Flexural Fatigue Behavior of Prestressed High-Performance Concrete Bridges with Double Mineral Fine Powder Admixture: An Experimental Study

Abstract

High-performance concrete (HPC) is commonly used in the main structures of bridges. HPC is widely applied in the main structures of bridges, yet some skepticism remains with integrating fly ash and mineral powder as admixtures into prestressed HPC bridges. To address this, this study conducted scaled-model experiments to analyze the flexural fatigue behavior of prestressed HPC bridges with double-mineral fine powder admixtures (PB-DA). This study derives the similarity criteria for a simply supported beam bridge under a concentrated load based on similarity theory. Subsequently, in following these criteria, a 30 m long actual bridge is scaled down to a 6 m PB-DA at a 1:5 scale. For this scaled PB-DA, the concentrated load is reduced to 1/25 of the actual bridge, while the strain remains the same as in the actual bridge. The double-mineral fine powder admixture (D-A) was produced and used to fabricate PB-DA by mixing fly ash and mineral powder. Five PB-DAs were constructed, with C50 and C80 concrete strength grades, and admixture ranges from 10% to 32%. Sinusoidal half-wave constant stress amplitude loading at 5 Hz frequency was applied, with 2 million fatigue loading cycles. After fatigue loading, a continuously increasing static load was applied until the PB-DA failed. The experimental results show that the upper part of the PB-DA is compressed, and the lower part is in tension. The PB-DA strain distribution from top to bottom generally conforms to the plane section assumption. During 2 million fatigue loading cycles, 200,000 cycles mark the beam strain and stiffness evolution boundary. Below 200,000 cycles, the PB-DA strain rapidly increases, and flexural stiffness quickly decreases. Beyond 200,000 cycles, the rate of increase in strain and the rate of decrease in flexural stiffness significantly slow down. After fatigue loading, the PB-DA displacement increases exponentially under a continuously increasing static load. The crack distribution is uniform across all PB-DA, with the cracks being sparsest at a 30% admixture. A comprehensive analysis shows that all PB-DAs demonstrate good flexural fatigue behavior. Notably, when D-A content reaches 30%, strain increases, but reductions in flexural stiffness and damage in PB-DA significantly decrease. This paper’s conclusions provide a reference for applying D-A at PB-DA.

1. Introduction

Since the concept of HPC was proposed by Norwegian scholars in 1986 [1], high-performance concrete has been widely used in various construction projects to enhance structural durability [2,3,4]. Traditional ordinary concrete is primarily composed of cement, sand, gravel, and water. Based on this, through replacing a portion of the cement with mineral additives and incorporating high-range water-reducing agents, it transforms into what is known as high-performance concrete. High-performance concrete imposes stringent technical requirements on all six of these raw materials, as detailed in the “Technical Specifications for Construction of Highway Bridges and Culverts” (JTG/T 3650-2020) [5]. However, due to the complex relationship between the performance of high-performance concrete and mineral admixtures, there have been concerns in early engineering projects regarding the use of mineral admixtures in large proportions [6]. Exploring the application of D-A in high-performance concrete, especially high-strength concrete, holds significant engineering value.

As railway and highway bridges are inevitably subjected to natural environments and cycle loads during the operational period, concrete faces challenges with durability. In 1994, 6137 railway bridges in China showed signs of deterioration, accounting for 18.8% of the total number of railway bridges that year. Incorporating mineral additives into the concrete to enhance bridges’ mechanical and durability properties has become vital to ensuring the service life of bridges [7,8,9].

