Degradation of Elastic Modulus of Ordinary Concrete Under Flexural Fatigue Loading

Abstract

To elucidate the degradation behavior of elastic modulus in normal-strength ordinary concrete under flexural fatigue loading, this study systematically examines its evolution in C50 concrete, which is widely used in engineering applications. Based on four-point bending fatigue test data of plain concrete (PC) and reinforced concrete (RC) beams, degradation curves of the relative residual elastic modulus as a function of the cycle ratio were established. To quantitatively characterize the fatigue degradation process, two integrated indicators—the area under the curve (AUC) and the stable-stage degradation slope (|$K_{mid}$|)—were introduced to represent the degree of cumulative damage and the degradation rate of elastic modulus, respectively. These indicators were subsequently employed to evaluate the effects of maximum stress level, stress ratio, and reinforcement on elastic modulus degradation. The results show that failed PC specimens exhibited a typical three-stage S-shaped degradation pattern, whereas RC specimens primarily exhibited a two-stage degradation behavior. However, the elastic modulus of runout PC specimens remained above 93% of its initial value throughout the entire loading process. For PC specimens, under the same maximum stress level, increasing the minimum stress level from 0.10 to 0.25 resulted in a 24% decrease in |$K_{mid}$| from 0.2505 to 0.1912. At the same minimum stress level, increasing the maximum stress level from 0.75 to 0.90 led to a 94% increase in |$K_{mid}$| from 0.1912 to 0.3705. The presence of reinforcement increased AUC by 3~15% and reduced |$K_{mid}$| by 54~74%, indicating that reinforcement not only mitigated overall damage accumulation but also significantly slowed the degradation rate of the elastic modulus during the stable fatigue stage. The degradation characterization approach proposed in this study provides a simplified and practical framework for fatigue analysis of concrete components based on damage mechanics.

1. Introduction

Concrete is one of the most widely used materials in modern civil engineering, and its mechanical performance has attracted sustained attention in both academic and engineering communities. With the rapid development of high-speed railway, airport construction, offshore energy exploitation, and wind power engineering, numerous concrete infrastructure facilities have entered service. Structures such as bridges, airport pavements, offshore platforms, and wind turbine foundations are often subjected to tens of millions of repeated load cycles during their service life, substantially increasing the risk of fatigue-induced damage [1,2]. Therefore, accurately characterizing the performance degradation of concrete under fatigue loading is essential for structural safety assessment and service-life management [3].

In fatigue research, many studies have examined damage evolution at the structural level using stiffness as a key indicator [4,5,6]. Reinforced concrete beams typically exhibit a three-stage stiffness evolution under fatigue loading: an initial rapid decline, a subsequent slow-varying stage, and an accelerated deterioration phase prior to failure. Although various stiffness-degradation and cumulative-damage models have been proposed [7,8], structural stiffness is influenced by reinforcement ratio, cross-sectional configuration, and boundary conditions in addition to material degradation. Consequently, stiffness evolution alone cannot fully reveal intrinsic material-scale damage mechanisms.

In contrast, the elastic modulus more directly reflects material-scale damage accumulation under cyclic loading [9]. Prior studies have demonstrated stage-dependent modulus degradation behavior and clarified its relationship with microcrack evolution and energy dissipation mechanisms. Birkner et al. compared stiffness degradation measured on fatigue-loaded concrete cylinders and large-scale beams and highlighted that stiffness-related indicators exhibit a stage-dependent evolution that is consistent across specimen scales [10]. Zhang et al. experimentally investigated stiffness degradation of RC beams under coupled corrosion–fatigue loading and discussed the associated evolution of structural responses, further supporting the link between stiffness/modulus degradation and member-level fatigue performance [11]. Existing studies have generally shown that the elastic modulus of ordinary concrete exhibits pronounced nonlinear degradation during fatigue loading, and its evolution typically proceeds through three stages: an initial rapid decline, a subsequent slow-degradation stage, and an accelerated deterioration phase toward the end of fatigue life [12,13,14,15]. However, Gan et al. [16] reported in their review of cementitious materials that the elastic modulus of ordinary concrete under flexural fatigue decreases continuously, with the early-stage reduction most sensitive to fatigue life. Meng et al. [17] calculated the dynamic elastic modulus from fatigue hysteresis loops and proposed a damage indicator based on the relative deformation modulus. Yadav et al. [18], from a continuum damage mechanics perspective, revealed the intrinsic relationships among modulus degradation, microcrack initiation, crack propagation, and plastic strain accumulation. Wang [19] further explained the mechanism underlying the three-phase degradation of secant stiffness from the viewpoints of material randomness and the evolution of energy release rate, thereby elucidating the physical essence of the fatigue-induced reduction in elastic modulus. Collectively, these studies highlight that the elastic modulus serves as a critical link between material-level damage mechanisms and the macroscopic degradation of structural performance. Nevertheless, research on elastic modulus degradation under flexural fatigue remains less extensive than under compressive or tensile fatigue, particularly for normal-strength concretes widely used in engineering practice. Reported degradation curves also exhibit noticeable variability.

Despite the substantial progress achieved to date, several issues remain insufficiently clarified for elastic modulus degradation under flexural fatigue loading. First, the effects of different stress ratios and maximum stress levels on the degradation curve still require systematic verification under comparable test conditions. Second, the influence of reinforcement on material-level degradation has not yet been quantitatively described using a unified set of measures, and comparative studies on the elastic modulus degradation of plain concrete and reinforced concrete under identical fatigue loading conditions remain limited. These gaps are particularly relevant for C50 concrete, which is among the most widely adopted strength grades in engineering practice.

To address these gaps, this study systematically investigates elastic modulus degradation of C50 concrete under flexural fatigue using four-point bending fatigue test data of plain and reinforced concrete beams [20]. First, secant modulus values are determined through batch data processing and analysis, and relative residual elastic modulus–cycle ratio curves are constructed. These results are then used to compare the evolution characteristics of elastic modulus degradation between PC and RC. Subsequently, four empirical functions are adopted to fit and evaluate the relative residual elastic modulus curves, and the most suitable mathematical model is identified. To quantitatively assess the degradation process, two integrated indicators—the area under the curve (AUC) and the mid-stage degradation slope (|$K_{mid}$|)—are introduced to characterize cumulative damage and degradation rate, respectively, and to quantify the effects of stress ratio, maximum stress level, and reinforcement on elastic modulus degradation. The resulting characterization, framed in a simplified damage-mechanics framework, provides a practical approach for fatigue assessment of concrete components and offers an efficient alternative when extensive fatigue testing is costly and time-consuming.

