Non-linear Analysis of Reinforced Concrete Deep Beams using Finite Element Method

Transcription

1 on-lnear Analss of Renforced Concrete Deep Beams sng Fnte Element Metod A tess sbmtted for partal flfllment of te reqrements of te degree of M.Sc. n strctres b LOUAY MAHMOUD ABDEL WAHAB Department Of Cvl Engneerng Faclt of Engneerng & Arctectre Unverst of Kartom, Sdan Spervsor: PROFFESOR/ ABU BAKAR ABDEL WAHAB MOHAMMED Marc 4

2 Abstract Te prpose of ts tess s to analze renforced concrete deep beams n lnear, non-lnear and ltmate ranges. Te fnte element metod s tlzed to std te beavor of deep beams nder monotoncall ncreasng loads. All te major factors casng materal non-lneart are consdered. Prmar consderaton s gven to te representaton of sear transfer mecansms, de to aggregate nterlock n cracked concrete and te dowel acton n renforcement. Epressons are derved from an analtcal model n conjncton wt epermental data to provde sear stress and stffness vales for specal elements sed to model te aggregate nterlock mecansm. Tese are epressed as fncton of crack wdt, concrete strengt and sear dsplacement. A comparable approac s sed to derve epresson for te dowel acton mecansm. Ts s epressed n terms of te renforcement dameter, eld stress, dowel dsplacement and crsng strengt of concrete. Te bond-slp penomenon between concrete and renforcement s acconted for b sng non-dmensonal sprng element. Stffness vales for sc ele me n ts ar e o bta n ed f r om epr esso n based on eper me n tal data. Improved soparametrc qadrlateral elements and tranglar elements are sed to represent concrete. Materal response s assmed to be ortotropc wt tangent stffness E, E derved from stress-stran relaton for concrete nder a general baal state of stress. Te renforcement s represented n a dscrete manner. One- dmensonal fleral elements and aal ones are sed for ts prpose. Materal response s assmed to be elastc-perfectl plastc. Two metods of crackng representaton are sed: te dscrete crackng approac and dstrbted crackng approac.

3 Te compter program wt combned ncremental-teratve metod s sed to solve te non-lnear problem. Te reslts of ts nvestgaton gave good agreement wt conclson reaced b te task commttee 46 on sear and dagonal tenson (,tat n renforced concrete deep beams bondslp mecansm takes te major part of sear transfer wle te oter mecansms(aggregate nterlock and dowel acton take less part. Ts s ndcated b comparng te reslts of ts nvestgaton wt epermental reslts obtaned b sreal (.

4 Acknowledgement Iwold lke to tank professor/ab baker Abdal Waab for s spport, constrctve crtcsm and sncere follow p trogot te std and preparaton of ts tess. And also I wold lke to tank teacng assstant/asm Elsanos for s gvng good dea abot te content of ts tess. 4

5 Table of Content Page o. Acknowledgment..( Abstract....( Abstract(arabc.. (v Capter( Introdcton. General.... Tess laot... Capter( Lteratre Revew.. Revew of epermental nvestgaton...5. Revew of analtcal nvestgaton...4 Capter( Modelng of Sear Mecansms and Post Crackng.. Aggregate nterlock mecansm Proposed analtcal model for aggregate nterlock.... Dowel acton mecansm..... Proposed analtcal model for dowel acton.... Bond-Slp mecansm

6 .. Proposed analtcal model for bond-slp mecansm..4 Capter(4 Fnte Element Modelng. 4. Basc concept on-lnear analss on-lnear nmercal tecnqes Incremental metod Iteratve metod Combned ncremental-teratve metod Selecton of elements and representaton of crackng Modelng of steel renforcement Dstrbted representaton Dscrete representaton Embedded representaton Modelng of concrete Plane soparametrc qaderlateral element Constant stran tranglar element Modelng of lnk element Modelng of crackng Dstrbted crackng Dscrete crackng...68 Capter(5 Concrete Constttve Relaton 5. Baal stress-stran relatonsp Baal stress falre envelope and correspondng strans Baal compresson-compresson Baal compresson-tenson Baal tenson-tenson

7 5. Falre crtera Falre tenson-tenson Falre tenson-compresson Falre compresson-compresson Comptaton of nbalanced load Convergence crtera Unbalanced loads convergence crteron Dsplacement ncrements convergence crteron...8 Capter(6 Applcatons and Reslts 6. Specmen descrpton and materal propertes Epermental response Analtcal response Conclsons ote Frter work...99 References Lst of compter program. Append 7

8 CHAPTER( ITRODUCTIO. General Renforced concrete deep members often form part of a more comple strctral sstem n mlt-stor bldngs. Te commonl take te form of sear walls, deep beams, corbels, bracket, and oter confgratons. Te rapd ncrease n te nmber of tall bldngs for bot resdental and commercal prposes as necesstated te searc for analtcal metods to aceve mproved nderstandng of te beavor of strctral components n sc bldngs. Te man factor affectng te dffnton of deep beams s spandept rato [ L l ] or [ n ] were L and L H d n are span and clear span of deep beam and H,d are dept and effectve dept of deep beam. l ACI code stplates tat a beam can be consdered deep f [ n ] <5. d Ero-nternatonal concrete commttee decded tat a beam cold 8

9 be consdered deep f [ L ] < or.5 for smpl spported and H contnos beams respectvel. some nvestgator ave decded tat te sear span-dept rato [ a ] d s more meanngfll to defne a deep beam, and tat a beam cold be consdered deep f [ a ] <.5. d Te dstrbton of stresses n vertcal secton n deep beams(non-lnear dstrbton sgnfcantl devates from tat of sallow beams(stragt lne dstrbton.ts propert was one of te mportant factors for classfng deep beams sc tat deep beam ma be defned as beam wose bendng stresses devate apprecabl from te stragt lne dstrbton assmed n elementar beam teor (te normal teor of flere based on aver-bernoll prncples s not applcable.for ts reason te ordnar beam desgn metods to calclate te member strengt are napplcable n case of deep beams. A meanngfl analss of renforced concrete deep beams reqres an analtcal model wc reprodces te beavor of te member as accratel as possble. some of dffcltes encontered n analtcal stdes of te beavor of deep beams arse from te followng: Te nonomogeneos natre of constrcton. Materal response to load s non-lnear. Te non-lneart of te stress-stran relatonsp for concrete. Te contnosl cangng topolog of concrete de to crackng. Epermental evdence ndcates tat te relaton between bond stress and bond-slp s non-lnear. 9

