INVESTIGATION ON APPLICABILITY OF SUBSTITUTE BEAM - COLUMN FRAME FOR DESIGN OF REINFORCED CONCRETE SWAY FRAMES

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1 INVESTIGATION ON APPLICABILITY OF SUBSTITUTE BEAM - COLUMN FRAME FOR DESIGN OF REINFORCED CONCRETE SWAY FRAMES Abrham Ewnti and *Girma Zrayohanns School of Civil and Environmntal Enginring, Addis Ababa Institut of Tchnology *Corrsponding Author s zrayohanns@yahoo.com ABSTRACT Th papr dals with th valuation of th sway fram momnt magnification provision for th dsign of slndr rinforcd concrt columns in sway frams according to EBCS 2: A spcial fatur of th momnt magnification mthod in EBCS 2: 1995 is th introduction of th concpt of substitut fram for th dtrmination of th story buckling load. Th valuation is carrid out by comparing th magnifid column momnt with th corrsponding valus obtaind from th mor rigorous scond ordr itrativ P-Δ scond ordr analysis. Th magnifid column momnts ar also compard with th corrsponding valus dtrmind using th ACI s sway momnt magnification provision. Th rsults of th valuation show that th sway momnt magnification mthod according to EBCS 2: 1995 yilds dsign momnts clos to th itrativ P-Δ solutions. Howvr, th rsults li on th unsaf sid and th prcntag dviation is found to b th highst for irrgular frams. Ky Words: Sway frams, slndr columns, substitut bam-column fram, ETABS, critical load, sway momnt magnification, scondordr analysis. INTRODUCTION Sinc th Ethiopian Building Cod Standard, EBCS-2: 1995 [4] is basd on Eurocod 2: ENV 1992 [5]; th two Cods ar vry similar with only fw xcptions such as provisions for th dsign of columns in sway frams. ENV 1992 [5] givs dtaild simplifid dsign provisions for slndr rinforcd concrt columns that may b considrd as isolatd columns. Ths ar individual columns with articulation in non-sway structurs, slndr bracing lmnts, and columns with rstraind nds in a non-sway structur. Corrsponding provisions for th dsign of columns in sway frams ar not providd by ENV 1992 [5]. According to ENV 1992 [5], such columns ar to b dsignd using th mor rigorous approach basd on th rsults of a scond ordr global analysis. Th EBCS-2: 1995 [4] sms to b mor complt in this rspct, bcaus it givs additional simplifid procdurs for th dsign of columns in sway frams. A closr look into th provisions rvals that thy ar basd on th corrsponding procdurs according to th Amrican Concrt Institut, ACI [1]. Th provisions in ACI and EBCS, howvr, hav significant diffrncs in th procdurs such as th concpt of th substitut fram adoptd by EBCS-2 [4] for column stiffnss computation. Thrfor th dsign of slndr rinforcd concrt columns in sway frams has long bn a controvrsial subjct among practicing structural nginrs with lack of consnsus with rgard to its suitability as a dsign tool or vn th validity of th rsults [10]. It is thus vry important to mak a dtaild invstigation on th validity of th rsults obtaind from th provision in EBCS-2 [4] by comparing thm with th corrsponding rsults using th mor rigorous scond ordr itrativ P-Δ analysis and ACI s sway momnt magnification mthod. DESIGN PROVISIONS Slndr Columns Dsign in Sway Frams According to ACI and EBCS: 1995 Cods Th Ethiopian Building Cod Standard, EBCS 2: 1995 sms to hav similar provisions for dsign of slndr columns in sway frams with th Amrican Concrt Institut (ACI). Howvr thy hav som clar diffrncs. On of ths is th introduction of th substitut bam-column fram in th EBCS 2: 1995 for th dtrmination of th ffctiv column stiffnss in sway frams to calculat th critical buckling loads. Th following is a summary of th stps followd in th momnt magnification 13 Journal of EEA, Vol. 33, Dcmbr 2015

