Crack constitutive model for the prediction of punching failure modes of fiber reinforced concrete laminar structures

Transcription

1 Crack constitutive model or the prediction o punching ailure modes o iber reinorced conete laminar structures A. Ventura-ouveia *, Joaquim A. O. Barros a, Álvaro F. M. Azevedo 3b Dept. o Civil Eng., ISISE, School o Technology & Management, Polytechnic Institute o Viseu, Viseu, Portugal Dept. o Civil Eng., ISISE, School o Engineering, University o Minho, uimarães, Portugal 3 Dept. o Civil Eng., Faculty o Engineering o the University o Porto, University o Porto, Porto, Portugal Abstract. The capability o a multi-directional ixed smeared ack constitutive model to simulate the lexural/punching ailure modes o iber reinorced conete (FRC) laminar structures is discussed. The constitutive model is implemented in a computer program based on the inite element method, where the FRC laminar structures were simulated according to the Reissner-Mindlin shell theory. The shell is disetized into layers or the simulation o the membrane, bending and out-o-plane shear nonlinear behavior. A stress-strain sotening diagram is proposed to reproduce, ater ack initiation, the evolution o the normal ack component. The in-plane shear ack component is obtained using the concept o shear retention actor, deined by a ack-strain dependent law. To capture the punching ailure mode, a sotening diagram is proposed to simulate the deease o the out-o-plane shear stress components with the inease o the corresponding shear strain components, ater ack initiation. With this relatively simple approach, accurate predictions o the behavior o FRC structures ailing in bending and in shear can be obtained. To assess the predictive perormance o the model, a punching experimental test o a module o a açade panel abricated with steel iber reinorced sel-compacting conete is numerically simulated. The inluence o some parameters deining the sotening diagrams is discussed. Keywords: Smeared Crack Model; Punching; Shear Sotening Diagram; Material Nonlinear Analysis; Steel Fiber Reinorced Sel-Compacting Conete; Inverse Analysis. * Corresponding Author, MSc, PhD Student, ventura@estv.ipv.pt a Associate Proessor, Aggregation, barros@civil.uminho.pt b Assistant Proessor, PhD, alvaro@e.up.pt

2 . Introduction The Reissner-Mindlin theory or shell structures is commonly used to predict the behavior o laminar conete structures up to ailure (Barros and Figueiras 00). The thickness o the laminar structure is disetized in layers that are assumed subjected to a plane stress state. The use o laws to simulate the nonlinear behavior, ater ack initiation, or the in-plane racture modes is appropriate in most cases, and the deormational response o the structure or load conigurations inducing lexural ailure modes can be predicted with suicient accuracy. However, the simulation o laminar structures ailing in punching is a much more complex task, being the treatment o the out-o-plane shear components o paramount importance. In the present work, in order to explore the use o a simple approach to simulate the material nonlinear behavior o conete laminar structures ailing in punching, a sotening law is proposed to model both out-o-plane shear components. This ack constitutive model has been implemented in the FEMIX computer program, which is based on the inite element method (Sena-Cruz et al. 007). Since the shell model only admits acks that are orthogonal to its middle surace, the inclined acks that are observed in the experimental punching tests cannot be accurately predicted. For this purpose a much more complex and time-consuming general 3D ack constitutive model must be used (Barzegar and Maddipudi 997). Steel iber reinorced sel-compacting conete (SFRSCC) is a relatively recent cement-based material that combines the beneits o the sel-compacting conete technology (Okamura 997) with the advantages o the addition o ibers to a brittle cementitious matrix (Pereira 006). The developed SFRSCC was used to manuacture the lightweight panel system schematically represented in Fig., which can be applied in building açades (Barros et al. 005a). The mix composition o the SFRSCC used to manuacture the panel is presented in Table. In the composition o the SFRSCC, 30 kg/m 3 o hooked ends steel ibers with a length (l ) o 60 mm, a diameter (d ) o 0.75 mm, an aspect ratio (l /d ) o 80 and a yield stress o 00 MPa were used. At seven days the average value o the compressive strength and modulus o elasticity o this SFRSCC was 5 MPa and 3 Pa, respectively. The lexural strength o this type o structural elements is a key aspect in their design, since, in general, the bending moments o the

3 wind load combination are an important actor in the design process o the panel. To assess the panel lexural behavior, representative modules o the SFRSCC panel system were tested, being the details o the experimental program desibed elsewhere (Barros et al. 007). The punching resistance is also a key aspect in the design o this type o panel, since its lightweight zones consist o a thin layer that is only 30 mm thick. To evaluate the punching resistance o these zones, representative modules o the panel system are submitted to a load coniguration that implies the occurrence o this type o ailure mode (Barros et al. 005a, Barros et al. 007). The results obtained in one o these tests were compared with the numerical simulations in order to assess the predictive perormance o the developed model. Several numerical simulations are carried out to assess the inluence o some parameters that deine the sotening diagrams. The objective o these simulations is to understand how each parameter aects the response o a laminar FRC structure ailing in punching. The inluence o the in-plane mesh and through-thickness reinement o the simulated structure is also analyzed. The possibility o deining the racture parameters that characterize the racture mode I strain-sotening diagram by perorming an inverse analysis (IA) (Barros et al. 005b) is also discussed. The IA is based on the results obtained in three point notched beam bending tests carried out according to the RILEM TC 6-TDF recommendations (Vandewalle et al. 00).. Crack constitutive model. - Introduction Presently, several inite element approaches are available to analyze the behavior o complex structures subject to arbitrary loads. The most recent ones are capable o modeling the behavior o conete structures presenting brittle ailure modes, and accurately predict ack ormation and progression. Disete cohesive racture models (disete) with ragmentation algorithms, strong discontinuity approaches (continuum) with the embedded discontinuities method and the extended inite element method are examples o advanced methodologies that, together with powerul mesh reinement algorithms, reveal great eiciency in modeling the conete racture initiation and propagation (Yu et al. 007). Alternative methods are based on damage models 3

