STIFFENED MULTI-BAY COUPLED SHEAR WALLS

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1 Iraa Joural of Scece & Techology, Trasacto B, Vol. 8, No. B Prted Islac Republc of Ira, 004 Shraz Uversty STIFFENED MUTI-BAY COUPED SHEAR WAS ON EASTIC FOUNDATION * M. BICE, O. ASOGAN ** AND H. M. ARSAN Dept. of Cvl Egeerg, Cuurova Uversty, Adaa, Turey, 00 Eal: asoga@al.cu.edu.tr Abstract I ths study, the statc aalyss of a ult-bay coupled shear wall o a elastc foudato, havg ay uber of stffeg beas s studed usg the cotuous coecto ethod (CCM). The shear wall s cosdered to be ade up of a fte uber of sectos the vertcal drecto, wth or wthout stffeg beas betwee each par of cosecutve sectos. Ths ethod of aalyss, whch accouts for thcess varato ad elastc bea-wall coectos, treats the dscrete coectg beas as a cotuous layered edu. To pleet the foregog aalyss a coputer progra has bee developed the MATHEMATICA coputer algebra syste. To support the valdty of the preset ethod coparsos have bee carred out wth the results of SAP000 structural aalyss progra ad those soe refereces. eywords Cotuous coecto ethod, ult-bay coupled shear wall, stffeg bea, fleble coecto. INTRODUCTION I tall buldgs, lateral loads due to wds ad earthquaes are geerally ressted by shear walls. A sold shear wall s easly desged, sce t ca be treated roughly as a catlever bea ad ore precsely eployg a fe fte eleet esh. However, the desg of shear walls weaeed by doors, wdows ad corrdors (Fg. ), ecesstates partcular atteto sce they are hghly deterate. - b a b b - a - b a b + b a b + A, I c, I c, A +, I, I +, -, ( ), -, ( - ( ) +,, ( ) +, H A, I c, I c, A +, I, I +, + Fg.. A ult-bay shear wall wth stffeg beas Receved by the edtors Aprl 8, 00 ad fal revsed for August 0, 00 Correspodg author

2 44 M. Bce / et al. Cotuous Coecto Method (CCM) s oe of the ost wdely used ethods for the aalyss of perforated shear walls. I the aalyss of perforated shear walls, a crease the uber of rows of opegs causes a crease the uber of uows ad the cosequet tedous coputato reders a closed for soluto possble. Hece, the uber of rows of opegs has bee restrcted to oe, two ad three the pertet lterature, the latter case beg applcable to syetrcal probles [-], oly. There are few papers the lterature o the aalyss of ult-bay shear walls. oo ad Cheug [4] have epressed the aal forces the pers of a ult-bay shear wall ters of soe coordate fuctos. However, they have foud a hgh percetage of errors ther results, especally low stress values copared to those of the fte strp ethod. Coull et al. [5] have used a atr progresso ethod for ult-bay hgh-rse buldgs havg ay dscotutes the vertcal drecto ad restg o a rgd foudato. Soae [6] has aalyzed hgh ult-bay shear walls cople buldgs usg aalog coputers. Although hs asserto about the approate soluto of three desoal buldgs ay be correct, he has o eaples for plaar ult-bay coupled shear walls. Hece, there s o proof of the valdty of hs results for such cases. The results of Elholy ad Robso [7] whch used the fte dfferece ethod to solve ult-bay ustffeed coupled shear wall probles, has bee used for checg purposes the preset study. If a coupled shear wall s very hgh, the thcess of the wall ca be vared alog the heght [8] ad ths ecesstates o ew feature copared to a slar aalyss wth dfferet propertes dfferet sectos alog the heght, such as the preset oe. Coull [9] obtaed closed for solutos for the proble of a osyetrc coupled shear wall wth oe row of opegs, whch has a stffeg bea at the top ad a fleble foudato at the botto. Cha ad uag [0, ] used the CCM to study a coupled shear wall wth oe row of opegs, a stffeg bea at ay heght ad a rgd or fleble foudato. Coull ad Besal [] studed the coupled shear wall o both a rgd or fleble foudato wth two stffeg beas at arbtrary heghts usg the CCM. The, research wors o the statc [-5] ad dyac [6-9] aalyses of coupled shear walls wth oe row of opegs cae oe after aother, ot to eto a host of others. However, sce ost buldgs have ore tha oe row of opegs, t s worthwhle to wor o the case of ult-bay shear walls, whch were ot studed suffcetly before. I the preset study, the statc aalyss of a ult-bay coupled shear wall o a elastc foudato wth a fte uber of stffeg beas alog the heght s carred out usg the CCM. The wdths of pers ad opegs are ept costat whle the thcess of the wall, the heghts of the storeys ad the geoetrc ad physcal propertes of the pers ca be vared fro oe secto to aother alog the heght. The eployet of the CCM aes the replaceet of the dscrete coectg beas by a laated edu wth equvalet stffess possble. Ths procedure, tur, reders possble the forulato of the proble as a oe desoal oe stead of two. The goverg equatos of ths approach are the copatblty equatos for the relatve vertcal dsplaceets at the d-pots of the coectg ad the stffeg beas. The syste of lear secod order coupled dfferetal equatos, thus obtaed, s trasfored to a egevalue proble. The, the latter syste of equatos s solved by a atr dagoalzato procedure whch reders the syste ucoupled [0]. To verfy the preset ethod, a coputer progra has bee prepared the MATHEMATICA coputer algebra syste to fd results for coparso wth the results gve the prevous lterature. Although soe results the prevous lterature dd ot atch wth those of the preset wor, the latter were good agreeet wth those of the SAP000 structural aalyss progra [] each ad every case cosdered.. ANAYSIS The prary assupto the CCM s that the legths of the coectg beas do ot chage,.e. ther aal stffess s ftely large. Ths assupto s equvalet to the wdely used rgd daphrag odel for storey floors, whch s ow to yeld rather good results. Ths assupto reders the lateral dsplaceets at the sae floor level equal for all pers. Cosequetly, the slope ad curvature at the sae level ca be assued to be equal, as well. Furtherore, ths ethod, t beg assued that every spa betwee two eghborg pers s costat throughout the total heght of the wall, ad the real coectg beas wth Iraa Joural of Scece & Techology, Volue 8, Nuber B Wter 004

