LRFD Software for Design and Actual Ultimate Capacity of Confined Rectangular Columns

Transcription

1 Report No. K-TRAN: KSU-11-3 FINAL REPORT Aprl 013 LRFD Sotware or Degn and Atual Ultmate Capat o Conned Retangular Column Ahmed Mohen ABD El Fattah, Ph.D., LEED AP Hader Raheed, Ph.D., P.E. Aad Emael, Ph.D., P.E. Kana State Unvert Tranportaton Center A ooperatve tranportaton reearh program between Kana Department o Tranportaton, Kana State Unvert Tranportaton Center, and The Unvert o Kana

2 Th page ntentonall let blank.

3 1 Report No. Government Aeon No. 3 Repent Catalog No. K-TRAN: KSU Ttle and Subttle 5 Report Date LRFD Sotware or Degn and Atual Ultmate Capat o Conned Retangular Column Aprl Perormng Organzaton Code 7 Author() Ahmed Mohen ABD El Fattah, Ph.D., LEED AP; Hader Raheed, Ph.D., P.E.; Aad Emael, Ph.D., P.E. 9 Perormng Organzaton Name and Addre Department o Cvl Engneerng Kana State Unvert Tranportaton Center 118 Fedler Hall Manhattan, Kana Sponorng Agen Name and Addre Kana Department o Tranportaton Bureau o Materal and Reearh 700 SW Harron Street Topeka, Kana Perormng Organzaton Report No. 10 Work Unt No. (TRAIS) 11 Contrat or Grant No. C Tpe o Report and Perod Covered Fnal Report Otober 010 September Sponorng Agen Code RE Supplementar Note For more normaton wrte to addre n blok 9. The anal o onrete olumn ung unonned onrete model a well etablhed prate. On the other hand, predton o the atual ultmate apat o onned onrete olumn requre pealzed nonlnear anal. Modern ode and tandard are ntrodung the need to perorm etreme event anal. There ha been a number o tude that oued on the anal and tetng o onentr olumn or lnder. Th ae ha the hghet onnement utlzaton ne the entre eton under onned ompreon. On the other hand, the augmentaton o ompreve trength and dutlt due to ull aal onnement not applable to pure bendng and ombned bendng and aal load ae mpl beaue the area o eetve onned onrete n ompreon redued. The hgher eentrt aue maller onned onrete regon n ompreon eldng maller nreae n trength and dutlt o onrete. Aordngl, the ultmate onned trength graduall redued rom the ull onned value (at zero eentrt) to the unonned value (at nnte eentrt) a a unton o the ompreon area to total area rato. The hgher the eentrt, the maller the onned onrete ompreon zone. Th paradgm ued to mplement adaptve eentr model utlzng the well known Mander Model. Generalzaton o the moment o area approah utlzed baed on proportonal loadng, nte laer proedure and the eant tne approah, n an teratve nremental numeral model to aheve equlbrum pont o P- and M- repone up to alure. Th numeral anal adapted to ae the onnng eet n retangular olumn onned wth onventonal lateral teel. Th model valdated agant epermental data ound n lterature. The omparon how good orrelaton. Fnall omputer otware developed baed on the non-lnear numeral anal. The otware equpped wth an elegant graph nterae that amlate nput data, detal drawng, apat dagram and demand pont mappng n a ngle heet. Opton or prelmnar degn, eton and renorement eleton are eamlel ntegrated a well. The otware generate 3D alure urae or retangular olumn and allow the uer to determne the D nteraton dagram or an angle between the -a and the reultant moment. Improvement to KDOT Brdge Degn Manual ung th otware wth reerene to AASHTO LRFD are made. Th tud lmted to tub olumn. 17 Ke Word Sotware, Column, Brdge, Certed Renored Conrete, LRFD, Load and Retane Fator Degn 18 Dtrbuton Statement No retrton. Th doument avalable to the publ through the Natonal Tehnal Inormaton Serve 19 Seurt Claaton (o th report) Unlaed Form DOT F (8-7) 0 Seurt Claaton (o th page) Unlaed 1 No. o page 0 Pre

4 LRFD Sotware or Degn and Atual Ultmate Capat o Conned Retangular Column Fnal Report Prepared b Ahmed Mohen ABD El Fattah, Ph.D., LEED AP Hader Raheed, Ph.D., P.E. Aad Emael, Ph.D., P.E. Kana State Unvert Tranportaton Center A Report on Reearh Sponored b THE KANSAS DEPARTMENT OF TRANSPORTATION TOPEKA, KANSAS and KANSAS STATE UNIVERSITY TRANSPORTATION CENTER MANHATTAN, KANSAS Aprl 013 Coprght 013, Kana Department o Tranportaton

5 PREFACE The Kana Department o Tranportaton (KDOT) Kana Tranportaton Reearh and New- Development (K-TRAN) Reearh Program unded th reearh projet. It an ongong, ooperatve and omprehenve reearh program addreng tranportaton need o the tate o Kana utlzng aadem and reearh reoure rom KDOT, Kana State Unvert and the Unvert o Kana. Tranportaton proeonal n KDOT and the unverte jontl develop the projet nluded n the reearh program. NOTICE The author and the tate o Kana do not endore produt or manuaturer. Trade and manuaturer name appear heren olel beaue the are ondered eental to the objet o th report. Th normaton avalable n alternatve aeble ormat. To obtan an alternatve ormat, ontat the Oe o Tranportaton Inormaton, Kana Department o Tranportaton, 700 SW Harron, Topeka, Kana or phone (785) (Voe) (TDD). DISCLAIMER The ontent o th report relet the vew o the author who are reponble or the at and aura o the data preented heren. The ontent do not neearl relet the vew or the pole o the tate o Kana. Th report doe not onttute a tandard, peaton or regulaton. v

6 Abtrat The anal o onrete olumn ung unonned onrete model a well etablhed prate. On the other hand, predton o the atual ultmate apat o onned onrete olumn requre pealzed nonlnear anal. Modern ode and tandard are ntrodung the need to perorm etreme event anal. There ha been a number o tude that oued on the anal and tetng o onentr olumn or lnder. Th ae ha the hghet onnement utlzaton ne the entre eton under onned ompreon. On the other hand, the augmentaton o ompreve trength and dutlt due to ull aal onnement not applable to pure bendng and ombned bendng and aal load ae mpl beaue the area o eetve onned onrete n ompreon redued. The hgher eentrt aue maller onned onrete regon n ompreon eldng maller nreae n trength and dutlt o onrete. Aordngl, the ultmate onned trength graduall redued rom the ull onned value (at zero eentrt) to the unonned value (at nnte eentrt) a a unton o the ompreon area to total area rato. The hgher the eentrt, the maller the onned onrete ompreon zone. Th paradgm ued to mplement adaptve eentr model utlzng the well known Mander Model. Generalzaton o the moment o area approah utlzed baed on proportonal loadng, nte laer proedure and the eant tne approah, n an teratve nremental numeral model to aheve equlbrum pont o P- and M- repone up to alure. Th numeral anal adapted to ae the onnng eet n retangular olumn onned wth onventonal lateral teel. Th model valdated agant epermental data ound n lterature. The omparon how good orrelaton. Fnall omputer otware developed baed on the non-lnear numeral anal. The otware equpped wth an elegant graph nterae that amlate nput data, detal drawng, apat dagram and demand pont mappng n a ngle heet. Opton or prelmnar degn, eton and renorement eleton are eamlel ntegrated a well. The otware generate 3D alure urae or retangular olumn and allow the uer to determne the D nteraton dagram or an angle between the -a and the reultant moment. Improvement to KDOT Brdge Degn Manual ung th otware wth reerene to AASHTO LRFD are made. Th tud lmted to tub olumn. v

7 Table o Content Abtrat... v Table o Content... v Lt o Fgure... v Lt o Table... Aknowledgement... v Chapter 1: Introduton Bakground Objetve Sope... Chapter : Lterature Revew Steel Connement Model Chronologal Revew o Model Duon Retangular Column Subjeted to Baal Bendng and Aal Compreon Pat Work Revew Duon Chapter 3: Retangular Column Subjeted to Baal Bendng and Aal Compreon Introduton Unonned Retangular Column Anal Formulaton Anal Approahe Reult and Duon Conned Retangular Column Anal Formulaton Numeral Formulaton Reult and Duon Chapter 4: Conluon and Reommendaton Conluon Reommendaton Reerene Append A: Ultmate Conned Strength Table v

8 Lt o Fgure FIGURE.1 General Stre-Stran Curve b Chan (1955)... 6 FIGURE. General Stre-Stran Curve b Blume et al. (1961)... 7 FIGURE.3 General Stre-Stran Curve b Solman and Yu (1967)... 9 FIGURE.4 Stre-Stran Curve b Kent and Park (1971) FIGURE.5 Stre-Stran Curve b Vallena et al. (1977) FIGURE.6 Propoed Stre-Stran Curve b Wang et al. (1978) FIGURE.7 Propoed Stre-Stran Curve b Muguruma et al. (1980) FIGURE.8 Propoed General Stre-Stran Curve b Shekh and Uzumer (198)... 0 FIGURE.9 Propoed General Stre-Stran Curve b Park et al. (198)... FIGURE.10 Propoed General Stre-Stran Curve b Yong et al. (1988)... 5 FIGURE.11 Stre-Stran Model Propoed b Mander et al. (1988)... 7 FIGURE.1 Propoed General Stre-Stran Curve b Fuj et al. (1988)... 9 FIGURE.13 Propoed Stre-Stran Curve b Saatoglu and Razv ( ) FIGURE.14 Propoed Stre-Stran Curve b Cuon and Paultre (1995) FIGURE.15 Propoed Stre-Stran Curve b Attard and Setunge (1996) FIGURE.16 Mander et al. (1988), Saatoglu and Razv (199) and El-Dah and Ahmad (1995) Model Compared to Cae FIGURE.17 Mander et al. (1988), Saatoglu and Razv (199) and El-Dah and Ahmad (1995) Model Compared to Cae FIGURE.18 Mander et al. (1988), Saatoglu and Razv (199) and El-Dah and Ahmad (1995) Model Compared to Cae FIGURE.19 Relaton between T and C b Anderen (1941) FIGURE.0 Relaton between and bbakhoum (1948) FIGURE.1 Geometr Dmenon n Crevn Anal (1948) v

9 FIGURE. Conrete Center o Preure veru Neutral A Loaton, Mkhalkn FIGURE.3 Steel Center o Preure veru Neutral A Loaton, Mkhalkn FIGURE.4 Bendng wth Normal Compreve Fore Chart np = 0.03, Hu (1955). In H Paper the Graph Were Plotted wth Derent Value o np 0.03,0.1, FIGURE.5 Lnear Relatonhp between Aal Load and Moment or Compreon Falure Whtne and Cohen FIGURE.6 Seton and Degn Chart or Cae 1(r/b = 0.005), Au (1958)... 7 FIGURE.7 Seton and Degn Chart or Cae, Au (1958)... 7 FIGURE.8 Seton and Degn Chart or Cae 3(d /b = 0.7, d /t = 0. 7), Au (1958) FIGURE.9 Graphal Repreentaton o Method One b Breler (1960) FIGURE.30 Graphal Repreentaton o Method Two b Breler (1960) FIGURE.31 Interaton Curve Generated rom Equatng and b Breler (1960) FIGURE.3 Fve Cae or the Compreon Zone Baed on the Neutral A Loaton Czernak (196) FIGURE.33 Value or N or Unequal Steel Dtrbuton b Pannell (1963) FIGURE.34 Degn Curve b Flemng et al. (1961) FIGURE.35 Relaton between and b Ramamurth (1966) FIGURE.36 Baal Moment Relatonhp b Parme et al. (1966) FIGURE.37 Baal Bendng Degn Contant (Four Bar Arrangement) b Parme et al. (1966) FIGURE.38 Baal Bendng Degn Contant (Eght Bar Arrangement) b Parme et al. (1966) FIGURE.39 Baal Bendng Degn Contant (Twelve Bar Arrangement) b Parme et al. (1966) FIGURE.40 Baal Bendng Degn Contant ( Bar Arrangement) b Parme et al. (1966) v

10 FIGURE.41 Smpled Interaton Curve b Parme et al. (1966)... 9 FIGURE.4 Workng Stre Interaton Dagram or Bendng about X-A b Mlona (1967) FIGURE.43 Comparon o Steel Stre Varaton or Baal Vendng When = 30 and q = FIGURE.44 Non Dmenonal Baal Contour on Quarter Column b Talor and Ho (1984) FIGURE.45 P u /P uo to A Relaton or 4bar Arrangement b Hartle (1985) (let) Non- Dmenonal Load Contour (Rght) FIGURE 3.1 a) Ung Fnte Flament n Anal b)trapezodal Shape o Compreon Zone FIGURE 3. a) Stre-Stran Model or Conrete b Hognetad b) Steel Stre-Stran Model FIGURE 3.3 Derent Stran Prole Due to Derent Neutral A Poton FIGURE 3.4 Denng Stran or Conrete Flament and Steel Rebar rom Stran Prole FIGURE 3.5 Flament and Steel Rebar Geometr Properte wth Repet to Cruhng Stran Pont and Geometr Centrod FIGURE 3.6 Method One Flowhart or the Predened Ultmate Stran Prole Method Fgure 3.7 D Interaton Dagram rom Approah One Beore and Ater Correton FIGURE 3.8 Tranerrng Moment rom Centrod to the Geometr Centrod FIGURE 3.9 Geometr Properte o Conrete Flament and Steel Rebar wth Repet to Geometr Centrod and Inelat Centrod FIGURE 3.10 Radal Loadng Conept FIGURE 3.11 Moment Tranerrng rom Geometr Centrod to Inelat Centrod FIGURE 3.1 Flowhart o Generalzed Moment o Area Method Ued or Unonned Anal

11 FIGURE 3.13 Comparon o Approah One and Two ( = 0) FIGURE 3.14 Comparon o Approah One and Two ( = 4.7) FIGURE 3.15 Comparon o Approah One and Two ( = 10.8) FIGURE 3.16 Comparon o Approah Oone and Two ( = 5) FIGURE 3.17 Column Geometr Ued n Sotware Comparon FIGURE 3.18 Unonned Curve Comparon between KDOT Column Epert and SP Column ( = 0) FIGURE 3.19 Degn Curve Comparon between KDOT Column Epert and CSI Col 8 Ung ACI Reduton Fator FIGURE 3.0 Degn Curve Comparon between KDOT Column Epert and SP Column Ung ACI Reduton Fator FIGURE 3.1 a) Ung Fnte Flament n Anal b)trapezodal Shape o Compreon Zone FIGURE 3. Aal Stre-Stran Model Propoed b Mander et al. (1988) or Monoton Loadng FIGURE 3.3 Eetvel Conned Core or Retangular Hoop Renorement (Mander Model) FIGURE 3.4 Eetve Lateral Conned Core or Retangular Cro Seton FIGURE 3.5 Conned Strength Determnaton FIGURE 3.6 Eet o Compreon Zone Depth on Conrete Stre FIGURE 3.7 Amount o Connement Engaged n Derent Cae FIGURE 3.8 Normalzed Eentrt veru Compreon Zone to Total Area Rato (Apet Rato 1:1) FIGURE 3.9 Normalzed Eentrt veru Compreon Zone to Total Area Rato (Apet Rato :1)

12 FIGURE 3.30 Normalzed Eentrt veru Compreon Zone to Total Area Rato (Apet Rato 3:1) FIGURE 3.31 Normalzed Eentrt veru Compreon Zone to Total Area Rato (Apet Rato 4:1) FIGURE 3.3 Cumulatve Chart or Normalzed Eentrt agant Compreon Zone Rato (All Data Pont) FIGURE 3.33 Eentrt Baed Conned Mander Model FIGURE 3.34 Eentr Baed Stre-Stran Curve Ung Compreon Zone Area to Gro Area Rato FIGURE 3.35 Eentr Baed Stre-Stran Curve Ung Normalzed Eentrt Intead o Compreon Zone Rato FIGURE 3.36 Tranerrng Moment rom Centrod to the Geometr Centrod FIGURE 3.37 Geometr Properte o Conrete Flament and Steel Rebar wth Repet to Cruhng Stran Pont, Geometr Centrod and Inelat Centrod FIGURE 3.38 Radal Loadng Conept FIGURE 3.39 Moment Tranerrng rom Geometr Centrod to Inelat Centrod FIGURE 3.40 Flowhart o Generalzed Moment o Area Method FIGURE 3.41 Hognetad Column FIGURE 3.4 Comparon between KDOT Column Epert wth Hognetad Eperment ( = 0) FIGURE 3.43 Breler Column FIGURE 3.44 Comparon between KDOT Column Epert wth Breler Eperment ( = 90) 17 FIGURE 3.45 Comparon between KDOT Column Epert wth Breler Eperment ( = 0). 173 FIGURE 3.46 Ramamurth Column FIGURE 3.47 Comparon between KDOT Column Epert wth Ramamurth Eperment ( = 6.5)

13 FIGURE 3.48 Saatoglu Column FIGURE 3.49 Comparon between KDOT Column Epert wth Saatoglu et al. Eperment ( = 0) FIGURE 3.50 Saatoglu Column FIGURE 3.51 Comparon between KDOT Column Epert wth Saatoglu et al. Eperment 1 ( = 0) FIGURE 3.5 Sott Column FIGURE 3.53 Comparon between KDOT Column Epert wth Sott et al. Eperment ( = 0) FIGURE 3.54 Sott Column FIGURE 3.55 Comparon between KDOT Column Epert wth Sott et al. Eperment ( = 0) FIGURE 3.56 T and C Merdan Ung Equaton 3.18 and ued n Mander Model or = 4.4 k FIGURE 3.57 T and C Merdan or = 3.34 k FIGURE 3.58 T and C Merdan or = 3.9 k FIGURE 3.59 T and C Merdan or = 5. k

14 Lt o Table TABLE.1 Lateral Steel Connement Model Comparon TABLE. Epermental Cae Properte TABLE 3.1 Data or Contrutng T and C Merdan Curve or Equal to 3.34 k TABLE 3. Data or Contrutng T and C Merdan Curve or Equal to 3.9 k TABLE 3.3 Data or Contrutng T and C Merdan Curve or Equal to 5. k

15 Aknowledgement Th reearh wa made poble b undng rom KDOT. Speal thank are etended to John Jone, Loren Rh, and Ken Hurt o KDOT, or ther nteret n the projet and ther ontnuou upport and eedbak that made t poble to arrve at uh mportant ndng v

16 Chapter 1: Introduton 1.1 Bakground Column are ondered the mot rtal element n truture. The unonned anal or olumn well etablhed n the lterature. Strutural degn ode dtate reduton ator or aet. It wan t untl ver reentl that degn peaton and ode o prate, lke AASHTO LRFD, tarted realzng the mportane o ntrodung etreme event load ae that neetate aountng or advaned behavoral apet lke onnement. Connement add another dmenon to olumn anal a t nreae the olumn apat and dutlt. Aordngl, onnement need peal non lnear anal to eld aurate predton. Neverthele the lterature tll lakng pealzed anal tool that take nto aount onnement depte the avalablt o all knd o onnement model. In addton the lterature ha oued on aall loaded member wth le attenton to eentr loadng. Although the latter more lkel to our, at leat wth malgnment tolerane, the eentrt eet not ondered n an onnement model avalable n the lterature. It wdel known that ode Speaton nvolve ver detaled degn proedure that need to be heked or a number o lmt tate makng the tak o the degner ver tedou. Aordngl, t mportant to develop otware that gude through the degn proe and altate the preparaton o relable anal/degn doument. 1. Objetve Th tud ntended to determne the atual apat o onned renored onrete olumn ubjeted to eentr loadng and to generate the alure envelope at three derent level. Frt, the well-known ultmate apat anal o unonned onrete developed a a benhmarkng tep. Seondl, the unonned ultmate nteraton dagram aled down baed on the reduton ator o the AASHTO LRFD to the degn nteraton dagram. Fnall, the atual onned onrete ultmate anal developed baed on a new eentrt model aountng or partal onnement eet under eentr loadng. The anale are onduted or retangular olumn onned wth onventonal tranvere teel. It mportant to note that the 1

17 preent anal proedure wll be benhmarked agant a wde range o epermental and analtal tude to etablh t aura and relablt. It alo the objetve o th tud to urnh nteratve otware wth a uer-rendl nterae havng anal and degn eature that wll altate the prelmnar degn o rular olumn baed on the atual demand. The overall objetve behnd th reearh are ummarzed n the ollowng pont: Introdue the eentrt eet n the tre-tran modelng Implement non-lnear anal or onderng the onnement eet on olumn atual apat Tet the anal or retangular olumn onned wth onventonal tranvere teel. Generate omputer otware that help n degnng and analzng onned onrete olumn through reatng three level o Moment-Fore envelope; unonned urve, degn urve baed on AASHTO-LRFD and onned urve. 1.3 Sope Th tud ompoed o our hapter overng the development o materal model, anal proedure, benhmarkng and pratal applaton. Chapter one ntrodue the objetve o the tud and the ontent o the derent hapter. Chapter two revew the lterature through two ndependent eton: Seton 1: Renored onrete onnement model Seton : Retangular Column ubjeted to baal bendng and Aal Compreon Chapter three preent retangular olumn anal or both the unonned and onned ae. Chapter three addree the ollowng ubjet: o Fnte Laer Approah (Fber Model) o Preent Connement Model or Conentr Column o Preent Connement Model or Eentr Column

18 o Moment o Area Theorem o Numeral Formulaton o Reult and Duon Chapter our tate the onluon and reommendaton. 3

19 Chapter : Lterature Revew Th hapter revew two derent top; lateral teel onnement model and retangular olumn ubjeted to baal bendng and aal ompreon..1 Steel Connement Model A omprehenve revew o onned model or onrete olumn under onentr aal ompreon that are avalable n the lterature onduted. The model revewed are hronologall preented then ompared b a et o rtera that ae onderaton o derent ator n developng the model uh a eetvel onned area, eldng trength and dutlt..1.1 Chronologal Revew o Model The onnement model avalable are preented hronologall regardle o ther omparatve mportane rt. Ater that, duon and ategorzaton o the model are arred out and onluon are made. Common notaton ued or all the equaton or the ake o onten and omparon Notaton A : the ro etonal area o longtudnal teel renorement A t: the ro etonal area o tranvere teel renorement A e: the area o eetvel onned onrete A : the area o ore wthn enterlne o permeter pral or hoop eludng area o longtudnal teel b: the onned wdth (ore) o the eton h: the onned heght (ore) o the eton : enter-to-enter dtane between longtudnal bar d : the dameter o longtudnal renorement d t : the dameter o tranvere renorement D: the dameter o the olumn 4

20 d the ore dameter o the olumn : the mamum onned trength : the peak unonned trength l: the lateral onned preure l: the eetve lateral onned preure h: the eld trength o the tranvere teel : the tre n the lateral onnng teel k e: the eetve lateral onnement oeent q: the eetvene o the tranvere renorement : te pang o: the vertal pang at whh tranvere renorement not eetve n onrete onnement o: the tran orrepondng to the peak unonned trength : the tran orrepondng to the peak onned trength ε : the tran at eldng or the tranvere renorement ε u: the ultmate tran o onned onrete ρ : the volumetr rato o lateral teel to onrete ore ρ l: the rato o longtudnal teel to the gro etonal area ρ: the volumetr rato o lateral + longtudnal teel to onrete ore Rhart, Brandtzaeg, and Brown (199) Rhart et al. (199) model wa the rt to apture the proportonal relatonhp between the lateral onned preure and the ultmate ompreve trength o onned onrete. k1 l Equaton.1 5

