A limit state design approach for hybrid reinforced concrete column‐supported flat slabs

Hybrid reinforced technology (combination of steel reinforcing bars and fibers) can be considered as a competitive alternative to the already existing solutions for the construction of column‐supported flat slabs. Constructed hybrid‐reinforced buildings prove that hybrid solutions have sufficient bearing capacity to maintain structural integrity despite being exposed to high stress levels, thereby providing a beneficial solution in terms of toughness, ductility, and sustainability performance. However, the lack of design‐oriented recommendations based on the accepted limit state format for dealing with both serviceability and ultimate limit states slows down the wider implementation of this technology. Considering the above‐mentioned, this article presents a simplified design‐oriented method that covers the evaluation of the structural response of hybrid reinforced concrete column‐supported flat slabs in terms of flexural strength, cracking, and instantaneous deformations. Two hybrid reinforced alternatives for a given flat slab are studied by means of the proposed approach. Furthermore, a nonlinear finite element analysis is carried out in order to evaluate the effectiveness of the developed simplified method. Based on the achieved results, its suitable accuracy and precision can be pointed out. This outcome may motivate current practitioners to consider hybrid reinforced concrete solutions as a possible alternative during the design of residential and office buildings.


| INTRODUCTION
During the last two decades, considerable scientific efforts have been made to identify and quantify the full potential of fiber reinforced concrete (FRC) and hybrid reinforced concrete (combination of steel reinforcing bars and fibers, HRC) slabs in statically redundant structures.
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Numerous laboratory tests were carried out, varying the fiber/rebar content, slab geometry, boundary, and load conditions. [1][2][3][4][5][6][7] The obtained results highlighted the enhanced structural response in terms of ductility, 3,6,7 cracking control, 1,2,5 and load redistribution. 1,5,7 Real-scale testing of elevated flat slabs confirmed the possibility of even total substitution of traditional reinforcement through using fibers in the concrete mix, this providing sufficient and even enhanced flexural and punching strength. [8][9][10][11][12][13] The promising capability of FRC technology was also confirmed in real projects: significant benefits in terms of construction time and costs were highlighted during the construction of the Ditton Nams shopping mall (Latvia), the Triangle office building (Estonia), the Rocca Al-Mare office tower (Estonia), and the LKS office building (Spain). [14][15][16] Despite the successful experiences of FRC slab construction, a number of shortcomings were revealed under special structural and/or construction conditions. The total substitution of traditional reinforcement by FRC can be inefficient if the bending and shear stresses magnitudes differ significantly throughout the slab. This variation requires different residual tensile strength (f R ) of the FRC and, as a consequence, different amount of fibers in different areas of the slab. In this sense, FRC has to be designed for guaranteeing the most demanding f R requirement; this strategy will provide over-reinforced areas and uncompetitive solutions.
Another drawback of FRC is the slight material heterogeneity within the depth of the element 8 ; the tendency of an increase in the fiber content towards the lower layers of the slab 9,17 can lead to insufficiently controlled cracks in the zones with high negative bending moments (in the vicinity of the columns). Considering these aspects, hybrid solutions might be even more attractive from technical and economic points of view since moderate values of f R can be imposed while most demanding bending and punching forces can be resisted by the combination of fibers and steel bars-enhancing also the workability of the concrete due to reduction of the fiber content in the mix.
Although the use of FRC and HRC in elevated flat slabs evidenced encouraging results, these alternatives are barely considered by engineers within the design process of residential and office buildings despite the acceptance of using FRC in structural elements by national and international codes along with specific guidelines. [18][19][20][21][22][23] One of the main reasons, apart from general concerns regarding the material capacity, is the lack of a detailed description on how to cover all the limit states for FRC/HRC flat slab design. Even scientific literature do not cover all required design aspects for this type of structural elements. In this regard, Table 1 shows that the majority of works are focused on the behavior of FRC/HRC elevated slabs at the ultimate limit state (ULS), providing the analytical and numerical approaches to assess the flexural and/or punching strength of the studied structural element.
In this context, the frequent questions of design engineers, which are hindering the widespread use of FRC/HRC in column-supported flat slabs, can be summarized as the following: • How should the cracking behavior of FRC/HRC elevated flat slab be estimated at the serviceability limit state (SLS)? • What is the most appropriate way to assess the deformations of FRC/HRC flat slabs? • Does the FRC/HRC approach provide adequate results with respect to long-term behavior? • What is a suitable method to carry out a comparison between traditional and FRC/HRC solutions?
