A four-exponential model to describe the behaviour of fibre reinforced concrete

Transcription

1 Materials and Strutures / Matériaux et Construtions, Vol. 37, August-September 004, pp A four-exponential model to desribe the behaviour of fibre reinfored onrete M. K. Lee and B. I. G. Barr Shool of Engineering, Cardiff University, UK ABSTRACT The paper reports on a proposed four-exponential (four-e) model to desribe the behaviour of fibre reinfored onrete (FRC) in flexure. The four-e model was found to be apable to model a variety of urves and has the advantage of being able to model the omplete load-deformation response of FRC with one ontinuous funtion. The four-e model is also mathematially effiient and differentiation and integration an be readily applied. In addition, a rational and objetive proedure by applying the model has been proposed inreasing the potential of the four-e model. A omparison was arried out between theoretial and experimental load-defletion urves obtained for steel fibre reinfored onrete beams under three point bending. It was found that the four-e model an satistifatorily desribe the experimental load-defletion urves. The priniples applied for the omparison may possibly be appliable to FRC subjeted to other types of loading. RÉSUMÉ La publiation rend ompte d'un modèle à quatre fontions exponentielles («four-e») ayant pour but de dérire le omportement du béton renforé de fibres (FRC) sous divers types de hargement. Le modèle «four-e» s est avéré apable de modéliser une variété de ourbes et a l'avantage de pouvoir modéliser la réponse omplète hargement-déformation de FRC ave une fontion ontinue. Le modèle «four-e» est également mathématiquement effiae ar il se manipule failement, dans le sens que les proédures mathématiques telles que différentiation et intégration peuvent être aisément appliquées. En outre, le proédé proposé est raisonnable et objetif augmentant ainsi le potentiel du modèle «four-e». Une omparaison a été effetuée entre ourbes théoriques et expérimentales de hargement-fléhissement obtenues pour des poutres en béton renforées par des fibres d aier soumises à une flexion 3 points. On a onstaté que le modèle «four-e» peut dérire de manière satisfaisante les ourbes expérimentales de hargement-fléhissement. Les prinipes appliqués pour la omparaison peuvent également être appliqués au FRC soumis à un hargement en ompression ou en tension.. INTRODUCTION Conrete is an attrative onstrution material sine it is heap, durable and possesses adequate ompressive strength and stiffness for strutural use. Unfortunately, it is a relatively brittle material and is inherently weak in tension. To endow the material with dutility and toughness, fibres have been inluded into the onrete matrix to produe fibre reinfore onrete (FRC). Typial appliations for FRC inlude highway and airfield pavements, hydrauli strutures, refratory onrete, tunnel linings, pre-ast produts and fibrous shotrete [, ]. Fibre addition modifies the frature proess of onrete and an be onsidered to at on the material and strutural sale [3, 4]. The mode of ation on a material sale involves the fibres ating as rak arrestors thus inreasing the required energy for rak propagation. On a strutural sale, the fibres span aross raks and transmit tensile stresses aross the raks even when the rak bridging apaity by aggregates have failed thus providing strutural stability [3]. The result of this mehanism is an inrease in the dutility of the struture [5]. The sudden brittle failure Editorial note Prof. Benjamin I.G. Barr is a RILEM Senior Member. He partiipates in RILEM TC HFC Hybrid fibre onrete /04 RILEM 464

