Flexural and longitudinal shear performance of precast lightweight steel–ultra-high performance concrete composite beam

Abstract

In this study, the flexural and longitudinal shear performances of two types of precast lightweight steel–ultra-high performance concrete (UHPC) composite beams are investigated, where a cluster UHPC slab (CUS) and a normal UHPC slab (NUS) are connected to a steel beam using headed studs through discontinuous shear pockets and full-length shear pockets, respectively. Results show that the longitudinal shear force of the CUS is greater than that of the NUS, whereas the interfacial slip of the former is smaller. Owing to its better integrity, the CUS exhibits greater flexural stiffness and a higher ultimate bearing capacity than the NUS. To further optimize the design parameters of the CUS, a parametric study is conducted to investigate their effects on the flexural and longitudinal shear performances. The square shear pocket is shown to be more applicable for the CUS, as the optimal spacing between two shear pockets is 650 mm. Moreover, a design method for transverse reinforcement is proposed; the transverse reinforcement is used to withstand the splitting force caused by studs in the shear pocket and prevent the UHPC slab from cracking. According to calculation results, the transverse reinforcement can be canceled when the compressive strength of UHPC is 150 MPa and the volume fraction of steel fiber exceeds 2.0%.

Graphical abstract

Keywords

precast steel–UHPC composite beam / flexural performance / longitudinal shear performance / parametric study / transverse reinforcement ratio

1 Introduction

Owing to its high strength, toughness, and durability, ultra-high performance concrete (UHPC) is used as a concrete slab material to fully exploit the high tensile strength of steel and the high compressive strength of UHPC [1–4]. Compared with conventional composite beams with a deck thickness of 250–400 mm, steel–UHPC composite beams can achieve a significantly smaller thickness and a lower self-weight. Meanwhile, considering the superior durability of UHPC, using the abovementioned combination in bridges can result in longer spans with better life cycle performance and a longer service life [5,6]. Hence, a promising opportunity for developing steel–concrete composite bridges arises, as 14 actual bridges in China span from 31.2 to 1933.6 m [7].

Steel–UHPC composite beams are generally categorized into two types (see Fig.1) based on the construction method adopted: the classical cast-in-place steel–UHPC composite beam with a monolithic UHPC slab (MUS) and a precast steel–UHPC composite beam, which offers accelerated construction, less environmental disturbance, less resource consumption, and reliable quality control. The precast UHPC slab of the normal UHPC slab (NUS) comprises a full-length shear pocket and is connected to a steel beam using uniformly distributed studs, as shown in Fig.1(b). However, the precast UHPC slab of the cluster UHPC one (CUS) comprises uniformly distributed shear pockets, which are connected to a steel beam using a cluster stud, as shown in Fig.1(c). Notably, the precast UHPC slab cast in a prefabricated factory with reserve shear pockets will be connected to a steel beam through these pockets, which is one of the pioneering trends.

Liu et al. [8] investigated the flexural performance of a precast steel–UHPC composite beam, and the test results showed that using a UHPC slab was conducive to improving the ductility of the abovementioned beam; furthermore, the thickness of the UHPC slab can be adjusted as required. Luo et al. [9] evaluated 40 steel–UHPC composite plates and eight steel–UHPC composite beams to investigate their cracking characteristics in the transverse and longitudinal directions. The results showed that the occurrence and growth of cracks in the UHPC layers were effectively restricted by increasing the reinforcement ratios, thereby reducing the thickness of the UHPC cover. Meanwhile, Hu et al. [10] performed a four-point bending test on a precast steel–UHPC composite beam. They observed no splitting cracks in the UHPC shear pockets as well as no concrete crushing on the surface of the UHPC slab when the beam specimens failed. Luo et al. [11] investigated the transverse bending behavior of steel–UHPC composite beams under a negative bending moment. They highlighted that the thickness of the UHPC layer did not significantly affect the crack spacing and cracking stress, whereas the stud spacing was the most important factor affecting the sliding performance of the specimen. Luo et al. [12] reported that the shear capacity of studs was sufficient even without reinforcement arranged in the UHPC slab and that steel fibers affected the flexural behavior of steel–UHPC composite beams. Zhu et al. [13] investigated the crack load and location of steel–UHPC composite beams with wet joints under a negative bending moment. Wang et al. [14] constructed nine steel–UHPC composite beams to investigate the failure mode, deflection, relative slip, and strain distribution with different interface treatments. Most of the existing research focused on steel–UHPC composite beams with monolithic slabs, and their flexural properties and crack behavior have been widely investigated. However, research pertaining to the flexural and longitudinal shear performances of precast steel–UHPC composite beams, where the UHPC slab is connected to a steel beam using post-poring shear pockets, is limited. Consequently, research pertaining to the relationship between the shear pocket arrangement and the bearing capacity of precast steel–UHPC composites is almost non-existent. In addition, few researchers are focusing on the development of methods for calculating the reinforcement in UHPC slabs with shear pockets.

