Simulation analysis for the ultimate behavior of full-scale lead-rubber seismic isolation bearings

Transcription

1 Smulaton analyss for the ultmate behavor of full-scale lead-rubber sesmc solaton bearngs M. Kuch Hoado Unversty, Japan I. D. Aen Sesmc Isolaton Engneerng Inc., USA A. Kasalanat Dynamc Isolaton Systems, USA SUMMARY: Ths paper presents a new three-dmensonal analytcal model for the accurate smulaton of the ultmate behavor of full-scale lead-rubber sesmc solatons bearngs. The model comprses multple shear sprngs at the md-heght and ncludes the nteracton between baxal shear and axal forces wth nonlnear hysteress. Severe loadng tests for a range of dfferent full-sze lead-rubber bearngs were conducted to obtan data to valdate the analytcal model. The test programs ncluded a varety of dfferent loadng patterns, such as monotonc, cyclc, and horzontal bdrectonal earthquae ground motons. Buclng or stffenng behavor was observed under large shear deformatons and hgh axal loads. Very good agreement between the expermental and analytcal results was obtaned, ndcatng that the model s useful for the predcton of the sesmc response of solated structures under severe earthquae ground motons. Keywords: Sesmc solaton, elastomerc solaton bearng, nonlnear analyss 1. INTRODUCTION Sesmc solaton s the most effectve technology for protectng structures from the damagng effects of earthquaes. The concept of sesmc solaton s to move the fundamental perod of a structure away from the predoant perod of the ground moton through the ntroducton of flexble supports at the foundaton level. It has been extensvely used worldwde over the last three decades. Numerous earthquaes have confrmed the good response of sesmcally-solated buldngs, ncludng the fully operatonal performance of solated hosptals n the 1994 Northrdge Earthquae, USA, and the 11 Great Tohou Earthquae, Japan, amongst others. Sesmc solaton has been proven to provde the hghest level of sesmc protecton possble for buldngs and other structures. The wdespread use of sesmc solaton has necesstated a better understandng of the ultmate behavor of solaton devces under large shear deformatons or hgh compressve stresses. Elastomerc solaton bearngs exhbt strongly nonlnear behavor, such as stffenng or buclng nfluenced by axal loadng under large deformatons. For the accurate predcton of the ultmate behavor of solaton devces, the authors have developed a seres of mechancal models. The ntal prototype model was developed n two dmensons, whch comprsed shear and axal sprngs at the md-heght and a seres of axal sprngs at the top and bottom boundares of the model. The latest model was developed by expandng the prototype model to three dmensons (Kuch et al., 1). It comprses multple shear sprngs at the md-heght and ncludes nteracton between baxal shear and axal forces wth non-lnear hysteress. In the early stage of development of the model t was valdated by showng that t could accurately predct the behavor of reduced-scale solaton bearngs, wth dameters of about cm. The valdaton of numercal models wth extreme loadng data from testng of full-sze solaton bearngs, however, has generally not often been done, because of the capacty lmtatons of most test facltes and the lmted opportuntes to perform such tests. The authors have conducted severe loadng tests of full-sze lead-rubber bearngs, and the data obtaned from these tests presented an unusual opportunty for further valdaton of the analytcal model. The dameter of the full-sze

2 bearngs tested was approxmately 1. meters. The test programs ncluded a varety of dfferent loadng patterns, such as cyclc and horzontal bdrectonal earthquae ground motons. Buclng or stffenng behavors were observed n the tests under large shear deformatons and hgh axal loads. The newly developed model ntally assumed a unform dstrbuton of the compresson modulus over the entre cross secton of a bearng. Whle the assumpton of unform dstrbuton was vald for the reduced-scale solaton bearngs; however, t faled to accurately predct the behavor for the full-scale bearngs. Therefore, n ths paper, a refnement s ntroduced for the calculaton of the compresson modulus to nclude the nfluence of the bul modulus compressblty and the shape of the bearng cross secton.. MECHANICAL MODEL Fg..1 shows the mechancal model proposed to smulate the behavor of elastomerc solaton bearngs. The model comprses multple shear sprngs (MSSs) and an axal sprng at the md-heght and two seres of axal sprngs at the top and bottom boundares. The MSS model s used n the mechancal model to represent the baxal behavor of the elastomerc solaton bearngs, whch conssts of a seres of dentcal shear sprngs arranged radally to represent sotropc behavor n the horzontal plane (Wada and Hrose, 1989). Each sprng n the seres of axal sprngs at the top and bottom boundares s a unaxal, nonlnear sprng and represents an ndvdual fber of the bearngs cross-sectonal area. When ths collecton of sprngs s combned n the model, the nonlnear nteracton behavor s acheved. The rgd colus, whch represent the heght of the bearng, are combned between the top and bottom seres of axal sprngs and md-heght MSSs and the axal sprng. Fgure.1. Three-dmensonal multple-sprng mechancal model The defnton of the forces and dsplacements on the model s shown n Fg... There are sx dsplacement degrees of freedom three translatons and three rotatons at the external nodes, a and b. The nternal nodes, m and n, have three dsplacement degrees of freedom: translaton A and rotatons B and C. The dsplacements for translatons B and C and rotaton A of the nternal node, m, are equal to those of the external node, a. The same defnton for nodes a and m s made for nodes b and n. By usng ncremental dsplacements of nodes a and m and assug that plane sectons reman plane, the relatonshp between the ncremental force vector, f am, and the ncremental dsplacement vector, u am, on nodes a and m can be obtaned as follows: where fam Kam u am (.1) T am Aa Ba Ca Am Bm Cm T am faa mba mca fam mbm mcm u f

