Experimental and Numerical Investigation of Fatigue Performance in Reinforced Concrete Beams Strengthened with Engineered Cementitious Composite Layers and Steel Plates

Abstract

Reinforcing concrete beams with adhesive steel plates is a widely adopted method for enhancing structural performance. However, its ability to significantly improve the load-carrying capacity of reinforced concrete (RC) beams is constrained and often leads to “over-reinforced” failure. To overcome these limitations, this study introduces a novel composite reinforcement strategy that integrates steel plates in the tensile zone with Engineered Cementitious Composite (ECC) layers in the compression zone of RC beams. Static and fatigue tests were conducted on the reinforced beams, and a finite element model was developed to perform nonlinear analyses of their structural behavior under cyclic loading. The model incorporates the nonlinear material properties of concrete and rebar, enabling accurate simulation of material degradation under cyclic conditions. The model’s accuracy was validated through comparison with experimental data, demonstrating its effectiveness in analyzing the structural performance of RC beams under cyclic loading. Furthermore, a parametric study demonstrated that increasing the thickness of steel plates and ECC layers substantially improves the beams’ ductility and load-carrying capacity. These findings provide effective reinforcement strategies and offer valuable technical insights for engineering design.

1. Introduction

With the rapid advancement of the construction industry, the aging and deterioration of reinforced concrete (RC) structures have emerged as pressing issues in recent decades [1]. Currently, the repair and reinforcement of structural elements are key areas of focus for researchers and engineers. Common reinforcement methods include adhesive steel plate, section enlargement, replacement of concrete reinforcement, and combined reinforcement techniques [2]. While these methods can enhance the load-carrying capacity of structural members to some extent, relatively little attention has been devoted to extending their service life. For instance, RC structures such as airport runways, highways, railway bridges, and marine facilities often experience fatigue-induced brittle failure under cyclic loading, even when the applied loads are below their static load capacity [3,4]. Furthermore, environmental and climatic factors significantly accelerate structural degradation over time [5]. To ensure the safety of these structures, it is crucial to evaluate their performance under fatigue loading conditions. Researchers have proposed various structural configurations to enhance the performance of structures under cyclic loading [6,7,8,9,10]. The primary mechanism for fatigue enhancement involves the reinforcing layer bearing part of the tensile stress or reducing strain localization and stress concentration in the longitudinal rebar [11]. Nie et al. [12] conducted fatigue tests on steel-concrete composite beams and observed that these beams exhibited improved ductility under fatigue loading, with the stress amplitude in the steel plate significantly influencing their fatigue performance. They also identified failure patterns in the composite beams, including fatigue fractures in the bottom steel plate and concrete crushing in the compression zone [13]. Furthermore, Rakgate [14] and Al-Hassani [15] investigated the strength and ductility of beams reinforced with bonded steel plates in greater detail. They found that increases in flexural capacity were limited because the tensile rebar remained unyielded when the compressive concrete was crushed. Recent studies have shown that ECC can strengthen the compression zone of RC beams, effectively reducing the tendency for brittle failure in over-reinforced beams [16]. Building on these findings, this paper proposes a combined reinforcement approach that integrates ECC and steel plates to strengthen reinforced concrete beams.

Engineered Cementitious Composite (ECC) is a high-performance cementitious material known for its pseudo-strain hardening behavior and exceptional crack control capabilities, first introduced by Professor Victor Li [17,18]. Renowned for its outstanding ductility and compressive toughness, ECC achieves a peak compressive strain two to three times higher than that of traditional concrete [19]. Furthermore, PVA-ECC specimens outperform standard concrete in fatigue strength, deformability, and fatigue lifespan [20,21,22]. Extensive research has demonstrated ECC’s effectiveness in enhancing the structural integrity of members. For instance, Leung et al. applied a 25 mm ECC layer to the tensile side of bending beams and conducted fatigue tests, which confirmed that ECC could endure significantly larger deflections under fatigue loading without damage, effectively controlling fatigue crack propagation [23]. Similarly, Zhang et al. [24] found that ECC overlays on base concrete effectively prevent reflection cracks in the pavement, thereby extending its service life. Liu et al. [25] developed a composite bridge deck incorporating ECC and large U-ribs. Through testing and simulations, they observed that the ECC overlay reduced stress at rib–deck and rib–bulkhead welded joints by 90% and 54%, respectively. Even after sustaining damage, ECC-covered bridge decks exhibited remarkable fatigue resistance. In another study, Qiu et al. [26] conducted multi-scale tests to investigate ECC’s fatigue behavior, concluding that mechanisms such as fiber debonding, interfacial hardening, and in-situ fiber strength reduction contribute to the deterioration of fatigue properties. Nonetheless, ECC’s crack-bridging capability significantly extends its fatigue life.

