PERFORMANCE-BASED SEISMIC DESIGN OF SUPPLEMENTAL DAMPERS IN INELASTIC SYSTEM

Transcription

1 4th Internatonal Conference on Earthquake Engneerng Tape, Tawan October -3, 6 Paper No. 8 PERFORMANCE-BASE SEISMIC ESIGN OF SUPPLEMENTAL AMPERS IN INELASTIC SYSTEM Bo L and Xng-wen Lang ABSTRACT A smplfed yet effectve desgn procedure for vscous dampers was presented based on mproved capacty spectrum method n the context of performance-based sesmc desgn. The amount of added vscous rured to meet a gven performance objectve was evaluated from the dfference between the total demand for effectve and nherent plus uvalent resultng from hysteretc deformaton of system. Applcaton of the method s llustrated by means of two examples, usng Chnese desgn response spectrum and mean response spectrum. Nonlnear dynamc analyss results ndcate that the maxmum dsplacements of structures nstalled wth supplemental dampers desgned n accordance wth the proposed method agree well wth the gven target dsplacements. The advantage of the presented procedure over the conventonal teratve desgn method s also hghlghted. Keywords: Vscous dampers; Equvalent ; Improved capacty spectrum method; Nonlnear dynamc analyss; Performance-based sesmc desgn. INTROUCTION Many modern buldngs, after attacked by a severe earthquake, faled to functon and rured costly structural and nonstructural repars, although they successfully protected the lves of the occupants. Thus questons could be rased regardng the tradtonal sesmc desgn practce of permttng nelastc deformaton of structural components to dsspate the nput sesmc octoral canddate, epartment of Cvl Engneerng, X an Unversty of Archtecture and Technology, Yanta Road 3 #, 755, X an, Chna. E-mal : lblbolb@sna.com Professor, epartment of Cvl Engneerng, X an Unversty of Archtecture and Technology, Yanta Road 3 #, 755, X an, Chna (correspondng author. E-mal: langxngwen@yahoo.com.cn. Fax: ).

2 energy. Energy dsspaton systems have been developed wth a desgn phlosophy dfferent than that of the conventonal sesmc desgn method. A structure nstalled wth dampers does not manly rely on plastc hnges to consume the sesmc energy. On the contrary, the dsspaton of nput energy s concentrated on some added dampers so that the damage to the prmary structure s reduced and the functons of the structure can then be possbly preserved. The structural passve control technques use dampers, rather than utlzng the hysteretc deformaton of structural members, to dsspate nput energy, and thus provde an alternatve method for sesmc retroft. The ATC-4 (ATC 996) document and the FEMA-73 (FEMA 997a) gudelne recommend techncal strateges such as ncreasng strength, alterng stffness, and reducng sesmc demand by employng base solaton or energy dsspaton devces to mprove sesmc performance of a structure. Utlzng energy dsspaton devces to retroft a structure has the advantage of reducng the sesmc demand of strength and ductlty on the system compared wth conventonal strengthenng schemes. The general procedure for sesmc retroft of a structure wth supplemental dampers s summarzed as follows: () evaluate the sesmc behavor of the structure for gven levels of earthquake ground motons; () assume an value of added rured to reduce the structural response to satsfy the prescrbed performance levels; (3) select the proper damper sze and the approprate confguraton scheme to realze the desred rato; (4) analyze the structure wth supplemental dampers the check the aduacy of the added. Task n step and 4 can be performed by nonlnear dynamc analyss, or by more smplfed approxmate method of analyss. For example, FEMA-73 suggests nonlnear statc procedure based on capacty spectrum method (CSM) be used to evaluate the nelastc response of damper-added structure to the earthquake ground moton. The successful accomplshment of the second and the thrd steps, whch plays a key role n the retrofttng process, largely depends on emprcal estmaton or try and error process performed by desgn professonals. After the tral values for dampers are obtaned, step 3 to step 4 are repeated untl the amount and confguraton of added dampers meet the gven performance crteron. However ths conventonal practce rures conductng a seres of try and error analyss, whch s onerous and tme-consumng. Therefore a more convenent yet effectve method s needed to reduce the amount of work done n step of and 3, partcularly n the prelmnary desgn stage. Although extensve research on performance-based sesmc desgn for typcal structures has been conducted the examnaton focused on that for energy dsspaton system s rarely carred out. In ths paper a smplfed procedure of desgn for vscous dampers was developed based on mproved CSM wthn the context of performance based sesmc desgn. The smple method, presented n the paper, s brefly summarzed as follows: frst predetermne a performance objectve for system and second evaluate the sesmc performance of the structure to be retroftted; f the exhbted behavor of the structure can not satsfy the acceptable performance crteron the total rured to meet the predetermned performance level can be evaluated usng sngle step method n combnaton wth the desgn response spectrum provded n Chnese code for sesmc desgn of buldng. The proposed method s smple, straghtforward and non-teratve compared wth the tradtonal one.

