Seismic Performance of Prestressed Prefabricated Concrete Frames with Mechanical Connection Steel Bars

Abstract

Seismic resilience is a critical concern in the development of prefabricated concrete structures. This study investigates the seismic performance of prestressed prefabricated concrete frames with mechanically connected steel bars through both experiment and finite element simulations using ABAQUS. The research aimed to evaluate the influence of prestressed and mechanical connections on structural stiffness, energy dissipation and failure mechanisms, and a restoring force model was developed based on the experimental and numerical results to provide a theoretical basis for seismic design. The parametric analysis based on the verified numerical model shows that the pretension can significantly enhance the bearing capacity, stiffness and deformation recovery ability of the prefabricated concrete frames. The peak load increased by 30.8%, the initial stiffness improved by 17.4%, the ductility coefficient reached 2.82, the residual deformation rate reduced by 40.7%, the emergence and development of cracks delayed, and the crack width reduced. Improving the effective prestress in a certain range can improve the bearing capacity and initial stiffness of the frame. Increasing the strength of concrete and the ratio of the longitudinal reinforcement of beam and column can effectively enhance the bearing capacity of the frame. With the increase of axial compression ratio in a certain range, the bearing capacity and initial stiffness of the frame increase significantly, but the ductility decreases. Based on the hysteresis curve and skeleton curve tested, the skeleton curve model and stiffness degradation law of the prestressed prefabricated concrete frames reinforced with mechanical connection steel bars were fitted, and the restoring force model was established. The predicted value was in good agreement with the experimental value, illustrating the validity of the model developed. These results offer valuable insights for optimizing the seismic design of prefabricated concrete frames, ensuring a balance between strength, stiffness, and ductility in earthquake-resistant structures.

1. Introduction

1.1. Research Background

The main reason for the collapse of prefabricated concrete frame structures under earthquake action is the destruction of the connection joints of each frame [1], so the beam–column joint is the weakness of the whole structure. A reliable connection can maintain the integrity of the structure, which is an important factor to improve the seismic performance of the prefabricated structure. The connection methods of fabricated concrete frame joints mainly include wet connection [2] and dry connection [3]. Their seismic performance is heavily influenced by the behavior of beam-column joints, which are critical load transfer points in a structure. Under earthquake loading, these joints experience high shear forces, leading to potential failure modes such as cracking, stiffness degradation, and bond slip of reinforcement. Ensuring the seismic resilience of beam–column joints is essential for the overall stability of reinforced concrete structures.

In traditional cast-in-place construction, monolithic connections provide sufficient integrity. However, in prefabricated reinforced concrete structures, the presence of mechanical connections introduces new challenges, such as stress concentration and potential joint failure under cyclic loading. The research of Nicoletti et al. [4] and Shen et al. [5] showed that inadequate joint detailing or insufficient confinement reinforcement can lead to premature joint failure, significantly reducing the ductility and energy dissipation capacity of the structure. Addressing these issues through innovative connection methods, such as prestressing and mechanically connected reinforcement, is a promising strategy to enhance the seismic performance of prefabricated frames.

Blandon et al. [6] conducted a reversed cyclic loading test on a two-story precast concrete frame structure, and focused on observing the changes in the joints of the specimens. The test showed that when the longitudinal bar with a bent hook in the bottom of the beam is less than eight times of its diameter in the joint, adhesive slip occurs easily and it is difficult to ensure its continuity. Priestley et al. [7] studied the seismic performance of a five-story prefabricated frame-shear wall structure system under pseudo-dynamic tests. The results showed that there is only slight damage in the direction of shear wall, and no significant strength degradation in the direction of frame despite the inter-story displacement reaching 4.5%. The structural system had good seismic performance. Ren et al. [8] designed and produced a clasp mechanical connection device for concrete-filled square steel tubular columns. Through one-way tensile tests on two mechanical connection forms, single clasp and double clasp, the failure mode and connection performance of the mechanical connection device were analyzed. The test showed that the mechanical connection device is simple in construction, fast in installation and high in bearing capacity, which is beneficial to shorten the project period. The strength and welding quality of the welded steel plate should be strengthened in practical application. Based on the above research, it is evident that traditional reinforcement anchorage and welded connections in prefabricated structures may face issues such as insufficient anchorage, complex construction procedures, or inadequate reliability. In contrast, mechanical connections can significantly improve construction efficiency, reduce on-site workload, and enhance connection quality while ensuring overall structural performance. Therefore, studying and promoting high-performance mechanical connection technologies suitable for prefabricated buildings is of great significance for increasing their engineering application value.

1.2. Research Significance

In recent years, prefabricated structure connection technology has been extensively studied, with significant progress made, particularly in enhancing the mechanical performance of connections and improving construction convenience. For example, Cheok et al. [9] proposed a prefabricated frame joint mixed with prestressed reinforcement and ordinary reinforcement, which provided energy dissipation through the yield of ordinary reinforcement and shear resistance through the friction generated by some unbonded prestressed tendons at the beam–column joints. The research showed that the lateral displacement angle and failure mode of the joint were basically equivalent to that of cast-in-place. However, early research primarily focused on traditional connection methods. In recent years, an increasing number of studies have shifted attention to the mechanical performance, seismic behavior, and durability of novel prefabricated connections. The latest research indicates that the optimized design of prefabricated connections can effectively enhance structural stability and safety. For example, Gong et al. [10] proposed an improved prefabricated connection method that significantly enhances load-bearing capacity and seismic performance, with its mechanical advantages validated through experiments and numerical simulations. Meanwhile, Liu et al. [11] investigated the deformation characteristics of prefabricated connections under cyclic loading and found that their ductility and fatigue resistance were significantly improved compared to traditional methods. Furthermore, Fang et al. [12] conducted shake table tests and nonlinear numerical analyses to systematically evaluate the seismic performance of innovative connection methods, demonstrating their promising application potential in high-seismic-intensity regions. Although these studies have made some progress in improving the performance of prefabricated structural connections, several key issues remain to be addressed, such as the balance between connection stiffness and construction efficiency, the mechanical performance of prefabricated connectors under extreme load conditions, and the long-term durability and reliability of connection joints.

