FEM of Time Effects in RC Structures CMM

UNCORRECTEDPROOF12 Time-dependent modelling of RC structures using the3 cracked membrane model and solidication theory4 Kak Tien Chong, Stephen J. Foster, R. Ian Gilbert *5 The School of Civil and Environmental Engineering, The University of New South Wales, Sydney, Australia6 Received 19 July 2007; accepted 30 August 200778 Abstract9 A non-linear nite element model is presented for the time-dependent analysis of reinforced concrete structures under service loads.10 For the analysis of members in plane stress, the model is based on the cracked membrane model using a rotating crack approach com-11 bined with solidication theory for modelling creep. The numerical results are compared with a variety of long-term laboratory measure-12 ments, including development of deections and cracking with time in a reinforced concrete beam, time-dependent change in support13 reactions of a continuous beam subject to support settlement and creep buckling of columns. The numerical results are in good agree-14 ment with the test data.15 2007 Published by Elsevier Ltd.16 Keywords: Concrete structures; Cracking; Creep; Finite elements; Serviceability; Shrinkage1718 1. Introduction19 Most design codes oversimplify design procedures for20 determining the mechanical response of reinforced concrete21 structures under service loads and focus, in the main, on22 instantaneous behaviour. Failure to adequately recognize23 and quantify the non-linear eects of cracking, creep and24 shrinkage can lead to excessive deections and crack25 widths and miscalculation of support reactions.26 The in-service behaviour of a reinforced concrete struc-27 ture depends on many factors, including the quality of28 bond between the reinforcing steel and the concrete. The29 composite interaction between the two materials is estab-30 lished and maintained by the bond stress, which eectively31 transfers load between the steel and concrete. The model32 developed in this research is an extension of the nite ele-33 ment implementation of the cracked membrane model34 (CMM-FE model) of Q1 Foster and Marti [10,11] to include35 the time-dependent deformation in the concrete caused36 by creep and shrinkage.37 When modelling time eects in concrete structures, the38 growth of the concrete tensile strength with time, the reduc-39 tion in volume with time due to shrinkage and the eects of40 creep under load are all important parameters in the estab-41 lishment of a rational model. For the cases of shrinkage42 and the development of concrete strength with time, rela-43 tively simple time functions may be adopted based on con-44 trol test measurements. The modelling of creep, however, is45 more complex as the loading history must also be consid-46 ered. In this study, the solidication model of Bazant and47 his colleagues is used. This creep model is capable of48 accounting for any loading history and includes creep49 recovery [16]. The application of the solidication50 approach to the cracked membrane model is described sub-51 sequently and numerical examples are presented.52 2. Background to the CMM53 Based on the tension chord model [16,18], Kaufmann54 and Marti [15] formulated the cracked membrane model0045-7949/$ - see front matter 2007 Published by Elsevier Ltd.doi:10.1016/j.compstruc.2007.08.005*Corresponding author. Tel.: +61 2 9385 6002; fax: +61 2 9313 8341.E-mail address: [email protected] (R.I. Gilbert).www.elsevier.com/locate/compstrucComputers and Structures xxx (2007) xxxxxxCAS 4159 No. of Pages 13, Model 5+22 September 2007 Disk UsedARTICLE IN PRESSPlease cite this article in press as: Chong KT et al., Time-dependent modelling of RC structures using the ..., Comput Struct (2007),doi:10.1016/j.compstruc.2007.08.005UNCORRECTEDPROOF55 (CMM) for the analysis of reinforced concrete membranes56 subject to in-plane stresses. Stresses on an orthogonally57 reinforced, cracked, element subjected to plane stress are58 shown in Fig. 1. In the CMM formulation, the crack faces59 are taken as stress free and able to rotate, that is60 rcn = scnt = 0 and the nt axes are aligned with the princi-61 pal 12 axes. Further, between the cracks, a portion of the62 tensile forces transfers from the steel to the concrete, via63 bond, and this results in a reduction in the stress in the rein-64 forcing steel and an increase in the tensile stress in the con-65 crete. This is known as tension stiening.66 With these assumptions, the stresses at the cracks are67 given by68rx rc2 sin2h rc1 cos2h qxrsx 1ary rc2 cos2h rc1 sin2h qyrsy 1bsxy rc1 rc2 sin h cos h 1c 70 7071 where h is the angle between a vector normal to the cracks72 and the global x-axis (p/2 6 h 6 p/2); rx, ry and sxy are73 the in-plane normal and shear stresses in the global xy-74 coordinate system, respectively; rc1 and rc2 are the concrete75 stresses in the principal 12 directions (ordered as r1Pr2);76 qx and qy are the steel reinforcement ratios in the global x-77 and y-directions; and rsx and rsy are the steel stresses at the78 cracks in the