Experimental and numerical analyses of FRC slabs on grade

Transcription

1 Eperimental and numerical analses o FRC slabs on grade B. Belletti & R. Cerioni Department o Civil and Environmental Engineering and Architecture, Universit o Parma, Ital A. Meda Department o Civil Engineering, Universit o Brescia, Ital G. A. Plizzari Department o Engineering Design and Technologies, Universit o Bergo, Ital ABSTRACT: An eperimental and numerical stud on the racture behavior o iber reinorced concrete slabs on grade or industrial pavements is presented. Four Fiber Reinorced Concrete (FRC) slabs with ibers having dierent aspect ratios and volume ractions are tested in laborator. In order to reproduce a Winkler subgrade, the slabs were placed on several steel springs. Numerical simulations have been carried b using a commercial inite element code based on nonlinear racture mechanics where user subroutines have been implemented. This etension concerns the use o a more realistic law or modeling the stiness and strength o FRC ater cracking o the concrete matri. Moreover, a stiness matri or FRC with primar and secondar cracks has been added to the model. The comparison between eperimental and numerical results has shown a good agreement and supplies useul inormation or design provisions. Kewords: iber reinorced concrete, industrial pavements, slabs on grade, non-linear racture mechanics, inite element. INTRODUCTION Slabs on grade represent one o the main applications o Steel Fiber Reinorced Concrete (SFRC). In these structures ibers can totall substitute the conventional reinorcement (rebars or welded mesh) with signiicant advantages in terms o toughness and strength under static and dnic loads. Furthermore, ibers can reduce cracking due to thermal or shrinkage eects. The use o iber reinorcement is oten economicall convenient with respect to conventional reinorcement (rebars or welded mesh) due to the labor cost reduction. SFRC slabs on grade are oten designed with elastic methods that cannot adequatel simulate the actual material behavior ater cracking o the concrete matri. In act, ater cracking SFRC has a remarkable non-linear behavior that should be correctl modeled with numerical analses based on Non-Linear Fracture Mechanics (Hillerborg et al. 976). In the present paper earl results o an etensive eperimental and numerical research progr on SFRC slabs on grade are presented. The behavior o slabs placed on an elastic subgrade are considered herein with regard to both serviceabilit and ultimate limit states. The slabs are subjected to a single point load in the slab center. The eperimental stud concerns our ull-scale slabs on grade loaded in the center up to ailure with the acquisition o the deormation ield and the crack pattern. The specimens were characterized b dierent iber geometries and iber contents. The numerical analses, based on Non-Linear Fracture Mechanics (NLFM), aim to take into account the actual behavior o iber reinorced concrete that becomes particularl important in hperstatic structures where a remarkable increment o the load ma occur ater cracking until a collapse mechanism occurs (Meda & Plizzari 3). FE analses have been carried out b commercial code ABAQUS (999) where user subroutines have been introduced to better describe FRC cracking behavior. A mesh o multi-laered isoparetric shell elements has been adopted to describe the midsurace shape o the slabs. Finall, the role plaed b secondar cracking on stiness and ultimate load is underlined.