2 Abrham Ewntu and Girma Z/Yohanns procdur for th dsign of slndr columns in sway frams basd on th EBCS 2: 1995 provision. It may b obsrvd that similar stps ar followd for th dsign of slndr columns in sway frams basd on th ACI provision. Momnt Magnification Procdur for Sway Frams According to EBCS Stp 1: Chck for Story Sway According to Sction of EBCS-2, 1995, a story in a givn fram may b classifid as non-sway story if: N N Sd cr 0.1 (1) Bam-and-column typ plan frams in building structurs with bams conncting ach column at ach story lvl may b classifid as non-sway story if: N 0.1 HL Whr, in both quations, (2) N Sd, N = total factord axial load in th story, N cr = story buckling load, H = total horizontal raction (shar) at th bottom of th story, = first-ordr rlativ dflction btwn th top and bottom of that story du to th dsign loads (vrtical and horizontal),plus th initial sway imprfction, L = story hight. Th displacmnt shall b dtrmind basd on stiffnss valus for bams and columns appropriat to Ultimat Limit Stat. Stp 2: Chck for Slndrnss (i) Gnrally, th slndrnss ratio of concrt columns should not xcd 140. (ii) According to sction of EBCS-2, scond ordr ffcts for columns in sway frams nd not b takn into account if: Max (25, 15/ d ) (3a) Whr d =N Sd / (f cd A c ), (3b) f cd = dsign comprssiv strngth of concrt, A c = gross cross-sctional ara of th columns Stp 3: Effctiv Buckling Lngth Factors Th ffctiv buckling lngth factors of columns in a sway fram shall b computd by using approximat quations givn in EBCS-2 Sction basd on EI valus for gross concrt sctions providd that th α valus do not xcd 10. For highr valus of or 1 2 mor accurat mthods must b usd. L 7.5 4( 1 2) k 1.15 L 7.5 Or consrvativly, 1 2 (4) L k m 1.15 (5) L Whr, for columns bing dsignd and bams and columns just abov thm, ( EcIc / lc ) 1 (6) ( E I / l ) b b For columns bing dsignd and bams and columns just blow thm b ( EcIc / lc ) 2 (7) ( E I / l ) b b b m (8) Stp 4: Magnifid Momnts Th magnifid sway momnts, s M s, ar computd using th amplifid sway momnts mthod givn in EBCS-2 Sction Th total dsign momnts M 1 and M 2 at th nds of th columns shall thn b obtaind by adding th unmagnifid non sway momnts, M ns, found by a first ordr analysis using mmbr stiffnss in EBCS-2, Sction 3.7.6, and th magnifid sway momnts δ s M s. M 1 = M 1ns + δ s M 1s M 2 = M 2ns + δ s M 2s (9a) (9b) Th sway momnt magnification factor δ s shall b computd from Journal of EEA, Vol. 33, Dcmbr

