Seismic Performance and Flexural Capacity Analysis of Embedded Steel Plate Composite Shear Wall Structure with Fiber-Reinforced Concrete in the Plastic Hinge Zone

Abstract

Due to its high axial bearing capacity and good ductility, the embedded steel plate composite shear wall structure has become one of the most widely used lateral force-resisting structural members in building construction. However, bending failure is prone to occur during strong earthquakes, and the single energy dissipation mechanism of the plastic hinge zone at the bottom leads to the concentration of local wall damage. To improve the embedded steel plate composite shear wall structure, the plastic hinge zone of the composite shear wall is replaced by fiber-reinforced concrete (FRC) and analyzed by ABAQUS finite element simulation analysis. Firstly, the structural model of the embedded steel plate composite shear wall structure with FRC in the plastic hinge zone is established and the accuracy of the model is verified. Secondly, the effects of steel ratio, longitudinal reinforcement ratio, and FRC strength on the bearing capacity of composite shear walls are analyzed by numerical simulation. Finally, a method for calculating the embedded steel plate composite shear wall structure with FRC in the plastic hinge zone is proposed. It is shown that the displacement and load curves and failure modes of the model are basically consistent with the experimental results, and the model has high accuracy. The axial compression ratio and FRC strength have a great influence on the bearing capacity of composite shear walls. The calculation formula of the normal section bending capacity of the embedded steel plate composite shear wall structure with FRC in the plastic hinge zone is proposed. The calculated values of the bending capacity are in good agreement with the simulated values, which can provide a reference for its engineering application.

1. Introduction

The embedded steel plate composite shear wall structure exhibits high axial load-bearing capacity, making it suitable for high-rise buildings subjected to significant axial pressure on the lower floors. However, it is susceptible to buckling and failure during severe seismic events, and the presence of a single energy dissipation mechanism in the plastic hinge region at the base results in localized damage, which hampers post-disaster repairs and reinforcement [1,2,3,4,5]. A small shear span ratio significantly diminishes its seismic performance and deformation capacity, and consequently, the earthquake-induced damage to the structure is more severe. Fiber-reinforced concrete (FRC) [6,7,8,9] is a composite material consisting of cement mortar or plain concrete mixed with a suitable amount of high-toughness reinforcing fibers, exhibiting high ductility under tensile and shear stresses, along with multi-fine cracking and pronounced strain hardening characteristics. Additionally, it demonstrates effective crack width control as deformation increases. Simultaneously, the shear capacity, deformation, and energy dissipation abilities, as well as damage tolerance of flexural and shear members can be enhanced, leading to significant improvements in the seismic performance of concrete structures [10,11].

Professor Li [12] from the University of Michigan pioneered experimental research and theoretical analysis on engineered cementitious composites (ECCs), utilizing micromechanical design methods. The matrix is blended with short fibers according to two strain-hardening criteria (initial cracking criterion and steady-state cracking criterion), with the fiber volume fraction not exceeding 2.5% and the fibers being randomly distributed within the matrix. Through uniaxial tensile testing, the matrix develops numerous micro-cracks along the tensile direction, with an ultimate tensile strain exceeding 3%, which is approximately 100 to 300 times greater than that of ordinary concrete. Building on this, Kabele [13] developed a constitutive model for the material, enabling finite element simulations of its structural properties, which facilitate selective applications in critical structural components. Shilang Xu [14] conducted systematic research on the fundamental mechanical and material properties of ultra-high toughness cementitious composites (UHTCCs). The research indicates that under uniaxial tensile stress, numerous micro-cracks are generated along the tensile direction due to the bridging effect of the fibers, with the ultimate tensile strain reaching or exceeding 2.5%, which is comparable to the tensile properties of reinforcing steel. To avoid confusion, and given that FRC shares the tensile strain hardening, small crack width, and high ductility characteristics of ECCs, both ECCs and UHTCCs are collectively referred to as FRC in this paper.

The incorporation of fiber-reinforced concrete into the potential plastic hinge regions of sheet-concrete composite beams and integrated sheet-concrete composite shear walls enhances their load-bearing capacity, deformation capacity, energy dissipation, and damage resistance, and facilitates rapid post-earthquake repairs. Quang [15] conducted experiments on six high-performance fiber-reinforced cement-based composite (HPFRCC) column specimens and found that the HPFRCC improved the ductility of the columns. Parra Montesinos [16] conducted tests on HPFRCC shear walls with shear span ratios of 3.45 and 3.7 to investigate the effect of fibers on the nonlinear behavior of the shear walls. The results indicate that, for walls with a fiber volume fraction of 2.0%, only minor damage occurs in the concrete matrix before the longitudinal reinforcement in the boundary elements fractures. Dazio [17] conducted experimental and numerical studies on the hysteretic behavior of three types of hybrid fiber-reinforced concrete structural walls. The results show that the hybrid fiber-reinforced concrete structural system exhibits significant nonlinear deformation capacity. Ying Zhou et al. [18] conducted tests on shear walls in the plastic hinge region using hybrid fiber-reinforced concrete or ECC fiber-reinforced concrete and found that the addition of fibers improved the deformation capacity of the plastic hinge region, potentially replacing the horizontal distribution reinforcement to some extent. The research group led by Xingwen Liang, under the guidance of ECC design theory, successfully developed a high-strength, high-toughness ductile fiber-reinforced concrete (DFRC) [19] through extensive mechanical performance tests, and applied it to the plastic hinge region at the base of shear walls to improve the seismic and shear resistance of high-performance concrete shear walls, thereby mitigating the extent of seismic damage to the structure. Abouzar Jafariton [20] studied the plastic hinge zone of the shear wall. Taking the axial compression ratio and coupling beam ratio of a coupling beam as variables, a parameter analysis was carried out, and the numerical simulation was established based on the laboratory data. The research shows that the coupled shear wall has a relatively high capacity and curvature compared with the hybrid coupled shear wall. In addition, increasing the axial compression ratio will change the behavior of the wall into a columnar component, thereby reducing the ultimate displacement, ductility, curvature, and plastic hinge length of the wall. Increasing the coupling beam ratio of the coupled shear wall will increase the bearing capacity of the wall and the risk of sudden shear failure, but will reduce the ductility, ultimate displacement, and plastic hinge length of the wall.

To investigate the failure mechanism and seismic performance of steel plate-reinforced concrete shear walls with FRC in the plastic hinge zone, the internal forces of these shear walls and those incorporating fiber-reinforced concrete under various parameters were simulated and analyzed. To verify the model’s accuracy, a finite element simulation of the test specimens was first conducted, and the skeleton curves obtained from the tests and simulations were compared to confirm the model’s validity. On the basis of the successful verification of the model, the finite element simulation analysis of the model was carried out. The effects of the steel and steel plate ratio, longitudinal reinforcement ratio, fiber-reinforced concrete strength, and axial compression ratio on the seismic performance of the steel plate composite shear wall structure with FRC in the plastic hinge zone were studied and analyzed. The optimal range of composite shear wall design was obtained by analyzing the simulation results. Finally, based on existing standards and the relevant literature, a calculation formula for the bending capacity of the cross-section in the plastic hinge region of a steel plate shear wall structure with FRC is proposed and verified, providing a valuable reference for engineering applications.

2. Verification of the ABAQUS Model

2.1. Material Constitutive Relation and Yield Condition

2.1.1. Constitutive Relation of Concrete

(1)

Constitutive relation of ordinary concrete

The numerical simulations in this study were conducted using ABAQUS software (DS SIMULIA Suite 2021 HF3). The constitutive model for normal concrete used the Concrete Plastic Damage Model, as shown in Figure 1.

