Study on the shear behavior of inter-module connection with a bolt and shear key fitting for modular steel buildings

Abstract

Due to the advantages of high integration, modular steel buildings have recently received extensive attention and research. The connection between modules (inter-module connection) plays an important role in ensuring the safety and integrity of modular steel buildings. However, most of the existing inter-module connections have problems such as insufficient construction space, difficulty in disassembly, lack of design specifications, etc. An innovative inter-module connection with bolt and shear key fitting was proposed. The connection separates the horizontal load-bearing component and the vertical load-bearing component, which greatly reduces the difficulty of analysis and design. The structure of the connection is convenient for construction and insensitive to installation errors. The structure and advantages of the connection were introduced in this paper, and the shear performance of the connection was studied by the monotonic static test. A finite element model verified with the experiment was proposed to simulate the shear performance of the connection. The influence of errors in fabrication on the performance of the connection was studied using the finite element model, and connection’s operating status in a modular frame was analyzed. Thus the allowance for the clearance between the shear key and slot was obtained. Based on the experimental and numerical study results, a formula was proposed to predict the shear capacity of the connection for the practical design of inter-module connection in modular steel buildings.

Keywords

modular steel buildings inter-module connection bolt and shear key fitting shear performance monotonic static test errors in fabrication

Introduction

As one of the most integrated prefabricated buildings, modular steel buildings have received widespread attention in many countries. It can achieve better quality control, less construction time, and recycling of modules (Luo et al., 2019). The disadvantages of modular steel buildings (Lacey et al., 2018) are that the structural span is relatively small, and the design lacks complete guidance and specification. However, due to their special advantages, various types of modular steel buildings have been developed and used in diverse applications such as hotels, hospitals and military facilities (Sanches et al., 2018).

Different from traditional steel buildings, modular steel buildings are constructed by stacking steel modules, which are manufactured off-site, transported and assembled on-site. Steel modules can be grouped into corner-supported and continuously supported. The corner-supported steel modules are more applicable in high-rise buildings for their larger vertical bearing capacity (Chen et al., 2017b). Connections in modular steel buildings play a vital role in maintaining the strength and stability of the overall structure. Connections in modular steel buildings can be divided into three types (Lacey et al., 2018) according to their different locations and functions: inter-module, intra-module and module-foundation.

Due to the importance of inter-module connection in construction efficiency and load transfer, various inter-module connections have been proposed and studied. Chen et al. (2017b) proposed a new type of design with beam-to-beam connections and studied its flexural and seismic performance. Results revealed that beams and connections had independent and individual bending behaviors. Annan et al. (2009a, 2009b) evaluated the hysteretic characteristics of modular steel building, using welding as the vertical connections and field-bolting of clip angles as horizontal connections between modules. By comparing the performance between a regular braced frame and a modular braced frame, the differences indicated that the detailing requirements of the system need to be incorporated in their design. Chen et al. (2017a) investigated the seismic performance of a modular frame with pretension inter-module connection, in which the columns are vertically connected by pre-stressed strands. It was found that the pretension assembled framed modular system had different internal stress distribution. Chen et al. (2020) studied the tensile and shear performance of a rotary inter-module connection, and simplified calculations were developed. Lee et al. (2017) proposed a new inter-module connection to form a rigidly connected modular system and verified the seismic performance of the proposed system. A splice connection (Li et al., 2019; Lyu et al., 2021) was proposed and studied in full-scale corner-supported modular building. The proposed connection was sufficient to transfer the vertical load and had a satisfactory tolerance for initial imperfections. 최경석 et al. (2016) adopted a bolted connection with an access hole opening at the end of the column in nonlinear static analyses of modular structures. It was found that the modelling of overlapped elements and the rotational behavior of connections can influence the structure's lateral stiffness.

These connections mentioned above include the following types: bolted, welded, pretensioned, and concreted. Welded and concreted connections may have better stiffness but cannot take advantage of the detachability of modular steel buildings. For bolted connections, although they are already widely researched and used, the problems of weakening members' sections and being sensitive to installation errors are still unsettled. Pretensioned connections that connected by pre-stressed strands possess great stiffness and strength but have high requirements for on-site construction and are prone to slip. Therefore, more reasonable connections, which have not only sufficient capacity, but are also convenient for on-site construction, need to be proposed (Dai et al., 2019).

This paper proposed an innovative inter-module connection, which can avoid the shortcomings of the existing connections. Comprised of bolt and shear key fitting, the proposed connection can bear the vertical loads and horizontal loads in the separate load transmission path. Under the combined action of vertical and horizontal loads, the bolt transmits tension and bending moment, and the shear key transmits shear force. Because the bolt is located inside the module instead of on the axes of the beam and column, the installation of connection can be completed in various complicated positions. And the bolt hole is a slotted hole, which is not sensitive to the processing and hoisting errors of the module. Therefore, the proposed connection is very practical and convenient, and it is suitable for popularization in low-rise and multi-story modular steel buildings.

This paper mainly studied the shear performance of the proposed connection, which was proven to play a significant role in ensuring the safety and integrity of modular steel buildings in several studies (Lawson et al., 2008; Lawson and Richards, 2010). The static monotonic test was carried out to research the mechanical performance of the connection. A finite element model was built to simulate the shear performance of the connection, and it was used for the parametric study to obtain the main parameters and failure modes of the connection under shear. The influence of errors in fabrication on the performance of the connection was studied using the finite element model, and the allowance for the clearance between the shear key and slot was obtained. In order to study the response of the connection under various loads, connection’s operating status in a modular frame was analyzed. Based on experimental and numerical results, a reasonable theoretical design formula for calculating the shear capacity of the connection was proposed.

Structural system of inter-module connection with bolt and shear key fitting

In order to facilitate the design and on-site installation, this paper proposed an inter-module connection with bolt and shear key fitting, in which the bolt bears most of the vertical loads, and the shear key mainly bears the horizontal loads. Figure 1 illustrates the details of the proposed connection at the corner. The connection is composed of four parts: floor steel casting, connecting plate, ceiling steel casting and bolt. The connecting plate is located between two steel castings, and the outer contour of the connecting plate is consistent with steel casting. Shear keys are welded on both sides of the connecting plate to transmit the horizontal load. As for the transmission of vertical load, a bolt connects two steel casting through the slotted hole on the triangular plate without contacting the connecting plate. The following will take the ceiling steel casting as an example to introduce the details of the steel casting. As shown in Figure 1(b), the ceiling steel casting consists of an L-shaped hollow box (yellow part) and a triangular plate (green part). The L-shaped box is welded to the column and beams on different surfaces. The shear key slot is opened on the bottom plate of the L-shaped box. The thickness of the bottom plate is the same as that of the triangular plate. And the other plates of the L-shaped box have the same thickness, which are collectively called the side plates.

Figure 1.

Details of the proposed connection. (a) Inter-module connection with bolt and shear key fitting (b) Ceiling steel casting.

Owing to its structure, the proposed connection has the following advantages in terms of mechanical performance and on-site construction: (1) The connection separates the vertical load-bearing and horizontal load-bearing components. Because there is no connection between the shear key and shear key slot, the shear key mainly bears the horizontal action caused by shear force. According to experimental and numerical analysis, when the errors in do not exceed the allowance, the bolt will not transmit shear force until the shear key fails; (2) Compared with opening bolt holes at the beam end or column end, placing the bolt on the triangular plate can avoid weakening members' section and ensure the integrity of beam and column; (3) During on-site construction, the main connection work is to install bolts after the module was positioned. The bolts are located inside the module rather than on the axis of the beam and column, overcoming the problem of insufficient construction space. Therefore, the connection can be completed inside the module through the reserved opening and can be applied to various positions in the structure.

Experimental study

Materials and specimen

Chinese steel Q345B with a nominal yield stress of 345 MPa was used to fabricate members and steel castings. High-strength bolts M24 with grade 10.9 were used in the experiment. In order to obtain the material properties, a series of tensile tests were carried out according to Metallic Materials-Tensile Testing-Part 1: Method of test at room temperature (GB/T 228.1-2010, 2010). Three round rod specimens (Bolt 1–3 and Steel 1–3) were processed for the two materials, respectively. The material properties and stress-strain curve are shown in Table 1 and Figure 2. The data in the table is the mean value of the original data. The standard deviation SD1 of the steel used for bolts and the standard deviation SD2 of the steel used for other components are also given in the Table 1. The yield strength of the steel does not reach the strength of the grade, but this will not affect the subsequent analysis.

Table 1.

Material properties.

Material	Elastic modulus (MPa)	Yield strength (MPa)	Ultimate strength (MPa)	Shrinkage (%)	Elongation (%)
Bolt	209,766.70	1019.80	1163.67	57.53	11.30
SD1	0.91	37.14	27.72	0.02	0.00
Steel	206,333.33	328.33	627.25	0.23	30.73
SD2	0.53	2.05	6.60	0.01	0.01

Figure 2.

Stress-strain curve of bolt and steel. (a) bolt (b) steel.

As shown in Figure 3, the shear specimen included the connection, upper column, bottom column, fixed beam and loading beam. The upper column and loading beam were welded to the floor steel casting, and the bottom column and fixed beam were welded to the ceiling steel casting. Details of the steel castings and connecting plate are shown in Figure 4. In order to facilitate production, steel castings were processed by welding. The cross-sectional shape of the shear key was square, making it convenient to measure the shear strain on the shear key. Due to the possible errors in the processing and installation of steel modules, the side length of the shear key slot was 4 mm larger than the side length of the shear key ( $d_{s k}$ =43 mm). To place the strain gauge on the shear key, the shear key slot leaving a gap of 10 mm in the direction perpendicular to the loading. A round hole with a diameter of 10 mm was opened on the steel castings to extend the wire connecting the strain gauge. The thickness of the bottom plate $t_{b}$ is 25 mm, and the thickness of the side plate $t_{s}$ is 18 mm. The bolt is half threaded and the diameter of bolt $d_{b}$ is 24 mm.

Figure 3.

Shear specimen.

Figure 4.

Details of connection's components (unit in mm). (a) Floor steel casting (b) Ceiling steel casting (c) Connecting plate.

Experiment setup

The shear test setup is shown in Figure 5(a). The specimen was fixed on the rectangular reaction frame through the fixed beam and bottom column. The sliding jack was set on the top of the upper column, and the other end was connected with the reaction frame. During the loading process, the actuator connected with the loading beam loaded the shear specimen at a speed of 0.5 mm/min. The jack slid to the right while applying a constant compression of 0.1 axial force ratio to the upper column (axial force = 212kN). And the loading will stop when the specimen is damaged, or the load-displacement curve enters the descending section.

Figure 5.

Test setup and measurement scheme. (a) Test setup (b) Measurement scheme.

The measurement scheme of the experiment is shown in Figure 5(b). Displacement meters SD1∼SD3 and SD4∼SD5 were used to measure the horizontal displacement of the floor steel casting and the ceiling steel casting, respectively. Strain gauges SS1∼SS4 were located at both ends of the shear key to measure the shear strain on it. Strain gauges SS5∼SS6 were placed on the middle of the screw to measure the tensile strain and shear strain of the bolt.

Experiment results and discussion

The phenomena in the test are shown in Table 2 and Figure 6. There was a significant relative displacement between the ceiling steel casting and the floor steel casting. After the loading of the test, both the shear key and the bolt were damaged.

Table 2.

Phenomena in the experiment.

Load (kN)	Displacement (mm)	Phenomena	Figure
719.9	9.20	Relative displacement occurred between the floor steel casting, the connecting plate and the ceiling steel casting.	Figure 6 (a) Figure 6 (b)
809.3	14.25	The shear key was severely deformed, and the bolt was broken.	Figure 6 (c) Figure 6 (d)

Figure 6.

Deforming state and failure behaviors in shear test.

The load-displacement curves of the floor steel casting measured by SD1∼SD3 are shown in Figure 7(a). When the load was between 32kN and 200kN, the stiffness decreased due to the closure of the gaps in the specimen. When the load was greater than 200kN, the gaps were completely closed, and the stiffness of the curve increased. When the load exceeded 400kN, the shear key yielded, and the stiffness decreased slowly. The load-displacement curve of the ceiling steel casting measured by SD4∼SD5 is shown in Figure 7(b). The horizontal displacement of the ceiling steel casting was small, and the gaps were closed in the early stage of loading. In order to reduce the influence of gaps and weld cracks on the performance of specimen, horizontal relative displacement $∆_{s}$ is calculated by the formula (1), ignoring the stiffness of the curve before the gap was closed. A new load-displacement curve can be obtained in Figure 8. In the section OA of the curve, the horizontal force was mainly transmitted by the shear key, which is the expected method of force transmission; for the section AB of the curve, the shear key yielded locally, and the stiffness of the curve began to decrease; for the section BC, the bolt and the shear key bore the horizontal force together until the bolt broke.

∆_{s} = \frac{∆_{s d 1} + ∆_{s d 2} ∆_{s d 3}}{3} - \frac{∆_{s d 4} + ∆_{s d 5}}{2}

(1)

where,

∆_{s d i}

is the horizontal displacement measured by displacement meter SDi.

Figure 7.

Data curves measured by displacement meters and strain gauges. (a) Displacement meters at the floor steel casting (b) Displacement meters at the ceiling steel casting (c) Strain on the shear key (d) Strain on the bolt.

Figure 8.

Load-displacement curve of the shear specimen.

The load-strain curve of the shear key measured by SS1∼SS4 is shown in Figure 7(c). Since the strain gauge on the shear key will be damaged when the load is larger, only the part within 450kN is shown in the figure. The shear key with the strain gauge SS3 was the first to close the gap, so the shear strain was also the first to develop. When the load was greater than about 330kN, the shear key with SS3 began to yield locally, and the shear strain increased rapidly with the increase of the load.

The load-strain relationship curve of the bolt measured by SS5 and SS6 is shown in Figure 7(d). When the load was less than 603.9kN, the shear strain on the bolt was very small, and the tensile strain increased rapidly. In fact, during the experiment, the connection was not only subjected to shear force but also to bending moment. There is only tensile strain on the bolt due to bending moment before the shear key failed. When the load was greater than 531.5kN, the shear strain of the bolt increased, and when the load was greater than 603.9kN, the tensile strain of the bolt decreased. At this time, the shear key fully entered the plastic stage, and a large relative displacement occurred between the floor steel casting and the ceiling steel casting. The bolt was inclined and subjected to the combined action of tensile and shear forces. This shows that the bolt is not the main shear member before the shear key is damaged.

Through experimental research, the mechanical performance of the connection and the stress development of key components under shear was obtained. In the initial stage of loading, the shear force was mainly transmitted by the shear key. When the shear key entered plasticity and undergone a large deformation, the bolt participated in the shear. But in practical applications, in order to separate the components that transmit vertical and horizontal forces, the bolt is not expected to transmit shear forces. Therefore, in the design, the yielding of the shear key can be used as the criterion for the failure of the connection under shear.

Numerical study

Finite element model

To better understand the shear performance of the connection with a bolt and shear key fitting, finite element software ABAQUS was used for the numerical simulation. The numerical model and its meshing result are shown in Figure 9. The components in the finite element model included the upper and bottom column, steel castings and the connecting plate. The mesh of the connection plate and bolt is finer than that of the other parts. In the experiment, the displacement meter measured the horizontal displacement of the connection, excluding the axial deformation of the beam. Therefore, to compare with the experimental data more conveniently, the loading beam and the fixed beam were not established in the finite element model. As shown in Figure 10, the material properties adopt the multi-segment linear model, and the key data points in the model were taken from Figure 2.

Figure 9.

Mesh of the finite element model.

Figure 10.

Multi-segment linear model of material properties.

To ensure the accuracy of the simulation results and the operation of the contact, all elements of the model adopt the eight-node linear hexahedral reduction element C3D8R. There were contacts between the steel casting and the connecting plate, between the shear key and the hole, and between the bolt and the triangular plate. The tangential behavior of contact was defined as surface contact with a friction coefficient of 0.15, and the normal behavior was defined as "hard” contact that allows separation. The solver used for the analysis is dynamic explicit solver.

In the model, a fixed constraint was applied to the bottom surface of the bottom column at point D, and the displacement of the surface at point B along the y-axis was constrained. A pressure of 212kN was applied to point C on the top surface of the upper column. Constrain the displacement along the x-axis and z-axis of the surface where point A is located, and apply the displacement load along the positive direction of the y-axis at point A.

Validation

The above numerical model is validated by the comparison with the experimental results. The comparison of load-displacement curve between the experimental and numerical study is shown in Figure 11. In the numerical study, the load-displacement curve is taken from point A in Figure 9. The shear capacity of the connection is defined as the load when the shear key yields, corresponding to the point with the greatest stiffness in the load-displacement curve. It can be found that the shape and key points' value of the two curves are relatively close. When the finite element model is loaded to 719kN, the stress contours of the connection is shown in Figure 12(a). Similar to the deformation observed in Figure 6, relative displacement occurred between the floor steel casting, the connecting plate and the ceiling steel casting. When the finite element model is loaded to 803kN, the stress contours of the connecting plate and bolt are shown in Figure 12(b) and (c), which is compared with the failure mode of the experiment in Figure 6. The failure mode of the shear key is basically the same as that in the experiment, which is the shear failure at the root. In the same way as the experiment, the screw is inclined and simultaneously subjected to tensile and shear forces.

Figure 11.

Load-displacement curve of the finite element mode.

Figure 12.

Stress contours of connection. (a) Relative displacement between the components (b) Stress contours of the shear key (c) Stress contours of the bolt.

The comparisons in Figures 11 and 12 indicate that the proposed numerical model can achieve a fine simulation on the practical behavior of the tested inter-module connection. Thus, it is reliable to use the numerical model to make further research on the performance of the connection.

Parametric study

The verified finite element model was used for parametric study. In order to make the connection closer to the pure shear condition, the loading point was changed from point A to point E in Figure 9, and the displacement of the top surface of the upper column along the z-axis was restricted.

The effect of dimensions of components

The dimensions of components that affect the shear performance of connection are the side length of the shear key

d_{s k}

, the thickness of the bottom plate

t_{b}

, the diameter of the bolt

d_{b}

, the thickness of the side plates

t_{s}

. As shown in Table 3, the values of the above parameters are changed according to the control variable method.

Table 3.

The values of parameters.

Constant (mm)	Variable (mm)
$t_{b}$ = 25, $d_{b}$ = 24, $t_{s}$ = 18	$d_{s k}$ = 33, 38, 43, 48, 50.
$d_{s k}$ = 43, $d_{b}$ = 24, $t_{s}$ = 18	$t_{b}$ = 10, 15, 20, 22, 25, 30, 35.
$d_{s k}$ = 43, $t_{b}$ = 25, $t_{s}$ = 18	$d_{b}$ = 20, 22, 24, 27, 30.
$d_{s k}$ = 43, $t_{b}$ = 25, $d_{b}$ = 24	$t_{s}$ = 8, 12, 16, 20, 24.

Figure 13.

Load-displacement curve of different dimensions' component. (a) Side length of the shear key d _sk (b) Thickness of the bottom plate t_b (c) Diameter of bolt d_b (d) Thickness of the side plates t_s.

Through parametric study, the load-displacement curves of different dimensions' components are shown in Figure 13, and shear capacity's change with the dimensions of component is shown in Figure 14. For parameter $d_{s k}$ : The shear stiffness and capacity of the connection increase with the increase of $d_{s k}$ , but the growth rate keeps decreasing. When the side length of the shear key is too large, the size of the shear key slot cannot match it. As shown in Figure 14(a), when $d_{s k}$ =33 mm, the hole wall only partially yields when the shear fails; when $d_{s k}$ = 58 mm, the hole wall has already entered plasticity in a large area when the shear key fails. For the parameter $t_{b}$ : When $t_{b}$ ≥ 25 mm, the load-displacement curve changes little with the increase of $t_{b}$ ; while when $t_{b}$ < 25 mm, the shear performance of the connection degrades with the decrease of $t_{b}$ . As shown in Figure 14(b), when the thickness of the shear key slot is not enough, the shear bearing capacity will decrease significantly. At this time, the failure mode will change from the shear failure of the shear key to the compression failure of the hole wall. The parameter $d_{b}$ and $t_{s}$ have little effect on the shear stiffness and capacity of the connection.

Figure 14.

Shear capacity and failure mode's change with the dimensions of component. (a) Side length of the shear key $d_{s k}$ (b) Thickness of the bottom plate $t_{b}$ .

Figure 15.

Clearance $∆_{0}$ .

The influence of errors in fabrication

As shown in Figure 15, the clearance $∆_{0}$ is set between the shear key and shear key slot. The clearance can provide tolerance to errors in installation and fabrication. But the clearance may have a bad influence on the performance of connection. The influence of the clearance was analyzed using the validated finite element model. To study the influence of the clearance in a single direction, the displacement of the floor steel casting along the other direction was restricted. Other information about the model is the same as above.

Figure 16.

Influence of clearance on the connection. (a) Load-displacement curve (b) Shear force of bolt-load curve.

The results of the analysis are shown in Figure 16 and Table 4. Load-displacement curve of different dimensions' clearance is shown in Figure 16(b). As the clearance $∆_{0}$ increases, the basic shape of the curve is the same: the stiffness of curve is large before the shear key yields(the yellow point in the figure), and the stiffness decreases gradually after the shear key yields. The clearance $∆_{0}$ will cause the curve having a slip stage at the beginning, and the trend of the slip stage of the curve is the same when $∆_{0}$ is different. It can be seen that there is a critical clearance $∆_{c}$ , and when the clearance is less than $∆_{c}$ , the bolt will not bear the shear force in the slip stage. The $∆_{c}$ is approximately exactly twice the distance between the bolt and the bolt hole, which is 4 mm.`

Figure 17.

Two-story modular frame.

Table 4.

The influence of the clearance on the connection.

$∆_{0}$ (mm)	Shear stiffness (kN/mm)	Shear capacity (kN)	Shear force of blot at capacity (kN)
0	441.61	442.65	4.63
4	397.00	469.12	57.27
8	368.97	468.65	127.80
12	308.39	561.85	150.90
16	208.47	596.14	158.90

Shear force of the bolt-load curve of different dimensions' clearance is shown in Figure 16(b). It can be found that the bolt bears most of the shear force in the slip stage. The larger the clearance, the longer the slip stage and the greater the shear force the bolt is subjected to before the shear key works. While when the clearance is too large, the shear force that the bolt can bear in the slip stage will be limited. It can be concluded that the shear force of the bolt increases mainly in the slip stage and the section after the shear key yields. And for different clearances, the increased shear force of the bolt during the working of the shear key is approximately the same.

Similar to the test, take the yellow point where the curve stiffness significantly decreases as the shear capacity, and take the slope between the two red points in Figure 16(a) as the shear stiffness. For different dimensions of clearance, the shear capacity and shear stiffness as well as the shear force of the bolt at capacity are shown in Table 4. As the clearance increases, the capacity increases because the bolts are pre-loaded with part of the shear force in the slip stage. However, the shear stiffness decreases because the closing of the oversized clearance will change the parallel relationship between the shear key and slot. But in reality the shear key is cylindrical and the decrease in stiffness will no longer be significant. When the clearance is greater than 4 mm, the bolt will be subjected to significant shear force before the shear key yields. Therefore, in order to ensure the normal operating condition of the component and the expected force transmission form of force separation, the clearance $∆_{0}$ should not be greater than the critical clearance $∆_{c}$ .

Operating status in the modular frame

The connection analyzed above is only subjected to shear force, which is not practical. In addition, the boundary conditions in the test did not take into account the restraint of out-of-plane beams, and the shear keys were not cylindrical in order to measure the strain. Therefore, it is necessary to study the operating status of the connection in the modular frame. A schematic diagram of the modular frame is shown in the Figure 17. The modular frame contains four inter-module connections (two CL and two CR). For the connection, the square shear key with a diameter of 43 mm are replaced with cylindrical shear key with a diameter of 50 mm, while the other parts are unchanged. The four corners at the bottom of the frame are hinged. A horizontal displacement load of H/50 is applied on top of the second layer. Conservatively, the loads on the floor and the constraints of the floor are not considered. The element type and material properties are the same as before.

Figure 18.

Influence of clearance on the modular frame. (a) Load-displacement (b) Shear force of bolt-load.

Load-displacement curve of different dimensions' clearance is shown in Figure 18(a). It can be found that the dimension of the clearance has little influence on the overall performance of the modular frame. Since the shape of the shear key is cylindrical, the stiffness of the curve has little difference on the whole. And the shear key does not yield overall at this time. When the clearance is larger than 4 mm, the stiffness of the curve decreases slightly and the curve fluctuates. Due to the overall constraint of the module, the closing of the clearance does not occur at the beginning. Instead, the clearance is closed while loading, so the curve fluctuates.

Figure 19.

Comparison of shear capacity between theoretical formula and parametric analysis. (a) Side length of the shear key dsk (b) Thickness of the bottom plate t_b.

Shear force of bolt-displacement curve of different dimensions' clearance is shown in Figure 18(b). It can be seen that when $∆_{0}$ =4 mm, the shear force of the bolt is very small. When $∆_{0}$ =12 mm, the shear force of the bolt is larger. This conclusion is consistent with the above: when the clearance is greater than 4 mm, the bolt will bear the shear force before the shear key yields.

In this part, a finite element model for simulating the shear performance of connection was proposed. The rationality of the finite element model was verified by comparison with the experimental results. Using the above model to conduct parametric study, it was found that the main parameters affecting the shear performance of the connection are the side length of the shear key $d_{s k}$ and the thickness of the bottom plate $t_{b}$ . The influence of errors in fabrication on the performance of the connection was studied using the finite element model, and the operating status of the connection in the modular frame was analyzed. It is concluded that when clearance between the shear key and slot is larger than 4 mm, the stiffness of the modular frame will decrease slightly and the bolt will bear shear force before the shear key yields, which is not in line with the previously assumed force transmission of separating vertical and horizontal forces.

Theoretical study

According to the experimental research and numerical analysis, the parameters and components that mainly affect the shear performance of the connection were obtained. Aiming at the failure mode of each bearing member, theoretical design formulas of shear capacity related to the parameters were given below.

1. Failure mode 1: the shear key yields under shear

The calculation formula of the shear capacity $P_{t 1}$ is given by

P_{v 1} = f_{y v} A_{s k} = f_{y v} d_{s k}^{2}

(2)

where,

f_{y v}

is the shear strength of the shear key;

A_{s k}

is the cross-sectional area of the shear key.

2. Failure mode 2: the shear key slot yields under compression

The formula for calculating the shear capacity $P_{t 2}$ is given by

P_{v 2} = f_{y} A_{s k h} = f_{y} t_{b} d_{s k}

(3)

where,

f_{y}

is the yield strength of the steel casting;

A_{s k h}

is the compression area of shear key slot.

By summarizing the above two failure modes, the shear capacity of the connection $P_{v}$ is given by

P_{v} = \min (P_{v 1}, P_{v 2}) = \min (f_{y v} d_{s k}^{2}, f_{y} t_{b} d_{s k})

(4)

In engineering applications, the expected failure mode of the connection under shear force is the yielding of shear key, which is convenient for replacement. In order to illustrate the practicability of the theoretical design formula of the shear capacity, the theoretical results are compared with the previous results. As shown in Table 5, the shear capacity obtained by the theoretical formula is conservative because the local stress is already greater than the yield stress in the experiment and numerical analysis when the shear key fails. As shown in Figure 19, the shear capacity of the theoretical formula and parametric analysis are compared. The theoretical formula can safely calculate the shear capacity of the connection and reflect the change of the failure mode.

Table 5.

Comparison of shear capacity between experimental, numerical and theoretical study.

Item	$P_{t}$ (kN)	Abs (error) (%)
Experimental	400.5
Numerical	400.89	0.10
Theoretical	350.50	12.48

In this section, based on the results of parameter analysis and experiment, a theoretical design formula for calculating the shear capacity of connection was proposed. Compared with the numerical and experimental results, the formula was proved to be able to calculate the shear capacity reasonably. In addition, this formula can also be used to separate the vertical load-bearing component from the horizontal load-bearing component, because the bolt will not transfer the shear force until the shear key yields.

Conclusions

The concluding remarks of the main research work presented in this paper are summarized as follows:

(1) To facilitate design and construction, the proposed inter-module connection separates the vertical load-bearing and horizontal load-bearing components, in which bolt bears most of the vertical action, and the shear key mainly bears the horizontal action. The on-site installation of bolts can be completed inside the module through the reserved opening. The slotted hole, as well as the clearance between the shear key and shear key slot can provide high tolerance to installation errors.

(2) At the end of the test, both the shear key and the bolt were damaged. Through analyzing the stress development of components, when the clearance between the shear key and slot is 4 mm, it is found that the bolt transmits shear force only after the shear key fully yields and large deformation occurs.

(3) By comparing with the experimental data and phenomena, a reasonable finite element model was proposed. Using the previous model to carry out the parametric study, the main parameters that affect the shear performance of the connection and different failure modes were obtained. The diameter of the bolt $d_{b}$ and thickness of the side plates $t_{s}$ have little effect on the shear performance of the connection. When the thickness of the bottom plate $t_{b}$ is constant, the shear stiffness and capacity increase with the increase of side length of the shear key $d_{s k}$ , but the growth rate keeps decreasing. When the parameter $d_{s k}$ is unchanged and $t_{b}$ is not large enough, the failure mode will shift from the failure of the shear key to the destruction of the hole wall, leading to a significant decrease in shear capacity. And when the parameter $t_{b}$ is large enough, the increase of $t_{b}$ will not improve the shear performance.

(4) The influence of errors in fabrication on the performance of the connection was analyzed using the finite element model. When the clearance between the shear key and shear key slot is larger than 4 mm, the bolt will bear shear force before the shear key fails, which will affect the previously assumed force transmission of separating vertical and horizontal forces. The operating status of the connection in a modular frame was studied. When cylindrical shear key is used, the clearance has little effect on the stiffness of the modular frame. In the modular frame, the closing of the clearance is carried out at the same time as the force is transmitted, rather than closing the clearance first and then transmitting the force.

(5) A theoretical design formula of the shear capacity of the connection was proposed. Comparing the theoretical results with the results of the experiment and parametric study, it is proved that the design formula can safely calculate the shear capacity and reflect different failure modes.

Footnotes

Acknowledgements

The research work presented hereinabove was supported by the National Key Research and Development Program of China (Project No.2017YFC0703803-04) and the National Science Foundation for Young Scientists of China (Grant No.51808068).

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Key Research and Development Program of China (Project No.2017YFC0703803-04) and the National Science Foundation for Young Scientists of China (Grant No.51808068).

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