CHARACTERIZATION Of SHEAR BONd STRESS fOR dESIgN Of COmpOSITE SlABS USINg AN ImpROVEd pARTIAl SHEAR CONNECTION mETHOd

Eurocode 4 design provisions specify two methods for the design of composite slabs, namely the m-k and the partial shear connection (PSC) methods. Currently, the m-k method includes the concrete thickness and the shear span of the slab as variables while the PSC method does not. This has resulted in a better accuracy for the m-k method when slabs with varying dimensions are considered. It is demonstrated in this paper that the horizontal shear bond stress varies with the ratio of shear span to effective depth of slab, defined as the slenderness. To include such an effect, a linear shear bond-slenderness equation is proposed. Using the proposed relationship, a linear interpolation of shear bond strength based on two configurations, determined from the outcomes of the bending tests for compact and slender slabs, has been satisfactorily performed. The shear bond strength obtained from this interpolation can be used in collaboration with the existing PSC method, such that the accuracy of the prediction of the composite slab capacity can be considerably improved, the validity of which has been verified with published results from literatures.


Introduction
Nowadays, the use of the steel-concrete composite slab system is a common practice especially in the steel framed buildings. The system comprises normal or lightweight concrete placed permanently over a profiled cold-formed steel deck. Compared to the construction of ordinary reinforced concrete slab, the steel-concrete composite slab offers more advantages because the system is lighter and the handling of the steel deck is easier since the laborious preparation and placement of reinforcement bars in the slab can be circumvent. During construction, the deck acts as the mold for the fresh concrete, hence no temporary formwork is needed. When the concrete hardens, the deck acts as reinforcement for the composite slab system, eliminating the need for positive reinforcement bars.
Horizontal shear bond is always the main governing parameter that determines the behavior and strength of the steel-concrete composite slabs. In studying the influence of interfacial property of composite slabs, Valivonis (2006) explored the contact behavior between concrete and profiled steel sheeting in three stages: elastic, plastic deformation, and formation of cracks. It was found that the contact strength and hence the overall behavior of composite slab is extensively affected by the shape of the profiled sheeting, the pre-compressing force acting perpendicularly to the contact plane, and the horizontal forces that restrain the transversal strains of concrete. It has been shown in Abdullah and Easterling (2009) that the horizontal shear bond depends on the geometry of the slabs, the most important parameter of which is the ratio of the shear span to the effective depth, L s /d, otherwise defined as the slenderness. Because the shear bond property is geometric dependent, its correct characterization is essential for design and numerical modeling purposes, so that an accurate prediction of the strength and behavior of the slab can be obtained. Such importance can be recognized through the influence of the slab slenderness on the maximum reaction force, as shown in Figure 1.