4 (de Borst and utiérrez 999), smeared ack models (Bazant and Oh 983)] and mioplane models (Bazant 984). These methods are less precise to predict the local phenomena related to ack propagation, but rom the computational eectiveness and the assessment o the global behavior o a conete structure point-o-views, are more appropriate to analyze complex structures with a large number o degrees o reedom. As shown by de Borst (00), ixed and rotating smeared ack models, but also mioplane models, can be conceived as a special case o (anisotropic) damage models, these three FEM-based solutions are closely related and produce similar results. Taking into account the main characteristics o all these approaches, the multi-directional ixed smeared ack model was selected and implemented in the scope o the present research, since it allows or the analysis o large scale SFRSCC structures (de Borst 987, Rots 988, Dahlblom and Ottosen 990), as long as an appropriate constitutive law is used to model the SFRSCC post-acking behavior.. - Formulation In the context o inite element material nonlinear behavior o conete shell structures, the developed ack constitutive model is implemented under the ramework o the Reissner-Mindlin theory adapted to the case o layered shells. The desiption o the ormulation is restricted to the case o acked conete, or a selected conete layer, and at the domain o an integration point (IP) o a inite element. According to the adopted constitutive law, stresses and strains are related by the ollowing equation co σ m Dm 0 ε m co σ = s 0 D ε s s () T where σ = { σ, σ, τ } and ε { ε, ε, γ } m m T = are the vectors o the inemental stress and inemental strain in-plane components, while σ { τ, τ } {, } 3 3 T s = and 3 3 ε = γ γ are the vectors o the total stress and total out-o-plane shear strain s components. T 4

5 Due to the decomposition o the total strain into an elastic conete part and a ack part co co ( ε = ε + ε ), in Eq. () the in-plane acked conete constitutive matrix, D co m, is obtained with the ollowing equation (Sena-Cruz 004) T ( ) T co co co co m, e m, e m, e m, e co Dm = D D T D + T D T T D () co where D m, e is the constitutive matrix o conete with a linear and elastic behavior D E ν co m, e = ν 0 ν ( ν ) (3) where E and ν are the elasticity modulus and the Poisson s ratio o conete, respectively. In Eq. (), T is the matrix that transorms the stress components rom the coordinate system o the inite element to the local ack coordinate system T cos θ sin θ sinθ cosθ = sinθ cosθ sinθ cosθ cos θ sin θ (4) and D is the ack constitutive matrix D DI 0 = 0 DII (5) In Eq. (4), θ is the angle between x and n (see Fig. ). In Eq. (5), D I and D II represent, respectively, the constitutive components relative to the ack opening mode I (normal) and mode II (in-plane shear). The ack opening propagation is simulated with the trilinear diagram represented in Fig. 3, which is deined by the parameters α i and ξ i, relating stress with strain at the transitions between the linear segments that compose this diagram. The ultimate ack strain, ε n, u, is deined as a unction o the parameters α i and ξ i, the racture energy, I, the tensile strength, ct = σ n,, and the ack bandwidth, b l, as ollows (Sena-Cruz 004), ε n,u = ξ + α ξ α ξ + α I l ct b (6) 5

6 where α = σ n, / σ n,, α = σ n,3 / σ n,, ξ = ε n, / ε n, u and ξ = ε n,3 / ε n, u. The racture mode II modulus, D II, is obtained with D II β = c (7) β where c is the conete elastic shear modulus and β is the shear retention actor. The parameter β is deined as a constant value or as a unction o the current ack normal strain, ε, and o the ultimate ack normal strain, ε n, u, as ollows, n ε β = ε n n, u p (8) When p is unitary, a linear deease o β with the inease o ε n is assumed. Larger values o the exponent p correspond to a more pronounced deease o the parameter β, in order to simulate a higher in-plane shear stress degradation with the ack opening process (Barros et al. 004). A sotening constitutive law to model the in-plane ack shear stress transer has also been implemented in the FEMIX code, but its adoption as an alternative to the shear retention concept does not contribute to an inease o the accuracy o the numerical simulations, and causes diiculties in the convergence o the Newton-Raphson procedure. The deinition o the out-o-plane () constitutive matrix, co D s in Eq. (), is based on the diagram represented in Fig. 4. When the conete associated with the IP changes rom unacked to acked state, the out-o-plane shear stresses are stored or later use and each out-o-plane shear stress-strain relation ( τ 3 γ 3 and τ 3 γ 3 ) ollows the sotening law depicted in Fig. 4. Thereore, the co D s matrix is deined by D co s 3 D,sec 0 = 3 0 D,sec (9) where D = τ = τ γ (0) 3 3,max 3,sec, D,sec γ 3,max 3,max 3,max 6

7 in accordance with the secant approach shown in Fig. 4. Each peak shear strain is calculated using the stored peak shear stress at ack initiation and c γ τ τ = = () 3, p 3, p 3, p, γ 3, p c c Each out-o-plane ultimate shear strain, γ u, is deined as a unction o the out-o-plane peak shear strain, energy, γ p, the out-o-plane shear strength,, and the ack bandwidth, l b, as ollows τ p, the mode (out-o-plane) racture γ = γ + γ = γ + () 3, u 3, p, 3, u 3, p τ 3, p lb τ 3, p lb The present approach assumes that the ack bandwidth used to assure mesh independence when modeling racture mode I can also be adopted in the out-o-plane racture process. 3. Evaluation o the mode I racture properties by inverse analysis This section desibes the inverse analysis (IA) methodology adopted to evaluate the racture mode I parameters o the SFRSCC used in the panel prototype that was experimentally tested and numerically simulated. Detailed inormation about this IA can be ound elsewhere (Barros et al. 005b, Sena-Cruz et al. 004). As already mentioned, in the implemented smeared ack constitutive model the post-acking behavior o SFRSCC under tension is desibed by a trilinear stress-strain sotening diagram (Fig. 3). This unction is deined by a set o racture parameters ( α i, ξ i, I, ct and l b ), being the accuracy o the FEM modeling largely dependent on the values that are assigned to these parameters. In this context, the experimental behavior o an element ailed in bending may be predicted by a FEM model, as long as the correct values o the material racture parameters are introduced in the constitutive model. The adopted strategy consists in the evaluation o the ξ i, αi and I parameters that deine the shape o the trilinear σ n ε constitutive law, based on n the minimization o the error parameter 7

