Experimental and Restoring Force Model of Precast Shear Walls with Steel Sleeve and Corrugated Metallic Duct Hybrid Connections

Abstract

This study proposes a novel hybrid connection method for precast concrete shear walls, where the edge walls are connected using grouting splice sleeves and the middle walls are connected using grouted corrugated metallic ducts. To investigate the effects of connection type and axial compression ratio on structural performance, five shear wall specimens were tested under low-cycle reversed loading, with detailed analysis of their failure modes and hysteretic behavior. Based on experimental results and theoretical derivation, a restoring force model incorporating connection type was developed. The results demonstrate that hybrid-connected specimens exhibit significantly improved load-bearing capacity, ductility, and seismic performance compared to those with only grouted corrugated metallic duct connections. A higher axial compression ratio enhances structural strength but also accelerates damage progression, particularly after peak loading. A three-line skeleton curve model was established to describe the load, displacement, and stiffness relationships at key characteristic points, and unloading stiffness expressions for different loading stages were proposed. The calculated skeleton and hysteresis curves align well with the experimental results, accurately capturing the cyclic behavior of the hybrid-connected precast shear walls.

1. Introduction

With the advancement of building industrialization, traditional on-site casting construction methods face numerous challenges, such as material storage, environmental pollution, long construction cycles, difficulty in quality control, and safety risks. As a result, prefabricated modular construction has gradually become a trend in the construction industry. In particular, prefabricated concrete shear wall (PCSW) structures, known for their excellent lateral stiffness and load-bearing capacity, are widely applied in high-rise buildings [1,2]. However, compared to modular frame structures, the seismic performance of prefabricated shear wall structures is limited by the design of vertical and horizontal connections, with the horizontal connection playing a critical role in the overall performance and seismic resistance of the structure. Therefore, the rational and reliable design of connections is a key factor in the structural design of prefabricated shear wall systems [3].

Currently, horizontal connections are classified into dry connections and wet connections. Dry connections typically refer to bolt, steel plate, or rebar lap connections. Liu et al. [4] have shown that high-strength bolt connections significantly improve the seismic performance of shear walls, with their seismic capacity matching that of the overall connection. Sun et al. [5] proposed a dry connection method using horizontal steel connectors (H-connectors) and high-strength bolts, demonstrating that this connection method effectively enhances the seismic performance of shear walls. Yu et al. [6] introduced a new steel plate-bolt (SPB) connection, and their research showed that SPB connections improved the horizontal load-bearing capacity of shear walls by 5.3%, energy dissipation by 14.8%, and ductility by 32%. Despite their ease of assembly and reusability, dry connections are often associated with complex design and installation processes, and the long-term durability of exposed mechanical components is vulnerable to environmental degradation, such as corrosion [7]. Wet connections offer a more effective and convenient method, primarily including grouting splice sleeve (GSS) connections and grouted corrugated metallic duct (GCMD) connections. The GSS connection joins two prefabricated sections of a shear wall through rebar and grouting materials and is widely used in the connection of precast components. Gu et al. [8] designed and studied the seismic performance of PCSW with GSS connections, proving that this connection method has good seismic performance, similar to that of cast-in-place shear walls. Xu et al. [9] investigated the seismic performance of PCSW with a single row of GSS connections, finding that the failure mode, ductility, and energy dissipation were similar to those of cast-in-place shear walls. Although the bond provided by grouting materials ensures strong connection strength and wall stability, GSS connections face challenges such as high cost, a complex grouting process, and difficulties in guaranteeing grouting quality [10]. GCMD connections use corrugated metallic ducts to connect PCSW, and research by Zhi et al. [11] showed that GCMD-connected shear walls excel in bearing capacity, crack patterns, energy dissipation, and stiffness, and their seismic performance matches that of cast-in-place shear walls. The corrugated metallic ducts effectively constrain the splicing bars and concrete, forming a strong connection after grouting. Nevertheless, the requirement for long lap lengths increases construction complexity and limits applicability in dense reinforcement regions. To address these limitations, the hybrid connection method combining GSS and GCMD merges the advantages of both. By using GSS for the edge walls and GCMD for the middle walls, the hybrid connection system provides high-strength joints, excellent seismic performance, and improved construction efficiency and durability. This approach also simplifies rebar detailing in congested regions and potentially reduces construction time and costs by optimizing material placement and anchorage design. Existing research shows that hybrid connections in PCSW outperform cast-in-place shear walls in terms of seismic performance [3,12]. Specifically, hybrid connection specimens exhibit fuller and wider hysteresis loops, reflecting improved hysteretic performance, which can be attributed to the enhanced confinement effect of the GCMD. This confinement delays the buckling of spliced reinforcement and the onset of structural failure. Moreover, the hybrid connection specimens maintain displacement ductility and stiffness degradation trends comparable to those of cast-in-place specimens, indicating that their energy dissipation is enhanced without compromising deformation capacity. However, most international standards currently provide general design requirements for GSS or GCMD connections for PCSW, but do not offer specifications for hybrid connections. Furthermore, research on hybrid connections remains relatively sparse in the available literature.

