Developing a Plastic Hinge Model for RC Beams Prone to Progressive Collapse

Transcription

1 Developing Plstic Hinge Model for RC Bems Prone to Progressive Collpse Frzd Rouhni A Thesis in The Deprtment of Building, Civil nd Environmentl Engineering Presented in Prtil Fulfillment of the Requirements for the Degree of Mster of Applied Science (Civil Engineering) t Concordi University Montrel, Quebec, Cnd November 2015 Frzd Rouhni, 2015

2 CONCORDIA UNIVERSITY School of Grdute Studies This is to certify tht the thesis prepred By: Frzd Rouhni Entitled: Developing Plstic Hinge Model for RC Bems Prone to Progressive Collpse nd submitted in prtil fulfillment of the requirements for the degree of Mster of Applied Science (Civil Engineering) complies with the regultions of the University nd meets the ccepted stndrds with respect to originlity nd qulity. Signed by the finl exmining committee: Luci Tirc Chir, Exminer Rmin Sedghti Exminer Ashutosh Bgchi Exminer Ln Lin Co-Supervisor Khled Gll Co-Supervisor Approved by Chir of Deprtment or Grdute Progrm Director November 2015 Den of Fculty

3 ABSTRACT Developing Plstic Hinge Model for RC bems prone to Progressive Collpse Frzd Rouhni The US Generl Service Administrtion (GSA) 2013 Guidelines specify the procedures nd the minimum requirements for the design nd evlution of the new nd existing buildings ginst progressive collpse due to n instntneous removl of verticl lod bering elements (i.e., columns). The objective of this study is to ssess the modeling prmeters for reinforced concrete (RC) bems specified in the GSA Three types of RC buildings locted in high, moderte nd low seismic zones in Cnd re designed ccording to the 2010 edition of the Ntionl Building Code of Cnd. They were designed to hve ductile, modertely ductile, nd conventionl seismic force resisting system (SFRS). In totl, 27 three-dimensionl finite element models re developed using ABAQUS by considering the design vribles, such s spn length, depth of the section, nd the reinforcement rtio. Nonliner pushdown nlyses re conducted by incresing the verticl displcement t the loction where the column is removed. The bending moment t the criticl section of the bems is monitored throughout the nlysis. Bsed on the nlysis results, moment-rottion curve for bem for ech type of the building is proposed. In ddition, it is found out in the study tht the detiling of the seismic design hs significnt effect on the progressive collpse resistnce. iii

4 Acknowledgements First nd foremost, I would like to thnk God, who hs lwys been blessing nd mking me whom I m tody. Also, I would like to thnk mny people who mde this thesis possible. I wish to express my sincere grtitude to my supervisors, Dr. Ln Lin nd Dr. Khled Gll, for their efforts nd guidnce during the course of this study. Throughout my thesis, they continuously supported me techniclly nd finncilly nd provided encourgement. I would hve been lost without them. There re no proper words to convey my deepest grtitude nd respect to my lovely prents, Zhr Ftemi nd Hossein Rouhni, for their unfiling emotionl support. They were the emotionl nd morl source of support. This thesis would certinly not hve existed without them. Therefore, I dedicted this thesis to them. I m indebted to mny of my friends for providing stimulting environment for me to lern nd grow. I especilly cknowledge my friend Mr. Hmid Arbzdeh who helped me get through the difficult times, nd for ll of his technicl supports. iv

5 Nottions Chord rottion of plstic hinge corresponding to the mximum bending moment cpcity Degrdtion of elstic stiffness in compression Degrdtion of elstic stiffness in tension e Chord rottion of plstic hinge Initil (undmged) modulus of elsticity of concrete Specified compressive strength of concrete Tensile strength Yielding stress of reinforcing brs Ultimte stress of reinforcing brs h Height of the concrete bem section IE Importnce fctor Ke Elstic stiffness of the bem MV Higher mode effect fctor Q Demnd resulting from the nlysis Cpcity of the member Rd Ductility-relted force modifiction fctor v

6 Ro Overstrength-relted force modifiction fctor S(T) Design spectrl ccelertion t the fundmentl lterl period T Fundmentl lterl period V Seismic bse sher force Vmin Minimum lterl erthquke sher force Vmx Mximum lterl erthquke sher force W Totl seismic weight Equivlent plstic strins in compression Crcking strin in compression Equivlent plstic strins in tension Crcking strin in tension Elstic strin in compression Elstic strin in tension ( ) Strin corresponding to the stress equl to 50% of the mximum strength of confined or un-confined concrete Yielding strin of reinforcing brs Ultimte strin of reinforcing brs Initil yield in compression vi

7 Ultimte compression stress Filure stress in tension Tensile stress Displcement t the bottom of the removed column η ρ Rtio of tension reinforcement ρ Rtio of compression reinforcement Blnced reinforcement rtio vii

8 TABLE OF CONTENTS LIST OF TABLES... x LIST OF FIGURES... xi Chpter 1: INTRODUCTION Motivtion Objective of the study Outline of the thesis... 4 Chpter 2: LITERATURE REVIEW Introduction Previous studies on progressive collpse Existing Guidelines Overview of 2003 GSA Overview of 2013 GSA Progressive collpse nlysis Summry Chpter 3: DESIGN OF RC MOMENT RESISTING FRAMES Description of studied buildings Design lods Grvity lods Seismic lods Design of frmes Summry Chpter 4: PROPOSED MODEL FOR RC BEAMS' PLASTIC HINGES Introduction Modelling techniques Elements Steel brs viii

9 4.2.3 Concrete Crcking nd filure of concrete Interction between concrete nd steel brs Meshing Anlysis results Response curves Development of prmeters for modelling plstic hinges in bems Prediction of the modelling prmeters Comprison with the prmeters proposed in 2013 GSA Summry Chpter 5: CONCLUSIONS AND RECOMMENDATIONS Introduction Conclusions Recommendtions for future reserch Appendix A: DESIGN OF REINFORCEMENT FOR BEAMS A.1 Design of flexure reinforcement A.1.1 Design singly reinforced sections A.1.2 Design doubly reinforced sections A.1.3 Minimum nd mximum reinforcement A.2 Design of sher reinforcement A.3 Additionl considertions for seismic design of bems A.4 Smple of design A.4.1 Flexure design A.4.2 Sher design A.4.3 Additionl considertion for seismic design REFERENCES ix

10 LIST OF TABLES Tble 2.1 Dmge control nd building performnce levels Tble 3.1 Design grvity lods Tble 3.2 Design prmeters for the frmes Tble 3.3 Dimensions nd the longitudinl nd trnsverse reinforcement of bems in Toronto Tble 3.4 Dimensions nd longitudinl nd trnsverse reinforcement of bems in Montrel Tble 3.5 Dimensions nd longitudinl nd trnsverse reinforcement of bems in Vncouver Tble 4.1 Men vlues for the modelling prmeters proposed x

11 LIST OF FIGURES Figure 2.1 Cses of collpsed nd non-collpsed buildings due to impcts... 9 Figure 2.2 Overll flow for considertion of progressive collpse dopted from GSA (2003) Figure 2.3 Applicbility flow chrt dopted from GSA (2013) Figure 2.4 Definition of the ctions doped from GSA (2013) () Deformtion-controlled ctions, (b) Force-controlled ctions Figure 3.1 Pln of floors nd elevtion of longitudinl frmes of the building (spn = 6.0 m) Figure 3.2 Seismic design spectr for Toronto, Montrel, nd Vncouver, site clss C, 5% dmping Figure 3.3 Distribution of the seismic force long the height of the building Figure 3.4 Lyout of the reinforcement of the bems in Toronto Figure 3.5 Lyout of the reinforcement of the bems in Montrel Figure 3.6 Lyout of the reinforcement of the bems in Vncouver Figure 4.1 Criticl frme with column removl Figure 4.2 Schemtic 3D ABAQUS model of the studied bem-column ssembly Figure 4.3 Stress-strin curve for the longitudinl nd trnsverse steel brs Figure 4.4 Stress-strin curve for concrete under compression nd tension Figure 4.5 Response of concrete to unixil loding: () tension, (b) compression dpted from ABAQUS Figure 4.6 Cses for the sensitivity nlysis on meshing Figure 4.7 Bem responses from the sensitivity nlyses xi

12 Figure 4.8 Bem response curves: () ductile bems, (b) modertely ductile bems, (c) conventionl bem Figure 4.9 Multi-liner bckbone curve for modeling plstic hinges in bem () ductile nd modertely ductile bems, (b) conventionl bems Figure 4.10 Comprison of the vlue for prmeter ' bsed on the proposed model nd 2013 GSA Figure 4.11 Comprison of the proposed model with 2013 GSA criteri () ductile/modertely ductile bems, (b) conventionl bems xii

13 Chpter 1 1. INTRODUCTION 1.1 Motivtion Since the prtil collpse of the Ronn Point Aprtment Building in London in 1968, design of building structures ginst progressive collpse hve brought ttention to reserchers round the world. Furthermore, comprehensive reserch work hs been conducted fter the collpse of the World Trde Center Twin Towers in Progressive collpse cn be initited by vriety of bnorml conditions, such s impct (e.g., ircrft/vehiculr collision), pressure lods (e.g., gs explosion) or overloding on the structure. These types of lods re mostly ssocited with the uncertinty in mgnitude while the durtion of impulse is significntly short, which might rnge from few milliseconds up to 1-2 seconds. Although the occurrence rte of these events is reltively low, they might cuse significnt dmge to structures nd ctstrophic losses. According to the current building design codes, including the Ntionl Building Code of Cnd (NBCC 2010) nd the Interntionl Building Code (IBC 2015), the typicl lods considered in the design of building structures re: ded, live, wind, nd erthquke lods. Moreover, most of the existing buildings were not designed for the lods due to disproportionte collpse, nd they might be vulnerble to progressive collpse under ny of the conditions mentioned bove. 1