Head studs are typically utilized as shear connectors to enhance structural integrity and hence the bearing capacity. Researchers have extensively investigated the shear performance of these studs. Wang et al. [15,16] tested 24 studs in UHPC, including six single studs and 18 grouped stud specimens, while considering shear pockets and then proposed a load–slip relationship. Tong et al. [17] conducted six push-out tests under static loads on grouped studs in high-strength steel–UHPC composite beams and emphasized the importance of an appropriate stud arrangement. Ding et al. [18] performed seven series of push-out tests while considering different parameters, including the diameter, aspect ratio, and spacing of studs, to investigate the shear behavior of grouped stud connectors embedded in UHPC. Fang et al. [19] conducted 18 push-out tests using various casting methods, stud diameters, slab thicknesses, shear pocket geometries, and shear stud arrangements. They discovered that the stud diameter and slab thickness significantly affected structural performance and that the appropriate shear pocket geometry is conducive to improving the shear behavior of connectors. In addition, researchers investigated the fatigue shear behavior of rubber-sleeved stud shear connectors to optimize the shear force distribution in connectors and improve the connection ductility in composite structures composed of high-strength concrete [20]. Numerous push-out tests were performed, and the shear behavior of single and grouped studs was investigated. However, the longitudinal shear performance of studs embedded along precast steel–UHPC composite beams, which affects the structural behavior in terms of strength, stiffness, and ductility, has not been investigated sufficiently.

In this study, the flexural and longitudinal shear performances of precast steel–UHPC composite beams, including the NUS and CUS, are investigated based on a comparison with numerical results. Additionally, a parametric study is conducted to analyze the relationship between the shear or flexural performance and several variables of the studs, such as the size, spacing, and configuration. Subsequently, a mathematical method for transverse reinforcement is proposed that allows the reinforcement to withstand the splitting force caused by studs and to prevent the UHPC slab from cracking.

2 Model establishment and verification

2.1 Finite element (FE) model

The flexural performance of a steel–UHPC composite beam is typically investigated via a four-point bending test [10], as shown in Fig.2. The setup for this test involves hinge supports arranged at both ends of the test beam. A load is applied directly to the transfer beam; thus, the beam specimens are loaded with a pure bending moment. Based on the test specimens of Liu et al. [8] and Hu et al. [10], which were loaded via the four-point bending test, an FE model of the NUS and CUS were established using the ABAQUS software to simulate the flexural performance of two types of precast steel–UHPC composite beams.