The degradation of concrete stiffness increases with plastic strain, and this degradation can be represented by the damage variable d_t for uniaxial tension and d_c for uniaxial compression [21,22].

$$d_{\mathrm{t}}=d_{\mathrm{t}}({\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{pl}},\theta ,f_i)$$

(1)

$$d_{\mathrm{c}}=d_{\mathrm{c}}({\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{pl}},\theta ,f_i)$$

(2)

In the equation, ${\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{pl}}$ and ${\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{pl}}$ represent the equivalent compressive and tensile plastic strains of concrete, respectively; ${\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{in}}$ and ${\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{ck}}$ represent the compressive crushing strain and cracking strain (unloading strain when undamaged), respectively; d_c and d_t represent the compressive and tensile damage parameters of concrete, respectively.

① Uniaxial compressive stress–strain relationship

The uniaxial compressive plastic damage model curve in ABAQUS consists of a linear segment, a curved segment, and a descending segment.

(a) Linear segment

This stage begins at the origin and represents the elastic phase, with the primary objective being the determination of the elastic failure point (ε_c,e0, σ_c,e0). Typically, the stress point is taken as 1/3 of f_c. This allows for the calculation of the initial tangent elastic modulus, E₀, during the elastic phase.

(b) Curved segment

Through comparative analysis, the curved segment is represented by the Saenz expression [23] as follows:

$$\sigma ={\displaystyle \frac{E_0\epsilon }{1+\left(\beta +\frac{E_0}{E_{\mathrm{S}}}-2\right)\left(\frac{\epsilon }{{\epsilon }_0}\right)+(1-2\beta ){\left(\frac{\epsilon }{{\epsilon }_0}\right)}^2+\beta {\left(\frac{\epsilon }{{\epsilon }_0}\right)}^3}}$$

(3)

In the equation, ε₀, Ε_u represent the strains corresponding to the peak stress σ₀ and the ultimate stress σ_u, respectively; E_s is the secant modulus at the maximum stress, E_s = σ₀/ε₀; and β is the corresponding coefficient, $\beta ={\displaystyle \frac{E_0/E_{\mathrm{s}}({\sigma }_0/{\sigma }_{\mathrm{u}}-1)}{{({\epsilon }_{\mathrm{u}}/{\epsilon }_0-1)}^2}}-{\displaystyle \frac{{\epsilon }_0}{{\epsilon }_{\mathrm{u}}}}$.

The compressive damage evolution equation for concrete is as follows:

$$D=1-\sqrt{\sigma /(E_0\epsilon )}=1-1/\sqrt{1+\left(\alpha +{\displaystyle \frac{E_0}{E_{\mathrm{S}}}}-2\right)\left({\displaystyle \frac{\epsilon }{{\epsilon }_0}}\right)+(1-2\alpha ){\left({\displaystyle \frac{\epsilon }{{\epsilon }_0}}\right)}^2+\alpha {\left({\displaystyle \frac{\epsilon }{{\epsilon }_0}}\right)}^3}$$

(4)

In the equation, D represents the damage variable.

② Uniaxial tensile stress–strain relationship

When the strain of concrete reaches the peak tensile strain, cracking begins to occur. To more realistically and effectively simulate the mechanical behavior of shear walls made of ordinary concrete, the stress–crack width relationship method is employed to define the uniaxial tensile behavior of concrete. The tensile softening modulus of concrete can be calculated using Equation (5) [24]:

$$E_{\mathrm{ts}}=f_{\mathrm{t}}/{\epsilon }_{\mathrm{tu}}=0.5f_{\mathrm{t}}^2{l}_{\mathrm{c}}/G_{\mathrm{f}}$$

(5)

In the equation, f_t represents the uniaxial tensile strength of concrete, l_c is the characteristic length of the concrete element, and ω denotes the crack width. The magnitude of the fracture energy can be calculated from the area enclosed by the stress–crack width curve and the horizontal axis during the tensile loading of concrete. In this study, the calculation follows the recommendations of the European Code CEB-FIP MC90.

$$G_{\mathrm{f}}=\alpha {\left({\displaystyle \raisebox{1ex}{$f_{\mathrm{c}}^{\prime }$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}\right)}^{0.7}\times {10}^{-3}(\mathrm{N}/\mathrm{mm})$$

(6)

In the equation, $\alpha =1.25d_{\max }+10$, $d_{\max }$ represents the particle size of the coarse aggregate, and f_ck denotes the compressive strength of the concrete. The peak tensile stress of the concrete with σ_t0 is calculated using Equation (7) [25].

$${\sigma }_{\mathrm{t}0}=0.26\times {\left(1.5f_{\mathrm{ck}}\right)}^{2/3}$$

(7)

When the initial elastic modulus of concrete is known, the uniaxial tensile damage stress–strain relationship is calculated using the following equation:

$${\sigma }_{\mathrm{c}}=(1-d_{\mathrm{c}})E_0({\epsilon }_{\mathrm{c}}-{\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{pl}})$$

(8)

$${\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{pl}}={\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{in}}-{\displaystyle \frac{d_{\mathrm{c}}}{1-d_{\mathrm{c}}}}{\displaystyle \frac{{\sigma }_{\mathrm{c}}}{E_0}}$$

(9)

(2)

The constitutive relationship of FRC

The compressive stress–strain relationship of FRC differs from that of ordinary concrete. In this study, the constitutive model proposed in References [26,27] is adopted, and its stress–strain relationship is calculated using Equation (10):

$$y=\left\{\begin{array}{ll}{\displaystyle \frac{Ax-x^2}{1+(A-2)x}}& \hfill 0\le x\le 1\\ {\displaystyle \frac{A_{1}x}{1+(A_1-2)x+x^2}}& \hfill x\ge 1\end{array}\right.$$

(10)

In the equation, $A=E_0/E_{\mathrm{c}}$, $x=\epsilon /{\epsilon }_{\mathrm{c}}$, $y=\sigma /{\sigma }_{\mathrm{c}}$, $A_1=0.001+1.672{\sigma }_3/f^{\prime }$, E₀ and E_c represent the initial tangent modulus and the secant modulus at the peak stress point, respectively. ε_c and σ_c are the strain and stress at the peak point, respectively. ${\sigma }_3/f_{\mathrm{c}}^{\prime }$ denotes the lateral microcompression, and $f_{\mathrm{c}}^{\prime }$ is the compressive strength of the fiber-reinforced concrete prism. The ultimate compressive strain of fiber-reinforced concrete, ε_1u, can be taken as 0.65%.

FRC exhibits fine micro-cracking and strain-hardening behavior under tensile loading. A bilinear model is typically used as a simplified representation of this behavior, and the specific relationship is given by Equation (11):

$$\sigma =\left\{\begin{array}{ll}{E}_{\mathrm{c}}\epsilon & \epsilon \le {\displaystyle \frac{{\sigma }_{\mathrm{ss}}}{E_{\mathrm{c}}}}\\ {\sigma }_{\mathrm{i}}+E_{\mathrm{ie}}\epsilon & \epsilon >{\displaystyle \frac{{\sigma }_{\mathrm{ss}}}{E_{\mathrm{c}}}}\end{array}\right.$$

(11)

In the equation, $E_{\mathrm{ie}}={\displaystyle \frac{{\sigma }_{\mathrm{tu}}-{\sigma }_{\mathrm{ss}}}{{\sigma }_{\mathrm{tu}}-\raisebox{1ex}{${\sigma }_{\mathrm{ss}}$}\!\left/ \!\raisebox{-1ex}{$E_{\mathrm{c}}$}\right.}}$, ${\sigma }_{\mathrm{i}}={\sigma }_{\mathrm{ss}}\left(1-{\displaystyle \frac{E_{\mathrm{ie}}}{E_{\mathrm{c}}}}\right)$, $E_{\mathrm{c}}=f_{\mathrm{cu}}^{0.596}\times {10}^3\mathrm{MPa}$, σ_ss represents the stable cracking stress, and σ_u and ε_u denote the ultimate tensile stress and ultimate tensile strain, respectively.

2.1.2. Concrete Failure Criteria

The failure criterion of concrete (the strength criterion) primarily refers to the representation of the failure envelope of concrete using an approximate function, which is employed to determine whether the concrete has failed. Currently, various failure criteria are available for concrete, among which the Drucker–Prager criterion, which best describes the plastic damage model in ABAQUS, is the most appropriate criterion.

2.1.3. The Constitutive Relationship of Steel

The stress–strain relationship of steel is represented by a bilinear model [28], as shown in Figure 2. The modulus of elasticity of steel after yielding is taken as 0.01E₀, where E₀ is the initial modulus of elasticity of the reinforcement. In addition, the initial yield stress is obtained from material property tests.

2.1.4. Yield Criterion for Steel

The von Mises yield criterion compares the von Mises stress with the material’s yield strength; if the stress exceeds the yield strength, the material is considered to have reached the yielding state. Steel typically experiences a multiaxial stress state; therefore, the von Mises yield criterion can be used to determine whether the steel has yielded.

2.2. Establishment of the ABAQUS Model

This study comprehensively considers the shear wall with FRC concrete in the plastic hinge zone and the steel plate-reinforced concrete shear wall, using specimens FRCSW-04 from Reference [29] and SW-F from Reference [30], respectively. The models were developed based on identical dimensions, material parameters, and loading conditions, with appropriate component types, boundary conditions, mesh sizes, and element types selected for the ABAQUS finite element model. The geometric model includes several components such as the concrete wall, steel plate, profile steel, and reinforcement. The concrete wall is composed of the foundation beam, wall body, end columns, and top beam (loading beam).

2.2.1. Model Size and Material Parameters

(1)

Size and material parameters of shear wall model with FRC concrete in the plastic hinge zone.

In Reference [29], in the specimen FRCSW-04, the wall section size is b × h = 100 mm × 1000 mm and the height H = 2000 mm. The plastic hinge zone of the specimen is made of FRC concrete, and the rest is made of ordinary concrete. The compressive cracking, axial compressive strength, and tensile peak strength of FRC concrete are 3.96 MPa, 65.2 MPa, and 4.16 MPa, respectively. The cracking strain and ultimate strain are 0.04 and 0.97, respectively. The volume content of PVA is 2.0%, and the axial compressive strength of ordinary concrete is 69.8 MPa.

The longitudinal reinforcement of the specimen is an HRB400 hot-rolled ribbed bar with a diameter of 12 mm and a 6.5 mm round bar with a diameter of 6.5 mm. The yield strength and ultimate strength of the steel are 497.1 MPa and 708.4 MPa, respectively. The horizontal distribution bars of the specimens are smooth round bars with a diameter of 6.5 mm. The arrangement form is 2D6.5@80, and the corresponding reinforcement ratio is 0.825%. The vertical distribution reinforcement of the wall adopts high-strength cold-drawn steel wire with a diameter of 4 mm, and the ultimate strength is 848.2 MPa. The edge constraint components are designed at both ends of the wall, and segmented constraint stirrups are used. High-strength cold-drawn steel wire with a stirrup diameter of 6 mm is used, and the ultimate strength is 696.3 MPa. The spacing of the dense section in the plastic hinge zone is 60 mm, the spacing of the non-dense section is 40 mm, and the stirrup characteristic value λ_v is 0.27.

(2)

Built-in steel plate concrete composite shear wall model size and specimen parameters

The test piece SW-F in Reference [30] is simulated and verified. The design values of the main parameters of the specimen and the material properties are detailed in Reference [30]. The shear wall specimens are all in the same shape, and the steel plates are arranged in a single layer. The yield strength and ultimate strength are 330 MPa and 457 MPa, and the thickness of the steel plate is 5 mm. The steel ratio of the column end steel is 5.7%. The steel bars distributed in the wall are D4@35 and D6@80, respectively. The yield strength and ultimate strength of D4 steel are 576 MPa and 677 MPa, respectively. The yield strength and ultimate strength of D4 steel are 498 MPa and 718 MPa, respectively. The axial compression ratio is 0.30, the section size is b × h = 100 mm × 1000 mm, and the height is H = 2000 mm. The shear span ratio is 2.0.

2.2.2. Element and Section Properties

(1)

Selection of concrete materials

In this study, the concrete materials are modeled as C60 concrete and fiber-reinforced concrete, both of which use the plastic damage model. The viscosity coefficient is set to 0.005, the eccentricity is 0.1, the compressive strength ratio is 1.16, the value of K is set to 2/3, the expansion angle ψ is set to 42°, and the Poisson’s ratio is 0.2 [31]. To minimize the influence of the Poisson effect on the simulation and improve post-processing efficiency, the elastic modulus of the loading beam and foundation beam is increased by a factor of 100 to reduce the deformation caused by the load.

(2)

Steel selection

The materials include rebar, steel plates, and H-section steel. Steel is modeled using a bilinear curve, and the constitutive curve is determined based on the measurements of the material’s elastic modulus, yield strength, and ultimate strength.

In addition, because the plastic hinge area of the specimen should be replaced with FRC concrete, the replacement height is 600 mm, so the wall should be cut, and the overall model of the specimen is shown in Figure 3.

2.2.3. Simulation of Boundary Conditions

In this study, ABAQUS finite element modeling is performed on the shear wall with FRC concrete in the plastic hinge zone and the steel plate-reinforced concrete shear wall. The contact interactions include the following: between the loading beam and the wall, between the FRC in the plastic hinge region and the normal concrete in the wall, between the foundation beam and the wall, between the reinforcement and the concrete, between the embedded steel and the concrete, and between the steel sections and the concrete. The boundary conditions include the following: the fixed constraint of the foundation beam and the constraint at the point of application of the horizontal loading force on the loading beam.

In actual components, studs are set on both sides of the steel plate to connect with concrete. This shear connection can better ensure the integrity of the structure. Therefore, an embedding method was used to embed steel plates and steel bars into concrete when defining constraints in the model. In the vertical jack test, devices such as slide plates and spherical hinges were set on the vertical jack to eliminate displacement constraints, so the top beam was considered not to be constrained by displacement when modeling. The lower surface and side of the ground beam are coupled with the reference point, and the fixed support is set to simulate the constraint effect of the anchor and the bottom surface on the structure in the actual test.

2.2.4. Simulation of Loading and Loading Procedures

This paper focuses on the bearing capacity and ductility of composite shear wall specimens. Therefore, a one-way horizontal displacement loading method was used for both simulated components to obtain the load–displacement curve, failure mode, and failure mode of the composite shear wall under a single push. During the unidirectional horizontal displacement loading simulation, a two-step loading procedure was employed. In the first step, a constant vertical uniformly distributed load is applied to the top of the loading beam. The magnitude of the uniformly distributed load is determined based on the average distribution of the axial force on the loading beam under the designed axial compression ratio. In the second step, a reference point is placed outside the center of the side of the loading beam, and the reference point is coupled with the side of the loading beam using a “Coupling” constraint. A unidirectional horizontal displacement (V) is then applied at this loading point.

2.2.5. Mesh Discretization

Since concrete in the model is a nonlinear material with complex constitutive behavior, the mesh size may influence the results. Thus, the impact of different mesh sizes for concrete on the computational results was evaluated. Considering both computational efficiency and accuracy, the mesh size for concrete was set to 50 mm, while the mesh size for rebar, steel plates, and I-beams was set to 40 mm. The meshing of each component and the overall model are shown in Figure 4.

