3, 6 and 8 storeys, 3m nter-storey heght) were accounted for. To assess the nfluence of dfferent re-centerng ratos, each wall was desgned usng values of.6,.7 and.8. Dfferent dsspater connecton stffness values were consdered as relatve connecton-dsspater stffness factor. The modfcaton factor, k ξ, was evaluated wth Equaton (1): k ξ ( 2 2β)( m 1) r ( ) ξnumercal 1 Ah = ; ξnumercal = ; ξanalytcal = ξanalytcal 2π Fm m mπ 1+ m 1 Where ξ numercal s the numercal area-based hysteretc dampng, A h the area wthn one complete cycle of stablzed force-dsplacement response, F m, Δ m the maxmum force and dsplacement acheved n the stablzed loops, ξ numercal the dealzed flag-shaped hysteretc dampng analytcal formula (Prestley et al. 27), μ the ductlty, β the re-centerng rato and r the post-yeld stffness factor. Fgure 1a shows an example comparson between the analytcal results and the numercal evaluated data. (1) 286

8 Area-Base Vsocus Dampng [-] Analytcal Force [kn] (a) Ductlty [-] (b) Numercal A analytcal A numercal Drft [-] Fgure 1. (a) Comparson of hysteretc dampng evaluated from analytcal and numercal data of a typcal wall base connecton; (b) Evaluaton of hysteretc areas. Fgure 2 shows a summary of the numercal results whch were ftted usng lnear functons reported n Equaton (2). k.6 w / sot. stran hard. =.83 + ( 1) (2).2 w / o sot. stran hard. ξ µ Modfcaton Factor, k ξ k ξ =.6(µ - 1) +.83 k ξ =.2(µ - 1) (a) Ductlty (b) Ductlty Fgure 2. EVD modfcaton factors (a) wth and (b) wthout sotropc stran hardenng effects. Fgure 2 and Equaton (2) consder the sotropc stran hardenng effect of the steel materal. Ths dstncton was necessary n order to properly account for the dssmlar behavour of dfferent devces. In fact, dsspater such as tenson-compresson yeld devce dsplay sgnfcant sotropc stran hardenng n compresson (Sart et al., 213), whle U-shaped Flexural Plates (UFPs) usually do not show any compressve yeld shft (Sknner et al., 1974). 5.2 Dsplacement profles and nter-storey drft The case study buldngs were desgned n accordance to a DBD approach (Prestley et al., 27) usng the above modfcaton factors for the hysteretc vscous dampng evaluaton and the followng desgn parameters: (a) Desgn drft θ d = 1.2%; (b) ductlty factor µ = 2.5; (c) post-yeldng stffness factor r =.2. It s to be clarfed that a lnear dsplacement dstrbuton was assumed n the desgn phase and was later compared to observed results. The black plots n Fgure 3 show the averaged peak dsplacement and nter-storey drft profles resultng from extensve NLTHA. The results hghlght that a lnear profle was not approprate for buldngs taller than three storeys. 287

9 8 Lnear Eq. (6) NLTHA 5 years 5 years 1 years 25 years 5 years 5 years 1 years 25 years 5 years 5 years 1 years 25 years 6 Floor (a) Normalzed Dsplacement (b) Inter-storey Drft Fgure 3. Case study buldng (a) average peak dsplacement and (b) nter-storey drft profles comparson to Eq. (3). The proposed nelastc dsplacement profle s gven by Equaton (3) and compared to dfferent sesmc ntenstes n Fgure 3 provng more effectve n fttng the numercal results. 2 h h h h sngle walls d = hn 2 hn 3hn hn m.3 coupled systems Where h and h n are the heght of the -th storey and the top storey respectvely. (3) 6 SEISMIC DESIGN COMPARISON A few of the case study buldngs was desgned n accordance to NZS (24) and adoptng ether a FBD or DBD approach provdng a comparson of the two methods. The case study buldngs plan confguraton shown n Fgure 3 was used (smlarly to the numercal analyses above) and a more comprehensve varaton of the buldng heght was accounted for (1, 2,, 8 storeys, nter-storey heght 3.6m). The buldngs were desgned n the Chrstchurch CBD (Z =.3), for a sol class D and a return perod factor R = 1.. The followng desgn parameters were used: () ductlty factor µ = 2.5; () storey mass W = 3t; () re-centerng rato β =.7. The desgn base shear was consdered as a comparson parameter for the analyss results and the resultng values are summarzed n Table 2. Dsplacement-Based Desgn resulted n smaller base desgn shear values than Force-Based Desgn. In partcular DBD values ranged from 85% to 64% of the equvalent FBD base shear values. Fgure 4 shows the plot of the dfferent desgn values and the average shear reducton of 7% was observed (hghlghted n the red dashed lne). The hgher values resultng from FBD can be justfed by the over-estmaton of the natural perod of the structure, whch was chosen n accordance to emprcal equatons of NZS117.5 (24). 288

10 Table 2. Sesmc desgn results and comparson. ID N story Inter- Storey Heght Total heght Base Shear V FBD V DBD V DBD V FBD (-) (m) (m) (kn) (kn) (-) DBD Shear [kn] FBD Shear [kn] Fgure 4. FBD vs DBD base shear comparson. 7 CONCLUSIONS The paper presented suggested amendments to Force-Based and Dsplacement-Based Desgn key desgn parameters focusng n partcular on the sesmc reducton factors for both desgn approaches. The perod estmaton for FBD and the nelastc dsplacement profle for DBD were also dscussed and amendments proposed. Based on extensve non-lnear numercal analyses carred out by the authors n accordance to the FEMA P-695 procedure (Sart et al. 214b), the numercal results were heren extended to the nelastc spectrum scalng factor (k μ ) whch resulted to be 2.5. The perod estmaton s a crucal step n the Force-Based Desgn of any structural system. Although classfed as other structures, emprcal equaton relatve to concrete frames n accordance to NZS117.5 better ftted the numercal data. The paper also provded a modfcaton factor on the Raylegh dampng equaton to account for the partcular decompresson behavour of post-tensoned rockng walls. As an alternatve desgn approach, DBD conssts n convertng the structure nto a sngle-degree-offreedom system characterzed by secant propertes. In the evaluaton of the sesmc demand two aspects were dscussed n the paper: the hysteretc vscous dampng and nelastc dsplacement profle. Based on statc numercal analyses the authors proposed a modfcaton factor for the current analytcal formula proposed by Prestley et al. (27) and account for partcular behavoural aspects of the system. The lnear dsplacement profle prelmnary assumed n the desgn phase dd not reflect the observed average peak dsplacement profle. The paper showed a non-lnear dsplacement profle whch better ftter the numercal results. Fnally, the paper brefly dscussed the outcomes of the Force-Based and Dsplacement-Based Desgn of several case study buldng. The results showed that DBD shear values are generally lower than FBD values. The DBD shear was n the range of 64%-85% when compared to FBD values (7% on average). 8 REFERENCES Amercan Socety of Cvl Engneers and S. E. Insttute. 21. Mnmum desgn loads for buldngs and other structures. Amercan Socety of Cvl Engneers and S. E. Insttute. Reston, (VA), Amercan Socety of Cvl Engneers, Structural Engneerng Insttute. Appled Technology Councl for the Federal Emergency Management Agency (ATC). 29. Quantfcaton of buldng sesmc performance factors. Appled Technology Councl for the Federal Emergency Management Agency (ATC). Washngton, D.C.], Washngton, D.C. : U.S. Dept. of Homeland Securty, FEMA. 289

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