14 U.S. Generl Service Administrtion (GSA) published guidelines for the progressive collpse resistnce of buildings in 2003 nd 2013, respectively. It should be noted tht no guidelines on progressive collpse nlysis re vilble in Cnd. According to GSA, progressive collpse is defined s n event triggered by the locl filure of the primry structurl members due to the column removl, which might, in turn, cuse the collpse of the djcent members. The ltest 2013 Guidelines were developed bsed on the seismic provisions of ASCE (2014) by considering the structurl integrity, ductility, nd nonliner behviour due to the sudden removl of column. Requirements for redundncy, overll structurl integrity nd resilience specified by the Americn Concrete Institution (ACI) were lso considered in GSA Guidelines. In ddition, United Sttes Deprtment of Defense (DoD) issued guidnce on protection of fcilities in cse of bnorml loding nd progressive collpse in 2001 nd 2005, respectively. Currently, GSA nd DoD re the two commonly used guidelines for evluting the progressive collpse resistnce of building structures. The typicl pproch considered in GSA nd DoD is designted s Alterntive Pth Method (APM). In APM, first, instntneous loss of verticl lod-bering element is ssumed, i.e., column is removed; then the cpbility of the bem elements supported by the column is evluted. More specificlly, the two mjor response prmeters considered in APM re the verticl deflection nd chord rottion of bems. It should be mde cler tht APM doesn't focus on the scenrio tht leds to the column removl itself. Therefore, this thret-independent method ims to provide redundncy to the structure in order to resist progressive collpse if it hppens, i.e., the cpbility of bems nd remining columns is exmined only fter the column is removed. 2

15 In order to comply with the requirements of APM, severl nlysis procedures re specified in GSA nd DoD, nmely, liner sttic, nonliner sttic, nd nonliner dynmic. The cceptnce criteri for progressive collpse nlysis re bsed on the response of bems; however, it is known tht the cpbility of other elements (e.g., columns) lso contributes to the overll resistnce of the building ginst progressive collpse. Although the results from the experimentl tests nd numericl nlyses hve mde significnt contribution to the ltest GSA Guidelines published in 2013, there is still lck of detiled implementtion rules for the numericl modeling of progressive collpse nlysis. In ddition, by compring the cceptnce criteri in 2013 GSA with 2003 GSA, it hs been found tht the level of the new criteri is much higher thn tht of the old ones. Since the evlution results of the building performnce ginst progressive collpse depends very much on the cceptnce criteri given in the Guidelines, they must be vlidted through experimentl or numericl studies. 1.2 Objective of the study The objective of this study is to investigte the performnce of reinforced concrete frme buildings ginst progressive collpse. To chieve this objective, the following tsks were crried out in the study, () Design 27 four-storey reinforced concrete moment-resisting frme buildings locted in Toronto, Montrel, nd Vncouver tht represent the low, moderte, nd high seismic hzrd zones in Cnd, respectively. (b) Propose moment-rottion curves to simulte the nonliner behviour of plstic hinges due to column removl. Three-dimensionl finite element nlyses were 3

16 conducted on bem-column ssembly using the structurl nlysis softwre ABAQUS. (c) Compre the proposed curves with those specified in 2013 GSA. (d) Equtions for prediction of the chord rottions corresponding to the mximum cpcity nd the first yielding of the bem were proposed. 1.3 Outline of the thesis The mteril in this thesis is presented in 5 chpters nd one ppendix. Chpter 2 serves s literture review; Chpter 3 provides bckground mteril (i.e., design of the buildings) tht is used in the reserch work presented in Chpter 4. The min conclusions from the reserch re given in Chpter 5. Chpter 2 summrizes the previous studies on progressive collpse nlysis. A comprison between the 2003 nd 2013 GSA Guidelines is lso presented in this chpter. Chpter 3 describes the design of the buildings used in this study. Twenty seven 4-storey reinforced concrete moment resisting buildings ssumed to be in Toronto, Montrel, nd Vncouver were designed ccording to the 2010 edition of the Ntionl Building Code of Cnd. More specificlly, the spn lengths considered re 4.0 m, 6.0 m, nd 8.0 m. The reinforcement rtios re the minimum nd mximum specified in the current Stndrd for design of concrete structures; nd the one in between. The detils of reinforcement of the designed buildings re presented in the Appendix A. Chpter 4 presents the nonliner finite element modeling of the bem-column ssembly of the 27 buildings subjected to the column removl using the ABAQUS softwre. This chpter focuses on the techniques for modeling the bem-column elements 4

17 with the loss of one column below the bem, nd the development of the moment-rottion curves to simulte the nonliner behviour of bems in the event. Finlly, Chpter 5 summrizes the min findings nd conclusions from this study, nd provides recommendtions for future reserch. 5

18 Chpter 2 2. LITERATURE REVIEW 2.1 Introduction The collpse of prt of the 22-storey Ronn Point Aprtment Building in London, Englnd in 1968 (Shnkr 2004) is normlly considered s the first exmple of the progressive collpse of building structures. The gs explosion t the 18 th floor of the building triggered collpse of the corner slbs t the upper floors (bove the 18 th floor) tht ws followed by collpse of ll corner slbs of the building (Fig. 2.1). From totl of 260 residents, four were killed nd seventeen were injured. The filure of the building ws due minly to the wekness of the joints connecting the wlls to the floor slbs, which could not provide n lterntive pth to trnsfer the lods during the event (Person nd Deltte 2005). L'Ambince Plz Building ws 16-storey residentil building in Bridgeport, Connecticut, the United Sttes. On April 23, 1987, the prtilly erect frme completely collpsed during construction (Fig. 2.1b), which killed 28 workers on site. High stresses in concrete on the slbs developed during the erection of the frme ws believed to be mjor cuse of the collpse (Heger 2006). This ccident led to ntion-wide investigtion on the sfety of lifting slbs in construction s well s temporry suspension on its use in the stte of Connecticut (Dusenberry 2002). 6

19 A typicl exmple of progressive collpse is the collpse of the Alfred P. Murrh Federl Building in Oklhom City, due to bomb explosion t the ground level in April 1995 (Ntionl Acdemy of Science 1995). A former soldier nd security gurd prked truck in front of the building with the intention of committing mss murder. Three columns t the first storey were highly dmged, which cused the totl collpse of lmost hlf of the building (Fig. 2.1c). The building ws demolished bout month fter the event. The collpse of the Smpoong Deprtment Store Building (Fig. 2.1d) on June 29, 1995 ws nother exmple of the progressive collpse. The ccident cused 502 deths nd injured 937 others, nd ws considered s the most significnt disster in the history of South Kore. The filure of the building ws minly ttributed to overloding due to chnge in the function of the building. More specificlly, the fifth floor of the originl building, which ws supposed to house skting rink, ws remodelled to hold eight resturnts. In ddition, it ws found tht the weight of the ir conditioners instlled on the roof incresed the design ded lod by bout three times more thn tht specified in the code. Due to the overloding on the roof nd the 5 th floor, the columns filed first followed by the collpse of the south wing of the building. With the incresing number of terrorist ttcks, sfety of government buildings hs become mjor concern fter the collpse of the Twin Towers of the World Trde Centre on September 11, Two hijcked irplnes hit the north nd south Towers within 15 minutes, which cused the collpse of the towers within two hours fter the ttck (Fig. 2.1e). Becuse of the redundncy of the tube-frme tht ws used s the min structurl system, the Towers sustined the dmge cused by the plne crshes, i.e., they did not collpse immeditely. According to the study on the performnce of the building published 7

20 by Federl Emergency Mngement Agency (FEMA 403), it ws believed tht fires triggered by the fuel of the jets wekened the connections of the steel trusses of the floor system, which led to the collpse of the towers. As described bove, building could collpse or prtilly collpse due to number of resons, such s gs explosion, bomb ttck, plne crsh, etc. However, there re cses tht the building survived, i.e., did not collpse, during the impct. A building beside Bnkers Trust Building collpsed in 1970 (Fig. 2.1f). The debris of the collpsed building hit the Bnkers Trust Building, nd cused severe dmge to severl columns. The zone of structurl dmge remined confined to one structurl by in the floors below. Therefore, progressive collpse ws not triggered in the Bnkers Trust Building given the extremely bd condition of the columns (Smilowitz et l. 2002). The Pentgon Building in Wshington D.C. ws lso ttcked by hijcked irplne on September 11, However, the building performed quite well during the event, nd the progressive collpse tht hppened to the World Trde Center Twin Towers ws not triggered in this cse (Fig. 2.1g). After n intensive investigtion, it ws reported tht sufficient structurl redundncy ws provided in the building for the progressive collpse resistnce (Mlker et l. 2003). 8

21 () (b) (c) (d) (e) (f) (g) Figure 2.1 Cses of collpsed nd non-collpsed buildings due to impcts, () Ronn Point prtment building ( (b) L Ambince Plz building ( (c) Alfred P. Murrh building ( (d) Smpoong Deprtment Store building ( (e) World Trde Center Tower ( (f) Bnkers Trust building ( (g) Pentgon building ( 9