3 Stffeed ult-bay coupled shear walls o 45 bedg stffess E I c are replaced by a laated edu wth E I c, / h per ut legth the vertcal drecto (Fg. ). I the foregog epresso, E, I c, ad h are, respectvely, the elastcty odulus, the oet of erta of the coectg beas bay of secto ad the storey heght secto. ewse, the dscrete shear forces the coectg beas are replaced by a cotuous shear force fucto q,, per ut legth of heght alog the d-pots of the coectg laae (.e. the pots of zero oet). Q -, + dq -, Q + dq q -,, q Q -, Q Fg.. The vertcal forces o a solated rego of the shear wall By applyg the vertcal force equlbru equato to a legth of per of secto, the followg equatos are foud: dq, dq q q,,,,..., +,,,..., () Here, t should be oted that Q Q q q 0,,,...,,,,..., () 0, +, 0, +, Moreover, each fucto Q, s the tegral of the egatve of the shear force fucto zero value at the top of the shear wall,.e. q, startg wth Q q,,,..., +,,,..., () H The oet-curvature relato for a cross-secto of the shear wall at heght ca be wrtte the followg for: d y E I M e Q,,,,,..., (4) Here, M e,, ad I are, respectvely, the oet of eteral forces above the cross-secto of cocer wth respect to ay pot the cross-secto, the dstace betwee the aes of the th ad (+) st pers ad + I, I,,,..., (5) It wll be assued that all rows of coectg laae wll be cut through the d-pots, whch are the pots of zero oet, thus eposg the shear forces the. The copatblty of the relatve vertcal dsplaceets at the eds, o the two sdes of the cut sectos ecesstates ther sus to be equal to zero for each laa,.e. ( Q + Q dy h a C cb +, q h a E I c q ) E A E ( Q + + Q A, ( Q + ) + A +,,...,,,,...,, Q, ( Q + ) + A + Q +,, ) δ 0 Ths copatblty equato ca be wrtte for all spas of the shear wall, havg d the prevously defed values Eq. (). I Eq. (6), a, C, I c,, A, ad δ 0, are, respectvely, ope legth of spa cb 0 (6) Wter 004 Iraa Joural of Scece & Techology, Volue 8, Nuber B

