SEISMIC ANALYSIS AND DESIGN OF BUILDING STRUCTURES WITH SUPPLEMENTAL LEAD DAMPERS

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1 SEISMIC ANALYSIS AND DESIGN OF BUILDING STUCTUES WITH SUPPLEMENTAL LEAD DAMPES X LIN 1, Peter J MOSS And Athol J CA 3 SUMMAY The response behavour of buldng structures ncorporatng a new type of dampng devce (lead shear damper developed by Pengun Engneerng Ltd) was under nvestgaton. For regular and symmetrcal frame structures, a satsfactory dstrbuton of these supplemental dampers n the stores has been determned. For such a dstrbuton of the dampers, the structure wth supplemental dampers wll behave predomnantly n ts frst mode. Ths leads to a smplfed method usng an equvalent SDOF system that s able to predct the response of the MDOF structure for prelmnary desgn. Optmal dampng levels due to supplemental dampers have been found. A dsplacement-based method to determne the strength levels of the dampers n the storeys sutable for prelmnary desgn s outlned. INTODUCTION A basc prncple n structural desgn when seekng to mnmse the effects of severe earthquake exctatons s to allow the structure to absorb and dsspate energy through structural ductlty. However, ductle structures may undergo very large nelastc deformaton so that they may be severely damaged after strong earthquake exctatons. ecently, more emphass has been gven to the development of cost-effectve devces for dsspatng sesmcally nduced energy n the structure whle keepng the structure s response as much as possble n the elastc range. These energy-dsspatng devces provde large supplemental dampng to the structure and sgnfcantly reduce the sesmc demand of the structure. The lead damper, Pengun Vbraton Damper (PVD), developed by Pengun Engneerng, s a compact dampng devce. The dampng of ths devce s acheved through deformaton of a lead core [Mont et al, 1996]. ANALYTICAL MODEL OF THE DAMPE AND THE STUCTUE All the results obtaned through the testng programme of ths devce have shown t to behave as an almost perfectly plastc devce [Mont et al, 1996]. A b-lnear model has been used to represent the force-deformaton relatonshp of the dampers [Ln, 1999]. A 1-storey 3-bay renforced concrete frame structure was used for ths study. The supplemental dampers were connected to the structure by means of dagonal braces (Fgure 1). DISTIBUTION OF THE YIELD STENGTHS OF THE DAMPES IN THE STUCTUE The purpose of the research outlned here was to fnd out a satsfactory dstrbuton of the dampers rather than the optmal dstrbuton. Two parameters were used to measure the structural demand and response. They are peak nterstorey drft and peak base shear. A code compatble earthquake El Centro 1940 NZS403 was adopted here for tme hstory analyses [Ln, 1999]. 1 3 Department of Cvl Engneerng, Unversty of Canterbury, Chrstchurch Department of Cvl Engneerng, Unversty of Canterbury, Chrstchurch. Emal: p.moss@cad.canterbury.ac.nz. Department of Cvl Engneerng, Unversty of Canterbury, Chrstchurch. Emal: a.carr@cvl.canterbury.ac.nz.