3 Invstigation on Applicability of Substitut Bam-Column Fram 1 s (10) 1 N Sd N cr Whr N Sd is th dsign valu of th total vrtical load N cr is its critical valu for failur in a sway mod Th amplifid sway momnts mthod shall not b usd whn th critical load ratio N Sd /N cr, is mor than Stp 5: Story Buckling, N cr Th approach usd in this stp is diffrnt from that of th ACI bcaus of th introduction of th concpt of substitut bam-column fram mthod for th dtrmination of th critical buckling load. Th approach allows a mor accurat appraisal of th bnding stiffnss of th columns bcaus th momnts of inrtia ar dtrmind including th contribution of stl dsignd using th substitut columns. Th substitut bam-column fram is a proppd half portal mad of substitut columns and bams as shown in Fig According to Sction (1) of EBCS-2, th buckling load of a story may b assumd to b qual to that of th substitut bam-column fram. EBCS-2 Sction (4) stats that th quivalnt rinforcmnt aras, A s,tot, in th substitut column ar obtaind by dsigning th column at ach floor lvl to carry th story dsign axial load and magnifid sway momnt at th critical sction. N cr EI (11a) 2 2 L Whr, th ffctiv stiffnss of a column EI shall b takn from Sction (1), EI Or altrnativly, Whr: EI 0.2Ec I c Es I 1 d d M bal /(1/ r 1 bal ) s (11b) (11c) Ec 1100 f cd (11d) E s = modulus of lasticity of th stl, I c =gross momnt of inrtia of th concrt sction about its cntroidal axis, I s = momnt of inrtia of th rinforcmnt about th cntroidal axis of th concrt sction, M bal = balancd momnt capacity of th column, (1/r bal ) = curvatur at th balancd load and may b takn as: 1 r bal = 5 d 10 3 (12) Th trm (1 + d ) in both quations rflcts th ffct of crp on th column dflctions as statd in Sction (4)). (a) Actual fram (b) Substitut fram Fig. 1: Substitut Multi-story Bam-Column Fram Stp 6: Location Chck for th Maximum Column Momnts EBCS-2 Sction (2) also rquirs chcking whthr momnt at som point btwn th nds of th column xcds that at th nd of th column but dos not giv any xplicit quation as in th ACI. Th chck is don by comparing th magnifid momnts for nonsway columns that ar dtrmind using th dsign procdur in EBCS Sction and with thos of th magnifid column nd momnts. Stp 7: Stability Chck undr Gravity s Only EBCS-2 sction (1) stats that all frams shall hav adquat rsistanc to failur in a sway mod, but it dos not plac any xplicit limit on s or th critical load ratio as in th ACI. E c = modulus of lasticity of th concrt, 15 Journal of EEA, Vol. 33, Dcmbr 2015

4 Abrham Ewntu and Girma Z/Yohanns RESULTS AND DISCUSSIONS Four diffrnt typs of frams hav bn analyzd according to th ACI and EBCS sway momnt magnification provisions. Th rsults obtaind hav bn compard with itrativ analysis rsults for th corrsponding load combinations. Th analysis outputs of ach fram hav bn summarizd and discussd in th following sctions. A) Fiv-Story Rgular Building Th rsults obtaind basd on th ACI and EBCS sway momnt magnification provisions as wll as th itrativ analysis ar summarizd in Tabl 1 blow. Th comparison of th rsults is shown in th tabl as a prcnt dviation. Fig. 2 also shows th rsults in graphical form. a) Plan b) Sction Fig. 2: Fiv-Story Building Dtail Tabl 1: Comparison of sway momnt magnification and itrativ analysis outputs ACI cas 1 cas 2 δ s Dsign Action Effcts Extrior Columns Itrativ s ETABS s Intrior Columns Itrativ s ETAB S s P (kn) M (kn-m) P (kn) M (kn-m) Journal of EEA, Vol. 33, Dcmbr

5 Dsign Momnt, knm Dsign Momnt, knm Invstigation on Applicability of Substitut Bam-Column Fram Tabl 1: Cont ACI cas 1 cas 2 δ s Dsign Action Effcts Extrior Columns Itrativ s ETABS s Intrior Columns Itrativ s ETAB S s P (kn) M (kn-m) P (kn) M (kn-m) Whr: δ s = Sway momnt magnification factor = Rsults of th Sway momnt magnifir mthod provisions, Itrativ = Rsults of itrativ analysis mthod (calculatd manually) Etabs = Rsults of Etabs softwar itrativ analysis cas 1 = gravity and wind loads = 1.05D L 1.3W, (according to ACI) 1.20D L 1.3W, (according to EBCS) cas 2 = gravity & arthquak loads = 1.05D L 1.0E, ( according to ACI) 0.975D L 1.0E, (according to EBCS) EBCS LOAD CASE 1 - EXTERIOR LOAD CASE 2 - EXTERIOR Fig. 3: Comparison of ACI and EBCS Rsults with Itrativ P-Δ Analysis Rsults 17 Journal of EEA, Vol. 33, Dcmbr 2015

