Finite elements for modeling of localized failure in reinforced concrete

Transcription

1 Fnte elements for modelng of localzed falure n renforced concrete Mha Jukc To cte ths verson: Mha Jukc. Fnte elements for modelng of localzed falure n renforced concrete. Other. École normale supéreure de Cachan - ENS Cachan, Englsh. <NNT : 2013DENS0064>. <tel > HA Id: tel Submtted on 27 May 2014 HA s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. archve ouverte plurdscplnare HA, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 ENSC-n d ordre) THESE DE DOCTORAT DE ECOE NORMAE SUPERIEURE DE CACHAN Présentée par Monseur JUKIC Mha pour obtenr le grade de DOCTEUR DE ECOE NORMAE SUPERIEURE DE CACHAN Domane : MECANIQUE GENIE MECANIQUE GENIE CIVI Sujet de la thèse : Fnte elements for modelng of localzed falure n renforced concrete Thèse présentée et soutenue à jubljana le 13/12/2013 devant le jury composé de : M. PETROVIC Dusan Professeur des Unverstés Présdent M. BICANIC Nenad Professeur des Unverstés Rapporteur M. JEENIC Gordan Professeur des Unverstés Rapporteur M. PANINC Igor Professeur des Unverstés Examnateur M. BRANK Bostjan Professeur des Unverstés Drecteur de thèse M. IBRAHIMBEGOVIC Adnan Professeur des Unverstés Drecteur de thèse MT-Cachan, ENS CACHAN 61, avenue du Présdent Wlson, CACHAN CEDEX France)

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4 Jukć, M Končn element za modelranje lokalzranh poruštev v armranem betonu. Doktorska dsertacja. jubljana, U, FGG. I IZJAVA O AVTORSTVU Podpsan Mha Jukć zjavljam, da sem avtor doktorske dsertacje z naslovom Končn element za modelranje lokalzranh poruštev v armranem betonu. Izjavljam, da je elektronska razlčca dsertacje enaka tskan razlčc, n dovoljujem njeno objavo v dgtalnem repoztorju U FGG. jubljana,

5 II Jukć, M Fnte elements for modelng of localzed falure n renforced concrete. Doctoral thess. Cachan, ENSC, MT. ERRATA Page ne Error Correcton

6 Jukć, M Končn element za modelranje lokalzranh poruštev v armranem betonu. Doktorska dsertacja. jubljana, U, FGG. III BIBIOGRAPHIC-DOCUMENTAISTIC INFORMATION AND ABSTRACT UDC : : ) Author: Mha Jukć Supervsor: prof. Boštjan Brank, Ph.D. Co-supervsor: prof. Adnan Ibrahmbegovć, Ph.D. Ttle: Fnte elements for modelng of localzed falure n renforced concrete Document type: doctoral dssertaton Notes: 200 p., 120 fg., 424 eq. Keywords: falure analyss, fnte element method, renforced concrete, localzed falure, embedded dscontnuty, stress-resultant, mult-layer, Euler-Bernoull beam, Tmoshenko beam Abstract In ths work, several beam fnte element formulatons are proposed for falure analyss of planar renforced concrete beams and frames under monotonc statc loadng. The localzed falure of materal s modeled by the embedded strong dscontnuty concept, whch enhances standard nterpolaton of dsplacement or rotaton) wth a dscontnuous functon, assocated wth an addtonal knematc parameter representng jump n dsplacement or rotaton). The new parameters are local and are condensed on the element level. One stress resultant and two mult-layer beam fnte elements are derved. The stress resultant Euler-Bernoull beam element has embedded dscontnuty n rotaton. Bendng response of the bulk of the element s descrbed by elasto-plastc stress resultant materal model. The cohesve relaton between the moment and the rotatonal jump at the softenng hnge s descrbed by rgd-plastc model. Axal response s elastc. In the mult-layer beam fnte elements, each layer s treated as a bar, made of ether concrete or steel. Regular axal stran n a layer s computed accordng to Euler- Bernoull or Tmoshenko beam theory. Addtonal axal stran s produced by embedded dscontnuty n axal dsplacement, ntroduced ndvdually n each layer. Behavor of concrete bars s descrbed by elasto-damage model, whle elasto-plastcty model s used for steel bars. The cohesve relaton between the stress at the dscontnuty and the axal dsplacement jump s descrbed by rgd-damage softenng model n concrete bars and by rgd-plastc softenng model n steel bars. Shear response n the Tmoshenko element s elastc. The mult-layer Tmoshenko beam fnte element s upgraded by ncludng vscosty n the softenng model. Computer code mplementaton s presented n detal for the derved elements. An operator splt computatonal procedure s presented for each formulaton. The expressons, requred for the local computaton of nelastc nternal varables and for the global computaton of the degrees of freedom, are provded. Performance of the derved elements s llustrated on a set of numercal examples, whch show that the mult-layer Euler-Bernoull beam fnte element s not relable, whle the stress-resultant Euler-Bernoull beam and the mult-layer Tmoshenko beam fnte elements delver satsfyng results.