Fig.3 shows the FE models of the two types of precast steel–UHPC composite beams. The UHPC precast slabs, shear pocket, headed stud, and steel beam were modeled using three-dimensional (3D) eight-node elements (C3D8R), and the reinforcement was simulated using a truss element (T3D2) embedded in the UHPC slab. Surface-to-surface contact was established to simulate the contact interaction between the UHPC and studs, UHPC and steel beam, and UHPC slab and shear pocket interface. A friction coefficient of 0.3 [21] was used in the tangential direction for the contact properties between the UHPC and studs, whereas a friction coefficients of 0.6 [22] was set in the tangential direction for the contact properties between the UHPC slab and shear pockets. A hard contact was defined in the normal direction. To improve the calculation accuracy and efficiency, mesh sizes of 50, 2, and 8 mm were set for the UHPC slab [14], studs, and steel beam [23], respectively. In addition, the boundary conditions of the FE model were set based on experiments [8,10]. The beam was set as a fixed hinge support and movable hinge support at the left and right ends, respectively, as shown in Fig.3. Moreover, a force control strategy selected based on tests conducted by Hu et al. [10] and Liu et al. [8] was applied to model the loading condition.

2.2 Material models and properties

The compressive/tensile strengths and elastic modulus of the UHPC are listed in Tab.1. The constitutive relations of the UHPC under tension and compression were determined based on studies by Zhang et al. [24] and Yang and Fang [25], respectively, as shown in Fig.4. The tension and compression constitutive relations of the UHPC are expressed as follows:

(1)

where f_ct is the average tensile stress at the hardening stage; f_c is the compressive strength. ɛ_ca is the strain deviating from the linear stage; ɛ_pca is the ultimate strain; w is the crack width; w_p is the crack width corresponding to the peak stress, which is 0.25 in this study; p is the fitting parameter, whose value is 0.95; ɛ₀ is the uniaxial peak strain, whose value is 3500; ξ = ɛ/ɛ₀; and n = E_c/E_cs, where E_c is the tangent modulus at the initial stage and E_cs is the secant modulus at peak stress.

The CDP model was used to describe the UHPC behavior. The plasticity parameters, including the flow potential eccentricity, dilation angles, ratio of the biaxial/uniaxial compressive strength, and viscosity coefficient input to the CDP model, and their values were 0.1, 30°, 1.16, 0.6667, and 0.0005, respectively. The tensile damage parameters were defined based on Eq. (2), as proposed by Krahl et al. [26]. The compressive damage parameters were defined based on Eq. (3), as proposed by Wang et al. [27].

(2)d_{\text{t}}\left({\epsilon }_{\text{t}}\right)=-0.114+0.872\left(1-{\text{e}}^{-{\epsilon }_{\text{t}}/0.0005}\right)+0.185\left(1-{\text{e}}^{-{\epsilon }_{\text{t}}/0.014}\right),

(3)d_{\text{c}}\left({\epsilon }_{\text{c}}\right)=\frac{0.3{\epsilon }_{\text{c}}{E}_{\text{c}}}{{\sigma }_{\text{c}}+0.3{\epsilon }_{\text{c}}{E}_{\text{c}}},

where d_t is the tensile damage factor, d_c is the compressive damage factor, ɛ_t is the tensile strain, ɛ_c is the compressive strain, and σ_c is the compressive stress.

The modulus of elasticity and Poisson’s ratio of the steel were 200000 MPa and 0.3, respectively. The stress–strain relationship of the stud, reinforcement, and steel girder was modeled based on a study by Esmaeily and Xiao [28], as shown in Fig.5. The stress–strain relationship in the hardening stage can be expressed as

(4)\begin{array}{rl}& \sigma =\\ & \frac{E_{\text{s}}\left(1-k_4\right)\left[{\epsilon }^2+2k_2\left(k_4-1\right)E_{\text{s}}\left|\epsilon \right|+E_{\text{s}}{\epsilon }_{\text{y}}\left(k_1^2{k}_4-2k_1{k}_2{k}_4+k_2^2\right)\right]\epsilon }{{\epsilon }_{\text{y}}\left|\epsilon \right|{\left(k_1-k_2\right)}^2},\end{array}

where k₁ is the ratio of the strain at the start of strain hardening to the yield strain, k₂ is the ratio of the strain at the peak stress to the yield strain, k₃ is the ratio of the ultimate strain to the yield strain (see Fig.5), and k₄ is the ratio of the peak stress to the yield stress.

