Study on the Bending Performance of Connection Joints in a New Type of Modular Steel Structure Emergency Repair Pier

Abstract

The pier-type repair equipment for bridges is a crucial branch of bridge emergency repair. However, the existing bridge pier repair equipment predominantly utilizes rod systems, which require substantial assembly work, hindering the rapid restoration of damaged bridges. Modular steel structure buildings, as a highly integrated form of prefabricated construction, can play a significant role in emergency rescue operations. Based on the modular architectural design concept, this paper proposes a new type of modular steel structure emergency repair pier joint that facilitates rapid assembly and connection between modular units. Using ABAQUS 2022 software to establish a finite element model of the joint, the bending performance under lateral displacement loads perpendicular to the joint opening direction (X-direction in the model coordinate system) and parallel to the joint opening direction (Z-direction in the model coordinate system) is analyzed. The influence of the width-to-thickness ratio of the upper corner piece base plate D/t₁ (where D is the width of the upper corner piece base plate and t₁ is the thickness of the upper corner plate), the height-to-thickness ratio of the lower corner piece top plate h/t₂ (where h is the height of the protrusion of the lower corner piece and t₂ is the thickness of the lower corner piece top plate), the height of the protrusion of the lower corner piece (h), and the bolt diameter (d) on the bending performance of the joint is investigated. Recommendations for the design values of the joint are provided. Then, the flexural behavior of the joint under 0.1, 0.2, and 0.3 axial compression ratios is studied, respectively. The results show that with the increase of axial compression ratio, the yield rotation angle and ultimate rotation angle of the joint decrease, and the bearing capacity decreases faster after the joint reaches the ultimate bearing capacity. When the joint is subjected to the X-direction horizontal lateral displacement load, the initial flexural stiffness and flexural capacity of the joint increase with an increase in the axial compression ratio. When subjected to the horizontal lateral displacement load in the Z-direction, the initial bending stiffness of the joint increases with an increase in the axial compression ratio, and the bending capacity does not change much. In addition, the joint is classified; from the perspective of load-bearing capacity, it is a partially resistant joint, and from the perspective of stiffness, it is a semi-rigid joint. Finally, a simplified calculation model for the joint is proposed based on the component method.

1. Introduction

Bridges, as key nodes in transportation networks, play an important role in ensuring smooth commutes in urban areas, facilitating the distribution of goods, and maintaining the continuity of vital infrastructure during emergencies and disasters [1]. However, according to incomplete statistics, approximately 40% of the bridges in China’s highway network have been in service for over 20 years. Most bridges in the Americas were built between 1960 and 1980, while most bridges in Europe were constructed between 1970 and 2001. Global bridge infrastructure is facing issues of aging and deterioration, with reduced strength [2]; in addition, bridges are susceptible to natural disasters, human damage, and wartime attacks [3,4,5,6], leading to sudden bridge failures. Therefore, rapid emergency repairs of bridges are particularly important.

The pier, as a supporting structure in a bridge, plays a critical role in the overall functionality of the bridge. Damage to the pier can result in the disruption of traffic across the entire bridge. In the early stages, bridge pier repairs in China primarily relied on timber piers and wooden frame structures. Later, the development of the Type 65 railway military bridge pier, the Type 83 railway lightweight military bridge pier, and improved steel piers based on these designs followed [7,8]. However, the existing bridge pier repair equipment is limited by the sudden and unpredictable nature of disasters, and the quantity and scale of reserves are constrained [9], which hampers effective responses to bridge repair needs. Bridge emergency repair should incorporate advanced concepts, accelerate the development of repair technologies, improve emergency repair assembly methods, and reduce the on-site assembly workload [10,11]. Modular steel structure buildings involve manufacturing modules or prefabricated components off-site, which are then transferred to the site for assembly to form the entire structure. This approach features a high degree of prefabrication [12] and is considered a highly integrated form of prefabricated construction [13]. There are similarities between the existing bridge pier type emergency repair equipment and modular buildings, as they all need to be prefabricated off-site and installed on-site. However, at present, emergency repair equipment mostly adopts the rod system structure, and the assembly workload is large, which is not conducive to the rapid repair of damaged bridges. Steel modular buildings are typically connected through joints, linking individual module units both vertically and horizontally. This approach improves on-site construction efficiency, reduces labor consumption [14,15], and can play a crucial role in emergency rescue operations [16].