2.3. Comparison Between Computational and Experimental Results

Figure 5a compares the experimental results for specimen FRCSW-04 with the skeleton curve of the shear wall with FRC concrete in the plastic hinge zone, as obtained from the ABAQUS simulation. Figure 5b compares the experimental results for specimen SW-F with the skeleton curve of the steel plate-reinforced concrete shear wall, as obtained from the simulation. As shown in the figures, the experimental and simulated curves for both specimens exhibit similar trends. The errors in the positive loading capacity are within 10%. The stiffness values show close agreement. However, the simulated loading capacity curve for the embedded steel plate composite shear wall is higher than that of the experimental curve.

The primary reasons for the overestimation of the loading capacity in the simulation are as follows:

(1)

Idealized boundary conditions were applied in the simulation. For example, the foundation beam was fully constrained in the simulation, whereas in the experiment, it was fixed with compression beams and shear bolts, which may loosen during cyclic loading.

(2)

The materials used in the experimental specimens (concrete and steel) inevitably contain defects, while in the simulation, they were treated as homogeneous, which does not accurately represent the material strengths observed in the experiment.

(3)

In the simulation, only monotonic horizontal displacement was applied, whereas the experiment involved low-cycle reversal loading with varying levels of damage at each stage. Consequently, the damage sustained by the specimen in the simulation and experiment at the same displacement differed.

(4)

Some relative slip between the reinforcement, steel plates, and concrete within the specimen is inevitable. However, in the finite element simulation, the interaction between steel and concrete was assumed to be embedded. Although shear bolts were placed at the top of the steel plates, the effect of this slip on the specimen’s strength cannot be neglected.

These factors contribute to the discrepancy between the simulated and experimental curves.

Figure 6 shows the comparison between the test and the simulated final shape of the two specimens. It can be seen from the comparison that (1) the DAMAGEC (concrete compression damage) cloud diagram of the specimen is basically consistent with the final shape comparison diagram of the test, and the concrete at the corner is seriously damaged. (2) For the shear wall structure with FRC in the plastic hinge zone, it can be seen from the simulation failure process that the initial cracks of the specimen appear at the bottom of the wall; with the increase in horizontal thrust, the shear oblique cracks appear earlier on the wall. At the same time, due to the higher FRC area, the distribution range of bending cracks on the wall is larger, the inclination angle of shear cracks decreases, and the formation of cross-oblique cracks occurs later. When the final failure occurs, the buckling deformation of the outermost steel bar is serious, which is basically consistent with the phenomenon of the experimental specimen. (3) For the built-in steel plate shear wall structure, the failure process is simulated during the observation period. At the initial stage of loading, horizontal tensile cracks begin to appear on the left side of the bottom of the wall. With the continuous increase in horizontal thrust, the cracks develop from bottom to top, and most of them are horizontal tension cracks. By the middle of the loading, the concrete area in the middle of the wall begins to show compressive oblique cracks; in the later stage of loading, the steel bar on the right side of the wall buckles, the concrete bulges, and damage occurs in the range of 500 mm on the right side; and when the load is loaded to the peak, a large range of tensile damage occurs at the bottom of the left side of the wall, and compressive damage occurs within the range of 500 mm at the bottom of the right side. Large blocks of concrete are peeled off and detached, and thus the wall is out of work and the loading simulation is completed. The simulated failure process is basically consistent with the experimental phenomenon. (4) The error of the peak bearing capacity corresponding to the positive peak point of each specimen is controlled within 10%, which indicates that the simulation effect is more accurate and the model has certain rationality.

3. Different Parameter Analysis of the Embedded Steel Plate Composite Shear Wall Structure with FRC in the Plastic Hinge Zone

3.1. Simulation Specimen Scheme

The embedded steel plate composite shear wall structure has large lateral stiffness and small lateral displacement, which is the main lateral resistance component of a high-rise structure. However, under the action of an earthquake, the damage at the bottom of the wall is the most serious. At this time, large irreversible plastic deformation will occur at the bottom of the shear wall, which is the plastic hinge zone of the shear wall. In order to enhance the ductility and bearing capacity of the composite shear wall and reduce the damage of the wall under earthquake, the plastic hinge zone at the bottom of the composite shear wall is replaced by FRC concrete, and the ABAQUS finite element simulation is established. A total of 19 specimens were simulated. The wall size and loading system of each specimen were the same as those of the above-simulated specimens. The effects of steel and steel plate ratio, longitudinal reinforcement ratio, FRC concrete strength, and axial compression ratio on the seismic performance of composite shear walls were comprehensively considered. The specific parameters of each specimen are shown in

Table 1. Simulated specimen parameters.

Specimen Number	μ	fc/MPa	Lp/mm	h × l × w/mm	tw/mm	ρ1/%	ρ2/%	ρ3/%	ρ4/%
FSW-2	0.22	C50	400	2000 × 1000 ×100	6	0.33	3.96	1.03	1.08
FSW-1	0.22	C60	400	2000 × 1000 ×100	6	0.33	3.96	1.03	1.08
FSW-3	0.22	C70	400	2000 × 1000 × 100	6	0.33	3.96	1.03	1.08
FSW-4	0.22	C80	400	2000 × 1000 × 100	6	0.33	3.96	1.03	1.08
FSW-S-1	0.22	C60	400	2000 × 1000 × 100	6	0.60	3.96	1.03	1.08
FSW-S-2	0.22	C60	400	2000 × 1000 × 100	6	0.91	3.96	1.03	1.08
FSW-S-3	0.22	C60	400	2000 × 1000 × 100	6	1.21	3.96	1.03	1.08
FSW-X-1	0.22	C60	400	2000 × 1000 × 100	6	0.33	3.96	1.03	0.84
FSW-X-2	0.22	C60	400	2000 × 1000 × 100	6	0.33	3.96	1.03	1.47
FSW-X-3	0.22	C60	400	2000 × 1000 × 100	6	0.33	3.96	1.03	1.68
FSW-G-1	0.22	C60	400	2000 × 1000 × 100	4	0.33	2.64	1.03	1.08
FSW-G-2	0.22	C60	400	2000 × 1000 × 100	8	0.33	5.28	1.03	1.08
FSW-G-3	0.22	C60	400	2000 × 1000 × 100	9	0.33	5.94	1.03	1.08
FSW-N-1	0.31	C60	400	2000 × 1000 × 100	6	0.33	3.96	1.03	1.08
FSW-N-2	0.4	C60	400	2000 × 1000 × 100	6	0.33	3.96	1.03	1.08
FSW-N-3	0.5	C60	400	2000 × 1000 × 100	6	0.33	3.96	1.03	1.08

Note: μ represents the ratio of axial compression stress to strength; f_c represents the strength of FRC concrete; L_p represents FRC concrete height; t_w represents the thickness of steel plate; ρ₁ represents wall longitudinal reinforcement ratio; ρ₂ represents steel ratio of central steel plate; ρ₃ represents longitudinal reinforcement ratio of concealed column; and ρ₄ represents steel ratio of concealed column steel.

.

