Relationship between Nonlinear Creep and Cracking of Concrete under Uniaxial Compression

Transcription

1 Journal of Advaned Conrete Tehnology Vol. 5, No. 3, -, Otober 7 / Copyright 7 Japan Conrete Institute Sientifi paper Relationship between Nonlinear Creep and Craking of Conrete under Uniaxial Compression Miguel Fernández Ruiz, Aurelio Muttoni and Pietro G. Gambarova 3 Reeived 4 Marh 7, aepted July 7 Abstrat This paper investigates the nonlinear reep behaviour of onrete in ompression and its relationship with raking under uniaxial ompression (raks developing parallel to the loading diretion. A physial model explaining the nature and the role of linear and nonlinear reep strains is presented, together with a failure riterion for onrete under sustained loads. The model assumes that all nonlinear reep strains are due to onrete miro-raking. The soundness of this assumption is heked against the experimental results obtained by the authors and by other researhers. The proposed model is shown to fit quite well the experimental results, for various load patterns and onrete ages. The model also proves that the affinity hypothesis between linear and nonlinear reep strains (usually taken for granted in the design for stress levels below 7% of onrete strength in ompression is no longer valid when onrete fails under a sustained load, beause of the unstable growth of raking. Conrete response in these ases is analyzed in detail and a simplified but realisti approah for the evaluation of the failure envelope in ompression is proposed.. Introdution The effets of high stress levels on onrete long-term behaviour in ompression are important with referene not only to the delayed strains, but also to the strength of the material. This topi was first studied by Rüsh (96, who identified two regimes in onrete subjeted to a sustained load, the first haraterized by a failure limit (when the speimen fails by onrete rushing after a ertain period after the appliation of the load and by a reep limit (below whih linear and nonlinear reep strains develop, but onrete does not fail, see Fig.. Researh on the reep limit inluding the development of nonlinear delayed strains has ontinued with several ontributions overing a number of experimental and modelling issues. For design purposes, the attention has mainly foused on how to orret the linear reep oeffiient (valid for /f < 4 %, taking advantage of the affinity hypothesis (i.e. proportionality between the linear and nonlinear reep oeffiients, see Fig., as shown for instane by Avram et al. (98, and Fernández Ruiz et al. (4 with satisfatory results. Formulae quantifying the influene of nonlinear reep strains and based on this approah were also adopted by some odes of pratie (see for instane CEB MC 9. Conerning the failure in ompression under a sustained load, its origin has been assoiated with the development and growth of miro-raking (Neville, 97, Post-dotoral fellow, Eole Polytehnique Fédérale de Lausanne, Switzerland. miguel.fernandezruiz@epfl.h Professor, Eole Polytehnique Fédérale de Lausanne, Switzerland. 3 Professor, Politenio di Milano, Italy. but it has been shown (Mazzotti and Savoia, 3 that only a fration of the total delayed strains developing inside the onrete is due to raking or, in other words, are related to material damage. A suitable approah for investigating the post-peak region has been presented in El-Kashif and Maekawa (4 based on a oupled plastiity-damage model. Suh approah an also inlude yli-loading effets (Maekawa and El-Kashif, 4 Similar onlusions on onrete strength under sustained loading or under inreasing loading (with various loading rates have been drawn for onrete subjeted to bending and to tension (Bažant and Gettu, 99; Bažant and Li, 997a,b. In these ases, satisfatory results have been obtained using models that assume linear visoelastiity for the reep in the undamaged onrete, and a rate-dependent formulation for rak development (van Zijl et al., ; Barpi and Valente,, even in the postpeak phase (Barpi and Valente, 5. Here, a model for studying both the reep and the failure limits in onrete under sustained ompression is presented, the aim being to investigate the effets of miro-raking on onrete delayed strains and failure. Creep limit Failure under a sustained load Monotoni behaviour Fig. Sustained-load envelope for onrete in uniaxial ompression: reep limit and failure limit aording to Rüsh (96.

