Mechanical properties and fatigue analysis of rubber concrete under uniaxial compression modified by a combination of mineral admixture

Cement: To enhance the overall strength of rubberized concrete, it is imperative to supplement the cement grade, as rubber aggregates possess a relatively low strength. Locally produced Ordinary Portland Cement (OPC) type II as per ASTM C136/C136M-19 [47], which is appropriate for general industrial and building structures, was utilized in this investigation, with particle size distribution as shown in Figure 1 (a). The physical, chemical, and mechanical properties of the cement in the substance are detailed in Tables 1 and 2.

Figure 1:

Particle size distribution: a) Cement. b) FA, c), and d) Grading curves of fine and coarse aggregates according to ASTM C136/C136M-19 grading standards.

Mineral admixture

The mechanical properties of rubber concrete modified by tire content are made with a particle size of 0.07 mm obtained from a sieve of 75 μm, which is employed as a replacement of the sand replacement 0 %, 8 %, and 25 %, silica fume cement at replacement stages of 5 %–15 % by weight of cement. The mechanical properties of fly ash are made with a particle size of 0.045 mm obtained from a sieve of 45 μm, which is employed as a replacement of cement at a replacement of 4 %–16 % by weight of cement. The physical, chemical, and mechanical properties of the cement, fly ash, and silica fume in the substance are detailed in Table 1.

Coarse aggregate: The coarse aggregate is natural stone particles. The particle distribution curve of the coarse aggregate is shown in Figure 1 (d); the aggregate surface is irregular due to the adhered concrete on the aggregate surface. Hence, before it is incorporated into concrete, the coarse aggregate should undergo a process of sieving [11], [12], [13]. As per the aggregate classification standards ASTM C136/C136M-19 [47], it is classified as a lightweight aggregate. In this study [32], the coarse aggregate particle size is graded from 5-20 mm and is divided continuously; the physical properties of coarse aggregate results are presented in Table 3.

Fine aggregate: The fine aggregate utilized in this investigation is natural sand sourced from the river region of Pakistan. With a uniformly graded particle size distribution, natural sand is categorized as Zone II sand and medium sand based on its fineness modulus. The results of the physical property analysis of the sand utilized in this investigation are shown in Table 4, following the “ASTM C136/C136M-19 [47].” The rubber utilized in this research is waste-tier rubber. Figure 1(c) shows the particle size distribution of the fine aggregate.

Rubber particles: The mechanical characteristics of rubber concrete, including crumb rubber (CR) with a particle size of 0.35 mm, derived from a 350-μm sieve, are evaluated as a substitute for fine aggregates at replacement levels of 8 %, 15 %, 20 %, and 25 % by weight of sand. Use mesh rubber particles to prepare rubber concrete specimens and conduct mechanical properties and fatigue tests on the cylinders. The rubber aggregate concrete with a fine mesh particle size was selected for the fatigue test investigation. The bulk density is 720 kg/m³. The mechanical properties are shown in Table 5, which is aligned with (ISO), 2018 [48]. The mechanical properties of the rubber particles align with the requirements of the “production line for mechanical preparation of rubber particles from waste tires at room temperature” (ISO, 2018).

The mix design for ordinary concrete was prepared based on the strength grade of the specimens, following the guidelines outlined in the ASTM C39-20 [49], 50]. A water-to-binder ratio of 0.45 and a sand content of 0.4 were selected. The experimental mix design program of all three rubber concrete groups can be seen in Figure 2. Crumb rubber particles were incorporated as a partial replacement for sand at volumetric replacement levels of 8 %, 15 %, 20 %, and 25 %. Similarly, fly ash (FA) and silica fume (SF) were replaced with cement at 0 %, 4 %, 8 %, and 12 % levels, respectively, 5 %, 10 %, and 15 %, while the crumb rubber content replaced fine aggregate at constant volumes of 10 %, 15 %, 20 %, and 25 %. The dosage of the water-reducing agent was adjusted to account for the increasing content of crumb rubber particles. The detailed mix proportions for the specimens are presented in Table 6.

Figure 2:

Design of rubber concrete mix, a combination of mineral admixture.