10 Recent development of te fnte element metod of analss permts consderaton of members wc are non omogeneos, defned b rrglar bonderes, arbtrarl spported and loaded. Te metod s sed to determne te nternal stresses and dsplacement for renforced concrete members sbjected to progressvel ncreasng load,wt recognton for several sorces of non-lneart. Te resltng model permts accontng for Te nflence of renforcement. Cangng topolog de to progressve crackng. Realstc bond stress transfer between concrete and renforcement. Incremental loadng permts std of member beavor trogt te entre range from zero to ltmate loads.. Tess Otlnes: A lteratre srve s presented n capter ( smmarzng te crrent knowledge related to epermental and analtcal prevos work n renforced concrete deep beams. Te beavor of renforced concrete deep beams nder dfferent tpe of loadng ncldng te effect of web renforcement, span/dept rato, sear span/dept rato, concrete strengt and oter factors related to sear capact s stded. In ts researc an analtcal model based on epermental reslts of Dlaska ( and Fenwck Pala (4 s sed to smlate dowel acton and aggregate nterlock mecansms ts work s descrbed n capter(. In tese models a dowel bar s treated as beam on elastc fondaton wle aggregate nterlock s modeled b dealzaton of rreglasr crack srface. Lnkage elements are emploed to model tese mecansms b releasng te pars of node defned along dscrete crack wen crackng s detected.

11 Ts nvestgaton also eplores varos fnte element models n order to aceve te non-lnear beavor of renforced concrete deep beam. All te sorces of non-lneart mentoned n capter (4 are acconted for. A dscretzed sstem consstng of plane stress elements and one-dmensonal elements, modelng te concrete and renforcement, respectvel, s constrcted for analss. Te plane stress elements sed are mproved soparametrc qadrlateral elements. Te man renforcement s represented b one-dmensonal fleral elements wle onedmensonal aal elements are sed to model te web renforcement. Elastc-perfectl plastc materal response s assmed for steel. Specal empass s placed on te representaton of crackng and modelng of te post-crackng sear transfer mecansm. Te members are assmed to be acted pon b monotoncall ncreasng lve loads. Two metods of crack modelng are consdered as mentoned n capter (4: dstrbted crackng approac and dscrete crackng approac. In some cases te se of dscrete crackng s sffcent. In oter cases te se of dscrete crackng n conjncton wt dstrbted crackng approac proved to be necessar f a realstc predcton of a member s response s sogt. A constttve relaton for concrete based on te analtcal and epermental nvestgatons of L and Tasj s sed to descrbe te beavor of concrete nder a dfferent state of stresses as mentoned n capter (5. Te compter program wt a combned ncremental-teratve nmercal metod s sed to solve te non-lnear problem. Ts conssts of applng te eternal load n a nmber of ncrements. Specfed nmber of teratons to brng mont of eqlbrm

12 volaton to a tolerable lmt follows eac load ncrement. Te stffness matr s pdated at te begnnng of eac load ncrement. Convergence of solton cecked b eter te nodal dsplacement crteron or te nbalanced load crteron. Te valdt of te proposed fnte element models s tested b comparng te reslts obtaned from analss wt te correspondng epermental ones as mentoned n capter(6. CHAPTER ( LIRERATURE REVIEW Te researc nvestgaton of sear strengt of renforced concrete deep beams s dvded nto two parts: epermental nvestgaton spported b statstcal correlaton s to get sear capact eqaton fttng te epermental data, and analtcal nvestgaton sng fnte element to smlate te non-lneart and crackng of concrete.. Revew of Epermental Investgaton De Pava and Sess (5,descrbed an epermental nvestgaton on te sear strengt and beavor of some moderatel renforced concrete deep beams b loadng tese beams at trd ponts. Te man factors consdered n ts nvestgaton are: Amont of tenson renforcement. Concrete strengt.

13 Amont of web renforcement. Span-dept rato. Te conclded tat for renforced concrete deep beams wtot web renforcement tere s a g load capact beond te dagonal crackng and tat te addton of vertcal strrp and nclned bars ave lttle effect on te ltmate strengt.also te sowed te fact tat te concrete strengt ad no effect on te ltmate strengt of beams falng n flere. Gergel (6 performed an epermental model to std te contrbton of aggregate nterlock and dowel acton to post crackng sear capact of renforced beams wt no web renforcement. Gergel estmated contrbton of aggregate terlock to be 4% to 6% of te total sear and tat of dowel acton was estmated to be % to 5% of te total sear. It was also conclded tat te dowel acton s te man factor casng splttng along man renforcement n beam wtot web renforcement. Talor (7 condcted several eperments to nvestgate te effect of aggregate nterlock and dowel acton b stdng te factors affectng te two mecansms. To smlate te aggregate nterlock Talor sed two tpes of specmens: Block tests. Beam tests. Te man factors nclded n tese tests to std ter nflences n aggregate nterlock mecansm are: - Te dsplacement rato / S, were s te dsplacement normal to te crack (crack wdt, and S s te orzontal dsplacement (sear dsplacement.

14 - Concrete strengt. - Aggregate sze. 4- Aggregate tpe. Te block test as te advantage tat t reqres less sopstcated set-p and measrng devces, and s also more economc and consmes less tme tan te beam test. Bt te beam test s sefl n obtanng more relable reslts abot aggregate nterlock mecansm. For dowel acton mecansm Talor consdered te followng factors: - Concrete strengt. - Sear span. - Crack wdt. 4- Concrete cover. Te man prpose of ts work was to establs complete dowel load-dsplacement crves, and to estmate te contrbton of sear resstng mecansm n renforced concrete beam wtot web renforcement. Ter reslts were as follows: Compresson zone - 4% Aggregate nterlock - 5% Dowel acton 5-5% Pala and Loeber (8, conclded tat te crack wdt s te most mportant factor affectng te aggregate nterlock sear stffness and tat te aggregate sape and sze ad ver lttle nflence on sear stress-dsplacement relatonsps. Kong et al (9,performed an epermental nvestgaton to std te effect of dfferent percentage of web renforcement n trt fve normal wegt concrete deep beams. Te beams were tested nder two-pont loadng. It was fond tat te effectveness of 4