2 NUMERICAL MODEL An etension o PARC model (Belletti et al., ) is ormulated herein to describe the non-linear mechanical behavior o FRC subjected to plane stresses (Belletti et al., 3a). In the uncracked stage, biaial stress state has been modeled b means o two equivalent uni-aial stress-strain curves (Darwin & Pecknold 977, Kuper et al. 969) while the racture energ is adopted to provide stress-strain relationship in the cracked stage. Normal and shear stresses along crack suraces are computed as unctions o the local crack opening and slip. The FRC post-cracking behavior is modeled in PARC b a realistic micromechanics-based constitutive relationship. The latter includes the eects o the aspect-ratio, volume raction and interace bond strength o ibers, as well as concrete strength, on the stress-crack opening law (Li 99, Li et al. 993). The aggregate bridging eect is separated rom the iber bridging eect and the state o stress that characterizes the ibers beore cracking is also taken into account. Material behavior in the uncracked and cracked stages is described in the ollowing Sections.. Uncracked stage Uncracked FRC is idealized as a nonlinear elastic orthotropic material, assuming as orthotrop aes the principal directions (, ) o strain and the ollowing material stiness matri (Darwin & Pecknold 977, Belletti et al. 3a): E c ν Ec Ec [ ] ( ) D ν Ec Ec E c ν ν G where E c, E c are concrete elastic secant moduli in the principal directions, and G is the shear modulus o concrete: ( E + E ν E E ) [ 4( ν )] G () c c c c E c and E c are evaluated through the respective uni-aial curves equivalent to the biaial state (Darwin & Pecknold 977, Belletti et al. 3a). The peak values σ ci,ma (i, ) o the uni-aial curves or compression can var according to a biaial strength envelope (Kuper et al. 969, Belletti et al. 3a).. Singl cracked FRC (primar cracking) When the maimum principal stress becomes greater than the concrete tensile strength, concrete cracks and PARC model is adopted or modeling the cracked FRC. Primar cracks are idealized as equall spaced (at a distance a m ) and oriented at right angles to the maimum principal tensile stress (direction remaining ied; Fig. a). The local coordinate sstem (, ) is assumed coincident with the principal stress directions at crack onset and it does not var during the entire loading process. The local strain vector is then epressed as unction o the crack opening w and slip v, as well as o the strain c in the concrete strut between two contiguous cracks (Fig. b): { γ } w v c. (3) The strength and stiness o FRC in this stage is due both to concrete and ibers. The total iber stress (σ ) is given b the superposition o two eects: () s σ () s σ () s σ + (4) b ps where σ b is due to iber bridging and σ ps is due to the initial prestress in the iber. The relationships between σ b, σ ps and s are assumed according to Li et al. (993) and Belletti et al. (3a). The total displacement is s w + v, having magnitude equal to s (w + v ) / (Belletti et al. ; Fig. b). σ τ (a) a m l σ σ ψ τ τ τ t σ l ψ (b) c ibres v s ω w σ ω Figure. (a) Cracked FRC membrane element subjected to inplane stresses; (b) kinematic pareters deining the local strain vector. The two components o σ, nel σ (at right angles with respect to the crack direction) and τ (parallel), are evaluated as ollows (Fig. b): w σ c σ cosω σ c w (5) w + v τ σ

3 v τ c σ sinω σ c γ v (6) w + v Stiness c is given b the ollowing relationship: σ () s c (7) s and contributes to the material stiness matri. For the aggregate bridging action (σ a) the relationship reported in Li et al. (993) has been assumed. This eperimental law was calibrated on the basis o man eperimental data as a unction o the crack opening w. The stiness term c t to be implemented into the local FRC stiness matri is: σ a ( w) ct. (9) w The superposition o concrete and iber eects in FRC sotening is shown in Figure through the relationship between the normal stress and the crack opening w (where v and s w). Thereore, the assumed cracked FRC stiening matri has the ollowing orm: [ D ] ( c + c ) t cv ( ) Ec c + ca where the secant modulus Ē c has been determined on the basis o the sotened model or cracked concrete in compression proposed b Vecchio & Collins (993). The evaluation o the aggregate interlock coeicients (c a and c v ) can be ound in Belletti et al. () and it is based on the approach proposed b Gbarova (983). The resulting local stiness matri in the global (, ) coordinate sstem (Fig. 3a) is obtained b operating through a transormation matri [T ], which takes into account the angle (ψ) between the local -ais and the global - ais: T [ D ] [ T ] [ D ][ T ]. ().3 Doubl cracked FRC (secondar cracking) When the stress ield between two cracks produces a maimum principal stress greater than tensile strength, a secondar crack orms (Fig. 3). In the case o double cracking, a generalization o PARC is here adopted. It is based on the ollowing assumptions: - when secondar cracks start, primar cracks are not subjected to closing and re-opening; - total strain is obtained b adding the strain due to primar cracks to that due to secondar cracks. - stress ield is the se both or primar and secondar cracks. The ollowing notation is introduced (Fig. 3):, is the local coordinate sstem relative to the primar cracks and, is the one relative to the secondar cracks, θ is the angle between and aes, while primar and secondar crack directions (with respect to direction) are indicated b the angles ψ and ψ, respectivel (Figs. 3c, d). Moreover, kinematic variables or primar cracking are w, v and c (Fig. 3c), while or secondar cracking the are w, v and () c (Fig. 3d). Thereore, the strains caused rom each o two cracks are deined as: { } { } ( ) w v a m c () { } { w a v ) }, () ( m a m c, (3) where crack spacing ( a m ) is assumed to be the se or primar and secondar cracks and equal to the iber length. Total strain, epressed in global coordinate sstem, is decomposed into the contribution due to the primar cracking { } and into a part due to secondar cracking { () } : () {} { } + { } being: { } [ ] { } ( ) T (4) and () () { } [ ] { } ( ) T (5) where [T ] and [T () ] are transormation matrices computed b ψ and ψ, respectivel. Primar and secondar cracks are assumed to act one b one, and so their behavior is hpothesized unrelated to each other. Thereore, stiness material matrices, [D ] and [D () ], have the se orm o that introduced or FRC with primar cracks onl (Eq. ), epressed in - and - local coordinate sstems, respectivel. The are unctions o local strain ields, { } and { () } and, when transerred to the global coordinate sstem -, the become: [ D ] t [ T ] [ D ][ T ], (6) () () t () () [ D ] [ T ] [ D ][ T ]. (7) The total stress in global coordinate sstem is the se or primar and secondar cracking: () () { σ } [ D ] { } [ D ] { }, (8) Thereore, due to strain-decomposition, the total strain can be also written as ollows: () σ D. (9) { } [ ] [{} { } ]