Three major groups of 150 × 300 mm test specimen cylinders were designed, with three cylinders in each group, to test the 7-day, 14-day, 28-day, and 90-day compressive strength better to understand the change of concrete strength with crumb rubber content and minral admixture (Figure 3). Three groups of 150 × 300 mm concrete test cylinders were prepared, with 18 cylinders in each group, of which three specimens were utilized for the split tensile strength test, and compressive strength test [49], 50]. At the beginning of the fatigue test, the test specimens were cured at room temperature in the curing tank. The compressive strength of the test piece at this time was determined as the basis for calculating the stress ratio in the fatigue test. Section 4 explains the fatigue test specimens preparation and instrumentation.

Figure 3:

Mechanical properties of the concrete.

The compressive strength of concrete was evaluated using varying amounts of fly ash (FA), silica fume (SF), and crumb rubber (CR). Then, the compressive strength of the concrete was compared to a control mix to determine the impact of the combination of mineral admixture materials on the performance of rubberized concrete. Testing was conducted at ages 7, 14, 28, and 90 days. As shown in Figure 4, the increase in strength is primarily due to two factors. The first aspect is the physical effect of SF, which is mainly manifested in the good FA effect and particle filling effect: the Bryan value of SF is greater, indicating that the content of its active ingredients is higher, and its particle size is about 1/100 of the particle size of the cement; compared to the FA, its mixing is less, the heat of hydration is small, and it doesn’t contain the active ingredients such as free silica and calomel components. The SF is highly dispersed, equal to the phenomenon of cement mixed with “low-graded aggregates,” and the cement particles form a good gradation complementary to fill in the cement gap, which improves the densification between rubber aggregate and cement paste, and increases the density between rubber aggregate and cement paste. The second aspect is the chemical effect of FA, and its main chemical component is silica, which can react with calcium hydroxide produced by cement hydration, and convert calcium hydroxide into calcium silicate gel [7], 8], which enhances the strength of the rubberized concrete, and thus improves the strength of rubberized concrete.

Figure 4:

Compressive strength development of rubber aggregate-based concrete mixtures: (a) Control mixes containing rubber aggregate only, (b) Mixes incorporating fly ash and rubber aggregate, (c) Mixes incorporating fly ash, silica fume, and rubber aggregate, and (d) Combined effects of mineral admixtures on the compressive strength of rubber aggregate-based concrete.

Observing Figure 4, the reason for this change in the strength law is that FA can be evenly distributed among cement particles, thereby preventing the agglomeration of the cement particles and releasing part of the free water to enhance its fluidity. Subject to the combined effect of FA activity effect and micro aggregate effect, the active silica and aluminum trioxide in FA reacted with the cement hydration product calcium hydroxide to generate hydrated calcium silicate and hydrated calcium aluminates, replacing the hexagonal plate-like calcium hydroxide, which reduced the crystal content of rubberized concrete, played an enhanced role in hardening the cement paste, and promoted the late-stage strength of rubberized concrete growth [9], 51]. The incorporation of SF and FA reduced the capillary porosity of concrete, effectively decreasing the number of macropores and reducing the number of harmless pores, while enhancing the interfacial layer. This increased the compressive strength of the concrete.

As shown in Figure 4, the compressive strengths of the control group at 7, 14, 28, and 90 days were 19.49 MPa, 31.4 MPa, 40.8 MPa, and 41.3 MPa, respectively. In contrast, the different percentages of fine aggregate replacement with rubber, mixed as RC1, RC2, RC3, and RC4, by weight at 8 %, 15 %, 20 %, and 25 %, respectively, were compared to the compressive strength of the control group. The maximum compressive strengths of RC-2 specimen rubberized concrete with a 15 % rubber ratio were 20.2 MPa, 32.4 MPa, 41.2 MPa, and 41.8 MPa, after 7, 14, 28, and 90 days, respectively. Compared with the compressive strengths of FA replacement as a cement with different ratios designated as RCFA1, RCFA2, and RCFA3, the difference in weight is 4 %. 8 %, 12 %, and 16 %; incorporating the rubber content of 15 % rubberized concrete, the maximum compressive strengths of RCFA-2 specimen rubberized concrete with a 15 % rubber ratio, and FA 8 %, were 21.1 MPa, 33.1 MPa, 42.1 MPa, and 42.4 MPa, after 7, 14, 28, and 90 days, respectively. The third group, with SF ratios of 5 %, 10 %, and 15 %, was used to compare the compressive strength of the concrete. A third group, RC-FASF-1, FASF-2, and FASF-3, was compared with the control group at 7, 14, 28, and 90-day compressive strengths, respectively. The compressive strengths of concrete after 7, 14, 28, and 90 days increased by 15 %, 8 %, and 10 %, respectively, the maximum compressive strengths of FASF-2 specimen rubberized concrete with a 15 % rubber ratio, and fly ash 8 %, and SF 10 % were 21.4 MPa, 34.1 MPa, 42.8 MPa, and 43.3 MPa, after 7, 14, 28, and 90 days, respectively. The rubber content was 15 %, FA 8 %, and SF 10 %, compared to the control group, which increased by 4 %. This suggests that combining with fly ash and silica fume can significantly improve the compressive strength of rubberized concrete.