3 K am symm. 4am am am 3 am 1 am am 3 am 4 am 5 am am 4 am 5 am 6 am 3 am 5 am 6 am 1 am am 3 am where Na s the tangental stffness of the -th axal sprng, and l C and l B are the dstances between the -th sprng and the centrod of the cross-sectonal area of the bearng along the B and C axes, respectvely (Fg..3). am am, 1am Na, am Na lc, 3am Na lb, 4am Na lc, l l, 5 am Na C B 6am Na lb Deformaton, restorng force Multple axal sprngs (top) Dsplacement, nternal force [node b (external)] δ Nb 1,,... f Nb 1,,... b θ Bb, m Bb δ Ab, f Ab δ Cb, f Cb δ Bb, f Bb θ Cb, m Cb Axal sprng (mddle) δ N, f N n n [node n (nternal)] δ An, f An θ Bn, m Bn Multple shear sprngs (mddle) δ S 1,,...j... f S 1,,...j... m m θ Cn, m Cn [node m (nternal)] δ Am, f Am Multple axal sprngs (bottom) δ Na 1,,... f Na 1,,... a A θ Bm, m Bm θ Cm, m Cm [node a (external)] δ Aa, f Aa B C θ Ba, m Ba δ Ba, f Ba δ Ca, f Ca θ Ca, m Ca Fgure.. Forces and dsplacements on the mechancal model l B Δδ Am Δδ Aa C Δθ Cm ( ) m Δθ Ca ( ) a A θ B C B Δδ Am Δδ Aa l C Δθ Bm m a Δθ Ba A B θ C Fgure.3. Axal sprngs at the top and bottom boundares The relatonshp between the ncremental forces and dsplacements on nodes b and n can be obtaned by replacng a by b and m by n n Eqn. (.1) as follows: fbn Kbn u bn (.)

4 where K bn s the stffness matrx, f bn s the ncremental force vector, and u bn s the ncremental dsplacement vector of nodes b and n. Now consder the force-dsplacement relatonshps for the multple shear sprngs and the axal sprng at the md-heght of the model. The force-dsplacement relatonshp on nodes n and m n Fg.., whch excludes the rgd colus, may be expressed as follows: where f K u (.3) T fam fbm fcm fan fbn fcn u T Am Bm Cm An Bn Cn f K N N symm. N 1 3, 1 j S j j S j j cos, sn 3 j S j cos sn, where j S s the tangental stffness, j s the angle to the B axs of the j-th shear sprng (Fg..4), and N s the stffness of the axal sprng at the md-heght. j S C jδ S, j fs B jϕ Fgure.4. Multple shear sprngs at the md-heght In order to convert the force-dsplacement relatonshp of nodes m and n, expressed by Eqn. (.3), to nodes m and n, whch ncludes the rgd colus, a transformaton matrx s used. Tang the geometrcal relatonshps of the deformatons, the force equlbrum condton, and the P effect nto account gves the transformaton matrx. Fg..5 shows the geometrcal relatonshps of the deformatons and the forces n the AC plane. Let AC T be the transformaton matrx n the AC plane, AC u and AC f be the dsplacements and forces on nodes m and n, and AC u and AC f be those on nodes m and n, respectvely. The transformaton of the dsplacements and forces may be expressed by u T u (.4) AC AC AC f T f (.5) T AC AC AC where T AC u Am Ca Bm An Cb Bn