2. Materials and Methods

2.1. Flexural Fatigue Tests on Concrete Specimens

The fatigue test data used in this study were derived from previously conducted four-point bending constant-amplitude fatigue experiments on plain concrete (PC) and reinforced concrete (RC) small beams made from C50 concrete. The detailed test procedures and loading scheme are reported in [20]. For clarity and to facilitate subsequent analysis of elastic modulus degradation, key information regarding material composition, specimen preparation, and test parameters is briefly summarized below.

2.1.1. Composition and Properties of Concrete

The commercial C50 ordinary concrete has a mix ratio of cement:sand:stone:water = 1:1.93:3.02:0.46. Ordinary 425 Portland cement, medium natural sand with a fineness modulus of 2.4, rubble, and cobble gravel with a maximum particle size of 25 mm were selected to produce the desired concrete. The target slump was 180 ± 20 mm. A high-performance water-reducing agent, STD-PCS (a polycarboxylic acid-type superplasticizer) at 1.99%, was added to improve workability. More information about the concrete mix can be found in Reference [20].

PC and RC specimens were made from two batches of the C50 concrete. The concrete batch used to make the PC specimens had an average 28-day compressive strength of 53.9 MPa with a coefficient of variation of 0.14, and those of the RC series were 51.1 MPa and 0.02, based on standard tests on six companion cubes for each series [20]. According to 33 standard four-point bending static tests, the average flexural strengths of the PC and RC specimens were 5.6 MPa and 5.1 MPa, respectively, and the coefficient of variations were 0.10 and 0.09 [20].

2.1.2. Test Specimens and Grouping

The flexural fatigue tests were conducted using standard flexural specimens specified in the Code for Test Methods of Mechanical Properties of Ordinary Concrete (GB/T 50081-2016) with a cross-section of 150 mm × 150 mm and a length of 550 mm, as shown in Figure 1 [20]. The geometries of the PC and RC specimens are identical, except that two longitudinal HRB400 steel bars with a diameter of 12 mm, representing a typical transverse reinforcement ratio for bridge decks, were continuously placed along the bottom of the RC beam, with a concrete cover thickness of 30 mm.

The specimens were categorized into PC and RC series, and the detailed test sets for 55 PC specimens and 42 RC specimens are presented in

Table 1. Test sets of PC and RC specimens in flexural fatigue testing [20].

Series	Set Designation	$\mathbf{Maximum} \mathbf{Stress} \mathbf{Level} {\mathbf{S}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$	$\mathbf{Minimum} \mathbf{Stress} \mathbf{Level} {\mathbf{S}}_{\mathbf{m}\mathbf{i}\mathbf{n}}$	Stress Ratio R	Number of Specimens
PC	S-65-10	0.65	0.10	0.154	3
	S-75-10	0.75	0.10	0.133	13
	S-75-25	0.75	0.25	0.333	12
	S-80-10	0.80	0.10	0.125	12
	S-90-10	0.90	0.10	0.110	5
	S-90-25	0.90	0.25	0.280	5
	S-90-40	0.90	0.40	0.440	5
RC	J-S-75-10	0.75	0.10	0.133	7
	J-S-75-25	0.75	0.25	0.333	7
	J-S-80-10	0.80	0.10	0.125	7
	J-S-80-25	0.80	0.25	0.313	7
	J-S-85-10	0.85	0.10	0.117	7
	J-S-85-25	0.85	0.25	0.294	7

[20]. Fatigue stress levels are defined as the ratios of applied fatigue stresses to the corresponding average flexural strengths. Specimen designation in PC series is S-a-b-x, where S indicates stress-controlled loading, a denotes the maximum stress level, b the minimum stress level, and x the sequence number within each test set. Similarly, RC specimens are labeled J-S-a-b-x, where J indicates reinforced concrete and the remaining parameters have the same meanings as in the PC series [20].

2.1.3. Loading and Instrumentation

The loading configuration of the four-point bending constant-amplitude fatigue tests is illustrated in Figure 2 [20].

A QBS-series electro-hydraulic servo fatigue testing machine, as shown in Figure 3, was employed to apply cyclic loading. The machine has a maximum load capacity of 50 kN and operates within a loading frequency range of 0.01~100 Hz. The maximum and minimum fatigue loads for each test set were determined according to the stress levels specified in Table 1. Detailed descriptions of the test setup, support configuration, and loading fixtures are reported in [20].

An extensometer with an initial gauge length of 6 mm and a measurement accuracy of 0.001 mm was mounted at the bottom surface of the concrete beam in the pure bending region. To ensure cracking occurred within the designated gauge length, rigid Z-shaped steel angles were bonded to the bottom surface of the specimen to enlarge the extensometer’s effective gauge length. After amplification, the effective gauge length $L_0$ was 90 mm for PC specimens and 146 mm for RC specimens. The configuration of the steel angles and the measurement setup is shown in Figure 4 [20].

2.2. Computation of Stress–Strain Curves

The fatigue testing machine automatically recorded the applied load, actuator displacement, and number of cycles. In the pure bending region of each specimen, an extensometer was installed to measure the axial deformation of the beam. Based on the recorded load–displacement data, the bending stress and strain were calculated according to classical mechanics-of-materials formulations.

2.2.1. Stress Calculation

The bending stress $\sigma$ was computed as follows:

$$\sigma = {\displaystyle \frac{My}{I}}.$$

(1)

In Equation (1), M denotes the bending moment, which is calculated as:

$$M=F{\displaystyle \frac{a}{2}},$$

(2)

where F is the applied load from the testing machine, and a is the distance between the loading point and the support, as shown in Figure 2. y is the distance from the neutral axis to the measurement point (for a rectangular section, y = h/2, where h is the section depth). I is the moment of inertia of the cross-section, which is given by:

$$I={\displaystyle \frac{b{h}^3}{12}},$$

(3)

where b is the width of the cross-section, as shown in Figure 1.

2.2.2. Strain Calculation

The strain $\epsilon$ was calculated using:

$$\epsilon ={\displaystyle \frac{\Delta L}{L_0}},$$

(4)

where $\Delta L$ is the axial deformation recorded by the extensometer, and $L_0$ is the effective gauge length as shown in Figure 4a.