22 2.2 Previous studies on progressive collpse Between 1968 nd 1975 Study on the vulnerbility of the building structures ginst progressive collpse ws prompted fter the collpse of the Ronn Point Aprtment Building in 1968 due to the gs explosion in one of the prtments. Most of the reserch work ws focused on the development of the reltionship between the bnorml lods nd progressive collpse, e.g., Astbury (1969), Burnett (1973), Minstone (1973), nd Burnett (1974). In order to prevent or to reduce the risk of the progressive collpse, severl methods for considering the bnorml lods in the design were specified in the British Building Design Code. The 1975 edition of the Ntionl Building Code of Cnd lso mde regultions on progressive collpse (Dusenberry 2002); however, no detiled guidnce ws provided. Menwhile, mny studies nd workshops were held in the US during this time fter the collpse of the Ronn Point Building. One of the studies tht were given in Breen (1975) focused on the detiled design configurtions of precst concrete structures ginst progressive collpse. Furthermore, development of the provisions of integrl ties throughout the structure (indirect design) for the progressive collpse nlysis ws described in Breen (1975). Ferhin (1971) reviewed the chnges mde in the British nd the Cndin codes on the progressive collpse nlysis, nd reported tht erthquke design lods would hve positive impct on protecting buildings from the progressive collpse. In ddition, Popoff (1975) reviewed vrious types of connections nd suggested the criteri for the minimum reinforcement of the design connections should be modeled in order to void progressive collpse. 10

23 Between 1976 nd 1995 A study conducted by Monsted (1979) showed the importnce of primry components of building (e.g., lod-bering wlls) nd connections in resisting the progressive collpse. He lso investigted the effects of the lterntive lod pth nd ctenry ction fter the filure of the lod-bering component on the collpse resistnce. Webster (1980) proposed methodology for determining the relibility of flt slbs in multi-storey building. It should be noted tht the objective of Webster's study ws to reduce the risk of the progressive collpse during construction. Peku (1982) performed study on evluting the behviour of the precst pnel sher wlls during progressive collpse. The results from the study showed tht filure of the exterior pnel would led to unexpectedly lrge sher forces, which would trigger the progressive collpse. Gross (1983) conducted the first study on progressive collpse nlysis of steel moment-resisting frme building using two-dimensionl finite element model. More specificlly, nonliner nlyses were performed in which the nonlinerity of bems, columns, nd connections were considered due to the filure of the columns. Sher effects of the infill pnels were lso tken into ccount in the numericl modelling. Csciti (1984) crried out similr study on reinforced concrete moment-resisting frme using 2D nlysis. In order to consider the cyclic behviour of the elements under seismic loding, Modified Tked model ws used to simulte the performnce of plstic hinges t the ends of members. The study performed by Pretlove (1991) showed tht frcture of given member my cuse overloding to the djcent members which, in turn, would trigger the progressive collpse. He lso questioned tht the stticlly sfe elements might not be relible if dynmic effects were tken into ccount. It should be noted tht sudden loss of 11

24 column, by nture, is dynmic mechnism. Unlike the reltively complicted nlyses conducted in the studies mentioned bove, Bennett (1988) proposed simplified method to evlute the potentil of structures' progressive collpse. The lterntive lod pth method ws used in the study to evlute the progressive collpse resistnce of spce truss; both liner nd nonliner nlyses were performed. Due to the lrge degree of sttic indetermincy nd sufficient redundncy in trusses, Bennet climed tht redistribution of forces would esily tke plce in truss system fter filure of n element. However, this conclusion hs not lwys proven to be true (Murth Smith 1988). It is necessry to mention herein tht most of the studies performed during this period were bsed on severl simplified ssumptions nd liner sttic nlysis. However, it hs been found lter on tht bnorml lods, which in most cses triggered the progressive collpse, were dynmic (i.e., not sttic) lods. In ddition, the methodology for nonliner nlysis ws not well developed t this time due to the lck of knowledge nd constrints of computer science. Therefore, most of the results from these studies were not correct s explined in Lim (2004). Between 1996 nd 2010 Astneh-Asl (2001) did n experimentl test on typicl steel building by removing middle column on the building s perimeter in order to evlute the progressive collpse resistnce of the structure. Results from the test showed tht the lods were well redistributed due to the ctenry ction of the steel deck nd girders. Mlkr (2003) prepred technicl report on the investigtion of the performnce of the Pentgon Building under the ttck of September 11, The results from the detiled finite element nlysis showed tht the building hd stisfctory performnce overll even 12

25 though the columns on the first floor were extensively dmged. Furthermore, severl fctors ttributed for preventing building structures from collpse were reported; some of them re summrized s follows, A frme system consisting of bems nd girders could provide sufficient redundncy nd lterntive lod pth in cse of the loss of the verticl lod-bering components. Shorter spns hve dvntges over longer spns in terms of progressive collpse resistnce. Higher design lods (i.e., 150 psf or 7.18 kp in excess of service lods) might be considered in the design of the building in order to resist the progressive collpse. Approprite design detilings, such s continuity of the bottom reinforcement in the bems nd girders extending into the supports, spirl reinforcement, could increse the progressive collpse resistnce of the building. During this period, bem element formultion nd solution ws introduced for the dynmic progressive collpse nlysis. According to this procedure, inelstic bemcolumn elements were formulted using the lumped plsticity pproch with the concentrted inelsticity t the element ends (Kewkulchi 2004). The results of this study showed tht both the cpcity of the structurl members nd the number of the plstic hinges could be underestimted if the dynmic effects were not considered in the progressive collpse nlysis. It ws lso concluded in Kewkulchi (2004) tht sttic nlysis might not provide conservtive results on estimting the potentil of the progressive collpse. Grierson et l. (2005) focussed study on developing qulittive criteri for progressive collpse nlysis. Bsed on the results from liner nlyses in the 13

26 study, Grierson proposed simplified methods tht cn be implemented in the structurl nlysis softwre. Moreover, the results were verified by compring them with those using nonliner nlyses. One yer lter, Kim nd Prk (2006) dvnced the progressive collspe nlysis by introducing concept of Energy Blnce, which ws considered to hve gret potentil to simplify the nlysis. A comprehensive study on the progressive collpse nlysis ws conducted by Kim et l. (2009). The exmined buildings were momentresisting steel frme buildings, nd they were 3-, 6-, nd 15-storey high, respectively. The buildings were designed for grvity lods only, nd s such, the combintion of the grvity nd seismic lods were ssessed in order to see the contribution of the seismic lods to the progressive collpse resistnce. Izzuddin et l. (2008) proposed frmework for progressive collpse nlysis of tll buildings. They suggested tht the ductility, redundncy nd energy dissiption in the structurl system should be considered in the event of sudden filure of column. Ygob et l. (2009) concluded tht the overll response of existing RC buildings ginst progressive collpse cn be improved by considering the locl filures. Also, need to review the vilble knowledge on the progressive collpse phenomenon hs been remrked in this study. Between 2011 nd 2015 Lin et l. (2011) conducted progressive collpse nlyses on reinforced concrete frme buildings designed ccording to the seismic provisions of the 2005 edition of the Ntionl Building Code of Cnd. In totl, six buildings were considered in the study, in which three were in Ottw nd three were in Vncouver with heights of 5, 10, nd 15 storeys, respectively. The performnce of the buildings ginst progressive collpse ws 14

27 evluted ccording to the 2003 GSA which ws vilble t tht time. The Ottw buildings were found to be more vulnerble to progressive collpse thn the Vncouver buildings. The results from this study showed tht the vulnerbility of the progressive collpse of seismiclly designed buildings depended gretly on the differences between the spns of the longitudinl nd the trnsverse frmes, i.e., lrger differences between the spns led to higher vulnerbility. Mirvld (2013) investigted the vulnerbility of progressive collpse of three steel moment-resisting frme buildings in Cnd, which were locted in different seismic hzrd zones. Nonliner dynmic nlysis ws conducted nd the vulnerbility of the buildings ws evluted bsed on 2003 GSA. Like the findings in Lin et l. (2011), Mirvld lso reported tht steel buildings in low seismicity zones re more vulnerble to progressive collpse. Furthermore, two methods for retrofitting of the buildings with high vulnerbility to progressive collpse were proposed in the study by using the top bemgirder system nd the grvity truss system. Trn nd Li (2014) studied the bckbone curves of reinforced concrete columns with light trnsverse reinforcement by conducting experimentl tests. Livingston et l. (2015) evluted the response of continuous bem by chnging structurl chrcteristics (e.g., yield strength of rebrs, spn length nd xil stiffness) using detiled finite element model. The collpse test of three-storey reinforced concrete frme (hlf scle) ws crried out by Xio et l. (2015); filure mechnisms in ddition to lod-trnsfer pth nd the dynmic response, were discussed in the study. They concluded tht the slbs nd bems directly connected to the filed columns hve significnt effect on disproportionte collpse resistnce. Moreover, the requirement of providing sufficient nchorge cpcity to the joints should be provided in the guidelines 15

28 for the progressive collpse nlysis in order to chieve the ctenry ction, which is beneficil for the collpse resistnce. 2.3 Existing Guidelines Overview of 2003 GSA In June 2003, the US Generl Service Administrtion (GSA) relesed "Progressive Collpse Anlysis nd Design Guidelines for New Federl Office Buildings nd Mjor Moderniztion Projects". The Guidelines were developed such tht the potentil of progressive collpse is tken into ccount in the design, plnning nd construction of new buildings nd mjor renovtion projects. More specificlly, the GSA Guidelines re intended to: Assist in the reduction of the potentil for progressive collpse in new Federl Office Buildings, Assist in the ssessment of the potentil for progressive collpse in existing Federl Office Buildings, Assist in the development of potentil upgrdes to fcilities, if required, Given this, the methodology proposed in GSA minly focuses on the subsequent effects of the bnorml loding on the structure, which is known s thret-independent. Moreover, the requirements specified in GSA were developed to meet the provisions of the Security Criteri on the progressive collpse developed by the Intergency Security Committee (ISC). As stipulted in GSA, the Guidelines pply to "In-house government engineers, rchitecturl/engineering (A/E) firms nd professionl consultnts under contrct to GSA s primry users. While mndtory for GSA fcilities, these Guidelines 16