2 Materials and Strutures / Matériaux et Construtions, Vol. 37, August-September 004 assoiated with unreinfored plain onrete is avoided and the struture is able to bear loads. When loaded, FRC first exhibits an approximately linear response before reahing its tensile strength. After raking, the load-deformation response beomes highly non-linear with the shape of the load-deformation urve dependent upon the type of fibre, fibre geometry, number of fibres, the orientation with respet to the rak faes, and the pull out behaviour. This is observed when arrying out losed-loop ompression tests, uni-axial tensile tests or flexural tests on FRC.. RESEARCH SIGNIFICANCE The shape of the load-deformation response of FRC is required when arrying out non-linear analyses of strutural members. The omplete stress-strain (-) urve for onrete in ompression is not usually quantified but is required to arry out non-linear analyses of strutural members to provide a sound basis for the derivation of design urves [6]. With the advent of non-linear frature mehanis (NLFM) in the analysis of onrete frature behaviour, inreasing interest has grown in determining the response of ementitious materials under uni-axial tensile loading. The first NLFM theory was oneived by Hillerborg et al. [7] for plain onrete and has sine been introdued for the analysis of FRC [8]. In general, the frature behaviour of onrete may be simulated using either a disrete rak approah [7] or a rak band approah [9]. For the former, the behaviour of the frature proess zone is governed by the stress-rak opening (-w) relationship whereas for the latter, the frature proess zone is assumed to be ative within a ertain rak band width and its behaviour is governed using - urves. The governing -w and - urves an be obtained diretly from uni-axial tensile tests [0, ] or indiretly evaluated using beam-bending tests [, 3]. In this paper, the onstitutive relationships desribing the material behaviour in the unraked and raked state will be olletively referred to as the stress-deformation (-u) relationship. inreasing the attrativeness of the double-e model. The shape parameter is related to the brittleness of the material and an be determined by using ratios of defletion. On the other hand, the values of and are determined diretly (as summarised in the Appendix) using the following equations: f and ln () x ( ) where f and x haraterises the peak of the f(x) urve as depited in Fig.. Further details regarding the double-e model are presented in the Appendix. By varying from approximately to, the double-e model modifies the f(x) response from an approximately brittle response to an elasti-plasti response, as depited in Fig.. This shows that the double-e model is apable of modelling a variety of urves by means of one ontinuous funtion. 3. The four-e model The behaviour of FRC generally exhibits more ompliated load-defletion urves relative to that observed for plain onrete. Therefore, an expression onsisting of 3. MULTIPLE-EXPONENTIAL MODELS Fig. Diagram showing the definition of f and x. 3. Double-e model In a ompanion paper [4] by the authors, a doubleexponential (double-e) model was introdued for plain onrete. This model an be expressed as follows: x x f (x) (e e ) () where, and are onstants ontrolling the shape of the double-e model (Note: This funtion is disussed in the Appendix and is illustrated in one of the graphs shown later in Fig. 3). The double-e model has been shown [4] to be suitable to model the load-defletion behaviour of plain onrete. It is both flexible and widely appliable sine a range of material responses varying from brittle to elasti-plasti may be desribed with one ontinuous funtion. Additionally, an objetive proedure was developed, thus Fig. Plots showing the different f(x) urves whih an be obtained by varying the ratio. 465

3 Lee, Barr Fig. 3 A shemati view of the four-e model. four exponential funtions is proposed whih is a superposition of two double-e models. This onept is illustrated in Fig. 3, and the general expression is as follows: x 3x 5x 6x e e e e f (x) (3) 4 where,, 3, 4, 5 and 6 are onstants ontrolling the shape of the f(x) urve. The first double-e expression may be regarded as representative of the ontribution of the onrete matrix (and is similar to that used by the authors [4] to desribe the failure urve obtained for plain onrete) whereas the seond double-e expression represents the inreased toughness due to fibre addition. Reognising the fat that the four-e model is the result of the superposition of two double-e models, the task of developing a suitable proedure in determining the model parameters may be arried out. By assuming x 3x F (x) (e e ) and 5x 6x F (x) 4 (e e ), we have the following simplified expression for f(x): f (x) F (x) F (x) (4) The parameters 3 and 6 may be expressed in terms of ratios of and 5 respetively, as follows: and 6 5 (5) 3 The motivation for superimposing two double-e models is to desribe urves with long tails suh as those exhibited by the loaddeformation response of FRC. As in the ase of the work arried out for the double-e model in desribing the load-deformation response of plain onrete [4], a study was arried out to determine whether the shape parameters ould be evaluated in an objetive manner. At first glane, the task of determining the six shape parameters seems a daunting one. However, this hallenge may be takled by first onsidering typial urves whih may be observed for FRC materials. Two general forms may be envisaged: ) Response A: The initial response is approximately linear up to a peak, f. Subsequently, in the postraking regime, the f(x) urve ahieves some sort of plateau as shown in Fig. 4a. ) Response B: The initial response is similar as that desribed above i.e. an approximately linear response up to a peak, f. Thereafter, a seond peak is ahieved in the post-raking regime. This seond peak may or may not be greater than the first peak. This is illustrated in Fig. 4b. 3.3 Response A Sine the main objetive of implementing the four-e model is for a better desription of the post-raking regime for FRC, the seond exponential funtion, F (x), will be assumed to be the dominant fator within the tail region. Thus, within the plateau region, the following ondition will prevail: d F x x 0 i.e. e e (6a) dx The above ondition is fulfilled if. Therefore, assuming >> 0, we obtain: 5x 4e F (x) (6b) Sine Equation (6b) has only two shape parameters, we an solve the equation by speifying two points, say at x a and x b. Hene, we have: f a f i.e. f 5 x a 4e 5 x 4 b and f e b a 5 (x b xa ) e (7a) b f Solving for gives a 5 ln (7b) x b x a f b Hene, 5xa 5x b 4 fae f be (7) Therefore, the speifiation of two points, (x a, f a ) and (x b, f b ), will be suffiient to define the plateau region for Fig. 4 Diagrams showing the general trends for two types of f(x) urves. 466

