On the general evaluation of the maximum allowable drift at the top of shear walls (constant and variable stiffness)

Transcription

1 Internatona Journa of Cv Engneerng and Constructon Scence 4; (): 8-5 Pubshed onne June, 4 ( On the genera evauaton of the maxmum aowabe drft at the top of shear was (constant and varabe stffness) Mche Fard Khour, Wassm Joseph Eas Kewords Renforced Concrete Structures, Shear Was, Maxmum owabe Drft, Constant and arabe Stffness, Maxmum Stran mts, atera and ertca Contrbuton Receved: pr 4, 4 Revsed: Ma, 4 ccepted: Ma 4, 4 Department of Cv Engneerng, Branch II, ebanese Unverst, Roumeh, ebanon Facut of Engneerng, Branch II, Research Engneer at Optma Engneerng Consutng & Contractng, ebanese Unverst Ema address mhur@u.edu.b (M. F. Khour), wassm.eas@optma-eng.com (W. J. Eas) Ctaton Mche Fard Khour, Wassm Joseph Eas. On the Genera Evauaton of the Maxmum owabe Drft at the Top of Shear Was (Constant and arabe Stffness). Internatona Journa of Cv Engneerng and Constructon Scence. o., o., 4, pp bstract The atest research on determnng maxmum aowabe dspacements at the top of a shear wa (or maxmum Drft) has contrbuted consderab to the mprovement of concrete structures. mng at mtng maxmum dspacement, varous nvestgators and sesmc codes use vaues rangng from H/5 to H/, (Hbudng heght). Dfference shows consderabe uncertant n these mts. Ths artce evauates maxmum aowabe drft and provdes genera formuae for constant and varabe stffness shear was b usng FEM aong wth structura dnamcs and renforced concrete desgn consderng craced shear wa sectons suggested b UBC and CI. comparson s made wth varous codes.. Introducton Stee structures are aowed to drft more than renforced concrete structures due to the fact that stee can accommodate tenson as we as compresson, whe concrete tense strength s genera ess than % of ts compressve strength. Under an tense movement concrete w crac when subected to wnd and sesmc reversas. Man nvestgators and sesmc codes [-9] suggest vaues for maxmum aowabe stor drft or maxmum aowabe atera dspacement but these vaues dffer sgnfcant. Suggested maxmum drft at the top of budngs var between H/5 and H/ where H s the heght of the budng. In addton to the fact that ths dfference s arge, the ueston s how the actua structura behavor can be used to determne maxmum aowabe drft? Some codes such as UBC secton 6.. consder that budngs are aowed to drft H/5 as ong as top foors occupants sta confortabe wthout feeng the swng; the aso base ther estmate on the nonnear behavor of the structure and materas. Other codes such as the ebanese code reduce the drft vaues to H/ because and ots are scarce and expensve conseuent, the code aows engneers and deveopers n certan areas to construct wthout setbacs. Ths reures two adacent budngs to be ver stff n order to emnate/reduce hammerng between structures when subected to sesmc or wnd oads. One of the frst attempts to evauate the maxmum aowabe drft based on rea shear wa behavor was done b the author [], where the maxmum aowabe drft

2 9 Mche Fard Khour and Wassm Joseph Eas: On the Genera Evauaton of the Maxmum owabe Drft at the Top of Shear Was (Constant and arabe Stffness) was determned b assumng a shear budng and mang use of the fnte eement anass aong wth the structura dnamcs and renforced concrete desgn. In that nta stud, a constant stffness was assumed from bottom to top of the shear wa. In a ater stud, drft mtatons n a shear wa was done consderng a craced secton [] as suggested b UBC and CI where shear wa nerta was reduced. nother stud was done to evauate the drft wth varabe stffness where the stffness n a shear wa can var from bottom to top; n addton, the effect of craced secton reduced nerta was consdered []. In ths artce, the prevous wor done s combned and the effect of the vertca oad contrbuton s added to present compete formuae that can be used to determne the maxmum aowabe drft for both constant and varabe stffness shear was and for craced as we as un-craced renforced concrete sectons.. Shear Budng Formuaton The obectve s to generate the stffness matrx for a shear budng and afterwards add the contrbuton of the vertca oad. s presented b references [-], and after appng boundar condtons, the stffness matrx of an eement on the shear wa s: From Fgure, and tang nto consderaton the boundar condtons presented n reference [], the structure stffness matrx can be assembed reatve to degrees of freedom 4, 7,. Usng constant and/or varabe stffness shear was where the atera stffness s constant or can var from the bottom to the top eve as desred, the setup of the stffness matrx for constant stffness shear wa where K () K () K () K and for a varabe stffness shear was where K () K () K () are presented n the foowng secton.. Base Shear, Dspacement and Reatve Dspacement To fnd a reaton between the tota base shear, stffness and maxmum dspacement for stores structure, a tranguar dstrbuton of s assumed. The base shear can be found b an procedure or from an sesmc code. Ths dstrbuton of gves a formua of the apped force F at ever eve of the structure as a functon of, F, where ( ) F. > - F can be wrtten as: ( ) EI [ K] Where E s the concrete moduus of eastct, I s the wa nerta and s the wa ength. () 4 7 ( ). Fgure. Representaton of an -stor budng. ( ) [ x x x... x... x ], but ( )( ) 6,