7 IV Jukć, M Fnte elements for modelng of localzed falure n renforced concrete. Doctoral thess. Cachan, ENSC, MT. BIBIOGRAFSKO-DOKUMENTACIJSKA STRAN IN IZVEČEK UDK : : ) Avtor: Mha Jukć Mentor: prof. dr. Boštjan Brank Somentor: prof. dr. Adnan Ibrahmbegovć Naslov: Končn element za modelranje lokalzranh poruštev v armranem betonu Tp dokumenta: doktorska dsertacja Obseg n oprema: 200 str., 120 sl., 424 en. Ključne besede: porušna analza, metoda končnh elementov, armran beton, lokalzrana poruštev, vgrajena nezveznost, rezultantn model, večslojn model, Euler- Bernoulljev noslec, Tmošenkov noslec Izvleček V dsertacj predlagamo nekaj formulacj končnh elementov za porušno analzo armranobetonskh noslcev n okvrjev pod monotono statčno obtežbo. okalzrano poruštev materala modelramo z metodo vgrajene nezveznost, pr kater standardno nterpolacjo pomkov al zasukov) nadgradmo z nezvezno nterpolacjsko funkcjo n z dodatnm knematčnm parametrom, k predstavlja velkost nezveznost v pomkh al zasukh). Dodatn parametr so lokalnega značaja n jh kondenzramo na nvoju elementa. Izpeljemo en rezultantn n dva večslojna končna elementa za noslec. Rezultantn element za Euler-Bernoulljev noslec ma vgrajeno nezveznost v zasukh. Njegov upogbn odzv opšemo z elasto-plastčnm rezultantnm materalnm modelom. Kohezvn zakon, k povezuje moment v plastčnem členku s skokom v zasuku, opšemo s togo-plastčnm modelom mehčanja. Osn odzv je elastčen. V večslojnh končnh elementh vsak sloj obravnavamo kot betonsko al jekleno palco. Standardno osno deformacjo v palc zračunamo v skladu z Euler-Bernoulljevo al s Tmošenkovo teorjo noslcev. Vgrajena nezveznost v osnem pomku povzroč dodatno osno deformacjo v posamezn palc. Obnašanje betonskega sloja opšemo z modelom elasto-poškodovanost, za sloj armature pa uporabmo elasto-plastčn model. Kohezvn zakon, k povezuje napetost v nezveznost s skokom v osnem pomku, opšemo z modelom mehčanja v poškodovanost za beton n s plastčnm modelom mehčanja za jeklo. Stržn odzv Tmošenkovega noslca je elastčen. Večslojn končn element za Tmošenkov noslec nadgradmo z vskoznm modelom mehčanja. Za vsak končn element predstavmo računsk algortem ter vse potrebne zraze za lokaln zračun neelastčnh notranjh spremenljvk n za globaln zračun prostostnh stopenj. Delovanje končnh elementov prezkusmo na več numerčnh prmerh. Ugotovmo, da večslojn končn element za Euler-Bernoulljev noslec n zanesljv, medtem ko rezultantn končn element za Euler-Bernoulljev noslec n večslojn končn element za Tmošenkov noslec dajeta zadovoljve rezultate.

8 Jukć, M Končn element za modelranje lokalzranh poruštev v armranem betonu. Doktorska dsertacja. jubljana, U, FGG. V INFORMATION BIBIOGRAPHIQUE-DOCUMENTAIRE ET RESUME CDU : : ) Auteur: Drecteur de thèse: Co-drecteur de thèse: Ttre: Type de document: Notes: Mots-clés: Résumé Mha Jukć prof. Boštjan Brank, Ph.D. prof. Adnan Ibrahmbegovć, Ph.D. Éléments fns pour la modélsaton de la rupture localsée dans le béton armé mémore de thèse de doctorat 200 p., 120 fg., 424 éq. rupture, méthode des éléments fns, béton armé, rupture localsée, dscontnuté forte, modèle en effort résultant, multcouche, poutre d Euler Bernoull, poutre de Tmoshenko Dans ce traval, dfférentes formulatons d éléments de poutres sont proposées pour l analyse a rupture de structures de type poutres ou portques en béton armé soumses a des chargements statques monotones. a rupture localsée des matéraux est modélsée par la méthode a dscontnuté forte, qu consste a enrchr l nterpolaton standard des déplacements ou rotatons) avec des fonctons dscontnues assocées a un paramètre cnématque supplémentare nterprété comme un saut de déplacement ou rotaton). Ces paramètres addtonnels sont locaux et condensés au nveau élémentare. Un élément fn écrt en efforts résultants et deux éléments fns multcouches sont développés dans ce traval. élément de poutre d Euler Bernoull écrt en effort résultant présente une dscontnuté en rotaton. a réponse en flexon du matérau hors dscontnuté est décrte par un modèle élastoplastque en effort résultant et la relaton cohésve lant moment et saut de rotaton sur la rotule plastque est, quant a elle, décrte par un modèle rgde plastque. a réponse axale est suppposée élastque. Pour ce qu concerne l approche mult-couche, chaque couche est consdérée comme une barre consttuée de béton ou d acer. a parte régulère de la déformaton de chaque couche est calculée en s appuyant sur la cnématque assocée a la théore d Euler Bernoull ou de Tmoshenko. Une déformaton axale addtonnelle est consdérée par l ntroducton d une dscontnuté du déplacement axal, ntrodute ndépendamment dans chaque couche. e comportement du béton est prs en compte par un modèle élasto-endommageable alors que celu de l acer est décrt par un modèle élastoplastque. a relaton cohésve entre la tracton sur la dscontnuté et le saut de déplacement axal est décrt par un modèle rgde endommageable adoucssant pour les barres couches) en béton et rgde plastque adoucssant pour les barres en acer. a réponse en csallement pour l élement de Tmoshenko est supposée élastque. Enfn, l élément mult-couche de Tmoshenko est enrch en ntrodusant une parte vsqueuse dans la réponse adoucssante. mplantaton numérque des dfférents éléments développés dans ce traval est présentée en détal. a résoluton par une procédure d operator splt est décrte pour chaque type d élément. es dfférentes quanttés nécessares pour le calcul au nveau local des varables nternes des modèles non lnéares ans que pour la constructon du système global fournssant les valeurs des dégrés de lberté sont précsées. es performances des éléments développés sont llustrées a travers des exemples numérques montrant que la formulaton basée sur un élément multcouche d Euler Bernoull n est pas robuste alors les smulatons s appuyant sur des éléments d Euler Bernoull en efforts résultants ou sur des éléments multcouche de Tmoshenko fournssent des résultats très satsfasants.