Finite element modeling (FEM) provides an efficient alternative to labor-intensive fatigue tests, enabling accurate predictions of structural performance under fatigue loading. Gencturk B et al. [27] employed ABAQUS software to simulate crack propagation in H-beams and damage evolution in steel and concrete using fracture mechanics theory. Their results indicated that fatigue life predictions for Steel-Reinforced Concrete (SRC) beams closely aligned with experimental data. Zhu et al. [28] proposed a novel finite element analysis method that incorporates material damage modeling to streamline the fatigue process, conducting a limited number of static analyses to represent specific load cycles with equivalent fatigue damage. Similarly, Al-Saoudi et al. [29] investigated the fatigue life of FRP-concrete joints under varying stress ratios, utilizing finite element simulations that demonstrated strong correlations with experimental data, thereby significantly reducing testing costs. Ganesh et al. [30] developed a finite element model to predict the performance of reinforced steel and concrete beams under both static and fatigue loading conditions, achieving a final error margin of approximately 12%.

To enhance the efficiency of reinforcement and mitigate the effects of fatigue loading on structures, this paper proposes a method for strengthening RC beams by integrating ECC with steel plates. Figure 1 presents the schematic diagram of the proposed reinforcement technique [31]. The initial phase involves simulating the response of the reinforced beam under static and fatigue loads using ABAQUS. The load–deflection curves and damage modes obtained from the simulations are compared with experimental results to validate the reliability of the finite element analysis. The model incorporates the degradation of concrete and ECC material properties under fatigue loading. Subsequently, the finite element method is utilized to analyze the influence of variations in ECC strength, concrete strength, and rebar strength on the fatigue behavior of reinforced beams. This study aims to provide valuable insights into the design and application of novel reinforced composite beams.

2. Summary of the Experimental Program

The test beams were 2100 mm in length, 200 mm in width, and 150 mm in height, as illustrated in Figure 2. Five beams were prepared for testing: two unreinforced RC beams (CB-1 and FCB-1) served as control specimens, while three reinforced beams (EB-1, FEB-1, and FEB-2) incorporated ECC layers and steel plates. Static load tests were conducted on beams without the letter ‘F’ in their labels, whereas fatigue tests were performed on those with ‘F’ in their labels. The detailed specifications of all test beams are provided in

Table 1. Detailed scheme of the test beams.

Specimen Beam Designation	ECC Layer Thickness (mm)	Stress Level	Steel Plate Thickness (mm)	Loading Scheme	Fatigue Load Amplitude	Number of Fatigue Cycles
CB-1	0	–	0	Monotonic static loading	–	–
EB-1	30	–	5	Monotonic static loading	–	–
FCB-1	0	0.6	0	Fatigue loading	0.2 PU1–0.6 PU1	200,000
FEB-1	30	0.6	5	Fatigue loading	0.2 PU3–0.6 PU3	200,000
FEB-2	30	0.6	5	Fatigue loading	0.2 PU3–0.6 PU3	2,000,000

.

Strengthening and Material Properties

Concrete was first poured into the mold, with steel pipes positioned at the bottom to create pre-formed holes before the concrete hardened. After a 28-day curing period, the top surface of the test beam was chiseled, washed with water, and coated with an ECC layer to enhance bonding at the interface. Following an additional 28 days of curing for the ECC layer, a steel plate was bolted to the bottom of the beam, U-shaped hoops were installed, and the assembly was secured using epoxy adhesive, as shown in Figure 3. The ECC mixture was prepared according to the ratio provided by Dan Meng et al. [32], with PVA fibers at a 2.0% volume fraction used for reinforcement. To evaluate compressive strength, three plain concrete cubes (150 mm) and three ECC cubes (100 mm) were fabricated alongside the test beams [33]. Dog-bone-shaped specimens were also prepared for tensile testing [34], as shown in Figure 4. The material properties obtained were as follows: The cubic compressive strength of plain concrete was 31 MPa, with a modulus of elasticity of 29,500 MPa; the compressive strength of ECC cubes was 55.7 MPa, with a modulus of elasticity of 20,100 MPa. The hoop bars, with a diameter of 8 mm, were made from HPB300 steel, which exhibited a yield strength of 382 MPa and an ultimate strength of 545 MPa. The longitudinal rebar was fabricated from HRB 400 steel, with measured yield and ultimate strengths of 445.2 MPa and 555.9 MPa, respectively.

The test setup is illustrated in Figure 5. In this setup, LVDT1 measures the mid-span displacement, while LVDT2 and LVDT3 are used to correct displacement errors. Strain gauges were affixed to the longitudinal rebar at the bottom of the test beam and sealed using epoxy resin adhesive prior to concrete pouring. Additionally, strain gauges were installed on the sides of the test beam to measure concrete strain, and an additional gauge was placed on the steel plate at the bottom to monitor its strain during loading.

Static and fatigue tests were performed using a ZH-SFS20 electro-hydraulic servo fatigue testing machine (Shanghai Chubang Company, Shanghai, China) via a 4-point bending method. The static test applied graded loading at a rate of 0.02 mm/s. For fatigue tests, preloading was conducted to ensure proper contact between the specimen and apparatus, followed by static loading to the maximum fatigue load (Pmax) at 0.5 mm/min, while strain and crack progression were recorded. After reaching Pmax, the beams were fully unloaded to complete the first fatigue cycle. Subsequent cycles involved equal-amplitude loading between Pmin and Pmax (Figure 6). FCB-1 and FEB-1 underwent 200,000 cycles prior to static failure loading, while FEB-2 was tested for 2 million cycles prior to static failure loading. At specified intervals—5000; 10,000; 20,000; 50,000; 100,000; 200,000; 500,000; 900,000; 1,100,000; 1,300,000; 1,500,000; 1,700,000; 1,900,000; and 2,000,000 cycles—the machine was stopped and unloaded, and a cyclic static test was conducted to measure beam deflections, strains in the concrete, rebar, and steel plates, as well as crack lengths and maximum crack width.