2 For a gven common yeld strength of the damper n the frst storey of the structure, four types of dstrbutons of the damper yeld strengths n the structure were compared. The yeld strength dstrbutons of the dampers n the structure for these cases are chosen to be proportonal to the storey shear due to four types of lateral load dstrbuton (whle the damper yeld strengths at the 1 st storey are the same for these cases). These four types of lateral load dstrbuton are: only one lateral force actng on the top level (case-i), parabolc load dstrbuton (case-ii), nverted-trangular load dstrbuton (case-iii), unform load dstrbuton (case-iv). These are shown n Fgure. The comparson of these four cases s shown n Table 1 and the values n brackets show the dfferences of the parameters of these cases compared to case-iii. The peak nterstorey drft s a mnmum for case-iii and ts peak base shear s also close to the mnmum, hence case III s close to the optmal case. Smlar results can also be found wth other earthquake exctatons [Ln, 1999; Ln et al, 1998a]. Thus a satsfactory dstrbuton of the yeld strengths of the dampers n the structure can be taken to be proportonal to the storey shear force due to an nverted-trangular lateral load pattern. There s no need to have the same yeld strength for all dampers n the structure. It has also been found that for ths type of dstrbuton of the strength levels of the dampers, all devces can reach ther nelastc (yeldng) stage smultaneously. Ths characterstc wll lead to maxmum energy dsspaton and cause the structure to behave manly n ts frst mode [Ln, 1999]. PUSHOVE ANALYSES OF THE STUCTUE WITH DAMPES Pushover analyses have been performed and compared for the structure wth and wthout dampers [Ln, 1999]. For the structure wth dampers, two types of damper yeld strength dstrbuton were consdered: case I (the same yeld strength for all dampers) and case III (satsfactory dstrbuton). For carryng out the smplfed nonlnear statc analyss and dsplacement-based desgn of the structure wth dampers, the deflected shapes of the structure wth dampers are necessary. These deflected shapes were obtaned from pushover analyses. The normalsed deflecton shape vectors of the structure wth the supplemental dampers (both case-i and III) and the orgnal structure wthout dampers are shown n Fgure 3. It can be seen that for the satsfactory dstrbuton of damper yeld strengths, the deflected shape of the structure wth the dampers s very close to that of the orgnal structure wthout dampers, hence the deflected shape can be taken as that of the orgnal structure wthout dampers. Ths makes analyss and desgn much easer. It can also be seen that for the dstrbuton of the same yeld strength for all the dampers n the structure, the dfference n the deflected shape of the structure wth dampers and the non-damped structure becomes much larger. Ths shows another advantage for the satsfactory dstrbuton. It has also been found that for a satsfactory dstrbuton of dampers n the structure, the dynamc peak response of the MDOF structure wth dampers can be predcted effectvely by ts equvalent SDOF system [Ln, 1999; Ln et al, 1998a]. The relatonshp between the characterstcs of a MDOF structure and ts equvalent SDOF system s shown n detal n the references [Fajfar et al, 1987; Ln, 1999 and Q et al, 1991]. THE EQUIVALENT VISCOUS DAMPING ATIO AND THE EFFECTIVE PEIOD OF THE STUCTUE WITH DAMPES Sdof System The SDOF system ncludes both the orgnal structural frame system and the supplemental dampng system. The two systems act n parallel and can be descrbed as a dual system (see Fgure 4). The relatonshp between energy dsspaton per cycle E d, equvalent vscous dampng ξ and maxmum elastc stran energy E s has been proposed by Clough [1993] as: E ξ d 4πE s The ntal elastc stffness of the orgnal structure s K s. The yeldng force of the orgnal structure s P y. The yeldng dsplacement of the orgnal structure s y0. P s the elastc force n the orgnal structure at a gven response dsplacement f the structure remans elastc. rk s s the post yeldng stffness of the orgnal structure. An elastc-perfectly-plastc hysteress model s adopted to represent the behavour of supplemental dampng system. The ntal elastc stffness of the supplemental dampng system s desgnated as S K s, and the

