ATENA Program Documentation Part 1

Transcription

1 Červenka Consulng s.r.o. Na Hrebenkach Prague Czech Republc Phone: E-mal: cervenka@cervenka.cz Web: hp:// ATENA Program Documenaon Par Theory Wren by Vladmír Červenka, Lbor Jendele, and Jan Červenka Prague, January 6, 08

2 Ackowledgemens: The sofware was developed wh paral suppor of Eurosars fundng program. The followng models were developed wh he fnancal suppor of TA ČR: CC3DNonLnCemenousHPRFC, CC3DNonLnCemenousFRC, CC3DNonLnCemenousSHCC, CCModelGeneral, CCTransporMaeral Durably Anslyss module has been developed durng he projec TA , Sofware for predcon and modellng of durably and safey of ransporaon srucures funded by TA ČR Trademarks: ATENA s regsered rademark of Vladmr Cervenka. Oher names may be rademarks of her respecve owners. Copyrgh Červenka Consulng s.r.o.

3 Conens CONTINUUM GOVERNING EQUATIONS. Inroducon. General Problem Formulaon.3 Sress Tensors 4.3. Cauchy Sress Tensor 4.3. nd Pola-Krchhoff Sress Tensor 4.4 Sran Tensors 5.4. Engneerng Sran 5.4. Green-Lagrange Sran 5.5 Consuve Tensor 6.6 The Prncple of Vrual Dsplacemens 6.7 The Work Done by he Exernal Forces 8.8 Problem Dscresaon Usng Fne Elemen Mehod 9.9 Sress and Sran Smoohng 0.9. Exrapolaon of Sress and Sran o Elemen Nodes.0 Smple, Complex Suppors and Maser-Slave Boundary Condons.. References 3 CONSTITUTIVE MODELS 5. Consuve Model SBETA (CCSbeaMaeral) 5.. Basc Assumpons 5.. Sress-Sran Relaons for Concree 8..3 Localzaon Lmers 4..4 Fracure Process, Crack Wdh 5..5 Baxal Sress Falure Creron of Concree 5..6 Two Models of Smeared Cracks 7..7 Shear Sress and Sffness n Cracked Concree 9..8 Compressve Srengh of Cracked Concree 9..9 Tenson Sffenng n Cracked Concree Summary of Sresses n SBETA Consuve Model 30 ATENA Theory

4 .. Maeral Sffness Marces 3.. Analyss of Sresses 3..3 Parameers of Consuve Model 33. Fracure Plasc Consuve Model (CC3DCemenous, CC3DNonLnCemenous, CC3DNonLnCemenous, CC3DNonLnCemenousUser, CC3DNonLnCemenousVarable, CC3DNonLnCemenousSHCC, CC3DNonLnCemenous3) 34.. Inroducon 34.. Maeral Model Formulaon Rankne-Fracurng Model for Concree Crackng Plascy Model for Concree Crushng Combnaon of Plascy and Fracure model 4..6 Varans of he Fracure Plasc Model Tenson Sffenng Crack Spacng Fxed or Roaed Cracks Fague 48.. Sran Hardenng Cemenous Compose (SHCC, HPFRCC) maeral 5.. Confnemen-Sensve Consuve Model 54.3 Von Mses Plascy Model 58.4 Drucker-Prager Plascy Model 6.5 User Maeral Model 6.6 Inerface Maeral Model 6.7 Renforcemen Sress-Sran Laws Inroducon Blnear Law Mul-lne Law No Compresson Renforcemen Cyclc Renforcemen Model 68.8 Renforcemen Bond Models CEB-FIP 990 Model Code Bond Model by Bgaj Memory Bond Maeral 73

5 .9 Mcroplane Maeral Model (CCMcroplane4) Equvalen Localzaon Elemen 74.0 References 78 3 FINITE ELEMENTS Inroducon Truss D and 3D Elemen Plane Quadrlaeral Elemens Plane Trangular Elemens D Sold Elemens Sprng Elemen Quadrlaeral Elemen Q Elemen Sffness Marx Evaluaon of Sresses and Ressng Forces Exernal Cable Renforcemen Bars wh Prescrbed Bond Inerface Elemen 3. Truss Ax-Symmerc Elemens Ahmad Shell Elemen Coordnae Sysems Geomery Approxmaon Dsplacemen Feld Approxmaon Sran and Sresses Defnon Serendpy, Lagrangan and Heeross Varan of Degeneraed Shell Elemen Smeared Renforcemen Transformaon of he Orgnal DOFs o Dsplacemens a he Top and Boom of he Elemen Nodal Coordnae Sysem Shell Ahmad Elemens Implemened n ATENA Curvlnear Nonlnear D Isoparamerc Layered Shell Quadrlaeral Elemens Geomery and dsplacemens Connecon of he shelld o an amben sold elemen 49 ATENA Theory

6 3.3.3 Green-Lagrange srans Curvlnear Nonlnear D Isoparamerc Layered Shell Trangular Elemens Curvlnear Nonlnear 3D Isoparamerc Layered Shell Hexahedral Elemens Geomery and dsplacemens Green-Lagrange srans Curvlnear Nonlnear 3D Isoparamerc Layered Shell Wedge Elemens Curvlnear Nonlnear 3D Beam Elemen Geomery and Dsplacemens and Roaons Felds Sran and Sress Defnon Marces Used n he Beam Elemen Formulaon The Elemen Inegraon Curvlnear Nonlnear 3D Isoparamerc Beam Elemen Curvlnear Nonlnear D elemen Connecon of he beamd o an amben sold elemen Inegraed forces and momens for shells Inegraed forces and momens for beams Global and Local Coordnae Sysems for Elemen Load References 89 4 SOLUTION OF NONLINEAR EQUATIONS 9 4. Lnear Solvers Drec Solver Drec Sparse Solver Ierave Solver Parallel Drec Sparse Solver PARDISO Full Newon-Raphson Mehod Modfed Newon-Raphson Mehod Arc-Lengh Mehod Normal Updae Mehod Conssenly Lnearzed Mehod Explc Orhogonal Mehod 06 v

7 4.4.4 The Crsfeld Mehod Arc Lengh Sep Lne Search Mehod Parameer Band Wdh Opmzaon 4.8 References 4 5 CREEP AND SHRINKAGE ANALYSIS 7 5. Implemenaon of Creep and Shrnkage Analyss n ATENA Basc Theorecal Assumpons 7 5. Approxmaon of Complance Funcons (, ') by Drchle Seres Sep by Sep Mehod Inegraon and Reardaon Tmes 5.5 Creep and Shrnkage Consuve Model References 3 6 DURABILITY ANALYSIS Carbonaon Example of Carbonaon Chlordes Dffuson coeffcen for chlordes MODELS for PROPAGATION PHASE Carbonaon durng propagaon phase Chlorde ngress durng propagaon phase Crackng of concree cover Spallng of concree cover Alkal-Aggregae Reacon Inroducon of alkal-aggregaea model for concree Model for ASR knecs Predcon of ASR swellng cal Influence of mosure F M 50 ATENA Theory v

8 6.5.5 ASR for 3D condons Valdaon on free expanson Implemenaon n Aena Commens References 58 7 TRANSPORT ANALYSIS 6 7. Numercal Soluon of he Transpor Problem Spaal Dscresaon Numercal Soluon of he Transpor Problem Temporal Dscresaon parameer Crank Ncholson Scheme Adams-Bashforh Inegraon Scheme Reducon of Oscllaons and Convergence Improvemen Maeral Consuve Model Fre Elemen Boundary Load Hydrocarbon Fre Fre Exposed Boundary Implemenaon of Fre Exposed Boundary n ATENA Mosure-Hea Elemen Boundary Load References 9 8 DYNAMIC ANALYSIS Srucural Dampng Specral analyss 99 9 EIGENVALUES AND EIGENVECTORS ANALYSIS Inverse Subspace Ieraon Raylegh-Rz Mehod Jacob Mehod Inverse Ieraon Mehod Algorhm of Inverse Subspace Ieraon Surm Sequence Propery Check References GENERAL FORM OF DIRICHLET BOUNDARY CONDITIONS 309 v

9 0. Theory Behnd he Implemenaon Sngle CBC Mulple CBCs 3 0. Applcaon of Complex Boundary Condons Fne Elemen Mesh Refnemen Mesh Generaon Usng Sub-Regons Dscree Renforcemen Embedded n Sold Elemens Curvlnear Nonlnear Beam and Shell Elemens References 39 INDEX 3 ATENA Theory v

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11 CONTINUUM GOVERNING EQUATIONS. Inroducon Ths chaper presens he general governng connuum equaons for non-lnear analyss. In general, here exs many varans of non-lnear analyss dependng on how many non-lnear effecs are accouned for. Hence, hs chaper frs nroduces some basc erms and enes commonly used for srucural non-lnear analyses, and hen concenraes on formulaons ha are mplemened n ATENA. I s mporan o realze ha he whole srucure does no have o be analyzed usng a full nonlnear formulaon. However, a smplfed (or even lnear) formulaon can be used n many cases. I s a maer of engneerng knowledge and pracce o assess, wheher he naccuraces due o a smplfed formulaon are accepable, or no. The smples formulaon,.e. lnear formulaon, s characerzed by he followng assumpons: The consuve equaon s lnear,.e. he generalzed form of Hook's law s used. The geomerc equaon s lnear ha s, he quadrac erms are negleced. I means ha durng analyss we neglec change of shape and poson of he srucure. Boh loadng and boundary condons are conservave,.e. hey are consan hroughou he whole analyss rrespecve of he srucural deformaon, me ec. Generally lnear consuve equaons can be employed for a maeral, whch s far from s falure pon, usually up o 50% of s maxmum srengh. Of course, hs depends on he ype of maeral, e.g. rubber needs o be consdered as a non-lnear maeral earler. Bu for usual cvl engneerng maerals he prevous assumpon s sasfacory. Geomerc equaons can be consdered lnear, f deflecons of a srucure are much smaller han s dmensons. Ths mus be sasfed no only for he whole srucure bu also for s pars. Then he geomerc equaons for he loaded srucure can hen be wren usng he orgnal (unloaded) geomery. I s also mporan o realze ha a lnear soluon s permssble only n he case of small srans. Ths s closely relaed o maeral propery because f srans are hgh, he sresses are usually, alhough no necessarly, hgh as well. Despe he fac ha for he vas majory of srucures lnear smplfcaons are que accepable, here are srucures when s necessary o ake n accoun some non-lnear behavor. The resulng governng equaons are hen much more complcaed, and normally hey do no have a closed form soluon. Consequenly some non-lnear erave soluon scheme mus be used (see Chaper Soluon of Nonlnear Equaons furher n hs documen). Non-lnear analyss can be classfed accordng o a ype of non-lnear behavor: Non-lnear maeral behavor only needs o be accouned for. Ths s he mos common case for ordnary renforced concree srucures. Because of servceably lmaons, deformaons are relavely small. However, he very low ensle srengh of concree needs o be accouned for. Deformaons (eher dsplacemens only or boh dsplacemens and roaons) are large enough such ha he equlbrum equaons mus use he deformed shape of he srucure. However he relave deformaons (srans) are sll small. The complee form of he geomerc ATENA Theory

12 equaons, ncludng quadrac erms, has o be employed bu consuve equaons are lnear. Ths group of non-lnear analyss ncludes mos sably problems. The las group uses non-lnear boh maeral and geomerc equaons. In addon, s usually no possble o suddenly apply he oal value of load bu s necessary o negrae n me ncremens (or loadng ncremens). Ths s he mos accurae and general approach bu unforunaely s also he mos complcaed. There are wo basc possbles for formulang he general srucural behavor based on s deformed shape: Lagrange formulaon: In hs case we are neresed n he behavor of nfnesmal parcles of volume dv. Ther volume wll vary dependen on a loadng level appled and, consequenly, on he amoun of curren deformaons. Ths mehod s usually used o calculae cvl engneerng srucures. Euler formulaon: The essenal dea of Euler's formulaon s o sudy he "flow" of he srucural maeral hrough nfnesmal and fxed volumes of he srucure. Ths s he favore formulaon for flud analyss, analyss of gas flow, rbulaon ec. where large maeral flows exs. For srucural analyss, however, Lagrangan formulaon s beer, and herefore aenon wll be resrced o hs. Two forms of he Lagrangan formulaon are possble. The governng equaons can eher be wren wh respec o he undeformed orgnal confguraon a me = 0 or wh respec o he mos recen deformed confguraon a me. The former case s called Toal Lagrangan formulaon (TL) whle he laer one s called he Updaed Lagrangan formulaon (UL). I s dffcul o say whch formulaon s beer because boh have her advanages and drawbacks. Usually depends on a parcular srucure beng analyzed and whch one o use s a maer of engneerng judgemen. Generally, provded he consuve equaons are adequae, he resuls for boh mehods are dencal. ATENA currenly uses Updaed Lagrangan formulaon, (whch s descrbed laer n hs chaper) and suppors he hghes,.e. 3 rd level of non-lnear behavor. Soon, should also suppor Toal Lagrangan formulaon.. General Problem Formulaon A general analyss of a srucure usually consss of applcaon of many small load ncremens. A each of hose ncremens an erave soluon procedure has o be execued o oban a srucural response a he end of he ncremen. Hence, denong sar and end of he load ncremen by and, a each sep, we know srucural sae a me (from he prevous seps) and solve for he sae a me. Ths procedure s repeaed as many mes as necessary o reach he fnal (oal) level of loadng. Ths process s depced n Fg. -. A me 0 he volume of srucure s 0 V, he surface area s 0 S and any arbrary pon M has coordnaes 0 X 0 0, X, X 3. Smlarly a he me he same srucure has a volume V, surface area S and coordnaes of pon M are X, X, X 3. Smlar defnon apples for me by replacng ndex by.

13 [ 0 X 3, X 3, + X 3 ] M 0 M M Confguraon + Confguraon 0 Confguraon [ 0 X, X, + X ] [ 0 X, X, + X ] Fg. - The movemen of body of srucure n Caresan coordnae sysem. For he dervaons of nonlnear equaons s mporan o use clear and smple noaon. The same sysem of noaon wll be used hroughou hs documen: Dsplacemens u are defned n a smlar manner o ha adoped for coordnaes, hence u s he -h elemen of he dsplacemen vecor a me, u X X s -h elemen of vecor of dsplacemen ncremens a me, The lef superscrp denoes he me correspondng o he value of he eny, he lef subscrp denoes he confguraon wh respec o whch he value s measured and subscrps on he rgh denfy he relaonshps o he coordnae axs. Thus for example 0 j denoes elemen, j of sress ensor a me wh respec o he orgnal (undeformed) confguraon. For dervaves he abbrevaed noaon wll be used,.e. all rgh subscrps ha appear afer a comma declare dervaves. For example: 0 u, j u (.) X The general governng equaons can be derved n he form of a se of paral dfferenal equaons (for example usng he dsplacemen mehod) or an energy approach can be used. The fnal resuls are he same. One of he mos general mehods of esablshng he governng equaons s o apply he prncple of vrual work. There are hree basc varans of hs: The prncple of vrual dsplacemens, The prncple of vrual forces, The Clapeyron dvergen heorem. Usng he vrual work heorems s possble o derve several dfferen varaon prncples (Lagrange prncple, Clapeyron prncple, Hellnger-Ressner prncple, Hu-Washzu prncple ec.). There are popular especally n lnear analyss. They can be used o esablsh equlbrum equaons, o sudy possble deformaon modes n fne elemen dscrezaon ec. Unforunaely n nonlnear analyss hey do no always work. ATENA Theory 3 j

14 In hs documen all he followng dervaons wll be presened n her dsplacemen form and consequenly he prncple of vrual dsplacemens wll be used hroughou. The followng secon deals wh he defnon of he sress and sran ensors, whch are usually used, n nonlnear analyss. All of hem are symmerc..3 Sress Tensors.3. Cauchy Sress Tensor Ths ensor s well known from lnear mechancs. I express he forces ha ac on nfnesmal small areas of he deformed body a me. Somemes hese are also called a "engneerng" sress. The Cauchy sress ensor s he man eny for checkng ulmae sress values n maerals. In he followng ex wll be denoed by. I s energecally conjugaed wh Engneerng sran ensor descrbed laer..3. nd Pola-Krchhoff Sress Tensor The nd Pola-Krchhoff ensor s a fcous eny, havng no physcal represenaon of as n he case of he Cauchy ensor. I expresses he forces, whch ac on nfnesmal areas of body n he undeformed confguraon. Hence relaes forces o he shape of he srucure whch no longer exss. The mahemacal defnon s gven by: Sj X, m mn X j, n (.3) where 0 s he rao of densy of he maeral a me 0 and, mn s he Cauchy sress ensor a me, 0 X s he dervave of coordnaes, ref. (.5)., m Usng nverse ransformaon, we can express Cauchy sress ensor n erms of he nd Pola- Krchhoff sress ensor,.e.: mn 0 0 Xm, 0Smn 0 Xn, j (.4) The elemens 0 X, where: m are usually colleced n he so-called Deformaon graden marx: X T 0 0 T X (.5) T X X X3 4,, X X, X, X T 3 T

15 The rao 0 can be compued usng: 0 de( ) (.6) Expresson (.6) s based on he assumpon ha he wegh of an nfnesmal parcle s consan durng he loadng process. Some mporan properes can be deduced from defnon of nd Pola-Krchhoff ensor (.3): a me 0,.e. he undeformed confguraon, here s no dsncon beween nd Pola-Krchhoff 0 and Cauchy sress ensors because 0 0 X E,.e. uny marx and he densy rao =., nd Pola-Krchhoff ensor s an objecve eny n he sense ha s ndependen of any movemen of he body provded he loadng condons are frozen. Ths s a very mporan propery. The Cauchy sress ensor does no sasfy hs because s sensve o he roaon of he body. I s energecally conjugaed wh Green-Lagrange ensor descrbed laer. They re some oher sress ensor commonly used for srucural nonlnear analyss, e.g. Jaumann sress rae ensor (descrbes sress rae raher han s fnal values) ec, however hey are no used n ATENA and herefore no descrbed n hs documen. 0 X.4 Sran Tensors.4. Engneerng Sran I s he mos commonly used sran ensor, comprsng srans ha are called Engneerng srans. Is man mporance s ha s used n lnear mechancs as a counerpar o he Cauchy sress ensor. e um X mn n.4. Green-Lagrange Sran Ths s energy conjugae of he nd Pola-Krchhoff ensor and s properes are smlar (.e. objecve ec.). I s defned as: ATENA Theory 5 un X m 0 j 0, j 0 j, 0 k, 0 k, j (.7) u u u u (.8) If we calculae he lengh of an nfnesmal fbber pror and afer deformaon n he orgnal coordnaes, we ge exacly he erms of he Green-Lagrange ensor. The followng equaon gves relaon beween varaon of Green-Lagrange and Engneerng sran ensors: Xm Xn 0j 0 0 emn (.9) X X These are he sran ensors used n ATENA. From he oher sran ensors commonly used n non-lnear analyss we can menon Almans sran ensor, co-roaed logarhmc sran, sran rae ensor ec. j

16 .5 Consuve Tensor Alhough he whole chaper laer n hs documen s dedcaed o he problem of consuve equaons and o maeral falure crera, assume for he momen ha sress-sran relaon can be wren n he followng form: where 0 Cjrs s he consuve ensor. S C (.0) 0 j 0 jrs 0 rs Ths form s applcable for lnear maerals or n s ncremenal form can be used also for nonlnear maerals. The followng mporan relaons apply for ransformaon from coordnaes o me 0 o coordnaes a me : or n he oher drecon (.) Cmnpq 0 0 xm, 0 xn, j 0Cjrs 0 xpr, 0 xqs, (.) Cjrs x, m xj, n Cmnpq x r, p xs, q Usng consuve ensor (.) and Almans srans, we can wre for Cauchy sresses (wh respec o coordnaes a me ): C (.3) j jrs rs Almans srans are defned (relaed o Green-Lagrange srans 0 or can be calculaed drecly: 0 0 mn, m j, n 0 j j by x x (.4) u u u u (.5) j, j j, k, k, j The equaon (.3) s equvalen o he equaon (.0) ha was wren for orgnal confguraon of he srucure. I s very mporan o know, wh respec o whch coordnae sysem he sress, sran and consuve ensors are defned, as he acual value can sgnfcanly dffer. ATENA currenly assumes ha all hese ensors are defned a coordnaes a me..6 The Prncple of Vrual Dsplacemens Ths secon presens how he prncple of vrual dsplacemen can be appled o he analyss of a srucure. For compleeness boh he Lagrangan Toal and Updaed formulaons wll be dscussed. In all dervaons s assumed ha he response of he srucure up o me s known. Now, a me we apply load ncremen and usng he prncple of vrual dsplacemen wll solve for sae of he srucure a. Vrual work of he srucure yelds followng. For Toal formulaon: for Updaed formulaon: d d d 0 Sj 0 j dv 0 R (.6) 0 V 6

17 d d d Sj j dv R (.7) V where 0 V, V denoes he srucure volume correspondng o me 0 and and d R s he oal vrual work of he exernal forces. The symbol denoes varaon of he eny. Snce energy mus be nvaran wh respec o he reference coordnae sysem, (.6) and (.7) mus lead o dencal resuls. Subsung expressons for sran and sress ensors he fnal governng equaon for srucure can be derved. They are summarzed n (.8) hrough (.9). Noe ha he relaonshps are expressed wh respec o confguraons a an arbrary me and an eraon (). Typcally, he me may by 0, n whch case we have Toal Lagrangan formulaon or ( ), n whch case we have Updaed Lagrangan formulaon, where some erms can be omed. ATENA also suppor sem Updaed Lagrangan formulaon, when conforms o me a he begnnng of me ncremen,.e. he begnnng of load sep. The followng able compares he abovemenoned formulaons: Table.6- Comparson of dfferen Lagrangan formulaon. Lagrangan formulaon Transform each eraon IP sae varables Maeral properes Transform each load ncremen IP sae varables Maeral properes Transform sress and sran for oupu Calculae u ( ), jfor e j Toal No No No No Yes Yes Updaed Yes Yes Yes Yes No No Sem - Updaed No No Yes Yes No Yes Governng equaons: () () S j j dv R (.8) V where nd Pola-Krchhoff sress and Green Lagrange sran ensor are: (.9) () () () () Sj x, m mn x j, n () () () () () j u, j u, j uk, uk, j (.0) The sress and sran ncremens: S S S (.) () ( ) () j j j () ( ) () j j j e () () () j j j where lnear par of he sran ncremen s calculaed by: (.) ATENA Theory 7

18 () () () ( ) () ( ) () ej u, j u, j uk, uk, j uk, j uk, (.3) and nonlnear par by: () () () j uk, uk, j (.4) Usng consuve equaons n form: () where jrs (.8) can be derved: S C (.5) () () j jrs rs C s angen maeral ensors and nong ha () () j j, an ncremenal form of () () () () () () () Cjrs ej j ej j dv Sj ej j dv R (.6) V Afer lnearsaon,.e. neglecng nd order erms n (.6): V () () () () () () Cjrs ej j ej j dv Cjrs e j ej dv (.7) V we arrve o he fnal form of he governng equaons: V () () ( ) () jrs rs j j j V V C e e dv S dv ( ) ( ) j j V () Noe ha he erm ej ej (.8). R S e dv s consan,.e. ndependen of.7 The Work Done by he Exernal Forces u () (.8), hence s on RHS of So far only he ncremenal vrual nernal work has been consdered. Ths work has o be balanced by he work done by he exernal forces. I s calculaed as follows: ( ) u (.9) R fb u dv fs u ds dv V S V where fb and fs are body and surface forces, S and V denoes negraon wh respec o he surface wh he prescrbed boundary forces and volume of he srucure (a me and ). The s negral n (.9) accouns for exernal work on surface (e.g. exernal forces), he second one for work done by body forces (e.g. wegh) and he las one accouns for work done by nera forces, whch are applcable only for dynamc analyss problems). A hs pon, all he relaonshps for ncremenal analyss have been presened. In order o proceed furher, he problem mus be dscrezed and solved by eraons (descrbed n Chaper Soluon of Nonlnear Equaons). 8

19 .8 Problem Dscresaon Usng Fne Elemen Mehod Spaal dscresaon consss of dscresng prmary varable, (.e. deformaon n case of ATENA) over doman of he srucure. I s done n ATENA by Fne Elemen Mehod. The doman s decomposed no many fne elemens and a each of hese elemens he deformaon feld s approxmaed by where j s ndex for fne elemen node, j... n, n s number of elemen nodes, j u h u (.30) j j h are nerpolaon funcon usually grouped n marx H h ( rs,, ), h( rs,, )... h( rs,, ) rs,, are local elemen coordnaes. j, The nerpolaon funcons h j are usually creaed n he way ha h j a node j and h j 0 a any oher elemen nodes. Combnng (.30) and equaon for sran defnon (.8) can be derved: () ( ) ( ) () L0 L NL U B B B (.3) where () s vecor of Green-Lagrange srans, () U s vecor of dsplacemens, ( ) ( ) BL0, BL, BNL are lnear sran-dsplacemens ransformaon marces (he s wo of hem) and nonlnear sran-dsplacemens ransformaon marx (he las one). Smlar equaon can be wren also for sress ensor. S C (.3) () () () where: S () s vecor of nd Pola-Krchhoff sress ensor and () s ncremenal sress-sran maeral properes marx. C Applyng he above dscresaon for each fne elemen of he srucure and assemblng he resuls he connuum based governng equaons n (.8) can be re-wren n he followng form: () ( ) () ( ) ( ) U M KL K NL U R F (.33) where K s he lnear sran ncremenal sffness marx, L n K ( ) NL s he nonlnear sran ncremenal sffness marx, M s he srucural mass marx, () U s he vecor of nodal pon dsplacemens ncremens a me + U () s he vecor of nodal acceleraons,, eraon ; ATENA Theory 9

20 R, F ( ) s he vecor of appled exernal forces and nernal forces, () ( ), superscrps ndcae eraon numbers. Noe ha (.33) conans also neral erm needed only for dynamc analyss. Fne elemen marces n (.33) and correspondng analycal expressons are summarzed: U C dv U C e e dv () T () () () KL L L jrs rs B B j V V U dv U S dv ( ) ( ) ( ) T ( ) ( ) ( ) ( ) ( ) KNL NL j NL j B S B j V V F S dv S e dv ( ) ( ) ( ) ( ) j j j V V (.34) R H f dv da H f dv R T A T B A V u u U dv U dv () () () T () M H H V V.9 Sress and Sran Smoohng All dervaons and soluon procedures n ATENA sofware are based on deformaonal form of fne elemen mehod. Any srucure s solved usng weak (or negral) form of equlbrum equaons. The whole srucure s dvded no many fne elemens and dsplacemen u a each parcular elemen (a any locaon) s approxmaed by approxmaon funcons h and elemen dsplacemens u as follows: u hu, ( s ndex of an elemen node). I s mporan o noe ha n order no o loose any nernal energy of he srucure, he dsplacemens over he whole srucure mus be connuous. The connuy whn fne elemens s rval. Use of connuous approxmaon funcons h j ensures hs requremen. A b more complcaed suaon s on boundares beween adjacen elemens, however, f he adjacen elemens are of he same ype, her dsplacemens are also connuous. Noe ha here exs are some echnques ha allevae he connuy requremen bu n ATENA hey are no used. Unlke dsplacemens, sress and sran feld s ypcally dsconnuous. Moreover, a srucure s nvesgaed whn so-called maeral (or negral) pons, whch are pons locaed somewhere whn each elemen. Ther poson s derved from requremen o mnmze he approxmaon error. In oher words, sandard fne elemen mehod provdes sress and sran values only a hose maeral pons and hese values mus be laer somehow exrapolaed no elemen nodal pons. Ofen, some sor of smoohng s requred n order o remove he menoned sress and sran dsconnuy. Ths secon descrbes, how hs goal s done n ATENA. There are wo seps n he process of sress and sran smoohng: / exrapolaon of sress and sran from maeral pons o elemen nodes and / averagng of sress n global node. The whole 0

21 echnque s descrbed brefly. All deals and dervaons can be found e.g. (ZIENKIEWICZ, TAYLOR 989) and ČERVENKA e. al Exrapolaon of Sress and Sran o Elemen Nodes The exrapolaon s done as follows (for each componen of srucural sress and sran ). Le us defne a vecor of sresses where he nd P P, P,... P xx xx, xx, xx, n The nodal value where: T xx a elemen nodes such as xx xx,, xx,,... xx, n, ndex ndcaes elemen node number. Le us also defne a vecor T, whose componen are calculaed P h d (.35) xx, xx e e xx (wh values of xx a nodes =..n ) s hen calculaed as follows: M xx nv M P (.36) xx hh d (.37) j j e e In he above xx s an exrapolaed feld of sress of xx calculaed by FEM. I s ypcally dsconnuous. n s number of elemen nods, e s volume of he nvesgaed fne elemen. The same sraegy s used also for remanng sress and sran componens. Ths smoohng echnque s called varaonal as s base on averagng energy over he elemen. In addon o ha ATENA suppors also anoher way of exrapolang vales from negraon pons o elemen nodes. In hs case (.37) s assumed o be a lumped dagonal marx, n order o elmnae he need for solvng a sysem of lnear equaons. The process of lumpng s characerzes as follows: M j h hk j de (.38) e k, n As mos elemen space approxmaons sasfy hk, he above equaon s smplfed o: k, n M h d (.39) j j e where j s Kronecker dela. Ths lumped formulaon ATENA uses by defaul. The above values are oupu as nodal elemen sress/sran values. I follows o calculae,,... n a global node ha s parcpaed by all averaged sress/san value xx yy xz elemens k wh ncdence a he global node. k k ek ek (.40) ATENA Theory

22 a a node, where s vecor of sresses xx, yy,... xz e k s volume of elemen k ha has ncdence of global node. I should be noed ha n ATENA he same exrapolaon echnques s used for oher negraon pon quanes as well such as: fracurng srans, plasc srans and ohers..0 Smple, Complex Suppors and Maser-Slave Boundary Condons. Smple suppor and complex suppor boundary condons represen boundary condons of Drchle ypes,.e. boundary condons ha prescrbe dsplacemens. On he oher hand, Smple load boundary condons s an example of von Neumann ype boundary condons, when forces are prescrbed. Le K s srucural sffness marx, u s vecor of nodal dsplacemens and R s a vecor of nodal forces. Furher le u s subdvded no vecor of free degrees of freedom u N (wh von Neumann boundary condons) and consraned degrees of freedom u D (wh Drchle boundary condons): u u u The problem governng equaons can hen be wren: N D (.4) K K u R NN ND N N DN DD u D R K K D (.4) ATENA sofware suppors ha any consraned degree of freedom can be a lnear combnaon of oher degrees of freedom plus some consan erm: u u u (.43),0 k D D k N k,0 where u D s he consan erm and k are coeffcens of he lnear combnaon. Of course, he l equaon (.43) can nclude also he erm u, however s ransformed no he consan erm. The free degree of freedom are hen solved by and he dependen R D are solved by l l N NN N ND D D u K R K R (.44) R K u K u (.45) D DN N DD D The ATENA smple suppor boundary condons mean ha he boundary condons use only,0 consan erms are u D, (.e. k 0 ). The complex suppor boundary condons use he full form of (.43). The boundary condons as descrbed above allow o specfy for one degree of freedom eher Drchle, or von Neumann boundary condon, bu no boh of hem he same me. I comes from he naure of fne elemen mehod. However, ATENA can deal also hs case of more

23 complex boundary condons by nroducng Lagrange mulplers. The dervaon of heory behnd hs knd of boundary condons s beyond he scope of hs manual. Deals can be found elsewhere, e.g. n (Bahe 98). To apply hs ype of boundary condons n ATENA, specfy for hose degree of freedom boh smple load and complex suppor boundary condon, he laer one wh he keyword RELAX keyword n s defnon. Nce feaure abou ATENA s ha a any me sores n RAM only K NN and all he elmnaon wh he remanng blocks of K s done a elemen level a he process of assemblng he srucural sffness marx. A specal ype of complex boundary condons of Drchle ype are so-called maser-slave boundary condons. Such a boundary condon specfes ha all (avalable) degrees of one fne node, (.e. slave node) are equal o degrees of freedom of anoher node (.e. maser node). If more maser nodes are specfed, hen hese maser nodes are assumed o form a fne elemen and degrees of freedom of he slave node s assumed o be a node whn ha elemen. Is (slave) degrees of freedom are approxmaed by elemen nodal (.e. maser) degrees of freedom n he same way as dsplacemens approxmaon whn a fne elemen. The coeffcens k n (.43) are hus calculaed auomacally. Ths ype of boundary condons s used for example for fxng dscree renforcemen bars o he suroundng sold elemen.. References BATHE, K.J. (98), Fne Elemen Procedures In Engneerng Analyss, Prence-Hall, Inc., Englewood Clffs, New Jersey 0763, ISBN ČERVENKA, J., KEATING, S.C., AND FELIPPA, C.A. (993), Comparson of sran recovery echnques for he mxed erave mehod, Communcaons n Numercal Mehods n Engneerng, Vol. 9, ZIENKIEWICZ, O.C., TAYLOR, R.L., (989), The Fne Elemen Mehod, Volume, McGraw-Hll Book Company, ISBN ATENA Theory 3

24 4

25 CONSTITUTIVE MODELS. Consuve Model SBETA (CCSbeaMaeral).. Basc Assumpons... Sress, Sran, Maeral Sffness The formulaon of consuve relaons s consdered n he plane sress sae. A smeared approach s used o model he maeral properes, such as cracks or dsrbued renforcemen. Ths means ha maeral properes defned for a maeral pon are vald whn a ceran maeral volume, whch s n hs case assocaed wh he enre fne elemen. The consuve model s based on he sffness and s descrbed by he equaon of equlbrum n a maeral pon: T x y xy x y xy sde, s,,, e,, (.) where s, D and e are a sress vecor, a maeral sffness marx and a sran vecor, respecvely. The sress and sran vecors are composed of he sress componens of he plane sress sae x, y, xy, Fg. -, and he sran componens x, y, xy, Fg. -, where xy s he engneerng shear sran. The srans are common for all maerals. The sress vecor s and he maeral marx D can be decomposed no he maeral componens due o concree and renforcemen as: s sc ss, DDc D s (.) The sress vecor s and boh componen sress vecors sc, s s are relaed o he oal cross secon area. The concree sress sc s acng on he maeral area of concree A c, whch s approxmaely se equal o he cross secon of he compose maeral Ac A (he area of concree occuped by renforcemen s no subraced). The marx D has a form of he Hooke's law for eher soropc or orhoropc maeral, as wll be shown n Secon... T Fg. - Componens of plane sress sae. ATENA Theory 5

26 Fg. - Componens of sran sae. The renforcemen sress vecor s s s he sum of sresses of all he smeared renforcemen componens: n ss s s (.3) where n s he number of he smeared renforcemen componens. For he h renforcemen, he, global componen renforcemen sress s s s relaed o he local renforcemen sress s by he ransformaon:, ss T p s (.4) As where p s he renforcng rao p, A s s he renforcemen cross secon area. The local Ac renforcemen sress s acng on he renforcemen area A s, s The sress and sran vecors are ransformed accordng o he equaons bellow n he plane sress sae. New axes u, v are roaed from he global x, y axes by he angle The angle s posve n he counerclockwse drecon, as shown n Fg. -3. Fg. -3 Roaon of reference coordnae axes. The ransformaon of he sresses: s T s (.5) ( u) ( x) 6

27 cos( ) sn( ) cos( )sn( ) T sn( ) cos( ) cos( )sn( ) (.6) cos( )sn( ) cos( )sn( ) cos( ) sn( ) T,,,,, s s ( u) u v uv ( x) x y xy The ransformaon of he srans: T e T e (.7) ( u) ( x) cos( ) sn( ) cos( )sn( ) T sn( ) cos( ) cos( )sn( ) (.8) cos( )sn( ) cos( )sn( ) cos( ) sn( ) T,,,,, T ( u) u v uv ( x) x y xy e e. The angles of prncpal axes of he sresses and srans, Fg. -, Fg. -, are found from he equaons: xy xy an( ), an( ) x y x y where s he angle of he frs prncpal sress axs and s he angle of he frs prncpal sran axs. In case of soropc maeral (un-cracked concree) he prncpal drecons of he sress and srans are dencal; n case of ansoropc maeral (cracked concree) hey can be dfferen. The sgn convenon for he normal sresses, employed whn hs program, uses he posve values for he ensle sress (sran) and negave values for he compressve sress (sran). The shear sress (sran) s posve f acng upwards on he rgh face of a un elemen.... Concep of Maeral Model SBETA The maeral model SBETA ncludes he followng effecs of concree behavor: non-lnear behavor n compresson ncludng hardenng and sofenng, fracure of concree n enson based on he nonlnear fracure mechancs, baxal srengh falure creron, reducon of compressve srengh afer crackng, enson sffenng effec, reducon of he shear sffness afer crackng (varable shear reenon), wo crack models: fxed crack drecon and roaed crack drecon. Perfec bond beween concree and renforcemen s assumed whn he smeared concep. No bond slp can be drecly modeled excep for he one ncluded nherenly n he enson sffenng. However, on a macro-level a relave slp dsplacemen of renforcemen wh respec o concree over a ceran dsance can arse, f concree s cracked or crushed. Ths corresponds o a real mechansm of bond falure n case of he bars wh rbs. The renforcemen n boh forms, smeared and dscree, s n he unaxal sress sae and s consuve law s a mul-lnear sress-sran dagram. (.9) ATENA Theory 7

28 The maeral marx s derved usng he nonlnear elasc approach. In hs approach he elasc consans are derved from a sress-sran funcon called here he equvalen unaxal law. Ths approach s smlar o he nonlnear hypoelasc consuve model, excep ha dfferen laws are used here for loadng and unloadng, causng he dsspaon of energy exhaused for he damage of maeral. The dealed reamen of he heorecal background of hs subjec can be found, for example, n he book CHEN (98). Ths approach can be also regarded as an soropc damage model, wh he unloadng modulus (see nex secon) represenng he damage modulus. The name SBETA comes from he former program, n whch hs maeral model was frs used. I means he abbrevaon for he analyss of renforced concree n German language - SahlBETonAnalyse... Sress-Sran Relaons for Concree... Equvalen Unaxal Law The nonlnear behavor of concree n he baxal sress sae s descrbed by means of he socalled effecve sress c ef, and he equvalen unaxal sran eq. The effecve sress s n mos cases a prncpal sress. The equvalen unaxal sran s nroduced n order o elmnae he Posson s effec n he plane sress sae. eq c (.0) E c The equvalen unaxal sran can be consdered as he sran, ha would be produced by he sress c n a unaxal es wh modulus E c assocaed wh he drecon. Whn hs assumpon, he nonlneary represenng a damage s caused only by he governng sress c. The deals can be found n CHEN (98). The complee equvalen unaxal sress-sran dagram for concree s shown n Fg. -4. Fg. -4 Unaxal sress-sran law for concree. The numbers of he dagram pars n Fg. -4 (maeral sae numbers) are used n he resuls of he analyss o ndcae he sae of damage of concree. 8

29 Unloadng s a lnear funcon o he orgn. An example of he unloadng pon U s shown n eq Fg. -4. Thus, he relaon beween sress and sran s no unque and depends on a load ef c hsory. A change from loadng o unloadng occurs, when he ncremen of he effecve sran changes he sgn. If subsequen reloadng occurs he lnear unloadng pah s followed unl he las loadng pon U s reached agan. Then, he loadng funcon s resumed. The peak values of sress n compresson f c ef and n enson f ef are calculaed accordng o he baxal sress sae as wll be shown n Sec...5. Thus, he equvalen unaxal sress-sran law reflecs he baxal sress sae. The above defned sress-sran relaon s used o calculae he elasc modulus for he maeral sffness marces, Sec.... The secan modulus s calculaed as s c E c (.) eq I s used n he consuve equaon o calculae sresses for he gven sran sae, Sec.... The angen modulus E c s used n he maeral marx D c for consrucon of an elemen sffness marx for he erave soluon. The angen modulus s he slope of he sress-sran curve a a gven sran. I s always posve. In cases when he slope of he curve s less hen he mnmum value E mn he value of he angen modulus s se E c = E mn. Ths occurs n he sofenng ranges and near he compressve peak. Deal descrpon of he sress-sran law s gven n he followng subsecons.... Tenson before Crackng The behavor of concree n enson whou cracks s assumed lnear elasc. E c s he nal 'ef elasc modulus of concree, f s he effecve ensle srengh derved from he baxal falure funcon, Secon..5.. ef eq ' ef E,0 f (.) c c c...3 Tenson afer Crackng Two ypes of formulaons are used for he crack openng: A fcous crack model based on a crack-openng law and fracure energy. Ths formulaon s suable for modelng of crack propagaon n concree. I s used n combnaon wh he crack band, see Sec...3. A sress-sran relaon n a maeral pon. Ths formulaon s no suable for normal cases of crack propagaon n concree and should be used only n some specal cases. In followng subsecons are descrbed fve sofenng models ncluded n SBETA maeral model. ATENA Theory 9

30 () Exponenal Crack Openng Law Fg. -5 Exponenal crack openng law. Ths funcon of crack openng was derved expermenally by HORDIJK (99). 3 w w w c ' exp c c exp c ef f wc wc wc w c G 5.4 f f ' ef 3, (.3) where w s he crack openng, w c s he crack openng a he complee release of sress, s he normal sress n he crack (crack coheson). Values of he consans are, c =3, c =6.93. G f s he fracure energy needed o creae a un area of sress-free crack, s he effecve ensle srengh derved from a falure funcon, Eq.(.). The crack openng dsplacemen w s derved from srans accordng o he crack band heory n Eq.(.8). () Lnear Crack Openng Law f 'ef Fg. -6 Lnear crack openng law. f G (.4) f w f ef ' c f ' wc w, w ef c ' c 0

31 (3) Lnear Sofenng Based on Local Sran Fg. -7 Lnear sofenng based on sran. The descendng branch of he sress-sran dagram s defned by he sran c 3 correspondng o zero sress (complee release of sress). (4) SFRC Based on Fracure Energy Fg. -8 Seel fber renforced concree based on fracure energy. f f G f Parameers: c, c ', w ef ' ef c f f f f (5) SFRC Based on Sran Fg. -9 Seel fber renforced concree based on sran. ATENA Theory

32 f f Parameers: c, c ' ef ' f f ef Parameers c and c are relave posons of sress levels, and c 3 s he end sran....4 Compresson before Peak Sress The formula recommended by CEB-FIP Model Code 90 has been adoped for he ascendng branch of he concree sress-sran law n compresson, Fg. -0. Ths formula enables wde range of curve forms, from lnear o curved, and s approprae for normal as well as hgh srengh concree. ef ' ef kx x Eo c fc, x, k (.5) ( k) x E c c Fg. -0 Compressve sress-sran dagram. Meanng of he symbols n he above formula n: ef c -concree compressve sress, f - concree effecve compressve srengh (See Secon..5.) 'ef c x - normalzed sran, - sran, c - sran a he peak sress f c ef, k - shape parameer, E o - nal elasc modulus, E c - secan elasc modulus a he peak sress, E c 'ef fc. Parameer k may have any posve value greaer han or equal. Examples: k=. lnear, k=. - parabola. As a consequence of he above assumpon, dsrbued damage s consdered before he peak sress s reached. Conrary o he localzed damage, whch s consdered afer he peak. c

33 ...5 Compresson afer Peak Sress The sofenng law n compresson s lnearly descendng. There are wo models of sran sofenng n compresson, one based on dsspaed energy, and oher based on local sran sofenng Fcous Compresson Plane Model The fcous compresson plane model s based on he assumpon, ha compresson falure s localzed n a plane normal o he drecon of compressve prncpal sress. All pos-peak compressve dsplacemens and energy dsspaon are localzed n hs plane. I s assumed ha hs dsplacemen s ndependen on he sze of he srucure. Ths hypohess s suppored by expermens conduced by Van MIER (986). Ths assumpon s analogous o he Fcous Crack Theory for enson, where he shape of he crack-openng law and he fracure energy are defned and are consdered as maeral properes. Fg. - Sofenng dsplacemen law n compresson. In case of compresson, he end pon of he sofenng curve s defned by means of he plasc dsplacemen w d. In hs way, he energy needed for generaon of a un area of he falure plane s ndrecly defned. From he expermens of Van MIER (986), he value of w d =0.5mm for normal concree. Ths value s used as defaul for he defnon of he sofenng n compresson. The sofenng law s ransformed from a fcous falure plane, Fg. -, o he sress-sran relaon vald for he correspondng volume of connuous maeral, Fg. -0. The slope of he sofenng par of he sress-sran dagram s defned by wo pons: a peak of he dagram a he maxmal sress and a lm compressve sran d a he zero sress. Ths sran s calculaed from a plasc dsplacemen w d and a band sze L ' d (see Secon..3) accordng o he followng expresson: wd d c (.6) ' L The advanage of hs formulaon s reduced dependency on fne elemen mesh Compresson Sran Sofenng Law Based on Sran. A slope of he sofenng law s defned by means of he sofenng modulus E d. Ths formulaon s dependen on he sze of he fne elemen mesh. d ATENA Theory 3

34 ..3 Localzaon Lmers So-called localzaon lmer conrols localzaon of deformaons n he falure sae. I s a regon (band) of maeral, whch represens a dscree falure plane n he fne elemen analyss. In enson s a crack, n compresson s a plane of crushng. In realy hese falure regons have some dmenson. However, snce accordng o he expermens, he dmensons of he falure regons are ndependen on he srucural sze, hey are assumed as fcous planes. In case of ensle cracks, hs approach s known as rack he crack band heory, BAZANT, OH (983). Here s he same concep used also for he compresson falure. The purpose of he falure band s o elmnae wo defcences, whch occur n connecon wh he applcaon of he fne elemen model: elemen sze effec and elemen orenaon effec. y 4 noded elemen crack drecon L Lc x Fg. - Defnon of localzaon bands...3. Elemen Sze Effec. The drecon of he falure planes s assumed o be normal o he prncpal sresses n enson and compresson, respecvely. The falure bands (for enson L and for compresson L d ) are defned as projecons of he fne elemen dmensons on he falure planes as shown n Fg Elemen Orenaon Effec. The elemen orenaon effec s reduced, by furher ncreasng of he falure band for skew meshes, by he followng formula (proposed by CERVENKA e al. 995). ' ' L L, L L max ( ) 45 d d, 0; 45 (.7) 4

35 An angle s he mnmal angle ( mn, ) beween he drecon of he normal o he falure plane and elemen sdes. In case of a general quadrlaeral elemen he elemen sdes drecons are calculaed as average sde drecons for he wo oppose edges. The above formula s a lnear nerpolaon beween he facor.0 for he drecon parallel wh elemen sdes, and max, for he drecon nclned a 45 o max. The recommended (and defaul) value of = Fracure Process, Crack Wdh The process of crack formaon can be dvded no hree sages, Fg. -3. The uncracked sage s before a ensle srengh s reached. The crack formaon akes place n he process zone of a poenal crack wh decreasng ensle sress on a crack face due o a brdgng effec. Fnally, afer a complee release of he sress, he crack openng connues whou he sress. The crack wdh w s calculaed as a oal crack openng dsplacemen whn he crack band. w L (.8) cr where cr s he crack openng sran, whch s equal o he sran normal o he crack drecon n he cracked sae afer he complee sress release. ' Fg. -3 Sages of crack openng. I has been shown, ha he smeared model based on he refned crack band heory can successfully descrbe he dscree crack propagaon n plan, as well as renforced concree (CERVENKA e al. 99, 99, and 995). I s also possble, ha he second sress, parallel o he crack drecon, exceeds he ensle srengh. Then he second crack, n he drecon orhogonal o he frs one, s formed usng he same sofenng model as he frs crack. (Noe: The second crack may no be shown n a graphcal pos-processng. I can be denfed by he concree sae number n he second drecon a he numercal oupu.)..5 Baxal Sress Falure Creron of Concree..5. Compressve Falure A baxal sress falure creron accordng o KUPFER e al. (969) s used as shown n Fg. -4. In he compresson-compresson sress sae he falure funcon s ATENA Theory 5

36 6 Fg. -4 Baxal falure funcon for concree. 3.65a f f, a ' ef ' c c c ( a) c (.9) where c, c are he prncpal sresses n concree and f c s he unaxal cylnder srengh. In he baxal sress sae, he srengh of concree s predced under he assumpon of a proporonal sress pah. In he enson-compresson sae, he falure funcon connues lnearly from he pon ', f no he enson-compresson regon wh he lnearly decreasng srengh: c 0 c c f f r, r ( ),.0 r 0.9 (.0) ' ef ' c c c ec ec ' ec fc where r ec s he reducon facor of he compressve srengh n he prncpal drecon due o he ensle sress n he prncpal drecon...5. Tensle Falure In he enson-enson sae, he ensle srengh s consan and equal o he unaxal ensle srengh f. In he enson-compresson sae, he ensle srengh s reduced by he relaon: ' ef ' f fr (.) e where r e s he reducon facor of he ensle srengh n he drecon due o he compressve sress n he drecon. The reducon funcon has one of he followng forms, Fg. -5. r 0.95 c e (.) ' fc A( A) B c re, B Kx A, x (.3) ' AB fc The relaon n Eq.(.) s he lnear decrease of he ensle srengh and (.3) s he hyperbolc decrease.

37 Two predefned shapes of he hyperbola are gven by he poson of an nermedae pon r, x. Consans K and A defne he shape of he hyperbola. The values of he consans for he wo posons of he nermedae pon are gven n he followng able. ype pon parameers r x A K a b Fg. -5 Tenson-compresson falure funcon for concree...6 Two Models of Smeared Cracks The smeared crack approach for modelng of he cracks s adoped n he model SBETA. Whn he smeared concep wo opons are avalable for crack models: he fxed crack model and he roaed crack model. In boh models he crack s formed when he prncpal sress exceeds he ensle srengh. I s assumed ha he cracks are unformly dsrbued whn he maeral volume. Ths s refleced n he consuve model by an nroducon of orhoropy...6. Fxed Crack Model In he fxed crack model (CERVENKA 985, DARWIN 974) he crack drecon s gven by he prncpal sress drecon a he momen of he crack naon. Durng furher loadng hs drecon s fxed and represens he maeral axs of he orhoropy. ATENA Theory 7

38 Fg. -6 Fxed crack model. Sress and sran sae. The prncpal sress and sran drecons concde n he uncracked concree, because of he assumpon of soropy n he concree componen. Afer crackng he orhoropy s nroduced. The weak maeral axs m s normal o he crack drecon, he srong axs m s parallel wh he cracks. In a general case he prncpal sran axes and roae and need no o concde wh he axes of he orhoropy m and m. Ths produces a shear sress on he crack face as shown n Fg. -6. The sress componens c and c denoe, respecvely, he sresses normal and parallel o he crack plane and, due o shear sress, hey are no he prncpal sresses. The shear sress and sffness n he cracked concree s descrbed n Secon Roaed Crack Model In he roaed crack model (VECCHIO 986, CRISFIELD 989), he drecon of he prncpal sress concdes wh he drecon of he prncpal sran. Thus, no shear sran occurs on he crack plane and only wo normal sress componens mus be defned, as shown n Fg. -7. Fg. -7 Roaed crack model. Sress and sran sae. If he prncpal sran axes roae durng he loadng he drecon of he cracks roae, oo. In order o ensure he co-axaly of he prncpal sran axes wh he maeral axes he angen shear modulus G s calculaed accordng o CRISFIELD 989 as c c G (.4) ( ) 8

39 ..7 Shear Sress and Sffness n Cracked Concree In case of he fxed crack model, he shear modulus s reduced accordng o he law derved by KOLMAR (986) afer crackng. The shear modulus s reduced wh growng sran normal o he crack, Fg. -8 and hs represens a reducon of he shear sffness due o he crack openng. Fg. -8 Shear reenon facor. 000 u ln c g c, g 3 (.5) c G r G r c c 7 333( p 0.005), c 0 67( p 0.005),0 p 0.0 where r g s he shear reenon facor, G s he reduced shear modulus and G c s he nal concree shear modulus G c Ec ( ) (.6) where E c s he nal elasc modulus and s he Posson's rao. The sran s normal o he crack drecon (he crack openng sran), c and c are parameers dependng on he renforcng crossng he crack drecon, p s he ransformed renforcng rao (all renforcemen s ransformed on he crack plane) and c 3 s he user s scalng facor. By defaul c 3 =. In ATENA he effec of renforcemen rao s no consdered, and p s assumed o be 0.0. There s an addonal consran mposed on he shear modulus. The shear sress on he crack plane uv G s lmed by he ensle srengh f. The secan and angen shear modul of cracked concree are equal...8 Compressve Srengh of Cracked Concree A reducon of he compressve srengh afer crackng n he drecon parallel o he cracks s done by a smlar way as found from expermens of VECCHIO and COLLINS 98 and formulaed n he Compresson Feld Theory. However, a dfferen funcon s used for he reducon of concree srengh here, n order o allow for user's adjusmen of hs effec. Ths funcon has he form of he Gauss's funcon, Fg. -9. The parameers of he funcon were derved from he expermenal daa publshed by KOLLEGER e al. 988, whch ncluded also daa of Collns and Veccho (VECCHIO a al.98) ATENA Theory 9

40 (8 ) ' ef ' u f r f, r c( c) e (.7) c c c c For he zero normal sran, here s no srengh reducon, and for he large srans, he ' ef ' srengh s asympocally approachng o he mnmum value f cf. c c 30 Fg. -9 Compressve srengh reducon of cracked concree. The consan c represens he maxmal srengh reducon under he large ransverse sran. From he expermens by KOLLEGGER e all. 988, he value c = 0.45 was derved for he concree renforced wh he fne mesh. The oher researchers (DYNGELAND 989) found he reducons no less han c=0.8. The value of c can be adjused by npu daa accordng o he acual ype of renforcng. However, he reducon of compressve srengh of he cracked concree does no have o be effeced only by he renforcng. In he plan concree, when he sran localzes n one man crack, he compressve concree srus can cross hs crack, causng so-called "brdgng effec". The compressve srengh reducon of hese brdges s also capured by he above model...9 Tenson Sffenng n Cracked Concree The enson sffenng effec can be descrbed as a conrbuon of cracked concree o he ensle sffness of renforcng bars. Ths sffness s provded by he uncracked concree or no fully opened cracks and s generaed by he sran localzaon process. I was verfed by smulaon expermens of HARTL, G., 977 and publshed n he paper (MARGOLDOVA e.al. 998). Includng an explc enson sffenng facor would resul n an overesmaon of hs effec. Therefore, n he ATENA versons up o..0 no explc enson sffenng facor s possble n he npu...0 Summary of Sresses n SBETA Consuve Model In he case of uncracked concree he sress symbols have he followng meanng: c - maxmal prncpal sress c - mnmal prncpal sress (enson posve, compresson negave) In he case of cracked concree, Fg. -6 sresses are defned on he crack plane: c - normal sress normal o he cracks - normal sress parallel o he cracks c

41 c - shear sress on he crack plane.. Maeral Sffness Marces... Uncracked Concree The maeral sffness marx for he uncracked concree has he form of an elasc marx of he soropc maeral. I s wren n he global coordnae sysem x and y. 0 E D c 0 (.8) 0 0 In he above E s he concree elasc modulus derved from he equvalen unaxal law. The Posson's rao s consan.... Cracked Concree For he cracked concree he marx has he form of he elasc marx for he orhoropc maeral. The marx s formulaed n a coordnae sysem m, m, Fg. -6 and Fg. -7, whch s concden wh he crack drecon. Ths local coordnae sysem s referred o he superscrp L laer. The drecon s normal o he crack and he drecon s parallel wh he crack. The defnon of he elasc consans for he orhoropc maeral n he plane sress sae follows from he flexbly relaon: 0 E E 0 E E (.9) 0 0 G Frs we elmnae he orhoropc Posson s raos for he cracked concree, because hey are commonly no known. For hs we use he symmery relaon E E. Therefore, n (.9) here are only hree ndependen elasc consans E, E,. Assumng ha s he Posson's rao of he uncracked concree and usng he symmery relaon, we oban E (.30) E The sffness marx L Dc s found as he nverse of he flexbly marx n (.30): L D c 0 H 0, 0 0 G E, H E( ) E (.3) ATENA Theory 3

42 In he above relaon E mus be nonzero. If E s zero and E s nonzero, hen an alernave E formulaon s used wh he nverse parameer. In case ha boh elasc modules are E zero, he marx L D c s se equal o he null marx. L The marx D c s ransformed no he global coordnae sysem usng he ransformaon marx T from (.8). D T L c T Dc T (.3) The angle s beween he global axs x and he s maeral axs m, whch s normal o he crack, Fg Smeared Renforcemen The maeral sffness marx of he h smeared renforcemen s 4 3 cos( ) cos( ) sn( ) cos( ) sn( ) 4 3 D s pe s cos( ) sn( ) sn( ) cos( )sn( ) (.33) 3 3 cos( ) sn( ) cos( )sn( ) cos( ) sn( ) The angle s beween he global axs x and he h renforcemen drecon, and E s s he elasc modulus of renforcemen. The renforcng rao p =A s /A c....4 Maeral Sffness of Compose Maeral The oal maeral sffness of he renforced concree s he sum of maeral sffness of concree and smeared renforcemen: n c DD D (.34) The summaon s over n smeared renforcng componens. In ATENA he smeared renforcemen s no added on he consuve level, bu s modeled by separae layers of elemens whose nodes are conneced o hose of he concree elemens. Ths corresponds o he assumpon of perfec bond beween he smeared renforcemen and concree. s...5 Secan and Tangen Maeral Sffness The maeral sffness marces n he above Subsecons...,...,...3,...4 are eher secan or angen, dependng on he ype of elasc modulus used. The secan maeral sffness marx s used o calculae he sresses for he gven srans, as shown n Secon... The angen maeral sffness marx s used o consruc he elemen sffness marx... Analyss of Sresses The sresses n concree are obaned usng he acual secan componen maeral sffness marx c s c s D e (.35) 3

43 s where D c s he secan maeral sffness marx from Secon.. for he uncracked or cracked concree dependng on he maeral sae. The sress componens are calculaed n he global as well as n he local maeral coordnaes (he prncpal sresses n he uncracked concree and he sresses on he crack planes). The sress n renforcemen and he assocaed enson sffenng sress s calculaed drecly from he sran n he renforcemen drecon...3 Parameers of Consuve Model Defaul formulas of maeral parameers: Parameer: Formula: ' ' Cylnder srengh f 0.85 f Tensle srengh ' ' 3 f 0.4 f Inal elasc modulus ' ' E ( f ) f Posson's rao 0. c cu cu c cu cu Sofenng compresson w mm Type of enson sofenng exponenal, based on G F Compressve srengh n cracked concree c = 0.8 Tenson sffenng sress s 0. Shear reenon facor Tenson-compresson funcon ype Fracure energy G f accordng o VOS 983 d varable (Sec...7) lnear G F f [MN/m] ' ef Orenaon facor for sran localzaon max.5 (Sec...3) The SBETA consuve model of concree ncludes 0 maeral parameers. These parameers are specfed for he problem under consderaon by user. In case of he parameers are no known auomac generaon can be done usng he defaul formulas gven n he able above. In such a case, only he cube srengh of concree f cu (nomnal srengh) s specfed and he remanng parameers are calculaed as funcons of he cube srengh. The formulas for hese funcons are aken from he CEB-FIP Model Code 90 and oher research sources. Used uns are MPa. The parameers no lsed n he able have zero defaul value. The values of he maeral parameers can be also nfluenced by safey consderaons. Ths s parcularly mporan n cases of a desgn, where a proper safey margn should be me. For ha reason he choce of maeral properes depends on he purpose of analyss and he fled of an applcaon. The ypcal examples of he applcaon are he desgn, he smulaon of falure and he research. ATENA Theory 33

44 In case of he desgn applcaon, accordng o mos curren sandards, he maeral properes for calculaon of srucural ressance (falure load) are consdered by mnmal values wh appled paral safey facors. The resulng maxmum load can be drecly compared wh he desgn loads. Accordng o some researchers, more approprae approach would be o consder he average maeral properes n nonlnear analyss and o apply a safey facor on he resulng negral response varable (force, momen). However, hs safey forma s no ye fully esablshed. In cases of he smulaon of real behavor, he parameers should be chosen as close as possble o he properes of real maerals. The bes way s o deermne hese properes from mechancal ess on maeral sample specmens.. Fracure Plasc Consuve Model (CC3DCemenous, CC3DNonLnCemenous, CC3DNonLnCemenous, CC3DNonLnCemenousUser, CC3DNonLnCemenousVarable, CC3DNonLnCemenousSHCC, CC3DNonLnCemenous3).. Inroducon Fracure-plasc model combnes consuve models for ensle (fracurng) and compressve (plasc) behavor. The fracure model s based on he classcal orhoropc smeared crack formulaon and crack band model. I employs Rankne falure creron, exponenal sofenng, and can be used as roaed or fxed crack model. The hardenng/sofenng plascy model s based on Menérey-Wllam falure surface. The model uses reurn mappng algorhm for he negraon of consuve equaons. Specal aenon s gven o he developmen of an algorhm for he combnaon of he wo models. The combned algorhm s based on a recursve subsuon, and allows for he wo models o be developed and formulaed separaely. The algorhm can handle cases when falure surfaces of boh models are acve, bu also when physcal changes such as crack closure occur. The model can be used o smulae concree crackng, crushng under hgh confnemen, and crack closure due o crushng n oher maeral drecons. Alhough many papers have been publshed on plascy models for concree (for nsance, PRAMONO, WILLAM 989, MENETREY e al 997, FEENSTRA 993, 998 ETSE 99) or smeared crack models (RASHID 968, CERVENKA and GERSTLE 97, BAZANT and OH 983, DE BORST 986, ROTS 989), here are no many descrpons of her successful combnaon n he leraure. OWEN e al. (983) presened a combnaon of crackng and vsco-plascy. Comprehensve rease of he problem was provded also by de BORST (986), and recenly several works have been publshed on he combnaon of damage and plascy (SIMO and JU 987, MESCHKE e al. (998). The presened model dffers from he above formulaons by ably o handle also physcal changes lke for nsance crack closure, and s no resrced o any parcular shape of hardenng/sofenng laws. Also whn he proposed approach s possble o formulae he wo models (.e. plasc and fracure) enrely separaely, and her combnaon can be provded n a dfferen algorhm or model. From programmng pon of vew such approach s well sued for objec orened programmng. The mehod of sran decomposon, as nroduced by DE BORST (986), s used o combne fracure and plascy models ogeher. Boh models are developed whn he framework of 34

45 reurn mappng algorhm by WILKINS (964). Ths approach guaranees he soluon for all magnudes of sran ncremen. From an algorhmc pon of vew he problem s hen ransformed no fndng an opmal reurn pon on he falure surface. The combned algorhm mus deermne he separaon of srans no plasc and fracurng componens, whle mus preserve he sress equvalence n boh models. The proposed algorhm s based on a recursve erave scheme. I can be shown ha such a recursve algorhm canno reach convergence n ceran cases such as, for nsance, sofenng and dlang maerals. For hs reason he recursve algorhm s exended by a varaon of he relaxaon mehod o sablze convergence... Maeral Model Formulaon The maeral model formulaon s based on he sran decomposon no elasc f and fracurng componens (DE BORST 986). j e j, plasc p j j (.36) e j p j f j The new sress sae s hen compued by he formula: n n p f j j Ejkl ( kl kl kl ) (.37) where he ncremens of plasc sran he used maeral models. p j and fracurng sran..3 Rankne-Fracurng Model for Concree Crackng Rankne creron s used for concree crackng f j mus be evaluaed based on f F f 0 (.38) I s assumed ha srans and sresses are convered no he maeral drecons, whch n case of roaed crack model correspond o he prncpal drecons, and n case of fxed crack model, are gven by he prncpal drecons a he onse of crackng. Therefore, denfes he ral sress and f ensle srengh n he maeral drecon. Prme symbol denoes quanes n he maeral drecons. The ral sress sae s compued by he elasc predcor. ATENA Theory 35 n j j Ejkl kl (.39) If he ral sress does no sasfy (.38), he ncremen of fracurng sran n drecon can be compued usng he assumpon ha he fnal sress sae mus sasfy (.40). F f n f f E f 0 (.40) Ths equaon can be furher smplfed under he assumpon ha he ncremen of fracurng sran s normal o he falure surface, and ha always only one falure surface s beng checked. For falure surface k, he fracurng sran ncremen has he followng form. j kl kl f f Fk j k (.4) Afer subsuon no (.40) a formula for he ncremen of he fracurng mulpler s recovered.