Most of the questions can actually be answered by means of nonlinear finite element analysis (NLFEA). The existing scientific literature (Table 1) evidences the capacity to properly predict the structural response of two-way slabs using nonlinear finite element models. 36,38,41 Moreover, a NLFE parametric studies were carried out in order to analyze the structural response of the different FRC/HRC alternatives. 37 However, the application of these models is infrequent among structural designers since: (1) appropriate software to perform these analyses are not meant for day-to-day design procedures and (2) the lack of thoroughly described approaches for nonlinear analysis in current codes and guidelines for this specific purpose. Taking this into account, this article presents a simplified method for the design of FRC/HRC elevated slabs which responds to the majority of above-presented questions. This method permits the evaluation of the flexural capacity of the structural elements at ULS and the analysis of their response in terms of cracking and instantaneous deformations at SLS. Moreover, the developed approach allows checking the required ductility of FRC/HRC column-supported flats slabs. Based on the proposed design procedure, HRC solutions were analyzed for the given structure and the obtained results were compared with those obtained by NLFEA. The Yield Line Method (YLM) is being frequently suggested to evaluate the flexural load-carrying capacity of FRC flat slabs owing to its representativeness and the ease of application (very few restrictions). 18 These restrictions, that exist for elasticity-based methods, are related to the boundary conditions, opening dimensions and load types along with the possibility of taking into account the ductility/rotation capacity of FRC-the essential parameter to use the plastic analysis methods. YLM is also a suitable analytical approach to design RC slabs, as long as the required ductility can be proven by limiting the area of tensile reinforcement so that the following limits are fulfilled: x/d ≤0.25 and x/d ≤0.15 for concrete strength classes ≤ C50/60 and ≥ C55/67, respectively. 19,42 Taking into account that flat slabs typically have low ratios of tensile reinforcement (ρ s = A s /(bÁd)), the abovementioned restrictions are to be generally fulfilled.
YLM is based on Johansen's theory [43][44][45] and claims that the ultimate load of a slab can be obtained by postulating a collapse mechanism (generated by yield lines) that is compatible with boundary conditions. The selection of the collapse mechanism is of paramount importance considering that its wrong estimation will lead to the theoretically unsafe results since the presented method gives upper bound solutions. However, yield line patterns have been developed and thoroughly studied for common geometries ( Figure 1). Additionally, the standard formulae are already developed for these geometries in order to directly calculate the relation between applied load and produced moments per unit length (m þ Ed,YL , m À Ed,YL ); otherwise, such ratios can be found by means of segmental equilibrium or method of virtual work. 18,45,47 In case of a column-supported flat slabs with a common column grid which is subjected to uniformly distributed load (UDL), three "applied load-produced design moment" relationships are to be assessed by means of standard formulae. Two of these refer to, so-called, global failure of internal (Equation 1) and corner (Equation 2) panels of the slab (Figure 1a), whereas the third case concerns local failure (Figure 1b)-the situation where the ultimate load is transferred to the column from the slab tributary area. 18 Equation 3 presents "concentrated loadproduced design moment" relationship for the internal column, whereas Equation 4 covers the general case for the perimeter columns with an unknown angle (ω, in [rad]) [46][47][48] which is equal to π or π/2 for edge and corner columns, respectively. Importantly, these equations permit to vary the ratio of support to mid span moments ( h , Equation 5) as long as the ductility requirement is respected.  19 suggests to evaluate the design resisting moment of FRC (m Rd,FRC ) by concentrating the whole compressive force in the top fiber of the section, disregarding the traditionally adopted compressive block. However, considering the appearance of relatively high stresses in certain zones of column-supported slabs, the neutral axis may be fixed at 10% of the element thickness for all studied sections in order to provide a safer design procedure; thus, The design resisting moment of HRC (m Rd,HRC ) could be evaluated complementing the above described sectional analysis by the presence of additional force provided by steel reinforcing bars (A s Áf yd ). However, the more convenient way is to estimate m Rd,HRC as depicted in Figure 2, i.e., the total flexural strength corresponds to the sum of separately calculated m Rd,FRC and m Rd,RC where the latter is the design resisting moment of traditionally reinforced concrete. 50 This approach permits to compute the maximum design moments (m Ed ) in the critical areas (Section 2.1.3) and after calculating the contribution of fibers (m Rd,FRC ), assess the requirement of reinforcing bars as if it were traditionally reinforced concrete in every section which satisfies the following condition: m Rd,RC = m Ed À m Rd,FRC ≥0.
Nevertheless, the neutral axis can be found below 10% of the element thickness which will lead to an overestimation of the flexural strength of the section using the simplified sectional model ( Figure 2). Therefore, for the cases where more accurate results are demanded, the essential sectional parameters for both FRC and HRC can be computed by imposing sectional force equilibrium and deformation compatibility conditions in accordance with models presented in Figure 3. These parameters permit to assess the bearing capacity of the two-way system (providing more accurate outcomes in comparison with the previously suggested approach, Figure 2) based on the procedure described in the Section 2.1.3.
The above-described models are currently used within the analytical design procedure of FRC two-way slabs. However, the recent studies revealed that the implementation of those could be complemented by the introduction of structural redistribution factor 51,52 and/or orientation factor [53][54][55] in order to ensure the more accurate evaluation of the structural response in flexure. The magnitude of the former (up to 1.4) depends on the redistribution capacity of the studied element, whereas the latter takes into account the orientation of fibers which, in turn, is influenced by the rheological properties of concrete, casting method, and the geometry of the element in question. 56