2.3. Calculation of Elastic Modulus

Chen et al. [14] employed the secant modulus to characterize elastic modulus degradation during fatigue tests, and they successfully developed a fatigue damage model that accurately captures the three-stage evolution of damage. Following this approach, the present study also adopts the secant modulus as the indicator of fatigue elastic modulus. Please note that the development of plastic strain may affect the representativeness of the secant modulus. Since pronounced plastic deformation tends to accumulate only toward the final several cycles of loading, the difference among secant modulus, loading modulus, and unloading modulus is minimal for the majority of fatigue cycles. The secant modulus was adopted in this study primarily for its practicality in batch data processing, enabling a simple procedure to be applied across specimens and sampling points for comparative analysis.

Secant modulus is defined as the slope of the line connecting the two extreme stress points (minimum and maximum stress) within a single loading cycle, as illustrated in Figure 5. In the figure, ${\epsilon }_{min}^1$ and ${\epsilon }_{max}^1$ correspond to the minimum and maximum fatigue strains in the initial loading cycle; ${\epsilon }_{min}^f$ and ${\epsilon }_{max}^f$ represent the minimum and maximum strains in the final cycle at fatigue failure. At any intermediate cycle n, the minimum and maximum fatigue strains are ${\epsilon }_{min}^n$ and ${\epsilon }_{max}^n$. The secant slope is taken as the elastic modulus, with E₀ and E_N denoting the initial elastic modulus and the elastic modulus at fatigue failure, respectively. The elastic modulus after n fatigue cycles E_n can be similarly calculated.

Given the large volume of fatigue test data, the raw stress–strain records were batch-processed using a MATLAB (version R2023b) program developed by The MathWorks, Inc. (Natick, MA, USA). To simplify the computation and avoid analyzing every individual cycle, the maximum fatigue life N of each specimen was used as the sampling reference. Representative cycle positions were selected at intervals of 0.1N within the range 0.1N to 0.9N. For each selected cycle number, a window of 10 consecutive cycles, including five cycles before and five cycles after the target cycle, was extracted as the stress–strain data segment for that position. At the beginning and end of the fatigue process, the first and last 10 cycles were used as endpoint data segments. Because the elastic modulus degrades more rapidly during the early and late stages of fatigue loading [15], the sampling density was increased within the intervals 0~0.1N and 0.9N~N. Additional samples were taken at cycle numbers corresponding to 0.025N, 0.05N, 0.075N, 0.925N, 0.95N, and 0.975N, again using 10 consecutive cycles for each sampling point.

Each data segment was fitted using linear regression, and the slope of the fitted line was defined as the elastic modulus corresponding to that cycle number. Figure 6 illustrates the stress–strain data and fitted regression line for a representative cycle.

By compiling the slopes obtained at all sampled cycles, the evolution curve of elastic modulus versus cycle count was constructed for each specimen, as shown in Figure 7.

2.4. Definition of Damage Variable

In fatigue damage research, the evolution of damage is typically described using parameters that reflect the degradation of material properties. During fatigue loading, damage accumulates progressively within a specimen, leading to a gradual reduction in its elastic modulus. Thus, the degradation of the elastic modulus serves as an indicator of the specimen’s damage state. To map the modulus degradation onto a unified, physically meaningful scale suitable for model development, the present study defines a damage variable, D, based on the attenuation of the elastic modulus. This variable is expressed as a function of the fatigue cycle ratio n/N, as follows:

$$D={\displaystyle \frac{E_0-E_n}{E_0-E_N}}=f\left(n/N\right),$$

(5)

where D ∈ [0, 1] is the damage variable; $E_0$ is the initial elastic modulus; $E_n$ is the elastic modulus after n fatigue cycles; and $E_N$ is the elastic modulus at fatigue failure.

Based on Equation (5), the relationship between the relative residual elastic modulus $E_n$ and the cycle ratio n/N can be further written as:

$${\displaystyle \frac{E_n}{E_0}}=1-\left(1-{\displaystyle \frac{E_N}{E_0}}\right)f\left(n/N\right).$$

(6)

2.5. Degradation Functions

Based on Equation (6), the attenuation of elastic modulus during fatigue loading is normalized into a monotonically evolving damage process as a function of the cycle ratio. Four empirical models have been commonly used in the literature to describe the fatigue damage of concrete.

Holmen [21] proposed a classical linear degradation model for concrete fatigue damage while studying deformation characteristics under constant- and variable-amplitude fatigue loading. This model has been widely adopted by subsequent researchers. Moreover, Zhao et al. [22] developed a cubic polynomial model to characterize the compressive fatigue damage in high-strength concrete following elevated-temperature exposure. Lian et al. [23] employed a damage model for the fatigue analysis of composite materials, demonstrating good predictive capability. Recently, Chen et al. [24] introduced a nonlinear fatigue damage evolution equation to describe the damage progression of concrete under cyclic loading.

The normalized functions of the above four models for the residual elastic modulus are expressed as:

$${\displaystyle \frac{E_n}{E_0}}=1-0.33 n/N,$$

(7)

$${\displaystyle \frac{{{\rm E}}_n}{{{\rm E}}_0}}=1-\left(1-{\displaystyle \frac{{{\rm E}}_{{\rm N}}}{{{\rm E}}_0}}\right)\left[a{\left(n/N\right)}^3+b{\left(n/N\right)}^2+c\left(n/N\right)+d\right],$$

(8)

$${\displaystyle \frac{E_n}{E_0}}=1-\left(1-{\displaystyle \frac{E_N}{E_0}}\right)\left[a{\left({\displaystyle \frac{a^b+1}{a^b+1-n/N}}-1\right)}^{{\displaystyle \frac{1}{b}}}\right],$$

(9)

$${\displaystyle \frac{E_n}{E_0}}=1-\left(1-{\displaystyle \frac{E_N}{E_0}}\right)\left[1-{\displaystyle \frac{1-(n/N{)}^a}{(1-n/N{)}^b}}\right],$$

(10)

where a, b, c, and d are material parameters fitted from regression analysis of test data. Equations (7)–(10) are from [19,20,21,22], respectively.

3. Results and Discussion

3.1. Prescreening of Specimens

Based on the procedure described in Section 2.2, continuous cyclic stress–strain curves were obtained throughout the fatigue loading process. Representative cyclic stress–strain responses for PC and RC specimens are shown in Figure 8.