9 VI Jukć, M Fnte elements for modelng of localzed falure n renforced concrete. Doctoral thess. Cachan, ENSC, MT. TABE OF CONTENTS BIBIOGRAPHIC-DOCUMENTAISTIC INFORMATION AND ABSTRACT III BIBIOGRAFSKO-DOKUMENTACIJSKA STRAN IN IZVEČEK IV INFORMATION BIBIOGRAPHIQUE-DOCUMENTAIRE ET RESUME V 1 INTRODUCTION Motvaton Theoretcal background Goals and outlne of the thess STRESS RESUTANT EUER-BERNOUI BEAM FINITE EEMENT WITH EMBEDDED DISCONTINUITY IN ROTATION Introducton Fnte element formulaton Knematcs Dervaton of operatorg Relatons between global and local quanttes Vrtual work equaton Consttutve models Computatonal procedure Computaton of nternal varables Computaton of nodal degrees of freedom Numercal examples Falure of a cantlever beam Falure of smply supported and clamped beams Four pont bendng test of a smply supported beam Two story renforced concrete frame Concludng remarks

10 Jukć, M Končn element za modelranje lokalzranh poruštev v armranem betonu. Doktorska dsertacja. jubljana, U, FGG. VII 3 MUTI-AYER EUER-BERNOUI BEAM FINITE EEMENT WITH AYER-WISE EMBEDDED DISCONTINUITIES IN AXIA DISPACEMENT Introducton Fnte element formulaton Knematcs Relatons between global and local quanttes Vrtual work equaton Dervaton of operatorsg R and G V Consttutve models Computatonal procedure Computaton of nternal varables Computaton of nodal degrees of freedom Numercal examples One element tenson and compresson tests Cantlever beam under end moment Cantlever beam under end transversal force Two story renforced concrete frame Concludng remarks MUTI-AYER TIMOSHENKO BEAM FINITE EEMENT WITH AYER-WISE EMBEDDED DISCONTINUITIES IN AXIA DISPACEMENT Introducton Fnte element formulaton Knematcs Dervaton of operatorg Relatons between global and local quanttes Vrtual work equaton Consttutve models Computatonal procedure Computaton of nternal varables Computaton of nodal degrees of freedom Numercal examples One element tenson and compresson tests Cantlever beam under end moment

11 VIII Jukć, M Fnte elements for modelng of localzed falure n renforced concrete. Doctoral thess. Cachan, ENSC, MT Cantlever beam under end transversal force Smply supported beam Renforced concrete portal frame Two story renforced concrete frame Concludng remarks VISCOUS REGUARIZATION OF SOFTENING RESPONSE FOR MUTI-AYER TIMOSHENKO BEAM FINITE EEMENT Introducton Vrtual work equaton Computaton of nternal varables Dscontnuty n concrete layer Dscontnuty n renforcement layer Computaton of nodal degrees of freedom Numercal examples One element tenson and compresson tests Tenson and compresson tests on a mesh of several elements Cantlever beam under end moment Concludng remarks CONCUSIONS 166 RAZŠIRJENI POVZETEK 170 BIBIOGRAPHY 195 APPENDICES 200

12 Jukć, M Končn element za modelranje lokalzranh poruštev v armranem betonu. Doktorska dsertacja. jubljana, U, FGG. IX IST OF FIGURES 2.1 Fnte element wth sx nodal degrees of freedom and embedded dscontnuty n rotaton Interpolaton functons for axal dsplacement left) and axal stran rght) Interpolaton functons for transversal dsplacement left) and curvature rght) Interpolaton functon ˆM and ts frst dervatve ˆM left). Heavsde and Drac-delta functons rght) Curvature-free deformaton of the beam when the moment n the hnge drops to zero Degrees of freedom at a node of the fnte element mesh Global left) and local rght) degrees of freedom, assocated wth a fnte element Internal forces, correspondng to degrees of freedom at a node of the fnte element mesh Contrbuton of a fnte element to nternal forces of the structure n global left) and local rght) coordnate system Moment - curvature dagram left). Blnear hardenng law rght). Only postve parts of the dagrams are shown. They are vald for constant value of axal force Moment at the hnge - rotatonal jump dagram Algorthm for phase A) of k-th teraton for fnte elemente) Cantlever beam under dfferent loads Moment - rotaton dagrams for cantlever beam under end moment: all fnte elements are the same left), one element s slghtly weaker rght) Moment - rotaton dagram for cantlever beam under end moment: wth and wthout axal force Moment at support - transversal dsplacement dagram for cantlever beam under end transversal force: all fnte elements are the same Smply supported beam: use of symmetry n computatonal model Clamped beam: use of symmetry n computatonal model Force - dsplacement under the force dagrams for smply supported and clamped beams. Elastoplastc left) and elastc rght) behavor n the hardenng phase Four pont bendng test of smply supported beam: computatonal model Force - dsplacement at the mddle of the beam dagrams for dfferent postons of the force: a = 0.96m left), a = 1.30m mddle), a = 1.60m rght) Two story frame: geometry, loadng pattern and cross-sectons Moment - rotaton dvded by length of FE dagrams for beam and column

