Size effect on the shear capacity of headed studs

Abstract

This study aimed to reveal the existence of size effect on the shear connectors used in the steel-concrete composite beams and slabs. The experimental study contains the monotonic tests of nine pushout specimens with the headed studs. Three-dimensional scaling was used for geometrically similar specimens of three sizes. High strength concrete slabs were used on both sides of the steel I-beam. The failure modes of the specimens include both concrete crushing and stud yielding. Finite element (FE) verification of the specimens was conducted using a realistic concrete damage constitutive model, Microplane Model M7. It is shown that there may be a non-negligible size effect based on the fracture patterns of the composite member. Bažant’s size effect law (SEL) can fit the size effect behavior of the shear connectors. The design equations which do not include a size effect term have high correction factors that overestimate the tested specimens. A new design equation can be drawn using the size effect factor for strength reduction of shear connectors.

Keywords

composite beam fracture headed stud shear strength size effect

Introduction

Composite beams and floor systems are extensively used in infrastructures and buildings. The wide-spread use of steel-concrete composites (SCC) in modern buildings and bridges is a result of their high load-bearing capacity to self-weight ratios. In SCC the composite action is provided with the mechanical shear connectors at the interface of components. The most widely used shear connectors are the headed shear studs. The principal role of the headed studs is to resist the longitudinal shear forces. Restraining the relative slip between the steel flange and concrete slab through the end-welded studs results in reduced deflections of the long-span beams and slabs. For the safe design margins, the behavior of SCC members, for all levels of connections and interactions, should accurately be predictable. However, because of the complex force transfer mechanism between the components, the real behavior of composite systems was not clearly understood. The capacity predictions of SCC beams, for instance, depend on many parameters such as the geometries of the structure, mechanical properties of the materials, the amount and outline of multiple stud systems, etc. Furthermore, size effect, which is the main objective of this study, also plays an important role on the strength capacity of the shear connectors.

The parameters affecting the shear capacity of studs are among the main purposes of recent studies (Chen et al., 2016; Xu and Liu, 2019; Yan and Xie, 2018). The shear load capacity of the headed studs has usually been provided by conducting the pushout tests since 1956 (Viest, 1956). In general, there are two different failure modes of pushout specimens, which are the concrete controlled failures ( $V_{uc}$ ) like concrete cracking, crushing, etc. and failure of the stud shank ( $V_{us}$ ) followed with a detaching mechanism of the stud, most likely from its weld collar. For the design purposes, the capacity of the headed studs is being calculated as the minimum of these two defined values, such as $V_{u} =$ min $(V_{uc}, V_{us})$ . Here $V_{uc}$ refers to the failure load capacity of concrete and $V_{us}$ to the failure due to the stud shank. The yield and tensile strengths of the stud are supposed to be known thoroughly. It suffices to apply a correction factor to the tensile strength ( $f_{su}$ ) of the stud material of which the cross-section is known. However, in this study, a small reservation is given about the accurateness of predicting the shear strength of steel connectors.

According to the test results, Viest defined the load capacity ( $V_{uc}$ ) of the headed studs, which have a diameter up to 25 mm, approximately proportional to the $\sqrt{f_{c}}$ , and $D^{2}$ , where $f_{c}$ is the concrete compressive strength and D stud diameter (shank diameter). For the studs with larger diameters, test data fit better when they are proportional to the D rather than the $D^{2}$ . Also, these instructions are for the studs with heights equal to or greater than 100 mm. The results of Viest’s (1956) experiments, was later somewhat supported by Thurlimann (1959). Slutter and Driscoll (1963) modified Viest’s semi-empirical formula by including the stud height-to-diameter ratio as in equation (1).

\begin{matrix} V_{u} = α_{1} hD \sqrt{f_{c}} (for h / D < 4.1); \\ V_{u} = α_{2} D^{2} \sqrt{f_{c}} (for h / D \geq 4.1) \end{matrix}

(1)

where $V_{u}$ is the ultimate load capacity per stud in Newtons, h the stud height [mm], and $α_{1}$ and $α_{2}$ are empirical constants, which equal to 18.27 and 77.22 respectively for SI (220 and 930 for USC). Equation (1) was also adopted by JSCE (2007) with different $α$ values and limitations. In JSCE the critical $h / D$ ratio is not 4.1 but 5.5, and $α_{1}$ and $α_{2}$ equal to 1.72 and 9.4 respectively. Ollgaard et al. (1971) modified the Slutter and Driscoll’s equation for design purposes and different codes used their equation with slightly modified versions. Eurocode-4 (Eurocode 4, 1994) recommends equation (2) for the loading capacity of shear connectors.