Before further data processing, all cyclic stress–strain records were carefully examined for completeness and physical validity. Several specimens exhibited clear abnormalities in the raw data, which can be categorized into four types, as illustrated in Figure 9. As shown in Figure 9a, some specimens displayed pronounced irregular fluctuations in the hysteresis loops during loading and unloading, indicating signal disturbances or sensor malfunction. As illustrated in Figure 9b, some specimens had missing segments in their stress–strain records, preventing the reliable calculation of the elastic modulus. As shown in Figure 9c, the measured strain exhibited an abnormal decrease during the mid-to-late stages of fatigue loading. The red-dashed region highlights the abnormal strain reduction, attributed to slippage or loosening of the extensometer during long-term cyclic testing. In addition, a small number of specimens, as shown in Figure 9d, failed within 10 loading cycles, which is indicative of low-cycle fatigue failure. Since the present study focuses on the degradation law of elastic modulus under high-cycle flexural fatigue, these specimens were not considered representative of the targeted fatigue regime and were therefore excluded from subsequent analysis.

The number of specimens corresponding to each type of abnormality is summarized in

Table 2. Summary of abnormal specimens excluded from further analysis of elastic modulus.

Type of Specimen Abnormality	PC Specimen	RC Specimen
Irregular fluctuations in stress–strain data	10	0
Missing stress–strain data	4	0
Abnormal strain decrease	3	1
Low-cycle fatigue failure	5	0
Total	22	1

. Among them, the most common cause is extensometer slippage, which typically manifests as abrupt changes in the strain monitoring signals. It should be noted that, if such discontinuities occur outside the sampling windows used for modulus calculation in this study, their influence on the computed secant modulus at the selected sampling points is limited. Nevertheless, these specimens were conservatively excluded. To ensure the accuracy and reliability of the elastic modulus calculations, only specimens with complete and physically sound stress–strain responses within the high-cycle fatigue regime were retained. After screening, 33 valid PC specimens (out of 55) and 41 valid RC specimens (out of 42) were obtained, still sufficient in number to provide an adequate sample size for subsequent analyses.

Table 3. Fatigue life test results of screened PC and RC specimens [20].

Set Designation	Specimen Designation Fatigue Life N (Cycle)
S-65-10	S-65-10-1 * 1,000,000	S-65-10-2 * 1,000,000	S-65-10-3 * 1,000,000
S-75-10	S-75-10-2 32,835	S-75-10-4 7574	S-75-10-7 528	S-75-10-8 63	S-75-10-9 14,048	S-75-10-10 224,931	S-75-10-11 11,328
	S-75-10-12 49,629	S-75-10-13 19,585
S-75-25	S-75-25-2 2518	S-75-25-3 * 365,759	S-75-25-4 34,187	S-75-25-5 5321	S-75-25-6 7308	S-75-25-8 32,646	S-75-25-9 26,049
	S-75-25-10 39,379	S-75-25-11 14,209	S-75-25-12 21,097
S-80-10	S-80-10-1 3872	S-80-10-3 496	S-80-10-4 6115	S-80-10-5 1391	S-80-10-8 * 61,960	S-80-10-9 1036	S-80-10-10 618
S-90-25	S-90-25-2 216	S-90-25-3 110	S-90-25-4 196	S-90-25-5 234
J-S-75-10	J-S-75-10-1 13,213	J-S-75-10-2 16,030	J-S-75-10-3 14,274	J-S-75-10-4 14,680	J-S-75-10-6 72,320	J-S-75-10-7 23,793
J-S-75-25	J-S-75-25-1 598	J-S-75-25-2 32,730	J-S-75-25-3 35,214	J-S-75-25-4 41,908	J-S-75-25-5 38,683	J-S-75-25-6 41,878	J-S-75-25-7 48,434
J-S-80-10	J-S-80-10-1 3782	J-S-80-10-2 5087	J-S-80-10-3 4517	J-S-80-10-4 5226	J-S-80-10-5 5857	J-S-80-10-6 4028	J-S-80-10-7 5459
J-S-80-25	J-S-80-25-1 9696	J-S-80-25-2 9900	J-S-80-25-3 10,954	J-S-80-25-4 4870	J-S-80-25-5 9307	J-S-80-25-6 8976	J-S-80-25-7 6780
J-S-85-10	J-S-85-10-1 1800	J-S-85-10-2 228	J-S-85-10-3 516	J-S-85-10-4 634	J-S-85-10-5 586	J-S-85-10-6 611	J-S-85-10-7 691
J-S-85-25	J-S-85-25-1 2331	J-S-85-25-2 1765	J-S-85-25-3 2544	J-S-85-25-4 2249	J-S-85-25-5 1976	J-S-85-25-6 4221	J-S-85-25-7 6185

Note: Specimens marked with “*” did not fail during the fatigue test and were termed runouts.

presents the fatigue life of the screened PC specimens and the fatigue cracking life of the RC specimens. Please note that the fatigue cracking life of the RC specimens is determined when a macroscopic concrete crack appears on the side surfaces. Fatigue loading was then continued after concrete cracking and surpassed the observed fatigue life of the PC counterparts, without any sign of reinforcement yielding. The predetermined number of cycles for fatigue test termination in the RC series ranged from 10,000 to 80,000, depending on the tested stress levels [20].

As shown in Table 3, the fatigue life exhibits pronounced scatter even under the same nominal loading condition. This phenomenon is due to the inherent heterogeneity of concrete and the stochastic nature of crack initiation under cyclic bending [21]. At the same time, minor differences in boundary conditions may further amplify the dispersion. Accordingly, normalization to represent the typical behavior and statistical treatment to reflect the scatter are justified for further investigation.

3.2. Elastic Modulus Degradation

The elastic modulus degradation of all screened specimens was obtained following the calculation procedures outlined in Section 2.2 and Section 2.3. Figure 10 and Figure 11 present representative results for PC and RC specimens, respectively. It can be observed that, even under identical loading conditions, noticeable dispersion exists among specimens in terms of fatigue life and initial elastic modulus.

To more accurately characterize the degradation behavior, the elastic modulus evolution curves of all valid specimens were normalized. The horizontal axis was expressed as the cycle ratio $n/N$, and the vertical axis was expressed as the relative residual elastic modulus $E_n/E_0$, where $E_n$ represents the elastic modulus at a given cycle ratio, calculated following the secant modulus definition described in Section 2.3, for each sampled fatigue cycle.

For specimens that did not fail during the fatigue tests (termed runouts), the calculated relative residual elastic modulus results are shown in

Table 4. Calculated relative residual elastic modulus in PC runout specimens.

Cycle Ratio	Specimen Designation
$\mathit{n}/\mathit{N}$	S-65-10-1	S-65-10-2	S-65-10-3	S-75-25-3	S-80-10-8
0	1.00	1.00	1.00	1.00	1.00
0.025	0.96	0.95	0.98	0.98	0.98
0.075	1.00	0.95	0.98	0.98	0.99
0.10	0.93	0.93	0.95	0.97	0.96
0.20	0.93	1.00	0.98	0.98	0.96
0.30	0.96	1.00	0.94	0.95	0.98
0.40	0.96	1.00	0.94	0.97	0.98
0.50	0.96	1.00	0.93	0.98	0.98
0.60	1.00	1.00	1.00	0.99	0.98
0.70	0.96	1.00	0.94	0.95	0.99
0.80	0.96	0.95	0.94	0.99	0.97
0.90	1.00	1.00	0.95	1.00	0.97
0.925	0.96	1.00	1.00	0.99	0.98
0.95	1.00	1.00	0.96	0.99	0.98
0.975	0.96	1.00	0.97	0.96	0.98
1.0	1.00	0.95	0.94	0.95	0.96

.

As shown in Table 4, the relative residual elastic modulus of the runout PC specimens remained above 0.93 throughout the entire loading process, indicating no noticeable degradation of elastic modulus.

For specimens that failed under fatigue loading, the degradation of elastic modulus is presented in Figure 12 and Figure 13 for PC and RC specimens, respectively.

As illustrated in Figure 12, the elastic modulus degradation of failed PC specimens exhibits a clear three-staged behavior compared with that of runout specimens. The degradation follows a monotonically decreasing “S-shaped” trend: a rapid reduction in the early stage of fatigue loading, followed by a gradual decrease during the mid-life region, and finally a pronounced drop as failure approaches.

However, the degradation behavior of RC specimens in Figure 13 seems different from that of PC. Under fatigue loading, the elastic modulus of RC specimens exhibits an apparent two-stage evolution: an initial rapid degradation stage, followed by a relatively stable, slowly degrading stage. This behavior is primarily because RC specimens were not loaded to complete fatigue failure (i.e., yielding and fracture of reinforcement), mainly due to constraints imposed by machine capabilities and operating conditions [20]. With the fatigue machine’s configuration and operational control scheme primarily tailored for low-cycle fatigue, practical limits on loading frequency and overall test efficiency made it difficult to conduct high-cycle fatigue tests on RC specimens capable of reaching several million cycles to failure. In addition, due to laboratory operational constraints, the machine had to be shut down at night, and tests could only be conducted during the day. When loading resumed the next day, a strain recovery process was observed in the strain–time records, disrupting the continuity of the strain data and adding uncertainty to the test results. Therefore, under the current experimental setup, the behavior of RC specimens close to fatigue failure was not captured and they showed a two-stage evolution similar to runout PC specimens.

3.3. Characterization of Elastic Modulus Degradation

As shown in Figure 12 and Figure 13, the elastic modulus degradation curves of specimens subjected to the same stress level exhibit noticeable dispersion, primarily due to the inherent heterogeneity of concrete and the stochastic nature of fatigue failure [21]. To obtain a more representative degradation trend and reduce the influence of extreme values, the median statistic method was used to synthesize the results from all specimens in each test set. Specifically, at each predefined cycle ratio, the relative residual elastic modulus values of all specimens within the test set were collected, and their median was calculated to represent the characteristic degradation level at that cycle ratio. The interquartile range (IQR), defined as the interval between the 25th and 75th percentiles, was used to quantify the dispersion among specimens under the same loading condition. Figure 14 illustrates the statistical degradation curves for representative sets of PC and RC specimens.

In the figure, the blue solid line represents the median degradation curve constructed from the median at each cycle ratio. In contrast, the shaded band denotes the IQR, capturing variability across specimens. To further provide a quantitative measure of the overall scatter level, an integrated IQR index was defined as:

$${IQR}_{int}={\int }_0^1\left[Q_3\left(x\right)-Q_1\left(x\right)\right]dx, x=n/N,$$

(11)

where $Q_1\left(x\right)$ and $Q_3\left(x\right)$ are the 25th and 75th percentiles of $E_n/E_0$ across specimens at the same cycle ratio. A larger ${IQR}_{int}$ indicates greater dispersion over the fatigue life.

Taking the PC series S-80-10 and the RC series J-S-80-10 as examples, the calculated ${IQR}_{int}$ are 0.255 for S-80-10 and 0.160 for J-S-80-10. This quantitative comparison is consistent with the intuitive observation from Figure 14, where the shaded IQR band of J-S-80-10 is narrower than that of S-80-10.

The median is less sensitive to outliers and data scatter [25], providing a stable representation of the typical degradation behavior at each cycle ratio. Therefore, the median degradation curve is adopted to process specimens tested under the same loading condition and to represent the degradation law of that specific test set.

3.4. Elastic Modulus Degradation Law

For each specimen set, the relative residual elastic modulus obtained above was fitted using Equations (7)–(10) to characterize the degradation of elastic modulus during fatigue loading. The degradation curves for PC and RC specimens are shown in Figure 15 and Figure 16, and the corresponding fitted parameters are summarized in

Table 5. Fitted model parameters describing elastic modulus degradation in PC specimens.

Set Designation	Equation	a	b	c	d	R2
S-75-10	(7)	/	/	/	/	0.554
	(8)	4.0548	−5.2404	1.9817	0.0119	0.910
	(9)	0.2403	3.3876	/	/	0.983
	(10)	0.7860	0.8242	/	/	0.943
S-75-25	(7)	/	/	/	/	0.617
	(8)	3.7970	−5.1059	2.0404	−0.0096	0.849
	(9)	0.2303	4.1502	/	/	0.993
	(10)	0.7612	0.8793	/	/	0.978
S-80-10	(7)	/	/	/	/	0.656
	(8)	2.7832	−3.6441	1.6927	0.0032	0.948
	(9)	0.2925	3.2873	/	/	0.991
	(10)	0.7411	0.7652	/	/	0.989
S-90-25	(7)	/	/	/	/	0.178
	(8)	4.1216	−5.7190	2.5438	0.0062	0.977
	(9)	0.3020	3.4969	/	/	0.975
	(10)	0.6467	0.7169	/	/	0.951

and

Table 6. Fitted model parameters describing elastic modulus degradation in RC specimens.

Set Designation	Equation	a	b	c	d	R2
J-S-75-10	(7)	/	/	/	/	−0.053
	(8)	2.8720	−5.3025	3.2751	0.1899	0.935
	(9)	9.8589	3.5363	/	/	0.993
	(10)	0.2826	0.0036	/	/	0.993
J-S-75-25	(7)	/	/	/	/	–0.967
	(8)	3.6815	−6.8646	3.9615	0.2796	0.859
	(9)	10.4555	5.34662	/	/	0.973
	(10)	0.1619	1.0 × 10−6	/	/	0.973
J-S-80-10	(7)	/	/	/	/	0.123
	(8)	4.3911	−7.9844	4.5452	0.0814	0.973
	(9)	12.2145	3.3233	/	/	0.921
	(10)	0.3009	4.2 × 10−6	/	/	0.921
J-S-80-25	(7)	/	/	/	/	−0.833
	(8)	3.8724	−7.1015	4.0179	0.2691	0.876
	(9)	16.435	5.8075	/	/	0.969
	(10)	0.1697	1.6 × 10−6	/	/	0.969
J-S-85-10	(7)	/	/	/	/	−0.963
	(8)	4.234	−7.4282	4.0176	0.2847	0.858
	(9)	15.7181	5.1691	/	/	0.981
	(10)	0.1935	1.0 × 10−6	/	/	0.981
J-S-85-25	(7)	/	/	/	/	−0.835
	(8)	3.2716	−6.2429	3.7976	0.2506	0.860
	(9)	16.3067	5.20951	/	/	0.962
	(10)	0.1994	1.1 × 10−6	/	/	0.962

. The fitting quality was evaluated using the coefficient of determination $R^2$, defined as:

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{(y_i-{\widehat{y}}_i)}^2}{{\sum }_{i=1}^n{(y_i-{\overline{y}}_i)}^2}},$$

(12)

where $y_i$ and ${\widehat{y}}_i$ denote the measured and predicted values of $E_n/E_0$ at the same cycle ratio, and ${\overline{y}}_i$ is the mean of $y_i$.

The fitting results indicate that Equation (7), which assumes a linear degradation trend, deviates substantially from the observed nonlinear degradation trajectories of the specimens, leading to pronounced offsets. The fixed linear form of Equation (7) produces residuals that exceed the intrinsic scatter of the experimental data. Such behavior demonstrates that a linear model cannot accurately capture the nonlinear evolution of flexural fatigue damage in ordinary concrete.

Equation (8) provides a good representation of the three-stage degradation behavior observed in PC specimens, yielding a close agreement with the experimental data. This observation suggests that the model is suitable for fatigue processes exhibiting a pronounced three-stage evolution. However, for RC specimens whose actual degradation typically follows a two-stage pattern consisting of an initial rapid decay followed by a stabilization phase, Equation (8) results in increasing deviations as fatigue progresses. Moreover, both PC and RC fitting curves occasionally show an increasing trend in $E_n/E_0$ with increasing cycle ratio, due to the unconstrained cubic term in the model, which may introduce local upward segments that violate the irreversibility requirement for fatigue damage.

In contrast, Equations (9) and (10) exhibit superior overall fitting performance. Both models adequately reproduce the nonlinear degradation of elastic modulus for PC and RC specimens, with a coefficient of determination consistently above 0.92, indicating that they capture the global trend well. Compared to Equation (10), Equation (9) provides a shorter transition duration for the accelerated damage in the first stage, as shown in Figure 15, thereby better reflecting the experimentally observed rapid damage accumulation in early fatigue [17], and it also achieves relatively higher fitting accuracy. Therefore, Equation (9) is selected as the baseline function for the subsequent fatigue damage evolution analysis.

3.5. Effects of Stress Ratio and Maximum Stress Level

As shown in Figure 17, the degradation of relative residual elastic modulus for each specimen set was fitted using Equation (9). To quantitatively evaluate the influence of stress ratio and maximum stress level on the fatigue degradation of elastic modulus in concrete, two representative integrated indicators were proposed in this study: the area under the curve AUC and the stable stage degradation slope |$K_{mid}$|. The AUC represents the integral of the relative residual elastic modulus–cycle ratio curve over the entire normalized fatigue life. A larger AUC indicates slower modulus degradation and lower cumulative fatigue damage throughout the loading process. The stable-stage degradation slope |$K_{mid}$| is obtained by performing linear regression on the portion of the degradation curve within the cycle-ratio range 0.1–0.9, corresponding to the stable fatigue stage. A larger absolute value of |$K_{mid}$| reflects a faster reduction in elastic modulus and a higher rate of damage accumulation. The calculated AUC and |$K_{mid}$| values for various cases are presented in

Table 7. Results of AUC and |$K_{mid}$| for various sets of PC and RC specimens.

Set Designation	AUC	$|{\mathit{K}}_{\mathit{m}\mathit{i}\mathit{d}}$|
S-75-10	0.8033	0.2505
S-75-25	0.8215	0.1912
S-80-10	0.7885	0.2757
S-90-25	0.6976	0.3705
J-S-75-10	0.8207	0.1187
J-S-75-25	0.8315	0.0745
J-S-80-10	0.7968	0.1330
J-S-80-25	0.8031	0.0784
J-S-85-10	0.7474	0.1145
J-S-85-25	0.7609	0.1085

.

As shown in Table 7, the stress ratio and maximum stress level exhibit distinct and significant effects on the elastic modulus degradation of both PC and RC specimens.

At the same maximum stress level, increasing the stress ratio (i.e., the minimum stress level) reduces the fluctuating fatigue stress range; thus, the degradation of the elastic modulus is expected to slow down. For PC specimens, comparing S-75-10 with S-75-25, the AUC increases from 0.8033 to 0.8215 (about 2%), while |$K_{mid}$| decreases from 0.2505 to 0.1912 (about 24%). A similar trend is observed in RC specimens: the AUC values of test sets J-S-75-25, J-S-80-25, and J-S-85-25 are all slightly higher than those of the corresponding J-S-75-10, J-S-80-10, and J-S-85-10 sets (with an average increase of about 2%), and |$K_{mid}$| decreases by an average of 28%. These results indicate that, for a given maximum stress level, a higher stress ratio yields a slightly larger AUC and a markedly lower |$K_{mid}$|, demonstrating that increasing the stress ratio effectively mitigates the rate of elastic modulus degradation during fatigue loading.