15 varos web renforcement depended on te span/dept rato [ L ] H and clear sear span/dept rato [ a ]. for low [ L ] and [ a ] d H d ratos, orzontal web renforcement gave te best reslts. Anoter detalng of web renforcement was stded epermentall and reported b Kong et al ( consstng of nclned bars. He fond tat web renforcement was an effectve tpe of renforcement to ncrease sear strengt of renforced concrete deep beams and also to control deflecton and crackng. Kong and Robns (,carred ot tests on smpl spported lgt wegt concrete deep beams, and te developed an eqaton tat calclates te ltmate load for normal wegt concrete deep beams. Ts eqaton s not necessarl stable for normal wegt concrete beams. Frter epermental work on lgtwegt concrete deep beams was reported b Kong and Robns (. Te revsed ter prevos eqaton n two factors: te [ a ] rato eplctl allowed for, and d te sed concrete clnder splttng tensle strengt, f t,as te tog tat te concrete contrbton to te ltmate sear strengt s mc more drectl related to f t tan clnder compressve strengt f c. Ter tests sowed tat te clear sear span/dept rato [ a ] ad greater effect on crackng and ltmate d loads tan span/dept rato [ L ]. H Smt and Vantsots ( carred ot tests for smpl spported renforced concrete deep beams nder two smmetrcal pont loads. 5

16 Tese tests revealed te followng reslts: -An ncrease n sear capact was observed wt ncreasng concrete strengt and decreasng sear span-dept rato. -Te ncrease n ltmate sear strengt and n dagonal crackng load was attrbted to te arc acton for specmens wt a sear span-dept rato less tan.5. -Vertcal strrp became more effectve wt te greater span/dept rato. 4-Horzontal web renforcement was more effcent n beams wt sear span/dept rato less tan.. 5-Web renforcement ad no effect n controllng te dagonal crackng load and crackng patterns for beam wt or wtot web renforcement. Kang-Ha tan et al (4 performed epermental tests on renforced concrete deep beams wt g strengt concrete nder two smmetrcal top loadng consderng two man varables; sear span-dept rato and span-dept rato. Te conclsons of ts test are : -Span dept rato as lttle nflence n falre load for beam wt [ a ] > d -Te fleral falre became domnant wt ncreasng span/dept rato -Te reslts of tese tests nsre safe desgn for ger strengt deep beams compared wt predcton based on ACI bldng code for concrete strengt less tan 4 Mpa. Asraf (5 carred ot tests on renforced concrete contnos deep beams consderng sear-span/dept rato, amont and tpe of web renforcement and amont of man longtdnal renforcement as man parameter of tese tests. 6

17 He conclded tat te vertcal web renforcement ad more nflence on sear capact tan orzontal web renforcement. He also compared s reslts wt ACI Bldng code and CIRIA code bt t sowed lttle agreement. Kang-Ha Tan et al (6 reported epermental tests for renforced concrete deep beams wt clnder compressve strengt of generall eceedng 55 Mpa and man longtdnal renforcement rato ρ and sear span/dept rato as man parameter of tese tests. Te beams were organzed nto for grops wt renforcement rato ρ.,.58, 4.8 and 5.8 percent. Te beams were tested for dfferent span-dept rato [ a ], rangng from.8 to.4. d He conclded tat : -Te falre mode was cefl nflenced b te [ a ] d rato and te effect of renforcement rato was not sgnfcant. Te beam wll fal n: Bearng mode for [ a ] <.8 d Sear compresson mode for.8< [ a ] <. d Dagonal tenson mode for.< [ a ] <.6 d Sear tenson mode for [ a ] >.6 d - Increasng of ρ beond % dd not sgnfcantl ncrease te sear strengt of HSC deep beams. Ts ts vale represents a practcal pper lmt n mamzng te man renforcement to agment te sear strengt. 7

18 Fenwck and Pala (4 condcted an etensve epermental nvestgaton to std te natre, sgnfcance and parameters affectng te aggregate nterlock mecansm. Te parameters nclded n ter work are concrete compressve strengt σ co and crack wdt, wc was kept constant at.75nc. Te observed tat te concrete strengt s a sgnfcant parameter nflencng te mamm sear stress level and sear stffness. Usng regresson analss of ter epermental reslts, sear stress s epressed as followng:.467 v ( 8.4(.75 σ.49( CO S Were v Sear stress S Sear dsplacement n nces Te sear stffness A, of te cracked concrete s obtaned b dfferentatng eq.(. wt respect to sear dsplacement s A v ( (.75 σ co S Rearrangng A.467( 8.(.75 σ co Accordng to ter reslts tese eqatons are vald for crack wdt range from.5 to.5 nc, σ co n (ks and A n (ks/n. Hode and Mrza (7 n ter std of sear strengt of renforced concrete beams, condcted an epermental program to establs force-dsplacement relaton-sp at crackng de to aggregate nterlock, te parameters consdered are: -Crack wdt -Concrete strengt σ co 8

19 -Mamm aggregate sze Hod and Mrza sggested te followng sear stressdsplacement relaton-sp: v.57 v A S. 5 S ks / n Were A, te sear stffness, s obtaned b dfferentatng eq.(.4 wt respect to S. A more general eqaton sggested b Hode and Mrza to cover te range of crack wdt below. nc. A G λ G G Were G s te elastc sear modls of ncracked concrete and λ s a constant. Bot of te above models are vald for lmted crack wdt and ts necesstates te searc for an analtcal aggregate nterlock model for wder range of crack wdts. In ter std of te post crackng sear resstance mecansm n renforced concrete beams wt dagonal tenson cracks go et al (8 modeled dowel acton b sng te concept of effectve dowel acton lengt, wc s te lengt over wc te bond assmed to be destroed, and te assmed an effectve dowel acton lengt to be two nces. Ts means sng constant dowel stffness over te entre post-crackng. Based on epermental reslts Talar sggested te followng eqaton for peak dowel force D f n kps: D f Were.4.[ ( Cs C ] σ...7 to 9

20 σ to Splttng tensle strengt of concrete. C s Sde cover to bars. C Dstance between te bars. Anoter epresson sggested b Talor to ft te dowel force dsplacement crves obtaned epermentall s: D f.48d D Dowel force (kps. D f Crackng force (kps. Dowel dsplacement (nc. Were Te dowel stffness D s ma be obtaned b dfferentatng eq.(.8 wt respect to : D D.87D s f Hode and Mrza based on epermental reslt sggested te followng epresson for Dowel crackng force D f : D.4( / f σ co [ ( Cs C ] Also te sggested te followng lnear dowel forcedsplacement relaton-sp: D D.... f Te dowel dsplacement, wc corresponds to te peak dowel force (crackng forcewas c 5 4 ncs. Te dowel stffness D S ma be obtaned b dfferentatng eq.(. wt respect to :