3 Internatona Journa of Cv Engneerng and Constructon Scence 4; (): 8-5 > () et the vector represent the dspacement n the partcuar case where the stffness of a stores s the same as computed and presented n references [-]. The obectve now s to determne the dspacement for varabe stffness shear was. The stffness matrx [] can be wrtten n terms of the atera stffness of each stor n the foowng manner: Where, () s the stffness of one stor, n I E. The stffness matrx for a constant stffness shear wa n the case where () () ()... (n), s as demonstrated n a reference []: B B t ths tme matrx C can be found such as: C x B > C x B -, > C The shear force F at an eve s computed as: F x, and C x B > F C x (B x ). Therefore, once the vaues of the dspacements are nown, the shear forces can be cacuated. ow, fnd matrx D as foows: B x D > D B - x, and D C T. D

4 Mche Fard Khour and Wassm Joseph Eas: On the Genera Evauaton of the Maxmum owabe Drft at the Top of Shear Was (Constant and arabe Stffness) ; - ; ; ; - ; ; ; ; ; D Where K (). K () K (). F x > - x F, but - D- x B- > D - x [B - x F], and [B - F] and snce s eua to: 7 4 > [ ] ; () The vaue of the reatve dspacement - s gven b:

5 Internatona Journa of Cv Engneerng and Constructon Scence 4; (): 8-5 ; whch can be reduced to: X X (4) 4. Computng Maxmum Stran Smar to what was presented n references [-] and usng references [] and [4], t was demonstrated that the stran n the x drecton aong a shear wa between the two eves (-) and s: - 6 x, and addng the effect of the vertca oad, the stran euaton becomes: - }.E P 6 x { (5) where s the agebrac dstance measured from the neutra axs to the extreme fber of the shear wa secton; s consdered to be postve n the opposte drecton of the defecton. The maxmum vaue of s obtaned when maxmum vaue of s repaced, x s the abscssa of the secton aong the shear wa between the two eves (-) and, and x s the stran n x drecton. ote that euaton (4) ncudes the effect of the vertca oad above the eve consdered where s the stor heght and P s the axa oad at eve apped at the center of the shear wa wth an area of and moduus of eastct E. B substtutng the vaues of ( - ) from e. () nto e. (4), the stran formua becomes: () X X }.E P 6 x { (6) The maxmum vaue of the stran between eves (-) and s found as foows: [ ] X X X () < The functon s decreasng whch means that the maxmum stran n a shear wa, between eves (-) and, occurs at the bottom of the shear wa (x, and ), where P s the axa oad at eve apped at the center of the shear wa wth an area of and ths maxmum stran s eua to:.e P 6 () x (7) 5. Maxmum Dspacement The maxmum drft at the top of the shear wa s reached when the stran of the renforcement n the tense zone at the crtca secton of the shear wa s eua to st (maxmum aowabe stran n stee), and the stran n the extreme fber of the compresson zone n the same secton s eua to c maxmum stran mt of concrete n compresson.. So the crtca secton n the shear wa s consdered to have the behavor descrbed n Fgure.[5] Fgure. Baanced Renforced Concrete Secton. From e. () we have the maxmum dspacement at the top of the shear wa s gven as foows: [ ] ; et