13 X Jukć, M Fnte elements for modelng of localzed falure n renforced concrete. Doctoral thess. Cachan, ENSC, MT Response of two story frame and materal state at dfferent stages of analyss left). Comparson wth experment and results of Pham et al. rght) Response of two story frame up to total collapse for dfferent materal data left). Comparson wth results of analyss wth mult-layer fnte element rght) Fnte element wth seven nodal degrees of freedom Interpolaton functons for axal left) and transversal dsplacement rght) Fnte element dvded nto layers, before and after occurrence of dscontnuty n -th layer, wth correspondng axal dsplacement n the layer Interpolaton functons for axal stran due to axal left) and transversal dsplacement rght) Degrees of freedom at nodes of the fnte element mesh Global left) and local rght) degrees of freedom, assocated wth a fnte element Internal forces, correspondng to degrees of freedom at nodes of the fnte element mesh Contrbuton of a fnte element to nternal forces of the structure n global left) and local rght) coordnate system Interpolaton of standard axal dsplacement n -th layer between nodal dsplacements of the fnte element left) and between nodal axal dsplacements of the layer rght) Doman and sub-domans of a cracked layer. Heavsde and Drac-delta functons Constructon of nterpolaton functon M n case of dscontnuty between nodes 1 and 3 left) and n case of dscontnuty between nodes 3 and 2 rght) Constructon of nterpolaton functon M n case of constant stran Operator Ḡ R for nterpolaton of addtonal real stran near stran and blnear stress n a structural element, modeled wth fve fnte elements Operator Ḡ V for nterpolaton of addtonal vrtual stran Stress - stran dagram for bulk of concrete layer Tracton - dsplacement jump dagram for dscontnuty n concrete layer Stress - stran dagram for bulk of renforcement layer Tracton - dsplacement jump dagram for dscontnuty n renforcement layer Algorthm for phase A) of k-th teraton for-th layer of fnte elemente) Seven possble lnear stress states n a layer Beam n pure tenson/compresson: geometry Axal force - dsplacement dagrams for concrete beam n pure tenson left) and pure compresson rght) Axal force - dsplacement dagram for steel beam layer) n pure tenson Axal force - dsplacement dagrams for renforced concrete beam n pure tenson left) and pure compresson rght)

14 Jukć, M Končn element za modelranje lokalzranh poruštev v armranem betonu. Doktorska dsertacja. jubljana, U, FGG. XI 3.26 near stress n -th layer left) and resultng unequal contrbutons of the layer to axal nternal forces of the fnte element at the two nodes rght) Indvdual layers out of balance left) and fnte element n balance rght) Axal force - dsplacement dagrams for renforced concrete beam n pure tenson left) and pure compresson rght) wth mposed locaton of dscontnuty at/ ocatons and szes of dscontnutes n layers of the beam n pure tenson, when transversal dsplacement of the free end of the beamv 2 s non-zero Axal force - dsplacement dagrams for renforced concrete beam n pure tenson left) and pure compresson rght): the case of non-zero transversal dsplacement Transversal dsplacement left) and rotaton rght) at the free end of RC beam n tenson Shear force left) and moment rght) at the support of RC beam n tenson Cantlever beam under end moment: geometry Moment - rotaton dagrams for cantlever beam under end moment: orgnal softenng modul left), softenng modul modfed accordng to length of FE rght) Moment - rotaton dagram for cantlever beam under end moment: weaker renforcement n one of the fnte elements Moment - rotaton dagrams for cantlever beam under end moment: mposed locaton of dscontnuty at/2. Orgnal softenng modul left), modfed softenng modul rght) Moment - rotaton dagram for cantlever beam under end moment: mposed locaton of dscontnuty at/2. Weaker renforcement n one of the fnte elements Cantlever beam under end transversal force: geometry Moment at support - transversal dsplacement dagram for cantlever beam under end transversal force: all fnte elements are the same Two story frame: geometry, loadng pattern and cross-sectons Consttutve dagrams for steel left) and concrete n compresson rght): comparson wth expermental curves Response of two story frame: comparson wth experment Fnte element wth sx nodal degrees of freedom Interpolaton functons for dsplacements left) and stran rght) Fnte element dvded nto layers, before and after occurrence of dscontnuty n -th layer, wth correspondng axal dsplacement n the layer Interpolaton of standard axal dsplacement n -th layer between nodal dsplacements of the fnte element left) and between nodal axal dsplacements of the layer rght) Doman and sub-domans of a cracked layer, Heavsde and Drac-delta functons left). Constructon of nterpolaton functonm rght) Degrees of freedom at a node of the fnte element mesh Global left) and local rght) degrees of freedom, assocated wth a fnte element