Modular construction provides a new design approach for the development and improvement of bridge repair piers. Specifically, repair pier module units are manufactured and assembled by connecting joints, expanding the modules in width and height to meet the needs of bridge repair operations. In modular construction, the connecting joints between modules are critical components that influence structural stability [17,18]. Currently, the connecting joints between modules primarily include bolted connections, welded connections, prestressed connections, and self-locking connections, among others [19]. Among these, bolted connections serve as an effective method to meet the rapid assembly requirements of modular construction. They eliminate the need for extensive on-site welding, reduce on-site workload, enhance the efficiency of steel structure installation, and facilitate disassembly and recycling [20,21]. Zhang et al. [22] proposed a detachable conical bolt connection method between modules. Through a four-point bending test, they investigated the bending behavior of the connection under static loads and conducted a numerical simulation to examine the influence of node parameters on bending performance. This study provides valuable references for the practical application of connectors in engineering projects. Jothiarun et al. [23] introduced a thin-walled beam–column modular connection. Through full-scale experiments and nonlinear finite element methods, they investigated the uniaxial and biaxial bending behavior of typical corner beam–column connections composed of steel components and connectors. This study provides valuable insights into the construction of auxiliary living facilities. Liu et al. [24] proposed new self-locking-unlockable joints of steel structure modules. Using ABAQUS 2022 numerical simulation software, they investigated the effects of the friction coefficient and node parameters (upper limit plate thickness, lower limit plate thickness, and upper slider width) on the tensile bearing capacity of the node. Liu et al. [25] conducted experimental and numerical simulation studies on the compression–bending–shear performance of column-to-column bolted-flange connections in prefabricated multi-high-rise steel structures under the combined action of axial compression, bending moment, and shear force, providing a theoretical basis for design. Wang et al. [26] proposed a new type of socket-type through-bolt connection for column-to-column joints. Through experimental and finite element simulations, they studied the mechanical properties of the joint under bending and shear. The results showed that this type of joint has high initial rotational stiffness and good bending performance, meeting the requirements for rigid connections in Eurocode 3, making it suitable for modular steel construction.

Under the action of the bridge self-weight and horizontal lateral displacement loads, the internal connection structure of the emergency repair pier experiences significant bending moments and shear forces. The distribution of the bending moment is related to the relative stiffness between the structural components [27]. During the design process, the connecting elements between the emergency repair piers should possess sufficient bending capacity and stiffness; otherwise, failure may occur, leading to an expansion of the damage. This paper proposes a new type of modular steel structure emergency repair pier connection joint, using bolt connections based on the modular building design concept, with the aim of achieving rapid connection and assembly between repair pier module units to improve the speed of bridge repair. In addition, ABAQUS 2022 software is used to simulate the static bending resistance of the new type of modular steel structure emergency repair pier connection joint, exploring the impact of the width-to-thickness ratio of the upper corner piece base plate, the height-to-thickness ratio of the lower corner piece top plate, the height of the protrusion of the lower corner piece, and the bolt diameter on the bending performance of the joint, and also to classify the joint. Finally, a simplified calculation model for the joint is proposed based on the component method.

2. Joint Structure

The new type of modular steel structure emergency repair pier joint consists of upper and lower corner pieces, a connection plate, and a bolt. The upper and lower corner pieces are welded to the bottom of the column and the top of the column in the upper and lower module units, respectively. The upper corner piece has holes at the top to reduce its weight. A hole is drilled on one side to provide operational space for installing the bolt. To prevent stress concentration, the holes are processed with rounded corners. For ease of installation, chamfer designs are applied to the base plate of the upper corner piece and the protrusion of the lower corner piece. Considering the alignment accuracy requirements when connecting the upper and lower modules, no threads are provided in the bolt holes on the base plate of the upper corner piece, while threads are provided in the bolt holes of the top plate of the lower corner piece. The bolt is selected as an internal hexagonal cylindrical head bolt, which facilitates quick installation using a torque wrench. The bolt is preloaded to ensure the stiffness and mechanical performance of the joint. This paper references the standard GB 50205-2020 [28], which sets a tolerance of ±1 mm for the bolt hole. In actual engineering, the specific tolerance depends on the assembly precision requirements. The emergency repair pier module unit and joint structure are shown in Figure 1.

The assembly of the module units proceeds as follows: first, install the lower module, then lift the upper module and align the corner pieces vertically, then pass the connection plate through the protrusion of the lower corner piece and install it on the lower corner piece, lower the suspended upper module to complete the assembly of the upper and lower corner pieces, and finally, install the bolt at the hole on the upper corner piece to complete the connection between the modules. When dismantling the module unit, simply unscrew the bolt, lift the upper module unit, and remove the remaining components. The assembly steps of the module unit are shown in Figure 2. The joint forms of different positions of the emergency repair pier module unit are shown in Figure 3.

3. Model Establishment

3.1. Constitutive Relationship of Steel

The constitutive relationship of the steel is based on a linear hardening elastoplastic model. The strength grade of the bolt is 8.8. The material properties were measured by a steel material test in reference [29]. The constitutive relationship of the steel is shown in

Table 1. Constitutive relationship of steel.

Component	Elastic Modulus/GPa	Yield Strength/MPa	Ultimate Strength/MPa	Plastic Strain/%
Module column	202.90699	354	508	0.208
Remaining components	174.06515	344	492	0.192

. The constitutive curve is shown in Figure 4.