2.3 Finite element verification

A comparison of the failure modes of the UHPC slabs obtained both experimentally and numerically is presented in Fig.6. The FE results show that the local principal tensile stress at the loading position exceeded than that at other positions of the UHPC slab, which is consistent with the experimental results. In addition, the cracking of the CUS in the side face and shear pockets of the UHPC slab is consistent with the experimental results.

The load mid-span deflection and load mid-strain obtained from the FE results were compared with those obtained from the test results. Fig.7 shows the comparison results. It was found that the FE results matched well with the test results in both the elastic and elastoplastic stages.

As shown in Fig.8 and Fig.9, the FE results of the steel strain at the bottom flange and those of the UHPC strain at the top surface of the UHPC slab matched well with the test results. The yield load and flexural stiffness of the NUS and CUS obtained from the FE simulation were 253 kN and 20.3 kN/mm, and 727 kN and 42.3 kN/mm, respectively, which were similar to the experimental results (Tab.2). The ultimate loads of the NUS and CUS obtained from the FE simulation were 463 and 940 kN, respectively, which were also similar to the test values of 470 and 935 kN, respectively. To further validate the FE method, the yield load, ultimate load, and flexural stiffness of the two specimens reported in Ref. [23] were compared. The test results agreed well with the FE results, and the comparison results are listed in Tab.2. The verification results above confirm that the FE modeling proposed herein is reasonable.

3 Comparison of two types of precast lightweight steel–ultra-high performance concrete composite beams

FE models of the NUS and CUS were established using the modeling method above to compare their flexural and longitudinal shear performances under positive and negative bending moments. As shown in Fig.10, the dimensions of the NUS and CUS, which can fully reveal the flexural and longitudinal shear performances, were used in the FE model by referring to Ref. [10]. Notably, the stud quantities and reinforcement ratios of the UHPC slabs used in these two types of composite beams were identical.

3.1 Positive flexural and longitudinal shear performances

3.1.1 Failure mode of ultra-high performance concrete slab

The maximum principal tensile stress of the UHPC slab under the ultimate loading is shown in Fig.11. The results show that the maximum principal tensile stress on the top surface of the UHPC slabs, which was at the loading position and the local position of the wet joint of the shear pocket, exceeded 6.9 MPa. The damaged area was on the bottom surface of the precast UHPC slab for both the CUS and NUS.

The main strain vector was obtained to compare the crack patterns of the UHPC slabs of the CUS and NUS, as shown in Fig.12. The results show that cracks developed in the former along the edges of the shear pockets in the horizontal and longitudinal directions; thus, the cracks were primarily distributed in the shear pocket. Because of the stress concentration, some of the cracks extended from the four corners of the shear pockets to the UHPC slab. Regarding the UHPC slab of the NUS, cracks were generated before the headed stud and then developed along the shear pocket edge near the loading area. The crack lengths at the bottom of the UHPC slabs of the CUS and NUS were approximately 2900 and 3100 mm, respectively.

3.1.2 Load–deflection response

A comparison of the load mid-span deflections of the NUS and CUS obtained from the FE analysis is shown in Fig.13. Both the NUS and CUS exhibited linear behavior when the applied load was smaller than the yielding load. In the elastic stage, the flexural stiffness (K_s) of the CUS and NUS was 42.3 and 34.5 kN/mm, respectively. The flexural stiffness of the former was 23% higher than that of the latter. Additionally, the yield and ultimate loads of the CUS were 727 and 940 kN, respectively, which were 1.05 and 1.12 times greater than those of the NUS, respectively.

3.1.3 Interfacial slip

The interfacial slip between the UHPC slab and steel beam at the yield and ultimate loads is listed in Tab.3. The overall slip value of the NUS exceeded that of the CUS. Moreover, the S₁ to S₂ ratio of the former exceeded that of the latter; specifically, the S₁ to S₂ ratios were approximately 0.53 and 0.31 at the mid-span of the NUS and CUS, respectively. The results above indicate that the slip of the CUS was primarily generated after the yield load.