8 err = A A A (3) exp num exp F δ F δ F δ where exp AF δ and num AF δ are the areas beneath the experimental and numerical load-delection curves corresponding to a three point notched beam bending test (Sena-Cruz et al. 004). The experimental curve corresponds to the average results observed in prismatic SFRSCC notched specimens, tested according to the RILEM TC 6-TDF recommendations at the age o 7 days (Vandewalle et al. 00), while the numerical curve consists o the results obtained by FEM analysis, where the specimen, loading and support conditions are simulated in agreement with the experimental lexural test setup (Fig. 5a). In this context, the specimen is modeled with a mesh o 8 node serendipity plane stress inite elements. The auss-legendre integration scheme with integration points is used in all elements, with the exception o the elements at the specimen symmetry axis, where integration points are used. With this particular integration point layout, the numerical results have a better agreement with the experimental observations, since a vertical ack may develop along the symmetry axis. Linear elastic material behavior is assumed in all the elements, with the exception o those above the notch, along the symmetry axis. In this region an elastic-acked material model in tension is adopted. The ack bandwidth, l b, is assumed to be equal to 5 mm, being this value coincident with the width o the notch and o the elements located above it. In Fig. 5b, the results experimentally obtained in the lexural tests are compared with the numerical results. The curve o the numerical simulation, obtained with the optimized racture parameters, is not perectly coincident with the experimental curve, suggesting that more parameters should be considered in order to obtain a better itting. The values o the racture parameters ξ i, α i and Table. I that lead to the numerical results represented in Fig. 5b are listed in 8

9 4. Numerical simulation 4. - Introduction The punching test o a module o the developed SFRSCC lightweight panel is used to assess the predictive perormance o the proposed multi-directional ixed smeared ack model. The test layout and the test setup are represented in Fig. 6. More details about the corresponding experimental program can be ound elsewhere (Barros et al. 007). The inluence o mesh reinement and some model parameters in the results o the numerical simulations is assessed and discussed in this section, namely: the values adopted or the racture mode I parameters used to deine the trilinear diagram and the values used to deine the out-oplane shear stress-strain diagram Analysis based on the values obtained rom the IA Inluence o the out-o-plane shear sotening diagram The results o the numerical simulations are compared with the experimental data obtained in the punching test o the panel module. The inite element idealization, load and support conditions used in the numerical simulations o the punching test are shown in Fig. 7a. Only one quarter o the panel is used in the simulations, due to double symmetry. The mesh is composed o 6 6 eight-node serendipity plane shell elements. The elements are divided into layers, each one being 0 mm thick. Since the panel has lightweight zones (shaded elements in Fig. 7a), materialized by the suppression o 80mm o conete in the central zone, null stiness is assigned to the 8 bottom layers o the corresponding inite elements (see Fig 7b). The material o the remaining three layers has an elastic-acked behavior, as desibed in Section.. This model is also used in the elements located outside the central lightweight zone. A trial-and-error procedure is required to estimate reasonable values or the out-o-plane components o the elastic-acked constitutive matrix, D co s, since their experimental evaluation is quite complex and beyond the scope o the present work. The out-o-plane shear racture energy that leads to the best agreement with the experimental results o the punching tests, 9

10 = 3.0 N mm, is determined with this procedure. The values o the mode I racture parameters that take part in the in-plane elastic-acked constitutive matrix or conete, co D m, are obtained by IA, as desibed in Section 3. In Fig. 8 the responses obtained with the numerical model are compared with the experimental results. A good agreement can be observed up to a delection o.5 mm. For larger delections, an overestimation o the load carrying capacity o the prototype panel occurs when a linear elastic behavior is assumed or the out-o-plane shear components. At a delection o about 3 mm the experimental curve suddenly alls, indicating the ailure o the panel by punching, as visually conirmed in the experimental test. This load decay that is not reproduced when assuming a linear elastic behavior or the out-o-plane shear components is, however, well captured when the bilinear diagram represented in Fig. 4 is used to model the sotening behavior o the out-o-plane shear components, with = 3.0 N mm, and assuming a ack bandwidth, l b, equal to the square root o the area associated with the corresponding IP. The abrupt load decay rom approximately 4 kn to 0 kn, which is observed in the experimental test, is accurately simulated by the numerical model, as well as the subsequent extended stage o residual load carrying capacity exhibiting a very small load decay. Up to a 0 kn load all the curves depicted in Fig. 8 are practically coincident. Aterwards, the straight line that represents the response assuming a linear-elastic behavior no longer ollows the curves that correspond to the experimental test and to the numerical analysis with material nonlinear model. These results suggest that some acks start to orm at a very early stage o the experimental test. The nonlinear numerical model accurately captures the ormation o bending acks at the top surace (see Fig. 9a), in agreement with the experimentally observed ack pattern. Fig. 9b shows the ack pattern at the top surace observed at the end o the test sequence. The numerical model also indicates the ormation o bending acks at the bottom surace, in the lightweight zone. These acks initiate at the center o the panel, beneath the loaded area, and then progress to the corners o the lightweight zone, showing some similarities 0