21 D D s D f Were D s s epressed n kps/n Te models vewed above ave scceeded n provdng sefl nformaton on te dowel mecansm bt are lmted to sallow beams, were te dowel falre s cased prmarl b splttng along te man renforcement.. Revew of Analtcal Investgaton Te frst attempt to se an analtcal approac (fnte element metod to model renforced concrete beams p to falre was made b go and Scordels (9. Te sed two dmensonal constant stran tranglar elements to model te concrete and steel as sown n fg.(c. Te bond-slp penomenon was represented b specal zero-lengt lnkage elements connectng te plane element of concrete to tose of steel at dscrete ponts as sown n fg..(b.dscrete crackng was consdered n ts model sc tat dagonal cracks were represented b separatng elements on eter sde of te crack (b assgnng two nodes, occpng te same coordnates n space, along te crack. Ts nvestgaton was not ntended to eplore te beavor of renforced concrete beams n te non-lnear and ltmate ranges and terefore, te materals propertes were consdered to be lnear. Te man sortcomng of ts model s: Lneart of materal. eglectng te effect of post-crackng sear resstng mecansm (aggregate nterlock and dowel acton n sear capact. Absence of te effect of web renforcement.

22 Dffclt to smlate bond slp mecansm de to te tpe of steel dealzaton. lson ( etended te work of go and Scordels b addng some consderatons to ter model n (materal propertes, concrete and steel modelng and fnte element tecnqe. Materal propertes were consdered to be nonlnear. Concrete and steel were dealzed b sng mproved qadrlateral plane stress fnte elements. Dscrete crackng was consdered n ts model to model te dagonal crackng. An ncremental metod s sed to reload te strctre after crackng.

23 go, Frankln and Scordels ( ntrodced some refnement to ts model b representng concrete and steel wt qadrlateral elements and ntrodcng te effect of strrps, represented b one dmensonal bar elements. Also te stded te post crackng sear mecansm b sng a lnkage element of zerodmenson to smlate aggregate nterlock mecansm and assmng tat sear forces transferred b dowel acton n man renforcement can be carred b porton of ts renforcement. Te man dsadvantage of ts model s tat non-lnear beavor of renforced concrete beam s not consdered. An mportant conclson of ts std s te sgnfcant contrbton b dowel acton and nterface sear transfer to te total sear capact of sallow beams, Tere s some reservaton de to te estence of predefned dagonal crack from te start of loadng. Hode and Mrza stded te non-lnear beavor of renforced concrete beam and te relatve contrbtons of sear forces transferred b aggregate nterlock and dowel acton to te total sear capact bt tere s no accont of bond-slp mecansm n ter model de to te tpe of dealzaton of renforcement. Basc consderatons of Hode and Merza model: For materal propertes a non-lnear naal stressstran relaton was sed for concrete n compresson and a lnear relaton n tenson. Concrete was dealzed b qadrlateral plane elements as well as tranglar ones wle te renforcement was represented b qadrlateral elements. Dscrete crackng was consdered n ts model for dagonal crackng. Frankln ( sggested two tpes of fnte element models:

24 In te frst model te beam s dvded nto nmber of specal tpe of frame elements wt dept eqal to te dept of te beam, Tese frame elements are ten sbdvded nto ten laers, and wt mamm nmber of renforcement laers eqal to for over ts dept. Basc consderatons of te frst model are: Progressve dstrbted crackng was assmed to occr over frame elements. Its ablt to represent bond slp mecansm and no accont s taken for te aggregate nterlockng and dowel acton mecansm. Ts model s effectve onl for vertcal cracks (frame element model was ncapable of prodcng dagonal cracks. An ncremental metod s sed n ts researc to reload te beam after crackng. In te second model materals are assmed to be non-lnear and bond-slp mecansm s consdered bt no accont s taken for te aggregate nterlock and dowel acton. Basc consderatons of te second model are: Concrete s modeled as qadrlateral plane stress elements and te steel as one dmensonal bar elements. Dstrbted crackng n ts work s sed to model te dagonal crackng. Frankln analtcal models were stffer tan te actal beams. 4

25 CHAPTER ( MODELLIG OF SHEAR MECHAISM AD POST CRACKIG Tere are tree man mecansms for sear transfer tat can develop after crackng of renforced concrete members: Te sear force mecansm tat s transferred across cracks (aggregate nterlock. Sear force tat develops n te renforcement (dowel acton. Te stress transfer mecansm, wc reslts n te nteracton of concrete and steel (bond-slp.. Aggregate Interlock For te aggregate nterlock mecansm te srfaces of crack, wc develop n concrete member, are sall rreglar and rog. Te occrrence of sear dsplacements parallel to te drecton of sc crack reslts n te transfer of sear forces b aggregate nterlock from one sde of crack to oter sde. Te man factors affectng te performance of ts mecansm are: -Sze of contact area between te srface of crack wc depends prmarl on crack wdt and sear dsplacement parallel to te crack.as sear dsplacement ncreases te contact area ncreases and as crack wdt decreases te contact area ncreases. -Te force dsplacement caracterstc of aggregate aspertes trog wc te sear forces are transferred depends on: Compressve strengt of concrete. 5

26 Tpe of aggregate. Sze of aggregate... Proposed Analtcal Model of Aggregate Interlock Mecansm In ts researc an analtcal model smlatng te aggregate nterlock mecansm s proposed. Te prmar parameters consdered n ts model are: Crack wdt Sear dsplacement S Compressve strengt σ C Te frst step n constrctng an analtcal model mst terefore be te dealzaton of te rreglar crack srface. Te possble dealzaton profle s sown n Fg.. below It conssts of toot lke aspertes avng nform geometrcal propertes along te entre crack. Ts dealzaton s cosen to 6

27 smplf te task of constrctng an analtcal model. A freebod dagram of te nternal stresses actng on te dealzed crack srface as sown n Fg..are of two tpes: Stress normal to te crack srface Sear stress component tangental to te crack srface Te orzontal eqlbrm condton s gven b: V k( σ scosθ σ ssnθ..... Were: s σ Sear stress along te crack srface. s σ Stress normal to te crack srface. s Contact srface area for one dealzed aspert. θ Angle between dealzed crack srface and orzontal as. k mber of moblzed aspertes along entre crack. Te vertcal eqlbrm condton s gven b: σ s ssnθ σ s cosθ.... or σ σ tanθ s B sbstttng eq.(. n eq.(. V kssecθσ s In ts researc te non-lnear relatonsp below s sggested(as tat of lson. m σ R ξ S were: σ Amont of slp along nclned crack srface. S R,m Parameters to be obtaned from epermental data. 7