\begin{matrix} V_{u} = min (0.29 β D^{2} \sqrt{f_{ck} E_{c}}, 0.8 A_{s} f_{su}) \\ β = 0.2 (h / D + 1) \geq 1.0 (for 3 \leq h / D \leq 4); \\ β = 1.0 (for h / D > 4) \end{matrix}

(2)

where $A_{s}$ is the cross-sectional area of stud, $f_{ck}$ the characteristic compressive strength of concrete, $E_{c}$ the modulus of elasticity of concrete, and $f_{su}$ tensile strength of the stud in MPa. The American Institute for Steel Construction (AISC, 2016) uses a similar prediction (for a single steel stud anchor) in their design recommendations with a modified version as in equation (3).

V_{u} = min (0.5 A_{s} \sqrt{f_{c} E_{c}}, 0.75 A_{s} f_{su})

(3)

If $E_{c}$ is unknown, the Young’s modulus of concrete can be calculated as $E_{c} = 0.043 w_{c}^{1.5} \sqrt{f_{c}}$ and the limits of the unit weight of concrete is $1500 \leq w_{c} \leq 2500 kg / m^{3}$ . Anderson and Meinheit (2005) proposed a different equation for PCI Design Handbooks (PCI, 2004), as in equation (4).

V_{u} = min (α_{3} λ \sqrt{f_{c}} h_{ef}^{0.5}, α_{4} A_{s} f_{su})

(4)

where $h_{ef}$ is the effective stud height calculated as the stud shank length without the thickness of stud head, $λ$ the concrete unit factor (1.0 for normal weight and 0.75 for lightweight concrete), and $α_{3}$ and $α_{4}$ the correction factors which have values of 26.4 (317.9 for kips and in.) and 0.75 respectively.

Many authors then verified the increase in capacity of the studs with enhanced mechanical properties of concrete and other geometrical aspects of the studs. In addition to the researches mentioned above, pushout tests of headed stud connectors were conducted to investigate the effect of different parameters. The number of studs (Seracino et al., 2003; Shim et al., 2004; Spremic et al., 2013; Xue et al., 2012), various loading conditions (Mirza and Uy, 2010; Oehlers and Park, 1992; Seracino et al., 2003), the effect of large stud sizes (Lee et al., 2005; Shim et al., 2004), the effect of $h / D$ , (Pallarês and Hajjar, 2010), and effects of the slab and its reinforcement (Baldwin, 1970; Liu and Alkhatib, 2013; Menzies, 1971; Oehlers and Park, 1992; Qian and Li, 2006; Topkaya et al., 2004) are among the investigations based on experimental results.

The first analyses about the size effect concept on the shear connectors were conducted by Bažant and Vitek (1994). The studies belong to Bažant and Vitek (1994, 1998, 1999a, 1999b) and Isik (2018) followed Bažant’s first work on this subject. According to these studies, the size effect of a composite beam can be sourced both by the shear strength of a single stud and the consecutive cracking of studs in a row under bending actions. These actions together result in a combined size effect. The combined size effect under appropriate circumstances can be significant. Moreover, these types of structures under brittle shear cracking might exhibit a more pronounced size effect than the strength reduction introduced by the linear elastic fracture mechanics (LEFM). So this critical aspect of a composite behavior should be analyzed in detail. As a part of the aforementioned combined size effect concept, this study specifically examines the effect of size on the shear strength of individual shear connectors.

Methods

The failure mode of shear connectors can vary according to the materials and geometrical characteristics. In a pushout specimen, the failure of a headed stud could be governed by the damage (or yielding) and fracturing of the stud shank, or crashing and cracking of the concrete, or by a combination of failures of both constituents at various scales. Usually, the latter is the most frequent type encountered in the pushout tests. Concrete strength and the ratio of $h / D$ play essential roles in failure types and maximum load capacities that the connector region can sustain. For small values of $h / D$ (like $h / D <$ 4.5), the failure occurs more likely at the concrete section (Pallarês and Hajjar, 2010), while for $h / D > 5.5$ the failure is mainly controlled by the stud yielding. On the other hand, the failures of studs, of which $h / D$ values stay in between these limits, generally exhibit a mixed-mode of failures. However, in all the cases, the concrete is a primary material for the overall response in SCC.