Fig.14 shows the interfacial slip of the CUS and NUS at loads of 420 and 840 kN, respectively. Because the constraints at both ends of the beam were not symmetric, the slips in the left and right spans were not symmetric. The interfacial slip of the NUS was significantly greater than that of the CUS. A greater slip indicates a weaker interaction between the UHPC slab and steel beam. Furthermore, a weaker interaction indicates lower integrity, which reduces the flexural bearing capacity of the composite beam.

3.1.4 Shear transfer response

The shear force at the root of the headed studs under the ultimate load is shown in Fig.15(a). The shear force exhibited the maximum and minimum values at the end of the beam and at the mid-span, respectively. The shear force distributions of the NUS and CUS were similar, although the overall shear force of the CUS was greater than that of the NUS. The maximum shear force of the CUS was 250 kN, which was 12.9% higher than that of the NUS. In addition, the local shear force distributions of the two composite beams were different. In particular, the shear force distribution of the NUS presented a wave shape from the beam end to the quarter span.

The distribution of the uplift force under the ultimate load is shown in Fig.15(b). The uplift force of the NUS presented a wave shape from the beam end to the quarter span. The maximum uplift force of the NUS in group II was 171 kN, whereas that of the CUS at the corresponding positions was 245 kN. Notably, the uplift and shear forces of the NUS near the loading position changed abruptly, and the abrupt change in the shear force was more conspicuous than that in the uplift force. This was primarily due to the local stress concentration. In addition, the effect of the loading point was indicated in an area, which may cause the influence range of the stress concentration at the loading point on the shear and uplift forces to differ slightly.

3.2 Negative flexural and longitudinal shear performances

3.2.1 Load–deflection response

A comparison of the load mid-span deflection of the NUS and CUS under a negative bending moment obtained from the FE analysis is shown in Fig.16. Both the NUS and CUS exhibited linear behavior when the applied load was smaller than the yielding load. In the elastic stage, the flexural stiffness (K_s) of the CUS and NUS was 43.8 and 37.6 kN/mm, respectively, which were similar to the positive bending stiffness of 42.3 and 34.5 kN/mm, respectively. In addition, the yield and ultimate loads of the CUS were 212 and 480 kN, respectively, which were 1.15 and 1.23 times greater than those of the NUS, respectively. Meanwhile, the yield and ultimate loads of the NUS and CUS under a negative bending moment were much smaller than those under a positive bending moment.

3.2.2 Shear transfer response

The shear force at the root of the headed studs under the ultimate load is shown in Fig.17(a). The shear forces of the NUS and CUS indicated maximum values at beam lengths of 0.5 and 0.9 m, respectively, whereas their minimum values were indicated in the mid-span. In particular, the stud shear force of the NUS in the mid-span region was extremely low. The shear force distributions for the NUS and CUS were similar, although the overall shear force of the CUS exceeded than that of the NUS. The maximum shear force of the CUS was 125 kN, which was 12.5% higher than that of the NUS.

The distribution of the uplift force under the ultimate load is shown in Fig.17(b). The uplift force of the NUS presented a wave shape from the beam end to the quarter span. The maximum uplift force of the NUS in group III was 78 kN; however, that of the CUS at the corresponding positions was 92 kN, which was 1.18 times greater than that of the former.

3.2.3 Interfacial slip

Fig.18 shows the interfacial slip of the CUS and NUS at loads of 180 and 360 kN under a negative flexural moment. Because the constraints at both ends of the beam were not symmetric, the slips in the left and right spans were not symmetric. The interfacial slip value of the former was significantly higher than that of the latter, as was the case under a positive flexural moment. Typically, a large slip indicates a weaker interaction. Furthermore, a weaker interaction indicates lower integrity, which reduces the flexural bearing capacity of the composite beam.

4 Evaluation of design parameter

To optimize the design parameters, the shape and size of the shear pockets, the distance between two shear pockets, and the arrangement of studs, which affect the longitudinal shear and flexural performances of the CUS under a positive flexural moment, were investigated.