2 M. F. Ruiz, A. Muttoni and P. G. Gambarova / Journal of Advaned Conrete Tehnology Vol. 5, No. 3, -, 7 The model is heked against the results of an experimental ampaign arried out on plain-onrete ylinders at the Strutural Conrete Laboratory of EPFL (Eole Polytehnique Fédérale de Lausanne, Switzerland. Speimens size Ø h is 6 3 mm. Referene is made to two different onrete ages and to various loading patterns. This researh is signifiant with referene to both the ultimate and the servieability limit states where reinfored and prestressed onrete strutures are loally subjeted to very large stresses. For instane, the onrete in ontat with the ribs of bonded bars or with the end-plates of the tendons in prestressed onrete strutures an loally be subjeted to very large stresses that ause sizable stress redistributions in the surrounding (less stressed onrete. These stress redistributions are favoured by the short term development of inelasti strains. Understanding the behaviour of the onrete in these zones requires the detailed assessment of short-term reep strains and of their interation with raking, damage and plastiity. The proposed model not only fits quite well the test results obtained by the authors and by other researhers, but it also provides a lear explanation of the nature of the different omponents of the strain. It has the further advantage of being rather simple.. Theoretial model Conrete exhibits a rheologial behaviour onsisting of delayed strains aused by different proesses, whose origin is to be found in the mirostruture of onrete. Conventionally, these strains are separated into shrinkage and reep strains, the former shrinkage omprising the strains that appear when no external loads are applied, and the latter reep omprising the delayed strains assoiated with the appliation of external loads (reep strains are defined as the differene between the total delayed strains and those aused by shrinkage. In spite of ertain inonsistenies, this definition enables to quantify the phenomena in a simple way. For instane, in a onrete loaded at the age t the strain at any given time t an be written as: σ σ σ + + t, ε, εs( t, ε t,, ( f f f where the reep strain an be obtained through a differene: t,, t,, s ( t, f f f ( As a rule, the reep strain is expressed in the following way: σ σ σ t,,, ϕ t,, (3 f f f σ where ϕ t, t, is the reep oeffiient of onrete, f whih omprises the effet of both drying and basi reep. By definition, shrinkage strains are independent of the stress state in the material. On the other hand, reep is diretly related to onrete stresses and to miro-raking. For any stress level below.4 f, reep strains an be desribed by means of a stress-independent formulation of the reep oeffiient: ϕ lin (t,t. Consequently, reep strains are linearly related to the stresses. However, at higher stress levels this linearity is lost and the reep oeffiient is no longer stress-independent (Fig.. Various relationships have been proposed to desribe the nonlinear effets of stresses on the reep oeffiient. Based on the tests performed at stress levels below 7% of onrete ompressive strength, a satisfatory fitting an be obtained by using the so-alled affinity hypothesis. This hypothesis assumes that the linear and nonlinear reep strains are related through the atual stress ratio /f (see Fig.. This hypothesis an be written as follows: (a (b ( Linear reep ( + ϕ lin (,t ( + ϕ(,t ϕ(t, t Inreasing /f >.4f.4f Nonlinear reep ϕ(t, t ϕ lin (t, t t /f [-] Fig. (a Linear and nonlinear reep strains; (b reep oeffiient for various values of the stress/strength ratio; and ( plot of the affinity oeffiient η (Eq. 5 for nonlinear reep, together with the test results by different authors (Fernández Ruiz, 3. 3 η [-] η ϕ/ϕ lin

3 M. F. Ruiz, A. Muttoni and P. G. Gambarova / Journal of Advaned Conrete Tehnology Vol. 5, No. 3, -, 7 3 σ σ ( ( ϕ t,, ϕlin ( t, η f f (4 where the affinity oeffiient η an be given a polynomial formulation, as reently proposed by the first author (3 with referene to the asending branh of the stress-strain urve: 4 σ η + (5 f As shown in Fig., Eq. (5 fits rather well the values worked out from several tests. This expression has satisfatorily been used for both reep and relaxation problems, using an extension of the aging-oeffiient method (Fernández Ruiz, 3. Going bak to the strains in the onrete, instantaneous plasti strains develop as a result of the loading proess, as shown in Fig. 3, where the uniaxial stress-strain response of a onrete speimen monotonially loaded up to A exhibits the plasti strain p,. Thereafter shrinkage strains develop, as well as reep strains if the load remains onstant over time (strain, from A to B in figure 3. Should the speimen be further loaded, the point C, whih is assumed to be on a shifted monotoni urve, would be reahed. In B, the total strain onsists in a number of ontributions: σ ε +, B p, + v + p E (6 B where v is the onrete visous strain (time-related strain not assoiated with onrete miro-raking linear-reep strain + shrinkage strain and p the inrease over time of the onrete plasti strains. The total strain inrease (,B,A is: v + p + σ EB E (7 A In the following, onrete miro-raking will be assumed to be the only soure of the nonlinear part of the reep strain. Consequently, the total strain an be written as follows: A E A ε ( p v ϕ E A lin ( t, +s ( t, ts E B v p, Fig. 3 Delayed strains aused by a sustained load. + p + σ EB E A (8, nl where the nonlinear reep strain (,nl representing the time-related inelasti strain onsists of the time-related plasti strain ( p plus a term ((/E B /E A whih represents the strain orresponding to the damage inrease aused by miro-raking (E B < E A. This behaviour an be represented by means of the mehanial model shown in Fig. 4a, that represents a oupled plastiity-damage model with visous strains where the damage is represented by the failure of the spring elements (a similar approah was followed by El-Kashif and Maekawa, 4. Fig. 4b shows the response of the various elements of the model for the previously-studied load pattern (see Fig. 3. B C v (a (b,el p v Fig. 4 Mehanial analogy: (a mehanial model; and (b response of the mehanial model at various loading states.