methods for analyzing and designing composite slab
Thus far, there exist several efforts in characterizing the interface shear bond behavior of steel-concrete composite slabs from numerous researchers in literature. To simplify the design and prediction of the behavior of composite slabs, Crisinel and Marimon (2004) proposed an approach that combines results from standard materials tests and small-scale tests with a simple calculation model to obtain the moment-curvature relationship at the critical crosssection. In their method, the load-carrying capacity of composite slabs was determined via the consideration of a three-phase moment-rotation behavior of the critical cross-sections, inputs of which include the geometric and material properties of steel and concrete and the characteristic description of the interface, as determined from the small-scale tests.
Several studies were devoted to the modeling of the shear bond behavior so far as the numerical method in particular the finite element (FE) approach is concerned. A special purpose FE procedure had been developed by Daniels and Crisinel (1993) with a material nonlinearity consideration, in which the shear interaction property was obtained from a pull out test, for analyzing composite slabs. Ferrer et al. (2006) modeled the pull-out tests of composite slabs, employing various coefficients of friction for the contact elements between the steel deck and the concrete. A calculation procedure was constructed by  to obtain the shear bond property from bending tests, the behavior of which was then adopted for the connector elements that were used to model the steel-concrete interface of composite slabs. Tsalkatidis and Avdelas (2010) employed the use of nonmonotone material laws for the interface shear bond interaction, treating it as a two dimensional contact model, in modeling a simply supported one-span composite slab. In similar modeling representation, Tzaros et al. (2010) used a nonconvex-nonsmooth optimization technique for the mechanical description of the shear bonding. More recently, Chen and Shi (2011) presented a universal FE model for composite slabs, in which the shear bond interaction was modeled as a contact element incorporating both adhesion and friction, employing the Coulomb friction model, and considering both geometrical and material nonlinearities. Focusing on an independent modeling consideration of concrete and rolled embossments on the steel surface, Seres (2012) analyzed the composite action of the composite slabs to determine the interfacial shear resistance, due to complex behavior of mechanical and frictional interlocks. Schuster (1970) pioneered the development of the ultimate shear bond equation for slabs exhibiting a shearbond failure mode. The method was adopted by ASCE (1992) in the form of the m-k equation: (1) where: V is the shear force; L s the shear span; f c ' the concrete strength; ρ the reinforcement ratio; and b and d are the width and effective depth of slab, respectively. m and k are the coefficients obtained from a linear regression.
By removing the concrete strength and substituting the reinforcement ratio with Eqn (1) becomes: (2) Fig. 1. Variation in the maximum reaction forces with respect to the slenderness of slab from the bending tests and FE analyses of simply supported composite slabs with a trapezoidal profiled steel deck  Eqn (2) is the version of the m-k method used in Eurocode 4 (2004). It reduces results dispersion produced by Eqn (1). This is in agreement with the findings of previous research in which the reason of dispersion is attributed to the concrete strength (Johnson 1994). It was found that Eqn (1) gives unsatisfactory m and k, if the concrete compressive strength varies widely within a series of tests. In addition, works reported by Luttrell (1987), Daniels (1988), Bode and Saurborn (1992) and Veljkovic (1995) had confirmed that the concrete compressive strength does not influence the properties of slab significantly.
According to Seleim and Schuster (1985), neither the reinforcement ratio nor the concrete compressive strength has a significant influence on the shear bond resistance, but the steel thickness is a governing parameter. It is due to these findings that the removal of f' c from Eqn (1) and therefore the use of Eurocode's equation for designing composite slab, based on a series of test data, are preferred.
The m-k method is thus far the most reliable tool for predicting composite slab strengths. Many researchers have used this method as a basis for comparison with the new design procedures that they developed. Notably new innovations include that incorporating perfobond (Jeong et al. 2009;Kim, Jeong 2010) as well as those with an introduction of additional concrete fillers, e.g. crumb rubber (Mohammed 2010) and palm oil clinker (Mohammed et al. 2011), all of which have implemented this regressed approach in the determination of the horizontal shear resistance of their structures. This exhibits the popularity and validity of the m-k method for the design of steel-concrete composite slabs. It should be however noted that the m-k method is a semi-empirical method that uses principally a statistical evaluation where the values of m and k are only applicable to specific slab configurations.
To overcome the deficiencies of the m-k method, especially the lack of a mechanical model that reflects the influence of composite slab parameters, and to reduce dependency on full size tests, researchers in Europe have developed the partial shear connection (PSC) method. This method was first proposed by Stark (1978) and subsequently improved by Stark and Brekelmans (1990), Bode and Sauerborn (1992), Bode et al. (1996), and Bode and Dauwel (1999). The method was adopted in Eurocode 4 (2004) as an alternative to the m-k method.