15 XII Jukć, M Fnte elements for modelng of localzed falure n renforced concrete. Doctoral thess. Cachan, ENSC, MT. 4.8 Internal forces, correspondng to degrees of freedom at a node of the fnte element mesh Contrbuton of a fnte element to nternal forces of the structure n global left) and local rght) coordnate system Stress - stran dagram for bulk of concrete layer Tracton - dsplacement jump dagram for dscontnuty n concrete layer Stress - stran dagram for bulk of renforcement layer Tracton - dsplacement jump dagram for dscontnuty n renforcement layer Algorthm for phase A) of k-th teraton for-th layer of fnte elemente) Stress n the bulk left) and tracton at the dscontnuty rght) of a concrete layer: value from the prevous step n), and tral and fnal values from the current step ) Beam n pure tenson/compresson: geometry Axal force - dsplacement dagrams for concrete beam n pure tenson left) and pure compresson rght) Axally loaded concrete beam: swtchng from softenng n tenson to compresson left) and back to tenson rght) Axally loaded concrete beam: swtchng from hardenng n compresson to tenson left) and back to compresson mddle). Swtchng from softenng n compresson to tenson rght) Axal force - dsplacement dagram for steel beam layer) n pure tenson Axally loaded steel beam: swtchng from hardenng n tenson to compresson left) and back to tenson mddle). Swtchng from softenng n tenson to compresson rght) Axal force - dsplacement dagrams for renforced concrete beam n pure tenson left) and pure compresson rght) Cantlever beam under end moment: geometry Moment - rotaton dagrams for cantlever beam under end moment: orgnal softenng modul left), softenng modul modfed accordng to length of FE rght) Moment - rotaton dagrams for cantlever beam under end moment: weaker renforcement n one of the fnte elements left), weaker concrete and renforcement n one of the elements rght) Cantlever beam under end transversal force: geometry Moment at support - transversal dsplacement dagrams for cantlever beam under end transversal force: orgnal softenng modul left), modul modfed accordng to length of FE rght) Smply supported beam: use of symmetry n computatonal model Force - dsplacement under the force dagrams for smply supported beam: results for dfferent meshes left), comparson of results for 8 FE wth results of Pham rght) Force - dsplacement under the force dagrams for smply supported beam: dfferent number of concrete layers n 5 FE mesh left), dfferent hardenng modulus of steel n 8 FE mesh rght) Smply supported beam: materal state at dfferent stages of analyss marked wth dots) Smply supported beam: dscontnutes cracks) at dfferent stages of analyss

16 Jukć, M Končn element za modelranje lokalzranh poruštev v armranem betonu. Doktorska dsertacja. jubljana, U, FGG. XIII 4.33 Pnned portal frame: geometry, loadng pattern and renforcement P w dagram: results for dfferent meshes of fnte elements f all elements to the rght of force P are the same left) and f renforcement s weakened n one of them rght) P w dagram: comparson to experment and results of Saje et al Moments at the jont of the beam and the column left) and n the mddle of the span rght): comparson to experment and results of Saje et al Portal frame: materal state at dfferent stages of analyss marked wth dots) Portal frame: dscontnutes cracks) at dfferent stages of analyss marked wth dots) Two story frame: geometry, loadng pattern and cross-sectons Stress - stran dagrams for steel left) and concrete n compresson rght) used by Veccho and Emara, compared to dagrams used n present analyss Response of two story frame: results for dfferent meshes left), comparson of results for 16 FE n a column and 14 FE n a beam wth experment and results of Pham rght) Response of two story frame: loadng and unloadng for a mesh of 16 FE n a column and 14 FE n a beam. Comparson to experment Two story frame: stages of analyss, correspondng to mages n Fgs and Two story frame: materal state at dfferent stages of analyss, marked n Fg Two story frame: dscontnutes at dfferent stages of analyss, marked n Fg Axal force - dsplacement dagrams for concrete beam n pure tenson left) and pure compresson rght) for dfferent values of vscosty parameter Axal force - dsplacement dagram for steel beam layer) n pure tenson for dfferent values of vscosty parameter Axal force - dsplacement dagram for concrete beam n pure compresson for dfferent meshes of fnte elements - wthout vscosty Axal force - dsplacement dagram for concrete beam n pure compresson for dfferent values of vscosty parameter 5 FE mesh) Axal force - dsplacement dagram for steel beam layer) n pure tenson for dfferent meshes of fnte elements - wthout vscosty Axal force - dsplacement dagram for steel beam layer) n pure tenson for dfferent values of vscosty parameter 5 FE mesh) Moment - rotaton dagram for cantlever beam under end moment for dfferent meshes of fnte elements - wthout vscosty Moment - rotaton dagram for cantlever beam under end moment for dfferent values of vscosty parameter 5 FE mesh)