11 with the classical yield lines ormed in square conete slabs ailing by lexure. These acks can also be observed in the experimental test (Pereira 006). In conclusion, the results indicate that lexure mechanisms prevail in the deormational behavior up to a delection o approximately.5 mm. For larger delections, the punching ailure mechanisms start to assume a greater relevance, and the overestimation o the panel out-o-plane rigidity components, when linear out-o-plane shear behavior is assumed, leads to a divergence between the numerical model and the experimental observations. With the adoption o a sotening law or the out-o-plane shear components, the numerical model becomes much more accurate in the prediction o the complete behavior o the panel ailing in punching, capturing the sudden load decay associated with punching ailure mechanisms. As already mentioned, the selection o a value or has no experimental support. In order to analyze its inluence on the results o the numerical simulation using a sotening law or both out-o-plane components, a parametric analysis is carried out consisting in the variation o its value rom.0 to 5.0 N/mm. The results depicted in Fig. 0 show that a value o = 3.0 N mm leads to a perect prediction o the abrupt load decay experimentally observed at a delection o about 3 mm. Ineasing or deeasing the value o implies the occurrence o the abrupt load decay at a larger or smaller delection, respectively. The conclusion o this study is that, independently o the value o, when using the model desibed in this work, it is essential to use a sotening law or the out-o-plane shear components in order to simulate the sudden load decay observed in the punching test Inluence o the through-thickness reinement o the panel In this section, the inluence o through-thickness reinement o the panel on the load-delection relationship is analyzed. The parameters used to simulate the racture mode I and the out-oplane shear sotening diagram are those that have best itted the experimental results, according to the strategy desibed in the previous section.

12 For this purpose, the ollowing two reinements are considered: 6 layers in the lightweight zone and layers in the remaining parts; 0 layers in the lightweight zone and 6 layers in the other zones. In Fig. the load-delection relationships o these numerical simulations are compared with the experimental one. In the legend o this igure, CM_jL represents the relationship obtained rom the simulation whose lightweight zone was disetized in j layers. It can be observed that by ineasing the number o layers rom 3 to 6 in the lightweight zone, the maximum load ineases in about 7% and the stiness corresponding to the branch between ack initiation and peak load also ineases. This behavior can be justiied by the act that the lexural stiness o the layers is not taken into account in the layered approach adopted to simulate the stiness o Mindlin shell inite elements, when a material nonlinear analysis is perormed. Thereore, the larger the number o layers disetizing the element, the higher the lexural stiness o the element is, resulting in a smaller deormability o the panel and a higher load carrying capacity. However, Fig. also shows that when the number o layers ineases rom 6 to 0, only a marginal inease o the maximum load is visible, which indicates that the inease ratio o the lexural stiness and load carrying capacity o the layered Mindlin-shell element deeases with the number o layers. It is also interesting to observe that the delection at the abrupt load decay, as well as the residual load carrying capacity o the panel are very similar in all three numerical analysis Inluence o the in-plane mesh reinement o the panel In order to assess the inluence o the in-plane mesh reinement on the load-delection relationship, an analysis with the reined mesh (RM) represented in Fig. is carried out. Eight-node serendipity plane shell elements are used, with 0 layers in the lightweight zone and 6 layers in the other zones. The load-delection relationship or the RM is represented in Fig. 3, which is compared with the one obtained with the previous coarse mesh (CM), and with the one experimentally registered. As expected, the deormability o the panel ineases with the mesh reinement,

13 causing the abrupt load decay to occur or a higher delection (3.3 mm). Due to the higher lexibility o the panel when perorming the analysis with the RM, a deease o about 5% in terms o load carrying capacity occurs. Thereore, the shape o the load-delection (F-u) curve or the RM is approximately the result o the rotation o the F-u curve or the CM in turn o the point that corresponds to the ack initiation. With the inease o the number o inite elements (and number o integration points), the conete in acked status and the corresponding consumed mode I racture energy also inease. This can be a possible justiication or this more deormable response o the in-plane RM numerical simulation Inluence o the racture energy ( ) used in the out-o-plane shear sotening diagram To assess the inluence o the racture energy used to deine the out-o-plane shear sotening diagram,, on the load-delection relationship, its value is varied between.0 N/mm and 5.0 N/mm. In these analyses the in-plane CM and the RM are used, with 0 layers disetizing the thickness o the panel in the lightweight zone. The obtained numerical curves are represented in Fig. 4a and 4b, respectively. It is observed that in the RM the value mainly aects the residual load carrying capacity ater the abrupt load decay. When using the CM, the value attributed to not only aects the residual load carrying capacity but also inluences the value o the delection corresponding to the abrupt load decay. This inluence, however, is less pronounced than when using an in-plane CM with 3 layers disetizing the thickness o the panel in the lightweight zone (see Fig. 0). Thereore it can be concluded that when a RM is used, suitable predictions can be obtained with =, but urther research I needs to be carried out or a more reliable estimation o. Figures 5a and 5b show the consumed out-o-plane racture energy (, c ) up to a delection o 3.5 mm or the in-plane CM and RM, respectively. In each integration point, this consumed racture energy receives the contribution o the two out-o-plane shear components in all layers 3