21 The average value or the oeent k 1, whh wa derved rom a ere o hort olumn pemen tet, ame out to be (4.1). The tran orrepondng to the peak trength (ee Mander et al. 1988) obtaned ung the ollowng unton: o 1 k l 5k1 k Equaton. where o the tran orrepondng to, k the tran oeent o the eetve lateral onnement preure. No tre-tran urve graph wa propoed b Rhart et al. (199). Chan (1955) A tr-lnear urve derbng the tre-tran relatonhp wa uggeted b Chan (1955) baed on epermental work. The rato o the volume o teel te to onrete ore volume and onrete trength were the onl varable n the epermental work done. Chan aumed that OA appromate the elat tage and ABC appromate the plat tage (Fgure.1). The poton o A, B and C ma var wth derent onrete varable. Chan aumed three derent lope E, E, E or lne OA, AB and BC repetvel. However no normaton about and wa provded. u p e 1E A 1E B E E C Stre O e p Stran u FIGURE.1 General Stre-Stran Curve b Chan (1955) 6

22 Blume, Newmark, and Cornng (1961) Blume et al. (1961) were the rt to mpoe the eet o the eld trength or the tranvere teel h n derent equaton denng the model. The model generated, Fgure., ha aendng traght lne wth teep lope tartng rom the orgn tll the plan onrete peak trength and the orrepondng tran ε o, then a le lope traght lne onnet the latter pont and the onned onrete peak trength and ε. Then the urve latten tll ε u A t h or retangular olumn Equaton.3 h p 10 p Equaton.4 o 6 5 Equaton.5 u 5 u Equaton Stre o u Stran FIGURE. General Stre-Stran Curve b Blume et al. (1961) 7

23 where ε the tran at eldng or the tranvere renorement, A t the ro etonal area o tranvere teel renorement, h the onned ro etonal heght, u the tran o tranvere pral renorement at mamum tre and u the ultmate onrete tran. Ro and Sozen (1965) Baed on ther epermental reult, whh were ontrolled b two varable; te pang and amount o longtudnal renorement, Ro and Sozen (1965) onluded that there no enhanement n the onrete apat b ung retlnear te. On the other hand there wa gnant nreae n dutlt. The propoed a blnear aendng-deendng tre tran urve that ha a peak o the mamum trength o plan onrete and orrepondng tran o wth a value o The eond lne goe through the pont dened b 50 tll t nteret wth the tran a. The tran ε 50 wa uggeted to be a unton o the volumetr rato o te to onrete ore ρ, te pang and the horter de dmenon b (ee Shekh 198). 3 b 50 Equaton.7 4 Solman and Yu (1967) Solman and Yu (1967) propoed another model that emerged rom epermental reult. The man parameter nvolved n the work done were te pang, a new term repreent the eetvene o te o, the area o te A t, and nall eton geometr, whh ha three derent varable; A the area o onned onrete under ompreon, A the area o onrete under ompreon and b. The model ha three derent porton a hown n Fgure.3. The aendng porton whh repreented b a urve tll the peak pont (, ε e ). The lat traght-lne porton wth t length varng dependng on the degree o onnement. The lat porton a deendng traght lne pang through (0.8, ε ) then etendng down tll an ultmate tran. q A 1.4 A 0.45 A At o 0.008B Equaton.8 t 8

24 q Equaton e 0.55 *10 Equaton (1 q) Equaton (1 0.85q) Equaton.1 where q reer to the eetvene o the tranvere renorement, o the vertal pang at whh tranvere renorement not eetve n onrete onnement and B the greater o b and 0.7 h. 0.8 Stre e Stran FIGURE.3 General Stre-Stran Curve b Solman and Yu (1967) Sargn (1971) Sargn onduted epermental work on low and medum trength onrete wth no longtudnal renorement. The tranvere teel that wa ued had derent ze and derent eld and ultmate trength. The man varable aetng the reult were the volumetr rato o lateral renorement to onrete ore ρ, the trength o plan onrete, the rato o te pang to the wdth o the onrete ore and the eld trength o the tranvere teel h. 9

25 A m 1 k 1 ( A ) m 3 Equaton.13 where m a ontant ontrollng the lope o the deendng branh: m Equaton Equaton.15 A E Equaton.16 k 3 k b h Equaton b h Equaton.18 k3 Equaton.19 where k 3 onentr loadng mamum tre rato. Kent and Park (1971) A Ro and Sozen (1965) dd, Kent and Park (1971) aumed that the mamum trength or onned and plan onrete the ame. The uggeted urve, Fgure.4, tart rom the orgn then nreae parabolall (Hognetad Parabola) tll the peak at and the orrepondng tran o at Then t deend wth one o two derent traght lne. For the onned onrete, whh more dutle, t deend tll the pont (0.5, ε 50 ) and ontnue deendng to 0. ollowed b a lat plateau. For the plan onrete t deend tll the pont (0.5, ε 50u ) and ontnue deendng to 0. a well wthout a lat plateau. Kent and Park aumed that onned onrete ould utan tran to nnt at a ontant tre o 0. : 10

26 o o or aendng branh o 1 Z or deendng branh Equaton u Equaton.1 h b A hb t Equaton. 50h 50 50u Equaton b 50h Equaton.4 Z 50h u o Equaton.5 where ρ the rato o lateral teel to the onrete ore, Z a ontant ontrollng the lope o deendng porton. Stre u Stran 50 0 FIGURE.4 Stre-Stran Curve b Kent and Park (1971) 11

27 Popov (1973) Popov ponted out that the tre-tran dagram nluened b tetng ondton and onrete age. The tre equaton : n n 1 n Equaton.6 3 n 0.4* Equaton.7 Equaton.8.7 * Vallena, Bertero, and Popov (1977) The varable utlzed n the epermental work onduted b Vallena et al. (1977) were the volumetr rato o lateral teel to onrete ore ρ, rato o longtudnal teel to the gro area o the eton ρ l, te pang, eetve wdth ze, trength o te and ze o longtudnal bar. The model generated wa mlar to Kent and Park model wth mprovement n the peak trength or onned onrete (Fgure.5). For the aendng branh: k 1 Z ( 1) 0. 3k Equaton k.3k 0 Equaton.30 Equaton.31 k Equaton.3 1

28 E E k 1 k Equaton.33 For the deendng branh: d t l d k h h Equaton h h Equaton.35 Z 3 4 h Equaton.36 where k oeent o onned trength rato, Z the lope o deendng porton, d and d t are the dameter o longtudnal and tranvere renorement, repetvel. Aal Stre 0.3k Aal Stran 0.3k FIGURE.5 Stre-Stran Curve b Vallena et al. (1977) 13

29 Wang, Shah, and Naaman (1978) Wang et al. (1978) obtaned epermentall another tre-tran urve derbng the behavor o onned renored onrete under ompreon, Fgure.6. The onrete teted wa normal weght onrete rangng n trength rom 3000 to p (0.7 to 75.8 MPa) and lght weght onrete wth trength o p (0.7 to 55 MPa). Wang et al. utlzed an equaton, wth our ontant, mlar to that o Sargn et al. Y AX BX 1 CX DX Equaton.37 where Y Equaton.38 X Equaton.39 The our ontant A, B, C, D were evaluated or the aendng part ndependentl o the deendng one. The our ondton ued to evaluate the ontant or the aendng part were dy/dx = E 0.45 /E e at X=0 E e = / Y = 0.45 or X = 0.45/(E 0.45 /E e ) Y=1 or X=1 dy/dx = 0 at X=1 wherea or the deendng branh: Y=1 or X=1 dy/dx = 0 at X=1 Y = / or X = / 14

30 Stre 0.45 Stran FIGURE.6 Propoed Stre-Stran Curve b Wang et al. (1978) where and are the tre and tran at the nleton pont, and reer to a pont uh that and E 0.45 repreent the eant modulu o elatt at 0.45 Y = / or X = / Muguruma, Watanabe, Katuta, and Tanaka (1980) Muguruma et al. (1980) obtaned ther tre-tran model baed on epermental work onduted b the model author, Fgure.7. The tre-tran model dened b three zone; Zone 1 rom 0-A: E o E o (kg/m ) 0 o Equaton.40 Zone rom A-D o (kg/m ) o Equaton.41 15

31 Zone 3 rom D-E u u u (kg/m ) u Equaton.4 S u (kg/m ) Equaton / 000 (kg/m ) Equaton.44 u C 1 0. W h 5 Equaton.45 where S the area urrounded b the dealzed tre-tran urve up to the peak tre and W the mnmum de length or dameter o onned onrete For rular olumn onned wth rular hoop: 1150C (kg/m ) Equaton.46 1 C o Equaton u 1 C u Equaton wherea or quare olumn onned wth quare hoop: 1 50C (kg/m ) Equaton.49 1 C o Equaton u 1 C u Equaton

32 Aal Stre u D u o u u Aal Stran FIGURE.7 Propoed Stre-Stran Curve b Muguruma et al. (1980) Sott, Park, Pretl (198) Sott et al. (198) eamned pemen b loadng at hgh tran rate to orrelate wth the em loadng. The preented the reult nludng the eet o eentr loadng, tran rate, amount and dtrbuton o longtudnal teel and amount and dtrbuton o tranvere teel. For low tran rate Kent and Park equaton were moded to t the epermental data k 0.00k 0.00k 0. 00k Equaton.5 k 1 Z ( 0.00k) 0. 00k m Equaton.53 where k 1 h Equaton.54 Z m b" 0.00k n MPa Equaton.55 17

33 where b the wdth o onrete ore meaured to outde o the hoop. For the hgh tran rate, the k and Z m were adapted to k h 1.5(1 ) Equaton.56 Z m b 0.00k n MPa Equaton.57 and the mamum tran wa uggeted to be: h u Equaton.58 It wa onluded that nreang the pang whle mantanng the ame rato o lateral renorement b nreang the dameter o pral, redue the een o onrete onnement. In addton, nreang the number o longtudnal bar wll mprove the onrete onnement due to dereang the pang between the longtudnal bar. Shekh and Uzumer (198) Shekh and Uzumer (198) ntrodued the eetvel onned area a a new term n determnng the mamum onned trength (Solman and Yu (1967) had tral n eetve area ntroduton). In addton to that the, n ther epermental work, utlzed the volumetr rato o lateral teel to onrete ore, longtudnal teel dtrbuton, trength o plan onrete, and te trength, onguraton and pang. The tre-tran urve, Fgure.8, wa preented parabolall up to (, ε ), then t latten horzontall tll ε, and nall t drop lnearl pang b (0.85, ε 85 ) tll 0.3, In that ene, t oneptuall mlar to the earler model o Solman and Yu (1967). and ε an be determned rom the ollowng equaton: 18

34 k p k p p k Equaton.59 p k.73b n t Po 5.5b b Equaton k *10 Equaton o b t Equaton b 85 Equaton.63 Z 0.5 Equaton b where b the onned wdth o the ro eton, t the tre n the lateral onnng bar, enter-to-enter dtane between longtudnal bar, 85 the value o tran orrepondng to 85% o the mamum tre on the unloadng branh, n the number o laterall upported longtudnal bar, Z the lope or the unloadng part, p the equvalent trength o unonned onrete n the olumn, and P o = K p (A - A ) 19

35 Stre 85 Stran FIGURE.8 Propoed General Stre-Stran Curve b Shekh and Uzumer (198) Ahmad and Shah (198) Ahmad and Shah (198) developed a model baed on the properte o hoop renorement and the onttutve relatonhp o plan onrete. Normal weght onrete and lghtweght onrete were ued n tet that were onduted wth one rate o loadng. No longtudnal renorement wa provded and the man two parameter vared were pang and eld trength o tranvere renorement. Ahmed and Shah oberved that the pral beome neetve when the pang eeed 1.5 the dameter o the onned onrete olumn. The onluded alo that the eetvene o the pral nverel proportonal wth ompreve trength o unonned onrete. Ahmad and Shah adapted Sargn model ountng on the otahedral alure theor, the three tre nvarant and the epermental reult: Y A X ( D 1) X 1 ( A ) X D X Equaton.65 p Y Equaton.66 pn 0

36 X Equaton.67 p where p the mot prnpal ompreve tre, pn the mot prnpal ompreve trength, the tran n the -th prnpal dreton and p the tran at the peak n the -th dreton. A E E p E p pn p E the ntal lope o the tre tran urve, D a parameter that govern the deendng branh. When the aal ompreon ondered to be the man loadng, whh tpall the ae n onentr onned onrete olumn, Equaton.65,.66, and.67 beome: Y AX ( D 1) X 1 ( A ) X DX Equaton.68 Y Equaton.69 X Equaton.70 A E Equaton.71 E e Park, Pretl, and Gll (198) Park et al. (198) moded Kent and Park (1971) equaton to aount or the trength mprovement due to onnement baed on epermental work onduted or our quare ull aled olumn (1.7 n ( mm ) ro etonal area and 10.8 t (39 mm) hgh (Fgure.9). The propoed equaton are a ollow: 1

37 k 0.00k 0.00k or aendng branh Equaton.7 k 1 Z 0.k m or deendng branh Equaton.73 o Z m b 0.00k Equaton.74 k 1 h Equaton.75 m C Aal Stre 0.00K Aal Stran FIGURE.9 Propoed General Stre-Stran Curve b Park et al. (198) Martnez, Nlon, and Slate (1984) Epermental nvetgaton wa onduted to propoe equaton to dene the tre tran urve or prall renored hgh trength onrete under ompreve loadng. The man parameter ued were ompreve trength or unonned onrete, amount o onnement and pemen ze. Two tpe o onrete where ued; normal weght onrete wth trength to about

38 1000 p. (8.75 MPa) and lght weght onrete wth trength to about 9000 p (6 MPa). Martnez et al. (1984) onluded that the degn peaton or low trength onrete mght be unae appled to hgh trength onrete. For normal weght onrete: (1 ) Equaton.76 4 l d t and or lght weght onrete:.8 (1 ) Equaton.77 1 l d t where d t the dameter o the lateral renorement. Fat and Shah (1985) Fat and Shah (1985) aumed that the mamum apat o onned onrete our when the over tart to pall o. The epermental work wa done on hgh trength onrete wth varng the onnement preure and the onrete trength. Two equaton are propoed to epre the aendng and the deendng branhe o the model. For the aendng branh: 1 A 1 0 Equaton.78 and or the deendng branh: 1.15 ep k Equaton.79 The equaton or the ontant A and k: A E Equaton.80 3

39 k 0.17 ep l Equaton.81 and ε an be ound ung the ollowng equaton: Equaton.8 l 7 l 1.07* Equaton.83 l repreent the onnement preure and gven b the ollowng equaton: l At h or rular olumn Equaton.84 d l At h or quare olumn Equaton.85 d e d the ore dameter o the olumn and d e the equvalent dameter. Yong, Nour, and Naw (1988) The model uggeted b Yong et al. (1988) wa baed on epermental work done or retangular olumn wth retangular te (Fgure.10). K Equaton h / 3 h Equaton.87 4

40 0.45 K h nd 8d t h l Equaton Equaton.89 K K Equaton Equaton.91 Mander, Pretl and Park (1988) Ung the ame onept o eetve lateral onnement preure ntrodued b Shekh and Uzumer, Mander et al. (1988) developed a new onned model or rular pral and hoop or retangular te (Fgure.11). In addton Mander et al. (1988) wa the eond group ater Bazant et al. (197) to nvetgate the eet o the l load de b de wth monoton one. Stre 0.45 Stran FIGURE.10 Propoed General Stre-Stran Curve b Yong et al. (1988) 5

41 7.94 l l Equaton Equaton.93 o l 1 ke h k e l Equaton.94 r r 1 r Equaton.95 r E Equaton.96 E E e Equaton.97 where k e the eetve lateral onnement oeent: A e ke Equaton.98 A A e the area o eetvel onned onrete, E e = / and A area o ore wthn enterlne o permeter pral or hoop eludng area o longtudnal teel. k e 1 d 1 For rular hoop Equaton.99 6

42 k e 1 d 1 For rular pral Equaton.100 k e 1 n 1 w 1 6b h b 1 1 h For retangular te Equaton.101 where the lear pang, the the rato o longtudnal renorement to the ore area w the um o the quare o all the lear pang between adjaent longtudnal teel bar n a retangular eton. Mander et al. (1988) propoed alulaton or the ultmate onned onrete tran u baed on the tran energ o onned onrete. E e Stre o Stran u FIGURE.11 Stre-Stran Model Propoed b Mander et al. (1988) Fuj, Kobaah, Magawa, Inoue, and Matumoto (1988) Fuj et al. (1988) developed a tre tran relaton b unaal tetng o rular and quare pemen o 150 mm wde and 300 mm tall (Fgure.1). The teted pemen dd not have longtudnal bar and no over. The propoed tre tran model ha our regon; 7

43 Regon 1 rom 0-A E o E o 0 o Equaton.10 Regon rom A-B 3 3 o o Equaton.103 Regon 3 rom B-C 0 Equaton.104 Regon 4 rom C-end 0. 0 Equaton.105 Fuj et al. (1988) dened three onnement oeent or mamum tre C, orrepondng tran C u and tre degradaton gradent C. For rular pemen, the peak trength and orrepondng tran are a ollow: 1.75C 1.0 Equaton.106 o C u Equaton C 574 Equaton.108 8

44 C h d Equaton.109 C d h u Equaton C 70 Equaton.111 C 1 h Equaton.11 C 1 h h u Equaton.113 h C 1/ / Equaton.114 The howed that the propoed model ha hgher aura than Park et al. (198) model ompared to the epermental work done b Fuj et al. (1988). B Aal Stre 0. A 0 o 0 Aal Stran FIGURE.1 Propoed General Stre-Stran Curve b Fuj et al. (1988) 9

45 Saatoglu and Razv (199) Saatoglu and Razv (199) onluded that the pave lateral preure generated b laterall epandng onrete and retranng tranvere renorement not alwa unorm. Baed on tet on normal and hgh trength onrete rangng rom 30 to 130 MPa, Saatoglu and Razv propoed a new model (Fgure.13) that ha eponental relatonhp between the lateral onnement preure and the peak onnement trength. The ran tet b varng volumetr rato, pang, eld trength, arrangement o tranvere renorement, onrete trength and eton geometr. In addton, the gnane o mpong the te arrangement a a parameter n determnng the peak onned trength wa hghlghted k1 or rular ro eton Equaton.115 l k Equaton ( l ) l A h or rular ro eton Equaton.117 d k1 or retangular ro eton Equaton.118 le le k l Equaton.119 l A h n or quare olumn Equaton.10 h h h l k Equaton.11 le leb leh or retangular olumn Equaton.1 b h 30

46 k 1 o 1 5 le Equaton.13 For the tre tran urve A b h Equaton.14 where 085 the tran at 0.85 or the unonned onrete k 1 le 1/ 1 Equaton.15 where pang o longtudnal renorement and the angle between the tranvere renorement and b Stre o 0 Stran FIGURE.13 Propoed Stre-Stran Curve b Saatoglu and Razv ( ) 31

47 Shekh and Tokluu (1993) Shekh and Tokluu (1993) tuded the dutlt and trength or onned onrete and the onluded that dutlt more entve, than the trength, to amount o tranvere teel, and the nreae n onrete trength due to onnement wa oberved to be between.1 and 4 tme the lateral preure. Karabn and Kou (1994) Karabn and Kou (1994) utlzed the theor o platt n evaluatng the development o lateral onnement n onrete olumn. However, no tre-tran equaton were propoed Hu and Hu (1994) Hu and Hu (1994) moded Carrera and Chu (1985) equaton that wa developed or unonned onrete, to propoe an empral tre tran equaton or hgh trength onrete. The onrete trength equaton : 1 or 0 Equaton.16 d Equaton.17 1 or 1 1 E Equaton.18 where andare materal properte.depend on the hape o the tre tran urve and depend on materal trength and t taken equal toand d the tran at 0.6. n the deendng porton o the urve. 3

48 Raheed and Dnno (1994) Raheed and Dnno (1994) ntrodued a ourth degree polnomal to epre the tre tran urve o onrete under ompreon. a a a a a Equaton.19 o The evaluated the ontant a o -a 4 ung the boundar ondton o the tre tran urve. Smlar to Kent and Park, the aumed no derene between the unonned and onned peak trength. k Equaton.130 where k 1 The ued epreon taken rom Kent and Park model to evaluate the lope o the deendng branh tartng at tran o A lat traght lne wa propoed when the tre reahe 0. up to C. where C the rato o mamum onned ompreve tran to. The ve boundar ondton ued are: = 0 at =0 d /d = E at =0 = at = o d /d = 0 at = o d /d = -Z at = 33

49 El-Dah and Ahmad (1995) El-Dah and Ahmad (1995) ued Sargn et al. model to predt analtall the behavor o prall onned normal and hgh trength onrete n one ere o equaton. The ued the nternal ore equlbrum, properte o materal, and the geometr o the eton to predt the preure. The parameter mpoed n the analtal predton where plan onrete trength, onnng renorement dameter and eld trength, the volumetr rato o lateral renorement to the ore, the dmenon o the olumn and pang. Y AX ( B 1) X 1 ( A ) X BX Equaton.131 where Y Equaton.13 X Equaton.133 k1 Equaton.134 l o k Equaton.135 l The value o A, B, k 1, k and l are dened b the ollowng equaton A E E E e Equaton

50 16.5 l d t 0.33 B Equaton.137 k h 0.5 d t 0.5 Equaton.138 k d t 66 Equaton l 0.5 h (1 ) Equaton d where d the ore dameter. Cuon and Paultre (1995) Unlke all the prevou work, Cuon and Paultre (1995) bult ther model baed on the atual tre n the trrup upon alure and the dd not onder the eld trength, a the epermental work have hown that the eld trength or the tranvere teel reahed n ae o well onned olumn. The aendng and the deendng branhe n the model urve are epreed b two derent equaton (Fgure.14). For the aendng porton: k k 1 k Equaton.141 k E E Equaton.14 35

51 For deendng one: k 1 ep k1 50 k 50 ln 0.5 k Equaton.143 Equaton l 1.4 k Equaton.145 where 50 aal tran n onned onrete when tre drop to 0.5. It oberved that equaton (.144) propoed b Cuon and Paultre dental to equaton (.95) uggeted b Mander et al. (1988). Followng the ame methodolog o Shekh and Uzumer (198) and Mander et al. (1988) Cuon and Paultre ondered the lateral onnement preure l. h A A b h l Equaton.146 where A and A are the lateral ro etonal area o the lateral teel perpendular to and ae repetvel and h the tre n the tranvere renorement at the mamum trength o onned onrete. k e 1 w 6bh b 1 1 h 1 Equaton.147 k l e Equaton.148 l where w the um o the quare o all the lear pang between adjaent longtudnal teel bar n a retangular eton. and an be ound b the ollowng equaton 36

52 0.7 l 1.1 Equaton l 0.1 o Equaton l 1.1 Equaton.151 Aal Stre 0.5 o Aal Stran 50 FIGURE.14 Propoed Stre-Stran Curve b Cuon and Paultre (1995) Attard and Setunge (1996) Attard and Setunge (1996) epermentall determned ull tre-tran urve or onrete wth ompreve trength o MPa and wth onnng preure o 1-0 MPa, Fgure.15. The man parameter ued were peak tre; tran at peak tre, modulu o elatt, and the tre and tran at pont o nleton. Attard and Setunge ollowed the ame equaton ued b Wang et al. (1978). and Sargn (1971): Y AX BX 1 CX DX Equaton.15 37

53 where Y Equaton.153 X Equaton.154 For the aendng branh, the our ontant are determned b ettng our ondton: d 1- at 0, E d d - at, 0 d 3- at, 4- at 0.45, E 0.45 The ontant are gven b: A E E E e Equaton.155 ( A 1) ) B Equaton.156 E E 0.45 E 0.45 E E A (1 E C A Equaton.157 D B 1 Equaton.158 whle or the deendng urve the our boundar ondton were 38

54 d 1- at, 0 d - at, 3- at, 4- at, where and reer to the oordnate o the nleton pont. The our ontant or the deendng urve are E 4 E A Equaton.159 B E E Equaton.160 C A Equaton.161 D B 1 Equaton.16 The ame out to be a unton o the onnng preure, the ompreve and tenle trength o onrete, l, t, and a parameter k that relet the eetvene o onnement. t l 1 k Equaton l k 0.06 MPa Equaton.164 o 1 l Equaton