46 f ) max kk k kk f ( wk max f and ( ˆ k L kk ) Ekkkk Ekkkk w (.4) Ths equaon mus be solved by eraons snce for sofenng maerals he value of curren max ensle srengh f ( w k ) s a funcon of he crack openng w, and s based on Hordjk s formula (defned n SBETA model). f The crack openng w s compued from he oal value of fracurng sran ˆ kk n drecon k, plus he curren ncremen of fracurng sran, and hs sum s mulpled by he characersc lengh L. The characersc lengh as a crack band sze was nroduced by BAZANT and OH. Varous mehods were proposed for he crack band sze calculaon n he framework of fne elemen mehod. FEENSTRA (993) suggesed a mehod based on negraon pon volume, whch s no well sued for dsored elemens. A conssen and raher complex approach was proposed by OLIVIER. In he presened work he crack band sze L s calculaed as a sze of he elemen projeced no he crack drecon, Fg. -0. CERVENKA V. e al. (995) showed ha hs approach s sasfacory for low order lnear elemens, whch are used hroughou hs sudy. They also proposed a modfcaon, whch accouns for cracks ha are no algned wh elemen edges. Fg. -0 Tensle sofenng and characersc lengh Equaon (.4) can be solved by recursve subsuons. I s possble o show by expandng max f ( ) no a Taylor seres ha hs eraon scheme converges as long as: w k f ( w w max k ) E L kkkk (.43) Equaon (.43) s volaed for sofenng maerals only when snap back s observed n he sresssran relaonshp, whch can occur f large fne elemens are used. In he sandard dsplacemen based fne elemen mehod, he sran ncremen s gven, herefore, a snap back on he consuve level canno be capured. Ths means ha he crcal regon, wh snap back on he sofenng curve, wll be skpped n a real calculaon, whch physcally means, ha he energy dsspaed by he sysem wll be over esmaed. Ths s of course undesrable, and fne E f (0) f elemens smaller hen L kkkk should be used, where (0) denoes he nal slope w w of he crack sofenng curve. 36

47 I s mporan o dsngush beween oal fracurng sran ˆ f j, whch corresponds o he f maxmal fracurng sran reached durng he loadng process, and curren fracurng sran j, whch can be smaller due o crack closure, and s compued usng (.44) derved by ROTS and BLAUWENDRAAD. f kl ( E E ) E, and jkl cr jkl klmn mn cr E jlk s defned by (.44) cr f j Ejkl kl The fourh order crack ensor E represens he crackng sffness n he local maeral drecons. cr jkl In he curren formulaon, s assumed, ha here s no neracon beween normal and shear componens. Thus, he crack ensor s gven by he followng formulas. cr jkl Mode I crack sffness equals cr f ( w E ˆ E 0 for k and j l (.45) max f ), (no summaon of ndces) (.46) and mode II and III crack sffness s assumed as: cr cr cr E s mn E, E, (no summaon of ndces) (.47) jj F jjjj where j, and s F s a shear facor coeffcen ha defnes a relaonshp beween he normal and shear crack sffness. The defaul value of sf s 0. Shear srengh of a cracked concree s calculaed usng he Modfed Compresson Feld Theory of VECHIO and COLLINS (986). j 0.8 f c 4 w 0.3 a 6 g, j (.48) Where f c s he compressve srengh n MPa, ag s he maxmum aggregae sze n mm and w s he maxmum crack wdh n mm a he gven locaon. Ths model s acvaed by specfyng he maxmum aggregae sze ag oherwse he defaul behavor s used where he shear sress on a crack surface canno exceed he ensle srengh. The secan consuve marx n he maeral drecon was formulaed by ROTS and BLAUWENDRAAD n he marx forma. s cr - E E - E(E E) E (.49) Sran vecor ransformaon marx T (.e. global o local sran ransformaon marx) can be used o ransform he local secan sffness marx o he global coordnae sysem. E s T s T E T (.50) I s necessary o handle he specal cases before he onse of crackng, when he crack sffness approaches nfny. Large penaly numbers are used for crack sffness n hese cases...3. Unloadng Drecon Crack closure sffness s conrolled by he unloadng facor (maeral parameer) 0 f U <. ATENA Theory 37

48 The value of 0 corresponds o unloadng o orgn (defaul value for backward compably), f U = means unloadng drecon parallel o he nal elasc sffness...4 Plascy Model for Concree Crushng New sress sae n he plasc model s compued usng he predcor-correcor formula. E ( ) E (.5) ( n ) ( n ) p p p j j jkl kl kl j jkl kl j j The plasc correcor algorhm. p j s compued drecly from he yeld funcon by reurn mappng p p p F ( ) F ( l ) 0 (.5) j j j j The crucal aspec s he defnon of he reurn drecon l j, whch can be defned as l j E jkl p G ( kl ) kl hen p G ( j ) p j (.53) j where G( j ) s he plasc poenal funcon, whose dervave s evaluaed a he predcor sress sae j o deermne he reurn drecon. The falure surface of MENETREY, WILLAM s used n he curren verson of he maeral model. where m 3 ' ' c ' ' fc f F p 3P f f e e 5. m r e f 6f (, ) 3f r(, e) ' ' ' c c c c 0 4 ( e )cos ( e), ( e )cos ( e) 4 ( e )cos 5e 4e (.54) In he above equaons (,, ) are Hegh-Vesergaard coordnaes, f c and f s compressve srengh and ensle srengh respecvely. Parameer e 050.,. defnes he roundness of he falure surface. The falure surface has sharp corners f e 05., and s fully crcular around he hydrosac axs f e 0.. The poson of falure surfaces s no fxed bu can move dependng on he value of sran hardenng/sofenng parameer. The sran hardenng s based on he equvalen plasc sran, whch s calculaed accordng o he followng formula. p p mn( ) (.55) eq For Menérey-Wllam surface he hardenng/sofenng s conrolled by he parameer c 0,, whch evolves durng he yeldng/crushng process by he followng relaonshp: j c p f ( c f c eq ) (.56) 38

49 p In he above wo formulas he expresson f ( ) ndcaes he hardenng/sofenng law, whch s c eq based on he unaxal compressve es. The law s shown n Fg. -, where he sofenng curve s lnear and he ellpcal ascendng par s gven by he followng formula: p c eq fco fc fco c (.57) f' c f c0= f c p =f' /E c eq p Fg. -. Compressve hardenng/sofenng and compressve characersc lengh. Based on expermenal observaons by VAN MIER. The law on he ascendng branch s based on srans, whle he descendng branch s based on dsplacemens o nroduce mesh objecvy no he fne elemen soluon, and s shape s based on he work of VAN MIER. The onse of nonlnear behavor f s an npu parameer as well as he value of plasc sran a compressve srengh. The Fg. - shows ypcal values ' of hese parameers. Especally he choce of he parameer f c0 should be seleced wh care, snce s mporan o ensure ha he fracure and plasc surfaces nersec each oher n all maeral sages. On he descendng curve he equvalen plasc sran s ransformed no dsplacemens hrough he lengh scale parameer L c. Ths parameer s defned by analogy o he crack band parameer n he fracure model n Sec...3, and corresponds o he projecon of elemen sze no he drecon of mnmal prncpal sresses. The square n (.56) s due o he quadrac naure of he Menéry-Wllam surface. Reurn drecon s gven by he followng plasc poenal G p p c ( j ) I J (.58) 3 where deermnes he reurn drecon. If 0 maeral s beng compaced durng crushng, f 0 maeral volume s preserved, and f 0 maeral s dlang. In general he plasc model s non-assocaed, snce he plasc flow s no perpendcular o he falure surface The reurn mappng algorhm for he plasc model s based on predcor-correcor approach as s shown n Fg. -. Durng he correcor phase of he algorhm he falure surface moves along he hydrosac axs o smulae hardenng and sofenng. The fnal falure surface has he apex locaed a he orgn of he Hagh-Vesergaard coordnae sysem. Secan mehod based Algorhm s used o deermne he sress on he surface, whch sasfes he yeld condon and also he hardenng/sofenng law. ' c0 ATENA Theory 39

50 Fg. - Plasc predcor-correcor algorhm. Fg. -3. Schemac descrpon of he erave process (.73). For clary shown n wo dmensons. Algorhm : (Inpu s Elasc predcor:,, ) ( n) ( n) p ( n) j j j E (.59) ( n) ( n) j j jkl kl Evaluae falure creron: f F, 0 (.60) p p ( n) p A ( j, j ) A p If falure creron s volaed.e. f 0 40 A

51 Evaluae reurn drecon: m j p G ( j ) j (.6) Reurn mappng: Evaluae falure creron: F p ( n) p ( j BEmj, j ) 0 B (.6) f F Em m (.63) p p ( n ) p B ( j B j, j B j ) Secan eraons () as long as A B e (.64) New plasc mulpler ncremen: p A f A (.65) f f B p B A p A New reurn drecon: p ( ) G ( ) () j E mj mj j (.66) Evaluae falure creron: f F E m m (.67) p p () ( n ) p () ( j j, j j ) New nal values for secan eraons: p p p f 0 f f, (.68) B B p p p p p f 0 f f,, f f, (.69) B End of secan eraon loop End of algorhm updae sress and plasc srans. A m, ( n) p ( n) p ( ) j j B j B A B B B ( n) ( ) j j B j B E m (.70)..5 Combnaon of Plascy and Fracure model The objecve s o combne he above models no a sngle model such ha plascy s used for concree crushng and he Rankne fracure model for crackng. Ths problem can be generally saed as a smulaneous soluon of he wo followng nequales. F p ( n) f p ( j Ejkl ( kl kl kl )) 0 solve for kl p (.7) F f ( ) ( n ( p f j Ejkl kl kl kl )) 0 solve for kl f (.7) Each nequaly depends on he oupu from he oher one, herefore he followng erave scheme s developed. Algorhm : Sep : F E b solve for p ( n) ( ) f ( ) cor ( ) p ( j jkl ( kl kl kl kl )) 0 () p kl Sep : F solve for f ( ) ( ) ( ) ( n ( p f j Ejkl kl kl kl )) 0 () f kl Sep 3: (.73) () cor () f () f j j j Ierave correcon of he sran norm beween wo subsequen eraons can be expressed as ATENA Theory 4

52 ( b) (.74) () cor f p () cor j j f where () f ( ) f j j p, () p ( ) p j j () p () p j j cor j and b s an eraon correcon or relaxaon facor, whch s nroduced n order o guaranee f p convergence. I s o be deermned based on he run-me analyss of and, such ha he f p convergence of he erave scheme can be assured. The parameers and characerze he mappng properes of each model (.e. plasc and fracure). I s possble o consder each model as an operaor, whch maps sran ncremen on he npu no a fracure or plasc sran ncremen on he oupu. The produc of he wo mappngs mus be conracve n order o oban a convergence. The necessary condon for he convergence s: f p ( b) (.75) If b equals 0, an erave algorhm based on recursve subsuon s obaned. The convergence can be guaraneed only n wo cases: f p One of he models s no acvaed (.e. mples or 0 ), There s no sofenng n eher of he wo models and dlang maeral s no used n he plasc par, whch for he plasc poenal n hs work means 0, (.58). Ths s a suffcen bu f p no necessary condon o ensure ha and. f p I can be shown ha he values of and are drecly proporonal o he sofenng rae n each model. Snce he sofenng model remans usually consan for a maeral model and fne elemen, her values do no change sgnfcanly beween eraons. I s possble o selec he scalar b such ha he nequaly (.75) s sasfed always a he end of each eraon based on f p he curren values of and. There are hree possble scenaros, whch mus be handled, for he approprae calculaon of b : f p, where s relaed o he requesed convergence rae. For lnear rae can be se o /. In hs case he convergence s sasfacory and b 0. as f p, hen he convergence would be oo slow. In hs case b can be esmaed f p b, n order o ncrease he convergence rae. f p, hen he algorhm s dvergng. In hs case b should be calculaed as b o sablze he eraons. f p Ths approach guaranees convergence as long as he parameers p, f do no change drascally beween he eraons, whch should be sasfed for smooh and correcly formulaed models. The rae of convergence depends on maeral brleness, dlang parameer and fne elemen sze. I s advanageous o furher sablze he algorhm by smoohng he parameer b durng he erave process: () ( ) ( )/ (.76) b b b 4

53 where he superscrp denoes values from wo subsequen eraons. Ths wll elmnae problems due o he oscllaon of he correcon parameer b. Imporan condon for he convergence of he above Algorhm s ha he falure surfaces of he wo models are nersecng each oher n all possble posons even durng he hardenng or sofenng. Addonal consrans are used n he erave algorhm. If he sress sae a he end of he frs sep volaes he Rankne creron, he order of he frs wo seps n Algorhm s reversed. Also n realy concree crushng n one drecon has an effec on he crackng n oher drecons. I s assumed ha afer he plascy yeld creron s volaed, he ensle srengh n all maeral drecons s se o zero. On he srucural level secan marx s used n order o acheve a robus convergence durng he sran localzaon process. The proposed algorhm for he combnaon of plasc and fracure models s graphcally shown n Fg. -3. When boh surfaces are acvaed, he behavor s que smlar o he mul-surface plascy (SIMO e al. 988). Conrary o he mul-surface plascy algorhm he proposed mehod s more general n he sense ha covers all loadng regmes ncludng physcal changes such as for nsance crack closure. Currenly, s developed only for wo neracng models, and s exenson o mulple models s no sraghforward. There are addonal neracons beween he wo models ha need o be consdered n order o properly descrbe he behavor of a concree maeral: (a) Afer concree crushng he ensle srengh should decrease as well (b) Accordng o he research work of Collns (VECHIO and COLLINS (986)) and coworkers was esablshed he also compressve srengh should decrease when crackng occurs n he perpendcular drecon. Ths heory s called compresson feld heory and s used o explan he shear falure of concree beams and walls. The neracon (a) s resolved by addng he equvalen plasc sran o he maxmal fracurng sran n he fracure model o auomacally ncrease he ensle damage based on he compressve damage such ha he fracurng srans sasfes he followng condon: f ˆ f p kk eq (.77) fc The compressve srengh reducon (b) s based on he followng formula based proposed by Collns: r f c c c r r r lm c, c c (.78) Where s he ensle sran n he crack. In ATENA he larges maxmal fracurng sran s used for and he compressve srengh reducon s lmed by no compresson reducon s consdered. lm r c. If..6 Varans of he Fracure Plasc Model The several ATENA maeral models are based on he above heores: CC3DCemenous, lm rc s no specfed hen ATENA Theory 43

54 CC3DNonLnCemenous, CC3DNonLnCemenous, CC3DNonLnCemenousVarable, CC3DNonLnCemenousFague (descrbed n secon..0), CC3DNonLnCemenousUser, CC3DNonLnCemenousFRC, CC3DNONLINCEMENTITIOUSSHCC, CC3DNONLINCEMENTITIOUSHPFRC (descrbed n secon..), and CC3DNonLnCemenous3 (descrbed n secon..), wh he followng dfferences: CC3DCemenous assumes lnear response up o he pon when he falure envelope s reached boh n enson and compresson. Ths means ha here s no hardenng regme n Fg. -. The maeral CC3DNonLnCemenous on he conrary assumes a hardenng regme before he compressve srengh s reached. The maeral CC3DNonLnCemenous s equvalen o CC3DNonLnCemenous bu purely ncremenal formulaon s used (n CC3DNonLnCemenous a oal formulaon s used for he fracurng par of he model), herefore hs maeral can be used n creep calculaons or when s necessary o change maeral properes durng he analyss. The maeral CC3DNonLnCemenousVarable s based on he maeral CC3DNonLnCemenous and allows o defne hsory evoluon laws for seleced maeral parameers. The followng maeral parameers can be defned usng an arbrary evoluon laws: young modulus E, ensle srengh ' ' ' f, compressve srengh f c and f c0. I s he responsbly of he user o defne he parameers n a meanngful way. I means ha a any me: f ' f (.79) ' c0 f f, f 0, f 0 (.80) ' ' ' ' c0 c0 c0 c The maeral CC3DNonLnCemenousUser allows for user defned laws for seleced maeral laws such as: dagrams for ensle and sofenng behavor (see Fg. -4 and Fg. -5), shear reenon facor (Fg. -6) and he effec of laeral compresson on ensle srengh (Fg. -7). /f.0 f ( ) L/L loc ch Fg. -4. An example of a user defned ensle behavor for CC3DNonLnCemenousUser maeral. loc f ~ 44

55 c /f c.0 p c c ( ) L/L eq loc c ch c loc p ~ Fg. -5. An example of a user defned compressve behavor for CC3DNonLnCemenousUser maeral. G/G c eq.0 f sh ( ) L/L loc ch sh loc Fg. -6. An example of a user defned shear reenon facor for shear sffness degradaon afer crackng. In he user defned maeral mode II and III crack sffness are evaluaed wh he help of he shear reenon facor r g as: f ~ r cr g G E jj (.8) r j where j, rg mn( rg, rg ) s he mnmum of shear reenon facors on cracks n drecons, j, and G s he elasc shear modulus. Shear reenon facor on a crack n drecon s evaluaed from he user specfed dagram as shown n Fg. -6. In he above dagrams L and Lc represens he crack band sze and crush band sze respecvely c as s defned Secon..3. L ch and Lch represens a sze for whch he ensle and compresson dagram respecvely s vald. For nsance represens he measurng base ha was used n an expermen o deermne he sran values n he dagrams above. loc represens he sran value, afer whch sran localzaon can be expeced. Usually, hs s he sran afer whch he dagram g ATENA Theory 45

56 s enerng no he sofenng regme. For nsance, he sran value ha s used o deermne he ensle srengh s calculaed based on he followng assumpons: f f f loc else f f f f f f L ( ) loc loc (.8) L ch The calculaon of he sran value for graphs n Fg. -5 and Fg. -6 s analogcal o Eq. (.8) bu he approprae values of loc, L and Lch should be used. I should be noed ha he f sran s he sran ha s calculaed from he sran ensor a he fne elemen negraon f pons, whle he sran s used o deermne he curren ensle srengh from he provded sress-sran dagram (see Fg. -4). The equaon (.8) hen represens a scalng ha akes no accoun he dfference beween he expermenal sze and he sze of he negraon pon. Ths approach guaranees ha he same amoun of energy s dsspaed when usng large and small fne elemens. I s also possble o defne a maeral law for he shear srengh of a cracked concree and for he compressve srengh reducon afer crackng. Compressve srengh of cracked concree ( f c rc ) f c (.83) Shear srengh of cracked concree ( f j f sh ) f (.84) I should be realzed ha he compressve srengh of he cracked concree.e. (.83) s a funcon of he maxmal fracurng sran,.e. maxmal ensle damage a he gven pon. The shear srengh should be a funcon of he crack openng. Because of ha he shear srengh s f specfed as a funcon of he fracurng sran afer he localzaon ransformaon (.8). The shear srengh law s specfed as a value relave o f. The compressve srengh reducon s specfed as a funcon relave o f c. 46

57 /f.0 Fg. -7. An example of a user defned ensle srengh degradaon law due o laeral compressve sress..0 3 /f c..7 Tenson Sffenng In heavly renforced concree srucures he cracks canno fully developed and concree conrbues o he seel sffness. Ths effec s called enson sffenng and n CC3DNonLnCemenous maeral can be smulaed by specfyng a enson sffenng facor c s. Ths facor represens he relave lmng value of ensle srengh n he enson sofenng dagram. The ensle sress canno drop below he value gven by he produc of c s f (see Fg. -8). The recommended defaul value for c s s 0.4 as recommended by CEB-FIP Model Code 990. f c s f ATENA Theory 47 Fg. -8: Tenson sffenng...8 Crack Spacng In heavly renforced concree srucures, or srucures wh large fne elemens, when many renforcemen bars are crossng each fne elemen, he crack band approach descrbed n Secon..3 wll provde oo conservave resuls, and he calculaed crack wdhs may be

58 overesmaed. Ths s he consequence of he fac ha he crack band approach assumes ha he crack spacng s larger han a fne elemen sze. In heavly renforced srucures, or f large fne elemens are used, may occur ha he crack spacng wll be smaller han fne elemen sze. Ths s especally rue f shell/plae elemens are used. In hs case, ypcally large fne elemens can be used, and hey usually conan sgnfcan renforcemen. In hese cases, s useful o provde he crack spacng manually, snce oherwse he program wll overesmae he crackng and due o ha also larger deflecons may be calculaed. The program ATENA allows he user o manually defne he crack spacng. Ths user defned spacng s used as crack band sze L n cases when he user defned crack spacng s smaller han he L ha would be calculaed by formulas presened n Secon Fxed or Roaed Cracks Smlarly o he SBETA maeral, he Cemenous maeral famly offers he choce of fxed and roaed crack models (see secon..6). The fxed crack maeral parameer deermnes a whch maxmum resdual ensle sress level he crack drecon ges fxed. In oher words, 0.0 means fully roaed crack model (as 0 n SBETA),.0 means fxed crack model (as n SBETA), values beween 0.0 and.0 deermne he crack drecon lockng level, e.g., 0.7 fxes he crack drecon as soon opens so far ha he sofenng law drops o 0.7 mes he [nal] ensle srengh...0 Fague For modellng fague behavor of concree (CEB 988 and SAE AE-4) under ensle load, a new maeral has been mplemened n ATENA. The new maeral (CC3DNonLnCemenousFague) s based on he exsng hree-dmensonal fracure plasc maeral (CC3DNonLnCemenous) and uses a sress based model (..0.). I has an addonal parameer, fague, and addonal daa arbues for base, N, and fague, used n he damage calculaon as descrbed n secon..0.. For deals and valdaon agans ess conduced by KESSLER-KRAMER (00) see ČERVENKA, PRYL (007) or PRYL, CERVENKA, PUKL (00). Modellng 3-pon bendng ess wh hs maeral s presened n PRYL, PUKL, CERVENKA (03) and PRYL, D., MIKOLÁŠKOVÁ, J., PUKL, R. (04)...0. Sress Based Models In hs approach he fague s represened by he so called S-N curves relang he appled sress, S, and he number of cycles, N, o falure. Such curves mus be deermned by ess, see Fg. -9. For seel renforcemen bars he performance can be normally expressed as a smple power law by BASQUIN (90). 48 m r N C (.85) where r s he sress range, N s he number of cycles o falure and m and C are consans. Ths means a lnear relaonshp beween and N n a full logarhmc dagram. The equaon (.85) s generally vald for he hgh-cycle range. For plan concree he performance can normally be expressed as a sragh lne n a semlogarhmc dagram of he form: max R log N (.86) f

59 where mn max s he maxmum sress, f s sac concree srengh, R, mn s mnmum max sress and s a maeral consan. The equaon (.86) holds for boh compressve and ensle sresses, however, he value of s no neccessarly he same for ensle and compressve behavor of a maeral. The value should be deermned from expermens. For example, =0.05 was used based on he expermenal resuls for load levels 0.7 and 0.9 F sa when modellng he es on a probe sealed durng curng wh a noch from secon of KESSLER-KRAMER (00) for valdaon. Fg. -9: Typcal S-N lne for concree n compresson (KLAUSEN (978)) The S-N relaons menoned above are manly obaned by consan amplude ess. However, n real srucures he sresses are varyng. One mehod whch can be of help n hs conex s he well-known Palmgren-Mner hypohess PALMGREN (94), MINER (945). k n (.87) N where n s he number of consan amplude cycles a sress level, N s he number of cycles o falure a sress level, and k s he number of sress levels. As a rough ool hs hypohess s useful, especally concernng seel. I can also be used for concree alhough some nvesgaons have suggesed ha a value lower han should be used. ATENA Theory 49

60 ..0. Fague Damage Calculaon In he mplemened model, fague damage consss of a conrbuon based on cyclc sress (..0..), and an addonal conrbuon from crack openng and closng n each cycle (..0..3). The former s domnan before crackng occurs, he laer n already cracked regons Sress Based Conrbuon The number of cycles o falure N s deermned from a smple sress based model, so called S- N or Wöhler curve as descrbed n he prevous secon..0.. upper f fague R logn,.e., upper f fague R N 0, where upper sands for he maxmum ensle or compressve sress and f for he correspondng srengh, f or f c, base R. upper Then, he damage due o fague afer n cycles s calculaed as an ncrease of he maxmum fracurng sran ˆ (see secon..3). The maxmum fracurng sran n each prncpal f j drecon s adjused by addng w n and he falng dsplacemen for he gven sress ElemSze N w nver _ sof _ law( ) (see Fg. -30). fague fague, where w fague w fal fal upper Fg. -30: Sofenng law and fague damage. 50

61 In ATENA 4.0, a sngle value of fague s used o calculae fague damage caused by boh ensle and compressve sresses. So far, here s also no specal provson mplemened for loads crossng zero,.e., changng from enson o compresson and back n each cycle, whch lead o faser damage accordng o expermenal resuls presened n CEB 988 and SAE AE-4. In ha suaon, he damage s calculaed separaely for cyclc loadng from 0 o max. compresson and from 0 o max. enson, and hen he worse of he boh damage values s consdered. I should be also noed ha he damage s only nroduced n form of maxmum fracurng sran, whch has no drec mpac on compressve maeral properes,.e., he fague damage effecvely only has nfluence on ensle behavour of he maeral Sress Based Conrbuon wh Trlnear Damage Hardcoded defnon of damage evoluon durng he fague process, wh he breakpons wf = wfr_ * w fal and wf = wfr_ * w fal. w fague = n * wf / N for n_o < N wf + ((n_o - N) * (wf - wf) / (N - N)) - wf_curr for N <= n_o < N wf + ((n_o - N) * (w fal - wf) / (N - N)) - wf_curr for N <= n_o < N w fal * n_o / N - wf_curr for N <= n_o where n_o = n + N_beg, N_curr = N - N_beg, N_beg = wf_curr * N / wf for wf_curr < wf N + (wf_curr - wf) * (N - N)/(wf - wf) for wf <= wf_curr < wf N + (wf_curr - wf) * (N - N)/( w fal - wf) for wf <= wf_curr 0 < N_beg < N and wfr_ = 0., Nr_ = 0., wfr_ = 0.5, Nr_ = 0.9, N = Nr_ * N, N = Nr_ * N Crack Openng Based Conrbuon The damage due o cracks ha open and close durng he cyclc loadng s deermned as wfaguecod faguecod, where wfaguecod n fague / RCOD cfaguecodload COD, R COD s he crack ElemSze openng rao (smlar o he cycle asymmery rao R used n he sress based conrbuon; wh a boom lm of 0.0), and COD denoes he dfference beween he maxmum and mnmum crack openng durng a cycle. The resulng faguecod s added o fague before he fague damage s nroduced no he maeral. Avalable snce verson In ATENA versons pror o 5..3 and 5.3.4: wfaguecod n fague cfaguecodload COD ATENA Theory 5

62 ..0.3 Brngng n Fague Damage I s recommended o nroduce he fague nduced damage no he unloaded srucure (.e., a he lower sress level). Several oher approaches of nroducng he damage no he model were also esed,.e., nroducng he damage a he upper load level or durng reloadng, bu hey usually brng more convergence problems, especally durng unloadng... Sran Hardenng Cemenous Compose (SHCC, HPFRCC) maeral The CC3DNONLINCEMENTITIOUSSHCC s suable for fbre renforced concree, such as SHCC (Sran Hardenng Cemenous Composes) and HPFRCC or UHPFRC (hgh and ulrahgh performance fber renforced concree) maerals. The heory of hs maeral model s dencal o hose descrbed n Secons The ensle sofenng regme (Fg. -3) and he shear reenon facor (Eq. (.94)) are modfed based on he model, proposed n KABELE, P. (00). Ths model s based on a noon of a represenave volume elemen (RVE), whch conans dsrbued mulple cracks (hardenng) as well as localzed cracks (sofenng) see Fg. -3. a) Mulple crackng regme b) Localzed crackng regme secondary crack se prmary crack se Fg. -3: Represenave volume elemen wh cracks.... Basc Assumpons a) mulple crackng regme (hardenng) 5 A se of parallel planar mulple cracks forms when maxmum prncpal sress max = fc (frs crack srengh). Crack planes are perpendcular o he drecon of max (-axs). The drecon of a crack se s fxed. Secondary crack se may form n drecon perpendcular o prmary se f he maxmum normal sress n he correspondng drecon (-axs) exceeds fc. Cracks may slde f he drecon of prncpal sress changes. Crack openng and sldng are ressed by fber brdgng. Crack openng and sldng dsplacemens are averaged over he RVE as crackng srans mc, mc, (noaon: lower ndces componens of ensor or vecor, upper ndces j, j

63 mulple or localzed crack mc, lc and assocaon wh prmary or secondary crack drecon, ) b) localzed crackng regme (sofenng) A localzed crack forms whn a se of mulple cracks f he correspondng normal crackng sran exceeds he level of mc mb (crackng sran capacy, a maeral consan). Openng and sldng dsplacemens of he, localzed cracks are reaed by he crack lc, lc, band model (.e. hey are ransformed no crackng srans j, j by dvdng hem wh correspondng band wdh w c or w c). The overall sran of he RVE s hen obaned as a sum of sran of maeral beween cracks (whch may possbly conan nonlnear plasc sran due o compressve yeldng), crackng srans due o mulple cracks, and crackng srans due o localzed cracks: j s j s where j represens he sran of he connuous maeral beween cracks. mc, j mc, j lc, j lc, j (.88)... Crack Openng Model The crack-normal sress componens are relaed o crackng srans correspondng o openng of mulple and localzed cracks by pecewse lnear relaons depced n Fg. -3 [alhough lnear hardenng and sofenng are shown, a user should be allowed o npu pecewse lnear curves]. Noe ha for mulple cracks, s assumed ha hey do no close unless exposed o crack-normal compresson (plascy-lke unloadng) whle a localzed crack s assumed o close so ha normal sress decreases lnearly o reach zero a zero COD [hese assumpons may need o be revsed n he fuure o some combnaon of plascy and damage-lke closure]. See also secon..3. mulple crackng regme localzed crackng regme loadng crack openng unloadng/ reloadng mc mb unloadng/ reloadng crackng sran mc crackng sran mc COD Fg. -3: Sress vs. crackng sran relaons n crack-normal drecon....3 Crack Sldng Model The model for crack sldng phenomena s mplemened by means of a varable shear reenon facor The shear reenon facor s defned as a rao of he maeral pos-crackng shear sffness G c o s elasc shear sffness G, ATENA Theory 53

64 c G. (.89) G Le us deermne sffness G c, whle consderng he mos general -D case of an elemen, whch conans wo perpendcular ses of mulple cracks and wo perpendcular localzed cracks. If he problem s defned n plane, hen he oal engneerng shear sran has only one non-zero componen, whch s obaned as: s mc, mc, lc, lc,, (.90) whch can be rewren wh use of he shear brdgng model (Kabele, 000) as: G M mc, M mc, w c L w c L G c (.9) Funcons M and L are defned by Vf kgf M ( ) (.9) Vf kgf L( ), for 0 0 4kG f 3E f d f L( ) 0, for 0 (.93) Here V f s he fber volume fracon, G f s he fber shear modulus, E f s he fber Young s modulus, d f s he fber dameer, and k s he fber cross-secon shape correcon facor. The quany and ndcaes he crack openng n drecon and respecvely. The parameer 0 represens he lmng value of he crack openng dsplacemen, when no ensle sress can be ransferred across he crack,.e. he pon when he sress-dsplacemen dagram n Fg. -3 drops o zero. These parameers are o be suppled by he user wh he excepon of he parameer 0, whch s auomacally exraced from he provded sress-sran law for enson. The shear reenon facor s hen expressed as G M M w L w L mc, mc, ( ) ( ) c ( ) c ( ) Noe ha for an elemen conanng only mulple cracks (before localzaon) /L erms approach zero. For an uncracked elemen, approach zero, gvng =. mc, mc, (.94) 0 and 0 and /M and /L.. Confnemen-Sensve Consuve Model The CC3DNonLnCemenous3 fracure-plasc consuve model s an advanced verson of he CC3DNonLnCemenous maeral ha can handle he ncreased deformaon capacy of concree under raxal compresson. I s suable for problems ncludng confnemen effecs 54

65 such as confned renforced concree members (columns, brdge pers), nuclear vessels and raxal compresson ess of plan concree. A dealed descrpon of he model formulaon s presened n PAPANIKOLAOU and KAPPOS (007). In hs secon, only he man dfferences beween he CC3DNonLnCemenous3 and he CC3DNonLnCemenous model are descrbed, whch are manly focused on he plascy par of he model (secon..4).... Hardenng and Sofenng Funcon The poson of falure surface can expand and move along he hydrosac axs (smulang he hardenng and sofenng sages), based on he value of he hardenng/sofenng parameer (κ). In he presen model, hs parameer denfes wh he volumerc plasc sran (GRASSL e al., 00) : dκ dε dε dε dε (.95) p p p p v 3 The nsananeous shape and locaon of he loadng surface durng hardenng s defned by a hardenng funcon (k), whch depends on he hardenng/sofenng parameer (κ). Ths funcon s drecly ncorporaed n he Menérey-Wllam falure surface equaons (.54), operang as a scalng facor on he compressve concree srengh (f c ). I has he same ellpc form wh CC3DNonLnCemenous (.57), bu heren n erms of he plasc volumerc sran : p p p εv, ε v k(κ) k(ε v) ko ko p ε v, (.96) where p ε v, s he plasc volumerc sran a unaxal concree srengh (onse of sofenng) and k o s he value ha defnes he nal yeld surface ha bounds he nal elasc regme (onse of plascy). A he end of he hardenng process, he hardenng funcon reans a consan value of uny and he maeral eners he sofenng regme, whch s conrolled by he sofenng funcon (c). Ths funcon smulaes he maeral decoheson by shfng he loadng surface along he negave hydrosac axs. I s assumed ha follows he sofenng funcon orgnally proposed by VAN GYSEL and TAERWE (996) for unaxal compresson: where : p c(κ) c(ε v) n n (.97) n ε / ε (.98) p p v v, n (ε ) / ε (.99) p p v, v, Parameer n equaon (.99) conrols he slope of he sofenng funcon and he oumos square s necessary due o he quadrac naure of he loadng surface. The sofenng funcon value sars from uny and complee maeral decoheson s aaned a c = 0. The evoluon of boh hardenng and sofenng funcons wh respec o he hardenng/sofenng parameer s schemacally shown n Fg ATENA Theory 55

66 k(κ) / c(κ).0 c k 0.8 k 0.6 c k o 0.0 ε p v, κ = ε p v Fg. -33: Evoluon of hardenng (k) and sofenng (c) funcons wh respec o he plasc volumerc sran.... Plasc Poenal Funcon The presen plascy model ncorporaes a non-assocaed flow rule usng a polynomal plasc poenal funcon (g), wh Lode angle (θ) dependency and adjusable order (n) : n ρ ρ ξ ga C (B C)( cos3θ) a k c f c k cfc k cfc (.00) Parameers A, B and C defne he shape of he plasc poenal funcon n sress space and her calbraon s based on he assumpon ha he nclnaon (ψ) of he ncremenal plasc sran vecor denfes wh he nclnaon of he oal plasc sran vecor a hree dsnc sress saes, namely he unaxal, equbaxal and raxal compressve concree srengh (Fg. -34). The aracon consan (a) s ncluded for mahemacal clary and s no a user parameer, due o plasc poenal funcon dfferenaon n he flow rule. 56

67 Fg. -34: Drecon (ψ) of he ncremenal (a) and oal (b) plasc sran vecors....3 Suggesed Model Parameers A dealed calbraon scheme for he plascy model parameers, based on and exensve expermenal daabase can be found n PAPANIKOLAOU and KAPPOS (007) and suggesed values (ncludng he fracure model parameers) for varous unaxal compressve concree srenghs (f c ) are shown n he followng able (see Aena Inpu Fle Forma documen for he maeral defnon deals): Table.- Suggesed parameers for he fracure and plascy models f c (ΜPa) Ε c (MPa) ν f (MPa) λ e f co (MPa) p ε v, A B C n G f (MN/m) ATENA Theory 57

68 f c (ΜPa) Ε c (MPa) ν f (MPa) λ e f co (MPa) p ε v, A B C n G f (MN/m) Von Mses Plascy Model Von Mses plascy model called also as J plascy s based only on one parameer k. The yeld funcon s defned as: p p F ( ) J k 0 (.0) j where J denoes he second nvaran of sress devaor ensor. The parameer p p k eq 3 y eq s he maxmal shear sress and y s he unaxal yeld sress. Ths parameer conrols he soropc hardenng of he yeld creron. p H p p p p 3 y eq y eq eq eq N nc, : ε ε (.0) y s he yeld sress, H he hardenng modulus and p eq s he equvalen plasc sran calculaed as a summaon of equvalen plasc srans durng he loadng hsory. In case of von Mses plascy he plasc poenal funcon s dencal wh he yeld funcon: p P G ) F ( ) (.03) ( j j The assocaed flow rule s assumed. The background nformaon can be found n (CHEN, SALEEB 98, Sec.5.4.). The Von Mses model could be used o model cyclc seel behavor ncludng Bauschnger effec. In hs case he yeld funcon s modfed as: 58

69 p eq σ X : σ X k ( r) k 0 (.04) 0 p where σ s he devaorc sress, k 0 s an nal value of k( eq ) accordng o (.0), X s he so called back sress conrollng he knemac hardenng: p p X 3 k ε k X eq (.05) In equaons (.04) and (.05) quanes rk,, k are maeral parameers for he cyclc response. If r s non-zero he cyclc model s acvaed and conrols he radus of he Von Mses surface. If r he yeldng wll sar exacly when s reached. For lower values he non-lnear behavor sars earler and he slope of he response s manly affeced by parameer k (larger value hgher slope). Parameer k on he oher hand affecs he memory of he cyclc response. Some examples of varous parameer combnaons are shown a Fg y ATENA Theory 59

70 Fg. -35: Effec of maeral parameer choce on cyclc response for E=0 GPa and y = 00 MPa. 60

71 .4 Drucker-Prager Plascy Model Drucker-Prager plascy model s based on a general plascy formulaon ha s descrbed n Secon..4. The yeld funcon s defned as: p FDP ( j ) I J k 0 (.06) Where and k are parameers defnng he shape of he falure surface. They can be esmaed by machng wh he Mohr-Coulomb surface. If he wo surface are o agree along he 0 compressve merdan,.e. 0, he formulas are: sn 6c cos, k 3 3 sn 3 3 sn (.07) Ths corresponds o a ouer cone o he Mohr-Coulomb surface. The nner cone, whch passes 0 hrough he ensle merdan where 60 has he consans gven by he followng expressons: sn 6c cos, k 3 3 sn 3 3 sn (.08) The poson of falure surfaces s no fxed bu can move dependng on he value of sran hardenng/sofenng parameer. The sran hardenng s based on he equvalen plasc sran, whch s calculaed accordng o he followng formula. p p mn( ) (.09) eq Hardenng/sofenng n he Drucker-Prager model s conrolled by he parameer k. Ths parameer s seleced such ha he surface a he peak passes hrough he unaxal compressve srengh, and changes accordng o he followng expresson. p f c ( eq ) k' k f c j (.0) The symbol k n he above formula replaces k n (.06). In he above wo formulas he p expresson f c ( eq ) ndcaes he hardenng/sofenng law, whch s based on he unaxal compressve es. The law s shown n Fg Fg Lnear sofenng n he Drucker-Prager maeral model Reurn drecon s gven by he followng plasc poenal: G p ( j ) I J (.) 3 ATENA Theory 6

72 where deermnes he reurn drecon. If 0 maeral s beng compaced durng crushng, f 0 maeral volume s preserved, and f 0 maeral s dlang. In general he plasc model s non-assocaed, snce he plasc flow s no perpendcular o he falure surface The reurn mappng algorhm for he plasc model s based on predcor-correcor approach as s shown n Fg. -. Durng he correcor phase of he algorhm he falure surface moves along he hydrosac axs o smulae hardenng and sofenng. The fnal falure surface has he apex locaed a he orgn of he Hagh-Vesergaard coordnae sysem. Secan mehod based Algorhm s used o deermne he sress on he surface, whch sasfes he yeld condon and also he hardenng/sofenng law..5 User Maeral Model In some suaons, none of he sandard maeral models avalable n ATENA can descrbe he behavor suffcenly. Many such cases can be handled by defnng user laws n he fracureplasc maeral model (see CC3DNonLnCemenousUser descrbed n secon..6), n he ohers he user can provde a dynamc lnk lbrary mplemenng hs own maeral model. The user maeral s based on he elasc soropc maeral, addng new maeral parameers and sae varables (boh lmed o floang pon values). See he User Maeral DLL Manual for descrpon and reference, and he CCUserMaeralExampleDLL drecory n Aena Scence Examples for an example projec ncludng he source code n C and a wndows help fle verson of he manual, AenaV4_UserMaeralDLL.chm. Please noe ha he behavor of he user model may have nfluence on convergence of he analyss..6 Inerface Maeral Model The nerface maeral model can be used o smulae conac beween wo maerals such as for nsance a consrucon jon beween wo concree segmens or a conac beween foundaon and concree srucure. The nerface maeral s based on Mohr-Coulomb creron wh enson cu off. The consuve relaon for a general hree-dmensonal case s gven n erms of racons on nerface planes and relave sldng and openng dsplacemens. K 0 0 v 0 K 0 v 0 0 K nn u (.) For wo-dmensonal problems second row and column are omed. The nal falure surface corresponds o Mohr-Coulomb condon (.3) wh ellpsod n enson regme. Afer sresses volae hs condon, hs surface collapses o a resdual surface whch corresponds o dry frcon. c, 0 (.3) c 0 f c, 0 c f c c f, c, 0 f c f 0, f 6

73 In enson he falure creron s replaced by an ellpsod, whch nersec he normal sress axs a he value of f wh he vercal angen and he shear axs s nerseced a he value of c (.e. coheson) wh he angen equvalen o. The parameers for he nerface model canno be defned arbrarly; here s ceran dependence of he some parameers on he ohers. When defnng he nerface parameers, he followng rules should be observed: c f, f c c0, f 0, 0 (.4) I s recommended ha parameers c, f, are always greaer han zero. In cases when no coheson or no ensle srengh s requred, some very small values should be prescrbed. Tral sress Fnal sress c f Inal surface Resdual surface Fg. -37: Falure surface for nerface elemens. In general hree-dmensonal case n Fg. -37 and equaon (.3) s calculaed as: (.5) ATENA Theory 63

74 c mn K K (a) v f K nn mn K nn (b) u Fg. -38: Typcal nerface model behavor n shear (a) and enson (b) The K nn, K denoe he nal elasc normal and shear sffness respecvely. Typcally for zero hckness nerfaces, he value of hese sffnesses correspond o a hgh penaly number. I s recommended no o use exremely hgh values as hs may resul n numercal nsables. I s recommended o esmae he sffness value usng he followng formulas E G Knn, K (.6) where E and G s mnmal elasc modulus and shear modulus respecvely of he surroundng maeral. s he wdh of he nerface zone. Is value can be seleced eher on he bass of he realy. For nsance for morar beween masonry brcks he value s ypcally 0-0 mm. Alernavely, can be esmaed as a dmenson, whch can be consdered neglgble wh respec o he srucural sze. For nsance n case of a dam analyss, where he dam dmensons are ypcally n he order of 00 meers, he wdh of he nerface zone can be esmaed o be

75 meers. I s suable due o numercal reasons f sffness s abou 0 mes of he sffness of adjacen fne elemens. There are wo addonal sffness values ha need o be specfed n he ATENA npu. They are denoed n Fg. -38 as K and K. They are used only for numercal purposes afer he mn nn mn falure of he elemen n order o preserve he posve defneness of he global sysem of equaons. Theorecally, afer he nerface falure he nerface sffness should be zero, whch would mean ha he global sffness wll become ndefne. These mnmal sffnesses should be abou 0.00 mes of he nal ones. I s possble o defne evoluon laws for ensle as well as shear sofenng by arbrary mullnear laws. Examples of such laws are shown n Fg The fgure descrbes b-lnear sofenng laws. The break pon of hs law can be deermned for nsance by he formula proposed by Bruehwler and Wman (990). f GF s, v 0.75 (.7) 4 f f cc 0 s G F I s c II G F u u v Fg. -39: Example of a sofenng law for enson and coheson. u eq f The evoluon law depends on he equvalen nonlnear nerface relave dsplacemen c v u eq f u u v v n 3D and f eq f f f u u v n D (.8) f eq f f Where u f and v f are he nelasc componens of he relave nerface dsplacemen on he bass of her decomposon no elasc and nonlnear,.e. fracurng par. u ue uf v v v f (.9) Ths approach ensures ha he degradaon n shear affecs also ensle srengh and vce versa. For nsance, when he nerface s damaged n shear, he ensle srengh s reduced as well. The ypcal behavor of he nerface model wh he sofenng evoluon laws s shown n Fg. -38 by he doed lnes. The defaul behavor when no sofenng law s gven s brle wh mmedae drop o zero n enson and o he resdual dry frcon n shear. The behavor s shown n Fg. -38 by he sold black lne. When user sofenng laws are defned for he nerface maeral, s recommended ha he sofenng law for coheson s always more ducle hen he one for ensle srengh,.e. he coheson should be hgher han he ensle srengh a any me durng he sofenng process. ATENA Theory 65

76 Fg. -40: Example of a cyclc response of he model n shear under consan normal pre-sress..7 Renforcemen Sress-Sran Laws.7. Inroducon Renforcemen can be modeled n wo dsnc forms: dscree and smeared. Dscree renforcemen s n form of renforcng bars and s modeled by russ elemens. The smeared renforcemen s a componen of compose maeral and can be consdered eher as a sngle (only one-consuen) maeral n he elemen under consderaon or as one of he more such consuens. The former case can be a specal mesh elemen (layer), whle he laer can be an elemen wh concree conanng one or more renforcemens. In boh cases he sae of unaxal sress s assumed and he same formulaon of sress-sran law s used n all ypes of renforcemen. More nfo abou dscree renforcemen s avalable n Secon 0..3 Dscree Renforcemen Embedded n Sold Elemens, locaed near he end of hs manual..7. Blnear Law The blnear law, elasc-perfecly plasc, s assumed as shown n Fg

77 Fg. -4 The blnear sress-sran law for renforcemen. The nal elasc par has he elasc modulus of seel E s. The second lne represens he plascy of he seel wh hardenng and s slope s he hardenng modulus E sh. In case of perfec plascy E sh =0. Lm sran L represens lmed ducly of seel..7.3 Mul-lne Law The mul-lnear law consss of four lnes as shown n Fg. -4. Ths law allows o model all four sages of seel behavor: elasc sae, yeld plaeau, hardenng and fracure. The mul-lne s defned by four pons, whch can be specfed by npu. Fg. -4 The mul-lnear sress-sran law for renforcemen. The above descrbed sress-sran laws can be used for he dscree as well as he smeared renforcemen. The smeared renforcemen requres wo addonal parameers: he renforcng rao p (see Secon...) and he drecon angle as shown n Fg ATENA Theory 67

78 Fg. -43 Smeared renforcemen. The spacng s of he smeared renforcemen s assumed nfnely small. The sress n he smeared renforcemen s evaluaed n he cracks, herefore should nclude also a par of sress due o enson sffenng (whch s acng n concree beween he cracks, secon..9). where ' ' scr s s (.0) ' s s he seel sress beween he cracks (he seel sress n smeared renforcemen), s he seel sress n a crack. If no enson sffenng s specfed s =0 and ' he dscree renforcemen he seel sress s always s..7.4 No Compresson Renforcemen ' scr ' s ' scr. In case of Normally all renforcemen maeral models n ATENA exhb he same behavor n enson as well as n compresson. The maeral ypes CCRenforcemen and CCSmearedRenforcemen nclude he capably o deacvae he compressve response of he renforcemen. Ths s somemes useful, f hs maeral model s used o smulae he behavor of renforcemen elemens ha have a very low bendng sffness, so can be assumed ha when he renforcemen s loaded by compressve forces, bucklng occurs and he srengh of he elemens n compresson s neglgble. Ths s conrolled by he command COMPRESSION 0 or, whch deacvaes and acvaes he compressve response respecvely (for more deals see ATENA Inpu Fle Forma)..7.5 Cyclc Renforcemen Model The renforcng seel sress-sran behavor can be descrbed by he nonlnear model of Menegoo and Pno (973). In ATENA hs model s exended o accoun of he soropc hardenng due o an arbrary hardenng law ha can be specfed for renforcemen (see Secons.7.,.7.3). The sress n he cyclc model s calculaed accordng o he followng expresson. where * * b * b * R * (.) R, 0 r r c, R R * r 0 0 r c (.) 68

79 where R 0, c and c are expermenally deermned parameers. The Fg. -44 shows he meanng of sran values r, 0, and sress values r and 0. These values changes for each cycle. The values wh he subscrp r ndcae he pon where he cycle sared, and he subscrp 0 ndcaes he heorehcal yeld pon ha would be reached durng he unloadng f he response would no have been modfed by he hyserec behavor. Durng he calculaon of hs pon he maeral sress-sran law s consdered (see Secons.7.,.7.3) f R eq *, eq N ncr. eq (.3) Fg. -44: Cyclc renforcemen model based on Menegoo and Pno (973)..8 Renforcemen Bond Models The basc propery of he renforcemen bond model s he bond-slp relaonshp. Ths relaonshp defnes he bond srengh (coheson) b dependng on he value of curren slp beween renforcemen and surroundng concree. ATENA conans hree bond-slp models: accordng o he CEB-FIB model code 990, slp law by Bgaj and he user defned law. In he frs wo models, he laws are generaed based on he concree compressve srengh, renforcemen dameer and renforcemen ype. The mporan parameers are also he confnemen condons and he qualy of concree casng. ATENA Theory 69

80 .8. CEB-FIP 990 Model Code b Fg. -45: Bond-slp law by CEB-FIP model code 990. b max s s, 0 s s (.4) b max b, max max f s s s (.5) s s s s 3, 3 s s s (.6) b, f s3 s (.7) 70

81 Table.8-: Parameers for defnng he mean bond srengh-slp relaonshp for rbbed bars Value Unconfned concree* Confned concree** Bond condons Bond condons Good All oher cases Good All oher cases S 0.6 mm 0.6 mm.0 mm S 0.6 mm 0.6 mm 3.0 mm S 3.0 mm.5 mm clear rb spacng max.0 f.0 C f.5 C f.5 C f C f 0.5 max * Falure by splng of he concree **Falure by shearng of he concree beween he rbs 0.40 max Table.8-: Parameers for defnng he bond srengh-slp relaonshp for smooh bars. Values Cold drawn wre Ho rolled bars s s s 3 Bond condons Bond condons Good All oher cases Good All oher cases 0.0 mm 0. mm max f 0. f 0.05 C f 0.3 C f 0.5 C f C.8. Bond Model by Bgaj The second pre-defned bond model avalable n ATENA s based on he work by BIGAJ 999. Ths model depends on he bond qualy, concree cubc compressve srengh f ' cu and renforcemen bar radus D. The slp law for hs model s shown n Fg ATENA Theory 7

82 3 4 Fg. -46: Bond law by BIGAJ 999 The ascendng par of he sress-slp law.e. par a s modeled by a b-lnear curve. The coordnaes of he four pons defnng hs sress-slp relaonshp are lsed n he able below. Table.8-3: Parameers for defnng he bond srengh-slp relaonshp for smooh bars. Concree Type ' f c < 60 ' f c > 60 7 Bond qualy Excelen Good Bad Excelen Good Bad Pon Pon Pon 3 Pon 4 s/ D / 0.8 f b ' cu s/ D / 0.8 f b ' cu s/ D / 0.8 f b ' cu s/ D / 0.88 f b ' cu s/ D / 0.88 f b ' cu s/ D / 0.88 f b ' cu

83 .8.3 Memory Bond Maeral The Memory Bond maeral s an mprovemen o beer capure he response durng cyclc loadng and unloadng n general. I can be used wh any of he above menoned bond sregh bond slp envelope funcons. The response only dfferes afer he bond sress sgn changes. Insead of followng he same envelope as durng loadng, he maxmum bond sress s deermned by he addonal paramaer, see Fg Admssble values are res max, where res s he resdual bond sress (las value from he bond sregh bond slp funcon) and max he maxmum bond sress (max. value from he bond srengh bond slp funcon). In he fgure, s s he curren slp value, s max he maxmum of he absolue slp value ever reached (damage varable), f () s s he bond srengh funcon. Fg. -47: Memory Bond workng dagram The response for a slp change s s s s defned separaely for cases: () Loadng range s smax f () s () Unloadng range -s max < s < s max s 0 s 0.9 Mcroplane Maeral Model (CCMcroplane4) The basc dea of he mcroplane model s o abandon consuve modellng n erms of ensors and her nvarans and formulae he sress-sran relaon n erms of sress and sran vecors on planes of varous orenaons n he maeral, now generally called he mcroplanes. Ths dea arose n G.I. Taylor s (TAYLOR 938) poneerng sudy of hardenng plascy of polycrysallne meals. Proposng he frs verson of he mcroplane model, BAZANT 984, n ATENA Theory 73