5 T ACu Am Ca Bm An Cb Bn T AC f fam fca mbm fan fcb mbn T ACf fam fca mbm fan fcb mbn AC Cb Ca 1 Bm h 1 1 T Cb Ca 1 Bn h 1 1 and h s the total heght of the bearng. n f An Rgd colu h/ δ An δ Cb m Bn f Cb θ Bn n δ An δ Cb f An m Bn θ B Rgd colu A C m m h/ δ Am δam δ Ca δ Ca m Bm f Am θ Bm m Bm θ Bm f Ca f Am θ Bn f Ca f Cb Fgure.5. Geometrcal relatonshps of the deformatons and forces n the AC plane The transformaton matrx, AB T, n the AB plane can be constructed usng the same procedure as for the AC plane, expressed by Eqns. (.4) and (.5). The transformaton matrx, T, s formed by assemblng the assocated components of AB T and AC T. Fnally, the force-dsplacement relatonshp on nodes m and n may be expressed as follows: f K u (.6) K T K T (.7) T The overall stffness matrx, K ab, s obtaned by arrangng the elements of the partal stffness matrces, K am, K, and K nb, nto an 18 by 18 matrx and addng a lnear torsonal stffness component. Fnally, the relatonshp of the forces and dsplacements at the external and nternal nodes n the model may be expressed as follows: where fex uex K ab fn un (.8)

6 T ex faa fba fca maa mba mca fab fbb fcb mab mbb mcb T n fam mbm mcm fan mbn mcn T ex Aa Ba Ca Aa Ba Ca Ab Bb Cb Ab Bb Cb T n Am Bm Cm An Bn Cn f f u u and f ex and u ex are the ncremental forces and dsplacements on the external nodes, a and b, and f n and u n are those on the nternal nodes, m and n, respectvely. 3. HYSTERESIS MODEL The shear hysteress model prevously developed by the authors, whch s capable of predctng the behavor of elastomerc solaton bearngs under large shear deformatons, s appled to each sprng n the MSS model (Kuch and Aen, 1997). In the proposed mechancal model, the reducton of horzontal stffness and eventual buclng behavor due to hgh axal loadng s represented by the nteracton between the shear and axal forces n the tlted MSS model and the axal sprngs. Therefore, the hysteress propertes to be used for the shear sprngs n the MSS model should be those under low compressve stress (deally under zero compressve stress). The shear force-shear stran relatonshp up to 4% stran used for the MSS model s shown n Fg. 3.1(a) (Kuch et al., 1). The stress-stran relatonshp for the seres of axal sprngs at the top and bottom boundares s shown n Fg. 3.1(b). In general, a laated rubber bearng exhbts hgh stffness and yeldng stress n the compresson regon and low stffness and yeldng stress n the tenson regon. The relatonshp between vertcal stran and stress s antsymmetrc. The behavor represented by the model shown n Fg. 3.1(b) s generally accepted for the nonlnear vertcal behavor of elastomerc solaton bearngs. tenson E ty σ ty o shear force axal stress E nt σ cy E cy shear stran compresson stran (a) Shear sprngs n the MSS (b) Axal sprngs at the top and bottom boundares Fgure 3.1. Hysteress models The prevously developed model assumed a unform dstrbuton of the ntal compresson modulus, E nt, over the cross secton of a crcular bearng, as shown n Fg. 3.(a), whch s gven by Eqn. (3.1), E nt E (1 S ) K E S K 1 (1 1 ) (3.1) where E s the Young s modulus of rubber, s a constant related to the rubbers hardness, S 1 s the rubbers shape factor, and K s the bul modulus of rubber. In the new model presented heren, an mprovement s made n the calculaton of the ntal compresson modulus. The dstrbuton of the compresson modulus s consdered as shown n Fg.