3.2. The Component Dimensions and Finite Element Simulation Parameters

The width-to-thickness ratio of the upper corner piece base plate D/t₁ (where D is the width of the upper corner piece base plate and t₁ is the thickness of the upper corner plate), the height-to-thickness ratio of the lower corner piece top plate h/t₂ (where h is the height of the protrusion of the lower corner piece and t₂ is the thickness of the lower corner piece top plate), the height of the protrusion of the lower corner piece (h), and the bolt diameter (d) were selected as parameters for static bending simulations to explore the influence of each parameter on the bending resistance of the joint. The component dimensions and finite element simulation parameters are shown in Figure 5.

3.3. Model Setup

3.3.1. Mesh Division

The finite element model of the joint was established using ABAQUS 2022 software, and each component was meshed using the eight-node reduced integral element C3D8R. Because the bolt has a great influence on the bearing capacity of the joint when the joint is bent, the results of the finite element model are related to the element mesh. The results become more accurate as the size decreases, but it also leads to a decrease in computational efficiency. Considering the accuracy and efficiency of the calculation results, the mesh size of the bolt element was determined by mesh convergence analysis. The process of mesh convergence analysis is to reduce the mesh size of the model continuously until the mesh convergence of the model is reached, i.e., when the difference between the calculation results of this time and the previous calculation results is less than 1% [30]. The mesh convergence simulation results of the bolt are shown in Figure 6. The final mesh size for the bolt was determined to be 5 mm; the mesh size for the upper and lower corner pieces, shim plate, and connection plate was 8 mm; and the mesh size for the upper and lower module columns was 10 mm. To ensure the accuracy of the calculation results, the upper and lower module columns, as well as the upper and lower corner pieces, were divided into two layers of mesh in the thickness direction. The mesh division is shown in Figure 7.

3.3.2. Interactions and Boundary Conditions

The bending model of the joint used a tie contact to simulate welding. The tangential characteristics of the contact between surfaces were set as isotropic, while the normal characteristics were set as “hard” contact, allowing separation after contact. The section size of the module column was 200 mm × 200 mm × 18 mm, the height of the upper module column was 1000 mm, and the height of the lower module column was 300 mm. The bottom surface of the lower module column was coupled to a point (RP-1), and the top surface of the upper module column was coupled to a point (RP-2). The three translational and three rotational movements of RP-1 were constrained, and horizontal lateral displacement was applied for loading through RP-2. The bending load model is shown in Figure 8.

The joint exhibits different mechanical behaviors under lateral displacement loads applied perpendicular to the joint opening direction (X-direction in the model coordinate system) and parallel to the joint opening direction (Z-direction in the model coordinate system). Hereafter, these directions will be referred to as the X-direction and Z-direction, respectively. Therefore, this paper studies the most unfavorable scenarios for the joint under horizontal lateral displacement loads in the X and Z-directions, respectively. The details are as follows:

Because the mechanical behavior of the joint is the same when the horizontal lateral displacement load is applied positively and negatively along the X-axis, this paper chooses the case of applying the load positively along the X-axis as an example.

Because the mechanical behavior of the joint is different when the horizontal lateral displacement load is applied positively and negatively along the Z-axis, this paper simulates the bending of the joint with the positive and negative horizontal lateral displacement load along the Z-axis to determine the most unfavorable situation of the bending. The stress cloud diagram of positive and negative loading along the Z-axis is shown in Figure 9. In our design, we take the yield strength as the maximum stress that the steel material can reach, so the maximum stress boundary in the stress cloud diagram is set to its yield strength value, with the unit in MPa. In this paper, the setting of the stress cloud diagram is the same, which will not be described in the following. The results show that the ultimate bearing capacity under positive loading along the Z-axis is 5.76% lower than that under negative loading, when loading in the positive direction along the Z-axis, the main failure model is compressive yielding on the side of the upper corner member with the hole. The hole, as a relatively weak point in the joint’s bearing capacity, makes this condition the most unfavorable case for bending in the joint. Therefore, this paper selects the forward loading along the X-axis and the forward loading along the Z-axis for research (hereafter referred to as X-direction and Z-direction).

3.4. Feasibility Verification of Finite Element Modeling and Analysis Methods

Reference [29] conducted a static performance analysis of the proposed connection joint through static loading tests and finite element simulations. The tests were performed using a 10 t hydraulic jack for unidirectional staged loading. The basic mechanical properties and failure modes of the connection joint were analyzed, and the simulation results were compared with the experimental results, verifying that the finite element simulation was feasible.

Using the modeling and finite element analysis methods in this paper to analyze the joint in reference [29], the simulation results in this paper were compared with the results in reference [29], as shown in Figure 10. The bearing capacity results are shown in

Table 2. Comparison of bearing capacity results.

	The Yield Bearing Capacity	The Ultimate Bearing Capacity
The simulation results in this paper/(kN·m)	33.47	38.96
The simulation results in reference [29]/(kN·m)	32.62	39.07
The experimental results in reference [29]/(kN·m)	32.51	39.38
The simulation results in this paper/The simulation results in reference [29]	1.03	1.00
The simulation results in this paper/The experimental results in reference [29]	1.03	0.99

. It can be observed that the joint load-bearing capacity results obtained using the finite element modeling and analysis methods in this paper are similar to the results in reference [29], and the component deformation results obtained using the finite element modeling and analysis methods in this paper match well with the experimental deformation results of the components in reference [29]. Therefore, the finite element modeling and analysis methods used in this paper are feasible.

4. Parameter Analysis

4.1. The Influence of the Width-to-Thickness Ratio of the Upper Corner Piece Base Plate

The static bending simulation of eight models with different width-to-thickness ratios D/t₁ (where D is the width of the upper corner piece base plate, taken as 200 mm in this paper, and t₁ is the thickness of the upper corner plate) and bending directions of the upper corner piece base plate was carried out. The bearing capacity of each model is shown in

Table 3. The bearing capacity of each model (B-1~B-8).

Model Number	Thickness of the Upper Corner Piece Base Plate/mm	D/t1	Bending Direction	Initial Bending Stiffness /(kN·m/rad)	Yield Bearing Capacity /(kN·m)	Ultimate Bearing Capacity /(kN·m)
B-1	16	12.50	X	3370.95	75.38	87.79
B-2	18	11.11	X	3652.15	82.20	93.17
B-3	20	10.00	X	3895.50	87.75	99.15
B-4	22	9.09	X	4162.83	94.12	102.66
B-5	16	12.50	Z	3220.00	73.92	80.57
B-6	18	11.11	Z	3488.16	78.49	86.48
B-7	20	10.00	Z	3724.80	84.31	92.76
B-8	22	9.09	Z	3955.17	88.88	97.51

. The bending moment–rotation curves of each model are shown in Figure 11.

The yield bearing capacity is taken as the average of the results from the geometric drawing method, the equal energy method, and the Park method [31]. The initial bending stiffness is the ratio of the moment to the rotation in the linear stage of the joint’s moment–rotation curve [32].

From the simulation results of Table 3 and Figure 11, it can be seen that the bending resistance of the joint loaded along the X-direction is better than that of the joint loaded along the Z-direction. The bearing capacity and initial bending stiffness of the joint both increase as the width-to-thickness ratio of the upper corner piece base plate decreases; however, the improvement in bearing capacity becomes less significant with a further reduction in the width-to-thickness ratio of the upper corner piece base plate. When the width-to-thickness ratio of the upper corner piece base plate decreases from 12.50 to 10.00, the bearing performance of the joint improves with the reduction of the width-to-thickness ratio. However, when it continues to decrease from 10.00 to 9.09, the improvement in the bearing capacity of the joint is not significant. Therefore, considering both the performance and cost-effectiveness of the joint, it is recommended that the thickness of the upper corner piece base plate should be 20 mm, with a width-to-thickness ratio of 10.0.

Figure 12 shows the stress cloud diagram under the ultimate bearing capacity of the joint when the width-to-thickness ratio of the upper corner piece base plate is 10.00 (model B-3, B-7). As can be seen from the figure, under the ultimate bearing capacity state of the joint, yielding mainly occurs at the base plate of the upper corner piece, the compressed sidewall of the upper corner piece, and at the bolt shank location, while the stress in the lower corner piece is relatively small. Based on the structural characteristics of the upper corner piece and the stress cloud diagram, it can be seen that during bending in the Z-direction, the joint primarily exhibits compressive deformation on the hole side of the upper corner piece. During bending in the X-direction, the pressure is mainly borne by the sidewall of the upper corner piece. The bearing capacity and initial bending stiffness in the Z-direction are both smaller than those in the X-direction. As the thickness of the upper corner piece base plate increases, the initial bending stiffness of the joint increases, and the ability of the joint to resist bending deformation is enhanced; therefore, the plastic development region at the base plate of the upper corner piece decreases, the area of the compressed region on the opening side of the upper corner piece decreases, and the difference in the bending capacity in the X-direction gradually decreases.

4.2. The Influence of the Height-to-Thickness Ratio of the Lower Corner Piece Top Plate

The bolt hole is arranged at the top plate of the lower corner piece, which is the key part for realizing the bolt connection. The larger the thickness of the upper corner piece base plate, the less likely it is to deform and the greater its bearing capacity. However, if the thickness is too large, it will not only fail to improve the bearing capacity but also increase its weight. Ten models with different height-to-thickness ratios h/t₂ (where h is the height of the protrusion of the lower corner piece, taken as 60 mm in this section and t₂ is the thickness of the lower corner piece top plate) and bending directions of the lower corner piece top plate were selected for static bending simulation. The bearing capacity of each model is shown in

Table 4. The bearing capacity of each model (B-3, B-7, B-9~B-16).