4 (σ a+ σ ) (MPa) Total response Fibre bridging σ s Aggregate bridging σ a Fibre prestress σps Through simple analtical computations the stiness matri, which links stress to strain in global coordinate sstem or doubl cracked FRC, is ormulated as ollows: () () D D D T D + [ ] ([ ] [ ] ([ ] ([( ] T () () () T + [ T ] [ D ] [ T ] ) ([ T ] [ D ] ))) 3 EXPERIMENTAL MODEL () s w (mm) Figure. Aggregate and iber roles in FRC sotening ( c 7 MPa, V %, s w ). σ a m τ a) a m b) ψ ψ σ σ l c) d) τ τ c τ ψ s ω t ψ σ v c () v w ω w s Figure 3. (a) Doubl cracked element subjected to plane stress; (b) notations or primar and secondar cracking; (c) kinematic pareters o primar cracking; (d) kinematic pareters o secondar cracking. l The behavior o SFRC slabs on grade was eperimentall studied b perorming tests on ull-scale specimens having a square geometr with a side o 3 m and a thickness o.5 m (Fig. 4). The adopted dimensions are signiicant or industrial pavements, where the presence o construction joints can divide the structures into independent slabs. Three dierent tpes o steel ibers have been adopted: hooked ibers having a length (l ) o 5 mm and a dieter (φ ) o mm (5/., aspect ratio5), hooked ibers l 3 mm and φ.6 mm (3/.6, aspect ratio5), and straight ibers having l mm and φ.8 mm (/.8, aspect ratio67). The slabs were made o a normal strength concrete (C5/3) tpical or pavement use. Table shows the average values o compressive strength c,cub (measured rom cubes having a side o 5 mm), tensile strength ct and Young s modulus o concrete E c (both measured on clinders with φ8 mm and h4 mm). These values were determined at the se da the slabs were tested. The volume raction o ibers introduced in the eperimental slabs is also shown in Table (slab P, the reerence one, is made o plain concrete). In order to determine the racture properties o the materials adopted in the slab tests, our point bending tests have been perormed on bes (5 5 6 mm), according to the Italian Standard (UNI 3). The bending tests were carried out b using a close loop hdraulic machine (Instron) with the crack opening displacement as eedback control. In order to reproduce a Winkler soil, the slabs were placed on small steel springs (Figs. 4a and 5) that were designed to simulate a tpical Winkler constant (k.8 N/mm 3 ) when the were placed at a distance o 375 mm in both directions. The springs were obtained through small steel plates supported along the border (Fig. 5). A thin laer o high-strength mortar was placed between the