3 ESIGN PROCEURE OF VISCOUS AMPERS FOR INELASTIC SYSTEM Modelng of vscous dampers The vscous dampers have the advantage n provdng wthout changng the dynamc characterstcs of structures. As stated n FEMA-74 (FEMA 997b), the force provded by vscous damper can be modeled to be proportonal to relatve velocty between the two ends of the damper wth the constant exponent rangng from.5 to.. Accordng to the study n ths paper, n prelmnary desgn stage the damper force s reasonably consdered to be proportonal to the velocty wth a constant exponent of. (lnear vscous damper) for convenence. Effectve of yeldng system wth supplemental dampers F F f u f y k ke ke αke f u f y c ω um ke k um k uy um u uy um u () Hysteretc energy () Equvalent vscous Fg.. Equvalent for Kowalsky that s based on Tekeda degradng stffness model Ln et al. (4) carred out pushover tests on RC columns and employed varous uvalent lnearzaton methods to evaluate the nelastc deformaton demand. Comparng the earthquake-nduced demands estmated by CSM usng varous uvalent lnearzaton methods, and those obtaned from pushover tests, they found that the method suggested by the ATC-4 document overestmates the energy dsspaton capacty of tested columns, and consuently underestmates ther nelastc dsplacement demands. When the Kowalsky hysteretc model was used and stffness degradaton slghtly consdered n the uvalent lnearzaton method, the values obtaned from CSM were found to be more consstent wth to the expermental results. So the Kowalsky hysteretc model was employed n ths paper to estmate the uvalent vscous for the structure. Kowalsky et al. (994) take the Taketa model for the hysteretc model and an unloadng n stffness factor of n, rangng from to.5. The unloadng stffness of k = k / μ and dsplacement ductlty rato of μ = um / uy are shown n Fg., where u y, u m, f y and f u are yeld dsplacement, maxmum dsplacement, yeld strength and ultmate strength for e

4 blnear system, respectvely; k and k are elastc stffness and uvalent stffness; α s e the postyeld stffness rato. The hysteretc energy, E H, dsspated n one cycle wth a deformaton ampltude ual to u m n both drectons s gven by (the area enclosed by the hysteretc loop shown n Fg. (a)): E H The stran energy stored n the system s: n α = μ ( α + ) f u μ m u () E S = fu m u () The uvalent rato s then gven by (Chopra ): EH n α ζ = = μ ( α ) 4π ES π + μ If energy dsspaton devces are added, the effectve becomes (FEMA 997b): where ζ eff ( 3 ) = E EH ζ + 4π E + 4π E ( 4 ) S E s the energy consumed by vscous dampers n one cycle of harmonc moton. The frst term n the rght hand sde of Eq. (4) represents the nherent rato of the system, the second one descrbes the vscous rato provded by supplemental vscous dampers, and the thrd one gves the uvalent rato resultng from the hysteretc deformaton of the system. As shown n Fg. (b), the energy dsspated by vscous dampers n one cycle of steady-state response of nelastc system to harmonc force s expressed as follows: T E f du c u udt c u k u π / ω = = ( ) = π ω m = πζv Te m ( 5 ) where c s the uvalent coeffcent of a lnear vscous damper ( c = mωζ v ); u s assumed to be u sn( ωt φ) m S, n whch ω and φ are crcular fruency of external exctaton and phase angle of system, respectvely; T and T are elastc perod and uvalent perod for system, correspondngly; ζ v s the vscous rato of the system wth added dampers vbratng n terms of the perod for the uvalent elastc system. In ths study the exctaton fruency s assumed to be ual to the ntal elastc natural fruency of e