3. Finite Element Modeling

To thoroughly analyze the static and fatigue performance of the newly reinforced beams, this study employed ABAQUS [35]. This paper details the load–deflection curves, ultimate load-carrying capacities, and fatigue performance modeling procedures for reinforced RC beams combined with ECC layers and steel plates. Initially, the beams were analyzed under static load using finite element analysis, and their results were validated against experimental data. Subsequently, the analysis considered the degradation of material properties, including concrete, ECC, and rebar under fatigue loading and simulated the structural behavior under fatigue conditions.

3.1. Element Used and Mesh Size Analysis

The CB-1 finite element model is illustrated in Figure 7a. An eight-node linear hexahedral solid element (C3D8R) was selected for simulating the reference beam. The model underwent a mesh sensitivity analysis to ascertain the optimal cell size. Mesh sizes from 10 to 50 mm were evaluated, and the final bearing capacity results of the numerical simulation are presented in Figure 7b. A mesh size of 15 mm was found to accurately predict the load-carrying capacity, balancing accuracy with computational efficiency. This mesh size was adopted for all subsequent analyses.

3.2. Material Models for ECC, Concrete, and Steel Under Static Load

3.2.1. Uniaxial Tension and Compression Nonlinear Behavior of ECC and Concrete

The cubic compressive strength of the ordinary concrete used in the experiment was 31 MPa. According to the Chinese Concrete Structure Design Code (GB50010-2010) [36], the stress–strain relationships for concrete under uniaxial tension and compression were established, as defined by Equations (1)–(4).

$$\sigma =\left(1-{\mathrm{d}}_c\right)E_c\epsilon$$

(1)

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

(2)

$$\sigma =\left(1-d_t\right)E_c\epsilon$$

(3)

$$d_t=\left\{\begin{array}{c}1-{\rho }_t\left[1.2-0.2x^5\right] x\le 1\\ 1-{\displaystyle \frac{{\rho }_t}{{\alpha }_t(x-1{)}^{1.7}+x}} x>1\end{array}\right\}$$

(4)

For pressurized: ${\rho }_{\mathrm{c}}={\displaystyle \frac{f_{c,r}}{E_c{\epsilon }_{c,r}}}$, $n={\displaystyle \frac{E_c{\epsilon }_{c,r}}{E_c{\epsilon }_{c,r}-f_{c,r}}}$, $\mathrm{x}={\displaystyle \frac{\epsilon }{{\epsilon }_{c,r}}}$, ${\alpha }_{\mathrm{c}}=0.157f_{c,r}^{0.785}-0.905$, ${\epsilon }_{c,r}=\left(700+172\sqrt{{\mathrm{f}}_c}\right)\times {10}^{-6}$.

For tensile: ${\rho }_{\mathrm{t}}={\displaystyle \frac{f_{t,r}}{E_c{\epsilon }_{t,r}}}$, $\mathrm{x}={\displaystyle \frac{\epsilon }{{\epsilon }_{\mathrm{t},r}}}$, ${\alpha }_{\mathrm{t}}=0.312f_{t,r}^2$, ${\epsilon }_{\mathrm{t},r}=65f_{t,r}^{0.54}\times {10}^{-6}$. The parameters ${\alpha }_{\mathrm{c}}$ and ${\alpha }_{\mathrm{t}}$ represent the values of the descending section of the uniaxial compressive and tensile stress–strain curves of concrete, respectively. Similarly, ${\epsilon }_{c,r}$ corresponds to the peak compressive strain, and ${\epsilon }_{t,r}$ to the peak tensile strain of concrete, each associated with their respective uniaxial strengths.

The intrinsic model of ECC was derived by fitting the model to data from ECC tensile tests and incorporating the model proposed by Meng et al. [33]. Figure 8 illustrates the stress–strain relationship curves of ECC under tension and compression. Key parameters defining the intrinsic model are listed in

Table 2. Key parameters of ECC uniaxial compression.

${\mathsf{\epsilon }}_{0.4}$	${\mathsf{\sigma }}_{0.4}\left(\mathbf{M}\mathbf{P}\mathbf{a}\right)$	${\mathsf{\epsilon }}_0$	${\mathsf{\sigma }}_0\left(\mathbf{M}\mathbf{P}\mathbf{a}\right)$	${\mathsf{\epsilon }}_{\mathbf{l}}$	${\mathsf{\sigma }}_{\mathbf{l}}$	${\mathsf{\epsilon }}_{\mathbf{m}\mathbf{a}\mathbf{x}}$	${\mathsf{\sigma }}_{\mathbf{c}}\left(\mathbf{M}\mathbf{P}\mathbf{a}\right)$
0.0015	21.75	0.00412	38.57	0.00619	21.44	0.01235	15.57

and

Table 3. Key Parameters for ECC Uniaxial Tension.