55 No lateral preure equaton wa provded Aal Stre FIGURE.15 Propoed Stre-Stran Curve b Attard and Setunge (1996) Manur, Chn, and Wee (1996) Manur et al. (1996) ntrodued atng dreton, the member at n plae(vertall) or pre-at (horzontall), a a new term among the tet parameter, or hgh trength onrete, whh were te dameter and pang and onrete ore area. The onluded that the vertall at onned ber onrete ha hgher tran at peak tre and hgher dutlt than the horzontall at pemen. In addton, vertall at onned non-ber onrete ha larger tran than that o horzontall at onrete wth no enhanement n dutlt. Manur et al. utlzed the ame equaton ound b Carrera and Chu or plan onrete wth ome modaton. For the aendng branh, the ued the eat ame equaton 1 Equaton.166 where a materal parameter dependng on the tre tran hape dagram and an be ound b : 40

56 1 1 E Equaton.167 k 1 and k are two ontant ntrodued n the equaton derbng the deendng branh: k 1 Equaton.168 k k 1 1 or onned horzontall and vertall at non-ber onrete: k h Equaton.169 k.19 h 0.17 Equaton.170 or horzontall at onned ber onrete k 3.33 h Equaton.171 k 1.6 h 0.35 Equaton.17 and the value o and an be obtaned rom the ollowng equaton or onned non-ber onrete: 41

57 h 1.3 Equaton.173 or onned ber onrete: h 1.3 Equaton.174 or vertall at ber onrete o 1 6. h Equaton.175 or horzontall at ber onrete and vertall at non-ber onrete o 1.6 h 0.8 Equaton.176 and or horzontall at non-ber onrete o h 1.5 Equaton.177 Hohkuma, Kawahma, Nagaa, and Talor (1997) Hohkuma et al. (1997) developed ther model to at brdge olumn eton degn n Japan. The model wa baed on ere o ompreon loadng tet o renored onrete olumn pemen that have rular, quare and wall tpe ro eton. The varable that vared n the epermental wok were hoop volumetr rato, pang, onguraton o the hook n the hoop renorement and te arrangement. 4

58 Hohkuma et al. (1997) aerted that the aendng branh repreented n eond degree parabola not aurate to at our boundar ondton: 1. Intal ondton = 0, ε =0.. Intal tne ondton d /d ε =E at ε =0. 3. Peak ondton = at ε = ε 4. Peak tne ondton d /dε =0 at ε = ε The unton that dene the aendng branh : E Equaton.178 E E Equaton.179 For the deendng branh: de E Equaton.180 where E de the deteroraton rate that ontrol lope o the deendng lne and an be ound ung the ollowng equaton E de 11. Equaton.181 h The peak tre and the orrepondng tran or the rular eton h Equaton.18 43

59 h Equaton.183 whle or the quare eton h Equaton Equaton.185 h Razv and Saatoglu (1999) Razv and Saatoglu moded ther model o Saatoglu and Razv (199) to t the hgh trength onrete ( MPa). The aendng zone dened b Popov equaton a ollow: r r 1 r Equaton.186 and or the deendng branh: 1 5k K Equaton.187 o 3 A 60 S B h 1 0.5k K3 k 1 Equaton.188 k k h 500 k K Equaton le Razv and Saatoglu (1999) howed the good agreement o the model wth ome epermental work avalable n the lterature. 44

60 Mend, Pendala, and Setunge (000) Mend et al. (000) moded Sott et al. (198) equaton to t hgh trength onrete. The emprall adjuted Sott et al. (198) equaton to the ollowng one: k or Equaton Zm ( re k ) or Equaton.191 re RK Equaton.19 K 1 3 Equaton.193 l Z m b" 0 n MPa Equaton.194 o 0.4K Equaton R 0 R Equaton Z Equaton.197 l alulated aordng to Mander equaton. Aa, Nhama, and Watanabe (001) A new model wa propoed or onrete onned b pral renorement baed on onrete-tranvere teel nteraton. The two man parameter were onrete trength and lateral tre-lateral tran relatonhp that repreent the repone haratert o the tranvere teel to the lateral epanon o onrete. Aa et al. (001) modeled a onnement mehanm and 45

61 lmted the lateral epanon o the onned onrete wth the mamum lateral epanon apat. Aa et al. (001) reahed ome relatonhp epreed n the ollowng equaton: Equaton.198 l o l Equaton.199 lu Equaton.00 l where lu the mamum lateral onrete tran. The propoed tre-tran urve ha one equaton: X ( 1) X 1 X X Equaton.01 X Equaton.0 where ontrol the tne o aendng branh and ontrol the lope o the deendng branh: E E Equaton.03 E e Equaton.04 46

62 where 80 the tran at 0.8. Lokuge, Sanjaan, and Setunge (005) A mple tre-tran model wa propoed baed on hear alure. The model wa baed on the epermental reult taken rom Candappa (000). Lokuge et al. (005) propoed a relatonhp between aal and lateral tran: l l v Equaton.05 l l a Equaton.06 where a tran at a pont where aal tran and lateral tran urve devate, the ntal Poon rato, and a a materal parameter whh depend on the unaal onrete trength v 8*10 Equaton a Equaton.08 where n MPa. 47

63 Bn (005) Bn (005) ntrodued a generalzed ormula derbng onrete under traal ompreon. The propoed tre tran urve dened b elat regon then non lnear urve. The aal ompreon epreed ung Leon-Paramono rteron a ollow k m ( 1 k) Equaton.09 l m t t Equaton.10 where t the unaal tenle trength, the otenng parameter and equal to one n hardenng regon and zero or redual trength and k the hardenng parameter and equal to one at ultmate trength and otenng regon and equal to 0.1 at the elat lmt. Bn (005) dened three equaton or determnng the tre n the elat, hardenng and otenng zone a ollow: For elat zone: E 1 e Equaton.11 For the hardenng zone: 1e 1e e 1 r 1e r r 1 1e 1e 1 e Equaton.1 r E E E e 1e 1 e E o Equaton.13 For the otenng zone: 48

64 49 r r E l G Equaton.14 where l the length o the pemen and G the ompreve alure energ and alulated a ollow: r r E d l G ep 1 1 Equaton.15 To ull dene the tre tran urve or ontant preure, Equaton.1 ued to dene the lmt tree. Thee tree are mpoed n Equaton.15 and.17 to ull dene the tre tran urve. The lateral preure alulated ung the lateral tran l ound b: l Equaton.16 o or e 1 Equaton ln ep e o or e 1 o p 1 1 Equaton.18 where the eant Poon rato p e 1 1 Equaton.19 wherea n ae o hangng lateral preure, the lateral preure olved b equatng the lateral tran n jaket to the lateral tran o onrete:

65 E l 0 Equaton.0 1 l j j where E j and j the modulu o elatt and volumetr rato o the jaket repetvel..1. Duon A tated b man reearh tude, lke Mander et al. (1988), Sott et al. (198), Shekh and Uzumer (1980) and Shuhab and Mallare (1993), the pral or rular hoop are more eent than the retangular hoop. The unorm preure generated b the rular hoop one o the reaon o rular pral advantage. Aordng to Ed and Danger (005), there are our man approahe or the modelng o onned onrete b lateral te 1. The empral approah: n whh the tre-tran urve generated baed on the epermental reult. Fat and Shah (1985) and Hohkuma et al. (1997) ollowed that approah.. Phal engneerng model baed approah: the lateral preure aung the onned behavor o the onrete ore, provded b the arh aton between the lateral renorement te. Th approah wa adopted b Shekh and Uzumer (1980), and wa ollowed b Mander et al. (1988). 3. The thrd approah baed ether on the rt approah or the eond one, but t doe not aume the lateral te eldng. Intead, It nlude omputaton o the teel tre at onrete peak tre, ether b ntrodung ompatblt ondton, olved b teratve proe a Cuon and Paultre (1995) dd, or b ntrodung empral epreon a Saatgolu and Razv (199) ollowed. 4. A platt model or onned onrete ore ntrodued b Karabn and Kou (1994). The hape o the onned ore baed on the arhng aton. 50

66 Baed on the revewed model, around 50% ollowed the empral approah, wherea 10% ued the phal engneerng approah, and the ret ombned between the empral and phal engneerng approah. Aordng to Lokuge et al. (005), the tre tran model an be laed a three ategore: 1. Sargn (1971) baed model: Martnez et al. (1984), Ahmad and Shah (198), Eldah and Ahmad (1995) Aa et al. (001).. Kent and Park (1971) baed model: Shekh and Uzumer (198), Saatgolu and Razv (199). 3. Popov (1973) baed model: Mander et al. (1988), Cuon and Paultre (1995) and Hohkuma et al. (1997). Mot o the onned model were developed b tetng mall pemen that dd not mulate the real ae or the atual olumn, and mall porton ued real olumn to ver ther work uh a Mander et al. (1988). Long. TABLE.1 Lateral Steel Connement Model Comparon pang Lateral Lateral Eetve Seton Lateral Lateral teel teel teel area geometr preure teel ze ong. tre Rhart * Chan * Blume * * * * Ro * * Solman * * Sargn * * * * 51

67 Long. pang Lateral Lateral Eetve Seton Lateral Lateral teel teel teel area geometr preure teel ze ong. tre Kent * * * Vallena * * * * Muguruma * * * Sott * * * * Shekh * * * * * * Ahmed * * Park * * * Martnez * * * Fat * * * * Young * * * * Mander * * * * * * * * Fuj * * * * * Saatoglu * * * * * * * * El-Dah * * * * Cuon * * * * * * * Attard * * * Manur * * * Fuj * * * * Razv * * * * * * * * Mend * * * * * * * 5

68 Long. pang Lateral Lateral Eetve Seton Lateral Lateral teel teel teel area geometr preure teel ze ong. tre Aa * * Bn * * Table.1 how that the mot ueul model onderng the lateral preure determnaton parameter are Mander et al. (1988) that le n the thrd group aordng to Lokuge et al. (005) omparon and Saatoglu and Razv (199), eond group (Razv and Saatoglu (1999) wa developed or hgh trength onrete). For the ake o omparng three model, one rom eah group, wth the epermental reult, El-Dah and Ahmad Model (1995) eleted rom the rt group a the model that ondered mot o the ontrbutng ator, Table.1, ompared to Attard and Setunge (1996), Manur et al. (1997), Martnez et al. (1984) and Sargn (1971) model. However El-Dah and Ahmad model wa developed or prall onned onrete, hene, t wa elmnated rom Retangular olumn omparon. The model eleted rom group Mander et al. (1988) and that hoen rom group 3 Saatoglu and Razv (199) a mentoned above. Length (n.) Wdth (n.) TABLE. Epermental Cae Properte F F Bar (k) (k) # Cover (n.) Bar dameter (n.) Lateral teel dameter (n.) Spang (n.) Fh (k.) Cae Crular Cae Crular Cae

69 The three model are ompared wth two epermental reult, ae 1 and ae or rular ro eton olumn, Table.. All the three model are ueull apturng the aendng branh. However, Mander model the bet n epreng the deendng one, Fgure.16 and.17. FIGURE.16 Mander et al. (1988), Saatoglu and Razv (199) and El-Dah and Ahmad (1995) Model Compared to Cae 1 FIGURE.17 Mander et al. (1988), Saatoglu and Razv (199) and El-Dah and Ahmad (1995) Model Compared to Cae 54

70 For the ae o retangular olumn omparon, Fgure.18, Saatoglu and Razv (199) better n apturng the ultmate ompreve trength. Wherea Mander derbe the otenng zone better than Saatoglu and Razv model. Baed on Table.1 and Fgure.16,.17, and.18, Mander model een to be the bet n epreng the tre tran repone or rular and retangular olumn. FIGURE.18 Mander et al. (1988), Saatoglu and Razv (199) and El-Dah and Ahmad (1995) Model Compared to Cae 3. Retangular Column Subjeted to Baal Bendng and Aal Compreon Retangular renored onrete olumn an be ubjeted to baal bendng moment plu aal ore. When the load at on one o the ro eton bendng ae the problem beome unaal bendng. However when the load appled eentrall on a pont that not along an o the bendng ae the ae beome baal bendng. The baal bendng ae an be ound n man truture nowada. Th ae vted etenvel n the lterature dregardng the onnement eet. The alure urae o retangular olumn 3D urae onted o man D nteraton dagram. Eah o the D nteraton dagram repreent one angle between the bendng moment about -a and the reultant moment. Man mplaton are ntrodued to jut the ompreve trapezodal hape o onrete, due to the two bendng ae etene. Th eton revew the prevou work onern retangular olumn ubjeted to baal bendng and aal load hronologall. Hene, the revew laed aordng to t author/. 55

71 ..1 Pat Work Revew..1.1 A Stud o Combned Bendng and Aal Load n Renored Conrete Member (Hogentad 1930) Hogentad laed onrete alure ubjeted to leure wth or wthout aal load to ve mode 1. Falure b eeve ompreve tran n the onrete wth no eld n tenoned teel (ompreon alure).. Tenon alure where the tenoned teel eld aue eeve tran n the onrete. 3. Balaned alure where tenoned teel eld at the ame tme ompreve onrete al. 4. Compreon alure where the tenoned teel pa the eld tre. 5. Brttle alure aued b tenoned teel rupture ater the rak developed n the ompreve onrete. 6. Hogentad (1930) uggeted degnng b the ultmate alure theor n h report a oppoed to the lnear elat theor (tandard theor) that wa wdel applable up to nearl t ear. He dued ome o the avalable nelat theore that were lmted to unaal tre aordng to hm. The theore dued were E. Suenon (191), L. Menh (1914), H. Don (19), F. Stu (193). C. Shreer (1933). S. Steuermann (1933). G. Kaznz (1933). F. Gebauer (1934) O. Baunmann (1934). E. Bttner (1935). A. Brandtzg (1935). F. Emperger (1936). R. Salger (1936). C. Wtne (1937), USSR peaton OST 90003, (1938). V. Jenen1943. R. Chambaud (1949). Alo Hognetad (1930) ntrodued h new theor o nelat leural alure. He at equaton or tenon alure and ompreon one...1. A Smple Anal or Eentrall Loaded Conrete Seton (Parker and Sanlon 1940) Parker and Sanlon (1940) ued elat theor. 56

72 P A M I M I Equaton.1 The developed a proedure b rt alulatng tree at the our orner, then hekng all tree are potve, no urther tep are needed, otherwe, alulatng enter o gravt and realulatng moment o nerta then realulatng tre and determnng the new poton o the neutral a. Thee tep are repeated tll the nternal ore onverge wth the appled one Renored Conrete Column Subjeted to Bendng about Both Prnpal Ae (Troell 1941) Troell (1941) Suggeted that porton o the appled aal load an be ued wth the bendng moment about one a to nd the mamum ompreve and tenle trength n the ro eton. Then the remanng load along wth the other bendng moment about the other a an be ued the ame wa, ung the method o uperpoton. The ummaton tree are the tree generated rom the eton. He alo uggeted takng equvalent teel area n eah de to altate the alulaton proedure Degn Dagram or Square Conrete Column Eentrall Loaded n Two Dreton (Andereen 1941) Anderen (1941) mplemented a new proedure or determnng mamum ompreve and tenle tree on ro eton wthout determnng the loaton o the neutral a. The lmtaton o th proedure that t jut appled on quare ro eton and the teel ha to be mmetr. Baed on the lnear elat theor and the perpendulart o the neutral a to the plane o bendng whh wa proven n a prevou tud, Anderen derved tree oeent equaton baall or ro eton renored wth our bar, and then repreented them graphall. Th dervaton wa et ater lang the problem nto three derent ae baed on the neutral a loaton. 57

73 C 1 3np k o o 6k o Equaton. T d o 1 1 D k Equaton.3 1 = a oeent that ull determned n h paper or eah ae o the three ae n = modular rato P = teel rato k = dtane rom ape o ompreon area to neutral a dvded b dagonal length D= dagonal length d = dtane rom orner to renorng bar Thee two value an be ubttuted n the ollowng equaton to determne the mamum ompreve trength and tenle trength repetvel P Ca Equaton.4 nt Equaton.5 where a the de length o ro eton and P ore magntude. 58

74 70 60 d/d= n p = Value o C n p = Value o T T urve urve Value o D/e FIGURE.19 Relaton between T and C b Anderen (1941) Andereen (1941) plotted graph relatng T and C; Fgure.19. It hould be noted that the graph der wth angle and the rato d/d varaton. Anderen adapted h proedure to t the 8 bar renorement, a well a 16 bar one. That wa done b ndng the loaton o the equvalent our bar n the ame ro eton that eld the ame nternal moment and moment o nerta Renored Conrete Column under Combned Compreon and Bendng (Weman 1946) Weman (1946) ntrodued algebra method under a ondton o the plane o the bendng onde wth the a o mmetr. Baed on the elat theor, Weman (1946) ound that the dtane between the appled load and the neutral a a: a I Q p p Equaton.6 59

75 I p = moment o nerta o the eetve area wth repet to the load a Q p = the rt moment o the eetve area. The proedure propoed ha ver lmted applablt ne t requred the appled load le on the a o mmetr, whh onder a ver peal ae. In addton t rele on the elat theor Anal o Normal Stree n Renored Conrete Seton under Smmetral Bendng (Bakhoum 1948) Ung the elat theor and equatng the nternal ore and moment to the appled one, Bakhoum (1948) developed proedure n loatng the neutral a. Th proedure wa et or unaal bendng. He alo ntened the mportane o takng the tenoned onrete nto aount whle analzng. H 1 1 Equaton.7 H H t Equaton.8 ni 3 bt p Equaton.9 ns bt p Equaton.30 H = dtane between the load and the neutral a N= modular rato b= eton wdth t = eton heght 60

76 I p = Moment o nerta o the total renorement teel about the lne parallel to the neutral a through the pont o applaton o the eternal ore. S p = Statal moment o the total renorement teel about the lne parallel to the neutral a through the pont o applaton o the eternal ore. The relaton between and plotted graphall; Fgure P t b Ip = bt 1 C Z 3 H 1 value o or Z = value o 10 or Z = Sp = bt Z = Z/t C = C/t 1.0 Value o 10 1 and 1 FIGURE.0 Relaton between and bbakhoum (1948) For the ae o unmmetral bendng, Bakhoum (1948) uggeted three oluton; method o enter o aton o teel and onrete, produt o nerta method and method o mathematal ueul tral. It noted that the rt two method are tral and error method, and all the three method were bult on the elat lnear theor. 61

77 ..1.7 Degn o Retangular Ted Column Subjeted to Bendng wth Steel n All Fae (Cervn 1948) The Portland ement aoaton publhed ontnut n onrete rame (thrd edton) that ha an equaton that relate the mamum load to the atual appled load and moment. It an be appled on a ro eton: P N CD M t Equaton.31 P = total allowable aal load on olumn eton N= atual aal load on olumn eton M = moment T = eton heght C a 0.45 Equaton.3 a = the average allowable tre on aall loaded renored onrete olumn D t R Equaton.33 R = radu o graton Th equaton lmted to renorement on the end ae. Crevn (1948) redened the term D n the equaton to t renorement n the our ae a ollow D n 1p z n 1pg Equaton.34 n E E Equaton.35 6

78 p = renorement rato g = rato between etremte o olumn teel and overall olumn depth = rato o total olumn teel at one end = rato o total olumn teel between entrod and one end z = arm rom etrod o teel rato to entrod o olumn b t gt gt X Z CENTROID OF COLUMN FIGURE.1 Geometr Dmenon n Crevn Anal (1948) He howed that +z var rom 0.5 to 0.5. The lmtaton o th equaton applablt that the rato e/t ha to be le than one The Strength o Renored Conrete Member Subjeted to Compreon and Unmmetral Bendng (Mkhalkn 195) Mkhalkn (195) perormed tude on determnaton o the allowable load and ultmate load o baall loaded retangular member. He developed degn and anal proedure or tenon and ompreon alure aordng to ultmate theor, a he generated hart or degn mplaton baed on the elat theor ung mple ompatblt equaton Fgure. and.3. Thee hart loate the onrete and teel enter o preure wth repet to the neutral a. 63

79 R=j k=0.5 J <K< K=3 K= X/b or Y/h FIGURE. Conrete Center o Preure veru Neutral A Loaton, Mkhalkn J 3 1 k=0.75 k=1 k= k=0.75 k=1 k= X 1/b1 or Y /h1 FIGURE.3 Steel Center o Preure veru Neutral A Loaton, Mkhalkn

80 ..1.9 Eentr Bendng n Two Dreton o Retangular Conrete Column (Hu 1955) Hu (1955) ollowed the elat aumpton n buldng h anal. He howed numerall that the lope o the neutral a or non homogeneou eton an be replaed b that o homogeneou one wth mall error perentage. He ound algebraall the equlbrum equaton N bd hk 1 1 np 1 6 h k Equaton.36 n 1 a 1 h 1 a k Equaton.37 N bd Q hk 1 1 e 1 m m b d 6 1 m nq nq np Qnp k Equaton.38 Equaton.39 N = the normal ompreve ore b = eton wdth d = eton heght = mamum onrete trength h and k dene the poton o the neutral a, b, d oeent (unton o h) A = over n dreton oeent A b = over n dreton oeent e = load eentrt rom the geometr entrod (n -dreton) e = load eentrt rom the geometr entrod (n -dreton) n E E 65 Equaton.40

81 0.3 p A A Equaton.41 q I I q I I Equaton.4 e m e Equaton.43 The prevou equaton are plotted graphall to obtan the unkown value k, n/bd, Fgure.4. Value o e bd N Q= =0.015 N bd 1. Value o e K= = Value o e K=0.3 Q= K= Value o e FIGURE.4 Bendng wth Normal Compreve Fore Chart np = 0.03, Hu (1955). In H Paper the Graph Were Plotted wth Derent Value o np 0.03,0.1,0.3 The rt obvou nteret n the ultmate trength o the trutural member appeared n the rt hal o the pat entur. Pror to that, there were ome degnaton to the mportane o degnng wth ultmate trength. Whle Thulle leural theor (1897) and Rtter ntroduton o the parabol dtrbuton o onrete tree (1899) were ntrodued pror to the 66

82 traght lne theor o Cognet and Tedeo (1900). The traght lne theor beame aepted due to t mplt and the agreement wth the tet requrement that tme. Cognet theor grew wdel tll t wa ontradted b ome epermental work done on beam b Le, Slatter and Zpprodt n 190, and on olumn b MMllan (191), a the onrete ontruton applablt wa preadng out (ACI-ASCE ommttee 37(1956)). Ater 1950 there wa a all to tart workng wth the ultmate trength degn a t wa adopted n everal ountre n Europe and other, a the renored onrete degn ha advaned. Th led the ACI-ASCE ommttee 37 to propoe the rt report on ultmate trength degn n 1956 (ACI-ASCE ommttee 37(1956)). The ommttee member howed n ther tude that the ultmate trength degn load an be ound auratel. The dened the mamum load apat or onentr load P o 0.85 ( A A ) A g t t Equaton.44 A g = the gro area o the eton. A t = teel bar area The ommttee ondered mnmum eentrt value to degn wth. For ted olumn the value wa 0.1 tme the eton depth. For ombned aal load and bendng moment P u 0.85 bdk k A A Equaton.45 u 1 P e 0.85 bd u k kuk1 1 k1 d k uk1 A d 1 d Equaton.46 P u = aal load on the eton e = eentrt o the aal load meaured rom the entrod o tenle renorement. = tre n the tenle renorement. dk u = dtane rom etreme ber to neutral a, where ku le than one 67