Assume that the stud shank governs the failure. In this case, the surrounding concrete would undergo very localized stress concentrations at the zones close to the weld collar. As the composite actions take place, the crushing of concrete or the microcracking and micro-slips of concrete start to define the boundary conditions of studs subjected to non-uniformly distributed shear loads. The micro deformations (micro-cracking and micro-slips) of surrounding concrete with the connector properties finally determine the ultimate shear capacity.

Since the concrete material is a quasibrittle material, the micro-cracking and micro-slips occurring inside the concrete material can be defined as a quasibrittle failure. The size effect is indispensable in quasibrittle failures. Thus the quasibrittle nature of this type of shear failure of stud connectors would induce a size effect. So the failure analysis of these structures should be conducted with the quasibrittle fracture mechanics (or with size effect analysis).

The pushout specimens with geometrically scaled sizes would exhibit a Type 2 size effect (Bažant, 1976, 1984; Bažant and Kim, 1984; Bažant and Planas, 1998). The size effect law (SEL) can be used to characterize the strength reduction of pushout specimens as well as the shear strengths of SCC beams and slabs under bending actions. The validation of the Bažant’s SEL is extensively verified in many of the previous researches in which the materials have quasibrittle nature (Caner et al., 2019; Dönmez and Bažant, 2017, 2019, 2020; Dönmez et al., 2020a; Dönmez et al., 2020b). Here the same phenomenon is applied to the shear failures of headed studs with experimental validation.

Geometry of the specimens

Experimental work consists of nine pushout specimens with a single headed shear stud welded on each flange of I-beams. Test specimens involve high strength concrete slabs on both sides of the I-beam. The adopted height-to-diameter ratio, $h / D$ , is 5.26. The size range for the size effect experiment includes three different (scaled) sizes, such as D = 4.75: 9.50: 19.00 (where D denotes the diameter of the headed studs). Each size-set has three specimens, that is, three specimens for small, medium, and large size sets.

In the production stage, headed studs were welded on each flange of the I-beams. The surfaces of the I-beam flanges were greased to reduce the friction coefficient between concrete and steel. The surface of the stud’s shank and stud’s head was cleaned to diminish the contamination effects in the concrete mix. The reinforcing bars, which are also scaled accordingly, were placed on the flange of I-beams. The minimum concrete cover of 2.5 mm for small size was scaled too for medium ( $2 D$ ) and large ( $4 D$ ) size specimens. After the necessary preparations, concrete slabs were set on the I-beam flanges. The specimens were preserved in the curing pool, underwater condition, for 20 days and tested when the age of concrete is 50 days.

The dimensions of the small specimens and a general view of the test plan are shown in Figure 1. For the medium and large size specimens, all the dimensions in Figure 1 should multiply with 2 and 4, respectively. The detailed information regarding the dimensions and material properties of the specimens are also given in Table 1.

Figure 1.

Test geometry for the small specimen (multiply by 2 or 4 for medium or large sizes respectively).

Table 1.

The geometrical parameters and failure modes of the specimens.

ID	$f_{c}$ /MPa	D/mm	h/mm	h/D	2 $P_{u}$ /kN^a	Failure mode
HPOS1	91.67	4.75	25	5.26	20.38	SY^b-CC^c
HPOS2	91.67	4.75	25	5.26	19.51	SY-CC
HPOS3	91.67	4.75	25	5.26	20.33	CC-WF^d
HPOM1	91.67	9.5	50	5.26	73.21	CC-CS^e
HPOM2	91.67	9.5	50	5.26	72.73	CC-CS
HPOM3	91.67	9.5	50	5.26	71.87	CC-SY
HPOL1	91.67	19	100	5.26	283.17	SY-CC
HPOL2	91.67	19	100	5.26	280.06	SY-CC
HPOL3	91.67	19	100	5.26	289.21	SY-CC

Values for two studs.

SY: stud yielding.

CC: concrete crushing.

WF: weld failure.

CS: concrete splitting cracking.

Materials

The scaled headed studs for each size-set are shown in Figure 2. The Nelson brand headed-studs were used as the shear connectors. The diameters of the studs were kept constant, which are 4.75, 9.50, and 19.00 mm for small, medium, and large sizes, respectively. The total heights of the studs, h, are 25, 50, and 100 mm for small, medium, and large sizes, respectively. The ratio of $h / D$ for all the specimens equals 5.26. To obtain the mechanical properties of the headed stud, a uniaxial test was conducted as shown in Figure 3(a). The attained yield strength of the headed stud is 460.26 MPa, and the ultimate tensile strength is 483.94 MPa. Young’s modulus was measured as 187.91 MPa. Note that there is a gradual and visible post-peak softening in the response of the uniaxial tensile test of the headed stud material which is not pure steel but consists of metal alloys. The gradual load drop as the post-yield behavior induces damage and fracturing mechanism different than perfectly plastic materials like mild steel. This type of characteristic response of the studs may be one of the (and probably the weakest) sources of the size effect. And this trend of the material response is expected to be more pronounced at relatively lower temperatures.