3.2. Finite Element Simulation Analysis

3.2.1. Comparative Analysis of Strength of FRC Concrete in the Plastic Hinge Zone

(1)

Comparison of Simulated Failure Modes and Patterns

Figure 7, Figure 8 and Figure 9 present the DAMAGEC (concrete compressive damage), steel plate and profile steel S, von Mises, and reinforcement cage S, von Mises contour plots for specimens FSW-1, FSW-2, and FSW-3. The steel reinforcement ratios for each specimen are 3.01%, 5.28%, and 5.94%, respectively. Based on the ABAQUS simulation of the failure process, taking specimen FSW-1 as an example, horizontal tensile cracks first appeared at the lower-left corner of the wall at the onset of loading. As the horizontal thrust progressively increased, the cracks developed from the bottom upwards, with the majority being horizontal tensile cracks. During the mid-stage of loading, oblique cracks began to appear in the central concrete region of the wall, while no significant cracks were observed at the bottom-right corner, indicating that the bonding effect of FRC concrete effectively inhibited crack development. In the late loading phase, the steel reinforcement on the right side of the wall buckled, the FRC concrete bulged, and damage occurred within a 500 mm range on the right side. When the load reaches its peak value, extensive tension damage occurs at the bottom-left corner of the wall, and compression damage occurs within 500 mm at the bottom-right corner. Concrete chunks peel off and detach, causing the wall to exit its service state, and the loading process concludes. The failure processes of the other specimens were generally consistent with that of specimen FSW-1. However, with increasing concrete strength, the compression damage at the bottom-right corner of the wall gradually diminished. Notably, the FSW-4 concrete underwent sudden crushing, exhibiting clear brittle failure characteristics. The von Mises cloud images revealed buckling at the bottom of the right longitudinal reinforcement in all four specimens. Comparison analysis indicated that the buckling range of the longitudinal reinforcement in FSW-4 was smaller than that in the other specimens, suggesting that as the FRC concrete strength increases, it effectively prevents reinforcement buckling. The von Mises cloud images of the steel plates and profiles showed significant strain damage at the bottom of both sides, which propagated to the middle of the plates. Comparison analysis shows that the range of strain damage in the steel plate gradually decreases with increasing strength, indicating that FRC concrete effectively inhibits internal strain damage in the steel plate as its strength increases.

(2)

Comparison of Load–Displacement Curves

Figure 10, Figure 11 and Figure 12 present the load–displacement curves and the corresponding load at key points for each specimen at different FRC concrete strengths. The load–displacement curves show that the development trends of the four specimens are generally consistent, each consisting of three phases: the elastic phase before cracking, the yield phase after cracking, and the significant descending phase followed by a plateau. Upon comparison, it was found that with increasing FRC concrete strength, the peak load capacity and ultimate load capacity of each specimen gradually increased compared to the others. In contrast to the other specimens, the descending phase of PSW-4 is less pronounced, exhibiting distinct brittle failure characteristics.

The load characteristics at key points of the specimens indicate that, compared to FSW-2, the yield load of FSW-1, FSW-3, and FSW-4 increased by 22.24%, 40.21%, and 30.06%, respectively, while the peak load increased by 13.23%, 36.6%, and 45.95%, respectively. This suggests that the strength of FRC concrete can effectively enhance the load-bearing capacity of the composite shear wall. However, when the strength becomes excessively high, the improvement effect significantly diminishes.

The ductilities of specimens FSW-2, FSW-1, FSW-3, and FSW-4 are 5.02, 5.13, 6.55, and 4.73, respectively. Based on the ductility curves of the specimens, it can be observed that with an increase in the strength of FRC (fiber-reinforced concrete), the ductility of the specimens initially increases and then decreases. This indicates that the toughness and crack suppression capability of FRC concrete significantly improved the ductility of the plastic hinge region, allowing the specimens to maintain a certain level of load-carrying capacity after undergoing large deformations, thereby preventing brittle failure. However, when the strength of FRC concrete is too high, the load-carrying capacity of the specimens becomes excessive, leading to a deterioration in ductility, and ultimately, potential brittle failure. Therefore, in practical engineering applications, it is essential to avoid excessively high FRC strength to prevent brittle failure of composite shear walls.

In summary, in practical engineering, the strength of FRC should not be too high, and its strength should be controlled within the range of C60-C70 (C60-C70 represents the standard value of concrete cube compressive strength 60–70 MPa), so as to ensure the bearing capacity and ductility of composite shear walls and prevent brittle failure.

3.2.2. Comparative Analysis of Steel Ratio of Central Steel Plate

(1)

Comparison of Simulated Failure Modes and Patterns

Figure 13, Figure 14 and Figure 15 show the DAMAGEC (concrete compressive damage), steel plate, profile steel S, von Mises, and reinforcement cage S, von Mises contour plots for specimens FSW-G-1, FSW-G-2, and FSW-G-3. The steel reinforcement ratios for these specimens are 3.01%, 5.28%, and 5.94%, respectively. The failure modes and patterns of the specimens are similar to those of SPW-1, as observed from the simulated damage cloud images. The concrete compression damage is concentrated within the 0–500 mm range on the right side, and slight compression damage occurs in the middle of the wall due to steel plate buckling. The comparison shows that as the steel plate reinforcement ratio increases, the range of compression damage on the right side gradually decreases, and the stiffness degradation in the lower right compression zone slows down. This indicates that increasing the reinforcement ratio of the middle steel plate can effectively alleviate the damage patterns of the concrete. The cracking patterns of the four specimens are generally the same, with cracks developing upward from the bottom, predominantly horizontal at the top, and diagonal at the bottom. Vertical cracks appear in the middle of the steel plate due to the effect of the steel plate, which reduces the tensile stress in the adjacent concrete, while increasing the tensile stress in the concrete on both sides, leading to cracking.

(2)

Comparison of Load–Displacement Curves

Figure 16, Figure 17 and Figure 18 present the load–displacement curves load at key points and ductility plots for specimens with different reinforcement ratios of the middle steel plate. As shown in the load–displacement curves, during the elastic phase before cracking, the curves of all specimens are similar, indicating that the reinforcement ratio has little effect on the stiffness of the composite shear wall. In the yield phase, both the peak load and ultimate load of all specimens gradually increase. Based on the load at key points for the specimens, compared to FSW-G-1, the yield load of FSW-1, FSW-G-2, and FSW-G-3 increased by 5.2%, 13.63%, and 17.3%, respectively, and the peak load increased by 7.07%, 15.17%, and 19%, respectively. This indicates that increasing the reinforcement ratio of the middle steel plate can enhance the bearing capacity of the composite shear wall, although the improvement is relatively small.

The ductilities of specimens FSW-G-1, FSW-G-2, and FSW-G-3 are 5.68, 4.70, and 4.52, respectively, based on the computational analysis. As shown in Figure 16, as the steel reinforcement ratio increases, the ductility of the composite shear walls shows a decreasing trend. However, the overall decrease is relatively small. In summary, this indicates that the central steel plate can dissipate the localized stress concentrations in the shear wall, thereby delaying the onset of localized yielding or failure, and improving the stability and ductility of the composite shear wall. However, when the steel plate thickness is excessive, it leads to increased stiffness of the composite shear wall, reducing the wall’s plastic deformation capacity, thereby decreasing its ductility and potentially leading to signs of brittle failure.

In summary, the steel ratio of the middle steel plate of the composite shear wall should be maintained at 3.96–5.28%. Within this range, the bearing capacity and ductility of the composite shear wall can be effectively guaranteed, and brittle failure can be prevented, resulting in the prevention of significant economic losses and safety accidents.

3.2.3. Comparative Analysis of Profile Flange Steel Plate Thickness

(1)

Comparison of Simulated Failure Modes and Patterns

Figure 19, Figure 20 and Figure 21 show the DAMAGEC (concrete compressive damage), steel plate, profile steel S, von Mises, and reinforcement cage S, von Mises contour plots for specimens FSW-X-1, FSW-X-2, and FSW-X-3. The steel reinforcement ratios of the profile steel flange plate for each specimen are 0.84%, 1.47%, and 1.68%, respectively. Under the same axial compression ratio, the simulation failure process of specimens with different profile thicknesses is similar to that of FSW-1. Comparing the failure cloud images, as the profile flange thickness increases, the strain damage degree and range of each specimen slightly increase. However, overall, the final failure modes, as well as the strain damage to the reinforcement, steel plate, and profile, are similar across all specimens. This indicates that changing the profile flange thickness has a minimal impact on the failure mode of the composite shear wall.