4 46 M. Bce / et al. bea-wall coecto stffess secto, the oet of erta of the coectg beas spa of secto, the cross-sectoal area of the th per secto ad relatve vertcal dsplaceet of the botto of per wth respect to that of per +. The ters of the copatblty Eq. (6) are the relatve vertcal dsplaceets of the two eds o the two sdes of the cut due, respectvely, to the bedg of the pers, relatve rotato of beas wth respect to pers, the bedg of the coectg beas due to shear forces, the aal deforatos of the parts of the pers betwee secto ad the foudato, the aal deforatos of the pers secto, ad the relatve vertcal dsplaceets the foudato. Dfferetatg (6) wth respect to, substtutg epressos () ad (4) ad splfyg the resultg equato, the followg ohoogeeous secod order lear dfferetal equato wth costat coeffcets s obtaed atr for [ Q, ] [ α ] [ Q ] M e γ,,,...,,,,..., (7) where the followg deftos apply: I α + +, ( ) A A+, I α, ( + ), A I α, ( - ) +, A +, α, ( < - ) (8) α, ( > + ) γ h a h a E I + Ccb E I,,,...,,,,...,,,,..., c The set of dfferetal Eqs. (7) s coupled ad oreover, as the uber of rows of opegs creases, a closed for soluto s ot feasble, f at all possble. I ths wor, the atr orthogoalzato ethod wll be used for solvg the set of dfferetal Eqs. (7). For ths purpose, usg the varable trasforato Z Z, Q Z Q Q,,,...,,,,..., (9) the equato set (7) ca be wrtte the followg for for the ew varables Z, : γ Z,, α α... α Z, M e Z Z M e,, α α... α, γ..., , γ , Z, α... Z M e α α,, Z Z M A B e,,..., (0) where A ad B are ad Z ad M e are desoal atrces. The hoogeeous part of ths atr equato, whch s a egevalue proble the followg for Iraa Joural of Scece & Techology, Volue 8, Nuber B Wter 004

5 Stffeed ult-bay coupled shear walls o 47 A Z + B Z 0 () s solved ad the egevectors correspodg to the egevalues all together yeld the trasforato atr T. Sce the coeffcet atrces A ad B are costat, Eq. (0) ca be dagoalzed. For ths purpose, the followg trasforato ca be used Z T Y, Z T Y () whch, whe substtuted Eq. (0), yeld A T Y + B T Y () Here, T ad Y are, respectvely, the trasforato atr of deso ad the vector of depedet fuctos of varable of deso. Multplyg both sdes of Eq. () by the traspose of T, the two coeffcet atrces A ad B are dagoalzed to yeld ~ ~ ~ A Y + B Y (4) whch s a ucoupled syste of dfferetal equatos. The set of epressos yeldg the soluto of Eq. (4) for all sectos s of the followg for: ~ ~ ~,, +, ~ + ( ),,, ~ B B B d Y C cosh ~ D sh ~ 0 ~ M e (5) A,, A B A,,...,,,,..., Ths soluto s oly vald for the polyoal fors of the eteral oet fucto M e. Hece, t apples for the specal cases of uforly dstrbuted forces, learly dstrbuted forces ad a pot force at the top a straghtforward aer. Whe M e s gve by other fuctos, the correspodg partcular solutos ust be foud. There are tegrato costats C, ad D, Eq. (5). To detere those costats, the boudary codtos at the top, botto ad betwee each par of cosecutve sectos are used. Before wrtg dow the boudary codtos, the shear forces the stffeg beas ust be detered. For ths purpose, copatblty Eq. (6) ust be wrtte both for secto at level ad the stffeg bea ad solved sultaeously. Thus, eployg the deftos M e M e h h a + Ccb 6 E I c η,,,...,,,,..., (6) H a + Csb 6 E I s the shear forces the stffeg beas are foud as follows: V dq η H,,...,,,,..., (7) The tegrato costats the equato set (5) are foud fro four dfferet types of boudary codtos: a) The frst type of equato s wrtte for the equlbru of vertcal forces at the top of each row of opegs Q V 0,,,..., (8), H As a specal case, f there s o stffeg bea at the top: Q 0,,,..., (9) H Wter 004 Iraa Joural of Scece & Techology, Volue 8, Nuber B