83 k 1 = rato o the average ompreve tre to 0.85, where k 1 not greater than 0.85 and to be redued at the rate o 0.5 per 1000 p or onrete trength over 5000 p. k = rato o dtane between etreme ber and reultant o ompreve tree to dtane between etreme ber and the neutral a. k hould not be taken le than 0.5. k 1 Ater ultmate trength degn wa releaed, the ACI ommttee 318 n ther Buldng ode requrement or renored onrete (ACI ) approved the uage o the ultmate trength method or degnng renored onrete member along wth the tandard method n The ondtoned that: F a a F bz b F b b 1 Equaton.47 Gven that the rato e/t doe not eeed /3 where b = the bendng moment about -a dvded b eton modulu o the tranormed eton relatve to -a. bz = the bendng moment about -a dvded b eton modulu o the tranormed eton relatve to -a. e = eentrt o the load meaured rom the geometr entrod t = overall depth o the olumn a = nomnal aal unt tre. 0.8* 0.5 pg = allowable bendng unt tre = b p g = teel rato to the gro area. = nomnal allowable tre n renorement Gude or Ultmate Strength Degn o Renored Conrete (Whtne and Cohen 1957) Followng th mave hange n paradgm, Charle Whtne and Edward Cohen releaed ther paper gude or ultmate trength degn o renored onrete whh erved a 68

84 a upplement to the ACI buldng ode (318-56). The uggeted a lnear relatonhp between the ae o the pure bendng and that o onentr load n the ollowng equaton M M u o P P o P o u Equaton.48 M u = total moment o the plat entrod o the eton. P o = ultmate dret load apat or a onentrall loaded hort olumn. P u = ultmate dret load apat or an eentrall loaded hort olumn. M o = the moment apat wthout thrut a ontrolled b ompren aumng enough tenle teel to develop t n ull and t equal to M o 0.333bd A d d Equaton.49 equaton The lmted the mamum moment allowed or degn to M u ung the ollowng M u d p 1 bd d Equaton Equaton.51 p A / bd Equaton.5 A = ompreve teel area. d = dtane rom etreme ompreve ber to entrod o tenon ore n tenle renorement. d = dtane rom etreme ompreve ber to entrod o tenon ore n ompreve renorement. b = olumn wdth. 69

85 Po Pu 0 Mu Mo Moment, Pu e LIMITING VALUE OF MOMENT FOR DESIGN FIGURE.5 Lnear Relatonhp between Aal Load and Moment or Compreon Falure Whtne and Cohen Ultmate Strength Degn o Retangular Conrete Member Subjeted to Unmmetral Bendng (Au 1958) Au (1958) generated hart to alulate the equvalent ompreve depth o the tre blok baed on aumed value o eton dmenon and bar arrangement. The degn equaton were reated omplng wth the ACI-ASCE aumpton. He howed that when a member ubjeted to ompreve ore a well a bendng, the eton an be ontrolled ether b tenon or ompreon alure dependng on the magntude o eentrte. H proedure to rt appromate the loaton o the neutral a that an be made b obervng that the appled load, the reultant o the tenle ore n teel and the reultant o the ompreve ore n ompreve teel and onrete mut all le n the ame plane. Th lae the problem a one o the three ae: 1. Neutral a nteret wth two oppote de. Neutral a nteret wth two adjaent de ormng a ompreon zone bgger than hal o the ro etonal area 70

86 71 3. Neutral a nteret wth two adjaent de ormng a ompreon zone maller than hal o the ro etonal area Equlbrum equaton plu ompatblt equaton are needed when the eton ontrolled b ompreon (onrete ruh). Wherea, equlbrum equaton are uent n tenon ontrolled ae. Tung peed two ondton baed on ACI-ASCE report, that are the average tre agned to eah tenoned bar and the reultant tenle ore ondered the tenle bar group entrod. Baed on that, the bar loe to the neutral a are gnored n omputaton. Havng equlbrum equaton, Tung denoted dmenonle varable, two or eah ae o the three ae mentoned above and plotted hart relatng eah two aoated varable Fgure.6,.7, and.8. The hart generated have an output o determnng the neutral a poton. The dmenonle varable utlzed are: t e t r bt P t d t d m p d t u Equaton.53 b e b r bt P b d b d p m d t u 0.85 Equaton.54 t e t r bt P t d t d m p d d bt Q u Equaton.55 b e b r bt P b d b d p m d d t b Q u 0.85 Equaton.56 t e t r bt P t d t d m p t r U u Equaton.57 b e b r bt P b d b d m p b r U u 0.85 Equaton.58

87 t d g go b/ r r T Pu e d d C a= K 1 h h r NA r e t/ g/d C1= C= b d go/d FIGURE.6 Seton and Degn Chart or Cae 1(r/b = 0.005), Au (1958) t n n d b/ r r T d n n b e d C Pu a=k 1h h d r r NA e t/ n/d Q1/Q= Q1= FIGURE.7 Seton and Degn Chart or Cae, Au (1958) n/d 7

88 b/ e r r d Pu 0.9 t w w d C T d w w b d r r a=k 1h h NA e t/ W/t U1/U= U1= W/b FIGURE.8 Seton and Degn Chart or Cae 3(d /b = 0.7, d /t = 0. 7), Au (1958) t = total depth o retangular eton d = dtane rom etreme ompreve orner to entrod o tenle renorement meaured n the dreton o -a p = A /bt b = wdth o retangular eton m = m-1, m = /0.85 d = dtane rom etreme ompreve orner to entrod o ompreve renorement meaured n the dreton o -a. P u = ultmate dret load apat or the member ubjet to bendng n two dreton r = dtane rom entrod o tenle renorement to -a. r = dtane rom entrod o tenle renorement to -a. e = eentrt o ultmate dret load meaured rom entrod o retangular eton n the dreton o -a d = dtane rom etreme ompreve orner to entrod o ompreve renorement meaured n the dreton o -a 73

89 d = dtane rom etreme ompreve orner to entrod o tenle renorement meaured n the dreton o -a e = eentrt o ultmate dret load meaured rom entrod o retangular eton n the dreton o -a Degn o Smmetral Column wth Small Eentrte n One or Two Dreton (Wenger 1958) Ung the eton moment o nerta and the eton modulu, Wenger (1957) ntrodued a new degnng equaton or the gro etonal area requred b degn or olumn ubjeted to mall eentrte n one dreton or two. Wenger (1957) propoed gro eton equaton: N Ne / t A g Q0.5 pg Fb g n 1g p g Equaton.59 and the apat o a gven olumn alulated ung the ollowng equaton N Q 0.5 p F n 1 g 1 b g e / t g p g Equaton.60 g I g t A g Equaton.61 I gt A Equaton.6 A g = gro area A = Steel area t = olumn length n the dreton o bendng 74

90 I g = gro moment o nerta n the bendng dreton I = moment o nerta o teel n the bendng dreton e = eentrt o the reultant load meaured to enter o gravt N= appled aal load Q= reduton ator = 0.8 or hort ted olumn p g = A /A g F b = allowable bendng unt tre that permtted bendng tre eted = 0.45 G = enter to enter teel n the dreton o bendng dvded b olumn length n the dreton o bendng Baall Loaded Renored Conrete Column (Chu and Pabaru 1958) In 1958 Chu and Pabaru ntrodued a new numeral proedure to determne the atual tree or a gve eton. Ther proedure wa baed on the nelat theor howed earler b Hogentad. Intall, the aumed the ro eton n the elat range, and aumed a loaton or the neutral a. Then ued the ollowng ormula that wa ound b Hard Cro (1930), to olve or tree " M o P" AE I M o I I o I " o o I o o X M " o I M o I I o I " o o I o o Y Equaton.63 = tre A e = Area o the elat porton. I o = moment o nerta about -a I o = moment o nerta about a I o = produt o nerta M o =moment o the elat porton about the a M o =moment o the elat porton about the a 75

91 P = aal ore taken b the elat porton. I the onrete and teel tree le n the elat range, the above equaton wa ued to loate a new poton or the neutral a, and omparng t wth the aumed one. The whole proe repeated tll the poton o the alulated neutral a onde wth the aumed one. On the other hand an o the onrete or teel are beond the elat range, the plat load and moment are alulatng, then deduted rom the total load and moment. The remnder ued, a the elat porton o the load, to loate the neutral a Degn Crtera or Renored Column under Aal Load and Baal Bendng (Breler 1960) Breler (1960) propoed a new approah o appromaton o the alure urae n two derent orm. He howed the magntude o the alure load a unton o prmar ator; olumn dmenon, teel renorement, tre-tran urve and eondar ator; onrete over, lateral te arrangement. He ntrodued two derent method. The rt method named reproal load method 1 P 1 P 1 P 1 P o Equaton.64 P =appromaton o P u P = load arrng apat n ompreon wth unaal eentrt. P = load arrng apat n ompreon wth unaal eentrt. P u = load arrng apat under pure aal ompreon The eond method the load ontour M M o M M o 1 Equaton.65 and th an be mpled to 76

92 o o 1 Equaton.66 B equatng and or more mplaton the nteraton dagram an be plotted a hown n Fgure.31 Breler (1960) well orrelated Equaton.70 to epermental tude ormed rom eght olumn, and analtall howed the trength rtera an be appromated b o o 1 Equaton.67 l/p S S A 1/P B C 1/Po X Y FIGURE.9 Graphal Repreentaton o Method One b Breler (1960) 77

93 P P M INTERACTION CURVES FAILURE SURFACE S3 PLANE P A CONSTANT LOAD CONTOUR Mo Mo Pu M M FIGURE.30 Graphal Repreentaton o Method Two b Breler (1960) Y/Yo 1.0 = = 3 = = = X/Xo FIGURE.31 Interaton Curve Generated rom Equatng and b Breler (1960) 78

94 Retangular Conrete Stre Dtrbuton n Ultmate Strength Degn (Mattok and Krtz 1961) Mattok and Krtz (1961) determned ve ae or the poton o the neutral a wth repet to the retangular ro eton; when the neutral a ut through two adjaent de wth mall and bg ompreon zone, the neutral a nteret wth the eton length or wdth and when t le outde the ro eton. The mplemented ormula or alulatng the poton o the neutral a baed on the load and moment equlbrum and the geometr o the ompreon zone Square Column wth Double Eentrte Solved b Numeral Method (Ang 1961) Ang (1961) ntrodued a numeral method to olve the problem. He propoed teratve proe to nd equlbrum between nternal ore and appled one, b aumng a poton or the neutral a. The loaton o the neutral a kept hangng tll equlbrum. However, he alulated tree baed on Bernoull plane theorem whh wa bult upon traght lne theor (elat theor). The tre o the etreme ompreon ber wa appromatel alulated aordng to the peaton o AASHTO 1957 Standard peaton or hghwa brdge Ultmate Strength o Square Column under Baall Eentr Load (Furlong 1961) Furlong (1961) analzed quare olumn that have equal renorement n the our de and renorement n two de onl, to vualze the behavor o retangular olumn that ha unmmetral bendng a. He ued a ere o parallel neutral a wth the ruhng ultmate tran o at one o the eton orner to develop a ull nteraton dagram at one angle. And b ung derent angle and loaton o the neutral a a ull 3D nteraton urae an be developed. He wa the rt to ntrodue th proedure. Furlong (1961) onluded that the mnmum apat o a quare olumn, havng equal amount o teel n all de, et when the load aue bendng about an a o 45 degree rom a major a. He alo onluded that m M m M 1 Equaton.68 79

95 M = moment omponent n dreton o major a. M = moment omponent n dreton o mnor a. M = moment apat when the load at along the major a. M = moment apat when the load at along the mnor a Te Requrement or Renored Conrete Column (Breler and Glbert 1961) Breler (1961) ntrodued the mportane o the te onnement n olumn a objet to hold the longtudnal bar n plae and prevent them rom buklng ater the over pallng o. No onrete trength mprovement wa dued Analtal Approah to Baal Eentrt (Czernak 196) Czernak (196) proved that the lope o the neutral a dependng on the relatve magntude o moment about the X a to the moment about the Y a and the geometr o the eton and t ndependent o the magntude o bendng moment and the appled ore or the elat range. Aordng to the eetve ompreve onrete, Czernak (196) determned ve ae baed on the loaton o the neutral a, Fgure.44. (a) (b) () (d) (e) FIGURE.3 Fve Cae or the Compreon Zone Baed on the Neutral A Loaton Czernak (196) He developed an teratve proedure or loatng the neutral a poton or a gven ro eton, b ung equaton.7 and.73 to determne the ntal poton o the neutral a a I YpQo I X pqo I o YpQo I o X pqo Q Y AI X Q Q X AI Y Q o p p o o p o p o Equaton.69 80

96 b I YpQo I X pqo I o YpQo I o X pqo Q X AI Y Q Q Y AI X Q o p p o o p o p o Equaton.70 a = -nterept o the neutral a lne b = -nterept o the neutral a lne I = elat produt o nerta o the area about the orgn I o =elat moment o nerta o the area about the -a I o =elat moment o nerta o the area about the -a Q o = moment area about -a (wthn elat regon) Q o = moment area about -a (wthn elat regon) A = area o tranormed eton (wthn elat regon) Y p = -oordnate o the appled eentr load X p = -oordnate o the appled eentr load then alulatng the new eton properte, eetve onrete and tranormed teel, and ndng the new value o X p and Y p. Y p r ro a b 1 a b Equaton.71 X p r o r a b 1 a b Equaton.7 and olvng or a, b agan and repeat the proedure up tll onvergene. A or ultmate trength degn, Czernak (196) proved wth ome mplaton that the neutral a parallel to onrete plat ompreon lne and teel plat tenon and 81

97 ompreon lne, o the an be ound b multplng the loaton o the neutral a b ome value. The ultmate eentr load and t moment about and a an be ound rom: P u o Au Q a o Q b o Equaton.73 M u o Q u I a I b o P Y u p Equaton.74 M u o Q u I a o I b P X u p Equaton.75 Q u Q o Q ( m 1) Q mq Equaton.76 Q u Q o Q ( m 1) Q mq Equaton.77 o = tre ntent at the orgn and the -a and -a nterept o the neutral a are ound: a b I YpQo I X pqo I o YpQo I o X pqo Q Y A I X P Q X A I Y Q u p u p o I YpQo I X pqo I o YpQo I o X pqo Q X A I Y Q Q Y A I X Q u p u p o u u p p u u o o p p o o Equaton.78 Equaton.79 where A u A " o A u m 1 A u ma u Equaton.80 8

98 Q u Q o o Q m 1 Q mq Equaton.81 Q u Q o o Q m 1 Q mq Equaton.8 Q = moment o area about -a o the plat porton o the onrete eetve eton Q = moment o area about -a o the plat porton o the onrete eetve eton Q = moment o area about -a o the plat porton o the elded tenle renorement Q = moment o area about -a o the plat porton o the elded tenle renorement renorement Q = moment o area about -a o the plat porton o the elded ompreve Q renorement = moment o area about -a o the plat porton o the elded ompreve A u = equvalent plat tranormed area. A u = area o onrete under plat ompreon A u = area o elded tenle renorement A u = area o elded ompreve renorement. P u = ultmate trength o eentrall loaded ro eton M u = moment o the ultmate load about -A M u = moment o the ultmate load about -A = mamum onrete tre at ultmate load (aumed a 0.85 ) m " 83

99 ..1.0 Falure Surae or Member n Compreon and Baal Bendng (Pannell 1963) Pannell mplemented a relaton between the alure moment about -a or a gven load and the omponent o radal moment wth the ame load. The ormula wa ound baed on devaton tud between the atual load ontour urve and an magnar urve ound rom the revoluton o the alure pont about a, wth the ame load, about the z a. The equaton ound or eton that have equal teel n eah ae: M M 1 N n Equaton.83 N 1 M M d Equaton.84 M = alure moment or ome load n plane = angle between and the tranormed alure plane He howed that h ormula more aurate and onervatve than that o Breler. He alo developed a hart or N or unequal teel dtrbuton; Fgure.33. / P/bd FIGURE.33 Value or N or Unequal Steel Dtrbuton b Pannell (1963) 84

100 ..1.1 Ultmate Strength o Column wth Baall Eentr Load (Meek 1963) Meek (1963) aumed ontant rato o moment about the -a and the -a. Conequentl, nreang the ore wll nreae the moment proportonall. M M e e ont. Equaton.85 Ung the above relaton a loaton o the neutral a eleted. Then th loaton adjuted untl the ollowng relaton ated P u A A A t t Equaton.86 He alo howed et o epermental pont orrelated well to the theortal nteraton dagram developed...1. Baal Eentrte n Ultmate Load Degn (Aa-Jakoben 1964) To ompl wth loal degn ode, Aa-jakboen (1964) replaed baall eentr load at on a regtangular ro eton wth an equvelant load at on the man a o mmetr wth an equvelant moment. He howed, ung moment and ore equlbrum, that the eqvelant moment M e : M Pe M e 1 m Equaton.87 e t e t Equaton.88 m 1 Equaton.89 85

101 The moment M 1 mall addtonal moment depend on alure mode and ome other ator. And n mot ae t equal to zero Degn o Column Subjeted to Baal Bendng (Flemng and Werner 1965) Flemng and Werner (1965) utlzed the ormula ound b Mattok (1961) or loatng the neutral a n the derent ae o the ompreon zone hape along wth Furlong (1961) method, b varng the loaton and nlnaton angle o the neutral a, to plot the nteraton dagram. Flemng and Werner (1965) plotted dmenonle nteraton dagram or a quare ro eton or ourteen ae ung parameter that ommonl ued e/e = P/ = 0.80 = p Pu/b 0.6 = FIGURE.34 Degn Curve b Flemng et al. (1961)..1.4 Invetgaton o the Ultmate Strength o Square and Retangular Column under Baall Eentr Load (Ramamurth 1966) Ramamurth (1966) propoed a new method or denng the load ontour or eton havng eght or more bar dtrbuted evenl. He mentoned that the avalable method o degn o baall loaded olumn are tral and error proedure and determnaton o ultmate load rom alure urae. He howed that olumn ontanng our bar behave derentl than thoe ontanng eght or more bar wth the ame renorement rato. He ound theoretall or quare olumn that the neutral a nlnaton angle and the angle ormed between the load ra 86

102 and -a are almot equal. And the relaton between the moment and the moment about -a n an load ontour level equal to M u M uo 3 1n Equaton.90 M u =ultmate moment about -A M uo = unaal moment on the ame load ontour o M u = nlnaton o the neutral a to -a angle Equaton.93 an be mpled to M u M uo 3 1 n e Equaton.91 M u =ultmate radal load about z a and wth plottng the prevou equaton agant ome atual load ontour he ound the ollowng relaton more aurate epeall or mall angle () M u M uo Equaton.9 Smlarl or retangular olumn, b ndng the tranormed hape o the retangular nteraton dagram to the quare one ung ome mlar trangle alulaton M u M uo o 45 K Equaton.93 n = tranormed equvalent angle o M K = tranormaton ator equal to M uo uo 87

103 Alo he howed that the upper equaton n good omparon wth epermental atual load ontour. He plotted the relaton between and or derent rato o length to wdth or retangular olumn. d/b = 1.0 d/b = 1.5 d/b =.0 FIGURE.35 Relaton between and b Ramamurth (1966)..1.5 Capat o Renored Retangular Column Subjeted to Baal Bendng (Parme, Neve, and Gouwen 1966) Parme et.al (1966). uggeted relatng the baal bendng to the unaal retane. The retated Breler equaton M M u log0.5 log M M u log0.5 log 1 Equaton.94 M = unaal ultmate moment apat about -a M u = unaal ultmate moment apat about -a M = baal bendng apat omponent about -a. M = baal bendng apat omponent about -a. 88

104 a unton o renorement poton, olumn dmenon and the materalt properte o teel and onrete. Parme et.al (1966) ued a omputer program to obtan value or. Then wa repreented graphall n our hart, Fgure.37,.38,.39, = M/Mu = Unaal Bendng Capat about a = Mu about a = Mu Baal Bendng Capat about a = M about a = M M/Mu FIGURE.36 Baal Moment Relatonhp b Parme et al. (1966) b t X Y gb X Y.6<g<1.0 3,000<<6, <t/b<4.0 q = Pt / Pt = At / tb.8.7 q =.1 q = = 60,000 p 1.3 = 60,000 p Pu/Po FIGURE.37 Baal Bendng Degn Contant (Four Bar Arrangement) b Parme et al. (1966) Pu/Po 89

105 1.0.9 b t X Y gb X Y.6<g<1.0 3,000<<6, <t/b<4.0 q = Pt / Pt = At / tb q =.1 q =1.3 q =.1 q = Pu/Po = 40,000 p 1.3 Pu/Po = 60,000 p FIGURE.38 Baal Bendng Degn Contant (Eght Bar Arrangement) b Parme et al. (1966) b t X Y gb X Y.6<g<1.0 3,000<<6, <t/b<4.0 q = Pt / Pt = At / tb q =.1.3 q = 1.3 q = q = = 40,000 p = 60,000 p Pu/Po Pu/Po FIGURE.39 Baal Bendng Degn Contant (Twelve Bar Arrangement) b Parme et al. (1966) 90

106 1.0.9 b t X Y gb X Y.6<g<1.0 3,000<<6, <t/b<4.0 q = Pt / Pt = At / tb q =.1 q = 1.3 q =.1 q = = 40,000 p = 60,000 p Pu/Po Pu/Po FIGURE.40 Baal Bendng Degn Contant ( Bar Arrangement) b Parme et al. (1966) Parme et al. (1966) howed agreement between the uggeted Equaton.97 and the theoretal one alulated wth equlbrum equaton. Furthermore, the mpled the eponental repreentaton o the upper equaton b ntrodung two equaton or two traght lne tartng rom M/Mu =1 and M/ Mu =1 nteretng at the pont o equal relatve moment Fgure.41. The equaton o the two traght lne are a ollow: M M M M u u 1 M u Equaton.95 M M M M u u 1 M u Equaton.96 91

107 1.0 eponental ontour (M/Mu)(1-B/B)+M/Mu = 1 M/Mu (M/Mu)(M.Mu)+(1-B/B) = M/M FIGURE.41 Smpled Interaton Curve b Parme et al. (1966)..1.6 Ultmate Strength Degn Chart or Column wth Baal Bendng (Weber1966) Baed on Furlong onluon that the mot rtal bendng ae the 45 degree one ater the major and mnor ae n the ae o baal bendng. Weber (1966) generated teen hart or the 45 degree nteraton dagram or quare olumn. the olumn are havng mmetral renorement wth derent amount o teel bar. Degn ad n the 1970 ACI SP- 17A Handbookl and the 197 CRSI Handbookl3 were baed on nteraton dagram developed be Weber (1966) Workng Stre Column Degn Ung Interaton Dagram (Mlona 1967) Mlona (1967) adapted the nteraton dagram hart generated n the ACI Degn handbook (1965), that were manl or olumn ubjeted to aal load and unaal bendng and the teel dtrbuted on two ae parallel to the bendng a, to t ae o baal bendng and teel dtrbuted along the our ae. Two reduton ator were ntrodued, one or eah zone (Fgure.4). 9

108 Regon 1 Fa Pa r.ma Ma Regon N r.mb D Mb D Compreon ontrol zone Pb N r.mb r.m Mb Regon 3 M Tenon ontrol zone o ko.mo Mo M FIGURE.4 Workng Stre Interaton Dagram or Bendng about X-A b Mlona (1967) or zone : r 1 k w 1 w Equaton.97 k the moment o the teel dtrbuted on two ae and equal to k a a 0.5 Equaton.98 w n 1g pg 3 Equaton.99 g =bar enter p g = teel rato a = eton area o arbtrar bar = bar dtane rom -a dvded b g t(eton heght) 93

109 For zone 3 r k o N P b r k o Equaton.300 K o the moment reduton ator or pure bendng about -a k o a a 0.5 Equaton.301 P b = load at balane alure N = normalzed aal load Mlona (1967) alo uggeted that the appled bendng moment hould be ompared to the redued moment apat, the moment apat ound rom unaal bendng nteraton hart, o the eton n orm r M M M r M 1 Equaton.30 M,M are the appled moment M,M moment apat..1.8 Comparon o Epermental Reult wth Ultmate Strength Theor or Renored Conrete Column n Baal Bendng (Brettle and Talor 1968) Brettle and Talor (1968) uggeted parttonng the ro eton nto mall ze area, and ung the lmtng tran and the neutral a poton n alulatng tree n eah lament ung urvlnear tre dtrbuton or retangular tre dtrbuton or trapezodal tre 94