| YLM for HRC flat slabs
The difference between FRC and HRC column-supported flat slab design by means of YLM consists of certain challenges in the correct proportioning of the stresses along F I G U R E 2 Simplified sectional model to assess flexural strength of HRC the yield lines in the latter case, taking into account the possible variety in reinforcement layouts. 41 However, drawing an analogy to the design of RC flat slabs by YLM, the procedure does not differ significantly. Once the geometry, reinforcement layout, and load conditions are established, the ratio of support to mid span moments ( h ) is to be chosen to calculate maximum design moments per unit length for the global failure mode.
The slight difference arises in the following step, when the required traditional reinforcement shall be computed in accordance with (1) considered layout and (2) maximum design moments along the yield lines. Firstly, the required amount of the reinforcing bars can be calculated for the bottom reinforcement. Taking into account that, in accordance with YLM, the curtailment is not advisable for the latter, the procedure is straightforward: the simplified approach ( Figure 2) or accurate sectional analysis ( Figure 3) can be applied. In turn, the top reinforcement is frequently found curtailed; therefore, the need of design moment proportioning should be considered.
In case of RC flat slabs (Figure 4), the task can be performed by means of Equation 6: multiplying the calculated moment (Equations 1 and 2) by the length of the yield line (L YL ) and dividing the obtained value by the length along the same yield line which is covered by the top reinforcement. 47 The presence of fibers in the concrete mix modifies the above described procedure; Equation 7 provides the consideration of its contribution to the flexural strength and, as a consequence, the design bending moment which shall be resisted by reinforcing bars in accordance with the simplified model presented in Figure 2. The more accurate procedure demands to find the resisting moment of the FRC section ( Figure 3a) with following proportioning of the design bending moment to the HRC zones by means of Equation 8 (Figure 4). The obtained magnitude of this moment permits to compute the required amount of the reinforcing bars using the sectional model presented in Figure 3b. Finally, the verification of the sufficient flexural strength against local flexural failure modes shall be performed for both cases in order to complete the analysis (the detailed description of the method implementation is provided in the Supporting Information available with the online version of the article, Appendix S1).
2.2 | Structural response of HRC flat slabs at SLS

| Internal forces and reinforcement distribution
In already constructed HRC flat slabs, generally, the absence of cracking under quasi-permanent load combinations was pointed out which permitted to evaluate the produced deformations as for an isotropic linear elastic material. 15,25 However, the presence of cracks in this type of elements is expectable-in accordance with ACI 421.3R-15, microcraking in twoway concrete slabs starts at an early level of approximately 10% of the service load, whereas the pattern of potential yield lines is almost fully developed at 30% of the same load. 57 Therefore, straightforward analytical methods are required to evaluate the response of HRC in terms of cracking and deformations. This is not a trivial task considering that even the behavior of RC two-way slabs at SLS still demands further studies and seems to be significantly more complex than the cases related to beams and/or one-way plates. Additionally, the proposed methods should be correlated with the current codes and guidelines in order to be used by practitioners. With this in mind, the approach based on the analogy with RC flat slabs design is presented herein; the overarching goal is to prove that HRC solutions comply with minimum requirements described in the fib MC 2010 19 and Eurocode 2 (EC2) 42 for traditionally reinforced alternative.
To this end, several additional steps, which are not directly dealing with the HRC flat slab design process, are to be carried out: (1) divide the given structure into column and middle strips ( Figure 5a) pursuant to current regulations 18,22,42 and (2)