5 springs and the slab to compensate or the out-o plane deormations o the concrete slab due to curling (Fig. 5). The stiness o all these steel springs was eperimentall determined with several loading ccles. A tpical result is shown in Figure 6. All the springs showed a similar behavior that was modeled b means o a bilinear curve. The load was applied in the center o the slab b means o a hdraulic jack (Fig. 4b). A kn load cell was placed under the jack and 6 Linear Variable Dierential Transormers (LVDTs) were used to measure the vertical displacements in dierent locations o the slab (Fig. 4b). Table. Mechanical properties o concrete and iber content. c,cub t E c V V V V (MPa) (MPa) (MPa) % % % % SLAB 5/ 3/.6 /.8 TOT P P P P Figure 5. Steel spring or simulating the elastic subgrade Vertical spring displacement (mm) - - Numerical curve Eperimental curve (a) Figure 6. Spring stiness -4 (b) Figure 4. (a) Spring position simulating the Winkler soil and (b) eperimental setup. The eperimental results underline the importance o iber reinorcement in concrete pavement (Fig. 7): Slab P, without ibers, ehibit an ultimate load that is signiicantl lower than that obtained in the iber reinorced slabs. In act, in FRC slabs the load can still increase ater the cracking due to the stress redistribution made possible b the material toughness and b the hperstatic structure. Moreover, the addition o short steel ibers (volume raction equal to.9%) slightl increases the bearing capacit o Slabs P and P3. This can be seen as a snergism since small ibers ma also reduce shrinkage cracking.

6 P P P P Central displacement (mm) Figure 7. Eperimental load versus central displacement curves Eperimental Central displacement (mm) P Proposed NLFEA P Eperimental Proposed NLFEA Central displacement (mm) Figure 8. Multi-laered shell element mesh adopted. 4 NLFE ANALYSES AND COMPARISON WITH EXPERIMENTAL RESULTS In order to better understand racture behavior o SFRC slabs on grade and to investigate the reliabilit o the numerical model here proposed, the previousl presented eperimental tests have been simulated b FE analses. These analses have been carried out b using multipurpose FE code ABAQUS (999) where the material constitutive relationships were deined b user subroutines. Figure 8 shows the mesh o isoparetric multilaered shell elements adopted. Laers describe the geometrical and mechanical eatures o shell cross section and behave as membrane elements subjected to plane stresses. The non-linear constitutive matri o laers can be determined b means o PARC model. In order to compute the stiness matri o the FE model, three Simpson s integration points in each laer and Gaussian reduced integration on the midsurace plane o shell element have been chosen. The non linear response o the springs, nel the no-tension behavior and the bilinear shape o the compression part, was taken into account b the numerical model as the trilinear law (Figure 6) Eperimental P Proposed NLFEA Central displacement (mm) Eperimental P3 Proposed NLFEA Central displacement (mm) Figure 9. Total load versus central displacement or P-P3 slabs: comparison between numerical and eperimental results.

7 (a) [mm] (b) [mm] (c) [mm] During numerical analses the displacement o the slabs center has been monotonicall increased, and the non-linear solution is reached b an iterative secant stiness procedure based on the Newton-Raphson convergence method. Figure 9 shows a comparison between eperimental and numerical results or the our eperimental slabs, in terms o total applied load versus central displacement. The numerical model can well predict the eperimental behavior both or plain (Slab P) and iber reinorced concrete slabs. In particular, the slab stiness in the initial branch o the curves and the ultimate load can be evaluated with good precision. These quantities are the most important pareters or the slab design because can predict the behavior o the structure at service limit state and at ultimate. Another important pareter in slab design is the prediction o the inal crack pattern since it evidences the collapse mechanism and allows other design methods (i.e. based on ield lines, Meda 3) to be used. The numerical crack pattern at the bottom side o Slab P3 at dierent values o the central displacement () is presented in Figure. It can be noticed that a secondar crack appears when the principal stress due to bending is equal to the tensile strength (Fig. b). Figures a, d show the values o the crack opening displacement o primar and secondar cracks. One should observe that, b increasing the central displacements, the size o the doubl cracked zone does not increase signiicantl, but the continuous opening o secondar and primar cracks causes a loss o the slabs stiness. Figure show the eperimental crack pattern obtained or Slab P3. It can be noticed that, as predicted in the numerical model (Fig. d), two main cracks develop along the median lines. (d) [mm] Figure. Crack pattern o P3 slab: primar cracks at (central displacement).5mm (a), at 3mm (c) and at 7.5mm (d); secondar cracks at.5mm (b). Figure. Slab P3: eperimental crack pattern.