4 4 M. F. Ruiz, A. Muttoni and P. G. Gambarova / Journal of Advaned Conrete Tehnology Vol. 5, No. 3, -, 7,nl Crushing Atual 3,in A (,in + v B Basedonaffinity 3,in Conerning the development of the nonlinear reep strain, it has to be noted that if the inelasti strain apaity of the material (,in is reahed, the onrete fails by rushing under a onstant load (B C, see Fig Appliability of the affinity hypothesis The development of the nonlinear reep strains an be evaluated in a simple way by means of Eq. (4, if the affinity hypothesis is introdued: ε v E A ε E A p ( t η ϕ ( t, E B Fig. 5 Failure under a sustained load., nl ( lin (9 As previously stated, this hypothesis provides good results for stress levels /f <.7. However, for larger stress levels (when onrete rushes under a sustained load, the atual development of nonlinear reep strains over time does not agree with the affinity hypothesis, see Fig. 6. As later shown by the tests presented in this paper, three different phases an be identified for the nonlinear t F t Fig. 6: Atual and affinity-based nonlinear reep strains. reep strains: ( rak formation, ( rak growth in a stable way, and (3 unontrolled rak propagation up to onrete failure. In the first and seond phases, the affinity hypothesis gives reasonable results (due to the onvex shape of the urve dereasing strain rate over time, whereas in the third phase unstable rak growth sizable deviations from the affinity law are observed (due to the onave shape of the urve inreasing strain rate over time. In order to desribe the development of the atual strain urve over time, that represents the onrete resistane to miro-raking, the results of the tests under yli loading are helpful (sine small-amplitude yles are a way to fore miroraking, whih in turn produes a pseudo-plasti behaviour and damage in onrete. Although there are some phenomenologial differenes between both phenomena (Shah and Chandra, 97 both phenomena an be treated in a unified manner as shown by Maekawa and El-Kashif (4. Fig. 7a shows the typial yli response of onrete. The plasti strain evolution with the number of yles (n exhibits three phases as in the ase of sustained loads. With referene to yli loading, the following analytial law is proposed to be adopted to haraterize rak growth from rak formation to rak unstable propagation: (a n p,a E A,in p E B 3 rd nd st Failure / ε in [-] (b Upper fratile Adopted law Lower fratile n/n F [-] ( 3 [ ] [ ] 3 σ max.7 f Adopted law σ max.85 f Adopted law Qingbin n/n F [-] Qingbin Fig. 7 Conrete response under yli loading: (a typial behaviour; (b plot of the proposed law and omparison with the fratile limits introdued by Pfanner et al. (; and ( plots of the total strain as a funtion of the yle numbers and omparison with the tests by Qingbin et al. (4.