${\mathsf{\epsilon }}_{\mathbf{t}0}$	${\mathsf{\sigma }}_{\mathbf{t}0}\left(\mathbf{M}\mathbf{P}\mathbf{a}\right)$	${\mathsf{\epsilon }}_{\mathbf{t}\mathbf{p}}$	${\mathsf{\sigma }}_{\mathbf{t}\mathbf{p}}\left(\mathbf{M}\mathbf{P}\mathbf{a}\right)$	${\mathsf{\epsilon }}_{\mathbf{t}\mathbf{u}}$
0.000219	3.17	0.03535	3.81	0.0386

. The peak front stress–strain relationship for the compression model is defined by Equations (5) and (6).

$$\sigma =\left\{\begin{array}{c}{E}_0\epsilon \left(0<\epsilon <{\epsilon }_{0.4}\right)\\ E_0\epsilon \left(1-\alpha \right) \left({\epsilon }_{0.4}<\epsilon <{\epsilon }_0\right)\end{array}\right.$$

(5)

$$\alpha =\mathrm{a}{\displaystyle \frac{\epsilon E_0}{f_{cr}^{\prime }}}-b$$

(6)

where E₀ represents the modulus of elasticity, $\epsilon$_0.4 denotes the strain at 40% of the ultimate strength, and ${\epsilon }_0$ is the strain at peak load. $\alpha$ is the discount factor for the modulus of elasticity in the interval between elastic strain and peak strain, with values of 0.301 for a and 0.109 for b based on curve fitting.

The post-peak stress–strain relationship is defined by Equation (7).

$$\sigma =\left\{\begin{array}{c}m\left(\epsilon -{\epsilon }_0\right)+f_{cr}^{\prime } \left({\epsilon }_0<\epsilon <{\epsilon }_l\right)\\ n\left(\epsilon -{\epsilon }_l\right)+{\sigma }_{l } \left({\epsilon }_l<\epsilon <{\epsilon }_{max}\right)\end{array}\right.$$

(7)

where m and n represent the slopes of the bi-linear curves, a linear regression analysis based on the test results yielded values of m = −8888.65, n = −952.5, with ${\epsilon }_l$ defined as 1.5${\epsilon }_0$.

For the tensile model, the stress–strain relationship is defined by Equation (8), where ${\sigma }_{t0}$ and ${\epsilon }_{t0}$ represent the first cracking strength and strain, ${\sigma }_{tp}$ and ${\epsilon }_{tp}$ denote the ultimate tensile strength and strain, and ${\epsilon }_{tu}$ is the tensile breaking strain.

$${\sigma }_{\mathrm{t}}=\left\{\begin{array}{ll}{\mathrm{E}}_0\epsilon & 0⩽\epsilon ⩽{\epsilon }_{\mathrm{t}0}\\ {\sigma }_{\mathrm{t}0}+\left({\sigma }_{\mathrm{t}\mathrm{p}}-{\sigma }_{\mathrm{t}0}\right)\left({\displaystyle \frac{\epsilon -{\epsilon }_{\mathrm{t}0}}{{\epsilon }_{\mathrm{t}\mathrm{p}}-{\epsilon }_{\mathrm{t}0}}}\right)& {\epsilon }_{\mathrm{t}0}⩽\epsilon ⩽{\epsilon }_{\mathrm{t}\mathrm{p}}\\ {\sigma }_{\mathrm{t}\mathrm{p}}\left(1-{\displaystyle \frac{\epsilon -{\epsilon }_{\mathrm{t}\mathrm{p}}}{{\epsilon }_{\mathrm{t}\mathrm{u}}-{\epsilon }_{\mathrm{t}\mathrm{p}}}}\right)& {\epsilon }_{\mathrm{t}\mathrm{p}}⩽\epsilon ⩽{\epsilon }_{\mathrm{t}\mathrm{u}}\\ 0& {\epsilon }_{\mathrm{t}\mathrm{u}}<\epsilon \end{array}\right.$$

(8)

Although the stress–strain relationship of concrete exhibits plastic behavior before reaching its peak stress, most scholars agree that both tensile and compressive damage occur only after the peak stress is surpassed. Prior to this point, concrete is assumed to remain undamaged and can fully return to its original state upon unloading. Thus, tensile and compressive damage are considered to initiate after the peak strength is achieved. The compressive and tensile damage parameters for ECC and plain concrete can be referenced from the model proposed by Birtel [37] and are calculated using Equations (9) and (10), respectively.

$$d_c=1-{\displaystyle \frac{{\sigma }_c{E}_c^{-1}}{{\epsilon }_c^{pl}\left(1/b_c-1\right)+{\sigma }_c{E}_c^{-1}}}$$

(9)

$$d_t=1-{\displaystyle \frac{{\sigma }_t{E}_c^{-1}}{{\epsilon }_t^{pl}\left(1/b_t-1\right)+{\sigma }_t{E}_c^{-1}}}$$

(10)

$${\epsilon }_k^{pl}=b_k{\epsilon }_k^{in}$$

(11)

$${\epsilon }_k^{in}={\epsilon }_k-{\sigma }_k{E}_c^{-1} (\mathrm{k}=\mathrm{c},\mathrm{t})$$

(12)

Here, ${\epsilon }_c^{pl}$ and ${\epsilon }_t^{pl}$ denote the compressive and tensile plastic strains, respectively, 0 $<b_k<$ 1, with values of $b_c$ = 0.7 and $b_t$ = 0.1 applied in this analysis.