To further optimize the prefabricated connection method, this paper proposes a new type of clasp type mechanical connection, as shown in Figure 1. Each clasp type mechanical connection contains two parts; one part is a rod and the other part is a card slot, and they are connected with two rebars, and the two rebars can be connected by inserting the rod into the card slot. The outer part of the plunger is a steel sleeve with thread inside; one end is connected with the steel bar after stripping, and the other end is connected with the plunger. The outer part of the card slot is the same as the steel sleeve of the insert rod; one end is connected with the steel bar, the other end is put into the spring, washer, and intermediate card in turn, and screwed into the nut without a cap. When the rod is inserted into the card slot, the spring below can provide a reacting force, and the card above can limit the card from coming out, so that the steel bar is effectively connected. For the study on the mechanical properties of this new type of clap-type mechanical connector, Sun et al. [13] designed and produced a clap-type mechanically connected prestressed concrete square pile, and then carried out a finite element analysis on the mechanical properties of its cap joints. The research showed that increasing the effective pre-compression stress can improve the horizontal bearing capacity at the joint, and setting ordinary reinforcement within 150 m of the pile end can effectively improve the ductility and horizontal bearing capacity of prestressed square piles.

In this study, three frame specimens were designed and fabricated, including two mechanically connected prefabricated integral frames and one conventional cast-in-place frame. The two mechanically connected prefabricated frames were modeled and analyzed using an ABAQUS finite element, with a focus on investigating their stiffness, strength, and energy dissipation characteristics. The reliability of the finite element model was verified by comparing the hysteretic curve, skeleton curve, and failure pattern with the test results. On the basis of this, the parameters affecting the prestressed mechanical joint assembly integral frame were analyzed, including effective prestress, concrete strength, axial compression ratio, and longitudinal reinforcement ratio of beam and column. Based on the hysteresis curves and skeleton curves tested, the skeleton curve model and stiffness degradation law of the prestressed mechanical connection assembly integral frame were fitted, and the restoring force model of the prestressed prefabricated concrete frames reinforced with mechanical connection steel bars was established according to the reasonable hysteresis rule. The findings provide valuable insights into the impact of prestressing on beam–column joints and contribute to the development of earthquake-resistant prefabricated structures.

2. Experimental Program

2.1. Specimen Design

The analysis focused on specimens ZK1, ZK2, and ZK3, as illustrated in Figure 2, where ZK2 is a prestressed specimen and ZK3 is a conventional cast-in-place specimen. All concrete used in this study was cast with C40 concrete. The design of the specimens followed the principle of “strong columns and weak beams, strong joints and weak components” [14].

2.2. Mechanical Property

The concrete was cast in two batches; the first batch was used for the cast-in-place specimens and the precast beams and columns of the assembled specimens, while the second batch completed the remaining parts of the assembled specimens. The concrete cubic compressive strength f_cu, axial compressive strength f_c, axial tensile strength f_t, and elastic modulus E_c were defined using the formulas specified in the Code for Design of Concrete Structures (GB50010-2010) [15]. The material properties of the concrete are presented in

Table 1. Material properties of concrete.

Concrete	fcu/MPa	fc/MPa	ft/MPa	Ec/GPa
First batch	46.6	31.2	1.96	34.0
Second batch	41.7	27.9	1.85	33.0

.

The mechanical properties of the reinforcement are shown in

Table 2. Mechanical properties of reinforcements.

Reinforcement	Type	D/mm	fy/MPa	fu/MPa	Er/GPa
HRB400	/	8	422	541	200
HRB400	/	18	442	618	200
HRB400	/	20	448	633	200
HRB400	/	25	440	618	200
HRB400	Mechanical connection	18	427	611	200
HRB400	Mechanical connection	20	436	633	200
Steel strand	/	15.2	1640	1930	195

Note: The elastic modulus of the corresponding material above is obtained from the Design Standard for Steel Structures (GB 50017-2017) [16]. The yield strength corresponding of the steel strand is the conditional yield strength, that is, the stress corresponding to the residual strain 0.2%, which is converted from 0.85 f_u.

. D is the diameter of the reinforcement; f_y is the yield strength; f_u is the ultimate strength; and E_r is the elastic modulus.

Taking a steel bar with a diameter of 18 mm as an example, Figure 3 presents the stress–strain curves of reinforcement and mechanical connectors. As observed in the figure, the stress–strain curve of reinforcement bars exhibits a rapid increase in stress during the initial phase. When the strain reaches a certain value (approximately 0.02), the stress of the reinforcement bar increases rapidly and tends to stabilize, demonstrating its high yield strength. The reinforcement bars maintain a high stress at larger strains, indicating strong tensile capacity and stable load-bearing performance. In contrast, the stress–strain curve of connectors is relatively flat, especially within the strain range of 0.02 to 0.05. The stress increase of the connector is slower, reflecting its excellent plastic deformation ability. This gradual curve indicates that the connector can adapt well to deformation under load and maintain a relatively stable stress value even under large deformations. Reinforcement bars exhibit high yield strength, especially when the strain approaches 0.05, where the stress value of the reinforcement bars is significantly higher. This makes them suitable for structural components that bear large loads. The high strength of reinforcement bars provides enhanced support capability under high stress, making them particularly suitable for building and structural applications that require high load-bearing capacity. Connectors, on the other hand, demonstrate good ductility, with a more gradual increase in stress, allowing them to maintain excellent deformation capacity even under larger strains. This gives connectors an advantage in connection points and areas subject to more complex loading conditions, as they can effectively absorb and adapt to the stress changes between different components, preventing structural failure due to premature yielding.

2.3. Test Setup

The loading device and loading protocol for this experiment are shown in Figure 4. A low-cycle reversed loading test was conducted, with the vertical load applied to the column tops using two hydraulic jacks. The column axial compression ratio was set to 0.15, corresponding to an axial force of 257.85 kN. Initially, 30% of the axial force was applied and then fully unloaded before reloading to the designed axial force, which was kept constant throughout the test. The loading process was terminated when the specimen’s load-bearing capacity dropped to 85% of the peak load or when significant concrete spalling and reinforcement exposure were observed. To facilitate crack observation, the joint core region and adjacent beam–column surfaces were pre-coated with diluted white latex paint. After drying, a 50 × 50 mm square grid was marked using ink lines. During each loading stage, the crack locations and propagation lengths were recorded with a marker pen, and a crack width gauge was used to measure crack width variations. At the end of the test, the final failure mode of the specimen was documented.