6 48 M. Bce / et al. Applyg the copatblty equato for the lowerost secto at ts lower ed, tag the rotato of the foudato to be dy ad the relatve vertcal dsplaceets to be the secod type of equato s wrtte as M Q e + 0, r v( + ), (0) Q Q, Q Q +, δ 0 +,,,..., () v ( Q ) h a h a Q M e,, Ccb E I c v, v( ) r 0 + () Q + Q Q 0,,,..., 0, + 0 +, 0 As a specal case, for rgd foudato v v( + ) Q 0 0,,,..., () b) Fro the equlbru of vertcal forces for the stffeg bea, coes the thrd type of boudary codtos as Q V Q 0,,,...,,,,..., (4),, As a specal case, f there s o stffeg bea at heght, V has to be tae as zero. c) The fourth type of boudary codtos are obtaed by equatg the slopes of two cosecutve sectos, - ad, usg the copatblty equatos appled at the boudary betwee the as h a C cb h a + E I c dq h a h a + Ccb E I c,,...,,,,...,, dq As a specal case, for two cosecutve sectos wth equal storey heghts ad oets of erta of coectg beas ad bea-wall coecto stffesses, the correspodg equato of the fourth type splfes to dq dq (6) Substtutg Q,, whch was obtaed usg the foregog boudary codtos ad Eqs. (9), () ad (5), the syste of Eqs. (4) ad tegratg twce wth respect to, M e EI y Q,, + H + G To fd the tegrato costats above, the followg codtos are used: a) The dsplaceets ad slopes betwee two eghborg sectos beg equal 0 (5),,..., (7) y y,,,...,, dy dy,,,..., (8) Iraa Joural of Scece & Techology, Volue 8, Nuber B Wter 004

7 Stffeed ult-bay coupled shear walls o 49 b) The horzotal dsplaceet at the botto beg W y 0 + (9) where W s the total horzotal force o the shear wall ad the su the deoator s the total equvalet horzotal stffess of the foudato. c) The rotato at the botto beg h dy M, r (0) If the foudato s rgd, Eqs. (9) ad (0) splfy to 0 dy y, (). NUMERICA RESUTS To verfy the foregog aalyss ad ts pleetato the MATHEMATICA coputer algebra syste, two eaples have bee solved for coparso purposes. Eaple s tae fro Elholy ad Robso [7] ad ca be see Fg.. Ths eaple has fve rows of opegs ad s solved for varous ds of sols, frst wthout stffeg beas as the lterature, ad the wth stffeg beas at oe fourth ad three fourths of the heght. The horzotal dsplaceets at the top of the wall are preseted Table ad Table, respectvely, for the ustffeed ad the stffeed walls. The results have bee copared wth those of the SAP000 structural aalyss coputer progra []. The heght of the wall s 00 ft, the storey heght s 0 ft, the wall thcess s ft, h, the heght of the coectg beas s ft, the oet of erta of the stffeg beas s 44 ft 4 ad the elastcty odulus s 0 5 p/ft. Fg.. Shear wall wth fve rows of opegs Table. Coparsos of the horzotal dsplaceet at the top of a ustffeed shear wall wth fve rows of opegs Sol propertes y H (ft) Dfferece* (%) v r Preset Ref. [7] Preset study Ref. [] Ref. [7] p/ft p-ft/rad study * Dfferece wth respect to SAP000 results [] Wter 004 Iraa Joural of Scece & Techology, Volue 8, Nuber B

8 50 M. Bce / et al. Eaple s a coupled shear wall o elastc foudato wth three stffeg beas ad four rows of opegs wth varous spas (Fg. 4). Foudato propertes are: Type (Rgd) : r, v, h Type (Stff) : r N-/rad, v N/, h N/ Type (Soft) : r N-/rad, v N/, h N/ Table. Coparsos for the horzotal dsplaceet of the top of a stffeed shear wall wth fve rows of opegs Sol propertes y H (ft) Dfferece * (%) v r p/ft p-ft/rad Preset study Ref. [] Preset study * Dfferece wth respect to SAP000 results [] Fg. 4. Shear wall wth four rows of opegs The results foud by the preset ethod ad SAP000 structural aalyss progra [] are preseted together Table. Table. Soe results foud by the preset ethod ad SAP000 Eaple Sol propertes Type Type Type Preset Preset Preset Ref. [] Ref. [] study study study Ref. [] T (N) T (N) T (N) T 4 (N) T 5 (N) M (N) y H () CONCUSIONS The ethod proposed the preset wor s advatageous two ways. The frst advatage stes fro the fact that the data preparato s easer tha the other ethods whch deal wth oe desoal ebers. Iraa Joural of Scece & Techology, Volue 8, Nuber B Wter 004