84 order o model sran sofenng, exended or modfed Taylor s model n several ways (n deal see BAZANT e al. 000), among whch he man one was he knemac consran beween he sran ensor and he mcroplane sran vecors. Snce 984, here have been numerous mprovemens and varaons of he mcroplane approach. A dealed overvew of he hsory of he mcroplane model s ncluded n BAZANT e al 000 and CANER and BAZANT 000. In wha follows, we brefly revew he dervaon of he mcroplane model ha s used n hs work. In he mcroplane model, he consuve equaons are formulaed on a plane, called mcroplane, havng an arbrary orenaon characerzed by s un normal n. The knemac consran means ha he normal sran N and shear srans M, L on he mcroplane are calculaed as he projecons of he macroscopc sran ensor j : N nn jj, M mn j mjnj, L ln j ljnj (.8) where m and l are chosen orhogonal vecors lyng n he mcroplane and defnng he shear sran componens. The consuve relaons for he mcroplane srans and sresses can be generally saed as: ( ) F ( ), ( ), ( ) N 0 N L M ( ) G ( ), ( ), ( ) M 0 N L M ( ) G ( ), ( ), ( ) L 0 N L M (.9) where F and G are funconals of he hsory of he mcroplane srans n me. For a dealed dervaon of hese funconals a reader s referred o BAZANT e al 000 and CANER and BAZANT 000. The macroscopc sress ensor s obaned by he prncple of vrual work ha s appled o a un hemsphere. Afer he negraon, he followng expresson for he macroscopc sress ensor s recovered (BAZANT 984): 3 L (.30) N m ( ) M j sj d6 w sj, wheresj Nnn j mn j mjn ln j ljn where he negral s approxmaed by an opmal Gaussan negraon formula for a sphercal surface; numbers label he pons of he negraon formula and w are he correspondng opmal weghs..9. Equvalen Localzaon Elemen The objecve of he equvalen localzaon elemen s o acheve equvalence wh he crack band model. Ths basc dea s ha he maeral properes and parameers of he sofenng maeral model are no modfed o accoun for he dfferences n he fne elemen sze, bu raher he sofenng crack band s coupled n seres wh an elascally behavng layer, n order o oban equvalence. For brevy, hs layer wll henceforh be called he `sprng. For large fne elemens, he effecve lengh of hs added elasc sprng, represenng he hckness of he added elasc layer havng he elasc properes of he maeral, wll be much larger han he sze (or hckness) of he localzaon zone (crack band). Thus, afer he crack naon, he energy sored n he elasc sprng can be readly ransferred o he localzaon zone and dsspaed n he sofenng (.e., fracurng) process. 74

85 Insde each fne elemen a each negraon pon, an equvalen localzaon elemen s assumed. The localzaon elemen s a seral arrangemen of he localzaon zone, whch s loadng, and an elasc zone (sprng), whch s unloadng. The oal lengh of he elemen s equvalen o he crack band sze L (wdh), and can be deermned usng he same mehods as descrbed n Secon..3 (see Fg. -). The wdh of he localzaon zone s gven eher by he characersc lengh of he maeral or by he sze of he es specmen for whch he adoped maeral model has been calbraed. The hree-dmensonal equvalen elemen s consruced by hree seral arrangemens of he elasc zone (sprng) and localzaon band. The sprng-band sysems are perpendcular o each oher, and hey are arranged parallel o he prncpal sran drecons (Fg. -48). The smplfed wo-dmensonal verson s shown n Fg In hs arrangemen of sprng-band sysems s possble o denfy he followng unknown sresses and srans:,,, and,,, b u u 3 u b u u 3 u j j j j j j j j where superscrp b denoes he quanes n he localzaon band and he symbol m x u wh superscrps u and m defnes he quanes n he elasc sprng n he drecon m., u j u j 3 L b j, b j, u j u j 3 h, u u j j h h L L Fg. -48: The arrangemen of he hree-dmensonal equvalen localzaon elemen. ATENA Theory 75

86 Fne elemen Localzaon elemen j u Elasc sprngs h j b j u L Localzaon band h L Fg. -49: The smplfed wo-dmensonal vew of he sprng-band arrangemen. Ideally, he chosen drecons should be perpendcular o he planes of falure propagaon. In ATENA, s assumed for hem o be algned wh he prncpal axes of he oal macroscopc sran ensor, whch n mos cases should approxmaely correspond o he above requremen. Alogeher here are 48 unknown varables. In he subsequen dervaons, s assumed ha hese sresses and srans are defned n he prncpal frame of he oal macroscopc sran ensor. The se of equaons avalable for deermnng hese varables sars wh he consuve formulae for he band and he elasc sprngs: b F( ) (.3) b j m u m u j jlk kl j D for m...3 (.3) The frs formula (.3) represens he evaluaon of he non-lnear maeral model, whch n our case s he mcroplane model for concree. The second equaon (.3) s a se of hree elasc consuve formulaons for he hree lnear zones (sprngs) ha are nvolved n he arrangemen a Fg Ths provdes he frs 4 equaons, whch can be used for he calculaon of unknown srans and sresses. The second se of equaons s provded by he knemac consrans on he sran ensors. 76

87 b u h L h L b u h L h L b 3 3 u h 33 L h L b u b u h L h h L h L L b u b 3 3 u h 3 L h 3 3 h 3 L h L L b u b 3 3 u h 3 L h 3 3 h 3 L h L L These 6 addonal equaons can be wren symbolcally as: b u b j j u j j j j h j L h j j h j L h L L (.33) (.34) The nex se of equaons s obaned by enforcng equlbrum n each drecon beween he correspondng sress componens n he elasc zone and n he localzaon band. For each drecon m, he followng condon mus be sasfed: e e for m...3 (.35) b m m u m j j j j where m e denoes coordnaes of a un drecon vecor for prncpal sran drecon m. Snce j he prncpal frame of he oal macroscopc sran ensor s used he un vecors have he followng coordnaes:, 0, 0, 0,, 0, 0, 0, e e e (.36) 3 j j j The remanng equaons are obaned by enforcng equlbrum beween racons on he oher surfaces of he band and he elasc zone (layer) magned as a sprng: e e where m..3, n...3, m n (.37) b m n u m j j j j The equaon (.37) s equvalen o a sac consran on he remanng sress and sran componens of he elasc sprngs. Formulas (.35) and (.37) ogeher wh he assumpon of sress ensor symmery represen he remanng 8 equaons ha are needed for he soluon of he hree-dmensonal equvalen localzaon elemen. These 8 equaons can be wren as: b m u j j for m...3 (.38) b Ths means ha he macroscopc sress mus be equal o j,.e., he sress n he localzaon elemen, and ha he sresses n all he hree elasc zones mus be equal o each oher and o he mcroplane sress. Ths mples also he equvalence of all he hree elasc sran ensors. b j Based on he foregong dervaons, s possble o formulae an algorhm for he calculaon of unknown quanes n he hree-dmensonal equvalen localzaon elemen. b u Inpu:,,, (.39) j j j j Inalzaon: (.40) b u j j j ATENA Theory 77

88 Sep : Sep : j j u() L h L h ( ) dj C j jklrkl (.4) L L u() u( ) u() j j d j (.4) Sep 3: j j j j b() L L L L L h L h u j j j j j j j L h L h L h L h (.43) Sep 4: () b() u() j j j r (.44) where Cjlk s he complance ensor. The above erave process s conrolled by he followng convergence crera; u () () () T u () dj rj rj dj e, e, e (.45) b b j j j j b The macroscopc sress s hen equal o he sress n he localzaon band j. More deals abou he dervaons of he above algorhm as well as varous examples of applcaon can be obaned from he orgnal reference CERVENKA e al I should be noed ha he descrbed equvalen localzaon elemen s used only f he calculaed crack band sze L (see Secon..3) n each prncpal sran drecon s larger han he prescrbed localzaon band sze h. For smaller elemen szes he equvalen localzaon approach s no used and mesh-dependen resuls may be obaned..0 References BASQUIN, H.O. (90), The exponenal law of endurance ess, Proc. ASTM, 0 (II). BAZANT, Z.P, OH, B.H (983) - Crack Band Theory for Fracure of Concree, Maerals and Srucures, RILEM, Vol. 6, BAŽANT, Z.P., (984), Mcroplane model for sran conrolled nelasc behavor, Chaper 3 n Mechancs of Engneerng Maerals (Proc., Conf. held a U. of Arzona, Tucson, Jan. 984), C.S. Desa and R.H. Gallagher, eds., J. Wlley, London, BAŽANT, Z.P., CANER, F.C., CAROL, I., ADLEY, M.D., AND AKERS, S.A., (000), Mcroplane Model M4 for Concree: I. Formulaon wh Work-Conjugae Devaorc Sress, J. of Engrg. Mechancs ASCE, 6 (9), BIGAL, A.J (999) - Srucural Dependence of Roaon Capacy of Plasc Hnges n RC Beams and Slabs, PhD Thess, Delf Unversy of Technogy, ISBN BRUEHWILER, E., and WITTMAN, F.H. (990), The Wedge Splng Tes, A New Mehod of Performng Sable Fracure-Mechancs Tess, Engneerng Fracure Mechancs, Vol. 35, No. -3, pp CANER, F.C., AND BAŽANT, Z.P., (000) Mcroplane Model M4 for Concree: II. algorhm and calbraon.", J. of Engrg. Mechancs ASCE, 9 (9), CEB-FIP Model Code 990, Frs Draf, Comee Euro-Inernaonal du Beon, Bullen d'nformaon No. 95,96, Mars. CEB 988, Bullen D Informaon No 88, Fague of concree srucures, Sae of he ar repor. 78

89 CERVENKA, V., GERSTLE, K. (97) - Inelasc Analyss of Renforced Concree Panels: () Theory, () Expermenal Verfcaon and applcaon, Publcaons IABSE, Zürch, V.3-00, 97, pp.3-45, and V.3-II,97, pp CERVENKA, V. (985) - Consuve Model for Cracked Renforced Concree, Journal ACI, Proc. V.8, Nov-Dec., No.6,pp CERVENKA, V., PUKL, R., ELIGEHAUSEN, R. (99) - Fracure Analyss of Concree Plane Sress Pull-ou Tess, Proceedngs, Fracure process n Brle Dsordered Maerals, Noordwjk, Holland, June 9-. CERVENKA, V., PUKL, R., OZBOLT, J., ELIGEHAUSEN, R. (995), Mesh Sensvy Effecs n Smeared Fne Elemen Analyss of Concree Srucures, Proc. FRAMCOS, 995, pp CERVENKA, V., PUKL, R. (99) - Compuer Models of Concree Srucures, Srucural Engneerng Inernaonal, Vol., No., May 99. IABSE Zürch, Swzerland, ISSN , pp CERVENKA, V., PUKL, R., OZBOLT, J., ELIGEHAUSEN, R. (995) - Mesh Sensvy Effecs n Smeared Fne Elemen Analyss of Concree Fracure, Proceedngs of FRAMCOS, Zurch, Aedfcao. CERVENKA, V., CERVENKA, J. (996) - Compuer Smulaon as a Desgn Tool for Concree Srucures, ICCE-96, proceedngs of The second Inernaonal Conference n Cvl Engneerng on Compuer Applcaons Research and Pracce, 6-8 Aprl, Bahran. CERVENKA, J, CERVENKA, V., ELIGEHAUSEN, R. (998), Fracure-Plasc Maeral Model for Concree, Applcaon o Analyss of Powder Acuaed Anchors, Proc. FRAMCOS 3, 998, pp ČERVENKA, J., BAŽANT Z.P., WIERER, M., (004), `Equvalen Localzaon Elemen for Crack Band Approach o Mesh Sensvy n Mcroplane Model, submed for publcaon, In. J. for Num. Mehods n Engneerng. ČERVENKA, J., PRYL, D., (007), `Fague Modellng of Crack Growh by Fne Elemen Mehod and Smeared Crack Approach, Inernal Repor DP, Cervenka Consulng. CRISFIELD, M.A., WILLS, J. (989)- The Analyss of Renforced Concree Panels Usng Dfferen Concree Models, Jour. of Engng. Mech., ASCE, Vol 5, No 3, March, pp CRISFIELD, M.A. (983) - An Arc-Lengh Mehod Includng Lne Search and Acceleraons, Inernaonal Journal for Numercal Mehods n Engneerng, Vol.9,pp CHEN, W.F, SALEEB, A.F. (98) - Consuve Equaons For Engneerng Maerals, John Wlley \& Sons, ISBN DARWIN, D., PECKNOLD, D.A.W. (974) - Inelasc Model for Cyclc Baxal Loadng of Renforced Concree, Cvl Engneerng Sudes, Unversy of Illnos, July. DE BORST, R. (986), Non-lnear analyss of frconal maerals, Ph.D. Thess, Delf Unversy of Technology, 986. DRUCKER, D.C., PRAGER, W., Sol Mechancs and Plasc Analyss or Lm Desgn, Q. Appl. Mah., 95, 0(), pp DYNGELAND, T. (989) - Behavor of Renforced Concree Panels, Dsseraon, Trondhem Unversy, Norway, BK-repor 989: ATENA Theory 79

90 FEENSTRA, P.H., Compuaonal Aspecs of B-axal Sress n Plan and Renforced Concree. Ph.D. Thess, Delf Unversy of Technology, 993. FEENSTRA, P.H., ROTS, J.G., AMESEN, A., TEIGEN, J.G., HOISETH, K.V., A 3D Consuve Model for Concree Based on Co-roaonal concep. Proc. EURO-C 998,, pp. 3-. ETSE, G., Theoresche und numersche Unersuchung zum dffusen und lokalseren Versagen n Beon, Ph.D. Thess, Unversy of Karlsruhe 99. FELIPPA, C. (966) - Refned Fne Elemen Analyss of Lnear and Nonlnear Two- Dmensonal Srucures, Ph.D. Dsseraon, Unversy of Calforna, Engneerng, pp GRASSL, P., LUNDGREN, K., and GYLLTOFT, K. (00) Concree n compresson : A plascy heory wh a novel hardenng law, Inernaonal Journal of Solds and Srucures, 39(0), VAN GYSEL, A., and TAERWE, L. (996) Analycal formulaon of he complee sresssran curve for hgh srengh concree, Maerals and Srucures, RILEM, 9(93), HARTL, G. (977) De Arbelne Engebeee Saehle be ers und kurz=belasung, Dssraon, Unvbersae Innsbruck HORDIJK, D.A. (99) - Local Approach o Fague of Concree, Docor dsseraon, Delf Unversy of Technology, The Neherlands, ISBN 90/ KABELE, P. (00) - Equvalen Connuum Model of Mulple Crackng, Engneerng Mechancs 00, 9 (/), pp.75-90, Assoc.for Engneerng Mechancs, Czech Republc KESSLER-KRAMER, CH., (00) Zugverhalen von Beon uner Ermüdungsbeanspruchung, Schrfenrehe des Insus für Massvbau und Bausoffechnologe, Hef 49, Karlsruhe. KLAUSEN, D. (978), Fesgke und Schadgung von Beon be haufg wederholer Beanschpruchung, PhD Thess, Unversy of Technology Darmsad, 85 pp. KOLLEGGER, J. - MEHLHORN, G. (988) - Expermenelle und Analysche Unersuchungen zur Aufsellung enes Maeralmodels für Gerssene Sahbeonscheben, Nr.6 Forschungsberch, Massvbau, Gesamhochschule Kassel. KOLMAR, W. (986) - Beschrebung der Krafueberragung über Rsse n nchlnearen Fne- Elemen-Berechnungen von Sahlbeonragwerken", Dsseraon, T.H. Darmsad, p. 94. KUPFER, H., HILSDORF, H.K., RÜSCH, H. (969) - Behavor of Concree under Baxal Sress, Journal ACI, Proc. V.66,No.8, Aug., pp MARGOLDOVA, J., CERVENKA V., PUKL R. (998), Appled Brle Analyss, Concree Eng. Inernaonal, November/December 998. MENETREY, P., WILLAM, K.J. (995), Traxal falure creron for concree and s generalzaon. ACI, Srucural Journal, 995, 9(3), pp MENETREY, Ph., WALTHER, R., ZIMMERMAN, Th., WILLAM, K.J., REGAN, P.E. Smulaon of punchng falure n renforced concree srucures. Journal of Srucural Engneerng, 997, 3(5), pp MIER J.G.M van (986) - Mulaxal Sran-sofenng of Concree, Par I: fracure, Maerals and Srucures, RILEM, Vol. 9, No.. MINER M.A. (945), Cumulave damage n fague. Transacons of he Amercan Socey of Mechancal Engneerng, 67:A59-A64. 80

91 OLIVIER, J., A Conssen Characersc Lengh For Smeared Crackng Models, In. J. Num. Meh. Eng., 989, 8, pp OWEN, J.M., FIGUEIRAS, J.A., DAMJANIC, F., Fne Elemen Analyss of Renforced and Pre-sressed concree srucures ncludng hermal loadng, Comp. Meh. Appl. Mech. Eng., 983, 4, pp PALMGREN, A. (94), De Lebensdauer von Kugellagern. Zeschrf Veren Deuscher Ingeneure, 68(4): PAPANIKOLAOU, V.K., and KAPPOS, A.J. (007) Confnemen-sensve plascy consuve model for concree n raxal compresson, Inernaonal Journal of Solds and Srucures, 44(), PRAMONO, E, WILLAM, K.J., Fracure Energy-Based Plascy Formulaon of Plan Concree, ASCE-JEM, 989, 5, pp PRYL, D., CERVENKA, J., and PUKL, R. (00) Maeral model for fne elemen modellng of fague crack growh n concree, Proceda Engneerng, (00) 03. PRYL, D., PUKL, R., and CERVENKA, J. (03) Modellng hgh-cycle fague of concree specmens n hree pon bendng, Lfe-Cycle and Susanably of Cvl Infrasrucure Sysems (Eds. Srauss, Frangopol & Bergmeser) PRYL, D., MIKOLÁŠKOVÁ, J., PUKL, R. (04) Modelng Fague Damage of Concree, Key Engneerng Maerals, 04 Vols , pp , ISSN: RAMM, E. (98) - Sraeges for Tracng Non- lnear Responses Near Lm Pons, Non- lnear Fne Elemen Analyss n Srucural Mechancs, (Eds. W.Wunderlch, E.Sen, K.J.Bahe) RASHID, Y.R. (968), Ulmae Srengh Analyss of Pre-sressed Concree Pressure Vessels, Nuclear Engneerng and Desgn,968, 7, pp ROTS, J.G., BLAAUWENDRAAD, J., Crack models for concree: dscree or smeared? Fxed, mul-dreconal or roang? HERON 989, 34(). SAE, AE-4, Fague Desgn Handbook SIMO, J.C., JU, J.W., Sran and Sress-based Connuum Damage Models-I. Formulaons, II- Compuaonal Aspecs, In. J. Solds Srucures, 987, 3(7), pp SIMO, J.C., KENNEDY, J.G., GOVINDJEE, S., (988), Non-smooh Mul-surface Plascy and Vsco-plascy. Loadng/unloadng Condons and Numercal Algorhms, In. J. Num. Meh. Eng., 6, pp TAYLOR, G.I., (938), Plasc sran n meal, J. Ins. Meals, 6, VAN MIER J.G.M. (986), Mul-axal Sran-sofenng of Concree, Par I: fracure, Maerals and Srucures, RILEM, 986, 9(). VECCHIO, F.J., COLLINS, M.P (986)- Modfed Compresson-Feld Theory for Renforced Concree Beams Subjeced o Shear, ACI Journal, Proc. V.83, No., Mar-Apr., pp 9-3. VOS, E. (983) - Influence of Loadng Rae and Radal Pressure on Bond n Renforced Concree, Dsseraon, Delf Unversy, pp.9-0. WILKINS, M.L., Calculaon of Elasc-Plasc Flow, Mehods of Compuaonal Physcs, 3, Academc Press, New York, 964. ATENA Theory 8

92

93 3 FINITE ELEMENTS 3. Inroducon The precedng chapers deal wh he general formulaon of he problem, geomerc and consuve equaons. All expressons were derved ndependenly of he srucural shape, he fne elemens used ec. Here, an nformaon abou fne elemens currenly mplemened n ATENA s gven h h 4 9 r 8 5 s 4 r h h s 8 s r 7 3 h r 8 5 s Fg. 3- Examples of nerpolaon funcon for plane quadrlaeral elemens. The avalable elemens can be dvded no hree groups: plane elemens for D, 3D and axsymmerc analyss, sold 3D elemens and specal elemens, whch comprses elemens for modelng exernal cable, sprngs, gaps ec. Wh few excepons all elemens mplemened n ATENA are consruced usng soparamerc formulaon wh lnear and/or quadrac nerpolaon funcons. The soparamerc formulaon of one, wo and hree dmensonal elemens belongs o he "classc" elemen formulaons. Ths ATENA Theory 83

94 s no because of s superor properes, bu due o he fac ha s a versale and general approach wh no hdden dffcules and, also very mporan, hese elemens are easy o undersand. Ths s very mporan parcularly n nonlnear analyss. For example s hghly undesrable o add elemen-relaed problems o problems relaed o e.g. maeral modelng. Bg advanage of ATENA soparamerc elemens s ha her nerpolaon funcons h ( r, s, ) are consruced n herarchcal manner. Take an example of plane quadrlaeral elemen. Some of s nerpolaon funcons are depced n Fg. 3-. The s four funcons,.e. funcons h (,,) r s o h (,,) 4 r s has o be always presen n he nerpolaon se, (o ensure blnear approxmaon). Then, any addonal funcon h 6 (,,) r s hrough h 9 (,,) r s can be added ndependenly. Ths would nvolve addng he new funcon self and also amendmens o he already presen nerpolaon funcons. Ths approach (and use of C++ emplaes) makes possble ha one elemen formulaon generaes quadrlaeral elemens wh nodes (,,3,4), (,,3,4,5), (,,3,4,6),... (,,3,4,8), (,,3,4,9), (,,3,4,5,6), (,,3,4,5, 7),... (,,3,8,9),... (,,3,4,5,6,7,8,9). Addonal md-sde pons are parcularly useful for changng mesh densy, (.e. elemen sze), see Fg. 3-, as hey allow change of mesh densy whou need rangular elemens. Alhough he concep of herarchcal elemens was descrbed for plane quadrlaeral elemens, n ATENA apples for plane rangular elemens, 3D brcks, erahedral and wedge elemens, oo. Always here s a se of basc nerpolaon funcon ha can be exended by any hgher nerpolaon funcon. Apar of nerpolaon funcons fne elemen properes depend srongly on numercal negraon scheme used o negrae elemen sffness marx, elemen nodal forces ec. In Aena, majory of elemens are negraed by Gauss negraon scheme ha ensure nn ( ) order accuracy, where n s degree of he polynomal used o approxmae he negraed funcon Soluon wh herarchcal elemens Sandard soluon Fg. 3- Change of fne elemen mesh densy. 84

95 3. Truss D and 3D Elemen D and 3D russ elemens n ATENA are coded n group of elemens CCIsoTruss<xx>... CCIsoTruss<xxx>. The srng n < > descrbes presen elemen nodes, (see Aena Inpu Fle Forma documen for more nformaon). These are soparamerc elemens negraed by Gauss negraon a or negraon pons for he case of lnear or quadrac nerpolaon,.e. for elemens wh or 3 elemen nodes, respecvely. They are suable for plane D as well as 3D analyss problems. Geomery, nerpolaon funcons and negraon pons of he elemens are gven n Fg. 3-3, Table 3.- o Table y s r 3 CCIsoTruss<xx> CCIsoTruxx<xxx> x Fg. 3-3 Geomery of CCIsoTruss<...> elemens. Table 3.- Inerpolaon funcons of CCIsoTruss<...> elemens. Node Funcon h Include only f node 3 s defned ( r ) ( r ) 3 ( r ) h 3 h 3 Table 3.- Sample pons for Gauss negraon of node CCIsoTruss<xx> elemen. Inegraon pon Coordnae r Wegh 0.. ATENA Theory 85

96 Table 3.-3 Sample pons for Gauss negraon of and 3 nodes CCIsoruss<xxx> elemens. Inegra on pon Coordnae r Wegh The elemen vecors and marces for Toal Lagrangan formulaon, confguraon a me and eraon () are as follows. Noe ha hey are equally applcable for Updaed Lagrangan formulaon upon applyng changes relaed o he elemen reference coordnae sysem (undeformed vs. deformed elemen axs.). The formulaon s presen for 3-nodes elemen opon. The -nodes varan s obaned by smply neglecng he erms for he elemen md-pon. An arbrary pon on he russ elemen has a reference me coordnaes X [ x, x, x] : x xh xh xh 3 3 x x h x h x h 3 3 (3.) x xh xh xh A me ( ) he same pon has coordnaes X ( ) : x ( x u ) h ( x u ) h ( x u ) h ( ) ( ) ( ) 3 3( ) 3 x ( x u ) h ( x u ) h ( x u ) h ( ) ( ) ( ) 3 3( ) 3 (3.) x ( x u ) h ( x u ) h ( x u ( ) ( ) ( ) 3 3( ) )h 3 and a me () coordnaes X () x ( x u ) h ( x u ) h ( x u ) h () () () 3 3() 3 x ( x u ) h ( x u ) h ( x u ) h () () () 3 3() 3 (3.3) x ( x u ) h ( x u ) h ( x u ) h () () () 3 3() () () ( ) Incremen of Green Lagrange sran (a me, eraon () wh o confguraon a me ) s calculaed: 86

97 where russ lengh dfferenals are () ( ) l l r r l r () (3.4) l x x x3 r r r r ( ) ( ) ( ) ( ) l x x x3 r r r r (3.5) () () () l x x r r r x r () 3 Subsung (3.5), (3.3) no (3.4) afer some mah manpulaon can be derved: l r h h h h h h3 3 x x x r r r r r r h h h h h h3 3 x x x r r r r r r h3 h h3 h h3 h 3 3 x x x r r r r r r h h h h h h 3 3 x x x r r r r r r h h h h h h r r r r r r h3 h h3 h h3 h3 3 x x x r r r r r r h h h h h h 3 3 x3 x3 x 3 r r r r r r h h h h h h 3 3 x3 x3 x3 r r r r r r h3 h h3 h h3 h3 3 x3 x3 x3 r r r r r r 3 3 BL0 x x x (3.6) ATENA Theory 87

98 and B l r ( ) L h h h h h h u u u r r r r r r h h h h h h u u u r r r r r r h3 h ( ) h3 h h3 h3 u u u r r r r r r h h h h h h3 u u u r r r r r r h h h h h h3 u u u r r r r r r h3 h ( ) h3 h h3 h3 u u u r r r r r r h h h h h h 3 u3 u3 u3 r r r r r r h h h h h h3 u3 u3 u3 r r r r r r h3 h ( ) h3 h h3 h3 u3 u3 u3 r r r r r r ( ) ( ) 3 3( ) ( ) ( ) 3 3( ) ( ) 3( ) ( ) ( ) 3( ) ( ) ( ) 3( ) ( ) 3( ) ( ) ( ) 3( ) ( ) ( ) 3( ) ( ) 3( ) ( n) 3 NL (3.7) h h h r r r h h h B (3.8) l r r r r h h h r r r The nd Pola-Krchhoff sress marx and ensor are: S , [ ] ( ) ( ) ( ) ( ) ( ) S S S S ( ) 0 0 S The formulaon s compleed by relaonshp for elemen deformaon graden yelds: X (), () l r l r () X, (3.9), whch (3.0) Noe ha -nodes russ elemen has consan srans along s lengh and hus he ncremen of Green Lagrange sran can be calculaed drecly, (.e. no usng dfferenals russ lengh as was he case of (3.4) ): 88

99 () ( ) l l l () (3.) Ths yelds a b smpler elemen formulaon (wh he same resuls). However, for he sake of preservng unfed approach o all russ elemens, ATENA uses even n hs case he equaon (3.4). 3.3 Plane Quadrlaeral Elemens Plane quadrlaeral elemens n ATENA are coded n group of elemens CCIsoQuad<xxxx>... CCIsoQuad<xxxxxxxxx>. The srng n < > descrbes presen elemen nodes (see Aena Inpu Fle Forma documen for more nformaon). These are soparamerc elemens negraed by Gauss negraon a 4 or 9 negraon pons for he case of blnear or b-quadrac nerpolaon,.e. for elemens wh 4 or 5 and more elemen nodes, respecvely. They are suable for plane D, axsymmerc and 3D problems. CCIsoQuad_5<...> elemens presen a smplfed 3D formulaon of he CCIsoQuad<...> elemens. Ther hgher execuon performance s acheved a cos of omng some nonlnear erms, see below. Geomery, nerpolaon funcons and negraon pons of he elemens are gven n Fg. 3-4 and n he subsequen ables. 3 y s 4 8 x r CCIsoQuad<xxxx> CCIsoQuad<xxxxx> CCIsoQuad<xxxx_x>... CCIsoQuad<xxxx_x_x_>... CCIsoQuad<xxxxxxxxx> Fg. 3-4 Geomery of CCIsoQuad<...> elemens. ATENA Theory 89

100 Table 3.3-: Inerpolaon funcons of CCIsoQuad<...> elemens. Node Funcon h Include only f node s defned = 5 I = 6 = 7 = 8 = 9 ( r )( s ) 5 4 h ( r )( s ) 5 4 h h 4 h 8 9 h 3 ( r )( s ) 6 4 h 4 h 6 9 h 4 ( r )( s ) 7 4 h 5 4 h 7 9 h 4 h 8 9 ( r )( s ) 9 h 6 ( s )( r ) 9 h 7 ( r )( s ) 9 h 8 ( r )( s ) 9 h 9 ( r )( s ) Table 3.3-: Sample pons for Gauss negraon of 4 nodes CCIsoQuad<...> elemen. Inegraon pon Coordnae r Coordnae s Wegh

101 Table 3.3-3: Sample pons for Gauss negraon 5 o 9 nodes CCIsoQuad<...> elemens. Inegra on pon Coordnae r Coordnae s Wegh Equaons (3.) hrough (3.) presen CCIsoQuad<...> axsymmerc elemen formulaon. D elemen formulaon s smply obaned by removng erms assocaed wh crcumferenal () () srans and sresses,. Incremenal srans: S () () ( ) () ( ) () () () u, u, u, u, u, u, u, () () ( ) () ( ) () () () u, u, u, u, u, u, u, u u () () (),, u u u u u u u u ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ),,,,,,,, u u u u () () () (),,,, () () () () () 33 x x x u u u u (3.) ATENA Theory 9

102 Dsplacemen dervaves: Srans and marces o calculae hem: B U () ( ) () L u () ( ) () u, j x j u ( ) ( ) u, j x j u (3.3),,, () () () () () 33 (3.4) () () ( ) () () () () n() n() U U U u, u, u, u,... u, u Lnear sran-dsplacemen marx: B B B (3.5) ( ) ( ) L L0 L Lnear sran-dsplacemen marx consan par: where h, 0 h, h, 0 h,... 0 h n, B 0 h, h, h, h,... h, h, (3.6) h h hn x x x L n n h h, j x j u u u () () ( ) (3.7) n k x hk x k 9

103 Lnear sran-dsplacemen marx non-consan par: l h l h l h l h ( ) ( ) ( ) ( ),,,, ( ) ( ) ( ) ( ) l h, l h, l h, l h, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) B L l h, l h, l h, l h, l h, l h, l h, l h, where l h x h 0 l 0 x ( ) ( ) l h n, l h n, ( ) ( ) l hn, l h n, ( ) ( ) ( ) ( ) l h n, l h n, l h n, l h n, ( ) hn l 33 x ( ) ( ) n ( ) k( ) l hk, u k (3.8) n ( ) k( ) l hk, u k n ( ) k( ) l hk, u k (3.9) n ( ) k( ) l hk, u k n l hk u x ( ) k( ) 33 k Nonlnear sran-dsplacemen marx h, 0 h, 0... hn, 0 h, 0 h, 0... h, 0 n ( ) 0 h, 0 h,... 0 h n, B NL (3.0) 0 h, 0 h,... 0 hn, h h h n x x x ATENA Theory 93

104 nd Pola-Krchhoff sress ensor and vecor S S ( ) ( ) ( ) ( ) S S ( ) ( ) ( ) S S S ( ) ( ) S S S S S S () ( ) ( ) ( ) S ( ) 33 S ( ) 33 (3.) In case of he smplfed 3D analyss,.e. elemens CCIsoQuad_5<...>, he equaons are furher exended as follows: All elemen marces and vecors are compued wh respec o elemen local coordnae sysem xlocal,, xlocal, usng equaons n (3.) hrough (3.). They are ransformed no 3D global coordnae sysem by means of smple ransformaon: M T M T T, v T v (3.) global local global local where M, M, v, v are global and local fne elemen marces and vecors, global local global local where: local, global, T s ransformaon marx from local o global coordnae sysem: cos( xlocal,, xglobal, ), cos( xlocal,, xglobal,) T cos( xlocal,, xglobal, ), cos( xlocal,, xglobal,) (3.3) cos( xlocal., xglobal,3 ), cos( xlocal,, xglobal,3) x, x are local and global coordnaes (n D and 3D space). The local elemen coordnae sysem (see Fg. 3-5) s defned by local xlocal,, xlocal,, x local,3 coordnaes. All of hem pass hrough orgn of he global (reference) coordnae sysem. The axes x local, and x local, consue a local coordnaes elemen plane ha s parallel o he elemen.. The axs xlocal,3 s perpendcular o he elemen and he axs x local, s defned as a projecon of global x axs o he local coordnae elemen plane. An excepon o ha s, when he elemen s normal o he global x. In hs case he local x local, concdes wh he global x axs. The presen defnon of local elemen coordnae sysem depends on plane of he fne elemen bu does no depend on s shape self. Ths s very mporan propery, as ATENA suppors use of local (nsead of global) nodal degrees of freedom and, (of course) hese degrees of freedom mus refer o a coordnae sysem common o all elemens of he plane, n whch hey le. 94

105 x 3 3 x lo ca l, 3 4 O X X x lo c a l, x x x lo c a l, Fg. 3-5 Local plane elemen coordnae sysem. Full 3D formulaon of he CCIsoQuad<...> elemens s much he same as ha for smplfed 3D elemens CCIsoQuad_5<...>. The only dfference s ha he marx 0 BNL wll nclude also erms relaed o he ou-of-elemen-plane drecon: B ( ) NL h, 0 h, 0 h3, 0 hn, h, 0 h, 0 h3, 0 hn, h, 0 h, 0 h3, 0 hn, h, 0 h, 0 h3, 0 hn, 0 0 h 0 0 h 0 0 h 0 0 h 0 0 h 0 0 h 0 0 h 0 0 h 3.4 Plane Trangular Elemens,, 3, N,,, 3, N, (3.4) Plane rangular elemens n ATENA are coded n group of elemens CCIsoTrangle<xxx>... CCIsoTrangle<xxxxxx>. The srng n < > descrbes presen elemen nodes (see Aena Inpu Fle Forma documen for more nformaon). These are soparamerc elemens negraed by Gauss negraon a or 3 negraon pons for he case of blnear or b-quadrac nerpolaon,.e. for elemens wh 3 or 4 and more elemen nodes, respecvely. They are suable for plane ATENA Theory 95

106 D, axsymmerc and 3D problems. Geomery, nerpolaon funcons and negraon pons of y 6 3 s 5 CCIsoTrangle<xxx>... CCIsoTrangle<xxxxxx> 4 r he elemens are gven n Fg. 3-6, Table 3-, Table 3-, and Table 3-3. x y 6 3 s 5 CCIsoTrangle<xxx>... CCIsoTrangle<xxxxxx> 4 r Fg. 3-6: Geomery of CCIsoTrangle<...> elemens. x Table 3-: Inerpolaon funcons of CCIsoTrangle<...> elemens. Node Funcon h Include only f node s defned = 4 = 5 = 6 r s r 3 s h h 4 6 h h 4 5 h h

107 4 4( r r s) 5 4rs 6 4( s r s) Table 3-: Sample pon for Gauss negraon of 3 nodes CCIsoTrangle<...> elemens. Inegraon pon Coordnae r Coordnae s Wegh /3 /3 / Table 3-3: Sample pons for Gauss negraon of 3 o 6 nodes CCIsoTrangle<...> elemens. Inegraon pon Coordnae r Coordnae s Wegh /6 /6 /6 /3 /6 /6 3 /6 /3 /6 All he above expressons for he formulaon for plane quadrlaeral elemens reman vald also for he rangular elemens, ncludng he exenson from D o smplfed and full 3D analyss. The expressons only use dfferen approxmaon funcons h ( r, s, ) and dfferen negraon pons [ rs,,], see Table 3-, Table 3-, and Table D Sold Elemens ATENA fne elemen lbrary ncludes he followng group of 3D sold elemens: erahedral elemens CCIsoTera<xxxx>... CCIsoTera<xxxxxxxxxx> wh 4 o 0 nodes, see Fg. 3-7, brck elemens CCIsoBrck<xxxxxxxx>... CCIsoBrck<xxxxxxxxxxxxxxxxxxxx> wh 8 up o 0 nodes see Fg. 3-8 and wedge elemens CCIsoWedge<xxxxxx>... CCIsoWedge<xxxxxxxxxxxxxxx> wh 6 o 5 nodes, see Fg The srng n < > descrbes presen elemen nodes (see Aena Inpu Fle Forma documen for more nformaon). These are soparamerc elemens negraed by Gauss negraon a negraon pons gven n he followng ables. Inerpolaon funcons for all varans of he elemens are also gven n he ables below. ATENA Theory 97

108 z s CCIsoTera<xxxx>... CCIsoTera<xxxxxxxxxx> x r y Fg. 3-7 Geomery of CCIsoTera<...> elemens. 4 z CCIsoBrck<xxxxxxxx>... CCIsoBrck<xxxxxxxxx...x> s 8 r x 5 y Fg. 3-8 Geomery of CCIsoBrck<...> elemens z s 6 CCIso Wedge<xxx xxx>... CCIso Wedge<xxxxxxxxxx xxxxx> x 4 0 r 5 y Fg. 3-9 Geomery of CCIsoWedge<...> elemens. 98

109 Table 3.5- Inerpolaon funcons of CCIsoTera<...> elemens. Node Funcon h Include only f node s defned = 5 = 6 = 7 = 8 = 9 = 0 rs r 3 s 4 5 4( r rs ) h h h h h h h h h h h h rs( ) 7 4( s rs ) 8 4 r( s) 9 4 s(- r ) 0 4(- r-s- ) Table 3.5- Sample pon for Gauss negraon of 4 nodes CCIsoTera<...> elemen. Inegraon pon Coordnae r Coordnae s Coordnae Wegh /4 /4 /4 /6 ATENA Theory 99

110 Table Sample pons for Gauss negraon of 5 o 0 nodes CCIsoTera<...> elemens. Inegraon pon Coordnae r Coordnae s Coordnae Wegh / / / /4 Table 3-4 Inerpolaon funcons of CCIsoBrck<...> elemens. Node Funcon h Include only f node s defned = 9 = 0 = = = 3 = 4 = 5 = 6 = 7 = 8 = 9 = 0 ( r )( s )( ) 9 8 h h h 7 ( r )( s )( ) 9 8 h h 0 h 8 3 ( r )( s )( ) 0 8 h h h 9 4 ( r )( s )( ) 8 h h h 0 5 ( r )( s )( ) 3 8 h h h ( r )( s )( ) 3 8 h h 4 h 8 7 ( r )( s )( ) 4 8 h h 5 h 9 8 ( r )( s )( ) 5 8 h h 6 h 0 00

111 9 ( r )( s )( ) 4 0 ( r )( s )( ) 4 ( r )( s )( ) 4 ( r )( s )( ) 4 3 ( r )( s )( ) 4 4 ( r )( s )( ) 4 5 ( r )( s )( ) 4 6 ( r )( s )( ) 4 7 ( r )( s )( ) 4 8 ( r )( s )( ) 8 9 ( r )( s )( ) 4 0 ( r )( s )( ) 4 ATENA Theory 0

112 Table Sample pons for Gauss negraon of 8 nodes CCIsoBrck<...> elemen. Inegraon pon Coordnae r Coordnae s Coordnae Wegh Table Sample pons for Gauss negraon of 9 o 0 nodes CCIsoBrck<...> elemen. Inegraon pon Coordnae r Coordnae s Coordnae Wegh

113 ATENA Theory 03

114 Table Inerpolaon funcons of CCIsoWedge<...> elemens. hh ( r s) hh r hh s 3 hh 4( r r s) hh rs hh 4( s r s) hv hv hv ( ) 04

115 Node I Funcon Include only f node s defned h = 7 = 8 = 9 = 0 = = = 3 = 4 = 5 hhhv h h 7 9 h 3 hhhv h h 7 8 h 4 3 hh3hv h h 8 9 h 5 4 hhhv h h 0 h 3 5 hhhv h h 0 h 4 6 hh3hv 7 hh4hv 8 hh5hv 9 hh6hv 0 hh4hv hh5hv hh6hv 3 hhhv 3 4 hhhv 3 5 hh3hv 3 h h h 5 Table Sample pons for Gauss negraon of 6 nodes CCIsoWedge<...> elemen. Inegraon pon Coordnae r Coordnae s Coordnae Wegh /6 / /6 /3 / /6 3 /6 / /6 4 /6 / /6 5 /3 / /6 6 /6 / /6 ATENA Theory 05

116 Table Sample pons for Gauss negraon of 7 o 5 nodes CCIsoWedge<...> elemen. Inegraon pon Coordnae r Coordnae s Coordnae Wegh /6 / /3 / /6 / /6 / /3 / /6 / /6 / /3 / /6 / Formulaon of 3D sold elemens s gven n he followng equaons: Incremenal srans: () () () ( ) () ( ) () () () j u, j uj, uk, uk, j uk, j uk, uk, uk, j (3.5) where ndces, j, k...3 Dsplacemen dervaves: Srans and marces o calculae hem: B U () ( ) () L u () ( ) () u, j x j u ( ) ( ) u, j x j u (3.6) () () () () () () () (3.7) () () ( ) U U U Lnear sran-dsplacemen marx: u u u u u u... u u u () () () () () () n() n() n() ( ) ( ) L L0 L Lnear sran-dsplacemen marx consan par: 06 B B B (3.8)

117 where B L0 h, 0 0 h, hn, h, 0 0 h, h, 0 n 0 0 h,3 0 0 h, hn,3 h, h, 0 h, h, 0... hn, hn, 0 0 h,3 h, 0 h,3 h,... 0 hn,3 h n, h,3 0 h, h,3 0 h,... hn,3 0 hn, (3.9) h h, j x j (3.30) u u u () () ( ) Lnear sran-dsplacemen marx non-consan par: l h l h l h l h ( ) ( ) ( ) ( ),, 3,, ( ) ( ) ( ) ( ) l h, l h, l3 h, l h, ( ) ( ) ( ) ( ) ( ) l3 h,3 l3 h,3 l33 h, 3 l3 h,3 BL ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) l h, l h, l h, l h, l3 h, l3 h, l h, l h, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) l h,3 l3 h, l h,3 l3 h, l3 h,3 l33 h, l h,3 l3 h, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) l h,3 l3 h, l h,3 l3 h, l3 h,3 l33 h, l h,3 l3 h, (3.3) where l h ( ) 3 n, ( ) l3 h n, ( ) l33 h n,3 ( ) ( ) l3 hn, l 3 h n, ( ) ( ) l3 h n,3 l33 h n, ( ) ( ) l3 h n,3 l33 h n, n ( ) k( ) lj hk, j u k (3.3) ATENA Theory 07

118 Nonlnear sran-dsplacemen marx B ( ) NL h, 0 0 h,... hn, 0 0 h, 0 0 h,... h, 0 0 n h,3 0 0 h,3... hn, h, hn, 0 0 h, hn, 0 0 h, hn, h, hn, 0 0 h, hn, 0 0 h, h n,3 (3.33) nd Pola-Krchhoff sress ensor and vecor S S S ( ) ( ) ( ) 3 ( ) ( ) ( ) S S S3 ( ) ( ) ( ) S3 S3 S33 ( ) ( ) ( ) S S S3 ( ) ( S ) ( ) ( ) S S S3 ( ) ( ) ( ) S3 S3 S S S S ( ) ( ) ( ) 3 ( ) ( ) ( ) S S S3 ( ) ( ) ( ) S3 S3 S33 ( ) ( ) ( ) ( ) ( ) ( ) ( ) S S S S33 S S3 S3 (3.34) 3.6 Sprng Elemen Sprng elemens n ATENA are used o model sprng-lke boundary condons,.e. suaon where exernal forces acng on boundary of he srucure are lnearly proporonal o he assocaed dsplacemens. Three elemens of hs ype are avalable, see also Fg. 3-0, Fg. 3-: CCSprng D and 3D elemen o model sprng-lke boundary condons a a pon, CCLneSprng D elemen o model sprng-lke boundary condons along a lne CCPlaneSprng 3D elemen o model sprng-lke boundary condons along a rangular area. All hese elemens are derved from D or 3D formulaon of he CCIsoTruss<xx> elemen descrbed earler n hs chaper. For example, CCSprng elemen consss of one CCIsoTruss<xx> elemen. The s node of each CCIsoTruss<xx> concdes wh one node of he CCSprng elemen, whereas he nd node of he CCIsoTruss<xx> s se by drecon vecor, see Fg Noe ha as he analyss s nonlnear, lengh of he drecon does maer. Ths vecor s specfed n ATENA &SPRING_GEOMETRY_SPEC command and s common for all sprng elemens ha use hs geomery. 08

119 CCLneSprng and CCPlaneSprng elemens were creaed o enable convenen defnon of unform sprng-lke condons along he boundares. The boundary force a a node of he sprng elemen s calculaed: uka R (3.35) n drecon where k s sprng maeral sffness parameer se by &MATERIAL SPRING command, (parameer k has characer of mul-lnear Young modulus), u s dsplacemen a sprng elemen node, A s he area of CCPlaneSprng elemen or lengh of CCLneSprng mulpled by hckness (whch defauls o f no specfed n elemen geomery) or he area defned n elemen geomery for CCSprng (smlarly, wh a defaul of f no specfed) for he respecve elemen, n s number elemen nodes,.e., or 3 for CCSprng, CCLneSprng or CCPlaneSprng elemen respecvely, drecon s Eucldean norm (.e. lengh) of he drecon vecor, see above. y CCSprng CCLneSprng x Fg. 3-0 Geomery of D CCSprng and CCIsoLneSprng. ATENA Theory 09

120 z CCPlaneSprng CCSprng area A y x Fg. 3- Geomery of 3D CCSprng and CCPlaneSprng. 3.7 Quadrlaeral Elemen Q Elemen Sffness Marx The quadrlaeral fne elemen Q-0 s derved from a sx-node rangle (CCQ0<xxxx>, CCQ0Sbea<xxxx>). The dervaon of he sffness marx s aken from FELIPPA 966. The poson of any nernal pon P n he elemen s defned by he rangular coordnaes (called also naural coordnaes). These coordnaes are expressed by means of areas whn he rangle as shown n Fg. 3-.Sub-areas A are subended by he pon P and wo corners. A s he area of rangular elemen. 0 Fg. 3- Coordnae sysems of he sx-node rangular elemen. A A A A A A 3,, 3 (3.36) 3

121 Usng he quadrac nerpolaon funcon, he dsplacemen componens u( ), v( ) s wren n he erms of rangular coordnaes and nodal dsplacemen vecors : T T u( ) F( ) u, v( ) F( ) v (3.37) The dsplacemen vecors u, v conan sx componens of he nodal dsplacemens and he vecor F ( ) conans he quadrac nerpolaon funcons n rangular coordnaes: T uuuuuu , vvvvvv u v (3.38) F ( ) ( ) ( ) ( ) (3.39) A general procedure o consruc he elemen sffness marx s descrbed by he se of followng equaons: (a) The consuve equaon: s De (3.40) (b) The sran-dsplacemen equaons n he Caresan coordnaes:,,,, u x y v x y u x y v x y x, y, x y y x T T (3.4) whch s wren n erms of he naural coordnaes and he nodal dsplacemens vecors u, v: The sffness marx: T u ( ) F v (3.4) T K F DF dv (3.43) V The marx F conans paral dervaves of he nerpolaon funcon F and he negral n he las equaon s made over he elemen volume V. The deals of he dervaon can be found n FELIPPA 966 and here only he fnal marx equaons are presened. Fg. 3-3 Quadrlaeral elemen (b) composed from wo rangular elemens (a). The quadrlaeral fne elemen s composed from wo 4-node rangular elemens, as shown n Fg Two degrees of freedom n a node are he horzonal and vercal dsplacemens. The rangular elemen s derved from he 6-node rangle by mposng knemac consrans on wo md-sde nodes. The resulng sran-dsplacemen marx relaon for he 4-node rangle s: ATENA Theory

122 e Bd e U O u e O V v g V U x y (3.44) where e x, e y are he normal sran vecors, g s he shear sran vecor (engneerng ype) and O s he null marx. The sran and dsplacemen vecors conan nodal componens: 3, 3, g 3 T T T e e (3.45) x x x x y y y y x x x T uuuu 3 4, vvvv 3 4 T u v (3.46) The sran nerpolaon funcon n he elemen s lnear and s unquely specfed by hree nodal values n he corners of he rangular elemen, whle he dsplacemen nerpolaon funcon s quadrac and s specfed by hree corners and one md-sde nodal dsplacemen. The componens u, v are he horzonal and vercal dsplacemens, respecvely, n he node. The ndexes, and 3 denoe he corner nodes of a sub-rangle and he ndex 4 s for he md-sde node, see Fg. 3-3 (a). The sran-dsplacemen sub-marces n (3.44) are 3bb3 b b3 4b U b 3b b3 b3 4b S b b b3. 3aa3 a a3 4a V a 3a a3 a3 4a S (3.47) a a a3. a x x b y y a x x b y y a x x b y y S a3b ab3 where x, y are he global Caresan coordnaes of he node n a sub-rangle, S s he area of he sub-rangle. The elemen sffness marx for he 4-node sub-rangle s K K uu uv K Kvu K (3.48) vv The sffness marx K has an order 8 and s so paroned ha he upper four rows correspond o he horzonal dsplacemen componens (ndex u) and he lower four rows correspond o he vercal dsplacemen componens (ndex v). The negraon of he sffness coeffcens s made exacly and he resulng sub-marces are: K 3 ( T uu S d A d H H ) d C 33 K 3 ( T vv S d C d H H ) d A 33 K T uv S d H d A 3 d C 3 d H (3.49) 33

123 where s he hckness of he elemen, d j are he coeffcens of he maeral sffness marx D, (3.40). The negraon n (3.43) s done explcly by he followng marx mulplcaon: Where he area negraon marx Q s: T T T A U QU, H U QV, C V QV (3.50) Q (3.5) The elemen sffness marx of he 5-node quadrlaeral, Fg. 3-3(b), s composed of he wo 4- node sub-rangles by summng he sffness coeffcens of he approprae nodes. The resulng marx of he 5-node quadrlaeral K 0 has he order 0. The coeffcens of he marx can be rearranged accordng o he exernal (ndex e) and nernal (ndex ) degrees of freedom: K 0 K K ee e K K e (3.5) The sub-marces correspondng o wo nernal degrees of freedom are elmnaed by condensaon procedure and he fnal elemen sffness marx K of he order 8 s obaned: ee e e K K K K K (3.53) Fg. 3-4 Subdvson of quadrlaeral elemen. The subdvson of he quadrlaeral elemen no he rangular elemens mus be done n an opmal way and s preformed auomacally by he program. The examples of he subdvsons are llusraed by Fg Due o hs mehod of he subdvson, a concave form of he quadrlaeral elemen s accepable. Ths elemen form could no be acheved by an soparamerc elemen Evaluaon of Sresses and Ressng Forces For he gven dsplacemen feld he srans and sresses are evaluaed n he cener of he quadrlaeral elemen. The sresses a hs pon are obaned from maeral laws as funcons of srans accordng o Secon... Also he consuve law for he elemen and he marx D are calculaed from he sresses and srans a he cener of he elemen. These sresses and srans are wren n he oupu fle as a par of he resuls. The calculaon of ressng nodal forces of he sub-rangle for a curren dsplacemen feld and a consuve law s done by he followng equaon: T R B Q9s 9 (3.54) where R s he vecor of nodal forces (same arrangemen and numberng as n he vecor d n (3.44)). The marx Q 9 conans hree negraon marces Q n he dagonal. The sress vecor s 9 ATENA Theory 3

124 (same numberng as he vecor e, (3.40), s calculaed from he curren srans and secan maeral marx, Secon... There are wo varaons of hs elemen n program ATENA: CCQ0<xxxx> and CCQ0Sbea<xxxx>. The man dfference beween hese wo elemens les n he way how he ressng forces are calculaed. In case CCQ0<xxxx>, hey are compued as descrbed by Equaon (3.54). In he second case, however, he maeral law s evaluaed only a he elemen cenrod. Based on he curren sae of damage a secan consuve marx s calculaed and s used o deermne he negraon pon sresses and resulng ressng forces. Ths elemen ype s almos dencal o he elemen ha was mplemened n he program SBETA,.e. he former verson of hs program. Due o hs approach here are some lmaons for usage of hs elemen wh respec o some maeral models. I can be only used wh maeral models ha are able o calculae and exac secan consuve marx. Ths means ha only he followng maeral models can be used wh he elemen CCQ0Sbea<xxxx>: CCElasIsoropc and CCSbeaMaeral. 4

125 3.8 Exernal Cable Exernal pre-sressng cables are renforcng bars, whch are no conneced wh he mos of he concree body, excep of lmed number of pons, so called devaors, as shown n Fg Ths elemen ype s denoed n ATENA as CCExernalCable. Fg. 3-5 Exernal cable model. Each cable has wo ends provded wh anchors. The anchor, where he pre-sressng force s appled s denoed as he acve anchor, he anchor on he oher sde s he passve anchor. The pons beween he anchors are called devaors (or lnks). Afer applyng pre-sressng he cable s fxed a anchors. In he devaors, cable can slde whle s movemens and he forces are governed by he law of dry frcon. The slps of he cable n he devaors (he relave dsplacemen of he cable ends wh respec o he devaors) are denoed as,, They are nroduced as varables o be deermned by he analyss. Fg. 3-6 Forces a he devaor. The forces, F and F acng on a devaor are he cable forces a he adjacen cable secons, Fg Ther dfference P = F -F, (F > F ) s he loss of he pre-sressng force due o frcon n he devaor. The relaon beween hese forces accordng o he law of frcon s expressed as: p F F e Q f ( ) f ( r) (3.55) r ATENA Theory 5

126 In hs expresson s he angular change of he cable drecon a he devaor and s he frcon coeffcen. The consan par of he frcon s Q = p k R, where k s he coheson (a consan par of he frcon ) of he cable of a un lengh and R s he radus of he devaor. p sands for renforcemen bar permeer. The produc R s he lengh on whch s he coheson k acng. If he consan par of frcon s negleced he erm Q s zero. f ( ), fr ( r) are user defned funcon ha enable change of devaor's properes dependng on value of slp s and devaor poson r (measured from s sarng pon). By defaul, hese funcons are se o one. a p Inroducng d e f ( ) f ( r) and d b pkr f ( ) f ( r) we can smplfy (3.55) o r r F F d d (3.56) a b Fg. 3-7 Forces and dsplacemens n he cable elemen (cable secon). A secon of he cable beween he devaors s consdered as he unaxal bar elemen, Fg The force F n he cable elemen depends on: he pre-sressng force P, he dsplacemens of ends u, u due o srucural deformaon and he cable slps, n he devaors. The slp s nroduced as an addonal varable for he exernal cables. The equlbrum equaon of he cable secon s: F Pk( u u ) (3.57) The elemen sffness k = E s A/L, where A, L are he cable cross secon and he lengh, respecvely, and E s s he acual secan or angen modulus derved n he same way as n case of oher renforcemen usng blnear or mul-lnear law. The cable forces F, F,, are deermned by applyng he above equaons n all cable devaors and by smulaneous soluon for slps. Inroducon of pre-sressng s accomplshed by applyng an nal slp (cable pull-ou) a he anchor end unl a prescrbed pre-sressng force s reached. Ths procedure reflecs a real process of pre-sressng and akes no accoun he loss of pre-sressng due o frcon devaors and deformaon of he srucure. 3.9 Renforcemen Bars wh Prescrbed Bond Renforcemen bars wh prescrbed bonds are devaon of he exernal cables descrbed n he prevous secon. The man dfference s ha hey are fully conneced o he surroundng concree body, however hs connecon has only lmed srengh. Ths bond srengh s specfed by so called coheson sress. Ths ype of elemen s denoed as CCBarWhBond n ATENA. Typcal renforcemen bar of hs ype s depced n he fgure below. The deal shows undeformed and deformed shape of a 6

127 segmen of he bar. The orgnal lengh l 0 wll change o l due o dsplacemen u of he surroundng body and bar slps. c - c m - + m+ undeformed russ deformed russ u l o u + + l Normal sress a elemen s calculaed by: Fg. 3-8 Renforcemen bar wh slps. ( u u ) E (3.58) l Is dervave s compared wh he coheson sress. If he coheson sress beween he bar and he surroundng concree s becomng oo hgh, he bar wll slp o reduce hs sress. Oherwse, he slps wll reman unchanged (or nally equal o zero), whch correspond o he case of perfec bond. The coheson sress can be consan or can be defned as a funcon of and r. Alernavely can accoun for so called wobble coheson, see f. xx w c r c0 xx w f () r f ( ) f (3.59) f ( ), fr ( r) are he same as hose descrbed for exernal cables, c0 s coheson sress due o slppng, c s oal coheson sress due o slppng and wobble coheson, p s permeer of he renforcemen bar, r s locaon a he bar. xx, f w are normal sress n he bar and wobble coeffcen. The wobble coheson s derved as follows: Presress losses are calculaed by: kx ( e ) xx p e xx p p x xx kx kx k pe k xx (3.60) ATENA Theory 7

128 In he above p s presressng sress, k follows fw are maeral consan. Comparng (3.60) o (3.59) k (3.6) The equlbrum condon for renforcemen bar wh prescrbed bond yelds: p c (3.6) x A where s sress n he bar, x s local coordnae axs n drecon of he bar and pa, means permeer and cross seconal area of he bar. Fg. 3-9 Forces a node. Dscrezed form of (3.6) for node, (consderng elemens, ), reads (he bars are of consan sran ype): F F A L A R R L R L l l l l d c k for ( F F ) : F F pc d l l l l d for ( F F ) : F F p d R L R L c k c (3.63) If hs elemen acs also as he exernal cable, see he prevous secon, hen for ( F F ) : F F d d R L L R a b F F F F d d R L R R a b R L R a b F F F d d ) for ( F F ) : F F d d R L R L a b F F F F d d R L L L a b F F F d d R L L a b (3.64) Assemblng (3.63) and (3.64) yelds fnal (n)equaons for force dfference a node : 8

