This paper examines the structural response of reinforced concrete flat slabs, provided with fully - embedded shear -heads, through detailed three -dimensional nonlinear numerical simulations and parametric assessments using concrete damage plasticity models . Validations of the adopted nonlinear finite element procedures are carried out against experimental results from t hree test series. After gaining confidence in the ability of the numerical models to predict closely the full inelastic response and failure modes , numerical investigations are carried out in order to examine the influence of key material and geometric parameters . The results of these numerical assessments enable the identification of three modes of fail ure as a function of the interaction between the shear -head and surrounding concrete. Based on the findings, coupled with results from previous studies, analytical models are proposed for predicting the rotational response as well as the ultimate strength of such slab systems . Practical recommendations are also provided for the design of shear -heads in RC slabs , including the embedment length and section size. The analytical expressions proposed in this paper , based on a wide -ranging parametric assessment, are shown to offer a more reliable design approach in comparison with existing methods for all types of shear -heads, and are suitable for direct practical application. Keywords: Non-linear numerical model ling, Concrete damage plasticity, RC flat slabs, Shear -heads, Punching shear, ___________________________________________________ *Corresponding author: Dr. D. V. Bompa - d.bompa@imperial.ac.uk A note on versions: the version presented here is the preprin t of the manuscript and should be cited using the reference at the top of the page; to access the Open Access published version use the doi. 2 1. Introduction Brittle failure s, which can occur at the slab –column connection , may typically be prevented through drop panels or by reinforcing the critical region with transverse bars (e.g. [1-3]) or structural steel inserts [4-11]. The performance of a flat slab with transverse reinforcement is however strongly dependent on the arrangement, type and siz e as well as bond characteristics of the rebars . On the other hand, shear -head system s typically translate the weak section, both in bending and shear, outside of the shear -head region [4, 1 2]. Early interest in shear -head configurations was shown with the development of high-rise structures in the United States (e.g. [13,14]). Early tests on such systems showed that they can offer reliable structural response at the connection [15]. The first tests on fully embedded shear -heads against punching shear w ere carried out tests on flat slabs with cruciform shear -head systems consisting of perpendicular Channel or I -sections (Figure 1a -c), placed between the longitudinal reinforcement layers . These showed an increase in punching shear strength by up to 75% in co mparison to reference cases without shear -heads [6]. In Europe, tests on 140 mm flat slabs provided with fully embedded cruciform I -section shear -heads with varying embedment lengths subjected to axial load and unbalanced moment with eccentricities around 0.25 showed an increase in ultimate capacity between 40-70% as a function of the shear -head embedment length [16]. Development tests on shear -head systems made of inverted wide tee steel beams positioned in a cruciform arrangement also showed enhancement of punching shear strength of 240 and 300 mm slabs in the range of 65% in comparison their non -reinforced counterparts . As for other cruciform shear -head systems, rupture occurred at the tip of the shear -head, showing an effective force flow as the load was transferred to the column by shear in the web and te nsion action in the connection plates of the shear -heads [17]. Other tests made use of fully and partially embedded closed -box shear -heads which led to an increase in punching shear strength by about four times and a two -fold reduction in the supporting moments [4, 18]. The crack patterns from tests on large -scale flat slabs provided with closed -box shear -heads and practical slab thicknesses , showed resp onses governed by tangential cracks around the shear -head, as well as straight radial cracks [19]. This indicated that the curvatures in the radial direction are concentrated around the relatively rigid shear -head acting as an enlarged column support , similar to a conventional column support (Figure 1d) . Moreover, t ests on 16 × 120 mm thick model slabs and 4 large -scale 3 m×3 m and 250 mm thick specimens investigate d the optimal shear -head configuration and their effect on punching shear capaci ty [20]. Other intricate shear -head systems exist, including for example composite cruciform systems consisting of vertical steel plates acting as shear -legs to 3 which headed studs are welded for composite action combined with special steel-endplates for strut support [21]. Non-linear finite element simulations can offer insight s into the structural behaviour and can predict the ultimate strength and deformational response of RC flat slabs and members (e.g. [22-28]). However, d etailed numerical studies on flat slabs with shear -heads are relatively limited and have typically focused on hybrid connections between steel columns and RC floors. Previous assessments on hybrid and RC have highlight ed the influence of key parameters whi ch govern the performance of shear -heads in hybrid flat slabs similar to the member depicted in Figure 1e [29, 30]. However, previous studies were limited in terms of scope and parameter ranges, and hence did not permit the development of analytical models and guidance suitable for practical application. Figure 1 Schematic of failure surfaces for specimens with and without cruciform shearheads [6]: a) AN -1, b) AC-1, c) AH -1; d) Crack distribution for closed box shear -heads [19]; Crack patterns for hybrid RC flat slabs with cruciform shearheads [12]. Although shea r-heads are widely considered as reinforcing systems against punching shear in practice [31-37], research studies on RC and hybrid configurations are c onstantly carried out [8,9,11,12, 30,38- 42] and such systems are typically recommended for practical application [4 3], codified guidance is comparatively limited. For example, no specific provisions are stipulated in Eurocode 2 [44], whilst the Model Code 2010 [45] offers some guidance for the design of flat slab s with closed -box shear - heads . In contrast, ACI318 -14 [46] offers detailed guidance for cruciform shear -heads, largely based on the limited tests data reporte d by Corley and Hawkins (1968) [6], whilst closed -box shear -heads are not addressed. Current guidelines are therefore limited to specific shear -head types , thus hindering a wider application . Observations from previous studies indicate that the punching shear resistance of flat slabs with shear -head s depend s not only on the typical shear -design parameters such as shear depth, concrete strength and support size but also on the geometry, deformation response and strength of the individual elements of the shear -head . 4 This paper deals with the ultimate behaviour of cruciform as well as closed -box shear -head systems fully embedded in RC flat slabs at interior RC columns through three-dimensional non-linear finite element assessments . After validating the numerical proced ures against three series [6,20, 47] (Figure 2a-f) which contained RC flat slabs with and without shearheads , the results of detailed parametric investigations are presented and discussed . The parameters examined include the type , shape, embedment length , section size and layout of the shear -head system as well as flat slab dependent parameters such as thickness, concrete strength, reinfo rcement ratio , and support siz e. The shearhead configurations investigated in this paper are depicted in Figure 2 g-i: cruciform shear -heads made of welded back -to-back channel or I section (CRH), cruciform shear heads made of two pairs of channels at the support region (CTP) and closed -box shear -heads (CBX). Figure 2 Schematic representation of a) full scale specimens PG1, PG2b, PG5, PG10, PG11, b) half scale specimens PG6, PG7, PG8, PG9, c) double scale specimen PG3 from Guandalini et al. (2009 [47]); d) FSSH series, e) SH series from Chana and Birjandi (1996) [20] ; f) A and B series (Corley and Hawkins, 1968) [6]; Cruciform shear -heads made of g) welded back -to-back channel or I section ( CRH); h) two pairs of channels at the support region ( CTP) shearheads; i) closed -box shear -heads ( CBX) 5 The numerical results obt ained enable a direct assessment of the ultimate behaviour in terms of both strength and deformation characteristics, and also enable an assessment of the failure surface as a function of the shear -head configuration. Based on the findings, coupled with re sults from previous studies, analytical models are proposed for predicting the rotational response as well as the ultimate strength of such slab systems. The proposed models are shown to offer a reliable and practical approach for a wide range of shear -head types and configurations. 