Figure 2.

The stud lengths used in the experimental campaign.

Figure 3.

Material tests of: (a) stud tension, (b) concrete compression (with M7 simulation), and (c) rebar tension.

High strength concrete materials were used in production. The mean compressive strength of the concrete is 91.67 MPa. Compressive tests were conducted at the age of testing using 15 × 30 mm cylinder specimens as shown in Figure 3(b). The calibration of the numerical model is included as M7 in Figure 3(b), which is described in the following sections. The elastic moduli were measured as 47.2 GPa for the high strength concretes. The maximum aggregate size of concrete was determined according to the shortest length of small specimens. To keep the fracture parameters of the material as the same inside each series, a constant aggregate size, $d_{a} = 4$ mm, was utilized in the concrete mixture with a water–cement ratio of 0.4.

The steel reinforcements, used in the slabs, were also scaled accordingly, to ensure the same reinforcement ratios for each direction and specimen sizes. The diameter of the smallest rebar is 2 mm, as shown in Figure 1. For the middle and large sizes, $ϕ 4$ and $ϕ 8$ rebars were used in accordance with the 3-dimensional scaling laws. The mechanical properties of the rebars were obtained by conducting the uniaxial tensile tests, as shown in Figure 3(c). The measured yield and tensile strengths are 613 MPa and 643 MPa, respectively. The fracture properties of ductile steel materials are different than their alloys. This may explain the difference in the stress-strain curves between the rebar and stud.

Test setup

All the tests were conducted using the Instron universal test machine with 500 kN load capacity. The same strain rate was used during the tests, which is 4.5 × $10^{- 4} s^{- 1}$ . The constant strain rate corresponds to a loading rate of 5 mm/min. for large specimens. One external load cell with a 500 kN compression load capacity was used (Figure 4). Two LVDTs were attached to the mid-height of concrete and top part of the steel beam, as shown in Figure 4. Displacements were monotonically increased until one of the studs completely detach from the beam flange.

Figure 4.

Test and measurement setup for the large specimens. The same design for medium and small sizes.

A 5 mm thin elastomer pad was used to homogenously distribute the jacking force on the I-beam sections. A small pre-load (0.05 kN for small size) was applied to the specimens, which was also scaled according to the specimen size. Results from the external load cell were used to compare the in-built load cell readings.

Results and discussion

The load slip curves of the tests are shown in Figure 5 and given in Table 1. The load values in Figure 5 correspond to two studs welded on both sides. The peak loads are considered as the limit load capacity which is also used in the strength calculations. Most of the specimens exhibited concrete crushing in the vicinity of the stud front and stud yielding at the maximum load. The failure types of weld failure and concrete splitting also contribute to the overall failure mechanism besides concrete crushing and stud yielding of some specimens. Because of the boundary conditions applied to the specimens, the tests were concluded when one of the studs detached from the flange of the I-beam after the post-peak.

Figure 5.

Load versus slip curves of pushout specimens for: (a) small (HPOSi), (b) medium (HPOMi), and (c) large (HPOLi) specimens (values are for two studs).

Because of the localized stress concentration regions in the vicinity of the stud front, the micro-damage or micro-cracks in these regions are unavoidable. The local crushing of concrete, when the I-beam slips relative to the slabs, was observed in all of the specimens. The examples of micro-cracking and micro-slip regions are shown in Figure 6(a) for each size sets. This type of local failure contributes to the pre-peak and post-peak nonlinearities of the load-deflection responses and the capacity of the composite member. The cracked regions, later, may lead to an abrupt load drop in the load-deflection curves when the micro-cracks coalescence to form a splitting cracking in the reinforced concrete slab. The load drops due to failures of the concrete slabs can be limited based on the strength levels of constituents and reinforcement. Concrete crashing is more pronounced in the small and medium specimens (Figure 5(a) and (b)), in which the load-slip response has a significant degradation.

Figure 6.

(a) The micro-cracking behavior of the concrete surrounding the stud shank, (b) the formation of the concrete wedge on stud front for each size, and (c) failure of the weld, and concrete splittings exhibited by the HPOS3, HPOM1, and HPOM2 specimens, respectively.