4.1 Shape and size of shear pocket

The effects of three types of shear pockets (square, circular, and rounded square shear pockets) on the bearing capacity of the CUS were investigated. To further clarify the effect of the shear pocket size on the shear performance and bearing capacity of the CUS, the optimal length from the center of the outermost stud to the edge of the shear pocket (a) was investigated.

The stud spacing was five times the stud diameter, and the length from the center of the outermost stud to the edge of the shear pocket (a) was 25, 35, and 45 mm. For a better comparison, the lengths from the center of the outermost stud to the edge of the shear pocket (a) of the three types of shear pockets were set to be equal, as shown in Fig.19.

The maximum principal tensile stress distributions for different types of shear pockets were different (see Fig.20). The maximum principal tensile stress of the UHPC slab with square pockets was significantly lower than that with a circular pocket. In addition, the stress influence area caused by the square shear pocket was smaller than that caused by the circular shear pocket.

The load mid-span deflection of the CUS with different sizes and shapes of shear pockets is shown in Fig.21. When a = 25 mm, the stiffness of the CUS with square and circular shear pockets was 42.3 kN/mm, whereas it was 38.5 kN/mm for the CUS with rounded square shear pockets (Fig.21(a)). Furthermore, the yield loads of the CUS with square, circular, and rounded square shear pockets were 727, 720, and 650 kN; meanwhile, the ultimate loads of the CUS with square, circular, and rounded square shear pockets were 920, 902, and 880 kN, respectively (Tab.4). When a ranged from 25 to 45 mm, the flexural stiffness and strength of the CUS with the square shear pocket changed slightly.

As shown in Tab.4, the CUS with square and circular shear pockets exhibited higher rigidity and bearing capacity when the three shear pocket sizes were identical. Although the flexural stiffness of the CUS with square shear pockets was identical to that of the circular shear pockets, the bearing capacity of the former was greater than that of the latter. Based on the comparison results, the square shear pocket is more applicable for the steel–UHPC composite beam, and the length from the center of the outermost stud to the edge of the shear pocket should exceed 25 mm.

4.2 Spacing between two shear pockets

In addition to the type and size of the shear pockets, the spacing between the two shear pockets affects the flexural performance of the precast steel–UHPC composite beam. Thus, the CUS (a = 45 mm) with shear pocket (square shear pocket) spacings l_s = 550, 600, 650, and 700 mm was investigated. Based on the load–deflection curves (Fig.22), the yield and ultimate loads increased as the spacing increased from 550 to 650 mm, whereas they decreased as the spacing increased from 650 to 700 mm.

Fig.23 shows the distribution of the shear force at the root of the studs in each shear pocket under the ultimate load. The shear force indicated a maximum and minimum value at the end of the beam and the mid-span, respectively. As the spacing of the shear pockets increased from 550 to 650 mm, the shear force of the studs in each shear pocket decreased gradually. The shear force of the studs in each shear pocket began to increase when the spacing was increased to 700 mm, where the end shear pockets were located at the constraints of the beam. Fig.24 shows the interfacial slip under different spacings at the ultimate load. Similar to the shear force trend, as the spacing of the shear pocket increased from 550 to 700 mm, the interface slip first decreased and then increased. The results above show that a spacing of 650 mm between the two shear pockets is feasible.

4.3 Arrangement of headed studs

Based on the previous section, the maximum and minimum shear forces were indicated at the end and mid-span of the beam, respectively. To closely match the distribution of the longitudinal shear force, a CUS with a non-uniform stud number in square shear pockets was investigated. The CUS beam, whose spacing between two shear pockets and a were 650 and 45 mm, respectively, was used as the reference beam. Five models with different cluster stud distributions in the shear pocket were analyzed, as listed in Tab.5.