The tensile strength of the concrete matrix is shown in Figure 5; the tensile strength of rubberized concrete is increased by conventional concrete or mineral admixture, which helps to improve the tensile strength [13]. The increase in splitting tensile strength is attributed to the presence of silica and aluminium trioxide in FA. These substances can undergo chemical reactions with the cement’s hydration products to yield low-calcium hydrated calcium silicate and hydrated calcium aluminate. The resulting mixture is then filled with a rubber content to minimize internal porosity in the concrete [10], 11]. The split tensile strength of rubberized concrete with a fine aggregate replacement ratio of 15 % CR, incorporating 10 % silica fume, and 8 % fly ash replacement cement, is greater than that of the control specimens. The reason for this is that the total amount of cementitious material does not change admixure content, but with the optimize range of the amount of fly ash and silica fume mineral admixture, the cement content decreases, resulting in the presence of a large amount of fly ash, which does not produce a hydration reaction, so the rubberized concrete splitting tensile strength decreases.

Figure 5:

Split tensile strength of rubber aggregate-based concrete mixtures: (a) Control mixes containing rubber aggregate only, (b) Mixes incorporating fly ash and rubber aggregate, (c) Mixes incorporating fly ash, silica fume, and rubber aggregate, and (d) Combined effects of mineral admixtures on the split tensile strength of rubber aggregate-based concrete.

As shown in Figure 5, the split tensile strengths of the control group at 7, 14, 28, and 90 days were 1.91 MPa, 3.14 MPa, 3.572 MPa, and 3.817 MPa, respectively. In contrast, the different percentages of fine aggregate replacement with rubber, mixed as RC1, RC2, RC3, and RC4, by weight at 8 %, 15 %, 20 %, and 25 %, respectively, were compared to the split tensile strength of the control group. The maximum split tensile strengths of RC-2 specimen rubberized concrete with a 15 % rubber ratio were 2.02 MPa, 3.43 MPa, 3.7 MPa, and 3.9 MPa, after 7, 14, 28, and 90 days, respectively. Compared with the split tensile strengths of FA replacement as a cement with different ratios designated as RCFA1, RCFA2, and RCFA3, the difference in weight is 4 %. 8 %, 12 %, and 16 %; incorporating with the rubber content of 15 % rubberized concrete, the maximum split tensile strengths of RCFA-2 specimen rubberized concrete with a 15 % rubber ratio, and FA 8 %, were 2.12 MPa, 3.72 MPa, 3.86 MPa, and 4.14 MPa, after 7, 14, 28, and 90 days, respectively. The third group, with SF ratios of 5 %, 10 %, and 15 %, was used to compare the split tensile strength of the concrete. A third group, RC-FASF-1, FASF-2, and FASF-3, was compared with the control group at 7, 14, 28, and 90-day split tensile strengths, respectively. The split tensile strengths of concrete after 7, 14, 28, and 90 days increased by 2 %, 3.5 %, and 5 %, respectively, the maximum split tensile strengths of FASF-2 specimen rubberized concrete with a 15 % rubber ratio, and fly ash 8 %, and SF 10 % were 2.25 MPa, 3.8 MPa, 3.94 MPa, and 4.26 MPa, after 7, 14, 28, and 90 days, respectively. The rubber content was 15 %, fly ash 8 %, and silica fume 10 %, compared to the control group, which increased by 5 %. This suggests that combining fly ash and silica fume with rubber content can significantly improve the split tensile strength of rubberized concrete.