129 R L R L R a b l l l l d c k for ( F F ) : F F F d d pc d (3.65) R L R L L a b l l l l d c k for ( F F ) : F F F d d p c d The above se of (n)equaons s calculaed n erave manner. Assume we know he forces a eraon ( k ), hen he forces a eraon k are: EA F ( u u ) Rk, k k l EA F ( u u ) Lk, k k l F F EA F ( ) Rk, Rk, k k l EA F ( ) Lk, Lk, k k l (3.66) and k k k k k k k k k R L for ( F F ) : EA EA l l d l l F ( ) F ( ) F d d p R k k L k k c k R a b c l l d EA k EA EA k EA k l l d c k R L F F F l l l l d l l d d p R a a c R L for ( F F ) : EA EA l l d l l F ( ) F ( ) F d d p R k k L k k c k L a b c l l d EA k EA EA k EA k l l d c k R L L a b l l F F F d d p c l l l l d (3.67) ATENA Theory 9

130 If he above equaon s wren for all nodes on he bar, we oban a se of nequales. I has o be solved n erave manner (whn each eraon of he man soluon loop). Aena also suppor so called CCBarWhMemoryBond D and 3D elemens. They dffer from her orgnal formulaon, (.e. elemens CCBarWhBond), n ha hey have dfferen funcon f ( ) for "loadng" and f, ( ) for "unloadng" regme. Ths means ( mn, max ) n he unload former and ( mn, max ) n he laer case. Fg. 3-0 Bond funcon for CCBarWhMemoryBond elemen In order o oban more realsc shape, he resulng coheson sresses are pror her oupu smoohed. The smoohng operaon for node s expressed as follows: rgh l l l l lef l l l l (3.68) 0 ( rgh lef ) A c pl The equaon (3.58) ogeher wh (3.63) complees he elemen descrpon. The elemen can be used o model realscally coheson beween renforcemen bar and concree. Such a model s

131 needed for analyss of pullou ess ec. Alhough he adoped soluon s farly smple, provdes reasonable resuls accuracy a low compuaon cos. A more elaborae model of coheson beween renforcemen bar and surroundng concree can be acheved by usng specal nerface elemens ha s descrbed n he nex secon. 3.0 Inerface Elemen The nerface elemens are used o model a conac beween wo surfaces. Currenly, he followng elemen ypes are avalable: CCIsoCCIsoGap<xxxx> and CCIsoGap<xxxxxx>, CCIsoGap<xxxxxxxx> for D and 3D analyss, respecvely. These elemens use lnear approxmaon of geomery. For he case of nonlnear geomery, use elemen ype CCIsoGap<xxxxxx> for D and CCIsoGap<xxxxxxxxxxxx> or CCIsoGap<xxxxxxxxxxxxxxxx> for 3D. The srng n < > descrbes presen elemen nodes, (see Aena Inpu Fle Forma documen for more nformaon). The elemens are derved from he correspondng soparamerc elemens (descrbed n secons 3.3 and 3.4),.e. hey use he same geomery and nodal ds ec. Geomery of he suppored gap elemens s depced n Fg. 3-. Lnear geomery v r,u(r) 4 3 CCIsoGap<xxxx> D Nonlnear geomery v u(r) r, CCIsoGap<xxxxx_x> r,u(r,s) 5 w D r,u(r,s) w s,v(r,s) s,v(r,s) CCIsoGap<xxxxxx> CCIsoGap<xxxxxxxxxxxx> r,u(r,s) w r,u(r,s) 8 7 s,v(r,s) CCIsoGap<xxxxxxxx> Fg. 3- CCIsoGap elemens 7 w s,v(r,s) CCIsoGap<xxxxxxxxxxxxxxxx> ATENA Theory

132 The nerface s defned by a par of lnes, (or surfaces n 3D) each locaed on he oppose sde of nerface. In he orgnal (.e. undeformed) geomery, he nerface lnes/surfaces can share he same poson, or hey can be separaed by a small dsance. In hs case we speak abou he nerface wh nonzero hckness. In he followng, he nerface behavor s explaned on a smple -dmensonal case, see secon.6 for a full descrpon of he nerface maeral. The nerface elemen has wo saes: Open sae: There s no neracon of he conac sdes. Closed sae: There s full neracon of he conac sdes. In addon, frcon sldng of he nerface s possble n case of nerface elemen wh a frcon model. Penaly mehod s employed o model he above behavor of he nerface. For hs purpose we defne a consuve marx of he nerface n he form: F K 0 u F u F 0 K D (3.69) nn v n whch u, v are he relave dsplacemens of he nerface sdes (sldng and openng dsplacemens of he nerface) n he local coordnae sysem rs, and K, Knn are he shear and normal sffness, respecvely. Ths coeffcen can be regarded as sffness of one maeral layer (real, or fcous) havng a fne hckness. I should be undersood ha he layer s only a numercal ool o handle he gap openng and closng. F, F are forces a he nerface, (agan a he local coordnae sysem). The acual dervaon of gap elemens s now demonsraed for he case of lnear D gap elemen CCIsoGap<xxxx>, see Fg. 3-. The oher elemens are consruced n a smlar way. The elemen has wo degrees of freedom defned n he local coordnae sysem, whch s algned wh he gap drecon. They are relave dsplacemens v, u and are defned as follows: h ( r), h ( r) u hu,4 hu,3 u v hv h v,4,3 u v u h 0 h 0 h 0 h 0 v u u 0 h 0 h 0 h 0 h B u 3 v3 u 4 v 4 (3.70)

133 The res of he elemen dervaon s he same as n case of any oher elemens,.e. he sffness T T marx K Β DB dv, vecor of nernal forces Q Β FdV ec. A numercal negraon n wo Gauss pons s used o negrae he nerface elemen sffness marx. The marx K a nd he vecor Q are n local coordnae sysem and herefore before hey are assembled n he problem governng equaons hey mus be ransformer n global coordnaes. The sffness coeffcens depend on he gap sae. The nerface s consdered open, f he normal force F >R (R s he nerface ensle srengh force) and he correspondng consuve law s (sress free nerface): F 0 (3.7) F 0 op op The sffness coeffcens are se o small, bu nonzero values K, K. The nerface elemen s consdered closed f F R. The sffness coeffcens are se o large cl cl values K, K nn. I should be noed ha he sffness coeffcens are defned only for he purpose of he numercal erave soluon. (Hn: The values of coeffcens n he closed sae (he large values) are based on hckness comparable o he sze of neghbor quadrlaeral elemens. The mnmum values n he open sae can be abou 000 mes smaller. ) The nerface hckness n he ou-of plane drecon s normally provded as an npu parameer. In he case of ax-symmerc analyss s however calculaed usng he formula: x (3.7) where x s he dsance from he axs of symmery. There are wo specal opons for processng he gap elemens: Inal gap openng I s possble o "open" gap a a parcular load sep, ypcally he frs sep of he analyss,.e. we can nroduce o he gaps somehng lke nal elemen srans n case of ordnary fne elemens. Ths s acheved by LOAD INITIAL GAP... INIT_STEP_ID sep_d command. Upon ha, durng calculaon of he (gap) elemen a he sep sep_d an arfcal openng of he nerface s nroduced. Is value s he dsance beween upper and lower elemen surfaces/lnes (wh reference o undeformed srucural shape). The GAP elemen load s ypcally used as follows: we have a srucure wh a base and upper blocks. The upper block falls down owards he base block ha s ypcally fxed. The srucure s solved by nroducng a layer of gap elemens beween he base and upper blocks and applyng he GAP elemen load (for hese gaps elemens) n he s sep. As a resuls, n he frs seps he gaps wll open o he dsance beween he blocks. I nvolves some ensonal forces, bu as he nerface maeral usually susans only compresson forces, hey can be negleced. In nex seps he upper block gradually s fallng down o he base block unl hs. A hs momen nerface gaps ge fully closed, hey change her regme form enson o compresson and he upper block ges fully suppored by he base block. Movng gaps 3 nn 3 Avalable sarng from ATENA verson ATENA Theory 3

134 Suppose we have a srucure has a base block and an upper block sng on he base block. The base block s fxed, he upper block s dragged on he upper surface of he base block. The blocks are no muually nerconneced, only some frcon and coheson forces exs beween hem. Such problems can be modelled by he RESET_DISPLS n flag for he CCDInerface / CC3DInerface. If hs flag s npu, hen he upper and boom surface/lnes for all correspondng elemens are realgned a he end of each sep as shown for D elemens n he followng pcure. The 3D gaps elemen are realgned n he same way. Of course, he boundary surface/lnes projecon of he gap nerface (and hus s "movng" can be used n more complex suaon, bu he essence of he descrbed echnque remans he same. The layer of nerface elemens are ypcally conneced o he boom/ upper block of srucure by MASTER SLAVE NODAL LISTS boundary condons, where we mus no forge o use he flag PROCESS_FLAG USE_CURRENT_COORDS. I wll assure ha afer realgnng he nerface ges properly conneced o he res of he (deformed) srucure. Fg. 3- Movng gap D elemen Noe ha he opon of he gap's nal openng and he rese dsplacemens flag can be combned. Boh hese specal processng opons are possble, because he ATENA sofware uses ncremenal approach o solve he srucure. Thus changng shape of he gap (a he end of he seps) wll no harm governng equlbrum equaons. 3. Truss Ax-Symmerc Elemens. In he followng a crcumferenal russ elemens for axsymmerc analyss are descrbed. The elemens call CCCrcumferenalTruss and CCCrcumferenalTruss and hey are amed manly for modelng srucural crcumferenal renforcemen. For radal renforcemen refer o CCIsoTruss<xx> and CCIsoTruss<xxx> elemens. The CCCrcumferenalTruss has one node only, whereas he CCCrcumferenalTruss has nodes wo. They behaves much he same, he dfference beng only n calculaon of her crossseconal area. In case of he CCCrcumferenalTruss elemen he area s enered drecly from npu daa. The CCCrcumferenalTruss elemen calculae he area as s hckness (defned n s geomery daa) mulpled by s lengh. Unlke soparamerc elemens hses elemens are derved and compued analycally. Geomery, nerpolaon funcons and negraon pons of he elemens are gven n Fg

135 y y CCCrcumferenalTruss x CCCrcumferenalTruss x Fg. 3-3 Geomery of CCCrcumferenalTruss and CCCrcumferenalTruss elemens. In he followng srucural vecors and marces for he CCIsoTruss elemen are derved. Developmen of he CCIsoTruss s much he same. In fac, s CCIsoTruss acng a he cenrepon of he CCIsoTruss elemen wh s cross-seconal area calculaed as explaned above. The elemen vecors and marces for Toal Lagrangan formulaon (TL), confguraon a me and eraon () are as follows. Noe ha hey are equally applcable for Updaed Lagrangan formulaon (UL) upon applyng changes relaed o he elemen reference co-ordnae sysem (undeformed vs. deformed elemen axs.). ( ) The russ elemen cener has a reference me and co-ordnaes X [ x, x] and ( ) ( ) ( ) X [ x, x], respecvely. The elemen lengh (a respecve me) s s lengh s ( ) ( ) l x and l ( x u ). () () ( ) Incremen of Green Lagrange sran (a me, eraon () wh o confguraon a me ) s calculaed: where russ lengh ( ) ( ) x x () ( ) l l l () l ( x u u ). Noe ha () ( ) () (3.73) ( ) u s co-ordnae ncremen ( ). Subsung expressons for elemen lengh no (3.73) yelds: Separang 4 x u u x u x ( ) ( ) ( ) () u u u u ( ) ( ) ( ) ( ) x x x ( ) u from (3.74) and rearrangng n marx form we oban: L0 x (3.74) B (3.75) ATENA Theory 5

136 and u B (3.76) ( ) ( ) L x ( n) NL The nd Pola-Krchhoff sress marx and ensor are: x B (3.77) ( ) ( ) ( ) S S [ S ] (3.78) The formulaon s compleed by relaonshp for elemen deformaon graden yelds: () X,, whch where engneerng sran 6 X ( ) x u e (3.79) () (), x () e s calculaed by (3.80) ( ) ( ) () 4 x u 4 x x u x () l e l 4 x x 3. Ahmad Shell Elemen Ths secon descrbes Ahmad shell elemen mplemened n ATENA, see Fg I can be used o model hn as well as hck shell or plae srucures. I accouns for boh plane and bendng srucural sffness. The elemen feaures quadrac geomery and dsplacemen approxmaon and herefore, he elemen s shape can be non-planar. I s possble o accoun for srucural curvaures. Bg advanage of hs elemen s ha s seamlessly connecble o rue 3D ATENA elemens. Three modfcaons of hs elemen are suppored and hese are characerzed by Lagrangan, Serendpy and Heeross varan of geomery and dsplacemen feld approxmaon. In order o avod or mnmze membrane and shear lockng of shell elemen s furher possble o use full negraon scheme, as well as reduced and/or selecve negraon. The problems concerned wh combnaon of parcular dsplacemen approxmaon and negraon scheme wh respec o lockng phenomena are dscussed. The elemen s derved n a way smlar as he oher fne elemens, whch are descrbed n hs manual. Hence, n he presen descrpon wll concenrae manly on feaures ha are specfc for hs elemen. Followng Toal Lagrangan formulaon of he problem, he prncple of vrual dsplacemen s used o assemble ncremenal form of governng equaons of srucure. The presen Ahmad elemen belongs o group of shell elemen formulaon ha s based on 3D elemens concep. Neverheless, uses some assumpons and resrcons, so ha he orgnally 3D elemen s ransformed no D space only. I saves compuaonal me and also avods some formulaon dffcules peranng o 3D elemens. The elemen s n-plane negraon s carred ou n usual way by Gauss negraon scheme, whls n he 3 rd dmenson (.e. perpendcular o md surface of elemen) he negraon can be

137 done n closed (analycal) form. However, n order o enable accounng for nonlneary of consuve equaons, he so called layer concep s used nsead. Hence, n he 3 rd dmenson smple quadrlaeral negraon s employed. The presen degenerae connuum elemen was orgnally proposed by Ahmad e al. (Ahmad, Irons e al. 970). Followng general shell elemen heory concep, every node of elemen has fve degree of freedom, e.g. hree dsplacemens and wo roaons n planes normal o mdsurface of elemen. In order o faclae a smple connecon of hs elemen wh oher rue 3D elemens, he (orgnal) fve degrees of freedom are ransformed no x,y,z dsplacemen of a op node and x,y dsplacemen of a boom node degrees of freedom. The wo nodes are locaed on he normal o md-surface passng hru he orgnal md-surface elemen s node, see Fg The essenal pon n he elemen s dervaon s ha dsplacemens and roaons felds are approxmaed "ndependenly", (see e.g. (Jendele 98), where smlar approach s used for plaes). Ths means ha hey are handled separaely. Unlke n rue Mndln heory our formulaon maches geomerc equaons auomacally. However, a specal echnque s used o mprove he elemen s shear behavour (Hnon and Owen 984). The frs formulaon of hs elemen proposed by Ahmad was lnear bu snce ha me many mprovemens have been acheved. The mos mporan s he applcaon of reduced or selecve negraon scheme ha reduces or oally removes lockng of he elemen. Also, many auhors exended he orgnal formulaon o geomercally and laer also maerally nonlnear analyss. One such an advanced form of he elemen s he formulaon mplemened n ATENA. On npu, he Ahmad elemen uses he same geomery as 0 nodes soparamerc brck elemen,.e. CCIsoBrck<xxxxxxxxxxxxxxxxxxxx>, see Fg Ths s needed, n order o be able o use he same pre- and posprocessors suppor for he shell and nave 3D brck (.e. hexahedron) elemens. Afer he s sep of he analyss, he npu geomery wll auomacally change o he exernal geomery from Fg As nodes 7 and 8 conan only so called bubble funcon, he elemen s pos-processed n he same way s would be he elemen CCIsoBrck<xxxxxxxxxxxxxxxx>. Inernally, all elemen s vecors and marces are derved based on he nernal geomery as depced also n Fg Wh shell elemens, he bes connecon a edges s o cu boh a 45 degrees, or a dfferen correspondng angle f he hcknesses are no he same, or f conneced a oher han rgh angle, see Fg. 3-4 (a). Anoher opon s o use a volume brck elemen a he corner, whch s he only feasble way when more han wo shells are conneced, see Fg. 3-4 (b). The nodes on he surface conneced o he volume elemen have o be lsed n he INTERFACE subcommand n he shell geomery defnon for correc behavor. Connecng lke n Fg. 3-5 s no recommended, as he maser-slave relaons nduced by he fxed hckness of he shell may cause numercal problems. Shell Brck Shell (a) Shell3 (b) ATENA Theory 7

138 Fg. 3-4: Ahmad Shell - recommended connecon (a) shells (b) 3 shells 3.. Coordnae Sysems. Fg. 3-5: Ahmad Shell - no recommended connecon The essenal pon n he elemen s dervaon s o undersand coordnae sysems ha are used whn he dervaon. These are as follows. Noe ha all vecors ndcang coordnae sysems axes are normalzed. Thus, any dreconal cosnes are smply compued as scalar producs ha need no be dvded by he vecors norm. Global coordnae sysem. I s used o defne he whole FE model. Global coordnaes are denoed by x, x, x 3, where he ndex referrers o me. Noe ha we are usng Modfed Lagrangan formulaon, n whch model confguraon s updaed afer each me sep, whle whn one sep (for erang) he confguraon from he sep begnnng s employed. Thus, 0 x 0 0, x, x3 are a pon global coordnaes pror any load has been appled. Nodal coordnae sysem Ths coordnae sysem s defned a each pon of elemen md-plane surface,.e. md-nodes -9. k k k A a node k s specfed by vecors V, V, V 3, see Fg

139 Fg. 3-6 Ahmad elemen coordnae sysems k k k The vecors V, V, V 3 are defned as follows: Frsly, wo auxlary vecors V, V 3 are calculaed. Vecor V 3 a a pon s defned as a lne jonng boom and op coordnaes a he node k (pror any deformaons,.e. a reference confguraon). The second vecor V s normal o V 3 and s parallel o plane of global 0 G X and 0 X 3 G. Hence: V V, V, V V V, V, V V 3 3,0, V 3 3 (3.8) If V 3 s parallel o 0 G X (.e. V 3 V 3 0 ), V s defned by 3 V V 3,0,0 (3.8) k k k k Afer ha, he coordnae sysem V, V, V 3 self s defned. The vecor V 3 s consruced n he same way as was he vecor V 3, however, curren,.e. deformed confguraon s used. The remanng wo vecors are defned as vecor produc: k k 3 V V V (3.83) k k k V V V3 (3.84) ATENA Theory 9

140 k k k The vecors V, V, V 3 defne local nodal shell coordnae sysem n whch he shell roaons are specfed. As already menon, he orgnal formulaon of he elemen has 5 DOFs per nodes. These are 3 dsplacemens, expressed n he global coordnae sysems and wo roaons,. k k They are roaons along he vecors V, V. I comes from defnon ha V 3 need no be normal o he elemen surface. I mus only connec he op and boom nodes of he shell. k Somemes, s advanageous o modfy he nodal coordnae sysem so ha V 3 remans k k unchanged bu V and V are roaed (along V 3 k ) o a ceran drecon. Noe however, ha k k k muual orhogonaly of V, V, V 3 mus no be damaged. Local coordnae sysem L L L Local coordnaes are denoed by x, x, x 3. The sysem refers o coordnae axes L L L,, 3 X X X. I s used manly a samplng (negraon) pons o calculae srans and L L L sresses. The vecor axes X, X, X 3 are defned by: X L 3 x x r s x x r s x3 x3 r s (3.85) L L k X X3 V L L L X X X3 (3.86) k k k As he nodal coordnae sysem V, V, V 3 can roae along V 3 k, he local coordnae sysem L L L L would X, X, X 3 roae smulaneously along X 3. Ths defnon allows for user defned shell local coordnae sysem ha s common for all shell elemens, rrespecve of her k L ncdences. Noe ha unlke V 3 he vecor X 3 s always normal o he elemen md-plane surface. Curvlnear coordnae sysem Ths sysem s used o calculae dervaves and negraon n elemen negraon pons. Is coordnaes are rs, for n-plane drecon and n drecon of elemen hckness, see Fg The n-plane dsplacemens are approxmaed by Lagrange, Heheross or Serendpy approxmaon smlar D soparamerc elemens. For he 3 rd drecon,.e. hrough he deph of he elemen. lnear approxmaon s used whn he frame of he shell layer concep. 30

141 Inpu geomery s 8 r Exernal geomery 5 7 s 8 r Inernal geomery s r Fg. 3-7 Geomery and he elemen s nodes ATENA Theory 3

142 3 Fg. 3-8 Degenerae shell Ahmad elemen coordnae sysems and degree of freedoms (DOFs)

143 3.. Geomery Approxmaon The coordnaes of he op and boom elemen surface s used o defne he elemen geomery: k, op k, bo x x x N k, op k, bo x x hk x x k k, op (3.87) k, bo x 3 x 3 x 3 where N=8 s number of nodes per elemen, (geomery s always nerpolaed by 8-nodes Serendpy nerpolaon, rrespecve of dsplacemen nerpolaon), h(r,s) s k-h nerpolaon k, op k, bo x x k, op k, bo funcon, r,s, are soparamerc coordnaes (see Fg. 3-7), x and x are vecor of k, op x k, bo 3 x 3 op and boom coordnaes of pon k, see Fg X [ x, x, x ] k, op k, op k, op k, op 3 node k X [ x, x, x ] k, md k, md k, md k, md 3 X [ x, x, x ] k, bo k, bo k, bo k, bo 3 Fg. 3-9 Approxmaon of he elemen geomery Usng he above he equaon (3.87) can be rewren n he followng form: where hck k k k, md V 3 x x N k md k x x h x V 3 hck k k, md x 3 x 3 k V3 3, k (3.88) k s elemen hckness n node k (.e. dsance beween op and boom pons) and md op bo x x x md op bo x x x md op bo x 3 x k 3 x k 3 k are coordnaes of md surface. (3.89) ATENA Theory 33

144 3..3 Dsplacemen Feld Approxmaon. The general concep of dsplacemen approxmaon s very smlar, (alhough no dencal) o geomery approxmaon. As already menoned he orgnal verson of Ahmad elemen uses 5 md md md degrees of freedom per node, see Fg These are u, u, u3,,, where md md md u, u, u are dsplacemens of he elemen s node a he md-surface and, are 3 roaons wh respec o vecors v k and v k respecvely. These degree of freedoms (DOFs) are used hroughou he whole elemen s developmen. However, n order o mprove compably of he presen shell elemen wh oher 3D elemens mplemened n ATENA, exernally he T op op op bo bo elemen uses u, u, u3, u, u DOFs,.e. dsplacemens a he op and boom of he elemen. The 6 h bo dsplacemen,.e. u 3 s elmnaed due o applcaon of shell heory ha assumes Approxmaon of he orgnal hree "dsplacemen" and wo roaon degrees of freedom s ndependen. Neverheless, he curvaures used n governng elemen equaons use all of hem n he sense dcaed by geomerc equaons. Ths approach enables o sasfy no only equlbrum equaons for membrane sresses and n-plane shear (n md-surface) as s he case of popular Krchhoff hypohess, bu also o sasfy equlbrum condon for ransversal shears (normal o md-surface). Noe ha n he followng dervaon of he elemen we wll deal wh he orgnal se of elemen s DOFs, see (0). Every pon hus has fve degree of freedom, md md md T u, u, u3,,. Dsplacemen vecor s calculaed by: k k k k, md V V u u N k k k md k k u u h u hck V V k k k, md u 3 u3 k k V V 3 3, k (3.90) k k md md md The orgnal dsplacemen vecor a pon k has he form u, u, u 3,,. Unlke n he case of geomery approxmaon, were N=8, dsplacemens approxmaon accouns also for dsplacemen n he elemen md-pon,.e. N=9. The nnh funcon h s so called bubble funcon. k V3 u [ u, u, u ] k, md k, md k, md k, md 3 T T node k k k k V k V Fg Dsplacemen approxmaon 34

145 3..4 Sran and Sresses Defnon. The nd Polla Krchhoff ensor and Green Lagrange sran ensor s used. They are calculaed and prned n he local coordnae sysem x ', x ' and x '3. Green - Lagrange ensor. The general defnon for Green-Lagrange sran ensor has he form (see eq. (.8)): u u u u (3.9) 0 j 0, j 0 j, 0 k, 0 k, j Usng he above equaon and applyng he Von-Karman assumpon, Eqn. (3.9) can be wren as: u x u 3 u x 0 x 3 0 u u u x 0 0 L 0 NL (3.9) x x u u 3 u u 3 x x 0 3 x3 x 0 u u3 0 x3 x The Von-Karman assumpons smplfy he calculaon of sran by accepng ha: 0 L All srans are relavely small, The deflecon normal o md surface of shell s of order of hckness, The boh curvaures are much smaller han., The n-plane dsplacemens are much smaller han ransverse dsplacemen and hus her dervaves n nd order erms can be negleced. and 0 represens lnear and nonlnear par of sran vecor, respecvely. More NL nformaon abou her calculaon s beyond he scope of hs publcaon. I s avalable e.g. n (Jendele 99). nd Polla Krchhoff ensor. Energecally conjugaed wh Green - Lagrange ensor s nd Polla Krchhoff ensor and hs ensor s used by he presen shell elemen. Remnd ha we accoun for all sresses wh excluson of normal sress whch s perpendcular o shell md surface (as s usual pracce n shell analyss). Ths s he reason, why we nroduced local coordnae sysem and all expresson are derved wh respec o. Obvously he local coordnae sysem vares dependng on elemen deformaon and hus s necessary o re-compue (each eraon) he ransformaon marx T (ha relaes local and global coordnae sysems). ATENA Theory 35

146 To compue nernal forces we wll use nd Polla Krchhoff ensor n vecor form (n a node k): S S S S S S (3.93) 0 k Noe ha ha s possble o abbrevae full 3 by 3 elemen ensor o he above vecor, because of adopng Von Karmann smplfyng assumpon. k 3..5 Serendpy, Lagrangan and Heeross Varan of Degeneraed Shell Elemen. Unl now no nformaon abou nerpolaon funcon h and number of negraon pons were gven. The presen shell elemen analyss uses Serendpy nerpolaon funcons. Noe ha bubble funcon h 9 (used n dsplacemen approxmaon only) represens relave deparure of approxmaed funcon wh respec o he funcon value calculaed by prevous egh approxmaon funcons. The nerpolaon funcons h read: h ( r, s) (- r)(- s)(-r-s-) 4 h (,) r s (-)(- s r ) h3 (,) r s ( r)(- s)( r-s-) 4 h4 (,) r s ( r)(- s ) h5 ( r, s) ( r)( s)( rs-) 4 h6 (,) r s ( s)(- r ) h7 (,) r s ( r)(-)( s r-s-) 4 h8 (,) r s (- r)(- s ) h9 ( rs, ) (- r)(- s) (3.94) The acual values n cener pon can be calculaed by: 8 9 ( 0, 0) 9 a a h r s a (3.95) where h are values of nerpolaon funcon a pon (0,0), a are correspondng node values, a 9 s deparure n he cener (.e. compued value correspondng o degree of freedom a cener) and a 9 s oal value n cener. Dependng on combnaon how many nodes and negraon pons are used, we dsngush he Serendpy, Lagrange and Heeross degeneraed elemen varans, see Fg

147 Serendpy elemen. Ths elemen was used n he orgnal Ahmad work. I comprses egh nodal pons (cener pon correspondng o bubble funcon s omed). Gauss negraon scheme s used for negraon. I can be negraed by full, reduced or selecve negraon procedure. Usng full negraon,.e. a hree by hree sample pons, elemen exhbs shear lockng for hn and even moderaely hck elemen. If reduced negraon s used. he problem of lockng s sgnfcanly mproved whou creang spurous energy modes on srucure level. However, n case of hn elemen here are wo non communcable spurous energy mode on elemen level. I should be noed ha here were repored some dffcules, f some unfavorable consrans are appled. Neverheless he elemen s popular. If reduced negraon s used he provded resuls are relavely good. Fg. 3-3 Node noaon for elemen varans of he Ahmad shell elemen Nne node Lagrangan elemen. The nne pon Lagrangan elemen s sll consdered o be he bes varan of he degeneraed elemen. Ths s especally because of s versaly. For full negraon scheme here s no problem wh membrane and shear lockng for very hn plae and shell applcaon. If elemen s moderaely hck, shear lockng can be mproved by reduced negraon scheme. However, n ha case he elemen exhbs rank defcency. ATENA Theory 37

148 Heeross elemen. The Heeross elemen s very smlar o Lagrangan elemen. The dfference s ha assumes frs hree DOFs a he elemen cenre o be consraned, (.e. only he roaons are reaned) The elemen exhbs beer behavor compared wh prevous quadrac elemens and especally n combnaon wh selecve negraon scheme no lockng s produced. Wh reduced negraon he elemen provdes resuls beer han Lagrangan elemen. In ha case here are some spurous mechansms, bu for praccal soluon here are no probable. Fg. 3-3 Summary of lockng and spurous energy modes Problem of membrane and shear lockng for lnear analyss are summarzed n Fg In he case of nonlneary, he suaon s much more complcaed and depends prmarly on he maeral sae a he samplng pons. For more nformaon refer o (Jendele, Chan e al. 99) Elemen s negraon In prevous paragraphs we menoned many me full, reduced and selecve negraon scheme. The sense of hese procedures s bes o demonsrae n Fg

149 Fg Inegraon schemes and samplng pon noaon The seps durng selecve negraon of shear can been explaned on example of negraon arbrary funcon (,) f rs : / Frs we calculae he value of f a samplng pons ha corresponds o wo by wo negraon rule,.e f( , ), f ( ,0.5773), f(0.5773, ), f (0.5773,0.5773) / Usng blnear approxmaon we calculae he values of f a pons ha correspond o hree by hree negraon rule. There are wo possbly o ha. The frs one s based on orgnal approxmaon area and he man dea s ha we calculae he value of funcon f a "corners" of soparamerc elemen (.e. r., s. ): ATENA Theory 39

150 f f f f 4 ' ( , ) fh( , ) k 4 ' ( , ) fh( , ) k 4 ' (0.5773, ) fh(0.5773, ) k 4 ' (0.5773, ) fh(0.5773, ) k (3.96) ' where f are elemen nodal values of funcon f and h are nerpolaon funcons correspondng o wo by wo nerpolaon and a node. Fg Exenson of blnear approxmaon funcon for arbrary recangular The se of equaon (3.96) can be solved for f. Havng hese value we can b-lnearly approxmae funcon f and compue funcon value a any pon,.e. also a samplng pons correspondng o hree by hree negraon rule. The second and more elegan soluon s drec approxmaon. The nerpolaon funcon h are presened for an square area of he sze x uns, bu hey can be exended o a recangular of any sze, as shown n Fg Snce he funconal values for he x samplng pons n he corner of he square ' lr ls x are avalable, he approxmaon funcons h can be used drecly o calculae he values of he funcon f a samplng pons correspondng o he 3x3 negraon rule. For negraon n drecon perpendcular o r - s plane, ha s n coordnae s also possble o use Gauss negraon, bu due o maeral nonlneary here s more advanages o use drec recangular negraon. Ths concep s called he Layered model, see Fg The man dea of s o dvde he elemen along he hckness o layers whereby n parcular layer he values of sran and sresses are expeced o be consan and equal o her value a wegh pon of layer. Hence he negraon n drecon s compued as a sum of negraed expressons mulplcaed by adequae area of layer for all layers from boom o op of elemen. I was found ha o acheve good accuracy s necessary o abou sx o en layers. Ths concep..e. layer model s advanageous because enables us o creae for example renforcng layers n elemen and also we can use fner dvson near op and boom of shell, where hgher sress level can be expeced. 40

151 3..6 Smeared Renforcemen Fg The layer model The ATENA mplemenaon of he Ahmad shell elemen suppors embeddng of smeared renforcemen layers. In hs concep, renforcemen bars wh he same coordnae z, (see Fg. 3-35), maeral and he same drecons are replaced by a layer of smeared renforcemen. Such a layer s placed a he same elevaon z as he orgnal renforcemen bars and s hckness s calculaed so ha sum of cross seconal area of he bars and he replacng smeared renforcemen layer s he same. The layer s usually supermposed over exsng concree layers and employs CCSmeardRenforcemen maeral law, whch makes possble o accoun for he orgnal renforcemen bars drecon. Because of he fac ha each layer of he Ahmad shell can use a dsnc maeral model, concree and smeared renforcemen layers are reaed n smlar way. (Consuve equaons,.e. maeral law are placed ousde of ATENA fne elemens code). Descrpon of synax of relaed npu commands s beyond scope of hs documen, bu can be found n he ATENA Inpu Fle Forma documen. Noe ha he suppor for smeared renforcemen does no exclude use of srucural dscree renforcemen. Boh he ype of renforcemen can be combned n one model o acheve he bes lkeness of he he real srucure wh s numercal model Transformaon of he Orgnal DOFs o Dsplacemens a he Top and Boom of he Elemen Nodal Coordnae Sysem Ths secon descrbes n deal he whole procedure of ransformng Ahmad elemens from s orgnal formulaon o he new one used by ATENA SW. Jus o remnd you: The orgnal formulaon (descrbed n he prevous secons) dffers from he new one n selecon of elemen degree of freedom, see Secon Le us sar o work n nodal coordnae sysem frs. The followng equaon saes ransformaon rules for ransformng hree global dsplacemens and wo nodal roaons a he elemen md-plane, (.e. he orgnal DOFs a a node k), o 6 dsplacemens a nodal coordnae sysem, hree a he op and hree a he boom surface of he shell. Noe he rgh superscrps N ha ndcae nodal coordnae sysem. ATENA Theory 4

152 k k k V V V hck 0 3 k, N, op u k k k V V V 0 hck k, md k, md k, N, op 3 u u u k k k k, N, op u k, md k, md V3 V 3 V u u 3 3 k, N, bo k, md k, md u3 T u3 (3.97) u k k k V,, V V hck 0 k k k N bo 3 u k k k, N, bo k k u k 3 V V V 0 hck 3 k k k V3 V 3 V Transformaon from nodal o global coordnaed sysem The nex sep n he elemen s dervaon s o wre ransformaon of he lef-hand sde vecor of (3.97) from nodal o global coordnae sysem. I reads: k k k V V V k, op k, N, op k, N, op u k k k V, V V u u k op u k, N, op k, N, op k k k u u k, op u V V 3 V k, N, op k, N, op u 3 u 3 k, bo k k u k k, N, bo T k, N, bo (3.98) V V V 3 k, bo u u u,,,, k k k N bo k N bo u k u k, bo V V V 3 k, N, bo k, N, bo u 3 u3 u3 k k k V V 3 V Complee ransformaon of he orgnal DOFs o he new elemen formulaon DOFs The fnal ransformaon from he orgnal o he new elemen DOFs a a node k s oban by subsung (3.97) no (3.98). Thus we can wre where T u k, op k, md k, md k, op u u u k, md k, md k, op u u u 3 k, md k, md k, bo u 3 u 3 u k k k, bo u k k k, bo u3 T T T (3.99) 4

153 k k k k k k k k k k k k k k k k k hck k hck k V V V 3 V V V V V 3 V 3 V 3 V V 3 V V 3 3 V 3 V 3 V k k k k k k k k k k k k k k k k k hck k hck k V V V V V 3 V 3 3 V V V3 V V V 3 V V 3 3 V 3 V 3 V k k k k k k k k k k k k k k k k k hck k hck k V V V 3 V V 3 3 V 3 V 3 V V 3 V V 3 3 V 3 3 V 3 V 3 V3 3 V V 3 3 k k k k k k k k k k k k k k k k k hck k hck k V V V3 V V V V V 3 V 3 V 3 V V 3 V V 3 3 V 3 V 3 V k k k k k k k k k k k k k k k k k hck k hck k V V V V V 3 V 3 3 V V V3 V V V 3 V V 3 3 V 3 V 3 V k k k k k k k k k k k k k k k k k hck k hck k V V V 3 V V 3 3 V 3 V 3 V V 3 V V 3 3 V 3 3 V 3 V 3 V3 3 V V 3 3 In a very smlar way we can defne nverse ransformaon,.e. from he new DOFs o orgnal one. Whou any dervaon he marx reads: where T T T3 T T T3 T T T3 T T T3 T3 T3 T33 T3 T3 T33 T' k k k k k k V V V V 3 V V 3 k k k k k k hck hck hck hck hck hck k k k k k k V V V V 3 V V 3 k k k k k k hck hck hck hck hck hck k, op u k, md u k, op k, md u u k, op u k, md 3 u T' 3 k, bo (3.00) u k k, bo u k k, bo u3 Consranng he redundan DOF o comply wh shell heory As noed earler, he orgnal se of DOFs a a node comprses 5 DOFs, whls he new one has sx DOFs. Consequenly, one DOF from fxed. The presened work prefers o consran canddaes, f k, bo 3 u, u, u, u, u, u k, op k, op k, op k, bo k, bo k, bo 3 3 k, bo 3 T mus be kbo u bu u, kbo or u, are also good u can no be fxed due some numercal problems, usually due o a specal poson of he elemen wh respec o global coordnae sysem. Dervaon of he consran s now demonsraed on he case of u. Usng (3.99) k, bo 3 ATENA Theory 43

154 u u ( u u ) k, bo k, op k, bo k, op k, op k, md k, md k u3 T6 T3 u T6 T3 u T56 T53 k, op k k k k k k u3 hck V hck V 3 3 (3.0) Now n (3.0) elmnae k and k usng (3.00). Thus we oban one equaon relang T k, op k, op k, op k, bo k, bo k, bo u, u, u3, u, u, u 3, whch s hen used o consran k, bo u 3 as a lnear combnaon of u, u, u, u, u : k, op k, op k, op k, bo k, bo 3 where: u c u c u c u c u c u (3.0) k, bo op k, op op k, op op k, op bo k, bo bo k, bo The DOFs u or kbo, kbo, c op c c c op op 3 bo c bo k k k k V V 3 + V V 3 k k V 3 V 3 k k k k V V 3 + V V 3 k k V V 3 3 k k V 3 V 3 k k V 3 V 3 k k k k V V 3 + V V k k V 3 V 3 yy k k k k V V 3 + V V 3 k k V V 3 3 z (3.03) u can be elmnaed n he same way. Durng he execuon of he elemen, s recommended o consran one of u, u, u based on whch soluon s k, bo k, bo k, bo 3 he mos sable, (.e. maxmum denomnaor n (3.03)). Consranng DOFs a he cenre of Heheross elemen A specal aenon needs o be pad o he 9 h md-plane node of Heheross elemen, when we have o addonally consran of hem. u, u, u. Thus, of he 6 DOFs we need o consran 4 k, md k, md k, md 3 For example, suppose we wan o keep free fx k, op k, bo k, bo k, bo 3 u and k, op u and we need o u, u, u, u. Equaon (3.0) from he prevous paragraph needs o be added by hree more equaons. These are: k, op 3 44

155 k, op u k, op k, md ' ' ' ' ' ' u u T T T3 T4 T5 T 6 0 k, op k, md ' ' ' ' ' ' u 3 u T T T3 T4 T5 T 6 0 k, bo k, md ' ' ' ' ' ' u u 3 T3 T3 T33 T34 T35 T 36 0 k, bo u k, bo u3 Equaons (3.0) and (3.04) are hen solved for combnaon of u and k, op u. k, op 3 k, op k, bo k, bo k, bo 3 k, op ' ' ' ' ' k, op ' k, op u T T4 T5 T 6 Tu T3u3 kbo, ' ' ' ' ' kop, ' kop, u T T4 T5 T6 Tu T3u3 kbo, ' ' ' ' ' kop, ' k, u T3 T34 T35 T op 35 T3u T33u3 k, bo op bo bo op k, op op k, op u3 c c c c u c3 u3 (3.04) u, u, u, u as a lnear (3.05) Agan, here are several alernaves regardng of whch of he 6 DOFs o keep and whch o elmnae. The bes opon s chosen he same way as descrbed n Secon Shell Ahmad Elemens Implemened n ATENA Several modfcaons he Ahmad shell elemens are mplemened n ATENA. They are lsed n he followng able: Table 3-5 Ahmad shell elemens. Elemen name Type of approxmaon Number of n-plane negraon pons per axs drecon for bendng Number of n-plane negraon pons per axs drecon for shear Commen CCAhmadElemen33L9 Lagrange 3 3 No spurous modes, lockng for hs shells CCAhmadElemen3L9 Lagrange 3 CCAhmadElemen33H9 Heeross 3 3 CCAhmadElemen3H9 Heeross 3 Good compromse beween lockng and spurous energy modes CCAhmadElemenS8 Serendpy Fas, bu spurous modes ATENA Theory 45

156 3.3 Curvlnear Nonlnear D Isoparamerc Layered Shell Quadrlaeral Elemens Ths secon descrbes shell elemens ha model a srucure by a curvlnear D surface. The elemen uses herarchcal geomery and dsplacemen nerpolaon. I can have from 4 o 9 nodes, each of hem havng 5 DOFs: 3 dsplacemens n drecon of global X,Y,Z axs and roaons along user defned vecors V, V. If he shell s locaed n he XY plane, hen ypcally V X, V Y. The elemen uses lnear geomery and dsplacemen nerpolaon n he drecon of s hckness and quadrac or lnear approxmaon n he elemen's plane. If quadrac approxmaon s used, behavour of he elemen resembles behavour of Ahmad shell elemen descrbed n he prevous secon. 4 nodes verson of hs elemen,.e. he elemen wh lnear approxmaon, does no perform well, (he elemen s oo sff), and hus s recommended only for some local lnks ec. On he oher hand, boh bendng and membrane behavour of 8-9 nodes verson of he elemens s grea. The elemens are derved based on he Shell heory, (smlarly o Ahmad elemen). As a resul, s assumed 0, s neglgble and he elemen canno change s hckness. ( ndcaes local axs n he shell's hckness). Dependng on number of elemen nodes hese fne elemens call CCIsoShellQuad<xxxx>... CCIsoShellQuad<xxxxxxxxx> Fg CCIsoShellD elemens 46

157 3.3. Geomery and dsplacemens The shell s geomery a he confguraon and d s defned by: k n k x hk X akv x h X a V x h X a V ( ) k,( ) nk,( ) k k () k,() nk,( ) k k (3.06) where =,,3 s ndex relang o global axes x, x, x 3, (.e. x,y,z), k... n G, n G = number of he elemen's nodes used o approxmae geomery, ypcally 8 or 9. Noe ha due o Shell heory he shell hckness a node k a a a a a. The symbol 0 ( ) ( ) k k k k k h coordnae, (,, 3 for coordnae x, yz, ), of he vecor eraon (). The vecor n ( V V V ) nk,( ) V n V a node k a me V n s normal o he shell. Laer we wll use also vecors. They wll consue base vecors for shell's bendng roaons,. Smlarly, dsplacemens a me, eraon (-) : s, V, V, Subsung (3.06) no (3.07) u x x (3.07) ( ) ( ) u h X a V X a V k,( ) k nk,( ) n k hk X X ak V V ( ) k,( ) nk,( ) k nk k k k (3.08) Noe ha n hs case k... n, n s number of nodes o approxmae dsplacemens. Curren mplemenaon of he shell elemens assumes ng Dsplacemen ncremens whn an eraon (a me n, (whch dffers for Ahmads elemens). ) are : u hk X X ak V V k nk,( ) nk,( ) hk U ak V V k,( ) k,( ) nk,( ) nk,( ) (3.09) A each node he elemen has 5 DOFs: 3 dsplacemens below: k k U and wo roaons, descrbed k ATENA Theory 47

158 Le us defne a each node of he shell a local coordnae sysem specfed by hree vecors V, V, V k,( ) k,( ) nk,( ), see Fg The las vecor s vecor normal o surface of he shell a node k and he frs and second vecors are calculaed as follows: V e V / e V k,( ) nk,( ) nk,( ) V V V k,( ) nk,( ) k,( ) (3.0) For he nex dervaon le us assume a general vecor vl vl, vl, v 3L wh un lengh ha s subjec o roaons [,, ] T L L L, (where he subscrp L ndcaes ha boh he vecor and he roaons are defned k,( ) k,( ) nk,( ) wh respec o he local coordnae sysem (defned by V, V, V ). The roaons of he vecor wll produce dsplacemens, (all n he local CS) ul 0 v3l v L L v3l vl L u L v3l 0 v L L v3ll vll (3.) u 3L vl vl 0 L vll vl L T Transformng he dsplacemens from local o global coordnae sysem k,( ) k,( ) nk,( ) u ul V V V ul k,( ) k,( ) nk,( ) u LG u L V V V u T L k,( ) k,( ) nk,( ) u 3 u 3L V 3 V 3 V 3 u 3L k,( ) k,( ) nk,( ) V V V v3ll vl L k,( ) k,( ) nk,( ) V V V v3ll vll k,( ) k,( ) nk,( ) V3 V3 V 3 vll vl L V v v V v v LL LL LL LL LL LL ) LL L L LL L L vll vll V v v,( k ) k,( ) nk,( ) 3L L L L 3L L L L L L L L V v v V v v V v v k,( ) k,( ) nk,( ) 3 3 V v v V v v V k,( ) k,( ) nk,( (3.) Now assume he same behavour for a vecor normal o he shell's surface (agan n he local CS n k,( ) v V 0,0,. When hs vecor ges roaed, produces and un lengh),.e. dsplacemens, (see (3.): L L k L L k L L k u V V,( k ),( ),( k ),( ) u V V,( k ),( ) u 3 V3 L V 3 L (3.3) 48

159 Subsung now, and L k,( ) k,( ) L nk,( u V ), k..3 we can wre fnal equaons for dsplacemens due o roaons, (for eraon (-) and () and he dfference): k V V V nk,( ) k,( ) k,( ) k,( ) k,( ) V V V nk,( ) k,( ) k,( ) k,( ) k,( ),( ),( ) k,( ) k,( ),( k ) V V V V nk,( ) nk,( ) k k k,( ) V V k k,( ) k k,( ) (3.4) Hence, hey represen roaon along wo user defned vecors ha he vecors nk,( ) V moves as he srucure deforms. nk,( ) V. I s mporan o noe Usng (3.4) n (3.09) yelds (and assumng shell hckness a a node k ) u h U a V V k k k,( ) k k,( ) k k (3.5) Noe ha he vecors V, V, V k,( ) k,( ) nk,( ) (3.3) should be used o connec dofs of shell and sold elemens. have o be normalzed. Noe also ha 3.3. Connecon of he shelld o an amben sold elemen Connecon of he shelld elemen o an amben srucure consss of wo par:. fx a FE node wh [ uvw,, ] dsplacemen whn he shelld elemen,. fx wo roaon dofs of he shelld elemen whn amben elemens Fxng a FE node wh [ uvw,, ] dsplacemen whn he shelld elemen Usng he shelld approxmaon he shell's dsplacemen a he boom are: bo u and a he op op u u a h U V V u a h U V V bo k k k k,( ) k k,( ) k op k k k k,( ) k k,( ) k u h a V V opbo k k,( ) k k,( ) k k (3.6) ATENA Theory 49

160 The ndex s..3 for x..z dsplacemens. Usng he shell3d approxmaon dsplacemen a he same locaons can be calculaed by: uu hh UU bo bo bo, l l uu hh UU op op op, l l uu uu uu opbo op bo (3.7) bo op where hh l and hh l are he sold's shell3d nerpolaon funcons a locaon op and boom bo, l op, l of he shell's a node, UU, UU are correspondng nodal dsplacemens of he sold elemen. Comparng (3.7) and (3.6) can be shown ha h hh hh op bo k k k h hh hh op bo k k k (3.8) Thus n order o fx [ uvw,, ] dofs of a node wh shelld elemens we frs calculae hh values for he case of shell3d approxmaon. Then, hese are used o ge h, (see (3.7), comprsed n he shelld approxmaon. I remans o compue shelld roaon, and hs s (agan) done by comparng D and 3D approxmaon n (3.6) and (3.7). Afer some mahemacal manpulaon we wll arrve o he fnal expressons: op bo ak k op bo ak k op bo hhk hhk 0 0 Vx hhk hhk Vx hhk hhk uk u v k w k k k k k (3.9) op bo ak k op bo ak k op bo u 0 hhk hhk 0 Vy hhk hhk Vy hhk hh k k u 3 op bo ak k op bo a op bo 0 0 hhk hhk Vz hhk hhk Vz hhk hhk Fxng wo roaon dofs of he shelld elemen whn amben elemens Dervaon of expressons o fx shelld roaons n amben elemens s based (smlarly o he prevous secon) on comparng he shelld and shell3d approxmaon of op and boom nodes. Wha we do s we frs we fx he op and boom n he amben elemen usng (sold) 3D approxmaon. I yelds expresson somehng lke 50 uu hh UU uu hh UU op op op, l l bo op bo, l l (3.0) Noe ha rhs of (3.0) may also nclude roaons. The resulng equaons for shelld roaon, are (3.)

161 D a ( V V V V V V V V V V k k, x k, x k, y k, x k, y k, x k, z k, x k, y k, x V V V V V V V V V V V kx, ky, kx, ky, kx, ky, kz, kx, kz, kx, kz, V V V V V V V V V V V V kx, kz, ky, kz, ky, kz, ky, kz, kz, kx, kz, ky, ) cf ( V V V V V V V V V V op k, kx, ky, kx, ky, kx, kx, kz, kz, kz, kx, cf ( V V V V V V V V V V op k, k, x k, y k, x k, y k, x k, y k, z k, z k, z k, y cf ( V V V V V V V V V V bo k, k, x k, y k, x k, y k, x k, x k, z k, z k, z k, x bo k, k, x k, y k, x k, y k, x k, y k, z k, z k, z k, y )/ D )/ D op cfk,3 ( Vk, x Vk, z Vk, xvk, yvk, z Vk, xvk, zvk, x Vk, xvk, zv, )/ D cf ( V V V V V V V V V V )/ D ) / D cf ( V V V V V V V V V, V, V, )/ D bo k,3 k, x k, z k, x k, y k, z k, x k, z k, x k x k z k y k y cf ( V V V V V V V V V V op k, k, x k, y k, x k, z k, y k, x k, y k, z k, x k, z cf ( V V V V V V V V V V op k, k, x k, x k, y k, y k, x k, y k, z k, z k, y k, z cf ( V V V V V V V V V V op k,3 k, x k, x k, z k, y k, y k, z kz, kx, kz, ky, cf ( V V V V V V V V V V bo k, k, x k, y k, x k, z k, y k, x k, y k, z k, x k, z cf ( V V V V V V V V V V bo k, k, x k, x k, y k, y k, x k, y k, z k, z k, y k, z cf ( V V V V V bo k,3 k, x k, x k, z k, y k, y V V V V V kz, kz, kx, kz, ky, / D / D / D / D / D / D cf ( hh UU...) cf ( hh UU...) op op op, l bo bo bo, l k l, l l, k cf ( hh UU hh UU k op l, op op, l bo bo bo, l l...) cfl, ( k...) (3.) k.. number of approxmaon shell D nodes..3, ( xz.. ) l.. number of approxmaon sold 3 D nodes k k where V V V V. k, k, Noe ha dsplacemen dofs are fxed by (3.9). If eher boom or op node ges ousde he amben elemen, he mddle pon s used nsead. Equaon (3.) s sll vald bu s necessary o use D Dha replaces D o calculae he op bo cf... cf coeffcens. k, k,3 ATENA Theory 5

162 3.3.3 Green-Lagrange srans The elemens are derved usng Green-Lagrange srans and nd Pola Krchhoff sresses. Green- Lagrange srans a (h eraon),,j-axs x,y,z are calculaed as follows : ( ) ( ) ( ) ( ) (,, ) (,, ) (,, )( u j u j uj uj um um um, j um, j ) ( ) ( ) ( ) () () () () j u, j uj, um, um, j ( ) j ej j (3.3) where: u u u u e u u u u u u e e ( ) ( ) ( ) ( ) ( ) j, j j, k, k, j ( ) ( ) j, j j, m, m, j m, j m, 0 j j e u u e u u u j um, um, j 0 j, j j, ( ) ( ) j m, m, j m, j u m, (3.4) Elemen's dsplacemens u are approxmaed by soparamerc nerpolaon. Hence, s smple o calculae her dervaves wh respec o local coordnae r,s,. Usng an arbrary funcon f(x,y,z) Eqn. (3.5) o (3.7) show, how o compue s dervaves wh respec o globalx,y,z axs. Calculaon of dervaves: f x y zf f r r r r x x f x y z f f s s s s y J y (3.5) f x y z f f z z f f x r f f y J (3.6) s f f z 5

163 Dervaves of coordnaes a wh respec o r,s, o calculae J: x hk k k X av n r r x h k k X av n s s x k h avn (3.7) Dervaves of dsplacemen ncremens a me... ( ) wh respec o r,s,: ( ) u hk k,( ) k nk,( ) n k X X ak V V r r ( ) u hk k,( ) k nk,( ) n k X X ak V V s s ( ) u nk,( ) n k hk ak V V ( ) u hk k,( ) nk,( ) U ak dv r r ( ) u hk k,( ) nk,( ) U ak dv s s ( ) u nk,( ) hk ak dv (3.8) (3.9) k,( ) k,( ) k U X X dv V V nk,( ) nk,( ) nk Dervaves of dsplacemen ncremens a me whn eraon wh respec o r,s,: u hk k k k,( ) k k,( ) U ak V V r r u hk k k k,( ) k k,( ) U ak V V s s u k k,( ) k k,( ) hk ak V V (3.30) U U U k k,( ) k,( ) ATENA Theory 53

164 In order o proceed furher n he dervaon of he 3D soparamerc elemen, we need o x x, x, x. Ths s calculae dervaves of he dsplacemen ncremens wh respec o 3 acheved usng (3.6) hru (3.30). Dervaves of dsplacemen ncremens a me... ( ) wh respec,, 3 x x x : u u u u ( ) ( ) ( ) ( ) nv, k nv, k nv, k J j J j J j3 x j r s u nv, k hk k,( ) nk,( ) J j U ak dv r ( ) xj nv, k hk k,( ) nk,( ) J j U ak dv s J h a dv h h U J J r s nv, k nk,( ) j3 k k k,( ) k nv k nv j j nk,( ) ak hk nv hk nv nv dv J j J j J j3hk r s k,( ) k nk,( ) ak k U hj dv Gj (3.3) (3.3) h h h J J r s k k nv, k k nv, k j j j G h J h k k nv, k j j j3 k Dervaves of dsplacemen ncremens a me whn eraon wh respec o x, x, x 3: u u u u J J J nv, k nv, k nv, k j j j3 x j r s (3.33) u nv, k hk k k k,( ) k k,( ) J j U ak V V x r j nv, k hk k k k,( ) k k,( ) J j U ak V V s J h a V V nv, k k k,( ) k k,( ) j3 k k Afer some rearrangemen Eqn. (3.57) yelds: (3.34) 54

165 u h h U J J k k nv, k k nv, k j j xj r s k ak k,( ) hk nv, k hk nv, k nv, k V j j j3 k J J J h r s k ak,( k ) hk nv, k hk nv, k nv, k V J j J j J j3 hk r s U k h k k,( k ) k k,( k ) k j j j g G g G (3.35) g a a V,( k ) k,( k ) g V k,( ) k k,( ) A hs place, we can derve fnal expresson o compue lnear and nonlnear srans ncremens. () Lnear srans e are calculaed as follows: j e () () () u e... () () e 33 L0 L L0 L L0 L () e B () B... Bk Bk... B n Bn uk e... () e 3 () u () n e3 h 0 0 g G g G k,( k ) k,( k ) k k,( k ) k,( k ) k 0 h 0 g G g G k,( k ) k,( k ) k L 0 0 h 0 3 g3 G3 g3 G3 Bk k k k,( ) k k,( ) k k,( ) k k,( ) k h h 0 g G g G g G g G k k k,( ) k k,( ) k k,( ) k k,( ) k 0 h3 h g G3 g3 G g G3 g3 G k k k,( ) k k,( ) k k,( ) k k,( ) k h3 0 h g G3 g3 G g G 3 g3 G (3.36) (3.37) where () k,() k,() k,() k,() k,() u k U, U, U3,, a node k. T The second par of L L ( ) B,.e B, s derved from e B u, a node k: L k k j k k ATENA Theory 55

166 e u u u u ( ) ( ) j m, m, j m, j m, u U h g G g G u U h g G g G ( ) k k k,( k ) k k,( k ) k m, m j m j m j ( ) k k k,( k ) k k,( k ) k m, j m m m ( ) ( ),( k ) ( ),( k ) Um um, j h um, j gm G um, j gm G k ( ) k k,( ),( ) Um u ( ) k k k ( ) k k m, hj um, gm Gj um, gm Gj k k k k k k Inroducng we can wre u ( ) ( ) m ( ) mj um, j x j l g k,( ),( k ) ( ) j m mj m g k,( ) k,( ) ( ) j m mj m e U l h G G U l h G G k ( ) k k k,( ) k k k,( ) k j m m j j j k ( ) k k k,( ) k k k,( ) k m mj j j l l (3.38) (3.39) l h l h l h G G ( ) k ( ) k ( ) k k,( ) k k,( ) k 3 ( ) k ( ) k ( ) k k,( ) k k,( ) k l h l h l3 h G G ( ) k ( ) k ( ) k k,( ) k k,( ) k L l 3 h3 l3 h3 l33 h3 3 G3 3 G3 B k ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k k,( ) k k,( ) k k,( ) k k,( ) k l h l h l h l h l3 h l3 h G G G G ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k k,( ) k k,( ) k k,( ) k k,( ) k l h3 l3 h l h3 l3 h l3 h3 l33 h G3 3 G G3 3 G ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k k,( ) k k,( ) k k,( ) k k,( ) k l h3 l3 h l h3 l3 h l3 h3 l33 h G3 3 G G3 3 G (3.40) The energy of nonlnear srans: Le S ( ) s a marx sorng sresses s j a me, eraon (-): 56

167 S 0 S 0 0 S SYM S 0 0 S S 0 S 0 0 S 0 0 S 0 0 S S3 0 0 S3 0 0 S33 0 S3 0 0 S3 0 0 S S 0 0 S 0 0 S ( ) (3.4) NL NL NL NL Then marx B B... Bk... B n s composed so ha (a a node k) T () NL T ( ) NL () j sj j uk Bk S Bk uk where saes for varaon of he followng eny. I can be shown ha he marx se n he followng shape: h 0 0 g G g G k,( k ) k,( k ) k k,( k ) k,( k ) k 0 h 0 g G g G k,( k ) k,( k ) k 0 0 h g3 G g3 G k,( k ) k,( k ) k h 0 0 g G g G NL k,( k ) k Bk k,( ) k 0 h 0 g G g G k,( k ) k,( k ) k 0 0 h g3 G g3 G k,( k ) k,( k ) k h3 0 0 g G3 g G3 k,( k ) k,( k ) k 0 h3 0 g G3 g G3 k,( k ) k,( k ) k 0 0 h3 g3 G3 g3 G3 (3.4) NL B can be (3.43) Havng he marces (3.37), (3.40), (3.4) and (3.43) hese are used o compue he elemen's sffness marx, mass marx, elemen loads ec. n exacly he same way as s done for oher Aena's elemen. 3.4 Curvlnear Nonlnear D Isoparamerc Layered Shell Trangular Elemens Ths secon descrbes rangular shell fne elemens. Ther properes and her dervaon are much he same as ha for quadrlaeral shell fne elemens CCIsoShellQuad<xxxx>... CCIsoShellQuad<xxxxxxxxx> descrbed n he prevous chaper. The only dference n ha hey feaure rangular shape. Ther geomery s depced n he fgure below Dependng on number of elemen nodes hese fne elemens call CCIsoShellTrangle<xxx>... CCIsoShellTrangle<xxxxxx>. ATENA Theory 57