17 XIV Jukć, M Fnte elements for modelng of localzed falure n renforced concrete. Doctoral thess. Cachan, ENSC, MT. KAZAO SIK 2.1 Končn element s šestm prostostnm stopnjam n vgrajeno nezveznostjo v zasuku Interpolacjske funkcje za osn pomk levo) n osno deformacjo desno) Interpolacjske funkcje za prečn pomk levo) n ukrvljenost desno) Interpolacjska funkcja ˆM n njen prv odvod ˆM levo). Heavsde-ova n Drac-delta funkcja desno) Deformrana lega brez ukrvljenost, ko moment v plastčnem členku pade na nč Prostostne stopnje v posameznem vozlšču mreže končnh elementov Globalne levo) n lokalne desno) prostostne stopnje, povezane s končnm elementom Notranje sle, k ustrezajo prostostnm stopnjam v vozlšču mreže končnh elementov Prspevek končnega elementa k notranjm slam konstrukcje v globalnem levo) n lokalnem desno) koordnatnem sstemu Dagram moment - ukrvljenost levo). Blnearno utrjevanje desno). Prkazana sta samo poztvna dela dagramov. Veljata za konstantno osno slo Dagram moment v členku - skok v zasuku Algortem za fazo A) k-te teracje za končn elemente) Konzola pod razlčnm obtežbam Dagram moment - zasuk za konzolo, obteženo z momentom: vs končn element so enak levo), en element je malce šbkejš desno) Dagram moment - zasuk za konzolo, obremenjeno z momentom: ob prsotnost n brez prsotnost osne sle Dagram moment ob podpor - prečn pomk za konzolo, obremenjeno s prečno slo: vs končn element so enak Prostoležeč noslec: uporaba smetrje v računskem modelu Togo podprt noslec: uporaba smetrje v računskem modelu Dagram sla - pomk pod slo za prostoležeč n togo podprt noslec. Elasto-plastčno levo) n elastčno desno) obnašanje v utrjevanju Štrtočkovn upogbn prezkus prostoležečega noslca: računsk model Dagram sla - pomk na sredn noslca za razlčne pozcje sle: a = 0.96m levo), a = 1.30m sredna), a = 1.60m desno) Dvoetažn okvr: geometrja, obtežba n prečn prerez Dagram moment - zasuk, deljen z dolžno KE za prečko n steber

18 Jukć, M Končn element za modelranje lokalzranh poruštev v armranem betonu. Doktorska dsertacja. jubljana, U, FGG. XV 2.24 Odzv dvoetažnega okvrja n stanje materala v posameznh fazah analze levo). Prmerjava z ekspermentom n z rezultat Pham et al. desno) Odzv dvoetažnega okvrja do popolne poruštve za razlčne materalne podatke levo). Prmerjava z rezultat analze z večslojnm končnm elementom desno) Končn element s sedmm prostostnm stopnjam Interpolacjske funkcje za osn levo) n prečn pomk desno) Na sloje razdeljen končn element pred n po nastanku nezveznost v -tem sloju ter prpadajoč osn pomk v sloju Interpolacjske funkcje za osno deformacjo zarad osnega levo) n prečnega pomka desno) Prostostne stopnje v vozlščh mreže končnh elementov Globalne levo) n lokalne desno) prostostne stopnje, povezane s končnm elementom Notranje sle, k ustrezajo prostostnm stopnjam v vozlščh mreže končnh elementov Prspevek končnega elementa k notranjm slam konstrukcje v globalnem levo) n lokalnem desno) koordnatnem sstemu Interpolacja standardnega osnega pomka v -tem sloju med prostostne stopnje končnega elementa levo) n med vozlščne osne pomke sloja desno) Domena n poddomen razpokanega sloja. Heavsde-ova n Drac-delta funkcja Konstruranje nterpolacjske funkcje M v prmeru nezveznost med vozlščema 1 n 3 levo) n v prmeru nezveznost med vozlščema 3 n 2 desno) Konstruranje nterpolacjske funkcje M v prmeru konstantnh deformacj OperatorḠ R za nterpolacjo dodatnh pravh deformacj nearne deformacje n blnearne napetost v konstrukcjskem elementu, modelranem s petm končnm element OperatorḠ V za nterpolacjo dodatnh vrtualnh deformacj Dagram napetost - deformacja za sloj betona Dagram napetost - skok v pomku za nezveznost v sloju betona Dagram napetost - deformacja za sloj armature Dagram napetost - skok v pomku za nezveznost v sloju armature Algortem za fazo A) k-te teracje za -t sloj končnega elementae) Sedem možnh lnearnh razporedov napetost v sloju Noslec v čstem nategu/tlaku: geometrja Dagram osna sla - pomk za betonsk noslec v čstem nategu levo) n čstem tlaku desno) Dagram osna sla - pomk za jeklen noslec sloj) v čstem nategu Dagram osna sla - pomk za armranobetonsk noslec v čstem nategu levo) n čstem tlaku desno)