28 ks Total contact area along te entre srface of crack. From Fg... ξ ( S / were : S Sear dsplacement. Crack wdt...6 A lnear relaton s assmed to est between ks and S S : ks R...7 were s,r Parameters to be obtaned from epermental data. B sbstttng eq.(.5and eq.(.7 n eq.(.4 n v C( Were: S S...8 8

29 R R secθ C A Sear and n.5 Te sear stffness A of cracked concrete s obtaned b dfferentatng eq.(.8 wt respect to v A S C([ n] S ( S n S :..... eq.(.8 and eq.(. epress te sear stress and sear stffness as non-lnear fncton of te crack wdt and sear dsplacement s. Te onl nknown parameters n ts eqaton are C and n, wc can be obtaned from epermental work. Te epermental data sed n ts researc are tese parameters adopted b Fenwck and pala. Te man parameters of ter epermental work are: -Crack wdt. -Compressve strengt. Frstl concrete strengt s assmed to be constant wt vale 4.8 ks and te crack wdt vares between (.5-.5nc. Te relaton between v and can be obtaned from Fg... 9

30 Were v Sear stress at falre. v C (..4 forσ co 4. 8ks Secondl crack wdt s assmed to be constant wt a vale of.75nc. Te relaton-sp between v and compressve strengtσ can be obtaned from Fg..4. C

31 Ts relaton can be epressed b v.576.σ co, for. 75nc... We can fnd parameter C b sbstttng mamm sear stress v correspondng and.75 nc te followng eqaton C /( v Wc gves C v /

32 B sbstttng eq (. n eq.(.4 C.6.4σ co....5 Ts eqaton s plotted n Fg..5 and te vales of parameter C based on sear stress-dsplacement crves correspondng to.75nc. Ts eqaton can be appled to oter crack wdts. Te relaton between te sear dsplacements s correspondng to mamm sear stress v can be obtaned from Fg..6 correspondng to crack wdt between (.5-.5 nc sng metod of least-sqare. Ts relaton can be wrtten as follow: 7.57*.68 4 s

33 B consderng bondar condtons of sear stress-dsplacement crves gven n Fg..7 n conjncton wt eqatons above te nknown parameters C and n can be obtaned as follows: a / wen v v S s Sbstttng te epresson above n eq.(.8 n v n s s s s v C( C (...7 b/ at ncepton of loadng S, A A for Sbstttng te epresson above n eq.(.7 C A n ( Eqatng eq.(.7 and eq.(.8

34 n ln( v A s s ln( Te vale of v and A can be obtaned b te followng procedre: /Calclate C from eq.(.5. /Calclate v b sbstttng C n eq.(.. /Calclate s from eq.(.6. 4/Calclate n b sbstttng v and s n eq.(.9. 5/Calclate C b sbstttng n n eq.(.8. Ten obtaned te vale of v and A b sbstttng n and C n eq.(.8 and eq.(. respectvel. Ts approac of aggregate nterlock modelng s sed n connecton wt dscrete crackng. Lnkage elements are sed to lock or release te node pars defned along dscrete crack.te two ortogonal fcttos sprngs stffness k and k v are assgned vales wen crackng s detected n elements adjacent to te node-pars, k a s A k v Were a s cracked srface area correspondng to a lnkage element. Ts modelng can not be sed n connecton wt dstrbted crackng becase of dffclt of obtanng and s de to te assmpton of dsplacement contnt across crack n ts approac, and eq.(. can not be appled to determne sear stffness. To solve ts problem man nvestgators se te concept of redced sear modls G. Ts concept ma be eplaned as 4

35 follow: wen crackng occrs, te sear term s redced b a certan factor to accont for capact of cracked concrete to carr sear b aggregate nterlock. Te coce of ts factor was determned b trng several redcton factors and selectng te factor, wc gves predcton closet to epermental reslts. In ts researc a proposal for an epresson for G sggested b lson s selected.te man concept of ts proposal can be eplaned as follow: Tensle stran normal to crack ε ma be sed to defne te redced sear modls G. Te cange of G wt ε sold be rapd and non-lnear. At falre of member te vale of G becomes ver small (ts s to reflect te fact tat at falre, te cracks become too wde to be able to transfer a sgnfcant amont of sear stresses. lson: Te followng relaton between G-ε s sggested b G G forε < ε... t ε G Were.4 G / forε ε t ε t ε Crackng tensle stran. t 5

36 Ts relaton assmes tat at ncepton of crackng (wen ε ε t te sear modls G s redced to 4% of sear modls of ncracked concrete G b sbstttng te vale of ε n eq.(.. 6

37 Wen ε s ncrease and reaces te vale (ε 5ε t at falre te sear modls redces to.8 G b sbstttng te vale of n eq.(. Te man fncton of ts model was to reac a relable estmate of te contrbton of aggregate nterlock to total sear resstance n beams. Factors affectng ts mecansm are: a- Concrete compressve strengt. b- Crack wdt. Te post-crackng sear resstance mecansm to aggregate nterlock as attracted consderable researc effort n recent ears. In ts regard, te fnte element metod as plaed a promnent role. Te mode of representaton of aggregate nterlock mecansm n fnte element model depends on te tpe of crack and tpe of model sed. Across dscrete crack, specal lnkage elements wt ortogonal fcttos sprng are sed. A nonlnear stffness caracterzaton for sc sprngs s derved from an analtcal model n conjncton wt epermental data. In cases were dstrbted crackng s emploed, a redced sear stffness G s emploed n te constttve relaton of cracked concrete. A nonlnear epresson relatng G to te tensle stran normal to te crack drecton s proposed. 7

38 8

39 .Dowel Acton Mecansm Te renforcement provded to carr tensle forces along ts as can also ressts forces normal to ts as. Ts occrs wen sc renforcement crosses sffcentl open crack. Ts addtonal resstance s known as dowel acton mecansm. Tere are man factors affectng ts mecansm: a- Renforcement rato A s /bd. b- Renforcement bar dameter. c- Yeld strengt of renforcement bar f. d - Angle between te bar and te normal to te crack srface. e- Concrete compressve strengt f c. f- Sear span... Proposed Analtcal Model of Dowel Acton Mecansm In ts researc an analtcal model smlar to tat of lson s sed. Ts model s based on epermental reslts of Dlacska wc smlate dowel acton mecansm as follows: A dowel bar s embedded n concrete and can be treated as a sem-nfnte beam on elastc fondaton as sown n Fg... 9

