Suggestions for a Collapse-Resistant Design for Frame–Masonry Hybrid Buildings Based on the Concept of Balancing Seismic Shear Forces

Abstract

Frame–masonry hybrid structures, though economically practical and widespread in rural China, face significant collapse risks during earthquakes due to shear imbalances from a component mismatch. A severe case was No. 7, Group 1, Detuo Town, after the 2022 Luding earthquake, where damage was concentrated on the ground floor. Numerical modeling revealed that the axis Ⓒ perforated wall, bearing 78% of the seismic shear due to its stiffness, suffered shear failure from geometric and structural factors, triggering a shear concentration–brittle failure chain reaction, pushing the building to near collapse. Meanwhile, the axis Ⓐ frame column, only sustaining 12% shear, sustained minor damage. Based on this typical seismic damage, this study proposes a collapse-resistant design using the deformation saturation theory to achieve balanced shear distribution by adjusting frame column sections. The results showed that compared to the prototype model, the collapse-resistant model (RE) under PGA = 0.4 g saw maximum displacement drop from 16.66 mm to 5.42 mm, which was reduced by 67.5%, the shear share of axis Ⓐ rose from 18% to 45%, the shear force of axis Ⓒ decreased from 70% to 46%, the shear ratio changed from 1:4 to 1:1, and maximum component damage was at 75% of the performance point, indicating significantly enhanced collapse resistance. These findings highlight the importance of balanced seismic shear distribution in preventing shear concentration and brittle failure, validate the deformation saturation theory, and offer a theoretical basis and design reference for the seismic reinforcement of similar hybrid structures.

1. Introduction

In recent years, the accelerated pace of urbanization has driven the emergence of a novel structural typology characterized by reinforced concrete columns and perforated masonry walls. This type of structure has been variously defined by scholars: classified as a masonry structure [1,2] based on its structural mechanics, termed a multi-story building with ground-floor commercial spaces [3] according to functional use, or referred to as a semi-frame structure [4] or frame–masonry composite load-bearing system [5,6]. Given the distinct composition and load-bearing behavior of this structural form, the authors designate this system as a frame–masonry hybrid structure [7]. While its economic efficiency and rapid construction capabilities have facilitated its widespread adoption in urban areas [8], post-earthquake investigations of destructive seismic events [9,10,11,12] reveal that these structures exhibit pronounced vulnerability to collapse, often resulting in severe casualties and economic losses. Consequently, elucidating their seismic collapse mechanisms and improving their collapse resistance have become critical research priorities.

These hybrid systems are commonly employed in mixed-use commercial–residential buildings, warehouses, and rural homestays [13], typically characterized by asymmetric planar configurations. Functional requirements, such as pedestrian access and ventilation, necessitate the use of reinforced concrete (RC) frame columns on street-facing elevations and perforated masonry walls on rear facades [14,15]. Field observations from seismic events [16,17,18,19,20] consistently highlight a stark disparity in damage patterns between these components. Specifically, perforated walls predominantly fail through shear-dominated damage at piers between windows, whereas frame columns exhibit only minor plastic hinge formation at member ends. This disproportionate damage distribution—particularly the catastrophic shear failure of perforated walls—triggers structural collapse toward the rear facade, underscoring the incompatibility between the two load-bearing systems under seismic action.

To address this critical vulnerability, research efforts have focused on the seismic capacity of perforated walls [21,22]. Nezhad et al. [23] combined experiments and numerical analyses to develop FRP-retrofitted wall models, significantly improving out-of-plane stability and in-plane seismic performance. However, this approach also influenced in-plane seismic performance [24]. Fares et al. [25] introduced SRG into shake-table tests, achieving similar reinforcement effects as observed in FRP studies. Additional materials, such as mortar, fiber-reinforced polymers, and engineered cementitious composites, have also been explored for their seismic retrofitting potential [26,27,28]. While Mantawy et al. [29] reached similar conclusions experimentally, effectiveness remains contingent on bond quality [30] and application techniques [31,32]. Alternative material systems have emerged as viable solutions, with field validations confirming their post-earthquake performance enhancements [33,34,35]. Celano et al. [36] numerically investigated fiber-reinforced cementitious matrix composites for masonry reinforcement, reporting enhanced shear resistance. Ban et al. [37] explored modified mortar formulations and reinforcement detailing, verifying their feasibility through component-level testing. Zhang et al. [38] compared polymer mortar–mesh systems, demonstrating measurable improvements in seismic resilience metrics, including in load capacity and ductility. Innovative external devices, such as damping systems and subsidiary substructures, have also shown promise in enhancing the seismic response [39,40,41,42]. Despite substantial progress, existing research predominantly focuses on component-level behavior, with limited attention to system-wide failure mechanisms and interaction effects under dynamic loading conditions.

Recent advancements in collapse mechanics and anti-collapse design have introduced novel theoretical frameworks [43,44]. Dong et al. [45] identified longitudinal displacement dominance in seismic damage investigations [46], a phenomenon corroborated by shake-table tests [47]. Complementary research by Luo et al. [48] employing acceleration–strain coupled monitoring revealed 3–5-fold variations in inter-story shear force distributions within hybrid systems—a phenomenon also manifesting in frame–masonry composites [7]. These observations informed Guo’s [49] deformation saturation theory, representing a novel perspective for explaining earthquake-induced collapse mechanisms. Due to functional requirements, buildings often feature multiple fully filled masonry walls in the transverse direction while containing numerous door and window openings in the longitudinal direction. This configuration creates greater transverse stiffness than longitudinal stiffness, paradoxically predisposing structures to longitudinal movements. Concurrently, variations in non-structural components or structural details produce significant differences in constitutive relationships among vertical load-bearing components with distinct characteristics, particularly when these heterogeneous components coexist on the ground floor. During seismic events, when vertical load-bearing elements at the base experience identical displacements, earthquake-induced shear forces are distributed according to the lateral stiffness of each distinct element. This differential distribution results in varying degrees and sequences of component damage, ultimately culminating in the collapse of the building.