4 Materials and Strutures / Matériaux et Construtions, Vol. 37, August-September 004 response type A, provided >> 0. For onveniene, this ondition is deemed to be fulfilled with a value = 0 4, as it has already been shown in Fig. that a plateau is ahieved at this value of. Now that F (x) is solved, the first exponential funtion, F (x), needs to be defined. To do so, it is assumed that the pre-plateau region is dominated by F (x). Therefore at the peak, we have: df dx x x 0 whih gives ln (8) x ( ) The value of an be found by onsidering two defletion ratios, as already shown for the double-e model [4] when used for plain onrete. For the four-e model, it is proposed that the ratio (x /x 0.9f ) be used where it is defined as the ratio of x to the value of x at 90% of f in the pre-peak regime. This is to inlude urves that have plateau levels near the peak, f. The relationship between the ratio and (x /x 0.9f ) is shown in Fig. 5. This relationship shown in Fig. 5 annot be expressed in a simple manner. Instead, it has been found that it is more suitable to mathematially desribe it using multiple expressions as given below: x j jo x0.9f 4 x k k 0 x f j k x for x 0.9f x for.63 x 0.9f x for.7 x x for x0.9f f The values of the onstants j and k are given in Tables a and b respetively. These were determined via normal urve fitting methods. The value of the shape parameter an then be determined, using the following: (9) f F (x ) (0) x x e e Table a Values of j for use with Equation (9) j 0 3 j Table b Values of k for use with Equation (9) k 0 3 k Response B The determination of the model parameters for Response B will again be arried out with the assumption that F (x) is dominant within the pre-raking regime whereas F (x) is dominant within the post-raking regime. Fig. 5 Plot showing the relationship between and (x /x 0.9f ). Fig. 6 Diagram showing the various points onsidered for the determination of the four-e model parameters for urves displaying Response B. Consideration is first given to F (x). The ratio between 6 to 5 i.e. an be determined by evaluating the deformation ratio (x* /x* 0.9f ). The term x* is the x-value orresponding to the seond peak, f*, whereas x* 0.9f is the x-value orresponding to 90% of f* in the tail region before x* as shown in Fig. 6. This value was hosen so as to inlude urves that possess signifiantly long tails and low gradients. After the evaluation of using Equation (9), the values of 4, 5 and 6 may be determined using the following equations: 5 4 ln ; 6 5 ; x * ( ) f * () After solving for F (x), the entire four-e model may be solved. As an approximation, the initial portion of the f(x) 467