To simulate the damage accumulation behavior of concrete, the Concrete Damaged Plasticity (CDP) model, which represents irrecoverable plasticity, was selected. This model, compatible with the finite element software ABAQUS, effectively captures the mechanical behavior of concrete under uniaxial and biaxial tension and compression. The uniaxial tensile and compressive damage plasticity properties of concrete as assumed by the CDP model are illustrated in Figure 9 [35].

In addition, five parameters are required for the CDP model: the dilatancy angle, flow potential eccentricity (e), the ratio of compressive strength under biaxial to uniaxial loading (fb0/fc0), the ratio of the second stress invariant on the tensile meridian to that on the compressive meridian (Kc), and the cohesion parameter. The parameter values for both concrete and ECC are provided in

Table 4. Key parameters of the ECC ontology model.

Parameters	$\mathsf{\psi }$	e	fb0/fc0	Kc	$\mathbf{V}$
NC	35°	0.1	1.16	0.667	0.0005
ECC	30°	0.1	1.17	0.7	0.001

. Notably, an increase in the viscosity coefficient facilitates the convergence of numerical simulations.

3.2.2. Mechanical Behavior of Steel Bars and Steel Plates

Based on the material property experiments of the steel bars, an isotropic ideal elastic–plastic model was adopted in the finite element model. The tensile and compressive stress–strain relationships are shown in Figure 10.

3.3. Finite Element Model for Fatigue Behavior

The fatigue process involves the accumulation of damage, with steel and concrete being the primary materials in reinforced concrete beams. It is essential to establish separate fatigue performance assessment models for these two materials and validate the models using fatigue test results.

3.3.1. Fatigue Properties of Concrete

Cyclic loading accelerates the inelastic deformation of concrete and increases the accumulation of internal damage, causing the elastic modulus of concrete to vary with continued fatigue loading. Holman [38] proposed an equation to calculate the effective elastic modulus of concrete after n loading cycles, as shown in Equation (13), based on experimental data.

$$E_N=\left(1-0.33N/N_f\right)E_0$$

(13)

Here, ${\mathrm{E}}_{\mathrm{N}}$ represents the effective modulus of elasticity of concrete at the n-th cycle, ${\mathrm{E}}_0$ is the initial modulus of elasticity of concrete, and ${\mathrm{N}}_{\mathrm{f}}$ is the number of loading cycles required to reach failure. The value of ${\mathrm{S}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ can be calculated using Equation (14) [39].

$${\mathrm{S}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.9885-0.0618 \lg {\mathrm{N}}_{\mathrm{f}}$$

(14)

where ${\mathrm{S}}_{\mathrm{m}\mathrm{a}\mathrm{x}}={\displaystyle \frac{{\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\mathrm{f}}_{\mathrm{c}}}}$ represents the maximum stress level, and ${\mathrm{f}}_{\mathrm{c}}$ denotes the uniaxial compressive strength of concrete.

Zhang et al. [40] employed the electron scattering spot interference (ESPI) technique to measure the three-point bending fracture process of test beams under cyclic loading. They observed that the load-displacement curve for monotonic loading serves as the envelope for the load-displacement curve under cyclic loading. This finding aligns with the principle that energy dissipation is consistent between fatigue and monotonic loading processes [41]. A schematic representation of these findings is presented in Figure 11.

The deterioration of concrete under fatigue loading is characterized by a continuous decay in its fatigue residual strength. When this residual strength declines to the stress level corresponding to the upper limit of the fatigue load, the concrete is considered to have reached fatigue failure. The relationship between fatigue residual strength and the number of loading cycles is presented in Figure 12.

When ${\left.x\right|}_{N=1}=x\left(1\right)=1,{\sigma }_{r,c}\left(1\right)=f_c$, the initial loading stress corresponds to the axial compressive strength of concrete. When ${\left.\mathrm{x}\right|}_{\mathrm{N}={\mathrm{N}}_{\mathrm{f}}}=\mathrm{x}\left({\mathrm{N}}_{\mathrm{f}}\right),{\sigma }_{\mathrm{r},\mathrm{c}}\left({\mathrm{N}}_{\mathrm{f}}\right)={\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}}$, indicating the cycles have reached fatigue life, the residual strength equals the upper fatigue stress of concrete. The fatigue residual strength of concrete at any given number of fatigue load cycles can be represented by Equations (15) and (16).

$$f_{r,c}\left(N\right)=\begin{array}{c}{f}_c{\displaystyle \frac{x\left(N\right)}{{\alpha }_c\left[x\right(N)-x(1){]}^2+x(N)}}\end{array}$$