Model Number	Thickness of the Lower Corner Piece Top Plate/mm	h/t2	Bending Direction	Initial Bending Stiffness /(kN·m/rad)	Yield Bearing Capacity /(kN·m)	Ultimate Bearing Capacity /(kN·m)
B-9	16	3.75	X	3135.89	74.72	85.32
B-10	20	3.00	X	3300.79	84.53	96.51
B-11	24	2.50	X	3403.42	87.31	98.74
B-12	28	2.14	X	3597.65	87.51	99.86
B-3	60	1.00	X	3895.50	87.75	99.15
B-13	16	3.75	Z	2985.93	70.47	82.16
B-14	20	3.00	Z	3189.46	78.66	91.04
B-15	24	2.50	Z	3334.82	81.78	92.74
B-16	28	2.14	Z	3446.93	81.26	93.61
B-7	60	1.00	Z	3724.80	84.31	92.76

, and the ultimate bearing capacity and stress cloud diagram of each model is shown in Figure 13. The comparison of the stress diagrams for the lower corner piece top plate with a height-to-thickness ratio of 2.50 and that with a height-to-thickness ratio of 1.00 (i.e., when the protrusion of the lower corner piece is solid) is shown in Figure 14.

It can be seen from the table and figures that both the initial bending stiffness and the bearing capacity of the joint increase as the height-to-thickness ratio of the lower corner piece top plate decreases. However, after the height-to-thickness ratio is reduced to 2.50, further reductions in the height-to-thickness ratio result in only a slight increase in initial stiffness, with negligible effects on the bearing capacity of the joint. The bearing capacity approaches that of the lower corner piece top plate with a height-to-thickness ratio of 1.00 (i.e., when the lower corner is solid). Additionally, when the height-to-thickness ratio of the lower corner piece top plate is 2.50, the material at the bolt holes reaches its yield strength, indicating a more efficient use of the material. Considering the performance and cost-effectiveness of the joint, it is recommended that the thickness of the lower corner plate be set at 24 mm, with a height-to-thickness ratio of 2.50.

4.3. The Influence of the Height of the Protrusion of the Lower Corner Piece

In the bending condition of the joint, the protrusion of the lower corner piece is one of the key areas. If the height of the protrusion of the lower corner piece is less than 30 mm, it will cause the “sliding out” phenomenon of the convex at the joint when the ultimate bearing capacity is reached, as shown in Figure 15. We selected eight models with different convex heights and bending directions for static bending simulation based on the premise that the convex height is greater than 30 mm. The bearing capacity of each model is shown in

Table 5. The bearing capacity of each model (B-11, B-15, B-17~B-22).

Model Number	Height of the Protrusion of the Lower Corner Piece/mm	Bending Direction	Initial Bending Stiffness /(kN·m/rad)	Yield Bearing Capacity /(kN·m)	Ultimate Bearing Capacity /(kN·m)
B-17	30	X	2449.47	62.57	70.08
B-18	40	X	2725.00	69.62	78.50
B-19	50	X	3031.17	76.62	87.27
B-11	60	X	3403.42	87.31	98.74
B-20	30	Z	2404.13	58.83	63.43
B-21	40	Z	2654.48	64.42	72.83
B-22	50	Z	2895.51	71.90	81.36
B-15	60	Z	3334.82	81.78	92.74

. The ultimate bearing capacity of each model is shown in Figure 16. As shown in Table 5 and Figure 16, the bearing capacity and initial bending stiffness of the joint both increase with the increase in convex height. This paper recommends that the height of the protrusion of the lower corner piece should be greater than 30 mm. However, it is important to note that an excessively large convex height may increase the difficulty of assembly at the construction site, affecting construction efficiency.

4.4. The Influence of Bolt Diameter

Six models with different bolt diameters and bending directions were selected to conduct static bending simulations of the joint. The bearing capacity of each model is shown in

Table 6. The bearing capacity of each model (B-11, B-15, B-23~B-26).

Model Number	Bolt Diameter /mm	Bending Direction	Initial Bending Stiffness /(kN·m/rad)	Yield Bearing Capacity /(kN·m)	Ultimate Bearing Capacity /(kN·m)
B-23	30	X	3236.20	76.88	86.55
B-11	36	X	3403.42	87.31	98.74
B-24	42	X	3674.95	95.99	107.32
B-25	30	Z	3159.02	73.91	83.21
B-15	36	Z	3334.82	81.78	92.74
B-26	42	Z	3526.18	88.72	99.23

. The stress cloud diagram of each model is shown in Figure 17.