reinforcement in the x- and y-directions,79 respectively. The principal tensile stress rc1 reduces to zero80 shortly after cracking but is included here to maintain the81 link to concrete fracture. From Eq. (1) and Fig. 2, the mean82 stresses in the steel and concrete are related to the stresses83 at the crack byqxrsx qx rsx rctsx; qyrsy qy rsy rctsy 2 85 8586 where rsx and rsy are the mean stresses in the reinforce-87 ment and rctsx and rctsy are the mean concrete tension sti-88 ening stresses in the x- and y-directions, respectively.89 In the determination of the concrete tension stiening90 components, consider a uniaxial tension chord with a sin-91 gle steel bar of diameter B, as shown in Fig. 2a. Sigrist92 [18] proposed that the bond between the bar and the con-93 crete be modelled using a stepped, rigid-perfectly-plastic94 bond shear stressslip relationship (Fig. 2b). From this95 Sigrist [18] and Marti et al. [16] developed the tension96 chord model and showed that the maximum spacing97 between cracks (which occurs when the concrete tensile98 stress midway between the cracks just reaches the tensile99 strength of concrete fct) is100srm0 fct1 q2sb0q 3 102 102103 If the spacing between two cracks were to exceed srm0, a104 new crack would form midway between the cracks. The105 minimum crack spacing is therefore srm0/2. The maximum106 concrete stress that can develop between two cracks spaced107 srm0/2 apart is fct/2. For any pair of cracks in a tensiontnxy1+x sxyyxxyxxy xy1cnt sin cnt cos cnt sin cnt cos cn sin ct sin ct cos cn cos +ab dcy syFig. 1. Orthogonally reinforced membrane subject to plane stress: (a)applied stresses; (b) axis notation; (c) and (d) stresses at a crack.bdxx AcsrmdxN Nc + dccbs + dssbb0byb1aFig. 2. Tension chord model: (a) dierential element; (b) bond shearstressslip relationship (after [16]).2 K.T. Chong et al. / Computers and Structures xxx (2007) xxxxxxCAS 4159 No. of Pages 13, Model 5+22 September 2007 Disk UsedARTICLE IN PRESSPlease cite this article in press as: Chong KT et al., Time-dependent modelling of RC structures using the ..., Comput Struct (2007),doi:10.1016/j.compstruc.2007.08.005UNCORRECTEDPROOF108 chord with a fully developed crack pattern, the crack spac-109 ing srm is limited by110srm ksrm0 4 112 112113 where 0.5 6 k 6 1.0. For a tension chord of sucient114 length, on average, k = 0.75 and the maximum tension sti-115 ening stress midway between adjacent cracks is kfct.116 In two dimensions, the tension stiening factors are117 given by [15]kx Drcxfct srmsrmx0 cos jhrj ; ky Drcyfct srmsrmy0 sin jhrj 5119 119120 where Drcx and Drcy are the x- and y-component stresses of121 the tension stiening stress; srm is the crack spacing mea-122 sured normal to the cracks and srmx0 and srmy0 are the max-123 imum crack spacings for uniaxial tension in the x- and y-124 directions, respectively, and are obtained by substituting125 qx or qy for q in Eq. (3), as appropriate.126 As the distribution of the bond shear stresses are known127 via the tension chord model, the stresses in the steel and the128 concrete between the cracks can be determined for any129 known stress in the steel at the cracks. Thus the mean130 and maximum steel stress can be expressed as functions131 of the mean (or average) strain, em. Before yielding of the132 reinforcing steel, the mean tension stiening stresses are133 rctsx sbxsb0kxfct2 ; rctsy sbysb0kyfct2 6135 135136 In Eq. (6), sb is the bond shear stress and is given by137sbx emxEs2srmx6 sb0; sby emyEs2srmy6 sb0 7139 139140 where Es is the elastic modulus of the reinforcing steel. The141 mean stress in the steel rsm is related to the mean stress in142 the concrete byrsm rsr rcts1 q=q 8 144 144145 where rsr is the steel stress at the crack. After yielding of146 the reinforcement, the bond strength is reduced and Eqs.147 (6) and (7) require modication. For service conditions,148 however, the steel is rarely at yield and the reader is re-149 ferred to Kaufmann [14] and Foster and Marti [11] for de-150 tails of the calculation of mean steel and concrete stresses151 post yield.152 As discussed by Foster and Marti [10], the crack spacing153 given by Vecchio and Collins [19] is a reasonable approxi-154 mation to the exact solution and is given bysrm cos jhcjsrmx0 sin jhcjsrmy0_ _19156 156157 where the principal stress angle midway between cracks is158 approximated as equal to the angle at the crack (that is159 hc % h). Lastly, with the crack spacing determined, the160 instantaneous crack width is calculated by considering elas-161 ticity across the continuum:162wcr srme1 m12e2 kfct=2Ec 10 164 164165 where e1 and e2 are strains in the major and minor principal166 directions, respectively, m12 is Poissons ratio for expansion167 in the 1-direction resulting from stress in the 2-direction, k168 is the uniaxial tension stiening factor and Ec is the initial169 elastic modulus for concrete. For a fully developed crack170 pattern in a tension chord of sucient length, the maxi-171 mum crack spacing is 1.33 times the average crack