The wedge formation is another consequence of the local crushing of concrete. The wedge-like failure formation can be seen after the detaching mechanism of studs as shown in Figure 5(b) for each size at the end of the yielding mechanism. This type of local crushing can be considered as another source of the size effect exhibited by pushout specimens. As one can see from the large (HPOLi) specimens, studs illustrate a pronounced yield plateau. However, in the response of the small (HPOSi) and medium (HPOMi) specimens, the concrete cracking controls the peak value and the load drops after a substantial micro-cracking and micro-slip behavior of the surrounding concrete.

The HPOS1 and HPOS2 specimens illustrated a pronounced stiffness degradation and gradual post-peak softening. These responses are due to the mixed mode of failure. The yielding of stud and concrete crushing controlled the behavior. HPOS3 has displayed a weld failure after an insignificant local failure at the weld-collar region as shown in Figure 6(c). The detachment of the stud from its welding point leads to the abrupt load drop. The responses of HPOM1 and HPOM2 are mainly controlled by the splitting cracking of concrete after the coalescence of the micro-cracks in the concrete. These splitting cracks are also shown in Figure 6(c). In these samples, there are significant post-peak softening and a few local abrupt load drops. However, for the HPOM3 specimen, the stud yielding is more effective on the load-slip curve than the other medium-sized specimens, which eventually resulted in a higher slip value for this specimen. The large size specimens (HPOLi) exhibited a pronounced yielding plateau without any abrupt or gradual decrease in the load values until the detaching of the stud from the weld collar area. In the specimens which do not show a gradual or steep softening behavior, it is assumed that the surrounding concrete is exposed to a limited amount of micro-cracking and concrete crushing. This type of behavior corresponds to the stud governed failure mechanisms. However, again, micro-cracking and micro-slip of concrete are inevitable as can be seen in Figure 6. The observed failure modes of all the specimens are also given in Table 1.

The difference between the competing failure modes of concrete crushing and stud yielding might be very low in the design of tests. Therefore both failure types can be seen in the load-slip responses of the specimens. This is mostly due to the high strength concrete might be too strong to get cracked by the pushout mechanism of studs. The ratio of $h / D$ of the studs is another significant influencing parameter on the responses and failure types of the pushout specimens. Long studs induce higher crushing areas and reduce the likelihood of concrete controlled failures. On the other hand, the variations of peak loads inside each size sets are negligible.

The strengths of the pushout specimens are calculated per stud. Therefore, the nominal strength is computed as the $P_{u} / (Dh)$ for each stud. In Figure 5 $P_{u}$ corresponds to half the peak load obtained from the load-deflection curves. D and h represent the stud diameter and total stud height, respectively. In the size effect analysis, for a three-dimensional scaling, the choice of the characteristic length is arbitrary. One will find the same results with the proper ordinate scale if a different dimension is chosen as the characteristic length. The three-dimensional scaling of the pushout specimens lets us choose any suitable length to use in the strength calculations. So in here, the stud diameter (D) is used as the characteristic length.

The strength values are plotted against the stud diameter in double-log scales in Figure 7(a). The Size Effect Law (SEL) is used to fit the test data. As a comparison, the test data by Isik (2018) is also demonstrated in Figure 7(b). In these tests, three-dimensional scaling was also adopted for three different sizes. The specimens’ concrete compressive strength is 55 MPa and the stud’s height-to-diameter ratio, ( $h / D$ ), equals 4.73 (Figure 7(b)). It can be seen that the concrete strength might significantly change the failure mechanisms which resulted in a more pronounced size effect trend.

Figure 7.

The strength values of pushout specimens and size effect fit in the double-log scale: (a) this study, (b) results of tests by Isik (2018), and (c) normalized strength values of both data sets.

The least-squares nonlinear regression analysis is employed to fit the SEL which is given in equation (5) (Bažant 1976, 1984; Bažant and Planas, 1998). The SEL in equation (5) corresponds to the asymptotic matchings of the nominal strength values for both infinitely large and small sizes. In SEL $σ_{N}$ refers to the nominal strength of structures, $σ_{0}$ to the strength of the infinitely small structures, $λ_{s}$ to the size effect factor, and $D_{0}$ (transitional size) to the fracture parameter related to the shape and microstructure of the material.

\begin{matrix} σ_{N} = σ_{0} λ_{s}; λ_{s} = \sqrt{\frac{1}{1 + D / D_{0}}} \end{matrix}

(5)