Mineral additives are widely used in designing high-performance electrically conductive cementitious composites, which are applied in traffic detection, structural health monitoring (SHM), and pavement deicing [10,11,12,13,14]. Research on incorporating mineral additives into concrete has evolved from micro to macro aspects. Microscopically, the focus has been on the role of mineral additives in concrete, which includes the following [15,16,17]: (1) the morphology of mineral additives can alter the rheological properties, initial structure, and multiple functions of the mixture, especially acting as a lubricant during concrete pumping; (2) mineral additives can chemically react with alkaline substances in humid environments to form cementitious materials, thereby enhancing the concrete’s corrosion resistance; and (3) the tiny beads and fragments in mineral additives can improve the concrete’s strength. These effects change the microstructure of the cement paste, inevitably leading to changes in the macroscopic mechanical behavior of concrete. Macroscopically, studies often use cubic, cylindrical, and prismatic specimens to analyze HPC’s mechanical and durability properties. Extensive research indicates that fly ash and mineral powder, as mineral additives, can effectively enhance HPC’s mechanical and durability properties [16,18,19,20,21,22,23,24]. However, the fatigue performance of bridges is related to their actual size [25], and these testing methods do not account for the actual size of bridges in engineering projects, thus failing to obtain the flexural fatigue behavior of bridge structures directly.

To address this issue, in recent years, some scholars have designed scaled-model experiments of HPC bridges based on similarity theory. The HPC bridges’ strain, displacement, and crack distribution were tested through cyclic loading, thus elucidating their fatigue behavior. These studies are listed in

Table 1. Scaled-model experiments for HPC bridge.

Year	Author (s)	Mineral Admixtures	Admixture and Concrete Strength Combination	Main Conclusions
2006	Xiao et al. [26]	Fly ash	C40: 29% C80: 25%	For flexural fatigue behavior, C80 HPC bridges are better than C40 HPC bridges, and HPC bridges outperform ordinary concrete beams.
2007	Song et al. [27]	Fly ash	C50: 0~40% C60: 25% C70: 25% C80: 25%	Properly incorporating fly ash does not adversely affect the fatigue behavior of bridges.
2007	Luo et al. [28]	Fly ash	C50: 0~40% C60: 25% C70: 25% C80: 25%	An admixture of 20% to 40% fly ash in prestressed simply supported beam bridges for railways is feasible.
2008	Song et al. [29]	Fly ash	C50: 0~40% C60: 25% C70: 25% C80: 25%	Prestressed concrete beam bridges with a 25% fly ash admixture exhibit good fatigue behavior and are suitable for use in railway bridges.
2010	Wang et al. [30]	Fly ash	C60: 17%	After fatigue loading, HPC bridges maintain good mechanical properties.

[26,27,28,29,30]. Table 1 shows that increasing the fly ash content and concrete strength positively affects HPC bridges’ fatigue behavior. However, existing scaled-model experiments mainly focus on HPC bridges with fly ash, with little research on PB-DA. Moreover, fly ash and mineral powder are industrial waste products from the power and steel industries. Previous research on prisms and other experiments indicates that utilizing fly ash and mineral powder can enhance the working performance of HPC and effectively solve the problem of industrial waste disposal. Therefore, using scaled-model experiments to explore the flexural fatigue behavior of PB-DA holds significant engineering value.

For this purpose, this paper derives the similarity criteria for simply supported beam bridges under concentrated load based on similarity theory and, according to this criterion, scales down a bridge with a 30 m span to a 6 m span. A total of 18 stages of loading schemes were designed, including (1) preloading to test equipment reliability; (2) cyclic loading to simulate repetitive loads during operation; (3) graded loading to test the PB-DA’s flexural fatigue behavior; and (4) static loading to test the mechanical properties of the flexural fatigue behavior after fatigue loading. During the cyclic loading process, the PB-DA’s flexural fatigue behavior is measured at specific intervals using graded loading. Three testing schemes were designed: (1) a strain gauge layout to measure the full-section strain variation from the top to the bottom of the PB-DA; (2) a dial indicator setup to measure the PB-DA’s maximum deflection; and (3) a grid drawing scheme to measure the crack distribution pattern of the beam. By constructing PB-DAs with concrete strength grades of C50 and C80 and admixture contents of 10% to 32%, this study obtains the strain and flexural stiffness evolution under fatigue loading, as well as the mechanical properties and crack distribution patterns after the fatigue loading of the PB-DA. The research conclusions provide a reference for the application of PB-DA.