29 my lso be used or dopt by ny gency, orgniztion or privte concern." Two methods were proposed in 2003 GSA, i.e., the simplified method nd the dvnced method, depending on the number of stories bove the ground. More specificlly, the simplified method is used for buildings less thn 10 stories. Otherwise, the dynmic method must be used. For ese of discussion, the simplified method nd the dvnced method re referred to s the liner nlysis method nd the nonliner nlysis method, respectively, in this chpter. It is well known tht nonliner nlysis method is more precise thn liner nlysis method since the nonlinerity of the mteril nd geometry during the event of the progressive collpse is tken into ccount in the nlysis. However, it should be noted tht nonliner modelling might be big chllenge for some of the projects. GSA lso ddresses the need to protect humn lives nd prevent injuries in ddition to protect the building nd its functions. In order to evlute the performnce level defined in these guidelines, ASCE (2014) specifies some structurl nd non-structurl criteri for Dmge Control nd Building performnce level (Tble 2.1). 17

30 Overll dmge Tble 2.1 Dmge control nd building performnce levels ASCE (2014) Trget building performnce levels Collpse prevention Life sfety Immedite occupncy Opertionl Severe Moderte Light Very light Little residul stiffness nd Some residul strength No permnent drift. Structure No permnent drift. strength, but lod bering nd stiffness left in ll substntilly retins originl strength Structure substntilly column nd wlls function. stories. Grvity lod nd stiffness. Minor crcking of retins originl strength nd Lrge permnent drifts. bering elements fcdes, prtitions, nd ceilings s stiffness. Minor crcking of Some exits blocked. Infill function. No out of well s structurl elements. Elevtors fcdes prtitions, nd nd un-brced prpets plne filure of wlls cn be restrted. Fire protection ceilings s well s structurl filure or t incipient filure. or tipping of prpets. operble. elements. All systems Generl Building is ner collpse. Some permnent drift. importnt to norml Dmge to prtitions. opertion re functionl. Building my be beyond economic repir. 18

31 Overll dmge Tble 2.2 Dmge control nd building performnce levels (Continued). Trget building performnce levels Collpse prevention Life sfety Immedite occupncy Opertionl Severe Moderte Light Very Light Extensive dmge Flling hzrds Equipment nd contents re generlly Negligible dmge occurs. mitigte but mny secure, but my not operte due to Power nd other utilities re Non-structurl rchitecturl, mechnicl filure or lck of utilities. vilble, possibly from components mechnicl nd stndby secure. electricl systems re dmged. Comprison with Significntly more dmge Somewht more Less dmge nd lower risk Much less dmge nd lower performnce nd greter risk. dmge nd slightly risk intended for higher risk. building designed under NEHRP provision, for design erthquke. 19

32 The mjor performnce levels defined in Tble 2.1 re summrized below, Immedite Occupncy Structurl Performnce Level (S-1): The risk of life-thretening injury s result of structurl dmge is very low, nd lthough some minor structurl repirs my be pproprite, these would generlly not be required prior to re-occupncy. Life Sfety Structurl Performnce Level (S-3): The overll risk of life-thretening injury s result of structurl dmge is expected to be low. It should be possible to repir the structure; however, for economic resons this my not be prcticl. While the dmged structure is not n imminent collpse risk, it would be prudent to implement structurl repirs or instll temporry brcing prior to reoccupncy. Collpse Prevention Structurl Performnce Level (S-5): Significnt risk of injury due to flling hzrds from structurl debris my exist. The structure my not be techniclly prcticl to repir nd is not sfe for re-occupncy, s ftershock ctivity could induce collpse. 20

33 To ssist in using the Guidelines, flow-chrt methodology (Fig. 2.2) ws given in GSA which helps to determine if the fcility under considertion might be exempt from detiled considertion for progressive collpse. As seen in the figure, number of questions should be nswered to identify whether the progressive collpse nlysis should be considered or not. These questions minly include building occupncy, building ctegory, seismic zone, number of stories, nd the detils bout connections. Progressive collpse nlysis nd design guidelines Exemption process (Fcility exemption considertion) No further considertion for progressive collpse is required Yes Is the fcility exempt from future considertion for progressive collpse? Report No The potentil for progressive collpse is high New Construction Design Anlysis - Liner sttic-dynmic -Nonliner sttic-dynmic New or existing construction? Existing construction Anlysis - Liner sttic-dynmic -Nonliner sttic-dynmic No Does the structure meet the requirements for minimizing the potentil for progressive collpse? Yes The potentil for progressive collpse is low nd the fcility hs met the requirements for minimizing the potentil for progressive collpse. Yes Does the structure meet the requirements for minimizing the potentil for progressive collpse? No The potentil for progressive collpse is high nd the fcility hs not met the requirements for minimizing the potentil for progressive collpse. Prepre report tht documents findings, recommendtions nd costs. Prepre report tht documents findings, recommendtions nd costs. Figure 2.2 Overll flow for considertion of progressive collpse dopted from GSA (2003). 21

34 In generl, the methods for progressive collpse nlysis re divided into direct method nd indirect method. The direct method includes "Alterntive Pth Method" nd "Specific Lod Resistnce Method", while the indirect method includes "Minimum Levels of Strength, Ductility nd Continuity" nd "Tie Force Method" (NISTIR 7396). Among the methods mentioned bove, GSA dopted the "Alterntive Pth Method". In this method, the demnd on the entire lod-bering elements including bems, columns, foundtions, etc., should be evluted for different scenrios of the column or wll removl. The objective of the nlysis is to mke sure tht the lternte lod pths re vilble in cse of n element(s) filure. In norml condition (i.e., the progressive collpse will not hppen) for building designed for grvity lods, the lods re first distributed over the slb, then they will be trnsferred to bems, nd further trnsferred to the columns; finlly, ll the lods will be trnsferred to the foundtion. However, in n bnorml condition (e.g., building is ttcked by bomb explosion), if column in building lost its cpcity, n lternte lod pth should be vilble such tht the lods cn still be trnsferred properly, i.e., no elements re overloded. Furthermore, the Alterntive Pth Method requires tht the structure be ble to bridge over verticl lod-bering elements to be removed t given loction for the progressive collpse nlysis. In order to conduct progressive collpse nlysis, criticl loctions for the column/wll removl should be defined first. This cn be determined by engineering judgement. Nevertheless, GSA specifies three scenrios cses for the column/wll removl for regulr structurl configurtions: Cse 1: the instntneous loss of column for one floor bove grde (1 st storey) locted t or ner the middle of the short side of the building. 22

35 Cse 2: the instntneous loss of column for one floor bove grde (1 st storey) locted t or ner the middle of the longer side of the building. Cse 3: the instntneous loss of column for one floor bove grde (1 st storey) locted t the center of the building. The potentil of progressive collpse of given element is ssessed by using the Demnd-Cpcity Rtio (DCR) if liner nlysis is considered. The DCR rtio cn be clculted using Eqution 1, Q DCR = Q UD CE (1) Where, QUD = demnd (i.e., moment, xil force, or sher force cting on the member) resulting from the nlysis, nd QCE = cpcity of the member (i.e., moment, xil force, or sher force tht the member cn resist). The lods used to determine the demnd QUD re: 2(DL+0.25LL) for sttic nlysis, nd DL+0.25LL for dynmic nlysis, where DL represents the ded lod, nd LL represents the live lod. The cpcity QUE is determined bsed on the geometry nd mteril properties of the section. The llowble DCR vlues for the structurl members re DCR 2.0 for regulr buildings, nd DCR 1.5 for irregulr buildings. If DCR rtios lrger thn the foregoing vlues re obtined, it indictes tht the building hs high potentil for progressive collpse. 23

36 Overview of 2013 GSA The 2003 GSA Guidelines were replced with new edition in October 2013, which is referred to s 2013 GSA in this thesis. More specificlly, the 2003 GSA ws updted in order to keep consistency between the Intergency Security Committee (ISC 2013) Stndrds nd GSA Guidelines in the level of building protection for progressive collpse. Furthermore, 2003 GSA Guidelines were modified such tht its methodologies re similr to those specified in the Design of Buildings to Resist Progressive Collpse prepred by the Deprtment of Defense (DoD 2005). Similr to 2003 GSA, the new 2013 Guidelines im to reduce the potentil for progressive collpse by bridging over the loss of the structurl elements, limiting the extent of dmge to loclized re (i.e., to mke Alterntive Pths vilble), nd providing redundnt structurl system long the height of the building. Moreover, 2013 GSA Guidelines ddress the need to sve lives nd prevent injuries s well. In 2013 GSA, progressive collpse is defined s severe dmge or collpse tht is disproportionte to the mgnitude of the inititing event. In fct, this definition focuses on the reltive consequence or mgnitude of collpse rther thn the mnner tht triggers progressive collpse, s specified in 2003 GSA. Therefore, in prctice, it is often referred to s "disproportionte" rther thn "Progressive". Two thret-dependent pproches re given in the Guidelines; The first pproch reduces the risk of progressive collpse for defined thret by directly limiting the initil dmge through hrdening of structurl elements. 24