At the same minimum stress level, increasing the maximum stress level widens the fluctuating fatigue stress range; therefore, cumulative fatigue damage is expected to increase. For PC specimens, comparing S-80-10 with S-75-10, the AUC decreases from 0.8033 to 0.7885 (about 2%), while |$K_{mid}$| increases from 0.2505 to 0.2757 (about 10%). For S-90-25 versus S-75-25, the AUC decreases from 0.8215 to 0.6976 (about 15%) and |$K_{mid}$| increases from 0.1912 to 0.3705 (about 94%). For RC specimens, the AUC values of J-S-75-10, J-S-80-10, and J-S-85-10 are 0.8207, 0.7968, and 0.7474, corresponding to reductions of 3% and 6% between adjacent sets. However, the influence of maximum stress level on the magnitude of elastic modulus degradation is more complicated. The |$K_{mid}$| values first increase by about 12% and then decrease by about 14%. It is not clear why set J-S-85-10 shows such a small |$K_{mid}$|. For test sets J-S-75-25, J-S-80-25, and J-S-85-25, the AUC values are 0.8315, 0.8031, and 0.7609, respectively, decreasing by about 3% and 5%, while |$K_{mid}$| increases by about 5% and 38%, respectively. These observations indicate that a higher maximum stress level generally leads to greater cumulative fatigue damage. Overall, for both PC and RC, increasing the stress ratio helps slow the degradation of fatigue elastic modulus, whereas increasing the maximum stress level exacerbates cumulative fatigue damage.

Across all tested stress conditions, AUC changes only slightly, whereas |$K_{mid}$| varies much more. This finding suggests that |$K_{mid}$|, the degradation rate, is more sensitive to changes in fatigue stress states, and AUC, as a life-integrated measure, is less responsive to stress levels or stress ratios. Therefore, although the two indices are complementary—AUC reflects the overall cumulative degradation, and |$K_{mid}$| captures the degradation rate during the stable fatigue stage—|$K_{mid}$| is recommended as the primary indicator of modulus degradation.

To quantitatively justify selecting |$K_{mid}$| as the primary indicator, a general linear model was applied using the indicator values summarized in Table 7. The fatigue stress state was represented by the stress range as defined in:

$$\Delta S=S_{max}-S_{min}.$$

(13)

For each series (PC and RC) and each indicator Y ∈ {AUC, |$K_{mid}$|}, an ordinary least squares linear regression was fitted as:

$$Y={\mathrm{\beta }}_0+{\mathrm{\beta }}_1\Delta S,$$

(14)

where β₀ is the regression intercept, and β₁ denotes the response slope of the indicator to variations in ΔS. In addition to β₁, the coefficient of determination R² was reported to quantify the fraction of the indicator variance explained by ΔS. To enable a direct comparison between AUC and |$K_{mid}$| despite their different numerical scales, the absolute standardized slope |β^∗| was further defined as:

$$|\mathrm{\beta }*|={|\mathrm{\beta }}_1|\frac{{\mathrm{\sigma }}_{\Delta \mathrm{S}}}{{\mathrm{\sigma }}_{\mathrm{Y}}},$$

(15)

where ${\mathrm{\sigma }}_{\Delta \mathrm{S}}$ and ${\mathrm{\sigma }}_{\mathrm{Y}}$ are the sample standard deviations of ΔS and Y, respectively. As summarized in

Table 8. Regression-based sensitivity metrics for AUC and |$K_{mid}$|.

Series	Indicator	β1	R2	|β∗|
PC	AUC	−0.2679	0.1773	0.4211
	$|K_{mid}$|	0.5468	0.4026	0.6345
RC	AUC	−0.2169	0.3760	0.6132
	$|K_{mid}$|	0.2137	0.7371	0.8585

, |$K_{mid}$| consistently exhibits higher R² and larger |β^∗| than AUC in both PC and RC series, indicating |$K_{mid}$| has a stronger dependence on fatigue stress state and a better discriminative capability with respect to stress-state changes. Therefore, within the investigated dataset, |$K_{mid}$| is recommended as the primary indicator for characterizing stable-stage modulus degradation, while AUC is retained as a complementary life-integrated measure reflecting cumulative degradation.

3.6. Influence of Reinforcement

To quantitatively assess the role of reinforcement during fatigue loading, PC and RC specimens subjected to identical loading conditions were compared, including S-75-10 versus J-S-75-10, S-80-10 versus J-S-80-10, and S-85-10 versus J-S-85-10. Since the RC specimens did not reach complete fatigue failure within the test duration, the comparison was performed over the overlapping observation range available for both series. In this comparison, the horizontal axis was expressed in terms of the actual fatigue life, and the corresponding normalized elastic modulus–fatigue life curves are presented in Figure 18. To facilitate a clearer observation of the initial rapid reduction stage, a zoomed-in inset was added to Figure 18 to magnify the variations in the relative residual elastic modulus E_n/E₀ in the range of 0.9–1.0.

To further characterize the differences in degradation rate and overall magnitude between the two series, the area under the curve AUC_n and the stable-stage degradation slope |$K_{mid,n}$| were recalculated on the cycle-count axis for a consistent comparison. Specifically, the indices were evaluated over the overlapping cycle-count range shared by the paired PC and RC sets shown in Figure 18, since the RC specimens did not reach complete fatigue failure within the test duration. The resulting AUC_n and |$K_{mid,n}$| values are summarized in

Table 9. Results of AUC_n and |$K_{mid,n}$|.

Set Designation	AUCn	$|{\mathit{K}}_{\mathit{m}\mathit{i}\mathit{d},\mathit{n}}$|
S-75-10	1.13 × 104	1.77 × 10−5
J-S-75-10	1.16 × 104	8.22 × 10−6
S-75-25	1.73 × 104	9.02 × 10−6
J-S-75-25	1.79 × 104	3.16 × 10−6
S-80-10	4.39 × 102	4.94 × 10−4
J-S-80-10	5.07 × 102	1.29 × 10−4

.

Based on Table 9, it is evident that under identical loading conditions, the degradation rate of elastic modulus in RC specimens is significantly lower than that of PC specimens. The AUC_n values of the RC specimens are consistently higher than those of the corresponding PC specimens, while their |$K_{mid,n}$| values are markedly reduced. Specifically, compared with S-75-10, the AUC_n of J-S-75-10 increases by about 3%, while |$K_{mid,n}$| decreases by approximately 54%. For the S-75-25 and J-S-75-25 comparison, the AUC_n increases by about 3% and |$K_{mid,n}$| decreases by roughly 65%. In the S-80-10 versus J-S-80-10 comparison, the AUC_n increases by around 15% and |$K_{mid,n}$| decreases by about 74%. These comparisons indicate that reinforcement both reduces the overall damage level and slows the degradation rate of elastic modulus under fatigue loading; in terms of the magnitude of change, the reinforcement effect is more clearly reflected in the pronounced reduction in |$K_{mid,n}$|, whereas the change in AUC_n is comparatively modest.