2. Numerical Simulations The numerical simulations described in this section were carried out using the non -linear finite element (FE) program ABAQUS [48] in order to obtain detailed insights into the structural response of RC flat slabs provided with cruciform and closed -box she ar-heads at connections to interior RC columns. After discussing the modelling procedures and constitutive parameters for the concrete, reinforcement and structural steel, validations are carried out against 36 flat slab specimens extracted from t hree expe rimental programmes. 2.1 Modelling procedures Three-dimensional (3D) models were constructed using double -symmetry to represent a quarter of an RC flat -slab specimens (Figure 3a,b) . Eight node brick elements with reduced integration (C3D8R) were employed for the concrete slab, steel shear -heads and reaction plates, whilst 3D truss elements (T3D2) were used for the longitudinal reinforcement. The contact between the shear -head arms a nd the concrete body was represented using an exponential decay friction law described later. In all models, the reinforcement was embedded in the concrete body and assigned with full bond conditions. The reaction plates, assigned with elastic steel materi al properties, were connected to the concrete body by considering full interaction. Pinned boundary conditions were assigned to reference points connected through link multi -point constraints to the reaction plates, whilst member displacements were applied in the same manner through tie multi -point constraints. The Newton -Raphson approach was adopted for the numerical integration procedure. The ‘concrete damaged plasticity’ model (CDP) was used to represent the tri -axial behaviour of concrete. For this, the potential yield function is controlled by the effective stress values, the bi -axial behaviour of concrete (f b0/fc=1.16) and K c=2/3 that governs the shape of the deviatoric plane. The CDP model accounts for a non -associated potential plastic f low, in which t he plastic volume expansion is not proportional to the increase in stresses. This is represented in the plastic potential flow function by the dilation angle φ measured in the p -q plane at high confining pressure and surface eccentricity ( ϵ=0.1). The constitutive model requires uniaxial stress (σ) - strain (ε) - damage (di) 6 relationships for the compression and tension behaviour. In this study, a plastic stiffness degradation scheme, typical for coupled damage -plastic concrete constitutive mo dels was adopted. Figure 3 Numerical models a) typical axisymmetric 3D representation, b) side view and mesh; c) uniaxial concrete model, d) steel model Before crushing in compression or cracking in tension, no degradation occurs and the plastic strains are equal to the inelastic strains. Beyond these points, the stiffness reduction enables the development of irreversible plastic strains that are directly proportional to the stress decrease [49]. The damage in compression d c and damage in tension d t parameters are also part of the variable field output of the FE environment. They allow direct interpretation of the stress state within different regions of th e members , illustrated by compression and tension damage patterns, and member stiffness degradation. Established on a continuum approach , in which no physical separation is created in the model mesh, tension damage mesh regions provide an effective way to illustrate the kinematic aspects of crack development. The compressive stress -strain σ -ε relationship illustrated in Figure 3c was defined using the Eurocode 2 [44] recommendation for non -linear structural analysis, whilst the compressive constitutive properties (elastic modulus E c, strain at crushing ε c1 and concrete tensile strength f ct) were assessed from the reported concrete strength f c. A factor of 0.8 was used to convert cube f c,cube to cylinder f c compressive strength where necessary , as typically considered in Eurocode 2 for mean concrete cylinder strengths below 68 MPa [ 44]. In tension, linear -elastic behaviour was considered up to cracking , followed by a bi -linear tension softening law extracted from a non -linear model [50]. The numerical input was converted into inelastic cracking strain ε ct,i using the concept of equivalent crack length in which the crack displacement w i up to w max=160 μm, is divided by the characteristic length of the element l ch. The characteristic length of a 3D element is the cubic root of the volume of the 7 mesh element V e. Both structural and reinforcement steels were modelled using bi-linear elasto - plastic material pr operties with hardening (Figure 3d) . A numerical investigation carried out previously by the authors on hybrid RC flat slabs connected to steel columns was used as a reference for the definition of the governing constitutive and geometrical parameters [30]. These parameters were first calibrated with the models of Specimen PG1 without shearheads reported by Guandalini et al . (2006) [47], Specimens SH -1 and SH -2 reported by Chana and Birjandi (1996) [20], and Specimens AN -1 and AC -1 from Corley and Hawkins (1968) test series [6], and then applied to all other validation cases as discussed in the following section. The parameters varied in the sensitivity analysis were the dilation angle φ, mesh size, and the uniaxial tension material post-peak representation. Detailed results of the sensitivity analysis are only presented in Figure 4 for Specimen PG1, whilst the complete load -displacement curves are given for all m odels of the three test series. In terms of mesh sensitivity, the results were largely independent of the size l m for the model of PG1 (Figure 4a), when l m varies from h/5 to h/10 (h - flat slab thickness ). The use of a fracture dependent post-peak tension behaviour for concrete allowed tension damage localization as a function of the chosen mesh size. The mesh sensitivity for models with low ρ l, provided with coarser mesh (l m=h/5), showed unrealistic rigid behaviour in the post-crack ing regime, primarily due to the delayed cracking as a result of poor propagation of tension damage in the continuum. A value of lm=h/7 was found to be satisfactory in the models described here for RC slabs without shear -heads w hile l m=h/10 was suitable for slabs with shear -heads. In addition to this, the influence of the accounted stress transfer through cracked interfaces was investigated by accounting for three post -cracking strain distributions (Figure 4b). Non -linear represe ntation of the post -peak tension behaviour was chosen as opposed to linear and bi -linear laws, since the latter two produced stiffer numerical response, particularly for members with low ρ l. In CDP, the material dilation angle φ is used to represent the stress increase at high confining pressures in the normal -shear stress p -q plane. In RC flat slabs, concrete in the column region is under bi-axial or tri -axial confinement. Hence, for these cases, parameters such as φ control the numerical predictions. Fi gure 4c illustrate the numerical response with φ varying from 10 ˚ to 55 ˚. As already noted in other studies on RC flat slabs [23,30,51,52 ], the most effective response is given for values around φ = 40 ˚-50˚, with an optimal value of φ=48° employed for the validations described below . 