The failure morphology of the prototype under fatigue loading is shown in Figure 6. Undergoing the action of fatigue load, control concrete produces compressive strain on the upper and lower pressure-bearing surfaces perpendicular to the load direction. Due to Poisson’s ratio effect, tensile strains are generated on the four sides parallel to the load direction. When the tensile strain of the concrete reaches the limit value, diagonal cracks perpendicular to the loading surface are generated, about two to three wide, deep, and penetrating; the specimen peels off and falls off, accompanied by a huge, crisp sound, which is a typical brittle failure. The different failure shapes of the sample are shown in Figure 6, in which it can be observed that under the action of fatigue load, the specimen’s shape remains intact after failure, and there is no peeling off or ballast. As the amount of rubber increases, the shape of multiple short and shallow cracks remains more complete, and the plastic deformation performance of concrete is significantly improved, manifesting as a ductile failure, and the sound of the damage becomes dull or even inaudible.

During the mixing and production process of concrete materials, original defects such as holes, bubbles, and microcracks can occur within them. Under the action of load, these weak parts will cause cracks due to stress concentration. Under the fatigue load, the main cracks of the control concrete specimens expand along the weak parts of the cement base, and the damage to the interface between the coarse aggregate and the mortar is almost invisible. After incorporating rubber particles, the specimen’s shape after damage is relatively complete, and the crack resistance is significantly improved. Combining rubber particles improves the crack resistance of concrete and enhances its deformation ability, and the ductile failure characteristics are obvious. With the accumulation of fatigue damage, macroscopic cracks occur in the specimen. The rubber particles currently exhibit an inhibitory effect at the crack tip, but lose their crack-arresting effect on wide and deep cracks. When fatigue failure occurs in rubber concrete, multiple rapid and thin cracks are distributed on the prototype’s surface. In summary, it can be observed that the deformation performance of rubber concrete is more advantageous than that of control concrete specimens. As the rubber proportion increases, the ductile failure characteristics become more prominent.

Table 7 shows the fatigue life test outcomes of the specimens under various mineral admixture contents, with rubber contents, and varying stress levels. The fatigue life test results in Table 7 demonstrate considerable variability. The discreteness of concrete fatigue test results is more common. The main reason is that numerous factors affect the fatigue life of concrete, including materials, loading age, loading stress level, etc. The S–N curve can be used to evaluate the fatigue life test results with large discreteness and express the fatigue life characteristics of concrete intuitively and clearly. The S–N curve is only a basic estimate of fatigue life. Combined with the reliability principle, fatigue life can be quantitatively analyzed.

Table 7 discloses that when the rubber content is constant, the fatigue life of rubber concrete decreases as the loading stress level increases. The reason is that rubber concrete is a stress-sensitive material with the characteristic that the fatigue life plunges as the stress level increases. Under the same stress level, the fatigue life of rubber concrete increases with the rise in the amount of rubber particles, indicating that the unification of rubber particles can conspicuously improve the plastic deformation of ordinary concrete. Compared with control concrete, the average fatigue life of rubber concrete with a dosage of 10–15 % increased by 0.18–1.12 times, 0.85 to 2.35 times, and 1.40 to 7.5 times, respectively, and the rubber concrete with a dosage of 15 % CR, 10 % SF, and 8 % FA, increased the longest fatigue life.

The cumulative fatigue damage is only related to the peak strain in the loading history [55]. The relationship between maximum strain and fatigue damage is examined. The maximum strain is used to estimate the cumulative damage of the specimen after fatigue, and the fatigue damage can be analyzed accordingly. Under the fatigue load, the crack growth process of rubber concrete is shown in Figure 6.