14 disetizing the thickness o the panel, and can be regarded as an indicator o damaged due to punching ailure mode. It can be observed that the punching ailure pattern is well predicted when using the RM. When using the in-plane CM reinement the shear ailure bandwidth is larger, which justiies the higher sensitivity o the delection corresponding to the abrupt load decay to the adopted value (Figure 4a). 4.3 Inluence o the parameters that deine the racture mode I In order to assess the inluence o the parameters that deine the racture mode I constitutive law (Fig. 3) on the load-delection relationship predicted by the numerical model, the values o these parameters are deeased and ineased by 50% relatively to those obtained by IA. The ack stress vs. ack strain ( σ - ε ) or these analyses and the corresponding load-delection n n relationships are depicted in Figs. 6 to 0. All these numerical analyses were perormed with the reined mesh and using 0 layers or the disetization o the thickness o the lightweight part o the panel. From the analysis o these graphs it can be concluded that the inclination o the irst branch o the σ -ε diagram ( D in Fig. 3) governs the point corresponding to the n n n irst drop in the load-delection relationship. In act, the less abrupt is this branch the higher is the load o this point. In consequence, the load carrying capacity o the panel is quite sensible to the slope o this branch. Direct tensile tests with SFRSCC similar to the one used in the tested panels showed, in act, an abrupt stress decay immediately ater ack ormation. Fig. 7b evidences that the numerically predicted load carrying capacity o the panel is quite dependent on the α parameter, since a pronounced sotening and a signiicant hardening delection are estimated when a value o α smaller or larger than the one obtained by IA is used (Fig. 7a). The higher strength n ( n ) σ ε o the second branch o σ -ε, when adopting higher values or the α parameter (Fig. 7a), also contributes to inease both the load carrying capacity o the panel and the delection corresponding to the punching ailure. However, Fig. 9 reveals that the strength n ( n ) σ ε corresponding to the irst branch o σ -ε diagram has a much higher n n n n 4

15 inluence on the load carrying capacity o the panel than the strength n ( n ) σ ε o the second branch. Nevertheless, Fig. 9 and Fig. 0 also demonstrate that the slope o the load-delection branch beore the punching ailure grows with the value o D n (Fig. 3). Finally, the deease o the racture energy is mainly relected on the point corresponding to the irst drop o the loaddelection relationship (Fig. 0b). This deease lead to a more abrupt decay o the irst branch o the σ -ε diagram (Fig. 0a), resulting a deease o the load at this point n n 5. Conclusions In the present work a model based on the inite element method is proposed to simulate iber reinorced conete (FRC) structures ailing in bending and shear. The Reissner-Mindlin theory, in the context o layered shells, is selected and special emphasis is dedicated to the treatment o the shear behavior. The model is based on a multi-directional ixed smeared ack concept. By considering the nonlinear behavior o each shell layer, ack propagation through the thickness o these structures can be simulated. Fracture mode I is modeled with a ack stress vs. ack strain trilinear diagram, whose deining parameters can be obtained by inverse analysis (IA) using the load-delection relationship obtained with three-point notched beam tests, carried out according to the RILEM TC 6-TDF recommendations. With this strategy the values o the racture parameters that deine the normal stress-strain ack constitutive relationship are obtained. Since this type o test is much simpler and aster to execute, it becomes an advantageous alternative to the direct tensile tests recommended to evaluate the racture mode I parameters o cement based materials. The adopted IA strategy is presented and discussed in the numerical simulation section. To simulate the out-o-plane strain gradient that occurs in punching tests, a sotening diagram is proposed to model, ater ack initiation, the out-o-plane shear components. The adequacy and accuracy o the model is appraised using the results obtained in the punching test o a panel prototype built with steel iber reinorced sel-compacting conete (SFRSCC). This numerical strategy allows or an accurate simulation o the load-deormational process o the experimentally tested panel, which exhibited a brittle punching ailure. 5

16 Several numerical simulations are presented and discussed. Mesh reinement, data obtained with inverse analysis to deine the trilinear diagram and a sotening out-o-plane shear diagram are alternatives whose inluence on the prediction o the experimental panel response is investigated. The use o sotening laws to simulate the mode I ack opening and the out-o-plane shear components is ucial in order to obtain accurate numerical simulations The numerical simulations carried out with the proposed model and its comparison with the results o the experimental test used in this work lead to the conclusion that the behavior o laminar SFRSCC structures ailing in punching can be numerically predicted by a FEM-based Reissner-Mindlin shell approach as long as a ack constitutive model that includes a sotening diagram or modeling both out-o-plane shear constitutive laws is used. 6

17 ACKNOWLEDEMENTS The authors wish to acknowledge the support provided by the Portuguese Science and Technology Foundation (FCT) by means o the project PTDC/ECM/73099/006 CUTINEMO - Carbon iber laminates applied according to the near surace mounted technique to inease the lexural resistance to negative moments o continuous reinorced conete structures. The irst author acknowledges the inancial support o FCT, PhD rant number SFRH/BD/336/005. Reerences Barros, J.A.O. and Figueiras, J.A. (00), Nonlinear analysis o steel ibre reinorced conete slabs on grade, Computers & Structures Journal, 79(), 97-06, January. Barros, J.A.O., ettu, R. and Barragan, B.E. (004), Material Nonlinear analysis o Steel ibre reinorced conete beams ailing in shear, 6th International RILEM Symposium on ibre reinorced conete - BEFIB 004, Edts. M. di Prisco, R. Felicetti,.A. Plizarri, Vol., p. 7-70, 0- September. Barros, J.A.O., Pereira, E.N.B., Cunha, V.M.C.F., Ribeiro, A.F., Santos, S.P.F. and Queirós, P.A.A.A.V. (005a), PABERFIA - Lightweight sandwhich panels in steel iber reinorced sel compacting conete, Technical Report 05-DEC/E-9, Dep. Civil Eng., School Eng. University o Minho, 63 pp, November. Barros J.A.O., Cunha, V.M.C.F., Ribeiro, A.F. and Antunes, J.A.B. (005b), Post-acking behaviour o steel ibre reinorced conete, RILEM Materials and Structures Journal, 38(75), Barros, J.A.O., Pereira, E.B. and Santos, S.P.F. (007), Lightweight panels o steel iber reinorced sel-compacting conete, Journal o Materials in Civil Engineering, 9(4), Barzegar, F. and Maddipudi, S. (997), Three-dimensional modeling o conete structures. I: plain conete, Journal o Structural Engineering, 3(0),