7 3.(b). The compresson modulus s computed from Eqn. (3.), whch s expressed as a functon of the dstance, r, from the centrod of the bearng cross secton. Eqn. (3.) s the exact soluton of the governng equaton of the pressure dstrbuton consderng the nfluence of the bul compressblty (Kelly, 1997), I( r) Ent () r K1 (3.) I( R) where 1G Kt and I s the modfed Bessel functon of the frst nd of order zero, G s the shear modulus of rubber, and t s the thcness of the rubber pad. (a) Unform (b) Bessel functon Fgure 3.. Dstrbuton of ntal compresson modulus The hysteress model shown n Fg. 3.1(b) defnes the tangent compresson modulus. The stffness of each axal sprng, Na, at the top and bottom boundares s obtaned from E A l Na (3.3) where E s the tangent compresson modulus of the -th sprng, A s the ncremental area correspondng to the -th sprng, and l s the magnary length of the sprng. One half of the total heght of the rubber n the bearng s usually used for ths magnary length. 4. SIMULATION ANALYSES Tests of large lead-rubber bearngs were conducted to nvestgate ther mechancal characterstcs under severe loadng condtons (Sherstobtoff et al., 8). Fg. 4.1 shows the dmensons of the lead-rubber bearngs used for the tests. The tests were conducted at the Unversty of San Dego SRMD Testng Faclty, whch s the most sophstcated faclty of ts type n the world for testng sesmc solaton devces. The test program ncluded a varety of dfferent loadng patterns, such as monotonc, cyclc, and horzontal bdrectonal earthquae ground motons. The results of two dfferent types of tests were selected to compare aganst the results gven by the analytcal model (Table 4.1). The parameters used for the smulaton analyses are summarzed n Table 4.. Some of these parameters were detered by ntally evaluated test results for other types of elastomerc solaton bearngs. From prevous analyses t was understood that the convergence of the model depends on the cross-secton dscretzaton selected for the analyss. A 5x5 grd of axal sprngs was selected as approprate for the bearng desgn. Three dfferent analyss cases for each test were performed n order to demonstrate the valdty of the refnement ntroduced to the mechancal model (Table 4.3). The

8 loadng sequence used for the analyses was exactly the same as for the tests. Intally, the axal load was appled to the top of the model, then the test shear dsplacement hstory was appled. Rubber dameter 116 mm Lead plug dameter mm Rubber layers 8 mm x 4 layers Steel shms 3.4 mm x 39 layers Shape factor, S Aspect rato, S 3.18 Fgure 4.1. Lead-rubber bearng desgn detals Table 4.1. Tests selected for analytcal comparson Test 1 Shear capacty lmt-state test Undrectonal, one cycle Constant axal load (stress): 14, N (17.3 MPa) Pea dsplacement (shear stran): 3 mm (1%), 64 mm (%) Pea velocty: 63.5 mm/s Test Bdrectonal, dynamc earthquae nput test Constant axal load (stress): 14, N (17.3 MPa) 1999 Kocael Earthquae X & Y dsplacement tme hstores, real tme Table 4.. Bearng parameters used for the smulaton analyses Increments n the top and bottom boundares 5 x 5 Number of sprngs n MSS model 8 Shear modulus of rubber, G.49 MPa Young s modulus of rubber, E 1.47 MPa Bul modulus of rubber, K 196 MPa Constant related to rubber hardness,.85 Tenson yeld stress, ty 1 MPa Tenson yeld modulus, ty E nt /1 Compresson yeld stress, cy 1 MPa Compresson post-yeld modulus, cy E nt / Table 4.3. Analyss cases Case 1 P- effect not consdered Case P- effect consdered, unform dstrbuton of compresson modulus Case 3 P- effect consdered, Bessel functon dstrbuton of compresson modulus The expermental results shown n Fg. 4. were obtaned from the shear capacty lmt-state tests as summarzed n Table 4.. The axal load appled to the solator n ths test was approxmately 6% greater than the crtcal load for the 64 mm shear dsplacement (11,95 N). Consequently, the solator hysteress exhbts negatve ncremental stffness at shear strans beyond about 1%. The analytcal results are also shown n Fg. 4.. The expermental hysteress loop at 1% pea shear stran doesn t exhbt negatve ncremental stffness, therefore all three analyss cases show good agreement wth the test result. However, the analytcal hysteress loops don t agree well wth the test results at% pea shear stran f the P- effect s not consdered, as shown n Fg. 4.(a). The hysteress loops obtaned from both of Cases and 3 exhbt negatve ncremental stffness. Case 3