110 dtrbuton or onrete. The generated ultmate trength degn hart relatng P u /P o to e r /b or derent t/b rato and derent nlnaton angle beween the lne onetng the load to the entrod and the -a b b +Pu 0.7 t 0.8 t e r = t =b q = 1.0 = p = p Pu/Po t = b 0.3 t = b t =b e / b r FIGURE.43 Comparon o Steel Stre Varaton or Baal Vendng When = 30 and q = 1.0 Brettle and Talor (1968) er = reultant eentrted t = eton heght b = eton wdth Po = theoretal ultmate load wth no eentrte Pu= theoretal ultmate load wth eentrte..1.9 Baal Fleure and Aal Load Interaton n Short Retangular Renored Conrete Column (Row and Paula 1973) Row and Paula (1974) ntrodued hart relatng the m to P u / bh to altate the degn proe. However thee hart are applable to lmted ae onl baed on the materal properte requred or degn 95

111 m M u 1 k bh Equaton.303 M k Equaton.304 M Baal Bendng Smpled (Gouwen 1975) Gouwen (1975) propoed mpled analtal equaton or degn olumn ubjeted to baal bendng. He utlzed Parme et al. (1966) mpled moment Equaton.98 and.99. He ound that approahe 1 or 0.5 bh b eamnng 67 olumn ae. Baed on that he propoed equaton a ollow: For P 0.5 C P 0.5 C C C 5 Equaton.305 For P < 0.5 C C P C C Equaton.306 C bh C A Equaton C 0.03 C C C Equaton C C 0. 5 C C Equaton

112 Anal o Short Retangular Renored Conrete Column Subjeted to Baal Moment (Sallah 1983) Sallah (1983) evaluated the Parameter, ound b Parme et al. (1966) and ound that t wa mot aeted b,, r, P u /P uo and le aeted b the number o bar. Sallah (1983) ntrodued number o hart mlar to Parme et al. (1966) or ndng..1.3 Degn Contour Chart or Baal Bendng o Retangular Renored Conrete Column Ung Breler Method (Talor and Ho 1984) Talor and Ho (1984) developed a omputer program to generate the two man nteraton dagram (wth unaal bendng-one or eah a). Thee two hart were ued to generate the whole baal alure urae (and the alure ontour)ung Breler equaton. Derent poton o parallel neutral a and ruhng tran o onrete were ued to generate tran prole. The tree were generated b tre blok or other aepted ormula. And ore and moment were alulated. The plotted hart howng the load trang on the ro eton. 1. Pn/Po= e /b.6 Pn/Po= Pn/Po= olumn Seton e /t FIGURE.44 Non Dmenonal Baal Contour on Quarter Column b Talor and Ho (1984) 97

113 Radal Contour Method o Baal Short Column Degn (Hartle 1985) Hartle (1985) propoed two degn proedure, one or ndng the ro etonal length and the other to alulate the teel renorement, gven all other den paramete. He howed an optmum pont to et on the 3 D. nteraton dagram that relate to the mallet area o the ro eton. Intall, he howed the relaton between the load and eentrt n the orm: Pu ln Po e C b Equaton.310 where a urve ontant,b eton length and e ore eentrt the ntal value o the ro eton length an be ound b 3.58b 3.6 ln Pu Po / A g b * M b t Pu 0 Equaton.311 Hartle (1985), ung omputer program, plotted graphall the relaton between the ro etonal area and the rato o P u /P o. Thee hart an be ued to determne the utable length n degn. = 300MPa = 0MPa = 300MPa >= 0.6 b/t >= 1 M/Mo 1.0 A R 1.0 M/Mo Pu/Puo FIGURE.45 P u /P uo to A Relaton or 4bar Arrangement b Hartle (1985) (let) Non-Dmenonal Load Contour (Rght) 98

114 Hartle (1985) alo howed the relaton between the R and n the load ontour b n R 1 An Equaton.31 where R and are howen n Fgure.45 (rght) Epert Interatve Degn o R/C Column under Baal Bendng (Sak nd Buukozturk 1986) Sak and Buukozturk (1986) developed omputer otware EIDOCC (Epert nteratve degn o onrete olumn) to anale and degn olumn ubjeted to baal bendng. The proedure a ollow 1. Fndng the neutral a loaton, aordng to Ramamurth proedure, uh that e tan e u u e tan e Equaton.313 e u = ultmate eentrt meaured parallel to -a e u = ultmate eentrt meaured parallel to -a e = eentrt meaured parallel to -a e = eentrt meaured parallel to -a. Ung the neutral a depth,, or the balaned alure a ntal value 3. Calulatng P u and teratng or ung moded eant numeral method tll the load ver loe to P u 4. Calulatng e u, e u and omparng them to e, e to hek eton adequa. 99

115 Interatve Degn o Renored Conrete Column wth Baal Bendng (Ro and Yen 1986) Ro and Yen (1986) developed a omputer program to analze and degn retangular olumn ubjeted to baal bendng. The proedure to hange the nlnaton angle o the neutral a to nd adequate relaton between M n, M n, and then hange the poton o the neutral a to olve or the aal load. The eton apat alulated ung a predned poton o the neutral a and ruhng tran equal to or onrete. The uggeted ung our bar ntall n the degn proe and keep nreang aordng to the appled load wth lmtng the number o bar a tated b ACI ode Degn o Column Subjeted to Baal Bendng (Horowtz 1989) Horowtz (1989) developed a omputer program or olumn wth an ro eton ubjeted to baal bendng. He reled on ndng the leat poble loaton o teel bar that make the eton apat more than the appled load Strength o Renored Conrete Column n Baal Bendng (Amrthandan 1991) Amrthandan et.al (1991) howed good orelaton between the epermental work done beore and the method proped n the autralan tandard or onrete truture AS 3600 or hort olumn. The load ontour n the tandard appromated b breler equaton. The adopted the beta value rom the Brth tandard N / 0. 6Nuo Equaton.314 N = degn aal ore N uo = ultmate aal load. 100

116 Computer Anal o Renored Conrete Seton under Baal Bendng and Longtudnal Load (Zak 1993) Zak (1993) propoed olvng the equlbrum equaton wth the modaton o the eant modulu method. The ultmate tran wa not determned. However, t wa ound ung mamzaton method Anal and Degn o Square and Retangular Column b Equaton o Falure Surae Hu (1994) Hu (1994) propoed equaton that over olumn ubjeted to baal bendng and aal ompreon or tenon. The propoed equaton a ollow: Pn P Po P nb nb M M n nb 1.5 M M n nb Equaton.315 Pn = nomnal aal ompreon or tenon Mn, Mn= nomnal bendng moment about and a Po = mamum nomnal aal ompreon or aal tenon Pnb = nomnal aal ompreon at balaned tran ondton Mnb, Mnb = nomnal bendng moment about and a at balaned tran ondton Baal Interaton Dagram or Short RC Column o An Cro Seton (Rodrguez and Ohoa 1999) Rodrguez and Ohoa (1999) propoed a general method or analzng an ro eton ubjeted to baal bendng. The developed loed orm oluton or nomnal total aal ore trength and nomnal bendng moment trength about the global X and Y-ae. Qua-Newton method wa ued to olve thee oupled nonlnear equaton to loate the neutral a poton. 101

117 P n m P n A n b A b b Equaton.316 M n n n t 1 M o n t 1 M Y n t a 1 P n t 1 A b Y b n b 1 A b Y b Equaton.317 M n o n t 1 M n n t 1 M X n t a 1 P n t 1 A b X b n b 1 A b X b Equaton.318 P n = Nomnal aal ore trength. M n = nomnal bendng moment trength about a M n = nomnal bendng moment trength about a X a, Y a = oordnate o orgn wth repet to global, ae = angle o nlnaton o neutral a wth repet to Xa; n = number o renorement bar; n b = number o rebar loated on ompreon de o ro eton; n t = number o trapezod ued to appromate onrete under ompreon; A b = area o teel rebar ; = onrete tre at renorement bar = teel tre at renorement bar P = ore or eah trapazod. M = Moment o eah trapazod about a. M = Moment o eah trapazod about a Short Renored Conrete Column Capat under Baal Bendng and Aal Load (Hong 000) Hong (000) dd not aume an ruhng tran lmt. He propoed two equaton rom equatng ore and moment: 10

118 M P n - e L 0 Equaton.319 M P n - e L 0 Equaton.30 where e l, e l the load eentrt to and ae repetvel.the two equaton ha three unknow; the urvature, neutral a nlnaton angle and the neutral a nterpt wth the - a. Hong (000) ued the equental quadrat programmng method to olve the ae a a nonlnearl ontraned optmzaton problem Relablt o Renored Conrete Column under Aal Load and Baal Bendng (Wang and Hong 00) Wang and Hong (00) evaluated the parameter (Parme et al. 1966) and ount that t nentve to the renored rato, t more entve to baal bendng than unaal bendng, t nreae wthload and onnrete ompreve trength Anal and Degn o Conrete Column or Baal Bendng: Overvew (Furlong, Hu, and Mrza 004) Furlong et al. (004) revewed man o the propoed ormula or anal. Thee ormula were ompared to epermental work. The onluded that the equaton o Breler (1960), although mple, are not ver onervatve, whle Hu equaton muh more onervatve. A Hu equaton an be ued n baal bendng and tenon a well. However, both Hu equaton and Breler reproal load equaton an not be ued n eletng ro eton, unlke Breler load ontour equaton New Method to Evaluate the Baal Interaton Eponent or RC Column (Bajaj and Mend 005) Bajaj and Mend (005) uggeted new equaton to evaluate the baal nteraton eponenet a ound b Breler (1960). The propoed equaton are a ollow 103

119 M M n no M M n no Equaton.31 log 0.5 K log Equaton.3 Bajaj and Mend (005) benhmarked ther equaton b omparng the reult wth epermental work done on 8 (150* 150 mm) olumn Anal o Renored Conrete Column Subjeted to Baal Load (Demagh, Chabl, and Hamzaou 005) Demagh et al. (005) uggeted olvng or the three equaton o equlbrum to nd the nomnal ore P n, the nlnaton angle o the neutral a and the depth o the neutral a b. The three equaton are: P n P A Equaton.33 M n P e n n M o M Y P A Equaton.34 M n P e n o M n M X P A Equaton.35 where the ubrpt reer to a onrete laer or teel bar element Analtal Approah to Falure Surae n Renored Conrete Seton Subjeted to Aal Load and Baal Bendng (Bonet, Mguel, Fernandez, and Romero 006) Bonet et al. (006) developed a new method or the urae alure baed on numeral mulaton. The numeral mulaton wa generated ung a omputer program apable o analng moment-urvature dagram or gven aal load and moment rato. The mamum value wa ued a a alure pont or the gven load. The alure urae dened b two 104

120 105 dretr urve and generatr urve. The dretr urve are the urve orrepond to zero aal ore and the one orrepond to balane alure.the generat urve are dened n Mu/Mu plane, the rt urve onnet the pure tenon aal load to balne alure load. Wheara the eond urve onnet the balne alure load to the pure ompreon load. The equaton or the our urve are a ollow Dretr 1 1.n.o 1, 1 1 1, 1 d d d d M M M M Equaton.36 Dretr 1.n.o, 1, d d d d M M M M Equaton.37 generat 1 0 * d u ut d ut u ut d d d ut u d u N N N N N N N N M M n N M M Equaton.38 generat d u d u u u ut d d d ut u d u M M N N N N N N M M n N M M Equaton.39

121 M d1 = abolute value o the nomnal bendng moment o the eton n mple leure orrepondng to angle beta Md 1,, Md 1, = nomnal bendng moment o the eton n mple leure or the and ae, repetvel. M d = abolute value o the nomnal bendng moment orrepondng to the mamum bendng apat o the eton or a partular angle Md,, Md, = nomnal bendng moment orrepondng to the mamum leure apat o the eton or the and the ae, repetvel., =eponent o the dretre = teel renorement N u = aal load appled N u = the ultmate aal load n pure ompreon N d =balane alure load. N N N u u 0.8* u N lm Equaton.330 N lm = nomnal aal ompreon at the balaned tran ondton Baal Bendng o Conrete Column: an Analtal Soluton (Cedoln, Cuat, Ehel, Roveda 006) Cedoln et al. (006) ntrodued analtal oluton o the alure envelope o retangular R/C ro eton ubjeted to baal bendng and to an aal ore b appromatng the retangle to equvalent quare eton. The anal wa or unonned onrete and the oluton outome wa dmenonle. 106

122 Comparatve Stud o Analtal and Numeral Algorthm or Degnng Renored Conrete Seton under Baal Bendng (Bonet, Barro, Romero 006) Bonet et al. (006) ntrodued analtal and numeral method or degnng rular and retangular ro eton ubjeted to b-aal bendng. The analtal method ue the hevde unton (Barro et al. 004) to dene the alure tran, then ntegrate the tre baed on that alure. The numeral method break the eton nto mult thk laer parallel to the nuteral a. The nternal ore are ound b numeral ntegraton o eah laer ung Gau- Legendre quadrature (Barro et al. 004). The onluded that the two method are eent or rular ro eton anal and the moded thk laer ntegraton more eent or the retangular ro eton anal Invetgaton o Baal Bendng o Renored Conrete Column through Fber Method Modelng (Lejano 007) Lejano (007) epanded the nte element method ound b Kaba and Mahn (1984). To predt the behavor o unonned retangular olumn ubjeted to baal bendng. The anal wa lmted to unorm emmetr quare olumn. Lejano (007) utlzed Bazant Endohron theor or onrete and Camp model or teel Varaton o Ultmate Conrete Stran at RC Column Subjeted to Aal Load wth B-Dretonal Eentrte (Yoo and Shn 007) Yoo and Shn (007) ntrodued the moded retangular tre blok (MRSB) to aount or non-retangular ompreon zone ndued b b-aal bendng. The howed epermentall that the ultmate tran o onrete epoed to b-dretonal eentrte an reah up to Baed on th ndng the ntrodued new equaton or the unonned ultmate tran a ollow: P n n u Po Po P Equaton

123 0.003 P n u Pn Po Po Equaton h tan Equaton h tan b b h 90 tan b tan 1 h 90 b Equaton.334 No denton or wa provded Capat o Retangular Cro Seton under Baall Eentr Load (Cedoln, Cuat, Ehel, Roveda 008) Cedoln et al. (008) utlzed the work o Cedoln et al. 006 to generate more aurate moment alure ontour through reatng one etra pont on the ontour. Th pont orrepond to the load atng on retangle dagonla and wa appromated b ung equvelant quare to benet rom mmetr. The developed moment ontour wa ued or better evaluatng the parameter ound b Breler (1960) Development o a Computer Program to Degn Conrete Column or Baal Moment and Normal Fore (Helgaon 010) Helgaon 010 developed a omputer program ung Matlab or degnng unonned retangular hollow or old olumn ubjeted to aal ore and bendng moment. Helgaon 010 ued the predened tran prole to generate the nteraton dagram and the equvelant tre blok equal to 80% o the ompreon zone depth. The outome wa ompared to Euroode. 108

124 .. Duon Aordng to the lterature revew, there are ve derent approahe that treated the olumn under aal load and bendng moment problem. Thee wa are ummarzed a ollow: 1. Tral or loatng the neutral a poton uh a Parker and Sanlon (1941), Ang (1961) and Czernak (196) work.. Implementng loed orm equaton or peal ae uh a Anderen (1941), Wenger (1958), Cedoln et al. (006) and Yoo and Shn (007) work. 3. Generatng hart that relate two or more varable to altate the degn proe, uh a Mkhalkn, Au (1958), Flemng and Werner (1965) and Brettle and Talor (1968) work. 4. Developng mpled Interaton dagram b ung oeent or urve denng. Th method wa adopted b ome reearher lke Whtne and Cohen (1957), Breler (1960), Furlong (1961), Parme (1966), Mlon (1967), Bonet et al. (006). 5. Generatng Set o read Interaton dagram to be ued dretl b degner, Weber (1966) and other. There are ome onluon that an be drawn a ollow The nte laer approah ueul n anal. Th approah wa adopted b ome author uh a Brettle and Talor (1968), Bonet et al. (006) and Lejano (007). The Breler Method one o the mot well known and ueul method n predtng the unonned nteraton dagram and load ontour. Th method wa utlzed and rened b man uh a Rammamurth (1966), Parme et al. (1966), Gouwen (1975), Sallah (1983) Amrthandan (1991), Wang and Hong (00) and Bajaj and Mend (005). However t ver onervatve or ome ae a hown b Furlong et al. (004) and other. Sotware applaton on olumn pread and beame popular n the begnnng o 1980.Talor and Ho (1984) developed omputer program 109

125 baed on Breler Method. Sak and Bugukoztruk (1986) developed ther program baed on teratng or neutral a and load onverge. Ro and Yen (1986) ued the predened tran prole n ther otware. Horowtz (1989) nremented the teel bar tll the olumn apat eeeded the load appled. Th tranton n relng on mahne or altate alulaton. Hene more aurate and pree anal needed to dene eatl the unonned and onned apat o derent eton. The predened tran prole een to be one o the mot eetve and at proedure oe unonned anal. Th method wa uggeted b Furlong (1961) and utlzed b man, uh a Ro and Yan (1986). There lak o onnement eet anal on olumn apat. Nowada, there a need n predtng olumn etreme event a tated b ome trutural ode lke AASHTO-LRFD. 110

126 Chapter 3: Retangular Column Subjeted to Baal Bendng and Aal Compreon 3.1 Introduton Retangular renored onrete olumn an be ubjeted to baal bendng moment plu aal ore. When the load at dretl on one o the ro eton bendng ae the problem beome o unaal bendng and aal ore. However when the load appled eentrall on a pont that not along an o the bendng ae the ae generall baal bendng and aal ore. The baal bendng ae an be ound n man truture nowada. Th ae vted etenvel n the lterature ade rom the onnement eet. The alure urae o retangular olumn 3D urae onted o man adjaent D nteraton dagram. Eah o the D nteraton dagram repreent one angle between the bendng moment about -a and the reultant moment. Man mplaton are ntrodued to jut the ompreve trapezodal hape o the onrete ompreon zone, due to the etene o the two bendng ae. Appromaton alo were preented to dept the 3D alure hape rom the prnpal nteraton dagram, n the two ae o mmetr. The mot eetve proedure ound n the lterature the predened ultmate tran prole that determne a ertan poton o the neutral a and agn ruhng ultmate tran (tpall 0.003) n one o the olumn orner. Wth the advane n tehnolog and the enormou peed o omputaton, anal needed to plot a more aurate alure nteraton dagram or both the unonned and onned ae. The methodolog n th tud baed on two derent approahe; the adjuted predened ultmate tran prole and the moment o area generalzaton approahe derbed below. The two method are ompared to benhmark the moment o area generalzaton method that wll be ued n the atual apat anal (Conned anal). Th anal ompared to epermental data rom the lterature. 111

127 3. Unonned Retangular Column Anal 3..1 Formulaton Fnte Laer Approah (Fber Method) The olumn ro eton dvded nto nte mall-area lament (Fgure 3.1a). The ore and moment o eah lament alulated and tored. The rebar are treated a drete objet n ther atual loaton. The advantage o that to avod naura generated rom ung the appromaton o the tre blok method, a a repreentatve o the ompreon zone and to well treat ae that have ompreve trapezodal or trangular hape generated rom the neutral a nlnaton (Fgure 3.1b). a) b) M M FIGURE 3.1 a) Ung Fnte Flament n Anal b)trapezodal Shape o Compreon Zone Conrete Model Conrete analzed ung the model propoed b Hognetad that wa adopted rom Rtter Parabola 1899 (Hognetad 1951). Hognetad model ued etenvel n numerou paper a t well eplan onrete tre-tran behavor n ompreon. In addton, t wa utlzed b wdel ued onrete model uh a Kent and Park model (1971). The tre-tran model epreed ung the ollowng equaton (Fgure 3.): 11

128 o o Equaton 3.1 = tre n onrete n ompreon. = mamum ompreve trength o the onrete. = tran at o = tran at A hown n Fgure 3.a onrete arre tenon up to rakng trength, then t negleted n alulaton beond that. a) b) Aal tre 0.75 Aal tre Aal tran k Aal tran FIGURE 3. a) Stre-Stran Model or Conrete b Hognetad b) Steel Stre-Stran Model Steel Model Steel aumed to be elat up to the eld tre then peretl plat a hown n Fgure 3.b. It aumed that there peret bond between the longtudnal teel bar and the onrete. Aordng to Bernoull Hpothe, tran along the depth o the olumn are aumed to be dtrbuted lnearl. 113

129 3.. Anal Approahe The proe o generalzaton o the moment-ore nteraton dagram developed ung two derent approahe; the adjuted predened ultmate tran prole and the generalzed moment o area method. The ommon eature o the two approahe are derbed a ollow: Approah One: Adjuted Predened Ultmate Stran Prole The rt approah the well known method that wa ued b man reearher and pratng engneer. The proedure to agn ompreve alure tran at one o the olumn orner (0.003) and to var the poton and the nlnaton angle o the neutral a that range rom zero degree, parallel to the wdth o the olumn, to nnet degree parallel to the heght a hown n Fgure FIGURE 3.3 Derent Stran Prole Due to Derent Neutral A Poton Eah et o the parallel neutral ae o a ertan orentaton repreent appromatel one D nteraton dagram, and all o the et rom zero to nnet degree repreent the 3D alure urae n one quadrant, whh dental to the other three quadrant due to the etene o 114

130 two a o mmetr wth repet o onrete and teel. The proedure derbed n the ollowng tep: 1. Denng the tran prole or eah neutral a poton and orner ultmate tran appled.. Calulatng tran and the orrepondng tre n eah lament o onrete and dong the ame or eah teel bar Fgure N.A FIGURE 3.4 Denng Stran or Conrete Flament and Steel Rebar rom Stran Prole 3. Calulatng the ore and the moment about the geometr entrod or eah lament and teel bar Fgure 3.5. or onrete: or teel P wt A P Equaton 3. M P * Y _ M P * Y _ Equaton 3.3 M P * X _ M P * X _ Equaton

131 B B X_ X X _S X H t w Y Y_ H Y Y_ FIGURE 3.5 Flament and Steel Rebar Geometr Properte wth Repet to Cruhng Stran Pont and Geometr Centrod Summng up the ore and moment, rom teel bar and onrete lament, to get the nternal ore and moment about -a and -a. The reultant ore and moment repreent one pont on the unonned nteraton dagram (Fgure 3.6). Start Dene Neutral a poton & orentaton Determne tran prole baed on at a orner Calulate,,, New Pont Sum up nternal ore and moment Fz,GM o,gm o,gm R End FIGURE 3.6 Method One Flowhart or the Predened Ultmate Stran Prole Method 116

132 The problem arng rom th proedure that the pont developed rom one et o parallel neutral ae are loe to but not lned up n one plane. However, the are attered tghtl near that plane (Fgure 3.7). To orret or that, an average angle o o 1 ( M / M ) alulated and another run etablhed b lghtl hangng the nlnaton angle o the neutral a o the eton wth repet to the -a and teratng tll the angle determned or eah pont onverge to the average angle. The average angle taken a the average o all angle obtaned or a ertan angle orentaton o the neutral a (Fgure 3.3 and Fgure 3.7). R P Plane o the average angle Pont beore orreton Pont ater orreton M M Fgure 3.7 D Interaton Dagram rom Approah One Beore and Ater Correton The teraton mentoned above onverge at n all ae. Th approah eld a ver at omputaton ne t dretl evaluate the ultmate unonned tran prole. However, no moment urvature or load-tran htor repone avalable wth th approah Approah Two: Generalzed Moment o Area Theorem Moment o Area Theorem The ver general aal tre equaton n an unmmetral eton ubjeted to aal ore P and baal bendng M and M (Hard Cro 1930): 117