spacing172 (k = 0.75 in Eq. (4)) and, similarly, the maximum crack173 width is 1.33 times the average crack width.174 For nite element (FE) implementation, the CMM-FE175 model employs a rotating crack concept [9] that assumes176 that cracks are normal to the principal tensile directions177 and are able to rotate accordingly. Therefore, crack faces178 are taken as free of shear stresses.179 The CMM-FE model has the advantage over previous180 models of being conveniently formulated in terms of equi-181 librium at the cracks, rather than average stresses across182 the element. This provides two distinct advantages over183 earlier compression eld models: rstly, by maintaining184 equilibrium at the cracks the link to limit analysis is pre-185 served; and, secondly, the eects of tension stiening and186 tension softening are decoupled with separate models for187 each. For details of the constitutive relationships used in188 the nite element modelling see Foster and Marti [11].189 3. Instantaneous behaviour of concrete and steel190 The bilinear softening stressstrain model of Petersson191 [17] is used for concrete in tension. It is dened by the ten-192 sion softening parameters a1, a2 and a3 as shown in Fig. 3c.193 The factors a1, a2 and a3 are adjusted for a specic fracture194 energy, Gf, for a given characteristic length. For the case of195 the distributed, smeared, cracking element used in thisccrfctfctuccEc1fcttpu2913+ccEc1fcttp 2 tp1fct 3 tpct rmff sG518=a bcuFig. 3. Concrete in tension: (a) linear-elastic model for uncrackedconcrete; (b) bilinear softening model for cracked concrete [17]; and (c)total tensile stressstrain relationship.K.T. Chong et al. / Computers and Structures xxx (2007) xxxxxx 3CAS 4159 No. of Pages 13, Model 5+22 September 2007 Disk UsedARTICLE IN PRESSPlease cite this article in press as: Chong KT et al., Time-dependent modelling of RC structures using the ..., Comput Struct (2007),doi:10.1016/j.compstruc.2007.08.005UNCORRECTEDPROOF196 study, the characteristic length is the crack spacing and the197 tension softening factors area1 13; a2 29a3 a1; a3 185EcGfsrmf2ct11199 199200 A tri-linear stressstrain model is adopted to model the201 reinforcing steel with the properties as dened in Fig. 4.202 4. Time-dependent behaviour of concrete203 The major factors aecting the time-dependent behav-204 iour of reinforced concrete structures are creep and shrink-205 age of the concrete. At any time t after rst loading, the206 total strain equals the sum of the instantaneous, creep207 and shrinkage strains. That iset ecit ecpt esht 12 209 209210 where eci, ecp and esh are the instantaneous, creep and211 shrinkage strains, respectively, and e ex ey cxy T.212 Before cracking, the instantaneous strain is equal to the213 concrete elastic strain, whereas, post-cracking the instanta-214 neous strain consists of components of the concrete elastic215 strain and the concrete cracking strain, dened in a total216 stressstrain relationship as described by the tension soft-217 ening parameters a1, a2 and a3.218 For nite element implementation, creep and shrinkage219 strains are treated as inelastic pre-strains updated with time220 and applied to the structure as equivalent nodal forces221 given by222P0

Ne_V eBTDe0dV 13224 224225 where Ne is the number of elements used to model the226 structure, B is the strain-displacement matrix, D is the227 material constitutive stiness matrix and e0 is the inelastic228 pre-strain which is the sum of the creep and shrinkage229 strains at time t.230 As time eects due to creep are signicantly aected by231 the age of the concrete at rst loading, the growth in the232 concrete tensile strength with time is an important consid-233 eration. In this study, the change of the tensile strength234 with time and the development of shrinkage strain of con-235 crete are modelled by a time-dependent function236F t AtB t 14238 238239 where A and B are empirically tted parameters for each240 material property and are obtained from test and control241 data and t is time after casting or the commencement of242 drying, as appropriate.243 4.1. Shrinkage244 Shrinkage is the time-dependent and load-independent245 strain resulting from the reduction in volume of concrete246 at constant temperature (due primarily to loss of water247 resulting from drying and hydration). Shrinkage is taken248 to be direction independent and the shrinkage shear strain249 is taken as zero. Thus for plane stress problems, the shrink-250 age strains in global axes areesht esht esht 0 T15 252 252253 where esh(t) is negative and the magnitude of the shrinkage254 strains is calculated by Eq. (14), with the appropriate255 parameters A and B determined from tests. In this study256 shrinkage is assumed to occur uniformly throughout the257 structure, the diusion process in the drying of concrete258 is not considered.259 4.2. Creep260 The creep model adopted here is the solidication creep261 aging model of Bazant and Prasannan [3] using Kelvin262 chains to describe the viscoelastic component (Fig. 5a).263 In this model, the aging aspect of concrete creep is due to264 growth, on the microscale, of the volume fractions v and265 h of the load-bearing solidied matter associated, respec-266 tively, with the viscoelastic strain ev and the viscous strains267 ef and is a consequence of hydration of the cement parti-268 cles. This process is described schematically in Fig. 5b in269 which creep strain is additive to the elastic and shrinkage270 strains and the volume fractions of he load-bearing solidi-271 ed matters v(t) and h(t) of the creep component grow with272 time t. Bazant and Baweja [1,2] proposed E0 = 1.6Ec.28,273 where Ec.28 is the elastic modulus at 28 days.274 The total creep strain at time t is decomposed asecpt evt eft 16 276 276277 where ev and ef are the viscoelastic (recoverable) and vis-278 cous strains (non-recoverable), respectively, as shown in279 Fig. 5b. The viscoelastic and viscous strain rates are given280 by281_ evt _ ctvt 17a_ ct _ t0_Ut t0 drt0 17b_ ef t rtg0ht rtgt 17c 283 283284 where t0 is the concrete age at application of the load; c(t) is285 the viscoelastic microstrain; U(t t0) is the microscopic286 creep compliance function of the solidied matter associ-fussEsEdEwEu1111fsyusyfwwFig. 4. Tri-linear stressstrain model for reinforcing steel.4 K.T. Chong et al. / Computers and Structures xxx (2007) xxxxxxCAS 4159 No. of Pages 13, Model 5+22 September 2007 Disk UsedARTICLE IN PRESSPlease cite this article in press as: Chong KT et al., Time-dependent modelling of RC structures using the ..., Comput Struct (2007),doi:10.1016/j.compstruc.2007.08.005UNCORRECTEDPROOF287 ated with the viscoelastic component; and g0 and g(t) are288 the eective viscosity of the solidied matter and the appar-289 ent macroscopic viscosity, respectively, associated with the290 viscous component. The viscous component is a linear291 function of stress calculated directly from Eq. (17c),292 whereas the viscoelastic component is evaluated by solving293 the superposition integral given by Eq. (17b). From these294 relationships, Bazant and Prasannan [3] derived the analyt-295 ical expression for the creep compliance of concrete296Ct; t0 q2Qt; t0 q3 ln 1 t t00:1_ _ q4 ln tt0_ _ 18298 298299 where q2, q3 and q4 are empirical parameters determined300 from control tests and Q(t, t0) is a binomial integral. As301 no closed formed solution exists for the integral Q(t, t0),302 an approximation is used (Ref. [3]).303 The analytical expression for the creep compliance given304 by Eq. (18) is used in this study for the determination of the305 empirical parameters q2, q3 and q4 for a given set of exper-306 imental creep data. In numerical implementation, the expli-307 cit calculation of Q(t, t0) is not required.308 To facilitate the numerical creep analysis, the integral-309 type equation given by Eq. (17b), is converted into a310 rate-type constitutive equation that allows the stress his-311 tory to be stored implicitly. For the creep model adopted,312 the rate-type equation is described using a Kelvin chain313 (Fig. 5a). For a constant stress, r, applied at time t0, biaxial314 viscoelastic microstrain vector is given by315ct r

Nj11Ej1 ett0=sj 19317 317318 where sj = gj/Ej is the retardation time of the jth Kelvin319 chain unit, Ej and gj are the viscoelastic microstrain, the320 elastic modulus and viscosity of the jth Kelvin chain unit,321 respectively, and N is the total number of Kelvin chains.322 Comparing Eqs. (19) and (17b) for a constant stress r323 applied at time t0, the microscopic creep compliance func-324 tion may be expressed in the form of a Dirichlet series325Ut t0

Nj11Ej1 ett0=sj_ _ A0 20327 327328 where the A0 term is added to include the negative innity329 area of the retardation spectrum in the discretization of the330 spectrum.331 Fig. 6 shows the numerical integration of the retardation332 spectrum using the trapezoidal rule with intervals D(lnsj).333 The relationship for the discretization of the Kelvin chains334 is given byAj LsjDln sj Lsj ln 10Dlog sj 21 336 336337 where Aj = 1/Ej and L(sj) describes the retardation spec-338 trum and is given by Bazant and Xi [5] asLs 0:023s2:80:9 3s0:1

1 3s0:1

3_ _3s32 q2 0:193s2:80:9 3s0:1 0:013s2:81 3s0:1

2_ _ 3s32 q2 22 340 340ln L( ) jRetardation spectrumA0ln2ln3ln4ln5ln6ln1A1 A2A3A4A5A6Fig. 6. Discretization of a continuous retardation spectrum.j =1j =2j =N12N/E0v( ) t-t = ( ) ( ) t-t d tfshE1E2EN12Nh t ( ) dh t ( )v t ( ) dv t ( )a bFig. 5. Solidication theory for concrete creep: (a) Kelvin chain description for viscoelastic component; (b) schematic representation of the solidicationcreep model [3].K.T. Chong et al. / Computers and Structures xxx (2007) xxxxxx 5CAS 4159 No. of Pages 13, Model 5+22 September 2007 Disk UsedARTICLE IN PRESSPlease cite this article in press as: Chong KT et al., Time-dependent modelling of RC structures using the ..., Comput Struct (2007),doi:10.1016/j.compstruc.2007.08.005UNCORRECTEDPROOF341 For a suciently smooth creep curve the retardation time342 discretization interval can be taken as D(logsj) = 1 for each343 adjacent Kelvin chain [5]. Lastly, the negative innity area344 is calculated asA0 q2 ln 1 t t00:1_ _