6 Dsign Momnt, knm Dsign Momnt, knm Abrham Ewntu and Girma Z/Yohanns EBCS LOAD CASE 1 - INTERIOR LOAD CASE 2 - INTERIOR Fig. 3: Cont From Tabl 1 and Fig. 3 on can s that: Th sway momnt magnification mthod provision of th EBCS givs a closr rsult to th itrativ P-Δ analysis rsults than th ACI provisions; numrically: o For load cas 1: vs dviation for xtrior columns, and vs for intrior columns. o For load cas 2: vs dviation for xtrior columns, and vs for intrior columns For load cas 2, howvr, th rsults of th EBCS provision ar smallr than th itrativ P-Δ analysis rsults; unsaf. B) Nin- Story Rgular Building Fram a) Plan b) Sction Fig. 4: Nin-Story Building Dtail Tabl 2: Comparison of sway momnt magnification and itrativ analysis outputs Journal of EEA, Vol. 33, Dcmbr

7 Dsign Momnt, knm Dsign Momnt, knm Invstigation on Applicability of Substitut Bam-Column Fram ACI cas 1 cas 2 EBCS cas 1 cas 2 δ s Dsign Action Effcts Extrior Columns Itrativ ETAB S Intrior Columns Itrativ ETAB S P (kn) M (kn-m) P (kn) M (kn-m) P (kn) M (kn-m) P (kn) M (kn-m) Whr: cas 1= 1.05D L 1.3W, (according to ACI 1.20D L 1.3W, (according to EBCS) cas 2 = 1.05D L 1.0E, (according to ACI) 0.975D L 1.0E, (according to EBCS) LOAD CASE 1 - EXTERIOR LOAD CASE 1 - INTERIOR Fig. 5: Comparison of ACI and EBCS provision rsults with itrativ P-Δ analysis rsults 19 Journal of EEA, Vol. 33, Dcmbr 2015

8 Dsign Momnt, knm Dsign Momnt, knm Abrham Ewntu and Girma Z/Yohanns LOAD CASE 2 - EXTERIOR LOAD CASE 2 - INTERIOR Fig. 5: Cont From Tabl 2 and Fig. 4 on can s that: Th sway momnt magnification mthod provisions of th EBCS giv a closr rsult to th itrativ P-Δ analysis rsults than th ACI provisions; numrically: o For load cas 1: vs dviation for xtrior columns, and vs for intrior columns. o For load cas 2: vs dviation for xtrior columns, and vs for intrior columns. In all cass abov, howvr, th rsults of th EBCS provision ar smallr than th itrativ P-Δ analysis rsults; unsaf. C) Fiv-Story Building with Plan Irrgularity From Tabl 3 and Fig. 6 on can s that: Th sway momnt magnification provision of th ACI givs a closr rsult to th itrativ P-Δ analysis rsults than th EBCS provisions. Numrically, vs dviation for xtrior columns, and vs for intrior columns. Th rsults of th EBCS provision ar smallr than th itrativ P-Δ analysis rsults; unsaf. Fig. 6: Plan and Sction of a Fiv-Story Building with Plan Irrgularity Journal of EEA, Vol. 33, Dcmbr

9 Dsign Momnt, knm Dsign Momnt, knm Invstigation on Applicability of Substitut Bam-Column Fram Tabl 3: Comparison of sway momnt magnification and itrativ analysis outputs ACI cas 1 EBCS cas 1 δ s Dsign Action Effcts Extrior Columns Itrativ ETABS Chan g Intrior Columns Itrativ ETABS P (kn) M (kn-m) P (kn) M (kn-m) Whr: cas 1 = 1.05D L 1.0E, according to ACI 0.975D L 1.0E, according to EBCS LOAD CASE 1 - EXTERIOR LOAD CASE 1 - INTERIOR Fig. 7: Comparison of ACI and EBCS provision rsults with itrativ P-Δ analysis rsults D) Nin-Story Building with Elvation Irrgularity Th figur in th right shows th sction of a 9-story building and from Tabl 4 and Fig. 8 on can s that: Th sway momnt magnification mthod provisions of th EBCS giv a closr rsult to th itrativ P-Δ analysis rsults than th ACI provisions. Numrically, vs dviation for xtrior columns, and vs for intrior columns. In both cass, howvr, th rsults of th EBCS provision ar smallr than th itrativ P-Δ analysis rsults; unsaf. 21 Journal of EEA, Vol. 33, Dcmbr 2015