168 Fg CCIsoShellD rangular elemens 3.5 Curvlnear Nonlnear 3D Isoparamerc Layered Shell Hexahedral Elemens A famly of 3D soparamerc shell elemens s presened, see he fgure below. Ther properes le beween degeneraed Ahmad shell elemens from Secon 3. and full 3D brck elemens from Secon 3.5. Shape and knemac behavour resembles ha of he shell's elemen. All pons hrough he shell's hckness reman locaed on a lne passng hru he correspondng op and boom nodes of he shell, however unlke n he classcal shell heory, her dsance can change. As for degrees of freedom, (DOFS), a ypcal 3D soparamerc shell elemen has 9 nodes a he op and nne nodes a he boom surface, each of hem havng 3 DOFS, (.e. 3 dsplacemens). A smlar D shell elemen would feaure 9 nodes locaed a he shell's mdplane, each of hem havng 5 DOFS, (3 dsplacemens plus roaons). The new elemens use full 3D sac equaons..e. he elemens consder all 6 componens of 3D sress and sran vecor. Geomercal and maeral nonlneary s suppored. The governng equaons are calculaed and negraed n maeral pons. Gauss negraon s used n shell's 58

169 plane drecon, whls layered concep s employed hroughou he hckness of he shells, (.e. recangur quadraure). As each layer can use dfferen maeral model, some layers can be employed for modellng of embedded renforcemen. The elemens ypcally use 3 x 3 x number_of_layers negraon (.e. maeral) pons. The elemens are suable for boh shallow and deep shells and are exremely smple for use, because hey can be npu and oupu as usual 3D sold hexahedral elemens wh 8, 0 or 7 nodes. Hence, hese shells can be hadled wh mos 3D pre- and pos-processors. They also use sandard 3D maeral models, elemen loads and oher boundary condons desgned for hexahedral elemens. The presened shell elemens are parcularly useful for srucures ha combne sold 3D elemens and shell elemens, because hey do no mply any addonal shell knemac consran ha would harm an anjancen 3D sold elemens. (Typcal shell elemens assume 0 ha enforces he same dsplacemens of he correspondng op and boom nodes n drecon of her connecng lne). They are desgned for ben shells and o analyze hese srucures (wh he same accuracy) hey requre far less fne elemens compared o a smlar analyss usng sandard hexahedral elemens. On he oher hand, he 3D behavour of hese elemens nvolves a small overhead, so ha sandard D shell elemens (wh only 5 sress/sran componens per maeral pon) can perform n some cases slghly beer. Neverheless, he overhead s well pad off by easy of use of he presened elemens, her nce 3D vsualzaon, smple connecon o adjacen 3D sold pars of he srucure ec. In addon, he hearchcal soparamerc space nerpolaon (used for he presened 3D shell elemens) ensures ha fner and coarser meshes are easy o connec. Of coarse, hs feaure mus be suppored by pre- and posprocessor beng used. ATENA Theory 59

170 Fg Isoparamerc 3D shell elemen - coordnae sysems Geomery and dsplacemens are approxmaed by hearchcal soparamerc spaal nerpolaon, (smlar o oher D and 3D elemens defned n prevous secons). The elemens have a mnmum 4 pons a s op and 4 pons a s boom surface. I corresponds o lnear approxmaon and he elemen's name CCIsoShellBrck<xxxxxxxx>. The mos accurae verson of he elemens uses nodes o 6 and,, see he fgure above. Is name s CCIsoShellBrck<xxxxxxxxxxxxxxxxxx>. Such elemen can have curvlnear shape and feaures quadrac dsplacemen approxmaon. Herarchcal concep he shell elemen s employed. Hence, he 3D shell elemen can have from 8 o 8 nodes. The nodes -8 are compulsory. Nodes 9-6 and, are oponal. Nodes 7 o are auomacally removed from he elemen's ncdences. They are consdered only for he sake of compably wh npu daa preprocessor. The <xxxxx..> srng n he elemen name (followng CCIsoShellBrck) specfes, whch of he elemen's node s (or s no) ncluded. An ncluded node s marke as "x", a node no ncluded s marked as "_", (underscore). The shell's nodes are maped no he srng as follows: <,,3,4,5,6,7,8,9,0,,,3,4,5,6,,>. For example CCIsoShellBrck<xxxxxxxxx_x_x_x_xx> uses nodes -8,9,,3,5,,. Noe ha he boom and op surface mus use he same number and locaon of he oponal nodes. Hence, f node 9 s ncluded, node 3 mus be ncluded, oo Geomery and dsplacemens The shell s geomery a he confguraon me and x h X X k, op k, bo k ( ) k, op( ) k, bo( ) x hk X X ( ) k, op( ) k, bo( ) x hk X X d, (eraon (-) and ()), s defned by: (3.44) where =,,3 s ndex relang o global axes x, x, x 3, (.e. x,y,z), hk hk( r, s) s k-h nerpolaon funcon, (see Table 3-4), k... n G s number of he shell's nodes, n G = number of he elemen's nodes used o approxmae geomery, ypcally 8 or 9. x represens -h coordnae of a node of he elemen (a he specfed me). Dsplacemens a me ( ), =,,3 for global axes x,y,z, a eraon ( ) reads : u x x (3.45) ( ) ( ) Subsung (3.44) no (3.45), =,,3 for global axes x,y,z, we can derve 60

171 u h X X h X X ( ) k, op( ) k, bo ( ) k, op k, bo k k k, op( ) k, op k, bo( ) k, bo hk X X X X k, op( ) k, bo( ) hk U U where (3.46) U X X, U X X k, op( ) k, op( ) k k, bo ( ) k, bo( ) k Dsplacemen ncremens whn -h eraon are calculaed as u x x : () () ( ) u h U U () k, op() k, bo() k (3.47) where and X kbo, k, bo( ) U U X X, U X X. In he above k, op( ) k, op( ) k( ) k, bo( ) k, bo( ) k( ) s op and boom nodal coordnae coordnae of node. Smlarly denoes dsplacemens a he same node. X kop, k, op( ) U, 3.5. Green-Lagrange srans The elemens are derved usng Green-Lagrange srans and nd Pola Krchhoff sresses. Toal Lagrangan formulaon s employed, bu afer each load sep we ransform he analysed model (and s sress and oher ensors) o he coordnae sysem defned by he curren shape of he model. (The sandard Toal Lagrangan formulaon calculaes all wh respec o he orgnal coordnae sysem whou any ransformaon; Updaed Lagrangan formulaon carres all he ransformaon each ransformaon, BATHE(98). ) The shell's oal srans a me () () () () () j, j j, k, k, j, -h eraon, are calculaed: (,j=..3 for axs x,y,z) u u u u e ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) u, j u, j uj, uj, uk, uk, uk, j uk, j ( ) ( ) ( ) j j j (3.48) ATENA Theory 6

172 where u (), j s dervave of dsplacemen () u wh respec o axs begnnng of me sep. () refers o eraon number. Smlarly ncremen a he curren eraon. (), j x j a me,.e. a he u denoes dsplacemen ( ) ( ) ( ) ( ) ( ) Subracng j u, j u j, u k, u k, j from (3.48) we can calculae () () lnear and nonlnear sran ncremens e and : j j e u u u u u u u u () () () ( ) () ( ) () j, j j, k, k, j k, j k, () () () j k, k, j (3.49) Dervaves wh respec o gloabal x x, x, x3 dervaves wh respec o curvlnear soparamerc coordnaes r r, s, r, r, r example dervaves of a funcon f ( x, x, x 3) s: f x x x f f are calculaed n sandard way from 3 r r r r x x f x x x x 3 f f f j f f,.e. Jj s s s s x x r r xj xj f x x x 3 f f x 3 x 3. For 3 J (3.50) f f x r f f f nv f J, e.. Jj x s x rj (3.5) f f x 3 The presened shell elemens employes soparamerc herarchcal nerpolaon. Hence, coordnaes x of a pon are calculaed by: x h X X k, op k, bo k where he nerpolaon funcons hk ( r, s ) are enlsed n Table -3- and her dervaves wh respec o o r,s, (o calculae J) are: (3.5) x r 6

173 x hk k, op k, bo X X r r x hk k, op k, bo X X s s x hk X X k, op k, bo (3.53) The above expressons are employed o oban dervaves of (oal) dsplacemens respec o r,s,. They are needed o calculae srans (3.49). ( ) u wh - ( ) u hk k, op( ) k, bo( ) U U r r ( ) u hk k, op( ) k, bo( ) U U s s u ( ) hk U U k, op( ) k, bo( ) (3.54) Dervaves of dsplacemen ncremens wh respec o r,s,: () u hk k, op( ) k, bo( ) U U r r () u hk k, op( ) k, bo( ) U U s s u () hk U U k, op( ) k, bo( ) (3.55) In order o proceed furher n he dervaon of he 3D soparamerc elemen, we need o x x, x, x. Ths s calculae dervaves of he dsplacemen ncremens wh respec o 3 acheved usng (3.5) and (3.55): u u u u u () () () () () nv nv nv nv J jl J j J j J j3 x j rl r s (3.56) u nv hk k, op( ) k, bo ( ) J j U U r () xj nv hk k, op( ) k, bo( ) J j U U s nv hk k, op( ) k, bo( ) J j3 U U (3.57) ATENA Theory 63

174 Afer some rearrangemen Eqn. (3.57) yelds: u h h h () k, op( ) nv k nv k nv k U J j J j J j3 x j r s h h h U J J J r s k, bo( ) nv k nv k nv k j j j3 h U h U op k, op( ) bo k, bo( ) k, j k, j (3.58) where op nv hk nv hk nv hk bo nv hk nv hk nv hk hk, j J j J j J j3, hk, j J j J j J j3 r s r s A hs place, we can derve fnal expresson o compue lnear and nonlnear srans ncremens. () Lnear srans e are calculaed as follows, see (3.49): j e () () () u e... () () e 33 L0 L L0 L L0 L () e B () B... Bk Bk... B n Bn uk e... () e 3 () u () n e3 (3.59) op bo hk, 0 0 h, 0 0 k op op 0 hk, 0 0 hk, 0 op op L hk,3 0 0 h k,3 B k op op op op (3.60) hk, hk, 0 hk, hk, 0 op op op op 0 hk,3 hk, 0 hk,3 h k, op op op op hk,3 0 hk, hk,3 0 hk, where () k, op() k, op() k, op() k, bo() k, bo() k, bo() u k U, U, U3, U, U, U 3 a node k. T Inroducng we can wre u ( ) ( ) ( ) j u, j x j l (3.6) u u u u ( ) ( ) ( ) ( ) m, m, j m, j m,,,,,, ( ) ( ) ( ) k, bo( ) ( ) bo ( ) bo m m k, j mj k, Um lm hk, j lmj hk, l h U h U l h U h U ( ) op k, op() bo k, bo() ( ) op k, op() bo k, bo() m k j m k j m mj k m k m U l h l h k op op op (3.6) 64

175 L Fnally, marx B yelds B L k l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l h l ( ) op ( ) op ( ) op ( ) bo ( ) bo ( ) bo k, k, 3 k, k, k, 3 k, ( ) op ( ) op ( ) op ( ) bo ( ) bo ( ) bo l hk, l hk, l3 hk, l hk, l hk, l3 hk, ( ) op ( ) op ( ) op ( ) bo ( ) bo ( ) bo l3 hk,3 l3 hk,3 l33 hk,3 l3 hk,3 l3 hk,3 l33 hk,3 ( ) op ( ) op ( ) op ( ) op ( ) op ( ) op ( ) bo ( ) bo ( ) bo ( ) bo ( ) bo ( ) bo k, k, k, k, 3 k, 3 k, k, k, l hk, l hk, l3 hk, l3 hk, ( ) op ( ) op ( ) op ( ) op ( ) op ( ) op ( ) bo ( ) bo ( ) bo ( ) bo ( ) bo ( ) bo k,3 3 k, k,3 3 k, 3 k,3 33 k, k,3 3 k, k,3 l3 hk, l3 hk,3 l33 hk, ( ) op ( ) op ( ) op ( ) op ( ) op ( ) op ( ) bo ( ) bo ( ) bo ( ) bo ( ) bo ( ) bo k,3 3 k, k,3 3 k, 3 k,3 33 k, k,3 3 k, k,3 3 hk, l3 hk,3 l33 hk, (3.63) Assemblng sresses a me, eraon (-) no marx () srans s, see (3.49) j S ( ), parcpaon of nonlnear S S S u, u, ( ) ( ) ( ) ( ) ( ) Sj uk, uk, j uk, uk, j ( ) ( ) ( ) ( ) ( ) ( ) ( ) j j j j j k k j S u u ( ) ( ) ( ) j k, k, j () T () u u T () NL ( ) NL () u k B S B u k () () u n un (3.64) B B... B... B NL NL NL NL k n B h 0 0 h 0 0 op bo k, k, op bo 0 hk, 0 0 hk, 0 op bo 0 0 hk, 0 0 hk, op bo hk, 0 0 hk, 0 0 NL op bo k 0 hk, 0 0 hk, 0 op bo 0 0 hk, 0 0 hk, op bo hk,3 0 0 hk,3 0 0 op bo 0 hk,3 0 0 hk,3 0 op bo 0 0 hk,3 0 0 hk,3 (3.65) ATENA Theory 65

176 s ( ) ( ) 0 s symmerc ( ) 0 0 s ( ) ( ) s 0 0 s ( ) ( ) ( ) S 0 s 0 0 s ( ) ( ) 0 0 s 0 0 s ( ) ( ) ( ) s3 0 0 s3 0 0 s 33 ( ) ( ) ( ) 0 s3 0 0 s3 0 0 s33 ( ) ( ) ( ) 0 0 s3 0 0 s3 0 0 s33 (3.66) Usng (3.50) and (3.5) follows o presen fnal expresson for compuaon of space dervaves of f ( x, x, x 3) : k f nv h k k ; (,, ) ˆk k J j Fk h h r s h ( r, s) h ( ) x r j ˆk k f nv ˆk k nv h k (,) () ˆk h J j Fk h r s h J j Fk h h x r j rj r j ˆk k nv h k ˆk h ˆk k ˆk k nv h k J Fk h h ˆk h nv h k ˆk h J r r Fk 3 k h h J F h h s s ˆk ˆk k nv h k nv h k nv ˆk h J Fk h J k F h J3 Fkh r s (3.67) Havng all he marces and relaonshps above, he res of dervaon of he presened L0 L NL soparamerc shell elemens s sraghforward. Smply use he marces B, B, B and ( ) S ( ) o calculae srucural sffness marces, K K, vecors of nodal forces L NL ( F ), and loads R as descrbed n he Secon Problem Dscresaon Usng Fne Elemen Mehod earler n hs documen. 66

177 3.6 Curvlnear Nonlnear 3D Isoparamerc Layered Shell Wedge Elemens Ths secon descrbes wedge shell fne elemens. Ther properes and her dervaon are much he same as ha for hexahedral shell fne elemens CCIsoShellBrck<xxxxxxxx>... CCIsoShellBrck<xxxxxxxxxxxxxxxxxx> descrbed n he prevous chaper. The only dference n ha hey feaure wedge shape. Ther geomery s depced n he fgure below Dependng on number of elemen nodes hese fne elemens call CCIsoShellWedge<xxxxxx>... CCIsoShellWedge<xxxxxxxxxxxx>. Fg CCIsoShell3D wedge elemens ATENA Theory 67

178 3.7 Curvlnear Nonlnear 3D Beam Elemen A curvlnear 3D beam fne elemen CCBeamNL s descrbed here. The elemen s based on a smlar beam elemen from BATHE(98). I s fully nonlnear, n erms of s geomery and maeral response. I uses quadrac approxmaon of s shape, so he can be curvlnear, wsed, wh varable dmensons of he cross-secons. Moreover, beam s cross-secons can be of any shape, oponally even wh holes. The elemen belongs o he group of soparamerc elemens wh Gauss negraon along s axs and rapezodal (Newon-Coes) quadraure whn he cross-secon. The negraon (or maeral) pons are placed n a way smlar o he layered concep appled o shell elemens, however, he layers are locaed n boh s, drecons Geomery and Dsplacemens and Roaons Felds Geomery of he elemen s depced n Fg The depced brck nodes specfcaon s employed o ensure compably of he elemen wh ATENA preprocessor. The beam 3D nodes defnon s used by ATENA posprocessor. The elemen response s compued whn he D beam geomery. Thus, on npu he elemen has 0 nodes, whle durng he calculaon has only 5 nodes,.e. nodes for 3D beam shape defnon and 3 nodes for he D beam geomery. Any of he 5 nodes can be subjec o a knemac or sac consran. The D beam nodes have 6 degrees of freedom (dofs) hree dsplacemens and hree roaons wh respec o global coordnae axes. The 3D beam nodes allocae only he hree dsplacemen dofs per node. The redundan brck nodes are gnored and hey allocae no dofs. The elemen uses hree confguraon. The reference confguraon corresponds o shape of he beam a he begnnng of he sep,.e. pror any load n he curren sep s appled was employed. I s used as a reference coordnae sysem for all calculaon whn a loadng sep, wh respec o whch all dervaves are compued. Ths confguraon s denoed by a superscrp lef o a referred symbol, e.g. x. The elemen shape afer all prevous eraons whn he curren sep and pror he curren eraon s denoed by d superscrp, d x. Incremens whn he curren eraon do no use any superscrp, e.g. x. 68

179 b a Geomery V b 3 a 3 a b V s s Brck nodes r Beam 3D nodes Beam D nodes w 3 z z s y r x x v y u Isoparamerc shape Global coord. sysem and elemen dofs Fg CCBeamNL elemen ATENA Theory 69

180 The beam s geomery a he confguraon and d s defned by: s x s x xh X av bv s y sy y h Y av bv s z s z z h Z av bv (3.68) In he above refers o axal nodes,.e...3 for he nodes 3,4,5, see he D beam nodes. h h() r s -h nodal nerpolaon funcon descrbed n Secon 3.. X, Y, Z are x y z sx sy sz global coordnaes of a node a me. The vecors V, V, V, V, V, V are he vecors V, V depced n Fg. 3-40, n a cross secon, a me, whch defne local s coordnae axs s,. The symbols a, b refers o dmensons of he cross secon, me ; see he fgure, oo. Geomery of he beam a me d s defned n a smlar way: s x s x xh X a V b V s y sy y h Y a V b V s z s z z h Z a V b V d d d d d d d d d d d d The elemen s dsplacemens a me d s calculaed as follows: d d u x x d d v y y d d w z z and dsplacemen ncremens whn a eraon: s x s x u hu av bv s y sy vhv av bv s z s z whw av bv T T T (3.69) (3.70) (3.7) s d s d s s In he above equaon he vecors V, V are V V V and V V V and hey are approxmaed by 70

181 x d z y d y z V V V y d x z d z x V V V z d y x d x y V V V sx d s s z y d y z V V V sy d sx z d sz x V V V sz d sy x d sx y V V V (3.7) x y z The parameers,, are roaons around he global axs, wh respec o begnnng of he curren load sep. Noe ha (3.7) s vald only approxmaely Sran and Sress Defnon The elemen uses Green-Lagrange sran and Pola-Krchhof sresses, see Secon.4. and Secon.3.. ransformed o he local soparamerc r,s, coordnae sysem. As he beam heory mples, only normal sran componen r and shear componens rs, r are consdered. The sress vecor ncludes he correspondng rr, rs, r enres, whereby he remanng srans have o reman zero. The procedure of calculaon sress-sran response s as follows:. Calculae all 6 componens of Green-Lagrange srans (.8) and her ncremens whn global coordnae sysems. The ncremens are compued wh respec o he begnnng of he curren load sep.. Transform he srans ncremens no local r,s, coordnae sysem. 3. Zerose componens,,. ss s 4. Execue maeral law o compue correspondng sresses. 5. Transform he sresses o he global coordnae sysem. The followng expressons are used o calculae dsplacemen dervaves needed for calculaon of he srans: df dx dy dz df df dx dr dr dr dr dr df dx dy dz df df J dy ds ds ds ds ds df dx dy dz df df dz d d d d d where f s a dsplacemen funcon o be dervaed. (3.73) Marces Used n he Beam Elemen Formulaon Subsung equaons (3.68) o (3.73) no he expressons for calculang elemen marces (.3) o (.34) all mporan marces and vecors of he beam elemen can be calculaed. Ther explc presenaon s beyond he scope of hs documen. Neverheless, he mos mporan ones are now gven: ATENA Theory 7

182 The Jacoban marx: x h s x s x J X av bv r r y h s y sy J Y av bv r r z h s z s z J3 Z av bv r r x s x J h bv s y sy J h bv s J 3 y s z h bv x x J3 h av y y J3 h av z z J33 h av (3.74) 7

183 The marx d B NL. : I s consruced n he way ha U u V x W u x y y u z z v U x V v W B y v z w U 3 x V 3 w W3 y x 3 w y 3 z z 3 d d NL x y z B NL U (3.75) The dealed expressons for calculang d BNL are gven n (3.78) and (3.79). The equaons are mporan because hey presen he way, how spaal dervaves of all he dsplacemens are calculaed. The enres n d d d BNL are hus used o seup also he marx BL0 and BL. These marces are compued as follows: B B d d L0(, ) NL(, ) B B d d L0(, ) NL(5, ) B B d d L0(3, ) NL(9, ) B B B d d d L0(4, ) NL(4, ) NL(, ) B B B d d d L0(5, ) NL(6, ) NL(8, ) B B B 3, ) d d d L0(6, ) NL(7, ) NL( (3.76) ATENA Theory 73

184 u v w x x x u v w y y y d d d d d d d BL (, ) BNL(, ) + BNL(4, ) + BNL(7, ) d d d d d d d BL (, ) BNL(, ) + BNL(5, ) + BNL(8, ) d d u v d w d + BNL(6, ) + BNL(9 ) z z z d d d u u v B (4, ) B (, ) B (, ) + B (5, ) + x y x d d d BL(3, ) BNL(3, ) d d d d L NL NL NL v w w y x y d d d d d d BNL(4,) + BNL(8,) + BNL(7,) u u v + + y z y d d d d d d d BL(5, ) BNL(3,) BNL(,) BNL(6,) v w w z y y d d d d d d BNL(5, ) + BNL(9, ) + z BNL(8, ) u u v z x z d d d v d w d w d B + B + B x z x d d d d d d d BL(6, ) BNL(, ) BNL(3, ) + BNL(4, ) NL(6,) NL(7,) NL(9,) + (3.77) 74

185 d d d d NL(,), NL(,) NL(,3) NL(,4) h s h B J a V b V J h b V J a V r r h d s d d h d BNL(,6) J, a V b V J, h b V J,3 a V r r h BNL(,) J, r B 0 d d z d sz d sz d z NL(,5),,,3 d d d d d B B B B B B NL(,) NL(,3) NL(,4) J 0 0 h r h s s z d sz h d z B J a V b V J, h b V J,3 a V r r h s h B (,6) J, a V b V J, h b V J,3 a V r r h BNL(3,) J3, r B 0 d d z d NL(,5), NL(3,) y sy sy y d d y d sy d sy d y NL d d d d B B NL(3,3) NL(3,4) 0 0 h s h B J a V b V J h b V J a V r r h s d s y h d B J a V b V J3, h b V J3,3 a V r r d d z d sz d sz d z NL(3,5) 3, 3, 3,3 d d y d sy NL(3,6) 3, d d d B B B NL(4,) NL(4,), NL(4,3) 0 J 0 h r h s h B J a V b V J h b V J a r r d d z d sx d sz d NL(4,4),,,3 d NL(4,5) d d x d sx d sx d x NL(4,6),,,3 d d d d d B NL(5,) NL(5,), NL(5,3) 0 h s h B J a V b V J h b V J a V r r B B B B B 0 J 0 J NL(5,4), NL(5,5) 0 h r h d s d s x d h d a V b V J, h b V J,3 a V r r z sz z h s h B J a V b V J h b V J a r r d d x d sx d sx d x NL(5,6),,,3 V V y z (3.78) ATENA Theory 75

186 d d d NL(6,) NL(6,) 3, NL(6,3) d d z d sx d sz d z NL(6,4) 3, 3, 3,3 d d d d d B B B NL(6,5) 0 J 0 h s h B J a V b V J h b V J a V r r B B B B 0 J NL(6,6) 3, NL(7,) NL(7,) 0 0 h r h d s d d h d a V b V J3, h b V J3,3 a V r r x sx sx x h BNL(7,3) J, r h s h r r h s h B (7,5) J, a V b V J, h b V J,3 a V r r d d y d sy d sy d y BNL(7,4) J, a V b V J, h b V J,3 a V d d x d sx d sx d x NL d d d d B B B NL(7,6) NL(8,) NL(8,) h BNL(8,3) J, r h s h r r h s s x h d B J a V b V J h b V J,3 a V r r d d y d sy d sy d y BNL(8,4) J, a V b V J, h b V J,3 a V d d x d sx d NL(8,5),, d d d d B B B NL(8,6) NL(9,) NL(9,) h BNL(9,3) J3, r h s h y B J a V b V J h b V J a V r r h s h B (9,5) J3, a V b V J3, h b V J3,3 a V r r d d y d sy d sy d NL(9,4) 3, 3, 3,3 d d x d sx d sx d x NL d B NL(9,6) 0 x (3.79) 76

187 The sress marx S j from (.34) has he form: xx xy xz yy yz zz xx xy xz Sj yy yz zz sym. xx xy xz yy yz zz (3.80) As already menoned, sress-sran relaons are calculaed n r,s, coordnae sysem, hence we need equaons for her ransformaons from global x,y,z coordnae sysem o he soparamerc sysem wh r,s, coordnaes and vce versa. d d Le us denoe T, T ransformaon marces for sran and sress ransformaon from global o soparamerc coordnae sysem, so ha: xx yy rr d zz rs T xy r yz xz xx yy rr d zz rs T xy r yz xz Then he ransformaon marces are calculaed by: (3.8) d rx d ry d r r z d rx d y d ry d rz d rx d rz V V V V V V V V V s x V d rx d r x d y d y d r z d sz d rx d y d ry d r x d y d z d rz d y d rx d z d rz d x V V V V V V V V V V V V V V V V V V d d rx d sx d ry d sy d r s z d sz d rx d y d ry d s r x d y d s s z d rz d y d rx d sz d rz d T V V V V V V V V V V V V V V V V V (3.8) ATENA Theory 77

188 d rx d ry d rz d rx d ry d ry d rz d rx d rz V V V V V V V V V r r x sx y sy r s z sz rx y ry s r x y s s z rz y rx sz rz d T V V V V V V V V V V V V V V V V V V d d d ry d y d r z d sz d d y d ry d d ry d z d rz d y d d z d rz d V V V V V V V V V V V V V V V V V V d d d d d d d d d d d d d d d d d d rx rx rx x x x sx (3.83) d s d s s x d y d sz d d y where vecors V V V V x d d z, V V V V are vecors of uny lengh from Fg The remanng vecor s calculaed as a vecor produc of he prevous wo vecors: V V V V V V Inverse ransformaon marces are calculaed as: The Elemen Inegraon d r d r r x d y d rz d s d T T d d T T T d d T (3.84) (3.85) The elemen s negraed numercally. Along s longudnal axs he elemen s negraed by sandard wo o sx nodes Gaussan negraon. The able below lss r coordnaes and assocaed weghs for ulzed negraon pons: Table 3-6: Gaussan negraon of he beam elemen along he longudnal axs Number of neg. pons Inegra on pon Coordnae r Wegh

189 Alhough he -nodes negraon may be suffcen, (wh respec o he quadrac dsplacemen ransformaon), a hgher order negraon scheme wll yeld beer resul n a case of hgh maeral nonlneary and/or n a case of a very curved beam geomery. As for negraon whn he cross-secon,.e. n s, coordnaes, rapezodal quadraure s used. The elemen cross-secon s subdvded no n, n srps as depced n he followng fgure. s d n s d d ds ds ds ns nd vdual wegh and maeral Fg. 3-4 The beam cross secon negraon The negraon s hen carred ou by summng funconal values n cenre of he all quadrlaerals mulpled by her area. Noe ha he elemen s negraed whn he soparamerc coordnae sysem, hence we have o use dx dy dz de( J ) dr ds d, see (3.73). Nce feaure of he ATENA s mplemenaon of he beam s ha each of he quadrlaerals n a cross secon adops an arfcal npu wegh facor. By defaul, such a wegh s equal o one, however, f we se s value o zero, essenally a hole s nroduced. Ths mechansm, ogeher wh possbly of defnng an cusomzed maeral law n each of he quadrlaerals faclaes o analyse beams ha have a arbrary shape of cross-secons. The presen beam mplemenaon suppors also smeared renforcemen. Ths s done n he same way as was for he Ahmad elemens descrbed n he prevous secon. ATENA Theory 79

190 3.8 Curvlnear Nonlnear 3D Isoparamerc Beam Elemen CCIsoBeamBrck_3D and CCIsoBeamBrck8_3D are beam curved soparamerc elemens smlar o he prevous CCBeamNL_3D elemen. They use smlar geomery, node numberng ec., bu dffer from CCBeamNL_3D n ha hey accoun for all 6 componens of 3D srans and sress vecors. They comply wh all 3D sac equaons and no addonal sac or knemac consrans are mposed. The comparson of CCBeamNL_3D vs. CCIsoBeamBrck_3D resembles ha of CCAhmad vs. CCIsoShell elemens descrbed above. The CCIsoBeamBrck_3D and CCIsoBeamBrck8_3D are easy o use, hey preserve her 3D volume and hey are ncely vsualzed durng pre and pos processng. They can be npu, loaded and oupu n he same way as CCIsoBrck hexahedral elemens. CCIsoBeam8_3D feaures lnear geomery and dsplacemen approxmaon, (.e. has nodes...8, see he fgure below), whls CCIsoBeam_3D has quadrac approxmaon, (.e. has nodes...). Geomery V V s s Brck nodes r Beam 3D nodes 3 4 w z z y r x x v u y Isoparamerc shape Global coord. sysem and elemen dofs Fg. 3-4 CCIsoBeamBrck_3D and CCIsoBeamBrck8_3D elemens s Shape of cross secon can be any quadrlaeral,.e. need no be only a recangle as depced above. The elemens are parcularly useful for analyses of srucures, where beam elemens mus be combned wh 3D sold and/or shell elemens. 80

191 Dervaon of he elemen s much he same as ha for CCIsoShell elemen,.e. Equaons (3.45) and (3.47) hru (3.67) reman vald. Geomery and dsplacemen approxmaon (3.46) s replaced by: s s x h X X X X s s ds s k, fron k, back k, op k, bo k ( ) k, fron( ) k, back ( ) k, op( ) k, bo ( ) x hk X X X X ( ) k, fron () k, back () k, op() k, bo() x hk X X X X hk hk() r are D nerpolaon funcons, see he nerpolaon funcon for CCIsoTruss elemens. The same noaon s used for CCIsoShell Elemens. The elemen s calculaed n negraon pons, (.e. maeral pons) ha are locaed smlar o CCBeamNL_3D elemens, refer o Fg The elemen can use any 3D maeral model. Dfferen maerals can be specfed for each maeral pons, (or pons n cross secon). Some of hem can be used for modellng of embedded renforcemen. (Bw. dscree renforcemen can be employed, oo). The elemens suppor boh maeral and geomerc nonlneary. (3.86) 3.9 Curvlnear Nonlnear D elemen The elemens CCIsoBeamBar<xx> and CCIsoBeamBar<xxx> are from he pon of vew of mechancs nearly dencal o he elemen descrbed n Secon 3.3, he dfference beng only n ha ha hese elemens are specfed by her axs as D beams. The frs elemen has nodes (and uses lnear nerpolaon of s geomery and dsplacemens). The laer elemen has 3 nodes (and uses quadrac nerpolaon of s geomery and dsplacemens, whch s dencal o CCBeamNL elemen referred above). The elemens can be curved and can have varable hegh, wdh and orenaon of her cross secon. All hese parameers are npu n CCBeamD geomery n form of algebrac expressons. The expresson are funcons of beam's coordnaes x,y,z. Smlar o CCBeamNL elemen, hese elemens are also negraed by Gauss negraon along he beam's axs whle grd quadraure s used for he remanng drecons (whn cross secons). The elemens suppors embedded renforcemens, holes dfferen maerals n dfferen negraon pons ec. n he same way as s he case of CCBeamNL elemen. They are suable for modelng of boh shallow and deep beams. Noe ha CCIsoBeamBar<xx> has far worse properes compared o CCIsoBeamBar<xxx>. Hence, he lnear elemen should be used only o model some lnks and connecons whn he srucures. ATENA Theory 8

192 Beam D nodes 3 w z z s x u x y y v r Isoparamerc shape Global coord. sysem and elemen dofs Fg CCIsoBeamNLBar<xxx> elemen 3.9. Connecon of he beamd o an amben sold elemen The procedure o connec beamd's dofs o an amben elemen s smlar o ha for shelld elemens, see Agan, consss of wo pars:. fx a FE node wh [ uvw,, ] dsplacemen whn he beamd elemen,. fx hree roaon dofs of he beamd elemen whn amben elemens Fxng a FE node wh [ uvw,, ] dsplacemen whn he beamd elemen Usng (3.7) and (3.7) wre expresson for beamd dsplacemens a he op bo u,.e. s 0, of a cross secon. Do he same for rgh u rgh s, 0. op u and boom and lef u lef pon,.e. Wre 3D sold approxmaon for he same 4 nodes. Then, f we compare he D and 3D approxmaon, afer some mahemacal manpulan we derve 8

193 hh hh ( r, s, ) hh ( r, s, ) hh ( r, s, ) hh ( r, s, ) 4 p58 k k k k k hh hh ( r, s, ) hh ( r, s, ) hh ( r, s, ) hh ( r, s, ) 4m58 k k k k k hh hh ( r, s, ) hh ( r, s, ) hh ( r, s, ) hh (, r s, ) 5m48 k k k k a b u k 4 p58 k 4m58 k 5m48 hhk hhk hhk vk u 4 p58 a w k 4m58 k u 0 hhk 0 hhk 0 0 k xk, u 3 4 p58 5m hhk hhk 0 0 yk, zk, k (3.87) Fxng hree roaon dofs of he beam3d elemen whn amben elemens Smlarly o he expressons for shelld he resulng equaons for beamd roaon x, y, z are x y z op op op bo bo bo rgh rgh rgh lef lef lef T MM UU UU UU3 UU UU UU3 UU UU UU3 UU UU UU 3 (3.88) ATENA Theory 83

194 MM T Vrx Vs VsxVry VrxVs x y VsxVrz VrxVs z / / / a a a a a VrxVsy VsxVry VryVsy VsyVrz VryVsz / / / a a a a a VrxVsz Vs Vr xvrz yvsz VsyVrz VrzVsz / / / a a a a a Vr Vs xvsx xvry VrxVsy VsxVrz VrxVsz / / / a a a a a VrxVsy VsxVry VryVsy VsyVrz VryVsz / / / a a a a a VrxVsz VsxVr Vr z yvsz VsyVrz VrzVsz / / / a a a a a Vr V xvx xvry VrxVy VxVrz VrxVz / / / b b b b b VrxVy VxVry VryVy VyVrz VryVz / / / b b b b b VrxVz VxVr Vr z yvz VyVrz VrzVz / / / b b b b b Vr V xvx xvry VrxV y VxVrz VrxVz / / / b b b b b VrxV y VxVry VryVy VyVrz VryV z / / / b b b b b Vr Vr xvz VxVrz yvz VyVr z VrzVz / / / b b b b b (3.89) rk sk where Vr, Vs, V, a a, b b k V k k k Noe ha dsplacemen dofs are fxed by(3.87). If eher boom or op node ges ousde he amben elemen, he mddle pon s used nsead. Equaons (3.88) and (3.89) are sll vald bu s necessary o use T MM (, j) MM (, j), j..6, MM (, j) MM (, j), j 7.. o calculae x y z. Smlarly, f eher rgh or boom node ges ousde he amben elemen, he mddle pon s used nsead. Then, s necessary o use T MM (, j) MM (, j), j..6, MM (, j) MM (, j), j 7.. o calculae x y z. 3.0 Inegraed forces and momens for shells Inegraed forces for shells are compued as follows: 84

195 N N N Q Q Q M M K / x' / x' x' / y' / y' y' z' x' z' / / / / / z' z' x' z' x' y' / x' y' / y' z' / y' z' / y' / x' x' / x' / y' y' / x' y' / x' y' dz' dz' dz' dz' dz' dz' zdz' ( zdz ) ' ( z) dz' (3.90) The above forces and momens ac on planes ndcaed below: yz ' ': Nx' Qx' y' Qx' z' Kx' y' M y' Kx' z' 0 x' z': Qy' x' Qx' y' Ny' Qy' z' Mx' Ky' x' Kx' y' Ky' z' 0 xy ' ': Qz' x' Qx' z' Qz' y' Qy' z' Nz' Kz' x' 0 Kz' y' 0 Mz' 0 The acual values of he forces and momens are calculaed by exrapolaon of sresses from IPs no fne elemen nodes, (please refer o Secon "Exrapolaon of Sress and Sran o Elemen Nodes" n Chaper CONTINUUM GOVERNING EQUATIONS. The process s as follows: Le us ake an example of N ha s calculaed by negraon of x ' x ' hru elemen's hckness. x ' The sress x ' x ' a elemen nodes s exrapolaed from sresses n IPs ' ' by nv x' x' M Px' x' P h ˆ dv xx, V x' x' e e M hh dv j V j e e ATENA Theory 85 ˆ x x (3.9) where V e sands for elemen volume. Usng (3.90) and wrng (3.9) for exrapolaon whn shell md-plane e, (.e. negraon rhu e nsead of V e ) we can wre nv x x / ˆ N MM PP x' x' ' ' PP h dz d h ˆ dv xx, x' x' e x' x' e e / Ve MM h h d h h dv e Ve j j e j e (3.9)

196 where ( r, s) s elemen hckness a r,s. The negraon for exrapolaon s carred ou over e, because he forces and momens are he same rhu shell hckness. Noe ha h k h k(,) r s s nerpolaon funcon n he shell md-plane and s ndependen of coordnae, (unlke h h(,,) r s n (3.9)). Therefore we can wre, (see he las equaon n (3.9): / h hj dve h (,) (,) Ve e / r s h j r s d d e Ve e h(,) r s h (,) r s / j / e h(,) r s h (,) r s (,) r s d h h d e j e j e e hh dv h h d MM e j e j e j d d (3.93) 3. Inegraed forces and momens for beams Inegraed forces for beams are compued as follows: N Q Q / x' / x' x' / x' y' / x' y' / x' z' / x' z' / y' z' x' y' x' z' / / y' / x' x' / z' / x' x' dy' dz' dy' dz' dy' dz' K ( ( z') y') dy' dz' M M ( z ') dy ' dz ' ( y ') dy ' dz ' (3.94) The forces and momens ac on he plane (x'y'). They are calculaed smlar way o (3.9), however, MM h h dr j l h h dv, where bhs area of he beam's cross secon and j j e e Ve bh l e s elemen lengh. 3. Global and Local Coordnae Sysems for Elemen Load Mos elemen loads can be defned n global or local coordnae sysem. Global coordnae sysem s always avalable, hence usng s usually he safes way o npu a desred elemen load. Neverheless, some elemens are nernally defned n a local coordnae sysem and can be employed for an elemen load defnon, oo. Locaon of such a local sysem, (f exss) has been descrbed ogeher wh descrpon of he assocaed fne elemen. For example, local coordnae sysems are defned for plane 3D soparamerc elemens, shell and beam elemens ec. On he oher hand, elemens such as erahedrons, brcks and ohers are defned n drecly n 86

197 global coordnae sysem and herefore a local elemen load s reaed as f were npu as a global elemen load. An excepon o he above are russ elemens. Alhough hey are defned n global coordnae sysem, hey do suppor local elemen load. Ther local coordnae sysem (for elemen loadng only) s defned as follows: local X axs pons n drecon of he russ elemen, local Y axs s normal o local X axs and les n he global XY plane, s posve orenaon s chosen so ha he local X and local Y forms a rgh-hand (D) coordnae sysem n he plane defned by hese local axes, local Z axs s vecor produc of he local X and local Y axes, (for 3D case only). f he russ s parallel o global z, hen local X pons n drecon of global Z, local Y concdes wh global Y and local Z has oppose drecon of he global X, (for 3D case only). D 3D Y G N Z G Y L Y G N N X L Y L Y G X G X G X L N Fg Local and global coordnae sysems for russ elemen N-N, (e.g. loaded elemen edge) Specfcaon of a boundary load deserves slghly more aenon. Frsly, s appled only o an elemen s edge or an elemen s surface, (see also he noe below), as opposed o e.g. an elemen body load ha s for he whole elemen. Local coordnae sysem s hus defned by locaon of he loaded edge or surface. Secondly, a boundary load defnon mus nclude a reference o a selecon, whch conans nodes o be loaded. Ther order n he ls s rrelevan, as wha really maers s he order n whch hey appear n he elemen ncdences. When processng a boundary load, ATENA loops hru all elemen s surfaces and edges, (n he order specfed n he able below) and checks approprae ncdenal nodes. If he esed node s presen n he ls of loaded boundary nodes, s pcked up and pu no ncdences of a new planar or lne elemen. Ths elemen s laer used o process he boundary load. I s s local coordnae sysem, ha s (possbly) used o deal wh local/global load ransformaons. The able below defnes he orders, n whch elemen surfaces and edges are esed for a surface or edge elemen load. (I s assumed ha elemen ncdences are ( n, n,... n num _ elem _ nodes ) ). I descrbes lnear elemens bu surfaces and edges of nonlnear elemens are reaed n he same order. Table 3-7: Order of elemen surface and nodes as hey are esed whn a boundary load defnon. ATENA Theory 87

198 Elemen shape Type Surface/node ncdences Truss Edge ( n, n ) Trangle Surface ( n, n, n 3) Edge ( n, n); ( n, n3); ( n3, n ) Quad Surface ( n, n, n3, n 4) Edge ( n, n); ( n, n3); ( n3, n); ( n4, n ) Hexahedron, (brck) Terahedron Wedge Surface n n n3 n4 n5 n6 n7 n8 n n4 n8 n5 n n3 n7 n6 (,,, ); (,,, ); (,,, ); (,,, ); ( n, n, n, n ); ( n, n, n, n ); Edge n n n n3 n3 n4 n4 n (, ); (, ); (, ); (, ); ( n5, n6); ( n6, n7); ( n7, n8); ( n8, n5); ( n, n ); ( n, n ); ( n, n ); ( n, n ) Surface ( n, n, n3); ( n, n, n4); ( n, n3, n4); ( n, n3, n 4) Edge n n n n3 n3 n (, ); (, ); (, ); ( n, n ); ( n, n ); ( n, n ) Surface ( n, n, n3); ( n4, n5, n6); ( n, n, n, n ); ( n, n, n, n ); ( n, n, n, n ) Edge n n n n3 n3 n (, ); (, ); (, ); ( n4, n5); ( n5, n6); ( n6, n4); ( n, n ); ( n, n ); ( n, n ); Noe ha only one surface or one edge of each elemen can be loaded n a sngle boundary load specfcaon. If more elemen s surfaces or edges are o be loaded, use more boundary load defnons. Volaon of hs rule causes an error repor and skppng of he offendng boundary load. 88

199 D edge load planar elemen 3D edge load planar elemen n Z G n X L Z L Y G X L Y L Y L n n 3 n n 3 Y G X G X G 3D edge load sold elemen Z G n Y L X L Z L n n3 3D surface load sold elemen Z G Z L n YL X L n n3 n 6 n 6 X G n 5 Y G X G n 5 Y G Fg Examples of posonng local coordnae sysem used by surface and elemen load for D and 3D elemens Transpor analyss does no dsngush beween local and global elemen loads. Hence, a local elemen load s reaed as beng a global load. The acual load value s always scalar, (unlke vecors n sacs) and s assumed posve for flow ou of he elemen. 3.3 References AHMAD, S., B. M. IRONS, ET AL. (970). "Analyss of Thck and Thn Shell Srucures by Curved Fne Elemens." Inernaonal Journal of Numercal Mehods n Engneerng : BATHE, K.J.(98), Fne Elemen Procedures In Engneerng Analyss, Prence-Hall, Inc., Englewood Clffs, New Jersey 0763, ISBN ATENA Theory 89

200 CRISFIELD, M.A. (983) - An Arc-Lengh Mehod Includng Lne Search and Acceleraons, Inernaonal Journal for Numercal Mehods n Engneerng, Vol.9,pp FELIPPA, C. (966) - Refned Fne Elemen Analyss of Lnear and Nonlnear Two- Dmensonal Srucures, Ph.D. Dsseraon, Unversy of Calforna, Engneerng, pp HINTON, E. AND D. R. J. OWEN (984). Fne Elemen Sofware for Plaes and Shells, Perdge Press. JENDELE, L. (98). Thck Plae Fne Elemen based on Mndln's Theory. Prague, suden research work. JENDELE, L. (99). Nonlnear Analyss of D and Shell Renforced Concree Srucures Includng Creep and Shrnkage. Cvl Engneerng Deparmen. Glasgow, Unversy of Glasgow: 393. JENDELE, L., A. H. C. CHAN, ET AL. (99). "On he Rank Defcency of Ahmad's Shell Elemen." Engneerng Compuaons 9(6): RAMM, E. (98) - Sraeges for Tracng Non- lnear Responses Near Lm Pons, Non- lnear Fne Elemen Analyss n Srucural Mechancs, (Eds. W.Wunderlch, E.Sen, K.J.Bahe) 90

201 4 SOLUTION OF NONLINEAR EQUATIONS The man objecve of hs chaper s o revew mehods for soluon of a se of nonlnear equaons. Several mehods, whch are mplemened n ATENA are descrbed laer n hs Chaper. However, all of hem need o solve a se of lnear algebrac equaon n he form A x b (4.) where A, x, b sands for a global srucural marx and vecors of unknown varables and rhs of he problem, respecvely. Hence, hs problem s dscussed frs. 4. Lnear Solvers Two ypes of he solvers are suppored: drec and erave, each of hem havng some pros and cons. Whou gong no deals, a drec solver s recommended for smaller problems or problems. I s more robus and manages beer ll-posed equaons sysems. On he oher hand, erave solvers are ypcally more effcen o solve large (well posed) 3D analyses. In addon, wo sparse drec solvers are provded. They nend o borrow advanages from boh drec and erave solvers. The wo approaches, (.e. drec and erave) dffer n a way hey sore he srucural marx A. I comes from he naure of FEM ha he srucural marces have sparse characer, wh mos of non-zero elemens locaed near he dagonal. The marx has banded paern and ATENA works wh band of varable wdh. If a drec solver s used, hen each column of marx A sores all enres beween he dagonal elemen and he las non-zero elemen n he column. Ths srucure s somemes called sky-lne profle srucure. The marx A a a a3 a5 a a a3 a4 a 5 a3 a3 a33 a34 a 35 A a4 a43 a44 a45 a46 (4.) a5 a5 a53 a54 a55 a 56 a64 a65 a66 a67 a76 a 77 s hus sored n hree vecors dul,, wh acual daa and one vecor p wh nformaon abou marx s profle: d a a a a a a a u a a a a a a a a a a a a a l a a a a a a a a a a a a a p T For each column of he marx A he vecor p sores locaon of a ( ) whn he array u, resp. l. If A s symmerc, hen u l and only l s sored. Noe he a drec solver we have o ATENA Theory 9 T T T (4.3)

202 sore all elemens whn he bandwdh, even hough some of hem may be equal o zero, because ha hey can become non-zero n he process of soluon, (.e. marx facorzaon). Ierave solver can sore only rue non-zero elemens, rrespecve of wheher hey are locaed above or below he skylne. Suppose he marx A from (4.) ha sores some zero elemens below he skylne a 0 a3 a5 0 a a3 a4 0 a3 a3 a33 a34 0 A a4 a43 a44 0 a46 (4.4) a a55 0 a64 a65 a66 a67 a76 a 77 All erave solvers would sore he marx A n hree vecors. All he daa are sored n a vecor a and locaon of he sored elemen s mananed n vecors r, c. The above marx s sored as follows: a a a a a a a a a a a... a a c r (4.5) The vecor a sores for each column of A frs dagonal elemen, followed by all non-zero elemens, from he op o he boom of he column. The vecor c sores row ndex of each enry n he vecor a. r sores locaon of all dagonal elemens a whn a appended by an arfcal poner o an n, where n dm( A ). 4.. Drec Solver The well known Cholesky decomposon s used o solve he problem. The marx A s decomposed no A LDU (4.6) where L, U s lower and upper marx and D s dagonal marx. The mehod o compue he decomposon s descrbed elsewhere, e.g. (Bahe 98). Equaon (4.) s hen solved n wo seps: v L b (4.7) x DU v Boh of he above equaons are compued easly, because he nvolved marces have rangular paern. Hence, he soluon of (4.7) represens back subsuon only. If A s symmerc, (whch s usually he case), hen 9

203 T U L (4.8) 4.. Drec Sparse Solver Drec sparse solvers are smlar o he above Drec solvers, however hey should work more economcally boh n erms of RAM and CPU requremens. They belong o a group of drec (.e. non-erave) soluon mehods. They are based on marx decomposon smlar o (4.6). The decomposon can be LU or LDU for nonsymmerc marces and/or LL T or LDL T decomposon for symmerc marces. The man dfference beween hese solvers and hose from Secon 4..3 s ha hey run so called pre-facorzaon procedure, before he acual facorzaon s execued. Such a pre-facorzaon has wo jobs:. Fnd ou, wha nally zero a j enres of he marx A (ha are sored below he skylne) become non-zero due o facorzaon of A. Such enres are called fll-n.. Per muae lnes and columns of A so ha he fllng ges mnmum. Once a map of fll-n s known, s added o he orgnally non-zero daa of A and only hese daa are o be sored and mananed n he nex operaons. Hence, as s no necessary o sore and work upon all daa below he skylne of A (as s he case of solvers n Secon 4..); we can use here a sparse marx sorage scheme. The ncurred savngs n boh RAM and CPU resources s sgnfcan and pays well of for a compuaon overhead caused by he prefacorzaon phase and a b more complcaed sorage scheme n use. I s beyond he scope of hs documen o descrbe all deals abou mplemenaon of hs solver. I s based on (Vondracek, 006) and (Davs e. al, 995). A number of opmzaon echnques are used o speed up he soluon procedure, such as he problem (4.6) can be solved usng a block srucure. Ths apply o pre-facorzaon, facorzaon as well as for backward/forward subsuon phases. Typcal sze of such a block s x.. 6x6. The bgger block sze, he smaller overhead for pre-facorzaon and mappng of he marx and he faser he operaon o acually facorze and solve he problem (4.6). Use of a bgger block, however, resuls also n a hgher wase of RAM because all non-zero daa and fll-n are rounded no a sorage wh block paern. Drec sparse solvers are a compromse beween Drec Solvers and Sparse Solvers. They ypcally need more RAM and CPU han Sparse solvers do, (and less han Drec Solvers), however, hey never dverge and brng unceranes as wha precodoner o use ec. Therefore, hey are recommended for mddle sze (may-be ll condoned) problems, he soluon of whch would no f no RAM subjec a Drec Solver s used, and for whch Sparse solvers are no suffcenly robus Ierave Solver The able below lss all solvers n ATENA ha can solve he problem (4.) eravely. Alhough he ls s long, from praccal pon of vew only a few of hem are recommended, see he column Descrpon. In addon, only he mehods DCG and ICCG are desgned o ake full advanage of symmery of A (f presen). The remanng solvers would sore only symmerc par of A, however hey wll operae on n he same way as s no symmerc. Therefore, for symmerc problems he solvers DCG and ICCG are preferable. Each of he erave solver ypcally consss of wo roune, one for preparaon of he soluon and he oher for he soluon self,.e. execuon phase. The former roune s parcularly ATENA Theory 93

204 mporan for he case of precondoned erave solvers. Ths s where a precondonng marx s creaed. The mos effcen precondonng roune are based on ncomplee Cholesky decomposon (Rekorys 995). The precondonng marx A' s decomposed n he same way as (4.6),.e. A' L'D'U' (4.9) Comparng A and A', can be wren for a 0 a ' a j j j for a 0 a ' a j j j (4.0) The ncomplee Cholesky decomposon s carred ou n he same way as complee Cholesky decomposon (4.6), however, enres n A, whch were orgnally zero and became nonzero durng he facorzaon are gnored,.e. hey say zero even afer he facorzaon. The ncurred naccuracy s he penaly for memory savngs due o usage of he erave solvers sorage scheme. For symmerc problem, use sscs roune, for nonsymmerc problems he sslus s T avalable o consruc A' L'D'(L') or A' L'D'U'. Las, bu no he leas noe ha each solver needs some emporary memory. Such requremens are ncluded n he able below. Typcally, he more advanced erave solver, he more exra memory needs and he less number of eraons needed o acheve he same accuracy. Type D/I Prep. phase Exec. phase Table 4.- SOLVER TYPES. Sym/N onsym Temporary requred memory Descrpon LU D S,NS For smaller or llposed probems JAC I ssds sr S,NS 4*()+8*(+4*n) Smple, no recommended GS I --- sr S,NS 4*(+nel+n+)+8*(+3 *n+nel) ILUR I sslus sr S,NS 4*(3+4*n+nu+nl)+8*( +4*n+nu+nl) DCG I ssds scg S 4*()+8*(+5*n) For large symmerc well-posed problems ICCG I sscs scg S 4*(+nel+n)+8*(+5*n +nel) For large symmerc problems, recommended DCGN I ssds scgn S,NS 4*()+8*(+8*n) For large nonsymmerc wellposed problems LUCN I sslus scgn S,NS 4*(3+4*n+nl+nl)+8*( +8*n+nl+nu) For large nonsymmerc problems, recommended 94

205 DBCG I ssds sbcg S,NS 4*()+8*(+8*n) LUBC I sslus sbcg S,NS 4*(3+4*n+nl+nu)+8*( +8*n+nu+nl) DCGS I ssds scgs S,NS 4*()+8*(+8*n) LUCS I sslus scgs S,NS 4*(3+4*n+nl+nu)+8*( +8*n+nu+nl) DOMN I ssds somn S,NS 4*()+8*(+4*n+nsave +3*n*(nsave+)) LUOM I sslus somn S,NS 4*(3+4*n+nu+nl)+8*( +nl+nu+4*n+nsave+3*n *(nsave+)) DGMR I ssds sgmres S,NS 4*(3)+8*(+n+n*(nsav e+6)+nsave*(nsave+3)) LUGM I sslus sgmres S,NS 4*(33+4*n+nl+nu)+8*( +n+nu+nl+n*(nsave+6)+ nsave*(nsave+3)) In he above: n s number of degree of freedom of he problem. nel s he number of nonzeros n he lower rangle of he problem marx (ncludng he dagonal). nl and nu s he number of nonzeros n he lower resp. upper rangle of he marx (excludng he dagonal). Table 4.-: EXECUTION PHASES. Phase name sr scg scgn sbcg scgs somn Descrpon Precondoned Ierave Refnemen sparse Ax = b solver. Roune o solve a general lnear sysem Ax = b usng erave refnemen wh a marx splng. Precondoned Conjugae Graden erave Ax=b solver. Roune o solve a symmerc posve defne lnear sysem Ax = b usng he Precondoned Conjugae Graden mehod. Precondoned CG Sparse Ax=b Solver for Normal Equaons. Roune o solve a general lnear sysem Ax = b usng he Precondoned Conjugae Graden mehod appled o he normal equaons AA'y = b, x=a'y. Solve a Non-Symmerc sysem usng Precondoned BConjugae Graden. Precondoned BConjugae Graden Sparse Ax=b solver. Roune o solve a Non-Symmerc lnear sysem Ax = b usng he Precondoned BConjugae Graden mehod. Precondoned Orhomn Sparse Ierave Ax=b Solver. Roune o solve a general lnear sysem Ax = b usng he Precondoned Orhomn mehod. ATENA Theory 95

206 sgmres Precondoned GMRES erave sparse Ax=b solver. Ths roune uses he generalzed mnmum resdual (GMRES) mehod wh precondonng o solve non-symmerc lnear sysems of he form: A*x = b. Table 4.-3: PREPARATION PHASES. Phase name ssds sslus sscs ssds Descrpon Dagonal Scalng Precondoner SLAP Se Up. Roune o compue he nverse of he dagonal of a marx sored n he SLAP Column forma. Incomplee LU Decomposon Precondoner SLAP Se Up.Roune o generae he ncomplee LDU decomposon of a marx. The un lower rangular facor L s sored by rows and he un upper rangular facor U s sored by columns. The nverse of he dagonal marx D s sored. No fll n s allowed. Incompl Cholesky Decomposon Precondoner SLAP Se Up. Roune o generae he Incomplee Cholesky decomposon, L*D*L-rans, of a symmerc posve defne marx, A, whch s sored n SLAP Column forma. The un lower rangular marx L s sored by rows, and he nverse of he dagonal marx D s sored. Dagonal Scalng Precondoner SLAP Normal Eqns Se Up. Roune o compue he nverse of he dagonal of he marx A*A'. Where A s sored n SLAP-Column forma. As for he soluon procedure,.e. he laer of he wo soluon phases, he mos commonly used mehod s Conjugae graden mehod (wh ncomplee Cholesky precondoner) (Rekorys 995). The flow of execuon s as follows: 96