9 Stffeed ult-bay coupled shear walls o 5 Moreover, tryg dfferet possbltes, or chages the data suffces for the purpose. The secod advatage les the fact that the coputer te eeded to solve a proble wth the preset ethod s roughly oe ffth of that eeded to solve t by the fte dfferece approach of Elholy ad Robso [7]. Hece, the preset ethod proves to be a effectve procedure for predesg purposes. I decdg o the propertes of the shear wall ad the postos of the stffeg beas, to be able to try a host of possbltes a short aout of te ad selectg the ost sutable syste, the preset ethod sees to wor ust fe. However, the fal soluto should preferably be carred out usg a ore precse ethod. NOMENCATURE a ope spa of th bay A cross-sectoal area of the th per secto b wdth of the th per C cb coectg bea-per coecto stffess secto C sb stffeg bea-per coecto stffess secto E elastcty odulus h storey heght secto H total heght of the wall I s oet of erta of the stffeer at heght I c oet of erta of the coectg beas the th bay of secto I oet of erta of the th per secto I su of the oets of erta of the pers secto ubers of sectos ad borderles betwee cosecutve sectos ubers of bays ad pers v, h, r equvalet vertcal, horzotal ad rotatoal stffess coeffcets of the foudato of the th per dstace betwee the aes of the th ad (+) st pers secto, M bedg oet the th per secto M e () oet of the load above w.r.t. heght total uber of bays total uber of sectos vertcal drecto q shear flow fucto the th bay secto Q tegral of the shear flow fucto the th bay, startg fro the top of the shear wall secto T trasforato atr V shear force the th bay of the stffeer at heght heght of the upperost pot of secto y horzotal dsplaceet fucto secto w () horzotal load fucto δ 0 relatve vertcal dsplaceet of the bases of the th ad (+) st pers Wter 004 REFERENCES. Rosa, R. (964). Approate aalyss of shear walls subect to lateral loads. J. A. Cocr. Ist., 6(6), 77.. Coull, A. & Pur, R. D. (968). Aalyss of coupled shear walls of varable cross-secto. Bldg. Sc.,,.. Coull, A. & Subed, N.. (97). Coupled shear walls wth two ad three bads of opegs. Bldg. Sc., 7, oo,.. & Cheug, Y.. (984). The statc aalyss of ult-bay coupled shear walls. Buldg ad Evroet, 9(), Coull, A., Pur, R. D. & Totteha, H. (97). Nuercal elastc aalyss of coupled shear walls. Proc. Ist. Cv. Egrs., 60, Soae, A. J. M. (967). The aalyss of tercoected shear walls by aalogue coputato. I Tall Buldgs, Elholy, N. S. & Robso, N. (97). Aalyss of ult-bay coupled shear walls. Bldg. Sc., 8, Coull, A. & Pur, R. D. (967). Aalyss of coupled shear walls of varable thcess. Bldg. Sc.,, Coull, A. (974). Stffeg of coupled shear walls agast foudato oveet. Struct. Egr., 5(),. 0. Cha, H. C. & uag, J. J. (988). Effect of a sgle deep bea o tw shear walls wth ratoal couplg. Proc. Ist. Cv. Egrs.,, 50. Iraa Joural of Scece & Techology, Volue 8, Nuber B

10 5 M. Bce / et al.. Cha, H. C. & uag, J. J. (989). Stffeed coupled shear walls. J. Struct. Egr., 5(4), Coull, A. & Besal,. (99). Stffeed coupled shear walls. J. Struct. Egr., 7(8), 05.. Asoga, O., Turer, H. T. & Osoue, A. V. (99). Stffeg of coupled shear walls at arbtrary uber of heghts. Advaces Cvl Egeerg, Frst Techcal Cogress, North Cyprus,, Choo, B. S. &, G. Q. (997). Structural aalyss of ult-stffeed coupled shear walls o fleble foudatos. Coputers ad Structures, 64(4), Asoga, O., Arsla, H. M. & Aavcı, S. S., (00). Stffeed coupled shear walls o elastc foudato wth fleble coectos ad stepwse chages wdth. Iraa J. Scece ad Techology 7(), 7. 6., G. Q. & Choo, B. S. (995). Natural frequecy evaluato of coupled shear walls. The Structural Egeer, 7(8), Arsla, H. M. & Asoga, O. (998). Dyac aalyss of stffeed coupled shear walls by cotuous coecto ethod. C. U. J. Fac. Eg. Arch., (),. 8. uag, J. S. & Chau, C.. (999). Dyac behavor of stffeed coupled shear walls wth fleble bases. Coputers ad Structures, 7, Asoga, O., Arsla, H. M. & Choo, B. S. (00). Forced vbrato aalyss of stffeed coupled shear walls usg cotuous coecto ethod. Egeerg Structures, 5, Merovtch,. (980). Coputatoal ethods structural dyacs. Netherlads, Sthoff ad Norrdhoff.. Wlso, E.. (997). SAP000 Itegrated fte eleet aalyss ad desg of structures. -, Calfora, Coputers ad Structures, Ic. Iraa Joural of Scece & Techology, Volue 8, Nuber B Wter 004