3. Experimental Result and Discussion

3.1. Failure Mode

The failure mode of the three mechanically connected prefabricated frames are shown in Figure 5. For the cast-in-place frame specimen XK, cracks first appeared in the tensile zone at the beam ends and subsequently extended to the column base and core region. Eventually, plastic hinges formed at the beam ends, leading to concrete crushing and spalling, which resulted in specimen failure. In contrast, for the mechanically connected prefabricated frame specimens ZK1 and ZK2, cracks were mainly concentrated at the connectors. The number of cracks on the beam top was relatively high and exhibited an asymmetric development trend. Ultimately, major cracks formed at the beam ends, causing concrete crushing and spalling. Furthermore, the prestressed specimen ZK2 demonstrated superior crack control performance, with delayed crack initiation, fewer cracks, and smaller crack widths. After unloading, the cracks could close effectively, and the beam ends were able to reposition, highlighting the positive role of prestressing in enhancing the seismic performance of prefabricated frames.

3.2. Hysteretic Curve

Hysteretic curve is a crucial indicator for evaluating the seismic performance of structures, as it comprehensively reflects key characteristics such as load-bearing capacity, energy dissipation ability, ductility, stiffness degradation, and restoring force behavior. Hysteretic curves of the three tested frames are shown in Figure 6. The displacement and load at the beam ends were automatically recorded by the actuator. The actuator’s pushing motion to the right was defined as the positive direction, while pulling to the left was defined as the negative direction.

By comparing the load-displacement hysteretic curves of the three frame specimens, it can be observed that in the initial elastic stage of loading, all three specimens exhibit linear growth in their hysteresis curves. During unloading, they return along the original path with minimal residual deformation, and no significant hysteresis loops are formed. As the displacement increases, the reinforcement at the beam ends yields, leading to an expansion of the hysteresis loops and an improvement in energy dissipation capacity. After reaching the peak load, concrete crushing occurs at the beam ends, resulting in a decrease in load-bearing capacity and an increase in residual deformation. The hysteresis curve of the cast-in-place frame specimen XK is relatively full and spindle-shaped, indicating strong energy dissipation capacity. In contrast, due to the presence of connectors, the hysteresis curves of the prefabricated specimens ZK1 and ZK2 exhibit varying degrees of pinching effects, transitioning from an arch shape to an inverse S-shape. This phenomenon is mainly attributed to the gap in the connectors, which causes slip between the concrete and the connectors. Compared to ZK1, the prestressed frame specimen ZK2 shows smaller residual deformation due to the elastic rebound effect of the prestressing tendons. However, the pinching effect is more pronounced, leading to a slightly lower energy dissipation capacity than ZK1.

3.3. Test Skeleton Curve

Based on the hysteretic curves of the specimens, the peak load points of the first cycle at each displacement level were sequentially connected to obtain the skeleton curve of the specimens, as shown in Figure 7.

By comparing the skeleton curves of the three frame specimens, it can be observed that all three curves are approximately centrosymmetric, with similar load-bearing capacities in both loading directions. This indicates good specimen quality and a reasonable loading setup. All specimens experienced an elastic stage, a yielding stage, and a failure stage. As the displacement increased, the specimens gradually entered the elastoplastic stage, leading to stiffness degradation. After reaching the peak load, the load-bearing capacity declined, with reinforcement at the beam ends yielding and concrete crushing and spalling. Comparing specimens XK and ZK1, their skeleton curves exhibit similar trends in the initial stage, with comparable initial stiffness. However, the peak load of ZK1 was 2.9% lower than that of XK, and ZK1 exhibited a more rapid load drop during failure, resulting in slightly lower ductility than XK. Comparing specimens ZK1 and ZK2, the introduction of prestressing improved both the initial stiffness and load-bearing capacity. The peak load of ZK2 increased by 17.4% compared to ZK1, while the slope of the descending branch remained nearly unchanged. This demonstrates that prestressing has a significant effect on enhancing the load-bearing capacity.

3.4. Ductility

In seismic design, structural ductility is equally crucial, as it determines the deformation capacity of a structure under extreme loads and its energy dissipation ability before failure. Greater ductility allows a structure to withstand larger deformations before its load-bearing capacity deteriorates, thereby enhancing overall seismic performance. Therefore, based on the analysis of the skeleton curves, it is necessary to further investigate the ductility differences among the specimens to evaluate the impact of prestressing and mechanical connections on the seismic performance of prefabricated concrete frames.

Structural ductility is a critical characteristic for evaluating seismic performance. The ductility of a specimen can be characterized by the ductility coefficient μ_Δ [17]. A larger ductility coefficient indicates greater deformation capacity and better seismic performance of the specimen. μ_Δ is shown in Equation (1), where Δ_u represents the ultimate displacement, and Δ_y represents the yield displacement. Δ_y is determined using the equal energy method [18].

$${\mu }_{∆}={\displaystyle \frac{{\mathrm{\Delta }}_{\mathrm{u}}}{{\mathrm{\Delta }}_{\mathrm{y}}}}$$

(1)

The calculated ductility coefficients of the specimens are shown in

Table 3. Specimen displacement ductility coefficients.

Specimen	Direction	Py/kN	Δy/mm	Pp/kN	Δp/mm	Pu/kN	Δu/mm	μΔ
ZK1	Average	165.32	19.11	192.69	32.48	147.89	57.41	3.01
ZK2	Average	195.79	20.16	226.22	42.56	189.75	57.24	2.84
XK	Average	170.82	19.28	198.33	32.13	159.77	65.88	3.42

. P_y is the yield load of the specimen; P_u is the ultimate load of the specimen; P_p is the peak load of the specimen; and Δ_p is the peak displacement of the specimen. All data represent the average values in both positive and negative directions.

By analyzing the above table, the following conclusions can be drawn:

(1)

The three frame specimens exhibited similar average ductility coefficients. Compared with the cast-in-place specimen XK, the ductility of specimens ZK1 and ZK2 was relatively lower. This was due to the concentrated crack development at the mechanical connectors, faster crack width propagation, and earlier occurrence of through-cracks at the upper and lower parts of the beam ends. These factors led to accelerated decline in load-bearing capacity during the later loading stages and reduced ultimate displacement, thereby lowering ductility;

(2)

The ductility of the prestressed specimen ZK2 was reduced by 5.6% compared to specimen ZK1. This reduction is attributed to the prestress enhancing the specimen’s overall integrity and increasing its load-bearing capacity, which resulted in a higher yield displacement and consequently a decrease in ductility.