5 system,.e. ω = ω. Substtutng Eq. (5) nto Eq. (4), the overall effectve ncludng e the contrbuton from the dampers can then be obtaned (Tsopelas et al. 997): T ζ = ζ + ζ + ζ ( 6 ) eff v Te where the relaton of fu = kum s utlzed n the process of dervaton. etermnaton of supplemental rured to satsfy performance objectve When estmatng the response of the structure, the total effectve obtaned from Eq. (6) can be used to reduce the sesmc demand on the structure. If the desred target dsplacement s predetermned n sesmc retroft, the amount of supplemental rured for structure to meet a gven performance objectve can then be obtaned by transformng Eq. (6) to the followng expresson: Te ζv = ( ζeff ζ ζ) ( 7 ) T where the effectve can be determned from the demand spectrum that crosses the capacty dagram at the gven target dsplacement. The procedure for evaluaton of the rured added s detaled as follows:. Set a performance objectve for the system wth supplemental dampers,.e., specfy the maxmum response of the damper-added structure to gven level of earthquake ground moton.. Perform pushover analyss of the structure to be retroftted, transform the pushover curve to blnear capacty dagram n the format of spectral acceleraton of S a and spectral dsplacement of S d (A- format), and determne the performance pont of the structure usng desgn response spectrum or response spectrum for earthquake ground motons. If the dsplacement at performance pont s less than the target one, t s mpled that the effect of hysteretc plus nherent aduately lmts the dsplacement demand of system to the gven target dsplacement, and n ths case no supplemental s needed. Otherwse, go to the thrd step. 3. Obtan the target ductlty by dvdng the target dsplacement by the yeld one, and compute the uvalent of system at the target ductlty n conjuncton wth postyeld stffness rato of α and unloadng stffness factor of n. 4. Evaluate n A- format the rato of demand spectrum that ntersects wth the capacty dagram at the target dsplacement, accordng to the reducton rules of demand spectrum. Ths corresponds to the total effectve rured for the structure to meet the gven performance objectve. 5. Compute the supplemental vscous needed to lmt the dsplacement demand to the target dsplacement usng Eq. (7).

6 The key pont n the above procedure s the estmaton of the effectve (demand from earthquake) and the uvalent (capacty of the structure). The dfference between them s provded by the supplemental dampers. It wll be shown later that the proposed procedure can smplfy the desgn process and reduces the amount of the work substantally snce the rured added s obtaned by a sngle computaton wthout teraton. etermnaton of performance pont usng mproved CSM The mproved CSM use the locus of performance ponts to determne the performance pont. Correspondng to each pont, heren termed ductlty pont, beyond the yeld one on the capacty dagram, the uvalent perod of system can be obtaned for the gven hysteretc model based on the prncple of uvalent lnearzaton. The ntersecton of the uvalent perod radal lne from the orgn and the demand dagram determned by uvalent plus nherent rato s here referred to as the demand pont correspondng to the gven ductlty pont on the capacty dagram. Increasng the value of the ductlty, a seres of demand ponts can then be ganed. The curve generated by connectng together these demand ponts s termed the locus of performance ponts (Guyader 4). The performance pont s located at the ntersecton of the locus of performance ponts and the capacty dagram as shown n Fg.. PSA Locus of p. p. capacty spectrum PSA Locus of p. p. capacty spectrum splacement range S (a) Large slope of locus of performance ponts splacement range S (b) Small slope of locus of performance ponts PSA PSA Locus of p. p. Performance pont Locus of p. p. Performance pont capacty spectrum capacty spectrum (c) Sngle performace ponts S S (d) Multple performance ponts Fg.. Possble cases for locus of performance ponts