18 Bazant, Z.P. and Oh, B.H. (983), Crack band theory or racture o conete, Materials and Structures, RILEM, 6(93), Bazant, Z.P. (984), Mioplane model or strain controlled inelastic behavior, In: Desai, C.S., allagher, R.H. (Eds.), Mechanics o Engineering Materials. J. Wiley, London, pp (Chapter 3). Dahlblom O. and Ottosen N.S. (990), Smeared ack analysis using generalized ictitious ack model, Journal o Engineering Mechanics, ASCE, 6(), de Borst, R. (987), Smeared acking, plasticity, eep and thermal loading - a uniied approach, Computational Methods in Applied Mechanics Engineering, 6, de Borst, R. and utiérrez, M.A. (999), A uniied ramework or conete damage and racture models including size eects, International Journal o Fracture, 95, de Borst, R. (00), Fracture in quasi-brittle materials: a review o continuum damage-based approaches, Engineering Fracture Mechanics, 69, 95-. Okamura, H. (997), Ferguson Lecture or 996: Sel-compacting high-perormance conete, Conete International, ACI 9(7), Pereira, E.N.B. (006), Steel ibre reinorced sel-compacting conete: rom material to mechanical behavior, Dissertation or Pedagogical and Scientiic Aptitude Proos, Department o Civil Engineering, University o Minho, 88 pp, Rots, J.. (988), Computational modeling o conete racture, PhD Thesis, Delt University o Technology. Sena-Cruz, J.M. (004), Strengthening o conete structures with near-surace mounted CFRP laminate strips, PhD Thesis, Department o Civil Engineering, University o Minho, Sena-Cruz, J.M., Barros, J.A.O., Ribeiro, A.F., Azevedo, A.F.M. and Camões, A.F.F.L. (004), Stress-ack opening relationship o enhanced perormance conete, 9th Portuguese Conerence on Fracture, ESTSetubal, Portugal,

19 Sena-Cruz, J.M., Barros, J.A.O., Azevedo, A.F.M. and Ventura-ouveia, A. (007), Numerical simulation o the nonlinear behavior o RC beams strengthened with NSM CFRP strips, Proceedings o CMNE/CILAMCE Congress, FEUP, Porto, Portugal, 3-5 June. Vandewalle, L. et al. (Technical Committee) (00), Test and design methods or steel ibre reinorced conete - Final Recommendation, Materials and Structures, 35(53), Yu, R.C., onzalo, R. and Chaves, E.W.V. (007), A comparative study between disete and continuum models to simulate conete racture, Engineering Fracture Mechanics, doi:0.06/j.engracmech

20 exp AF δ num AF δ d NOTATION = areas beneath the experimental load-delection curves = areas beneath the numerical load-delection curves = steel iber diameter D = ack constitutive matrix D = opening racture mode stiness modulus or the i th trilinear ni stress-strain sotening branch D = ack constitutive matrix component relative to the ack I normal opening mode (mode I) D = ack constitutive matrix component relative to the ack inplane sliding mode (mode II) II ij D = secant constitutive matrix ij component relative to the ack,sec out-o-plane sliding mode (mode ) D = secant stiness relative to the ack out-o-plane sliding mode,sec (mode ) co D = constitutive matrix o in-plane membrane and bending m, e components or conete in elastic regime co D = constitutive matrix o in-plane membrane and bending m components or acked conete co D = constitutive matrix o out-o-plane shear components or s acked conete E c, E = conete elasticity modulus FRC = iber reinorced conete c = conete elastic shear modulus = mode I (in-plane) racture energy I = mode (out-o-plane) racture energy = consumed mode (out-o-plane) racture energy, c IA = inverse analysis IP = integration point SFRSCC = steel iber reinorced sel compacting conete T = transormation matrix rom the coordinate system o the inite element to the local ack coordinate system c = compressive strength ct = tensile strength l b = ack bandwidth l = steel iber length n, t = ack local coordinate system = out-o-plane p = shear degradation actor p = parameter deining the mode I racture energy available to the new ack s = sliding displacement w = opening displacement x = element local coordinate system i α = racture parameters used to deine the trilinear stress-strain i sotening diagram α = threshold angle th β = shear retention actor ε m = vector containing the in-plane membrane and bending strain inemental components co ε = elasto-acked strain 0

21 ε = ack strain co ε = elastic conete strain ε = ack normal strain n ε = ack normal strain used to deine point i in the trilinear n, i stress-strain sotening diagram ε = ultimate ack normal strain n, u ε = vector containing the out-o-plane shear strain components s γ = out-o-plane shear strain ij component ij γ = out-o-plane peak shear strain p = out-o-plane shear strain γ γ = out-o-plane peak shear strain or the ij component ij, p γ = out-o-plane ultimate shear strain u ij, u γ = out-o-plane ultimate shear strain or the ij component γ = out-o-plane maximum shear strain in the sotening branch max γ = out-o-plane maximum shear strain ij component in the ij,max sotening branch ν = Poisson s ratio θ = angle between the element local coordinate system and the ack local coordinate system ξ = racture parameters used to deine the trilinear stress-strain i sotening diagram σ m = vector containing the in-plane membrane and bending stress inemental components σ = ack normal stress n σ = ack normal stress used to deine point i in the trilinear n, i stress-train sotening diagram σ = vector containing the out-o-plane shear stress components s τ = in-plane ack shear stress t τ = out-o-plane shear strength p τ ij, p = out-o-plane shear stress τ = out-o-plane shear strength or the ij component τ ij max ij,max = out-o-plane shear stress or the ij component τ = out-o-plane maximum shear stress in the sotening branch τ = out-o-plane maximum shear stress ij component in the sotening branch

22 TABLE CAPTIONS Table - Composition or m 3 o SFRSCC including 30 kg/m 3 o ibers. Table - Values o the parameters o the constitutive model used in the numerical simulations o the punching test.