10 Abrham Ewntu and Girma Z/Yohanns Fig. 8: Ground and First Floor Plan (lft) and Typical Floor (right) of a Nin-story Building with Elvation Irrgularity Tabl 4: Comparison of sway momnt magnification and itrativ analysis outputs ACI cas 1 EBCS cas 1 δ s Dsign Action Effcts Extrior Columns Itrativ ETAB S Outpu t Intrior Columns Itrativ ETABS P (kn) M (kn-m) P (kn) M (kn-m) cas 1 = 1.05D L 1.0E, according to ACI 0.975D L 1.0E, according to EBCS Journal of EEA, Vol. 33, Dcmbr

11 Dsign Momnt, knm Dsign Momnt, knm Invstigation on Applicability of Substitut Bam-Column Fram LOAD CASE 1 - EXTERIOR LOAD CASE 1 - INTERIOR Fig. 9: Comparison of ACI and EBCS provision rsults with itrativ P-Δ analysis rsults Conclusions CONCLUSIONS AND RECOENDATIONS From this rsarch th following conclusions hav bn mad. 1. Gnrally, th ACI provisions giv mor consrvativ rsults (highr dsign axial load and dsign momnt) than thos of th EBCS provisions rflcting th diffrncs in load combinations usd in th two cods. Howvr, whn dsigning structurs for gravity and wind loads, th axial loads obtaind from EBCS provisions ar highr than thos from ACI provisions. 2. In all th building frams considrd, xcpt th cas with planar irrgularity, th EBCS provision givs rsults closr to th itrativ analysis than th ACI provision, although th rsults ar, almost always, on th unsaf sid. 3. Unlik th ACI provision, th sway momnt magnification provision of th EBCS givs dsign momnts smallr than th itrativ analysis outputs, with maximum dviation of for th nin story fram with vrtical irrgularity. 4. Rsults of th dsign xampls also show that th sway-momnt magnification factors from EBCS provision ar slightly lss than th ACI sway momnt magnification factors in all cass. 5. Whil using th sway momnt magnification provision of th EBCS for dsigning slndr columns in sway frams, on has to rcall that th sway-momnt magnification factor is diffrnt for diffrnt load conditions. This is bcaus of th introduction of th substitut fram which has to b dsignd for th load combination undr considration to dtrmin th ffctiv stiffnss, critical load and hnc th sway momnt magnification factor. 6. Th provision in EBCS dos not giv any xplicit limit as in th ACI for chcking fram stability undr gravity loads only; though it rquirs th chck to b mad. Rcommndations 1. Whn using th sway momnt magnification mthod provisions of th ACI and th EBCS for th dsign of slndr columns of sway frams with irrgularitis, prcaution should b mad sinc th rliability of th rsults dcrass with irrgularitis. 2. Th author rcommnds th following limits for chcking th possibility of sidway buckling undr gravity loads only, which ar quivalnt to th limits in ACI i) Whn s M s is computd from scond-ordr lastic analysis, th ratio of scond-ordr latral dflctions to first-ordr latral dflctions for factord dad and liv loads plus factord latral loads applid to th structur shall not xcd 2.5; ii) Whn s M s is computd using th sway momnt magnification procdur, s computd by Equ. (10) using N Sd for 1.3D + 1.6L and N cr basd on 23 Journal of EEA, Vol. 33, Dcmbr 2015