| Cracking control
Generally, the indirect control of cracking is widely applied for RC slabs subjected to bending without significant axial tension, i.e., respecting the established values of maximum permitted steel stresses and reinforcing bar diameter/spacing. 19 In case of RC flat slabs, specific suggestions regarding the detailing of the reinforcement can be found in different studies and regulations. For instance, EC2 42 states that at internal columns, 50% of the reinforcement, which is required to resist the full negative moment from the sum of the two half panels each side of the column, should be placed in a width equal to the sum of 0.125 times the panel width on either side of the column. Similarly, the works of Jofriet 59 and Brotchie et al. 60 propose the division of the negative column strip into two strips of equal length. Further, the inner negative column strip is to be designed for two-thirds of the total negative column strip moment in order to meet SLS requirements.
The demand for more rigorous calculations can be accompanied with a number of constraints. ACI Codes, 57,61,62 for instance, consider that crack control equations for beams underestimate the crack widths developed in two-way slabs, drawing on the scientific works developed, mainly, by Nawy. 63,64 However, the development of this expression was formulated for cracks produced mainly in positive-moment regions, i.e., the regions with clear two-way bending. In turn, cracks tend to appear in the vicinity of columns under the quasipermanent combination of actions in column-supported flat slabs, 65 therefore, it might be important to note that the one-way bending is predominant at faces of the columns.
Referring the above described information to HRC flat slabs, it is evident that further experimental tests along with the validation of existing models are necessary for the accurate assessment of maximum crack widths in column-supported flat slabs. Therefore, being conservative, the methods related to indirect control of cracking in RC column-supported flat slabs can be adapted to HRC solutions, assuring that the latter is not inferior to the RC alternative.
For that purpose, first, the maximum design moment above the columns computed by DDT/EFM (Figure 5c) is to be modified in accordance with EC2 42 (Figure 5d). In fact, EC2 42 provides this re-proportioning in terms of required top reinforcement area; however, the same relationship can be presented in terms of ultimate flexural moments for the sake of more direct design procedure (with minor difference in the overall result). Based on the modified moment, the required amount of the reinforcement for RC solution is to be assessed-the obtained result permits to omit the subsequent verification of the structure at limit state of cracking according to EC2. 42 Eventually, the steel stresses can be evaluated in the RC section above the columns as for a continuous beam with a unit width (1000 mm) and the depth equal to the thickness of the slab (h).
A similar procedure should be repeated for HRC solution. Basing on the FRC properties along with the reinforcement layout, the required amount of steel bars is to be found by means of YLM (Section 2.1). Thereafter, the steel stresses in the critical sections should be estimated, considering the same dimensions (h Â 1000 mm). For more detailed analysis of FRC contribution to the structural response of the HRC section at SLS, the multilinear constitutive model in tension ( Figure 6) shall be applied. 19 The majority of variables for this model depend on the mean residual tensile strengths at CMODs of 0.5 mm (f R1m ), 2.5 mm (f R3m ) according to EN 14651 49 and can be computed as follows: R1m ); (4) ε SLS = CMOD 1 /l cs ; (5) ε ULS = w u /l cs = min (ε Fu , 2.5/l cs = 2.5/y); (6) ε Fu = 20‰ (considering variable strain distribution along the cross section).
The computed sectional parameters for both RC and HRC should be used to check the sufficient cracking control of the latter by means of two approaches: (1) comparing the achieved stresses (more conservative method) or (2) computing the crack widths as for one-way elements.
F I G U R E 6 Sectional model to assess steel stresses in cracked sections (adapted from Reference 66) Conservatively, the steel stresses in each section should be compared and, in case of σ s,HRC ≤ σ s,RC , HRC solution presents at least the same performance in terms of cracking control in comparison with RC alternative owing to the fact that the presence of fibers in the material enhances the bond between steel bars and concrete. This phenomenon leads to the reduction of the bond transfer length and, as a result, crack spacing. [67][68][69] The reduction of crack spacing will ensure a minor crack width in the HRC alternative for the same magnitudes of steel stresses (strains).
Otherwise (σ s,HRC ≥ σ s,RC ), the chosen reinforcement layout for ULS (YLM) can be unevenly distributed within the column strip pursuant to EC2 42 in order to reduce the steel stresses up to the required magnitude. However, it is important to consider the necessity to keep the required flexural strength in the column strip outside the mentioned area ( Figure 5d). The described case is not likely to occur in the HRC solution with moderate fiber content since YLM allows to concentrate the rebar layout in the vicinity of columns ( Figure 4) and, as a consequence, lead to considerable values of negative moments in these zones. 47 Nevertheless, by increasing the fiber content, the negative moments along the yield line will be more uniformly distributed, i.e., less traditional reinforcement in the form of steel bars is required. Therefore, the verification of sufficient crack control might be necessary and can be assured even by placement of the additional steel bars in the centre part of the column strip.
Alternatively, the crack widths can be assessed in the critical sections as for one-way elements. 65 The design crack width for the RC solution can be evaluated by means of Equations 9-13 in accordance with the fib MC 2010 (Clause 7.6.4.4). 19 The same equations are valid for the analysis of HRC-for that purpose, provided f ctm is to be substituted by f ctm -f Ftsm in Equations 10 and 12, taking into account the effect of fibers on the transfer length. Thereafter, the comparative procedure is identical with the one related to the analysis of the produced steel stresses. Summarizing, Figure 7 highlights the essential steps to justify the appropriate structural response of HRC columnsupported flats slabs at limit states of cracking for both approaches.

| Instantaneous deformations
The design of elevated slabs is typically governed by the deflection control 70 and the correct evaluation of this aspect is of a paramount importance given that this structural element constitute 80-90% of the total cost of a concrete frame. 71 Moreover, it is also important to highlight that the flexural strength for the studied case is to be assessed by means of YLM-approach that provides suitable results for ULS, but does not provide information for serviceability design. 65 With this in mind, simplified analysis methods for RC solutions are adapted herein in order to evaluate the magnitude of deflections in HRC columnsupported flats slabs (Figure 8). Several methods were studied [72][73][74][75][76][77] and the approach to calculate deflections by means of crossing beam analogy was chosen as this providing: (1) relatively simple design procedure and (2) significant correlation with experimental data. 71,75,76 Importantly, the presence of fibers as reinforcing material is compatible with this design procedure. Moreover, that has a potential to be extended in order to assess the long-term response of the FRC structural elements-the aspect of the paramount importance that still requires further research 78 and is beyond the scope of this article. Having calculated a distribution of moments (Figure 5c), the total instantaneous deflection at the center of certain panel (δ tot ) can be computed as the sum of the average deflection of two parallel column strips and the deflection of the middle strip spanning at right angle to the column strips (Figure 8b or Figure 8c). Among the possible approaches to assess the deflections of these strips, the procedure proposed by Ghali et al. 72,79 was found to be convenient for HRC solutions owing to the assumption that the variation of curvature follows a second degree parabola for the continuous straight members and, therefore, can be easily defined as per Figure 8d. As a result, this method is based on defining between six and nine curvatures depending on the symmetry of the bay considered.
Moreover, the mid-span sections can be referred to as the "determinant" sections (representing the stiffness of the zone exposed to positive bending moments and having the largest overall effect on deflections), and thus, the simplified approach permits 72,79 to evaluate the influence of cracking on the produced curvatures only in these sections, i.e., curvatures at the end sections can be studied as per homogenous material with established properties (moment of inertia and modulus of elasticity). In case of RC solutions, the curvatures of "determinant" sections can be computed considering two extreme states: the uncracked condition in which concrete and steel are assumed to behave elastically and exhibit compatible deformations, and the fully cracked condition with the concrete in tension ignored. 79 Thereafter, the mean value of curvatures is to be obtained by means of interpolation. Adapting this approach to HRC solution, the section model described previously ( Figure 6) along with the multi-layer sectional approach (inverse analysis) 80 may be applied to estimate the curvatures at the governing sections subjected to the given bending moments with further evaluation of the produced deflections in accordance with Figure 8d.