8 Figure 4 Identification of constitutive and numerical parameters for PG1 [47]: a) mesh size, b) tension stiffening reponse, c) dilation angle φ As mentioned above, to mode l the steel -concrete interface behaviour, the contact between the steel profile and the concrete body is modelled using an exponential decay law . In a sensitivity analysis on hybrid RC members provided with shear -heads carried out by authors [30], μ was var ied from low fiction (0.2) to relatively high friction (0.6). For μ = 0.2, the load -strain response show ed softer behaviour in comparison to experimental results , activated by an earlier slip , whilst μ = 0.4 -0.6 these were in good agreement with test strain recordings. Hence a friction coefficent μ = 0.4 was used in all the analysis described in th is paper . 2.2 Validation Studies 2.2.1 Guandalini et al. (2009) The main aim of the tests was to assess the influence of ρ l (0.25 -1.5%) and specimen size on the punching shear strength of RC flat slabs without shear reinforcement or shear heads [47]. Double size and half scale members to the Reference Specimen (PG1) were reported (Figure 2a -c). The plan dimensions of PG1 were 3 m × 3 m, whereas its thickness was h = 250 mm. The double size specimen (PG3) was 6 m × 6 m in plan and 500 mm thick. Half scale specimens (PG6 -PG9) were only 1.5 m × 1.5 m in plan with thickness h = 150 mm. The specimen details and material parameters of the specimens considered herein are summarised in Table 1. For specimens with high ρ l (PG1, PG6, PG7, PG11), failure occurred in punching due to the dislocation of a conical body out of the slab. In two cases, some reinf orcement yielded over the column. None of these specimens reached their plastic regime [47]. On the other hand, specimens with low ρ l (PG2b, PG4, PG5, PG8, PG9, PG10) showed clear ductile behaviour with a visible plastic plateau. Despite the low ρ l, the d ouble scale specimen PG3, failed in punching shear at a lower load than its flexural strength [ 47]. Identical failure modes were obtained through the numerical models adopting the procedures described above. In most cases both the elastic and cracked stiff ness were accurately predicted, and the ultimate strength was well captured (Table 1 and Figure 5). The predicted strength of several members with low ρ l = 0.25% (e.g. PG2b, PG4) was 9 slightly overestimated by about 6% on average primarily due to the late cracking in the numerical model in comparison with the tests. On the other hand, the ultimate strength of member PG5, also provided with low ρ l = 0.33%, was underestimated with an average V test/Vu,num=1.04. Additionally, the cracking load of the doubl e-size Specimen PG3 was underestimated by about 30% leading to a premature development of inelasticity compared to the test. However, there is close agreement in the numerical and test results in terms of the cracked stiffness and the ultimate strength. Sp ecimens with high ρ l (e.g. PG1 and PG6) showed faithful strength predictions with a discrepancy within 3% from the test values . Figure 5 Numerical validation: Guandalini et al (2009) test series [47] a) PG2b, b) PG4, c) PG5, d) PG10, e) PG3, f) PG1, g) PG6, h) PG 11 (continuous curves - V-δ from numerical simulations, whilst dashed black curves depict V -δ from tests) 2.2.2 Chana and Birjandi (1996) test series In this test series, two sets of tests were c arried out for the development of a range of structural shear - heads capable of carrying large loads [20]. The first phase involved 16 scaled specimens with or without shear -heads to assess the influence of the shear -head configuration on the ultimate slab behaviour. Circular flat slabs (SH series ) of 1.55 m diameter , depicted in Figure 2 e, were loaded around their perimeter through eight floor ties and supported by 140 mm square columns, and had a thickness of 120 mm and two levels of flexural reinforcement (ρl=0.79% and ρl=1.51%). The shear - heads were made of structural channel sections welded in a cruciform or closed -box arrangement. The cruciform shear -heads were made of welded back -to-back channel section (CRH) or two pairs of channels around the support region (CTP) . On the other hand, closed -box shear -heads (CBX) were made from two pair s of channels which had end beams welded at their outer ends. 10 Figure 6 Numerical validations: Chana & Birjandi (1996) test series [20] a) SH1, b) SH2, c) SH3, d) SH5, e) SH7, f) SH8, g) SH0, h) SH11, i) FSSH1, j) FSSH2, k) FSSH3, l) FSSH4; (continuous black curves represent the V -δ from numerical simulations, whilst dashed b lack curves depict V -δ from tests) Figure 7 Tension damage patterns for slabs with and without shear -heads, and stresses in shear -heads for Specimens: a) SH2, b) SH3, c) SH5, d) SH1; e) crack pattern from tests (Chana & Birjandi , 1996) [20] 11 The results from the first set of tests enabled the choice of an optimal shear -head configuration, employed further in the second phase which involved tests on 3.0 m square slabs (FSSH series) of 250 mm depth supported by 300 mm wide columns . As illustrated in Figure 2 f, these were loaded in a similar manner through equally spaced points at 1.2 m radius from the centre of the specimen. All specimens were made of normal concrete with f c,cube=38.1 -46.6 MPa . The shear -head dimensions (as indicated in Table 1) , were C76x38x7 mm for the SH slabs, C127x64x15 mm for the large -scale slabs with cruciform shear -heads (FSSH2 and FSSH3) and C102x51x10 for the FSSH4 with closed -box shear -head. The shear -head yield strength was f v=355 MPa and the rei nforcement yield strength fy=500 MPa were considered for modelling, as suggested in the test report. From a total of 20 tests, 13 were reported to develop punching shear failures. The scaled tests with CBX shear -heads (SH5 and SH11) showed the high est punching shear capacity characterised by failures outside the shear -head area, followed by the CTP shear -heads (SH2 and SH8). Specimens with shear -head arms placed further than a distance of column size plus half the slab effective depth had columns pu nching through the slab through the centre of the shear -head at lower strengths. The test results showed that shear -heads that produce a larger punching shear perimeter exhibit a larger punching shear capacity with failure surfaces outside or near the edge s of the shear -heads. Figure 6 and Table 1 depict the predicted response against the test results as obtained from the test report. The test-to-predicted strength ratio obtained is Vtest/Vnum=1.05 and the coefficient of varia tion is 6.9% for the model SH slabs, and Vtest/Vnum=1.02 and COV of 7.7% for the large scale FSSH slabs . Both the stiffness and ultimate capacity of the specimens with shear -heads are well predicted (ρ=0.79% for SH2, SH3, SH5 ; ρ=1.51% for SH8, SH9, SH11 ; ρ=1.00% for FSSH2, FSSH3 and FSSH4) , offering confidence in the numerical procedures employed. For members without shear - heads (SH1, SH7 and FSSH1) , although the cracked stiffness is stiffer compared to the tests, Vtest/Vnum=1.01 and the failure mode is correctly pred icted. The tension damage scalar maps are depicted in Figure 7a-d for model tests with shear -heads and ρ=0.79% (SH2, SH3, SH5) and for the corresponding slab without shear -heads (SH1) that failed in punching , along with an image with a typical top crack pattern from the large tests (Figure 7e). These show that the top crack patterns are governed by flexural cracking with cracks extending to the slab edge whilst , regardless the shear -head type, the punching shear crack ini tiates from the bottom tip of the shear -head arm and extends through the flexural reinforcement to the tension face of the slab. This suggests a force transfer through the shear -head bottom -flange supported struts, similar to the case of Specimen SH1 without shear -head in Figure 7d in which the strut is supported at the column - slab connection. 12 In terms of shear -head behaviour, Figure 7 shows that the cruciform shear -heads (SH2 and SH3) developed some yielding at the top flang e, whilst the closed -box she ar-head (SH5) was well below the yield stress. These stress levels are combined with stronger tension damage for SH5 as evident in the 3D representations around the shear -head region. This indicates higher reinforcement stresses above the shear -head that are correlated to a shift in the weak bending section at the shear -head tip. Similarly, for slabs with cruciform shear -heads (SH2 and SH3), the highest flexural damage is outside of the shear -heads indicating an influence from th e shear -head on moment distribution , whilst for SH1 it occurs at the symmetry line following the weak bending axis. These observations indicate that the weak section, both in terms of flexure and punching shear, is translated away from the column face, outside of the shear -head region. 