7 The mproved CSM s a graphcal soluton procedure and can be used to perform the senstvty analyss of the dsplacement predcton. The procedure clearly reveals how varaton n both the capacty or demand wll affect the locaton of the performance pont. If the strength of capacty dagram were ncreased or decreased, the locaton of the performance pont changes, but by how much? The answer depends on the slope of the locus of performance ponts near the performance pont. Some possble cases of locus of performance ponts are gven n Fg.. Fg. (a) shows a case where locus s nearly 9 degrees. In ths case, rasng or lowerng the capacty dagram has very lttle effect on the performance pont dsplacement. Ths dsplacement range s very small. However, Fg. (b) gves a case where the slope of locus s relatvely small. So rasng or lowerng the capacty dagram has sgnfcant effect on the performance pont dsplacement and the correspondng dsplacement range s large. There may be one, several or zero ntersectons of the locus of performance ponts and capacty dagram. Extendng locus beyond the frst ntersecton wll reveal f multple ntersecton ponts exst as seen n Fg. (d). Serous attenton should be pad to multple performance ponts case. A conservatve approach s to use the performance pont at the largest dsplacement. The conventonal teratve method for obtanng the performance pont s laborous and may leads to the dvergence of soluton n some cases or the erroneous soluton, although converged (Chopra ). When usng locus of performance ponts to evaluate performance ponts, the above case can be precluded, and addtonally possble multple performance ponts can be dentfed. ANALYSIS OF INELASTIC SOF SYSTEM WITH SUPPLEMENTAL AMPERS Ths secton wll llustrate how to use the smplfed method, n conjuncton wth Chnese desgn response spectrum (Mnstry of Constructon ) and the response spectrum of recorded ground motons, to evaluate the amount of supplemental rured for the structure to meet a gven performance objectve. Evaluaton of rured supplemental usng desgn response spectrum etaled steps for determnaton of rured added usng Chnese desgn response spectrum are gven as follows: () Set the performance objectve for system wth supplemental dampers. () Plot the elastc demand dagram wth 5% rato n A- format for the gven spectral acceleraton, α max, whch s ual to S / g, and the characterstc perod of the ste consdered. The yeld dsplacement can be obtaned by dvdng the elastc demand dsplacement by strength reducton factor. Plot the capacty dagram n the same A- format. (3) Compute the uvalent rato at the specfed ductlty ponts on the capacty dagram. The ntersecton of uvalent elastc demand curve determned by the effectve wth the uvalent perod radal lne from the orgn s here defned as the uvalent elastc demand pont correspondng to the uvalent perod. The locus of performance ponts s then developed by connectng together these demand a

8 ponts. (4) If the uvalent elastc demand pont on the uvalent perod lne s above (below) ts capacty pont, and the demand pont on the next uvalent perod lne s below (above) ts capacty pont, then the performance pont s defntely located between the two adjacent capacty ponts,.e., check the product of the dfference between the spectral demand acceleraton and spectral capacty acceleraton n the ( +) th step. If the product s less than zero the performance pont exsts between the two adjacent ponts. Otherwse the performance pont does not exst between the two ponts. (5) If the demand dsplacement at the performance pont s greater than the target one, go to the next step. Otherwse there s no need for sesmc retroft. (7) Accordng to the spectral reducton factor of perod of T, the characterstc perod, th η, the uvalent T g, of the ste consdered and spectral acceleraton of α max n the horzontal drecton, the rured effectve rato can be computed where the demand dagram ntersects wth the capacty dagram at the target dsplacement. (8) Evaluate ζ from the Eq. (7). v The desgn response spectrum wth the parameters of α max =.9 and T g =.35 sec, and the system wth Te =.5 sec, R =3, α =.5, and n = were selected as an example, where R uals to the rato of elastc demand strength to yeld strength. Table and Table descrbe the detaled process for dentfyng the performance pont and evaluatng the rured supplemental, respectvely. The graphcal demonstratons are seen n Fg. 3 and Fg. 4. Table Evaluaton of performance pont for system wthout added usng locus of performance ponts Capacty spectrum Elastc demand spectrum Check for Step Capacty Effectve emand exstence of Structural capacty pont Response demand pont No. ductlty ductlty soluton μ ( ζ + ζ ) c (%) C ( mm ) A C ( m /sec ) ( mm ) A ( m /sec ) μ P+ ( ) d ( =, ) Performance pont of system wthout added 3, (, A ) (, A ) P+ ( ) Note: and are the coordnates of capacty pont and demand pont n A- format, respect vely ; s C C ual to ( A ( + ) A ( + ))( A ( ) A ( )) C C