When the bolt diameter increases from 30 mm to 36 mm, the initial bending stiffness in the X and Z-directions of the joint increases by 5.17% and 5.57%, respectively; the yield bearing capacity increases by 13.57% and 10.65%, respectively, and the ultimate bearing capacity increases by 14.08% and 11.45%, respectively. When the bolt diameter increases from 36 mm to 42 mm, the initial bending stiffness in the X and Z-directions of the joint increases by 7.98% and 5.74%, respectively; the yield bearing capacity increases by 9.94% and 8.49%, respectively, and the ultimate bearing capacity increases by 8.69% and 7.00%, respectively. As shown in Table 6 and Figure 17, both the bearing capacity and initial bending stiffness of the joint increase with the bolt diameter. However, as the bolt diameter increases, the improvement in the joint’s bearing capacity becomes less pronounced. This is because the joint’s ability to resist bending deformation strengthens as the bolt diameter increases. However, the ability of the components in the joint region to work together decreases. Under the ultimate bearing capacity state of the joint, the plastic deformation areas at the base plate of the upper corner piece and the top plate of the lower corner piece gradually increase, leading to a reduced improvement in bearing capacity. Therefore, the selection of the bolt diameter should be coordinated with the other components in the joint region. This paper suggests that the bolt diameter be chosen as either 36 mm or 42 mm.

4.5. Development of Bending Stress in the Joint

Based on the above parameter analysis results, models B-11 and B-15 can be used as examples to illustrate the stress development of the joint under bending conditions [24], as shown in Figure 18.

4.6. The Influence of Axial Compression

For the forces acting on the emergency repair pier in actual conditions, the joint mainly bears axial pressure and bending moment. This paper studies the bending performance of the joint under axial compression ratios of 0.1, 0.2, and 0.3, respectively. The calculation formula for axial compression ratio [33] is shown in Equation (1). The bearing capacity of each model is shown in

Table 7. The bearing capacity of each model (B-11, B-15, B-27~B-32).

Model Number	Axial Compression Ratio	Bending Direction	Initial Bending Stiffness /(kN·m/rad)	Yield Load Point		Ultimate Load Point
				Yield Rotation/rad	Yield Bearing Capacity /(kN·m)	Ultimate Rotation/rad	Ultimate Bearing Capacity /(kN·m)
B-11	0	X	3403.42	0.041	87.31	0.116	98.74
B-27	0.1	X	11,461.35	0.020	89.02	0.049	105.26
B-28	0.2	X	13,677.88	0.015	97.56	0.039	120.01
B-29	0.3	X	13,283.93	0.013	116.18	0.032	135.40
B-15	0	Z	3334.82	0.040	81.78	0.088	92.74
B-30	0.1	Z	7784.97	0.018	72.70	0.042	88.76
B-31	0.2	Z	8220.58	0.013	77.55	0.029	91.06
B-32	0.3	Z	8721.32	0.013	83.96	0.023	93.41

. The moment–rotation curves of each model are shown in Figure 19.

$$u={\displaystyle \frac{N}{A\times f_{\mathrm{y}}}}$$

(1)

where u is the axial compression ratio, N is the vertical axial pressure of the steel column, f_y is the measured effective yield strength of the steel column, and A is the cross-sectional area of the specimen’s steel column.

As the axial compression ratio increases, the yield rotation and ultimate rotation of the joint decrease, while the initial bending stiffness of the joint increases significantly. This is because when the joint is subjected to lateral horizontal displacement loads, the axial pressure applied at the top of the column makes it more difficult for the joint to deform. As the axial compression ratio increases, the rate of decrease in the joint’s bending capacity accelerates after it reaches its ultimate load capacity. This is because, with the increase in axial pressure, the plastic strain in the column intensifies, leading to an accelerated rate of decline in the structural load-bearing capacity.

When subjected to lateral horizontal displacement loads in the X-direction, the bending capacity of the joint increases with the increase in axial compression ratio. However, when subjected to lateral horizontal displacement loads in the Z-direction, the bending capacity of the joint does not change significantly with the increase in axial compression ratio. This is because the axial pressure applied at the top of the column makes it more difficult for the joint to deform when subjected to lateral horizontal displacement loads in the X-direction, thereby allowing the joint to withstand larger lateral horizontal displacement loads, which increases the joint’s load-bearing capacity. When the joint is subjected to lateral horizontal displacement loads in the Z-direction, the upper corner piece experiences compressive yielding on the side with the hole. The hole is a weaker point in the joint’s load-bearing performance. A larger axial pressure also causes compressive yielding on the hole side of the upper corner piece, leading to failure. As a result, the joint’s load-bearing capacity does not change significantly or may even decrease with the increase in axial compression ratio.

5. Joint Property Determination

The classification method of the joint in reference [29] is used to assess the attributes of the new type of modular steel structure emergency repair pier joint. The connection classification is shown in Figure 20.

In the figure, both m and θ are dimensionless quantities, and the calculation formula is given by Equation (2).

$$m={\displaystyle \frac{M}{M_{\mathrm{bp}}}}, \theta ={\displaystyle \frac{{\theta }_{\mathrm{r}}E{I}_{\mathrm{b}}}{M_{\mathrm{bp}}{L}_{\mathrm{b}}}}, M_{\mathrm{bp}}=W_{\mathrm{bp}}{f}_{\mathrm{y}}$$

(2)

where M_bp is the full plastic bending moment of the steel beam, ${\displaystyle \frac{E{I}_{\mathrm{b}}}{L_{\mathrm{b}}}}$ is the linear stiffness of the steel beam, EI_b is the section’s elastic bending stiffness of the steel beam, L_b is the span of the steel beam, W_bp is the plastic section modulus of the steel beam, f_y is the yield strength value of the steel beam, M is the finite element simulation result of the joint moment, and θ_r is the finite element simulation result of the joint rotation angle.