Based on the relationship curve between restoring force and deformation obtained from experiments, a simplified mathematical model, referred to as the restoring force model, is proposed. This model effectively reflects the ability of a structure or component to return to its original state after the external force is removed [13], serving as a crucial basis for the analysis of internal forces and deformations of structures under seismic actions. The restoring force model has been widely applied in nonlinear finite element analysis, with significant research conducted by scholars in this field [14,15]. Penzien J [16] first proposed a bilinear restoring force model that does not consider the degradation of component strength and stiffness. Clough [17] further introduced a degraded bilinear restoring force model that accounts for the degradation of stiffness. Takeda [18], building upon this, developed a trilinear restoring force model incorporating concrete damage, a three-segment backbone curve model, and hysteretic rules. Following these foundational studies, extensive research has been conducted on restoring force models for shear walls. Zhao et al. [13] defined the characteristic points of the restoring force model to establish a three-segment restoring force model for steel-fiber reinforced concrete shear walls. Zuo et al. [19] studied the seismic performance of high-strength foam concrete (HFC) shear walls and developed a four-linear restoring force model using MATLAB fitting. Hu et al. [20] proposed the Pinching4 restoring force model for new cold-formed steel (CFS) walls through experimental and numerical analysis, focusing on the energy dissipation capacity of CFS walls. Xue et al. [3] established a three-segment restoring force model for GCMD-connected shear walls through parameter fitting. Li et al. [21] performed numerical and theoretical studies on the seismic behavior of concrete-filled double steel plate (CFDSP) composite shear walls and proposed a three-line restoring force model to describe the hysteretic behavior of CFDSP composite shear walls.

This study proposes a hybrid connection method for precast concrete shear walls, where the GSS connection is applied to the edge walls and the GCMD connection is used for the middle walls. Five shear wall specimens were experimentally studied under low-cycle loading, with connection type and axial load ratio as variables. Based on the experimental results and theoretical analysis, a restoring force model was developed to describe the hysteretic behavior of shear walls with hybrid connections. This model not only provides theoretical support for elastic-plastic analysis but also offers a reliable basis for promoting the widespread application of precast shear wall structures.

2. Experiments

2.1. Specimen Design

This experiment involved the design and fabrication of five full-scale concrete shear wall specimens. One specimen was a cast-in-place concrete shear wall (CIP), and the other four were precast shear wall specimens (PGCW1, PGCW2, PGSW1, and PGSW2). All specimens were derived from a reinforced concrete shear wall structure located in Urumqi, with a height of approximately 70 m, seismic fortification intensity of 8 degrees, and a basic seismic acceleration value of 0.20 g. The structure is classified into the second group, with a seismic design classification of Level II. The precast specimens utilized two different connection methods: the single grouted corrugated metallic ducts (GCMDs) connection and the hybrid connection of GCMD with a grouted splice sleeve (GSS). Each connection type was tested under two different axial load ratios to investigate the impact of axial load ratio on the seismic performance of the precast specimens. The axial compression ratio is defined as the ratio of the applied vertical axial load to the product of the concrete compressive strength and the cross-sectional area of the shear wall. Based on the maximum loading capacity of the vertical hydraulic jack, the highest achievable axial compression ratio in the experiment was calculated to be 0.28. To ensure that the test conditions reflect typical service scenarios and moderate seismic demands, and with reference to relevant design codes and engineering practice [22], two representative axial compression ratios of 0.1 and 0.2 were selected.

The specific parameters of the specimens are shown in

Table 1. Specimen design parameters.

Specimen	Category	Axial Compression Ratio	Connection Type		Connection Rows
			Middle of the Wall	Boundary Element	Middle of the Wall	Boundary Element
CIP	Cast-in-place	0.1	—	—	One row	Two rows
PGCW1	Precast	0.1	GCMD	GCMD
PGSW1	Precast	0.1	GCMD	GSS
PGCW2	Precast	0.2	GCMD	GCMD
PGSW2	Precast	0.2	GCMD	GSS

. Each shear wall was composed of three parts: the top beam, the middle wall, and the bottom beam. The dimensions of the wall were 1500 mm (length) × 200 mm (width) × 2500 mm (height), the top beam dimensions were 1600 mm × 300 mm × 300 mm, and the bottom beam dimensions were 2600 mm × 600 mm × 600 mm. The cross-section dimensions of the edge components of the wall were 200 mm × 200 mm, with a concrete cover thickness of 20 mm. These dimensions were selected to reflect typical full-scale shear wall configurations in mid- to high-rise buildings, ensuring engineering relevance. The edge component sizes and pipe layouts were based on common construction standards to meet the anchorage requirements of both GSS and GCMD connections.

The CIP specimen was cast in a single pour for the top beam, middle wall, and bottom beam, with continuous reinforcement maintained within the top and bottom beams. The vertical reinforcement in the wall was anchored at both ends, with the top ends embedded into the cast-in-place top beam and the bottom ends anchored into the base beam. The anchorage lengths were designed in accordance with the requirements specified in the GB50010-2010 [23]. For the PGCW specimens, single-row steel bars were reserved at the lower part of the middle wall, and double-row steel bars were reserved at the edge components. Metal corrugated ducts (outer diameter 74 mm and inner diameter 70 mm) were embedded at corresponding positions in the bottom beam. The PGSW specimens differ from the PGCW specimens in that a post-cast region was reserved at the corner of the edge components, with double-row steel bars. The geometric dimensions and reinforcement details of the specimens are shown in Figure 1. Since the reinforcement in the upper part of the shear walls was identical, only the full reinforcement for the CIP specimen is presented. For the PGCW and PGSW specimens, only the reinforcement at the connection points is shown, with the detailed cross-sectional reinforcement displayed on the right.

2.2. Materials Properties

In this experiment, the concrete grade for the post-cast region was C50, while the remaining sections were C40. According to GB/T50081-2019 [24], uniaxial compressive tests were performed on concrete specimens. The standard cubic compressive strength, f_cu,k, represents the basic reference value for the mechanical properties of concrete. The axial compressive strength standard value, f_cu,k is calculated as f_ck = 0.88α_c1α_c2f_cu,k. Since the concrete used in this experiment was C50 and lower grades, α_c1 is taken as 0.76 and α_c2 as 1.00. The axial tensile strength standard value, f_tk, is given by f_tk = 0.26f_cu,k^2/3. The design values for axial compressive strength, f_c, and axial tensile strength, f_t, are taken as f_c = f_ck/1.4 and f_t = f_tk/1.4, respectively. The results of the uniaxial compressive tests and calculations for concrete are shown in

Table 2. Mechanical indexes of concrete.

Category	Strength Grade	fcu,k/MPa	fck/MPa	ftk/MPa	fc/MPa	ft/MPa
Precast concrete	C40	52.6	46.3	3.7	35.2	2.6
Post-poured concrete	C50	64.3	56.6	4.2	43	3.0

.

The grout used in this experiment was CGM-type grout, produced by Zhongde Xinya Company. According to the inspection report from the National Building Materials Testing Center, this grout meets the experimental requirements. The mechanical performance results of the grout are shown in

Table 3. The mechanical indexes of the grouting material.

Categories		Standard Requirements	Measured Values
Fluidity	Initial	≥300 mm	310 mm
	30 min	≥260 mm	293 mm
Compressive strength	1 d	≥35 MPa	44.4 MPa
	3 d	≥60 MPa	63.2 MPa
	28 d	≥85 MPa	96.4 MPa
Vertical expansion rate	3 h	≥0.02%	0.08%
	Between 24 h and 3 h	0.02%~0.50%	0.04%
Chloride ion content		≤0.03%	0.018%
Hydrogenic		0	0

.

The longitudinal and stirrup reinforcement used in this experiment were HRB400 and HPB300, respectively. Uniaxial tensile tests were conducted on all steel bars of different diameters according to the specifications of GB/T 228.1-2021 [25]. Three specimens of each diameter were tested, and the average value was used as the representative result. The tensile test results for the steel bars are shown in

Table 4. Measured mechanical properties of steel bar materials.

Types	d/mm	A/mm2	fy/MPa	fu/MPa	Elongation/%
HPB	8	50	342.0	470.6	15.5
HRB	10	78	443.6	617.3	19.9
	12	113	441.3	622.4	19.1
	14	154	434.6	618.0	19.3
	16	201	432.8	601.8	20.4
	18	254	464.8	658.2	21.0

Notes: d represents the diameter; A represents the cross-sectional area; f_y represents the yield strength; f_u represents the ultimate strength;.

.

2.3. Loading Device and Loading System

The experimental loading system consists of a vertical loading device, a horizontal loading device, and a distribution beam. The shear walls were fixed to the ground using four anchor bolts, with a vertical hydraulic jack capable of applying a maximum load of 3000 kN. The hydraulic jack was mounted on a steel reaction frame and allowed to move horizontally along the beam using a sliding track, ensuring that the vertical load remained vertically aligned with the centroid of the specimen throughout the test. A 1000 kN horizontal actuator mounted on a rigid frame was responsible for applying the horizontal load. The actuator was hinged at the loading head via a spherical bearing and connected to the shear wall through a high-strength threaded rod and steel clamping fixture. This configuration ensured consistent force transfer and prevented unintended moment transmission during cyclic tension-compression cycles. The diagram of the loading system is shown in Figure 2. The loading scheme employed a load-displacement hybrid control system [26], applying low-cycle repeated loading. Initially, the load was controlled until the cracking load was reached and concrete cracks appeared. Subsequently, displacement control was applied. Before yielding, the displacement increment was 0.2Δ_y, with one cycle for each level. After yielding, Δ_y was used as the displacement increment, with three cycles for each level. The loading ceased when the horizontal load dropped to 85% of the peak load. Δ_y was set to 8 mm. The loading protocol is shown in Figure 3. Throughout the test, data including load, displacement, and strain were recorded at a frequency of 1 Hz.

3. Analysis of Test Results

3.1. Failure Modes

The failure processes of the five shear wall specimens exhibit distinct regularities. Initial cracks in all specimens appeared at a lateral displacement of 6 mm, with the crack locations being approximately the same. As the loading cycles progressed, 3–4 primary horizontal cracks initially developed. By a lateral displacement of 16 mm, all visible cracks had formed. Subsequently, the cracks extended laterally across the wall surface, and their widths increased progressively. At the ultimate displacement, the maximum crack width ranged from 0.35 mm to 3 mm. The failure modes of all specimens displayed crushing of the concrete at the wall corners and tensile failure of the outermost longitudinal reinforcement at the structural edges, resulting in bending failure. Based on the test phenomena and final failure modes, the failure characteristics of the specimens can be divided into two categories. The first category includes the cast-in-place specimen CIP and the specimens with single grouted corrugated metallic duct (GCMD) connections, PGCW1 and PGCW2. The failure of these specimens is mainly caused by the development of the primary cracks on both sides of the wall and the cracking of the horizontal joints, leading to a reduction in load-carrying capacity and eventual failure. The second category involves the specimens with hybrid connections, PGSW1 and PGSW2, whose failure is primarily caused by the complete damage of the concrete in the post-poured regions on both sides, resulting in a decrease in load-carrying capacity and ultimate failure. The failure diagrams of the specimens are shown in Figure 4, with the inset image on the left being a magnified view of the wall corner damage, highlighting the maximum crack width in the specimens.