9 Target dsplacement TAR ( mm ) Table Estmaton of supplemental rured to meet target dsplacement Target ductlty TAR Equvalent Inherent Effectve Supplemental μ ζ (%) ζ %) ζ (%) ζ (%) ( eff v The target dsplacement was set to be 8% of the demand dsplacement at the performance pont. The mass was kept constant of a unt value and the nherent rato was taken as 5%. Improved Capacty Spectrum Method Improved Capacty Spectrum Method 6 ARS(μ ) ζ =5% 6 T e =.5sec 5 T sec (μ ) Locus of Performance Ponts 5 7.% ζ =5% ARS(μ ) ζ eff =5.7% Locus of Performance Ponts PSa (m/sec ) 4 3 ζ=7.% T sec (μ ) Performance Pont (.53m,.4m/sec ) PSa (m/sec ) 4 3 Target Pont for System wth Added Vscous ampng Performance Pont for System wthout Supplemental Vscous ampng Reducton ue to ampng (.53m,.4m/sec ) T sec (μ 6 ) ARS(μ 6 ) (.4m,.34m/sec ) Sd (m) Sd (m) Fg. 3. Evaluaton of performance pont for system wthout added usng locus of performance ponts Fg. 4. Estmaton of added for system based on mproved capacty spectrum method Evaluaton of rured supplemental employng response spectra for recorded ground motons To obtan more generalzed results the mean response spectra constructed from horzontal earthquake records were used n the evaluaton of rured supplemental. The target dsplacement of a structure added wth dampers was assumed to be 6% of the demand dsplacement of the correspondng structure wthout dampers. The supplemental needed to lmt the structural response wthn the target dsplacement was estmated usng Eq. (7). Fnally the maxmum response of the structure nstalled wth the supplemental dampers was obtaned by nonlnear dynamc analyss and the result was compared wth the target value to verfy the accuracy of the proposed method.

10 Model consdered The system wth the followng desgn parameters was analyzed: () elastc perod: () strength reducton factor: stffness rato: Input ground motons T e =.4 sec; R =4; (3) postyeld stffness rato: α =.5; (4) unloadng n =; (5) mass: m= and nherent rato: ζ =5%. The famous earthquake ground motons recorded on the ste of class Ⅱ and Ⅲ class were taken for dynamc analyss. The selected ground motons conssted of 5 earthquake events wth each one ncludng the north-south component and the west-east component. These earthquake records are lsted as follows: El centro (94), Tang shan-be jng Hotel (976), Ch-Ch (999), Whttrer (987) and Northrdge (994). The records were scaled n such a way that the fruency content of each record was preserved and an ual contrbuton of these records to the mean spectrum was ensured (FEMA 997b; Tsopelas et al.). The peak acceleraton of these records were scaled up to 4 gal prescrbed by the code for nonlnear dynamc response analyss (Mnstry of Constructon ), n the case of the rare earthquake event correspondng to the sesmc fortfcaton of ntensty of 8 degrees. Fg. 5(a) shows the % damped response spectrum for scaled motons. Fg. 5(b) presents a famly of mean response spectrum for varous ratos, whch were later converted to the A- format together wth capacty dagram to obtan the performance pont. 4 Mean Response Spectra for % ampng Average of Scaled Motons Mean Spectra for Varous ampng Pesudo-Spectra Acceleraton (m/s) Pesudo-Spectra Acceleraton (m/s ) ζ=5% ζ=% ζ=% ζ=3% ζ=4% Perod (sec) Perod (sec) (a) Response spectra for % (b) Mean spectra for varous Fg. 5. Response spectra for the earthquake records Analyss of model The procedure for obtanng the rured rato was the same as that presented above, except that the desgn spectrum was replaced by the mean demand spectrum of recorded ground motons. For verfcaton of the results obtaned from the smplfed method nonlnear dynamc analyss were performed on the system wth the supplemental plus the nherent. The process of dentfyng the performance pont and computng