details of the pSC method
In the PSC method (Eurocode 4 2004), the slab is assumed to fail by horizontal shear where the concrete can slip relatively to the steel deck without losing the load carrying capacity. In other words, the horizontal shear stress at the slip surface remains constant. This behavior requires a ductile shear failure.
The procedure for the PSC method begins with the generation of a theoretical partial interaction curve that is expressed using the relationship of versus η, as shown in Figure 2(a). The curve is produced using the measured dimensions and strengths of concrete and steel components where η is the intensity of interaction, for which the values are chosen at fixed intervals from 0 to 1. M and M p,Rm are the moment resistances of composite slabs with partial interaction and full interaction, respectively. Points A and C on the curve shown in Figure 2(a) are two extreme cases, corresponding to no interaction and full interaction where M = 0 and M = M p,Rm , respectively. The corresponding stress blocks are depicted in Figure  2(b). The partial interaction state lies between these two points, for example at point B where M is calculated following the stress block B in Figure 2 ( 3) where: N c -concrete compressive force under partial interaction; z -moment arm whose value depends on the intensity of shear interaction; M pr -reduced moment capacity of the steel deck. Under partial interaction, the concrete compressive force constitutes only a fraction of that under full interaction, N cf , and the amount depends on the intensity of interaction: (4) The moment arm takes the following form: where: h t -total slab thickness; x -depth of concrete compressive zone; e p -distance from the plastic neutral axis of the steel deck to its extreme bottom fiber; e -distance from the centroid of the effective area of the steel sheeting to its extreme bottom fiber.
The steel deck carries dual functions under partial composite interaction. Firstly, it serves as the tensile reinforcement in composite action, which is considered in the first term of Eqn (3). Second, it serves as an independent bending element where the deck bends about its own axis. The bending capacity is however reduced from the full capacity, depending on the amount of composite action. The reduced moment, M pr , is defined as: where: M pa is the plastic moment of the effective cross section of the steel deck. The depth of the concrete compressive zone, x, in Eqn (5) is given by: where: f cm -concrete compressive strength; h c -thickness of the concrete cover above the steel deck.
M p,Rm in the partial interaction curve is given by: where is the depth of the concrete compressive zone at full interaction. Here, A p and f yp are the effective area of the steel deck and the yield strength of the steel sheeting, respectively. N cf , the concrete compressive force under full interaction can be expressed as: Eqns (3) to (9) are applicable to under-reinforced sections where the neutral axis lies in the concrete, which is the case for shallow depth profiles.
Having established all important parameters, the intensity of interaction can be next obtained from the partial interaction curve. In this step, full size tests are required from which the maximum bending moment, M test , is determined. Using M test , the intensity of interaction, h test , can be obtained from the curve as shown by path 1-2-3 in Figure 2(a). Knowing the intensity of interaction, the ultimate shear bond stress, τ u , can be next calculated using: where: L s -shear span; L 0 -overhanging length beyond support.
To satisfy the required level of confidence, Eurocode 4 suggests at least six full size tests to be performed to obtain the characteristic value of shear strength, τ u,Rk . It also recommends the use of the minimum value of all tests, with a reduction of 10%. The design strength of the shear connection, τ u,Rd , is then obtained by dividing the characteristic strength with the partial safety factor for full shear resistance, γ v : Once the design shear strength is known, the design partial interaction curve or more exactly the design moment envelope for the particular deck profile can be drawn using Eqns (12) to (17), as given below: where: M Rd -design value of the resisting bending moment in partial interaction mode, i.e. the moment envelope; M p,Rd -maximum resisting moment for the particular profile at full interaction; L x -distance between supports, representing the beam length; L sf -shear span required for full shear connection; γ ap -partial safety factor for profiled steel sheeting; γ c -partial safety factor for concrete; f ck -characteristic compressive strength of concrete.
One example of such a plot is illustrated in Figure 3. The design partial interaction curve is independent of loading type and magnitude. Hence, the allowable loads of, for examples, beams A and B in Figure 3, can be determined easily by plotting the applied moment diagram, M sd , below or just touching the envelope of the design curve.
One advantage of the PSC method is that it is based on a clear mechanical model where the effect of other parameters such as the end anchorage, additional reinforcement and the friction at the support can be incorporated separately in the equations. The use of the plastic design for continuous composite slabs is also possible (Bode 1996;Bode, Dauwel 1999). A procedure to incorporate the effects of end anchorage, frictional interlock, and mechanical interlock in the PSC method was proposed by Calixto et al. (1998). They suggested a plot of test data using for the horizontal axis and for the vertical axis such that a regression line can be formed for the interaction curve. The corresponding equation is: where: τ um -mechanical shear bond strength; µ − friction coefficient, values of which are given by the intercept and the slope of the regression line, respectively. This method shows a much better correlation with the test data when compared to that of the m-k method.