37 The second one reduces the risk of progressive collpse by limiting the propgtion of initil dmge, without explicit considertion the cuse of the initil event, through implementtion of Guidelines. According to 2013 GSA, the ppliction of the progressive collpse design depends on the required level of protection, which should be determined bsed on the number of stories nd the Fcility Security Levels (FSL) in ccordnce with ISC. There re five FSL levels specified in ISC, to ensure tht security becomes n integrl prt of the plnning, design nd construction of new federl office buildings. It should be noted tht progressive collpse design is not required for FSL I & II given the low occupncy nd risk level ssocited with these types of fcilities. However, the design is mndtory for FSL III & IV, nd V; the detils re s follows, FSL III & IV: for the buildings with four or more stories mesured from the lowest point of exterior grde to the highest point of elevtion. These fcilities should implement both the Alterntive Pth nd Redundncy Design Procedures. FSL V: 2013 GSA is pplicble for ll FSL V buildings regrdless of the number of stories while the Redundncy Design Procedures do not need to be pplied to these fcilities. Once fcility's FSL level hs been determined, Guidelines cn be pplied by following the flow chrt illustrted in Fig

38 Applicbility of Progressive collpse guidelines New building or building ddition Existing building Fcility security level (FSL) Is the building undergoing mjor moderniztion? FSL V FSL III & IV FSL I & II Yes No Applicble Is the building 4- stories or greter? Not Applicble Not Applicble Yes No Applicble Not Applicble Figure 2.3 Applicbility flow chrt dopted from GSA (2013). One of the mjor differences between 2003 GSA nd 2013 GSA is the lods pplied for the nlysis. In order to simulte the behviour of the structure fter loss of lod-bering elements, both Guidelines suggested tht the grvity lods should be incresed for the purpose of the progressive collpse nlysis. It is necessry to mention tht the lod incresing fctor is used to tke into ccount the dynmic effect during the column removl in sttic nlysis GSA recommends using fctor of 2 to mplify the grvity lods 26

39 on the res mostly ffected by the column removl for the sttic nlysis. However, this fctor ws replced by fctor in 2013 GSA. This fctor is not fixed number; insted, it depends on the type of the structure, mteril, nd nlysis method. 2.4 Progressive collpse nlysis According to the 2013 GSA Guidelines, three types of nlyses cn be used in the ssessment of the potentil for progressive collpse of buildings, i.e., liner sttic nlysis, nonliner sttic nlysis, nd nonliner dynmic nlysis. For the purpose of the nonliner nlysis, including both dynmic nd sttic, which is the most relible method for the progressive collpse nlysis, ll the ctions of the elements should be clssified s either deformtion- or force-controlled ctions ccording to the Guidelines. The typicl curves for the bove-mentioned ctions re illustrted in Fig. 2.4, in which the horizontl xis represents the deformtion (i.e., rottion or displcement) while the verticl xis represents the moment or force. For better understnding, Tble 2.2 lists the typicl exmples of the deformtion- nd force-controlled ctions defined in 2013 GSA. A detiled description of these two types of ctions is given below, Deformtion-controlled ctions: significnt ductile behviour is expected for these ctions. More specificlly, n element or component behves elsticlly until its yielding strength is reched, which is normlly designted s fy for steel element or reinforcing steel under tension. When the yielding strength is exceeded, plstic hinges would be formed t the ends of the bems nd/or columns. Referring to Fig. 2.4, the elstic nd plstic rnges re represented by the rnge between Points 0 nd 1, Points 1 nd 3, respectively. As illustrted in Figure 2.4, the plstic 27

40 rnge includes strin-hrdening (between Points 1 nd 2) nd strength deteriortion (between Point 2 nd 3). According to 2013 GSA, the ction of primry component is defined s deformtion-controlled if e g, where g nd e represent the deformtions t Points 1 nd 2, respectively. Otherwise, the ction should be clssified s force-controlled, minly becuse of the limited inherent ductility in such primry element. On the other hnd, the ction of secondry component is considered to be deformtioncontrolled for ny e/g rtio. Force-controlled ctions: brittle (i.e., non-ductile) behviour is expected for these ctions s shown in Fig. 2.4b. The component will lose its cpcity once the yielding point (Point 1) is reched. Strictly speking, this type of response should be voided in the design of ny type of structures. The components with forcecontrolled ctions re just required to hve strength cpcity equl or lrger thn the demnd, while neither rottion nor deformtion is needed to be checked. Furthermore, performnce level is not defined in GSA for these types of ctions except for the end of the elstic stge. 28

41 Tble 2.2 Exmples of deformtion- nd force-controlled ctions (GSA 2013). Components Deformtion-controlled ction Force-controlled ctions Moment frmes Bems Moment (M) Sher (V) Columns M Axil lod (P), V Joints -- V 1 Sher wlls M,V P Brced frmes Brces P -- Bems -- P Columns -- P Sher links V P, M Connections P, V, M 2 P, V, M Note: 1 Sher my be deformtion-controlled ction in steel moment frme construction. 2 Axil, sher, nd moment my be deformtion-controlled ctions for certin steel nd wood connections. Figure 2.4 Definition of the ctions doped from GSA (2013), () Deformtion-controlled ctions, (b) Force-controlled ctions. 29

42 According to 2013 GSA, the cceptnce criterion for progressive collpse for deformtion-controlled ctions is tht the mximum deformtion should be within the elstic nd plstic rnges, nmely, between ordintes e nd g s shown in Fig The llowble deformtions for the two ordintes re specified in GSA, nd they depend on the expected performnce level of the building fter the event, nmely, life sfety nd collpse prevention s described Section Summry Studies on the progressive collpse re reviewed in this chpter. More specificlly, comprehensive reserch on the experimentl tests nd nlyticl works between 1968 nd 2015 is summrized in order to develop the originlity of the work proposed in the thesis. The mjor requirements specified in the Guidelines published in 2003 nd 2013 in the United Sttes (2003 GSA nd 2013 GSA) re lso described in this chpter. The 2013 GSA Guidelines is commonly used by reserchers nd prctitioners in North Americ for progressive collpse nlysis. 30

43 Chpter 3 3. DESIGN OF RC MOMENT RESISTING FRAMES 3.1 Description of studied buildings Typicl 4-storey reinforced concrete office buildings in ech of the following loctions, i.e., Toronto, Montrel, nd Vncouver, were designed for the purpose of the study. These loctions were selected to represent the low, medium, nd high seismic hzrd zone in Cnd, respectively. In ech loction, three spn lengths were considered, nmely, 4.0 m, 6.0 m, nd 8.0 m in order to sttisticlly nlyze the reltion between force nd deformtion of bems' end sections, which will be discussed in Chpter 4. The storey heights of ll the buildings re 4.0 m. There re four spns in both the longitudinl nd trnsverse directions. For illustrtion, Figure 3.1 shows the pln nd elevtion views of one exmined building with spn length of 6.0 m. The lterl lod resisting system consists of moment-resisting reinforced concrete frmes in both the longitudinl nd the trnsverse directions. There re five frmes in the longitudinl direction (designted Le nd Li in Fig. 3.1; Le exterior frmes, nd Li interior frmes) nd six frmes in trnsverse direction (Te nd Ti). The floor system consists of two-wy slb supported by the bems of the longitudinl nd trnsverse frmes. The slb is cst integrlly with the bems. 31

44 T e T i T i T i T i T e L e L i L i L i \ L e Figure 3.1 Pln of floors nd elevtion of longitudinl frmes of the building (spn = 6.0 m). 32

45 3.2 Design lods Grvity lods For the purpose of design, one of the interior longitudinl frmes (Li) of the buildings ws considered. The grvity lods were determined ccording to the 2010 edition of the Ntionl Building Code of Cnd (NBCC 2010). More specificlly, the superimposed ded lod considered in the design ws 2.0 kp, which included the lods due to floor finishing, mechnicl services, prtitions, nd suspended ceiling. The design live lods were 1.0 kp nd 2.4 kp for the roof nd floors, respectively. It is necessry to mention tht snow lods were lso considered in the design. As n exmple, Tble 3.1 provides the design grvity lods for the frme with the spn length of 6.0 m. Tble 3.1 Design grvity lods (kn/m 2 ). Ded lod Weight of slb 3.75 Roof Weight of bems Weight of columns 1.5 Superimposed 1.5 Totl Weight of slb 3.75 Weight of bems Floor Weight of columns 1.5 Superimposed 2.0 Totl Live lod Seismic lods The lterl lods due to erthquke ground motions were determined in ccordnce with NBCC using the equivlent sttic force procedure. 'Reference' ground conditions, represented by site clss C in NBCC, were ssumed t the building loctions. The seismic 33

46 bse sher force for ech building, V, ws computed ccording to the code formul (Eqution 3.1): V = S(T ) M V I E W R d R o (3.1) The minimum lterl erthquke sher force for moment-resisting frmes, Vmin, should not to be less thn tht provided by Eqution 3.2, V min S(2.0) MV I E W = (3.2) R R d o The mximum lterl erthquke sher force, Vmx should be clculted ccording to Eqution 3.3, V = 2 3 S(0.2) I E W R d R o (3.3) where, S(T) is the design 5% dmped spectrl response ccelertion t the fundmentl lterl period of the building, MV is the higher mode effect fctor, IE is the importnce fctor, W is the totl seismic weight s defined by NBCC ssocited with the frme, Rd is the ductility-relted force modifiction fctor, nd Ro is the overstrength-relted force modifiction fctor. The fundmentl period of the frmes ws computed ccording to the code formul for reinforced concrete moment-resisting frmes, T = 0.075hn 3/4, where hn is the height of the frme bove the bse in meters. The design spectrl ccelertion, S(T), ws determined from the seismic design spectrum for the building loction (Fig. 3.2). The vlues of the other prmeters used in Eqution (3.1), s specified in NBCC, re: MV = 1, IE = 1. Given the seismicity of the building loction, the frmes in Toronto were designed s conventionl frmes (i.e., Rd = 1.5, Ro = 1.3); in Montrel they were designed s 34