| Geometry and reinforcement layout
The selection of the geometry for the case study was oriented to reproduce common dimensions of slab panels that could be representative for office and residential buildings. Additionally, the number of three successive panels in each direction was chosen in order to involve The reinforcement layout (Figure 9b) was adopted taking into consideration the advantages of using YLM for flat slabs: the bottom reinforcement (R B1 , R B2 ) may be placed across whole bays, generally without curtailment. The top reinforcement (R T1 , R T2* , and R T3* , Figure 9b) was concentrated in the vicinity of columns in order to enhance the structural response at service loads in accordance with following ratios: the areas of 0.5 Á L x Â 0.5 Á L y , 0.5 Á L x/y Â (0.2 Á L y/x + E.D.), and (0.2 Á L x + E.D.) Â (0.2 Á L x + E.D.) were, respectively, placed over the internal, edge, and corner columns-E.D. in the presented expressions was equal to the distance between the centreline of column to edge of the slab, i.e., edge distance. Additionally, the selected types of reinforcement R T2* and R T3* should be explained: YLM suggests to provide similar positive and negative bending capacities to sections at the vicinity of columns once the possible local failures are to be analyzed; therefore, the "U" bars were provided in these zones.
The top reinforcement between concentrations over column heads is omitted-this approach can be applied even for RC solution, 47 even though cracking can develop in these areas (with minor effect on the structural performance). In case of HRC solutions, the possibility of cracking is significantly reduced due to presence of fibers. However, when necessary, the need of cracking-control mesh in the above discussed zones can be assessed: the minimum reinforcement for the crack control may be estimated in accordance with the Clause 7.7.4.3 of the fib MC 2010. 19

| Design actions
The loads specified in the Spanish Building Code for residential buildings 81 were considered as a reference to compute load combinations for both ULS and SLS. Apart from the self-weight (q SW ) of 4.8 kN/m 2 , a dead load (q G ) and variable load (q Q ) of 2.0 and 3.0 kN/m 2 , respectively, were assumed. Load partial safety factors γ G = 1.35 and γ Q = 1.50 were assumed to evaluate the design load at ULS: q Sd = γ G Á (q SW + q G ) + γ Q Á q Q = 13.7 kN/m 2 . The quasi-permanent load combination (q k,ψ2 = q SW + q G + ψ 2 Á q Q ; ψ 2 = 0.3) for residential and office buildings was adopted for deflection and crack control checks. In the present study, this load combination resulted in a UDL of 7.7 kN/m 2 .

| Material properties
Two different types of FRC were analyzed in the presented study for a more comprehensive analysis. FRC of 3c and 4d strength classes were selected-both types of concrete fulfilled the established requirements to be capable of substituting conventional reinforcement at ULS in accordance with the fib MC 2010: f R1k /f Lk >0.4 and f R3k / f R1k >0.5. 19 Table 2 gathers all mechanical properties of the selected materials which were considered in both analytical and numerical design procedures. The magnitudes of the compressive strength and modulus of elasticity were adopted based on the experimental campaign which involved the analysis of 15 selfcompacting FRC mixes with fiber content up to 120 kg/ m 3 in order to establish suitable material for the construction of full-scale SFRC flat slab. 82 The mean values of the residual tensile strengths (f R1m , f R3m ) were established by (1) assuming the normal distribution of f R and (2) imposing values of the coefficient of variation (CV fR ) for the latter. In this regard, taking into consideration the extensive experimental programme 83 and values of CV fR derived from a numerical study on the intrinsic scatter of FRC, 84 CV fR of 25.0% and 20.0% were assumed for FRC3c and FRC4d, respectively.
The presented residual tensile strengths were not sufficient to provide the required bearing capacity of the structure in question. Therefore, the reinforcing steel B500C (classification was taken from the fib MC 2010, 19 Clause 5.2) was assumed for the reinforcing bars ( Table 3). The numerical analysis also demands the mean yield/tensile strengths of steel to compute the global safety factor (Section 6.1); for this purpose, the established characteristic values were increased by 10% in conformity with the fib MC 2010 and EC2. 19,85 4 | IMPLEMENTATION OF THE PROPOSED METHOD

| Required flexural capacity
Based on YLM along with the established design UDL at ULS and selected geometry, the maximum design bending moments can be computed for both corner and internal panels by means of Equations 1 and 2. For this purpose, the ratio of negative to positive flexural capacities ( h ) was considered as 1.0. Additionally, the distance between the negative yield line and the centreline of external columns was adopted as 5.85 m for the corner panels and spacing between two negative yield lines for the internal panels ( Figure 1) was assumed to be 5.70 m.
As a result, m þ Ed,YL ¼ m À Ed,YL ¼ 40.2 and 27.8 kNm/m for corner and internal panels, respectively (the detailed calculations can be found in the Supporting Information available with the online version of the article, Appendix S1). Thereafter, the flexural strengths of FRC 3c and FRC 4d were calculated by using the sectional model presented in Figure 3 followed by (1) distribution of the negative moments (Equation 7) and (2) assessment of the required amount of reinforcement in order to provide the sufficient bearing capacity demanded by the global failure mode. Finally, the local failure mode was analyzed; for this purpose, the loads transferred to each column from the slab tributary areas (under UDL of 13.7 kN/m 2 ) were calculated (Table 4), this permitting to check the already computed reinforcement and to estimate the reinforcement for the edge and corner columns by Equation 3. Table 5 presents the required amount of reinforcing steel bars for both FRC 3c and FRC 4d in accordance with the reinforcement layout depicted in Figure 9b; the T A B L E 2 Mechanical properties of FRC used in the design procedure  rebar detailing was not provided and the same amount of reinforcement per meter was used in the nonlinear analysis (Section 5.2) in order to carry out a more accurate comparison.