3.7. Limitations and Future Work

This study has two limitations that should be acknowledged. First, when calculating stress in accordance with Equations (1)–(3), the moment of inertia I was assumed to remain constant, corresponding to an uncracked rectangular section. However, during flexural fatigue loading, once macrocracks develop in the tensile zone, the sectional response gradually transitions from an uncracked to a partially cracked state. Crack propagation shifts the neutral axis and changes the effective moment of inertia. Therefore, the observed stiffness reduction reflects the combined effects of sectional evolution and material degradation. As a result, adopting a constant I throughout the fatigue process introduces deviations in the post-cracking stress evaluation, which in turn affects the stress–strain slopes used to determine the secant modulus. The obtained elastic modulus of concrete is thus merely an equivalent value for the sake of comparison.

Even though uncracked state accounts for a majority of fatigue life in plain concrete specimens, this is apparently not the case for reinforced concrete specimens. It should be noted that, within the cycle range achieved in the present study, the upward extension of bottom cracks in the RC specimens remained limited, and the crack did not develop above the concrete cover of the reinforcement. Therefore, the influence of sectional geometry changes on the moment of inertia was relatively small, and using the uncracked-section inertia for estimating an equivalent modulus has a limited impact on the overall trend interpretation of the current observation. However, when RC specimens approach fatigue failure, crack growth and cracked-section effects become much more pronounced. The error associated with the uncracked inertia assumption would no longer be negligible.

Second, the calculated nominal stress in RC specimens did not account for reinforcement. For a typical reinforcement ratio of 1%, the difference amounts to approximately 7% when the ratio of elastic moduli between steel reinforcement and concrete is taken into account. Moreover, the RC specimens did not reach complete fatigue failure during the test. This limitation makes it difficult to fully capture the potential accelerated degradation behavior that may occur near the final stage of fatigue life.

Nevertheless, the degradation of concrete’s elastic modulus obtained from this study provides important input for constitutive relations between stress and strain in modeling the fatigue evolution of structural components. In this context, recent fatigue-life assessment and prediction frameworks for RC members commonly combine material degradation descriptors with structural-level fatigue models to support remaining-life evaluation. Therefore, the present findings are presented as providing inputs for degradation characterization in such assessment frameworks, while the near-failure degradation stage of RC is essential for accurately characterizing performance degradation under cyclic loading [26,27].

Regarding future research, three directions are particularly important. First, the evolution of residual strain under flexural fatigue should be systematically quantified. Similar to elastic modulus, residual strain can reflect the accumulation of fatigue damage from a deformation-based perspective. Second, further investigation is needed to examine a common assumption in practical fatigue analysis, whether the peak strain at the extreme fiber at flexural fatigue failure corresponds to, or intersects with, the uniaxial strain envelope. Establishing such consistency would help bridge flexural fatigue observations with material constitutive descriptions and provide a clearer basis for simplified damage-mechanics-based fatigue assessment of concrete members. Third, additional flexural fatigue tests on RC specimens, together with cracked-section correction methods, should be conducted to fatigue failure under improved experimental conditions, so that the near-failure degradation stage can be captured. The RC-related conclusions in this study may need further validation, especially with respect to the role of reinforcement and the potential final accelerated degradation stage.

4. Conclusions

This study examined the degradation of the elastic modulus of C50 concrete under flexural fatigue loading through four-point bending fatigue tests on plain concrete (PC) and reinforced concrete (RC) beams. Degradation curves of the relative residual elastic modulus versus cycle ratio were developed, and two integrated indicators—the area under the curve (AUC) and the absolute value of the stable-stage degradation slope (|$K_{mid}$|)—were introduced to quantify the cumulative degradation level and the degradation rate during the stable fatigue stage, respectively. Based on these analyses, the following conclusions are drawn:

(1)

Under fatigue loading, the elastic modulus degradation of failed plain concrete specimens exhibited a typical three-stage, S-shaped evolution. In contrast, for PC specimens that did not fail within the test duration, the elastic modulus remained essentially stable, with the relative residual elastic modulus exceeding 0.93 throughout the loading process.

(2)

Within the tested range, reinforced concrete specimens primarily exhibited an apparent two-stage behavior consisting of an initial rapid reduction followed by a relatively stable phase. Due to practical limitations of the loading system and testing conditions, RC specimens did not reach complete fatigue failure; therefore, the reported degradation characteristics are based on available observations and do not include the near-failure accelerated stage.

(3)

The integrated indicators AUC and |$K_{mid}$| offer complementary measures of modulus degradation. AUC represents the overall cumulative degradation over the normalized fatigue life, whereas |$K_{mid}$| quantifies the degradation rate during the stable fatigue stage, enabling consistent comparisons among different stress conditions and specimen types. Because |$K_{mid}$| demonstrated greater sensitivity to variations in fatigue loading and specimen configuration, it is recommended as the primary indicator of modulus degradation.

(4)

Stress ratio and maximum stress level exerted distinct influences on elastic modulus degradation. Under the same maximum stress level, increasing the stress ratio mitigated degradation. For PC specimens, AUC increased from 0.8033 (S-75-10) to 0.8215 (S-75-25), while |$K_{mid}$| decreased from 0.2505 to 0.1912. Under the same minimum stress level, increasing the maximum stress level intensified degradation: for PC specimens, AUC decreased from 0.8215 (S-75-25) to 0.6976 (S-90-25), accompanied by an increase in |$K_{mid}$| from 0.1912 to 0.3705.

(5)

Under identical fatigue loading conditions, RC specimens exhibited slower elastic modulus degradation than PC specimens. Compared with the corresponding PC specimens, the AUC_n of RC specimens was consistently higher, whereas |$K_{mid,n}$| was markedly lower. For example, relative to S-75-10, J-S-75-10 showed an AUC_n increase of about 3% and a |$K_{mid,n}$| reduction of approximately 54%; relative to S-80-10, J-S-80-10 showed an AUC_n increase of approximately 15% and a |$K_{mid,n}$| reduction of about 74%. Overall, reinforcement reduced cumulative degradation and slowed the stable-stage degradation rate of elastic modulus.