2.2.3 Corley and Hawkins (1968) test series In this test series, twenty -one 2.1×2.1 m square 146 mm thick flat slab specimens with reinforcement ratios between 1.8 -3.3% and 45° cut cruciform shear -heads , supported on square columns with sides of 254 mm or 204 mm , were loaded symmetrically at the edges of the slab [6]. Cruciform shear -heads made either from Standard American I-sections (I 3" ×7.5 or I 3" ×5.7 equivalent of I 76.2 mm × 11.2 kg/m or I 76.2 mm × 8.48 kg/m, respectively ) or two pairs of channels (2 × C 3" 7.1 or 2 × C 3" 4.1 equivalent of 2 × C 76.2 mm 10.6 kg/m or 2 × C 76.2 mm 6.10 kg/m, respectively ) running above the columns , were fully embedded in normal or lightweight concrete flat slab. The reported concrete strength of the slabs varied between f c=18.1 -22.8 MPa, whilst the flexural reinforcement had f ys=403 - 444 MPa (Table 1) . The shear -head yield strength was assumed as fv=300 MPa as reported by the same authors in a companion study on edge connections [7]. Reference specimens (AN-1 and BN -1) without shear -heads failed in punching with the failure surface extending from the column -slab compression face intersection to the tension face of the slab at inclinations between θ=20° -30°. For slabs with relatively stiff shear -head s, the failure surface generally followed the perimeter of the shear -head with θ=20° -45°, whilst for members with relatively flexible shear -head s, the failure surface developed inside the shear -head with inclinations of about θ=30°. The results indicated two distinct responses, depending on the stress state of the shear -head . For elastic shear -head behaviour, the failure surface initiate d from its tip . For cases where yielding occurred, a more flexible response was observed with the failure surface crossing the shear - head within 25% of its length from the edge. The V -δ response obtained from the numerical simulations for the specimens with or without shear -heads are illustrated in Figure 8, whilst the main parameters are given in Table 1 . Good agreement was obtained between the predictions and test strengths with a Vtest/Vnum=1.05 and coefficient of variation COV of 6.3%. 13 Figure 8 Numerical validations for Corley and Hawkins (1968) test series [6] a) AN -1, b) AC -1, c) AC-2, d) AC -3, e) AH -1, f) AH -2, g) AH -3, h) BN -1, i) BC -1, j) BH -1, k) BH -2, l) BH -3 (continuous black curves represent the V -δ from numerical simulations, whilst da shed black curves depict V -δ from tests) Although a coarser mesh (l m=h/7) was used for these models to reduce the computational time in comparison to SH and FSSH specimens (with l m=h/10), both the predicted strengths and tension damage patterns offer a cle ar insight into the governing ultimate behaviour. The V-δ plots show mostly sudden drops in capacity after reaching ultimate, whilst the crack patterns at the quarter slab symmetry line in Figure 9 clearly depicts inclined damage tension fields, outside of the shear -head tip, suggesting punching shear failures. Along its section size, the embedment length of the shear - head plays a significant role in the ultimate slab response as it determines the location of the failure surface. Figure 9 Influence of the embedment length on the tension damage and compression fields 14 Table 1 Comparison between ultimate strengths obtained from tests and numerical simulations Specimen Shearhead Section Shearhead Type Shearhead length l v (mm) Column size bc (mm) Slab radius rs (mm) Slab thickness h (mm) Effective depth d (mm) Shear depth d0 (mm) Reinforcement ratio ρ l (%) Rebar yield strength f ys (MPa) Concrete strength f c (MPa) Test strength Vtest (kN) Prediction Vnum (kN) Vtest/ Vnum (-) PG3 N/A - - 520 2850 500 456 456 0.33 520 32 2153 2218 0.97 PG1 N/A - - 260 1500 250 210 210 1.50 573 28 1023 991 1.03 PG11 N/A - - 260 1500 250 210 210 0.75 570 32 763 775 0.98 PG2b N/A - - 260 1500 250 210 210 0.25 552 41 440 458 0.96 PG4 N/A - - 260 1500 250 210 210 0.25 541 32 408 448 0.91 PG5 N/A - - 260 1500 255 210 210 0.33 555 29 550 528 1.04 PG10 N/A - - 260 1500 255 210 210 0.33 577 29 540 533 1.01 PG6 N/A - - 130 752 125 96 96 1.50 526 35 238 230 1.04 PG7 N/A - - 130 752 125 100 100 0.75 550 35 241 220 1.09 PG8 N/A - - 130 752 140 117 117 0.33 525 35 140 149 0.94 PG9 N/A - - 130 752 140 117 117 0.25 525 35 115 125 0.92 AN-1 N/A - - 254 914 146 111 111 1.50 403 19 334 314 1.06 AC-1 2 × C 3" 7.1 CTP 330 254 914 146 111 82 1.50 404 18 544 530 1.03 AC-2 2 × C 3" 7.1 CTP 483 254 914 146 111 82 1.50 413 18 479 465 1.03 AC-3 2 × C 3" 4.1 CTP 406 254 914 146 111 84 1.50 407 21 458 432 1.06 AH-1 I 3"×7.5 CRH 381 254 914 146 111 85 1.30 438 23 426 408 1.04 AH-2 I 3"×5.7 CRH 381 254 914 146 111 85 1.30 436 22 443 425 1.04 AH-3 I 3"×5.7 CRH 178 254 914 146 111 85 1.30 440 22 406 414 0.98 BN-1 N/A - - 203 914 146 111 111 2.20 444 20 266 273 0.97 BC-1 2 × C 3" 7.1 CTP 254 203 914 146 111 82 0.80 420 20 319 342 0.93 BH-1 I 3"×7.5 CRH 432 203 914 146 111 85 1.00 437 20 394 337 1.17 BH-2 I 3"×5.7 CRH 127 203 914 146 111 85 1.00 423 18 301 276 1.09 BH-3 I 3"×5.7 CRH 356 203 914 146 111 85 1.00 439 22 402 355 1.13 SH1 N/A - - 140 563 120 100 100 0.80 500 37 256 226 1.13 SH2 C 76×38×7 CTP 300 140 563 120 100 71 0.80 500 35 405 388 1.04 SH3 C 76×38×7 CRH 275 140 563 120 100 71 0.80 500 36 420 421 1.00 SH5 C 76×38×7 CBX 200 140 563 120 100 71 0.80 500 34 437 406 1.08 SH16 C 76×38×7 CTP 300 140 563 120 100 71 0.80 500 31 376 411 0.92 SH7 N/A - - 140 563 120 98 98 1.50 500 36 312 339 0.92 SH8 C 76×38×7 CTP 300 140 563 120 98 69 1.50 500 32 542 505 1.07 SH9 C 76×38×7 CRH 275 140 563 120 98 69 1.50 500 35 592 516 1.15 SH11 C 76×38×7 CBX 200 140 563 120 98 69 1.50 500 31 580 564 1.03 FSSH1 N/A - - 300 1200 250 205 205 1.00 500 34 976 1008 0.97 FSSH2 C127×64×15 CRH 615 300 1200 250 205 134 1.00 500 31 1484 1515 0.98 FSSH3 C127×64×15 CTP 615 300 1200 250 205 134 1.00 500 33 1745 1533 1.14 FSSH4 C102×51×10 CBX 410 300 1200 250 205 123 1.00 500 34 1611 1611 1.00 Average 1.02/ COV 0.07 Notes: the geometrical characteristics of the steel profiles depicted in the table are: hv×bv×tw×tf a) I 3"×5.7 - 76.2×59.2×4.32×6.60 mm, b) I 3"×7.5 - 76.2×63.7×8.86× 6.60 mm , c) C 3" 4.1 - 76.2×35.8×4.32×6.93 mm, d) C 3" 7.1 - 76.2×49.2×7.92×8.91 mm , e) C 76×38×7 - 76.2×38.1×5.1×6.8 mm , f) C102×51×10 - 101.6×50.8×6.1×7.6 mm , g) C127×64×15 - 127×63.5×6.4×9.2 mm 15 Overall, the deformational response and failure modes obtained from the numerical simulations were in good agreement with reported tests results. The top face and cross -section damage patterns obtained from the analysis resemble the top crack patterns and punching shear failures obtained from tests. Having gained confidence in the numerical procedures employed in this investigation through the validations of the test results from 36 specimens of which 20 were provided with a wide range of shear -heads, the f ollowing section describes parametric assessments which were carried out in order to provide more detailed insights into the behaviour of RC slabs with shear -heads. 3. Parametric Assessments The tests modelled in Section 2.2 focused on the response of relat ively thin flat slabs with shear - heads, having an effective depth varying in the range d=98 -111 mm (h=120 -146 mm). From the 2 0 specimens with shear -heads simulated , only 3 had practical slab effective depths with d=205 mm (h=250 mm). Also , these slabs were provided with shear -head length -to-depth ratios of lv/hv=1.33 - 6.33, reinforcement ratios of ρ l=0.5-1.5% and concrete strengths f c=16.7 -36.2 MPa. Although these experimental studies may appear to cover a wide range of lv/hv, ρl, fc, slab spans and depths , only FSSH specimens (e.g. d=205 mm) can be considered within practical ranges. Existing assessment models are similarly based on a limited number of test ranges, and their reliability is therefore limited to these ranges. Parametric investigations covering parameters outside of the available test range s are therefore required to provide improved assessment models with wider applicability . A total of 122 three-dimensional models , including 44 CTP, 44 CRH and 34 CBX shear -heads, were constructed using the numerical procedures described in Section 2.1, in which RC flat slabs were connected to RC columns by means of full -embedded shear -heads. The parametric investigations were undertaken to provide detailed insights