Nj11Ej1 ett0=sj_ _ 23346 346347 4.3. FE implementation of creep348 By Eq. (17a), the viscoelastic creep component is349 expressed in the formDevti1 Dci1vi1=224351 351352 where the subscripts i and i + 1/2 indicate the reference to353 time ti and the time in the middle of a logarithmic time step354 ti+1/2, respectively, whereti1=2 t0 ti1 t0ti t00:525 356 356357 and t0 is the age at rst loading.358 By Eqs. (17b) and (20), and the relationship c = Ncj,359 the change in viscoelastic microstrain Dc is obtained asDci1

Nj1cji1 cji GDrA0 26361 361362 where G is the biaxial volumetric growth matrix and is gi-363 ven byG 1 m 0m 1 00 0 21 m____ 27365 365366 The viscoelastic microstrain at time ti+1 for the jth Kelvin367 chain is a modied form of that derived by Bazant and Pra-368 sannan [4] and iscji1 cji eDyj Gri1Ej1 eDyj 1 kjEjGDr 28370 370371 whereDyj Dtsj; kj 1 eDyjDyj; Dr ri ri1 29373 373374 The volume of the solidied matter at mid-time of a loga-375 rithmic time step, vi+1/2, is then given byvi1=2 1ti1=2 q3q2_ _130377 377378 The change in the viscous, non-recoverable, component of379 creep is evaluated from Eq. (17c). Considering the change380 over a nite time step, we write381Def ti1Dt Gri1=2gi1=231383 383384 where ri1/2 = ri1 + Dr/2. Substituting the apparent mac-385 roscopic viscosity, dened as gi1=2 q14 ti1=2, into Eq.386 (31), the change in viscous strain is then written asDefti1 Gri1=2q4Dtti1=232 388 388389 Lastly, the changes in viscoelastic and viscous strain com-390 ponents are added to the creep strain components obtained391 from the previous converged time step giving392evti1 evti Devti1 33aef ti1 ef ti Def ti1 33b 394 394395 The sum of the creep strain components from Eq. (33) are396 then added to the shrinkage strains to give the total inelas-397 tic pre-strains, e0. The inelastic pre-strains are then con-398 verted to equivalent nodal forces by Eq. (13) and applied399 to the nodes of the FE model.400 5. Time-dependent crack widths401 In a time-dependent analysis, the tension in the concrete402 between cracks (tension stiening) induces tensile creep403 deformation and drying shrinkage causes shortening of404 the concrete between the cracks. Adding these components405 to the instantaneous crack widths of Eq. (10) gives the406 time-dependent crack width407wcrt srm e1t rcts:crtE0 ecpt esht m12e2t_ _ _ _34 409 409410 where rcts:cr is the tension stiening stress component in the411 crack opening direction. In Eq. (34), the tension stiening412 and creep components contribute to the expansion of the413 concrete between the cracks and, hence, reduce the crack414 opening; whereas, drying shrinkage causes a volume reduc-415 tion in the concrete between the cracks resulting in a wid-416 ening of the cracks. As the inuence of shrinkage417 dominates the behaviour, the cracks widen with time.418 6. Experimental corroboration419 Three examples are used to demonstrate the FE formu-420 lation. The rst example consists of two simply-supported421 beams subjected to third point sustained loading. The422 time-dependent cracking of the beam was investigated with423 particular attention given to the development of exural424 cracks within the constant moment region. The second425 example is to simulate the time-dependent change in the426 reactions at the supports of a series of continuous beams427 subjected to support settlements at dierent ages of the428 concrete. This example demonstrates the eects of varying429 load-histories on load redistribution in continuous beams.430 Finally, a series of eccentrically loaded columns subjected6 K.T. Chong et al. / Computers and Structures xxx (2007) xxxxxxCAS 4159 No. of Pages 13, Model 5+22 September 2007 Disk UsedARTICLE IN PRESSPlease cite this article in press as: Chong KT et al., Time-dependent modelling of RC structures using the ..., Comput Struct (2007),doi:10.1016/j.compstruc.2007.08.005UNCORRECTEDPROOF431 to sustained loading and prone to creep buckling are ana-432 lysed including the eects of geometric non-linearity.433 6.1. Example 1: four-point bending test under sustained load434 Gilbert and Nejadi [13] tested a series of beams and one-435 way slabs under sustained load for a period of 380 days to436 investigate the growth of cracks with time. Beam 1a is437 modelled using the formulation described above with the438 dimensions of the beam shown in Fig. 7. The beam was439 loaded at age 14 days.440 Material tests on standard 150 mm diameter cylinders at441 28 days gave the mean concrete strength as fcm = 24.8 MPa442 and modulus of elasticity Ec = 24950 GPa. The tensile443 strengths were obtained from indirect tension (Brazil)444 tests on 150 mm diameter cylinders tested at 14, 21 and445 28 days and were fct.14 = 2.0 MPa, fct.21 = 2.6 MPa and446 fct.28 = 2.8 MPa, respectively. For FE modelling, the447 growth of concrete tensile strength with time was taken448 from Eq. (14) with Afct 4 MPa and Bfct 