The load mid-span deflection curves are shown in Fig.25. The flexural stiffness and yield load were identical. The bearing capacities of SP-1–SP-5 were 940, 923, 885, 927, and 912 kN, respectively. The bearing capacity of SP-3 was the minimum; meanwhile, the bearing capacities of SP-1, SP-2, SP-4, and SP-5 were similar. Fig.26 shows the distribution of the shear force at the root of the studs in each shear pocket under the ultimate load. The shear force distribution associated with the ultimate load was associated closely with the stud arrangement. For SP-1 and SP-4, the shear force decreased from the end to the mid-span of the beam. Because of the staggered sequence of the number of studs in the shear pockets, the shear force distributions of SP-2, SP-3, and SP-5 presented a wave shape from the beam end to the mid-span. Although the total number of studs of SP-5 was the least, its flexural strength and shear force distribution were not affected significantly, which indicates that the stud arrangement for SP-5 is the most reasonable.

5 Calculation method of splitting reinforcement in ultra-high performance concrete slab

The shear force of the studs caused a splitting stress σ_split in the longitudinal direction of the UHPC slab and shear pocket [30], as shown in Fig.27. The equation for calculating the splitting force T_split generated by splitting stress σ_split is

(5)T_{\text{split}}=t{\int }_0^h{\sigma }_{\text{split}}\text{d}x,

where t is the thickness of the UHPC slab and h is the length of the tension zone.

Several calculation methods for the splitting force T_split proposed by Johnson and Oehlers [31], Shen et al. [32], and Badie et al. [33] are listed as follows. If the tensile strength of UHPC cannot withstand the splitting force, then transverse reinforcement must be arranged in the UHPC slab to prevent the UHPC slab from cracking.

Johnson and Oehlers [31]:

(6)T_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}=0.318P\left(1-a/\phantom{ab}b\right),

Shen et al. [32]:

(7)\begin{array}{c}{T}_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}=0.35P\left(1-2.43a/\phantom{ab}b\right),a/\phantom{ab}b⩽0.2,\\ T_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}=0.318P\left(1-a/\phantom{ab}b\right),a/\phantom{ab>0.2}b>0.2,\end{array}

Badie et al. [33]:

(8)T_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}=0.25P\left(1-a/\phantom{ab}b\right),

where P is the concentrated force of the studs in the shear pocket, a is the width of the shear pocket, b is the effective width of the UHPC slab.

Based on Eqs. (6)–(8), the splitting force can be solved if P is solved. According to Ref. [34], the shear strength of a headed stud embedded in UHPC is better than that embedded in the NUS, which can be calculated as

(9)P=A_{\text{sc}}{f}_{\text{u}}+0.16\left(0.0119f_{\text{c}}-0.983\right)f_{\text{c}}{d}_{\text{b}}^2,

where f_u is the tensile strength of the headed stud, A_{\mathrm{s}\mathrm{c}} is the area of the stud shank, f_c is the compressive strength of the UHPC, and d_b is the nominal diameter of the stud shank.

The splitting force caused by the studs can be solved by substituting Eq. (9) into Eqs. (6)–(8). To prevent the UHPC slab from cracking, the resistance of the transverse reinforcement should be greater than the splitting force. When the tensile contribution of the UHPC is not considered, the relationship between the splitting force and reinforcement ratio can be expressed as

(10)T_{\text{split}}⩽\frac{{\rho }_{\text{s}}\cdot e\cdot s\cdot f_{\text{t}}}{2},

where s is the spacing between two shear pockets; f_t is the tensile strength of the steel bar; e is the effective thickness of the slab, which can be set as 1.9d_s (d_s = diameter of stud connector), as suggested by Johnson and Oehlers [31]; and ρ_s is the transverse reinforcement ratio.

Thus, the critical transverse reinforcement ratio ({\rho }_{\text{sw}}) can be expressed as

(11){\rho }_{\text{sw}}⩾\frac{2T_{\text{split}}}{f_{\text{t}}\cdot s\cdot e}.