into the behaviour of a wide range of configurations outside of the existing test database, hence enabling a direct assessment in terms of str ength and deformation characteristics , and to provide offer a wider dataset for developing practical analytical models. The non-linear parametric assessments may be grouped into five main studies in which the following key parameters were investigated: - Shear-head embedment length -to-depth ratio (lv/hv=0.5-5.0): in each case, the reinforcement ratio ρ l was varied from 0.3 1% to 1.54%. All other parameters were kept constant including the effective depth d, slab radius r s, concrete strength f c and shear -head s ection size ( hv×bv×tw×tf = 100×100×6×10 mm, for CRH shear -heads ; total h v×bv =100×100 mm from two parallel flange channel sections (C) 100×50×10 mm for CTP ; total h v×bv =100× 200 mm from four C 100×50×10 mm for CBX ) 16 - Slab radius (rs/d=5.4 -11.8): in conjunction with ρ l=0.65%-1.05%, and a constant l v, d, f c, rs/d, and h v×bv - Slab bending effective depth (d=1 70-370 mm): in conjunction with ρ l=0.42%-1.18% with constant l v/hv, fc, rs, and h v×bv - Shear -head section size with varying h v×bv for lv/hv=1.0 and 3.0, and constant f c, rs, d o hv×bv×tw×tf = 100×100 ×6×10 mm, 160×160 ×8×13 mm, 200×200 ×9×15 mm, referred to as CRH 100, CRH160, CRH200, resp ectively ; o hv×bv = 100×100, 150×150, 200×150 mm from two C 100×50×10, C 150×75×18, PFC 200×75×23, respectively for CTP; o hv×bv = 100×200, 150×350, 200×350 mm from four C 100×50×10, C 150×75×18, PFC 200×75×23, respectively for CBX - Concrete strength fc = 30 and 50 MPa and ρ l = 0.63%-0.98%, with constant l v/hv, d, r s and shear - head section size Selected results from the parametric assessments are plotted in Figures 10 to 12 in terms of rotational response (V -ψ), in which the load V is normalised against the control perimeter b 0 and concrete strength f c, as described in Equation 1 and 2 and illustrated in Figure 13. For CRH , b0 is defined for each shear -head by an arc -length with a radius equal to the in -plane half strut projection d 0/2 plus connecting lines (Figure 13b). As illustrated in Figure 13c, the shape of the critical perimeter for CTP shear -heads follows the same configuration, considering the distance between t he steel profiles. It is worth noting, that Equation s 1a-c are limited to shear -head lengths for which t he control perimeter does not lie within d 0/2 from the column face . Hence, fo r short shear -heads, b 0 should be constructed by accounting for the column , noting that b0 can be evaluated using Equations 1a ,b for straight -cut CRH and CTP shear -heads, respectively. For CBX shear -heads, the critical perimeter is located at d 0/2 from the edge of the bottom flange, throughout the length of all shear -head sides considering rounded corners (Figure 13d). The length of a CBX critical perimeter can be assessed using Equation (1c). It is worth noting that the cut type of the shear -head influences the location of the geometry of the force transfer mechanism. Hence , for shear -heads that have a 45 ° cut [6] (Figure 9), the critical perimeter reduces with due account for the location of the force transferring strut that tends to form closer to the root of the shear -head , in comparison to the case of straight -cut shear -heads. In a simplified manner , lv should be reduced with hv/2 in Equations 1a,b. Besides the obtained V -ψ, Figures 10 to 12 indicate the failure criteria (FC) for punching shear in RC flat slabs (Equation 3 [53]) that account s for a shear -head dependent support of the governing strut, and provide key observations on the member behaviour. These include envelopes of the loads 17 corresponding to yielding in the steel reinforcing materials, namely: longitudinal reinforcement (RY), flange of shear -head (FY), web of shear -head (WY) and the ultimat e envelope (U). In the case of RY, the points on the graph correspond to the initiation of yielding in one truss element in the analysis. For FY and WY, they correspond to the case in which yielding spreads in a band (i.e. at least two mesh elements for FY , typically at the edges of the flanges; or a band of mesh elements in which WY occurs from the top to bottom flange). 00 0,CTP v vd 2 db 4 l sin 2 ( 2b a )2 8 8 = + + + +  for CTP (1a) c 0 0 0,CRH vb d 2 db 4 l sin 22 2 8 8 = + + +  for CRH (1b) ()0 0,CBX v vdb 4 2l 2b a4 = + + + for CBX (1c) 0 vfb fd d d t / 2= − − (2) () 00 0.75 / 1 15 /gg k d d d = +   + (3) 3.1 Shear -head embedment length Figures 10a-e illustrate the rotational response V -ψ from selected CTP, CRH and CBX numerical models in which the embedment length l v was varied against the flexural reinforcement ratio ρ l. For low to intermediate reinforcement ratios (ρ l = 0.3 -0.9%), regardless of the embedment length and shear -head type, yielding of the longitudinal reinforcement bars (RY) triggers the failure , with a more pronounced effect for low ρ l combined with low l v/hv. For shear -head length ratios l v/hv≤2.0 (lv/rs≤0.15) the shear -head remains largely elastic . As l v increases, inelastic strains develop at the shear -head flange edges (FY). For l v/hv≥3.0 (l v/r≥0.22 ) and ρ l=0.9%, (FY) is triggered immediately after (RY). For intermediate CRH shear -head lengths (l v/hv=2.0-3.0; lv/rs=0.15 -0.22) some yielding at the bottom flange due to strut support was observed. As depicted in Figures 10f,i, for relatively long CRH and CBX shear -heads (l v/hv=5.0; l v/rs=0.36) and relatively high ρ l = 0.9 -1.5%), flange yielding (FY) is accompanied by web yielding (WY), resulting in full yielding of the shear -head cross -section. As illustrated in Figures 11 compare d to Figures 12, an increase in slab radius from r s/d=8.18 (rs=1390 mm) to r s/d=11.76 (rs=2000 mm), for constant l v/hv= 1.0 and 3.0 mm (l v/rs=0.72 and 0.22, and lv/rs=0.05 and 0.15, respectively ), leads to an increase in rotations ψ at ultimate, which indicates a more flexible slab behaviour. In the latter case, even if relatively high ρ l>1.1% is used, for CTP shear - 18 heads with l v/hv= 3.0, full yielding of the steel profiles occurred, due to the higher bending moment carried by the slab strips containing the shear -heads. A direct comparison between the three shear -head types investigated indicate that when the same l v/hv is considered, the flat slab behaviour in terms of kinematics and yielding sequence is virtually identical. The geometry of CBX shear -heads involve s an increased number of steel profiles in comparison to CTP and CRH, hence implicitly increasing the local stiffness of the slab. The above observations indicate that for relatively short l v/hv, the behaviour of a flat slab with shear -heads tends towards conventional RC behaviour, whilst for relatively long shear -heads this tends towards steel - concrete composit e response. A relatively long shear -head benefits from an increased steel -concrete contact area, leading to an enhanced composite behaviour in comparison to case s of relatively short shear -heads. In addition to the numerical damage maps indicating inclined failure surfaces, the ultimate envelopes (U) in proximity to the punching shear failure criterion (FC) show that punching occurred at ultimate for cases with intermediate to high ρ l although shear -head flange yielding (FY) was also recorded. 3.2 Shear -head cross -section and slab thickness The influence of the ratio between the shear -head cross -section, shear -head type (CTP, CRH, CBX) and slab thickness was examined in three parametric studies. Initially, the slab effective depth d was varied from 170 to 370 mm for two embedment length ratios l v/hv = 1.0 and l v/hv = 3.0 . Each of the two ratios was also used in conjunction with three reinforcement ratios ρ l between 0.4 and 1.1 %. In addition, the shear -head cross -section was varied from C 100×50×10 to 200×7 5×23 mm for CTP and CBX and 100×100×6×10 to 200×200×9×15 mm for CRH shear -heads. For compactness, only selected results of these variations are depicted in Figure 12a-f which need to be considered in conjunction with the results given in Figure 10. Similar to other observations made before, members with low ρ l=0.4-0.5% developed inelastic strains in the longitudinal bars (RY). Members with short shear -heads (l v/hv=1.0; l v/rs=0.08) exhibited elastic shear -head behaviour with failure triggered by crush ing at the shear -head tip. For longer CTP and CRH shear -heads (l v/hv=3.0; l v/rs=0.22) and relatively thin slabs (d=170 mm), the behaviour was governed by punching at ultimate with inelastic strains occurring in the flexural reinforcement (RY) or shear -head flanges (FY), depending on the slab configuration (Figures 10b,e). As the slab thickness increases (d=270 –370 mm) some differences occur in the response of CBX shear -heads in comparison to cases with d=170 mm (Figures 12e,f and 10h). For these cases, the ultimate envelopes (U) show trends below (FC), which identify failure mechanisms different than those described above. 