47 modertely-ductile frmes (i.e., Rd = 2.5, Ro = 1.4), nd in Vncouver, were designed s ductile frmes (i.e., Rd = 4.0, Ro = 1.7) in ccordnce with NBCC. The weight W includes the self-weight of the frme nd the ded lods corresponding to the tributry res cting on the frme t ll floors, nd 25% of the snow lod is lso dded in the weight W. The design vlues for the fundmentl periods of the building, T, the spectrl ccelertions, S(T), nd the bse sher coefficients, V/W, re listed in Tble Vncouver Montrel Toronto Spectr Respons Accelertion (g) Period (s) Figure 3.2 Seismic design spectr for Toronto, Montrel, nd Vncouver, site clss C, 5% dmping. 35

48 Tble 3.2 Design prmeters for the buildings. Design Prmeter Seismicity Ductile Modertelyductile Conventionl Fundmentl period, T(s) S(T) (g) V/W Mx. Drift (%) For illustrtion of the results of the equivlent sttic force procedure, Figure 3.3 shows the distribution of the seismic sher force long the height of the frmes in Toronto, Montrel, nd Vncouver for the spn length of 6.0 m. As expected, the sher force used for the design of the frme in Vncouver is more decresed thn in Toronto nd Montrel due to its reltively lrger vlue for RdRo. 4 Toronto Montrel Vncouver 3 Storey Bse sher forces (kn) Figure 3.3 Distribution of the seismic force long the height of the building (spn = 6.0 m). 36

49 3.3 Design of frmes The member forces for use in the design were determined by elstic nlyses of the frmes subjected to the combintions of grvity nd seismic lods s specified in NBCC. The computer progrm ETABS ws used in the nlysis. Rigid zones were used t the bem-column joints of the structurl model. The lengths of the rigid zones were selected to be the sme s the depths of the bems nd columns. The effects of crcking were included by using reduced member stiffnesses, i.e., 35% nd 70% of the gross EcI for bems nd columns respectively, where Ec is the modulus of elsticity of concrete (Ec = MP in this study), nd I is the moment of inerti of the member section. The gross EcI for the bems includes the slb thickness s specified in the Cndin stndrd CSA A (CSA 2014). Lod-deflection (P- ) effects were tken into ccount in the nlysis. As specified in NBCC, mximum inelstic inter-storey drifts were clculted s RdRo times the drift obtined from the elstic nlyses. The mximum clculted drifts for the frmes re given in Tble 3.2. It cn be seen tht the clculted drifts re smller thn the design drift of 2.5% llowed by NBCC. The member forces obtined from the elstic nlyses were used in the design of the frmes. The design ws conducted in ccordnce with the requirements for the seismic design of frmes specified in CSA stndrd A (CSA 2014). These requirements re bsed on the cpcity design method. The cpcity method intends to provide strong column - wek bem frme structure in which the inelstic deformtions due to strong seismic motions occur in bems rther thn in columns. In the design, compressive strength of concrete is fc' = 40 MP, nd yield strength of reinforcement is fy = 400 MP. The sher modulus of steel nd concrete re defined to be 76.9 GP nd 12.1 GP, respectively. The 37

50 dimensions of the longitudinl nd trnsverse reinforcement of bems of buildings locted in Toronto, Montrel, nd Vncouver re shown in Tbles 3.3, 3.4, nd 3.5, respectively. A smple of the design is given in Appendix A. Furthermore, Figurers 3.4, 3.5, nd 3.6 illustrte the lyout of the reinforcement in the bems for the three loctions mentioned bove. 3.4 Summry For the purpose of the study, three types of RC moment frme buildings ginst seismic lods were designed ccording to 2010 NBCC, nmely, conventionl (Toronto buildings), modertely-ductile (Montrel buildings), nd ductile (Vncouver buildings). In ech loction, nine frmes were designed with the spn lengths of 4.0 m, 6.0 m, nd 8.0 m, nd different reinforcement rtios (minimum, mximum, nd the verge between these two). Therefore, 27 frmes were designed in order to propose moment-rottion curves for wide rnge of the bems tht will be discussed in Chpter 4. 38

51 Tble 3.3 Dimensions nd the longitudinl nd trnsverse reinforcement of bems in Toronto. Bem Frme type Spn length (m) Bem Trnsvers rebr dimension plstic hinge region (mm) Trnsvers rebr non hinge region Longitudinl rebr in support Longitudinl rebr in mid-spn Top (Tension) Bottom (Compression) Top (Compression) Bottom (Tension) Rebr ρ Rebr ρ Rebr ρ Rebr ρ C1 * Conventionl M@200mm 10M@220mm Minimum 0.5% Minimum 0.2% Minimum 0.0% Minimum 0.2% C2 (Design) Conventionl M@200mm 10M@220mm 1-15M+2-25M 0.5% 2-15M 0.2% 1-15M 0.1% 2-15M+1-25M 0.4% C3 Conventionl M@200mm 10M@220mm 1-30M+2M45 1.5% 2-30M 0.6% 1-30M 0.3% 2-30M+1-30M 0.9% C4 Conventionl M@200mm 10M@220mm Minimum 0.7% Minimum 0.3% Minimum 0.1% Minimum 0.5% C5 (Design) Conventionl M@200mm 10M@220mm 1-10M+2-15M 0.3% 2-10M 0.1% 1-10M 0.1% 2-10M+1-15M 0.3% C6 Conventionl M@200mm 10M@220mm 1-20M+2M45 2.2% 2-25M 0.7% 1-20M 0.2% 2-25M+1-35M 1.3% C7 Conventionl M@200mm 10M@220mm Minimum 0.3% Minimum 0.2% Minimum 0.3% Minimum 0.2% C8 (Design) Conventionl M@200mm 10M@220mm 1-10M+2-15M 0.4% 2-10M 0.2% 1-10M+2-15M 0.4% 2-10M+1-15M 0.3% C9 Conventionl M@200mm 10M@220mm 2-35M 1.7% 2-15M 0.3% 2-35M 1.7% 2-15M+1-30M 0.9% * C stnds for the Conventionl bems. 39

52 Tble 3.4 Dimensions nd longitudinl nd trnsverse reinforcement of bems in Montrel. Bem Frme type Spn length (m) Bem Trnsvers rebr dimension plstic hinge region (mm) Trnsvers rebr non hinge region Longitudinl rebr in support Longitudinl rebr in mid-spn Top (Tension) Bottom (Compression) Top (Compression) Bottom (Tension) M1 * M2 (Design) M3 M4 M5 (Design) M6 M7 M8 (Design) M9 Modertely ductile Modertely ductile Modertely ductile Modertely ductile Modertely ductile Modertely ductile Modertely ductile Modertely ductile Modertely ductile Rebr ρ Rebr ρ Rebr ρ Rebr ρ M@100mm 10M@220mm Minimum 0.5% Minimum 0.2% Minimum 0.1% Minimum 0.3% M@100mm 10M@220mm 1-35M+2-20M 0.7% 2-25M 0.4% 1-35M 0.4% 2-25M 0.4% M@100mm 10M@220mm 2-45M+1-45M 1.9% 2-45M 1.3% 1-45M 0.6% 2-45M 1.3% M@100mm 10M@220mm Minimum 0.3% Minimum 0.1% Minimum 0.1% Minimum 0.1% M@100mm 10M@220mm 1-20M+2-25M 0.9% 2-20M 0.4% 1-20M 0.2% 2-20M+1-10M 0.5% M@100mm 10M@220mm 1-30M+2-35M 1.8% 2-25M 0.7% 1-30M 0.5% 2-25M+1-30M 0.7% M@100mm 10M@220mm Minimum 0.3% Minimum 0.2% Minimum 0.1% Minimum 0.2% M@100mm 10M@220mm 1-10M+2-20M 0.6% 2-20M 0.5% 1-10M+2-20M 0.6% 2-20M 0.5% M@100mm 10M@220mm 1-25M+2-35M 2.1% 2-25M 0.8% 1-25M 0.4% 2-25M+1-20M 1.1% * M stnds for the Modertely ductile bems. 40

53 Tble 3.5 Dimensions nd longitudinl nd trnsverse reinforcement of bems in Vncouver. Bem Frme type Spn length (m) Bem Trnsvers rebr dimension plstic hinge region (mm) Trnsvers rebr non hinge region Longitudinl rebr in support Longitudinl rebr in mid-spn Top (Tension) Bottom (Compression) Top (Compression) Bottom (Tension) Rebr ρ Rebr ρ Rebr ρ Rebr ρ D1 Ductile M@100mm 10M@220mm Minimum 0.3% Minimum 0.2% Minimum 0.2% Minimum 0.2% D2 (Design) Ductile M@100mm 10M@220mm 1-30M+2-25M 0.7% 2-25M 0.4% 1-30M 0.3% 2-25M 0.4% D3 Ductile M@100mm 10M@220mm 1-45M+2-45M 1.9% 2-35M 0.8% 1-45M 0.6% 2-35M+1-30M 1.1% D4 Ductile M@100mm 10M@220mm Minimum 0.5% Minimum 0.1% Minimum 0.1% Minimum 0.2% D5 (Design) Ductile M@100mm 10M@220mm 1-25M+2-20M 0.7% 2-20M 0.4% 1-25M 0.3% 2-20M+1-10M 0.5% D6 Ductile M@100mm 10M@220mm 1-30M+2-35M 1.8% 2-30M 0.9% 1-30M 0.5% 2-30M+1-15M 1.1% D7 Ductile M@100mm 10M@220mm Minimum 0.3% Minimum 0.2% Minimum 0.1% Minimum 0.2% D8 (Design) Ductile M@100mm 10M@220mm 1-25M+2-15M 0.8% 2-15M 0.3% 1-25M 0.4% 2-15M 0.3% D9 Ductile M@100mm 10M@220mm 25M+2-35M 2.1% 2-30M 1.2% 1-25M 0.4% 2-30M 1.2% * D stnds for the Ductile bems. 41