5 Lee, Barr urve is assumed to be dominantly governed by F (x). Thus the value of may be alulated using the deformation ratio (x /x 0.9f ) in Equation (9). Subsequently, the values of and 3 may be determined using the following equations: ln and 3 (a) x ( ) The value of an then be omputed as follows: f F (x ) (b) x x e e 3.5 Applying the four-e model and disussions In Setions.3 and.4, a rational proedure has been developed whereby the four-e model parameters may be evaluated in an objetive manner. Following this, an illustration of the appliation of the four-e model, using some experimental results obtained during a round-robin test programme [5] for steel fibre reinfored onrete (SFRC) is reported here. Two different onrete strengths were tested whih were C5/30 and C70/85, whih were normal strength onrete (NSC) and high strength onrete (HSC) respetively. The letter C indiates the harateristi ompressive strength and the numerals indiate the ompressive strengths in MPa measured for ylinders and ubes respetively. Three types of fibres were onsidered in the test programme and were ollated, bound by water-soluble adhesive. The fibre dosages employed within the round robin test programme were 5kg/m 3, 50kg/m 3 and 75kg/m 3. The mix details used in the study are given in Table. Beam speimens with dimensions of mm were ast and allowed to ure for seven weeks before testing was arried out. The beams were nothed perpendiular to the top surfae at asting using diamond impregnated rotating saws to introdue a uniform ligament depth of 5mm. The nothed beams were subjeted to three point loading aording to RILEM TC 6-TDF [6] and the test set-up is shown in Fig. 7. Some slight modifiations to the RILEM reommendations were introdued in the Fig. 7 Shemati diagram showing the beam bending test set-up (after RILEM [6]). experimental work: whereby the tests were arried out under rak mouth opening displaement (CMOD) ontrol. All tests were arried out under losed looped onditions with applied load, mid-span defletions and CMOD measured eletronially at regular intervals. Typial experimental load versus mid-span defletion (P-) urves for the four SFRC types are given in Fig. 8. The C5/30 with 5kg/m 3 of fibres an be lassified as displaying Response A-type behaviour. The other three types of SFRC beams, on the other hand, may be lassified as Response B-type behaviour. The experimentally determined parameters for the four-e model for the C5/30 (with 5kg/m 3 of fibres, i.e. Response A) SFRC beam are given in Table 3a and the model parameters are given in Table 4a. A similar ompilation for the other three SFRC types (exhibiting Response B) is presented in Tables 3b and 4b. Using the model parameters, theoretial P- urves may be evaluated and the omparison between the experimental and theoretial urves is shown in Fig. 9. From the plots shown in Fig. 9, it an be onluded that the four-e model shows good agreement Con. grade (Fibre dosage in kg/m 3 ) Table Details of mix proportions Cement Fine aggregate Coarse aggregate Water Silia fume C5/30 (50) C5/30 (5) C5/30 * (75) C70/85 ** (5) Note: * best estimate for well-graded aggregate ** 7.75 litre/m 3 of superplastiiser was added. Dramix 80/60 BN was used (aspet ratio = 80, length = 60mm) Dramix 65/60 BN was used (aspet ratio = 65, length = 60mm) 3 Dramix 80/60 BP was used (aspet ratio = 80, length = 60mm) (a) C5/30 with 5kg/m 3 of fibres Fig. 8 Typial P- urves for the SFRC beams. 468

6 Materials and Strutures / Matériaux et Construtions, Vol. 37, August-September 004 (a) (b) () (d) Fig. 9 Plots showing a omparison between the experimental and four-e model P- urves. (a) C5/30 with 5kg/m 3 of fibres; (b) C5/30 with 50kg/m 3 of fibres; () C5/30 with 75kg/m 3 of fibres; (d) C70/85 with 5kg/m 3 of fibres. Table 3a The experimental values required for the determination of the four-e model parameters for the C5/30 with 5kg/m 3 of fibres SFRC beam Conrete type C5/30 (5) P (kn) (mm) P 0.9P (kn) (mm) P (kn) (mm) Table 3b The experimental values required for the determination of the four-e model parameters (for P- urves showing Response B) First peak Seond peak Conrete P P* * * type (kn) (mm) (kn) (mm) C5/30 (50) C5/30 (75) C70/85 (5) 0.9P * 0.9P with the experimental P- urves for the SFRC beams, espeially when the P- urve exhibits a plateau in the post-peak regime (Response A). In the ase of Response B-type behaviour, a ompromise must be made between desribing the behaviour in the servieability range and in approahing the ultimate behaviour at large defletions. Suitable adjustments an be made depending on the requirements of the desriptive model. Further work is in progress to deal with this limitation. 4. CONCLUSIONS A four-e model has been developed with the intent of desribing the omplex behaviour typial of FRC. The four-e model is attrative sine mathematial proedures suh as differentiation and integration an be arried out with ease. Furthermore, entire load- 469