12 days. The449 shrinkage constants were calculated from measurements450 on shrinkage companion specimens giving Ash = 950 le451 and Bsh = 45 days in Eq. (14). The bond shear stress sb0452 was taken as that determined by Gilbert and Nejadi [13]453 for their tests and was sb0 = 4.5 MPa. The concrete fracture454 energy Gf was taken as 75 N/m and Poissons ratio was455 assumed to be m = 0.2. The reinforcing steel was taken as456 elastic-perfectly plastic with a yield strength of 500 MPa457 and elastic modulus of 200 GPa. The self-weight of the458 beam was included in the analysis using gravity loading459 with the weight of the reinforced concrete taken to be460 23.5 kN/m3.461 For the solidication creep modelling, the asymptotic462 elastic modulus of concrete was taken to be E0 =463 1.6Ec.28 = 40 GPa. The empirical material constants q2, q3464 and q4 were determined by tting the compliance data465 obtained from a creep test under a 5 MPa sustained stress466 undertaken in conjunction with the laboratory tests. The467 calculated values of the constants were q2 = 186.5 le/468 MPa, q3 = 1.0 le/MPa and q4 = 23.7 le/MPa. The Dirich-469 let series was discretized into eight Kelvin chain units for470 storing the deformation history of the viscoelastic strain.471 The corresponding elastic moduli for each link in the chain472 Ej and retardation times sj are given in Table 1. The nega-473 tive innity area is A0 = 52.8 MPa1.474 The FE mesh (Fig. 7c) consisted of 199 nodes and is475 made up of 108 plain concrete elements, 54 reinforced con-476 crete elements and two sti elastic support elements. The477 characteristic lengths were taken to be the crack spacingsAAL/3L = 3500L/3 L/318.6 kN 18.6 kN ab cx = 0.01676 y = 0 x = 0 y = 0 250 482N16bars348Fig. 7. Details for Gilbert and Nejadis [13] beams: (a) elevation; (b) cross-section; and (c) FE mesh.Table 1Kelvin chain data used for model corroborationjthunitsj(days)Ej (MPa)Gilbert and Nejadi[13]Ghali et al.[12]Bradford[7]1 0.0001 0.08480 0.11054 0.196122 0.001 0.07214 0.09404 0.166853 0.01 0.06209 0.08094 0.143614 0.1 0.05411 0.07054 0.125165 1 0.04778 0.06229 0.110516 10 0.04276 0.05574 0.098897 100 0.03877 0.05054 0.089668 1000 0.03560 0.04641 0.08234K.T. Chong et al. / Computers and Structures xxx (2007) xxxxxx 7CAS 4159 No. of Pages 13, Model 5+22 September 2007 Disk UsedARTICLE IN PRESSPlease cite this article in press as: Chong KT et al., Time-dependent modelling of RC structures using the ..., Comput Struct (2007),doi:10.1016/j.compstruc.2007.08.005UNCORRECTEDPROOF0.000.050.100.150.200.250.300.350.400.450 100 200 300 400Time (days)Crack width (mm)ExperimentalFEMBeam B1-amaximumaverageFig. 9. Comparison of changes in crack width with time for beam B1-a[13].Beam B1-a02468101214160 100 200 300 400Time (days)Midspan deflection (mm)ExperimentalFEM ( =0.75)Fig. 8. Midspan displacement with time for beam B1-a [13].(b) 914.42133.6914.4A ADialgauge25.4 25.4209.6212.7 212.7Stirrups6.35 at 152.4Dial gaugeThreaded bar Calibrated rod Section A-Aa bcdApplied deflection x = 0.049090 y = 0.004091 x = 0 y = 0.004091 Fig. 10. Details of the Ghali et al. [12] continuous beams: (a) longitudinal layout of a beam set; (b) section of the test; (c) cross-section of the beam; and (d)FE mesh.8 K.T. Chong et al. / Computers and Structures xxx (2007) xxxxxxCAS 4159 No. of Pages 13, Model 5+22 September 2007 Disk UsedARTICLE IN PRESSPlease cite this article in press as: Chong KT et al., Time-dependent modelling of RC structures using the ..., Comput Struct (2007),doi:10.1016/j.compstruc.2007.08.005UNCORRECTEDPROOF478 as calculated by the tension chord model with k = 0.75.479 Due to symmetry, only half the beam was modelled.480 The measured and calculated variations of mid-span481 deection with time are plotted in Fig. 8. The average482 and maximum crack spacings, in the constant moment483 region, calculated by the FE analysis were 220 mm and484 293 mm, respectively, and compare reasonably with the485 measurements of 190 mm and 210 mm, respectively, con-486 sidering the experimental variation of crack spacings in487 reinforced concrete beams. Lastly, the maximum and aver-488 age crack widths versus time curves are plotted in Fig. 9 in489 the constant moment region. Overall, good agreement was490 obtained between the FE results and the measured test491 data.492 6.2. Example 2: time-dependent forces induced by settlement493 of supports of continuous beams494 Ghali et al. [12] undertook a series of tests to investigate495 time-dependent changes in support reactions for continu-496 ous beams subjected to support settlement. The tests con-497 sisted of four pairs of two-span continuous beams with498 each beam pair tested in a vertical position so as to elimi-499 nate the bending caused by self-weight. Rollers were placed500 at each free end of the specimen with the mid-support set-501 tlement introduced by means of threaded bars tying the502 beams together. Details of the tests and the testing arrange-503 ments are shown in Fig. 10. Each pair of the beams was504 