| Crack control at SLS
Firstly, the simplified elastic distribution of moments at ULS was evaluated by means of DDT ( Figure 10). Table 6 reports the coefficients assumed to distribute the total static moment (M 0 = q u ÁL y ÁL n 2 /8) to supports and midspan for negative moments and positive moments, respectively. Table 7 gathers the information regarding the distribution of the negative and positive moments transversely to the column and middle strips.
Additionally, the obtained values should be modified in accordance with EC2 42 to indirectly guarantee the cracking control (Figure 5d, Section 2.2.1)-the internal column B2 is taken as reference: 50% of the full negative moment from the sum of the two half panels (118.5 kNm) should be established in a width equal to the sum of 0.125 times the panel width on either side of the column (1.5 m), resulting in the overall bending moment of 79 kNm/m. Considering the material properties reported in Tables 2 and 3, the RC section of 200 Â 1000 mm requires 1117 mm 2 in order to provide the adequate flexural strength under the moment of this magnitude (taking into account the traditionally adopted partial factors at ULS; γ s = 1.15, γ c = 1.5).
Finally, the structural response of three section alternatives should be studied at SLS in terms of the stresses to which the most demanded steel rebars are subjected under the quasi-permanent load combination. These sections of 200 Â 1000 mm differ only in the type of reinforcement: (1) RC section with 1117 mm 2 of steel reinforcing bars, (2) FRC 3c with 804 mm 2 , and (3) FRC 4d with 592 mm 2 ( Table 5).
The moment-steel stress relationship ( Figure 11) was computed for the sections in question in order to provide better visualization of the structural response of the studied alternatives. The material properties gathered in Tables 2 and 3 (represented by mean values) along with the constitutive model depicted in Figure 6 were taken into consideration for the sectional analysis. As a result, the influence of the bending moment on the steel stresses was estimated by varying the magnitude of the bending moment from 30 to 90 kNm/m.
The structural response of the RC section can be a reference for the evaluation of HRC solutions as it should ensure the sufficient cracking control of the studied twoway slab subjected to the previously described loads according to EC2. 42 The better performance of FRC 3c

| Instantaneous deflections at SLS
The quasi-permanent load combination resulted in a UDL of 7.7 kN/m 2 . The flexural moments in column and middle strip under this magnitude of UDL can be estimated by reducing (proportionally) those computed for ULS ( Figure 10). Figure 12 presents the magnitudes of the bending moments to evaluate the instantaneous deformation in the centre of the corner panel. For this purpose, the deflections of two column strips in X direction (AB, CD) and the deflection of the middle strip at right angle to the column strips (EF) are to be calculated.
It is important to remark that only two of the bending moments presented in Figure 12 exceeded the cracking moment (m cr = f ct ÁbÁh 2 /6 = 27.1 kNm/m)-this is owed to the selected concrete mix, i.e., the required self-compacting behavior of FRC demands the increased content of cement and fine aggregates along with the additives (to reduce the water-cement ratio) as it was evidenced in. 8,82 These modifications lead to the increment of the material tensile strength which, in turn, can be beneficial in terms of serviceability performance; especially, considering that RC solutions for the flat slabs are usually designed/constructed with concrete classes C25-C30 (classification was taken from the fib MC 2010, 19 Clause 5.1.2).
From a design perspective, the increased concrete tensile strength favors the use of a simplified approach to calculate the produced deformations-less cracking tends to keep at a certain degree the elastic distribution of moments, i.e., cracking of the slab would lead to moment redistribution and, thereby, it would have an effect on deflections. Therefore, a traditional linear elastic analysis will produce a minor error.
Taking this into account, the elastic analysis was also carried out and compared with the results obtained by means of the adopted simplified method and NLFEA (Section 5.2). Realizing that the produced deformations should not differ significantly for a UDL of 7.7 kN/m 2 , it was decided to extend the study and estimate the instantaneous deformations up to a UDL of 17 kN/m 2 ( Figure 12) by means of the elastic analysis and the proposed method in order to: (1) compare in more detail the computed output with the NLFEA and (2) to prove the possibility of evaluating the required ductility in bending of FRC / HRC column-supported flat slabs by the proposed method.
The abovementioned requirement, in accordance with the fib MC 2010, 19 must satisfy at least one of the following conditions: (1) δ u ≥ 20 Á δ SLS and (2) δ peak ≥ 5 Á δ SLS , where δ u is the displacement corresponding to the ultimate capacity (P u ), δ peak is the deflection at the maximum load (P max ), and δ SLS is the displacement at SLS computed by performing a linear elastic analysis with the assumption of uncracked concrete. The assessment of δ u by considering the elastic distribution of moments is excessively conservative due to the significant redistribution capacity of two-way HRC elements. However, it is possible to prove that the structure does not reach the maximum load at the displacement of 5 Á δ SLS , augmenting the UDL and computing the produced deformations up to the required value (5 Á δ SLS ) relying on the estimated curvatures.
As a result, the UDL-displacement relationship was estimated by means of elastic analysis (the software SAP2000 86 was used) and proposed method for both HRC solutions ( Figure 13) varying the load from 0 to 17 kN/m 2 . The produced displacement in the centre of the corner panel at SLS (UDL of 7.7 kN/m 2 ) is 3.2 and 4.2 mm according to the elastic analysis and proposed method, respectively-these values are to be compared with the NLFEA output in the following sections. Additionally, it is important to remark that the proposed method evidenced sufficient ductility of the structural system: the displacement under the UDL of 17 kN/m 2 (with the capability of further load increment) was found to, respectively, be 19.7 and 17.0 mm for HRC 3c (FRC 3c + reinforcing bars) and HRC 4d (FRC 4d + reinforcing bars), whereas 5 Á δ SLS = 16 mm.