7 Lee, Barr Table 4a The four-e model parameters for the P- urve for the C5/30 with 5kg/m 3 of fibres SFRC beam Con. type 4 5 C5/30 (5) Table 4b The four-e model parameters for the P- urves showing Response B Con. type 4 5 C5/30 (50) C5/30 (75) C70/85 (5) deformation responses may be desribed with one ontinuous funtion. Additionally, it has been shown that the four-e model parameters an be alulated using rational and objetive proedures. A omparison has been arried out between the four-e model and experimental P- urves for SFRC beams subjeted to three point bending obtained during a roundrobin test programme. Satisfatory agreement was found between the four-e model and experimental P- urves. Although omparisons have been arried out using only P- urves, the same priniples may be applied for other types of loading. REFERENCES [] Keer, J.G., Fibre reinfored onrete in New Reinfored Conrete (Surrey University Press, 984) [] Skarendahl, Å., Development and use of steel fibre shotrete, in Conrete Tehnology: New Trends, Industrial Appliations; Proeedings of the International RILEM Workshop on Tehnology Transfer of the New Trends in Conrete (E&FN Spon, 995) [3] Rossi, P., Aker, P. and Malier, Y., Effet of steel fibres at two different stages: the material and the strutures, Mater. Strut. 0 (987) [4] Rossi, P. and Harrouhe, N., Mix design and mehanial behavior of some steel-fibre-reinfored onretes used in reinfored onrete strutures, Mater. Strut. 3 (990) [5] Rossi, P. and Chanvillard, G., A new geometry of steel fibre for fibre reinfored onretes in High Performane Fiber Reinfored Cement Composites, Proeedings of the International RILEM/ACI Workshop (E&FN Spon, 99) [6] Van Gysel, A. and Taerwe, L., Analytial formulation of the omplete stress-strain urve for high strength onrete, Mater. Strut. 9 (996) [7] Hillerborg, A., Modéer, M. and Petersson, P.E., Analysis of rak formation and rak growth in onrete by means of frature mehanis and finite elements, Cement and Conrete Researh 6 (976) [8] Hillerborg, A., Analysis of frature by means of the fititious rak model, partiularly for fibre reinfored onrete, International Journal of Cement Composites (4) (980) [9] Bažant, Z.P. and Oh, B.H., Crak band theory for frature of onrete, Mater. Strut. 6 (983) [0] Gopalaratnam, V.S. and Shah, S.P., Softening response of plain onrete in diret tension, ACI Journal 8(3) (985) [] van Mier, J.G.M. and van Vliet, M.R.A., Uniaxial tension test for the determination of frature parameters of onrete: state of the art, Engineering Frature Mehanis 69 (00) [] Chuang, T.-J. and Mai, Y.W., Flexural behavior of strainsoftening solids, International Journal of Solids and Strutures 5() (989) [3] Guo, X.H., Tin-Loi, F. and Li, H., Determination of quasibrittle frature law for ohesive rak models, Cement and Conrete Researh 9 (999) [4] Barr, B.I.G. and Lee, M.K., Modelling the strain-softening behaviour of plain onrete using a double-exponential model Magazine of Conrete Researh 55(4) (003) [5] Barr, B.I.G., Lee, M.K., de Plae Hansen, E.J., Dupont, D., Erdem, E., Shaerlaekens, S., Shnütgen, B., Stang, H. and Vandewalle, L., Round-robin analysis of the RILEM TC 6-TDF beam-bending test: Part test method evaluation, Mater. Strut. 36 (63) (003) [6] RILEM TC 6-TDF, RILEM TC 6-TDF: Test and design methods for steel fibre reinfored onrete. Bending tests. Reommendations, Mater. Strut. 33 (5) (000) 3-5. APPENDIX The double-e model is the result of the superposition of two exponential funtions as shown in Fig. A. It an be expressed as follows: f (x) (e (e x x e e 3x ) x ) (A) where = 3 / Differentiating equation (A), the gradient at any point an be easily alulated: df dx x x ( e e ) (A) Fig. A The double-e model is the result of two exponential funtions. 470

8 Materials and Strutures / Matériaux et Construtions, Vol. 37, August-September 004 At the peak of f(x) i.e. (x, f ), the gradient is zero. Thus, by equating equation (A) to zero, the value of x in terms of is as follows: x ln (A3a) ( ) Rearranging equation (A3a), the model parameter is a funtion of and x : ln (A3b) x ( ) Also, onsidering the point (x, f ) and using equation (A), we have: f x x (e e ) (A4) Substituting equation (A3b) in equation (A4), we obtain: ln ln f e e (A5a) Solving equation (A5a) in terms of, we have: f (A5b) Therefore, the model parameter is a funtion of and f. Paper reeived: May 6, 003; Paper aepted: July 0,