3.5. Energy Dissipation

Under seismic action, a structure must not only have sufficient load-bearing capacity and ductility but also exhibit excellent energy dissipation capability. The energy dissipation capacity of a structure is typically evaluated using the energy dissipation coefficient E or the equivalent viscous damping coefficient h_e [19]. The calculation formulae are shown in Equations (2) and (3). In the equation, S_ABCD represents the area enclosed by the hysteresis loop of the specimen during one loading cycle, while S_(OBE+ODF) represents the sum of the areas of triangles OBD and ODF corresponding to the upper and lower vertices of the hysteresis loop. The larger the values of the energy dissipation coefficient E and the viscous damping coefficient h_e, the fuller the shape of the hysteresis curve, indicating stronger energy dissipation capacity of the specimen. The schematic diagram is illustrated in Figure 8.

$$E={\displaystyle \frac{S_{\mathrm{ABCD}}}{S_{\mathrm{OBE}+\mathrm{ODF}}}}$$

(2)

$$h_{\mathrm{e}}={\displaystyle \frac{S_{\mathrm{ABCD}}}{2\mathrm{\pi }·(S_{\mathrm{OBE}+\mathrm{ODF}})}}$$

(3)

The energy dissipation coefficient E and viscous damping coefficient h_e of the specimen during the final loading cycle under horizontal loading are shown in

Table 4. Energy dissipation index coefficient of specimens.

Specimen	E	he
ZK1	0.94	0.15
ZK2	0.84	0.13
XK	1.35	0.21

. The relationship curve between the energy dissipation coefficient E and the loading displacement is illustrated in Figure 9.

Analysis of the graphs indicates that the energy dissipation coefficients of all three specimens increase with displacement. In the initial loading stage, the specimens remain in the elastic phase, resulting in relatively small hysteresis loop areas and an energy dissipation coefficient of approximately 0.4. When the displacement reaches 16.5 mm, the reinforcement at the beam ends yields, plastic hinges form, and the deformation capacity of the specimens improves, leading to a significant increase in energy dissipation capacity. Comparing specimens XK and ZK1, ZK1 exhibits slightly higher initial energy dissipation due to the rapid crack development at the connectors. However, after reaching a displacement of 16.5 mm, the energy dissipation of XK gradually surpasses that of ZK1. The curves of ZK1 and ZK2 exhibit similar trends, but the presence of prestressing tendons in ZK2 results in greater stiffness and stronger deformation recovery capacity. This leads to a more pronounced pinching effect in the hysteresis loops and a 13.3% reduction in the equivalent viscous damping coefficient, ultimately resulting in a slightly lower energy dissipation capacity compared to ZK1.

4. Finite Element Model

4.1. Establishment of Finite Element Model

Due to the variety of material types, in order to obtain more accurate analysis results, this study adopted a separate model for modeling, selected a constitutive model similar to the material of the specimen, and applied the same boundary conditions and loads as in the test. The schematic diagram of the established prestressed mechanical connection assembly integral frame model [20,21] is shown in Figure 10.

Element Type: The three-dimensional solid reduction integral element (C3D8R) [22,23] was used for simulation concrete. The characteristic of this type of element is that one less integral point is used in each direction compared with the fully integrated element, which can greatly reduce the time cost required for calculation, ensure the accuracy of simulation results, and avoid the problem of shear locking.

The three-dimensional two node truss element (T3D2) was used for the steel bar and strand. The beam element or truss element is more suitable for the component whose length direction is much larger than other direction, such as the steel bar and strand. Since the beam element can transfer the bending moment and shear force, but the truss element can only transfer tension, considering the stress characteristics of the steel bar and strand, the truss element was used.

Interactions & Restrains: ABAQUS needs to manually set the constraint relationship between the contact surfaces of different components. For the integrated prestressed frame model assembled by the mechanical connection steel bar, the interactions involved the interface between new and old concrete, steel bar and concrete, steel strand and concrete, and rigid pad and concrete.

The interaction between the pre-cast part and the post-cast part was divided into two parts, that is normal behavior and tangential behavior. The normal behavior of the interface between the new and old concrete was set to “hard” contact, which means that the contact pressure is generated when the gap between the two surfaces is zero, and the contact surface is separated when the contact pressure between the contact surfaces becomes zero or negative. The friction model with penalty function was used to set the friction behavior between two surfaces by the friction coefficient μ. The ends of the prefabricated beam and column were provided with keyways, and the other new and old concrete interfaces were chiseled. European Union regulations state that, for the joint surface of the tooth groove, μ = 0.9 [24]. When the surface treatment is about 6 mm chisel, μ = 1.0 [25]. Therefore, the friction coefficient μ set in this model was 0.9 at the keyway and 1.0 at the horizontal overlapping surface of the beam.

For the interaction between the unbonded prestressed tendons and concrete, in order to better reflect the actual conditions, no contact was defined between the concrete duct and the prestressed tendon. Instead, rigid cushions were placed at both ends of the beam, binding them to the concrete. The prestressed tendon ends were then connected through a magnesium phosphate cement restraint, thus transferring the prestress to the specimen via the rigid cushions.

The frame model was subjected to vertical and horizontal loads in the experiment, which were applied through three analysis steps: (1) prestress was applied to the steel strands using a cooling method; (2) axial pressure was applied at the top of the model column; and (3) low-cycle cyclic load was applied at the beam ends. The boundary conditions and loading methods are shown in Figure 4 above.

To prevent stress concentration at the loading points, rigid cushions were placed at both ends of the beam and column, and the cushions were bound (tied) to the concrete. At the center of the rigid cushion surface, a reference point was set, and the reference point was coupled with the surface of the cushion. Both the load and displacement were applied at the reference point.