3 yeldng force of the supplemental dampng system s desgnated as F P. (Fgure4). F and S are the force and stffness factors respectvely. The equvalent vscous dampng rato and the effectve perod of the dual system can be expressed as [Ln, 1999 and Ln et al, 1998a,b]: E ξ d 4πE s F F 1 S π 1 1 F + + r 1 µ µ (1) M µ T eff π T0 () K eff µ F r( µ 1) The term F 1 1 n Equaton (1) s very close to 1 for lead dampers. S µ The orgnal vscous dampng ξ 0 and the equvalent vscous dampng due to the nelastc deformaton of the structure ξ 0 can be estmated by [Shbata et al, 1975]: ξ s ξ + ξ / µ (3) Mdof Structure The modal stran energy method has been adopted to estmate the amount of equvalent structural dampng provded by the supplemental dampers [Zhang et al, 1989]. The equvalent vscous dampng can be estmated accordng to ths formula (only the fundamental mode s of nterest): Ed ξ (4) 4πE s where E d s the energy dsspated by the supplemental dampers per cycle for the th vbraton mode, E s s the stran energy of the structure wth the supplemental dampers for the th vbraton mode, ξ s the equvalent vscous dampng rato for the th vbraton mode. The effectve perod (frst mode) of the MDOF structure wth the supplemental dampers at the target dsplacement x t can be calculated from the aylegh method: T eff ( ) N m φ W u 1 π x ( where u x ) ( ) N t φ t (5) g F u F φ 1 For regular frame structure wth a satsfactory dstrbuton of dampers (the damper yeld strengths n the structure are proportonal to the shear forces due to the nverted-trangular lateral load pattern), the equvalent vscous dampng rato and the effectve perod of the structure wth dampers can be expressed as [Ln, 1999]: ξ π F F r 1 µ µ (6) 3

4 T eff (7) T 0 µ F µ r ( µ 1) where F the force factor for the MDOF structure F yd 1 cosθ / F0 (8) F yd1 the damper yeld strength at the 1 st floor, and F 0 elastc base shear of the MDOF undamped structure f the structure remaned elastc at the target dsplacement. N N P ψ / φψ 1 1 (9) F n Equatons (6) and (7) for a MDOF structure s equvalent to F n Equatons (1) and () for the SDOF F system. The only dfference between Equatons (6) and (1) s the term 1 1. Ths dfference s very small S µ for lead dampers, hence t can be concluded that the equvalent vscous dampng rato and the effectve perod of the structure wth dampers can be easly calculated by ts equvalent SDOF system through Equatons (6) and (7) [Ln, 1999 and Ln et al, 1998b]. SIMPLIFIED STATIC METHOD OF ANALYSIS OF STUCTUES WITH SUPPLEMENTAL DAMPES From above t can be seen that the peak dynamc response of the MDOF structure wth the supplemental dampers can be predcted effectvely by ts equvalent SDOF system. And the equvalent vscous dampng rato and the effectve perod of the structure wth dampers can be easly calculated by ts equvalent SDOF system. Hence, SDOF system s a good tool for smplfed analyss and desgn purposes. The SDOF method makes t possble to adopt spectral analyss for desgn. Based on these results, a smplfed statc analytcal method can be adopted to predct the dynamc response of the MDOF structure wth the supplemental dampers. Due to the fact that the equvalent vscous dampng of supplemental dampers and the effectve perod of structures wth supplemental dampers vary wth respect to the dsplacement response of structures, some teratons mght be needed to obtan the response of structures snce the dsplacement response s not known pror to analyss. A pushover analyss s needed to obtan the base shear-roof dsplacement relatonshp durng structural nelastc deformaton. The analyss procedure can also be obtaned as follows: Step 1. Conduct a pushover analyss of the undamped structure. The base shear and the roof dsplacement of the orgnal structure at yeld can be obtaned. The rato of the post-yeld stffness to the ntal stffness can be obtaned as well. Step. Make an ntal estmate for the roof dsplacement (x t0 ) of the structure wth the supplemental dampers. The deflected shape { φ} can be obtaned by the dsplacement