Based on energy release evaluations adjusted to quasibrittle fracture mechanics, the energetic size effect law (SEL) can be applied to failures occurring after stable growth of a long crack, as typical of shear failure of RC beams, was formulated in Bažant (1984); Bažant and Kim (1984). Subsequently, equation (5) (SEL) was shown to apply to many types of failure in all quasibrittle materials that do not follow the conventional mechanics of fracture. Aside from concrete, SEL can also correspond to the fracture behavior of tough ceramics, fiber composites, rocks, stiff soils, sea ice, wood, stiff foams, bone, etc. The SEL captures the transition from a nearly ductile behavior in small concrete structures to a nearly brittle behavior in large ones (Dönmez and Bažant, 2019). The reason for this ductile-brittle transition of structural response is the material heterogeneity, which causes a larger fracture process zone (FPZ) (0.5 m in concrete vs. micrometers in metals), and non-negligible compared to the cross-section size (Dönmez and Bažant, 2017).

Figure 7(a) illustrates the size effect fit for the pushout specimens. The size effect trend presented by the pushout specimens matches the energetic type of the size effect law, which is called as Type 2 size effect (Bažant and Planas, 1998). As can be seen from Figure 7(a), the SEL closely fits the strength values of the pushout specimens. The value of $D_{0}$ is high because of the stud governed failures of the large specimens. A higher $D_{0}$ results in a higher value for the length of FPZ. Many microcracks, pre-exist in the concrete, merged to form a macroscopic FPZ (Bažant and Planas, 1998). A larger FPZ would lead to higher energy dissipation as the crack grows. Therefore a more pronounced strength reduction is visible between the small and medium size samples. The transitional size is lower in Figure 7(b), in which the specimens have a lower $h / D$ ratio and concrete compressive strength. These parameters are considered as the secondary parameters and may change the governing failure mode in a pushout specimen. The combined data set in Figure 7(c) include the specimens of the current study and the tests by Isik (2018). All of the specimens with their average values, in Figure 7(c), are normalized by the corresponding $σ_{0}$ values. The size effect trend of pushout specimens is more obvious in Figure 7(c).

The predictions of the design standards of Eurocode 4 and AISC (given in equations (2) and (3), respectively) are shown in Table 2. The test results of all the specimens are about 20%–50% lower than the $V_{uc}$ specified by the code standards. On the other hand, obtained strength values are 20~50% higher than the $V_{us}$ estimations according to the Eurocode and AISC specifications. The difference might be due to the individual calculations of failure strength due to concrete crushing and stud shank failure under shear actions, suggested by the codes. Whereas, the failure of shear connectors mainly involves the combination of local (or global) concrete crushing and yielding of the stud shank. Furthermore, the mechanical properties of metals under shear actions deviate from the case of tensile loading. The fracture parameters, mechanical responses, and damage mechanism might be significantly different than these features under uniaxial tension. Finally, the size effect is not included in the design codes. The accuracy of the estimate starts to recede as the size increases. On the other hand, according to the Eurocode 4 and AISC, the failure types of the tested samples are defined as the yielding of the stud. The Chi-square values of the Eurocode 4 and AISC proposals are 31.61 and 45.57, respectively. If one isolates only the large sizes, the chi-square statistics results in 22.08 and 32.43 for Eurocode 4 and AISC, respectively. The code provisions underestimate the size effect specimens.

Table 2.

Comparison of the test results with the code predictions of Eurocode 4 (or equation (2)) and AISC (or equation (3)).

Experiment		Eurocode 4-prediction			AISC-prediction			Prediction score
ID	$P_{u}^{a}$	$V_{uc}$	$V_{us}$	$V_{u}$	$V_{uc}$	$V_{us}$	$V_{u}$	$P_{u} / V_{E}^{b}$	$P_{u} / V_{A}^{c}$
HPOS1	10.19	13.65	6.86	6.86	18.49	6.43	6.43	1.49	1.58
HPOS2	9.75	13.65	6.86	6.86	18.49	6.43	6.43	1.42	1.52
HPOS3	10.16	13.65	6.86	6.86	18.49	6.43	6.43	1.48	1.58
HPOM1	36.60	54.61	27.44	27.44	73.95	25.73	25.73	1.33	1.42
HPOM2	36.36	54.61	27.44	27.44	73.95	25.73	25.73	1.33	1.41
HPOM3	35.93	54.61	27.44	27.44	73.95	25.73	25.73	1.31	1.40
HPOL1	141.58	218.46	109.77	109.77	295.82	102.91	102.91	1.29	1.38
HPOL2	140.03	218.46	109.77	109.77	295.82	102.91	102.91	1.28	1.36
HPOL3	144.60	218.46	109.77	109.77	295.82	102.91	102.91	1.32	1.41

Values for one stud.