9 predcts the behavor more accurately than Case, and shows good agreement wth the test result. Ths shows that the refnements to the analytcal model have mproved the capablty to accurately predct bearng behavor under extreme loadng condtons. The dstrbuton of the compresson modulus over the cross-secton leads drectly to the geometrc moment of nerta. The accurate representaton of compressve behavor s mportant, because the bendng stffness has a strong nfluence on the nclnaton of the shear sprngs at the md-heght of the model. The Bessel functon dstrbuton of the compresson modulus as shown n Fg. 3.(b) produces lower bendng stffness than does the unform dstrbuton of the compresson modulus. Consequently the refned compresson modulus dstrbuton mproves the accuracy of the smulaton analyses. Experment Analyss Shear stran Shear stran Shear stran (a) Case 1 (b) Case (c) Case 3 Fgure 4.. Comparson of analytcal and test results for un-drectonal cyclc loadng wth hgh compressve stress (17.3 MPa), Test 1 Experment Analyss X-dr. 1 1 Shear stran 1 X-dr. 1 1 Shear stran 1 X-dr. 1 1 Shear stran Y-dr. 1 1 Shear stran 1 Y-dr. 1 1 Shear stran 1 Y-dr. 1 1 Shear stran (a) Case 1 (b) Case (c) Case 3 Fgure 4.3. Comparson of analytcal and test results for b-drectonal dynamc earthquae nput, wth hgh compressve stress (17.3MPa), Test

10 The expermental results shown n Fg. 4.3 were obtaned from the b-drectonal, dynamc earthquae tests descrbed n Table 4.. An axal load of 14, N was appled, then the bearng was subjected to b-drectonal horzontal dsplacement tme hstores detered from the sesmc response analyses of an solated tower structure subjected to ground motons from the 1999 Kocael, Turey, M W 7.4 Earthquae (Sherstobtoff et al., 8). The pea test dsplacement and velocty were 58 mm and 785 mm/s, respectvely, and the total duraton of loadng was s. Such as loadng can categorzed as a severe, hgh-speed loadng test. Slght negatve ncremental stffness s observed n the X-drecton hysteress. The analytcal results are also shown n Fg. 4.3, and t can be seen that Case 3 accurately predcts the test b-drectonal hysteress behavour. 5. CONCLUSIONS A three-dmensonal mechancal model for sesmc solaton bearngs under large shear deformatons and hgh axal loads has been developed. The model comprses MSSs at ts md-heght and a seres of axal sprngs at the top and bottom boundares. The model can capture the nteracton between mult-drectonal shear dsplacement and axal force, non-lnear hysteress, and the dependence on vertcal load by consderng both materal and geometrcal nonlneartes n ts formulaton. The present wor has ntroduced a refnement for bearng compresson modulus to nclude the nfluence of the bul modulus compressblty and the shape of the bearng cross secton. The model ncludes the dstrbutons of the compresson modulus at the top and bottom boundares expressed as Bessel functons. The results of two types of severe loadng tests of full-scale lead-rubber solaton bearngs were used to show the valdty of the refned model. The tests were a shear capacty lmt-state test and a b-drectonal, dynamc earthquae test. Negatve ncremental stffness was observed n the hysteress loops obtaned from both tests, due to hgh axal load and large shear dsplacement. Smulaton analyses were conducted for the tests. The nfluence of the P- effect and compresson modulus dstrbuton on the bearng shear force-dsplacement hysteress was exaed. The best agreement between the expermental and analytcal results was obtaned when both the P- effect and Bessel functon compresson modulus dstrbuton were consdered. The results comparsons ndcate that the proposed model s useful for the predcton of the response of sesmcally-solated structures under severe earthquae ground motons. ACKNOWLEDGEMENT The contrbuton of full-scale lead-rubber bearng test data by BC Hydro and Ausenco Sandwell s gratefully acnowledged. The smulaton analyses were performed usng a nonlnear structural analyss program, IDAC, developed by Shmzu Corporaton, Japan. The research was partally supported by the Mnstry of Educaton, Scence, Sports and Culture, Grant-n-Ad for Scentfc Research (B), 11, The authors would le to express ther thans to all of those who contrbuted to the research presented n ths paper. REFERENCES Kuch, M., Naamura, T. and Aen, I.D. (1). Three-dmensonal analyss for square sesmc solaton bearngs under large shear deformatons and hgh axal loads. Earthquae Engneerng and Structural Dynamcs. 39: Wada, A. and Hrose, K. (1989). Buldng frames subjected to D earthquae moton. Structures Congress 89. ASCE: Kuch, M. and Aen, I.D. (1997). An analytcal hysteress model for elastomerc sesmc solaton bearngs. Earthquae Engneerng and Structural Dynamcs. 6: Kelly, J.M. (1997). Earthquae-resstant Desgn wth Rubber. Sprnger: London, U.K. Sherstobtoff, J., Reza, M., Aen, I. and Yan, L. (8). Sesmc upgrade of an ntae tower usng underwater base solaton, prelary desgn; Vancouver Island, Canada, The 14 th World Conference on Earthquae Engneerng: