Keywords

1 Introduction

With the increasing demand of submarine tunnels in transportation, the role of immersed tunnels has gradually become significantly important. Engineering challenges of immersed tunnels include large back-silting, high water pressure, complex geological conditions, super-long longitudinal span and super-wide cross-section. Long length of variable-section curved pipe sections, and types of immersed tunnel and the longitudinal structural system have become the key technical issues in tunnel construction [9, 10, 16]. The optimized structural design of connectors can significantly improve bonding between ribs and surrounding concrete to improve bearing conditions.

In recent years, many scholars have researched mechanical properties of steel–concrete composite connectors, and proposed many types of shear connectors [1, 2, 6, 11]. A flexural test [14] used a steel–concrete composite beam, and its result showed that the composite structure had better elasticity, and also performed better under the condition of large deformation. Structural failure was usually caused by yielding and tearing of flange plate and web. A paper researching shear connectors [5] indicated that performances of shear connectors by evaluating the influences of thickness, size, number and loading conditions of connectors would lead to changing patterns of bearing capacities of different structural designs. According to a flexural loading tests report [7] on thin-walled steel–concrete composite structure with bend-up bars and steel studs, bending capacity of steel- concrete composite structure was significantly higher than the reinforced concrete beams. The shear resistances of the different designs differs significantly. According to paper researching U-shaped steel–concrete composite members [17], bearing mechanisms would be impacted by different shear-span ratios, height-to-thickness ratios, connector spacings, and wing plate thicknesses. Accordingly, the bearing capacity, ductility of each specimen and their shear force transfer model were investigated and established. Another research [3] compared and contrasted the failure modes of connectors with different thicknesses, studs and spacings, and obtained respective effects on the shear performance of composite structures. Additionally, they also proposed a new shear strength calculation method of connectors with more approachable calculation theory for the shear bearing capacity. One published paper [13] indicated the shear mechanical properties of short steel bar connectors based on static push-out loading tests. Results show that failure modes of composite members could be generally divided into two categories. The bearing capacity of the connector is affected by welding length. Research results solved the problem that thin UHPC layer generally needs conventional shear connectors.

Based on backgrounds above, plenty of researches for reinforced concrete composite members, and investigations of influencing factors of steel ribs and studs were conducted. However, the calculation methods and theoretical models for shear resistances, designing guidance and bearing capacities were rarely researched. Nowadays, applications of steel connectors, which work as reinforced ribs on steel plates, have been widely used by immersed tunnel projects [12, 15]. In this paper, structural tests were carried out for steel connectors to research bonding forces between concrete and steel plates, and the influence of each designs on the performance of connectors. The combination of loading test and numerical calculations is used to research the overall tension and compression stressing state, voids and connector openings.

2 Loading Tests

2.1 Testing Apparatus

All specimen tested were full-scale models including 11 groups of test pieces in total, and 3 identical specimen are casted and tested in each group, with a total of 78 specimen. The controlled variables of specimen include: void size, tension/compression state of the connector and opening holes. The detailed controlled parameters of the shear connector are listed in Table 1. Size of the concrete specimen is 300 × 450 × 600 mm, and cement used in specimen is C60. The void shape is a triangular prism (as shown in Fig. 1), the void heights are 10 and 20 mm, dimensions in Table 1 are length × width × height, and the void length to height ratio is 10.

Table 1 Detailed parameters of shear connectors
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Fig. 1
A schematic of a steel shell tube connector. It consists of a steel pad with a loading device on top. Void push out test is carried on the specimen.

Schematic diagram of specimen void testing

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For the specimen of which concrete is under compression, the traditional push-out test were conducted. For the specimen of which concrete is under tension, the inner side of the concrete slab that pushes out the specimen is padded, and the outer side is restrained by steel bars, as shown in Fig. 2.

Fig. 2
2 schematics of a steel shell tube connector explain compression and tension. It consists of a steel pad with a loading device on top.

Stress form of connector

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2.2 Experiment Designs

In the loading test, connectors are in the form of L shape steel ribs. Double fillet welding were used for H-beam flange plate, connector web, and flange plate design. Experimental specimen including concrete casting were shown in Figs. 3 and 4.

Fig. 3
A close-up view of an L-shaped steel structure labeled L-ribs on its right is a T-shaped structure labeled welding details.

Steel structure diagram under test

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Fig. 4
A closer view of concrete casting. On its right is the casted concrete specimen.

Pouring process and formed connections

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The tested specimen were casting in four batches. A 150 mm standard cube concrete reference block is reserved for each pouring, and is maintained under the same conditions to obtain concrete strengths. The concrete strength adopts the compressive strength of the 150 mm cube test. Steel used for tested connectors and I beam were Q345 steel. There are four types of connectors with different thicknesses used, including 6 mm, 8 mm, 10 mm, and 12 mm. Standard tests results are carried out respectively [4]. Table 2 shows the results of each batch of concrete test and steel mechanical property test.

Table 2 Test results of mechanical properties of concrete and steel
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2.3 Test Data Analysis

Effects of stress conditions of connectors

The differences of shear-slip curves between connectors under tension and compression are significant. The results of the shear-slip curves of the specimens T1 and T3 were used as examples to discuss loading patterns and mechanisms due to similar trends of all other comparisons. The general trends of the tensile and compressive shear-slip curves are similar. Before the maximum shear limit is reached, there were two loading stages: a quasi-elastic stage and a nonlinear stage as Fig. 5 shows.

Fig. 5
A graph plots shear versus displacement. It plots 2 increasing curves for compression and tension test. Compression has more shear compared to tension.

Comparison of shear slip curves of tension compression connectors (T1, T3)

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The figure above shows relationship between the strain changes of upper/lower surfaces of connectors and the stress state. A great difference between tensile and compressive ones also exist. Test results of T1-3 and T6-3 could be shown as examples, Fig. 6 shows the strain comparison of stress and compression connectors. The strain of upper and lower surfaces of compressive connector are basically identical at initial stage. With increasing shear loading, strains of upper and lower surface begins to differ.

Fig. 6
2 graphs, a and b plot shear versus strain for compressive and tensile connectors. A, plots 2 curves for the bottom and top tension. The curve for top tension increases initially and then drops down with a sudden decrease. B, plots 2 curves for bottom tension and up compression. Both curves have a high shear.

Strain comparison between stressed and compressed connectors

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The strain variation pattern of upper and lower surfaces of the tension connector is basically identical. Before the maximum shear force is reached, the strain increases slowly and increase rate is relatively small. After the maximum shear force is reached, the strain increases rapidly and the increase rate would remain unchanged.

The curvature pattern of web for tension and compression connectors is significantly impacted by stress state. And trend would be different with the increase of the height. The specimen tests of T1-1 and T3-1 were used to discuss changing pattern of the web curvature. Figure 7 is the comparison of the variation of the web curvature for tensile and compressive connectors.

Fig. 7
2 graphs plot height versus curvature for the compressive and tensile specimens. The graphs plot 3 curves for 0.2, 0.5, and 1.0 P subscript U. In the first graph, the curve for 1.0 P subscript U has a greater curvature of approximately negative 200.

Comparison of web curvature changes between tension and compression connectors

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Research on mechanism by voids

Experiments tested three different connectors with voids: L150 × 90 × 10 (L150 for short, same below), L180 × 110 × 10 (L180, same below) and L200 × 125 × 12 (L200, same below). The void between steel and concrete is simulated by using the EVA material with low elastic modulus. The failure mode is discussed by taking L50 as an example and loading tests of crack start, development and failure were shown in the Fig. 8.

Fig. 8
9 specimens are arranged in 3 rows and 3 columns. The Rows are for non-void specimens, void with a height of 10 and 20 millimeters. The columns are for crack generation, crack development, and limit state.

Failure process of different void connectors

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Failure modes of specimen with void and without void are similar. First, inclined cracks are formed at tips of steel ribs, then vertical cracks are developed at bottom of steel ribs, horizontal cracks are developed inside specimen, and finally concrete at bottom zone of steel ribs is crushed.

The results of shear resistance capacity of each group of connectors are shown in Table 3. According to Table 3, it is certain that bearing capacity of specimens with void is significantly lower than specimens without void. In addition, for different types of large-size connectors, effects of different void height on shear resistance are similar. The larger void heigh generally lead to significant reduction of its bearing capacity. When void height reaches 10 mm, the bearing capacity is reduced by a maximum of 16% and a minimum of 9%. When the void height reaches 20 mm, the bearing capacity is reduced by a maximum of 36% and a minimum of 28%. The influences of the void depth by loading tests were similar to conducted results of literature investigating mechanical properties of connectors [8].

Table 3 Comparison of bearing capacity of hollow connectors
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Besides impacts on ultimate bearing capacity of connectors, the void height also affects the shear-slip curves. Figure 9 shows the comparison between the shear-slip curve and stiffness of the void and non-void connectors. From the comparison of the shear-slip curves, it could be certain that voids has crucial effects on bearing capacity of three groups of connectors.

Fig. 9
3 graphs and a bar chart. The graphs plot for shear versus displacement. The column chart plots stiffness versus voids. The bar chart plots 3 columns for non-void, 10 and 20 millimeters void with the highest for non-void equal to 1680. The graphs for non-void have high shear.

Comparison of shear slip curve and stiffness between void and non-void connectors

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Figure 10 shows differences of the web curvature of with or without void connectors. The greater the curvature of the web, the greater the bending moment it bears. The change of the web curvature reflects the change of the bending moment. At the height of 50 mm, the non-empty connector has a large difference in curvature and bending moment, while the difference in curvature and bending moment of the hollow connector at different heights is large, and reaches the maximum at 50 mm. The curvature value between the two reverse bending points of the plate is significantly larger than that of the specimen without void.

Fig. 10
2 graphs plot height versus curvature for non-void and 20 millimeters connectors. The graphs plot 3 curves for 0.2, 0.5, and 1.0 P U.

Comparison of curvature pattern of non-void and with void connector webs

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Research on mechanisms of opening holes

The mechanical properties of the connector with open-holes and connector without holes in the push-out test are different. Figure 11 shows the comparison of failure modes of two settings with similar failure mechanisms. First, inclined cracks occur at the tip of steel ribs, then vertical cracks and horizontal cracks occur in the concrete around steel ribs, and finally concrete at bottom of the angle steel is crushed to fail. The failure mode of connectors with open holes are relatively severe with more and larger cracks on both sides.

Fig. 11
6 specimens are arranged in 2 rows and 3 columns. The Rows are for without opening and with opening connectors. The columns are for oblique crack, crack development, and limit state.

Comparison of failure process between open and non-open connections

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Figure 12 shows the comparison of the shear-slip curves of the open-hole and non-open-hole connectors. From the analysis of the figure, it could be certain that “shear force fluctuation phenomenon” occurs in the curve of the open-hole connector which was caused by easily damaged open-hole part. The damage of open-holes aggravates differential stress distribution. Compared with non-opening connectors, the bearing capacity of connectors with holes is reduced by about 10%. Taking the secant stiffness of 0.5 mm as the stiffness of the connector, it can be found that the opening can lead to a 20% reduction in stiffness. The main reason is that openings reduced effective contact area between the web of connector and concrete, so is the bonding area of surface of the connector and concrete. When area of opening hole at bottom of steel ribs reaches 20%, the bearing capacity is reduced by 10%, indicating that concrete at the opening is partially involved in compression but the compressive strength is not fully achieved. The main reason could be that concrete at the opening can also be constrained by the surrounding concrete to a certain extent to provide compressive strength.

Fig. 12
A graph plot shear versus displacement. It plots 2 increasing curves for with void and none void. The curve for none void has high shear.

Comparison of shear slip curves of with open-holes and without open connectors (T1, T5)

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3 Numerical Analysis of Connectors

3.1 Finite Element Modelling and Settings

Finite element analysis software MSC.MARC was used to simulate loading test. The concrete, steel ribs and connecting flange plate are simulated by the solid element SOLID75. According to the symmetry of structure, only one side structure is calculated for analysis. The FEM model is shown in Fig. 13, in which light red part is concrete, the dark red part is connecting steel ribs, the yellow part is the flange plate connected with the steel ribs, and the green part is the right side plate. Contact algorithm is used for interactions between concrete and the connector, and 0.1 friction coefficient was taken between the steel and the concrete according to reference specification.

Fig. 13
2 F E M models, A and B with concrete, steel connecting ribs, and flange plate. A is a non-void model and B has a void.

Finite element model of connector

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To simulate actual contact interaction at the bottom of the connector, the right backing plate (green one) is used to simulate the actual supporting surface of actual pressure testing machine. Accordingly, the right side of the supporting surface is set as fixed constraints. With considerations of uniformities of the force along the length of the connector, the plane strain and plane stress models were established separately to conduct calculation and comparison.

The non-linear material of concrete element has compressive linear plasticity and tensile non-linear cracking behavior. The material of the steel rib elements is an ideal elastic–plastic material with tested yield strength. And the von Mises yield criterion is used.

3.2 Numerical Calculation Results Analysis

The non-void connector and void connector (void height of 20 mm) were calculated and simulated respectively. The stress calculation results of each model at the limit state are shown in Figs. 14 and 15.

Fig. 14
9 stress distribution models arranged in 3 rows and 3 columns. The Rows are for concrete distribution and steel ribs distribution. The columns are for a solid model, plain strain, and plain stress.

Finite element simulation of non-void components

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Fig. 15
9 F E M distribution models arranged in 3 rows and 3 columns. The Rows are for concrete distribution and steel ribs distribution. The columns are for a solid model, plain strain, and plain stress.

Finite element simulation of with-void components

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The results of three finite element models show that compressive stress distributions of concrete in three FEM models are similar, non-void model concrete compressive stress is higher at a height of about 20 mm from the bottom where the maximum compressive stress has reached design strength. For FEM models with voids, concrete compressive stress is more concentrated and the high stress zone is smaller, which leads to reduction of bearing capacity. The tensile and compressive stress distributions of the connectors of the three models are similar in which tensile stress is greater than compressive stress, showing a mode of tensile-bending stress, and a part of the bottom of connector is yield. The bending stress at the bottom of connector with void is larger than that of the non-void connector, while the deformation and yield range are also larger. The bottom shear stress for solid model and plane strain model is more fully developed, and shear stress decreases rapidly with height. The plane stress model only has a large shear stress in a small zone near the bottom, and the shear stress above the bottom is relatively small. However, all shear stress in the three models are only half of the yielding stress, indicating that for structural design of connectors, bending capacity and deformation are governing.

The comparisons of bearing capacity and stiffness between FEM calculations and loading test results are shown in Tables 4 and 5. In terms of bearing capacity, the solid model and the plane strain model are basically identical, both of which are larger than the plane stress model. In terms of stiffness, the solid model is larger than the planar model. Both the finite element model and the experimental results show that voids have a significant effects on the bearing capacity and stiffness of connectors. For stiffness calculations, results of plane strain model can better simulate the experimental results, whereas the stiffness of the solid model is significantly greater than the tested results, which is mainly because the stiffness of the solid element is enlarged by using contact algorithm.

Table 4 Comparison results of bearing capacity (kN)
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Table 5 Stiffness comparison results (kN/mm)
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4 Theoretical Calculation Method Proposal

According to the analysis of FEM calculation results and experimental verifications, main factors impacting bearing capacity of connector include: (1) the tensile and compressive state of the concrete at the connector, that is, the connector under tension and compression; (2) the concrete void; (3) the opening hole. Compared with the compression state, the bearing capacity of the connector under tension state is reduced by less than 10%. The comparison of the push-out test results of connectors with different of void heights shows bearing capacity of the connector is reduced by 9–15% when the void height reaches 10 mm; the bearing capacity is reduced by 28–36%, when the void is 20 mm. The comparison of the push-out test results of connectors with different degrees of void shows that when void is 10 mm, the bearing capacity of the connector is reduced by 9–15%; when the void is 20 mm, the bearing capacity is reduced by 28–36%.

Both opening holes of connector and voids of between concrete would reduce contact area between bottom of connector and surrounding concrete. Besides these holes also reduced effective area of steel ribs. By comparison, it can be concluded that when connectors designed with no holes and with 10 mm void, bearing capacity is reduced by 15%; when connectors designed with opening hole rate of 20% and without voids, the bearing capacity is reduced by 10%; when connectors designed with hole rate of 20% and with void height of 10 mm, the bearing capacity is reduced by 20%  as Table 6 shows.

Table 6 Two parameter analysis of bearing capacity of connectors with holes and voids
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In project, opening hole rate is generally less than 20%, and the maximum void height is no more than 20 mm. Under such circumstance, calculation formula of bearing capacity considering multiple factors is determined through the above analysis as follows:

$$\begin{aligned} V = & 5.6l_{c} h_{c} \sqrt {f_{c} } k_{1} k_{2} k_{3} \eta \times \left( {1 - \frac{{1.5h_{e} }}{{100}} - \frac{{0.5l_{h} }}{{l_{c} }} + \frac{{1.5h_{e} }}{{100}} \times \frac{{l_{h} }}{{l_{c} }}} \right) \\ & \, \le \left( {l_{c} - l_{h} } \right)t_{c} f_{y} /\sqrt 3 \\ \end{aligned}$$
(1)

where: V = shear resistance with unit of kN; \(\eta\) = stress factor with 1 for compressive elements and 0.9 for tensile elements; \(k_{1}\) is dimensional factor in which \(k_{1} = 2.2\left( {t_{c} /h_{c} } \right)^{2/3} \le 1\); \(k_{2}\) is bottom stress factor in which \(k_{2} = 0.4\left( {t_{f} /t_{c} } \right)^{0.5} + 0.43 \le 1\); \(k_{3}\) is spacing factor, in which \(k_{3} = \left( {s_{c} /h_{c} /10} \right)^{\frac{1}{2}} \le 1\); \(l_{c}\) = length of connector with unit of mm; \(h_{c}\) = height of connector with unit of mm; \(s_{c}\) = spacing between connectors with unit of mm; \(t_{c}\) = the thickness of connector with unit of mm; \(t_{f}\) = thickness of flange of connector with unit of mm; \(l_{\text{h}}\) = length of part with opening holes with unit of mm; \(h_{e}\) = void height with unit of mm; \(f_{\text{c}}\) = strength of concrete with unit of MPa; \({\text{f}}_{{\text{y}}}\) = strength of steel connector with unit of Mpa.

Figure 16 shows the comparison between results calculated by the proposed Formula (1) and the test results. It can be concluded that most of results calculated are within the 15% error rate, and differences between test results and theoretical results relatively unsignificant. Considering the error of loading tests and dispersion of results, the proposed calculation formula could lead to accurate results.

Fig. 16
A scatter plot for bearing capacity. It plots an increasing line from the origin. Most of the plots are concentrated above and below the line. A dotted line for a 15% error extends from the origin above and below this line.

Comparison between new calculation method and test results

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5 Conclusion

In this paper, the mechanical properties of shear connectors with different factors are analyzed by combination of experimental verifications, theoretical analysis and numerical calculations. Finally a calculation method for the bearing capacity of shear connectors with considerations of multiple factors is established. The following conclusions can be determined:

  1. (1)

    The overall trends of the shear-slip curve under tension and compression state are all similar, which indicates that stress state has minor effect on bearing capacity for connectors. The bearing capacity in the compression state is slightly higher, whereas stiffness in compression is significantly higher than the one in tension.

  2. (2)

    The failure modes of with-void and non-void connectors are also similar, whereas crushed damages of connectors with void are more severe. With increased height for void, the failure of connector are more apparent. Cracks development for non-void specimen are more regular, whereas cracks for with-void specimen were caused by multiple failure mechanisms. The curvature of steel ribs will increase significantly when void exists, which is mainly due to the lack of supportive bottom concrete.

  3. (3)

    The failure modes of connectors with and without opening holes are similar. The yielding damage of with-opening connector is relatively more obvious. Cracks of the non-opening connector develops regularly from the inward and extends outward gradually, while the development of cracks of with opening connector is complex which will develop back inward. For connectors with opening, the stress distribution is uneven, and shear force fluctuation phenomenon will appear in the shear-slip curve, which is caused by the decreased contact area of concrete and steel and reduced effective area of steel plate of connectors.

  4. (4)

    The numerical calculation results of the non-void and with-void connector show that the compressive stress of concrete for the with-void model is more concentrated, the crushing area is smaller, the bending stress at the bottom of steel connector is larger than that of the non-void. Additionally, the deformation and yield range is also large, and shear stress of with-void connector decreases rapidly with height. The stiffness and bearing capacity of with-void connector will decrease with the increase of height, and calculated stress on the model will also be unevenly distributed.

  5. (5)

    According to the test and numerical calculation results, with considerations of tensile and compressive states of concrete and connector with or without voids and openings, the influence of three factors above on the bearing capacity of the shear connector is discussed and determined. In addition, the shear capacity formula with considerations of three factors is established and proposed. Accordingly, the proposed formula is also verified by loadings tests above.