40 Te bar s assmed to be fed freel over crsed porton of concrete. Te dmenson C defnes te etent of ts crsng and s epressed as follows: σ C γ σ co φ snδ.... Were C and φ are epressed n nces and σ and σ co n ks and γ s a dmensonless constant to be obtaned from epermental data. Ts epresson s based on te followng assmptons: / Te ger concrete strengt te lower amont of crsng C. / Te amont of crsng C ncreases as qantt φσ (wc reflect te bar rgdt ncreases. / Drect dependenc s assmed to et between C and angle of nclnaton δ epressed n te form snδ. Te mplcaton of ts s tat no crsng occrs wen δ,wen te bar s perpendclar to te sear plane, and tat as δ ncreases te dmenson C ncreases. Te stress dstrbton beneat te dowel bar s sown n Fg... It s governed b followng epresson: t q D π π e cosπ..... t t Were D Dowel force appled at te sear plane. t Dstance along te bar defned n Fg... 4

41 t s possble to replace te actal stress dstrbton n Fg..(b b smpler one sng eqvalent rectanglar stress blocks as sown n Fg..(c te stress ntenst can be obtan from te followng epresson: t qd t q e 4

42 for to t b sbstttng eq.. nto eq..4 π q t t cos.8d e D π e π d ( k / n t t t t for t to t π q t t cos.7d e D π e π d ( k / n t t t t t for t to 5t q.9d e ( k / n t (Ts vale s ver small compared wt oter vales; terefore stresses n regon t ma be neglected. In ts modelng Dlascka neglected all te stresses n te regon t and consdered te stress block n regon t M D * q e Mamm bendng moment occrs at a pont were te sear (VdM/d. V D q * D (.8 D e t Wc gves.8t Te dstrbton of radal stresses along te crcmference of te dowel bar s sown n Fg.. Ts dstrbton s governed b te followng eqaton: σ σ cosθ..... r Were θ Angle defned n Fg... σ Radal stress at angleφ, n ks σ r Ma. radal stress wc occrs at θ,n ks 4

43 q e π φ σ cosθdθ r.8d π σ φ cos θdθ t were t.65 D.... φσ ACI Code specfes tat σ sold not eceed.445σ, and p to eq.(. becomes D f t φσ c were D f Te dowel force at falre. Ma. eternal moment can be obtan b calclatng te area nder crve n Fg.. M area of rectanglar area of tranglar D f M C * D.8t * D ( C.4t..5 f f Ts moment s ressted nternall b plastc eld moment M P M 6 p φ σ p Z σ were Z p Plastc modls of secton E q a t n g e q. (. 5 t o e q. (. 6 g v e s D f ( c.4 t φ σ Sbstttng te vales of C, t gven b eq.(. and eq.(.4 respectvel n eq.(.7 te followng qadratc eqaton can be obtaned.5d f ( γσ φ snδ D f.67φ σ σ c..8 Dlascka ( estmated te parameter γ to be eqal to.5. 4

44 Te solton of eq.(.8 s: D f 54.67σ co.σ φ snδ[ ] σ sn δ For δ, and D f.9φ σ σ..4 co Te dowel force dsplacement relaton s: D D π.5 [ tan( ] φ σ C D f Several determnate forms of D- relatons ma be sed to ft te epermental response of D-. In ts researc te followng polnomal s sggested: m D C C..4 ( Were C, C and m are parameters to be obtaned from epermental data.te dowel stffness D s ma be obtaned b dfferentatng te above epresson wt respect to : m DS D C C m ( B consderng te bondar condton of a tpcal dowel forcedsplacement obtan epermentall, te followng can be notced: / at te ntaton D S D S wen / wen te dowel force reaces te mamm vale D f te dowel dsplacement reaces a vale f / wen dowel force eceeds D f te D- becomes flat, te dowel stffness D S redces to zero: B sbstttng tese tree condton n eq.(.4 and eq.(.4 te vales of C,C and m can be obtaned as follows: C DS 44

45 m DS D Were Sf D Secant dowel stffness at falre. sf D sf D f f D s 667k / n. Te ntal dowel stffness D s ma be estmated to be 667 k/n C 667 C 8 m D ( Ts epresson s plotted n Fg.. and gves good reslts compared wt several epermental crves done b Dlacska. Lnkage elements smlar to tose sed n connecton wt aggregate nterlock mecansm are emploed to model te dowel 45

46 acton mecansm. Upon te formaton of dscrete crack, a dowel stffness D s defned b eqaton s assgned to te fcttos sprng parallel to crack K D s K v Ts t can be conclded tat te stffness of te dowel acton can be calclated from eq.(.4. 46

47 . Bond Slp Mecansm: Te nteracton between te renforcement and concrete n renforced concrete strctres occrs b means of sear transfer mecansm known as bond. Ts bond between te steel and concrete at te nterface does not reman ntact, resltng n relatve movement (slp between te two components. Te mportance of ncldng sc a bond-slp mecansm n a fnte element model s de to ts drect nflence on te wdt and spacng of tensle cracks as well as te dstrbton of stress n partall cracked concrete. Factors affectng ts mecansm: - Compressve strengt of concrete σ co. - Peak bond stress (ks. -Dstance from te end of specmen to te face of crack C as sown n Fg Proposed Analtcal Model of Bond-slp Mecansm Te transfer of stress b bond between concrete and steel s most dffclt to model realstcall. In ts researc an analtcal model smlatng te bond slp mecansm proposed b lson s sed. Closel spaced sprng lnkages are sed. Eac lnkage contan two sprngs, one actng parallel to te bar as and te oter actng perpendclar to t as sown n Fg..5, Ever sort 47

48 segment of te bar s drectl joned to te adjacent bar segments, and s connected to te adjacent concrete b te lnkage sprngs. Te dmenson of eac sprng ma be redced to zero (dmensons less lnkage. In te practcal cases, t s sall convenent to specf one sc lnkage at te top of a bar segments, and one at te bottom, consderng bond n ts sal sense, onl longtdnal sprng needs to be nclded. Basc assmptons of ts model: on lnear materals propertes on omogenet on lnear bond slp relatons Local bond destrcton Local slp mst be obtaned ndrectl. It s defned as te dfference between steel dsplacement and concrete dsplacement n te drecton of te bar as at an secton. Steel dsplacement s fond from measred strans, wle concrete dsplacement s 48