(2)

Comparison of Load–Displacement Curves

Figure 22, Figure 23 and Figure 24 present the load–displacement curves and load values at key points for the specimens under different steel reinforcement ratios of the profile steel flange plate. According to the load–displacement curves, the stiffness of all specimens is similar during the elastic stage; as the flange steel plate thickness increases, the load-bearing capacity of the composite shear wall continuously improves. The load at key points indicates that compared to FSW-X-1, the yield load of FSW-1, FSW-X-2, and FSW-X-3 increased by 10.27%, 12.84%, and 19.26%, respectively, and the peak load increased by 12.85%, 14.4%, and 21.05%, respectively. The comparison shows that increasing the profile flange steel plate thickness can enhance the load-bearing capacity of the composite shear wall, but the improvement is marginal. Moreover, as the steel plate thickness increases, the rate of improvement in load-bearing capacity diminishes.

The load at key points shows that, compared to FSW-X-1, the yield load of FSW-1, FSW-X-2, and FSW-X-3 increased by 10.27%, 12.84%, and 19.26%, respectively, while the peak load increased by 12.85%, 14.4%, and 21.05%, respectively. The comparison indicates that increasing the thickness of the profile flange steel plate can enhance the load-bearing capacity of the composite shear wall; however, the magnitude of the improvement is relatively small. Furthermore, as the steel plate thickness increases, the rate of improvement in load-bearing capacity diminishes.

The ductility coefficients of specimens FSW-X-1, FSW-X-2, and FSW-X-3 are 5.05, 5.35, and 5.01, respectively. As shown in Figure 23, the ductility of the composite shear wall initially increases and then decreases. This suggests that a moderate increase in the profile steel thickness can enhance the load-bearing capacity, seismic performance, and ductility of the shear wall. However, if the profile steel is too thick, it may lead to excessive wall stiffness, limiting plastic deformation and thereby reducing ductility. Therefore, in practical engineering design, the profile steel thickness should be selected appropriately based on the seismic requirements and load-bearing capacity of the structure. This ensures that the wall provides sufficient strength while maintaining good ductility and seismic performance, thus preventing brittle failure.

It can be seen from the above that when designing the composite shear wall, the steel ratio of the steel flange plate should be guaranteed to be in the range of 1.08–1.47%, so that it has good bearing capacity and ductility.

3.2.4. Comparative Analysis of Longitudinal Reinforcement Ratio in the Non-Restrained Zone

(1)

Comparison of Simulated Failure Modes and Morphologies

Figure 25, Figure 26 and Figure 27 show the DAMAGEC (concrete compressive damage), steel plate, profile steel S, von Mises, and reinforcement cage S, von Mises contour plots for specimens FSW-S-1, FSW-S-2, and FSW-S-3. The reinforcement ratios of the longitudinal reinforcement for each specimen are 0.60%, 0.91%, and 1.21%, respectively. The simulation of the failure process in ABAQUS indicates that the failure modes and morphologies of these specimens are similar to those of FSW-1. A comparison of the failure cloud plots shows that the extent of damage and the range of strain damage are essentially consistent across all specimens. The strain damage in the wall is concentrated in the 0–450 mm range at the bottom right side, and the strain damage of the reinforcement is focused at the bottom of both sides. This suggests that the longitudinal reinforcement ratio in the non-restrained zone has little effect on the failure mode and morphology of the composite shear wall.

(2)

Comparison of Load–Displacement Curves

Figure 28, Figure 29 and Figure 30 present the load–displacement curves, load values at key points, and ductility plots for the specimens under different longitudinal reinforcement ratios. From the load–displacement curves, it is evident that as the longitudinal reinforcement ratio increases, the load-bearing capacity of the composite shear wall improves. However, compared to FSW-S-1, the increase in load for FSW-S-2 and FSW-S-3 is much smaller. Notably, FSW-S-3 exhibits a slower decrease in the post-peak load region, indicating that while the longitudinal reinforcement ratio may not significantly enhance the load-bearing capacity of the composite shear wall, it has a noticeable effect on its ductility.

Regarding the characteristic point loads, it is observed that compared to FSW-X-1, the yield loads of FSW-1, FSW-X-2, and FSW-X-3 increase by 10.27%, 12.84%, and 19.26%, respectively, while the peak loads increase by 12.85%, 14.4%, and 21.05%. These results suggest that increasing the thickness of the steel plate in the profile flange can enhance the load-bearing capacity of the composite shear wall, although the improvement is relatively small. Furthermore, the increase in load-bearing capacity diminishes as the thickness of the steel plate increases.

The ductility coefficients of specimens FSW-S-1, FSW-S-2, and FSW-S-3 are 5.25, 5.60, and 4.52, respectively. The overall trend of the specimens, as shown in Figure 25, initially increases and then decreases. This indicates that an increase in the longitudinal reinforcement ratio has a dual effect on the ductility of the embedded steel plate shear wall. In summary, a moderate increase in the longitudinal reinforcement ratio helps improve the strength, seismic performance, and crack control capacity of the wall, while also enhancing its post-cracking ductility. However, if the longitudinal reinforcement ratio is too high, it will increase the wall stiffness, potentially leading to earlier yielding of the composite shear wall, restricting its plastic deformation capacity, which in turn reduces ductility and increases the risk of brittle failure. In design, the effect of the longitudinal reinforcement ratio on wall stiffness, load-bearing capacity, ductility, and seismic performance should be comprehensively considered to achieve an optimal performance balance.

In summary, when designing the built-in steel plate shear wall with FRC concrete in the plastic hinge zone, the longitudinal reinforcement ratio of the wall should be maintained at 0.60–0.91%, so that it has good bearing capacity and ductility, inhibits wall damage, and prevents wall collapse.

3.2.5. Comparative Analysis of Axial Load Ratio

(1)

Comparison of Simulated Failure Modes and Forms

Figure 31, Figure 32 and Figure 33 show the DAMAGEC (compressive concrete damage), steel plate, profile steel S, von Mises, and reinforcement cage S, von Mises contour plots for specimens FSW-1, FSW-N-1, FSW-N-2, and FSW-N-3. The axial loads for each specimen are 2180 kN, 2610 kN, and 3050 kN, respectively. During the initial loading stage, horizontal tensile cracks first appeared at the bottom of the right side of the specimen, and oblique compressive cracks simultaneously appeared at the same location. As the horizontal thrust increased, the tensile cracks progressively developed upward from the bottom, while the oblique compressive cracks extended towards the center of the wall. In the later loading stages, the reinforcement and steel plate buckled, and the concrete at the bottom of the wall exhibited severe strain damage, leading to spalling. Ultimately, the bottom of the right side of the specimen was crushed, marking the end of the loading. Compared to specimen FSW-1, the failure modes of the specimens with high axial load ratios were significantly different, with more severe damage.

The comparison of the damage contour plots indicates that as the axial load ratio increased, the extent of compressive damage in the wall gradually extended upward, and large-scale spalling of regular concrete occurred, indicating that the axial load ratio significantly influenced the failure modes of the specimens.

(2)

Comparison of Load–Displacement Curves

Figure 34, Figure 35 and Figure 36 present the load–displacement curves, load values at key points, and ductility plots for the specimens under different axial load ratios. From the load–displacement curves, it can be observed that as the axial load ratio increases, the bearing capacity of the composite shear wall continuously improves, with a significant increase. During the descending stage, the higher the axial load ratio, the faster the load decreases, indicating that the damage to the composite shear wall becomes more severe.