Guided by this theory, this study proposes a balanced seismic shear collapse-resistant design concept. A representative case (No. 7, Group 1, Detuo Town) from the 2022 Luding earthquake was selected. A corresponding numerical analysis model was established using Perform-3D software (version 7), providing clear insights into the collapse process through motion patterns and component damage analysis. Furthermore, the optimal retrofitting scheme for the case model was determined using the pushover method based on the concept of balancing seismic shear force and compared with the prototype model. This proved the rationality of the deformation saturation theory and necessity of a balanced seismic shear design concept for future designs.

2. Earthquake Damage Investigation of the Prototype Building

No. 7, Group 1 of Detuo Town represents a four-story (partially five-story) frame–masonry hybrid structure, as illustrated in Figure 1a,b. For systematic structural analysis, a coordinate system was established with three longitudinal axes (x-direction: Ⓐ–Ⓒ) and six transverse axes (y-direction: ①–⑥). Longitudinal axes Ⓐ and Ⓑ comprise bare frame columns, while axis Ⓒ incorporates load-bearing masonry walls with openings (original axes Ⓒ and Ⓓ were consolidated as new axis Ⓒ for analytical clarity). Transverse fully grouted walls exist only along axes ①, ②, and ⑥. Seismic damage investigations revealed concentrated damage at the ground level, while the upper floors remained largely intact. Consequently, structural documentation focused on the ground floor plan (Figure 1c) and three-dimensional configuration (Figure 1d).

Field observations from No. 7, Group 1, Detuo Town, revealed distinct damage patterns, as documented in Figure 2. The masonry wall along axis Ⓒ exhibited severe shear failure (Figure 2a,b), with window piers partitioned into four segments by diagonal cracks. Spalling of the concrete cover on tie columns exposed buckled internal reinforcements, indicating significant slab settlement. Figure 2c reveals concentrated damage at the column ends of axis Ⓑ, where upper concrete sections were crushed. The insufficient reinforcement cover at the column top resulted in exposed rebars, while annular cracks formed at the column base accompanied by compressive concrete bulging. This column exhibited longitudinal residual displacement (52 mm). Similar damage patterns were observed in axis Ⓐ columns, yet the damage severity was comparatively lower than that of the perforated walls (Figure 2d). As observed in Figure 2e,f, horizontal cracks emerged at the junction of the transverse wall and beam in axis ② of the building. The shear damage in axis ⑥ was more severe than in axis ② because the bay between axes ② and ⑥ was larger with fewer transverse walls, causing the building to experience slight torsion. Since the transverse wall in axis ⑥ was farther from the structure’s center of rigidity, its damage was more severe. Comparative assessment of the damage and the residual displacement of frame columns in both structural directions revealed more pronounced longitudinal displacement.

3. Modeling and Analysis of the Prototype Building

In this study, two experimental configurations from prior studies [50,51] were selected to establish numerical analysis models. The results were compared with experimental data to validate the numerical models, providing a basis for constructing 3D models of real buildings. Considering the emphasis on global seismic behavior, Perform-3D was adopted to develop practicable macro-element models that balance accuracy and computational efficiency.

3.1. Elements and Materials Definition

Field observations of the target building revealed bending-dominated failure in frame columns, characterized by circumferential cracking at member ends. Over the past decades, numerical simulation has emerged as a widely adopted method for studying column damage progression and failure mechanisms. Spacone et al. [52] first proposed the concept of fiber beam–column elements, successfully applied to the seismic response analysis of reinforced concrete structures. Subsequently, McKenna and Fenves [53] implemented this approach in the OpenSees software. Building on this work, Qu et al. [54] investigated the progressive collapse of concrete frames under mid-column and side-column failure scenarios. With advancements in computational power, microscopic simulation methods have gained traction. Li et al. [55] developed ABAQUS-based numerical models calibrated against experimental data, identifying the axial load ratio and the partial prestressing ratio as critical parameters for enhancing seismic resilience in RC frames. While solid-element models (e.g., ABAQUS) offered high-fidelity predictions of global and local deformations, their computational expense was prohibitive for large-scale analyses, and convergence issues may have compromised result accuracy. In contrast, fiber-section elements in Perform-3D effectively simulated this coupled axial–flexural behavior through discrete material integration. Each frame member was discretized into five segments along its length: two rigid zones at ends sandwiching a fiber-dominated region, with transitional elastic segments intervening (Figure 3a). The fiber regions incorporated uniaxial material models for longitudinal rebars and confined/unconfined concrete, discretized into several fibers across the section [56,57,58].

A bilinear model (Equation (1)) was chosen for the steel constitutive relationship, with parameters from the Chinese Code of Concrete [59]. The concrete in the frame was categorized into confined and unconfined types based on its position relative to the stirrups. Confined concrete within stirrups adopted a confined concrete constitutive model, while unconfined concrete outside stirrups used an unconfined model. For hybrid frame–masonry systems, constitutive models were required to balance computational efficiency, physical fidelity, and experimental validation. Three widely adopted concrete models were considered: the Mander model [60], the Saatcioglu–Razvi model [61], and the modified Kent–Park model [62]. The latter was selected for the following reasons. (i) Parameter simplicity: It required only 4–5 key parameters (e.g., peak strain, residual strength, and curvature coefficient), enabling straightforward calibration via uniaxial test data. (ii) Numerical robustness: It avoided convergence issues inherent to complex plastic potential functions (e.g., in the Mander model) during non-monotonic loading. (iii) Stirrup effect integration: It incorporated a reinforcement coefficient K to simultaneously account for stirrups’ contributions to both strength enhancement and ductility improvement, critical for distinguishing core and cover concrete behaviors. The modified Kent–Park model’s simplicity and accuracy made it widely applicable. Notably, its backbone curve under compression aligned with the Concrete01 and Concrete02 models in OpenSees [53], which were also based on a modified Kent–Park framework. To suit the computational efficiency and accuracy of macro-elements in Perform-3D, the modified Kent–Park model was formatted into a five-segment format, as shown in Figure 3 and Equations (1)–(3).