5 M. F. Ruiz, A. Muttoni and P. G. Gambarova / Journal of Advaned Conrete Tehnology Vol. 5, No. 3, -, 7 5, in 3 4 ζ n, y + ζ n (4 ζ n n where : ζ n.5 N F ( where,in is the maximum allowable inelasti strain for a given load level. Many test data by various authors (Pfanner et al., ; Qingbin et al. 4 have been suessfully fitted with this equation (Figs. 7b,. Equation ( an also be used to haraterize the development of nonlinear reep strains over time provided that the time to failure t F (Fig. 6 is known. This parameter an be estimated by adopting the hypothesis that, at failure, /3 of the inelasti strain due to onrete miro-raking provides from the unstable rak-growth phase (Fig. 6. Consequently, the maximum allowable inelasti strain an be obtained from the nonlinear reep strains based on the affinity hypothesis (Eq. (9 as follows: 3, in [ ε ( ( η ϕlin ( t F, ] ( where the value of t F is the only unknown in Eq. ( and an thus be evaluated. In spite of its simpliity, this formulation leads to a very good fitting of the test data. As a result, for the ases where onrete fails in ompression under a sustained load, the following expression for the development of nonlinear reep strains over time is proposed:, nl where : ζ t 3 4 ζ t + ζ t (4 ζ t, in t t F.5 ( whereas in the ases when onrete does not fail under a sustained load, Eq. (9 based on the affinity hypothesis should be used to evaluate the nonlinear reep strains. 4. Experimental ampaign 4.. Objetives This setion presents the results of an experimental ampaign performed by the authors at the Strutural Conrete Laboratory of EPFL (Eole Polytehnique Fédérale de Lausanne, Switzerland. The ampaign has been arried out on plain-onrete, ylindrial speimens (size Ø h6 3 mm. Speimens were ast in two bathes with the same mix design, see Table. A first set of speimens was tested 7-49 days after onreting, while a seond set was tested eight months after onreting. Of ourse, the seond set exhibited muh smaller rheologial effets. The properties of the speimens are summarized in Table. The tests were performed using a Shenk Hydroplus 5 kn, as well as two pairs of ordinary shrinkage and linear reep frames (Fig. 8. All tests were performed under ontrolled environmental onditions (relative humidity 6 %, temperature C. Table Speimens, onrete age and type of test/onrete grade. (* Ordinary laboratory onditions: / t -5 se - Speimen t [days] Type of test/conrete grade 4 Failure under sustained load 7 Failure under sustained load 35 Failure under sustained load 36 Failure under sustained load 8/34 Nonlinear reep + reloading to failure 45 Nonlinear reep + reloading to failure 4 Nonlinear reep + reloading to failure 43 Nonlinear reep + reloading to failure 48 Relaxation steps 49 Relaxation steps 7 Virgin urves* (f.6 MPa 6 Virgin urves* (f 3. MPa 34 Virgin urves* (f 34.3 MPa 44 Virgin urves* (f 33.7 MPa 47 Prefixed loading rate ( δ 3 mm/se 45 Prefixed loading rate ( δ mm/se 45 Prefixed loading rate ( δ 3 mm/se 45 Relaxation steps 45 Relaxation steps Table Composition of ubi meter and results of tests on fresh onrete. Sand (-4 mm [kg] Gravel (4-8 mm [kg] Gravel (8 mm [kg] Cement [kg] Water [kg] Slump test [mm] Flow table test [mm]

6 6 M. F. Ruiz, A. Muttoni and P. G. Gambarova / Journal of Advaned Conrete Tehnology Vol. 5, No. 3, -, 7 (a (b (a -4 ➀ ➂ [MPa] ➁ ➋ ➃ ➍ ( (b -3 [ ] ➂ [ ] ➁ ➀ [ ] ( ➃ + v Fig. 8 Experimental set-ups: (a Shenk Hydroplus with a apaity of 5 kn; (b linear-reep frames; and ( shrinkage frames. 4.. Nonlinear reep Several speimens were subjeted to a sustained load up to failure, with stresses ranging from.8 f to.9 f, as shown in Fig. 9. Speimens, and (sustained stress.9 f failed rapidly. Consequently, it may be assumed that almost all inelasti strains were due to onrete miro-raking. These speimens exhibited a well-developed rak pattern at the end of the loading proess all along the speimen. Furthermore, the width of the longitudinal raks inreased regularly during the test until failure. Regarding the strain rate (Fig. 9b, the above-mentioned three different phases an be easily identified: at first a rapid strain inrease (and rak development, followed by a stable phase where the strain rate is approximately onstant; finally on the unstable phase, in whih both the strains in the solid onrete and the raks beome unontrolled. Speimen exhibited the same failure pattern, but sine the stress level was lower and the load was applied for a muh longer period linear reep strains developed as well. These visous strains (evaluated on the basis of the results obtained by the authors on onrete speimens exhibiting only linear-reep strains, <.4 f, are plotted in Fig. 9 (dashed line, where one an note that the urves