6 Mche Fard Khour and Wassm Joseph Eas: On the Genera Evauaton of the Maxmum owabe Drft at the Top of Shear Was (Constant and arabe Stffness) R [ ] ; (8) But R, and repacng n euaton 6 > 6 P ( x ) () (9) R.E number of shear was wth sma d than to use fewer shear was wth arge d; eepng n mnd that the nerta of a shear wa s ncreased cubca as a functon of d, and a bgger d w ncrease the stffness sgnfcant. From smar tranges of the baanced secton of Fgure, the foowng can be wrtten: x st d () st c st The maxmum stran n the shear wa presented n e. (9) at the eve of stee shoud be smaer than (the ed stran of stee): 6 P ( x ) () st () R.E Repace from e. () > R () ( st 6. c st )(.d st c P ).E () If the stran n the stee s consdered to sta wthn (the edng stran of stee), the maxmum aowabe dspacement at the top of a varabe stffness shear wa obes the foowng euaton: v R () ( )( 6. c.d P ).E () For constant stffness shear was from bottom to top, e. () becomes: P ( )( c)( ).E c 8..( d) (4) In ths case, the crtca secton of the shear wa behaves as a baanced secton; the mts are reached n the renforcement n tenson and n the concrete n compresson at the same tme, and at that pont, the maxmum aowabe dspacement at the top of the shear wa s reached. otce that, n both euatons () and (4), as the heght of a stor ncreases, the maxmum aowabe dspacement ncreases, and as d ncreases the maxmum aowabe dspacement decreases snce an sma movement tends to cause arger stran at the crtca secton of the shear wa. On the other hand, and as far as maxmum dspacement s concerned and dsregardng economca and archtectura ssues, t s better to use arger 6. Maxmum Drft Consderng a Craced Secton ccordng to UBC, Modeng Reurement Secton 6.., Stffness Propertes of Renforced Concrete and masonr eements sha consder the effects of craced sectons.[] Ths means that when a desgner uses craced secton anass then accordng to CI 8, secton..4., the shear wa moment of Inerta woud have to be reduced b a factor.5 such that the nerta woud become.5ig.[] Ths w conseuent reduce the stffness of the atera oad resstng eements and w eua ncrease the drft b the same factor, whch means that the drft shoud be mutped b a factor of (/.5) to get a vaue tang nto consderaton the effects of a craced secton suggested b UBC-97 and CI-8-8. Euaton for a craced secton of varabe stffness shear was becomes: vm v (5).5 and euaton for constant stffness shear was becomes: cm c (6).5 The formuae of ens. (5) and (6) suggested b the authors gves the maxmum aowabe drft as a functon of the heght of one stor, the number of stores, the effectve depth of the shear was used, and the eastc propertes of the materas used n the shear wa (stee and concrete). 7. Exampes In order to see how these formuas wor, consder an exampe of a stor budng wth shear was havng varabe atera stffness as we go up the budng as foows: - from eve to 5 the stffness s, - from eve 6 to the stffness s.9, - from eve to 5 the stffness s.8, - and from eve 6 to the stffness s.7. Yed Strength, f 44 Mpa, Moduus of Eastct, E s.5 Mpa, Renforcement Stee Yed Stran,.7, Stor Heght, m; Concrete Crushng Stran c.. For the sae of smpct of cacuatons, oad P s taen such that the nfuence area around the shear wa s (h)

7 Internatona Journa of Cv Engneerng and Constructon Scence 4; (): wth a sab thcness of.5 m, but n rea evauaton t can be cacuated exact, where h s the shear wa ength. Tabe presents the vaues of the maxmum aowabe dspacement for constant and varabe stffness shear was wth ength h m and effectve depth d.9h.7m, and for h4 wth d.6m. The shear was have a constant wdth b.m. Stor Tabe. owabe drft for Exampe usng Ens (4)&(5). Heg ht (m) Dm (Constant Stffness) Dm (arabe Stffness) Dm (Constant Stffness) Dm (arabe Stffness) h, d.7 m h4, d.6 m Resuts show that the effect of addng the contrbuton of the axa oad P on the drft mtaton vaue for a stor budng s around an addton of % ncrease n the stor drft dependng whether the structure s wth constant or varabe stffness. Ths percentage ncreases progressve wth ncreasng the number of stores due to the ncrease n the axa oad P. 7.. Comparson of Resuts wth arous Codes On performng a comparson for the maxmum aowabe atera dspacement between en.(6) and seected sesmc codes as presented n Tabe beow, t can be observed that resuts of ths euaton e between the formua suggested b Mar Fnte [6] and the French codes PS9 [], whe UBC97 [] gves a arger vaue. The mportant pont that needs to be made s that the suggested ens. (5) and (6) consders the geometr of the shear wa and the effectve depth of the renforcng stee. The aso consder the heght of a stor, whe the code formuas consder on the heght of the budng dsregardng an structura and matera propertes. Stor Heght (m) Stffness) Khour- Eas Tabe. owabe drft comparson between En. (5) and varous codes. Stffness) ebanese Code H/ Stffness) M. Fnte H/5 Stffness) PS9 H/5 Stffness) UBC97- IBC6 H/5 h, d.7 m Concuson In ths stud, shear budng was anazed usng the fnte eement method for both constant and varabe stffness shear was and the contrbuton of the vertca oad to the stran was consdered. The shear was obtaned as a functon of the dspacement. vaue for the dspacement at an stor was obtaned, and from whch a functon for the reatve dspacement between two stores was then