$$\sigma =\left\{\begin{array}{ll}E\epsilon \hfill & 0<\epsilon <{\epsilon }_y\hfill \\ E{\epsilon }_y\hfill & {\epsilon }_y\le \epsilon \hfill \end{array}\right.$$

(1)

$$f_c=\left\{\begin{array}{ll}K{f}_c^{\prime }\left[2{\epsilon }_c/K{\epsilon }_0-{\left({\epsilon }_c/K{\epsilon }_0\right)}^2\right]\hfill & {\epsilon }_c\le K{\epsilon }_0\hfill \\ K{f}_c^{\prime }\left[1-Z_m\left({\epsilon }_c-K{\epsilon }_0\right)\right]\hfill & {\epsilon }_c<K{\epsilon }_0\le {{\epsilon }_{0.2}}_c\hfill \\ 0.2K{f}_c^{\prime }\hfill & K{\epsilon }_0>{{\epsilon }_{0.2}}_c\hfill \end{array}\right.$$

(2)

$$\left\{\begin{array}{l}K=1+{\displaystyle \frac{{\rho }_s{f}_{yh}}{f_c^{\prime }}}\hfill \\ Z_m={\displaystyle \frac{0.5}{\frac{3+0.29f_c^{\prime }}{145f_c^{\prime }-1000}+\frac{3}{4}{\rho }_s\sqrt{\frac{h^{″}}{s_h}}}}\hfill \end{array}\right.$$

(3)

In these equations, σ and ε represent steel stress and strain; E is the elastic modulus; ε_y indicates the yield strain; f_c and ε_c denote concrete compressive stress and strain; f_c′ and ε₀ signify cylinder compressive strength and corresponding strain; K symbolizes the strength enhancement factor; ρ_s stands for the volumetric stirrup ratio; f_yh is the stirrup yield stress; and h″ indicates stirrup spacing.

Extensive numerical studies have investigated masonry wall failure mechanisms under seismic loading. Current modeling approaches for masonry walls are broadly categorized into micro-scale modeling and macro-scale modeling [63,64]. Micro-scale modeling approaches divide walls into individual masonry units and mortar, accounting for their anisotropic and heterogeneous properties and interfacial interactions. Alternatively, walls are treated as homogeneous entities with nonlinear material properties derived from block–mortar composites [65]. These methods typically employ finite element (FEM) and discrete element (DEM) analyses with solid elements. For instance, Mughal et al. [66] utilized ANSYS solid elements to explore damage evolution in multiple wall models, emphasizing the influence of opening ratios and boundary constraints. Tekeli et al. [67] investigated failure mechanisms using ABAQUS, focusing on the impact of opening locations on wall collapse modes. However, micro-scale modeling requires highly refined meshes relative to component dimensions, resulting in computational complexity and high costs, which restricts its application to component-level studies. With advancements in seismic modeling, macro-scale modeling approaches—represented by the story mechanism model with concentrated mass elements connected via shear springs [68], the diagonal strut model [69], and the macro-element model with rigid interfaces and vertical springs [70]—have enabled large-scale simulations while maintaining computational efficiency [71,72]. Among these, the simplified equivalent frame method (EFM) [73,74] has gained prominence due to its balance between accuracy and efficiency, being widely adopted by researchers and codes. Therefore, this study adopted EFM to model frame–masonry hybrid systems.

Field observations and experimental data identified shear-dominated failure in window piers. To replicate these mechanisms accurately using this method, plastic shear hinges [75,76,77] were incorporated into the modeling framework. These hinges exhibited high initial stiffness and remained elastic until a certain deformation was reached, after which their stiffness degraded as damage initiated. When the peak shear force was attained, irreversible damage caused a continuous reduction in load-carrying capacity until it equaled the friction force. Upper and lower walls, with higher stiffness than window piers, were modeled as rigid zones. Although piers between windows eventually failed via shear, their bending moment contribution could not be fully disregarded. Thus, bending moment contributions from window piers were retained through composite wall–column elements. These elements integrated fiber zones, elastic zones, and plastic shear hinges to simulate perforated walls. The constitutive model for the wall–column elements followed a five-segment framework, with the parameters [75] detailed in Figure 4 and Equation (4), while the shear hinge models are shown in Figure 5 and Equations (5)–(8). Model simplification and element division are illustrated in Figure 6.

$${\displaystyle \frac{{\sigma }_m}{f_m}}=\left\{\begin{array}{ll}1.96\left({\displaystyle \frac{\epsilon }{{\epsilon }_0}}\right)-0.96{\left({\displaystyle \frac{\epsilon }{{\epsilon }_0}}\right)}^2\hfill & 0<{\displaystyle \frac{\epsilon }{{\epsilon }_0}}\le 1\hfill \\ 1.2-0.2{\displaystyle \frac{\epsilon }{{\epsilon }_0}}\hfill & 1<{\displaystyle \frac{\epsilon }{{\epsilon }_0}}\le 1.6\hfill \end{array}\right.$$

(4)

$$\left\{\begin{array}{l}{V}_y={\alpha }_y{V}_m\hfill \\ V_m=2.15{\lambda }^{0.256}\left[{\displaystyle \frac{1}{1.2}}\sqrt{1+{\eta }_a{\displaystyle \frac{{\sigma }_0}{f_{v0}}}}+\mu \left(1-{\eta }_a\right){\displaystyle \frac{{\sigma }_0}{f_{v0}}}\right]f_{v0}A\hfill \\ V_s=0.2V_m\hfill \end{array}\right.$$

(5)

$${\alpha }_y=\left\{\begin{array}{ll}{\displaystyle \frac{{\sigma }_0+0.0049f_m}{0.86{\sigma }_0+0.15f_m}}\hfill & \mathrm{Unperforated}\mathrm{wall}\hfill \\ {\displaystyle \frac{{\sigma }_0-0.067f_m}{0.81{\sigma }_0+0.043f_m}}\hfill & \mathrm{Perforated}\mathrm{wall}\hfill \end{array}\right.$$

(6)

$$\lambda =\left\{\begin{array}{ll}{\displaystyle \frac{{\sigma }_0}{f_m}}{e}^{-{\displaystyle \frac{H}{L}}}\hfill & 0<{\displaystyle \frac{H}{L}}\le 0.675\hfill \\ {\displaystyle \frac{{\sigma }_0}{f_m}}{e}^{-{\displaystyle \frac{1.55H}{L}}}\hfill & {\displaystyle \frac{H}{L}}>0.675\hfill \end{array}\right.$$

(7)

$${\eta }_{\mathrm{soft}}=\left\{\begin{array}{ll}-0.168{\displaystyle \frac{{\sigma }_0}{f_m}}-0.0168\hfill & \mathrm{Unperforated}\mathrm{wall}\hfill \\ -0.365{\displaystyle \frac{{\sigma }_0}{f_m}}-0.01\hfill & \mathrm{Perforated}\mathrm{wall}\hfill \end{array}\right.$$

(8)

where σ_m and ε_m represent the stress and strain of masonry; f_m and ε₀ denote the peak stress and corresponding strain of masonry, with strain set at 0.003; V_m indicates the peak force of the shear hinge; α_y signifies the relative parameter between the yield strength and peak strength; V_y symbolizes the yield strength of the shear hinge, equal to the product of α_y and V_m; λ represents the correction factor; σ₀ stands for the vertical compressive stress; H and L denote the height and width of the wall; η_soft is the stiffness degradation coefficient.

3.2. Verification and Establishment of the Numerical Model

To verify the numerical modeling approach, experimental configurations from prior research [50,51] were replicated using the proposed methodology. According to the experiments, the elastic modulus (E) and yield strength (σ_y) of the steel reinforcement in the constitutive model required by the software were set to 197,000 MPa and 190 MPa, respectively. For the modified Kent–Park model of the core concrete, parameters K and Z_m were set to 1.03 and 80.81, respectively. For the sake of brevity, the remaining parameters are not listed in detail, but these parameters have been tested and verified. As shown in Figure 7, close agreement between the simulated and experimental hysteresis curves validated the modeling strategy, confirming its reliability for analysis of different components.

The numerical model of the case-study building was constructed following the workflow in Figure 8. Frame–column axes (axes Ⓐ and Ⓑ) were modeled using the method depicted in Figure 6a, incorporating combined elements for frame columns. Perforated walls (axis Ⓒ), with damage concentrated at window piers, were simulated by shear hinges. Transverse walls, sharing the same deformation mechanism as the perforated walls, were modeled using the same method. As shown in Figure 2, most damage occurred in the vertical load-bearing components, with no significant damage to beams, which were therefore connected using members with high stiffness. The material properties adhered to the Chinese Code [59], with concrete strength specified as C30, beams and columns reinforced with HRB400 steel, and walls constructed from clay bricks and mortar with a thickness of 240 mm. Considering the walls, finishes, and equipment, loading was calculated at 1 t/m² in accordance with the relevant codes [78], with loads distributed to each node based on tributary area ratios.

According to the Chinese Seismic Code [79], Luding County in Garzê Tibetan Autonomous Prefecture, Sichuan Province, has a seismic fortification intensity of 8 degrees, a basic design earthquake acceleration of 0.2 g, and is in the second earthquake design group. Three seismic station records from Luding County were selected for analysis. An elastic–plastic time–history analysis of the example building was performed with peak ground accelerations (PGA) adjusted to 0.1 g (service level earthquake, SLE), 0.2 g (design-basis earthquake, DBE), and 0.4 g (maximum considered earthquake, MCE) [79,80]. To enhance computational efficiency, records were truncated to remove invalid segments while maintaining ≤0.01 spectral acceleration tolerance. These seismic station data are presented in

Table 1. Seismic motion record.

No.	Station Code	PGA/g			Duration/s
		EW	NS	UD	Record	After Clipping
1	SC.T2271	0.91	0.68	0.18	120	46
2	51LDJ	0.11	0.31	0.16	139	29
3	51LDL	0.31	0.20	0.21	120	40

and Figure 9.

3.3. Damage State of the Prototype

To evaluate the failure states of components along various axes in the model, the damage criteria for masonry walls and frame columns were defined in accordance with experimental studies and relevant codes. For perforated walls, research by the National Institute of Standards and Technology (NIST) [81] indicated that collapse was assumed when the strain in 30% of the wall’s cross section reached 0.01. The Masonry Standards Joint Committee (MSJC) [82] recommended that shear failure occurred when the applied shear force exceeded the wall’s maximum shear resistance. Lagomarsino et al. [83] further proposed using the deformation corresponding to the peak shear force as the performance point, which aligns with drift ratio thresholds in codes [84] for different damage states. Experimental observations [7,50] showed that masonry walls exhibited limited ultimate deformation, with minimal lateral drift between the peak load and collapse states, making them prone to brittle shear failure. Thus, deformation corresponding to the peak force of perforated walls was defined as the performance point for these components. For frame columns, damage was evaluated using longitudinal reinforcement strain, as these components typically undergo ductile bending failure under seismic loading [85,86]. Post-earthquake investigations confirmed localized damage concentrations at column ends with limited global drift, prompting the adoption of strain-based criteria over displacement-based methods to capture inelastic deformation. The correspondence between strain thresholds and damage states for column components is summarized in

Table 2. Correspondence between frame column damage and rebar strain.

Type	Damage State
	Minor Damage	Moderate Damage	Severe Damage	Collapse
Frame column	0.004–0.007	0.007–0.016	0.016–0.024	>0.024

.