207 r bax z M r rz r z p z p p rz Ap x x p r r Ap z M r Ths soluon procedure s mplemened n scg roune. The erave solvers n ATENA are based on SLAP package (Seager and Greenbaum 988) ha where modfed o f no ATENA framework. The auhors of he package refer o (Hageman and Young 98), where all of he mplemened soluon echnques are fully descrbed. (4.) 4..4 Parallel Drec Sparse Solver PARDISO 4 Ths solver uses PARDISO parallel drec sparse solver from he Mah Kernel Lbrary (MKL) provded by Inel ogeher wh Inel Composer XE 0. The solver has been developed whn he PARDISO Projec, (see for example hp:// I s amed for large sparse symmerc and un-symmerc lnear sysems wh shared memory. I offers drec or erave solver algorhms. The solver s well esablshed and used by many sofware packages. A lo of leraure s relaed he PARDISO projec. For more nformaon refer o hp://fgb.nformak.unbas.ch/people/oschenk/ndex.hml. Also, a basc nformaon s gven n he Inel Composer XE 0 manuals. A smplfed verson of hs solver s ncluded also n Aena. For he sake of smplcy mos soluon parameers are kep wh her defaul value. The excepon o ha s he parameer "PARDISO_REQUIRED_ACCURACY". I s npu va he Aena "SET" npu command. I specfes, wheher use of drec mehod wh LU decomposon or erave mehod wh CGS precondonng s preferred. In he laer case, also se a requred soluon accuracy. (For more nformaon refer o he Aena Inpu Fle Manual). The followng solver descrpon s aken from he MKL manual provded by wh Inel Composer XE 0, (also a hp://sofware.nel.com/ses/producs/documenaon/hpc/ mkl/mklman/guid-7e fef-46b a09346.hm. Symmerc Marces: The solver frs compues a symmerc fll-n reducng permuaon P based on eher he mnmum degree algorhm (Lu, 985) or he nesed dssecon algorhm from he METIS package (Karyps, 998) (boh ncluded wh Inel MKL), followed by he parallel lef-rgh lookng numercal Cholesky facorzaon (Schenk, 000) of PAPT = LLT for symmerc 4 Avalable sarng from ATENA verson 5. ATENA Theory 97

208 posve-defne marces, or PAPT = LDLT for symmerc ndefne marces. The solver uses dagonal pvong, or x and x Bunch and Kaufman pvong for symmerc ndefne marces, and an approxmaon of X s found by forward and backward subsuon and erave refnemens. Whenever numercally accepable x and x pvos canno be found whn he dagonal supernode block, he coeffcen marx s perurbed. One or wo passes of erave refnemens may be requred o correc he effec of he perurbaons. Ths resrcng noon of pvong wh erave refnemens s effecve for hghly ndefne symmerc sysems. Furhermore, for a large se of marces from dfferen applcaons areas, hs mehod s as accurae as a drec facorzaon mehod ha uses complee sparse pvong echnques(schenk, 004). Anoher mehod of mprovng he pvong accuracy s o use symmerc weghed machng algorhms. These algorhms denfy large enres n he coeffcen marx A ha, f permued close o he dagonal, perm he facorzaon process o denfy more accepable pvos and proceed wh fewer pvo perurbaons. These algorhms are based on maxmum weghed machngs and mprove he qualy of he facor n a complemenary way o he alernave dea of usng more complee pvong echnques. The nera s also compued for real symmerc ndefne marces. Unsymmerc Marces: The solver frs compues a non-symmerc permuaon PMPS and scalng marces Dr and Dc wh he am of placng large enres on he dagonal o enhance relably of he numercal facorzaon process (Error! Hyperlnk reference no vald.. In he nex sep he solver compues a fll-n reducng permuaon P based on he marx PMPSA + (PMPSA)T followed by he parallel numercal facorzaon QLUR = PPMPSDrADcP wh supernode pvong marces Q and R. When he facorzaon algorhm reaches a pon where canno facor he supernodes wh hs pvong sraegy, uses a pvong perurbaon sraegy smlar o (Error! Hyperlnk reference no vald.. The magnude of he poenal pvo s esed agans a consan hreshold of alpha = eps* A nf, where eps s he machne precson, A = P*PMPS*Dr*A*Dc*P, and A nf s he nfny norm of he scaled and permued marx A. Any ny pvos encounered durng elmnaon are se o he sgn (lii)*eps* A nf, whch rades off some numercal sably for he ably o keep pvos from geng oo small. Alhough many falures could render he facorzaon well-defned bu essenally useless, n pracce he dagonal elemens are rarely modfed for a large class of marces. The resul of hs pvong approach s ha he facorzaon s, n general, no exac and erave refnemen may be needed. Drec-Ierave Precondonng. The solver enables o use a combnaon of drec and erave mehods (Error! Hyperlnk reference no vald.) o accelerae he lnear soluon process for ransen smulaon. Mos of applcaons of sparse solvers requre soluons of sysems wh gradually changng values of he nonzero coeffcen marx, bu he same dencal sparsy paern. In hese applcaons, he analyss phase of he solvers has o be performed only once and he numercal facorzaons are he mporan me-consumng seps durng he smulaon. PARDISO uses a numercal facorzaon A = LU for he frs sysem and apples he facors L and U for he nex seps n a precondoned Krylow-Subspace eraon. If he eraon does no converge, he solver auomacally swches back o he numercal facorzaon. Ths mehod can be appled o unsymmerc and srucurally symmerc marces n PARDISO. For symmerc marces 98

209 Conjugae-Gradens mehod s appled. You can selec he mehod usng only one npu parameer. Separae Forward and Backward Subsuon. The solver execuon sep can be dvded no wo or hree separae subsuons: forward, backward, and possble dagonal. Ths separaon can be explaned by he examples of solvng sysems wh dfferen marx ypes. A real symmerc posve defne marx A s facored by PARDISO as A = L*LT. In hs case he soluon of he sysem A*x=b can be found as sequence of subsuons: L*y=b (forward subsuon) andlt*x=y (backward subsuon). A real unsymmerc marx A s facored by PARDISO as A = L*U. In hs case he soluon of he sysem A*x=b can be found by he followng sequence: L*y=b (forward subsuon) and U*x=y (backward subsuon). Noe ha dfferen pvong (x, x...) produces dfferen LDLT facorzaon. Therefore resuls of forward, dagonal and backward subsuons wh dagonal pvong can dffer from resuls of he same seps wh Bunch and Kaufman pvong. Of course, he fnal resuls of sequenal execuon of forward, dagonal and backward subsuon are equal o he resuls of he full solvng sep regardless of he pvong used. Sparse Daa Sorage. Sparse daa sorage n PARDISO follows he scheme descrbed above. 4. Full Newon-Raphson Mehod Usng he concep of ncremenal sep by sep analyss we oban he followng se of nonlnear equaons: K ( p) p q f( p) (4.) where: q s he vecor of oal appled jon loads, f ( p) s he vecor of nernal jon forces, p s he deformaon ncremen due o loadng ncremen, p are he deformaons of srucure pror o load ncremen, K ( p) s he sffness marx, relang loadng ncremens o deformaon ncremens. The R.H.S. of (4.) represens ou-of-balance forces durng a load ncremen,.e. he oal load level afer applyng he loadng ncremen mnus nernal forces a he end of he prevous load sep. Generally, he sffness marx s deformaon dependen,.e. a funcon of p, bu hs s usually negleced whn a load ncremen n order o preserve lneary. In hs case he sffness marx s calculaed based on he value of p peranng o he level pror o he load ncremen. The se of equaons (4.) s nonlnear because of he non-lnear properes of he nernal forces: f ( kp) kf ( p) (4.3) ATENA Theory 99

210 and non-lneary n he sffness marx K( p) K ( pp) (4.4) where k s an arbrary consan. The se of equaons represens he mahemacal descrpon of srucural behavor durng one sep of he soluon. Rewrng equaons (4.) for he -h eraon whn a dsnc loadng ncremen we oban: K ( p ) p q f( p ) (4.5) All he quanes for he (-)-h eraon have already been calculaed durng prevous soluon seps. Now we solve for p a load level q usng: p p p (4.6) As poned ou earler, equaon(4.5) s nonlnear and herefore s necessary o erae unl some convergence creron s sasfed. The followng possbles are suppored n ATENA ( k marks k -h componen of he specfed vecor): T p p T p p rel. dsp ( q f( p )) ( q f( p )) f p T T ( ) f( p) T p ( q f( p )) T p f( p ) rel. energy rel. force (4.7) max(( q f ( p ))) max(( q f ( p ))) k k k k k k max( f ( p)) max( f ( p)) abs. force The frs one checks he norm of deformaon changes durng he las eraon whereas he second one checks he norm of he ou-of-balance forces. The hrd one checks ou-of-balance energy and he fourh condons checks ou-of-balanced forces n erms of maxmum componens (raher hen Eucld norms). The values of he convergence lms are se by defaul o 0.0 or can be changed by npu command SET. The concep of soluon nonlnear equaon se by Full Newon-Raphson mehod s depced n Fg. 4-: 00

211 q Loadng Loadng ncremen p p p 0 Deformaon Fg. 4- Full Newon-Raphson mehod. 4.3 Modfed Newon-Raphson Mehod The mos me consumng par of soluon (4.5) s he re-calculaon of he sffness marx K ( p ) a each eraon. In many cases hs s no necessary and we can use marx K ( p0) from he frs eraon of he sep. Ths s he basc dea of he so-called Modfed Newon-Raphson mehod. I produces very sgnfcan me savng, bu on he oher hand, also exhbs worse convergence of he soluon procedure. The smplfcaon adoped n he Modfed Newon-Raphson mehod can be mahemacally expressed by: K( p ) K( p ) (4.8) 0 The modfed Newon-Raphson mehod s shown n Fg. 4-. Comparng Fg. 4- and Fg. 4- s apparen ha he Modfed Newon-Raphson mehod converges more slowly han he orgnal Full Newon-Raphson mehod. On he oher hand a sngle eraon coss less compung me, because s necessary o assemble and elmnae he sffness marx only once. In pracce a careful balance of he wo mehods s usually adoped n order o produce he bes performance for a parcular case. Usually, s recommended o sar a soluon wh he orgnal Newon- Raphson mehod and laer,.e. near exreme pons, swch o he modfed procedure o avod dvergence. q Loadng Loadng ncremen p 0 p p p 3 p 4 Deformaon Fg. 4- Modfed Newon-Raphson mehod ATENA Theory 0

212 4.4 Arc-Lengh Mehod Nex o he Modfed Newon-Raphson mehod, he mos wdely used mehod s he Arc-lengh mehod. Ths mehod was frs employed abou ffeen years ago o solve geomercally nonlnear srucures. Because of s excellen performance, s now que well esablshed for geomerc non-lneary and for maeral non-lneary as well. Many workers have been neresed n usng and mprovng Arc-lengh procedures. In Aena, can be used whn CCSrucures module,.e. for sac analyss. The man reason for he populary of hs mehod s s robusness and compuaonal effcency whch assures good resuls even n cases where radonal Newon-Raphson mehods fal. Usng an Arc-lengh mehod sably problems such as snap back and snap hrough phenomena can be suded as well as maerally non-lnear problems wh non-smooh or dsconnuous sress-sran dagrams. Ths s possble due o he changng load condons durng eraons whn an ncremen. The man dea of hs mehod s well explaned by s name, arc-lengh. The prmary ask s o observe complee load-dsplacemen relaonshp raher hen applyng a consan loadng ncremen as s n he Newon-Raphson mehod. Hence hs mehod fxes no only he loadng bu also he dsplacemen condons a he end of a sep. There are many ways of fxng hese, bu one of he mos common s o esablsh he lengh of he loadng vecor and dsplacemen changes whn he sep. From he mahemacal pon of vew means ha we mus nroduce an addonal degree of freedom assocaed wh he loadng level (.e. a problem has n dsplacemen degrees of freedom and one for loadng) and n addon a consran for he new unknown varable mus be nroduced. The new degree of freedom s usually named. There are many possbles for defnng consrans on and hose mplemened n ATENA are brefly revewed n he followng secons. To derve he Arc-lengh mehod we rewre he se of equaons (4.) n form of (4.9), where defnes he new loadng facor: K ( p) pq f( p) (4.9) Now re-wrng (4.9) n a form suable for erave soluon: K ( p ) p q f( p ) q f (4.0) p p p p (4.) p p (4.) (4.3) The noaon s explaned n Fg The marx K can be recompued for every eraon (smlar o Full Newon-Raphson mehod) or can be fxed based on he s eraon for all subsequen eraons (Modfed Newon Raphson mehod). The vecor q does no mean n hs case he oal loadng a he end of he sep bu only a reference loadng "ype". The acual loadng level s a mulple of hs. 0

213 The scalar s an addonal varable nroduced by he Lne-search mehod, whch wll be dscussed laer. The scalar s used o accelerae soluons n cases of well-behaved loaddeformaon relaonshps or o damp possble oscllaons, f some convergence problems arose, e.g. near bfurcaon and exreme pons. q 0 q 0 q q q 3 q g 0 g R Load ncremen Loadng R T 0 q sar 0 p p p p 0 p p Deformaon Fg. 4-3 The Arc-lengh mehod Addonal noaon s defned as follows: Ou-of-balance forces n -h eraon: g( p ) g f q f ( ) q (4.4) R.H.S vecor n -h eraon: RHS q f q g (4.5) Subsung (4.) hrough (4.5) no (4.0), he deformaon ncremen can be calculaed from: K RHS q g (4.6) Hence: where (4.7) T T K K q g (4.8) I remans only o se he addonal consran for and and he whole algorhm s defned. Thus compared o he Newon-Raphson mehods n whch we solve n dmensonal nonlnear problem, he Arc-lengh mehod need o solve a (n + ) dmensonal problem, where he frs n unknowns correspond o deformaons and he las wo are and. ATENA Theory 03

214 If we se, hen we deal wh an (n + ) dmensonal problem ha correspond o pure Arclengh mehod, oherwse a combnaon of Arc-lengh and Lne search mus be employed. The Lne search mehod s dscussed laer n hs chaper. Noe ha all vecors ncludng, T are of order (n + ). Ther (n + )-h coordnae corresponds o he loadng dmenson and s se o zero. Now, nroduce wo new vecors and n as shown n Fg There are defned by: p ( ) (4.9) sar where: n (4.30) s scalar ha relaes dmensons of o sze of deformaon space, s a (n + ) dmensonal vecor wh s frh n coordnaes se o zero (deformaon space) and s (n + )-h coordnae equal o. sar s a (n+) dmensonal vecor smlar o o sar., however s (n + )-h coordnae equal n n n 3 3 p I s hen obvous ha Defnng he resdual R : Fg. 4-4 The vecors and n n and scalar. n (4.3) R n (4.3) equaons (4.0) hrough (4.3) lead o he fnal expresson for he unknown (nong ha T T p p ): 0 04

215 R p (4.33) ( ) T T p T sar To oban by (4.33) he resdual R mus be defned. In fac, also defne ype of Arclengh consran beng used. The ypes suppored n ATENA are descrbed below Normal Updae Mehod Vecor and n are normals n hs case, hence resdual R 0, see Fg n n n 3 3 Fg Normal updae mehod. The man advanage of hs mehod s s smplcy. The Normal updae plane s relavely relable, bu can fal f he l-p dagram suddenly changes s slope or urns back or down (snap back and snap hrough). Neverheless f hese specal condons are reaed by hs mehod hen a very sgnfcan reducon n sep lengh s unavodable Conssenly Lnearzed Mehod The resdual R s defned n hs case by R n n cos( ) ( s) (4.34) T p The sep lengh s and angle are depced n Fg The norm of he vecor calculaed usng (4.9): s T p p ( sar) (4.35) ATENA Theory 05

216 curren s requred s - n - s = sep lengh p Fg Conssenly lnearzed mehod. Subsung (4.34) and (4.35) n (4.33) we oban he fnal expresson for. I should be noed ha he scalar s s se 'a pror' and governs he acual sep lengh. Of course, he proper choce of hs parameer s essenal for he soluon and herefore wll be dscussed laer n more deal. Ths mehod s especally suable for soluons ha embrace p dagrams wh sudden breaks and dsconnues, e.g. for maerally nonlnear problems Explc Orhogonal Mehod The basc consran for n hs case s ha s, where s s some dsnc 'a pror' se sep lengh. Smlar o he prevous mehod we also have o evaluae he resdual R : R n n cos( ) r (4.36) T Based on he smlar rangles (see Fg. 4.4-), he followng can be derved: r ' s ' l l l l (4.37) R s ( s) (4.38) ' ' n (4.39) ' ' ' ' (4.40) ' The vecor s calculaed usng (4.35). By subsung he above equaons no (4.33) fnal expresson for s obaned. From he above dervaon s clear ha n pracce we a frs employ Normal Updae Mehod ' ' (Chaper 4.4.) o solve for and n and hereafer we correc he n order o sasfy he consran s. 06

217 r - n n - r - = - s s s = sep lengh p Fg Explc orhogonal mehod. Ths mehod s usually ulzed o analyze geomercally nonlnear srucures, parcularly sably problems. Is man feaure s robusness and compared wh he "classcal" Crsfeld cylnder mehod (see below) avods he problem of he choce of he proper roo (he condon s whle expressng vecor lengh analycally). As for convergence, he mehod s comparable o he mehod 4.4.3, bu has he advanage ha preserves he sep lengh The Crsfeld Mehod. The Crsfeld mehod s derved drecly from he consran of consan sep lengh s The resdual R s no used n hs case and we subsue equaons (4.0) hrough (4.3) sragh no he above consran. I leads o he followng equaon for : where: a a a (4.4) 3 0 a T T T a ( ) T sar T (4.4) a ( ) s T 3 sar Equaon (4.4) has generally wo roos and hence we mus decde whch of hem o use. There exs several sraeges bu ATENA chooses ha roo, for whch cos(, ) 0 (or hgher of hem),.e. drecon of new ncremen as close as possble o drecon of he prevous ncremen (whn he same sep). ATENA Theory 07

218 4.4.5 Arc Lengh Sep The proper sep lengh s of essenal mporance for good execuon performance. I drecly nfluences he convergence radus on he one hand and he number of requred seps on he oher. ATENA uses he followng procedure o se (or opmze) s : () Se loadng vecor q and hus defne a reference loadng level (whn one load ncremen). () Srucural response o hs load n he s execuon sep, he s eraon defnes sep lengh s n he s sep. In he subsequen seps he sep lengh s kep fxed or opmzed (based on SET ATENA npu command, subcommand &ARC_LENGTH_OPTIMISATION: s s n n s (4.43) n n 4 s (4.44) where n s s (4.45) n s and s s Arc lengh sep lengh n he curren and he prevous load ncremen, respecvely. n and n s desred number of eraons and number of eraons n he prevous sep. n s ypcally Lne Search Mehod The objecve of hs mehod s o calculae he parameer ha was already nroduced n he Chaper 4.4 Arc-Lengh Mehod The mehod can be used eher ndependenly or n combnaon wh Arc lengh mehod. The prmary reason for nroducng a new parameer (.e. a new degree of freedom o he se of equaons) s o accelerae or o damp he speed of analyss of he loaddsplacemen relaonshp. The basc dea behnd s o mnmze work of curren ou-of-balance forces on dsplacemen ncremen. Le us assume ha we have already solved already wo pons p 0 and p0 ' p and hus we have also calculaed ou-of-balance forces g( p 0 ) and g( p0 ' ) a hese pons. The am of hs mehod s o se he parameer so ha he work beng done by ou-of- balance forces a pon p0 s mnmum. The work of ou-of-balance forces s: Hence: p T ( p) ( p0) g( p) dp mnmum (4.46) po 08

219 ( p) p p p T T 0 g( p) g( p) 0 d p (4.47) p0 Inerpolang lnearly ou-of-balance forces beween pons p 0 and p0 ' g( p0 ' ) g( p0) g( p0 ' ) g( p0) g( p0 ) g( p0) p0 p0 g( p0) p0 ' p0 ' (4.48) and usng : p p 0 p (4.49) The fnal expresson for ' can be derved: T g( p0 ) ' T g( p ) g( p ' ) T 0 0 (4.50) Thus, he Lne search mehod can be summarzed: Use any mehod o calculae dsplacemen ncremen, (see Fg. 4-3 and (4.8)). The parameer ' can be se from he las load ncremen or smply o uny. Calculae ou-of-balance forces for boh g( p 0 ) and g( p0 ' ). Use (4.50) o calculae new value for. As all he above equaons are nonlnear, he parameer mus be solved by eraons unl g( p 0 g( p ) 0 a specfed energy drop, ypcally < >. Praccal experence suggess ha he value of parameer should be kep n nerval < 0. 5>. 4.6 Parameer The parameer scales he deformaon space p o he loadng dmenson. If 0, he soluon for s searched on an area of a cylndrcal shape of radus equal o sep lengh s (Crsfeld mehod) and he axs normal o he p (deformaon) space. The soluon s he pon of nersecon of hs area and he lne, defned by he energy gradens of srucure and by he appled load a pon p. If 0, he soluon s carred ou n he same way on ellpsodal or sphercal space. The hgher value of, he hgher "wegh facor" for changes n loadng space compared o dsplacemen ncremens. ATENA currenly suppors he followng formulae for seng and opmzaon of (for curren sep j ). They are revewed below. ATENA Theory 09

220 The frs sraegy requres he load o dsplacemen ncremen rao (4.5) s consan hroughou all seps, (e.g. npu value ) req ( p) req (4.5) Then, a he end sep j- we can calculae Ths value (due o nonlneares) wll no mach as follows: j j j (4.5) ( p) j req. Therefore, for sep j we wll modfy j j req j req j ( p) req req j j j j j req j j j j ( p) j (4.53) The above opmsaon process s nalzed n he frs sep by assumng ha 0, 0, ( jp) T, where T s dsplacemen correspondng o maser Arc-lengh load ncremen defned earler n hs chaper. Hence req req req req T j j j ( p) T 0 j (4.54) The parameers j n all subsequen seps are calculaed usng (4.53). If rao of dsplacemens changes ( ) o load changes ( ) n he las load sep ncrease, hen he equaon (4.54) j p j (4.55) ncreases n he curren sep, hereby pus hgher wegh facor on loads compared o dsplacemens. Hence, he equaon (4.54) ends o keep consan mporance of loadng space rrespecve of dsplacemens. Noe ha he equaon (4.54) corresponds o BETA_FORCES_DISPLS_RATIO_CONSTANT. The second suppored sraegy s dfferen. In ATENA s referred o as BETA_RATIO_CONSTANT mehod and res o keep consan coeffcens, whls managng he coeffcens. Thus, works n oppose way as compared o he frs sraegy descrbed above. From (4.5) we can wre for seps (j-) and j 0

221 j j j ( p) j j j ( p) j j Now requrng j j we have ( p) ( p) j j j j j j j ( p) j j ( p) j j j (4.56) and f we assume j j ( p) ( p) j j, hen j j j j and he above equaon yelds j j ( p) j j ( p) j j (4.57) If ( p) j j ( p) j j n subsequen seps changes, he procedure s ryng o compensae for ha by readjusng he coeffcens. In oher words, hs sraegy s ryng o keep relave mporance of load vs. dsplacemen spaces). 4.7 Band Wdh Opmzaon ( p) consan, (.e. The way n whch ndvdual srucural degrees of freedom (dofs) are mapped no he global srucural marces has sgnfcan mpac on her sze and cos of he soluon n erms of requred CPU and RAM resources. Le us assume he D example of he 3 bars elemen from Fg The srucure consss of hree beam elemens,,3. I has four global nodes wh hree degrees of freedom n each of hem,.e. wo dsplacemens and one roaon. Suppose he srucure s solved by a drec solver,.e. we use half-band skylne sorage scheme (4.4). By defaul,.e. whou any opmzaon, he srucural degree of freedom are allocaed sequenally sarng from he node up o he las node n,.e. 4. Hence, he jh degree of freedom a he node has number ndof ( ) j, where ndof s number of dofs per node. ATENA Theory

222 If he srucural nodes are numbered as ndcaed, hen he beam, and 3 has nodal ncdences -3, 3-4 and 4-, respecvely and he fnal sffness marx K has he paern from he lefboom par of Fg Noe ha he marx K mus sore also he enres depced as crcles whou fllng. Alhough hey are nally zero, hey may urn non-zero durng he marx decomposon needed o solve he problem,.e. we mus sore he marx wh 69 enres and maxmum half-band wdh 9. On he oher hand, f nodal degrees of freedom are numbered as shown n he rgh-boom par of Fg. 4-5, hen he marx K mus sore only 5 enres and has maxmum half bandwdh only 6. The wo examples documen, how mporan effcen numberng of he degrees of freedom of he srucure s. If he srucure (o be solved) s smple, hen a suable dofs' numberng can be done manually by approprae numberng of he srucural nodes. However, n he more complex cases (and n parcular f a model of he srucure s generaed auomacally), an opmal dofs mappng mus be calculaed. There are number of algorhms ha delver more or less effcen dofs mappng. Probably he bes esablshed algorhm of ha knd s Cuhll-McKee algorhm (Cuhll, McKee 969). Ths s no due o s superor propery, bu due has been developed as frs. The algorhm produces an ordered n-uple R of verces whch s he new order of he srucural verces. I numbers he verces accordng o a parcular breadh-frs raversal, where neghborng verces are vsed n order from lowes o hghes verex order. The reverse Cuhll McKee algorhm (RCM) s he alernave of he Cuhll-McKee algorhm, n whch he verces are vsed n reverse order,.e. form he hghes o he lowes verex. ATENA mplemens Gbbs and Sloan dofs opmzaon algorhms:

223 Fg. 4-5 Opmzaon of dofs numberng The Sloan algorhm (Sloan, Randolf (983) In an effor o oban an opmum elmnaon order, he algorhm frs renumbers he nodes, and hen uses hs resul o resequence he elemens. Ths nermedae sep s necessary because of he naure of he fronal soluon procedure, whch assembles varables on an elemen-byelemen bass bu elmnaes hem node by node. To renumber he nodes, a modfed verson of he Kng algorhm s used. In order o mnmze he number of nodal numberng schemes ha need o be consdered, he sarng nodes are seleced auomacally by usng some conceps from graph heory. Once he opmum numberng sequence has been asceraned, he elemens are hen reordered n an ascendng sequence of her lowes-numbered nodes. Ths ensures ha he new elmnaon order s preserved as closely as possble. For meshes ha are composed of a sngle ype of hgh-order elemen, s only necessary o consder he verex nodes n he renumberng process. Ths follows from he fac ha mesh numberngs whch are opmal for low-order elemens are also opmal for hgh-order elemens. Sgnfcan economes n he reorderng sraegy may hus be acheved. ATENA Theory 3

224 The Gbbs e. al. algorhm (Gbbs e. al. 976) Ths algorhm ypcally produces bandwdh and profle, whch are comparable o hose of he commonly-used reverse Cuhll McKee algorhm, ye requres sgnfcanly less compuaon me. Neverheless, delvers dofs mappng ha s usually slghly less effcen han ha by he Sloan algorhm and herefore, s less preferred opon he opmzaon. Noe ha he above algorhms opmze dofs numberng by reorderng he srucural nodes. They do no accoun for possble dfferen number of dofs whn a parcular node. Noe also ha n order o mnmze cos of he dofs remappng, he opmzaon s carred ou before assemblng he srucural global marces and vecors. Thus, hey are assembled drecly no her fnal, opmzed locaon. Ierave solvers use daa sorage scheme (4.3). As he sorage scheme sores only non-zero elemens, he soluon s less sensve o a bad dofs mappng. For huge analyses s neverheless suggesed o carry ou a dofs mappng opmzaon, as ypcally yelds ndvdual elemens enres sored closer o each oher wh posve effec on soluon convergence and RAM daa managemen. A dealed descrpon of he above algorhms s above scope of he publcaon. For more nformaon he reader s suggesed o sudy he gven references. 4.8 References BATHE, K.J.(98), Fne Elemen Procedures In Engneerng Analyss, Prence-Hall, Inc., Englewood Clffs, New Jersey 0763, ISBN CRISFIELD, M.A. (983) - An Arc-Lengh Mehod Includng Lne Search and Acceleraons, Inernaonal Journal for Numercal Mehods n Engneerng, Vol.9, pp CUTHILL, E. and J. MCKEE (969). Reducng he Bandwdh of Sparse Symmerc Marces. Proc. 4h Na. Conf. ACM. DAVIS, T., AMESTOY, P., DUFF, I.S (995) - An Aproxmae Mnmum Degree Orderng Algorhm, Comp. and Informaon Scence Dep., Unversy of Florda, Tech. Repor TR DUFF, I. S. and KOSTER, J. (999) - The Desgn and Use of Algorhms for Permung Large Enres o he Dagonal of Sparse Marces. SIAM J. Marx Analyss and Applcaons, 0(4): FELIPPA, C. (966) - Refned Fne Elemen Analyss of Lnear and Nonlnear Two-Dmensonal Srucures, Ph.D. Dsseraon, Unversy of Calforna, Engneerng, pp GIBBS, N. E., W. G. POOLE, e al. (976). "An Algorhm for Reducng he Bandwdh and Prole of a Sparse Marx." SIAM Journal of Numercal Analyss 3(). KARYPIS, G. and KUMAR, V. (998) - A Fas and Hgh Qualy Mullevel Scheme for Paronng Irregular Graphs. SIAM Journal on Scenfc Compung, 0(): LIU, J.W.H. (985) - Modfcaon of he Mnmum-Degree Algorhm by Mulple Elmnaon. ACM Transacons on Mahemacal Sofware, ():4-53. LI, X.S., DEMMEL, J, W. (999) - A Scalable Sparse Drec Solver Usng Sac Pvong. In Proceedng of he 9h SIAM conference on Parallel Processng for Scenfc Compung, San Anono, Texas, March

225 RAMM, E. (98) - Sraeges for Tracng Non- lnear Responses Near Lm Pons, Non- lnear Fne Elemen Analyss n Srucural Mechancs, (Eds. W.Wunderlch,E.Sen, K.J.Bahe) REKTORYS, K. (995). Přehled užé maemaky. Prague, Promeheus. SLOAN, S. W. and M. F. RANDOLF (983). "Auomac Elemen Reordenng for Fne Elemen Analyss wh Fronal Soluon Schemes." In. Journal for Numercal Mehods n Eng. 9: SEAGER, M. K. and A. GREENBAUM (988). A SLAP for he Masses, Lawrence Lvermore Naonal Laboraory SCHENK, O., GARTNER, K. and FICHTNER, W. (000) - Effcen Sparse LU Facorzaon wh Lef-rgh Lookng Sraegy on Shared Memory Mulprocessors. BIT, 40(): SCHENK, O. and GARTNER, K. (004) - On Fas Facorzaon Pvong Mehods for Sparse Symmerc Indefne Sysems. Techncal Repor, Deparmen of Compuer Scence, Unversy of Basel, submed. SONNEVELD, P. (989) - CGS, a Fas Lanczos-Type Solver for Nonsymmerc Lnear Sysems. SIAM Journal on Scenfc and Sascal Compung, 0:36-5. VONDRACEK, R. (006) - The Use Of The Sparse Drec Solver In The Egneerng Applcaons Of The Fne Elemen Mehod. Theses for Ph.D. Czech Techncal Unversy, Prague. ATENA Theory 5

226

227 5 CREEP AND SHRINKAGE ANALYSIS Creep and shrnkage are undoubedly feaures ha have sgnfcan nfluence on concree behavour. Alhough creep and shrnkage analyss can be negleced n desgn of mos cvl srucures, here exs cases, when hese phenomena have o be accouned for. The Ref. (Bazan and Baweja 999) provdes a fve levels classfcaon of srucures ha can serve as a smple gudelnes for makng decson, when creep and shrnkage analyss s needed and when s no needed. The recognzed levels of srucures are as follows: Level : Renforced concree beams, frames and slabs wh span under 0m and heghs of up o 30m, plan concree foongs, reanng walls. Level. Presressed beams or slabs of spans up o 0m, hgh-rse buldng frames up o 00m hgh. Level 3. Medum-span box grder, cable-sayed or arch brdges wh spans of up o 80m, ordnary anks, slos, pavemens. Level 4. Long-span presressed box-grder, cable-sayed or arched brdges; large brdges bul sequenally n sages by jonng pars, large gravy, arch or buress dams, coolng owers, large roof shells, very all buldngs. Level 5. Record span brdges, nuclear conanmens and vessels, large offshore srucures, large coolng owers, record-span hn roof shells, record-span slender arch brdges. Full creep and shrnkage analyss s mandaory for desgn of srucures level 4 and 5 and s recommended also for he level 3 srucures. 5. Implemenaon of Creep and Shrnkage Analyss n ATENA ATENA sofware provdes a powerful mehod for creep and shrnkage analyss for mos problems from engneerng pracce. I s based on so called cross-seconal approach, meanng ha he analyss bulds upon creep and shrnkage behavour of he whole cross secon raher han behavour of ndvdual maeral pons only. The reason for choosng hs mehod s ha a hs momen, here s avalable numerous models for predcng creep and shrnkage behavour of a concree cross secon, whereas here s very low evdence abou he same behavour a maeral pon level. The second reason s ha s accuracy suffces for mos analyses from engneerng pracce and s much less expansve n erms of compuaonal cos. 5.. Basc Theorecal Assumpons The mplemened creep and shrnkage analyss s based on assumpon of lnear creep, whch n oher words means ha maeral complance funcon (, ') and accompanyng funcon for shrnkage 0 () depends only on maeral composon, emperaure, shape and me a observaon and a loadng '. I does no depend on sress-sran condons. In spe of he smplfcaons he provded analyss s n mos praccal cases suffcenly accurae and s fas and effcen. On he oher hand, s applcable only for srucures, where sress value does no exceed abou 60% of ulmae srengh of concree. For hgher load levels he maeral ATENA Theory 7

228 nonlneary becomes sgnfcan and a more elaborae soluon has o be employed. The above smplfcaon apples o me dependen (.e. long erm) maeral behavour only. For shor-erm behavour of he maeral model reans s nonlneary,.e. accouns for phenomena such as cracks, plascy. The creep and shrnkage analyss s based on assumpon of Seljes negral, whch s s wren for he case of D analyss n he followng form: where: = observaon me, ' = loadng me, () =sress a me, 0 () (, ') d () (4.58) ' 0 () = nal sress-ndependen sran such as concree shrnkage, (, ') = complance funcon of concree. Fg. 5- Decomposon of sress hsory no sress seps (lef) or mpulses (rgh). The sense of Seljes negral s gven n he above fgure. Equaon (4.58) has o be modfed for he case of and 3D analyses for praccal analyses. Ths s done below. I s mporan o noe ha (4.58) apples for any sress and sran hsory and s defned n ncremenal form. I means ha a a parcular me sress a depends only on curren maeral sae a me and sress ncremen a me,.e. d. The fnal form of he above equaons reads: B( ( )) 0 () (, ) ( ( )) ( ) d () ' B (4.59) where: () = s sress vecor a me, (noe he bar aop of a symbol ndcaes vecor), 0 () = vecor of nal srans, such as shrnkage, B ( ( )) = marx accounng for mulaxal sress-sran condons, ncludng all maeral shor-erm nonlneares. 8

229 Noce he way he equaon (4.59) s wren. Long-erm and shor-erm maeral behavour s separaed. The former s encapsulaed n complance funcon (, '), whereas he shor-erm behavour s comprsed n he marx B ( ( )). Ths assumpon brngs sgnfcan smplfcaon of he creep and shrnkage analyss and s beleved ha for mos praccal analyss he nduced naccuracy s accepable. Subsung ', 0 no (4.59) and applyng load ncremen (') (') (.e. loadng from zero level) a me ', can be derved 0 ( ' ) ( ', ') B ( ( ')) ( ') ( ' ) (4.60) Comparson of (4.60) wh smlar equaons for consuve relaons for shor-erm loadng condons,.e. ' ', yelds nsananeous secan maeral rgdy marx: D (') = B ( (')) (',') (4.6) The marx D ( ') corresponds o recprocal value of he well known secan Young modulus E (') n he case of D sress-sran condons. In he case of plane sress condons, he marx B( ( )) reads (4.6), ec. 0 B= 0 (4.6) sym. ( ) 5. Approxmaon of Complance Funcons (, ') by Drchle Seres. Ref. (Bazan and Spencer 973) and ohers show ha sgnfcan mprovemen of compuaonal effcency can be obaned, f he orgnal maeral complance funcon (, ') s durng he creep soluon approxmaed by Drchle seres '(, ') as follows: where : = are so called reardaon mes, ' n '(, ') e (4.63) E (') E (') n = number of approxmaon funcons,.e. hs parameer s relaed o he npu parameer number of reardaon mes. E (') = nsan Young modulus a me ', E (') =coeffcens for he approxmaon funcons. ATENA Theory 9

230 Fg. 5- Approxmaon of complance (or reardaon) funcon curve a age a loadng by a sum of exponenals used as shape funcons of Drchle seres The effec of use of Drchle seres approxmaon s depced n he above fgure. A sngle approxmaon exponenal s drawn n sub-fgure (a), whle he whole process of decomposon of complance and reardaon curves s depced n he sub-fgures (b), (c), respecvely. The ncorporaon of Drchle seres '(, ') brngs he followng benefs: - Creep analyss s ndependen of maeral creep predcon model. - Tme negraon s exac; hence, less emporal ncremens are necessary. - Less demand of compuer sorage needed for sorng daa from he prevous emporal seps of he analyss. I suffces o sore daa from he prevous analyss sep only, raher han he complee sresses-sran hsory of he analyzed srucure. 5.3 Sep by Sep Mehod Equaon (4.59) (upon subsuon (4.63) s solved numercally. The srucure s dscrezed n space by fne elemen mehod, (descrbed elsewhere n hs documen). As for me, he soluon s carred ou by Sep-by-sep mehod (SBS) (Bazan 988). The srucural behavour s analysed n several me seps,.e. n me ncremens, as corresponds o (4.59). Afer some mahemacal manpulaons (Jendele and Phllps 99) he fnal soluon equaons read: r E r/ Br-/ ( r - r ) (4.64) ( ) (4.65) r r r r ( ) (4.66) r r r r n, r (4.67) E E E r/ r/, r/ r e r r (4.68) 0

231 n * 0 r e r r (4.69) ( ) (4.70) r r r r r E r/ r r e r (4.7) E * * r In he above he followng noaon s used: r/ r = denfcaon of emporal ncremens, r.. N ncremens for he analyss, = me ncremen, r r r, where N s number of me r r r = sress ncremen n me r * * ( ) r r =nernal varables a me r,, 0 0 r ( r) = shrnkage a me r, Er / E( r) E( r ) = consan average secan Young modulus a me ncremenen r, Er / E( r) E( r ) E E r r = consan average value of Drchle coeffcen E a r, Br / B( r) B ( r ) = average value of he marx B a r Equaon (4.64) hru (4.7) defnes all necessary relaons o complee he creep and shrnkage analyss n ATENA. Of course, hey are supplemened by relaons used by shor-erm maeral consuve model,.e. equaons for calculang he marx B. A each me ncremen, a ypcal shor-erm alke analyss s carred. Dfference beween he shor-erm analyss and he descrbed analyss of one sep of he creep and shrnkage s ha he laer one uses especally adjused Young modulus E r / and nal sran ncremens r o accoun for creep and shrnkage. Afer each sep hese have o be updaed. I nvolves manly updae of and r. Wh hese values a new E r / s calculaed and he nex emporal r analyss sep s carred ou. 5.4 Inegraon and Reardaon Tmes Approprae selecon of reardaon and negraon mes s of crucal mporance for accurae and effcen creep and shrnkage analyss. The choce of reardaon mes has drec mpac on accuracy of approxmaon an orgnal complance funcon by Drchle seres, see Equaon ATENA Theory.

232 (4.63) and Fg. 5-, whls he choce of negraon mes affecs accuracy of he approxmaon of loadng funcon of he srucure, see Equaon (4.58) and Fg. 5-. If number of he mes s oo low, some mporan feaures of concree behavour can be dsregarded. The oppose exreme,.e. usng oo many of reardaon or negraon mes resuls n worhless lenghy soluon of he problem. The ATENA sofware respecs recommendaon n (Bazan and Whman 98). Reardaon mes are spread unformly n log( ) space and hey are auomacally calculaed as follows: m n 0 ( ),,.., 0, 0 m 0 ( ),,.. n (4.7) In he above m s number of reardaon mes per log( ) un, m. By defaul hs consan s n ATENA se o. If requred, a more dealed approxmaon s possble,.e. any value m can be used. In he program hs parameer s npu as a number of reardaon mes per me un n logarhmc scale. For a ypcal concree creep law a ceran opmal value can be deermned and s ndependen of a srucure beng analyzed. Noe however, ha he value depends on he choce of me uns. Example: If he reardaon mes parameer s se o, he creep law wll be approxmaed by wo approxmaon pons for he me nerval beween 0 - day, wo pons for he nerval - 0 days, hen wo pons for 0-00 days, ec. Therefore, he proper values wll depend on he choce of me uns. If he me un s a day, he recommended value s -. Sar me mus be chosen suffcenly low, so ha Drchle seres can accoun for processes n very young concree, rgh afer s loadng has been appled. As a defaul, ATENA uses he expresson 0. '. As for he upper lm for, s requred: n (4.73) The above lms are applcable for he case, when he coeffcens E (') of Drchle seres n (4.63) are calculaed by Leas-square mehod (Jendele and Phllps 99). ATENA also suppors alernave way of calculaon of he coeffcens E (') of Drchle seres n (4.63). In hs case, Inverse Laplace ransformaon (Bazan and X 995) s used nsead. Ths mehod requres 0, ypcally E-3 and (4.74) n Comparng he above wo approaches, can be sad ha Leas-square mehod yelds approxmaon of he complance funcon a dscree mes, whereby Inverse ransformaon s based on connuous approach. In some cases Leas-square mehod resuls n beer convergence behavour, however somemes suffers from numercal problems durng calculaon due o ll-

233 posed problem for soluon of E ('). I s lef o experence and engneerng judgmen o decde, whch of he mehod s more approprae for a parcular soluon. Inegraon mes or sample mes r are calculaed n smlar way. In hs case, he mes are unformly spread n log( ') me scale. They are generaed sarng from he s loadng me '. Hence, we can wre r l r 0, ' (4.75) where l s number of me ncremens per un of log( ') and ' days. Each new major load ncremen or decremen causes he generaon procedure (4.75) mus sar agan from small me ncremens. Ths parameer defnes he number of me seps, he program wll use o negrae he srucural behavor. Creep or oher nonlnear effecs wll cause a redsrbuon of sresses nsde he srucure. In order o properly capure such processes a suffcenly small me seps are needed. Is defnon depends on he ype of he analyzed srucure as well as on he choce of me uns. For ypcal renforced concree srucures and for he me un beng a day, s recommended o se hs parameer o. Ths wll mean ha for each load nerval longer hen day, wo sub-seps wll be added. For a load ha s nerval longer hen 0 days, 4 sub-seps wll be added. For an nerval longer han 00 days, wll be 6 sub-seps, ec. The creep and shrnkage analyss n ATENA requres ha he user se number of reardaon mes m and number of me ncremens l per un of log me, (unless he defaul values are OK). He/she also specfes me span,.e. and n. Then, reardaon mes are generaed,.e. an approprae command s ssued. I follows o se sop me of he analyss. Usual npu daa descrbng srucural shape, maeral ec. are gven hereafer, however, here are hree mporan dfferences from me-ndependen analyss:. Maeral model for concree conans daa for long erm as well as for shor erm maeral model.. Sep daa mus nclude nformaon abou me, a whch he sep s appled. 3. I s recommended o npu daa for all nended load me seps pror he seps are execued. I helps he generaon of negraon (nermedae) mes Inermedae me seps,.e. mes r as well reardaon mes are generaed auomacally. The analyss proceeds unl he sop me s reached. If no sop me s specfed, s assumed o be me of he las load sep. If he me span for reardaon mes does no covered sep load mes, he soluon s abored, gvng an approprae error message. 5.5 Creep and Shrnkage Consuve Model In he above secons, was slenly assumed ha long-erm par of he maeral model,.e. complance funcon (, ') and shrnkage funcon 0 r for concree, s known and was shown, how s ulzed whn creep and shrnkage analyss. I s prmary nenon of hs ATENA Theory 3

234 secon o descrbe wha long-erm creep and shrnkage predcon models are mplemened n ATENA and how hey should be used. Generally speakng, ATENA apples no resrcon on knd and shape of boh (, ') and 0 r, as adops SBS mehod soluon algorhm, n whch complance funcon s approxmaed by Drchle seres. Hence, mos wdely recognzed creep predcon models could be mplemened. The CCSrucureCreep module currenly suppors he followng models:. CCModelACI78 (ACI_Commee_09 978), recommended by ACI, by now already obsolee,. CCModelCEB_FIP78 (Beon 984), recommended by CEB commee, by now already obsolee, 3. CCModelB3 (Bazan and Baweja 999), developed by Bazan and Al Manaseer n 996, very effcen model recognzed world-wde, 4. CCModelB3Improved, same as he above, mproved o accoun for emperaure hsory, probably he bes model avalable n ATENA, 5. CCModelCSN730, model developed by CSN 730 Code of pracce n Czech Republc, 6. CCModelBP_DATA (Bazan and Panula 978; Bazan and Panula 978; Bazan and Panula 978; Bazan and Panula 978), relavely effcen and complex model; now s superceeded by CCModelBP_KX or CCModelB3, 7. CCModelBP_DATA (Bazan and Panula 978), smplfed verson of he above model, 8. CCModelBP_KX (Bazan and Km 99; Bazan and Km 99; Bazan and Km 99; Bazan and Km 99), powerful model wh accouns for humdy and emperaure hsory ec., for praccal use may-be oo advanced, 9. CCModelGeneral general model no whch expermenally obaned (, ') and 0 r funcon can be npu. 0. CCModelEN99- Eurocode model for creep, (EN99),. CCModelFIB_MC00- creep model based on CEB-FIP FIB Model Code 00. The followng daa summarzed npu parameers for he suppored models. Noe ha some models allow mproved predcon based on laboraory daa. If s he case, he model npu he correspondng expermenally measured values. Also, some model can accoun for maeral pon hsory of humdy h ( ) and emperaure T. ( ) Agan, a model suppors hs feaure, f can npu adequae daa. Table 5.5- : Ls of maeral parameers for creep and shrnkage predcon defnon and descrpon Parameer name Descrpon Uns Defaul 4

235 Concree. ype Type of concree accordng o ACI. Type s Porland cemen ec. Types,3 acceped for sac analyss, ypes -4 acceped for ranspor analyss. Cemen class Type of cemen, see e.g. hp:// : 4,5 Srengh classes of cemen Cemens are accordng o sandard srengh grouped no hree classes, hey beng: Class 3,5 Class 4,5 Class 5,5 Three classes of early srengh are defned for each class of sandard srengh: Class wh ordnary early srengh N Class wh hgh early srengh R Class wh low early srengh L Class L can be appled only on CEM III cemens. Aggregae Type of aggregae. One of BASALTDENSELIMESTONE, QUARTZITE, LIMESTONE, SANDSTONE, LIGHTWEIGHTSANDSTONE Thckness V / S Cross secon hckness defned as raon of secon s volume o surface Srengh f Maeral cylndrcal srengh n compresson cyl 8 a me 8 days Srengh f Srengh a onse of nonlnear behavour n cyl 0,8 compresson a me 8 days QUART ZITE lengh m sress sress 35.MPa Consan from he base maeral Fracure energy G f,8 Fracure energy a me 8 days sress Consan from he base maeral ATENA Theory 5

236 Srengh f 8 Maeral ensle srengh a me 8 days sress Consan from he base maeral Young m. E Shor-erm maeral Young modulus a 8 8 days,.e. nverse complance a 8.0 days loaded a 8 days sress F( f cyl 8) Amben humd. h Amben relave humdy. Acceped range (0.4..) Rao a Toal aggregae/cemen wegh rao c Rao w Waer/cemen wegh rao c Rao a s Toal aggregae/fnd sand wegh rao. a s. s a.8 Rao s a Fne/oal aggregae wegh rao. s a 0.4 a s Rao g Coarse gravel/fne aggregae wegh rao..3 s Rao s Fne aggregae/cemen wegh rao..8 c Shape facor Cross secon shape facor. I should be,.5,.5,.3,.55 for slab, cylnder, square prsm, sphere, cube, respecvely..5 Slump Resul of maeral slump es. lengh 0.0m Ar conen Maeral volumerc ar conen. % 5 Cemen mass Wegh of cemen per volume of concree mass/ Concr. densy Curng ype Thermal expanson coeffcen T Maeral densy used o evaluae srengh and Young modulus a 8 days.. Curng condons. I can be eher n waer (.e. WATER) or ar under normal emperaure (.e. WATER) or seamed curng (.e. STEAM). Thermal expanson coeffcen T lengh 3 mass/ lengh 3 /emp eraure 30kg/ m 3 5kg/ m 3 AIR Consan from he base maeral 6

237 End of curng a, Tme a begnnng of dryng,.e. end of curng. Auogenous shrnkage a nfny me, (ypcally negave!) s a a, (0.99 mn(0.99, ha, ) anh a days 7-0 I/D Improvem. Half-me of auogenous shrnkage. days 30 a Tme of fnal se of cemen days 5 s ha, Fnal self-desccaon relabe humdy Curren me Curren me days 0 Load me Load me days 0 To.waer loss w Waer loss w() Shrnk. Compl. Toal waer loss (up o zero humdy and nfne me). I s measured n an oven n a laboraory and s used o enhance predcon of shrnkage nfne sh (Bazan and Baweja 999). Ths value s n urn used o elaborae dryng creep and shrnkage predcon of he model. If s no specfed, he model predcon enhancemen s no acvaed. I can be used, f waer loss w() are npu as well. Waer losses a me ; measured a a laboraory. I s used o enhance dryng creep and shrnkage predcon. See also descrpon of oal waer loss w. 0 () Measured shrnkage a me. I s used o enhance dryng creep and shrnkage predcon. See also descrpon of oal waer loss w. (, ') Measured maeral complance a me. I s used o mprove overall creep and shrnkage predcon of he model. kg kg /sress N/A N/A N/A N/A ATENA Theory 7

238 Hs. Humdy h ( ) Hsory of humdy n a maeral pon. Value a me. Some maeral models can use hese values o accoun for real emporal humdy and emperaure condons. Alhough he daa can be npu manually,.e. o group maeral pons wh smlar humdy and emperaure hsory no a group and dedcae a dsnc maeral for ha group, s prepared for full auomac processng beng currenly n developmen. I wll auomacally lnk hea and humdy ranspor analyss wh he sac analyss usng one of avalable creep and shrnkage predcon model. Applcable range (0.4..). N/A Tempera. T ( ) Hsory of emperaure n a maeral pon. See also descrpon of h ( ) Celsa Drec Compl. (, ') Measured complance a me loaded a me. Ths and he nex wo parameers should be used, f known (measured) complance funcons are o be employed n ATENA creep and shrnkage analyss. Hence, no predcon s done and he gven daa are only used o calculae he parameers of Drchle seres approxmaon. / sress Shrnk. 0 () Measured shrnkage a me. See he parameer above. N/A Srengh ( ) cyl f Measured shrnkage a me. See he parameer above 8

239 Table 5.5-: Inpu parameers needed by ndvdual creep and shrnkage predcon models Model name B3 B3- mpr BP- KX CEB ACI CSN BP BP Gen eral EN 99 Model No Concree. Type x x x x x x x Cemen class x x Aggregae x x MC 00 Thckness S / V x x x x x x x x x x Srengh f x x x x x x x x x x cyl 8 Srengh f cyl0,8 Fracure energy G f,8 x x x x x x Srengh f x x x 8 Young m. E x x x x x x x 8 Amben humd. h x x x x x x x x x x Rao a x x x x x x c Rao Rao a s w x x x x x x c Rao s x x a Rao g x x s Rao s x x c Shape facor x x x x x Slump Ar conen Cemen mass Concr. densy x x x x x x x x x ATENA Theory 9

240 Curng ype x x x x x x End of curng x x x x x x x x x x Thermal expanson coeffcen T a, a s ha, Curren me x x x x x x x x x x x x x x x x xx x Load me x x x x x x x x x xx x To.waer loss w Waer loss w() Shrnk. 0 () Compl. (, ') Humdy h () Tempera. T () Compl. (, ') Shrnk. 0 () Srengh () fcyl x x x x x x x x x x x x x xx x x x x x x x xx x x x x xx x x x x The above parameer Concree ype acually referes o a cemen ype accordng o he ACI classfcaon. I used n he creep analyss. The followng able brngs descrpon of wdely recognzed cemen ypes. Noe ha only ypes,3 are suppored n Aena sac analyss. The 30

241 ranspor analyss n Aena recognzes ypes -4. The remanng ypes are descrbed jus for nformaon. Table 5.5-3: Cemen ypes accordng o ACI classfcaon ATENA Concree ype Cemen ype Descrpon I and Type IA 5 General purpose cemens suable for all uses where he specal properes of oher ypes are no requred. II and Type IIA 5 alumnae (C 3 A) for moderae sulfae ressance. Some Type II cemens mee he moderae hea of hydraon Type II cemens conan no more han 8% rcalcum opon of ASTM C III and Type IIIA 5 excep hey are ground fner o produce hgher early Chemcally and physcally smlar o Type I cemens srenghs. 4 IV Used n massve concree srucures where he rae and amoun of hea generaed from hydraon mus be mnmzed. I develops srengh slower han oher cemen ypes. 5 V Conans no more han 5% C 3 A for hgh sulfae ressance. 6 IS (X) 6 Porland blas furnace slag cemen 7 IP (X) 6 Porland-pozzolan cemen. 8 GU 7 General use 9 HE 7 Hgh early srengh 0 MS 7 Moderae sulfae ressance HS 7 Hgh sulfae ressance MH 7 Moderae hea of hydraon 5 Ar-enranng cemens 6 Blended hydraulc cemens produced by nmaely and unformly nergrndng or blendng wo or more ypes of fne maerals. The prmary maerals are porland cemen, ground granulaed blas furnace slag, fly ash, slca fume, calcned clay, oher pozzolans, hydraed lme, and pre-blended combnaons of hese maerals. The leer X sands for he percenage of supplemenary cemenous maeral ncluded n he blended cemen. Type IS(X), can nclude up o 95% ground granulaed blas-furnace slag. Type IP(X) can nclude up o 40% pozzolans. 7 All porland and blended cemens are hydraulc cemens. "Hydraulc cemen" s merely a broader erm. ASTM C 57, Performance Specfcaon for Hydraulc Cemens, s a performance specfcaon ha ncludes porland cemen, modfed porland cemen, and blended cemens. ASTM C 57 recognzes sx ypes of hydraulc cemens. ATENA Theory 3

242 3 LH 7 low hea of hydraon 5.6 References ACI_COMMITTEE_09 (978). Predcon of Creep, Shrnkage and Temperaure Effecs n Concree Srucures. Dero, nd draf, ACI. BATHE, K. J. (98). Fne Elemen Procedures n Engneerng Analyss. Englewood Clffs, New Jersey 0763, Prence Hall, Inc. BAZANT, Z. AND T. SPENCER (973). "Drchle Seres Creep Funcon for Agng Concree." ASCE Journal of Engneerng and Mechancal Dvson: BAZANT, Z. P. (988). Mahemacal Modelng of Creep and Shrnkage of Concree. New York, John Wley & Sons. BAZANT, Z. P. AND S. BAWEJA, EDS. (999). Creep and Shrnkage Predcon Model for Analyss and desgn of Concree Srucures: Model B3. Creep and Shrnkage of Concree, ACI Specal Publcano. BAZANT, Z. P. AND J. K. KIM (99). "Improved Predcon Model for Tme-Dependen Deformaon of Concree: Par - Shrnkage." Maerals and Srucures 4: BAZANT, Z. P. AND J. K. KIM (99). "Improved Predcon Model for Tme-Dependen Deformaon of Concree: Par - Basc Creep." Maerals and Srucures 4: BAZANT, Z. P. AND J. K. Km (99). "Improved Predcon Model for Tme-Dependen Deformaon of Concree: Par 3- Creep a Dryng." Maeral and Srucures 5: -8. BAZANT, Z. P. AND J. K. KIM (99). "Improved Predcon Model for Tme-Dependen Deformaon of Concree: Par 4- Temperaure Effecs." Maeral and Srucures 5: BAZANT, Z. P. AND L. PANULA (978). "Praccal Predcon of Tme-dependen Deformaons of Concree; Par : Shrnkage." Maeral and Srucures (65): BAZANT, Z. P. AND L. PANULA (978). "Praccal Predcon of Tme-dependen Deformaons of Concree; Par 3: Dryng Creep." Maeral and Srucures (65): BAZANT, Z. P. AND L. PANULA (978). "Praccal Predcon of Tme-dependen Deformaons of Concree; Par 4: Temperaure Effec on Basc Creep." Maeral and Srucures (66): BAZANT, Z. P. AND L. PANULA (978). "Praccal Predcon of Tme-dependen Deformaons of Tme-dependen Deformaon of Concree; Par : Basc Creep." Maeral and Srucures (65): BAZANT, Z. P. AND L. PANULA (978). Smplfed Predcon of Concree Creep and Shrnkage from Srengh and Mx. Sruc. Engng. Repor No. 78-0/6405. Evanson, Illnons, Norhwesern Unversy, Dep. of Cv. Engng. BAZANT, Z. P. AND F. H. WHITTMAN (98). Creep and shrnkage n Concree Srucures. New York, John Wley & Sons. BAZANT, Z. P. AND Y. XI (995). "Connous Reardaon Specrum for Soldfcaon Theory of Concree Creep." Journal of Engneerng Mechancs ():

243 BETON, C. E.-I. D. (984). CEB Desgn Manual on Srucural Effecs on Tme Dependen Behavour of Concree. San Saphorn, Swzerland, Georg Publshng Company. HAGEMAN, L. AND D. YOUNG (98). Appled Ierave Mehods. New York, Academc Press. JENDELE, L. AND D. V. PHILLIPS (99). "Fne Elemen Sofware for Creep and Shrnkage n Concree." Compuer and Srucures 45 (): 3-6. REKTORYS, K. (995). Přehled užé maemaky. Prague, Promeheus. SEAGER, M. K. AND A. GREENBAUM (988). A SLAP for he Masses, Lawrence Lvermore Naonal Laboraory. ATENA Theory 33

244

245 6 DURABILITY ANALYSIS 8 The durably analyss n ATENA can currenly assess deeroraon of srucures due o carbonaon and chlordes ngress. I s avalable for sac and creep analyses. A each me sep, an approprae D ranspor analyss s carred ou o nvesgae, how far he polluon, (.e. carbonaon and/or chlordes) penerae from loaded surfaces nsde he srucure. Man resuls of he analyses are nducon mes,.e. mes a whch he polluon concenraon reaches crcal values ha are already for he srucure unaccepable, (e.g. he renforcemen corroson begns ec.). They are always gven wh respec o me 0 0. In addon, polluon concenraon a mes (correspondng o he ndvdual seps) are also compued. Noe ha sac analyss n ATENA ypcally doesn' care abou me, (or more precsely each analyss sep ncremens he srucural age by un me). A each sep yelds sor of arfcal age of he srucure. Hence, f he durably analyss s carred ou, hs arfcal age mus be somehow mapped ono real srucural age. I s done n ATENA wh help of a mullnear funcon. Such a funcon corresponds o loadng funcons used o defne varable BCs and s npu n exacly he same way. The followng ex descrbes heory behnd he D ranspor analyss of he carbonaon and chlordes polluon and, a he end, some nformaon regardng he ranspor parameers are gven. Servce lfe of a srucure l has usually he form of l c p r (4.76) where c s he consrucon phase, naon (nducon) perod, p propagaon perod and r pos-repar perod. We am a predcng he naon perod, whou gong no propagaon or pos-repar phases. Carbonaon and chlorde ngress are wo leadng mechansms conrbung o renforcemen corroson. Boh of hem are descrbed furher. The naon phase ends wh he begnnng of renforcemen corroson. Fg. 6- brngs more dealed descrpon of naon and propagaon phases and her relaonshp o concree evens. Predcon of naon perod represens a prevenve measure whch s affeced above all by concree cover hckness, concree composon, and envronmen. I makes sense o change desgn n he begnnng raher han mgang renforcemen corroson laer. Acceleraon of carbonaon and chlorde ngress on crack appearance s aken no accoun. 8 No avalable n ATENA verson 5. and older. Developmen/esng mplemenaon of CARBONATION, CHLORIDES, and ASR n verson 5.3. ATENA Theory 35