| Model validation
The finite element software ATENA 5.7.0 87 was used to model the structure under study, considering both hybrid alternatives. The structural response of FRC in tension was reproduced by means of the multi-linear constitutive diagram depicted in Figure 6 which was based on the mechanical properties of the selected materials ( Table 2). The compressive behavior of concrete was modeled using the stress-strain relationship for short-term loading in accordance with the Clause 5.1.8.1 of the fib MC 2010. 19 The behavior of steel was represented by the bilinear constitutive diagram following the Clause 3.2.7 of EC2 42 and taking into account the mechanical properties of reinforcing steel ( Table 3).
The geometry of the HRC column-supported flat slab was modeled by means of 3D shell and solid elementsalmost the entire structure was comprised of 3D shell elements considering that the analyzed element was mainly subjected to bending stresses; only the zones near the column-slab connections were modeled by solid 3D elements. Additionally, only a quarter of the studied element was modeled due to a double symmetry resulting in a total of 13,300 hexahedral elements ( Figure 14). This approach required to impose the displacements in both symmetry planes. The column supports were reproduced by the rigid constraints in vertical direction-it is important to remark that the post-cracking tensile strength was increased in the regions above the columns (where vertical displacements were actually restricted) in order to properly reproduce the column-slab connection, avoiding the tensile local failures of these zones.
Before the verification of the proposed simplified method by means of NLFEA, the numerical model was validated by simulating two experimental tests 11,13 on full-scale column-supported flat slabs of the same geometry as that of the case study (Figure 9a). The first test was reported by Gossla,11 which consisted in evaluating the structural response of the FRC slab (fiber content of 100 kg/m 3 ) up to failure. The ultimate bearing capacity of the structure (named Bissen slab) was estimated by Prior to the full-scale test, the material characterization of the FRC considered was performed by means of four point bending test, square panel and round panel tests. The experimental outcome obtained by testing the round panels permitted to estimate the post-cracking behavior of FRC using the inverse analysis. 39 These properties (f ct = 2.5 MPa, f 1 = 1.75 MPa, f 2 = 1.06 MPa, f 3 = 0 MPa and w ct = 0 mm, w 1 = 0.25 mm, w 2 = 1.25 mm, w 3 = 2 mm), along with the information related to the measured compressive strength (f cm = 35 MPa) and modulus of elasticity (E cm = 32,-300 MPa) were used in the simulation of the Bissen slab with minor modifications: ATENA permits to modify the pre-and post-cracking tensile models by means of stressstrain constitutive diagrams and, therefore, the presented crack widths were transformed to corresponding strains in accordance with the crack band method and the smeared crack approach. 88 The second FRC full-scale test (fiber content of 70 kg/ m 3 , named Limelette slab) was tested under the same conditions up to failure-applying the point load at the centre of the slab. Therefore, only the material properties were reintroduced according to the information presented in the previous studies. 41,52 Post-cracking behavior was modeled considering the following tensile and residual tensile strengths at established crack widths: f ct = 2.2 MPa, f 1 = 1.2 MPa, f 2 = 1.2 MPa, f 3 = 0 MPa and w ct = 0 mm, w 1 = 0.05 mm, w 2 = 1.5 mm, w 3 = 6 mm, whereas the compressive strength and modulus of elasticity were adopted as 35 and 29,000 MPa, respectively. In both cases, the self-weight was imposed and then the point load was applied at a circular steel As a result, both simulations of the full-scale tests showed an accurate prediction of the structural behavior of FRC flat slabs tested experimentally ( Figure 15). The computed maximum loads differed from those observed during the real testing by 6.3 and 3.6% in case of Bissen and Limellette slabs, respectively. Moreover, the simulations led to almost identical structural response up to deflections of 27 and 38 mm for the abovementioned FRC slabs-this aspect being of a paramount importance from the perspective of further analysis of the hybrid solutions at SLS. Additionally, the estimation of the developed cracks should be pointed out-considering that the crack patterns in both cases were similar, Figure 16 only presents the comparison of the produced cracks during the experimental test of the Limelette slab 13,52 with those predicted by the numerical simulation. Consequently, based on the obtained results, it can be stated that the numerical model was capable of predicting the structural response of the HRC solutions from low to high levels of applied loads.