Model Meshing: In finite element analysis, mesh division significantly impacts computational accuracy and numerical convergence. Excessively large mesh sizes can lead to uneven stress distribution in critical regions, affecting the accuracy of local damage simulation, while excessively small mesh sizes increase computational cost and may even cause convergence difficulties. Therefore, this study achieves a balance between computational accuracy and efficiency by employing varying mesh sizes for different materials. Specifically, a mesh size of 50 mm was utilized for concrete to ensure adequate accuracy in regions with significant stress gradient variations. For reinforcement, steel strands, and steel plates, a mesh size of 100 mm was adopted, which reduces computational costs while maintaining an appropriate simulation of the interaction at the reinforcement–concrete interface. The meshed model is shown in Figure 11.

4.2. Material Constitutive

The concrete damage plastic model can express the inelastic behavior of concrete through isotropic elastic damage and tensile and compressive plastic theory [26], so the concrete damage plastic model was selected. The constitutive model of concrete in the Code for Design of Concrete Structures (GB50010-2010) was used for calculation.

The equation for uniaxial compression of concrete is from Equations (4) to (8), where σ_c is compressive stress in concrete; d_c is the evolution parameters of concrete uniaxial compression damage; E_c is the elastic modulus of concrete; ε_c is the compressive strain in concrete; α_c is the parameter value of the descending section of the uniaxial tensile stress–strain curve; f_c,r is the compressive strength of concrete; and ε_c,r is the peak compressive strain of concrete corresponding to f_c,r.

$${\sigma }_{\mathrm{c}}=(1-d_{\mathrm{c}})E_{\mathrm{c}}\epsilon$$

(4)

$$d_{\mathrm{c}}=\left\{\begin{array}{c}1-{\displaystyle \frac{{\rho }_{\mathrm{c}}\mathrm{n}}{\mathrm{n}-1+x^{\mathrm{n}}}}, x\le 1\\ 1-{\displaystyle \frac{{\rho }_{\mathrm{c}}}{{\alpha }_{\mathrm{c}}{\left(x-1\right)}^2+x}}, x>1\end{array}\right.$$

(5)

$${\rho }_{\mathrm{c}}={\displaystyle \frac{f_{\mathrm{c},\mathrm{r}}}{E_{\mathrm{c}}{\epsilon }_{\mathrm{c},\mathrm{r}}}}$$

(6)

$$n={\displaystyle \frac{E_{\mathrm{c}}{\epsilon }_{\mathrm{c},\mathrm{r}}}{E_{\mathrm{c}}{\epsilon }_{\mathrm{c},\mathrm{r}}-f_{\mathrm{c},\mathrm{r}}}}$$

(7)

$$x={\displaystyle \frac{{\epsilon }_{\mathrm{c}}}{{\epsilon }_{\mathrm{c},\mathrm{r}}}}$$

(8)

The equation for uniaxial tensile stress of concrete is from Equations (9) to (12), where σ_t is the tensile stress in concrete; d_t is the evolution parameters of concrete uniaxial tensile damage; ε_t is the tensile strain in concrete; α_t is the parameter values of the descending section of the uniaxial tensile stress–strain curve; f_t,r is the tensile strength of concrete; and ε_t,r is the peak tensile strain of concrete corresponding to f_t,r. The uniaxial compressive and tensile stress–strain curves of concrete are shown in Figure 12.

$${\sigma }_{\mathrm{t}}=(1-d_{\mathrm{t}})E_{\mathrm{c}}\epsilon$$

(9)

$$d_{\mathrm{t}}=\left\{\begin{array}{c}1-{\rho }_{\mathrm{t}}\left[1.2-0.2x^5\right], x\le 1\\ 1-{\displaystyle \frac{{\rho }_{\mathrm{t}}}{{\alpha }_{\mathrm{t}}{\left(x-1\right)}^{1.7}+x}}, x>1\end{array}\right.$$

(10)

$$x={\displaystyle \frac{{\epsilon }_{\mathrm{t}}}{{\epsilon }_{\mathrm{t},\mathrm{r}}}}$$

(11)

$${\rho }_{\mathrm{t}}={\displaystyle \frac{f_{\mathrm{t},\mathrm{r}}}{E_{\mathrm{c}}{\epsilon }_{\mathrm{t},\mathrm{r}}}}$$

(12)

Here, the bifold line model under the monotonic load [27] was used for the steel bar, which was divided into the elastic section and strengthened section, and this could simulate the ideal elastic–plastic material with isotropic strain strengthening considered. The mechanical behavior of rebar under monotonic loading typically consists of an elastic phase and a plastic phase. The two-line model simplifies the mechanical response of rebar and can effectively describe the stiffness in the elastic phase and the hardening characteristics in the plastic phase. This model, when simulating the tensile behavior of rebar, can accurately capture the yield point and the increasing plastic deformation, while also having a low computational complexity. The expression is given by Equation (13). In the equation, σ_s is the stress of the reinforcement, E_s is the elastic modulus of the reinforcement, ε_s is the strain of the reinforcement, f_y is the tensile yield strength of the reinforcement, f_s,u is the ultimate strength of the reinforcement, ε_y is the tensile yield strain of the reinforcement, ε_s,u is the strain corresponding to the ultimate strength f_s,u, and k is the elastic modulus of the hardening segment of the reinforcement; k = (f_s,u − f_y)/(ε_s,u − ε_y). The stress–strain relationship curve is shown in Figure 13.

$${\sigma }_{\mathrm{s}}=\left\{\begin{array}{c} E_{\mathrm{s}}{\epsilon }_{\mathrm{s}}, {\epsilon }_{\mathrm{s}}\le {\epsilon }_{\mathrm{y}}\\ f_{\mathrm{y}}+k({\epsilon }_{\mathrm{s} }-{\epsilon }_{\mathrm{y}}), {\epsilon }_{\mathrm{y}}<{\epsilon }_{\mathrm{s} }\le {\epsilon }_{\mathrm{s},\mathrm{u}}\\ 0, {\epsilon }_{\mathrm{s} }>{\epsilon }_{\mathrm{s},\mathrm{u}}\end{array}\right.$$

(13)

The modeling method for the snap-fit mechanical connector was based on the double linear model of reinforcement. A uniaxial tensile test was conducted on reinforcement connected with mechanical connectors, resulting in the monotonic tensile stress–strain curve of the mechanically connected reinforcement. As shown in Figure 3, the stress–strain characteristics of the mechanically connected reinforcement are similar to those of reinforcement under uniaxial tension, exhibiting distinct yield and ultimate points. Therefore, the double linear model of reinforcement can be used as a reference. The calculation expression for the snap-fit mechanical connector is the same as Equation (13), and its stress–strain curve is shown in Figure 13.