133 P M I M A I I I M I I I M z I I I Equaton 3.5 z = normal tre at an pont (a) n ro eton P = appled load. A = ro etonal area. M = bendng moment about the geometr -a M = bendng moment about the geometr -a = dtane between the pont (a) and -a = dtane between the pont (a) and -a I = moment o nerta about the geometr -a I = moment o nerta about the geometr -a I = produt moment o nerta n plane Rewrtng Equaton 3.5 to determne the tran at an pont n the ro eton: P M EI M EI EA EI EI EI M EI M EI EI EI EI z Equaton 3.6 In ae o lnear elat anal, E n EA or EI epreon ontant (E=E ). However, the eton ha lnear tran but nonlnear tre prole, t wll amount to varable E prole (per laer or lament) n nonlnear anal. Aordngl, the eton parameter mut nlude E A, EI or a more generalzed theor (Raheed and Dnno 1994). Note that the lnear tran prole o the eton rom Equaton 3.6 eld two dtnt ontant urvature: M EI M EI Equaton 3.7 M EI M EI Equaton

134 = urvature about the -a = urvature about the -a EI EI EI To prove Equaton 3.7 and 3.8 above, nvoke the oupled equaton o moment about the atual or urrent entrod (Bkord 1998). M EI EI Equaton 3.9 M EI EI Equaton 3.10 In a matr orm: M M EI EI EI EI Equaton 3.11 Invertng Equaton EI EI EI EI M M Equaton 3.1 whh reprodue Equaton 3.7 and 3.8. Rewrtng Equaton 3.6 n term o and z P EA Equaton 3.13 Fndng z at the atual or urrent entrod, ne = = 0. o = P EA Equaton 3.14 Fndng z at the geometr entrod, 119

135 o P = EA Equaton 3.15 Solvng or P at the geometr entrod; P EA EA EA o Equaton 3.16 But EAM EA EAM EA Y G Y X G X Y G the vertal dtane to the geometr entrod meaured rom bottom, X G the dtane to the geometr entrod meaured rom the ro eton let de, Y the vertal dtane to the nelat entrod meaured rom the bottom and X the horzontal dtane to the nelat entrod meaured rom the ro eton let de (Fgure 3.8). Thu, P EA EAM EAM Equaton 3.17 o The general ormula o the moment about the geometr -a and the geometr -a derved a ollow: when the moment tranerred rom the entrod to the geometr entrod (Fgure 5.8a) M M P Equaton 3.18 Subttutng Equaton 3.9 and 3.16 n 3.17 eld: M EI EI EA EAM EAM o Equaton

136 M EAM o EI EAM EI EAM Equaton 3.0 Smlarl (Fgure 3.8b): a) b) G M C M G P P M C G G M FIGURE 3.8 Tranerrng Moment rom Centrod to the Geometr Centrod M M P Equaton 3.1 M EI EI EA EAM EAM o Equaton 3. M EAM o EI EAM EI EAM Equaton 3.3 The term EI EAM and EI EAM repreent the EI and geometr entrod repetvel ung the parallel a theorem. And the term and EI about the EI EAM EI EAM are equal gven that: EAM EA and EAM EA. Ung Equaton 3.16, 3.19, and 3. eld the etended generalzed moment o area equaton: P EA M EAM M EAM EAM EI EI EAM EI EI o Equaton

137 Sne the moment o area about the nelat entrod vanhe (Raheed and Dnno 1994), Equaton 3.3 redue to a partall unoupled et when t appled bak at nelat the entrod ne EAM and EAM vanh about that entrod. P M M EA EI EI 0 EI EI o Equaton 3.5 whh mpl Equaton 3.9, 3.10, and Method Two Th approah mulate the radal loadng o the ore and moment b keepng the relatve proporton between them ontant durng the loadng. Aordngl, all the pont omprng an nteraton dagram o angle wll be eatl on that D nteraton dagram. In addton to the ultmate pont, the omplete load deormaton repone generated. The ro eton analzed loaded nrementall b mantanng a ertan eentrt between the aal ore P and the reultant moment M R. Sne M R generated a the reultant o M and M, the angle = tan -1 (M /M ) kept ontant or a ertan D nteraton dagram. And ne nreang the load and reultant moment proportonall aue the neutral a to var unpredtabl, the generalzed moment o area theorem deved. Th method baed on the general repone o retangular unmmetral eton ubjeted to baal bendng and aal ompreon. The ammetr tem rom the derent behavor o onrete n ompreon and tenon. The method developed ung nremental teratve anal algorthm, eant tne approah and proportonal or radal loadng. It eplaned n the ollowng tep. (Fgure 3.1 preent a lowhart o the outlned proedure): 1. Calulatng the ntal eton properte: Elat aal rgdt EA: 1

138 13 Equaton 3.6 = ntal modulu o elatt o the onrete = ntal modulu o elatt o the teel rebar The depth o the elat entrod poton rom the bottom ber o the eton Y and rom the let de o the eton X Equaton 3.7 EA X B A E E X B w t E X ) ( ) ( ) ( Equaton 3.8 where Y and Y are meaured to the top etreme ber, X and X are meaured to the rght mot etreme ber, ee Fgure 3.9. Elat leural rgdt about the elat entrod EI: Equaton 3.9 ) ( ) ( ) ( X X B A E E X X B w t E EI Equaton 3.30 X X B Y Y H A E E X X B Y Y H w t E EI ) ( ) ( ) ( Equaton 3.31 Tpall the ntal elat Y = H/, X = B/ and EI = 0 A E E w t E EA ) ( E E EA Y H A E E Y H w t E Y ) ( ) ( ) ( ) ( ) ( ) ( Y Y H A E E Y Y H w t E EI

139 0.003 X B X B Y Y H C t w G H C G Y G Y Y G Y X X G X X G FIGURE 3.9 Geometr Properte o Conrete Flament and Steel Rebar wth Repet to Geometr Centrod and Inelat Centrod The depth o the geometr eton entrod poton rom the bottom and let ber o the eton Y G, X G: H Y G Equaton 3.3 X G B Equaton Denng the eentrt e, whh pee the radal path o loadng on the nteraton dagram. Alo, denng the angle n between the reultant moment GM R and GM X 14

140 e Aal Fore Load Step GP Reultant Moment FIGURE 3.10 Radal Loadng Conept 3. Denng the loadng tep GP a a mall porton o the mamum load, and omputng the aal ore at the geometr entrod. GP new GP old GP Equaton Calulatng the moment GM R about the geometr entrod. GM R e GM R e * GP GP Equaton 3.35 GM X GM R o Equaton 3.36 GM Y GM X tan Equaton Tranerrng the moment to the nelat entrod and alulatng the new tranerred moment TM X and TM Y : TM X GM X GP Y ( G Y ) Equaton

141 TM Y GM Y GP ( X G X ) Equaton 3.39 The advantage o tranerrng the moment to the poton o the nelat entrod to elmnate the ouplng eet between the ore and the two moment, ne EAM EAM 0 about the nelat entrod P TM TM EA EI EI 0 EI EI o Equaton 3.40 Y G Y X XG FIGURE 3.11 Moment Tranerrng rom Geometr Centrod to Inelat Centrod 6. Fndng: Curvature X and Y X TM X TM Y * EI Y * EI XY Equaton 3.41 Y TM Y TM X * EI X * EI XY Equaton

142 EI X EIY EI Equaton 3.43 Stran at the nelat entrod, the etreme ompreon ber tran, and tran at the etreme level o teel n tenon are ound a ollow: e o e GP o EA Equaton 3.44 e o ( H Y ) ( B X ) Equaton 3.45 ( Y Cover) ( X Cover) Equaton 3.46 e o where over up to enter o bar 1. Calulatng tran and orrepondng tre n eah lament o onrete eton b ung Hognetad model (Equaton 3.1) n ae o unonned anal TM B X X TM H Y Y GP TM H Y Y TM B X X EI EI EI EA EI Equaton Calulatng tran and orrepondng tre n eah bar n the gven eton b ung the teel model hown n Fgure 3.b. TM B X X TM H Y Y GP TM H Y Y TM B X X EI EI EI EA EI Equaton Calulatng the new eton properte: aal rgdt EA, leural rgdte about the nelat entrod EI,, EI, EI moment o aal rgdt about nelat entrod EAM, EAM, nternal aal ore F z, nternal bendng moment about the nelat entrod M o,m o: 17

143 EA E w t ( E E ) A Equaton 3.49 EAM Ewt ( H Y Y ) ( E E ) A ( H Y Y ) Equaton 3.50 EAM E wt ( B X X ) ( E E ) A ( B X X ) F w t ( ) A z Equaton 3.51 Equaton 3.5 EI E w t H Y Y E E H Y Y ( ) ( ) ( A ) Equaton 3.53 EI E w t B X X E E H X X ( ) ( ) ( A ) Equaton 3.54 EI E w t ( H Y Y ) B X X ( E E ) A ( H Y Y ) B X X Equaton 3.55 M M o o wt H Y Y ) ( ) A H Y Y ( Equaton 3.56 wt B X X ) ( ) A B X X ( Equaton 3.57 where E = eant modulu o elatt o the onrete lament. E = eant modulu o elatt o the teel bar.. Tranerrng bak the nternal moment about the geometr entrod GM o M o GP Y ( G Y ) Equaton 3.58 GM o M o GP ( X G X ) Equaton

144 3. Chekng the onvergene o the nelat entrod TOL EAM / EA/ Y Equaton 3.60 TOL EAM / EA/ X Equaton Comparng the nternal ore to appled ore, nternal moment to appled moment, and makng ure the moment are alulated about the geometr entrod : GP F z 1*10 5 Equaton 3.6 1*10 5 GM GM o 5 1*10 GM o GM Equaton 3.63 TOL 1*10 5 TOL 1*10 5 Equaton 3.64 I Equaton 3.61, 3.6, and 3.63 are not ated, the loaton o the nelat entrod Updated b EAM /EA and EAM /EA and tep 5 to 1 are repeated tll Equaton 3.61, 3.6 and 3.63 are ated. Y new Y old EAM EA Equaton 3.65 X new X old EAM EA Equaton 3.66 One equlbrum reahed, the algorthm hek or ultmate tran n onrete teel not to eeed and 0.05 repetvel, then t nreae the loadng b GP and e run the anal or the new load level ung the latet eton properte. Otherwe, 19 e and e

145 equal or e equal 0.05, the target ore and reultant moment are reahed a a pont on the alure urae or the amount o eentrt and angle ued. Start C alulate ntal eton properte EA,EI, EI,EI Y,Y G,X,X G Input P, e & Traner moment to nelat entrod TM =GM +P(Y G -Y ) TM =GM +P(X G -X ) Calulate ø,ø, o, e, e Calulate,,, Calulate new eton properte EA,EI, EI, EAM, EAM,EI C alulate nternal ore and m om ent F z,m o,m o Traner moment bak to G.C. GM o=m o -P(Y G -Y ) GM o=m o -P(X G -X ) No onvergene aheved. top no P += P no EAM EA*Y < EAM EA*X < e G P-F z < 10-5 G M - GM o <10-5 G M - GM o < 10-5 e e > e > 0.05 no Y += EAM/EA X += EAM/EA e e New entrt P & M aheved e no New a no End FIGURE 3.1 Flowhart o Generalzed Moment o Area Method Ued or Unonned Anal 130

146 3..3 Reult and Duon Comparon between the Two Approahe The two approahe are ompared to eah other n the ollowng. The olumn ued n omparon ha the ollowng properte: Seton Heght = 0 nhe Seton Wdth = 10 nhe Clear Cover = nhe Steel Bar n dreton = 3 # 4 Steel Bar n dreton = 6 # 4 Hoop #3 = 4 k = 60 k. Fore (k.) Reultant Moment (k.n) Method 1 Method FIGURE 3.13 Comparon o Approah One and Two ( = 0) 131

147 Fore (k.) Reultant Moment (k.n) Method 1 Method FIGURE 3.14 Comparon o Approah One and Two ( = 4.7) Fore (k.) Reultant Moment (k.n) Method 1 Method FIGURE 3.15 Comparon o Approah One and Two ( = 10.8) 13

148 Fore (k.) Reultant Moment (k.n) Method 1 Method FIGURE 3.16 Comparon o Approah One and Two ( = 5) The eellent orrelaton between the two approahe appear n Fgure 3.13 through The reultant moment angle hown below eah graph. Th evdene that approah two eetvel ompared to the well known predened ultmate tran prole approah. Aordngl, method two an be ued n the onned anal or analzng the atual apat o the retangular olumn Comparon wth Etng Commeral Sotware KDOT Column Epert ompared wth CSI Col 8 o omputer and truture In. and SP olumn Sotware o truture pont LLC. The ae eleted rom Eample 11.1 n Note on ACI Buldng ode Requrement or trutural onrete b PCA. The olumn detal are a ollow (Fgure 3.17): Seton Heght = 4 nhe Seton Wdth = 4 nhe Clear Cover = 1.5 nhe Steel Bar = 16 # 7 evenl dtrbuted Hoop #3 = 6 k = 60 k. 133

149 FIGURE 3.17 Column Geometr Ued n Sotware Comparon Aal Fore k Reultant Moment k t. Unonned urve(sp Column) Unonned urve(kdot Column Epert) FIGURE 3.18 Unonned Curve Comparon between KDOT Column Epert and SP Column ( = 0) Fgure 3.18 how the math between the two program n aal ompreon alulaton and n tenon ontrolled zone. However KDOT Column Epert how to be lghtl more onervatve n ompreon ontrolled zone. Th mght be due to ung nte laer approah n alulaton that ha the advantage o aura over other appromaton lke Whtne tre blok. 134

150 Aal Fore k Reultant Moment k t. Degn urve (KDOT Column Epert) Degn urve (CSI Col 8) FIGURE 3.19 Degn Curve Comparon between KDOT Column Epert and CSI Col 8 Ung ACI Reduton Fator The degn urve n Fgure 3.19 and Fgure 3.0 were plotted ung ACI reduton ator that ue a reduton ator o 0.65 n ompreon ontrolled zone a oppoed to 0.75 ued b AASHTO. There a good orrelaton between the KDOT Column Epert urve and CSI Col 8 and SP Column urve a hown n Fgure 3.19 and Fgure 3.0. Aal Fore k Reultant Moment k t. Degn Curve (KDOT Column Epert) Degn urve (SP Column) FIGURE 3.0 Degn Curve Comparon between KDOT Column Epert and SP Column Ung ACI Reduton Fator 135

151 3.3 Conned Retangular Column Anal Formulaton Fnte Laer Approah (Fber Method) The olumn ro eton dvded nto nte mall-area lament (Fgure 3.1a). The ore and moment o eah lament alulated and tored. The rebar are treated a drete objet n ther atual loaton. The advantage o that to avod naura generated rom ung the appromaton o the tre blok method, a a repreentatve o the ompreon zone and to well treat ae that have ompreve trapezodal or trangular hape generated rom the neutral a nlnaton (Fgure 3.1b). a) b) M M FIGURE 3.1 a) Ung Fnte Flament n Anal b)trapezodal Shape o Compreon Zone Connement Model or Conentr Column Mander Model or Tranverel Renored Steel Mander model (1988) wa developed baed on the eetve lateral onnement preure, l, and the onnement eetve oeent, k e whh the ame onept ound b Shekh and Uzumer (198). The advantage o th proedure t applablt to an ro eton ne t dene the lateral preure baed on the eton geometr. Mander et al. (1988) howed the adaptablt o ther model to rular or retangular eton, under tat or dnam loadng, ether wth monotonall or lall appled load. In order to develop a ull tre-tran 136

152 urve and to ae dutlt, an energ balane approah ued to predt the mamum longtudnal ompreve tran n the onrete. Mander derved the longtudnal ompreve onrete tre-tran equaton rom Popov model that wa orgnall developed or unonned onrete (1973): r r 1 r Equaton 3.67 Equaton 3.68 r E Equaton 3.69 E E e E 473 n MPa Equaton 3.70 E e Equaton 3.71 and a uggeted b Rhart et al. (198) the tran orrepondng to the peak onned ompreve trength,, : Equaton 3.7 o The derent parameter o th model are dened n Fgure

153 Compreve Stre Conned Conrete Unonned Conrete E Ee o o p u Frt Hoop Frature FIGURE 3. Aal Stre-Stran Model Propoed b Mander et al. (1988) or Monoton Loadng A hown n Fgure 3. Mander et al. (1988) model ha two urve; one or unonned onrete (lower urve) and the other or onned onrete (upper one). The upper one reer to the behavor o onned onrete wth onentr loadng (no eentrt). It hown that t ha aendng branh wth varng lope tartng rom E dereang tll t reahe the peak onned trength at (, ). Then the lope beome lghtl negatve n the deendng branh repreentng dutlt tll the tran o u where rt hoop rature. The lower urve epree the unonned onrete behavor. It ha the ame aendng branh a the onned onrete urve tll t peak at (, o ). Then, the urve deend tll 1.5- o. A traght lne aumed ater that tll zero trength at pallng tran p. Mander et al. (1988) utlzed an approah mlar to that o Shek and Uzumer (198) to determne eetve lateral onnement preure. It wa aumed that the area o onned onrete the area wthn the enterlne o permeter o pral or hoop renorement A a llutrated n Fgure

154 Eetvel Conned Core Eetvel Conned Core Smallet Length or Conned Core FIGURE 3.3 Eetvel Conned Core or Retangular Hoop Renorement (Mander Model) Fgure 3.3 how that eetvel onned onrete ore A e maller than the area o ore wthn enter lne o permeter pral or hoop eludng longtudnal teel area, A, and to at that ondton the eetve lateral onnement preure l hould be a perentage o the lateral preure l : k l e l Equaton 3.73 and the onnement eetvene oeent k e dened a the rato o the eetve onned area A e to the area enloed b enterlne o hoop eludng the longtudnal bar A : A e ke Equaton 3.74 A A A Al Equaton 3.75 A A A l 1 Equaton 3.76 A A A 1 Equaton

155 where A the area o the eton ore enloed b hoop, A l the area o longtudnal teel and the rato o longtudnal teel to the area o the ore. B b w Eetvel Conned Core d-/ d H Ineetvel Conned Core b-/ b FIGURE 3.4 Eetve Lateral Conned Core or Retangular Cro Seton The total neetve onned ore area n the level o the hoop when there are n bar: A n 1 w 6 Equaton 3.78 Gven that the arhng ormed between two adjaent bar (Fgure 3.4) eond degree parabola wth an ntal tangent lope o 45 o, the rato o the area o eetvel onned onrete to the ore area at the te level: n et w A et 1 6 Equaton 3.79 A 140

156 141 where A = b * d, The area o onned onrete n the mdwa eton between two oneutve te: emt d b d b d b A 1 1 Equaton 3.80 Hene, the eetve area at mdwa: n e d b d b w d b d b d b d b A Equaton 3.81 n e d b w d b A Equaton 3.8 Ung Equaton 3.73 n e d b d b d b w d b k Equaton 3.83 n e d b d b w k Equaton 3.84 and the rato o the volume o tranvere teel n an dreton to the volume o onned ore area and dened a: d A d b b A Equaton 3.85 b A d b d A Equaton 3.86

157 A, A are the total area o lateral teel n and dreton repetvel. The eetve lateral onnng preure n and dreton are gven b: l k e h Equaton 3.87 l k e h Equaton 3.88 l/o 0 Conned Strength rato /o Larget onnng Stre rato Larget onnng Stre rato l1/o FIGURE 3.5 Conned Strength Determnaton 14

158 Fgure 3.5 wa developed numerall ung multaal tre proedure to alulate ultmate onned trength rom two gven lateral preure. The numeral proedure ummarzed n the ollowng tep: 1. Determnng l and l ung Equaton 3.86 and Convertng the potve gn o l and l rom potve to negatve to repreent the major and ntermedate prnpal tree (Thee value are reerred to a 1 and o that 1 > ). 3. Etmatng the onned trength ( 3 ) a the mnor prnpal tre 4. Calulatng the otahedral tre ot, otahedral hear tre ot and lode angle a ollow: ot Equaton 3.89 ot Equaton o ot ot Equaton Determnng the ultmate trength merdan urae T,C (or =60 o and 0 o repetvel) ung the ollowng equaton derved b Elw and Murra (1979) rom data b Skert and Wnkler (1977): T Equaton 3.9 ot ot C Equaton 3.93 ot ot / Equaton 3.94 ot ot 143

159 . Determnng the otahedral hear tre ung the nterpolaton unton ound b Wllam and Warnke (1975): ot C D T C D T TC D T C 0.5 / o Equaton 3.95 D 4 C T o Equaton 3.96 Equaton 3.97 ot ot 3. Realulatng ung the ollowng equaton (ame a Equaton 3.89) but olvng or 3 : ot Equaton I the value rom Equaton 3.97 loe to the ntal value then there onvergene. Otherwe, the value rom Equaton 3.97 reued n tep 4 through 8. Equaton 3.91 and 3.9 that dene the tenon and ompreon merdan are ompared wth derent equaton or derent unonned ompreve trength. The reult are hown n eton Mander et al. (1988) propoed an energ balanng theor to predt the ultmate onned tran, whh determned at the rt hoop rature. The tated that the addtonal dutlt or onned onrete reult rom the addtonal tran energ tored n the hoop U h. Thereore rom equlbrum: U h U g U o Equaton

160 where U g the eternal work done n the onrete to rature the hoop, and U o the work done to aue alure to the unonned onrete. U h an be repreented b the area under the tenon tre tran urve or the tranvere teel between zero and rature tran. U h A 0 d Equaton whle Ug equal to the area under the onned tre tran urve plu the area under the longtudnal teel tre tran urve: U g u 0 A d u 0 A d l Equaton mlarl, t wa proven epermentall that U o equal to: U o A d pall 0 A n MPa Equaton 3.10 and U h A d 110 A Equaton Subttutng Equaton 3.100, 3.101, and 3.10 nto Equaton 3.98: u 110 d ld u 0 Equaton where l the tre n the longtudnal teel. Equaton an be olved numerall or u. 145

161 Connement Model or Eentr Column Unlke onentr loadng, the eentr loadng generate bendng moment n addton to aal loadng. Column ubjeted to eentr loadng behave derentl rom thoe onentrall loaded, a the hape o the tre tran urve or ull onned renored onrete (onentr loadng) how hgher peak trength and more dutlt than the unonned one (nnte eentrt). Mot o the prevou tude were baed on the unorm dtrbuton o ompreve tran aro the olumn eton. FIGURE 3.6 Eet o Compreon Zone Depth on Conrete Stre Fgure 3.6 llutrate three derent eton under onentr load, ombnaton o aal load and bendng moment and pure bendng moment, the hghlghted ber n the three ae ha the ame tran. An urrent onnement model eld the ame tre or thee three ber. So the depth or ze o ompreon zone doe not have an role n predtng the tre. Hene, t more realt to relate the trength and dutlt n a new model to the level o onnement utlzaton and ompreon zone ze. 146

162 FIGURE 3.7 Amount o Connement Engaged n Derent Cae B denton, onnement get engaged onl when member ubjeted to ompreon. Compreed member tend to epand n lateral dreton, and onned, onnement wll prevent th epanon to derent level baed on the degree o ompreve ore and onnement trength a well. For ull ompreed member (Fgure 3.7), onnement beome eetve 100% a t all at to prevent the lateral epanon. Wherea member ubjeted to ompreon and tenon, when the neutral a le nde the eton permeter, onl onnement adjaent to the ompreon zone get engaged. Aordngl, member beome partall onned. In lterature, varou model were mplemented to ae the ultmate onned apat o olumn under onentr aal load. On the other hand the eet o partal onnement n ae o eentr load (ombned aal load and bendng moment) not nvetgated n an propoed model. Thereore, t pertnent to relate the trength and dutlt o renored onrete to the degree o onnement utlzaton n a new model. The two urve o ull onned and unonned onrete n an propoed model are ued n the eentrt-baed model a upper and lower bound. The upper urve reer to onentrall loaded onned onrete (zero eentrt), whle the lower one reer to pure bendng appled on onrete (nnte eentrt). In between the two boundare, nnte number o tre-tran urve an be generated baed on the eentrt. The hgher the eentrt the maller the onned onrete regon n ompreon. Aordngl, the ultmate onned trength graduall redued rom the ull onned value to the unonned value a a unton o eentrt to dameter rato. In addton, the ultmate tran graduall 147