(15)

$$f_{r,t}\left(N\right)=f_t\left(1-{\displaystyle \frac{\lg n}{10.954}}\right)$$

(16)

Here, ${\mathsf{\alpha }}_{\mathrm{c}}$ is derived from Equation (2), while ${\mathrm{f}}_{\mathrm{c}}$ and ${\mathrm{f}}_{\mathrm{t}}$ represent the compressive and tensile strengths of concrete, respectively. The longitudinal residual strain of concrete beams with expanding internal defects under cyclic loading can be calculated using Equation (17).

$$\mathsf{\Delta }{\epsilon }_r\left(N\right)={\epsilon }_r\left(1\right)+{\displaystyle \frac{0.00105{\epsilon }_{max}^{1.98}{\left(1-{\epsilon }_{min}/{\epsilon }_{max}\right)}^{5.27}}{{\epsilon }_{\mathrm{unstab} }^{1.41}}}{N}^{0.395}$$

(17)

Here, ${\epsilon }_{\mathrm{r}}\left(1\right)$ represents the initial residual strain, $\mathsf{\Delta }\epsilon \left(1\right)=0.25{\left({\epsilon }_{\mathrm{m}\mathrm{a}\mathrm{x}}/{\epsilon }_{\mathrm{k}}\right)}^2$. While ${\epsilon }_{min}$ and ${\epsilon }_{max}$ denote the initial instantaneous strains corresponding to the upper and lower limits of fatigue loading, respectively. ${\epsilon }_k$ is the peak strain of concrete under a single static load, a value dependent solely on the material properties. Fatigue damage in concrete occurs when the residual strain reaches a specific threshold. The fatigue damage of concrete is quantified by Equation (18).

$$\Delta {\epsilon }_r\ge 0.4{\displaystyle \frac{f_c}{E_0}}$$

(18)

3.3.2. Fatigue Properties of ECC

ECC specimens subjected to uniaxial compression fatigue are deemed to have an infinite lifetime if they show no damage after enduring 2 × 10⁶ cycles [40]. The load stress level in this test is comparable, and the strain behavior at the upper edge of FEB-2 over 2 million fatigue cycles remains stable, as presented in Figure 13. Consequently, the CDP model was applied to simulate the stress–strain relationship of ECC under fatigue conditions.

3.3.3. Fatigue Properties of Steel Bar

During high circumferential fatigue loading of bent beams, the reinforcement stresses remain within the elastic range, and the elastic modulus shows minimal degradation [8,43]. Consequently, the intrinsic equations for reinforcement are formulated based on a rational elastic–plastic model. Typically, the fatigue strength of concrete beams is governed by the fatigue strength of their reinforcement, allowing the fatigue life of the beams to be predicted using the S-N curve of the reinforcement. The residual strength of the reinforcement for any given number of fatigue cycles can be calculated based on its fatigue life and the number of cycles applied.

In this study, the S-N equation for steel bars proposed by the China Academy of Railway Science [44] is used to estimate the fatigue life of the steel bars, as shown in Equation (19).

$$\left\{\begin{array}{c}\lg N=15.135-4.383\lg \Delta \sigma \left(N<{10}^7\right)\\ \lg N=18.847-6.383\lg \Delta \sigma \left(N\ge {10}^7\right)\end{array}\right.$$

(19)

The performance of steel reinforcement bars gradually degrades under fatigue loading, with most fatigue test failures leading to bar fracture. The strength degradation of steel bars during fatigue is characterized according to Miner’s linear cumulative fatigue damage theory, as shown in Equation (20).

$$D_{\mathrm{s}}={\sum }_{i=1}^j{\displaystyle \frac{n_i}{N}}$$

(20)

The residual fatigue strength of the reinforcement is given in Equation (21).

$${\mathrm{f}}_{\mathrm{y}}(N)=f_y\left[1-{\displaystyle \frac{N}{N_f}}\left(1-{\displaystyle \frac{{\sigma }_{\max }}{f_y}}\right)\right]$$

(21)

where ${\mathrm{N}}_{\mathrm{f}}$ represents the fatigue cycle life, ${\mathrm{D}}_{\mathrm{s}}$ is the damage factor, and ${\mathrm{n}}_{\mathrm{i}}$ denotes the number of the ith cycle. Failure occurs when the remaining fatigue strength of the reinforcement reaches its peak fatigue load stress.