Here, the three-fold model [28] was used for the prestressed steel strand. This model was chosen because it better captures the nonlinear behavior of steel strands under different loading conditions, especially in terms of the transfer of initial prestress, the relaxation of the steel strands, and their interaction with the concrete, as shown in Figure 14. The stress–strain relationship is defined by Equation (14), where E_p1, E_p2 and E_p3 are the elastic stiffness in the elastic, plastic, and post-conditional yielded states. Respectively, f_p1 and ε_p1 are stress and strain in the elastic state, f_p2 and ε_p2 are stress and strain in the yield state, and f_p3 and ε_p3 are stress and strain in the post-conditional yielded state.

$${\sigma }_{\mathrm{p}}=\left\{\begin{array}{c}{E}_{\mathrm{p}1}{\epsilon }_{\mathrm{p}}, {\epsilon }_{\mathrm{p}}\le {\epsilon }_{\mathrm{p}1}\\ f_{\mathrm{p}1}+E_{\mathrm{p}2}({\epsilon }_{\mathrm{p}}-{\epsilon }_{\mathrm{p}1}), {\epsilon }_{\mathrm{p}1}\le {\epsilon }_{\mathrm{p}}\le {\epsilon }_{\mathrm{p}2} \\ f_{\mathrm{p}2}+E_{\mathrm{p}3}({\epsilon }_{\mathrm{p}}-{\epsilon }_{\mathrm{p}2}), {\epsilon }_{\mathrm{p}2}\le {\epsilon }_{\mathrm{p}}\le {\epsilon }_{\mathrm{p}3}\end{array}\right.$$

(14)

4.3. Model Verification

Figure 15 presents the Mises stress contour of the reinforcement obtained from the ABAQUS simulation. The results indicate that the stress distribution patterns of the reinforcement in specimens ZK1 and ZK2 are similar, with the longitudinal reinforcement at the beam ends yielding in both cases. The maximum stresses in ZK1 and ZK2 were 478.54 MPa and 462.37 MPa, respectively, suggesting that the application of prestressing slightly reduced the stress level of the beam-end longitudinal reinforcement. Additionally, the longitudinal reinforcement at the column base also yielded, whereas the yielding range of the beam-end longitudinal reinforcement was larger and more severe. In contrast, the maximum stress of the longitudinal reinforcement at the column top and the stirrups in the core region was only 142.26 MPa, significantly lower than the yield stress, indicating that these regions remained in the elastic stage without damage. Overall, the simulated reinforcement stress distribution closely aligns with the experimental observations, verifying the accuracy and reliability of the finite element model.

The finite element simulation results indicate that the compression damage distribution in the concrete of specimens ZK1 and ZK2 is similar, primarily concentrated in the post-cast concrete sections at the beam ends and column bases. In contrast, the prefabricated concrete, due to its higher strength, exhibited less damage. In specimen ZK2, the application of prestressing introduced compressive forces, leading to a slightly larger compression damage area compared to ZK1. The damage range at the column base was smaller than that at the beam ends, while the core region and column top remained almost undamaged. Overall, the simulated failure mode aligns with the “strong column-weak beam, strong joint-weak member” design principle and closely matches the experimental observations, verifying the accuracy of the model. The compression damage distribution of the concrete is shown in Figure 16.

In order to further verify the validity of the finite element model, the simulated skeleton curve was compared with the tested skeleton curve, as shown in Figure 17. The simulated skeleton curves of the two frames are consistent with the skeleton curves tested. In the initial loading stage, the simulated initial stiffness of the frame is slightly larger than the tested value. The difference results from the difference between the actual test device and the loading mode defined by the simulation. In the actual test, there is an inevitable gap between the steel plate sandwiching both sides of the beam end and the specimen, and the bolts for transferring force may be slightly loosened in the loading process, resulting in a larger gap. As a result, the initial stiffness of the test is slightly less than the simulated value. When the displacement load reaches the peak load, the skeleton curve begins to decline, and the slope of the descending section of the finite element model is smaller than that of the value tested.

The comparison of the simulated and tested peak load of specimens ZK1 and ZK2 in positive and negative directions is shown in

Table 5. Comparison of simulated and tested peak load.

Specimen Number	Loading Direction	Simulation Result (Kn)	Average (kN)	Test Result (kN)	Average (kN)	Deviation (%)
ZK1	Positive	221.9	210.4	202.4	192.7	9.2
	Negative	198.8		183.0
ZK2	Positive	255.6	244.2	231.7	226.2	8.0
	Negative	232.8		220.7

. The deviations between the simulated and tested peak load of specimen ZK1 and ZK2 are 9.2% and 8.0%, respectively, both less than 10%, indicating good accordance.

4.4. Parametric Analysis

Based on the finite element models of specimens ZK1 and ZK2, an extended parameter analysis was conducted, primarily focusing on prestress, material properties, and reinforcement ratios. The selected parameters include effective prestress, concrete strength grade, longitudinal reinforcement ratio of beams, longitudinal reinforcement ratio of columns, and axial compression ratio, as detailed in

Table 6. Analysis parameter selected.

Analysis Parameter	Foundation Specimen	Parameter Selection
Prestress level	ZK2	0.2 fptk, 0.4 fptk, 0.6 fptk, 0.8 fptk
Concrete strength	ZK1	C30, C35, C40, C45, C50
Reinforcement ratio of beam	ZK1	0.68%, 0.89%, 1.12%, 1.39%
Reinforcement ratio of column	ZK1	2.26%, 2.79%, 3.38%, 4.36%
Axial compression ratio n0	ZK1	0.15, 0.3, 0.45, 0.6

.

Changes in the parameters has a more pronounced impact on the skeleton curves. The analysis yielded a comparison of skeleton curves under different levels of prestress, concrete strength grades, longitudinal reinforcement ratios of beams, longitudinal reinforcement ratios of columns, and axial compression ratios, as shown in Figure 18.