The strain data selection process excludes specimens where the fracture intersects with the strain gauge’s position during the test. Choose a representative specimen that exhibits little influence from bias voltage and has comparable strain data on both sides, and then choose the greater value from each side. The maximum fatigue strain E $\mathcal{E}$ augments with enhancing relative fatigue times and develops in three stages [44], 56]. The first stage is the crack initiation stage, and the fatigue strain grows rapidly, accounting for about 8–12 % of the complete fatigue process, as shown in Figure 6. The crack propagation stage is the next stage, after the first phase. While cracks appear, the fatigue strain development rate progressively becomes steady. The fatigue strain emerges sequentially and develops relatively monotonously, accounting for about 60–70 % of the entire fatigue process, as shown in Figure 7. The tertiary phase is the macroscopic development stage of cracks, with the accumulation of fatigue damage, cracks that are visible to the naked eye generated, and the fatigue strain grows expeditiously, in the end leading to damage, considering for 10–15 % of the whole fatigue action, as shown in Figure 7.

Figure 7:

Maximum fatigue strain curve.

Figure 7 shows that the higher the mineral admixture and rubber quantity, the greater the fatigue strain in the first and third stages; when the CR, SF, and FA, within the optimum proportion, are constant, the higher the stress level, the greater the strain at the same life cycle ratio. The amount of damage is defined by Miner’s linear cumulative damage theory [57]. Under constant amplitude loading conditions, its expression is: (1) M = e F L $$M=\frac{e}{FL}$$ In the formula: e the number of cycles; FL is the fatigue life.

The development of concrete fatigue damage is not related to the stress amplitude but to the damage state or life cycle ratio. It is assumed that the equation for the fatigue strain of rubber concrete is: (2) σ = g B 3 − h B 2 + i B + u C R + t $$\sigma =g{{B}^{3}-hB}^{2}+iB+uCR+t$$ In the formula, CR is the rubber content; g,h,i the test reflects the material fatigue performance parameter, u is the influence coefficient of rubber content on fatigue strain; t is the damage threshold strain when concrete is not mixed with rubber.

By performing nonlinear regression on the fatigue damage and strain during the test process, the relationship between fatigue strain and damage amount of rubber concrete can be obtained: (3) S L = 0.6 , σ = 1950 B 3 − 2390 B 2 + 935 B + 1030 S L + 560 $$SL=0.6,\sigma =1950{B}^{3}-2390{B}^{2}+935B+1030SL+560$$ (4) S L = 0.7 , σ = 2790 B 3 − 4210 B 2 + 2025 B + 965 S L + 625 $$SL=0.7,\sigma =2790{B}^{3}-4210{B}^{2}+2025B+965SL+625$$ (5) S L = 0.8 , σ = 2270 B 3 − 2850 B 2 + 1415 B + 740 S L + 685 $$SL=0.8,\sigma =2270{B}^{3}-2850{B}^{2}+1415B+740SL+685$$ (6) S L = 0.9 , σ = 3788 B 3 − 4394 B 2 + 1855 B + 865 S L + 760 $$SL=0.9,\sigma =3788{B}^{3}-4394{B}^{2}+1855B+865SL+760$$

The correlation coefficients are 0.925, 0.887, 0.905, and 0.892. From the fitting correlation coefficient and fatigue strain curve, it can be concluded that the damage development law of rubber concrete is the same as that of ordinary concrete. By parameter u it can be noticed that adding rubber can enhance the fatigue deformation capacity of concrete to a certain extent, and the degree of improvement has nothing to do with the stress level. Taking the blending of 10 % rubber as an example, the difference in fatigue strain increment under different stress levels is small, with a maximum difference of 7.4 μs from the average value, and the fatigue strain is greater than 1,000 μs. It can be examined that the fatigue effect of rubber on concrete at various stress periods is almost the same degree of strain improvement. The fatigue damage and leftover fatigue life of the specimen can be estimated through fatigue strain, which provides a basis for the fatigue application of rubber concrete.

It is predominantly considered to be true that the fatigue life of concrete conforms to the lognormal distribution [58], [59], [60], and its probability density function is: (7) m r u = 1 2 π ln 10 σ k u exp − l g u − μ k 2 2 σ k 2 $${m}_{r}\,\left(u\right)=\frac{1}{\sqrt{2\pi }\mathrm{ln} 10{\sigma }_{k}u}\mathrm{exp}\left[-\frac{{\left(lgu-{\mu }_{k}\right)}^{2}}{2{\sigma }_{k}^{2}}\right]$$

Its cumulative distribution function is: (8) M r u = P B U ≤ u = Φ l g u − μ k σ k $${M}_{r}\left(u\right)=PB\left(U\le u\right)={\Phi}\frac{\left(lgu-{\mu }_{k}\right)}{{\sigma }_{k}}$$

The function value proportional to the cumulative distribution function is the cumulative failure probability. It is recognized that the cumulative failure probability is P and Φ It is a strictly monotonically increasing function with an inverse function. Take the inverse functions on both sides of equation (8) Φ⁻¹, and then its corresponding percentile value is: (9) lg u P B = μ k + Φ − 1 p σ k $$\mathit{lg} {u}_{PB}={\mu }_{k}+{{\Phi}}^{-1}\left(p\right){\sigma }_{k}$$

Make X = lgu _PB,W = Φ⁻¹(p), transform equation (9) into the following linear relationship expression: (10) W = g X + h $$W=gX+h$$

It can be seen that if the logarithm of fatigue life lgV random variables with a standard normal distribution X there is a straight connection, indicating that the fatigue life follows the lognormal distribution, where μ ˆ k = − g ˆ / h ˆ ${\hat{\mu }}_{k}=-\hat{g}/\hat{h}$ , μ ˆ k = 1 / g ˆ ˆ ${\hat{\mu }}_{k}=1/\hat{\hat{g}}$ , the distribution parameters can be determined by regression analysis of the test data according to Eq. (10). Cumulative failure probability P is estimated using the average rank.

The fatigue life test data in Table 7 were analyzed and processed, and the P – N diagram of the probability model with varying rubber content and assorted stress levels was achieved, as demonstrated in Figure 8.

Figure 8:

Lognormal distribution of fatigue life.

Figure 8 shows that the fatigue life of rubber concrete at various dosages obeys a logarithmic normal distribution, and the fitting effect is good. The distribution parameters of each stress level are shown in Table 8.

Table 8:

Serial number	SL	μ ˆ k ${\hat{\mu }}_{k}$	σ ˆ k ${\hat{\sigma }}_{k}$	R2
CS-0	0.6	4.69395	0.16625	0.8607
	0.7	4.35575	0.14915	0.9158
	0.8	3.9653	0.3154	0.95
	0.9	3.4409	0.32205	0.8512
RC-FASF 1	0.6	4.66581	0.05766	0.81933
	0.7	4.49655	0.06231	0.92907
	0.8	4.18035	0.24645	0.92628
	0.9	3.44193	0.20925	0.91884
RC-FASF 2	0.6	4.89646	0.05922	0.93906
	0.7	4.68684	0.06016	0.8413
	0.8	4.4321	0.12596	0.91744
	0.9	3.80324	0.15886	0.92496
RC-FASF 3	0.6	5.11584	0.0624	0.95808
	0.7	4.9008	0.03744	0.93024
	0.8	4.75776	0.04896	0.94656
	0.9	4.38528	0.04224	0.9456

The Kolmogorov test was performed on the fatigue life distribution. The specific method is for random statistics X, its distribution function M(x)unknown X ₁,X ₂,….,X _n to follow M simple random sample drawn from M ₀(x)is a given distribution function if H ₀:M(x) = M ₀(x) through the sample, M(x) the empirical distribution function is: (11) M n x = 0 , j / N , 1 , x ≤ X 1 X i < x ≤ X j + 1 x > X N , j = 1 , 2 , … , N − 1 $${M}_{n}\left(x\right)=\begin{array}{c}0\text{,}\\ j/N\text{,}\\ 1\text{,}\end{array}\,\begin{array}{c}x\le {X}_{\left(1\right)}\\ {X}_{\left(i\right)}{< }x\le {X}_{\left(j+1\right)}\\ x{ >}{X}_{N}\text{,}\end{array}\text{\ }j=1,2,\dots ,N-1$$

Calculate statistics D _n  = sup _{−∞<x<+∞} |F _n (x)-F ₀(x)|, that is, in each sequence sample x _k find the maximum deviation value between the empirical and theoretical distribution functions. If you specify a threshold α (or called the inspection level), it corresponds to a constant D _n,α , making P(D _n  > D _n,α /H ₀) = α then when D _n  > D _n,α negate H ₀, otherwise, accept H ₀. The investigation findings are illustrated in Table 9.

Statistics D _n (0.1) the critical value is 0.642. The test data in Table 9 are all less than the critical value, so it is accepted, H ₀ that is to say, the fatigue life probability model of rubber concrete is considered to obey the lognormal distribution.