Taking the model subjected to the SC.T2271 station data with a PGA of 0.4 g as an example, the plastic states of the model at the moment when a certain axis reached its maximum performance point and when the PGA reached its peak under this condition were extracted. As is shown in Figure 10a, all x-direction axes were damaged to varying degrees. Specifically, the load-bearing members of axis Ⓒ reached their performance limit points, indicating shear failure. In contrast, the strain of the rebars in the frame columns of axes Ⓐ and Ⓑ was less than 0.004, meaning no significant damage had occurred. Among the load-bearing components in the y-direction, only axes ⑥ and ④ were damaged, with damage levels at 25–50% of their performance points. At peak ground acceleration, the strain of the rebars of axis Ⓐ was between 0.004 and 0.007, indicating minor damage. While the strain for axis Ⓑ was less than 0.004, the wall in axis ⑥ reached its performance point and failed in shear. The damage to the opposite wall in axis ② exhibited minor damage, suggesting torsional effects during dynamic motion. Comparing the damage at these stages, when all the components in axis Ⓒ reached their performance limits, damage to the frame columns on the other x-direction axes was minor, indicating a high seismic reserve. The numerical results closely matched the actual seismic damage, further validating the overall numerical model.

3.4. Displacement Response

3.4.1. Inter-Story Drift Ratio

The inter-story drift ratio (IDR) of each floor was analyzed using the SC.T2271 station data (Figure 11). At a PGA of 0.1 g, IDR curves for each floor were similar, with no significant phase differences, indicating that the model primarily vibrates in the first mode. The maximum IDR occurred on the ground floor, with a value of 0.0005. The IDR decreased with height: 0.0002 on the second floor, and 0.0001 on the third floor and above. This phenomenon occurred because the seismic shear force on each floor was the sum of the inertial forces of that floor and all floors above it. As the number of floors increased, the seismic shear force on each floor decreased, leading to smaller IDRs. Generally, the maximum IDR occurred on the ground floor. Comparing the IDRs of the five floors indicated that the damage was mainly concentrated on the ground floor, and the deformation on the second floor and above could be largely ignored. At a PGA of 0.2 g, the IDRs for floors 1–5 were 0.0011, 0.0003, 0.0002, 0.0002, and 0.0001, respectively, with the first floor being about four times that of the second floor, as shown in Figure 11b. When the PGA reached 0.4 g, the ground-floor IDR reached 0.0046, while the maximum IDR for the other floors was only 0.0005, an eightfold difference. This confirmed the deformation concentration on the ground floor with increasing PGA, consistent with both the experimental observations and numerical results. The deformation curves and maximum values of Model PT under other conditions are shown in

Table 3. IDR of Model PT under different conditions (%).

Condition/g		Floor
		F1	F2	F3	F4	F5
SC.T2271	0.1	0.05	0.02	0.01	0.01	0.01
	0.2	0.11	0.03	0.02	0.02	0.01
	0.4	0.46	0.05	0.03	0.02	0.02
51LDJ	0.1	0.04	0.02	0.01	0.01	0.01
	0.2	0.09	0.03	0.02	0.01	0.01
	0.4	0.28	0.05	0.03	0.03	0.02
51LDL	0.1	0.04	0.02	0.01	0.01	0.01
	0.2	0.09	0.03	0.02	0.02	0.01
	0.4	0.19	0.05	0.04	0.03	0.02

.

3.4.2. Lateral Displacement

Based on the seismic damage and numerical results, the prototype building’s damage was mainly concentrated on the ground floor. Therefore, ground-floor displacements of critical axes (x-direction axes Ⓐ and Ⓒ and y-direction axes ① and ⑥) were extracted from Model PT (SC.T2271 data, Figure 12). As shown in Figure 12a, at a PGA of 0.1 g, the displacement curves of x-direction axes Ⓐ and Ⓒ were similar in trend and amplitude, with maximum values of 2.05 mm and 1.78 mm at 12.24 s. At this moment, the y-direction displacements were 0.66 mm and 0.84 mm. The displacement ratio of axes ①:⑥:Ⓐ:Ⓒ was 1.00:1.27:3.11:2.70. The different displacements of the same-direction axes were attributed to the model’s non-uniform stiffness. The center of mass was in the middle of the model, while the center of rigidity was closer to axes ① and Ⓒ, causing a slight torsional effect. This explained why the displacement of axis Ⓐ was greater than that of axis Ⓒ and axis ⑥ was greater than axis ①, and why the damage to the wall on axis ⑥ was more severe than that on axes ① and ②. Based on the displacement distribution at this moment, the model exhibited coupled horizontal and torsional motions. The torsional effect coefficient κ was defined as the ratio between the displacement difference to the sum in the y-direction of axes ① and ⑥ to assess the torsional impact on the model’s motion, calculated using Equation (9). In this condition, the torsional effect coefficient was 0.12, indicating a minor torsional impact and predominantly x-direction motion. The displacement responses of the model under other conditions are shown in Figure 12 and

Table 4. Displacement of Model PT under different conditions.

Conditions/g		y/mm		x/mm		Ratio	Torsion Effect Coefficient
		①	⑥	Ⓐ	Ⓒ
SC.T2271	0.1	0.66	0.84	2.05	1.78	1.00:1.27:3.11:2.70	0.12
	0.2	1.44	1.70	4.38	3.99	1.00:1.18:3.04:2.77	0.08
	0.4	−3.08	−5.46	−17.59	−16.66	1.00:1.77:5.71:5.41	0.28
51LDJ	0.1	0.76	0.98	1.55	1.39	1.00:1.29:2.04:1.83	0.13
	0.2	−1.38	−2.25	3.42	3.13	1.00:1.63:2.48:2.27	0.24
	0.4	−2.94	−4.56	10.85	9.78	1.00:1.55:3.69:3.33	0.22
51LDL	0.1	0.63	0.89	1.66	1.45	1.00:1.41:2.63:2.30	0.17
	0.2	1.32	1.74	3.51	3.12	1.00:1.32:2.66:2.36	0.14
	0.4	1.96	3.86	−7.50	−6.71	1.00:1.97:3.83:3.42	0.33

.

$$\kappa =\left|{\displaystyle \frac{{\delta }_6-{\delta }_1}{{\delta }_6+{\delta }_1}}\right|$$

(9)

where κ stands for the torsional effect coefficient, and δ₁ and δ₆ represent displacements of axes ① and ⑥ in the x-direction.