For the specimens CIP, PGCW1, and PGCW2, the walls were either cast in place or fabricated as a single precast unit, resulting in favorable overall integrity. During cyclic lateral loading, as the applied load gradually increased, the bending moment along the wall height increased accordingly, leading to the uniform formation of horizontal cracks on both sides of the wall. These cracks exhibited continued propagation in the later stages of loading. Compared with the CIP specimen, the horizontal joint regions of PGCW1 and PGCW2 developed slightly wider cracks; however, the maximum widths remained significantly below the ultimate elongation capacity of the longitudinal reinforcement, and no signs of interface failure or extensive crack openings were observed. Although the horizontal joints are expected to be critical zones experiencing high bending moments—and thus are theoretically more prone to bar slip—the measured slip of the longitudinal reinforcement at these joints was minimal throughout the test. This observation indicates that the connection provided sufficient anchorage and mechanical performance. The effectiveness of this behavior is primarily attributed to the use of GCMD, which offers reliable confinement and anchorage to the embedded reinforcement. As confirmed in previous studies [27], the mechanical interlock provided by the corrugated geometry, in combination with the high compressive strength of the grout, substantially enhances the bond strength between connected elements and facilitates efficient force transfer across the joint. Such configurations have been shown to improve the flexural and ductile performance of joints subjected to cyclic loading. In addition, experimental investigations [28] have demonstrated that reinforcing bars embedded in GCMDs can achieve full anchorage within relatively short embedment lengths, thereby effectively mitigating bar slip.

The failure characteristics of specimens PGSW1 and PGSW2 were more pronounced. During the test, cracks first appeared at the wall corners, and damage gradually accumulated and expanded, especially at the corners of the specimens, where the damage was most severe. Compared with CIP, PGCW1, and PGCW2, the failure of PGSW1 and PGSW2 was more concentrated in the corner areas. After the concrete at the corners was damaged, the concrete at the bottom of the wall’s central region began to deteriorate. At the end of the test, when clearing the broken concrete, it was found that the grouting splice sleeve connection at the corners remained intact, with no observed steel bar slippage. This indicates that, although the failure in PGSW1 and PGSW2 was relatively concentrated, the GSS connection played a crucial role in the edge region of the shear wall, effectively maintaining the stability of the structure. It significantly increased the load-carrying capacity of the region, preventing premature buckling and fracture of the reinforcement. The strong confinement at the edges effectively absorbed large bending moments and shear forces, preventing local failure and delaying the onset of damage. On the other hand, the GCMD connection helped disperse stress when the shear wall was subjected to lateral loads, reducing stress concentration in the central region and avoiding local failure in the middle part of the shear wall. The hybrid connection of GSS and GCMD combined the advantages of both methods, with GSS providing high-strength confinement at the edges and GCMD optimizing stress distribution in the central region. This further reduced localized stress concentration and delayed the occurrence of damage.

The variation in axial load ratio did not lead to a fundamental change in the failure modes of the specimens. All walls primarily exhibited horizontal cracking concentrated in the middle to lower regions, accompanied by concrete crushing and spalling at the base. Specimens PGCW2 and PGSW2, which were subjected to higher axial load ratios, showed more extensive concrete crushing and more pronounced crack development, both in terms of number and width. These observations suggest that higher axial loads may accelerate the progression of localized damage. Nevertheless, the overall failure mechanism remained consistent across all specimens and was characterized by a typical shear-compression mixed mode, without any substantial alteration in the global failure pattern.

Overall, compared to the cast-in-place reference specimen and single connection specimens, the hybrid connection specimens exhibited earlier crack initiation and more severe damage at the wall corners on both sides. However, in terms of load-carrying capacity, the peak load of the hybrid connection specimens was significantly higher than that of the cast-in-place and single connection specimens, indicating that precast shear walls with hybrid connections have stronger lateral resistance. Additionally, the hybrid connection precast specimens and single connection precast specimens had the same maximum loading displacement, further demonstrating the advantage of hybrid connections in terms of ductility. The increase in axial load ratio had a minimal impact on the overall failure process of the specimens; while a higher axial load ratio effectively enhanced the lateral resistance of the shear walls, it also intensified the damage at the final failure stage.

3.2. Hysteresis Curve

The hysteretic load-displacement curves of the five concrete shear wall specimens are shown in Figure 5, displaying different failure characteristics. At an axial compression ratio of 0.1, the hysteretic curve of the specimen CIP exhibits a pinched shape, with the load maintaining around 600 kN after reaching the peak, indicating good ductility. During the positive loading process, the load variation for each cycle, from 16 mm to 56 mm, is relatively consistent, with a noticeable decrease in load capacity at 64 mm. Compared to the CIP specimen, the hysteretic curves of the precast specimens PGCW1 and PGSW1 transition from a pinched shape to a bow shape, with a pinching effect observed in the later stages. Specifically, PGCW1 reaches the peak load at 32 mm, after which the load dropped sharply due to longitudinal rebar fracture, indicating a brittle failure mechanism; whereas PGSW1 reaches the peak load at 24 mm, with the load decreasing slowly, and the load at the maximum displacement is significantly higher than that of PGCW1, and far higher than the CIP specimen. This indicates that PGSW1 exhibits greater overall strength, benefiting from the optimized design of the GCMD and GSS hybrid connection. The hybrid connection provides strong confinement in the edge regions with GSS and optimizes stress distribution in the central region with GCMD, reducing stress concentration. This configuration effectively improves connection strength and overall seismic resistance, thereby significantly enhancing the specimen’s load-bearing capacity and ductility, and demonstrating the clear advantages of hybrid connections in structural seismic applications.

When the axial compression ratio is 0.2, the load-bearing capacity of the PGCW2 and PGSW2 specimens is significantly enhanced. PGCW2 reaches its peak load at 32 mm, while PGSW2 reaches its peak load at 24 mm, and after reaching the peak load, the load gradually decreases. Both specimens have the same number of hysteretic cycles after positive loading and stop the test after the first cycle at 48 mm. During the negative loading process, the negative peak load of PGCW2 occurs at 32 mm, while PGSW2 reaches its negative peak load at 40 mm. The negative load of PGSW2 decreases quickly, with the load dropping to below 85% of the peak load, indicating significant damage to the concrete in the right-side corner. Although PGSW2 uses the hybrid GSS and GCMD connection, which effectively enhances the connection strength, the rapid decline in load after peak load indicates substantial concrete damage, particularly under high axial compression conditions. This indicates that although a higher axial compression ratio contributes to improving the load-bearing capacity of shear walls, the capacity degrades more rapidly when concrete damage intensifies, particularly when damage occurs in the connection regions.