From the characteristic point loads, it is evident that compared to FSW-1, the yield loads for FSW-N-1, FSW-N-2, and FSW-N-3 increased by 17.18%, 30.13%, and 37.41%, respectively, while the peak loads increased by 12.75%, 26.63%, and 33.2%. The comparison clearly indicates that modifying the axial load ratio significantly improves the bearing capacity of the composite shear wall.

The ductility coefficients of specimens FSW-N-1, FSW-N-2, and FSW-N-3 are 4.15, 3.76, and 3.28, respectively. As shown in Figure 33, the ductility of the composite shear wall generally decreases as the axial load increases. This indicates that as the axial load increases, it causes the steel plate to buckle and the material to yield, thereby reducing the ductility of the wall. Therefore, when designing composite shear walls, it is essential to consider the effect of axial load on the wall’s performance, conduct appropriate mechanical analysis, and ensure that the wall maintains sufficient shear strength, stability, and seismic capacity within the designed axial load range.

In summary, when designing the steel plate shear wall structure with FRC concrete in the plastic hinge area, the axial compression ratio should be controlled between 0.3 and 0.4 to prevent brittle failure due to an excessive axial compression ratio, which results in major safety accidents and economic losses caused by wall collapse.

4. Calculation of Bending Capacity for the Cross-Section

For FRC shear wall components subjected to bending failure, considering the effects of edge restraint components and tensile forces in FRC, the analysis can be conducted based on the eccentric compression member theory under the combined actions of axial force N and bending moment M [32,33]. A calculation method for the bending capacity of the shear wall’s cross-section is proposed for walls with symmetric reinforcement design.

When calculating the flexural capacity of the shear wall, it is generally assumed that it is a uniform material and conforms to the plane section assumption. However, as a composite material, FRC concrete is composed of different components and fibers. These components may be unevenly distributed on the micro scale, resulting in performance prediction errors of materials in different regions, which cannot reflect the local changes in the mechanical properties of concrete. The homogeneous material model ignores the interaction effects of different phase interfaces (such as bonding between fiber and matrix, interface slip, interface strengthening, etc.), which may lead to misjudgment of material behavior. As a common assumption in structural mechanics, the plane section assumption is that any cross-section of the beam remains straight during the deformation process during the bending process of the beam, and the relative position relationship between the cross-sections before and after bending remains unchanged. However, this assumption also has some limitations. When the following situations occur, the plane section assumption may no longer hold: (1) when the bending deformation of the specimen is large, especially when large-angle bending or buckling occurs, the cross-section no longer maintains a plane state, and warping or other nonlinear deformation may occur, resulting in the plane section assumption being no longer applicable; (2) as FRC fiber-reinforced concrete is a nonlinear material, during material yield or failure, the plane section assumption may not hold; and (3) the plane section assumption usually ignores the influence of shear deformation. Under the action of high shear force, shear deformation may have a significant effect on bending deformation. At this time, the plane section assumption may no longer be applicable.

In the calculation and analysis of the bearing capacity, the scale of the analysis is much larger than the microstructure of the concrete material, and the uniform material assumption can be applied. Therefore, the calculation assumption of the flexural capacity of the normal section of the specimen is as follows: (1) The section of the composite shear wall approximately satisfies the plane section assumption. (2) Before the peak load, the confinement effect of the concrete encased by the hidden column remains unchanged, and the ultimate compressive strain ε_cu is taken as 0.0035 [34]. (3) Before steel yielding, the stress and strain of the steel are proportional; after steel yielding, the stress of the steel remains constant.

For FRC composite steel plate shear walls with confined edge components, when the strain of the concrete at the compressed edge of the wall reaches the peak compressive strain ε_p of the FRC confined by stirrups, the section of the shear wall reaches its maximum bearing capacity, and the corresponding peak bending moment of the wall is M_p. As shown in Figure 36, the key characteristics of this stage are as follows:

(1)

After the yielding of the wall, due to the increasing influence of bond slip deformation, when the peak load is reached, the strain distribution in the tensile region of the section no longer conforms to the plane section assumption. However, considering that the fiber bridging effect of FRC significantly improves the tensile ductility after matrix cracking, the strain distribution in the section is still assumed to conform to the plane section assumption [35].

(2)

When the maximum compressive strain at the edge of the FRC under compression reaches the peak strain of the confined FRC (0.0096), the constrained end of the compression zone will be in a stirrup-confined state. Therefore, the concrete in the compression zone is divided into two parts: the stirrup-confined dark column region and the unconfined region, both of which are simplified into rectangular stress distribution patterns. The increase factor for compressive concrete strength in the stirrup-confined dark column region, α₁, is taken as 1.3 [36]. For the unconfined region, the equivalent stress distribution factors are calculated as α_c = 0.91, β_c = 0.73 [37].

(3)

Consider the effect of fiber-reinforced concrete (FRC) in the tensile zone. When the longitudinal reinforcement at the tensile-constrained end reaches the design limit tensile strain (ε_su = 0.01) [38], the edge of the tensile zone FRC also reaches the limit tensile strain (ε_tu = 0.0097). The tensile zone begins to exhibit a transition from the stable development of fine cracks to the concentration of localized cracks. At this stage, the stress distribution in the tensile zone is bilinear. The equivalent stress distribution factors are calculated as α_t = 0.91, β_t = 0.73, and the coefficient λ = x_t/x_p = ε_tu/ε_p [39] is introduced to calculate the effective height of the FRC tensile zone.

(4)

Typically, under peak load conditions, both the tensile and compressive constrained end longitudinal reinforcements have yielded and may even reach the strengthened region (ε_s ≥ ε_tu). For safety and simplified calculation purposes, the stress in the reinforcement is assumed to remain at the yield strength. For walls with a higher distribution of vertical reinforcement, only the contribution of the tensile vertical reinforcement T_sw is considered. The separated reinforcement distributed along the height h_wt of the unconfined zone can be converted into an area-equivalent but continuous steel web A_sw = t_sh_wt. It is specified that within a certain range (η_xp) on both sides of the neutral axis, the stress in the reinforcement reaches the yield strength. The value of η can be determined according to the plane section assumption [40].

$$\eta ={\displaystyle \frac{f_{\mathrm{yw}}}{E_{\mathrm{sw}}{\epsilon }_{\mathrm{p}}}}$$

(12)

Based on the above assumptions and Figure 37, the relationships for the resultant forces on the section are as follows:

$$C_{\mathrm{cc}}={\beta }_1{f}_{\mathrm{c}}{A}_{\mathrm{cc}}$$

(13)

$$C_{\mathrm{cw}}={\alpha }_2{f}_{\mathrm{c}}\left(x_{\mathrm{p}}-h_{\mathrm{c}}\right)$$

(14)

$$T_{\mathrm{c}}={\alpha }_{\mathrm{t}}{f}_{\mathrm{tu}}{\beta }_{\mathrm{t}}\lambda x_{\mathrm{p}}$$

(15)

$$C_{\mathrm{s}}=T_{\mathrm{s}}=f_{\mathrm{y}}{A}_{\mathrm{s}}$$

(16)

$$T_{\mathrm{sw}}=f_{\mathrm{yw}}{t}_{\mathrm{s}}\left[h_{\mathrm{w}}-\left(1+\eta \right)x_{\mathrm{p}}-h_{\mathrm{c}}\right]$$

(17)

$$C_{\mathrm{pw}}=\left(1+{\displaystyle \frac{x-{\beta }_1{h}_{\mathrm{w}}}{0.5\beta h_{\mathrm{pw}}}}\right)$$

(18)

In the equation, C_cc, C_cw, C_s, C_pw, and C_pw represent the axial forces carried by the compressed zone of the FRC concrete with edge column confinement, non-confined wall concrete, distributed reinforcement in the edge column confinement zone, and embedded steel plates, respectively. T_c, T_s, and T_sw represent the axial forces carried by the tensile zone FRC concrete, distributed reinforcement in the edge column confinement zone, and non-confined distributed reinforcement, respectively. f_c, f_tu represent the compressive strength and tensile strength of FRC concrete, while f_y, f_yw represent the yield strengths of the distributed reinforcement in the edge column confinement zone and non-confined reinforcement, respectively. A_cc, x_p, h_c, A_s, h_w, and h_pw represent the area of the compressed FRC concrete in the confined zone, the compressive height, the effective height of the edge column confinement zone, the cross-sectional area of distributed reinforcement in the edge column confinement zone, the effective section height, and the height of the embedded steel plate, respectively.