246 6. Carbonaon Fg. 6- Imporan evens n servce lfe (III 000). Carbonaon deph of a sound (uncracked) concree reads (Papadaks and Tsmas 00) D CO eco, xc A 0.8( C kp) (4.77) where x c s he carbonaon deph, D e,co s he effecve dffusvy for CO, C s he Porland cemen conen n kgm -3, k<0.3,.0> s he effcency facor of supplemenary cemenous maeral (SCM-slag, slca, fly ash), P s he amoun of SCM n kgm -3, CO s he volume fracon of CO n he amosphere aken as 3.6e-4 and s he me of exposure. The effecve dffusvy n m s - s gven by he emprcal equaon (Papadaks and Tsmas 00) D eco, ( W 0.67 CkP) /000) 6.0 ( RH) C kp W c (4.78) where W s he waer conen n concree n kgm -3, c s he cemen densy n kgm -3 assumed as 350 kgm -3 and RH s he relave humdy of amben ar. Eqs. (4.77)(4.78) allow predcng eher carbonaon deph or nducon me of uncracked concree. Relave humdy mus be hgher han 0.50 for carbonaon o proceed. Cracked concree leads o faser carbonaon. Ths acceleraon s gven n he form (Kwon and Na 0) x ( ) (.86 w ) A (4.79) c where w s he crack wdh n mm, A s he carbonaon velocy accordng o Eq.(4.77). Eq. (4.79) allows compung carbonaon deph or nducon me. Noe ha crack 0.3 mm ncreases carbonaon deph by a facor of.54. Ths also means ha nducon me s 6.46 mes shorer compared o a sound concree. 36

247 In realy, crack may grow durng any servce me. Thus, Eq. (4.79) needs o be recas o ncremenal form. An ncremen of carbonaon deph n a gven me sep s evaluaed from he oal dervave by dfferenang Eq. (4.79).86 w A.86A xc w (4.80) w where w + s he crack wdh a he end of he me sep, +0.5 s he md-me. I s assumed ha nonzero w a a frozen me has no effec on carbonaon deph, hus he erm w can be lef ou. Eq. (4.80) allows predcng eher carbonaon deph or nducon me of gradually crackng concree. 6.. Example of Carbonaon Le us consder frs a regular concree made from ordnary Porland cemen, w/b=0.45, C=400 kgm -3, W=0.5 kgm -3, P=50 kgm -3. The supplemenary cemenous maeral s fly ash wh almos zero calcum conen hence k=0.5. Concree s exposed o relave humdy Consder concree cover of 30 mm. A crack s always nroduced n he begnnng of exposure. The second concree s made from ordnary Porland cemen, w/b=0.45, C=00 kgm -3, W=90 kgm -3, P=0 kgm -3. Table 6.- compares boh concrees n erms of nducon me. Crack (mm) wdh Inducon me for concree w/b=0.45, C=400 kgm -3, P=50 kgm -3 (years) Inducon me for concree w/b=0.45, C=00 kgm -3, P=0 kgm -3 (years) Table 6.-. Inducon me for carbonaon, wo concrees, cover hckness 30 mm. 6. Chlordes Implemened model for chlorde ngress s based on (Kwon, Na e al. 009). Le us consder D ransen problem of chlorde ngress n concree wh nally free chlorde conen x Cx, C S erf Dm f( w) (4.8) where C S s he chlorde conen a surface n kgm -3, D m s he averaged dffuson coeffcen a me n mm s -, x s he poson from he surface n mm and f(w) gves acceleraon by crackng and equals o one for a crack-free concree. C s and C can be relaed o concree volume or o bnder volume, however, he uns mus be kep conssenly hrough he compuaon. Dffuson coeffcen D() s assumed o decrease over me accordng o he power law ATENA Theory 37

248 D ref Dref m (4.8) where m s a decay rae (somemes called an age facor). If m=0, a consan value of D()=D ref s recovered. Ths model was proposed by (Collepard, Marcals e al. 97). Nowadays became clear ha hs assumpon s oo conservave and s no generally recommended. The mean dffuson coeffcen D m s obaned by averagng D() over me of neres m m ref Dref ref m() ref, R m 0 D D d (4.83) R m ref D () D, m R m ref R m (4.84) where R s me when dffuson coeffcen s assumed o be consan and s generally aken as 30 years. ref corresponds o me when he dffuson coeffcen was measured. Fg. 6- shows characersc evoluon of dffuson coeffcens over me. The mean dffuson coeffcen ncreases when cracks are presen n he concree. Based on recen resuls, he followng scalng funcon s proposed (Kwon, Na e al. 009) f( w) 3.6w 4.73w (4.85) where w sands for crack wdh n mm. The crack wdh 0.3 mm ncreases mean dffuson coeffcen by a facor of 5.6. In realy, crack wdh evolves and ncremenal soluon needs o be formulaed. The mean coeffcen D m,w () ncorporang crack wdh s evaluaed from a crack ncremen D mw, () ( ) ( ) D w dw D w w w w n w w m( ) m { ( ) ( )} 0 0 If las values of f(w) and w are sored, Eq. (4.86) can be evaluaed only n he acual me sep. Ths speeds up he soluon. (4.86) 38

249 Fg. 6-. Evoluon of acual and mean dffuson coeffcens for sandard concree, based on daa from (Kwon, Na e al. 009). 6.3 Dffuson coeffcen for chlordes Proper deermnaon of dffuson coeffcen s no a rval subjec, consderng varous concrees, cemens, models and exposure condons. (Papadaks 000) presened a model for esmang nrnsc effecve dffusvy for concrees made from blended cemens, however, recalculaon o D a s no sraghforward. DuraCree model (III 000) provdes usable daa for esmang apparen dffuson coeffcen n he form 0 D k k D ( ) a e c Cl 0 Da m (4.87) where k e <0.7,3.88> s he envronmen facor, k c <0.79,.08> s he curng facor, D cl ( 0 ) s he measured dffuson coeffcen deermned a me 0, m<0.,0.93> s he age facor and Da <.5,3.5> s he paral facor. In our noaon, D a ()=D m () and 0 = ref. To our opnon, he mos relevan and well documened feld daa come from 0 years exposure ess (Lupng, Tang e al. 007). Fg. 6-3 shows he apparen dffuson coeffcen n dependence of waer-bnder rao. In hs parcular case, ref =0 years, m s uknown, D ref =(-m)d a, R can be assumed as 30 years. ATENA Theory 39

250 Fg Fed apparen dffuson coeffcens from 0-years exposure of concree (Lupng, Tang e al. 007). The nex fgure shows apparen dffusvy coeffcen a 0 years from Fg They can be used as sarng pon for esmang D ref. 40

251 Fg Apparen dffuson coeffcens from 0-years exposure of concree (Lupng, Tang e al. 007). 3.. Example of chlorde ngress Le us consder regular concree made from ordnary Porland cemen, w/b=0.45. Accordng o Fg. 6-3, D a s abou e- m s - a ref =0 years. Accordng o Duracree model, he age facor for concree submerged n sal waer corresponds o m=0.30 (Table 8.6 n DuraCree). In such case, D ref =(-m)d a =.4e- m s -. Fg. 6-5 shows evoluon of dffuson coeffcens for hs parcular case. ATENA Theory 4

252 Fg Evoluon of dffuson coeffcens for chlordes n an example. Le us assume characersc value C s 0.3% of chlordes per bnder for submerged concree whou furher reducons (Table 8.5 n DuraCree). The crcal level for corroson s.85 % per bnder (Table 8.7 n DuraCree). The concree cover s aken as 00 mm. Compued nducon me accordng o Eq. (4.8) s summarzed n Table Crack wdh s consdered snce he begnnng of exposure. Table 6.3- Inducon me for chlorde corroson of submerged concree, n dependence on orgnal crack wdh. Crack (mm) wdh MODELS for PROPAGATION PHASE 6.4. Carbonaon durng propagaon phase Inducon me (years) The corroson rae for he carbonaon depends on he corroson curren densy corr [µa/cm ], whch ranges beween 0.-0 (passve corroson-hgh corroson) and depends on he qualy and he relave humdy of he concree (Page CL, 99). Ths model predcs amoun of corroded seel durng he whole propagaon perod p. The corroson rae s based on Faraday s law (Rodrguez, 996), deermned as follows: corr corr x ( ) 0.06 (4.88) 4

253 where x corr s he average corroson rae n he radal drecon [m/year], corr s corroson curren densy [µa/cm ] and s calculaed me afer he end of nducon perod [years]. By negraon of Eq. (), s obaned he corroded deph for D propagaon: x ( ) 0.06 R d (4.89) corr corr corr n where x corr s he oal amoun of corroded seel n radal drecon [mm] and R corr s parameer, depends on he ype of corroson [-]. For unform corroson (carbonaon) R corr =, for png corroson (chlordes) R corr = <; 4> accordng o (Gonzales a.al., 995) or R corr = <4; 5.5> accordng o ( Darmawan &, 007). Effecve bar dameer for boh ypes of corroson s obaned from: ( ) d d x (4.90) n where d() s evoluon of bar dameer n me, d n s nal bar dameer [mm], ψ s uncerany facor of he model [-], mean value ψ = and x corr s he oal amoun of corroded seel accordng o () Chlorde ngress durng propagaon phase corr The corroson rae for chlordes s more complcaed because s affeced by concenraon of chlordes n he concree. Calculaon of corroson curren densy was formulaed by Lu and Weyer s model (Lu, Weyers, 998): corr 0.96*exp ln.69C RC.4 T (4.9) where corr s corroson curren densy [µa/cm ], C s oal chlorde conen [kg/m 3 of concree] on renforcemen whch s deermned from D nonsaonary ranspor, T s emperaure a he deph of renforcemen [K] and R c s ohmc ressance of he cover concree [Ω] (Lu, 996) and s me afer naon [years]: C C R exp ln. 69 (4.9) The average corroson rae n radal drecon s deermned furher when pluggng(4.93),(4.94) o (). The oal amoun of corroded seel n radal drecon sems from () and he effecve bar dameer from (3) Crackng of concree cover Crackng of concree cover for boh carbonaon and chlordes can be esmaed from DuraCree model whch provdes realsc resuls (DuraCree, 000). The crcal peneraon deph of corroded seel x corr,cr s formulaed as: C (4.95) x corr, cr a a a3 f, ch dn ATENA Theory 43

254 where parameer a s equal 7.44e-5 [m], parameer a s equal 7.30e-6 [m], a 3 s [-.74e-5 m/mpa], C s cover hckness of concree [m], d n nal bar dameer [m], f,ch s characersc splng ensle srengh of concree [MPa] Spallng of concree cover The crcal peneraon deph of corroded seel x corr,sp for boh carbonaon and chlordes s calculaed from (DueaCree, 000) as: x w w (4.96) b d 0 corr, sp xcorr, cr where parameer b depends on he poson of he bar (for op renforcemen 8.6 µm/µm and boom 0.4 µm/ µm), w d s crcal crack wdh for spallng (characersc value mm), w 0 s wdh of nal crack (known from prevous ATENA compuaon) and x corr,cr deph of corroded seel a he me of crackng [m]. Afer spallng of concree cover, corroson of renforcemen akes place n drec conac wh he envronmen. To deermne he rae of corroson of renforcemen afer spallng, Error! Reference source no found. gves raes of renforcemen corroson (Spec-ne, 05). Table : Corroson raes of seel under amospherc exposon Corrosvy zone (ISO 93) Caegory Descrpon Typcal envronmen Corroson rae for frs year (µm/yr) Mld seel Znc C Very low Dry ndoors,3 0, C Low Ard/Urban nland >,3 a 5 >0, a 0,7 C3 Medum Coasal and ndusral >5 a 50 >0,7 a, C4 Hgh Calm sea-shore >50 a 80 >, a 4, C5 Very Hgh Surf sea-shore >80 a 00 >4, a 8,4 CX Exreme Ocean/Off-shore >00 a 700 >8,4 a Alkal-Aggregae Reacon 6.5. Inroducon of alkal-aggregaea model for concree 44

255 In mos concree, aggregaes are more or less chemcally ner. However, some aggregaes reac wh he alkal hydroxdes n concree, causng expanson and crackng over a perod of many years. Ths alkal-aggregae reacon has wo forms: alkal-slca reacon (ASR) and alkalcarbonae reacon (ACR). Alkal slca reacon (ASR), one of hose common deleerous mechansms, consss n a chemcal reacon beween unsable slca mneral forms whn he aggregae maerals and he alkal hydroxdes (Na, K OH) dssolved n he concree pore soluon. I generaes a secondary alkal slca gel ha nduces expansve pressures whn he reacng aggregae maeral(s) and he adjacen cemen pase upon mosure upake from s surroundng envronmen, hus causng mcro crackng, loss of maeral's negry (mechancal/durably) and, n some cases, funconaly n he affeced srucure. Several aggregae ypes n common use, parcularly hose wh a slceous composon, may be aacked by he alkalne pore flud n concree. Ths aack, essenally a dssoluon reacon, requres a ceran level of mosure and alkals (leadng o hgh ph) whn he concree o ake place. Durng he reacon, a hygroscopc gel s produced. When mbbng waer, he gel wll swell and hus cause expanson, crackng, and n wors case dsrupon of he concree (Lndgar 0). Thus, he degree of reacon of an aggregae s a funcon of he alkalny of he pore soluon. For a gven aggregae, a crcal lower ph-value exss below whch he aggregae wll no reac. Consequenly, ASR wll be prevened by lowerng ph of he pore soluon beneah hs crcal level where he dssoluon of alkal-reacve consuens (slca) n he aggregaes wll be srongly reduced or even prevened, as dscussed n (Rodrguez a.al, 996). No absolue lm s defned, because he crcal alkal conen largely depends on he aggregae reacvy [3], bu from many expermenal es we can esmae hreshold value (Lndgar 0), (Poye, 003). Many sudes carred ou over he pas few decades have shown ha ASR can affec he mechancal properes of concree as a maeral. Usually, ASR generaes a sgnfcan reducon n ensle srengh and modulus of elascy of concree. These wo properes are much more affeced han compressve srengh, whch begns o decrease sgnfcanly only a hgh levels of expanson. Several ASR models were developed over he years o predc expanson and damage on boh ASR affeced maerals (mcroscopc models) (Mulon a.al., 009), (Bazan, Seffens, 009), (Comby-Pero, 009) and ASR affeced srucures/srucural elemens (macroscopc models) (Ulm a.al., 999), (Saouma, Pero,006), (Com, Fedele, Perego, 009). The frs group has a goal of modelng boh he chemcal reacons and he mechancal dsresses caused by ASR or even he couplng of he wo phenomena. The second group ams a undersandng he overall dsress of srucures/srucural concree elemens n a real conex, smulang her lkely n su behavor (Farage e al.,000) seem o have fnally brdged he gap beween scenfc rgor and praccal applcably o real srucures. In erms of mechancal effecs, s known ha ASR expansons occur over long me perods. Durng hs process, ASR affeced concrees are subjeced o a progressve sress bul up ha s very lkely o cause creep on he dsressed maerals. ATENA Theory 45

256 AAR depends on he avalably of hree facors: alkals lberaed from cemen durng hydraon, slceous mnerals presen n ceran knds of aggregaes and waer. Several mcroscopc and random facors are nvolved n AAR expanson, such as concree porosy, amoun and locaon of reacve regons n he maeral and permeably (Farage e al.,000). These parameers, added o concree s nrnsc heerogeney, urn smulang AAR expanson no a raher complex ask. Even hough AAR process has no been well explaned so far, he commonly acceped heory for descrbng s wo dsnc phases need o be consdered: gel formaon and waer absorpon by he gel, causng expanson. Accordng o hs mechansm, reacon does no always lead o expanson. As long as here s enough vod space o be flled by he gel,.e. pores and cracks, concree volume remans unchanged. Due o he lack of model whch s able o ncorporae effecs of relave humdy, alkal/slca conen n he mxure, amben emperaure, auhors sugges o combne ASR knecs proposed by (Ulm e. al., 999) wh nfluence of mosure, publshed by (Léger e al., 996) and nfluence of alkal/slca conen proposed by Mulon e al. Implemenaon of modelng expanson due o ASR consss of modelng engensans n meseps on enre srucure. Funcon for volumerc egensran reads ASR cal F M (4.97) where cal s volumerc sran of ASR swellng a nfny me, 0, s he chemcal exen of ASR and F M s he coeffcen reflecng mosure nfluence. I s descrbed laer n he ex. In he case of varyng he relave humdy, Equaon (4.97) changes o ncremenal form, for me 6.5. Model for ASR knecs ASR ASR cal FM (4.98) For he complee 3D consuve model we consder frs-order reacon where c d 0 c ξ,ξ ξ (4.99),ξ k / A s he characersc me. I has been found ha c depends on emperaure [ K] and he ASR exen ξ. Referng o (4.99) he mplemenaon of he chemoelasc maeral law n he consuve laws s relavely sraghforward and asuable negraon scheme s gven n (Ulm ea., 999). Consder an sohermal sress-free ASR expanson es carred ou a consan emperaure 0. In hs es, he volumerc sran ASR s recorded as a funcon of me ha and ASR exen s calculaed as ξ ASR ASR (4.00) 46

257 For macroscopcally sress-free sample, (4.99) n (4.00) yelds ASR ASR ( ξ) ASR c,ξ ASR ξ c,ξ,ξ ASR ASR ASR ASR ASR c ASR (4.0) Wh ASR( ) and ASR() beng measurable funcons of me, he characersc me c can be deermned from a sress-free expanson es. In a recen exensve seres of sress-free expanson ess carred ou a dfferen consan emperaures( Larve, 998), c has been found o depend on boh emperaure [ K] and reacon exen ξ [-] n he form (4.0) c c, exp L / C, ξ exp L / C In hs expermenally deermned knecs funcon, L c (4.03) s a characersc me [day] and s a laency me [day] The use of (4.03),(4.0) n (4.0) yelds afer negraon ξ exp / C (4.04) exp / / C L C For varable emperaure, crackng ec. s dffcul o solve for ξ analycally and a numercal negraon s needed. Suable soluon scheme s derved n (Ulm a.al., 006) whch s mplemened n n ATENA. Fg. 6-6 shows he shape of (4.00), ogeher wh he me consans, c and L, whch sand for he characersc me and he laency me of ASR swellng, respecvely. Furhermore, proceedng as n physcal chemsry (Akns, 994), we explore he emperaure dependence of he me consans c and L from sress-free expanson ess carred ou a dfferen consan emperaures. The plos of ln( c ) and ln( L) agans / are gven n Fg I s remarkable ha he expermenal values algn (almos) perfecly along a sragh lne, machng he Arrhenus concep. CC0exp U c 0 (4.05) where exp U C L L 0 L L (4.06) 0 U K; U K (4.07) ATENA Theory 47

258 I s explored (Akns, 994) ha he emperaure dependence of he me consans c L carred ou a dfferen consan emperaures (3, 33, 38 and 58 C), see Fg. 6-6.Defaul values are days and L(3.5C) 45 days [0], see Fg. 6-8., Fg Accordng o Larve s expermenal daa from waer sauraed ess [4] days and days, days and days. Under dryng condons, he values for L roughly ncrease by a facor of 4; and C by.5 (Larve, 998), (Ulm a.al, 999) and Fg Larve s es daa of emperaure dependency of ASR me consans and. Slope of rendlnes represens acvaon energy consans Uc = 5,400 K and UL = 9,700 K. Fg Defnon of Laency Tme and Characersc Tme n Normalzed Isohermal Expanson Curve = 48

259 Fg Parameer Analyss of Characersc Tme ( = 3 K) of ASR Swellng wh Regard o Hydral Amben Condons, reproduced from 0. Fg Parameer Analyss of Laency Tme ( = 3 K) of ASR Swellng wh Regard o Hydral Amben Condons, reproduced from Predcon of ASR swellng cal cal [-] s he predced volumerc expanson a nfny me obaned by model proposed by (Mulon e al., 008). I s calculaed based on reacve aggregaes, amoun of reacve slca n he aggregaes and value of measured sress-free expanson es done n Poye s sudy (Lndgar, 0) on samples conanng reacve parcles only. cal s defned as follows ATENA Theory 49

260 cal A C s pac F (4.08) AR where F [-] s measured ASR sran expanson on samples conanng reacve parcles only wh enough suffcency of alkal. See Table 3 for more deals. [kg/m 3 Na O eq ] and [kg/m 3 Na O eq ] are amouns of consumed and requred alkal respecvely. AC s oal aggregae conen n [kg/m 3 ]. One of he man assumpons of he model s ha he maxmum expanson of morar s acheved f here s enough alkal o reac wh all he reacve slca of he mxure. Ths amoun of requred alkal conen [kg/m 3 Na O eq ] s defned as AR rsp AC (4.09) where s s he proporon of quany of soluble slca [-], p s he proporon of reacve aggregae [-]. r saes for he amoun of requred alkal per kg of reacve slca and s a consan value r = 5.4 %. Value s defned as mn(, ). s he avalable amoun of alkal for ASR reacon. s defned as dfference beween nal amoun of avalable alkal [kg/m 3 Na O eq ] and alkal conen hreshold [kg/m 3 Na O eq ] when ASR reacon sars. A A A (4.0) A T I should be noed ha hs model does no consder any alkal flow hrough boundares nsde he srucure durng he servce lfe. By defaul, s equal o 3.7 kg/m 3 Na O eq (Poye, 003), bu oher values n range of 3 5 kg/m 3 Na O eq can be found n he leraure (Lndgar, 0) 0 Table 3: Mxures and ASR expansons of morars suded by (Poye, 003) and (Mulon, 008). F-F3 are sze fracons 80 μm-3.5 mm. Value of p depends on he mx rao of reacve aggregae. Value s depends on amoun of reacve slca n aggregaes, moreover common values are: p =,% (Mulon, 008) 0or 9,4% and,4% (Mulon, 009) Influence of mosure F M Approxmaely 75% relave humdy (RH) whn concree s necessary o nae sgnfcan expanson, whch s assumed o vary lnearly beween 75% RH and 00% RH as shown n Fg

261 Fg Parameer Facor of RH nfluencng ASR concree expanson, reproduced from (Mulon, Toulemonde, 00). The coeffcen reflecs nfluence of mosure h. The funcon for FM s approxmaed as F M ( h) h mn ( h h ) mn (4.) where hmn s relave humdy hreshold where ASR begns o appear, 0.75 by defaul. Oher varables wll be explaned n furher ex ASR for 3D condons Expanson of free concree specmens due o ASR has been summarzed n (Červenka, Jendele, Šmlauer, 06). The laer predcs ASR under unresraned condons,.e. under free expanson. The expanson model akes no accoun reacon knecs, alkal conen, reacve amoun of aggregaes, relave humdy and emperaure. The model has been valdaed on 4 examples found n leraure. Degradaon of maeral due o ASR reacon, (Saouma, 06, eqs. 8,9), --, E E (4.) 0, --, f f,0 f E, --, f f,0 G (4.3) G G (4.4) ATENA Theory 5

262 where β E,f,G are resdual values of E/E 0, f/f 0, G f /G f0. Defaul values are E 0., 0.6 (Esposo, Hendrks, 0) and 0.6 s esmaed. f G The general equaon for he ncremenal volumerc AAR sran s gven by, (Saouma, 06, (5)) V I II III c, f c FM h cal c, f c FM h cal,, f c, F M, (4.5) Only consdered n mplemenaon where c reflecs effec of compressve sresses (Saouma, 06, eq. 0), accouns for he nfluence of ensle crackng, ( assumed here as ), F M s he effec of relave humdy whch s already accouned for n (4.) and equals o one. (4.5) consders furher only he mos relevan frs erm and s rewren n ncremenal form as, f F h V c c M cal ( ( ))/ (4.6) Reducon c due o compressve sress s consdered as follows: f 0 c e f <0 e I II III 3 f c Tenson Compreson (4.7) ' where he shape facor s by defaul (Saouma, 06, Tab.) and f c s compressve srengh. Under consraned condons, ASR expanson develops dependng on he sress sae. I s known ha compressve sress beyond approxmaely -0 MPa sops ASR expansons, whch needs o be refleced for sran redsrbuon no prncpal drecon. Smlarly o (Saouma, 06, Fg. 5), wegh facors are assgned o hree drecons. Le us assume ha drecons of prncpal sresses σ I, σ II, σ III are known. Expanson s hen assgned o each prncpal sress drecon accordng o a wegh facors W, W, W 3. When compressve sress reaches -0.3 MPa, he wegh facor decreases unl maxmum sress -0 MPa s reached n ha drecon. Ths suaon s depced n Fg

263 Fg. 6-. Wegh facor for ASR expanson For compressve sress σ under -0.3 MPa, he followng decay funcon s used, accordng o (Leger, Coé, Tnaw, 995), where σ L -0.3 MPa and σ u -0 MPa, see Fg. 6-. : W log / L for 0.3MPa log L for 0.3MPa u (4.8) Wegh facors need o be normalzed as Three prncpal srans from ASR are assgned as W W (4.9) ' 3 ' W W (4.0) ASR, V Ths new approach smplfes he procedure oulned by (Saouma, 06, Fg. 5) where several sress sae cases were hreaed ndvdually Valdaon on free expanson The followng Fg. - and Fg. 6 3 valdae expermenal daa for free expanson. The followng maeral parameers were used, summarzed n Table Varable Symbol Value Source (Mulon, Cyr, REQUIRED ALKALI PER REACTIVE r 5.4 % Seller, SILICA Leklou, & ATENA Theory 53

264 PROPORTION REACTIVE SILICA s.8 % PROPORTION REACTIVE PARTICLES IN SAND p 30 % Pe, 008) (Mulon, Cyr, Seller, Leklou, & Pe, 008) (Mulon, Cyr, Seller, Leklou, & Pe, 008) SAND MASS AC 833 kg/m 3 Yasuda, & Kawamura, (Kagmoo, 04) ASR MEASSURED ASR STRAIN Ɛ F %/kg (Poye, 003) AMOUNT OF REQUIRED ALKALI A R 8.39 kg/m 3 (Poye, 003) TOTAL ALKALI IN MORTAR for Ca-5.4 (for Ca-9.0) A T 5.4 (9) kg/m 3 Yasuda, & Kawamura, (Kagmoo, 04) THRESHOLD ALKALI IN CONCRETE A kg/m 3 (Poye, 003) CHARACTERISTIC TIME τ C 0 day LATENCY TIME for Ca-5.4 (for Ca-9.0) τ L 55 (45) day ELASTIC MODULUS COMPRESSIVE STRENGTH E fc 7 GPa 6 MPa (Kagmoo, Yasuda, & Kawamura, 04) (Kagmoo, Yasuda, & Kawamura, 04) 54

265 Table Summarzed parameers for valdaon. Fg. 6. Valdaon of free expanson (Kagmoo, Yasuda, & Kawamura, 04) Fg Valdaon of free expanson, (Kagmoo, Yasuda, & Kawamura, 04) Implemenaon n Aena Dfferenal Equaon (4.99) represens knecs of developmen of ASR exen ξ. In case of consan emperaure n he srucure can be solved analycally, see (4.04). Oherwse mus be solved numercally. The followng lnes and equaons descrbe he procedure o solve ξ ha s mplemened n ATENA. Le's sar from (4.99) and rewre he equaon no s dfferenal form. We expec o now all a me and solve for me. We do n an erave manner,.e. we know all a eraon k and compue ξ a eraon k : ATENA Theory 55

266 ξ ξ k k k c, ξ =0 (4.) The unknown ξ k ASR exen s searched for n form k k ξ ξ ξ, where ξ s correcon of ξ resulng from he k -h eraon. Denong and k ξ ξ ξhe above equaon can be wren k ξξ ξ ξ =0 (4.) k c, from whch afer some mahemacal manpulaon we can calculae ξ ξ ξ ξ k c, k c, (4.3) k k k k and ξ ξ ξ. Noe ha n (4.) hru (4.3) we used c, alhough c, should be employed, as c c(, ) s a nonlnear funcon. Therefore afer each eraon k we updae k k o and recalculae (4.): c, c, ξξ ξ ξ = E (4.4) k k k c, k I yelds an error E ha s furher compared agans some maxmum accepable error. If s oo hgh, he nex eraon s carred ou, oherwse we are done. Noe however, ha for he sake of convergency speed he hrd and furher eraons are n ATENA compued n a dfferen way. Usng lnear nerpolaon beween eraon k and k k and requrng error E 0 n eraon k value ξ k s calculaed by k k k k k k k E E ξ ξ E ξ ξ 0 E k k ξ ξ ξ ξe ξ ξ 0 k k k k k k (4.5) k k k k k ξ E ξ E ξ k k E E and checked by (4.4) wren for eraon k. The erang process connues hs (laer) way unl a suffcen accuracy s obaned. The me sep s npu by used, bu auomacally lmed by 0.0 c requremen o ensure a reasonable accuracy and convergence of he soluon. ASR loadng resuls n developmen of ASR sran and deeroraon of maeral properes lke Young modulus E, enson srengh f and fracure energy G f. For each sep we can wre: E ( ) ( E E ) (4.6) 56

267 The above equaon calculaes sress a (curren) me sep based on sress from he prevous me sep and a curren changes of Young modulus E and srans. The srans represens "mechancal srans,.e. srans producng sresses n unresraned maeras. They are oal geomercal srans mnus nal sran ha corresponds o ASR expanson srans ASR,. The dfferenal formulaon corresponsnds o ncremenal soluon used n Aena and he case of lnear elasc maeral law. (More advanced maerals are reaed n smlar way). Usng ASR Ek E0 cfe ( k), (4.6) can be wren ASR ASR E0cf E ( ) E0 cf E (, ) ASR ASR ASR E0cf E ( )( ) E0(cf E ( ) cf E ( )) ASR ASR ASR cf E ( ) E0cf E ( )( ) E0cf E ( ) ASR cf E ( ) E cf ( ) cf ( )( ) cf ( ) ASR ASR E 0 E ASR E (4.7) Noe ha srans are srans ha are faclaed n maeral law,.e. geomercal srans afer subracng ASR swellng srans. The ASR srans are mplemened by elemen nal srans and he erm s ncorporaed n he soluon n he form of elemen nal sresses. Also, a each sep we updae f and G f. Alernave soluon o (4.7) s E cf ( )( ) E (cf ( ) cf ( )) ASR ASR ASR 0 E 0 E E ASR ASR ASR cf E ( ) E0cf E ( )( ) E0cf E ( ) ASR cf E ( ) ASR ASR cf E ( ) E0cf E ( ) ( cf ASR E ( ) (4.8) ASR cf E ( ) The erm cf ASR s hen added o ASR swellng nal elemen srans (4.0) E ( ) calculaed earler. For lnear maeral law boh equaon (4.7) and (4.8) are equvalen. For he case of nonlnear law hey can slghly dffer. By defaul, Aena preffers approach accordng o (4.8). For he sake of smplcy he above dervaon has been presened for unaxal sress-sran condons. Is exenson o 3D condons s obvous Commens The proposed model s derved from free expanson ess. The model works n D and 3D sress sae by lmng expanson when compressve load s presen n a prncpal drecon. In case of ATENA Theory 57

268 hydrosac compresson above -0 MPa, no ASR expanson occurs and no reducon of mechancal properes happens (E, f, Gf). Ths s jusfed by he fac ha ASR gel grows no cracks and no macroscopc cracks occur. The majory of srucures s exposed o hermal feld hence ASR usually proceeds faser close o he surface due o hgher average emperaure. Snce he surface s ofen unloaded, man expanson happens perpendcular o surface whch nduces small compressve load and delamnaon of layers. 6.6 References Akns, P. W. (994). Physcal chemsry, 5h Ed., Oxford Unversy Press, Oxford, U.K. Berra, M. e al., Influence of sress resran on he expansve behavor of concree affeced by alkal-slca reacon, Cemen and concree research 40, , 00 Bazan, Z.P., Seffens, A. Mahemacal model for knecs of alkal slca reacon n concree. Cem Concr Res 000;30:49 8.M. S. Darmawan & M. G. Sewar, Effec of Png Corroson on Capacy of Presressng Wres, Magazne of Concree Research, 59(), 3-39, 007. Collepard, M., A. Marcals, e al. (97). "Peneraon of Chlorde Ions no Cemen Pases and Concree." Journal of he Amercan Ceramc Socey 55: Comby-Pero, I., Bernard F., Bouchard P.O., Bay, F., Garca-Daz, E. Developmen and valdaon of a 3D compuaonal ool o descrbe concree behavour a mesoscale. Applcaon o he alkal slca reacon. Compu Maer Scence 009;46(4): Červenka, J, Jendele, L., Šmlauer, V: Repor I TAČR - TA , 06 Com, C., Fedele, R., Perego, U. A chemo hermo-damage model for he analyss of concree dams affeced by alkal slca reacon. Mech Maer 009;4:0 30. III, T. E. U. B. E. (000). DuraCree Fnal Techncal Repor. General Gudelnes for Durably Desgn and Redesgn. Doc. BE95-347/R7, 000 Esposo, R., Hendrks, M.A.N., Degradaon of he mechancal properes n ASR-affeced concree: overvew and modelng, Sraeges for Susanable Concree Srucures, 0 Farage, M. C. R. Modelagem e mplemenaca o nume rca da expansa o por reaca o a lcal agregado do concreo, DSc hess, Deparmen of Cvl Engneerng, COPPE, Unversdade Federal do Ro de Janero, Ro de Janero, 000 (n Poruguese). J. A. Gonzales, C. Andrade, C. Alonso & S. Felu, Comparson of Raes of General Corroson and Maxmum Png Peneraon on Concree Embedded Seel Renforcemen, Cemen and Concree Research, 5(), 57-64, 995. Kwon, S.-J. and U.-J. Na (0). "Predcon of Durably for RC Columns wh Crack and Jon under Carbonaon Based on Probablsc Approach." In. Journal of Concree Srucures and Maerals 5(): -8. Kwon, S. J., U. J. Na, e al. (009). "Servce Lfe Predcon of Concree Wharves wh Earlyaged Crack: Probablsc Approach for Chlorde Dffuson." Srucural Safey 3():

269 Larve, C. (998). Appors combnés de l expérmenaon e de la modélsaon á la comprehenson de l alcal-reacon e de ses effes mécanques. Monograph LPC, OA 8, Laboraores des Pons e Chaussées, Pars (parally ranslaed no Englsh). Leger, P., Coé, P., Tnaw, R., Fne elemen analyss of concree swellng due o alkalaggregae reacons n dams, Compuer and srucures, 995 Léger, P., Coé, P. and Tnaw, R. (996) Fne Elemen Analyss of Concree Swellng due o Alkal-aggregae Reacons n Dams, Comparers & Srucures Vol. 60. No. 4. pp Lndgar, J., (0), Alkal slca reacons (ASR): Leraure revew on parameers nfluencng laboraory performance esng. Cemen and Concree Research, 4, Lu Y, Weyers R.E., Modelng he Dynamc Corroson Process n Chlorde Conamnaed Concree Srucures, Cemen and Concree Research, 8(3), , 998. Lu Y., Modellng he Tme-o-corroson Crackng of he Cover Concree n Chlorde Conamnaed Renforced Concree Srucures, Ph.D. dsseraon, Vrgna Polyechnc Insue, 996. Lupng, Tang, e al. (007). Chlorde Ingress and Renforcemen Corroson n Concree under De-Icng Hghway Envronmen A Sudy afer 0 Years Feld Exposure, SP Sverges Teknska Forsknngsnsu. Vol. SP Repor 007:76. Mulon, S., Cyr, M., Seller, A., Leklou, N., Pe L. (008) Coupled effecs of aggregae sze and alkal conen on ASR expanson, Cemen and Concree Research 38, Mulon, S., Toulemonde, F. (00) Effec of mosure condons and ransfers on alkal slca reacon damaged srucures, Cemen and Concree Research 40, Mulon, S., Seller, A., Cyr, M. Chemo mechancal modelng for predcon of alkal slca reacon (ASR) expanson. Cemen Concree Research 009;39: Lndgar, J., (0), Alkal slca reacons (ASR): Leraure revew on parameers nfluencng laboraory performance esng. Cemen and Concree Research, 4, Papadaks, V. G. (000). "Effec of Supplemenary Cemenng Maerals on Concree Ressance Agans Carbonaon and Chlorde Ingress." Cemen Concree Research 30(): Papadaks, V. G. and S. Tsmas (00). "Supplemenary Cemenng Maerals n Concree. Par I: Effcency and Desgn." Cemen and Concree Research 3(0): Page CL, Naure and properes of concree n relaon o envronmen corroson, Corroson of Seel n Concree, Aachen, 99. S. Poye, Eude de la dégradaon des ouvrages en béon aens par la réacon alcal-slce: Approche expérmenale e modélsaon numérque mul-échelles des dégradaons dans un envronnemen hydro-chemomécanque varable, Ph.D. Thess (n French), Unversé de Marne-La-Vallée, 003 J. Rodrguez, L. M. Orega, J. Casal & J. M. Dez, Corroson of Renforcemen and Servce Lfe of Concree Srucures. In Proc. of In. Conf. on Durably of Buldng Maerals and Componens, 7, Sockholm, :7-6, 996. Saouma, V., Pero, L., 006, Consuve model for alkal-aggregae reacon. ACI Maeral Journal 03, Spec-ne, Corrosvy zones for seel consrucon [onlne] avalable from: hp:// ne.com.au/press/04/gaa_0804/corrosvy-zones-for-seel-consrucon- Galvanzers-Assocaon, 05. ATENA Theory 59

270 Ulm, F. J., Coussy, O., L, K., Larve, C. Thermo chemo mechancs of ASR expanson n concree srucures. J Eng Mech 999;6(3):33 4. Saouma, V., Pero, L. Consuve model for alkal-aggregae reacons. ACI Maer J 006;03(3):

271 7 TRANSPORT ANALYSIS As poned ou n he prevous secon, creep maeral behavour of concree srongly depends on mosure and emperaure condons. Some consuve models for creep n ATENA can pay regards o hese facors and based on prevously compued mosure and emperaure hsores whn he srucure hey can predc concree behavour more accuraely. Ths secon descrbes a module called CCSrucuresTranspor ha s used o calculae he hsores. A more accurae creep analyss hen ypcally consss of wo seps: frsly execue CCSrucuresTranspor module and calculae he mosure and humdy hsores of he srucure and secondly execue CCSrucuresCreep module o carry ou he acual sac analyss. Of course, for boh analyses we have o prepare an approprae model. Expor/Impor of he resuls beween he modules s already done by ATENA auomacally. To be exac, boh he ranspor and sac analyss should be execued smulaneously bu as mosure and emperaure ranspor does no depend sgnfcanly on srucural deformaons,.e. couplng of he analyses s low, he mplemened saggered soluon yelds suffcenly accurae resuls. The governng equaons for mosure ranspor read (for represenave volume REV] : where: w ( we wn) dv( J w) ATENA Theory 6 (5.) w s oal waer conen defned as a rao of wegh of waer a curren me o wegh of waer and cemen a me 0 0 n REV, [mass/mass], e.g. [kg/kg] w, w = sands for he amouns of free and fxed (.e. bound) waer conens, [mass/mass], e n J w = mosure flux, [lengh*mass/ (me*mass)]. e.g. [m/day], =me, [me], e.g. [day]. The mosure flux s compued by where D w s mosure dffusvy ensor of concree [m /day], s graden operaor. Jw Dw we (5.) Noe ha n (5.) only dffuson of waer vapor s consdered. Mosure advecon s neglgble. The equaons (5.) and (5.) can be also wren as beng dependen on w or relave mosure h. A relaonshp beween h and w s gven by Usng (5.3) Equaon (5.) can be wren as follows w w( h) (5.3) J D h (5.4) w h

272 A specal aenon mus be pad o calculaon of he above me dervaves and negraon of he governng equaons. For example, n case of usual Gauss negraon and use of exac me dervaves he soluon may suffer from mass losses. To remedy he problem he CCSrucuresTranspor module negraes he srucure,.e. all he ndvdual fne elemens n nodes and me dervaves are calculaed numercally (Jendele 00). Ths negraon s smlar o use of fne volume mehod, whch s also known o be robus agans he mass losses. Hea ransfer s governed by smlar equaon Q T CT ( T Tref ) CT dv( JT) where Q s oal amoun of energy n a un volume [J/m 3 ] (5.5) C T s hea capacy [J/(K.m 3 )], J T s hea flux [J/(day.m )]. If hydraon we wan o add hea Qh ( ), whch expr whn un volume.e, J Qh m 3, Equaon (5.5) changes o J Hea flux JT, m s s calculaed by and T Qh CT ( T Tref ) QhCT dv( JT) K T sands for hea conducvy, e.g. [J/(day.m.K)]. J T esses amoun of hydraon hea (5.6) K grad( T ) (5.7) T Noe ha Equaon (5.5) accouns for hea ranspor due o conducon only. Hea advecon s neglgble. In (5.5) we can also neglec hydraon hea, because n large mes s mpac for hea ransfer s small. On he oher hand, we canno neglec concree mosure consumpon due o hydraon process. Accordng o (Bazan and Thonguha 978; Bazan 986) hydraon waer conen w can be calculaed by: h where e wn wh 0. c e e 3 (5.8) e = 3 days, e s equvalen hydraon me n waer a emperaure 5 0 C ha corresponds o he same degree of hydraon subjec o real age, mosure and emperaure condons of he maeral. The parameer c relaes o he amoun of cemen and s calculaed by(5.53). If emperaure ranges from 0 o 00 0 C, e s compued by 6

273 d (5.9) e h T where d s me ncremen afer he mould has been removed and coeffcens, are calculaed by h (5.0) 4 ( h) T h In he fracon U h T exp R T 0 T (5.) U h R he symbol U h sands for acvaon energy of hydraon and R s gas consan. Accordng o (Bazan 986) emperaure expressed n 0 K. The reference emperaure s gven by U h K R. T, T 0 are real and reference concree T (5.) The followng fgure depcs relaonshp beween real and equvalen me e for he case of consan emperaure T 0 5 C and mosure h 0.8. In pracce, hs relaonshp s rarely lnear, because wh ncrease of me he amoun of fxed waer (due o hydraon) ncreasng as well and nvolves gradual decrease of relave mosure h. w h s Fg. 7- Equvalen vs. real me relaonshp ATENA Theory 63

274 The amoun of waer ha was needed for hydraon of concree accordng o Equaon (5.8) for he case of c 300 kg s shown below: Fg. 7- Mosure consumed by hydraon as a funcon of equvalen me 7. Numercal Soluon of he Transpor Problem Spaal Dscresaon The ranspor governng equaons for a ypcal engneerng problem are oo complex for analycal soluon. Hence, smlar o oher ATENA engneerng modules, fne elemen mehod s used also for he CCSrucuresTranspor module. The ranspor problem ges spaally and emporarly dscrezed and hen he resulng se of nonlnear algebrac equaons s solved by a specal erave solver. Ths secon s dedcaed o dealed descrpon of he former ype of dscresaon. The soluon s based on Equaons (5.) hru(5.7). Noe ha he unknown varables are h h ( ); TT( ); wwht (, ); w w( ht,, ) (5.3) and hey are o be dscrezed. Le he lef-hand sde par of (5.) and (5.4) s denoed LHSh, LHS T, respecvely. The subscrp h and T ndcaes mosure and emperaure flux. Smlar subscrps are also used for rgh-hand-sde of he equaons, RHSh, RHS T. Noce ha RHS expressons do no nclude he dvergence operaor! LHSh w w h h h (5.4) T Q h LHST CT (5.5) 64

275 RHSh J w Jh (5.6) RHS T J (5.7) The srp over an eny n he above equaons means ha he eny s vecor. (Scalar enes do no have he srp). The fluxes Jw Jh are dencal,.e. he subscrp w ndcaes also mosure phase. Usng he above noaon Equaons (5.) and (5.5) can be wren as follows The LHS LHS h T T dv( RHS ) h dv( RHS ) T (5.8) LHS h ncludes me dervave of mosure. I s compued usng he followng expressons: w w ( ) h h e e h T (5.9) wh wh e wh h T For he nex dervaon, le us wre Equaons(5.4), (5.5) n a general form: and equaons(5.6), (5.7) e h w T LHS c c c c h hh hw ht h0 h w T LHS c c c c T Th Tw TT T0 e (5.0) RHS k h k w k T k h hh hw ht h0 RHS k h k w k T k T Th Tw TT T0 where square bracke ndcaes ha he enclosed eny s a marx [ ]. Comparng (5.0) wh (5.) and (5.5) we fnd ha c c 0; c ; c 0 hh ht hw h0 Qh cth ctw 0; ctt 0; ct 0 0 (5.) (5.) ATENA Theory 65

276 The parameer c TT s n ATENA an npu maeral parameer, c h0 s compued from(5.9),.e. c w w w. The soluon also ncludes expressons 0;. Ther values depend on a h T h h0 h T e consuve model beng used n he soluon. For more nformaon please refer o Secon Maeral Consuve Model. For rgh-hand sdes we can wre n smlar manner: The parameer TT see he nex secon. k k 0; k 0; k 0 hw ht hh h0 k k 0; k 0; k 0 Th Tw TT T 0 k s a maeral npu parameer, hh (5.3) k s calculaed from a consuve model, For he nex dervaon, le us assume dscresaon of he unknown varables as follows. Remnd ha hese are n he governng equaons negraed n fne nodes, (Cela, Boulouas e al. 990; Cela and Bnnng 99). T h N h; h N h T T w N w; w N w T (5.4) where T T N T; T N T T h, w, T sands for vecors of he correspondng enes. The vecors have dmenson n equal o number of fne nodes of he problem. N s vecor of nerpolaon, (.e. shape) funcons, N N Nn... x x x T N N Nn N... y y y N N Nn... z z z Usng (5.4) Equaons (5.0) and (5.) can be wren n he form 66

277 and h w T LHS c N c N c N c T T T h hh hw ht h0 h w T LHS c N c N c N c T T T T Th Tw TT T0 T T T RHS k N h k N w k N T k h hh hw ht h0 T T T RHS k N h k N w k N T k T Th Tw TT T0 (5.5) (5.6) The resulng se of equaons are solved eravely usng fne elemen mehod, see (Zenkewcz and Taylor 989), (weak formulaon, n whch he shape funcons N are used as wegh funcon): V V N LHS dv( RHS ) dv 0 h N LHS dv( RHS ) dv 0 T h T (5.7) where V s volume of he analyzed srucure. Each of he above equaons represens a se of equaons wh dmenson equal o number of fne nodes n. Noe ha dv( RHS h) and dv( RHS T ) are scalars! In he nex dervaon he wo pars of (5.7) are deal wh separaely. V T h T w T T N LHShdV N chhn chwn chtn ch0 dv V h w c NN dv c NN dv... c NdV (5.8) T T hh hw h0 V V V h w cchh cchw... cc h0 V T h T w T T N LHSTdV N cthn ctwn cttn ct0 dv V h w ccth cctw... cc T 0 (5.9) and he marces cc are calculaed by ATENA Theory 67

278 T T ; ;... h cc c NN dv cc c NN dv cc c NdV hh hh hw hw 0 h0 V V V T T ; ;... T cc c NN dv cc c NN dv cc c NdV Th Th Tw Tw 0 T 0 V V V The second par of (5.7) are calculaed usng Green heorem (5.36): T ( ) N dv RHS dv N n RHS ds N RHS dv h s h h V S V (5.30) V S T T T T s hh hw ht h0 Nn k N h k N w k N Tk ds T T T N khh N h khw N w kht N T kh dv 0 (5.3) where S s srucural surface (wh possbly defned boundary condons). In case of hea ransfer we can derve all he equaons n a smlar way. In analogy o (5.30) le us nroduce marces kk hh hh kk N k N dv V hw hw kk N k N dv V T T kk N k dv... h0 h0 V (5.3) TT TT kk N k N dv V T and also kk N k dv T 0 T 0 V 68

279 T hh s hh J N n k N ds S T hw s hw J N n k N ds S T T... T TT s TT J N n k N ds S T J N n k ds T h0 s h0 S J N n k ds (5.33) T T0 s T0 S Usng (5.8) o (5.33) he orgnal governng equaons (5.7) can be wren as follows: h w T cchh cchw ccht cc 0 kkhh h kkhw w kkht T kk h0 h J h J w J T J hh hw ht T Th Tw TT h0 h w T ccth cctw cctt cc 0 kkth h kktw w kktt T kk T0 J h J w J T J T 0 (5.34) Afer sorng he unknown varables h, T by fne nodes no a sngle vecor, Equaon (5.34) wll read cc kk cc kk J J (5.35) The rgh-hand sde (5.35) s non-zero only for non-zero prescrbed boundary condons and hence has characer of load vecor n a sac analyss. In (5.3) we used Green heorem. I saes: where T u dv( v ) dv u ns v ds u v dv V S V T u dv( v ) dv u ns v ds u v dv V S V (5.36) ATENA Theory 69

280 u u u u x y z u u u x y z u u u u x y z un un un x y z (5.37) 7. Numercal Soluon of he Transpor Problem Temporal Dscresaon The hea and mosure ransfer governng equaons (5.35) can be wren n he form: K + C J (5.38) where K, C = are unsymmercal problem marces defned n he prevous secon, J =vecor of concenraed nodal fluxes (boh mosure and hea) and s vecor of unknown varables. All of hese apply for me. Equaon (5.38) s solved eravely..e. he vecor s searched for n he ncremenal form: (5.39) () ( ) () where ndex () and eraon () : ndcaes number of eraon and () s ncremen of he unknowns for me () ( ) - () ( ) - The marx and vecor K () and J s derved from on emporal negraon mehod beng used: K J (5.40) K, C and based ( ) - ( ) - CCSrucureTranspor module currenly suppors Crank Ncholson (Wood. 990) and Adams- Bashforh (Dersch and Perroche 998) negraon scheme. The former scheme s lnear,.e. s a frs-order negraon procedure. The laer scheme s a second-order negraon procedure. I s supposed o be more accurae, however, s also more CPU and RAM expensve and s more dffcul o predc s real behavour. Hence, he Crank Ncholson scheme s ypcally preferred. I has been more esed and verfed n he CCSrucureTranspor module and hereby s more recommended. 70

281 7.. -parameer Crank Ncholson Scheme Ths scheme comprses a number of well esablshed negraon procedures. I depends, wha value of he parameer s used. The se of equaons (5.38) s solved for me, whereby he vecor of unknown varables s calculaed as a lnear combnaon of he correspondng vecors a me and. Hence ( ) (5.4) Dependng on a parcular value of he parameer we ge he well known Euler mplc negraon (for =), rapezodal Crank Ncholson scheme (for =0.5), Galerkn negraon mehod (for =/3) or even Euler explc scheme (for =0), whch s only condonally sable. Soluon predcor: Soluon correcor: (5.4) Usng he above afer some mahemacal manpulaon we derve fnal expressons for These read: K= K C (5.43) K, J. C J J K (5.44) K J 7.. Adams-Bashforh Inegraon Scheme Soluon predcor: prev prev prev (5.45) where ndex prev ndcaes ha he eny comes from me precedng me Noe ha we assume ha all enes from me are already known and we solve for her values a me. Soluon correcor: (5.46) ATENA Theory 7

282 prev prev prev prev prev (5.47) Smlar o (5.44) we have here for K, J : K= K C n n n n n n n n n n n n n n n n n n J K C J n n n n (5.48) K J 7..3 Reducon of Oscllaons and Convergence Improvemen The ranspor governng equaons are prone o suffer from oscllaons. As repored n (Jendele 00) hs can be mproved by nroducng a sor of Lne Search mehod dampng. The basc dea s ha Equaon (5.39) ges replaced by (5.49) () ( ) () where s a new dampng facor. The facor s ypcally se o somehng n range dependng on curren convergence behavour of he problem. 7.3 Maeral Consuve Model The prevous secon referred o a maeral consuve model,.e. was assumed ha we know how o compue maeral dffusvy marx D h, (see(5.4)), and maeral capacy w w( h), (see(5.). Calculaon of hese enes s descrbed here. Currenly, ATENA has only wo consuve model avalable for he ranspor analyss. The frs one,.e. CCModelBaX94 s characerzed as follows and he second one,.e. CCTransporMaeral s brefly descrbed laer n hs secon. CCModelBaX94 For hea ranspor a smple consan lnear model s mplemened. For mosure ranspor a nonlnear model based on he model (X, Bazan e al. 993; X, Bazan e al. 994) has been developed. 0 I can be used for emperaures n range T C and mosure H I s mporan o noe ha he model was orgnally wren only for morar hence, s naccurae for concree wh an aggregae havng hgher permeably (.e. dffusvy) and/or absorpon. The model has he followng man parameers 7

283 Type of cemen Waer-cemen rao w wc c As already poned ou, he model does no accoun for aggregae,.e. predc mosure move only n pores flled by waer-cemen pase. w The man eny of he model s waer conen w w( h,, T, ). I s defned as follows: c G w w G w,0 G c (5.50) where G w s waer conen n morar a me, 3 G w,0 s waer conen a me zero, 3 G c s amoun of cemen a me zero, 3 kg m of morer, kg m of morer, kg m of morer. Morar here sands for mxure of waer and cemen. If concree maeral s o be consdered, hen w can be calculaed by V V where concree morar w G w,0 Vconcree Gw V morar G w (5.5) V concree Vconcree Gw,0 Gc G c V V morar ATENA Theory 73 morar s rao of oal volume o (only) volume of morar (.e. waer and cemen) and G are correspondng amouns of waer and cemens n concree, (.e. no only n kg morar!) 3 m of concree. w The model self already accouns for mosure used by hydraon process..e. 0. As a resul, w h accordng o (5.9) need no be mplemened. On he oher hand, f mosure losses due o hydraon are o be compued by he model based on w (5.9), s mporan o fx 0 and o modfy w h, so ha predcs relave mosure conen w used hroughou whole dervaon CCSrucuresTranspor. The orgnal funcon for w h was wren for absolue wegh of waer and hence, for relave conen of waer Equaons (5.8) mus be rewren no

284 w h 3 e 0. Gc e e Gc e G c e Gw,0 Gc Gw,0 Gc e e Gw,0 Gc e e 3 3 (5.5) and he consan c from (5.8) becomes equal o G c Gc c G G G w,0 G w,0 c c (5.53) More dealed descrpon of he model s beyond scope of hs documen and he reader s referred o n (X, Bazan e al. 993; X, Bazan e al. 994). CCTransporMaeral CCTranspor maeral s a smple consuve law ha allow users o ener laboraorly measured mosure and hea characerscs. Referng o Equaons (5.) and (5.5) hea and mosure flow governng equaons can be wren n he followng general form: Hea : Q h T w C C C C K x grad( h) K grad( T) K grad( w) K Th TT Tw T Th TT Tw Tgrav Mosure : w h T w C C C C D x grad( h) D grad( T ) D grad( w) D (5.54) The parameers C Th, C TT K wgrav are calculaed as: wh wt ww w wh wt ww wgrav 74

285 C C f ( h) f ( T) f ( ) 0 h T Th Th CTh CTh CTh C C f ( h) f ( T) f ( ) 0 h T TT TT CTT CTT CTT C C f ( h) f ( T) f ( ) 0 h T Tw Tw CTw CTw CTw C C f ( h) f ( T) f ( ) 0 h T T T CT CT CT C C f ( h) f ( T) f ( ) 0 h T wh wh Cwh Cwh Cwh C C f ( h) f ( T) f ( ) C 0 h T wt wt CwT CwT CwT ww C 0 ww h T f ( h) f ( T) f ( ) Cww Cww Cww C C f ( h) f ( T) f ( T) 0 h T w w Cw Cw Cw K K f ( h) f ( T) f ( ) 0 h T Th Th KTh KTh KTh K K f ( h) f ( T) f ( ) 0 h T TT TT KTT KTT KTT K K f ( h) f ( T) f ( ) 0 h T Tw Tw KTw KTw KTw K K f ( h) f ( T) f ( ) D 0 h T Tgrav Tgrav KTgrav KTgrav KTgrav D f ( h) f ( T) f ( ) 0 h T wh wh Dwh Dwh Dwh D D f ( h) f ( T) f ( ) 0 h T wt wt DwT DwT DwT D D f ( h) f ( T) f ( ) 0 h T ww ww Dww Dww Dww D D f ( h) f ( T) f ( ) 0 h T wgrav wgrav Dwgrav Dwgrav Dwgrav (5.55) and he consan parameers C 0 Th hru 0 h T D wgrav and funcons fc ( h) hru f ( ) Th D T are npu wgrav parameers, (o be possbly obaned from some expermens). The funcons are defned as mullnear funcons and only her ds are npu no CCTransporMaeral model defnon. Noe ha gravy erms n RHS of (5.54) have a lle physcal jusfcaon n hea and mosure dffuson gahered ranspors, neverheless, hey are ncluded o allow usng hs maeral law for soluon of oher knds of ranspor problems. CCTransporMaeralLevel7 maeral CCTranspor maerallevel7 s an exenson of he above CCMaeralTranspor maeral n he way auomacally compues mosure and emperaure capacy and conducvy/dfussvy ncl. "snk" erms regardng hydraon, (.e. rae of hydraon hea and mosure consumpon durng connree hydraon). In erms of he above nomenclaure hs upper maeral level calculaes CTT, KTT, CT, Cwh, Dwh, C w. As already menned, he presened maeral adds on s boom level,.e. CCMaeralTranspor. All parameers and characersccs from he boom level, (e. hose from CCMaeralTranspor) can sll be npu and used. They ypcally serve for a refnemen/addon of parameers generaed by he upper maeral level. The resul from he boom an upper level are smply added o form fnal characerscs of he maeral model CCTransporMaeralLevel7. Noe ha defaul values of CTT, KTT, CT, Cwh, Dwh, Cw n he boom level are by defaul se o zero. Hydraon hea and affny hydraon model ATENA Theory 75