2. PB-DA Design and Fabrication

2.1. Derivation of Similarity Criteria

This section derives the similarity criteria for a simply supported beam proportional model under concentrated load based on dimensional analysis. The similarity criteria are established to describe the bending state of a simply supported beam bridge under concentrated load, as shown in Equation (1):

$$\pi ={\sigma }^a{P}^b{M}^c{l}^d{W}^e$$

(1)

where σ represents the normal stress of the bridge’s cross-section, l is the span of the bridge, M is the bending moment of the bridge, P is the load applied on the bridge, and W is the section modulus of the bridge against bending. The fundamental dimensions corresponding to these physical quantities are shown in

Table 2. Fundamental scales of physical quantities.

Physical Quantities	Fundamental Scale
σ	[F][L]−2
P	[F]
M	[F][L]
l	[L]
W	[L]3

.

Through substituting the fundamental dimensions from Table 2 into Equation (1), the dimensional equation can be obtained, as shown in Equation (2).

$$\left[\pi \right]=\left[L^0{F}^0\right]={\left[L^{-2}F\right]}^a{\left[F\right]}^b{\left[LF\right]}^c{\left[L\right]}^d{\left[L^3\right]}^e$$

(2)

Based on the principle of dimensional homogeneity, the following set of equations can be derived from Equation (3):

$$\left\{\begin{array}{l}-2a+c+d+3e=0\\ a+b+c=0\end{array}\right.$$

(3)

Equation (3) represents two equations with five unknowns. Thus, the unknowns cannot be directly solved. Therefore, in assuming a, b, and d as known values, c and e are determined as follows:

$$\left\{\begin{array}{l}c=-a-b\\ e=a+{\displaystyle \frac{1}{3}}b-{\displaystyle \frac{1}{3}}d\end{array}\right.$$

(4)

In dimensional analysis, a, b, and d can be assumed as three sets of numbers, allowing for the solution of the five unknowns using Equation (4). Although different assumed values lead to different results, all conform to the similarity criteria. For an easy solution, a, b, and d are assumed to be the following three sets of numbers:

$$\left\{\begin{array}{l}a=1,b=0,d=0\\ a=0,b=1,d=0\\ a=0,b=0,d=1\end{array}\right.$$

(5)

Substituting each of the three sets of numbers in Equation (5) into Equation (4) and the results obtained into Equation (1), the following three similarity criteria can be obtained:

$$\left\{\begin{array}{l}{\pi }_1=\sigma M^{-1}W\\ {\pi }_2=P{M}^{-1}{W}^{{\displaystyle \frac{1}{3}}}\\ {\pi }_3=L{W}^{-{\displaystyle \frac{1}{3}}}\end{array}\right.$$

(6)

Based on the similarity criteria of Equation (6), the similarity indices can be obtained as

$$\left\{\begin{array}{l}{\displaystyle \frac{C_{\sigma }{C}_W}{C_M}}=1\\ {\displaystyle \frac{C_P{C}_W^{\frac{1}{3}}}{C_M}}=1\\ {\displaystyle \frac{C_l}{C_W^{\frac{1}{3}}}}=1\end{array}\right.$$

(7)

When the materials of the proportional model and the prototype are the same, it can be ensured that their failure stresses are identical; hence, C_σ = 1. In assuming that the geometric scale between the proportional model and the original is 1: k, then C_l = 1/k. By substituting C_σ = 1 and C_l = 1/k into Equation (7), the values of the other similarity constants can be obtained:

$$\left\{\begin{array}{l}{C}_W={\displaystyle \frac{1}{k^3}}\\ C_M={\displaystyle \frac{1}{k^3}}\\ C_P={\displaystyle \frac{1}{k^2}}\end{array}\right.$$

(8)

Additionally, since C_σ = C_E = 1, it follows that C_ε = 1. Through designing the scaled model according to the similarity criteria derived in this section, the relationship between the proportional model and the prototype can be established, allowing the experimental results of this paper to reflect the actual bridge’s flexural fatigue behavior directly.

2.2. Dimension and Reinforcement Design of PB-DA

This section focuses on designing a PB-DA based on a 30 m long bridge prototype. The PB-DA in this study was designed using fully prestressed concrete, ensuring that no tensile stress is generated under the maximum design load, thus preventing cracking. The maximum static design load for this model bridge is 43 kN, and within this load range, the concrete should not crack. According to existing research [27,28,29], a 1:5 geometric scale can accurately simulate the fatigue behavior of bridges. This paper also adopts a 1:5 geometric scale, with the scaled model’s materials being the same as the prototype’s. Based on the derivations in Section 2.1, when k = 5, the concentrated force is reduced to 1/25 of the prototype, resulting in the scaled model strain being the same as that of the prototype.

The detailed dimensions of the scaled model are shown in Figure 1, with the scaled model being 0.4 m high and 6 m long, with an effective span of 5.7 m. The scaled model’s cross-section is T-shaped, with a total flange width of 0.43 m, the outer edge of the flange at 36 mm, and the thickness at the juncture of the flange and web at 54 mm. The reinforcement layout and the reinforcement detail list are shown in Figure 2 and

Table 3. Reinforcement details of the PB-DA.

Serial Number	Reinforcement Shape	Length/mm	Diameter/mm	Numbers
1		5980	8	2
2		785	8	4
3		6060	6	2
4		6069	6	10
5		1034	6	60
6		330	6	60
7		166	6	120
8		413	6	60
9		5980	6	2

, respectively.

In this experimental model, the ordinary reinforcement in the beams consists of HPB235 grade steel bars, and the prestressed reinforcement consists of steel strands. These strands were produced by Tianjin Metallurgical Group Zhongxing Shenda Steel Co., Ltd., China, with a nominal diameter of 9.53 mm and a nominal area of 54.8 mm². To ensure the lateral balance of the beam, the following tensioning sequence was adopted: 50% N2 → 100% N3 → 100% N2 → 100% N1. Here, the tension control force for N1 was 62.05 kN, and for N2 and N3, it was 69.8 kN. After tensioning the prestressed tendons, the beam exhibited an upward deflection between 10 mm and 15 mm.

2.3. Design of D-A Content

Following the research objectives of this study, fly ash and mineral powder, which were the same as those used in practical applications, were used as mineral admixtures. In line with the GB/T 18736-2017 [31], I-grade fly ash and I-grade mineral powder were selected for high-grade concrete. Based on the review of Table 1, the most common concrete strength in scaled experiments is C50, with the maximum being C80; hence, this study also produced C50 and C80 PB-DA. Detailed mass information of the materials used in C50 and C80 can be found in

Table 4. Detailed mass information of the concrete components.

Serial Number	Concrete Strength Grade	Cement/kg	Fly Ash/kg	Mineral Powder/kg	Sand/kg	Crushed Stone/kg	Water/kg
L1	C50	436.5	24.25	24.25	687	1100	148
L2	C50	388.0	48.50	48.50	687	1100	148
L3	C50	339.5	72.75	72.75	687	1100	148
L4	C80	440.0	78	58	640	1090	144
L5	C80	390.0	70	116	640	1090	144

, and the corresponding mix design information is provided in

Table 5. Design of concrete strength grade and D-A content.

Serial Number	Concrete Strength Grade	Fly ash Content/%	Mineral Powder Content/%	D-A Content/%
L1	C50	5	5	10
L2	C50	10	10	20
L3	C50	15	15	30
L4	C80	14	10	24
L5	C80	12	20	32

. As shown in Table 4 and Table 5, for both C50 and C80 strength grades, the total amount of cementitious materials (cement, fly ash, and mineral powder) and the quantities of other components remained unchanged. Different D-A contents are achieved solely by varying the amounts of fly ash and mineral powder. This study aimed to explore the potential application of D-A in prestressed HPC bridges. Therefore, the subsequent analysis focuses on the impact of changes in D-A content on the fatigue performance of the bridge.

After careful selection, the cement used in the experimental model beams was 52.5-grade ordinary Portland cement produced by Quzhai Cement Co., Ltd. in Luquan, China. The mineral powder was S95 grade from the Hebei Xingtai Huayang Mineral Powder Processing Plant, China, with a measured specific surface area of 400 m²/kg. The remaining physical performance indicators comply with the “Ground Granulated Blast Furnace Slag Used for Cement, Mortar, and Concrete” (GB/T 18046-2017) standard [32]. The fly ash used was F-1 grade produced by Handan Power Plant, with physical performance indicators largely meeting the requirements of the “Fly Ash Used for Cement and Concrete” (GB/T 1596-2017) standard [33]. Additionally, the fine aggregate was river sand with a particle diameter ranging from 0.16 mm to 5 mm, provided by Shilingqi Sand Field in Lingshou, Shijiazhuang, China. The coarse aggregate was crushed stone with a maximum particle diameter of no more than 20 mm, produced by Yinjing Stone Factory in Shijiazhuang, China. The physical performance indicators of these two materials meet the requirements of the “Technical Specifications for Construction of Highway Bridges and Culverts” (JTG/T 3650-2020) [5]. A polycarboxylate superplasticizer was used as the water-reducing agent, with a water reduction rate of no less than 20%.

2.4. Fabrication of PB-DA

The manufacturing process of the double-finely powdered material prestressed concrete beam is similar to that of onsite bridges, as referenced in the JTG/T 3650-2020 [5]. The PB-DA was steam-cured, and the finished structural entity is shown in Figure 3.

To ensure the compressive strength of the concrete, five groups of 150 × 150 × 150 mm³ cubic specimens were made according to the GB/T 50107-2010 [34], with three identical cubic specimens per group, as shown in Figure 4. After 28 days of standard curing, the compressive strength of the cubic specimens was determined using a pressure-testing machine, with results presented in

Table 6. The 28-day compressive strengths of the cubic specimens.

Serial Number	Design Strength Grade	Compressive Strength/MPa	D-A Content/%
L1	C50	59.9	10
L2	C50	60.03	20
L3	C50	61.5	30
L4	C80	80.2	24
L5	C80	81.9	32

. According to the “Standard for Test and Evaluation of Concrete Compressive Strength” (GB/T 50107-2010) [34], Table 6 presents the average compressive strength of three specimens for each group. Both the C50 and C80 specimens met the required strength grade standards.

3. Flexural Fatigue Test Design for PB-DA

3.1. Loading Scheme

The loading scheme aims to test two aspects: (1) the full-time variation in flexural fatigue behavior during the cyclic loading process and (2) the mechanical properties and crack distribution of the PB-DA after cyclic loading. The experiment was controlled in a pure bending state, using two-point concentrated loading, with the specific loading positions shown in Figure 5. The test equipment of this paper was a 1000 kN three-dimensional multi-point coordinated electro-hydraulic servo dynamic loading test system produced by Tianshui Red Shan Testing Machine Co., Ltd., Tianshui, China. The test data acquisition system adopts the DH3818 static strain test system and DHDAS dynamic signal acquisition and analysis system, which can realize the acquisition of strain, displacement, and acceleration.

Based on the testing objectives of this paper, 18 loading stages were designed, with the loading schematic and each stage’s loading method shown in Figure 6 and Table 6, respectively. The first loading stage is the preloading stage, aimed at testing the reliability of the experimental equipment. From the second stage onward, the cyclic loading stage begins. The upper limit of the fatigue load is 25 kN, determined through a depressurization loading test. The lower limit of the fatigue load is the same as the lower limit of the testing machine, set at 15 kN. Sinusoidal half-wave constant-amplitude stress cyclic loading was used, with 2 million cyclic loading cycles at 5 Hz. To capture the full-time evolution of flexural fatigue behavior, graded loading was performed after a certain number of cycle loads to obtain the PB-DA’s flexural fatigue behavior after varying numbers of cycle loads. Finally, after completing all the cyclic loading numbers, a static load was applied to the PB-DA until cracks appear and the loading was stopped.