Prediction of Eurocode 4.

Prediction of AISC.

The size effect should be considered in the design codes. Most of the strength criteria about the SCC systems may provide realistic results for relatively small structures used in most laboratory testing. At maximum load, the distributed cracking cannot localize into one dominant crack before maximum load and, thus, the size effect becomes negligible for small structures. The size effect appears only when the structure is sufficiently larger than the FPZ. Consequently, extending the strength limit criteria to large sizes is relatively easy; it suffices to multiply the structural strength according to the current code, based on limit analysis, with the proper size effect factor (equation (5)). The size effect factor, $λ_{s}$ , is already adapted in ACI (2019) standard code for beam and punching shear equations. Regarding the shear connectors, the size effect factor can be used either in $V_{cu}$ or $V_{u}$ values. The value for $D_{0}$ in a code-like equation should be defined based on a sufficiently large filtered database. It can be deduced from this study the $D_{0}$ depends on the secondary parameters like $h / D$ ratio, cement type, aggregate sizes, etc. Also, note that multiple stud strength is not proportional to the strength of the single isolated stud. In the case of the stud rows, the failure due to the shear forces produces another source of size effect which leads to much lower load capacities as mentioned above and in Bažant and Vitek (1994).

The $λ_{s}$ as the size effect factor in equation (5) can be recommended for design purposes. The $σ_{0}$ in equation (5) can be substituted by $V_{u} / D^{2}$ instead of $V_{u} / (hD)$ . One of the most important parts is to define the $V_{u}$ , which is the ultimate load capacity, and can be found by using equation (3) or (4) (or any other strength prediction for small sizes). The transitional size $D_{0}$ in the size effect factor ( $λ_{s}$ ) can be used between the range of 10 to 50 mm which would suffice to capture the size effect for single stud strengths and one row. However, using a sufficiently large database and performing regression for the values of $D_{0}$ and other (secondary) parameters (such as $f_{c}$ , $h / D$ , etc.) other than the characteristic size, would lead to optimum results.

Finite element verification

To verify the experimental results, the FE analysis was conducted based on realistic concrete material models. The microplane model M7 was used as the reliable damage constitutive model for concrete. The M7 is the latest in a series of microplane models for concrete developed at Northwestern University (Caner and Bažant, 2013a, 2013b). The properties of concrete are characterized by a relation between the stress and strain components (or forces and displacements) on the microlevel (Bažant and Ožbolt, 1992) (or, more precisely, the mesolevel). The stress–strain relations are defined not in terms of the macrolevel continuum tensors, but in terms of the stress and strain vectors on planes of all possible orientations within the material, called the microplanes.

The microplane model, M7, which is designed for an explicit numerical algorithm, has been shown capable of realistically predicting the concrete damage behaviors over a broad range of loading scenarios (Caner and Bažant, 2013a, 2013b; Dönmez and Bažant, 2017, 2019; Dönmez et al., 2020a; Dönmez et al., 2020b; Rasoolinejad and Bažant, 2019). It is always convergent, robust, and has been applied in dynamic problems with more than $3 \cdot 10^{7}$ unknowns. The present simulations are performed on ABAQUS, where M7 is inserted as a user’s material subroutine VUMAT.

To prevent spurious mesh sensitivity, the crack band model is used as the localization limiter (Bažant and Oh, 1983). To minimize accuracy loss due to scaling of the post-peak, a constant element size $h_{c}$ (approximately equals to the maximum aggregate size of concrete) is adopted for specimens of all sizes as shown in Figure 8. All the computations include four-node linear tetrahedron elements for concrete and stud, eight-node brick elements for I-beam, and two-node beam elements in 3D space for steel reinforcement. A perfect bonding feature between the concrete and rebar is used. The frictionless contact behavior might be incorrect in the case of modeling a composite beam under bending actions. Because the normal stress acting on the steel beam-concrete interface amplifies the influence of the friction on the contact. However, in the pushout tests, there is no such confinement on the interface, which makes it possible to neglect the friction between the steel beam and concrete. Therefore, frictionless contact is defined between the concrete and steel I-Beam flange. The same contact features are employed between the surfaces of stud and concrete since there are significant separations between the studs and the surrounding concrete at failure (see Figure 6(a)). To calibrate the free parameters of M7, The results of concrete compressive strength tests were used. The adjustable parameters of M7 sufficed to match the main properties, that is, post-peak behavior, fracture energy, and strength values. The steel I-beam is modeled as bilinear elasto-plastic material. The result of the calibration at the material level is also shown in Figure 3(b), where the simulation of the concrete compression tests is shown as the dark solid line. Studs and reinforcing steel materials are modeled according to their stress-strain behavior shown in Figure 3. The load-deflection curves as the results of the FE analysis are shown in Figure 9.