54 2-25M 1-10M 10M 400 mm 8.0 m 2-15M 1-10M 8.0 m 600mm 25M 10M 15M 600mm BEAM ELEVATION CASE C1 TYPICAL BEAM SECTION M 1-15M 15M 400 mm 8.0 m 2-15M 1-25M 8.0 m 600mm 25M 10M 15M 600mm BEAM ELEVATION CASE C2 TYPICAL BEAM SECTION M 1-30M 30M 400 mm 8.0 m 2-30M 1-30M 8.0 m 600mm 45M 10M 30M 600mm BEAM ELEVATION CASE C3 TYPICAL BEAM SECTION - Figure 3.4 Lyout of the reinforcement of the bems in Toronto. 42

55 2-25M 1-10M 10M 300 mm 6.0 m 2-15M 1-20M 6.0 m 500mm 25M 10M 15M 500mm BEAM ELEVATION CASE C4 TYPICAL BEAM SECTION M 1-10M 10M 300 mm 6.0 m 2-10M 1-15M 6.0 m 500mm 15M 10M 15M 500mm BEAM ELEVATION CASE C5 TYPICAL BEAM SECTION M 1-20M 20M 300 mm 6.0 m 2-25M 1-35M 6.0 m 500mm 45M 10M 25M 500mm BEAM ELEVATION CASE C6 TYPICAL BEAM SECTION - Figure 3.4 Lyout of the reinforcement of the bems in Toronto (Continued). 43

56 2-15M 300 mm 4.0 m 2-10M 4.0 m 400mm 15M 10M 10M 400mm BEAM ELEVATION CASE C7 TYPICAL BEAM SECTION M 1-10M 10M 300 mm 4.0 m 2-10M 1-15M 4.0 m 400mm 15M 10M 10M 400mm BEAM ELEVATION CASE C8 TYPICAL BEAM SECTION M 300 mm 4.0 m 2-15M 1-30M 4.0 m 400mm 35M 10M 15M 400mm BEAM ELEVATION CASE C9 TYPICAL BEAM SECTION - Figure 3.4 Lyout of the reinforcement of the bems in Toronto (Continued). 44

57 1150mm 2-25M 1-15M 1150mm 1150mm 15M 400 mm 8.0 m 2-15M 1-15M 8.0 m 600mm 25M 10M 15M 600mm BEAM ELEVATION CASE M1 TYPICAL BEAM SECTION mm 2-20M 1-35M 1150mm 1150mm 35M 400 mm 8.0 m 2-25M 8.0 m 600mm 20M 10M 25M 600mm BEAM ELEVATION CASE M2 TYPICAL BEAM SECTION mm 2-45M 1-45M 1150mm 1150mm 45M 400 mm 8.0 m 2-45M 8.0 m 600mm 45M 10M 45M 600mm BEAM ELEVATION CASE M3 TYPICAL BEAM SECTION - Figure 3.5 Lyout of the reinforcement of the bems in Montrel. 45

58 1150mm 2-15M 1-10M 1150mm 1150mm 10M 300 mm 6.0 m 2-10M 1-10M 6.0 m 500mm 15M 10M 10M 500mm BEAM ELEVATION CASE M4 TYPICAL BEAM SECTION mm 2-25M 1-20M 1150mm 1150mm 20M 300 mm 6.0 m 2-20M 1-10M 6.0 m 500mm 25M 10M 10M 500mm BEAM ELEVATION CASE M5 TYPICAL BEAM SECTION mm 2-35M 1-30M 1150mm 1150mm 30M 300 mm 6.0 m 2-25M 1-30M 6.0 m 500mm 35M 10M 25M 500mm BEAM ELEVATION CASE M6 TYPICAL BEAM SECTION - Figure 3.5 Lyout of the reinforcement of the bems in Montrel (Continued). 46

59 1150mm 2-10M 4.0 m 1-10M 2-10M 1150mm 4.0 m 1150mm 400mm 10M 10M 10M 10M 300 mm 400mm BEAM ELEVATION CASE M7 TYPICAL BEAM SECTION mm 2-20M 4.0 m 1-10M 2-20M 1150mm 4.0 m 1150mm 400mm 20M 10M 10M 20M 300 mm 400mm BEAM ELEVATION CASE M8 TYPICAL BEAM SECTION mm 2-35M 4.0 m 1-25M 2-25M 1150mm 11-20M 4.0 m 1150mm 400mm 25M 35M 10M 25M 300 mm 400mm BEAM ELEVATION CASE M9 TYPICAL BEAM SECTION - Figure 3.5 Lyout of the reinforcement of the bems in Montrel (Continued). 47

60 1150mm 2-20M 8.0 m 1150mm 1-15M 2-15M 1-10M 8.0 m 1150mm 600mm 15M 20M 10M 15M 400 mm 600mm BEAM ELEVATION CASE D1 TYPICAL BEAM SECTION mm 2-25M 8.0 m 1-30M 2-25M 1150mm 8.0 m 1150mm 600mm 30M 25M 10M 25M 400 mm 600mm BEAM ELEVATION CASE D2 TYPICAL BEAM SECTION mm 2-45M 8.0 m 1150mm 1-45M 2-35M 1-30M 8.0 m 1150mm 600mm 45M 45M 10M 35M 400 mm 600mm BEAM ELEVATION CASE D3 TYPICAL BEAM SECTION - Figure 3.6 Lyout of the reinforcement of the bems in Vncouver. 48

61 1150mm 2-20M 1-15M 1150mm 1150mm 15M 300 mm 6.0 m 2-10M 1-10M 6.0 m 500mm 20M 10M 10M 500mm BEAM ELEVATION CASE D4 TYPICAL BEAM SECTION mm 2-20M 1-25M 1150mm 1150mm 25M 300 mm 6.0 m 2-20M 1-10M 6.0 m 500mm 20M 10M 10M 500mm BEAM ELEVATION CASE D5 TYPICAL BEAM SECTION mm 2-35M 1-30M 1150mm 1150mm 30M 300 mm 6.0 m 2-30M 1-15M 6.0 m 500mm 35M 10M 15M 500mm BEAM ELEVATION CASE D6 TYPICAL BEAM SECTION - Figure 3.6 Lyout of the reinforcement of the bems in Vncouver (Continued). 49

62 1150mm 2-10M 1-10M 1150mm 1150mm 10M 300 mm 4.0 m 2-10M 4.0 m 400mm 10M 10M 10M 400mm BEAM ELEVATION CASE D7 TYPICAL BEAM SECTION mm 2-15M 1-25M 1150mm 1150mm 25M 300 mm 4.0 m 2-15M 4.0 m 400mm 15M 10M 15M 400mm BEAM ELEVATION CASE D8 TYPICAL BEAM SECTION mm 2-35M 1-25M 1150mm 1150mm 25M 300 mm 4.0 m 2-30M 4.0 m 400mm 35M 10M 30M 400mm BEAM ELEVATION CASE D9 TYPICAL BEAM SECTION - Figure 3.6 Lyout of the reinforcement of the bems in Vncouver (Continued). 50

63 Chpter 4 4. PROPOSED MODEL FOR RC BEAMS' PLASTIC HINGES 4.1 Introduction It is known tht nonliner dynmic nlysis is the most suitble method for evluting the performnce of building structures ginst progressive collpse. To perform nonliner nlysis, prmeters for modeling the plstic hinges tht might be developed due to the column removl should be defined in ccordnce with 2013 GSA. As described previously, the objective of this study is to investigte ppropriteness of the modelling prmeters provided in GSA. Given this, three-dimensionl (3D) finite element models of the designed buildings described in Chpter 3 were developed using the commercil softwre ABAQUS to ccurtely ssess the geometric nd mteril nonlinerity tht occurred in the bems during the removl of the column. More specificlly, nonliner pushdown nlysis ws conducted on the interior frme C by incresing the displcement (downwrd) t the joint C3 (Fig. 4.1) where the column is removed until the bem collpses. It should be noted tht ABQAUS hs been used in numerous studies on evluting the performnce of RC bems (e.g., Sinei et l. 2012, Deng et l. 2015, Liu et l. 2015, etc.). Other softwre, such s ANSYS hs been lso used to simulte nonliner behviour of concrete bems subjected to column removl (e.g., Ssni nd Kropelnicki 2008, Vlipour nd Foster 2010, etc.) 51

64 Figure 4.1 Criticl frme with column removl. 4.2 Modelling techniques Elements Figure 4.1 schemticlly shows the 3D model developed using ABAQUS for the bem-column ssembly, in which one of the columns is removed. Bems nd columns were modelled using 3D deformble homogeneous solid element C3D8R (i.e., Continuum, 3-D, 8-node, Reduced integrtion). The C3D8R element, with good mesh, provides results with high level of ccurcy nd less computtion time (ABAQUS Anlysis user s mnul). Ech node hs three degrees of freedom long the x, y, nd z xes. 52

65 It is necessry to mention tht filure of side columns, nd the bem-column joint, is not considered in the current ABAQUS model minly becuse of the strong columnwek bem criteri dopted in current seismic design stndrds. Accordingly, the plstic hinges re expected to form in the bems, not in the columns. All the columns re modelled s fixed-fixed in xil, flexurl nd sher rections. Bem-column joints re lso ssumed to be rigid, i.e., the joint undergoes the equl rottion of the corresponding bem. Equl sizes of the meshes re defined in bem-column fces in order to provide the connection between mesh edges. Figure 4.2 Schemtic 3D ABAQUS model of the studied bem-column ssembly Steel brs Steel brs re modeled using 3D truss element T3D2 defined in ABAQUS, which cn tke tensile or compressive lods. T3D2 is 3D spr element hving two nodes with three-degrees of freedom t ech node (i.e., trnsltion in the x, y nd z direction). A perfect 53