3.5. Seismic Shear Force Response

According to the base shear method, the maximum seismic shear force occurred on the ground floor, which was confirmed by both seismic earthquake investigations and numerical analyses. Using data from station SC.T2271 as an example, the shear forces of the ground floor in the x-direction are plotted in Figure 13. Figure 13a shows the shear force curves of the three axes, when the PGA reached 0.1 g. The curves were well synchronized without significant phase differences. At 12.24 s, the seismic shear forces reached their maximum values: 219.25 kN, 174.69 kN, and 1393.50 kN for axes Ⓐ, Ⓑ, and Ⓒ, respectively. Axes Ⓐ and Ⓑ had similar shear forces, while axis Ⓒ had a force about 6–8 times larger. This indicated that the seismic shear force on the ground floor was highly concentrated in axis Ⓒ. This caused premature failure of its vertical load-bearing components, triggering a shear concentration–brittle failure chain reaction. The shear force curves, peak values, and ratios under other conditions are detailed in Figure 13 and

Table 5. Shear force of each axis of Model PT under different conditions.

Condition/g		Shear Force/kN
		Ⓐ	Ⓑ	Ⓒ
SC.T2271	0.1	219.25	174.69	1393.50
	0.2	457.79	335.10	2269.80
	0.4	791.45	538.39	3073.30
51LDJ	0.1	169.51	136.55	1092.00
	0.2	365.98	285.83	1975.00
	0.4	1054.5	656.85	2979.90
51LDL	0.1	179.37	146.44	1134.80
	0.2	368.23	281.56	1968.20
	0.4	−751.33	−554.42	−2980.50

.

4. Establishment of a Collapse-Resistant Model Based on Shear Force Balance

4.1. Model Scheme Validation

According to the deformation saturation theory [49], an optimal load-bearing state is achieved when shear forces are uniformly distributed across all structural axes. Collapse-resistant retrofitting schemes were developed to achieve this equilibrium. As illustrated in Figure 14, to control the experimental variables during model calibration, the cross-sectional thickness of the frame columns in axis Ⓐ was maintained at 400 mm while exclusively adjusting their width. For example, in scheme S1, the frame columns in axis Ⓐ were widened by 100 mm on each side in the x-direction, expanding them from 400 mm to 600 mm. The pushover method was employed to identify the optimal retrofitting scheme, with detailed parameters provided in

Table 6. Collapse-resistant design schemes for No. 7, Group 1, Detuo Town.

Type	Width-Increase Length
Length	Prototype	100	200	300	400	500
Scheme	PT	S1	S2	S3	S4	S5

for the prototype structure.

Figure 15 presents the capacity curves of axis Ⓒ and axis Ⓐ with varying sectional widths. Systematic enhancement of the frame column width produced measurable improvements in the seismic performance of axis Ⓐ. Notably, intersection points between the retrofitted axis Ⓐ and axis Ⓒ systems exhibited characteristic strength enhancement–deformation reduction behavior. Comparative analysis of S1–S5 revealed that scheme S4 achieved the most effective equilibrium between structural capacities and demands (force-displacement response of axis Ⓒ), thereby fully harmonizing the seismic performance of both axes. This synergistic utilization of both axial systems’ seismic capacities justified the selection of scheme S4 for constructing the collapse-resistant numerical model (Model RE). Model RE differed from Model PT solely in the modified axis Ⓐ parameters, with all other structural attributes preserved. Identical seismic loading protocols were applied to both models to facilitate comparative response analysis.

4.2. Damage State of Model RE

Figure 16 presents the damage state of Model RE subjected to seismic excitation from station SC.T2271 (PGA = 0.4 g). Consistent with Model PT, damage localization occurred primarily on the ground floor. Section enlargement of axis Ⓐ intensified shear-dominated failure modes, as evidenced by numerical simulations. The damage progression in axis Ⓐ was governed by both rebar strain and shear hinge formation. At low-intensity ground acceleration, the y-direction components remained in the elastic regime, while only a portion of axis Ⓐ components reached 25% of their performance points. When ground acceleration peaked, the strain in the y-direction components increased, particularly in axis ⑥, exceeding 50% of the design capacity. In the x-direction, all the primary load-bearing components in axes Ⓐ and Ⓑ reached 50% of their performance limits, with some components approaching 75%.

4.3. Displacement Response

4.3.1. Inter-Story Drift Ratio

Employing the SC.T2271 seismic records, Figure 17 illustrates IDR distributions across five stories of Model RE. The synchronous vibration patterns across all levels confirmed the first-mode dominated response. Maximum IDR manifested on the ground floor (0.0003), attenuating to 0.0002 on the second/third floors and 0.0001 on upper levels. With increasing PGA, the IDR of the ground floor escalated to 0.0007, while the maximum on the second floor and above only increased by 0.0002. This disproportionate deformation pattern persisted under PGA = 0.4 g, confirming that scheme S4 preserved the original weak-story mechanism. Consequently, upper stories maintained rigid-body behavior assumptions. The detailed IDR metrics under other seismic conditions are compiled in

Table 7. IDR of Model RE under different conditions (%).

Condition/g		Floor
		F1	F2	F3	F4	F5
SC.T2271	0.1	0.03	0.02	0.02	0.01	0.01
	0.2	0.07	0.05	0.04	0.03	0.02
	0.4	0.15	0.07	0.05	0.03	0.02
51LDJ	0.1	0.03	0.02	0.01	0.01	0.01
	0.2	0.04	0.03	0.02	0.02	0.01
	0.4	0.11	0.05	0.04	0.03	0.02
51LDL	0.1	0.03	0.03	0.02	0.02	0.01
	0.2	0.07	0.05	0.04	0.03	0.02
	0.4	0.08	0.05	0.03	0.02	0.17

.