8 5 CONCLUDING REMARKS Eperimental tests have shown that steel-iber reinorcement signiicantl improves slab behavior in terms o strength, ductilit and toughness. Fibers are eicient at both serviceabilit limit state, b reducing cracking phenomena due to loads or shrinkage eects, and at the ultimate limit state, b increasing the ultimate load. Following the concept o strain-decomposition, the irst version o PARC secant matri has been improved b considering ied primar and secondar cracks starting when maimum principal stress eceeds the tensile strength o concrete. The etension o PARC model to iber reinorced concrete under primar and secondar cracks stage has allowed a more realistic prediction o the nonlinear behavior o slabs on grade. Both in the eperimental tests and in the numerical simulations, primar and secondar cracks were observed under service loads. For this reason, ied-crack models considering onl primar cracks results in stier response than eperimental observations. In order to better predict the behavior o FRC slabs on grade, adequate non linear numerical methods should be adopted. Contraril to the elastic theor, methods based on non linear racture mechanics take into account the stress redistribution made possible b the material toughness and the hperstatic structure (as the slab). These methods can be used or a more accurate pavement design. REFERENCES Bazant, Z. & Gbarova P. 98. Rough cracks in reinorced concrete. Journal o Structural Division, Proceedings o ASCE, 6(4): Belletti B., Bernardi P., Cerioni R. & Iori I. 3b. Non-linear analsis o reinorced concrete slabs. Proceedings o ISEC- Conerence, 3-6 September 3: : Belletti, B., Bernardi, P., Cerioni, R. & I. Iori. On the role o ibers on the behaviour o FRC bes. Proceedings o XXXI National Smposium AIAS, Parma, Ital, 8- September. Belletti, B., Bernardi, P., Cerioni, R. & Iori, I. 3a. On a Fibre Reinorced Concrete Constitutive Model or NLFE Analsis. STUDIES AND RESEARCHES 4, Milan, Ital. Belletti, B., Cerioni, R. & Iori, I.. A phsical approach or reinorced concrete (PARC) membrane elements. ASCE Journal o Structural Engineering 7(): Darwin, D. & Pecknold, D.A Non-linear biaial stressstrain law or concrete. Proceedings o ASCE Journal o the Engineering Mechanics Division 3 (EM): 9-4. De Borst, R. & Nauta, P Non-orthogonal cracks in a smeared inite element model. Engng. Computation : Gbarova, P Sulla trasmissione del taglio in elementi bidimensionali piani di c.a. essurati. Atti delle Giornate AI- CAP, Bari: ABAQUS 999. Theor manual. Hibbitt, Karlsson & Sorensen, Pawtucket, R.I. Hillerborg, A., Modèer, M. & Petersson, P.E Analsis o crack ormation and crack growth in concrete b means o racture mechanics and inite elements. Cement and Concrete Research 6: Kuper, H., Hilsdor, H.K. & Rusch, H Behavior o Concrete Under Biaial Stresses. Proceedings o ACI Journal 66(8): Li, V. C. 99. Postcrack Scaling Relations or Fiber Reinorced Cementitious Composites. Journal o Materials in Civil Engineering 4: Li, V.C. Stang, H. & Krenchel, H Micromechanics o crack bridging in iber-reinorced concrete. Materials and Structures 6: Meda, A. 3. On the etension o the ield-line method to the design o SFRC slabs on grade. STUDIES AND RE- SEARCHES 4, Milan, Ital. Meda, A. & Plizzari, G.A. 3. A new design approach or SFRC slabs on grade based on racture mechanics. ACI Structural Journal. In press Riggs, H.R. & Powell, G.H Rough crack model or analsis o concrete. J. Engng. Mech. ASCE, (5): Rots, J.G. & Blaauwendraad, J Crack models or concrete: discrete or smeared? Fied, multi-directional or rotating?. HERON 34: -59, The Netherlands. UNI Steel ibre reinorced concrete - Part I: Deinitions, classiication speciication and conormit - Part II: test method or measuring irst crack strength and ductilit indees. Italian Board or Standardization. Vecchio, F.J. & Collins M.P Compression Response o Cracked Reinorced Concrete. Journal o Structural Engineering, American Societ o Civil Engineers 9(): ACKNOWELEDGEMENTS The eperimental work was inanced b Oicine Maccaerri S.p.A. (Bologna, Ital). The understanding and cooperation o Ing. Bruno Rossi (Project Supervisor) is kindl acknowledged. The Authors would like to thank Dr. Luca Sorelli and Ing. Paolo Martinelli or their assistance in carring out the eperiments and Dr. Patrizia Bernardi and Ing. Martino Cerri or their assistance in running the numerical analses.