Weakness of the present pSC method
As indicated in Figure 1, the slab slenderness, L s /d, affects substantially the behavior of composite slab. The PSC method, which is adopted in Eurocode 4 (2004), does not properly address the effect of slab slenderness when the intensity of interaction, η, is determined from a test, as illustrated in Figure 2(a). Here, it is obvious that the ultimate moment obtained from the test, which is required to determine the intensity of interaction, η, depends on the slab thickness and the shear span. Slender slabs, i.e. with large L s /d yield low ultimate moment whereas the opposite is true for compact slabs (small L s /d). As a result, various intensities of interaction, η, and accordingly several shear bond strengths, τ u , as given in Eqn (10), exist for the slab specimens built with the same deck type. Abdinasir et al. (2012) have proposed and quantified L s /d = 7.0 as the ratio that divides the definition for composite slabs into slender and compact based on the observation first made by Abdullah and Easterling (2009) in regard to the recommendation given in Annex B (B.3.2 Testing arrangement) of Eurocode 4 in which at least 3 groups of specimens must be conducted each in regions A and B (Fig. 4), the chief parameter of which is the slenderness ratio, L s /d. In this case, they have defined regions A and B for slender and compact slabs, respectively. On close inspection, they observed in their work that reaction force of composite slab experiences an exponential drop within a certain range of L s /d as this ratio increases. The reaction force shows however little change even though L s /d is increased when a certain L s /d is reached, the value of which has been quantified as 7.0 by Abdinasir et al. (2012), i.e. L s /d < 7 for compact while L s /d ≥ 7 for slender slabs. Note that the dividing ratio, L s /d = 7.0, is characterized through observation for definition convenience. This implies that if the shear strength obtained from the tests of compact slabs is used to design a slender slab, the resulting design can be unsafe for practical use.
On the other hand, if a compact slab is designed based on the shear strength obtained from the tests of slender slabs, an overly conservative design may be produced. Eurocode 4 specifies that the test specimens for determining the design value of the longitudinal shear strength, τ u , should be as long as possible (as slender as possible) such that the failure by longitudinal shear can be formed. As a result, the evaluation of a slab with slenderness lower than the test specimen by the PSC method is always conservative and potentially uneconomical. In this paper, an improvement in the determination of the ultimate shear bond strength, τ u , used in the PSC method, which takes into account the slab slenderness, is therefore proposed.

Establishing a relationship between shear bond stress and slenderness
Since the shear bond strength varies with the slenderness, a single value of τ u cannot represent the overall range of slab slenderness accurately. Therefore, the improvement should be made such that τ u is represented as a function of slab slenderness.
Consider a free body diagram in Figure 5 for a typical shear bond failure as assumed in the partial shear connection theory. By taking moments about the compressive force, C, and considering that the moment arm differs very slightly from the slab effective depth, the moment equilibrium equation can be estimated as: where M r is the remaining moment strength in the deck.
Eqn (20) is an approximation because the lever arm is always less than the effective depth. Nevertheless, for composite slabs, the difference is insignificant because after the initiation of shear slip, the crack tip grows upward quite rapidly, bringing the composite neutral axis close to the top fiber and hence the lever arm to the effective depth, d.
Rearranging Eqn (20) and substituting it into Eqn (2), a new equation relating shear bond stress to the slab geometry is obtained: If b is taken as the length of the steel sheeting, then is equal to the thickness, t, of the steel deck. We therefore have: It should be noted that the derivation of Eqn (22) is based on the consideration that a uniform distribution of shear bond stress, τ, is formed throughout the surface area of the steel-concrete interface and along the length measured from the applied load to the slab's end.
When deriving the original m-k equation, Schuster (1970) neglected the contribution of the remaining moment strength, M r in the steel deck and so did Patrick and Bridge (1994), Veljkovic (1996) and Widjaja and Easterling (1996) in their modified PSC methods. It was shown that the second term of the right hand side of Eqn (22) acts collectively as a constant regardless of the value of M r and therefore a detailed determination of this term is not needed. Following this reasoning, Eqn (22) can be simplified to: .
Also, the overhanging length, L o , is usually short. Therefore, it is assumed here that L o is not a determining factor for the slab behavior and it can be removed from the equation, to give: .
It should be noted that the contribution of L o to the horizontal shear resistance cannot be neglected when τ is calculated using Eqn (10). Eqn (24) is a linear equation, written in a similar manner as that of the m-k equation. To facilitate extensive reference to Eqn (24) in the remainder of this paper, and to differentiate its variables m and k from the m-k method, Eqn (24) is subsequently called the shear bond-slenderness equation and the slope and the intercept obtained from the linear regression are changed to p and s replacing respectively m and k. The term d/L s is defined as the slab compactness while its inverse, L s /d is readily known as the slab slenderness. The shear bondslenderness equation is now expressed as: .

Application of the shear bond-slenderness equation
The method to assess the moment capacity of a slab using the shear bond-slenderness equation follows the same procedure as that of the m-k method. Firstly, a sufficient number of bending tests of slabs (in accordance with the testing procedure given in Eurocode 4) should be conducted for two configurations: slender and compact configurations (regions A and B, respectively, in Fig. 4).
From the tests, the maximum shear bond stress, τ u , for each test is calculated in accordance with Figure 2(a) and Eqn (10) such that a relationship between τd and td/L s can be formed as shown in Figure 4, using a linear regression line. The line is used to interpolate the shear bond strength of slabs with different spans, and different concrete and sheeting thicknesses. As suggested in the testing procedure of Eurocode 4, the shear span is to be taken as a quarter of the total length of the slab. The shear bond is then used in the determination of the moment resistance of the slab in accordance with the PSC method specified in Eurocode 4. Also, a reduced regression line may be considered adopting a safety factor, as implemented in the m-k method.