Figure 8.

FE discretization of the pushout specimens using constant element size for the concrete and stud and the corresponding total element numbers.

Figure 9.

Results of the simulations of the pushout specimens with M7 and corresponding maximum principal strain ( $ε_{I}$ ) pattern at peak load, $P_{u}$ , in the vicinity of studs for: (a) small, (b) medium, and (c) large size.

The constructed numerical model fit the test data for all sizes. The peak loads of the numerical results match the test results with negligible errors. Initial stiffness values of the test and FE results have small differences. The failure modes obtained from the numerical results are very similar to tests. The governing failure mode attained from the computations is a mixed-mode of failure which is the concrete crushing and stud yielding. The micro-cracking (or local crushing) behaviors of concrete attained at the maximum load are demonstrated in Figure 9 by using the maximum principal logarithmic strain components. The yielding of the stud is shown in Figure 10 including a comparison to test results. The deformation of stud displays a similar pattern with the test results. Using tetrahedron elements for both concrete and stud yields better results compared to the hexahedron elements concerning the initial stiffness and mixed-mode failure behavior of the specimens. There are of course some drawbacks too. The displacements at the maximum loads have deviations from the test data for the large size. However, the stud yielding controlling the overall responses of the pushout specimens is more pronounced for the large-sized specimens similar to the test data.

Figure 10.

Stud yielding of the small and large size specimens and logarithmic strain components of the numerical simulation after the stud yielding (for small size).

Conclusion

This study is aimed to reveal the size effect behavior of single stud welded on the I-beam flange. This kind of size effect sourced by the individual failure of connectors can be considered as the failure of the substructure (meso-level). It is known that doubling the stud number does not double the entire shear capacity. The consecutive failure of connectors in a group of stud rows also induces another kind of size effect as a whole (macro-level) because of the fracture energy and released energy imbalance of each single stud row. The superposition of these two size effects, sourced by different mechanisms, would be significantly pronounced in design respects. This mechanism requires further research in terms of the size effect analysis. In the current examination, experimental and numerical procedures to demonstrate the variation of the nominal strength of pushout specimens are presented. The main results are listed as follows:

The pushout tests on headed shear studs exhibit Type 2 size effect. The slope of the size effect curve, which fits the experimental results, has an asymptotic slope of −1/2. But failure modes can change the significance of strength reduction. And concrete strength as one of the main parameters affecting the failure mode change can influence the transitional size parameter, $D_{0}$ , between the plastic analysis and the brittle failure regarding the size effect curve.

The source of the size effect is mainly the micro-damage and macro-cracking of the concrete material. The damage of the concrete has a quasibrittle nature. Moreover, the response of stud material is not purely elasto-plastic. The uniaxial tensile tests illustrate a pre-peak nonlinear behavior which usually reflects the stiffness degradation because of the damage in the microstructure of the metal alloy. Moreover, the post-peak response exhibits a gradual softening, which is an indication of damage and fracturing of the material. Similar results were obtained in Liu and Alkhatib (2013). This type of response is a common behavior in quasibrittle materials that presents a size effect. Therefore, the response of the stud might be another source of the overall size effect demonstration of the connector.

Depending on the secondary parameters (such as the aspect ratio ( $h / D$ ), concrete compressive strength ( $f_{c}$ ), the rows and numbers of the used studs, etc.) a size effect factor is needed in the design codes in order to maintain the safety design margins and avoid excessive safety factors.

The FE analysis, including a realistic concrete damage model, is implemented to validate the test results. Microplane model M7 is capable of modeling the concrete material in pushout specimens (or steel-concrete composite beams and slabs) on where the singularity plays an important role.

The combination of the strength reduction with a larger size of studs and the capacity reduction due to the consecutive failures of the studs on the beam itself may induce a significant size effect for SCC beams and studs, which has analytically shown in Bažant and Vitek (1999a, 1999b). However, this phenomenon requires a comprehensive numerical validation.

Footnotes

Acknowledgements

The author would like to thank Lokman Isik, Prof. Senol Ataoğlu, Dr. Tuğrul Turan, and Cengiz Sengül for their help during the experimental work.

Declaration of conflicting interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research described in this paper was financially supported by the Istanbul Technical University, BAP Unit with a project number of 40860.

ORCID iD

Ahmet Abdullah Dönmez

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