3.3.4. Fatigue Properties of Steel Plate

The maximum stress of the steel plate does not reach the yield point under normal amplitude high circumferential fatigue, though its properties gradually degrade during the fatigue process. This degradation leads to the residual yield strength model of the steel plate, as described in Equation (22) [45].

$$f^{\prime }\left(n\right)=f-\left(f-{\sigma }_{s,max}\right){\left({\displaystyle \frac{n}{N_S}}\right)}^{c_1}$$

(22)

In this context, f represents the initial tensile strength of the steel plate, ${\sigma }_{\mathrm{s},\mathrm{m}\mathrm{a}\mathrm{x}}$ is the peak stress of the steel plate under cyclic loading, and ${\mathrm{N}}_{\mathrm{S}}$ denotes the fatigue life of the steel plate. The fatigue life of Q345 steel in this test can be calculated using Equation (23).

$$\left[\mathsf{\Delta }\sigma \right]={\left({\displaystyle \frac{C}{Ns}}\right)}^{1/\beta }$$

(23)

where the parameters $C$ and $\mathsf{\beta }$ are selected according to the Code for Design of Steel Structures [46], $\mathsf{\Delta }\sigma$ represents the constant amplitude of the fatigue stress, and Ns denotes the fatigue life. Failure occurs when the remaining yield strength falls below the peak fatigue strength of the steel plate. This condition must satisfy Equation (24).

$${\displaystyle \frac{f^{\prime }\left(n\right)}{{\sigma }_{s,max}}}<1$$

(24)

3.4. Numerical Simulation

ABAQUS was used to develop the finite element model of the test beams. The main assumptions include a perfect bond between the RC beams and the ECC layer, as confirmed by experimental observations, and the continuous attachment of steel plates to the bottom of the concrete beams to define the contact conditions. Concrete, bearing plates, and rigid pads were modeled using C3D8R solid elements, while reinforcement bars were modeled using T3D2 elements and were coupled at the loading point to avoid stress concentrations. Bond-slip between reinforcement and concrete was not considered, and the contact relationship was instead simulated using built-in embedding. The finite element model of the test beam is presented in Figure 14.

Fatigue Numerical Simulation Method

For high-cycle fatigue, conducting hundreds of thousands or even millions of simulations for a single stress amplitude in ABAQUS would require substantial computational time. Therefore, this paper proposes a method to simulate the performance at various stages of high-cycle fatigue using static loading at specified intervals. This approach accounts for the fatigue-related damage to the ECC, reinforcement, and steel plate, as well as the residual strain in the concrete. The specific calculation procedure is illustrated in Figure 15 below.

4. Results and Discussion

4.1. The Test Beams Under Static Loads

Static finite element analyses of the experimental beams CB-1 and EB-1 were conducted using the recommended cell sizes from Section 3.1 to validate the accuracy of the finite element models by comparing them with the experimental results.

4.1.1. Load–Deflection Curve

Figure 16 presents the load–deflection curves of the test beams alongside those predicted by the finite element model for comparison. In Figure 16a, the load–deflection curve from the finite element model turns at 12.5 kN, closely approximating the cracking load of CB-1, which is 14.65 kN. The simulation curve for CB-1 follows the same trend as the actual test load-displacement curve, and its ultimate load capacity is comparable to the test beams. The ultimate load capacities are 67.5 kN and 60.7 kN for the model and test beams, respectively, with a difference of less than 3%.

Figure 16b presents the load–deflection curve of EB-1, which shows a two-fold increase in load capacity compared to CB-1, demonstrating the feasibility of the new reinforcement method. The initial slopes of the load–deflection curves from the test beams are low, possibly due to the presence of micro-cracks within the concrete and gaps at contact surfaces. As the load increases, the curve’s slope gradually flattens, reaching the ultimate load capacity at 134 kN, after which the curve declines, indicating failure in EB-1. The ultimate load capacity predicted by the FEM closely matches the experimental results, confirming the model’s accuracy. However, the slope of EB-1’s FEM curve is slightly steeper, and its deflection is smaller compared to the test data. The main reasons for this discrepancy are as follows: (1) The finite element model assumes the concrete is homogeneous and dense, without internal micro-cracks or voids; (2) ideal boundary conditions could not be fully achieved in the testing process; (3) the model assumes perfect bonding between the steel reinforcement and concrete, as well as between the steel plate and the bottom of the concrete beam, neglecting bond-slip effects. This leads to the finite element model exhibiting higher overall stiffness compared to the actual test beam.

4.1.2. Failure Mode

The damage modes of CB-1 and EB-1 are illustrated in Figure 17. In CB-1, pure bending damage occurred, with the tensile reinforcement yielding and the concrete at the top of the beam being crushed. In contrast, no cracking was observed in the top concrete of EB-1, aligning with the finite element model results. Due to the high symmetry in the finite element model, no uneven crack propagation occurred during loading. The DAMAGET variable in the finite element model effectively recorded crack distribution and growth, and upon reaching the ultimate load capacity, EB-1 exhibited more cracks than CB-1 and experienced damage. It was primarily due to the steel plate positioned at the bottom of the beam, which limited additional cracking of the concrete and improved its load-carrying capacity.

4.2. The Test Beams Under Fatigue Loads

Finite element simulations of FCB-1 and FEB-1 under fatigue loading were conducted. Figure 18 compares the test results with the finite element modeling results across different fatigue cycles. Both show consistent overall trends, with the mid-span deflection at the upper fatigue load limit increasing gradually as the number of fatigue cycles rises, and the deflections of the test beams exhibit a linear change. The slopes of the curves from the finite element model are steeper than those from the test beams, likely due to stricter boundary conditions in the finite element model compared to the test setup. Overall, however, the stiffness of the test beams decreases progressively with increasing fatigue cycles. The error in reaching the maximum load amplitude for the same number of fatigue cycles is within 1.5 mm.