Verification of the proposed shear bondslenderness equation
A set of bending test data for the composite slabs constructed using the trapezoidal shape steel deck shown in Figure 6, as reported in , is used to verify the proposed shear bond-slenderness equation. Results from the finite element analyses based on the same test data that were reported in Abdullah and Easterling (2009) and presented in Figure 1 are also analyzed. The finite element results are for the slabs whose geometries (span and concrete thickness) differ from those of test specimens. They are for slabs built on 76 mm deck with 1.5 mm and 1.2 mm sheeting thicknesses and on 51 mm deck with 0.9 mm sheeting thickness. The details of the finite element analysis can be referred in . Using the maximum loads obtained from the bending tests and the finite element analyses, the maximum shear bond stress of each slab is calculated in accordance with the PSC method ( Fig. 2(a) and Eqn (10)). The parameters required for plotting the shear bond-slenderness relationship are tabulated in Table 1 and Table 2 for the test and finite element results, respectively. The main relationships are presented in Figure 7 to Figure 10. Clearly, all results fall along a straight line, which confirms the validity of the proposed equation.
It should be noted that the incorporation of sheeting thickness, t, in the shear bond-slenderness equation enables all slabs to be built on the same deck profile but with different sheeting thicknesses for the bending test and assessment of the shear bond. This is shown in Figure 7 to Figure 10 where the plots are from a combination of tests on slab specimens using decks with 0.9 mm, 1.2 mm and 1.5 mm sheeting thicknesses. Using the data from the tests for slabs that are built on the same deck but with the thickness different from the test specimen, the determined shear bond is in agreement with the findings of Seleim and Schuster (1985) and the results produced using the Canadian Sheet Steel Building Institute (CSSBI) specification (1996).

design example
For convenience, we demonstrate next the design procedure for composite slabs that are studied in this paper. In this example, test specimens #5 and #9 for slabs using 76 mm deck as shown in Table 1 are used to calculate the intensity of interaction, η ( Fig. 2(a)), and shear bond stress, τ (Eqn (10)), respectively. These values are plotted to form the relationship of τd versus td/L s (Eqn (25)), from which p and s are obtained from a regression line (Table 3). With these parameters, τ u for other slabs with different thicknesses and shear spans are obtained by linear interpolation in accordance with Eqn (10). Once τ u for the corresponding slabs are obtained, the ultimate loads for these slabs are calculated according to Eqns (12) to (17), as shown graphically in Figure 3. A partial safety factor of 1.0 is used in this exercise. The same procedure is repeated for 51 mm deck slabs based on tests #10 and #11 of Table 1 where the calculation outputs are shown in Table 3. For comparison, the ultimate load predictions adopting the m-k method (Eqn (2)), using the data from the same specimens are also performed ( Table 4). The results from both methods are compared with the elemental and full size test data reported in Easterling (2007, 2009) by computing the ratio of testto-calculated values. The results are tabulated in Table 5.
For the improved PSC method, the mean of the ratios of elemental test-to-calculated values is 1.08 with a 10% average variation of the test values from those For the m-k method, the same comparisons of the ratios give 1.13 and 0.11 for the mean and standard deviation, respectively. For full size specimens, the PSC method gives a mean and a standard deviation of 1.07 and 0.17, respectively. They are 1.11 and 0.12, respectively, for the m-k method. Graphical comparison between the calculated ultimate loads and the tests data for elemental and full size tests are depicted in Figure 11 and Figure 12, respectively. It can be seen that both methods are relatively conservative compared to the test results in the region represented by compact slabs, where the It can be demonstrated that the ultimate load calculated using the improved PSC method employing the shear bond-slenderness equation is as accurate as that calculated using the m-k method. Such comparison is depicted in Figure 13. This implies that the PSC method has been greatly improved using the proposed shear bond-slenderness equation. Hence, the procedure can be used to design slabs of various slendernesses with better accuracy using only two sets of test data. A direct comparison between the improved PSC and the m-k methods is now possible because the shear bond-slenderness equation used in the PSC method employs the concept of linear interpolation similar to the m-k equation.