It can be seen from Figure 18a that the trend of the skeleton curves of frames with a different prestress level is similar. When the prestress level is less than 0.6 f_ptk, the bearing capacity and initial stiffness of the frame increases with the increase of the prestress level. When the prestress level is greater than 0.6 f_ptk, the skeleton curve of the frame is almost unchanged, indicating that when the prestress level is large, the increase of the prestress has little contribution to the enhancement of the bearing capacity and stiffness of the frame. The reduction in the effectiveness of prestress beyond 0.6 f_ptk can be attributed to the diminishing returns associated with the prestressing force applied to the frame. As the prestress level increases, the concrete in the frame is subjected to higher initial compression, which leads to a stiffer response and an enhanced load-bearing capacity. However, after a certain threshold, the increase in prestress results in less significant improvements. This is because the concrete has already reached its optimal level of compression, and further increasing prestress mainly affects the behavior of the reinforcement rather than the concrete itself. Additionally, when the prestress level exceeds 0.6 f_ptk, the effect on crack control diminishes as the frame has already exhibited enhanced stiffness and capacity in the earlier stages. On a micro-mechanical level, the concrete’s compressive strength reaches a plateau as it can only carry so much additional load before reaching its ultimate stress–strain limit. Beyond this, the reinforcing bars are more effectively engaged in carrying the load, but the overall contribution to bearing capacity enhancement becomes less pronounced. The contribution of the prestressed tendons, although still beneficial, starts to saturate, and the frame’s overall performance becomes more dependent on the reinforcement ratio and the concrete’s ultimate strength. It can be concluded that proper increase of prestress level can improve the bearing capacity and initial stiffness of the frame, but the improvement is not obvious when the prestress level is large.

It can be seen from Figure 18b that the skeleton curves of frames with different concrete strength have similar development trends. With the increase of concrete strength, the initial stiffness of the frame increases slightly and the bearing capacity increases to a certain extent, but the displacement corresponding to the peak load does not change. It can be concluded that increasing the concrete strength can improve the bearing capacity of the frame. The peak load of the C50 concrete frame is about 15% higher than that of the C30 concrete frame.

It can be seen from Figure 18c that the development trend of the skeleton curve of the frame with a different beam reinforcement ratio is consistent. In the initial stage of displacement loading, the slope of skeleton curve increases with the increase of the reinforcement ratio, indicating that the initial stiffness of the frame increases with the increase of the beam reinforcement ratio. When the displacement gradually increases to the peak displacement, it can be found that the increase of the beam reinforcement ratio will lead to a significant increase in the peak load, but it has a little effect on the slope of the descending section. Therefore, increasing the ratio of the beam reinforcement ratio can improve the bearing capacity of the frame in a certain range, but has little effect on other seismic performance indexes.

It can be seen from Figure 18d that the skeleton curves of the four frames with different column reinforcement ratios are consistent. In the elastic stage, the stiffness of the frame increases slightly with the increase of the column reinforcement ratio. After the frame reaches the yield displacement, the peak load of the frame increases with the increase of the column reinforcement ratio. In the later loading process, the velocity of the descending section of the skeleton curve is relatively uniform, and the slope is basically the same.

It can be seen from Figure 18e that the development trends of the skeleton curves of specimens with different compression ratio are quite different. In terms of stiffness, with the increase of compression ratio, the early-stage stiffness of the frame increases rapidly, and the initial stiffness increases. The influence is more obvious when the compression ratio is small, but it decreases when the ratio is greater than 0.45. With the increase of the compression ratio, the decline rate of the bearing capacity accelerates obviously after the load reaches the peak value. In terms of bearing capacity, the peak load of the frame increases with the increase of the compression ratio. When the compression ratio is greater than 0.45, the peak load decreases obviously. In terms of ductility, it is obvious that the increase of the compression ratio makes the peak displacement of the frame move forward, causing the frame to yield in advance. In the later loading period, due to the rapid reduction of the bearing capacity, the ultimate displacement is greatly reduced, and the ductility of the frame is eventually reduced. It can be seen that increasing the compression ratio in a certain range can improve the initial stiffness and bearing capacity of the frame, but the decline rate of the bearing capacity in the later period will be significantly accelerated.

Based on the parametric analysis of prestress, concrete strength grade, beam longitudinal reinforcement ratio, column longitudinal reinforcement ratio, and axial compression ratio as variables, the effects on the seismic performance of the frame are summarized as follows:

• The comparison of the simulated and tested Mises stress cloud map and concrete stress damage map show all specimens’ failure at the beam end. The results show that the Mises stress and concrete damage at the beam end are lower than those at the column foot, and no damage occurred in the core area and the top of the column, which is consistent with the results tested. The skeleton curve development trend and peak load extracted from the simulation results are also in good agreement with the test results, which verifies the rationality and accuracy of the finite element model established;
• Increasing the effective prestress enhances the load-bearing capacity and initial stiffness of the frame; however, when the effective prestress exceeds 0.6 f_ptk, the improvement becomes negligible;
• Changes in the concrete strength grade, longitudinal reinforcement ratio of beams, and longitudinal reinforcement ratio of columns significantly impact the frame’s skeleton curve. An increase in these parameters enhances the load-bearing capacity and initial stiffness of the frame to varying degrees;
• The increase in the axial compression ratio significantly alters the trend of the skeleton curve of the frame. As the axial compression ratio increases, the load-bearing capacity and initial stiffness of the frame gradually improve. When the axial compression ratio is less than 0.45, the enhancement effect is pronounced, whereas for axial compression ratios greater than 0.45, the improvement diminishes. Additionally, an increase in the axial compression ratio causes the peak displacement of the structure to occur earlier.

5. Restoring Force Model

5.1. Skeleton Curve

The skeleton curves of specimens ZK1 and ZK2 exhibit three distinct stages during the loading process, which can be simplified into a trilinear model consisting of the elastic segments OA and OD, the strengthening segments AB and DE, and the descending segments BC and EF. Points A and D represent the yield points of the skeleton curves in the positive and negative directions, determined using the equal energy method. Points B and E correspond to the peak points of the skeleton curves in the positive and negative directions, while points C and F denote the failure points of the skeleton curves, defined as the positions where the load decreases to 85% of the peak load. These six characteristic points are used to segment and fit the skeleton curves, and the resulting skeleton curve model is shown in Figure 19. To represent the different skeleton curves using a unified formula and graphical representation, the curves are normalized with Δ/|Δ_m| as the horizontal axis and P/|P_m| as the vertical axis, where Δ_m and P_m represent the peak displacement and peak load of the skeleton curve, respectively.