12 Abrham Ewntu and Girma Z/Yohanns 0.2Ec I c Es I s EI, shall b positiv 1 d and shall not xcd 2.5. iii) Th critical load ratio N Sd /N cr, N Sd computd using N Sd for 1.3D + 1.6L and 0.2Ec I c Es I s N cr basd on EI shall 1 not xcd 0.60, which is quivalnt to s = 2.5. In i), ii) and iii) abov, d shall b takn as th ratio of th total sustaind axial loads to th total axial loads. iv) As in ACI318-08, th abov thr chcks can b ignord simply by limiting th ratio of th total momnt including scondordr ffcts to first-ordr momnts to REFERENCES 1. ACI Committ 318, Building Cod Rquirmnts for Structural Concrt (ACI ) and Commntary, Amrican Concrt Institut, Farmington Hills, MI, ACI Committ 318, Building Cod Rquirmnts for Structural Concrt (ACI ) and Commntary (ACI 318R-05), Amrican Concrt Institut, Farmington Hills, MI, Bkl M., Effctiv Lngth and Rigidity of Columns, ACI Journal, Procdings V. 84, No. 3, pp , July-August Ethiopian Building Cod Standard, EBCS 2-Part 1, Structural Us of Concrt, Ministry of Works and Urban Dvlopmnt, Eurocod 2: Dsign of concrt structurs - Part 1-1: Gnral Ruls and Ruls for Buildings, McGrgor, J. G., J.K. Wight, Rinforcd Concrt Mchanics and Dsign, 4th dition in SI units, Prntic-Hall, 2006, pp McGrgor, J. G., Brn, J. E. and Pfrang, E. O., Dsign of Slndr Concrt Columns, ACI Journal, Procdings V. 67, No. 2, Jan. 1970, pp McGrgor, J. G., Dsign of Slndr Concrt Columns-Rvisitd, ACI d Journal, Procdings V. 90, No. 3, May- Jun. 1993, pp Zrayohanns, G., Ethiopian Building Cod Standard, EBCS 2-Part 1 and Part 2, Dsign Aids for Rinforcd Concrt Sctions, Ministry of Works and Urban Dvlopmnt, Zrayohanns G., Influnc of ACI Provisions for th Dsign of Columns in Sway Frams on EBCS-2:1995, ACI- ETHIOPIA CHAPTER Journal, procdings, 2009, pp Ewnti A., Invstigation on Applicability of Substitut Bam-Column Fram for Dsign of Rinforcd Concrt Sway Frams, MSc Thsis, ACI: A s,tot : Notations Amrican Concrt Institut Thortical ara of rinforcmnt rquird by th dsign D: Dad (prmannt) load E: Earthquak load EBCS: Ethiopian Building Cod Standard E c : E s : EI: EI : Modulus of lasticity of concrt Elastic modulus of rinforcmnt stl Flxural Stiffnss Effctiv flxural stiffnss f cd : Dsign comprssiv strngth of concrt I c, I b : I g : Gross Momnt of inrtia of column & bam cross sctions rspctivly Gross momnt of inrtia of a mmbr I s, I s : Momnt of inrtia of rinforcmnt stl of th column with rspct to th cntroid of th concrt sction k: Effctiv buckling lngth factor L: Liv (variabl) load l b : l c, L: L : M 1 : M 2 : Lngth of bams Story hight Effctiv buckling lngth Th algbraically smallr of M top and M bottom Th algbraically biggr of M top and M bottom Journal of EEA, Vol. 33, Dcmbr

13 Invstigation on Applicability of Substitut Bam-Column Fram M bal : Balancd momnt capacity of a column M bottom : Momnt at th bottom of a column M ns : M s : M top : δ s M s : N cr : Nonsway momnt Sway momnt Momnt at th top of a column Magnifid sway momnt Critical buckling load, story buckling load N, N Sd : Total factord axial load in th story P-: Scond ordr momnts which rsult from latral dflctions,, of th bam column joints from thir original un-dflctd locations Q: Stability indx U, S d : Factord load combination V u, H: Total factord shar in all frams in th story undr considration W: Wind load (1/r bal ): Curvatur at th balancd load δ s : Sway momnt magnification factor for th sam load combination : Rlativ dflction btwn th top and bottom of a story : Slndrnss ratio α 1, α 2 : Rlativ stiffnss of columns to bams at th top and bottom of a story 25 Journal of EEA, Vol. 33, Dcmbr 2015

14 Abrham Ewntu and Girma Z/Yohanns Journal of EEA, Vol. 33, Dcmbr

4.2 Design of Sections for Flexure

4.2 Design of Sections for Flexure 4. Dsign of Sctions for Flxur This sction covrs th following topics Prliminary Dsign Final Dsign for Typ 1 Mmbrs Spcial Cas Calculation of Momnt Dmand For simply supportd prstrssd bams, th maximum momnt