163 redued rom the ultmate tran u or ull onned onrete to the ultmate tran or unonned onrete 1.5 o. The relaton between the ompreon area to whole area rato and normalzed eentrt omplated n ae o retangular ro eton due to the etene o two bendng ae. The ore loaton wth repet to the two ae aue the ompreon zone to take a trapezodal hape ometme the ore appled not along one o the ae. Hene the relaton between the ompreon area and the load eentrt need more nvetgaton a oppoe to the ae o rular ro eton whh wa hown to be mpler. The normalzed eentrt plotted agant the ompreon area to ro etonal area rato or retangular ro eton havng derent apet rato (length to wdth) at the unonned alure level. The apet rato ued are 1:1, :1, 3:1, 4:1 a hown n Fgure 3.8, 3.9, 3.30, and Eah urve repreent pe angle (tan = M/M) rangng rom zero to nnet degree. It een rom thee gure that there nverel proportonal relaton between the normalzed eentrt and ompreon zone rato regardle o the angle ollowed. Compreon area to the ro etonal area rato Apet rato 1:1 angle eentrt/(bh)^0.5 FIGURE 3.8 Normalzed Eentrt veru Compreon Zone to Total Area Rato 148

164 (Apet Rato 1:1) Compreon area to the ro etonal area rato Apet rato :1 angle eentrt/(bh)^0.5 FIGURE 3.9 Normalzed Eentrt veru Compreon Zone to Total Area Rato (Apet Rato :1) Compreon area to the ro etonal area rato Apet rato 3:1 angle eentrt/(bh)^0.5 FIGURE 3.30 Normalzed Eentrt veru Compreon Zone to Total Area Rato (Apet Rato 3:1) 149

165 Compreon area to the ro etonal area rato Apet rato 4: eentrt/(bh)^0.5 FIGURE 3.31 Normalzed Eentrt veru Compreon Zone to Total Area Rato (Apet Rato 4:1) 10 Compreon area to the ro etonal area rato eentrt/(bh)^0.5 FIGURE 3.3 Cumulatve Chart or Normalzed Eentrt agant Compreon Zone Rato (All Data Pont) 150

166 In order to nd aurate mathematal epreon that relate the ompreon zone to load eentrt, the data rom Fgure 3.8 through 3.31 are replotted a atter pont n Fgure 3.3. The bet ttng urve o thee pont baed on the leat quare method ha the ollowng equaton: e 0.* 0. 1 bh C R e bh Equaton where C R reer to ompreon area to ro etonal area rato. Eentr Model baed on Mander Equaton The equaton that dene the peak trength aordng to the eentrt : 1 1 Equaton C 1 R C R wherea the equaton developed or rular ro eton 1 e 1 bh 1 1 bh e Equaton where e the eentrt, b and h the olumn dmenon and eentrt (e). The orrepondng tran gven b the peak trength at the Equaton o 151

167 and the mamum tran orrepondng to the requred eentrt wll be a lnear unton o tre orrepondng to mamum tran or onned onrete u and the mamum unonned onrete uo at uo = 0.003: u E E u e e, u r r r E e, u u u uo Equaton u E e, u * E e, u E e r E E E e Equaton In order to ver the aura o the model at the etreme ae, the eentrt rt et to be zero. The oeent o wll be zero n Equaton and Equaton 3.105, and wll redue to be: Equaton Equaton 3.11 u u Equaton On the other hand, the eentrt et to be nnt the other oeent wll be zero, and the trength, orrepondng tran and dutlt equaton wll be: Equaton o Equaton Equaton u 15

168 Compreve Stre Conned Conrete Unonned Conrete Partall Conned Conrete o p u u Compreve Stran FIGURE 3.33 Eentrt Baed Conned Mander Model An pont on the generated urve the tre-tran unton an be alulated ung the ollowng equaton: r r 1 r Equaton where: Equaton r E Equaton E E e E e Equaton 3.10 To how the dtnton between the Eentr model degned or retangular ro eton, Fgure 3.34 and that o rular ro eton, Fgure 3.35, Equaton to

169 and to are ued n plottng a et o Stre-Stran urve wth eentrt rangng rom 0 nhe to. The olumn ro etonal properte ued to plot thee urve 36 n *36 nhe, teel bar are 13 #11, pral bar # 5, pang 4 nhe, equal to 4 k, equal to 60 k and h equal to 60 k. Th ae ued n plottng the Eentr Stre-Stran urve that are developed or retangular ro etonal onrete olumn; Fgure 3.34 whle the ame ae ued n plottng the eentr Stre-Stran urve that are developed or rular ro eton, Fgure The eentr tre-stran urve n Fgure 3.35 are almot parallel and equdtant to eah other. Wherea, the leap rom one urve to the net one n Fgure 3.34 varng. Th due to the eet o the oeent C R, that ued n Equaton 3.105, whh ha non lnear mpat on the ompreon zone a oppoed to the lnear relaton between the eentrt and ompreon zone or rular ro eton (Fgure 3.35) 6 5 Aal Stre (k) Aal Stran e = 0 n e = 1 n e = n e = 3 n e = 4 n e = 5 n e = 6 n e = 7 n e = 8 n e = 9 n e = nnt FIGURE 3.34 Eentr Baed Stre-Stran Curve Ung Compreon Zone Area to Gro Area Rato 154

170 6 Aal Stre (k) e = 0 n e = 1 n e = n e = 3 n e = 4 n e = 5 n e = 6 n Aal Stran FIGURE 3.35 Eentr Baed Stre-Stran Curve Ung Normalzed Eentrt Intead o Compreon Zone Rato Generalzed Moment o Area Theorem The ver general aal tre equaton n an unmmetral eton ubjeted to aal ore P and baal bendng M and M (Hard Cro 1930): P M I M A I I I M I I I M z I I I Equaton 3.11 z = normal tre at an pont (a) n ro eton P = appled load. A = ro etonal area. M = bendng moment about -a M = bendng moment about -a = dtane between the pont (a) and -a 155

171 = dtane between the pont (a) and -a I = moment o nerta about -a I = moment o nerta about -a I = produt moment o nerta n plane Rewrtng Equaton to determne the tran at an pont n the ro eton: P M EI M EI EA EI EI EI M EI M EI EI EI EI z Equaton 3.1 In ae o lnear elat anal, E n EA or EI epreon ontant (E=E). However, the eton ha lnear tran and nonlnear tre prole, t wll amount to varable E prole E A (per lament) n nonlnear anal. Aordngl, the eton parameter mut nlude, E I or a more generalzed theor (Raheed and Dnno 1994). Note that the lnear tran prole o the eton rom Equaton eld two dtnt ontant urvature: M EI M EI Equaton 3.13 M EI M EI Equaton 3.14 = - urvature = - urvature EI EI EI To prove Equaton 3.10 and 3.11 above, nvoke the oupled equaton o moment about the entrod (Bkord 1998). M EI EI Equaton 3.15 M EI EI 156 Equaton 3.16

172 In a matr orm: M M EI EI EI EI Equaton 3.17 Invertng Equaton EI EI EI EI M M Equaton 3.18 whh reprodue Equaton 3.10 and Rewrtng Equaton n term o and z P EA Equaton 3.19 Fndng z at the entrod, ne = = 0. o = P/EA Equaton Solvng or P at the geometr entrod; P EA o EA EA Equaton o the aal tran at the geometr entrod But EAM EA EAM EA Y G Y X G X 157

173 Y G the vertal dtane to the geometr entrod meaured rom bottom, X G the dtane to the geometr entrod meaured rom the ro eton let de, Y the vertal dtane to the nelat entrod meaured rom the bottom and X the horzontal dtane to the nelat entrod meaured rom the ro eton let de Thu, P EA EAM EAM Equaton 3.13 o The general ormula o the moment about the geometr -a and the geometr -a derved a ollow: when the moment tranerred rom the entrod to the geometr entrod (Fgure 3.36a) M M P Equaton Subttutng Equaton 3.1 and 3.19 nto eld: M EI EI EA EAM EAM o Equaton M EAM o EI EAM EI EAM Equaton Smlarl, Fgure 3.36b: 158

174 a) b) G G M C M P C P M G G M FIGURE 3.36 Tranerrng Moment rom Centrod to the Geometr Centrod M M P Equaton M EI EI EA EAM EAM o Equaton M EAM o EI EAM EI EAM Equaton EI EAM The term EI EAM EI EI and repreent the and about the EI EAM geometr entrod repetvel ung the parallel a theorem. And the term EI EAM EAM EA and are equal gven that: EAM EA and. Ung Equaton 3.19, 3.13 and eld the etended general moment o area equaton: P EA M EAM M EAM EAM EI EI EAM EI EI Equaton Sne the moment o area about the entrod vanhe (Raheed and Dnno 1994), Equaton redue to a partall unoupled et when t appled bak at the entrod ne EAM and EAM vanh about the entrod. 159

175 P M M EA EI EI 0 EI EI o Equaton whh mpl Equaton 3.1, 3.13, and Numeral Formulaton Th approah mulate the radal loadng o the ore and moment b keepng the relatve proporton between them ontant durng the loadng. Aordngl, all the pont wll be eatl on the D nteraton dagram. In addton to the ultmate pont, the omplete load deormaton repone generated. The ro eton analzed loaded nrementall b mantanng a ertan eentrt between the aal ore P and the reultant moment M R. Sne M R generated a the reultant o M and M, the angle = tan -1 (M /M ) kept ontant or a ertan D nteraton dagram. Sne nreang the load and reultant moment aue the neutral a to var nonlnearl, the generalzed moment o area theorem deved. Th method baed on the general repone o retangular unmmetral eton ubjeted to baal bendng and aal ompreon. The ammetr tem rom the derent behavor o onrete n ompreon and tenon. The method developed ung the nremental teratve anal algorthm, eant tne approah and proportonal or radal loadng. It eplaned n the ollowng tep Fgure 3.40: 1. Calulatng the ntal eton properte: Elat aal rgdt EA: EA E w t ( E E ) A Equaton E E = ntal eant modulu o elatt o the onrete = ntal modulu o elatt o the teel rebar 160

176 161 The depth o the elat entrod poton rom the bottom ber o the eton Y and rom the let de o the eton X, Fgure 3.37 Equaton 3.14 EA X B A E E X B w t E X ) ( ) ( ) ( Equaton Elat leural rgdt about the elat entrod EI: Equaton ) ( ) ( ) ( X X H A E E X X B w t E EI Equaton X X B Y Y H A E E X X B Y Y H w t E EI ) ( ) ( ) ( Equaton Tpall the ntal elat Y = H/, X = B/ and EI = 0 EA Y H A E E Y H w t E Y ) ( ) ( ) ( ) ( ) ( ) ( Y Y H A E E Y Y H w t E EI

177 0.003 X B X B Y Y H C t w G H C G Y G Y Y G Y X X G X X G FIGURE 3.37 Geometr Properte o Conrete Flament and Steel Rebar wth Repet to Cruhng Stran Pont, Geometr Centrod and Inelat Centrod The depth o the geometr eton entrod poton rom the bottom and let ber o the eton Y G, X G, Fgure 3.37: H Y G Equaton X G B Equaton Denng eentrt e, whh pee the radal path o loadng on the nteraton dagram. Alo, denng the angle n between the reultant moment GM R and GM X 16

178 e Aal Fore Load Step GP Reultant Moment FIGURE 3.38 Radal Loadng Conept 3. Denng loadng tep GP a a mall porton o the mamum load, and omputng the aal ore at the geometr entrod. GP new GP old GP Equaton Calulatng moment GM about the geometr entrod. GM R e GM R e * GP GP Equaton GM X GM R o Equaton GM Y GM X tan Equaton Tranerrng moment to the urrent nelat entrod and alulatng the new tranerred moment TM X and TM Y : TM X GM X GP Y Y ( G ) Equaton

179 TM Y GM Y GP X X ( G ) Equaton The advantage o tranerrng the moment to the poton o the nelat entrod to elmnate the ouplng eet between the ore and moment, ne EAM EAM 0 about the nelat entrod P TM TM EA EI EI 0 EI EI o Equaton Y G Y X XG FIGURE 3.39 Moment Tranerrng rom Geometr Centrod to Inelat Centrod 6. Fndng: Curvature X and Y X TM X TM Y * EI Y * EI XY Equaton Y TM Y TM X * EI X * EI XY Equaton EI X EIY EI Equaton

180 Stran at the nelat entrod, the etreme ompreon ber tran, and tran at the etreme level o teel n tenon are ound a ollow: e o e GP o EA Equaton e o ( H Y ) ( B X ) Equaton ( Y Cover) ( X Cover) Equaton e o where over up to the enter o bar 7. Calulatng tran and orrepondng tre n eah lament o onrete eton b ung Eentr Baed Model (Mander Equaton) TM GP EA TM H Y Y TM B X X B X X TM H Y Y EI EI EI EI Equaton Calulatng tran and orrepondng tre n eah bar n the gven eton b ung the teel model hown n Fgure 3.b. TM GP EA TM H Y Y TM B X X B X X TM H Y Y EI EI EI EI Equaton Calulatng the new eton properte: aal rgdt EA, leural rgdte about the nelat entrod EI,, EI, EI moment o aal rgdt about nelat 165

181 entrod EAM, EAM, nternal aal ore F z, nternal bendng moment about the nelat entrod M o,m o: EA E w t ( E E ) A Equaton EAM Ewt ( H Y Y ) ( E E ) A ( H Y Y ) Equaton EAM E wt ( B X X ) ( E E ) A ( B X X ) Equaton F w t ( ) A z Equaton EI E w t ( H Y Y ) ( E E ) A ( H Y Y ) Equaton EI E w t ( B X X ) ( E E ) A ( H X X ) Equaton EI E w t ( H Y Y ) B X X ( E E ) A ( H Y Y ) B X X Equaton M o wt ( H Y Y ) ( ) A ( H Y Y ) Equaton M o wt ( B X X ) ( ) A ( B X X ) Equaton 3.17 where E = eant modulu o elatt o the onrete lament. E = eant modulu o elatt o the teel bar. 10. Tranerrng bak the nternal moment about the geometr entrod 166

182 GM o M o GP Y ( G Y ) Equaton GM o M o GP( Y X ) G Equaton Chekng the onvergene o the nelat entrod TOL EAM / EA/ Y Equaton TOL EAM / EA/ X Equaton Comparng the nternal ore to appled ore, nternal moment to appled moment, and makng ure the moment are alulated about the geometr entrod : GP F z 1*10 5 Equaton GM 1*10 GM o 5 1*10 GM o GM Equaton TOL 1*10 5 TOL 1*10 5 Equaton I Equaton 3.174, and are not ated, the loaton o the nelat entrod updated b EAM /EA and EAM /EA and tep 5 to 11 are repeated tll Equaton 3.174, and are ated. Y new Y old EAM EA Equaton X new X old EAM EA Equaton

183 teel e One equlbrum reahed, the algorthm hek or ultmate tran n onrete not to eeed and 0.05 repetvel. Then t nreae the loadng b the anal or the new load level ung the latet eton properte. Otherwe, or e and and run equal equal 0.05, the target ore and reultant moment are reahed a a pont on the alure urae or the amount o eentrt and angle ued. Th method an be ued ombned wth Approah One n the unonned anal, eton 3...1: Predened Ultmate Stran Prole, or proeng tme optmzaton. Intall unonned anal utlzed. The etonal properte, EA, EI, EI, EI, EAM, EAM,Y,X F z, M o and M o are alulated rom the unonned alure pont and ued a eton properte or the ollowng tep. So ntead o loadng the eton rom the begnnng, The equlbrum ought at unonned alure pont, Then, knowng the nternal ore apat o the eton, P added and the ro eton analzed ung the propoed numeral ormulaton o th eton untl alure o the onned eton. GP e e 168

184 Start C alulate ntal eton properte EA,EI, EI,EI Y,Y G,X,X G Input P, e & Traner moment to nelat entrod TM =GM +P(Y G -Y ) TM =GM +P(X G -X ) Calulate ø,ø, o, e, e Calulate,,, Calulate new eton properte EA,EI, EI, EAM, EAM,EI C alulate nternal ore and m om ent F z,m o,m o Traner moment bak to G.C. GM o=m o -P(Y G -Y ) GM o=m o -P(X G -X ) No onvergene aheved. top no P += P no EAM EA*Y < EAM EA*X < e G P-F z < 10-5 G M - GM o <10-5 G M - GM o < 10-5 e e >= u e >= 0.05 no Y += EAM/EA X += EAM/EA e e New entrt P & M aheved e no New a no End FIGURE 3.40 Flowhart o Generalzed Moment o Area Method 169

185 3.3.3 Reult and Duon Interaton dagram generated b KDOT Column Epert Sotware are plotted and ompared to the orrepondng epermental work ound n the lterature. Interaton dagram are generated ung the numeral ormulaton derbed n eton Comparon wth Epermental Work Cae 1 A Stud o ombned bendng and aal load n renored onrete member (Evnd Hogentad) Seton Heght = 10 nhe Seton Wdth = 10 nhe Clear Cover = nhe Steel Bar n dreton = Steel Bar n dreton = 4 Steel Dameter = nhe Te Dameter = 0.5 nhe = 5.1 k = 60 k. h = 61.6 k. Spang = 8 nhe FIGURE 3.41 Hognetad Column 170

186 FIGURE 3.4 Comparon between KDOT Column Epert wth Hognetad Eperment ( = 0) Comparon wth Epermental Work Cae Degn rtera or renored olumn under aal load and baal bendng (Bor Breler) Seton Heght = 8 nhe SetonWdth = 6 nhe Clear Cover = nhe Steel Bar n dreton = #5 Steel Bar n dreton = #5 Te Dameter = 0.5 nhe 171

187 FIGURE 3.43 Breler Column = 3.7 k = 53.5 k. h = 53.5 k. Spang = 4 n FIGURE 3.44 Comparon between KDOT Column Epert wth Breler Eperment ( = 90) 17

188 FIGURE 3.45 Comparon between KDOT Column Epert wth Breler Eperment ( = 0) Comparon wth Epermental Work Cae 3 Invetgaton o the ultmate trength o quare and retangular olumn under baall eentr load (L.N. Ramamurth) Seton Heght = 1 nhe Seton Wdth = 6 nhe Clear Cover = nhe Steel Bar n dreton = 3#5 Steel Bar n dreton =3#5 Te Dameter = 0.5 nhe = 3.8 k = k h = k. Spang = 6 nhe 173

189 FIGURE 3.46 Ramamurth Column FIGURE 3.47 Comparon between KDOT Column Epert wth Ramamurth Eperment ( = 6.5) Comparon wth Epermental Work Cae 4 Conned olumn under eentr loadng (Murat Saatoglu. Amr Salamat and Salm Razv ) 174

190 Seton Heght = 8.7 nhe Seton Wdth =8.7 nhe Clear Cover = 0. 5 nhe Steel Bar n dreton = 3 Steel Bar n dreton = 3 Steel Area = nhe Te Dameter = nhe = 5.1 k = 75 k. h = k. Spang = 1.97 nhe FIGURE 3.48 Saatoglu Column Fore (k.) Epermental pont Interaton dagram " e relaton" Interaton dagram "CR relaton" Moment (k.n.) FIGURE 3.49 Comparon between KDOT Column Epert wth Saatoglu et al. Eperment ( = 0) 175

191 Comparon wth Epermental Work Cae 5 Conned olumn under eentr loadng (Mural Saatoglu. Amr Salamat and Salm Razv ) Seton Heght = 8.7 nhe Seton Wdth =8.7 nhe Clear Cover = 0. 5 nhe Steel Bar n dreton = 4 Steel Bar n dreton = 4 Steel Area = n. Te Dameter = nhe = 5.1 k = 75 k. h = k. Spang = 1.97 nhe FIGURE 3.50 Saatoglu Column 176

192 Fore (k.) Epermental pont Interaton Dagram " e relaton" Interaton Dagram " CR relaton" Moment (k.n.) FIGURE 3.51 Comparon between KDOT Column Epert wth Saatoglu et al. Eperment 1 ( = 0) Comparon wth Epermental Work Cae 6 Stre tran behavor o onrete onned b overlappng hoop at low and hgh tran rate: (B. Sott, R Park and M. Pretl) Seton Heght = 17.7 nhe Seton Wdth =17.7 nhe Clear Cover = nhe Steel Bar n dreton = 4 Steel Bar n dreton = 4 Steel Area = 0.49 nhe Te Dameter = nhe = 3.67 k = 63 k. h = 44.8 k. Spang =.83 nhe 177

193 FIGURE 3.5 Sott Column Fore (k.) Moment (k.n.) Epermemtal pont Interaton dagram "e relaton" Interaton dagram "CR relaton" Interaton dagram "CR relaton" wth over pallng FIGURE 3.53 Comparon between KDOT Column Epert wth Sott et al. Eperment ( = 0) 178

194 Comparon wth Epermental Work Cae Cae 7 Stre tran behavor o onrete onned b overlappng hoop at low and hgh tran rate (B. Sott, R Park and M. Pretl) Seton Heght = 17.7 nhe Seton Wdth =17.7 nhe Clear Cover = nhe Steel Bar n dreton = 3 Steel Bar n dreton = 3 Steel Area = 0.7 nhe. Spral Dameter = nhe = 3.67 k = k. h = 44.8 k. Spang =.83 nhe FIGURE 3.54 Sott Column 179

195 Fore (k.) Moment (k.n.) Epermental pont Interaton dagram "e relaton" Interaton dagram "CR relaton" Interaton dagram "CR relaton" wth over pallng FIGURE 3.55 Comparon between KDOT Column Epert wth Sott et al. Eperment ( = 0) The analzed even ae over the three Interaton dagram zone o; ompreon ontrolled, tenon ontrolled and balaned zone. There good agreement between the theoretal nteraton dagram and the orrepondng epermental data a hown n Fgure 3.4, 3.44, 3.45, 3.47, 3.49, 3.51, 3.53, and It hown rom Fgure 3.49, 3.51, 3.53, and 3.55 that nteraton dagram plotted ung Equaton that repreentatve o the ompreon zone area are more aurate ompared to thoe plotted ung Equaton 3.105a that a unton o eentrt. Alo the epermental data orrelate well to t aoated nteraton dagram. Fgure 3.53 and 3.55 how more aura and onervatve nteraton dagram when the anal aount or the over pallng when the unonned ruhng tran ondered. Th repreented b the mot nner urve n Fgure 3.53 and Alo n Fgure 3.53 and 3.55 the epermental pont 1 and are havng the ame eentrt but the loadng tran rate derent. The loadng tran rate or pont , wherea t or pont. Pont 3 and 4 alo have the ame loadng tran rate. It een that the loadng tran rate or pont 1 180

196 and 3 are etremel mall. Hene pont and 4 are more realt and the are aptured well b the theoretal nteraton dagram. In onluon, the tran rate a parameter that need urther nvetgaton Comparon between the Surae Merdan T&C Ued n Mander Model and Epermental Work The ultmate trength urae merdan equaton or ompreon C and tenon T derved b Elw and Murra (1979) rom the data o Skert and Wnkler (1977), that are utlzed b Mander et al. (1988) to predt the ultmate onned aal trength ung the two lateral onned preure, are ompared heren to ome epermental data ound rom Mll and Zmmerman (1970). The equaton ued b Mander are developed orgnall or onrete that ha unonned trength o 4.4 k. The have the ollowng ormula T ot ot Equaton 3.18 C ot ot Equaton ot 0.4 = = T ( theta = 0) C (theta = 60) Pol. (T ( theta = 0)) Pol. (C (theta = 60)) ot FIGURE 3.56 T and C Merdan Ung Equaton 3.18 and ued n Mander Model or = 4.4 k 181

197 The T and C merdan adopted b Mander rom Elw and Murra (1979) work are reported on n Fgure Mll and Zmmerman (1970) developed three et o multaal tet or onrete wth unonned trength o 3.34, 3.9 and 5. k. For eah et, the value o and are etrated at unonned trength, the rakng tenle trength t, equbaal ompreve trength b and two etra pont; one on eah o the merdan. Thee ve pont are ued to plot the T and C merdan a hown n Fgure 3.57, 3.58 and ot 0.4 = = T ( theta = 0) C ( theta = 0) Pol. (T ( theta = 0)) Pol. (C ( theta = 0)) ot FIGURE 3.57 T and C Merdan or = 3.34 k 18