Table 7. Control feature points.

Feature Point	Positive Direction			Negative Direction
	A	B	C	D	E	F
P/|Pm|	0.869	1.015	0.850	−0.852	−1.010	−0.850
Δ/|Δm|	0.463	0.965	1.508	−0.502	−0.979	−1.486

and

Table 8. Regression equation in skeleton curve model.

Sector	Fitting Equation
BC	P/|Pm| = −0.303Δ/|Δm| + 1.307
AB	P/|Pm| = 0.289Δ/|Δm| + 0.735
OA	P/|Pm| = 1.878Δ/|Δm|
OD	P/|Pm| = 1.697Δ/|Δm|
DE	P/|Pm| = 0.332Δ/|Δm| − 0.685
EF	P/|Pm| = −0.316Δ/|Δm| − 1.319

present the coordinates of the control characteristic points and the regression equations for the skeleton curve model.

Based on the skeleton curve model, the skeleton curves of specimens ZK1 and ZK2 were calculated and compared with the experimental skeleton curves, as shown in Figure 20. The results indicate good overall agreement, demonstrating that the calculation formula is highly accurate and can serve as the foundation for subsequent studies on restoring force models.

5.2. Hysteresis Rule

Before reaching the yield displacement, the structure remains in the elastic state, where the loading stiffness and unloading stiffness are identical, and the loading and unloading in both positive and negative directions follow straight lines OA and OD. Once the structure reaches the yield point A but has not yet attained the peak point B, the loading continues along the backbone curve segment AB, where a noticeable stiffness degradation occurs. Starting from the positive unloading point 1, the unloading proceeds along straight line 12 to the negative loading point 2, with the slope of line 12 defined as K1. From point 2, the structure is loaded along straight line 23 until it reaches the negative unloading point 3, with the slope of line 23 defined as K2. Then, unloading occurs from point 3 along straight line 34 to the positive loading point 4, with the slope of line 34 defined as K3. The loading then resumes from point 4 along straight line 41 back to point 1, where the slope of line 41 is defined as K4, thereby completing one cycle. After point 1, the loading continues along the backbone curve. Upon reaching the peak point B, loading proceeds along segment BC. From unloading point 5, the unloading occurs to point 6, then loading continues to point 7, followed by unloading to point 8, and finally returning to point 5. The structure continues along the segment BC in subsequent cycles, as illustrated in Figure 21.

5.3. Stiffness Degradation

Based on the experimental hysteresis curves, it was observed that both loading and unloading stiffness exhibit degradation under cyclic loading. To determine the stiffness degradation behavior of the structure, the forward unloading stiffness K₁, reverse loading stiffness K₂, reverse unloading stiffness K₃, and forward loading stiffness K₄ of specimens ZK1 and ZK2 at different displacement cycles were statistically analyzed. Scatter plots were generated and regression analysis was performed to obtain the degradation curves of K₁, K₂, K₃, and K₄. The schematic diagram of loading and unloading stiffness, as well as the stiffness degradation curves, are shown in Figure 22.

5.4. Verification

The calculated hysteretic curves of the restoring force model for the prestressed mechanically connected assembled frame were compared with the experimental hysteretic curves, as shown in Figure 23. It can be observed that the calculated hysteretic curves closely match the experimental results, exhibiting a similar trend with a high degree of agreement. This indicates that the developed restoring force model for the prestressed mechanically connected assembled frame is both reasonable and accurate.

6. Conclusions

Through the seismic performance analysis of prestressed prefabricated concrete frames reinforced with mechanical connection steel bars, the main conclusions are as follows.

(1)

The presence of prestressed tendons in specimen ZK2 results in a higher initial stiffness and better deformation recovery capacity, which leads to a noticeable curve-pinching phenomenon. As a result, the hysteresis loop of specimen ZK2 is slightly less full compared to specimen ZK1, and h_e decreases by 13.3%, indicating a reduction in energy dissipation capacity. Increasing the concrete strength and reinforcement ratio of beam and column improve the bearing capacity of the frame. Compared with frame made of C30 concrete, the peak load of frame made of C50 concrete improves by 25%;

(2)

The numerical models of specimens ZK1 and ZK2 were established by using ABAQUS finite element software (2022), and the interaction, mesh and boundary conditions were reasonably set by selecting reasonable material constitutive and element types. The failure modes obtained by numerical simulation analysis are consistent with the test results, and the development trends of the skeleton curve and peak load are also in good agreement with the test results. The rationality and accuracy of the finite element model were verified;

(3)

Parameter analysis was carried out based on the verified finite element model. Through comparison, it was found that when the prestress level is less than 0.6, increasing the prestress level can improve the bearing capacity and initial stiffness of the frame. The application of prestress can reduce the ductility of the component. The prestressed specimen ZK2 exhibits a 5.6% lower ductility than specimen ZK1. This is because the prestress improves the overall integrity of the specimen, increases its bearing capacity, and leads to an increase in the yield displacement, which, in turn, reduces the ductility of the specimen;

(4)

The increase in the axial compression ratio causes a significant change in the skeleton curve of the frame. As the axial compression ratio increases, the bearing capacity and initial stiffness of the frame gradually increase. When the axial compression ratio is less than 0.45, the improvement effect is significant; however, when the axial compression ratio exceeds 0.45, the improvement effect becomes relatively weak. The increase in the axial compression ratio shifts the peak displacement of the frame to an earlier point, accelerates the decrease in the bearing capacity during the later stages of loading, and significantly reduces the ultimate displacement, ultimately leading to a reduction in frame ductility;

(5)

Based on the experimental hysteresis and skeleton curves, the dimensionless experimental data were fitted to establish a three-segment skeleton curve model. Regression analysis was performed to obtain the stiffness degradation law of the specimens under cyclic loading. Combined with the hysteretic characteristics, a restoring force model for the prestressed mechanically connected prefabricated frame was developed. A comparison revealed that the calculated results of the restoring force model closely match the experimental results, providing a basis for structural design and engineering applications.