| Modeling of HRC case studies
The geometry of Bissen and Limelette slabs was the same as that of the case study. Therefore, the previously described model (Figure 14) was used in order to analyze numerically the HRC alternatives with minor modifications-the onedimensional reinforcing steel bars with a perfect bond were introduced to this model in accordance with (1) the established material properties (Table 3), (2) reinforcement layout (Figure 9), and analytical design output (Table 5). Additionally, the loading of the model was updated-the structure was subjected to the gradually increased UDL up to a failure. The failure criterion was related to the flexural response of the analyzed HRC column-supported flat slabs in terms of the produced deflections-further increment of UDL of 0.1 kN/m 2 should not have led to disproportionate deformations at the corner panel. Figure 17 presents similar UDL-deflection relationship for studied HRC solutions using both mean and characteristic values of material properties, despite the differences of both residual tensile strengths and reinforcing steel bars amounts. Therefore, it can be concluded that, varying the residual tensile strength, the proposed simplified approach based on the YLM permits to compute the required amount of the reinforcing steel bars, assuring almost identical structural response in flexure of HRC alternatives.
However, the next question that may arise is "Does this approach provide safe solutions in terms of design resistance?" To answer this question, the design global resistance of the structure was calculated in accordance with the fib MC 2010 19 (Clause 4.6.2.2) and compared with the initially established q Sd = 13.7 kN/m 2 .

| Global resistance of the studied cases
The structural response of both HRC solutions was predicted ( Figure 17) by nonlinear analysis using the mean values of selected material properties (Tables 2 and  3). Nonetheless, the global safety factor should be quantified by complying with the design condition (Equation 14). For this purpose, the method of estimation of a coefficient of variation of resistance (ECOV) was adopted.
In this simplified probabilistic approach, the underlying assumption states that the coefficient of variation of resistance (V R ) can be assumed as lognormally distributed and can be expressed by means of Equation 15, i.e., may be estimated based on the mean (R m ) and characteristic (R k ) values of global resistance. 89 The computed coefficient of variation provides the possibility of evaluating the global safety factor (γ R ) using the Equation 16, which also takes into account the sensitivity factor (α R ) for the reliability of resistance and the reliability index (β). These two values (α R , β) can be considered to be, respectively,  19 In the current study, the typically adopted value of 1.06 was assumed.
Applying the abovementioned approach, the global design resistance for the HRC alternatives was evaluated ( Table 8). The design values of the global strength (R d ) are 1.4 and 4.8% higher than the required global resistance (E d ) for the alternatives FRC 3c and FRC 4d, respectively, i.e., the required flexural capacity was achieved by means of the proposed simplified method.

| Structural response of studied cases at SLS
The instantaneous deflections were computed by means of the developed approach ( Figure 13). The numerical analysis, which was posteriorly conducted, permits to evaluate the accuracy of this approach. Figure 18 corroborates the precision of the proposed method-the estimated deflections at 7.7 kN/m 2 (quasi-permanent load combination) differ from those obtained numerically by 0.1 mm (2.4%). Moreover, the analytically assessed deflections were also in line with those calculated numerically up to the UDL which exceeded the established magnitude at ULS (13.7 kN/m 2 ).
Based on the abovementioned, the proposed method can be considered as suitable for verifying the ductility requirements in bending of the HRC alternatives. It must be highlighted that the numerical outcomes also allow confirming the viability of indirect method to guarantee the sufficient crack control since the maximum crack width at SLS is inferior to 0.3 mm, which is the value generally established as a maximum crack width (w max ) for several exposure classes under quasi-permanent load combination of actions. 19,22,42 7 | CONCLUSIONS In this article, a design-oriented approach to evaluate the flexural capacity along with the estimation of instantaneous deflections and crack-width governing parameters is proposed for HRC column-supported flat slabs. Two HRC alternatives were studied by means of the proposed approach for a given geometry and boundary conditions. Additionally, a NLFE analysis was conducted in order to compare and validate the results obtained with the analytical approach. From this analysis, the following conclusions may be derived: • The Yield Line Method permits to evaluate the overall flexural capacity of the HRC flat slabs with a possibility of distributing moments while accounting for the presence of FRC and HRC sections along the same yield line. Based on nonlinear analyses, the analytical approach provides a reliable results in terms of overall bearing capacity in flexure in case of the given structure subjected to UDL. • The proposed approach, based on the crossing beam analogy along with indirect control of cracking for RC flat slabs, proved to be promising for dealing with the design at both SLS (crack width control and instantaneous deflection control). • Additionally, the proposed method permits to check the ductility requirements for bending (δ peak ≥ 5 Á δ SLS ) by evaluating the expected deflections at ULS.
Even though the numerical analyses showed a considerable accuracy and precision of the proposed designoriented approach, certain aspects are still to be studied for the consideration of this technological alternative (partial substitution of reinforcing steel bars by fibers in HRC flat slabs) by practitioners in the design phase. In this regard, the potential modification of the current FRC constitutive models for two-way elements, the estimation of the long-term behavior of FRC/HRC flat slabs, and the structural effect of induced holes into the slab are among the topics that should be further investigated. Furthermore, the parametric study which involves the variation of geometry, load type (UDL, point load) and magnitude of the latter can complement the presented investigation,

ACKNOWLEDGMENTS
This study was financially supported by the Spanish Ministry of Science and Innovation under the scope of project  α R sensitivity factor β reliability index γ c partial safety factor for concrete properties γ F partial safety factor for FRC γ G partial safety factor for permanent actions γ Q partial safety factor for variable actions γ R global resistance safety factor γ Rd model uncertainty factor γ s partial safety factor for reinforcing steel properties δ displacement δ tot displacement at the centre of the panel ε cm average concrete strain ε cs shrinkage strain at concrete ε sm mean steel strain η r coefficient to consider the shrinkage contribution ρ s ratio of tensile reinforcement ρ s,ef effective reinforcement ratio