198 1. ot 0.4 = = T ( theta = 0) C ( theta = 60) Pol. (T ( theta = 0)) Pol. (C ( theta = 60)) ot FIGURE 3.58 T and C Merdan or = 3.9 k ot = = T ( theta = 0) C ( theta = 60) Pol. (T ( theta = 0)) Pol. (C ( theta = 60)) ot FIGURE 3.59 T and C Merdan or = 5. k The T and C equaton or Fgure 3.57 through 3.59 are a ollow: or = 3.34 k: 183

199 T ot ot Equaton C ot ot Equaton or = 3.9 k: T ot ot Equaton C ot 0.11 ot Equaton or = 5. k: T ot ot Equaton C ot ot Equaton Equaton through are ued n generatng onned trength value or derent lateral preure a hown n Append A. Equaton and are ued alo n developng onned trength value or the ame lateral preure value. It een rom the table that Equaton and gve onervatve value ompared to Equaton through Aordngl, Equaton and are ued heren to predt the ultmate onned aal trength value or an gven unonned trength ( ) value. TABLE 3.1 Data or Contrutng T and C Merdan Curve or Equal to 3.34 k Control Parameter Ot Ot = 3.34 k t b traal on C traal on T

200 TABLE 3. Data or Contrutng T and C Merdan Curve or Equal to 3.9 k Control Parameter Ot Ot = 3.9 k t b traal on C traal on T TABLE 3.3 Data or Contrutng T and C Merdan Curve or Equal to 5. k Control Parameter Ot Ot = 5. k t b traal on C traal on T

201 Chapter 4: Conluon and Reommendaton 4.1 Conluon Th tud aomplhed everal objetve at the anal, materal modelng, degn mplaton and otware development level. It ma be onluded that: 1. Baed on the etenve revew o the onned model avalable n the lterature, Mander Model ound to be the mot utable onentr loadng model epreng the tre-tran behavor o rular and retangular olumn onned wth onvenent lateral teel and teel tube a well.. The eentr baed tre-tran model developed n th tud provde more aura ompared to the avalable onentr onned model n the lterature a t hown through omparon wth epermental data. 3. For retangular olumn, the rato o the area o ompreon zone to the etonal gro area more repreentatve than the normalzed alone eentrt n orrelatng eentr behavor. 4. The non-lnear numeral proedure ntrodued, ung the eentr model and the nte laer approah, ueull predted the ultmate apat o retangular renored onrete olumn onned wth teel. 5. A omputer program named KDOT Column Epert developed baed on the non-lnear approah mplemented or analzng and degnng retangular olumn onned wth lateral teel hoop. 6. The unonned onrete anal arred out b KDOT Column Epert benhmarked ueull agant well-etablhed ommeral otware or a range o degn parameter. 7. The onned onrete anal mplemented b KDOT Column Epert well orrelated to epermental data. 4. Reommendaton Th work hould be etended to addre the ollowng area: 1. Model the eet o FRP wrappng on onnement or retangular olumn. 186

202 . Model orroon o longtudnal and tranvere teel or rular and retangular olumn. 3. Model CFST or rular olumn. 4. Model CFST or retangular olumn. 5. Epand the otware applaton to nlude the CFST olumn. 187

203 Reerene Aa-Jakoben, A Baal Eentrte n Ultmate Load Degn. ACI Journal 61 (3): ACI-ASCE Commttee Ultmate Strength Degn. ACI Journal, Proeedng 5: Ahmad, S. H., and Shah, S. P Complete Traal Stre-Stran Curve or Conrete. ASCE Journal 108 (ST4): Ahmad, S. H., and Shah, S. P Stre-Stran Curve o Conrete Conned b Spral Renorement. ACI Journal 79 (6): Anderen, Paul Degn Dagram or Square Conrete Column Eentrall Loaded n Two Dreton. ACI Journal, Proeedng 38: 149. Ang, Cho-Lm Charle Square Column wth Double Eentrte Solved b Numeral Method. ACI Journal, Proeedng 57: 977. Aa B., Nhama, M., and Watanabe, F New Approah or Modelng Conned Conrete. I: Crular Column. Journal o Strutural Engneerng, ASCE 17 (7): Attard, M. M., and Setunge, S Stre-Stran Relatonhp o Conned and Unonned Conrete. ACI Materal Journal 93 (5): Au, Tung Ultmate Strength Degn Chart or Column Controlled b Tenon. ACI Journal, Proeedng 54: 471. Au, Tung Ultmate Strength Degn o Retangular Conrete Member Subjeted to Unmmetral Bendng. ACI Journal, Proeedng 54: 657. Bajaj, S., and Mend, P New Method to Evaluate the Baal Interaton Eponent or RC Column. Journal o Strutural Engneerng, ASCE 131 (1): Bakhoum, Mhel Anal o Normal Stree n Renored Conrete Seton under Smmetral Bendng. ACI Journal, Proeedng 44: 457. Bazant, Z. P., and Bhat, P. D Endohron Theor o Inelatt and Falure o Conrete. Journal o the Engneerng Mehan Dvon, ASCE 10 (4): Bazant, Z. P., and Bhat, P. D Predton o Htere o Renored Conrete Member. Journal o the Strutural Dvon, ASCE 103 (1): Bn, B An Analtal Model or Stre Stran Behavor o Conned Conrete. Engneerng Struture 7 (7):

204 Blume, J. A, Newmark, N. M., and Cornng L. H Degn o Mult Store Renored Conrete Buldng or Earthquake Moton. Chago. Portland Cement Aoaton. Bonet, J. L., Barro, F. M., Romero, M. L Comparatve Stud o Analtal and Numeral Algorthm or Degnng Renored Conrete Seton under Baal Bendng. Computer and Struture 8:31. Bonet, J. L., Mguel, P. F., Fernandez, M. A., and Romero, M. L Analtal Approah to Falure Surae n Renored Conrete Seton Subjeted to Aal Load and Baal Bendng. Journal o Strutural Engneerng 130 (1): Buldng Code Requrement or Renored Conrete ACI Ameran Conrete Inttute (ACI). Detrot, Mhgan. Braga, F., Gglott, R., Laterza, M Analtal Stre Stran Relatonhp or Conrete Conned b Steel Strrup and/or FRP Jaket. Journal o Strutural Engneerng ASCE 13 (9): Breen, J. E The Retraned Long Conrete Column a a Part o A Retangular Frame. Ph. D. The Preented to the Unvert o Tea, Autn, TX. Breler, B Degn, Crtera or Renored Column under Aal Load B-Aal Bendng. ACI Journal Nov.: Breler, B., and Glberg, P. H Te Requrement or Renored Conrete Column. ACI Journal 58 (5): Brettle, H. J.; Talor, J. M Comparon o Epermental Reult wth Ultmate Strength Theor or Renored Conrete Column n Baal Bendng. Tranaton o the Inttuton o Engneer, Autrala, Cvl Engneerng Aprl 11 (1): Candappa, D The Conttutve Behavor o Hgh Strength Conrete under Lateral Connement. PhD The, Monah Unvert, Claton, VIC, Autrala. Candappa, D. P., Sanjaan, J. G., and Setunge, S Complete Traal Stre-Stran Curve o Hgh-Strength Conrete. Journal o Materal n Cvl Engneerng 13 (3): Candappa, D. P., Setunge, S., and Sanjaan, J. G Stre Veru Stran Relatonhp o Hgh Strength Conrete under Hgh Lateral Connement. Cement and Conrete Reearh 9 (1): Carrera, D J., and Chu, K. H Stre-Stran Relatonhp or Plan Conrete n Compreon. ACI Journal 83 (6): Cedoln, L.; Cuat, G.; Ehel, S.; Roveda, M Baal Bendng o Conrete Column: an Analtal Soluton. Stude and Reearhe 6. Cedoln L, Cuat G, Ehell S, Roveda M Capat o Retangular Croeton under Baall Eentr Load. ACI Strutural Journal 105 ():

205 Cervn, D. R Degn o Retangular Ted Column Subjeted to Bendng wth Steel n All Fae. ACI Journal 19: Chan, W. W. L. 00. The Ultmate Strength and Deormaton o Plat Hnge n Renorement Conrete Framework. Magazne o Conrete Reearh (London) 18 (1): Chu, Kuang-Han and Pabaru, A Baall Loaded Renored Conrete Column. ASCE Proeedng 84 (ST 8): Paper Cro, H The Column Analog: Anal o Elat Arhe and Frame. Urbana, IL: Unvert o Illno. Cuon, D. and Paultre, P Stre-Stran Model or Conned Hgh-Strength Conrete. ASCE Journal o Strutural Engneerng 11 (3): Czernak. E Analtal Approah to Baal Eentrt. Proeedng o the ASCE Journal the Strutural Dvon 88: Davter, M. D A Computer Program or Eat Anal. Conrete Internatonal: Degn and Contruton 8 (7): Demagh, K. Chabl, H.; Hamzaou, L Anal o Renored Conrete Column Subjeted to Baal Load. Proeedng o the Internatonal Conerene on Conrete or Tranportaton Inratruture Ehan, Mohammad R CAD or Column. Conrete Internatonal: Degn and Contruton 8 (9): El-Dah, K. M., and Ahmad, S. H A Model or Stre- Stran Relatonhp o Sprall Conned Normal and Hgh-Strength Conrete Column. Magazne o Conrete Reearh 47 (171): Elw, A., and Murra, D. W A 3D Hpoelat Conrete Conttutve Relatonhp. Journal o Engneerng Mehan 105: Emal, A., and Xao, Y Behavor o Renored Conrete Column under Varable Aal Load. ACI Strutural Journal 101 (1): Everand, Noel J., and Cohen, Edward Ultmate Strength Degn o Renored Conrete Column. ACI Speal Publaton SP-7: 18. Everand, Noel J Aal Load Moment Interaton or Cro Seton Havng Longtudnal Renorement Arranged n a Crle. ACI Journal 94 (6). Fat, A., and Shah, S.P Lateral Renorement or Hgh-Strength Conrete Column. ACI Speal Publaton SP 87 (1):

206 Flemng, J. F., and Werner, S. D Degn o Column Subjeted to Baal Bendng. ACI Journal, Proeedng 6 (3): Fuj, M., Kobaah, K., Magawa, T., Inoue, S., and Matumoto, T A Stud on the Applaton o a Stre-Stran Relaton o Conned Conrete. Proeedng, JCA Cement and Conrete 4, Japan Cement. Furlong. R. W Ultmate Strength o Square Column under Baall Eentr Load. Journal o the Ameran Conrete Inttute 57 (53): Furlong, R. W., Hu, C. T. T, Mrza, S. A Anal and Degn o Conrete Column or Baal Bendng-Overvew. ACI Strutural Journal 101 (3): Gakoumel, G., Lam, D Aal Capat o Crular Conrete-Flled Tube Column. Journal o Contrutonal Steel Reearh 60 (7): Gouwen, Albert J Baal Bendng Smpled, Renored Conrete Column, SP-50, Ameran Conrete Inttute, Detrot, Gunnn, B. L., Rad, F. N., and Furlong, R. W A General Nonlnear Anal o Conrete Struture and Comparon wth Frame Tet. Computer and Struture 105 (): Helgaon, V Development o a Computer Program to Degn Conrete Column or Baal Moment and Normal Fore. M.S. The. Lund Inttute o Tehnolog, Lund Unvert, Lund, Sweden. Hartle, G.A Radal Contour Method o Baal Short Column Degn. ACI Journal 8 (6): Hognetad, E A Stud o Combned Bendng and Aal Load n Renored Conrete Member. Unvert o Illno, Urbana. Hong, H. P Short RC Column Capat under Baal Bendng and Aal Load. Canadan Journal o Cvl Engneerng 7 (6): Horowtz, B Degn o Column Subjeted to Baal Bendng. ACI Strutural Journal 86 (6): Hohkuma, J., Kawahma, K., Nagaa, K., and Talor, A. W Stre-Stran Model or Conned Renored Conrete n Brdge Per. Journal o Strutural Engneerng 13 (5): Hu, C. T. T Anal and Degn o Square and Retangular Column b Equaton o Falure Surae. ACI Strutural Journal 85 (): Hu, L. S., and Hu, C. T. T Complete Stre-Stran Behavor o Hgh-Strength Conrete under Compreon. Magazne o Conrete Reearh 46 (169):

207 Hu, Lu-Shen Eentr Bendng n Tow Dreton o Retangular Conrete Column. ACI Journal 51: 91. Karabn, A. I., and Kou, P. D Eet o Connement on Conrete Column: Platt Approah. ASCE Journal o Strutural Engneerng 1099: Kent, D., and Park, R Fleural Member wth Conned Conrete. Journal o Strutural Dvon, Proeedng o the Ameran Soet o Cvl Engneer 97 (ST7): Kroenke, W. C., Gutzwller, M. J., and Lee, R. H Fnte Element or Renored Conrete Frame Stud. Journal o Strutural Dvon, ASCE 99 (ST7): Lazaro, A. L., and Rhard, Jr., R Full Range Anal o Conrete Frame. Journal o Strutural Dvon, ASCE 100 (ST1): Lejano, B., A Invetgaton o Baal Bendng o Renored Conrete Column through Fber Method Modelng. Journal o Reearh o Sene and Computer Engneerng 4 (3): L, G Epermental Stud o FRP Conned Conrete Clnder. Journal Engneerng Struture 8: Lokuge, W. P., Sanjaan, J. G., and Setunge S Stre Stran Model or Laterall Conned Conrete. Journal o Materal n Cvl Engneerng, ASCE 17 (6): Luo, K Analtal Perormane o Renored Conrete Seton ung Varou Connement Model. Mater The, Kana State Unvert. Mander, J. B., Pretle, M. J. N., and Park, R Theoretal Stre-Stran Model or Conned Conrete. Journal o Strutural Engneerng, ASCE 114 (8): Mander, J. B., Pretle, M. J. N., and Park, R Oberved Stre Stran Behavor o Conned Conrete. Journal o Strutural Engneerng, ASCE 114 (8): Mander, J. B Sem Degn o Brdge Per. Ph.D. the. Unvert o Canterbur, New Zealand. Manur, M. A., Chn, M. S., and Wee, T. H Stre-Stran Relatonhp o Conned Hgh Strength Plan and Fber Renorement. Journal o Materal n Cvl Engneerng 9 (4): Mattok, Allan H., and Krtz, Ladlav, B Retangular Conrete Stre Dtrbuton n Ultmate Strength Degn. ACI Journal Proeedng 57: 875. Duon b P. W. Abele, Homer M. Hadle, K. Hanjnal-Kon, J. L. Meek, Lu Saenz, Ignao Martn, Raael Tamargo, ACI Journal, September, 1961, p Martnez, S., Nlon, A. H., and Slate, F. O Sprall Renored Hgh-Strength Conrete Column. ACI Journal 81 (5):

208 Mazzott, C., and M. Savoa. 00. Nonlnear Creep, Poon Rato and Creep-Damage Interaton o Conrete n Compreon. ACI Materal Journal 99 (5): Medland, I. C., and Talor, D. A Fleural Rgdt o Conrete Column Seton. Journal o Strutural Dvon, ASCE 97 (ST): Meek, J. M Ultmate Strength o Column wth Baall Eentr Load. ACI Journal 60 (Augut): Mend, P., Pendala, R. and Setunge, S Stre Stran Model to Predt the Full Range Moment Curvature Behavor o Hgh Strength Conrete Seton. Magazne o Conrete Reearh 5 (4): Mkhalkn, B The Strength o Renored Conrete Member Subjeted to Compreon and Unmmetral Bendng. M.S. The, Unvert o Calorna. Mll, L. L., and Zmmerman, R. M Compreve Strength o Plan Conrete under Multaal Loadng Condton. ACI Materal Journal 67 (101): Moran, D. A., Pantelde, C. P. 00. Stre Stran Model or Fber-Renored 665 Polmer- Conned Conrete. Journal o Computng n Contruton, ASCE 6 (4): Mlona, G. A Workng Stre Column Degn Ung Interaton Dagram. Volume 64, Augut 1. Najam, A., and Taem, A Degn o Round Renored Conrete Column. Journal o Strutural Engneerng, ASCE 1 (9): Pantazopoulou, S. J., Mll, R., H Mrotrutural Apet o the Mehanal Repone o Plan Conrete. ACI Materal Journal 9 (6): Pannel, F. N Falure Surae or Member n Compreon and Baal Bendng. ACI Journal, Proeedng 60 (1): Park, R., Pretle, M. J. N., and Gll, W. D Dutlt o Square Conned Conrete Column. Proeedng o ASCE 108 (ST4): Parker, L. G., and Santon, J. J A Smple Anal or Eentrall Loaded Conrete Seton. Cvl Engneerng 10 (10): 656. Parme, A. L., Neve, J. M., and Gouwen, A Capat o Renored Retangular Column Subjet to Baal Bendng. Journal o the Ameran Conrete Inttute 63 (9): Popov, S A Numeral Approah to the Complete Stre-Stran Curve o Conrete. Cement and Conrete Reearh 3 (5):

209 Ramamurth, L. N Invetgaton o the Ultmate Strength o Square and Retangular Column under Baall Eentr Load. Smpoum on Renored Conrete Column, Detrot, ACI-SP-13, Paper No. 1, Raheed, H. A., and Dnno, K. S An Eent Nonlnear Anal o RC Seton. Computer and Struture 53 (3): Razv, S., and Saatoglu, M Connement Model or Hgh-Strength Conrete. Journal o Strutural Engneerng 15 (3): Rhart. F. E., Brandtzaeg, A., and Brown R. L The Falure o Plan and Sprall Renored Conrete n Compreon. Bulletn No. 190, Engneerng Staton, Unvert o Illno, Urbana. Rodrguez, J. A., and Ohoa, J. D. A Baal Interaton Dagram or Short RC Column o An Cro Seton. Journal Strutural Engneerng ASCE 15 (6): Row, D. G., and Paula, T. E Baal Fleure and Aal Load Interaton n Short Retangular Column. Bulletn o the New Zealand Soet or Earthquake Engneerng 6 (3): Ro, H. E. H., and Sozen, M. A Dutlt o Conrete. Fleural Mehan o Renored Conrete SP-1, Ameran Conrete Inttute/Ameran Soet o Cvl Engneer, Detrot, Saatoglu, M., and Razv, S. R Strength and Dutlt o Conned Conrete. Journal o Strutural Engneerng 118 (6): Saatoglu, M., Salamt, A. H., and Razv, S. R Conned Column under Eentr Loadng. Journal o Strutural Engneerng 11 (11): Sak, R., Buukozturk, O Epert Interatve Degn o R\C Column under Baal Bendng. ASCE Journal o Computer n Cvl Engneerng 1 (). Saenz, L. P Equaton or the Stre-Stran Curve o Conrete. ACI Journal 61(9): Sallah, A. Y Anal o Short Retangular Renored Conrete Column Subjeted to Baal Moment. Mater o Engneerng the, Department o Cvl Engneerng, Carleton Unvert, Ottawa, 14 pp. Sargn, M Stre-Stran Relatonhp or Conrete and the Anal o Strutural Conrete Seton. Sold Mehan Dvon, Unvert o Waterloo, Stud No. 4. Sott, B. D., Park, R., and Pretle, N Stre-Stran Behavor o Conrete Conned b Overlappng Hoop at Law and Hgh Stran Rate. ACI Journal 79 (1): Shekh, S. A A Comparatve Stud o Connement Model ACI Journal 79 (4):

210 Shekh, S. A., and Uzumer, S. M Analtal Model or Conrete Connement n Ted Column. Journal o Strutural Engneerng, ASCE 108 (ST1): Shekh, S. A., and Tokluu, M. T Renored Conrete Column Conned b Crular Spral and Hoop. ACI Journal 90 (5): Solman, M. T. M., and Yu, C. W The Fleural Stre-Stran Relatonhp o Conrete Conned b Retangular Tranvere Renorement. Magazne o Conrete Reearh 19 (61): 3 8. Talor, M. A., and Ho, S. W Degn Contour Chart or Baal Bendng o Retangular Renored Conrete Column Ung Breler Method. Strutural Engneerng Prate (4): Troel, G. E Renored Conrete Column Subjeted to Bendng about Both Prnpal Ae. Cvl Engneerng 11 (4): 37. Vallena, J., Bertero, V. and Popov, E Conrete Conned b Retangular Hoop and Subjeted to Aal Load. Reearh Center Report UCB/EERC-77-13, Unvert o Calorna at Berkele. Wang, W., and Hong, H. 00. Relablt o Renored Conrete Column under Aal Load and Baal Bendng. Proeedng, Annual Conerene - Canadan Soet or Cvl Engneerng Wang, G. G., and Hu, C. T Complete Baal Load-Deormaton Behavor o RC Column. Journal o Strutural Engneerng, ASCE 118 (9): Wang, P. T., Shah, S. P., and Naaman, A. E Stre-Stran Curve o Normal and Lghtweght Conrete n Compreon. ACI Journal 75 (11): Weber, D.C Ultmate Strength Degn Chart or Column wth Baal Bendng. ACI Journal 63(11), , (1966). Wee, T. H., Chn, M. S., and Manur, M. A Stre-Stran Relatonhp o Hgh Strength Conrete n Compreon. Journal o Materal n Cvl Engneerng 8 (): Weman, Harold E Renored Conrete Column under Combned Compreon and Bendnd. ACI Journal, Proeedng 43: 1. Whtne, C. S. and Cohen, E Gude or the Ultmate Strength Degn o Renored Conrete. ACI Journal, Proeedng 53 (November), Wenger, Frederk P. T. Degn o Smmetral Column wth Small Eentrte n One or Tow Dreton. ACI Journal, Proeedng 55 (Augut): 73. Yong, Y.-K., Nour, M. G., and Naw, E. G Behavor o Laterall Conned Hgh- Strength Conrete under Aal Load. Journal o Strutural Engneerng 114 ():

211 Yoo, S. H., Shn, S. W Varaton o Ultmate Conrete Stran at RC Column Subjeted to Aal Load wth B-Dretonal Eentrte. Ke Engneerng Materal : Zak, M Computer Anal o Renored Conrete Seton under Baal Bendng and Longtudnal Load. ACI Strutural Journal 90 ():

212 Append A: Ultmate Conned Strength Table Table A.1 developed or o 3.3 ung Equaton and Table A. or o 3.9 ung Equaton and Table A.3 developed ung Mander proedure that utlze Skert and Wnkler (1977) ormula. Table A.4 or o 5. ung Equaton and Table A.5 through A.7 how the onned value or the ame lateral preure ung Skert and Wnkler (1977) equaton. Table A.5 through A.7 gve onervatve value ompared to table A.1, A. and A.4. Th ndate that Equaton and ound b Skert and Wnkler (1977) and utlzed b Mander et al. (1988) are onervatve enough to be ued n the anal 197

213 TABLE A.1 Ultmate Conned Strength to Unonned Strength Rato or = 3.3 k 1* *

214 1* * TABLE A. Ultmate Conned Strength to Unonned Strength Rato or = 3.9 k

215 1* * TABLE A.3 Ultmate Conned Strength to Unonned Strength Rato or = 4.4 k (ued b Mander et al. (1988))

216 TABLE A.4 Ultmate Conned Strength to Unonned Strength Rato or = 5. k 1* *

217 TABLE A.5 Ultmate Conned Strength to Unonned Strength Rato or = 3.3 k (ung Skert and Wnkler (1977)) 1* * 0

218 TABLE A.6 Ultmate Conned Strength to Unonned Strength Rato or = 3.9 k (Ung Skert and Wnkler (1977)) 1* * 03

219 Table A.7 Ultmate Conned Strength to Unonned Strength Rato or = 5. k (Ung Skert and Wnkler (1977)) 1* *

220