Overall, higher axial compression ratios significantly enhanced the load-bearing capacity of the specimens, particularly in the PGCW2 and PGSW2 specimens. However, after reaching the peak load, the load-bearing capacity of the specimens sharply declined, and the number of hysteretic cycles decreased significantly, indicating that higher axial compression ratios accelerated the failure process of the specimens. The specimens with the GCMD and GSS hybrid connections significantly improved the overall strength and seismic performance due to the optimized connection design, playing a key role in enhancing load-bearing capacity and ductility. However, under higher axial compression ratios, despite the enhanced lateral resistance due to the connection design, concrete damage became inevitable, leading to a pinching effect in the hysteretic curve and reduced energy dissipation capacity. This suggests that although higher axial compression ratios enhance the initial load-bearing capacity of the specimens, they also accelerate concrete damage, thus affecting the final hysteretic behavior.

3.3. Skeleton Curve

The skeleton curves of the five specimens are shown in Figure 6, and

Table 5. Feature points of each specimen.

Specimens	Yield Point			Peak Point			Limit Point
	Δy/mm	Py/kN	Ky(+)	Δm/mm	Pm/kN	Km(+)	Δu/mm	Pu/kN	Ku(-)
CIP	10.50	498.22	47.4	40.00	611.80	15.3	65.40	515.16	7.9
	−18.00	−529.78	29.4	−40.00	−612.00	15.3	−64.00	−528.30	8.3
PGCW1	9.84	486.61	49.5	32.00	605.40	18.9	50.00	514.98	10.3
	−18.13	−496.05	27.4	−32.00	−568.20	17.8	−49.80	−487.00	9.8
PGSW1	10.97	536.62	48.9	24.00	635.80	26.5	39.61	541.91	13.7
	−15.11	−570.79	37.8	−24.00	−627.60	26.2	−48.56	−538.24	11.1
PGCW2	14.72	738.06	50.1	32.00	861.50	26.9	42.59	731.78	17.2
	−21.34	−759.74	35.6	−32.00	−837.20	26.2	−48.00	−752.90	15.7
PGSW2	10.60	752.71	71.0	24.00	889.68	37.1	35.16	755.53	21.5
	−17.85	−747.56	41.9	−40.00	−825.00	20.6	−47.12	−699.24	14.8

Note: Δ_y represents yield displacement, Δ_m represents peak displacement, and Δ_u represents ultimate displacement; P_y represents yield load, P_m represents peak load, and P_u represents ultimate load. K_y, K_m, and K_u represent stiffness.

summarizes the load, displacement, and stiffness between adjacent characteristic points for the five specimens. The comparison and analysis of the skeleton curves for specimens CIP, PGCW1, and PGSW1 are shown in Figure 6a. Before a horizontal displacement of 8 mm, the curves of the three specimens largely overlap, with the only difference being that the positive load of PGSW1 is slightly lower than that of CIP and PGCW1. As the horizontal displacement increases, from 8 mm to 48 mm, the bearing capacity of PGCW1 is similar to that of CIP, indicating that the precast specimen with the metal corrugated pipe grout anchor connection has a bearing capacity comparable to the cast-in-place specimen. However, after 48 mm displacement, the bearing capacity of PGCW1 decreases significantly, and the gap in the skeleton curve between PGCW1 and CIP gradually increases, suggesting a decline in the load-carrying capacity with increased displacement for this connection method. In contrast, PGSW1 demonstrates significantly higher bearing capacity than both PGCW1 and CIP within the range of 8 mm to 32 mm, particularly when the material strength of the post-poured region at the wall corners is higher, allowing PGSW1 to achieve a higher bearing capacity. However, when the displacement exceeds 32 mm, the bearing capacity of PGSW1 sharply declines due to the degradation of the post-poured concrete’s integrity. This indicates that the hybrid connection demonstrates strong load-bearing capacity in the early stages; however, as the displacement increases, the influence of the post-cast concrete becomes more pronounced, resulting in a sharp decline in load-bearing performance.

For specimens PGCW2 and PGSW2, the trend of the skeleton curve is similar to that of PGCW1 and PGSW1, as shown in Figure 6b, indicating that the axial compression ratio has little effect on the overall trend of the skeleton curve. However, within the lower range of horizontal displacements, specimen PGSW2 exhibited a higher load-bearing capacity. For instance, at a displacement of 16 mm, PGSW2 achieved a lateral load of 882 kN, which is approximately 14% higher than the 774 kN recorded for PGCW2. This suggests that the hybrid connection can provide greater initial stiffness and stronger resistance during the early stages of loading. The peak load of PGSW2 was 889.7 kN at a displacement of 24 mm, whereas PGCW2 reached its peak load of 861.5 kN at 32 mm. Although the difference in peak loads between the two specimens is relatively small, the earlier occurrence of the peak in PGSW2 indicates a faster seismic response and a more stable stiffness degradation behavior. Moreover, PGSW2 showed a smoother descending branch throughout the loading process, without sudden load drops, implying that the hybrid connection demonstrates more reliable energy dissipation and ductility control.