From the sectional stress distribution and vertical force equilibrium conditions, the following equation can be derived:

$$N=C_{\mathrm{cc}}+C_{\mathrm{cw}}+f_a{A}_a^{\prime }+C_{\mathrm{pw}}-{\delta }_{\mathrm{a}}{A}_{\mathrm{a}}-T_{\mathrm{sw}}-T_{\mathrm{c}}$$

(19)

In the equation, f_a and A_a^′ represent the yield strength and cross-sectional area of the compressed zone edge column steel, while δ_a and A_a represent the yield strength and cross-sectional area of the tensile zone edge column steel.

By solving Equation (13) to (19) for the compressed zone height x_p, the centroidal bending moment M of the section can then be determined [41]:

$$\begin{array}{l}M=\left(C_{\mathrm{cc}}+f_{\mathrm{a}}{A}_{\mathrm{a}}^{\prime }\right)\left(h_{\mathrm{w}}-{\displaystyle \frac{h_{\mathrm{c}}}{2}}\right)+C_{\mathrm{cw}}\left(h_{\mathrm{w}}-h_{\mathrm{c}}-{\displaystyle \frac{x_{\mathrm{p}}}{2}}\right)+C_{\mathrm{s}}\left(h_{\mathrm{w}}-h_{\mathrm{c}}\right)+\left[0.5-{\left({\displaystyle \frac{x-{\beta }_1{h}_{\mathrm{w}}}{{\beta }_1{h}_{\mathrm{pw}}}}\right)}^2\right]f_{\mathrm{p}}{A}_{\mathrm{p}}{h}_{\mathrm{pw}}\\ -T_{\mathrm{c}}\left({\displaystyle \frac{{\beta }_{\mathrm{t}}\lambda x_{\mathrm{p}}}{2}}+h_{\mathrm{w}}-2x_{\mathrm{p}}-{\displaystyle \frac{h_{\mathrm{c}}}{2}}\right)-N\left({\displaystyle \frac{h_{\mathrm{w}}-h}{2}}\right)-T_{\mathrm{sw}}\left({\displaystyle \frac{h_{\mathrm{w}}-\eta x_{\mathrm{p}}-x_{\mathrm{p}}}{2}}\right)\end{array}$$

(20)

By considering the second-order effects of axial forces under large horizontal displacements, the horizontal bearing capacity V can be calculated using Equation (21):

$$V={\displaystyle \frac{M}{H-L_{\mathrm{p}}}}$$

(21)

In the equation, L_p represents the equivalent plastic hinge length, which can be calculated using the following formula:

$$L_{\mathrm{p}}=0.277h_{\mathrm{w}}\left(1-1.148n\right)H^{0.877}\left(1-4.11\times {10}^{-5}{f}_{\mathrm{a}}{f}_{\mathrm{c}}\right)\left(1-3.83{\rho }_{\mathrm{sh}}\right)\left(1+5.455{\rho }_{\mathrm{b}}\right)$$

(22)

In the equation, H represents the distance from the shear wall loading point to the top of the foundation beam; n is the design axial load ratio; and ρ_sh and ρ_b represent the horizontal reinforcement ratio of the wall and the steel ratio of the edge column H-section, respectively.

Table 2. Comparison of simulated values and experimental values.

Parametric Variable		Specimen Number	Calculated Value	Value of Simulation	Specific Value
FRC strength/MPa	C60	FSW-1	729	722	1.01
	C50	FSW-2	605	631	0.96
	C70	FSW-3	865	842	1.03
	C80	FSW-4	976	951	1.03
Longitudinal reinforcement ratio/%	0.60	FSW-S-1	884	846	1.04
	0.91	FSW-S-2	892	889	1.00
	1.3	FSW-S-3	921	941	0.98
Shape steel ratio/%	0.84	FSW-X-1	643	646	0.99
	1.47	FSW-X-2	773	739	1.05
	1.68	FSW-X-3	816	782	1.04
Steel ratio of steel plate/%	2.64	FSW-G-1	659	682	0.97
	5.28	FSW-G-2	791	782	1.01
	5.94	FSW-G-3	843	808	1.04
Axle pressure/kN	2180	FSW-N-1	811	822	9.89
	2610	FSW-N-2	932	921	1.01
	3050	FSW-N-3	989	971	1.02

presents the results of the formula calculations. As shown in Figure 38, the calculated bearing capacities are in good agreement, indicating that this formula can be used for calculating the bending capacity of FRC shear walls with embedded steel plates in the plastic hinge zone.

5. Conclusions

This paper presents a finite element simulation of the shear wall specimens with embedded FRC steel plates in the plastic hinge zone. The analysis investigates the behavior of these specimens, and based on the successful establishment of the model, parameter analyses were conducted to draw the following conclusions:

(1)

Finite element analysis of the specimens shows that the curves obtained from numerical simulations are in good agreement with the experimental results, with the error in the positive bearing capacity controlled within 10%. Therefore, the model is considered successfully established and exhibits a high level of accuracy.

(2)

At the peak load point, the steel plates at the beam–wall interface all experienced yielding, which is consistent with the observed buckling of the steel plates at the beam–wall interface after the pseudo-static tests. The entire cross-section of the steel plates at the beam–wall interface yielded, with a relatively large yielding area. This is consistent with the large bending moments experienced by the specimen, indicating that the use of fiber-reinforced concrete (FRC) significantly enhances the performance of the steel plates.

(3)

With the increase in reinforcement ratio, H-section thickness, and the strength of fiber-reinforced concrete (FRC), the bearing capacity of shear walls with embedded FRC steel plates in the plastic hinge zone increases steadily. Overall, the strength of fiber-reinforced concrete (FRC) has a relatively significant impact on seismic performance.

(4)

By comparing and analyzing the seismic requirements such as failure mode, bearing capacity, and ductility of the composite shear wall obtained by numerical simulation, the optimal design values of FRC concrete strength, steel plate steel ratio, steel ratio, and longitudinal reinforcement ratio in the plastic hinge area are C70, 3.96%, 1.08%, and 0.6%, respectively, which can provide design reference for its practical engineering application.

(5)

In the current relevant design standards, the design of the plastic hinge zone usually depends on the characteristics of traditional concrete materials, which fails to fully consider the superior performance of FRC concrete. Moreover, the lack of special design formulas for FRC concrete may lead to the failure to give full play to the advantages of FRC in the design of the plastic hinge zone. The formula for calculating the flexural capacity of the normal section of the steel plate composite shear wall structure with FRC in the plastic hinge zone is proposed in this paper. The calculation results are in good agreement with the numerical simulation results. Therefore, the formula proposed in this paper can be referred to, and the relevant provisions of FRC concrete can be added to supplement the relevant design standards of crack resistance and toughness of FRC.