3.2. Testing Scheme

To analyze the flexural fatigue behavior and crack distribution pattern of the PB-DA, it is necessary to design a testing scheme for the PB-DA’s strain, displacement, and cracks.

Strain gauges were used to test the strain on the PB-DA’s top, bottom, and sides, as shown in Figure 7. The strain gauges on the side of the beam are numbered from top to bottom as I, II, III, IV, and V, with gauge I on the beam flange and gauges II to V equally spaced on the web.

As the PB-DA ends are supported on a solid foundation and will not sink during loading, a dial indicator was placed at the mid-span of the PB-DA to measure the maximum deflection after loading, as shown in Figure 8. Its spring needle was protected with tape to prevent damage to the dial indicator due to vibrations during cyclic loading, and deflection measurements were made only during graded loading.

To facilitate the recording of the distribution of cracks on the PB-DA, a grid with dimensions of 50 mm × 50 mm was drawn on the sides of all PB-DAs, as shown in Figure 9.

4. Flexural Fatigue Behavior of PB-DA

4.1. Strain Development Pattern of PB-DA

4.1.1. Overall Characteristics of Strain Development

Test data for the L3 PB-DA, based on commonly used concrete strength grades and the maximum admixture of double finely powdered materials, were analyzed to analyze the overall characteristics of strain development during cyclic loading.

Figure 10 shows the strain of the L3 PB-DA under different cyclic loading cycles, where Figure 10a represents the change in strain of strain gauges I to V with the number of cyclic loading cycles, and Figure 10b shows the change in strain from top to bottom of the PB-DA under different cyclic loading numbers. In Figure 10, a strain value greater than 0 indicates compression, while less than 0 indicates tension.

Figure 10a shows that the closer to the top and bottom of the PB-DA, the greater the compression and tension experienced. Compressive and tensile strains exhibit two-phase change characteristics with increasing cyclic loading cycles. The first phase is a rapid development stage, where both compressive and tensile strains increase linearly and quickly. After exceeding 200,000 cyclic loadings, it enters the second phase, the slow development stage, where both compressive and tensile strains continue to increase linearly but at a significantly reduced rate. This two-phase strain development characteristic reflects the cumulative feature of internal damage in the beam.

A further analysis of the change in PB-DA strain from top to bottom under different cyclic loading cycles was conducted based on Figure 10b. It shows that the change in strain from the top to the bottom of the PB-DA generally conforms to the assumption of plane sections. As the number of cyclic loading cycles increases, the strains in both the compression and tension zones increase, showing plane sections tilting around the neutral axis.

4.1.2. Influence of D-A Content and Concrete Strength

After obtaining the overall characteristics of the PB-DA strain development under cyclic loading, this section further analyzes the effects of D-A content and concrete strength on PB-DA strain.

The maximum compressive and tensile strains of each PB-DA under different cyclic loading cycles are extracted, as shown in Figure 11. It reveals that the maximum tensile strain for all PB-DAs is slightly more significant than the maximum compressive strain, not showing complete symmetric loading. However, increasing the D-A content and concrete strength can reduce the maximum compressive and tensile strains. The maximum compressive and tensile strain change is minimal when the D-A content increases from 10% to 20%. However, when the D-A content reaches 30%, or the concrete strength level is raised to C80, both the maximum compressive and tensile strains significantly decrease, indicating that a higher D-A content can substantially enhance the working performance of the PB-DA in both C50 and C80 concrete.

4.2. Development Pattern of Flexural Stiffness for PB-DA

During cyclic loading, continuous damage accumulation may lead to a variation in the flexural stiffness of the PB-DA. This section analyzes the evolution law of PB-DA flexural stiffness.