When the tensile contribution of the UHPC is considered, the relationship between the splitting force and reinforcement ratio can be expressed as

(12)T_{\text{split}}⩽\frac{{\rho }_{\text{s}}\cdot e\cdot s\cdot f_{\text{t}}}{2}+\frac{e\cdot s\cdot f_{\text{spl}}}{2},

where the f_spl is the splitting tensile strength of the UHPC.

When the compressive strength of the UHPC ranges from 120 to 200 MPa, its splitting tensile strength can be expressed as [35]

(13)f_{\text{spl}}=\left(\frac{0.94V_{\text{f}}{l}_{\text{f}}{b}_{\text{f}}}{d_{\text{f}}}+0.67\right)\sqrt{f_{\text{c}}},

where V_f is the volume fraction of the steel fibers, l_f is the length of the steel fibers, d_f is the diameter of the steel fibers, and b_f is the bond factor based on the fiber type (b_f = 0.5 for fibers with a circular section; b_f = 0.75 for hooked or crimped fibers).

Thus, the critical transverse reinforcement ratio can be expressed as

(14){\rho }_{\text{sc}}⩾\frac{2T_{\text{split}}}{f_{\text{t}}\cdot s\cdot e}-\frac{f_{\text{spl}}}{f_{\text{t}}}.

Because of the three different methods for calculating the splitting force, three transverse reinforcement ratios were calculated based on the steel fiber length and diameter of 13 and 0.2 mm, respectively, as shown in Tab.6 [10]. The result shows that the maximum transverse reinforcement ratio of the UHPC slab was 3% when the tensile contribution of the UHPC was not considered. However, the transverse reinforcement ratio reduced significantly when the tensile strength of the UHPC was considered. Under the same fiber volume content, the calculated transverse reinforcement ratio decreased as the compressive strength of the UHPC increased. Based on the calculation results above, one can conclude that the transverse reinforcement is not necessary when the compressive strength of the UHPC and the volume fraction of steel fibers exceed 150 MPa and 2%, respectively.

The volume fraction of the steel fibers of the unreinforced UHPC slab under different compressive strengths of the UHPC is shown in Fig.28. Based on the calculation results, the volume fraction of steel fiber in the UHPC with a compressive strength exceeding 120 MPa should be greater than 2.3%, whereas those exceeding 150 and 180 MPa should be greater than 1.8% and 1.5%, respectively. Notably, the calculation method is only applicable to the design of transverse splitting reinforcement for the UHPC slab, which is subjected to a positive bending moment. Furthermore, the reinforcement in the transverse bending and shear should be considered separately in the calculation.

6 Suggestion and conclusions

The flexural and longitudinal shear performances of the NUS and CUS of two types of precast steel–UHPC composite beams were investigated via comparison to figure out the superior one in structural performance. The following conclusions were inferred based on the result of the comparison as well as that from a parametric study.

1) The longitudinal shear and uplift force of the CUS were greater than those of the NUS, whereas the interfacial slip of the former is smaller for better integrity. Thus, the longitudinal shear and flexural performances of the CUS, which is subjected to positive and negative bending moments, are superior when the flexural stiffness is 21% and 28% higher, respectively, which caused the ultimate bearing capacity to increase by 1.12 times and 1.23 times, respectively.

2) Regarding the CUS, a square shear pocket with a length from the center of the outermost stud to the shear pocket edge (a) that exceeded 25 mm improved the flexural performance. In addition, the recommended shear pocket spacing was 650 mm when the headed studs inside were arranged in a decreasing number from the beam end to the mid-span.

3) A mathematical method for transverse reinforcement was proposed to withstand the splitting force caused by the studs and to prevent the UHPC slab from cracking. Specifically, the splitting reinforcement can be canceled when the volume fraction of the steel fiber exceeded 2.3%, 1.8%, and 1.5% in the UHPC with compressive strengths of 120, 150, and 180 MPa, respectively. It is recommended that the transverse reinforcement is negligible when the compressive strength of the UHPC and the volume fraction of the steel fiber exceeded 150 MPa and 2%, respectively.

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