As shown in Figure 6c, the positive and negative peak loads of PGCW1 and PGCW2 occur at the same horizontal displacement. The positive peak load of PGCW2 is 861.5 kN, and the negative peak load is −837.2 kN, which represents an increase of 42.3% and 47.3%, respectively, compared to PGCW1. Similarly, in Figure 6d, the positive and negative peak loads of PGSW2 reach 889.7 kN and −825.0 kN, which are 39.9% and 31.5% higher than those of PGSW1. Although the negative peak load of PGSW2 appears at a slightly larger displacement, both directions exhibit load enhancements exceeding 30%, confirming the improved lateral capacity and seismic performance provided by the hybrid connection. These results indicate that a higher axial compression ratio, within a reasonable range, can significantly enhance the lateral strength of shear walls. This improvement is likely due to the confining effect of axial pressure, which delays crack initiation and propagation and strengthens the compressive zone. Although, as noted in Section 3.1, higher axial loads may accelerate localized damage such as concrete crushing and cracking, the overall seismic capacity and structural stability are clearly enhanced.

The analysis of the skeleton curves reveals that specimens with higher load-bearing capacity tend to exhibit a more stable post-peak load degradation and improved deformation capacity throughout the loading process. This observation corresponds with the hysteretic behavior, where specimens experiencing slower load reduction and more stable cyclic responses generally demonstrate better energy dissipation and ductility. Although this study does not include dedicated sections on energy dissipation or ductility analysis, the consistency among results from different tests highlights the intrinsic relationships between various seismic performance indicators. These findings also offer empirical support for the selection of key parameters in the restoring force model, thereby enhancing its accuracy in representing structural behavior under seismic loading.

4. Restoring Force Model

The restoring force model consists of the skeleton curve and hysteretic rules [29]. Many researchers have established restoring force models through regression analysis to define hysteretic behavior [13,30]. However, most existing models are based on cast-in-place or wet connection structures, with limited research on hybrid connection structures. To address this gap, this study proposes a skeleton curve model and stiffness degradation relationship for hybrid connection shear walls, based on regression fitting and theoretical derivation, utilizing experimental results from previous studies. By integrating these models with hysteretic rules, a restoring force model considering the connection type is established, providing a theoretical basis for seismic performance analysis of hybrid connection shear walls.

4.1. Skeleton Curve Model

In the experiment, the skeleton curve is simplified into a tri-linear model, represented using dimensionless coordinates, as shown in Figure 7. The vertical axis is P/P_m, and the horizontal axis is Δ/Δ_m. Although the specimens adopted different connection types, all had identical geometric dimensions and similar overall weight, ensuring comparable boundary conditions. It is assumed that the simplified model for the negative load phase is the same as that for the positive load phase. The skeleton curve consists of the yield phase, peak phase, and failure phase, corresponding to the yield points A₁ (A₂), peak points B₁ (B₂), and ultimate points C₁ (C₂). At the initial stage of the curve, the yield point A₁ represents the specimen in the elastic stage, with the slope indicating the specimen’s relative elastic stiffness (K_y). As loading continues, the specimen enters the peak phase, where the material begins to undergo significant plastic deformation. The slope of the segment A₁B₁ (A₂B₂) indicates the specimen’s plastic stiffness (K_m). The segment B₁C₁ (B₂C₂) represents the failure phase, where the slope indicates the unloading stiffness of the specimen (K_u). This model provides a simplified theoretical framework for further analysis and simulation of the hysteretic behavior of shear walls.

The following assumptions must be satisfied when calculating the characteristic points of the backbone curve of the shear wall: (1) The specimen maintains the plane section assumption from the elastic to the yield stage, i.e., the strain distribution perpendicular to the section remains linear when the section is under tension and compression. (2) The tensile strength of concrete and grout is not considered. (3) The stress distribution in the compressed concrete zone is assumed to be equivalent to a rectangular distribution. (4) The calculation of the load-carrying capacity is based on the condition when the reinforcement reaches the yield state, without considering the force condition in the ultimate stage of the reinforcement.

4.1.1. Yield Point

(1)

Yield load P_y

Under cyclic loading, there exists a proportional relationship between the yield load and the peak load in shear walls [31]. To highlight the effects of GCMD connections and the hybrid connection of GSS and GCMD, this study introduces the sum of the cross-sectional areas of the metallic corrugated pipes (A_D) as an important parameter in the model. The equation is as follows:

$${\displaystyle \frac{P_m}{P_y}}=1.295-0.6n+0.021\lambda +0.051{\lambda }_v+0.011{\gamma }_a+0.006{\displaystyle \frac{A_D}{A}}$$

(1)

where P_m represents the peak load, n is the axial compression ratio, λ is the shear span ratio, λ_v is the characteristic value of the hooping reinforcement for the edge components, taken as 0.12 [32], γ_a is the ratio of the edge component’s cross-sectional area to the wall’s cross-sectional area [33], and A is the cross-sectional area of the shear wall.

(2)

Yield displacement Δ_y

In the early elastic deformation stage under loading, the deformation of the shear wall is primarily composed of bending deformation and shear deformation. Specifically, the shear wall exhibits significant bending stiffness and shear stiffness during this stage. The deformation calculation expression for the shear wall can be obtained from Equation (2), as shown in Figure 8. The deformation characteristics in this stage mainly depend on the geometric dimensions of the shear wall, material properties, and loading conditions. The elastic response of the shear wall determines the performance in the subsequent loading stages.

$${\Delta }_y={\Delta }_{by}+{\Delta }_{sy}$$

(2)

In the equation, Δ_by represents the bending displacement, and Δ_sy represents the shear deformation displacement.