This paper classifies the joint in this study using the moment–rotation curves of models B-11 and B-15 as examples. The joint in this paper is used to constrain the modular column, so the parameters related to the steel beam should be calculated according to the modular column parameters. The cross-section of the modular column is 200 mm × 200 mm × 8 mm, with a height of 3000 mm and a yield strength of 354 MPa. The calculation is as follows:

$$\begin{array}{c}{M}_{\mathrm{bp}}=W_{\mathrm{bp}}{f}_{\mathrm{y}}={\displaystyle \frac{b{h}^2}{4}}{f}_{\mathrm{y}}=156.7\mathrm{kN}\cdot \mathrm{m};\hfill \\ {\displaystyle \frac{E{I}_{\mathrm{b}}}{L_{\mathrm{b}}}}={\displaystyle \frac{E\frac{b{h}^3}{12}}{L_{\mathrm{b}}}}=2557.59\mathrm{kN}\cdot \mathrm{m};\hfill \\ \mathrm{That} \mathrm{is}: m={\displaystyle \frac{M}{156.7\mathrm{kN}\cdot \mathrm{m}}},\theta ={\displaystyle \frac{{\theta }_{\mathrm{r}}E{I}_{\mathrm{b}}}{M_{\mathrm{bp}}{L}_{\mathrm{b}}}}=16.3\times {\theta }_{\mathrm{r}}\hfill \end{array}$$

The joint stiffness criterion is shown in Figure 21. From the perspective of load-bearing capacity, it is classified as a partially resistant joint, and from the perspective of stiffness, it is classified as a semi-rigid joint.

6. Calculation of the Initial Bending Stiffness of the Joint

This study references the component method for calculating the bending stiffness of joints. The basic principle of the component method [34] involves decomposing the structural node into several fundamental components, with each component modeled as a linear or nonlinear spring to simulate its mechanical properties. The overall mechanical performance of the joint is then derived by performing mechanical calculations through series and parallel combinations of the springs. Based on the finite element analysis mentioned above, it is observed that the joint exhibits poor bending performance when subjected to lateral displacement loads parallel to the direction of the opening. This paper analyzes the bending stiffness of the node under such loading conditions as an example.

The components that undergo deformation under the action of bending moments include the upper corner piece side plate (Plate I and Plate II), the lower corner piece side plate (Plate III), the end plate (Plate IV) of the upper corner piece, and the bolt. The overall bending stiffness of the joint can be simplified and calculated as the stiffness generated by four equivalent springs arranged in series [35,36]. The simplified model is shown in Figure 22.

In the figure, K_rsp is the stiffness of the side plates (Plate I and Plate II) after being combined, K_rlp is the stiffness of the lower side plate (Plate III), K_rep is the stiffness of the end plate (Plate IV), and K_rbl is the stiffness of the bolt; z is the internal lever arm of the rotational system, which is the distance between the centerline of the tensioned bolt and the rotation point. The calculation of the joint’s bending stiffness is given by Equation (3).

$$K={\displaystyle \frac{M}{\theta }}={\displaystyle \frac{z_{eq}^2}{\frac{1}{K_{rsp}}+\frac{1}{K_{rlp}}+\frac{1}{K_{rep}}+\frac{1}{K_{rbl}}}}$$

(3)

To simplify the analysis, the following assumption is made.

• The rotational center of all equivalent springs is assumed to be point O.
• Based on the position of the opening, the upper corner piece of the joint is divided into the side plates (Plate I and Plate II), the lower side plate (Plate III), and the end plate (Plate IV).
• Since the deformation of the lower corner piece is relatively small during the bending process of the joint, the influence of the lower corner piece on the overall bending stiffness of the joint is neglected. The overall bending stiffness of the joint is obtained by the combination of the stiffness of the upper corner piece and the bolt.

  (1)

  Calculation of the stiffness of the side plate (Plate I and Plate II) of the upper corner piece.

The bending stiffness of the side plate (Plate I and Plate II) of the upper corner piece is obtained by the combination of the tensile stiffness of Plate I, the tensile stiffness of Plate II, and the shear stiffness of Plate II. The tensile stiffness of Plate I is calculated as shown in Equation (4), and the tensile stiffness of Plate II is calculated as shown in Equation (5).

$$K_{rsp1}={\displaystyle \frac{Et{b}_{s1}}{h}}$$

(4)

$$K_{rsp2}={\displaystyle \frac{Et{b}_{s2}}{h}}$$

(5)

where E is the elastic modulus of the upper corner piece, t is the thickness of the side plate, b_s1 is the effective width of Plate I, b_s2 is the effective width of Plate II, and h is the height of the side plate.