19 Figure 10 Influence of the embedment length and reinforcement ratio on the slab capacity and rotation a) CTP, d=170 mm, h v=100 mm, l v/hv=1.0, f c=30 MPa; b) CTP, d=170 mm, h v=100 mm, lv/hv=3.0, f c=30 MPa; c) CTP, d=170 mm, h v=100 mm, l v/hv=5.0, f c=30 MPa; d) CRH, d =170 mm, hv=100 mm, l v/hv=1.0, f c=30 MPa; e) CRH, d=170 mm, h v=100 mm, l v/hv=3.0, f c=30 MPa; f) CRH, d=170 mm, h v=100 mm, l v/hv=5.0, f c=30 MPa; g) CBX, d=170 mm, h v=100 mm, l v/hv=1.0, f c=30 MPa; h) CBX, d=170 mm, h v=100 mm, l v/hv=2.0, f c=30 MPa; h) CBX, d= 170 mm, h v=100 mm, lv/hv=5.0, f c=30 MPa; For these cases with ρ l up to 1.4%, the failure was triggered by a more rapid propagation of yield in the steel profiles than in the steel rebars, leading to a flexurally -governed behaviour by the shear - head. Similarly, CTP members with (d=270 -370 mm), having shear -heads made of parallel flange channels as CBX, indicated a response governed by complete yielding of the steel profiles in which the yielding propagated rapidly from the flange to the web . 20 Figure 11 Influence of the slab radius on the slab capacity and rotations a) CTP, d=270 mm, hv=150 mm, ρ l=0.7% f c=30 MPa; b) CTP, d=370 mm, h v=200 mm, ρ l=0.5% fc=30 MPa; c) CRH, d=290 mm, h v=160 mm, ρ l=0.7%, f c=30 MPa; d) CRH, d=370 mm, h v=200 mm, ρ l=0.5%, f c=30 MPa; a) CBX, d=270 mm, h v=150 mm, ρ l=0.9% f c=30 MPa; b) CBX, d=370 mm, h v=200 mm, ρl=0.7% f c=30 MPa However, for these specimens, the failure envelope (U) was in the vicinity or above the failure criterion (FC) indicating a flexurally -governed punching shear failure triggered by the shear -head flange and web yielding (Figures 10b and 12). It was previously shown that for relatively short shear - heads with relatively thin webs, in hybrid flat slabs connected to ste el columns, failure may be triggered by web yielding (WY) . Such web yielding was produced by a sudden slip of the shear -head from the embedding concrete which led to a premature punching shear failure [30]. Although , for the cases illustrated in Figure 12, web yielding occurred, this was a consequence of progressive yielding of the steel profile from the top flange through the web and into the bottom flange. 3.3 Concrete strength The influence of the concrete properti es on the response of flat slabs with shear -heads was examined by modifying the compressive strength from f c=30 MPa to 50 MPa for all CTP, CRH and CBX models with l v/hv=1.0-3.0; ρ l=0.6-0.9%, d=170 -370 mm, whilst the slab geometry was kept constant. For com pactness, only brief discussions are presented here, but the V u and Ψu obtained for all the cases considered are used for the subsequent analytical assessments. 21 Figure 12 Influence of the slab depth on the slab capacity and rotation a) CTP, d=270 mm, l v/hv=3.0, hv=150 mm, f c=30 MPa; b) CTP, d=370 mm, l v/hv=3.0, h v=200 mm, f c=30 MPa; c) CRH, d=290 mm, lv/hv=3.0, h v=160 mm, f c=30 MPa; d) CRH, d=370 mm, l v/hv=3.0, h v=200 mm, fc=30 MPa; e) CBX, d=270 mm, l v/hv=3.0, h v=150 mm, f c=30 MPa; f) CBX, d=370 mm, l v/hv=3.0, h v=200 mm, f c=30 MPa Close inspection on the results indicated that for CTP shear -heads, an increase of f c from 30 to 50 MPa led to an increase in Vu in the range of 2-11%. The slab rotation Ψu was between 4 -26% larger for shear -heads with l v/hv=1.0, whilst this decreased by up to 8% for slabs with l v/hv=3.0. On the other hand, for the models with CRH shear -heads and f c=50 MPa, V u was 5 -15% higher than the case with lower f c, whilst Ψu was mostly the same with an Ψu,C50/ Ψu,C30=1.00. For CBX cases , an increase of 3.3% in V u combined with a reduction in Ψu of up to 20% was obtained, particularly for cases with lv/hv=3.0 for which the complete shear -head cross -section yielded. The results of the parametric studies identified three modes of failure as a function of the interaction between the shear -head and surrounding concrete: steel reinforcement flexural failure due to yielding of the rebars ( A); flexural failure governed by the complete shear -head yielding (B) , and punching shear due to crushing with or without yielding of the reinforcement or top flange of the shear -head – referred to here as a controlled failure (C). These observations, with the complementary key findings from the numerical investigations described in Sections 2 and 3, enable the validation of analytical models for a wide range of RC slab systems with shear -heads as described below . 22 4. Analytical Models and Design Consideration s 4.1 Punching shear strength An axisymmetric rotational model , which considers that relatively stiff shear -heads in conjunction with reinforcement transfer the load from steel columns to the RC flat slab through cruciform shear - heads, was proposed previously for hybrid systems by authors [12] . The model can be directly empl oyed for assessing the complete V-ψ response for the configurations investigated in this paper. Since the slab flexibility is the governing factor for the assessment of flat slab punching shear strength [53], the V-ψ prediction must have a significant degree of reliability. Hence, a focal point in the parametric investigations described in Section 3 was the prediction of V-ψ for a wide range of configurations. Based on the numerical results, a bilinear model, derived f rom the axisymmetric model described above , and compliant with current design procedures [45], is proposed herein . A factor λψ in Equation ( 4) that accounts for slab flexibility through the reinforcement ratio ρ l and shear -head embedment length l v was derived from regression to numerical results and is validated against the results from the parametric assessments presented in Section 3. It is worth noting that for connections between steel columns and hybrid RC flat slabs by means of shear -heads , the contribution of r s/lv to the slab flexibility parameter λ ψ is significantly higher in the case of RC columns to RC flat slabs in the form in vestigated in this paper, as the force is carried through shear -heads and flexural reinforcement only, without the contribution of the column support. The rotation ψ of the RC member with shear -heads, estimated using Equation (4), depends on the section utilisation factor V i/Vflex, in which V flex is the flexural strength (Equation 5) of the slab and V i is the shear action. Other parameters include the slab radius r s, yield strength f ys and elastic modulus E s of the longitudinal reinforcement, the slab effective depth d, and the slab flexibility factor λψ,. The flexural strength V flex is a function of η which accounts for the in -plane distribution of the shear -heads within the slab , the plastic moments of the composite sectors m Rk that include the s hear-heads (Equation 6a,b) and the concrete sectors m Rc (Equation 6c), as well as the slab configuration (l v, rs, re, rc). The plastic moments per unit width in Equation 6 may be determined from assumptions of linear strain compatibility in the cross - section, in which yielding occurs first in the tension reinforcement. 