The formula for calculating the fatigue life of rubber concrete is obtained from equation (9): (12) F L p = 10 μ k + Φ − 1 p σ k $${FL}_{p}={10}^{\left[{\mu }_{k}+{{\Phi}}^{-1}\left(p\right){\sigma }_{k}\right]}$$

Substitute five different failure probabilities of 0.1, 0.2, 0.3, 0.4, and 0.5 and the distribution parameters corresponding to the concrete rubber content and stress level in Table 8, into equation (12) to obtain the fatigue life under different failure probabilities, such as shown in Figure 9.

Figure 9:

The fatigue life of rubber concrete under various failure probabilities.

In studying concrete fatigue properties, exponential and power function formulas often express the change rules of its S–N and P–S–N curves [37]. After logarithmic processing, the exponential function model is converted into a single logarithmic form. S–IgN, the power function model is transformed into the logarithmic form IgS–IgN since the logarithmic equation cannot satisfy the boundary conditions [38], the scope of use is limited, and it is only suitable for the fatigue life of the main part, and extension is not allowed. The logarithmic fatigue equation can satisfy the boundary conditions and better fit the fatigue test results, and its applicable range can be slightly extended. This study uses the double logarithmic fatigue equation for research and analysis. That is, the fatigue life of rubber concrete samples under a certain failure probability N and corresponding stress levels S there is the following expression relationship between: (13) I g S = A I g N + B $$IgS=AIgN+B$$

The two S–N curves are of greatest concern in fatigue design and engineering applications of concrete structures. One is the probability of failure P = 0.05 when the P–S–N the curve generated by joining the points under each stress level condition is obtained, thus obtaining N = 2 × 10⁶ the conditional fatigue ultimate strength is the upper limit stress corresponding to the condition. This value makes arrangements for a base for the fatigue design of concrete structures; the other is the failure probability. P = 0.5 when the P–S–N the curve generated by connecting the points under each stress level condition is obtained, thus obtaining N = 2 × 10⁶ the upper limit stress corresponding to the time is the fatigue ultimate strength. This value is the fatigue ultimate bearing strength of the concrete structure and sets a standard for verification [39]. According to Eq. (13), I double-log linear regression is performed on the data to obtain the failure probability P = 0.5 h IgS–IgN linear relationship, as shown in Figure 10.

Figure 10:

Failure probability P = 0.5 double logarithmic fatigue equation of rubber concrete.

When the failure probability P = 0.05 Hour: (14) C S − 0 : I g S = − 0.135 I g N + 0.452 $$CS-0:IgS=-0.135IgN+0.452$$ (15) RC − FASF 1 : I g S = − 0.137 I g N + 0.478 $$\text{RC}-\text{FASF\,}1:IgS=-0.137IgN+0.478$$ (16) RC − FASF 2 : I g S = − 0.144 I g N + 0.537 $$\text{RC}-\text{FASF\,}2:IgS=-0.144IgN+0.537$$ (17) RC − FASF 3 : I g S = − 0.153 I g N + 0.652 $$\text{RC}-\text{FASF\,}3:IgS=-0.153IgN+0.652$$

The correlation coefficients are 0.961, 0.853, 0.841, and 0.845, respectively. The correlation coefficient of rubber concrete is lower than R ² = 0.9, indicating that the fatigue life data of rubber concrete during the test was relatively discrete. It is generally required to ensure a fatigue life of more than 2 million times under cyclic loading, fatigue life FL = 2 × 10⁶ substituting into Eqs. (14)–(17), the fatigue strengths. It can be examined from the above data that compared with ordinary concrete, the fatigue ultimate strength of rubber concrete has been improved. If a certain amount of rubber particles is mixed into a concrete structure subjected to cyclic loads, its fatigue performance will be improved. In this article, the stress cycle eigenvalues are small, rhoequals 0.1. Therefore when the stress level is lower than the ultimate strength of fatigue mentioned above, it is considered that fatigue failure will not occur. This paper presents the fundamental rules governing the change in compressive fatigue life of rubber concrete, addressing the shortcomings of existing research in this field. The number of test samples is insufficient, and the research results still need more experimental research to supplement and improve.