are not proportional espeially in the phase of unstable rak growth. Sine the linear-reep strains do not ontribute to the raking of the speimen, the development of these strains seems to be the reason why, ontrary to the other speimens, the failure of speimen oured outside the monotoni urve (whih in this ase is represented by speimen. Fig. shows the results obtained with speimen, whih was loaded to failure after being loaded for six days at the stress level /f.6. The speimen developed appreiable nonlinear reep strains at the beginning of the loading proess, with an inrease in the number and width of the raks. However, its response beame stable at a later stage, and no additional raks t [se] 8 t [hours] Fig. 9 Conrete failure under a sustained load: (a stress-strain diagrams for speimens - ; and (b, plots of the strains as a funtion of the time: (b /f.9, and ( /f [MPa] -3 [ ] -35 [MPa] (a ➌ ( (e ➄ [ ] t [days] ➄ + v 6 ➅ 6 η [-] -4 [MPa] (b (d ➄ Expeted t ➍ t [days] ➅ ➆ [ ] - [ ] -4 Fig. (a Stress-strain diagram of speimen ; (b affinity oeffiient (ηϕ/ϕ lin, speimen ; ( development of the strains over time of speimen ; (d stress-strain urves of speimens - reloaded after a period of sustained loading; and (e detail of the previous plot. ➇ ➆ ➍ ➇ 6

7 M. F. Ruiz, A. Muttoni and P. G. Gambarova / Journal of Advaned Conrete Tehnology Vol. 5, No. 3, -, 7 7 appeared (neither did their width signifiantly inrease. Comparing the total strains with the linear-reep strains (obtained from some linear-reep speimens loaded in the same day, see the dashed line in Fig. it an be seen that between and 6 days (onset of reloading the delayed strains of the speimen were almost those orresponding to linear reep. The time-related evolution of the affinity oeffiient η an also be observed in Fig. b, where the values are very large at the beginning, and then quikly derease and stabilise. The expeted asymptote (for time would be lose to η.3, if the remaining part of the nonlinear strain is negleted ompared to the linear-reep strain, as suggested by the trend shown over the last days. Suh value is alulated assuming that the linear reep oeffiient at time infinite is ϕ(,8. (obtained by adjusting the reep expression of the MC-9 to the test results and by introduing (t and,nl.77 (obtained from Fig. into eq. (9: η ϕ ϕ lin ( ϕlin + ε ( t ϕ lin, nl (3 The value η.3 is in good agreement with other theoretial preditions based on the affinity hypothesis (η + ( /f 4.5, see Eq. (5. Finally, several tests were arried out with the same loading history as speimen (sustained load + reloading, but with higher initial stress levels, in order to shorten the time to failure under sustained loading. These speimens were left to develop nonlinear reep strains for less than five minutes. The results shown in Figs. d,e refer to three speimens, where the load was inreased at different phases of the nonlinear reep proess, so as to observe the typial behaviour of Phase (rak development, speimen, Phase (stable rak growth, speimen and Phase 3 (unstable rak propagation, speimen. The results show that after the development of some nonlinear reep strains, the speimen remains apable of arrying additional loads. However, the previous load history is remembered and the stress inrease depends on the amount of inelasti strains developed during the nonlinear reep phase. Furthermore, during the reloading proess, the stiffness is similar to that in the elasti domain, with the onrete still undamaged or only slightly damaged. However, when onrete is onsiderably raked, nonlinear-reep strains develop during the reloading proess and the apparent stiffness is smaller Influene of the loading rate An analogy may be established between the effets of a sustained load and of the loading rate on the failure of the material. Loading-rate effets were studied by Rüsh (96 and are urrently onsidered by some odes (for instane CEB MC 9. Some tests have been arried by the authors, by hanging the rate of the imposed displaement (δ on several speimens. These tests were performed on the same onrete, but at different ages (.5 months and 8 months, set and set in the following and at different displaement rates ( 3 δ mm/se and δ mm/se. The results are shown in Fig.. Both sets exhibited a similar behaviour haraterized by a smaller peak stress ( strength when the loading -5 (a (b -5 [MPa] ➎ Set ➍ δ [mm/s] [MPa] ➐ Set ➏ δ [mm/s] δ [mm] -4.5 δ [mm] -4.5 (. Creep-failure tests ➀.. ➃ f /f [-] Loading-rate tests Creep tests ➎ - ➐ ➄ - ➇ - 3 t [min] 5 MC - 9 Fig. Test results of the influene of the loading-rate on onrete strength: (a set ; (b set ; and ( plot of onrete non-dimensional ompressive strength as a funtion of the loading time.