4.3.2. Lateral Displacement

From the prototype and numerical models, the deformation was most pronounced on the ground floor. Consequently, second-floor displacements served as reliable indicators of the global structural response. Figure 18 shows the x- and y-direction displacements for Model RE under SC.T2271 excitation. At a PGA of 0.1 g, a synchronous in-phase response was observed between axes Ⓐ and Ⓒ in the x-direction, yielding peak displacements of 1.05 mm and 0.96 mm, respectively. Corresponding values for axes ① and ⑥ reached 0.85 mm and 0.92 mm. The torsional effect coefficient was 0.04, confirming the translational dominance. At a PGA of 0.2 g, displacements in the x-direction for axes Ⓐ and Ⓒ amplified to 2.57 mm and 2.32 mm, while y-direction responses at axes ① and ⑥ measured 1.40 mm and 1.62 mm. The torsional effect coefficient increased marginally to 0.07. At a PGA of 0.4 g, displacement amplitudes exhibited proportional growth to 5.64 mm and 5.42 mm in the x-direction, with y-direction values reaching 2.13 mm and 3.54 mm, while the torsional effect coefficient was 0.25. Despite a 250% increase in the PGA from 0.1 g to 0.4 g, the translational response remained dominant. Comprehensive displacement data across all loading protocols are compiled in

Table 8. Displacement of Model RE under different conditions.

Condition/g		y/mm		x/mm		Ratio	Torsion Effect Coefficient
		①	⑥	Ⓐ	Ⓒ
SC.T2271	0.1	0.85	0.92	1.05	0.96	1.00:1.08:1.24:1.13	0.04
	0.2	−1.4	−1.62	2.57	2.32	1.00:1.16:2.84:1.66	0.07
	0.4	2.13	3.54	−5.64	−5.42	1.00:1.66:2.65:2.54	0.25
51LDJ	0.1	0.67	0.82	0.58	0.63	1.00:1.22:0.87:0.94	0.10
	0.2	1.3	1.74	1.3	1.38	1.00:1.34:1.00:1.06	0.14
	0.4	−3.48	5.32	3.82	4.25	1.00:1.53:1.10:1.22	0.21
51LDL	0.1	0.7	0.79	−1.25	−1.19	1.00:1.13:1.79:1.70	0.06
	0.2	1.45	1.66	−2.76	−2.58	1.00:1.14:1.90:1.78	0.07
	0.4	3.6	4.32	−6.39	−6.38	1.00:1.20:1.78:1.77	0.09

.

4.4. Seismic Shear Force Response

Figure 19 shows the shear force responses of Model RE under SC.T2271 excitation. At a PGA of 0.1 g, all axes exhibited almost identical vibration frequencies without an obvious phase difference. Maximum shear values reached 1051.40 kN (axis Ⓐ) and 843.32 kN (axis Ⓒ), contrasted with 104.19 kN at axis Ⓑ. When the PGA increased to 0.2 g, the x-direction shear forces of Model RE were 1646.50 kN, 216.06 kN, and 2027.50 kN. It could be observed that the shear forces of axis Ⓐ and axis Ⓒ were similar, and the shear forces of each axis increased with the increase in PGA, with the increase factor being approximately the same as the increase factor of the ground acceleration. Moreover, the shear force vibration waveforms in Figure 19a,b were basically the same, indicating that Model RE was still in the elastic stage overall. When the PGA reached 0.4 g, shear forces reached 2915.20 kN, 574.17 kN, and 3004.10 kN for the three axes. Comparative analysis revealed that axes Ⓐ and Ⓒ shared approximately equal shear loads, validating the design intent of scheme S4. Maximum shear force responses under additional loading scenarios are documented in

Table 9. Shear force of each axis of Model RE under different conditions.

Condition/g		Shear Force/kN
		Ⓐ	Ⓑ	Ⓒ
SC.T2271	0.1	−1051.40	−104.19	−843.32
	0.2	1646.50	261.06	2027.50
	0.4	−2915.20	−574.17	−3004.10
51LDJ	0.1	1522.00	146.60	1300.30
	0.2	1521.50	147.32	1302.30
	0.4	2649.10	426.14	2649.00
51LDL	0.1	−1168.20	−137.06	−1115.50
	0.2	−1977.10	−288.08	−2118.70
	0.4	−3151.60	−659.24	−3288.30

.

5. Comparison of the Model Seismic Response

To evaluate the efficacy of scheme S4, displacement and shear force responses were compared between Models PT and RE, with data extracted from the top node of the ground-floor component A1.

5.1. Displacement Comparison

Figure 20 contrasts the maximum displacements in the x-direction of the two models. At a PGA of 0.1 g, the maximum displacement of Model PT was 1.78 mm while that of Model RE was 0.96 mm, approximately half Model PT’s response. At a PGA of 0.2 g, the displacement of Model PT increased to 3.99 mm, whereas Model RE exhibited a displacement of 2.32 mm, a 41.8% reduction. At a PGA of 0.4 g, Model PT’s displacement peaked at 16.66 mm compared to Model RE’s 5.42 mm, a 67.5% decrease. These results demonstrated that scheme S4 significantly reduced lateral deformation, particularly under major seismic events. Detailed displacement responses under additional conditions are provided in

Table 10. Maximum displacement under different conditions (mm).

Condition	SC.T2271		51LDJ		51LDL
PGA/g	PT	RE	PT	RE	PT	RE
0.1	1.78	0.96	1.39	0.63	1.45	1.19
0.2	3.99	2.32	3.13	1.38	3.12	2.58
0.4	16.66	5.42	9.78	4.25	6.71	6.38

.