profle correspondng to the estmated dsplacement from the result of step 1. The ntal assumed target ductlty µ s also obtaned. Step 3. The force factor F of the structure wth the supplemental dampers can be calculated. From Equaton (6) the equvalent vscous dampng ξ d due to the supplemental dampers can be calculated. The equvalent vscous dampng rato (ξ 0 + ξ 0 ) due to nelastc deformaton and orgnal dampng can also be estmated by Equaton (3) for the guessed dsplacement. The total equvalent vscous dampng rato ξ t (ξ d +ξ 0 + ξ 0 ) s known. Step 4. The effectve perod T eff of the structure wth the supplemental dampers can be obtaned from Equaton (7). 4

5 Step 5. The dsplacement for the equvalent SDOF system (or spectral dsplacement) x* for the effectve perod T eff and equvalent vscous dampng rato of ξ t can be obtaned drectly from the dsplacement spectra. Step 6. The target roof dsplacement x t1 can be obtaned [Ln, 1999 and Q et al, 1991] by: L * N N x x * where M * t1 m φ, L* m φ, and x* s the spectral dsplacement. M * 1 1 xt1 + xt0 Let the new target roof dsplacement x t be: xt Step 7. Compare x t wth x t0. If they are close enough, they are the target roof dsplacement. Then go to step 8. If the dfference s large, teraton s needed. We need to use a new estmated roof dsplacement (x t0 x t ) and go back to step 3. Step 8. For the target dsplacement obtaned n step 7, the effectve perod of the structure T eff and the total equvalent vscous dampng rato ξ t can be obtaned. The spectral acceleraton value S a can be obtaned from the acceleraton spectra for the T eff and ξ t. The peak base shear of the structure wth the supplemental dampers (MDOF) can be calculated from the spectral acceleraton of ts equvalent SDOF system as follows: N ψ Q L * 1 S N a φ ψ 1 (, ξ ) T eff t where Q s the peak base shear, S a (T eff,ξ t ) s the spectral acceleraton for the effectve perod T eff and the equvalent vscous dampng rato ξ t, whle ψ s the normalsed vector of the nverted-trangular lateral load pattern. The fnal dsplacement shape of the structure at the target dsplacement can be obtaned from the result of the pushover analyss of the orgnal structure n step 1. Then the peak nterstory drft ndex IDI max can be calculated as follows: φ φ IDI 1 max h f max The 1-storey model structure shown n Fgure1 was used as the example. The ntal perod of the undamped structure s 1.99s. The damper yeld strength at the 1 st level s 11kN. The yeld strengths of dampers n upper storeys are taken to be satsfactory dstrbuton. S 10, cosθ The smplfed nonlnear statc analyss of ths model structure was carred out followng the procedure mentoned above. Tme hstory analyss of the same structure was also performed. The comparsons of the peak structural response of these two methods are shown n Table. The results of the smplfed nonlnear statc analyss are very close to those of the tme-hstory analyss except for the peak base shears. Ths s because the base shear has a sgnfcant contrbuton from the hgher modes whle the smplfed method s based on the frst mode response. However, for the desgn of the structure wth supplemental dampers the man concern s the dsplacement. For a prelmnary desgn ths method of analyss of the structure wth the supplemental dampers should be suffcently accurate. THE OPTIMAL DAMPING LEVEL DUE TO THE SUPPLEMENTAL DAMPES From the equvalent SDOF system t can be seen that when the strength levels of the dampers ncrease, the effectve perod of the structure wth the supplemental dampers wll decrease and the equvalent vscous dampng due to the dampers wll ncrease. Ths wll lead to a reducton of the spectral dsplacement. However, the nfluence of the strength levels of the dampers on the response spectral acceleraton s not that straghtforward. When the strength levels of the dampers ncrease, on the one hand, the equvalent vscous dampng wll tend to ncrease, ths wll lead to a lower value of the acceleraton response; on the other hand, the effectve perod of the structure wth the supplemental dampers wll tend to reduce, ths wll result n a hgher value of the acceleraton response. The acceleraton response of the structure reflects the response level of the base shear of the structure. There exsts a certan level of dampng to mnmse the acceleraton response. For dfferent ductlty the structure mght experence durng earthquake exctatons, t has been show that ths optmal dampng rato due to the supplemental dampers s 15%-17% [Ln, 1999 and Ln et al, 1998b]. 5