49 estmated from epermental work b measrng slp at te face of te tested specmens. In ts work a bond slp eqaton s derved ndrectl from eperments reported b Bresler and Bertero (4. lson as condcted sc an epermental program and obtaned nformaton of drect relevance to fnte element model. lson fond tat te peak bond stress U p s proportonal to te dstance from te end of te specmen C (or te face of crack as sown n Fg..4. Ts relaton was presented b: U σ P C.6(.4C A bond stress crve for C6n establsed b lson and te same crve for dfferent vale of C obtaned b Hode and Mrza s plotted n Fg..6 49

50 Bt for prpose of smplct, te nflence of C be ma neglected and te bond stress-slp crve of lson wt C 6nc s sed and te two followng relatons are obtaned: U σ c.6(7.5α 5α 7.5α for α < U σ c.6(5α 5α.5α forα U bond stress, ks α P.49 were αormalzed slp Slp, nc. p Slp at peak bond stress, nc. Te dotted lne n Fg.6 s a plot of tese two polnomals and s ver close to epermental crve. To establs te non lnear bond stffness Eq.(.47 s dfferentated wt respect to α B s U σ c.6 (.5α 5α P eqvalent sprng lnkage stffness can be obtaned b mltplng te above epresson b te srface area trbtar to one lnkage to obtan stffness n proper nts. K B A..5 s s Were A s srface area of renforcement bar In ts modelng t s assmed no apprecable relatve dsplacement normal to te bar occrs. Terefore stffness vale K v K s sed n ts researc to mpose ts condton. 5

51 CHAPTER(4 FIITE ELEMET MODELIG 4. Basc Concepts Te fnte element metod s a nmercal tecnqe for obtanng appromate solton to varet of engneerng problems. Drng te last fft ears man problems n aeronatcal, cvl, mecancal and nclear engneerng, wc reqred te determnaton of statc as well as dnamc response, ave been solved b te fnte element metod. Te basc concept n te fnte element metod s to fnd te solton of a complcated problem b replacng t b a smpler one. Ts s done b modelng analtcall a contnos strctre and sbdvdng t nto regons or elements. Eac element s descrbed b a separate fnctonal representaton. Tese sbdvsons consdered to be nterconnected at specfed jonts, wc are called nodes or nodal ponts sall lng on te element bondares were adjacent elements consdered to be connected. In totalt, tese sbdvsons follow te beavor of te real contnos strctre wt a certan margnal error nerent n te metod tself, lke an oter nmercal metod. Te amont of sc an error depends prmarl on te fnctonal representaton (modelng cosen to best resemble te contnos beavor of actal strctre, and pon te sze of element relatve to te sze of te regon stded. Te applcaton of te fnte element metod to non-lnear problems reslts n more nmercal operatons compared to lnear problems. However, recent developments of te dgtal compters overcome ts pont. Snce te fnte metod ses a sbsttte strctre wose parts are, n a sense, peces of actal strctre, soltons of complcated problems are epected to be appromate.in most complcated problems appromate soltons are acceptable as te 5

52 eact soltons are too elaborate or rater mpossble to obtan. Te non-lnear problems mst satsf te fndamental condtons of eqlbrm, compatblt and constttve relatons of materal. Te non-lnear solton s obtaned b solvng a seres of lnear problems sc tat at te fnal stage approprate nonlnear condton are satsfed. 4. on-lnear Analss on-lneart wc occr n strctres are of two tpes: Geometrc non-lneart Materal non-lneart Te frst s cased b large dsplacements n te strctre, te second reslts from non-lneart n materals (stress-stran relatonsp. Te geometrc non lneart s of ver lttle mportance n renforced concrete strctres becase te dsplacements n sc strctres are generall small. In ts researc ts tpe of non-lneart s neglected and onl te effects of materal non-lneart are consdered. Te man sorces of materal non-lneart n renforced concrete strctres are: Crackng of concrete Yeldng of renforcement on-lnear stress-stran relaton for concrete Concrete-renforcement nteracton (bondslp, and nterface sear transfer de to aggregate nterlockng and dowel acton. 5

53 4. on-lnear mercal Tecnqes sng Fnte Element Metod Te solton of non-lnear problems b te fnte element metod s sall attempted b one of tree basc tecnqes: - Incremental, or stepwse procedres - Iteratve or ewton metods - Step-teratve or med procedres Te bass of ncremental procedre s te dvson of load nto man small ncrements, wc are not necessarl eqal. Drng te applcaton of eac load ncrement, te stffness matr s assmed to be fed, lnear relaton between load and dsplacement s assmed. Te dsplacement resltng from load ncrements are accmlated to gve te total dsplacement at stage of loadng. Ts process s repeated ntl te total load s reaced. Usng small ncrements mprove te accrac of te metod bt reslts n more comptatonal effort. In te teratve procedre te fll load s appled to te strctre and te nternal stresses are compted. A load sstem eqvalent to te compted stresses s evalated and cecked aganst te appled loadng sstem. Snce an appromate stffness matr s sed, resdal forces reslt de te lack of eqlbrm. Tese resdal forces are ten appled to te strctre to restore eqlbrm and addtonal dsplacements are compted. Te process s repeated ntl te resdals are sffcentl small. Te med procedre tlzes a combnaton of te ncremental and teratve scemes. Te load s appled ncrementall, bt after eac load ncrement sccessve teratons are performed ntl eqlbrm s aceved to an acceptable level of accrac. Ts procedre tends to mnmze te dsadvantages of prevos 5

54 two metods and elds ger accrac bt needs more comptatonal effort. Te non-lnear nmercal metod sed n ts nvestgaton can be dentfed as combned ncremental-constant stffness teratve metod (med procedre. It s evdent from te procedre otlned above tat te global stffness matr s pdated onl at te begnnng of eac load ncrement and mantaned naltered trogot te teratve process. Te oter alternatve s to pdate te stffness matr after eac teraton bt need more comptatonal effort. 4.4 Selecton of Element and Representaton of Crackng Te coce of te tpe of elements to dealze a certan strctre depends not onl on ts load-carrng caracterstcs, bt also n ts geometr, degree of accrac and admssblt condtons. Te prmar featres of strctres to be stded n ts researc are: Composte natre of renforced concrete. Te nteracton between te renforcement and concrete (bond slp. Te sear resstance mecansm, wc develops after crackng (Aggregate nterlock dowel acton. A proper representaton of tese caracterstcs reqres te emploment of tree tpes of elements: - Plane elements to represent concrete. - Lne elements to represent te renforcement. -Lnkage elements to represent te renforcement-concrete nteracton and post crackng sear mecansm. 54