5.2. Shear Force Comparison

Figure 21 illustrates the shear force ratio of Models PT and RE based on SC.T2271 data. At a PGA of 0.1 g, axis Ⓒ of Model PT sustained the largest proportion of the total shear force (78%), with the remaining axes contributing only 12% and 10%, respectively. In contrast, Model RE exhibited a redistributed pattern: axis Ⓐ’s contribution increased from 12% to 53%, while axis Ⓒ’s decreased from 78% to 42%. The shear force ratio between axes Ⓐ and Ⓒ transitioned from 1:6.5 to 1.3:1, indicating a more uniform distribution. This transformed the phenomenon of concentrated seismic shear force on one axis into a more uniform contribution of shear force between axes. Comparable trends were observed at 0.2 g and 0.4 g, where Model RE maintained a more homogeneous shear distribution, suggesting enhanced seismic resistance. The complete shear force distribution data under varied conditions are presented in

Table 11. Maximum shear ratio of each axis under different conditions (%).

Model	Axis	PGA
		SC.T2271			51LDJ			51LDL
		0.1 g	0.2 g	0.4 g	0.1 g	0.2 g	0.4 g	0.1 g	0.2 g	0.4 g
PT	Ⓐ	12	15	18	12	14	22	12	14	17
	Ⓑ	10	11	12	10	11	14	10	11	13
	Ⓒ	78	74	70	78	75	64	78	75	70
RE	Ⓐ	53	42	45	51	51	46	48	45	45
	Ⓑ	5	6	9	5	5	8	6	7	9
	Ⓒ	42	52	46	44	44	46	46	48	46

.

Due to the negligible shear force contributions and damage in axis Ⓑ in both models, this axis was excluded from further analysis. Using SC.T2271 data, the maximum displacements and corresponding shear forces of axes Ⓐ and Ⓒ under different PGAs were extracted and marked on capacity curves. In Model PT, the ultimate displacement of axis Ⓐ was when its bearing capacity dropped to 85%, while axis Ⓒ in Model PT and axis Ⓐ in Model RE reached their ultimate displacements at maximum shear capacity.

As shown in Figure 22, the minimum ultimate displacement in both models was in axis Ⓒ, making it the collapse trigger point. Under SLE conditions, the maximum displacements recorded in axes Ⓐ and Ⓒ in both models remained minimal (2.05 mm and 1.78 mm for Model PT; 1.05 mm and 0.96 mm for Model RE), indicating both models remained in the elastic stage. When subjected to DBE conditions, Model PT exhibited plastic displacements of 4.38 mm and 3.99 mm, whereas Model RE maintained elastic behavior (2.57 mm and 2.32 mm) with mitigated torsional effect. Under MCE conditions, Model PT’s displacement in axis Ⓒ (16.66 mm) surpassed its ultimate displacement, demonstrating inadequate shear resistance and catastrophic failure mode. This conclusion was substantiated by (i) the localized damage patterns in Figure 10, (ii) abrupt frequency shifts in displacement time histories (Figure 12), and (iii) permanent residual deformation accumulation. In contrast, Model RE’s displacements (5.64 mm and 5.42 mm) remained below the critical collapse trigger value (defined as displacement at peak shear force in axis Ⓒ), indicating sustained structural integrity and enhanced PGA tolerance.

Seismic force distribution analyses [7,49] revealed that the load-resisting elements partitioned the inertial forces proportionally to their stiffness contributions. Comparing Figure 22, Model PT’s failure under MCE conditions was due to the suboptimal global bearing capacity and axis Ⓒ bearing most of the seismic shear, leading to large lateral displacements and severe shear failure. Conversely, Model RE achieved higher total resistance and a more uniform distribution of seismic shear between axes Ⓐ and Ⓒ. Notably, the MCE condition-induced deformations in Model RE mirrored the DBE condition responses in Model PT yet remained below axis Ⓒ’s performance limit, confirming robust performance margins.

Due to the presence of perforated walls, this type of hybrid frame–masonry structure could not effectively dissipate seismic energy through ductility. However, rational modification of component cross-sectional dimensions enabled the transformation of original components from a low load-carrying capacity with large ultimate deformation to a high load-carrying capacity with reduced ultimate deformation. By improving the seismic bearing capacity and lateral stiffness, the total structural resistance was elevated while maximum deformation was mitigated, ultimately converting vulnerable structures into structures with improved seismic performance.

6. Conclusions

No. 7, Group 1, Detuo Town, struck by a 6.3 magnitude earthquake in Luding, was selected for numerical analysis, revealing its seismic performance. Based on the deformation saturation theory, this article proposes a seismic design concept to balance shear forces among different components, enhancing the seismic performance of frame–masonry hybrid structures. The validity of this concept is verified using the same modeling method. The following conclusions are drawn:

(1)

In the damaged prototype, piers between two windows in axis Ⓒ suffered shear failure, while cracked concrete and significant x-direction residual deformation were observed at the column ends of the other two axes. A comparative analysis of the damage severity and residual displacements in two horizontal directions revealed that the building’s movement during the earthquake involved some torsion, with the most significant displacement in the x-direction. This conclusion is corroborated by numerical simulations, highlighting the need to balance seismic shear forces in the x-direction.

(2)

The damage in the prototype building is primarily concentrated on the ground floor and intensifies progressively with increasing peak ground acceleration. Comparative damage between the ground-floor components of the numerical model and the actual building reveals a close match between the numerical result and the observed damage. In terms of seismic bearing capacity, axis Ⓒ exhibits disproportionate shear force concentration (around 78% of the total), indicating that the ground-floor seismic shear force is concentrated in axis Ⓒ. This causes it to reach its ultimate displacement first, resulting in severe shear failure. This shear concentration–brittle failure chain sequentially propagates to other x-direction components, culminating in x-direction collapse.

(3)

The quantitative assessment of Model PT and Model RE under varying seismic intensities demonstrates significant performance improvements. Under SLE conditions, the shear force ratio between axes Ⓐ and Ⓒ decreases from 1:6.5 to 1.3:1, and the maximum displacement is reduced from 1.78 mm to 0.96 mm, achieving a 46% reduction. These results indicate a reduction in ground-floor lateral displacement and shear force concentration in axis Ⓒ, effectively enhancing the building’s collapse resistance. The deformation saturation theory can guide the transformation of buildings from a low bearing capacity with large ultimate deformation to a high bearing capacity with low ultimate deformation, thereby improving the collapse resistance of the prototype building.