66 bond is ssumed between concrete nd steel brs. The longitudinl nd trnsverse reinforcement developed in ABAQUS re shown in Fig Note tht the detiling of the steel brs, e.g., loction of longitudinl nd trnsverse reinforcement, nd the corresponding dimeters were defined to be the sme s those given in the design illustrted in Figs. 3.4 to 3.6 in Chpter 3. At the loction where longitudinl nd trnsverse rebrs intersect, penetrtion of rebrs is ssumed; therefore, no extr interction is required to connect them t points of intersection. Previous reserchers (e.g., Bo et l. 2015, Ahmed 2014, Yu nd Tn 2013, etc.) used similr pproches in modeling the bem reinforcement. Figure 4.3 Stress-strin curve for the longitudinl nd trnsverse steel brs. Figure 4.3 presents the stress-strin curve for the steel brs (for both tensile nd compressive) used in developing ABAQUS models in this study. It cn be seen in the figure tht the curve consists of three segments tht represent the behviour of the steel t three stges, nmely elstic (AB), plstic (BC), nd strin hrdening (CD). Elstic strin is developed before steel reches its yielding strength fy (i.e., Point B), nd it will be fully recovered while unloding. Once the strin exceeds the yielding strin εy, steel will go into 54

67 the plstic stge, i.e., uncovered deformtions re developed. In generl, permnent deformtions re generted in steel followed by strin hrdening. In this study, frcture of steel brs is ssumed to occur when filure strin of 0.2 is reched Concrete The typicl stress-strin reltionship used for concrete mteril under tension nd compression is shown in Fig The curve for compressive concrete is dopted from the model developed by Kent nd Prk (1971) nd it is pplicble for both unconfined nd confined concrete. As seen in the figure, the curve for both unconfined nd confined concrete hs two brnches, i.e., n scending brnch nd descending brnch. Moreover, the curve for the scending brnch is the sme for unconfined nd confined concrete, which is represented by second-order prbolic function. However, the descending brnch is different, i.e., the unconfined concrete follows liner function while the confined concrete follows prbolic function. The stress of concrete for the scending brnch cn be determined using Eqution 4.1 while tht for the descending brnch cn be determined using Eqution 4.2. The prmeter Z in Eqution 4.2 should be clculted bsed on Equtions 4.3 to 4.6 depending on the type of the concrete, i.e., unconfined nd confined concrete. 55

68 Figure 4.4 Stress-strin curve for concrete under compression nd tension. f c 2 ' 2ε c ε f c = c ε co ε co (4.1) [ 1 ( ] f = f Z ε ε ' c c c co (4.2) Z 0.5 = ε ε 50h co (Unconfined concrete) (4.3) Z 0.5 = ε + ε ε 50h 50u co (Confined (concrete) (4.4) ε ' fc = u ' fc (4.5) ε = ε ε 50h 50c 50u (4.6) 56

69 CEB-FIP (1991) proposed biliner curve for uncrcked concrete subjected to tension s defined in Eqution 4.7, which cn be used to the scending brnch of the tensile curve. The crcked concrete follows stright descending line in order to pproximte the descending brnch of tensile behviour. f E t ciε ct < ε ct < E σ ct = ε ct f ε ct f t f t t ft ft f < ε ct < f Eci t t E cie ci ci (4.7) Crcking nd filure of concrete It is known tht RC structures crck t smll lods due to the low tensile strength of concrete. Furthermore, structures become soft when crcking occurs. To consider the effects of crcking on the performnce of structures, three models re defined in ABAQUS, which re the Concrete Smered Crck model (CSC), Brittle Crcking model for Concrete (BCC) nd Concrete Dmged Plsticity model (CDP). In this study, the CDP model ws selected to simulte crcking nd post filure of concrete. The CDP model follows the concepts of isotropic dmge elsticity with isotropic tensile nd compressive plsticity. The two mjor filure mechnisms considered in the model re concrete compressive crshing nd tensile crcking. Figures 4.5 nd 4.5b illustrte the response of concrete subjected to unixil loding in tension nd compression, respectively, which re specified in ABAQUS. As seen in Fig. 4.5, under unixil tension, the stress-strin response of concrete follows liner (i.e., elstic) reltionship until the 57

70 filure stress,, is reched, which corresponds to the onset of micro-crcking in concrete. Beyond, micro crcks become mcro (i.e., reltively lrger) with softening stressstrin response, nd it leds to the formtion of strin locliztion. Wheres under unixil compression (Fig. 4.5b), the response is elstic until the yield stress,. Then the response is plstic, chrcterized by strin hrdening followed by strin softening beyond the ultimte stress,. σ c0 Figure 4.5 Response of concrete to unixil loding: () tension, (b) compression dpted from ABAQUS. It cn be seen in Fig. 4.5 tht when strin softening occurs, the initil elstic stiffness is degrded. In nother word, concrete is dmged. As presented in the figure, the moduli of dmged concrete under tension nd compression cn be estimted by (1 dt)e0 nd (1 dc)e0, respectively. Note tht E0 is the initil (elstic) modulus of undmged concrete while dt nd dc re designted s dmge vribles for concrete under tension nd compression, respectively; nd both re less thn 1.0. According to ABAQUS, the stressstrin reltions for concrete under tension nd compression re expressed in Equtions

71 nd 4.9, respectively. The prmeters re used to define the post filure behviour of concrete. They re referred to s the equivlent tensile plstic strin nd equivlent compressive plstic strin, nd cn be clculted using Equtions 4.10 nd The prmeter represents the crcking strin of the undmged concrete while stnds for the crushing strin of the concrete. pl ( 1 d ) E0 ( ) σ = ε ε t t t t pl ( 1 d ) E0 ( ) σ = ε ε c c c c (4.8) (4.9) ε d σ = ε t 1 d E pl cr t t t t 0 (4.10) d ε = ε c 1 d σ pl in c c c c E0 (4.11) Interction between concrete nd steel brs Mintining the composite ction, which is required to trnsfer lods between concrete nd steel brs, hs been numericlly modeled. This lod trnsfer mechnism is referred to s bond nd cn be idelized continuous field of stress long the steel brs. In this study, the fully bonded interction between concrete nd steel brs hs been considered. It should be mentioned tht the fully bonded flexurl nd sher reinforcement detiling bsed on CSA supports this ssumption Meshing In this study, reltively dense meshes were defined ner the bem supports where plstic hinges re expected to form in order to mke the concrete nd rebr meshes well 59

72 intercted. The contcts between longitudinl nd trnsverse rebrs were defined in such wy tht penetrtion nd sliding re prevented in two orthogonl directions. It should be noted tht the steel brs were modeled s embedded nd fully bonded elements within the concrete block t their cut-off points. Since the size of meshes is reltively sensitive in ABAQUS, sensitivity nlyses were conducted in this study to select the most pproprite size for meshing. The selected mesh sizes of the bem elements rnged from 50 mm to 200 mm with n spect rtio of less thn 4.0 in the orthogonl directions. For the truss elements (for rebrs), the mximum mesh size is 50 mm, nd the meshes re distributed long the xil direction of the elements. Figure 4.6 shows the five cses for the sensitivity nlysis on meshing. The bem considered is the ductile one for the building locted in Vncouver (i.e., bem D5, Tble 3.3, Chpter 3). For this purpose, two-spn frme ws considered. It should be mentioned tht similr configurtion ws lso used by other reserchers to investigte structurl response due to column removl (e.g., Bo et l. 2015, Ssni nd kropelnicki 2008, etc.). In prticulr, the mesh of the concrete bem contins 56 elements in the bem s crosssection, i.e. 7 elements long height nd 8 elements long the width. Efforts hve been mde to use smller mesh in the expected plstic hinge zone (i.e., end of the bems) where reltively hevy trnsverse reinforcement is required for ductile nd modertely-ductile bems ccording to NBCC (2010). More specificlly, the numbers of elements considered in the plstic zone re 5, 8, 11, 22, nd 33 for Cses 1 to 5, respectively. The numbers of bem s mesh elements t the side of the missing column re selected to be the sme s tht in the plstic hinge zone in order to chieve symmetricl meshing bout the middle spn of the bem. Accordingly, the divisions for the remining prts of the bem re 13, 16, 26, 60

73 52, nd 26 for Cses 1 to 5, respectively. In ddition, the preliminry results showed tht meshing of the region outside the plstic hinge zone did not hve significnt effects on the bem response. During the nlysis, verticl (downwrd) lod ws pplied t the loction of the removed column. This lod ws grdully incresed; in the mentime the displcement t the loction of lod pplied ws monitored until the bem(s) filed. Note tht the mplitude of the lod itself ws not importnt in the nlysis. The bending moment t the left fce of the column nd the verticl displcement of the bem t the plce where the lod ws pplied were recorded, nd the results re illustrted in Fig Plese note tht the displcement in Fig. 4.7 is normlized with respect to the bem spn length (6.0 m). The results in the figure clerly show tht the responses provided by Cses 3, 4, nd 5 re lmost the sme while the response given by Cse 1 is the smllest mong the five cses considered. The response provided by Cse 2 might be considered s n verge. Given this, the size of meshing defined in Cse 3 ws selected for further nlyses due to the significntly reduced computtion time compred to Cses 4 nd 5. 61

74 Sym. Plne Sym. Plne Sym. Plne (Cse 1) (Cse 2) (Cse 3) Sym. Plne Sym. Plne (Cse 4) (Cse 5) Figure 4.6 Cses for the sensitivity nlysis on meshing. (Note: only prt of the ssembly is shown, the full ssembly is given in Fig. 4.2) 62