The residual strength of concrete and steel gradually decreases with increasing fatigue cycles, highlighting the importance of studying deformation throughout the process. Finite element simulations were conducted for 200,000 and 2,000,000 fatigue cycles on FCB-1 and FEB-1, respectively, to compare their load–deflection curves, as presented in Figure 19.

As illustrated in Figure 19a, the load–deflection curves from the finite element model exhibit steeper slopes compared to the experimental results. After 200,000 cycles of fatigue loading, the load–deflection curves align more closely with those from the finite element model, with a slight decrease in ultimate load-carrying capacity. The finite element model’s load–deflection curves after 2,000,000 fatigue cycles also follow a similar trend to the test results, supporting the model’s feasibility. The ultimate load-carrying capacity decreases by approximately 6.7% after 2,000,000 cycles of fatigue loading, further validating the model’s accuracy.

4.3. Parameter Analysis

To investigate the factors affecting the load-carrying capacity of reinforced beams, parametric analyses were conducted on variables including NSC (Normal Strength Concrete), ECC layer thickness, and steel plate thickness. The detailed parameters for each group of specimens are presented in

Table 5. Details of specimen parameters.

Group	Specimen	NSC (MPa)	ECC (MPa)	TECC (mm)	TS-P (mm)
G1	N1	30	50	30	5
	N2	35	50	30	5
	N3	40	50	30	5
	N4	50	50	30	5
G2	T1	30	50	10	5
	T2	30	50	20	5
	T3	30	50	30	5
	T4	30	50	40	5
	T5	30	50	50	5
G3	TS1	30	50	30	3
	TS2	30	50	30	4
	TS3	30	50	30	5
	TS4	30	50	30	6

Notes: TS-P means the thickness of the steel plate; TECC means the thickness of the ECC.

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4.3.1. Strength of NSC

With all other material properties held constant and only the strength of plain concrete varying, the resulting load–deflection curves are presented in Figure 20. The load–deflection curves of the test beams can be broadly categorized into three distinct stages: cracking, elastic–plastic, and damage. An increase in concrete strength leads to a corresponding rise in the load capacity of the test beam, albeit marginally, with N4 exhibiting only a 4.5% increase compared to the load capacity of N1. Due to its lower cracking strength, N1 exhibits the greatest deformation capacity, resulting in the lowest overall stiffness of the beam.

4.3.2. Thickness of the ECC Layer

The load–deflection curves obtained by varying only the thickness of the ECC layer are presented in Figure 21. Increasing the thickness of the ECC layer notably enhances both the mid-span deflection and the ultimate load-carrying capacity of the test beams. When the ECC layer thickness increases from 10 mm to 50 mm, the ultimate load-carrying capacity increases by 16.2%, while the mid-span deflection at the ultimate load increases by 37.9%. However, increasing the ECC layer thickness does not significantly enhance the overall stiffness of the beams; it primarily affects their overall deformation capacity.

4.3.3. Thickness of Steel Plate

Figure 22 presents the load–deflection curves for varying steel plate thicknesses in the finite element model. The thickness of the steel plate has a direct impact on the stiffness of the test beam, with flexural stiffness increasing proportionally with the steel plate thickness. A thicker steel plate results in a greater ultimate load-carrying capacity; for instance, when the steel plate thickness increases from 3 mm to 6 mm, the load-carrying capacity increases by approximately 25.2%, while the mid-span deflection increases by about 17.2%.

5. Conclusions

This study investigated the static and fatigue performance of concrete beams reinforced with a combination of ECC layers and steel plates, focusing on load-carrying capacity, deformation capacity, and damage patterns through both experimental testing and finite element analysis. The findings demonstrate strong alignment between the test and finite element analysis results, validating the feasibility and accuracy of the finite element approach. Building on these results, additional parametric studies were conducted. The conclusions are as follows:

• The load-carrying capacity of the RC beam reinforced with an ECC layer and steel plate is 134.04 kN, which is double that of an ordinary concrete beam, demonstrating a clear reinforcement effect.
• A finite element model incorporating the fatigue damage of both concrete and steel was established and compared with test results. The load–deflection curves and damage modes of the test beams from the finite element model closely matched the test results, further validating the model’s feasibility.
• Parametric analysis indicates that increasing the strength of ordinary concrete has a limited impact on enhancing the load-carrying capacity of beams, whereas increasing the thickness of the ECC layer significantly enhances the ductility of composite beams. Additionally, increasing the thickness of the steel plate effectively strengthens the overall structural load-carrying capacity. These findings provide theoretical guidance for the application and design of composite reinforcement methods in practical projects, making them particularly suitable for the reinforcement design of concrete structures under cyclic loading.
• Although this study has demonstrated the significant effectiveness of the ECC and steel plate composite reinforcement method in improving the fatigue performance and load-carrying capacity of concrete beams, the finite element model assumes a perfect bond at the interface without fully accounting for the fatigue characteristics associated with bond-slip between the concrete, ECC layer, and steel plate. Future research should focus on further investigating the mechanisms of fatigue behavior related to bond-slip.