11 the rured are presented n Table 3 and Table 4, respectvely. Graphcal llustratons are gven n Fg. 6 and Fg. 7. As seen from the last column n the Table 4, the result derved from the proposed method matches statstcally well wth that obtaned from nonlnear dynamc analyss Table 3 Evaluaton of performance pont for system wthout added usng locus of performance ponts Capacty spectrum Mean demand spectra for records Check for Step Capacty Effectve emand exstence of Structural capacty ductlty Response demand pont No. ductlty ductlty soluton μ ( ζ ) c + ζ (%) C ( mm ) A C ( m /sec ) ( mm ) A ( m /sec ) μ P+ ( ) d ( =, ) Performance pont of system wthout added 3, (, A ) (, A ) P+ ( ) Note: and are the coordnates of capacty pont and demand pont n A- format, respectvely ; s C C ual to ( A ( + ) A ( + ))( A ( ) A ( )) C C Table 4 Estmaton of supplemental rured to meet target dsplacement Target dsplacement Target ductlty Equvalent Inherent Effectve Supplemental Total uvalent vscous Mean value for THA Comparson TAR ( mm ) μ ζ (%) ζ (%) ζ (%) ζ (%) ( ζ + ζ )(%) THA ( mm ) / THA TAR eff v v TAR Improved Capacty Spectrum Method Improved Capacty Spectrum Method 9 8 ARS(μ ) T e =.4ses T sec (μ ) Locus of Performance Ponts ζ =5% T e =.4sec PS a (m/sec ) ARS(μ ) ARS(μ 3 ) T sec (μ ) T sec (μ 3 ) Performance Pont (.468m, 3.m/sec ) T sec (μ 6 ) ARS(μ 6 ) PS a (m/s) Target Pont for System wth Added Vscous ampng 8.3% (.8m,.89m/sec ) ζ eff =44.% Locus of Performance Ponts Performance Pont for System wthout Supplemental Vscous ampng (.468m, 3.m/sec ) S d (m) S d (m)

12 Fg. 6. Evaluaton of performance pont for system wthout added usng locus of performance ponts wth mean response spectra Fg. 7. Estmaton of added for system based on mproved capacty spectrum method wth mean response spectra CONCLUSIONS In ths paper a smplfed procedure was presented usng mproved CSM to determne the amount of supplemental rured to meet a gven performance objectve. The uvalent rato was computed employng Kowalsky hysteretc model, performance pont was determned usng the locus of performance ponts, and the supplemental was evaluated by the smplfed method. The proposed procedure have two advantages over the tradtonal one: () the graphcal performance pont soluton procedure s a smple, straghtforward, and non-teratve, and addtonally can dentfy the possble multple performance ponts; () the supplemental rured to satsfy a gven performance objectve s evaluated through a sngle step wthout teraton. The proposed procedure has been appled to SOF system and can also be extended to MOF system for determnaton of the total amount of supplemental rured to meet a gven performance objectve. REFERENCES Appled Technology Councl ATC. (996). Sesmc Evaluaton and Retroft of Concrete Buldngs. ATC-4, Redwood Cty, Calforna. Chopra, A. K., and Goel, R. K. (). Evaluaton of NSP to estmate sesmc deformaton: SF system. Journal of Structural Engneerng, 6(4), Chopra, A. K. (). ynamcs of Structures: Theory and Applcatons to Earthquake Engneerng, Prentce Hall: New Jersey. Federal Emergency Management Agency (FEMA) (997a). NEHRP Gudelnes for the Sesmc Rehabltaton of Buldngs. FEMA-73, Washngton,.C. Federal Emergency Management Agency (FEMA) (997b). NEHRP Commentary on the Gudelnes for the sesmc rehabltaton of buldngs. FEMA-74, Washngton,.C. Guyader, A. C. (4). A statstcal approach to uvalent lnearzaton wth applcaton to performance-based engneerng. Techncal report. EERL 4-4. Calforna Insttute of Technology. Kowalsky, M. J., Prestley, M. J. N., MacRae, G.A. (994). splacement-based desgn, a methodology for sesmc desgn appled to SOF renforced concrete structures. Techncal Report SSRP-94/6, Structural System Research Project, Unversty of Calforna, San ego, La Jolla, CA. Ln, Y. Y., Chang, K. C., and Wang, Y. L. (4). Comparson of dsplacement coeffcent method and capacty spectrum method wth expermental results of RC columns. Earthquake Engneerng and Structural ynamcs, 33,

13 Mnstry of Constructon (). Code for sesmc desgn of buldngs, GB5-. Chna Archtecture & Buldng Press. (n Chnese) Tsopelas, P., Constantnou, M. C., Krcher, C. A., and Whttaker, A. S. (997). Evaluaton of smplfed method of analyss for yeldng structures. Techncal Report NCEER-97-, Natonal Center for Earthquake Engneerng Research, State Unversty of New York at Buffalo.