Δ_by is calculated using Equation (3):

$${\Delta }_{by}=\eta {\displaystyle \frac{P_m{h_{w0}}^3}{3E{I}_w}}$$

(3)

In the equation, h_w0 represents the effective height of the shear wall, where h_w0 = h_w-a_s′, (h_w is the total height of the shear wall section, and a_s′ is the distance from the centroid of the compressive reinforcement to the edge of the compressive zone). E is the elastic modulus of concrete, and I_w is the bending moment of inertia of the specimen’s cross-section. To account for the influence of the corrugated metallic duct on the specimen’s displacement, the coefficient η is introduced to correct the equation.

Based on the theoretical study of the displacement of the shear wall before yielding [33], the calculation formula for Δ_sy is determined as shown in the following equation:

$${\Delta }_{sy}={\displaystyle \frac{\mu P_{m}H}{G{A}_w}}$$

(4)

where μ represents the shear non-uniformity coefficient, which is taken as 1.2 for rectangular sections, H is the height of the specimen, G is the shear modulus, typically taken as 0.4E, and A_w is the cross-sectional area of the shear wall.

Therefore, the yield displacement of the shear wall is expressed by the following equation:

$${\Delta }_y=\eta {\displaystyle \frac{P_m{h_{w0}}^3}{3E{I}_w}}+{\displaystyle \frac{\mu P_{m}H}{G{A}_w}}$$

(5)

The yield displacement Δ_y marks the transition from elastic to nonlinear behavior. A larger Δ_y indicates better capacity to withstand minor or moderate earthquakes without damage, supporting structural integrity and faster post-earthquake recovery.

(3)

Yield stiffness K_y

Based on the yield load and displacement, the yield stiffness of the shear wall is the slope of the load-displacement curve and can be expressed as:

$$K_y={\displaystyle \frac{P_y}{{\Delta }_y}}$$

(6)

4.1.2. Peak Point

(1)

Peak load P_m

Since all five shear wall specimens in this experiment failed under bending, the peak load calculation considers only the bending failure case. According to the regulations in JGJ3-2010 [32], for simplification, the vertical reinforcement within 1.5 times the height of the compression zone (denoted as x) is assumed not to have yielded and is not considered in the calculation. Figure 9 shows the calculation model for the bending capacity of the shear wall, which is used for the mechanical analysis based on this simplification assumption.

It follows that ΣN = 0.

$$N={\alpha }_1{f}_c{b}_{w}x-f_{yw}{\displaystyle \frac{A_s}{h_{w0}}}\left(h_{w0}-1.5x\right)$$

(7)

where α₁ is the concrete strength coefficient, taken as 1, b_w is the thickness of the shear wall, f_yw is the design tensile strength of the vertical distribution reinforcement, A_s is the cross-sectional area of the longitudinal reinforcement, and h_w0 is the effective height of the shear wall.

$$P_{m}H+N\left(h_{w0}-0.5h_w\right)={\displaystyle \frac{1}{{\gamma }_{RE}}}\left[f_y{A}_s\left(h_{w0}-a_s^{\prime }\right)-M_{sw}+M_c\right]$$

(8)

where M_sw and M_c are calculated by Equation (9) and Equation (10), respectively, γ_RE is the seismic adjustment coefficient, taken as 0.85, and f_y is the design tensile strength of the edge compression reinforcement.

$$M_{sw}=0.5{\left(h_{w0}-1.5x\right)}^2{b}_w{f}_{yw}{\rho }_w$$

(9)

$$M_c={\alpha }_1{f}_c{b}_{w}x\left(h_{w0}-0.5x\right)$$

(10)

where ρ_w represents the reinforcement ratio of the vertical distribution reinforcement in the shear wall.

(2)

Peak displacement Δ_m

There is typically a proportional relationship between Δ_m and Δ_y in the shear wall, which is related to the specimen parameters. Based on relevant studies [34], the following formula is fitted:

$${\displaystyle \frac{{\Delta }_m}{{\Delta }_y}}=3.001-3.223n+0.543{\lambda }_v+0.019{\gamma }_a+0.415\lambda -2.416{\displaystyle \frac{A_D}{A}}$$

(11)

The Δ_m characterizes the maximum deformation capacity of the structure under strong seismic events. If the structure can maintain overall stability at large displacement levels, it typically retains a higher residual load-bearing capacity and reparability, thereby enhancing its post-earthquake recovery potential and functional resilience.

(3)

Peak stiffness K_m

Since the fitted model is a piecewise linear model, the reinforced stiffness K_m can be obtained through the calculation of the dimensionless slope.

$$K_m={\displaystyle \frac{P_m-P_y}{{\Delta }_m-{\Delta }_y}}$$

(12)

4.1.3. Limit Point

(1)

Limit load P_u

According to GB 50010-2010 [23], the ultimate load is 85% of the peak load [12], and the experimental values of the five shear wall specimens also comply with this requirement.

$$P_u=0.85P_m$$

(13)

(2)

Limit stiffness K_u

There is a linear relationship between K_u and the initial stiffness K_y, as shown below:

$$K_u=\beta K_y$$

(14)

The value of β is determined to be 0.3 based on the fitting of experimental data.

(3)

Limit displacement Δ_u

The ultimate displacement can be calculated based on the peak load, displacement, ultimate load, and ultimate stiffness, as shown below:

$${\Delta }_u={\displaystyle \frac{P_u-P_m}{K_u}}+{\Delta }_m$$

(15)

4.2. Determination of Restoring-Force Model

4.2.1. Unloading Stiffness K_i

As shown in Figure 5, before yielding of the shear wall, the hysteretic curve is approximately linear. After yielding of the specimen, stiffness degradation occurs during the unloading process. Therefore, it is necessary to determine the rule of stiffness degradation during unloading to establish the hysteretic behavior. The unloading stiffness parameters were obtained from the slopes of the unloading segments in the experimental hysteresis curves, specifically by extracting the first unloading segment of each loading cycle after yielding. A nonlinear regression analysis was then performed based on the unloading stiffness values from the first cycle at each loading level to establish the degradation trend. The fitted equation is shown in Equation (16), and the stiffness degradation curve is presented in Figure 10.

$$K_i/K_y=0.392\ln \left({\Delta }_i/{\Delta }_y\right)+0.8416$$

(16)

where K_y is the yield stiffness; Δ_y is the yield displacement of the specimen; Δ_i is the absolute counts of the maximum displacement of the ith hysteresis loops after yield load.