3/2 ys si s flexfrV d E V=  where ()1 1 20 321003s l vr l=  (4) 2v v e flex Rk Rc s s s cl l rV m mr r r r   = + − −   where ()18sin 0.5 /vcbr −= (5) 23 ()()() / 2 /2k Rk ys s vij vij k vij k kcm f A d A d c d c d c = − + − − −  (6a) ,0 ,0 ,0 ,0 ,0 ,0vft k vfb k vw k ys s vft vw vfb k k k k ccd c d c dcf A A A Ad c d c d c cfb −− −+ + + − − −= (6b) 2c Rc ys scm f A d=− where ys s c ccfAcfb= and ( ) : 0 0; 0xx =  = (6c) Figures 10-12 illustrate the V-ψ predictions using Equations (4 -6) for the selected numerical models plotted against the results from the simulations. Considering the limitations of a bilinear approach, the predicted results are in good agreement with the stiffness of RC flat slabs obtained from simulations, particularly for the CTP and CRH cases. In the case of CBX flat slabs, the results show slight flexibility compared to that obtained from numerical simulations, mostly due to the more robust form of CBX shear -heads . However, a lower predicted stiffness in conjunction with the failure criterion (Equation 3 [53]) results in a conservative prediction of the punching shear strength Vu. The predicted V flex, attained in members with relatively low reinforcement ratios ρl = 0.3 -0.6 %, is in good agreement with all numerical result s. The analytical results show consistency with the numerical results since the full flexural capacity is reached only for models with very low levels of ρl. In terms of code provisions, the North American guidel ines ACI318 -14 [46] are largely based on the design procedure proposed by Corley and Hawkins (1968) [6], following the test results on members with cruciform shear -heads (Figure 8 and 9 ). In this model , Vu, estimated with Equation (7), is a function of f c, d, a limiting factor (0.33) and a control perimeter b0, which considers that t he critical slab section s for punching shear intersect each shear -head at three -quarters the distance 0.75l v from the column fac e, but not closer than d/2 from the column face (in this investigation , lv is the distance from the column face). 0 0.33ccV f b d= (7) No specific provisions are given in Eurocode 2 (EC2) [44] to assess the punching shear strength of members with shear -heads. The value of Vu for conventional RC flat slabs without shear reinforcement and without pre -stressing is dependent on the size effect [1+(200/d)1/2<2.0], ρ l, fc, d and b 0 situated at 2d fro m the column face. As observed in Figure 9 and in previous tests [12,42] , a relatively stiff shear -head shifts the critical zone outside the shear -head region. Hence, the strength assessment of flat slabs with shear -heads can be comparable to the verification for failure outside of the shear -reinforced region for members provided with transverse bars. The control perimeter 24 accounts for a rounded control section situated at k ×d from the shear -head tip (k = 1.5) extended in both sides of the shear -head by a distance of 1.0d . ()()1/3 0 0.18 1 200 / 100c l cV d f b d  =+ (8) On the other hand, the Model Code 2010 [45] offers recommendations for the design of flat slabs with CBX shear -heads by considering it as a rigid support region . It also accounts for the shear -head penetration to the shear effective depth d0 considered in the Vu assessments (Equation 9) . The parameters involved in strength assessments are the k ψ factor (Eq uation 10a), representing a conservative function of the failure criterion in Eq uation (3) [53]. This is dependent on ψ, d and a k dg parameter as a function of the maximum aggregate siz e dg. For Level II Approximation (LoA II) , the rotation of a RC slab at a specific shear force is given by Eq uation (10b) (where m S is the design bending moment, m R is the plastic moment of a RC cross -section and r s=0.22L is a function of the member moment span L). The critical section is at d 0/2 from the strut support and considers rounded corners. 00 ccV k f b d= (9) ( ) 1/ 1.5 0.9 0.6dg k d k  = +     where () 32 / 16 0.75dg gkd= +  (10a) 1.5 1.5ys ss sRfrm d E m=  (10b) Figure 13 a) Strut transfer scheme for slabs with shear -heads; control perimeter for: b) slabs with cruciform H/I shear -heads (CRH), c) cruciform with two parallel channels (CTP), d) closed box shear -heads (CBX) Figure 14 summarises the ratio between the test or numerical strength and predicted strength (Vnum/test /Vcalc) versus the ratio between the test or numerical result and the flexural strength, from the application of the existing codified guidelines (Equations 7 -10) and Equations ( 1-6) proposed in this paper. Values of V num/test /Vcalc above unity depict conservative predictions, whereas those below unity represent unconservative estimates . As indicated in Figure 14a, the ACI318 strength assessments give an average of 1. 12 (with a COV of 2 1%) between the punching shear strength obtained from numerical models or tests and Vu predicted. The influence of the slab flexibility is captured reasonably 25 well, with a slight tendency of under -estimation of the capacity for flexible slabs with relatively low reinforcement ratio and Vnum/test /Vflex ≈1.0. The modified assumptions regarding the definition of b0, as described previously for E C2, show conservativ e predictions with an average of 1.52 and a COV of 0.25, with a tendency for offering conservative estimates for most configurations (Figure 14b). It is worth noting that for the assessments using the MC2010 guidelines, ψ and m R were assessed without consi dering the influence of the shear -head to slab rotation and plastic moment, as prescribed by the code. Consequently , as depicted in Figure 14c, LoA II of MC 2010 shows large scatter primarily because a key parameter is a more flexible ψ , hence leading implicitly to overly - conservative estimate s with an average of 1.30 and a COV of 0.28. Figure 14 Strength predictions for existing models: a) ACI318, b) Eurocode 2, c) Model Code 2010; proposed models d) bilinear, e) modified Model Code 2010 In contrast, the assessment models proposed in this paper , in which the punching shear strength is the result of the intersection between the V -ψ response, from a bi-linear representation (Equation 4) and the failure criterion (Equation 3) [53] and accounts for a shear -head dependent control perimeter, offer generally improved and consistent strength estimates. The ratios between the predicted strength s and the test or numerical strengths indicate an average of 1. 06 and a COV of 0.13 (Figure 14d). This indicates that the suggested assessment approach can predict accurately the ultimate strength, for both low and high reinforcement ratios represented by the wide range of V num/test /Vflex. Additionally, an appraisal of the MC2010 design approach by considering the influence of the shea r-head on ψ and mR (Equations 4 and 11) , whilst the slab radius is a function of the position of the supports, shows 26 improved predictions in comparison to the unmodified conventional approach with an average of 1.23 and COV of 0.21 (Figure 14e). R,avg R,c R,c R,km (1 / 2)m ( m m ) / 4 = − + + (11) For these assessments, t he section utilisation factor mS/mR can be estimated using an average plastic moment ( Equation 11) for the case when no eccentricity is acting on the member , by replacing the (Vi/Vflex) in Equation 4 . The results above indicate that the proposed model offers a simple and practical method for design purposes, whilst considering a more realistic approach compared to existing code procedures. 4.2 Shear -head properties The embedment length l v of a shear -head may be determined using Equations ( 12) based on the assumption that the critical section is situated at d0/2 from the shear -head tip . This results from the length of the control perimeter b 0,req as a function of the shear action V i in whic h kψ is assessed using Equation ( 3). Equation (10a) may also be used for assessing kψ, yet this result s in a longer embedment length , due to the more conservative nature of the expression . Figure 15a illustrates the relationship between the ratio of b0,req, resulting from Equation (12a) , and b 0 using the layout indicated in Figure 13b-d, against the l v/rs ratio, for all numerical models and test s failing in punching shear. As the average b 0,req/b0 is 1.06 , this indicates that the assumption regarding t he shape of b0, along with the suggested approach for assessing b0,req, is reliable and may be used to determine the required embedment length l v,req. (12a) 0 v,CTP,req 0,CTP,req 0 vdl b 4 d sin ( 2b a ) 4 288  = − + + +    for CTP (12b) c0 v ,CRH ,req 0,CRH ,req 0b 2 dl b 4 d sin 4 22 8 8  = − + +     for CRH (12c) 0 v,CBX ,req 0,CBX vdl b 4 2b a 84 = − + +  for CBX (12d) The results of the numerical simulations also showed that short shear -heads, in the range of l v/hv ≤ 1.0, are unlikely to be able to support the force -transferring struts, which may lead to compression yielding of the bottom flange and potential slip. As observed in Figure 10, as the shear -head length increases composite action may develop and yielding of the shear -head may occur (e.g. l v/hv>3.0). Depending on ρl, one of the steel tension members may yield with punching shear eventually governing. For relatively long shear -heads (l v/hv=5.0), the yielding of the flange shear -head (FY) is () 0, 0 /req i cb V k f d 27 close to the ult