8 8 M. F. Ruiz, A. Muttoni and P. G. Gambarova / Journal of Advaned Conrete Tehnology Vol. 5, No. 3, -, 7 rate was lower. This seems reasonable, sine lower rates imply that the load is applied over a greater period of time, thus giving the raks the possibility to open and to propagate in the same way as in pure-reep tests, up to the failure of the material (this is also an explanation of the similar behaviours of the young onrete set and of the older onrete set. However, the development of miro-raking (whih was onsidered in the proposed model to depend on the ratio /f is not as fast under an inreasing load as in a pure reep test in whih the load is sustained (i.e. applied at its maximum value sine the beginning of the test. This is logial sine raks propagate from the beginning of the test in pure reep tests but not in loading rate tests. The results obtained with different loading rates are ompared in Fig., along with those onerning nonlinear reep tests and the MC-9 loading rate formulation. One should note that in the nonlinear reep tests a lower strength is obtained than in the orresponding loading-rate tests. This onfirms previous onsiderations on the development of miro-raking Role of raking To study the development of raking in a nonlinear rheologial proess, four speimens were subjeted to multiple relaxation steps (i.e. unloading at onstant displaement between the plates of the hydrauli jak. At eah step, the number of raks and their width were reorded by measuring the rak widths at the surfae of the speimen using a manual rak omparator. The results (Fig. show that raking has a sizeable influene at stress levels above 6 % of the ompressive strength. The relaxation due to nonlinear reep is remarkable and pseudo-plasti strains develop as raking progresses, as onfirmed by the investigation on the onrete tested at different ages (48 days and eight (a -5 σ [MPa] 4 w [mm] Set (b ➍ Set ➏ ➈ [ ] ➉ Set [ ] Set ➑ ➒ Fig. Multiple relaxation steps: stress-strain diagrams and plots of the maximum rak width as a funtion of the mean strain, for: (a set ; and (b set. months. As already observed, the influene of onrete age is minimal, beause nonlinear reep strains developed during the relaxation proess have mainly to do with onrete raking. 5. Fitting of test results In Fig. 3, the proposed model is shown to fit rather well the results of the tests. The monotoni envelope (omprising both the pre- and post-peak branhes is obtained using the expressions detailed in Appendix of this paper. The points on the reep limit (omprising linear and nonlinear reep strains are obtained assuming a linear reep oeffiient equal to. (see setion 4. for all speimens and using eq. (5 for estimating nonlinear reep strains. The points representing failure under sustained load (omprising also linear and nonlinear reep strains are obtained onsidering the available inelasti strain of the monotoni envelope and, as previously stated, assuming that failure develops when the nonlinear reep strain equals two thirds of the available inelasti strain. Thus, sine the value of oeffiient η is known from eq. (5, in light of eq. ( the linear reep strains developed at failure an be estimated and onsequently the total (linear + nonlinear reep strains. As shown in Fig. 3, the points representing onrete failure are very lose to the envelope onsisting of two urves, the upper with a softening branh (failure under sustained load and the lower with a single inreasing branh (reep limit. The development of the strain over time as obtained with Eq. ( is further ompared in -4 [MPa] ➀ ➁ [ ] ➂ ➃ [ ] Fig. 3 Stress-strain urves for speimens : ompliane with the proposed sustained-load envelope. /,in [-] Proposed model Tests t/t F [-] Fig. 4 Development of the delayed strains in speimens to, and omparison with the proposed theoretial model.

9 M. F. Ruiz, A. Muttoni and P. G. Gambarova / Journal of Advaned Conrete Tehnology Vol. 5, No. 3, -, 7 9 Fig. 4 with similar results. For the part of the envelope related to the reep limit, the results shown in Fig. 5, with the reloading of speimen, exhibit a very satisfatory agreement between the visous delayed strains and the strain inrement obtained by shifting the monotoni urve to the right. This result perfetly agrees with the proposed theoretial model (Fig. 3. Furthermore, for the reloading behaviour, the model is also in aordane with the results presented in Figs. a,d,e and. The reloading modulus is similar to the elasti modulus whenever the inelasti strains (and subsequently the miro-raking and damage of the material are rather small, but it dereases at large inelasti strains. The proposed model was also used to desribe experimental results by Rüsh (96. To this end, the affinity oeffiient was evaluated by means of Eq. (5 and the analytial law desribing the monotoni urve is detailed in Appendix. The fitting of the test results (Fig. 6 is very good indeed. 