23 Table - Composition or m 3 o SFRSCC including 30 kg/m 3 o ibers Paste total volume Cement CEM I 4.5R Limestone iller Water Superplasticizer * Fine sand Coarse sand Crushed aggregates (%) (kg) (kg) (dm 3 ) (dm 3 ) (kg) (kg) (kg) * Third generation based on polycarboxilates (lenium 77SCC) 3

24 Table - Values o the parameters o the constitutive model used in the numerical simulations o the punching test. Poisson s ratio ν = 0.5 Initial Young s modulus E = N mm c Compressive strength Trilinear tension sotening diagram o SFRSCC (used in the numerical simulations o section 4.. Parameters values obtained rom inverse analysis) Trilinear tension sotening diagram o SFRSCC (used in the numerical simulations o section 4.3. Parameter values obtained by modiying in ± 50% the ones obtained rom inverse analysis) Fracture energy (mode ) used in the out-o-plane shear stress-strain diagram Parameter deining the mode I racture p = energy available to the new ack = 5.0 N mm c I ct = 3.5 N mm ; 4.3 N mm = ; ξ = ; α = 0.5 ; ξ = 0.5 ; α = 0.59 ct = 3.5 N mm ; = 50% 4.3 N mm ; I ξ = ± 50% ; α = ± 50% 0.5 ; ξ = ± 50% 0.5 ; α = ± 50% 0.59 ( ± - depends on the numerical simulation) rom =.0 N mm to = 5.0 N mm (depends on the numerical simulation) Shear retention actor Exponential ( p = ) Crack bandwidth Threshold angle α th = 30º Square root o the area o the integration point α = σ n, / σ n,, α = σ n,3 / σ n,, ξ = ε n, / ε n, u, ξ = ε n,3 / ε n, u (see Fig. 3) 4

25 FIURE CAPTIONS Fig. - Concept o a lightweight steel iber reinorced sel-compacting conete panel (dimensions in mm). Fig. - Stress components, relative displacements and local coordinate system o the ack. Fig. 3 - Trilinear stress-strain diagram to simulate the racture mode I ack propagation o SFRSCC ( σ = α σ, σ n, n, = α σ, ε n, = ξ ε n, u, ε n,3 = ξ ε n, u ). n,3 n, Fig. 4 - Diagram to simulate the relationship between the out-o-plane () shear stress and shear strain components. Fig. 5 - Three-point notched beam lexural test at 7 days: (a) FEM mesh used in the numerical simulation, and (b) obtained results. Fig. 6 - (a) Test panel module, and (b) test setup (dimensions in mm). Fig. 7 - (a) eometry, mesh, load and support conditions used in the numerical simulation o the punching test Coarse Mesh (CM); (b) Properties o the layered oss section. Fig. 8 - Relationship between load and delection at the center o the test panel. Fig.9 - Punching test simulation: (a) top surace acks predicted by the numerical model (using a FEM mesh with eight-node serendipity plane shell elements), and (b) photograph showing the acks at the top surace o the panel, at the end o the test sequence. Fig. 0 - Inluence o, using the in-plane coarse mesh and 3 layers in the lightweight zone, on the numerical relationship between load and delection at the center o the test panel. Fig. - Inluence o the number o layers disetizing the thickness o the panel (the number indicated is restricted to the lightweight zone o the panel). Fig. - eometry, mesh, load and support conditions used in the numerical simulation o the punching test Reined Mesh (RM). Fig. 3 - Inluence o the in-plane reinement on the numerical relationship between load and delection at the center o the test panel. Fig. 4a - Inluence o on the numerical relationship between load and delection at the center o the panel, when using the in-plane coarse mesh and 0 layers in the lightweight zone. 5

26 Fig. 4b - Inluence o on the numerical relationship between load and delection at the center o the panel when using the in-plane reined mesh and 0 layers in the lightweight zone. Fig. 5a - Representation o the consumed out-o-plane racture energy,, c, when using the in-plane coarse mesh and 0 layers in the lightweight zone, or a delection o 3.5 mm. Fig. 5b - Representation o the consumed out-o-plane racture energy,, c, when using the in-plane reined mesh and 0 layers in the lightweight zone, or a delection o 3.5 mm. Fig. 6 - Inluence o the ξ parameter: (a) trilinear sotening diagrams and (b) relationship between load and delection at the center o the test panel. Fig. 7 - Inluence o the α parameter: (a) trilinear sotening diagrams and (b) relationship between load and delection at the center o the test panel. Fig. 8 - Inluence o the ξ parameter: (a) trilinear sotening diagrams and (b) relationship between load and delection at the center o the test panel. Fig. 9 - Inluence o the α parameter: (a) trilinear sotening diagrams and (b) relationship between load and delection at the center o the test panel. Fig. 0 - Inluence o delection at the center o the test panel. I : (a) trilinear sotening diagrams and (b) relationship between load and 6

27 50 00 S Polystyrene 300 S' S-S' 50 Polystyrene Fig. - Concept o a lightweight steel iber reinorced sel-compacting conete panel (dimensions in mm). 7

28 θ x s t n τ t σ n Crack σ n τ t w x Fig. - Stress components, relative displacements and local coordinate system o the ack. 8

29 σ n σ n, σ n, σ n,3 D n D nsec D n I g = / l b l b D n3 ε n, ε n,3 ε n,u ε n Fig. 3 - Trilinear stress-strain diagram to simulate the racture mode I ack propagation o SFRSCC ( σ = α σ, σ n,3 n, n, n, = α σ, ε n, = ξ ε n, u, ε n,3 = ξ ε n, u ). 9