19 XVI Jukć, M Fnte elements for modelng of localzed falure n renforced concrete. Doctoral thess. Cachan, ENSC, MT nearen potek napetost v -tem sloju levo) n rezultrajoča razlčna prspevka k osnm notranjm slam končnega elementa v obeh vozlščh desno) Neuravnotežen posamezn sloj levo) n končn element v ravnotežju desno) Dagram osna sla - pomk za armranobetonsk noslec v čstem nategu levo) n čstem tlaku desno) ob vsljen nezveznost pr/ okacje n velkost nezveznost po slojh pr noslcu v čstem nategu, ko je prečn pomk prostega konca noslcav 2 razlčen od nč Dagram osna sla - pomk za armranobetonsk noslec v čstem nategu levo) n čstem tlaku desno): prmer, ko je prečn pomk razlčen od nč Prečn pomk levo) n zasuk desno) na prostem koncu AB noslca v nategu Prečna sla levo) n moment desno) ob podpor AB noslca v nategu Konzola, obremenjena z momentom: geometrja Dagram moment - zasuk za konzolo, obteženo z momentom: orgnaln modul mehčanja levo), modul mehčanja prrejen glede na dolžno KE desno) Dagram moment - zasuk za konzolo, obteženo z momentom: malce šbkejša armatura v enem od končnh elementov Dagram moment - zasuk za konzolo, obteženo z momentom: vsljena nezveznost pr/2. Orgnaln modul mehčanja levo), modul mehčanja prrejen glede na dolžno KE desno) Dagram moment - zasuk za konzolo, obteženo z momentom: vsljena nezveznost pr/2. Malce šbkejša armatura v enem od končnh elementov Konzola, obremenjena s prečno slo: geometrja Dagram moment ob podpor - prečn pomk za konzolo, obremenjeno s prečno slo: vs končn element so enak Dvoetažn okvr: geometrja, obtežba n prečn prerez Konsttutvna zakona za jeklo levo) n beton v tlaku desno): prmerjava z ekspermentalnm krvuljam Odzv dvoetažnega okvrja: prmerjava z ekspermentom Končn element s šestm prostostnm stopnjam Interpolacjske funkcje za pomke levo) n deformacje desno) Na sloje razdeljen končn element pred n po nastanku nezveznost v -tem sloju ter prpadajoč osn pomk v sloju Interpolacja standardnega osnega pomka v -tem sloju med prostostne stopnje končnega elementa levo) n med vozlščne osne pomke sloja desno) Domena n poddomen razpokanega sloja, Heavsde-ova n Drac-delta funkcja levo). Konstruranje nterpolacjske funkcjem desno) Prostostne stopnje v posameznem vozlšču mreže končnh elementov Globalne levo) n lokalne desno) prostostne stopnje, povezane s končnm elementom

20 Jukć, M Končn element za modelranje lokalzranh poruštev v armranem betonu. Doktorska dsertacja. jubljana, U, FGG. XVII 4.8 Notranje sle, k ustrezajo prostostnm stopnjam v vozlšču mreže končnh elementov Prspevek končnega elementa k notranjm slam konstrukcje v globalnem levo) n lokalnem desno) koordnatnem sstemu Dagram napetost - deformacja za sloj betona Dagram napetost - skok v pomku za nezveznost v sloju betona Dagram napetost - deformacja za sloj armature Dagram napetost - skok v pomku za nezveznost v sloju armature Algortem za fazo A) k-te teracje za -t sloj končnega elementa e) Napetost v sloju levo) n v nezveznost sloja betona desno): vrednost z prejšnjega koraka n) ter testna n končna vrednost z trenutnega koraka ) Noslec v čstem nategu/tlaku: geometrja Dagram osna sla - pomk za betonsk noslec v čstem nategu levo) n čstem tlaku desno) Osno obremenjen betonsk noslec: prehod z mehčanja v nategu v tlak levo) n nazaj v nateg desno) Osno obremenjen betonsk noslec: prehod z utrjevanja v tlaku v nateg levo) n nazaj v tlak sredna). Prehod z mehčanja v tlaku v nateg desno) Dagram osna sla - pomk za jeklen noslec sloj) v čstem nategu Osno obremenjen jeklen noslec: prehod z utrjevanja v nategu v tlak levo) n nazaj v nateg sredna). Prehod z mehčanja v nategu v tlak desno) Dagram osna sla - pomk za armranobetonsk noslec v čstem nategu levo) n čstem tlaku desno) Konzola, obremenjena z momentom: geometrja Dagram moment - zasuk za konzolo, obteženo z momentom: orgnaln modul mehčanja levo), modul mehčanja prrejen glede na dolžno KE desno) Dagram moment - zasuk za konzolo, obteženo z momentom: malce šbkejša armatura v enem od končnh elementov levo), šbkejša armatura n beton v enem od elementov desno) Konzola, obremenjena s prečno slo: geometrja Dagram moment ob podpor - prečn pomk za konzolo, obremenjeno s prečno slo: orgnaln modul mehčanja levo), modul mehčanja prrejen glede na dolžno KE desno) Prostoležeč noslec: uporaba smetrje v računskem modelu Dagram sla - pomk pod slo za prostoležeč noslec: rezultat za razlčne mreže končnh elementov levo), prmerjava rezultatov za 8 KE s Phamovm rezultat desno) Dagram sla - pomk pod slo za prostoležeč noslec: razlčno števlo slojev betona v mrež s 5 KE levo), razlčen modul utrjevanja jekla v mrež z 8 KE desno) Prostoležeč noslec: stanje materala v posameznh fazah analze označene s pkam) Prostoležeč noslec: nezveznost razpoke) v posameznh fazah analze