8 5 Mche Fard Khour and Wassm Joseph Eas: On the Genera Evauaton of the Maxmum owabe Drft at the Top of Shear Was (Constant and arabe Stffness) determned. Usng the above, an euaton for the maxmum stran was resoved. mtng vaue for the maxmum dspacement wthn the eastc mts was obtaned as a functon of the heght of a stor, the stffness of a stor, number of stores, effectve depth d of a shear wa, the ed stran of stee and the maxmum aowabe concrete stran c. On the other hand, the vaue h/5 suggested b UBC97 [] and IBC 6 [] generates arge strans at the bottom of a shear wa; t s mportant to note that even though UBC and other codes consder the non-near neastc behavor of the structure and hgh drft vaues correspond to a fexbe structure thereb ower atera forces, such arge dspacements mts ma be dangerous when the structure depends on shear was for atera stffness. Wth ths n mnd, t s pertnent to menton that UBC provdes a restrcton on neastc nter-stor drft whch w be evauated n a future wor. It s now eft for the desgnng engneer to evauate hs structure and decde/choose a maxmum aowabe stran mt for concrete and stee, and determne the correspondng maxmum aowabe dspacement vaues. Fna, the formuae suggested b the authors can serve as a startng pont after whch the desgnng engneer woud now that the shear wa n ueston has passed the eastc mt. cnowedgements The author thans Optma Engneerng Consutng and Contractng, (OECC) for sponsorng the maor part of ths wor and speca thans to the ebanese Unverst, Facut of Engneerng, Branch II for sponsorng part of ths on-gong research. References [] Unform Budng Code, b Internatona Conference of Budng Offcas, Whtter, Caforna, 997. [] Internatona Budng Code, IBC-6 Edton, Pubshed b the Internatona Code Counc, IC., 6. [] Rège de Constructon Parassmue, Rège PS appcabes aux bâtments PS 9 ormes F P 6-, 99. [4] Standards for Sesmc Cv Engneerng Constructon n Japan, Earthuae Resstant Reguatons for Budng Structures n Japan, Too, Japan, 98. [5] ebanese Code, Genera Sesmc Desgn Gudenes and Reguatons, Berut, ebanon, 997. [6] Mar Fnte, Handboo of Concrete Engneerng, Pubshed b an ostrand Renhod Compan, nd edton, ew Yor, 985. [7] ew Zeand Standard ZS 4, Genera Structura Desgn and Desgn oadng for Budngs, Daft DZ 4, Standards ssocaton of ew Zeaand, Wengton, ew Zeaand, 99. [8] Règes Parassmues gérennes, RP-88, Reguaton of geran Sesmc Code, Pubcaton OPU, gers, gera, 988. [9] Gar Searer, Poor Worded, I-Conceved, and Unnecessar Code Provsons, 6 nnua Meetng of the os ngees Ta Budng Structura Desgn Counc, pp. 7-85, os ngees, 6. [] Khour, M. F., Drft mtatons n a Shear Wa Consderng a Craced Secton, Internatona Journa of Reabt and Safet of Engneerng Sstems and Structures, (), pp. 8. [] Khour, M.F., Evauatng the maxmum aowabe drft n a Shear Wa wth arabe Stffness, Romanan Journa of coustcs and bratons, oume II, Issue I, pp. -9, Romana,. [] Khour, M. F. Estmaton of the Maxmum owabe Drft at the Top of a Shear Wa (wthn Eastc mts), Presented and Pubshed n Earthuae Resstant Engneerng Structures Internatona Conference, (ERES), pp.5-6, Cprus 9. [] Bathe K.J., Fnte Eement Procedures n Engneerng nass, Pentce Ha, Engewood Cffs, ew Jerse, 98. [4] Crag, Ro, Structura Dnamcs- n Introducton to Computer Methods, John We and Sons, 98. [5] rthur son and George Wnter, Desgn of Concrete Structures, Mc-Graw H Internatona, th Edton, ew Yor, 99. [6] Mar Fnte, Handboo of Concrete Engneerng, an ostrand Renhod Compan, ew Yor, 985.