In the calculation of the shear stiffness of Plate II, the plate is divided into upper and lower parts along the rotation center. The shear stiffness of the upper part of Plate II is calculated as shown in Equation (6), and the shear stiffness of the lower part of Plate II is calculated as shown in Equation (7).

$$K_{rspv1}={\displaystyle \frac{G{A}_w}{(1-\rho )h_0}}$$

(6)

$$K_{rspv2}={\displaystyle \frac{G{A}_w}{(1-\rho )h_0}}$$

(7)

where G is the shear modulus, A_w is the shear area, (1 − ρ) is the reduction factor introduced to account for the beneficial effects of shear in the beam [37,38], and ρ is equal to h₀/h, where h₀ is the height of the shear region.

In summary, the stiffness of the side plate (Plate I and Plate II) of the upper corner piece is calculated as shown in Equation (8).

$$K_{rsp}={({\displaystyle \frac{2}{K_{rspv1}}}+{\displaystyle \frac{2}{K_{rspv2}}}+{\displaystyle \frac{2}{K_{rsp2}}}+{\displaystyle \frac{1}{K_{rsp1}}})}^{-1}$$

(8)

• (2)

  The compressive stiffness of the lower side plate (plate III) of the upper corner piece is calculated as shown in Equation (9).

$$K_{rlp}={\displaystyle \frac{Et{b}_l}{h}}$$

(9)

where b_l is the effective compressive width of the lower side plate (plate III).

• (3)

  The bending stiffness of the end plate (plate IV) of the upper corner piece is calculated as shown in Equation (10).

$$K_{rep}={\displaystyle \frac{EA}{l}}$$

(10)

where A is the bending area and l is the effective length of the end plate (plate IV).

• (4)

  The tensile stiffness of the bolt is calculated as shown in Equation (11).

$$K_{rbl}=2{\displaystyle \frac{(1+\alpha )E_b{A}_e}{l_b}}$$

(11)

where l_b is the calculation length of the bolt shank, equal to the thickness of the clamped steel plate plus the thickness of the washer; E_b is the elastic modulus of the bolt material; A_e is the effective cross-sectional area of the bolt shank; and α is the ratio of the preload distribution area between the clamped steel plates of the high-strength bolt to the cross-sectional area of the bolt shaft. According to reference [39], α is taken as 10.

The calculation results of the simplified model in this paper are compared with the simulation results, as shown in

Table 8. Comparison of the initial bending stiffness of the joint.

Model Number	FEM/(kN·m/rad)	Cal/(kN·m/rad)	Cal/FEM
B-5	3220.00	3042.07	0.94
B-6	3488.16	3110.61	0.89
B-7	3724.80	3169.52	0.85
B-8	3955.17	3221.06	0.81

. The accuracy of the theoretical calculation model in this study is within 20%, which provides certain guidance for structural design and engineering applications.

7. Conclusions

The main conclusions drawn from the above study are as follows:

• A new type of modular steel structure emergency repair pier joint is proposed, aiming to achieve rapid assembly and connection between modular units, thereby improving the bridge repair speed.
• The width-to-thickness ratio of the upper corner piece base plate D/t₁ (where D is the width of the upper corner piece base plate and t₁ is the thickness of the upper corner plate), the height-to-thickness ratio of the lower corner piece top plate h/t₂ (where h is the height of the protrusion of the lower corner piece and t₂ is the thickness of the lower corner piece top plate), the height of the protrusion of the lower corner piece (h), and the bolt diameter (d) are selected as parameters to study the bending performance of the new type of modular steel structure emergency repair pier joint. The recommended values are as follows: t₁ is 20 mm, D/t₁ is 10.00; t₂ is 24 mm, h/t₂ is 2.50; h is greater than or equal to 30 mm, and d is either 36 mm or 42 mm.
• The bending performance of the joint under axial compression ratios of 0.1, 0.2, and 0.3 is studied separately. The results show that as the axial compression ratio increases, the yield rotation angle and ultimate rotation angle of the joint decrease, and the rate of decline in the bearing capacity increases after the joint reaches its ultimate load-bearing capacity. Under the action of the X-direction horizontal lateral displacement load, the initial bending stiffness and bending capacity of the joint increase with the axial compression ratio. Under the action of Z-direction horizontal lateral displacement load, the initial bending stiffness of the joint increases with the axial compression ratio, while its bending bearing capacity does not change significantly.
• The new type of modular steel structure emergency repair pier joint is classified as a partially resistant joint from the perspective of bearing capacity and as a semi-rigid joint from the perspective of stiffness.
• A simplified calculation model for the joint was proposed based on the component method. The accuracy of the theoretical calculation model in this study is within 20%, which provides certain guidance for structural design and engineering applications.

8. Future Work

• This paper only analyzes the bending performance of the joint. Future research will continue to focus on the tensile performance, compressive performance, shear performance, and the overall structure of the repair pier.
• This paper investigates the bending performance of the joint through finite element numerical simulation, and future studies will continue with experimental research for further investigation.