6. Parametri study The influene of onrete short-term strength (f and linear-reep oeffiient on onrete long-term strength (f *, see Fig. 7a is investigated in this setion, taking advantage of the proposed model. Conrete short-term strength is introdued with values ranging from to MPa. The orresponding -4 [MPa] (a ➌ [ ] ➄,lin +s -3 [ ] (b ➄ t [days] Fig. 5 Speimen : (a stress-strain diagram; and (b strain development over time. /f [-] [ ] -8 Fig. 6 Comparison of the proposed theoretial model with the test results by Rüsh (96.,lin +s 6 stress-strain diagrams (Fig. 7b are obtained by using the analytial laws detailed in Appendix (assuming the same loading rate as in ordinary laboratory onditions: / t -5 se -. Although developed for normal-strength onrete, the proposed model is applied here to high-strength onrete to study the effet of brittleness on onrete long-term strength (in other words, the behaviour of high-strength onrete is assumed to be similar to that of normal-strength onrete onerning nonlinear reep and miro-raking. Two ases are studied, with the same equivalent thikness (e A /u 8 mm, where A is the onrete ross-setional area and u its perimeter and loading time (t 8 days. The relative humidities are assumed to be 95% and 6% respetively; as a result, the linear reep oeffiients are equal to.3 and.9 respetively for a onrete ompressive strength of 3 MPa. The results are ompared with those by Rüsh (96 and Fouré (985 in Fig. 7, where one should note that the ratio f */ f is rather stable. Furthermore, the soundness of the value f */ f.85, taken for granted in pratie and adopted from the experimental results of Rüsh (96, is onfirmed for f up to 3-4 MPa. It should be noted that this limit orresponds to average value and that a statistial analysis should be performed for determining the harateristi value of this limit. Although the model has not been developed for (a σ ( f */f [ ] f Rüsh Fouré f - [MPa] f [MPa] (b [ ] Fig. 7 Long-term strength of onrete: (a definition of short- and long-term strengths, f and f * respetively (the dotted line is the monotoni post-peak envelope; (b plots of the analytial stress-strain diagrams used in the parametri study; and ( plots of f */ f aording to the proposed model and omparison with the experimental results of Rüsh (96 and Fouré (985; equivalent thikness 8 mm, loading time 8 days, relative humidity 95% (urve A and 6% (urve B. A B

10 M. F. Ruiz, A. Muttoni and P. G. Gambarova / Journal of Advaned Conrete Tehnology Vol. 5, No. 3, -, 7 high-strengh onrete, a good agreement was also obtained with the test by Fouré for strengths up to 8 MPa. The f */ f ratio learly dereases in suh ases, and limiting its value to.65 or.7 seems very reasonable. A reason for the derease of the f */ f ratio with f is found in the more brittle post-peak behaviour of high-strength onrete, whih redues the value of the inelasti strains (,in that an be developed for a given stress/strength ratio. To limit this effet, the inelasti strain apaity of onrete should be inreased (for instane by adding fibres or by introduing a onfining pressure, see Fig. 8. In this way, the ratio f */ f is inreased as well. In any ase, more investigation on high strength onrete behaviour at high stress levels is required to hek the appliability of the assumed hypotheses to other mixes sine the rak initiation and nonlinear reep strain development may be different to those assumed in this paper for other matries. 7. Conlusions The relationship between nonlinear-reep strains and raking in onrete is investigated in this paper to desribe the possible failure of the onrete under sustained loading. A physial model is proposed to desribe the nonlinear reep strains that are assumed to have an inelasti nature and to be related to miro-raking. A failure riterion is also proposed. Aording to this riterion, onrete rushes under a sustained load when no additional inelasti strains an be developed within the material for a given level of the stress. An experimental ampaign was arried out to validate the previous assumption on the nature of nonlinear-reep strains, and to hek the reliability and auray of the proposed model. The relationship between nonlinear reep and miro-raking was onfirmed by the measurements performed on the speimens subjeted to a sustained load and to multiple-relaxation steps. Furthermore, the proposed failure riterion leads to a quite satisfatory theoretial-experimental agreement for various onrete ages, loading paths and loading-reloading proesses. The soundness of the affinity hypothesis between the Confined or fibre reinfored onrete Plain onrete Fig. 8 Inrease of the inelasti strain apaity of onrete by adding fibres or by introduing a onfining pressure. linear and nonlinear reep strains (ommonly adopted in strutural design is also the subjet of this study, that onfirms the validity of the affinity hypothesis in the rak-development phase and in the stabilized-rak phase, where the shape of the strain-time urve is onvex. However, this hypothesis annot be introdued in the phase of unstable rak-growth, where the strain-time urve has a onave shape. In