6 POCEDUE FO THE DISPLACEMENT-BASED METHOD FO CHOOSING PAAMETES FO THE SUPPLEMENTAL DAMPES It s known that of the two characterstc parameters of the supplemental dampers (the force factor F and the stffness factor S ), the force factor F domnates the dynamc behavour of the structure wth the supplemental dampers as long as the stffness factor S exceeds some partcular level. Hence n the dsplacement-based method we focus on the choce of the force factor F for the supplemental dampers. We already know that a satsfactory dstrbuton of the dampers occurs when the dstrbuton of the strength of the dampers n every storey along the heght of the structure s proportonal to the shear dstrbuton developed due to an nverted trangular dstrbuton of lateral load. It s assumed that the orgnal structures are regular and symmetrcal n mass and stffness dstrbuton. Hence the peak response of the MDOF structure wth the supplemental dampers can be effectvely predcted by ts equvalent SDOF system. Ths gves good grounds for the dsplacement-based method [Prestley, 1995 and Q et al, 1991] to be adopted n the choce of the force factor F of the supplemental dampers at the prelmnary desgn stage. The procedure for the dsplacement-based method can be establshed as follows [Ln, 1999 and Ln et al, 1998a]: Step 1. Check the orgnal structure to see whether the maxmum nterstorey drft meets the requrements of the desgn or not (tme-hstory analyss or nonlnear statc analyss can be adopted for ths purpose). If yes, there s no need of any supplemental dampers. If not, go to step. Step. The ntal dsplacement shapeφ 0 can be obtaned from the nonlnear pushover analyss of the orgnal structure (at the yeldng dsplacement of the orgnal structure). The yeldng base shear and the yeldng roof dsplacement of the orgnal structure (converted to ts equvalent SDOF system) can be obtaned. Step 3. Gven the requred maxmum nterstorey drft rato, the frst target dsplacement at roof level ( f ) of the structure wth the supplemental dampers can be obtaned (from the ntal dsplacement shape). Compare ths frst target dsplacement wth that of the ntal dsplacement shape. If there s a bg dfference, some teraton s needed untl the target dsplacement obtaned from nonlnear pushover analyss also meets the maxmum nterstorey drft requrement. Then the target dsplacement and the constant dsplacement shape φ (at the target dsplacement) can be obtaned. The target ductlty µ of the orgnal structure can also be calculated. The target spectral dsplacement (for the equvalent SDOF system) can be obtaned. The elastc force of the orgnal structure P at the target dsplacement (converted to ts equvalent SDOF system) can be calculated. Step 4. Choose the optmal dampng ξ d of 15-17% of crtcal due to supplemental dampers. Calculate the ntal vscous dampng and the effectve dampng of the structure due to nelastc deformaton of the orgnal structure (ξ 0 + ξ 0 ) at the target ductlty. The total equvalent vscous dampng rato ξ t (ξ 0 + ξ 0 +ξ d ) can then be obtaned. Step 5. From the generated dsplacement spectra, knowng the equvalent vscous dampng ξ t and the target spectral dsplacement, the maxmum effectve perod of the structure wth supplemental dampers T max can be obtaned to meet the requrement of the maxmum nterstory drft rato. Step 6. For the gven equvalent vscous dampng rato ξ d (15-17%) due to the supplemental dampers and the stffness factor S, the force factor F can be obtaned from ξ F relatonshp (Equaton (6)). Step 7. The effectve perod of the structure wth supplemental dampers correspondng to the F factor from Step 6 and the target ductlty µ can be calculated. Step 8. Compare T max and T eff : f T eff T max, the assumed optmal equvalent vscous dampng rato ξ and the force factor obtaned meet the desgn requrement. If T