subjected to a nal mid-support settlement of 1.65 mm505 but with varying deection increments at varying times inTable 2Details of application of deections for Ghali et al.s [12] beamsTestno.Age (days) of applicationof deection incrementDeectionincrement(mm)Duration for eachincrement (min)1 2 3 4 51 9 1.65 302 12 1218 1212 1314 1414 0.33 103 12 13 1514 20 2613 0.33 104 1112 15 2714 4114 7214 0.33 100.02.04.06.08.010.012.014.016.018.020.00 50 100 150 200 250Age (days)Reaction at mid-support (kN)ExperimentalFEMTest 10.02.04.06.08.010.012.014.016.018.00 50 100 150 200 250Age (days)Reaction at mid-support (kN)Experime ntalFEMTest 20.02.04.06.08.010.012.014.016.018.00 50 100 150 200 250Age (days)Reaction at mid-support (kN)ExperimentalFEMTest 30.02.04.06.08.010.012.014.016.00 50 100 150 200 250Age (days)Reaction at mid-support (kN)ExperimentalFEMTest 4a bc dFig. 11. Comparisons of the FEM and experimental time-dependent reaction at the mid-support for the Ghali et al. [12] controlled deection specimens.K.T. Chong et al. / Computers and Structures xxx (2007) xxxxxx 9CAS 4159 No. of Pages 13, Model 5+22 September 2007 Disk UsedARTICLE IN PRESSPlease cite this article in press as: Chong KT et al., Time-dependent modelling of RC structures using the ..., Comput Struct (2007),doi:10.1016/j.compstruc.2007.08.005UNCORRECTEDPROOF506 the loading history. The testing period was up to 300 days507 with details of the age at application of the settlements508 given in Table 2.509 The FE mesh for the beams is shown Fig. 10d, with only510 one half of each beam being modelled due to symmetry.511 The mesh consisted of 374 nodes and contained 320 rein-512 forced concrete elements and three sti elastic support513 elements.514 The time-dependent properties of the concrete were not515 reported by Ghali et al. [12]. The creep compliance and516 shrinkage functions recommended in the CEB-FIP Model517 Code [8] were used in the nite element analysis assuming518 a relative humidity of 65%. Ghali et al. tested a number519 of concrete cylinders to obtain the compressive strengths520 at ages from 7 days to 190 days and approximated the521 growth of concrete compressive strength at age t byfcmt 37:97=t 0:75 35 523 523524 The tensile strength at time t was taken to be fctt 525 0:4fcmt_ . The concrete parameters used in the analysis526 were: E0 = 1.6Ec.28 = 46.3 GPa, fcm = 38 MPa, fct.28 = 2.5527 MPa, Afct 2:8 MPa; Bfct 3:5 days, sb0 = 4 MPa, m =528 0.2, Ash = 450 le, Bsh = 60 days, q2 = 142.3 le/MPa,529 q3 = 6.4 le/MPa, q4 = 16.3 le/MPa and A0 = 40.4 MPa1.530 The elastic moduli and corresponding retardations times of531 the Kelvin chain units are given in Table 1 and the concrete532 fracture energy Gf was taken to be 75 N/mm. The reinforc-533 ing steel was assumed to be elastic-perfectly plastic with534 yield stress = 400 MPa and elastic modulus = 200 GPa.535 The changes in the mid-support reactions with time for536 the FE modelling results are compared with the measured537 data in Fig. 11 for each of the four beams tested. Generally538 good agreement is observed with the key trends being cap-539 tured well.A A50001250125012501250eTBeStrong wall Dial gauge Dial gauge Dial gauge Steel channel section Eccentric loading Tensioning cable Test column Hydraulic jack Load cell I-section loading arm 150Section A-AStirrups 10 at 1502N122N12Clear cover 15 mmFig. 12. Details and testing arrangements for columns [7].Table 3Loading details for Bradford [7] columnsSpecimen C1 C2 C3 C4 C5eT (mm) 50 50 50 50 50eB (mm) 50 25 0 25 50Load (kN) 70.0 70.0 80.0 80.0 85.010 K.T. Chong et al. / Computers and Structures xxx (2007) xxxxxxCAS 4159 No. of Pages 13, Model 5+22 September 2007 Disk UsedARTICLE IN PRESSPlease cite this article in press as: Chong KT et al., Time-dependent modelling of RC structures using the ..., Comput Struct (2007),doi:10.1016/j.compstruc.2007.08.005UNCORRECTEDPROOF540 6.3. Example 3: creep buckling of RC columns541 Bradford [7] tested a series of eccentrically loaded slen-542 der reinforced concrete columns loaded at various eccen-543 tricities for 280 days. Five identically reinforced ve544 metre long columns were tested with varying end eccentric-545 ities. The test arrangement and the cross-section details of546 the specimens are shown in Fig. 12 with details of the load-547 ing given in Table 3.548 The columns were modelled using 1200 four-node rein-549 forced concrete elements (200 rows each consisting of six550 elements across the column width) with 1407 nodes (see551 Fig. 13). As the geometric non-linearity due to second552 order eects has a considerable impact on the stresses in553 the column, the analyses were undertaken using an updated554 Lagrangian formulation to model the geometric non-555 linearity.556 The compressive strength, shrinkage and creep proper-557 ties were taken as those measured by Bradford. That is:558 fcm = 29.3 MPa, Ash = 420 le; Bsh = 90 days; q2 = 80.2559 le/MPa, q3 = 2.5 le/MPa, q4 = 38.5 le/MPa and A0 =560 22.8 MPa1. The remaining material properties were taken561 as: E0 = 1.6, Ec.28 = 35.4 GPa, fct.28 = 2.2 MPa, Afct 2:2562 MPa; Bfct 12 days, sb0 = 3.5 MPa, m = 0.2 and Gf =563 75 N/mm. The self-weight of the columns was included in564 the analysis using gravity loading with the weight of the565 reinforced concrete taken to be 23.5 kN/m3. The stress566 strain relationship for reinforcing steel was taken to be567 elastic-perfectly plastic with an elastic modulus of568 200 GPa and a yield strength of 500 MPa.569 The calculated deections at mid-height and top and570 bottom quarter points with time are compared with the571 measured results in Fig. 14. Overall, a good correlation is572 observed between the numerical results and the laboratory573 measurements.574 7. Conclusions575 A non-linear nite element model has been developed to576 calculate the response of two dimensional reinforced con-577 crete structures subjected to time-dependent deformations.578 The model builds on the nite element implementation of579 the cracked membrane model of Foster and Marti [11]580 incorporating the time-dependent deformations of creep581 and shrinkage in the concrete. Creep was modelled using582 the solidication creep aging model of Bazant and Prasan-583 nan [3].584 Numerical results obtained using the nite element585 model have been compared with the measurements taken586 in a variety of laboratory tests, including a simply-sup-587 ported beam tested under sustained loading for a period588 of 380 days by Gilbert and Nejadi [13], a series of two-span589 continuous beams tested by Ghali et al. [12] and subjected590 to support settlements over a period of 250 days; and a ser-591 ies of slender reinforced concrete columns tested by Brad-592 ford [7] subjected to a sustained eccentric compressionFig. 13. FE mesh and reinforcement details for columns [7].K.T. Chong et al. / Computers and Structures xxx (2007) xxxxxx 11CAS 4159 No. of Pages 13, Model 5+22 September 2007 Disk UsedARTICLE IN PRESSPlease cite this article in press as: Chong KT et al., Time-dependent modelling of RC structures using the ..., Comput Struct (2007),doi:10.1016/j.compstruc.2007.08.005UNCORRECTEDPROOF593 loading for 280 days and prone to creep buckling. Overall,594 good correlations were observed between the results of the595 FE models and the laboratory measurements, including the596 growth of cracks, the changes in deection and the redistri-597 bution of load with time.598 Acknowledgements599 This research was funded by the Australian Research600 Council Discovery Grant No. DP0210039. The support601 of the ARC is gratefully acknowledged.602 References603 [1] Bazant ZP, Baweja SCollaboration with RILEM Committee TC 107-604 GCS. Creep and shrinkage prediction model for analysis and design605 of concrete structures model B3 (RILEM recommendation). Mater606 Struct 1995;28:35765.607 [2] Bazant ZP, Baweja S. Justication and renements of Model B3 for608 concrete creep and shrinkage 1. Statistics and sensitivity. Mater609 Struct 1995;28:41530.610 [3] Bazant ZP, Prasannan S. Solidication theory of concrete creep. I:611 formulation. J Eng Mech ASCE 1989;115(8):1691703.612 [4] Bazant ZP, Prasannan S. Solidication theory of concrete creep. II:613 verication and application. J Eng Mech ASCE 1989;115(8):170425.0.05.010.015.020.025.030.035.040.045.050.00 100 200 300Time since loading (days)Deflections (mm)Exp (top) FEM (top)Exp (mid) FEM (mid)Exp (bot) FEM (bot)Column C10.05.010.015.020.025.030.00 100 200 300Time since loading (days)Deflections (mm)Exp (top) FEM (top)Exp (mid) FEM (mid)Exp (bot) FEM (bot)Column C20.05.010.015.020.025.00 100 200 300Time since loading (days)Deflections (mm)Exp ( top) FEM (top)Exp ( mid) FEM (mid)Exp ( bot) FEM(bot)Column C30.02.04.06.08.010.012.00 100 200 300Time since loading (days)Deflections (mm)Exp (top) FEM (top)Exp (mid) FEM (mid)Exp (bot) FEM (bot)Column C4-6.0-5.0-4.0-3.0-2.0-1.00.01.02.03.04.05.00 100 200 300Time since loading (days)Deflections (mm)Exp (top) FEM (top)Exp (mid) FEM (mid)Exp (bot) FEM (bot)Column C5Fig. 14. Comparison of computed deections with test data for columns [7].12 K.T. Chong et al. / Computers and Structures xxx (2007) xxxxxxCAS 4159 No. of Pages 13, Model 5+22 September 2007 Disk UsedARTICLE IN PRESSPlease cite this article in press as: Chong KT et al., Time-dependent modelling of RC structures using the ..., Comput Struct (2007),doi:10.1016/j.compstruc.2007.08.005UNCORRECTEDPROOF614 [5] Bazant ZP, Xi Y. Continuous retardation spectrum for solidication615 theory of concrete creep. J Eng Mech ASCE 1995;121(2):2818.616 [6] Bazant ZP, Hauggaard AB, Baweja S, Ulm F-J. Microprestress-617 solidication theory for concrete creep (I: aging and drying eects, II:618 algorithm and verication). J Eng Mech ASCE 1997;123(11):619 1188201.620 [7] Bradford MA. Viscoelastic response of slender Q2 eccentrically loaded621 reinforced concrete columns. Mag Concrete Res 2005 [in press].622 [8] CEB-FIP. Comite Euro-International du BetonFederation Interna-623 tional de la Precontrainte, Model Code 1990. 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