imate envelopes (U), yet relatively distant from the failure criterion, indicating a failure governed by the shear -head behaviour. Figure 15 Comparative assessment: a) required control perimeter and shear -head length, b) moment response. Based on these observations, the shear -head section size should be determined from Equation ( 13a), considering that only half of the force may be transferred by the shear -head whilst the remaining half is transferred directly to the columns (e.g. Figure 10a). A conservative of hv/d≥0.5 limit should also be applied to avoid possible shear -head web triggered failure s [12,30] . Since the force is transferred from t he slab to the column through struts supported on the bottom flange, to ensure a smooth transfer, the bottom flange should be relatively stiff in order to avoid failure in compression in the steel insert. The minimum shear -head width b v may be determined using Equation ( 13b), based on the assumption that the transferring struts are supported through the entire l v, in which σ c,max is the strut crushing strength [45]. Figure 15b on the other hand show s the relation ship between the moment demand to capacity Mv,i/Mv,i,Rd for each shear -head for numerical models in which the flange yielding (FY) was recorded, against the l v/rs ratio. The moment demand M v,i was assessed considering the force distribution illustrated in Figure 16 resulting from integration of bending stresses for CRH shear -heads along the cracked cross -section. This indicated that the shear -head flexibility influences the position of the reaction force with longer shear -heads pushing the force towards the column . In contrast, shorter shear -heads translate the reaction force towards the shear -head tip. To accommodate the influence of the shear -head flexibility on the moment demand in the shear -head, a λm parameter is introduced (Equation 13c). In design, the moment capacity of a shear -head, estimated using Equation (13d) has to be higher than the demand (Equation 13c). As indicated in Figure 15b, for the majority of cases in which (FY) was obser ved, Mv,i/Mv,i,Rd is above 1.0, indicating that the above assumptions may be employed for practical assessment . 28 i vv vy1 V 3A2 n f (13a) i v v c,max vV1bnl where ()1/3 c,max c c 0.55 30 / f f= (13b) i v,i m v vVMln= where ()m v s0.75 1 l / r=− (13c) v,i v,pl yvM W f (13d) The results from the numerical assessments presented in Sections 2 and 3 enabled the definition of the above expressions for the design of shear -heads in RC flat slabs. As a general guide, after the assessment of the shear -head section size using Equations ( 13), hv should be at least d/2, whilst the maximum shear -head depth would be limited by practical aspects including slab thickness, amount of longitudinal reinforcement and concrete cover . For a reliable strut support, it is recommended that the bottom flange of the shear -head is located within the compres sion zone of the slab. Members provided with low to intermediate reinforcement (e.g. ρl=0.75% ) generally reach their flexural strength when small shear -head section sizes are employed ( as identified in Figures 10-12). Figure 16 Assumed force distribution for shearheads as a function of embedment length: a) l v/hv=5.0, lv/rs=0.36; b) l v/hv=3.00, l v/rs=0.29; c) l v/hv=2.00, l v/rs=0.14; d) l v/hv=1.00; l v/rs=0.07 Typically, irrespective of lv, high ρl (>1.4% ) produce elastic reinforcement behaviour. The numerical results showed that for short shear -heads ( lv/rs<0.1), a reliable force transfer may not be realised due to yielding initiation at the bottom flange. In contrast, cases with 0.1< lv/rs<0.36 showed effective 29 behaviour and seem generally practical. These observations point to an effective use in design of shear -heads with embedment length -to-slab radius ratios within the range of lv/rs=0.2-0.4 or above, mainly due to their practicality an d stable structural behaviour. In terms of shear -head type, CTP and CRH can provide equally reliable performance in terms of bending and punching shear, as long as the distance between the two parallel profiles in CTP is lower than the column side, to avo id possible penetration of the column through the shear -head. CTP shear - heads may also interfere with the longitudinal column reinforcement and may require a more intricate layout. On the other hand, CBX shear -heads benefit from a more robust configuration , particularly for relatively stiff steel inserts, offering a continuous strut support along the shear -head edges. In terms of bending performance, CBX resemble CTP as the maximum bending action is taken by the central profiles of the CBX. Generally, shear -head s act as an equal -sized rigid support, with a concentration of deformation around its circumference, translating the weak section in terms of bending and punching shear outside of the shear -head edges. The slab rotation combined with a relatively flex ible shear -head, may progressively translate the failure surface within the length of the shear -head. 5. Concluding Remarks This paper inve stigated the ultimate behaviour of cruciform and closed -box shear -head systems fully embedded in RC flat slabs at interior RC columns by means of non -linear finite numerical simulations employing concrete damage plasticity models . After carrying out sensitivity studies on relevant constitutive and geometrical parameters, simulations were carried out and validat ed against three available test series consisting of 36 flat slab specimens with and without shear -heads. The numerical results showed good agreement with both sets of test results, indicating the reliability of the employed modelling procedures. Subsequently , a total of 122 p arametric assessments were carried out by directly varying the shear -head type , shape, embedment length and section size as well as the flat slab thickness, concrete strength, reinforcemen t ratio , and support size. The results of the parametric investigation enabled a direct assessment of the ultimate behaviour in terms of strength and deformation characteristics, as well as a qualitative assessment of the shape of the failure surface. Based on the findings of the parametric studies , three modes of failure were identified, depending on the interaction between the shear -head and surrounding concrete: flexural failure due to yielding of the rebars, flexural failure governed by complete she ar-head yielding, and punching shear due to crushing with or without yielding of the re bars or top flange of the shear -head. It was observed that comparatively stiff shear -heads act as a relatively rigid support, with a concentration of deformation 30 around their tips, translating the weak section in terms of bending and punching shear outside of the shear -head edges. More flexible shear -heads allow increased slab rotations that influence t he inclination of the governing strut , eventually leading to failure surfaces developing at lower angles , with their root moving inside the shear -head . Closed -box shear -heads tend to be more robust as these offer continuous strut support along the shear -head bottom flanges , with the shear -head encased concrete being in a multi -axial state of stress. Cruciform shear -heads are lighter and equally reliable , offering a force transfer through their length in conjunction with the column support. In terms of bendi ng, all types contribute to the flexural capacity and bending stiffness within the shear -head region. The rotational responses and ultimate strengths obtained from the numerical assessments were used for the modification and improvement of a bilinear rotational model for conventional slabs which , in conjunction with an established failure criterion , may be adopted for reliable assessment of punching shear strength. Importantly, the observations from the numerical studies permitted the definiti on of the shear -head dependent parameters required for design, with focus on the shear -head embedment length and section size , as well as to assess the bending moment at which yielding occurs in the shear - head. Considering the wide range of relevant parame ters accounted for in the parametric assessments, the expressions proposed in this paper offer a more reliable design method in comparison with existing approaches , for all shear -head types. The proposed models are also suitable for direct practical applic ation and implementation in codified procedures.