this ase, an analytial law based on the resistane of the material to rak propagation is proposed in order to desribe the development of the inelasti strains over time. Finally, a parametri study based on the proposed model has shown that the long-term strength of onrete (urrently assumed as 85% of the short-term strength is likely to be unsafe for high-strength onrete. In suh a ase, the long-term strength of onrete should be dereased to 65-7% of the orresponding short-term strength, unless the inelasti strain apaity of the material is inreased by adding fibres or by introduing a onfining pressure. Appendix The analytial stress-strain diagrams used in this paper have been obtained by using the following equation proposed by the authors: E ε ε + ε with ε : ε and α: E α α f ( α α [ MPa] ( [ MPa] (4 (5 f.5 f α + + (6 5 Notation The following symbols are used in this paper: E modulus of elastiity of onrete E A,B modulus of elastiity of onrete (at A,B N F number of yles at failure f onrete ylindrial strength in ompression n number of yles t time t time at loading,in maximum allowable inelasti strain for any given load level δ imposed displaement δ displaement rate onrete strain,el onrete elasti strain onrete shrinkage strain s

11 M. F. Ruiz, A. Muttoni and P. G. Gambarova / Journal of Advaned Conrete Tehnology Vol. 5, No. 3, -, 7 p onrete plasti strain total reep strain in onrete ( linear + nonlinear reep strains,lin onrete linear-reep strain,nl onrete nonlinear reep strain (time-dependent strain assoiated with onrete miro-raking plasti strain + damage strain v onrete visous strains (time-dependent strain not assoiated with onrete mi ro-raking linear reep strains + shrinkage strains η affinity oeffiient ϕ reep oeffiient ϕ lin linear-reep oeffiient onrete stress Referenes Avram, C., Faaoaru, I., Filimon, I., Mirsu, O. and Tertea, I. (98. Conrete strength and strain. Elsevier Sientifi Publishing Company, Amsterdam Oxford New York, 558 p. Barpi, F. and Valente, S. (. Creep and frature in onrete: a frational order rate approah. Engineering Frature Mehanis, 7, Barpi, F. and Valente, S. (5. Lifetime evaluation of onrete strutures under sustained post peak loading. Engineering Frature Mehanis, 7, Bažant, Z. P. and Gettu, R. (99. Rate effets and load relaxation in stati frature of onrete. ACI Materials Journal, 89 (5, Bažant, Z. P. and Li, Y N. (997a. Cohesive rak with rate dependent opening and visoelastiity: I. mathematial model and saling. International Journal of Frature, 86, Bažant, Z. P. and Li, Y N. (997b. Cohesive rak with rate dependent opening and visoelastiity: II. Numerial algorithm, behavior and size effet. International Journal of Frature, 86, CEB FIP (993. Model ode for onrete strutures. Comité Euro International du Béton, Lausanne, Switzerland, Thomas Telford Ltd., London, 46 p. El-Kashif, K. F. and Maekawa, K. (4. Time-dependent nonlinearity of ompression softening in onrete. Journal of Advaned Conrete Tehnology, (, Fernández Ruiz, M. (3. Nonlinear analysis of the strutural effets of the delayed strains of steel and onrete. (in Spanish, Evaluaión no lineal de los efetos estruturales produidos por las deformaiones diferidas del hormigón y el aero, PhD. Thesis, Universidad Politénia de Madrid, Ed. ACHE, Madrid, Spain, 75 p. Fernández Ruiz, M., Del Pozo Vindel, F. J. and Arrieta Torrealba, J. M. (4. Nonlinear reep of onrete. analytial modelling and agreement with test results and previous theoretial models. (in Spanish, Estudio sobre el omportamiento no lineal de la fluenia. Propuesta de modelo y omparaión on resultados experimentales y modelos teórios, Hormigón y Aero, 3, Madrid, Spain, Fouré B. (985. Long term strength of onrete under sustained loading. (in Frenh, Résistane potentielle à long terme du béton soumis à une ontrainte soutenue, Annales de l Institut Tehnique du Bâtiment et des Travaux Publis, 43, Paris, Frane, Maekawa, K. and El-Kashif, K. F. (4. Cyli umulative damaging of reinfored onrete in post-peak regions. Journal of Advaned Conrete Tehnology, (, Mazzotti, C. and Savoia, M. (3. Nonlinear reep damage model for onrete under uniaxial ompression. ASCE, Journal of Engineering Mehanis, 9 (9, Neville, A. M. (97. Creep of onrete: Plain, reinfored and prestressed. Noth Holland, Amsterdam, 6 p. Pfanner, D., Stangenberg, F. and Petryna, Y. S. (. Probabilisti fatigue damage model for reinfored onrete. Institute for reinfored and prestressed onrete strutures, Ruhr Universität Bohum, Bohum, 8 p. Qingbin, L., Peiyin, L. and Lixiang, Z. (4. Damage degradation of onrete due to ompressive fatigue loading. Key Engineering materials, 74 76, 3 8. Rüsh, H. (96. Researh toward a general flexural theory for strutural onrete. ACI Journal, 57 (, 8. Shah, S. P. and Chandra, S. (97. Frature of onrete subjeted to yli and sustained loading. ACI Journal, 67 (, van Zijl, G. P. A. G., Borst, R. and Rots, J. G. (. The role of rak rate dependene in the long term behaviour of ementitious materials. International Journal of Solids and Strutures, 38,