4.2.2. Hysteretic Rule

The hysteretic rule reflects the loading and unloading paths of the hysteresis curve, with the external envelope curve representing the skeleton curve. The C₂—B₂—A₂—A₁—B₁—C₁ line in the hysteretic rule curve represents the shape of the skeleton curve. The hysteretic rule for the three-line backbone model of the mixed connection precast shear wall specimen is shown in Figure 11. The specimen remains in an elastic state between point o and the positive yield point A₁ and the negative yield point A₂, with no residual unloading deformation. The unloading stiffness is equal to the yield stiffness K_y, and the hysteresis path is O—A₁—O—A₂—O.

When the load exceeds the yield load but has not yet reached the peak load, taking point 1 as an example, in segment A₁1, the stiffness decreases during unloading, and the unloading path moves from 1 to 2, with the corresponding displacement at point 2 being the residual deformation. During reverse loading, the loading curve moves from point 2 to the symmetric point 3, and the reverse unloading curve moves from point 3 to the symmetric point 4 at point 2. The hysteresis path is O—A₁—1—2—3—4—1. Subsequently, the curve starts the second cycle from point 1.

When the load exceeds the peak load but has not yet reached the ultimate load, the hysteretic rule follows a similar pattern to that of the path to point 1. Taking point 5 as an example, in segment B₁5, the stiffness decreases during unloading, and the curve unloads along line 5—6 to the zero load point 6, with the displacement at point 6 representing the residual deformation. The reverse loading and unloading rules are similar to those of point 1, with the curve moving along 6—7—8, ultimately returning from point 8 to point 5 to start the next cycle.

4.3. Verification of Restoring Force Model

4.3.1. Validation of Skeleton Model

The comparison between the calculated theoretical skeleton curve and the experimental skeleton curve is shown in Figure 12. It can be observed from the figure that the theoretical skeleton curve closely matches the experimental skeleton curve. Particularly, at key characteristic points such as the yield point, peak point, and ultimate point, the load and displacement values calculated theoretically are very close to the experimental results, further validating the accuracy and applicability of the established restoring force model. Through these comparisons, it can be confirmed that the theoretical model effectively reflects the stress and deformation characteristics of the mixed connection shear walls under seismic action, providing a reliable theoretical foundation for further analysis and engineering applications.

4.3.2. Verification of Hysteresis Model

The comparison between the hysteretic curve theoretical model and the experimental data is shown in Figure 13. The figure presents the theoretical curves of five specimens, where the yield point, peak point, ultimate point, and pinching phenomena align well with the experimental curves. This indicates that the restoring force model proposed in this study effectively reflects the seismic hysteretic performance of the mixed connection shear walls under seismic actions.

5. Conclusions

In this study, low-cycle repeated loading tests were conducted on five precast concrete shear wall specimens, with connection type and axial compression ratio as the primary variables. The failure modes and hysteretic responses under different connection methods and axial loads were systematically analyzed. Based on experimental observations and theoretical derivation, a restoring force model incorporating connection type effects was developed. However, we acknowledge that the proposed restoring force model has inherent limitations. It was primarily derived from quasi-static cyclic loading tests and has yet to be validated under dynamic seismic excitations, complex loading paths, or realistic boundary conditions. In addition, critical nonlinear behaviors—such as grout degradation and concrete softening—are not explicitly modeled, which may compromise the model’s accuracy under severe or long-duration seismic demands. In addition, the model has only been verified against the five specimens tested in this study, and its generalizability to broader structural configurations or boundary conditions remains to be demonstrated. Future studies should introduce variables such as connection detailing, joint configuration, and ground motion characteristics, and conduct comprehensive parametric analyses and full-scale tests to improve the model’s applicability and reliability in complex engineering contexts. Based on the analysis results, the following conclusions are drawn:

(1). Experimental and theoretical analyses confirm that precast shear walls with hybrid connections of grouted sleeve (GSS) and grouted corrugated metal duct (GCMD) exhibit significant advantages in terms of load-bearing capacity, ductility, and seismic performance. Compared to conventional cast-in-place specimens and those using only corrugated duct connections, the hybrid connection specimens demonstrated fuller hysteresis loops and more than a 5% increase in peak load.

(2). An increased axial compression ratio significantly improves structural load-bearing capacity but also accelerates damage evolution, particularly resulting in a sharper decline in strength after reaching the peak load. While hybrid connections effectively enhance the structural capacity, concrete damage under higher axial compression ratios may still lead to pinching in the hysteresis curve, which impairs the energy dissipation capacity.

(3). Taking into account the influence of shear-span ratio, axial load ratio, and connection type, the yield point, peak point, and ultimate point were identified and analyzed. Load, displacement, and stiffness equations for these characteristic points were obtained through parameter fitting and theoretical derivation, leading to the development of a three-line skeleton curve model.

(4). Formulas for unloading stiffness at different loading stages were proposed, and a restoring force model that incorporates the influence of connection type was established and validated. Comparisons between the model and experimental results show a high degree of agreement in both the skeleton and hysteresis curves. This confirms that the proposed model can accurately capture the cyclic response of precast shear wall specimens and provides a reliable theoretical foundation for analyzing the seismic performance of similar structures.