30 τ τ p linear behavior -γ u -γ max -γ p τ max c γ p D,sec γ max l b γ u γ -τ max -τ p Fig. 4 - Diagram to simulate the relationship between the out-o-plane () shear stress and shear strain components. 30

31 (a) 8 node serendipity plane stress elements integration points elastic behavior 8 node serendipity plane stress elements integration points elastic-acked behavior 5 (b) 0 Load [kn] scatter o experimental results experimental average results numerical simulation Delection [mm] Fig. 5 - Three-point notched beam lexural test at 7 days: (a) FEM mesh used in the numerical simulation, and (b) obtained results. 3

32 Steel plate (00x00x0) 50 Q-Q' Actuator Q Q' 300 Polystyrene Steel plate (00x00x0) SFRSCC (a) (x300) (b) Fig. 6 - (a) Test panel module, and (b) test setup (dimensions in mm). 3

33 x (a) Point load mm A + Supports A' Thickness - 0 mm Thickness - 30 mm 300 mm x (b) Cross section: A-A' 3 0 mm 8 0 mm Layers with SFRSCC properties Layers with null stiness Fig. 7 - (a) eometry, mesh, load and support conditions used in the numerical simulation o the punching test Coarse Mesh (CM); (b) Properties o the layered oss section. 33

34 60 50 Load [kn] Experimental Linear behavior Linear out-o-plane shear (CM_3L) Sotening out-o-plane shear (CM_3L) Delection [mm] Fig. 8 - Relationship between load and delection at the center o the test panel. 34

35 (b) Fig.9 - Punching test simulation: (a) top surace acks predicted by the numerical model (using a FEM mesh with eight-node serendipity plane shell elements), and (b) photograph showing the acks at the top surace o the panel, at the end o the test sequence. 35

36 Load [kn] Experimental Sotening out-o-plane shear (CM_3L_) Sotening out-o-plane shear (CM_3L_) Sotening out-o-plane shear (CM_3L_3) Sotening out-o-plane shear (CM_3L_4) Sotening out-o-plane shear (CM_3L_5) Delection [mm] Fig. 0 - Inluence o, using the in-plane coarse mesh and 3 layers in the lightweight zone, on the numerical relationship between load and delection at the center o the test panel. 36

37 60 50 Load [kn] Experimental Sotening out-o-plane shear (CM_3L) Sotening out-o-plane shear (CM_6L) Sotening out-o-plane shear (CM_0L) Delection [mm] Fig. - Inluence o the number o layers disetizing the thickness o the panel (the number indicated is restricted to the lightweight zone o the panel). 37

38 x Point load 300 mm Supports Thickness - 0 mm Thickness - 30 mm 300 mm Fig. - eometry, mesh, load and support conditions used in the numerical simulation o the punching test Reined Mesh (RM). x 38

39 60 50 Load [kn] Experimental Sotening out-o-plane shear (CM_0L) Sotening out-o-plane shear (RM_0L) Delection [mm] Fig. 3 - Inluence o the in-plane reinement on the numerical relationship between load and delection at the center o the test panel. 39

40 Load [kn] Experimental Sotening out-o-plane shear (CM_0L_) Sotening out-o-plane shear (CM_0L_) Sotening out-o-plane shear (CM_0L_3) Sotening out-o-plane shear (CM_0L_4) Sotening out-o-plane shear (CM_0L_5) Delection [mm] Fig. 4a- Inluence o on the numerical relationship between load and delection at the center o the panel, when using the in-plane coarse mesh and 0 layers in the lightweight zone. Load [kn] Experimental Sotening out-o-plane shear (RM_0L_) Sotening out-o-plane shear (RM_0L_) Sotening out-o-plane shear (RM_0L_3) Sotening out-o-plane shear (RM_0L_4) Sotening out-o-plane shear (RM_0L_5) Delection [mm] Fig. 4b- Inluence o on the numerical relationship between load and delection at the center o the panel when using the in-plane reined mesh and 0 layers in the lightweight zone. 40

41 Fig. 5a- Representation o the consumed out-o-plane racture energy,, when using the in-plane coarse mesh and 0 layers in the, c lightweight zone, or a delection o 3.5 mm. Fig. 5b - Representation o the consumed out-o-plane racture energy,, when using the in-plane reined mesh and 0 layers in the, c lightweight zone, or a delection o 3.5 mm. 4

42 Crak stress [MPa] Inverse analysis Modiy Czi Modiy Czi Load [kn] Experimental Sotening out-o-plane shear (RM_0L_3) Sotening out-o-plane shear (RM_0L_3_Modiy_Czi ) Sotening out-o-plane shear (RM_0L_3_Modiy_Czi-0.035) Crack strain Delection [mm] (a) (b) Fig. 6 - Inluence o the ξ parameter: (a) trilinear sotening diagrams and (b) relationship between load and delection at the center o the test panel. 4

43 Crak stress [MPa] Crack strain Inverse analysis Modiy Alpha Modiy Alpha Delection [mm] (a) (b) Fig. 7 - Inluence o the α parameter: (a) trilinear sotening diagrams and (b) relationship between load and delection at the center o the test panel. Load ]kn] Experimental Sotening out-o-plane shear (RM_0L_3) Sotening out-o-plane shear (RM_0L_3_Modiy_Alpha-0.5) Sotening out-o-plane shear (RM_0L_3_Modiy_Alpha-0.75) 43

44 Crak stress [MPa] Inverse analysis Modiy Czi Modiy Czi Load [kn] Experimental Sotening out-o-plane shear (RM_0L_3) Sotening out-o-plane shear (RM_0L_3_Modiy_Czi-0.075) Sotening out-o-plane shear (RM_0L_3_Modiy_Czi-0.5) Crack strain Delection [mm] (a) (b) Fig. 8 - Inluence o the ξ parameter: (a) trilinear sotening diagrams and (b) relationship between load and delection at the center o the test panel. 44