Initially, taking L3 as an example, the force–displacement curves under different cyclic loading cycles were analyzed, as shown in Figure 12. It is evident from Figure 12 that, with an increase in cyclic loading numbers, the force–displacement curves undergo shifts and tilts. The starting point of the force–displacement curves slightly shifts to the right with increasing cyclic loading numbers, indicating a certain degree of fatigue damage in the PB-DA, which in turn causes the slope of the force–displacement curves to tilt, indicating a slight decrease in PB-DA flexural stiffness. The force–displacement curves for 1 million and 2 million loading cycles almost coincide, demonstrating the PB-DA’s good fatigue load resistance capacity when loaded to the maximum number of cycles.

An expression for flexural stiffness is derived to intuitively analyze the stiffness degradation law of the PB-DA during cyclic loading. When a simply supported beam bridge is subjected to two concentrated forces, the expression for the mid-span displacement (S_max) is given as follows [35]:

$$S_{\max ,N}{\displaystyle \frac{Fm{n}^2}{48{\left(EI\right)}_N}}\left(3-4{\alpha }^2\right)$$

(9)

where m is the distance from the concentrated load to the adjacent support; n is the distance between the two supports; F is the resultant force of the two concentrated forces; α = m/n; EI represents the flexural stiffness; and N represents the number of load cycles.

From Equation (9), the expression for flexural stiffness can be derived, as shown in Equation (10).

$${\left(EI\right)}_N={\displaystyle \frac{Fm{n}^2}{48S_{\max ,N}}}\left(3-4{\alpha }^2\right)$$

(10)

For ease of analysis, the flexural stiffness degradation factor D is defined as the ratio of the flexural stiffness at different load cycles to the initial flexural stiffness, with a smaller D value indicating more significant stiffness degradation.

$$D={\displaystyle \frac{{\left(EI\right)}_N}{{\left(EI\right)}_0}}$$

(11)

Substituting Equation (10) into Equation (11) gives the following:

$$D={\displaystyle \frac{S_{\max ,0}}{S_{\max ,N}}}$$

(12)

The variation curve of D with N can be obtained by inputting the raw data from the measurement points into Equation (12), as shown in Figure 13. Figure 13 indicates that the PB-DA’s flexural stiffness exhibits degradation as the N increases. D shows a nonlinear rapid degradation when the N does not exceed 200,000. After surpassing 200,000 cycles, D approximates a linear decay. Additionally, increasing the D-A content and the concrete strength grade can increase the D value overall, indicating that increasing the D-A content and concrete strength grade can enhance the flexural stiffness of the PB-DA under cyclic loading.

5. Mechanical Behavior and Crack Distribution Pattern of PB-DA after Fatigue Loading

Since no cracks appeared in all PB-DAs during cyclic loading, to further investigate the mechanical performance and crack distribution of the PB-DA after cyclic loading, a continuously increasing static load is applied to the PB-DA until cracks emerge.

Figure 14 shows the relationship between force and displacement during the crack formation process in the PB-DA. To further clarify this relationship, Equation (13) is used to perform regression analysis on force and displacement, with the regression analysis results presented in Figure 14 and

Table 7. Details of 18 loading stages.

Stage Number	Loading Content	Cyclic Loading Numbers
Stage 1	Preloading	—
Stage 2	Cyclic loading	N=1×104
Stage 3	Graded loading	—
Stage 4	Cyclic loading	N=5×104
Stage 5	Graded loading	—
Stage 6	Cyclic loading	N=10×104
Stage 7	Graded loading	—
Stage 8	Cyclic loading	N=20×104
Stage 9	Graded loading	—
Stage 10	Cyclic loading	N=50×104
Stage 11	Graded loading	—
Stage 12	Cyclic loading	N=100×104
Stage 13	Graded loading	—
Stage 14	Cyclic loading	N=150×104
Stage 15	Graded loading	—
Stage 16	Cyclic loading	N=200×104
Stage 17	Graded loading	—
Stage 18	Continuously increasing Static loading	—

:

$$F=F_0+A_1\exp (-S/A_2)$$

(13)

where F₀, A₁, and A₂ are fitting parameters.