55 4.5 Modelng of Steel Renforcement In developng a fnte element metod of a renforced concrete member, one of te followng tree alternatve representatons of renforcement ma be sed: - Dstrbted - dscrete - embedded In te case of a dstrbted representaton, te steel assmed to be dstrbted over te concrete elements, wt partclar orentaton angle θ. Perfect bond between steel and concrete s assmed so tat a composte concrete-renforcement constttve relaton can be derved. Dstrbted representaton, tog eas to mplement, s ver nrealstc as te renforcng bars are no longer naal members embedded nsde te concrete and bonded to t. 55

56 A dscrete representaton of renforcement, sng onedmensonal elements as been wdel sed. Bot aal and beam elements ma be sed. Aal elements are assme to be pn connected wt two degrees of freedom at nodal ponts wle for te beam elements tree degrees of freedom are assgned at te ends. Te dscrete representaton, n addton to ts smplct, as a sgnfcant advantage of accontng for possble dsplacement of te renforcement wt respect to srrondng concrete. Te man dsadvantage of ts tecnqe s tat t reqres a ver fne mes to enable eac bar to le on a bondar of te brck element. An embedded representaton ma be sed n connecton wt g order soparametrc elements. Te renforcng bar s consdered to act as an aal member blt nto te soparametrc element sc tat ts dsplacements are consstent wt tose of te basc element. Te man advantage of ts tecnqe s n placng te steel n ts eact poston rrespectve of te coce of te mes. 56

57 In ts researc one-dmensonal dscrete elements are emploed to model renforcement. Ts dscrete form of representaton s cosen de to ts smplct and ts ablt to accont adeqatel for dowel acton and bond slp mecansm. Te man renforcement s dealzed b fleral elements (beam elements snce t s capable of resstng bendng and drect sear. Te secondar renforcement (sde bars and web renforcement s dealzed b aal elements (trss elements te -sectonal area of sc renforcement s small and terefore ts fleral rgdt mabe neglected. 57

58 4.6 Modelng of Concrete A strctral member wose tckness s small compared to ts span or dept and s prmarl loaded b n-plane forces can be model b two-dmensonal elements. Te stress state n sc elements s defned n term of tree stress components; σ, σ and σ. Stresses, normal to te plane of te elements, are consdered neglgble. Several tpes of plane stress elements are commonl sed. - Constant tranglar element - Plane soparametrc qadrlateral - Hg order soparametrc element Constant tranglar elements ave enjoed consderable appeal for ter smplct as well as ter ablt to dealze strctre wt rreglar bondares. Te man dsadvantage of tese elements stem from: Ter nform stran feld and ence stress feld; te stress reslts obtaned sall reqre nterpretaton and averagng Mes-dependenc, a strctral response obtaned from certan mes confgraton ma dffer from tat obtaned from oter meses. Isoparametrc qadrlateral element ave enjoed for ncrease te mes sze and redces te nmber of degrees of freedom, wc reslts n sgnfcant redcton of comptatonal effort. Also mproved stress vales can be obtaned snce te strans, and terefore te stresses, var lnearl wtn eac element. 58

59 59 Te man dsadvantage of tese elements s ter poor beavor n representng pre bendng Plane Isoparametrc Qadrlateral: Te qadrlateral consdered ere as egt degrees of freedom; eac node as two degree of freedom, and. Te element as local coordnate sstem s-t, wc s non-dmensonal: s and t var from - to. Fg. 4.4 Plane Stress Lnear Isoparametrc qadrlateral Element Te local and global coordnate sstems are related as follows: Y X Were X,Y Te global coordnates., Te local coordnates. Interpolaton fncton Y X Y X

60 s defned as: 4 ( s ( t 4 ( s ( t 4 ( s ( t 4 ( s ( t 4 General representaton for nterpolaton fncton can be wrtten as follows: ( s * s ( t* t Were (s,t are te coordnates of node Te above nterpolaton fnctons can also be sed to defne dsplacement wtn te element (Local sstem n term of nodal dsplacements (global sstem. U U U U 4 X 4 Y X Y

61 6 Y X U U Formlaton of stran-dsplacement for two-dmensonal deformaton can be wrtten as: U U U U Y X Y X,,,, ( ( ( ( ε ε ε ε ε ε

62 6 Sbseqentl, we wll need to epress te dervatves of fncton n, coordnates n terms of ts dervatves n s,t coordnates. Ts s done as follows: A fncton f(, can be consdered to be an mplct fncton of s and t as ff[(s,t,(s,t]. Usng te can rle of dfferental, we ave ( ( ( ( ( ( ( ( t f t f t f s f s f s f f f t f s f J Were JJacoban matr to transfer varable from - plane to s-t plane. ( ( ( ( t t s s J J t t t t s s s s. 4.7 J J J J J ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( s s s s s s s s t t t t t t t t J t f s f J f f J J J J det...4. * ( * ( det j j j j j...4. jdsdt dd det. 4.

63 A major dsadvantage of lnear soparametrc qadrlateral s ts poor beavor nder pre bendng as sown n fg.4.5. Ts nvolves te addton to dsplacement felds, defned b eq.(4. and eq.(4.4 correctve dsplacement fncton of te form: U X α ( s α ( t U Y α ( s α 4 ( t

64 f 64

65 Complete dsplacement feld of te mproved element s as follows: U X 4 α ( s α ( t U Y 4 α ( s α ( t Were α,α,α,α 4 Te constants wc consttte addtonal degrees of freedom. Te addtonal degrees of freedom ncrease te sze of stffness matr to.snce α, α 4 consttte dsplacement ampltdes relatve to te dsplacements of te corner nodes; te can be condensed b mnmzng te stran energ on te element level. To aceve ts, te statc condensaton metod can be sed b parttonng te stffness matr: P K K U P K K α 65

66 [ P ] K ] U [ ][ α] [ K [ P ] K ] U [ [ ]α] [ K Te stran energ of te element s: T U K K U S. E. 4. α K K α For mnmm stran energ:. E α [ K ] U [ K ]α S α [ K ] [ K ] U B sbstttng eq.(4. nto eq.(4.8 we can obtan te followng epresson: P K ] [ K ][ K ] [ K ] U ([ P [ K] U Were [K]8 8 modfed stffness matr T K ] w w j J[ B( s, t ] [ C][ B( s, t ] j [ Were s, t Integraton ponts. w, w j Wegt factors correspondng to t C Materal stffness matr. B s, t Stran dsplacement matr. ( s,. 66