286 The mos mporan par of he presened model s compuaon of concree hydraon maury facor. I s accompaned by calculaon of generaed hydraon hea and consumed hydraon mosure. The analyss s based on he affny hydraon model, whch provdes a framework for accommodang all sages of cemen hydraon. Consder hydrang cemen under sohermal emperaure 5 o C a relave humdy h. A hs emperaure, he rae of hydraon maury facor, 0... can be expressed by chemcal affny A A ( ) 5 5 : A 5 (5.56) where A sands for he chemcal affny, [ s ], The expresson already nclude coeffcen Ea exp RT. Hence A 5 s no normalzed and refers o emperaure 5 o C. For dfferen emperaure s replaced by A, see (5.60). R s gas consan [K] and J kmol K, T s emperaure, E a s 40 kj/mol. I s worhy o noe he ncorporaon of he maury mehod no (5.56). A characersc me mgh be nroduced o express an affny A (Bernard, Ulm e al. 003). The affny propery can be obaned expermenally or analycally. Usng expermenal approach, hea flow q () ha corresponds o he hydraon hea Q Q ( ) s meassured n an sohermal calormery. Alernavely, he hydraon maeral parameers are compued by an analycal mcro-scale model ha accouns for he majory of underlyng chemcal reacons as well as opology of cemen grans (wh he consequence o hydraon knecs). The soluon sems from (Smlauer and Bnar 006) and employs dscree hydraon model CEMHYD3D (Benz 005) allowng o accoun for parcle sze dsrbuon of cemen, chemcal composon of cemen, emperaure and mosure hsory n concree ec. h h Havng hsory of Q (for T , ), he approxmaon of parameer s gven by h hpo, Q Q h hpo, (5.57) Qh A 5 (5.58) Q where Q h, po s poenal hydraon hea, [J/kg]. Hence he normalzed hea flow sohermal 5 o C equals o chemcal affny A 5. Q Q h h, po under Cervera e al. (Cervera, Olver e al. 999) developed an analycal form of he affny whch was refned n (Gawn, Pesaveno e al. 006). A slghly modfed formulaon s proposed here 76

287 B Ea A 5 B exp exp RT (5.59) where B,[ s ], B, [-] are coeffcens o be calbraed, s he ulmae hydraon degree, [-], and represens mcrodffuson of free waer hrough formed hydraes, [-]. (5.59)express sohermal hydraon a 5 C. The parameers n When hydraon proceeds under varyng emperaure, maury prncple expressed va Arrhenus equaon scales he any o arbrary emperaure T Ea A T A 5 exp R T (5.60) For example, smulang sohermal hydraon a 35 o C means scalng A 5 wh a facor of.65 a a gven me. Ths means ha hydrang concree for 0 hours a 35 o C 35 C releases he same amoun of hea as concree hydrang for 6.5 hours under 5 C. Noe ha seng Ea 0 gnores he eec of emperaure and proceeds he hydraon under 5 C. Gawn e al. (Gawn, Pesaveno e al. 006), among ohers, added he effec of relave humdy. The exenson of (5.58) leads o Qh A Th (5.6) Q h, po h (5.6) a ah 4 where h h( h ) accouns for he reducon of capllary mosure. h s relave humdy r, (Bazan and Najjar 97). a s maeral parameer, ypcally a 7.5. Dependng on curng condons s calculaed as follows: Sealed curng: w/ c, (5.63) 0.4 Sauraed curng: w/ c, (5.64) 0.36 w/ c s waer-cemen rao. Subsung (5.59) and (5.6) no (5.6) yelds fnal equaon o predc developmen of hydraon hea. As s dffcul o express funcon analycally (from (5.59), (5.6)), he above equaons are negraed numercally. ATENA Theory 77

288 Noe ha Q h s calculaed n he same un as s enered he parameer Q h, po. If he governng equaons are wren for un volume and Q h, po s gven per cemen un wegh, hen Q h mus be mulpled by fracon of cemen mass Hea capacy m cemen and oal volume of concree V o. The model assumes he followng componens of concree: aggregae, fller, waer and cemen. Toal mass of concree n one cubc meer resuls from ndvdual masses of componens: m m m m concr aggregae fller pase m m m pase cemen waer (5.65) where m concr s mass of concree per a un volume. Smlarly for mass of aggregae m aggregae, mass of fller m fller, mass of waer m waer and mass of cemen m cemen. Correspondng volumes are Vaggregae maggregae / aggregae, Vfller mfller / fller ec. sands for mass densy of he phase. Havng oal volume V concr V aggregae V fller V waer V cemen, we can calculae phase fracons f V / V and smlarly for he remanng phases. aggregae aggregae concr Hea capacy and s evoluon of cemen pase (cemen+waer) was suded n (Benz 007) a 3 0 C for w/c beween 0.3 and 0.5. The capacy of fresh cemen pase yelds where pase. The las erm,.e. where C f C f C Cˆ (5.66) concree aggregae aggregae fller fller pase Cconcree s concree capacy (per un volume) and akn for aggregae, fller and cemen C pase depends also on degree of hydraon and s calculaed by C ˆ ( f C f C ) 0.6( exp(.9 )) (5.67) pase cemen cemen waer waer C cemen s cemen capacy a me zero. The hea capacy of srucural concree spans he range beween 0.8 and.7 Jg - K -. A former Czech sandard CSN 7308 declares 840 and 870 Jkg - K - for dry and sauraed maure concree, respecvely. C aggregae s approxmaely 840 Jkg - K - for basal and lmesone, 790 Jkg - K - for grane, 800 Jkg - K - for sand. C cemen s abou 750 Jkg - K - and Cwaer s 480 Jkg - K -. Hea conducvy The hermal conducvy of cemen pase was found o reman n he range Wm - K - for arbrary degree of hydraon, for boh sealed and sauraed curng condons, and for w/c from 0:3 o 0.4 (Benz 007). Waer n he capllares has he hermal conducvy Wm - K - (Benz 007). The hermal conducvy of hardened concree vares beween 0.85 and 3.5 Wm - K - (Nevlle 997) p.375, dependng srongly on an aggregae ype. Thermal conducvy also depends on he sauraon sae of concree. For example, a srucural concree made from normal-wegh aggregae wh a un mass of 40 kg/m 3 yelds =.696 Wm - K - for proeced and.904 for weaher exposed condons (Nevlle 997), p

289 Fgure 7-. Thermal conducvy of concree accordng o he Czech code CSN Fgure 7- summarzes hermal conducves for ordnary concree dependng on concree un mass and sauraon condons, accordng o (Nevlle 997) and a former Czech sandard CSN The laer consders.5 for a dry concree and.7 Wm - K - for a waer-sauraed concree. Fara e al. (Fara, Azenha e al. 006) appled he evoluon of concree conducvy wh regards o where s he conducvy of fully hardened concree,.e. n nfne me. The model mplemened n Aena,.e. CCTransporMaeralLevel3 sems from homogenzaon heores. Consder conducvy of cemen pase pase and aggregaes aggregae such ha. Correspondng volume fracons are f, f. Hashn-Shrkman lower pase concree, low aggregae and upper bounds, are (Benz 007) concree upper pase aggregae concree, low, concree, upper, pase 3 f 3 f aggregae The presened model uses average conducvy,.e. aggregae pase aggregae pase pase pase aggregae pase 3 f 3 f pase aggregae pase aggregae aggregae aggregae pase aggregae (5.68) concree. low, concree, upper, concree (5.69) ATENA Theory 79

290 Fgure 7- consders pase =.0 Wm - K - and aggregae =.0 Wm - K -. Volume fracon of aggregaes vares from 0 o. Imporan hermal conducves: lmesone , sandsone.7, grane Wm - K -. The above equaons for homogenzaon are wren for phases pase-aggregaes. In aena he homogenzaon s carred ou as follows:. homogenze phases cemen - waer -> phase pase.. homogenze phases pase - fller -> phase pase wh fller 3. homogenze phases pase wh fller - ar -> phase pase wh fller and ar 4. homogenze phase pase wh fller and ar - aggregaes -> concree Noe ha fller and aggregae s n hs case reaed as one componen and he same apples for waer and cemen, (beng he componen pase). Volume averagng echnque s used o calculae correspondng properes of pase and mxed aggregae. Fgure 7- Predced hermal conducves of concree from bounds. Mosure consumpon due o hydraon I s assumed ha kg of cemen (n concree) approxmaely consumes durng full hydraon process abou Q w, po of waer. Typcally Q w, po =0.4 kg of waer per kg of cemen. Thus, e.g. concree mxure wh 300kg cemen per m 3 of concree needs 300*0.4=6kg o waer per m 3 of concree. Assumng lnear dependence of waer hydraon consumpon w h on concree hydraon level, ( 0 for fresh concree and for fully hydraed concree) he waer snk erm due o hydraon s C h, wh wh where c sands for wegh of cemen n m 3 of concree. (5.70) wh Qw, po c,[kg] (5.7) 80

291 Mosure capacy Mosure conen a un volume where -3 f,[kgm ] w -3,[kgm ] s calculaed a smple expresson b h w wf b h (5.7) w s he free waer sauraon and b s dmensonless approxmaon facor, whch mus always be greaer han one. I can be deermned from he equlbrum waer conen w a relave humdy h 0.8 by subsung he correspondng numercal values n equaon 80 (5.7): hw ( f w80) b whw f -3 Mosure capacy C, kgm s calculaed as dervave of mosure conen wh respec o h : C h 80 f w w b b h b h (5.73) (5.74) The above expresson s applcable for analyses usng reference un volume. If reference un C C/, kg/kg, where wegh of he srucure s preferred, hen we employ mosure capacy s concree densy, Mosure dffuson kg/m 3. The presen model accouns for dffusvy mechansm of mosure ranspor. I s vald for dense concree, whch has no muually conneced pores and mosure convecon hru pores (beng drven by waer pressure) can be negleced. Hence, mosure flux, kg qh m s s calculaed by he kg equaon qh D hh, where oal mosure dffusvy Dh, ms s calculaed as sum of waer w D h and waervapour Waer lqud dffusvy wv Dh dffusvy: w Dh s calculaed D D D (5.75) w wv h h h where waer dffusvy D w w, m / s s w w w Dh Dw h (5.76) D w w w w 3.8 A 000 f (5.77) w ATENA Theory 8 f

292 kg and A s waer absorpon coeffcen, 0.5 m s. Waer vapour permeably s compued from waer vapour pressure drven dffusvy wv kg Dp, mspa : wv D p (5.78) where s he waer vapor dffuson ressance facor and s he vapor dffuson coeffcen n ar kg mspa pa T 73.5 R 73.5 T 73.5 p a M w.8 (5.79) Amospherc pressure waer s M 8.058kgkmol w p 035Pa, gas consan a R 834.4Jkmol K and molar mass of As n he presened model relave humdy h s he prmary varable used o analyze mosure ranspor, wv D p mus be ransformed o wv D h. Ths s done by: wv wv p wv ( psah) wv D D D D p h h h p p p sa (5.80) Any expresson o calculae pressure of sauraed waer vapour can be used. The presened model uses psa at T0 T 6 e, Pa (5.8) In he above T s emperaure [ o C ] and he remanng parameers are o T0 : T 34.8 Ca, 7.08; T0 : T 7.44 Ca,.44 o 0 0 Some gudelnes owards he model's parameers Fed parameers for cemen pase hydraon need o be consdered for each concree separaely. Due o hgh cemen varably, s mpossble o assgn one parcular cemen o one concree grade. The user needs frs o selec he cemen parameers from he followng able: 8

293 Table 7.3- Parameers of affny hydraon model used for CEM I. The above able s based on fng predced resuls from CEMHYD3D analyss by (5.59), see Table and Fgure 7-3. The smulaons were carred ou on CEMHYD3D s mcrosrucures µm and wh he acvaon energy 38.3 kj/mol. Sauraed curng condons were assumed, snce sealed condons wll be obaned from couplng wh mosure ranspor. Table specfes npu daa for seleced Porland cemens. Fgure 7-3 F of seleced cemens o he affny model, w/c = 0.4 The majory of concrees s produced from blended cemens (CEM II - CEM V), hence s necessary o scale down Q po by approxmaely 30 %. Ths s a common Porland clnker subsuon n he majory of blended cemens n Europe. There are oher defaul parameers, whch are no specfed here: QW POT= 0.4, TH INIT = 0, ALPHA INIT = 0, TEMPERATURE INCR MAX =0., H80 = 0.8, TEMP0 = 34.8, A WV = 7.08, TEMP0 ICE = 7.44,A WV ICE =.44 The parameer A 7.5 expresses hydraon slow-down wh regards o relave humdy. The hydraon praccally sops a 0.8. ATENA Theory 83

294 Parameers n Fgure 7- are compued for sauraed sae. When =, he hydraon proceeds as here s sauraed waer envronmen around concree. Under sandard crcumsamces, hydraon consumes waer, whch decreases relave humdy n he calculaon. Three parameers are relaed o mosure ranspor and are gven for ordnary srucural concree: W80 expresses oal mass of free waer a =80%. Sandard value s 50 kg/m 3 for srucural concree. A W s waer absorpon coecen, whose value spans he range kgm h 0.5 ]. MI WV s he waer vapor duson ressance facor, spannng 0-60 [-] for srucural concree. Parameers specfyng specfc hea capacy for concree componens are summarzed n Table Values are obaned from hp:// hp:// Parameers specfyng specfc hea conducvy for concree componens are summarzed n Table Sources from hp://www-odp.amu.edu/publcaons/9_sr/09/09_5.hm Concree srengh classes srongly depend on he amoun of cemen n concree. Table specfes approxmae composons for major concree classes used n EN 06-. The calculaon assumes 5 % of enraned ar n he concree, cemen densy 30 kg/m 3 and aggregae densy 800 kg/m 3. Table 7.3- Parameers specfyng densy and specfc hea capacy for concree componens 84

295 Table Parameers specfyng specfc hea conducvy for concree componens Ready-mx concree s assumed, whch requres raher hgher w/c due o workably and pumpng ssues. The parameers CEMENT DENSITY, WATER DENSITY, AGGREGATE DENSITY, FILLER DENSITY are provded n Table 7.3- n he uns [kg/m 3 ]. Table Approxmae composon for major concree classes used n EN06- ATENA Theory 85

296 86 Table CEMHYD3D parameers for fong of affny hydraon model.

297 7.4 Fre Elemen Boundary Load When underakng hea ransfer calculaons s mporan ha relevan hermal properes of maerals and hea ransfer coeffcens a he boundares are defned for he enre emperaure nerval of he load Hydrocarbon Fre Hydrocarbon fres are hose susaned by hydrocarbon-based producs, such as chemcals, gas and peroleum. Dependng on he hea load dfferen HC-curves can be derved n accordance wh Equaon (5.8). The magnude of he maxmum emperaure of he radaon source ( T ) s crucal for he me emperaure developmen. The nomnal HC-curve s represened by he hea load 00 kw/m and reaches maxmum emperaure of 00 C. The curve represenng 345 kw/m s called he "modfed" or "ncreased" HC-curve for unnel applcaons. I reaches a maxmum 300 C. where: T T e e e (5.8) ( ) ( ) T= () emperaure of radaon source as funcon of me [ C], T = maxmum emperaure of radaon source [ C] accordng o (5.8) = me [mnues] Tme developmen of emperaure of radaon source s depced n he fgure below. For me 0 Equaon (5.8) yelds T (0) 0 and hence, s necessary o supplemen (5.8) by requremen T () Tamban, n, where Tamban, n s nal amben emperaure pror he fre broke up, (ypcally somehng abou 0 C). Fg. 7-3 Temperaure of radaon source ATENA Theory 87

298 7.4. Fre Exposed Boundary The naure of he srucural amben condons s essenal for he deermnaon of he emperaure felds. Dependng on he geomery, vew facors and amben condons, varous ypes of boundary condons may be consdered. Fre exposed boundary The hea s ransferred from he fre gas o he exposed srucure hrough radaon and convecon. A hgh emperaures he radaon domnaes. The radaon s expressed by he resulng emssvy facor, whch akes no accoun emssvy of he fre source,, and absorpvy of he heaed surface,. The convecon s calculaed from he emperaure dfference beween he srucure and amben gas, dependng on he gas velocy. Emssvy and convecon facors used for exposed surfaces are shown below 0.56, [ ] h r c 50, W mk The convecon and emssvy hea flux on a boundary exposed o fre s calculaed as follows: where 4 4 n c( g b) r ( g b ) = Sefan-Bolzmann consan [5.67x0-8 W/m K 4 ], T g = absolue emperaure of radaon source [K], T b =boundary emperaure of he srucure, (5.83) q h T T T T (5.84) r = resulng emssvy facor of he radaon source and he heaed surface [-], q n = hea flow a he fre exposed boundary [W/m ], h c = convecon hea ransfer coeffcen [W/m K]. Adabac boundary Adabac boundary surface refers o a boundary surface, where no hea can pass n (and/or ou) he srucure. Srucural symmery lnes and areas are good example of hs boundary condons Implemenaon of Fre Exposed Boundary n ATENA The descrbed fre boundary load condons are ATENA modelled by CCFreElemenBoundaryLoad load. I s essenally an elemen boundary load ha apples he hea flow q n a he elemen boundary,.e a a surface exposed o fre. Unlke oher loads n ATENA (ha are of ncremenal naure and consan whn one load sep) hs load s consdered varable and has knd of a oal load. Four ype of hea source defnons are mplemened: 88

299 Nomnal HV fre Temperaure of he hea source s calculaed by (5.8) and T (unless s manually npu as emp_g_ref) s se o 00 [ C]. Modfed HC fre Ths defnon s much he same as he above wh he only dfference ha defaul value for T s 300 [ C]. Generc fre, (also refered o as User curve fre) - Temperaure of he hea source s assumed consan and s se value of emp_g_ref. If emp_g_ref s no npued, hen 00 [ C] s used. In any case, he generaed (or drecly npued) curve for T () can be addonally modfed n me by a user suppled funcon me_d. The funcon akes one parameer, whch s me of he fre and specfes a coffecen, by whch he orgnally generaer (or npued) boundary condons should be mulpled. Of course, load varaon n space can be modfed by coeff_x, coeff_y coeffcens ec. n he same way as for any oher generaed elemen load, (for more deals see Aena Inpu fle manual). 7.5 Mosure-Hea Elemen Boundary Load 9 Ths ype of boundary load s used for modellng hea and mosure fluxes from he srucure o he amben ar. Hence, s ypcally appled as a boundary elemen load on exernal surfaces of he srucure. I resembles fre boundary load descrbed above and s mplemened n smlar way. The mosure flux consss of hree pars. qh, hcw( hg hb) q ( x x ) h, g b,5,5 ' h,3.38 Cb 3 g Ck q E T h T v [kg/m s] (5.85) 5 5 ' qh,3 ransform _ uns( q h,3) q q q q h h, h, h,3 The frs par q h, ncludes mosure flux drven by graden beween amben and surface relave humdy hg and h b. hcw sands for mosure convecon coeffcen of he concree-ar nerface. The second par q h,, accouns for mosure flux due evaporaon drven by graden of humdy ar rao a he nerface,.e. x b and kg (59 v), ms xg wh he evaporaon coeffcen. By defaul, where v s amben ar velocy, [m/s]. For more nformaon see hp:// 9 Avalable sarng from ATENA verson 5. ATENA Theory 89

300 The humdy ar rao, [-] s calculaed as follows, ( reflecs condons n amben ar,.e. =g, or n surface of he srucure.e. =b): x m m (5.86) a a I s calculaed a sae varables h, T,.e. relave humdy and emperaure a condons. In he above m,, m, s mass and densy of waer vapour n REV correspondng o a a condons and mass and densy of dry ar, [kg/m 3 ], respecvely. where a a R M a a ( T 73.5) p (5.87) M s wegh of kmol of dry ar, (assumed M 8.96 kg/kmol). R s gas consan, (R=833JK - ), T s emperaure n o C. a pa s paral pressure of dry ar, [Pa] p p h p (5.88) a vw. sa Here p sands for oal ar pressure, (ypcally normal ar pressure p=035pa), h s relave humdy and pvw. sa s paral pressure of sauraed waer vapour a T, (see hp://en.wkpeda.org/wk/densy_of_ar) p wv sa 7.5T T 37.3, (5.89) The hrd par s mosure flux evaporaed from concree calculaed by CEMSTONE, see hp:// I yelds nearly he same values as provded by ACPA calculaor, see hp:// TCg, T Cb s amben and surface emperaure n Celsa. The hea flux consss also from wo pars. q h T T T T qt qhhwe q q q 4 4 T ct( g b) rt( Kg Kb) T T T (5.90) The frs par of he hea flux q T represens usual flux due o hea convecon and emsson. Is compuaon resembles (5.84). h ct sands for hea convecon coeffcen of he concree-ar W nerface mk, rt s hea emssvy coeffcen [-], TKg, T Kb are amben and surface W emperaures n Kelvns and s Sephan-Bolzmann consan, 5.67E 8, 4 mk. The second par akes no accoun hea consumpon due o evaporaon of mosure flux. By defaul 90

301 kj hwe 70, kg s assumed. More nformaon avalable a hp:// Boh mosure and hea fluxes are ypcally compued usng only her frs or second par. Therefore, he relaed ATENA npu commands allows o read some boolean flags ha specfy, whch pars of he above fluxes should by accouned for and whch should be skpped. For more nformaon refer o he ATENA npu fle manual. 7.6 References BAZANT, Z. P. (986). Mahemacal Modellng of Mosure Dffuson and Pore Pressure, Chaper 0. Concree a Hgh Temperaure. Z. P. Bazan: BAZANT, Z. P. and W. THONGUTHAI (978). Pore Pressure and Dryng of Concree a Hgh Temperaure. Proceedngs of he ASCE. CELIA, M. A. and P. BINNING (99). "A Mass Conservave Numercal Soluon for Two- Phase Flow n Porous Meda wh Applcaon o Unsauraed Flow." Waer Resour. Res 8(0): CELIA, M. A., T. BOULOUTAS, e al. (990). "A General Mass-Conservave Numercal Soluon for he Unsauraed Flow Equaons." Waer Resour. Res 7(7): DIERSCH, H. J. G. and P. PERROCHET (998). On he prmary varable swchng echnque for smulang unsauraed-sauraed flows, hp:// swpool/swpool.hm#fef_manuals. HUGHES, J. R. (983). Analyss of Transen Algorhms wh Parcular Reference o Sably Behavour. Compuaonal Mehods for Transen Analyss, Elsever Scence Publshers B.V. JENDELE, L. (00). ATENA Polluan Transpor Module - Theory. Prague, Eded PIT, ISBN X. JENDELE, L. and D. V. PHILLIPS (99). "Fne Elemen Sofware for Creep and Shrnkage n Concree." Compuer and Srucures 45 (): 3-6. REKTORYS, K. (995). Přehled užé maemaky. Prague, Promeheus. SEAGER, M. K. and A. GREENBAUM (988). A SLAP for he Masses, Lawrence Lvermore Naonal Laboraory. WOOD., W. L. (990). Praccal-Tme Seppng Schemes. Oxford, Clarenon Press. XI, Y., Z. P. BAZANT, e al. (993). "Mosure Dffuson n Cemenous Maerals, Adsorbon Isoherms." Advn. Cem. Bas. Ma. : XI, Y., Z. P. BAZANT, e al. (994). "Mosure Dffuson n Cemenous Maerals, Mosure Capacy and Dffusvy." Advn. Cem. Bas. Ma. : ATENA Theory 9

302 ZIENKIEWICZ, O. C. and R. L. TAYLOR (989). The Fne Elemen Mehod, Volume : Basc Formulaon and Lnear Problem. London, McGraw-Hll, 4h edon. 9

303 8 DYNAMIC ANALYSIS ATENA sofware suppor four mehods o execue dynamc analyses. These are: Newmark s mehod, Hughes mehod (Hughes 983), Wlson Modfed Wlson. Noe ha Hughes mehod wh 0 reduces o Newmark s mehod and Modfed Wlson s jus an exenson o Wlson. The governng equaons for dynamc analyss read: Hughes mehod: Mu C α u αu K α u αu α R αr Newmark mehod: Mu Cu Ku R where (Modfed) Wlson mehod: Mu Cu Ku R (5.9) u, u, u s acceleraon, velocy and dsplacemen a me, (smlar for me and ), MCK,, s mass, dampng and sffness marx respecvely, R s vecor of exernal forces,.e. concenraed loads, s Hughes dampng parameer. They are s solved for dsplacemen a me. The dsplacemen, acceleraon and velocy a me s calculaed as funcons of (already known) u, u, u and dsplacemen ncremens u u. If l-h eraon s solved, hen we solve for dsplacemen ncremen u and l u uk k ATENA Theory 93

304 Hughes mehod: Newmark mehod: u u u u v u u u γ u u β β u u u u u u β γ _ γ β β Modfed Wlson mehod: Wlson mehod: u u u u 3u 3( u u u) u u u 6u 6 v_ 6( u u u) u u (5.9) Subsung (5.9) no (5.9) and afer some mahemacal manpulaon he requesed dsplacemen ncremen a eraon l can be calculaed: u K R (5.93) where (for usng srucural dampng C MM KK) effecve sffness and RHS vecor are: nv eff eff K M K eff M K M K R eff ( M M ) ( K K ) 0 (5.94) The coeffcens above are calculaed usng he followng expressons. They are summarzed (by soluon mehod): 94

305 Hughes mehod: α M γ αm γ M α M u M u β β β β u β αu u β α Kγ K α K u K u β β γ M α Kγ αu u β α M γ α M γ M α M u M u β β β β u αγm γm M β α Kγ α Kγ K α K u K u β β u K 0 α γ β R ( α) R α F (+α)+ F α K γ K (5.95) Newmark mehod: Mγ Mγ M M u M u β β β β u β u β γ Kγ Kγ u K K u K u β β β M γ Mγ Mγ M M u M u β β β β u γ M M β 0 Kγ Kγ K K u K u β β u γ K K β R F K (5.96) ATENA Theory 95

306 Wlson mehod and Modfed Wlson mehod: 3 M 6 M θ θ 3 K K θ 6θ 3θ θ M 6 6θ M M u u θ θ θ θ 3u θ M M θ 3 3θ θ K θ θ K u 3 3 θ θ 3u K K θ Wlson mehod : R θ θ θ θ Modfed Wlson mehod 0 R F F 3 3 F R θ θ θ θ 0 F 3 3 6θ θ 3 K θ u 3 3 θ θ (5.97) The parameers, are negraon parameers used by Newmark and Hughes mehod. Ther value s essenal for convergence of he hs me marchng scheme. I can be shown ha, corresponds o lnear acceleraon whn he me sep. Values, 6 4 yelds consan acceleraon. The negraon scheme s uncondonally sable, f, 0.5( ) and s only condonally sable for, 0.5( ) provded ha he sably lm s fulflled: cr where s modal dampng parameer. (5.98) The above defnes he condon for me ncremen for a lnear condonally sable case: 96

307 Tn (5.99) As for Wlson and Modfed Wlson mehod hey use parameer. Is value s and he scheme s uncondonally sable for.4. I essenally specfes me, for whch me we calculae he governng equaons (5.9),.e. For Wlson and Modfed Wlson mehod yeld he same soluon expressons and equaons and hese are also he same as hose for Newmark and Hughes mehods wh,, 0. 6 Modfed Wlson mehod assemble he governng equaons for me. As a resul, all Von Neumann boundary condons mus be gven for, (e.g. concenraed load, load by MASS_ACCELERATION ec). I does no apply o Drchle boundary condons ha are (as usually) npu for, (e.q. prescrbed dsplacemen, acceleraon ec.). The fac ha Modfed Wlson mehod execues for also affecs oupu/draw of resuls n srucural maeral pons. Whn eraons, (e.g. for monors a eraons) hey are prned/drawn for. Afer he eraons process has been compleed, hey are prned/drawn for as usually. Inernal forces are always prned for and he same for exernal forces. As descrbed above Modfed Wlson mehod behaves n a b nonsandard way. Parcularly npu of R s unpraccal. To allevae hese dffcules and nconvenence Aena also offers Wlson mehod. Alhough sll solves he governng equaons for me, uses several exrapolaon, (e.g. R R ( R R ), F F ( F F )) so ha suffces wh R and F only. Consequenly, npus all boundary condons and prn/draw all resul for akn o any oher soluon mehod for dynamc analyss. On he oher hand s a prce of accuracy because he exrapolaon s lnear whereby he loadng and nernal forces s no! Remnd ha for dynamc analyss concenraed forces, elemen body/boundary load.. s npu n ncremenal form and s "cummulaed" n he srucure. The same apples for prescrbed dsplacemens. Prescrbed veloces, acceleraons... mus be npu as oal load. MASS_ACCELERATION mus be also npu n oal values(and n each sep s also recalculaed from scrach. More deals on he mehods convergency can be found n (Hughes 983) and (Wood. 990). 8. Srucural Dampng As far as dampng marx C s concerned, Aena uses he well known proporonal dampng: ATENA Theory 97

308 C M K (5.00) M where, are user defned dampng coeffcens. These coeffcens can be drecly se as M K user npu daa or hey can be generaed based on knowledge of modal dampng parameers. The parameers are defned by: where: s -h srucural egenvecor, s -h srucural egenmode, K T T C ( M K ) (5.0) M K s modal dampng parameer assocaed wh and. T T Usng he fundamenal properes of egenmodes M, K we can rewre (5.0): (5.0) M K Equaons (5.00) nroduces parameers for dampng and hus, f only values of are o be used, hey are drecly subsued n (5.0), (resp. (5.0)) and solved for from hs se of equaons. However, n pracce srucural dampng s more complcaed and some sor of compromse mus be done. In hs case srucural dampng properes are ypcally measured for more egenmodes and opmal values of coeffcens, are calculaed by leas square mehod,.e. we are seekng mnmum of he expresson equaons M K. I yelds he followng se of M K w w M Kw w w w M K whch s used o calculae he requred dampng parameers,. M K (5.03) There exss oher assumpons o accoun for srucural dampng, however her use s ypcally sgnfcanly more complex and more cosly n erms of boh RAM and CPU. 98

309 8. Specral analyss A proper selecon of he soluon me ncremen d s essenal for each dynamc analyss. If s oo large, he compued resuls wll suffer from unaccepable naccuraces. We wll probably mss some mporan peaks n he loadng hsory and he analyss as a whole may even dverge. On he oher hand, use of a oo small value of d wll yeld an analyss ha s ponlessly expensve n erms of execuon me and s demands owards CPU/RAM resources. In addon, s posprocessng s more laborous and prone o errors. The specral analyss descrbed n hs secon s desgned o asss he engneer n seng suable d. The man dea of he procedure s based on approxmaon of he srucural loadng f () by Fourer seres fft ( ),.e. f () fft (), refer e.g. o hp://en.wkpeda.org/wk/fourer_seres. Boh f ( ), fft ( ) have one ndependen varable, whch s srucural me. The funcon f ( ) s assembled n he followng form: FT N a0 fft () an sn nbn cos n (5.04) n T T where N denoes number of harmoncs used for he approxmaon, n s harmonc-h d and sn n T and cos n are n-h approxmaon funcons, (.e. n-h harmncs). Eqn. T (5.04) s suable for approxmaon of a funcon (e.g. f () ) n nerval 0... T. Is Fourer coeffcen are calculaed as follows, see hp://selweb.asu.cas.cz/~slecha/fourer/fourer.hml, hp:// T a0 f( ) d 0 T an f( )sn n d T 0 T bn f( )cos n d T 0 (5.05) Now le us nroduce a coeffcen c a b and creae a specrum dagram of he loadng. n n n ' For each harmonc from (5.04) plo a pon, whose coordnaes are n, cn T. Such a pon ATENA Theory 99

310 shows, how much mporan s he nh harmonc (.e. he harmonc wh crcular frequency n ) for he loadng funcon,.e. how much s exced by he load funcon f ( ). T The recommended soluon me ncremen should be se so ha he hghes mporan harmoncs are negraed n abou 0 seps,.e. T d mn( nsgnfcan ) (5.06) 0 By defaul he FFT analyss uses full modal specrum,.e. n.. N n (5.04). However, he modal specrum can be flered, e.g. n n.. m, n.. m,... nk.. mk,... nl.. m. In hs case only values n from whn he L nervals are used. Ths echnque can be used o fler ou some nose sgnals, ec. Le's ake an example: Assume a smplfed ElCenro accelerogram loadng condons, whose acceleraon n me are depced by he green lne n he fgure below: Le's approxmae hs funcon by Fourer seres. In he frs case we use 300 harmoncs,,e, N 300. The approxmaed acceleraons are shown by he blue lne, see he fgure above. In he second case, we use only 50 harmoncs and he correspondng approxmaon funcon s drawn by he red lne. Plong he funcons n more deal, can be seen ha he approxmaon wh N 300 s farly accurae whls he approxmaon wh N 50 s raher crude, see he fgure below. Ths concluson s endorsed by calculaed average relave absolue error of he approxmaons. These are respecvely and

311 The specrum dagram below shows conrbuon of ndvdual approxmaon harmoncs. I deecs wha harmoncs are or are no mporan. Lookng a he dagram we see ha he hghes mporan harmoncs s he one wh log0( Tn ),.e. Tmn 0.. Therefore, he recommended Tmn soluon me ncremen s d 0.0. Ths d should ensure reasonably accurae resuls 0 from dynamc analyss of a srucure ha s loaded by he nvesgaed accelerogram. ATENA Theory 30

312 The descrbed specrum analyss s fully suppored by Aena, (ncl. all he plos). Is use s smple as requres only a few npu commands. For more deals please refer o he examples of commands for npu of a mullnear funcon (n he Aena npu fle documenaon). 30

313 9 EIGENVALUES AND EIGENVECTORS ANALYSIS Ths secon descrbes mehods used by ATENA sofware o calculae srucural egenvalues and egenvecors. Good knowledge of egenmodes of a srucure s n many cases essenal for undersandng s behavour and selecon of a mehod for s furher analyss. I apples o sacs and parcularly o dynamc analyses, n whch case helps choosng a proper me ncremen n subsequen loadng seps. I also help n avodng or reducng oscllaon of he srucure. 9. Inverse Subspace Ieraon Currenly ATENA uses Inverse subspace eraon mehod o compue he egenvalues and egenvecors. The mehod s n deals descrbed n (Bahe 98) and hence, only s man feaures are presened here. The curren mplemenaon can be used only of symmerc marces. The same apples abou Jacob and Raylegh-Rz mehod ha are menoned laer n hs secon. I consns of hree mehods, each of hem s capable of solvng egenvalue problem on s own. However, f hey are used smulaneously, hey yeld a very effcen scheme for solvng egenvalues and egenvecors of large sparse srucural sysems. The sgnfcan advanage of hs approach s ha s possble o search for a seleced number of he lowes egenmodes only. The lowes egenmodes are ypcally he mos mporan for behavor of he srucure because hey represen he hghes energy ha he srucure can absorb. On he oher hand, he hghes egenmode are of low mporance, can be negleced and hereby save a lo of CPU me and oher compuaonal resources. The Inverse subspace eraon consss of Inverse eraon mehod Raylegh-Rz mehod Jacob mehod I solves general egenvalues and egenvecor problem of he followng form: where K,M s sffness and mass marx of srucure, u s vecor of egenvecor s nodal dsplacemens, s crcular egenfrequency Ku M u (8.) We are lookng for a non-rval soluon, so ha we solve for ha comes from de( ) 0 K M (8.) ATENA Theory 303

314 9.. Raylegh-Rz Mehod Ths mehod s used o ransform he orgnal egenproblem of dmenson n no an assocaed egenproblem of dmenson m<<n. The soluon s search for n a space Vm Vn. Le vecors k consue lnearly ndependen bases n V n. An egenvecor u s compued as a lnear combnaon c of he base vecors k,.e. where Ψ s marx of base vecors k, k.. m, u Ψ c (8.3) c s vecor of coeffcens of he lnear combnaon. Raylegh quoen s defned as T u Ku ( u ) T u Mu I can be proved ha ( u ) converges from upper sde o he correspondng crcular frequency. The condon of mnmum of ( ) yelds: u (8.4) ( u ) 0, k.. m c k, (8.5) where c k, s k componen of he vecor c If we nroduce he condon (8.5), afer subsung (8.6), resuls n T T A Ψ KΨ, B Ψ MΨ (8.6) Ac B c (8.7) whch s an equaon of egenproblem of marces A,B. Ths problem has dmenson m, whch s sgnfcanly smaller han he orgnal dmenson n. 9.. Jacob Mehod Jacob mehod s used for soluon of full symmerc egensysems of lower dmenson. In he frame of Inverse subspace eraon mehod s used o solve (8.7). (Noe however, ha ha he egenproblem (8.7) can be used by any oher mehod). The Jacob mehod s based on he propery ha f we have a marx A, a orhogonal marx C and a dagonal marx D, whereby T CACD (8.8) 304

315 hen he marces A and D have dencal egenvalues and hey are dagonal elemens of he marx D. The ransformaon marx C s calculaed n erave manner where he ndvdual C S S... S, k.. (8.9) S k has he followng form k Sk 0 cos( ) 0 sn( ) sn( ) cos( ) The enres cos( ), sn( ) are pu n,j rows and columns and hey are consruced n he way ha hey wll zeroze a j afer he ransformaon. The oher dagonal elemens are equal o and he remanng off-dagonal elemens are 0. In he case of general egenproblem he whole procedure of consrucng S k s very smlar. The marces S k now adop he shape Sk a b Noce ha he marx S k s no orhogonal anymore. The wo varables a,b are calculaed o zeroze off-dagonal elemens,j of he boh marces K and M. Egenmodes of he problem are hen calculaed as ' ' (8.0) (8.) a (8.) b where a, b are dagonal elemens of ransformed (and dagonalzed) marces A, B. ' ' Egenvecors of he problem are columns of he ransformaon marx C Inverse Ieraon Mehod Inverse eraon mehod s carred ou as follows: Sarng wh an nal ransformaon of egenvecor u, we calculae vecor of correspondng nera forces, (sep ) ATENA Theory 305

316 f M u (8.3),, Knowng f,, we can compue a new approxmaon of u, (sep ) u K f (8.4),, and repea he sep. Hence, for eraon k we have f M u k, k, u K f k, k, (8.5) and he erang s sop, when uk, uk,. The above descrbed algorhm ends o converge o he lowes egenmodes. If any of hese are o be skpped, he nal egenvecor u, mus be orhogonal o he correspondng egenvecors. In pracce, he vecor u k, mus orhogonalzed wh respec o he skpped egenvecors even durng he erang procedure, as he nal orhogonaly may ge (due o some round-off errors) los Algorhm of Inverse Subspace Ieraon Havng brefly descrbed he above hree mehods we can now proceed o he acual soluon algorhm of Inverse subspace eraon mehod self: Sep-Inverseeraon mehod: KU MU k k Sep -Raylegh quoen mehod: T Ak U kku k B U MU T k k k (8.6) Sep3 Jacob mehod: A C B C Δ k k k k In he above Sep 4- Correc he egenvecors: U U C T T k k k m s number of projecon egenmodes, (reasonably hgher han he number of requred egenmodes), U k s marx of columnwse arranged m envecors afer k- h eraon, A k, B k are ransformed sffness and mass marces of he problem, (havng dmenson m<<n), C k s marx of egenvecors of k A, B k, see (8.9) 306

317 Δ s marx wh egenmodes (on s dagonal). Noce ha egenmodes for ransformed and he orgnal egenmode problem are he same. The seps hru 4 are repeaed unl he dfference beween he wo subsequen operaons s neglgble. The soluon algorhm (8.6) s n ATENA a b modfed n order o reduce CPU me and RAM resources and s descrbed below: Sep-Inverseeraon mehod: Uˆ MU k KU Uˆ k k k Sep -Raylegh quoen mehod: A U KU U Uˆ Uˆ T T k k k k k MU k k B U MU U Uˆ T T k k k k k Sep3 Jacob mehod: A C B C Δ k k k k Sep 4-Correc he egenvecors: T T U U C k k k The advanage of hs procedure over he one defned n (8.6) s ha now you don need o sore he orgnal and facorsed form of he marx K. Only he facorsed form s needed durng he eraons. A specal ssue n hs mehod s how o seup he nal vecors U. Ths s wha we do n ATENA. The frs vecor conans he dagonal elemens of M. The nex vecors are consruced n he way ha hey have zeros everywhere excep one enry. Ths enry m correspond o maxmum and s se o. k The procedure as s, (because of Inverse eraon mehod), canno solve for zero egenmodes. Ths may be a problem, especally f we wan o analyze srucural rgd body moons or spurous energy modes. If hs s he case, shf marx K by an arbrary value s,.e. solve he assocaed egenproblem s s s s The orgnal egenvalues and egenvecors are hen calculaed by (8.7) ( K M) u M u (8.8) ATENA Theory 307

318 u u s s Anoher problem of Inverse subspace eraon s o compue mulple egenvecors. Unforenaly, s no ha rare case and happens e.g., f he srucure has an axs of symmery. Occurrence of mulple egenmodes n he srucure may yeld non-orhogonal egenvecors and hus some egenmodes can be mssed. There are some echnques for resolve hs problem (Jendele 987), however, hey have no been mplemened n ATENA ye. Good news s ha n realy no egenmodes are usually que dencal due o some round-off errors. The case of mulple srucural egenmodes hus ypcally causes only some worsenng of accuracy and no egenmode ges mssed. Neverheless, f we wan o be sure ha no egenmode was mssed, we can assess by Surm sequence propery es. s (8.9) 9..5 Surm Sequence Propery Check Ths propery says (Bahe 98) ha f we have an egenproblem (8.), perform a shf s and facorse ha marx, (.e. D s dagonal marx, L s lower rangular marx ), T K s M LDL (8.0) hen number of negave dagonal elemens n D equal o he number of egenvalues smaller han he shf. Ths way we can smply es, wheher we mssed an egenvalue wh he calculaed se of m egenmodes or no There are oher mehods ha can be used o compue egenvalues and egenvecors of large sparse egensysem. Parcularly popular s e.g. Lanczosh mehod (Bahe 98). There exs also several enhancemens for he presen Inverse subspace eraon mehod. For nsance usng shfng echnque may sgnfcanly mprove convergency of he mehod, (especally f some egenvalues are close each oher). These mproved echnque may be mplemened n he fuure. In any case, he curren ATENA mplemenaon of egenmodes analyss proves o solve he egenmodes problem n mos case que successfully. 9. References BATHE, K. J. (98). Fne Elemen Procedures n Engneerng Analyss. Englewood Clffs, New Jersey 0763, Prence Hall, Inc. JENDELE, L. (987). The Orhogonalzaon of Mulple Egenvecors n Subspace Ieraon Mehod. IKM - XI. Inernaonaler Kongress ueber Anwendungen der Mahemak n der Ingeneurwssenschafen, Wemar. WOOD., W. L. (990). Praccal-Tme Seppng Schemes. Oxford, Clarenon Press. 308

319 0 GENERAL FORM OF DIRICHLET BOUNDARY CONDITIONS A unque feaure of ATENA sofware s he way, n whch mplemens Drchle boundary condons. I suppors o consran any degree of freedom (DOF) by a lnear of any number of oher srucural DOFs. The proposed mehod of applyng and processng he boundary condons s compuaonally effcen and memory economcal, because all consran degrees of freedoms (DOFs) are elmnaed already durng assembly of srucural global sffness marx and load vecors. The adoped concep has wde range of use and several s possbles are dscussed. A he end of he Secon a few samples are gven. 0. Theory Behnd he Implemenaon A crucal par of a ypcal fne elemen analyss, (wheher lnear or nonlnear) s soluon of a se of lnear algebrac equaons n he followng form where n Kj uj r,.. n (9.) j K j s an elemen, j of a predcor marx K, (.e. usually srucural sffness marx), r s an exernal force, (or unbalanced force) appled no -h srucural degree of freedom (DOF) and fnally u s dsplacemen (or dsplacemen ncremen) a he same DOF. Such a se of equaons s always accompaned by many boundary condons (BCs). They can be one of he followng: Von-Neumann boundary condons, (also called rgh-hand sde (RHS) BCs). Number and ype of hese BCs has no mpac on dmenson n of he problem (9.). They are accumulaed n he vecor r. Ths vecor s assembled on he per-node bass for concenraed nodal forces and/or per-elemen bass for nodal forces beng equvalen o elemen loads. The second ype of boundary condons are Drchle boundary condons, (also called lefhand sde (LHS) BCs). ATENA mplemenaon of hs ype of BCs s now descrbed. A smple form of such BCs reads: ul 0, l, n (9.) u u, l, n l l0 These knds of BCs ypcally represen srucural suppors wh no dsplacemens, (he frs equaon) or wh prescrbed dsplacemensu l0, (he second equaon). Alhough mos LHS BCs are of he above form, (and only a few fne elemen packages offer anyhng beer), hey are cases, when a more general LHS BC s requred. Therefore, ATENA sofware provdes soluon for mplemenng a form of Drchle BCs, where each degree of srucural freedom can be a lnear combnaon of any oher degrees of freedom. Mahemacally, hs s expressed by: ul ul0 lkuk, l, n (9.3) k, n ATENA Theory 309

320 There are many cases, n whch he above form of Drchle condons proves helpful. Some examples are dscussed laer n he Chaper. The mporan pon abou mplemenng Equaons (9.3) s ha hey are ulsed already durng assemblng of he problem (9.). I means ha, f we have m of hese BCs, hen fnal dmenson of he marx K becomes only ( n m). Ths fac sgnfcanly reduces requremens owards compuer sorage. In he followng we shall call such boundary condons as Complex Boundary Condons, or CBCs, (see also ATENA Inpu fle manual, where he same name s used). 0.. Sngle CBC The procedure of mplemenng Drchle BCs of he form (9.3) s now presened. Le us sar wh jus one BC equaon (9.4). I says ha u l equals o a consan prescrbed dsplacemen u l 0 plus lk mulple of a dsplacemen u k. Subsung (9.4) no he Equaon (9.) yelds n n u u u (9.4) l l0 lk k K u K u K u K ( u u ) r,.. n (9.5) j j l l j j l l0 lk k j, jl j, jl whch afer some manpulaon can be smplfed no he form n j j l lk kj j l l0 K K u r K u,.. n (9.6) The above se of equaons could be already used o solve for he unknown dsplacemens (or dsplacemen ncremens) u. sands for kj. Kronecker dela ensor. The rouble s, j kj however, ha even hough he marx K mgh be symmerc, he se of equaons (9.6) s no symmerc anymore. Thus, o preserve he symmery, add an lk mulple of he row l,.e. o he row k,.e. n lk Klj Klllkkj uj lk rl Kllul 0 (9.7) j n j Ths resuls n he row k geng he form: n j n Kkj Kkl lk kj uj rk Kklul 0 (9.8) Kkj lk Klj Kkllk lklk Kll kj uj (9.9) j K K K K u kj kl lk kj lk lj ll lk kj j r K u r K u k kl l0 lk l lk ll l0 30

321 Hence, he fnal form of he governng se of equaons wll read: n j K K K K u j l lk kj k lk lj k kj lk ll j r K u r K u l l0 k lk l ll l0 The above equaons can be wren as (9.0) where K n K j uj r,.. n (9.) j K... K... K k K llk... K j... K n K... K... Kk Kllk... Kj... K n Kklk Kl... Kk lk Kl... Kkk Kkllk kkkklk Kll... Kkj lk Klj... Kkn lk K ln K j... K j... K jk K jllk... K jj... K jn Kn... Kn... Knk Knllk... Knj... K nn (9.) r r K lul0... r Klul0... r K u r K u... rj K jlul0... rn Knlul0 k kl l0 lk l ll l0 (9.3) Provdng he orgnal marx K s symmerc,.e. Kj K j, hen he marx K s now also symmerc,.e. K j K. j The dsplacemen u l consraned by Equaon (9.4) has a consan par u l 0 and a varable par lk u k, n whch u l depends only on a sngle u k. A more general form of hs BC would be, f u l depends on more dsplacemens. I corresponds o he followng form of he boundary condon: ATENA Theory 3

322 u u u (9.4) l l0 lk k k In hs case, he dsplacemen u l s calculaed as a consan par u l 0 plus a lnear combnaon lk of dsplacemens u k. k can be any dsplacemen,.e. k.. n. Replacng BC defned by Equaon (9.4) by he above Equaon (9.4), he equaon wll change o he form n Kj Kl lk kj k lk Klj k kj lk Kll uj j k, kl r K u r K u l l0 k lk l ll l0 kk, l (9.5) 0.. Mulple CBCs The prevous paragraph derved all he necessary relaons for mplemenng a sngle boundary condon. Now we wll proceed o he case of mulple boundary condons. Each parcular BC s agan wren n he form (9.4). ul ul0 lkuk, l, n, l { l, l,... lr} (9.6) k The problem s, however, ha dsplacemens u k n (9.6) need no be free bu fxed by anoher BC, k can run also hrough l, (resulng n a recursve formulaon), more BCs can be specfed for he same u l, a parcular BC can be specfed more mes and n more forms ec. For example, we may have a se boundary equaons ha conans BCs or can conan u u, u u (9.7) u u, u u, u (9.8) Boh of hese are correc. Unforunaely, he se can also conan u u, u 0.003, u u, u (9.9) whch s defnely wrong. Therefore, before any use of such se of BCs s necessary o deec and fx all redundan and conradcory mulple BCs presen n. I s easly done n case of a smple se of BCs lke he one above, bu n real analyses wh housands of BCs n he form (9.6), (some of hem que complex,.e. k runs hrough many DOFs) he only way o proceed s o rea (9.6) as a se of equaons o be solved pror her use n (9.3). Redundan BCs are gnored and conradcory BCs are fulflled afer her summaon. Le us suppose ha all srucural consrans are specfed n he se of equaon (9.6). Ths can be wren n marx form u u A u (9.0) l l0 k The above relaonshp represens a sysem of algebrac lnear equaons. The sysem s ypcally non-symmerc, sparse and has dfferen number of rows (.e. number of BCs) and columns, (.e. number of maser and slave DOFs). Moreover, s ofen ll-condoned, wh a 3

323 number of equaons beng lnear combnaons of he ohers, e.g. see he example n (9.7). A he begnnng s ofen no known, whch DOF s dependen, (.e. slave) and whch s ndependen, (.e. maser), (e.g. see also (9.7)). Based on he above properes he followng procedure has been developed o solve he problem (9.0):. Allocae "columns" for all slave and maser DOFs,.e. loop hrough all BCs n (9.6) and allocae DOFs for boh slave (.e. LHS) and maser (.e. RHS) dsplacemens u.. Allocae sorage for he marx A and vecors ul, u l0 n (9.0). The marx has l r number of rows (see (9.6)) and l c number of columns. l c s dmenson of he DOFs map creaed n he pon add.. 3. Assemble he marx A and he vecors ul, u l0. 4. Deec consan BCs,.e. ul ul0 and swap rows of A so ha he rows correspondng o consan BCs are pushed o he boom. 5. Deec consan fxed DOFs,.e. hose wh lk 0 and varable fxed DOFs,.e. ha are hose dependen on oher (maser) DOFs and havng lk Swap columns of A, so ha he former DOFs are pushed o he rgh and he laer DOFs o he lef. The operaons descrbed a he pon 5 and 6 are needed o assure order, n whch he consraned DOFs are elmnaed. Ths s mporan for laer calculaon of he srucural reacons. 7. Usng Gauss mehod rangulze he se of BC equaons. The rangulzaon s carred ou n he sandard way wh he followng dfferences. a. Before elmnang enres of A locaed n column below a kk, check for a nonzero enry n he row k. If all s enres are zero, hen gnore hs row and proceed o he nex one. (I s he case of mulple BCs havng he same conen). b. Check, wheher he row k specfes BC for consan or varable DOF, (see explanaon n he pon 5 above). In he former case push he row k o he boom and proceed o he nex row. c. Swap columns k... l c so ha abs( akk ) becomes maxmum. d. If akk 0, swap lnes k... l r swap columns k... l c o fnd a non-zero enry n a kk o fnd maxmum a kk. e. Elmnae enres below a kk as usually.. Thereafer, As was already menoned, he marx A s ypcally very sparse. Hence, a specal sorage schemes s used ha sores only non-zero enres of A. The daa are sored by rows. Each row has a number of daa seres,.e. sequences or chunks of consecuve non-zero daa (whn he row). The daa are n a hree-dmensonal conaner. For each such chunk of daa we also need o sore s frs poson and lengh. Ths s done n wo wo-dmensonal conaners. ATENA Theory 33

324 As an example, suppose ha we have he followng marx A: a a a3 0 0 a6 a a A 0 a4 0 a (9.) a55 a56 a 57 0 a a a 77 I s sored as follows ( A. daa sores he acual daa, A. rowbase sores base ndces for nonzero enres n rows, Arowlengh. conans dmenson of non-zero daa chunks; all arranged by rows): A. daa()()() a A. daa()()() a, A. daa()()() a, A. daa()()() a, A. daa()()() a A. daa(3)()() a33, A. daa(4)()() a4, A. daa(4)()() a44... A. rowbase()() A. rowbase()(), A. rowbase()() 6 A. rowbase(3)() 3 A. rowbase(4)(), A. rowbase(4)() 4... A. rowlengh()() A. rowlengh()(), A. rowlengh()() A. rowlengh(3)() A. rowlengh(4)(), A. rowlengh(4)() A number of opmsaon echnques are used o speed up he process of rangularzaon of he marx A. These are summarzed below: The daa are sored by rows and he elmnaon s also carred ou by rows. (Row-based sorage s also more convenen durng assemblng he A from (9.6)). All he operaons needed for he elmnaon are carred ou only for nonzero daa. Ther horzonal poson s sored n A. rowbase and A. rowlengh, hence s no problem o skp all zero enres. A ypcal oal number of columns l c, see (9.6), s of order from housands o hundred housands DOFs. On he oher hand a. rowlengh s on average only of order of ens. Ths s where he CPU savngs comes from. By he way, he same mappng of non-zero enres s also used for columns. Ths s acheved by addonal arrays A. columnbase and Acolumnlengh. ha are also ncluded n he sorage scheme A. (Ther consrucon s smlar o A. rowbase and Arowlengh. ; nsead by rows (9.) 34

325 hey are arranged by columns). These wo addonal arrays make possble o skp all zero enres durng column-base operaons. The resulng sgnfcan ncrease of rangularzaon speed pays off for a small amoun of an exra RAM ha s needed o sore Acolumnbase. and Acolumnlengh.. The adoped procedure of rangularzaon many mes swaps lnes and/or columns of A. In vew of he adoped sorage scheme can be que expensve procedure. To allevae hs problem, he sorage scheme ncludes four addonal arrays, namely A. rowndex, A. rownversendex, Acolumnndex. and A. rownversendex. A he begnnng A. rowndex( ) and smlarly A. rownversendex( ),... l c. When a row r should be swapped wh a row r, he daa n A. daa remans unchanged and we swap only correspondng row ndces n A. rowndex, (and accordngly also enres n he array for nverse mappng A. rownversendex ). The same sraegy s used for swappng he columns. As a resul any swappng operaon does no requre any movng of acual daa, (excep of swappng correspondng ndces for mappng he rows and columns) and hus s exremely fas. On he oher hand, n order o access a j we mus use a ' j', where ' rowndex( ) and j ' columnndex( j). The ncurred CPU overhead s well accepable, because he marx A s very sparse. 0. Applcaon of Complex Boundary Condons Ths secon presens several examples, where he developed Drchle boundary condons are advanageously used. In each case he correspondng fne elemen model explos he general form of BC defned by Equaon (9.6). 0.. Fne Elemen Mesh Refnemen Suppose we need o refne a mesh as shown n Fg. 0-. The mesh should refne from 5 elemens per row o 0 elemens per row. The fgure depcs hree possble echnques o acheve he goal. In he case A he fne and coarse pars of he mesh (conssng of quadrlaeral elemens) are conneced by a row of rangular elemens. Ths way of mesh refnemen s used he mos ofen. However, mxng quadrlaeral and rangular elemens s no always he bes soluon. In he case B he refnemen s acheved by usng herarchcal fne elemens, see (Bahe 98). The coarse mesh near he nerface employs fve nodes herarchcal elemens. Ths refnemen s superor o he ohers, however, requres a specal fne elemens and specal mesh generaor; boh of hese rarely avalable n a ypcal fne elemen package. In he case C he fne and coarse pars of he mesh are generaed ndependenly. Afer generaon of all nodes and elemens he nerface nodes are conneced by complex boundary condons. For example, we can use u um, uk un, uj 0.5um 0.5un. The man advanage of hs approach s ha s smple for boh fne elemen pre/posprocessor and fne elemen modeller, (namely s fne elemen lbrary). Hence s preferable! ATENA Theory 35

326 Fg. 0- Mesh refnemen Noe ha all he above echnques are suppored n ATENA fne elemen package, he las one requrng mplemenaon of CBCs n he form (9.4). 0.. Mesh Generaon Usng Sub-Regons Ths example demonsraes anoher advanage of usng he proposed CBCs: I s possble o generae meshes whn sub-regons whou requremen of nodes concdence on her nerfaces. Because mesh srucure on he sub-regons surfaces s no prescrbed, hs approach provdes more flexbly o mesh generaon. Ths feaure s heavly used by ATENA 3D preprocessor. Compable meshes on he conac beween he blocks Fg. 0- Mesh generaon from smple blocks 36

327 Incompable meshes on he conac beween he blocks usng CBCs Fg. 0- (con) Mesh generaon from smple blocks In he above example wo blocks are conneced o form a srucure, where he op (smaller) block s placed aop of he boom (larger) block. Poson of he op block s arbrary wh respec o he boom block. Unless he concep of CBCs s used, he meshes on nerface of he wo blocks mus be compable, (see op of Fg. 0-). On he oher hand, he proposed CBCs allow usng of ncompable meshes, (see he boom of Fg. 0-). In hs case he mesh n each block s generaed ndependenly, whch s sgnfcanly smpler. Afer hey are done, he proposed CBCs are appled o connec he nerface nodes. (Typcally he surface wh he fner mesh s fxed o he surface wh he coarse mesh). The laer approach also demonsraes possbly of a mesh refnemen whle sll usng well-srucured and ransparen meshes. Ths s parcularly useful n he case of complex numercal models Dscree Renforcemen Embedded n Sold Elemens In hs example, he descrbed boundary condons are used o smplfy modellng of renforced concree beam, see Fg The procedure o creae he model s as follows. Frsly, he mesh for solds,.e. concree elemens s generaed. I poses no problem, as s a regular mesh conssng of 48 quadrlaeral elemens. A hs pon, no aenon needs o be pad o he geomery of renforcng bars presen n he beam. Thereafer, he renforcng bars are nsered and her meshes are generaed based on he exsng mesh of sold elemens. Frs sep s o fnd all nodes, where he bar changes drecon. These nodes are called prncpal nodes; see e.g. node n n Fg Then, nersecon of all sragh pars of he bar wh underlyng sold elemens are deeced, e.g. he nodes m,p. Thus, all end nodes of embedded bar elemens are defned. The las sep s o lnk dsplacemens of he nodes of he bar o he underlyng sold elemens. ATENA Theory 37