Figure 14 and

Table 8. Regression coefficients and R² values.

Serial Number	F0	A1	A2	R2
L1	108.445	−106.237	30.953	0.994
L2	116.910	−114.051	31.968	0.993
L3	109.573	−100.200	25.278	0.995

show that the force–displacement curve of the PB-DA during the crack formation process generally conforms to an exponential function, with an R² value reaching at least 0.993, indicating good regression analysis results. As the load increases, the rate of rise in PB-DA displacement decreases, suggesting that the PB-DA will not undergo brittle failure and exhibit good ductility. When the D-A content is increased, the displacement of the PB-DA under the same load decreases, indicating that the D-A content plays a beneficial role in the process of crack formation in the PB-DA.

Figure 15 shows the crack distribution pattern during the PB-DA failure. Figure 15 shows that the crack distribution across the PB-DA is a typical bending stress failure pattern, representing tensile cracking after prestressing loss, with cracks symmetrically and evenly distributed from the loading center to both sides. The crack distribution is denser in L1 and L2, while it appears sparser in L3, indicating that an increase in double finely powdered material admixture improves the PB-DA’s strength performance. All of the PB-DA demonstrated good ductility during the failure process, and with the increase in the D-A content, the rate of PB-DA failure was slower, and the extent of failure was reduced.

6. Conclusions and Discussion

The main conclusions are as follows:

(1) Based on similarity theory and utilizing dimensional analysis, this paper derived the similarity criteria for a simply supported beam bridge under concentrated load and designed the proportional model accordingly. The results show that when the geometric scale is 1:k, and the materials of the proportional model and prototype are the same, the concentrated force on the proportional model is 1/k³ of the prototype, and the strain in the proportional model is the same as in the prototype. A 1:5 scale model designed using these similarity criteria can ensure that the experimental results intuitively reflect the actual bridge’s flexural fatigue behavior.

(2) Under cyclic loading, the upper part of the PB-DA is in compression, and the lower part is in tension. The variation in the PB-DA strain from top to bottom is linear, generally conforming to the plane section assumption. With the increase in cyclic load numbers, the beam strain exhibits a two-stage development characteristic, accelerating first and then slowing down, with the boundary point at 200,000 cycles. Increasing the D-A content and concrete strength reduces the PB-DA strain. Compared to the C50 PB-DA with 10% D-A content, increasing the D-A content to 30% or enhancing the concrete strength to C80 significantly reduces both maximum compressive and tensile strains. This indicates that a high proportion of the D-A content can ensure the C50 and C80 PB-DA’s work performance.

(3) Under cyclic loading, the flexural stiffness of the PB-DA degrades. When the number of loading cycles does not exceed 200,000, the flexural stiffness shows a nonlinear rapid decay. Once the loading cycles exceed 200,000, the rate of stiffness degradation significantly decreases and shows a linear trend. Increasing the D-A content across different concrete strength grades can reduce the overall degradation of flexural stiffness, indicating that the D-A content is beneficial in C50 and C80 concrete.

(4) Under continuously increasing static load, the PB-DA displacement increases exponentially, with a constantly decreasing rate, indicating that the PB-DA will not undergo brittle failure. The larger the D-A content, the smaller the displacement of the PB-DA under the same load. After cracks appear in the PB-DA, the distribution of cracks is uniform. When the D-A content is at 10% and 20%, the cracks are more densely distributed, but at 30%, the crack distribution is sparser. This indicates that a higher proportion of the D-A content can slow down the process of PB-DA failure and reduce the extent of damage.

This paper focuses on the feasibility of applying a large proportion of the D-A content in C50 prestressed concrete bridges, with C80 prestressed concrete bridges as a comparison. Thus, the analysis of the D-A content in prestressed concrete bridges of more strength grades was not conducted. Additionally, this study did not include a control group without D-A in the experimental design, which will be improved in future work.