21 XVIII Jukć, M Fnte elements for modelng of localzed falure n renforced concrete. Doctoral thess. Cachan, ENSC, MT Vrtljvo podprt portaln okvr: geometrja, obtežba n armatura Dagram P w: rezultat za razlčne mreže končnh elementov, če so vs element desno od sle P enak levo) n če je v enem od njh armatura oslabljena desno) Dagram P w: prmerjava z ekspermentom n z rezultat Saje et al Moment na stku stebra n prečke levo) ter na sredn razpona desno): prmerjava z ekspermentom n z rezultat Saje et al Portaln okvr: stanje materala v posameznh fazah analze označene s pkam) Portaln okvr: nezveznost razpoke) v posameznh fazah analze označene s pkam) Dvoetažn okvr: geometrja, obtežba n prečn prerez Dagrama napetost - deformacja za jeklo levo) n beton v tlaku desno), k sta ju uporabla Veccho n Emara, v prmerjav z dagramoma, uporabljenma v tej analz Odzv dvoetažnega okvrja: rezultat za razlčne mreže končnh elementov levo), prmerjava rezultatov za 16 KE v stebru n 14 KE v prečk s Phamovm rezultat desno) Odzv dvoetažnega okvrja: obremenjevanje n razbremenjevanje za mrežo s 16 KE v stebru n s 14 KE v prečk. Prmerjava z ekspermentom Dvoetažn okvr: faze analze, k ustrezajo stanjem materala na slkah 4.44 n Dvoetažn okvr: stanje materala v fazah analze, označenh na slk Dvoetažn okvr: nezveznost v fazah analze, označenh na slk Dagram osna sla - pomk za betonsk noslec v čstem nategu levo) n čstem tlaku desno) za razlčne vrednost vskoznega parametra Dagram osna sla - pomk za jeklen noslec sloj) v čstem nategu za razlčne vrednost vskoznega parametra Dagram osna sla - pomk za betonsk noslec v čstem tlaku za razlčne mreže končnh elementov - brez vskoznost Dagram osna sla - pomk za betonsk noslec v čstem tlaku za razlčne vrednost vskoznega parametra mreža s 5 KE) Dagram osna sla - pomk za jeklen noslec sloj) v čstem nategu za razlčne mreže končnh elementov - brez vskoznost Dagram osna sla - pomk za jeklen noslec v čstem nategu za razlčne vrednost vskoznega parametra mreža s 5 KE) Dagram moment - zasuk za konzolo, obteženo z momentom, za razlčne mreže končnh elementov - brez vskoznost Dagram moment - zasuk za konzolo, obteženo z momentom, za razlčne vrednost vskoznega parametra mreža s 5 KE)

22 Jukć, M Končn element za modelranje lokalzranh poruštev v armranem betonu. Doktorska dsertacja. jubljana, U, FGG. 1 1 INTRODUCTION In the ntroductory chapter, the motvaton for research on numercal modelng of localzed falure of materal, wth emphass on renforced concrete, s presented. Prevous achevements n ths feld of research are brefly revewed, and the goals and the outlne of the thess are explaned. 1.1 Motvaton ocalzed falure s a common phenomenon n varety of materals, used n cvl engneerng. At a certan load level, materals often exhbt hghly localzed deformatons before falng. Typcal examples are cracks n brttle materals, such as concrete, stone, brck or ceramc, and shear bands n metals or sols, see [1] and references theren. Growth of localzed deformatons s accompaned by reducton of stress, a process called softenng of materal. Adequate descrpton of ths phenomenon s essental for a comprehensve materal model, whch allows for a more accurate numercal modelng of structures and structural elements, made of such materal. In ths work, we focus on renforced concrete beams and frames, whch are one of the most wdespread structural forms. It has been observed n expermental tests, as well as on actual buldngs, damaged n earthquakes, that most of materal damage s concentrated at several crtcal locatons n the structure. ocalzed falure of renforced concrete comprses crackng and crushng of concrete, yeldng of renforcement and bond slp between the two components. Ths leads to the concept of plastc hnge n the lmt load and push-over analyses, see e.g. [2 4]. In the classcal lmt load analyss, the lmt capacty of each plastc hnge s kept constant, whle addtonal hnges develop wth the ncreasng load. Ths approach restrans the accuracy, wth whch the lmt load of the structure s determned, and prevents the computaton of structure s ductlty and post-peak response. In hghly statcally undetermned structures, falure of a crtcal element does not jeopardze ther ntegrty. It s therefore essental for an accurate analyss to be able to descrbe the softenng response of the crtcal element, assocated wth the localzed falure. Ths leads to the concept of softenng plastc hnge, whch allows for computaton of ductlty and post-peak response of the analyzed structure. There are many dfferent approaches to modelng of softenng hnges n numercal analyss, see e.g. [5, 6]. In earthquake engneerng, researchers often deal wth large scale models of complex structures under rather complcated loads. Effectve analyss of such problems can only be performed by usng relatvely smple fnte elements, e.g. fnte element wth lumped plastcty, see [7, 8], where all plastc deformatons are concentrated n the nodes, whle the rest of the fnte element stays elastc. Plastc hardenng and softenng of the element are descrbed by the moment-rotaton relatonshp of the nodes. Another way to model a softenng hnge s to use a short crackband fnte element, n whch localzaton s smeared over the whole element, see [9 11]. Snce the softenng s descrbed on stran level, a fxed length of the crack-band element has to be computed, whch s then consdered a materal property. In contrast to these two typcal approaches, we decde to use lately establshed strong dscontnuty concept, man characterstc of whch s ncorporaton of dscontnuous dsplacement felds nto standard dsplacement based fnte elements. The am s to develop precse, effectve and robust fnte elements, capable of accurate descrpton of localzed falure n renforced concrete beams and frames.