eff > T max, the assumed optmal equvalent vscous dampng rato (thus the correspondng force factor F ) s too small. Lettng T eff T max, a modfed force factor F and the correspondng equvalent vscous dampng rato ξ d can be obtaned. Step 9. From the force factor F (or F ) obtaned n Step 8 and the elastc force P at the target dsplacement of the orgnal structure obtaned n Step 3, the yeldng force of dampng system for the equvalent SDOF system can be calculated as: F F P yd Step 10. The strength level of the damper at the 1 st storey can be obtaned by the relatonshp between the base shear of the MDOF structure and ts SDOF system. Through Equatons (8) and (9), t can be obtaned as: 6

7 F yd1 * Fyd F P cosθ cos ψ φ ψ ψ θ φ ψ The strength levels of the dampers at the upper storeys can be determned by the satsfactory dstrbuton mentoned above. An example for determnng the parameter of the dampers n a structure followng the above procedure s shown n Ln et al [1998a] and Ln [1999]. ACKNOWLEDGEMENT The fnancal support provded by Pengun Engneerng Ltd s gratefully acknowledged. The authors would also lke to thank John Zhao and Trevor Kelly for ther knd cooperaton and suggestons. EFEENCES Carr,A.J. (1996) UAUMOKO - nelastc dynamc analyss program, Unversty of Canterbury, Department of Cvl Engneerng, Chrstchurch, New Zealand. Clough,.W. and Penzen J. (1993) Dynamcs of Structures, McGraw-Hll, New York. Fajfar, P. and Fschnger, M. (1987) Non-lnear Sesmc Analyss of C Buldngs: Implcatons of a case study, European Earthquake Engneerng, No.1. Ln, X. (1999) Analyss and Desgn of Buldng Structures wth Supplemental Lead Dampers under Earthquake and Wnd Loads, Ph.D Thess, Department of Cvl Engneerng, Unversty of Canterbury, Chrstchurch, New Zealand. Ln, X., Moss, P.J. and Carr, A.J. (1998a) Analyss and Sesmc Desgn of Buldng Structures wth Supplemental Lead Dampers, Proc. NZNSEE Conference, Warake, Taupo, New Zealand, March 1998, pp Ln, X., Moss, P.J. and Carr, A.J. (1998a) Sesmc Analyss and Desgn of Buldng Structures wth Supplemental Lead Dampers, Proc. Australasan Structural Engneerng Conference, Auckland, New Zealand, 30 September- October, Vol., pp Mont, M. D. and obnson, W. H. (1996) A Lead Shear Damper Sutable for educng the Moton Induced by Wnd and Earthquake, 11 WCEE, Acapulco, Mexco, Paper 71. Prestley, M.J.N. (1995) Dsplacement-Based Sesmc Assessment of Exstng enforced Concrete Buldngs, Proc. Pacfc Conference on Earthquake Engneerng, Australa, 0- Nov.. Q, X. and Moehle, J. P. (1991) Dsplacement Desgn Approach for enforced Concrete Structures Subjected to Earthquakes, Earthquake Engneerng esearch Center eport, No. EEC 91/0, Unversty of Calforna, Berkeley, Jan., Scholl,. E. (1993) Desgn Crtera for Yeldng and Frcton Energy Dsspatons, Proc. Semnar on Sesmc Isolaton, Passve Energy Dsspaton, and Actve Control, Appled Technology Councl eport No. ATC-17-1, edwood Cty, CA. Shbata, A. and Sozen, M. A. (1975) Substtute-Structure Method for Sesmc Desgn n /C, Journal of the Structural Dvson, Vol.10, No.ST1, Jan., 1976, pp Zhang, -Hu and Soong, T.T. (1989) Sesmc esponse of Steel Frame Structures wth Added Vscoelastc Dampers, Earthquake Engneerng and Structural Dynamcs, Vol. 18, pp Table 1. Comparson of the response for the structure wth four dfferent dstrbutons of the yeld strength of the dampers under El Centro 1940 NZS403 compatble earthquake Peak nterstorey drft (cm) peak base shear (kn) Case-I (.39%) (7.98%) Case-II (8.58%) (3.6%) Case-III 1.61 (0) (0) Case-IV 1.84 (13.63%) (-6.07%) 7

8 Table. Comparson of the peak responses of the structure wth the supplemental dampers for a smplfed non-lnear statc analyss and tme hstory analyss under the El Centro 1940 NZS403 compatble earthquake oof dsplacement (cm) Base shear (kn) Interstorey drft (cm) Smplfed statc analyss Tme hstory analyss Dfference.37% 17.51% 0.14% Fgure Four Types of Lateral Load Dstrbuton Fgure 1 1-storey Frame Structure wth Supplemental Dampers Fgure 3 Comparson of the Deflected Shapes of the Structure wth and wthout Dampers Fgure 4 Force-Dsplacement elatonshp for the Supplemental Dampng System and the Undamped Orgnal Structure 8