Front. Mater. Frontiers in Materials Front. Mater. 2296-8016 Frontiers Media S.A. 1216747 10.3389/fmats.2023.1216747 Materials Original Research A new experimental method to measure and calculate the tensile strength of concrete Chai et al. 10.3389/fmats.2023.1216747 Chai Qihui 1 2 Huang Shasha 1 Wan Fang 1 * Wu Feng 1 Feng Lingyun 1 1 School of Water Resources, North China University of Water Resources and Electric Power, Zhengzhou, China 2 Henan Key Laboratory of Water Resources Conservation and Intensive Utilization in the Yellow River Basin, Zhengzhou, China

Edited by: Wei Ge, Zhengzhou University, China

Reviewed by: Wei Dongle, Dalian University of Technology, China

Zhang Lei, Yellow River Institute of Hydraulic Research, China

 *Correspondence: Fang Wan, wanxf1023@163.com 20 07 2023 22 05 2026 2023 10 1216747 04 05 2023 28 06 2023 Copyright © 2023 Chai, Huang, Wan, Wu and Feng. 2023 Chai, Huang, Wan, Wu and Feng

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 Introduction

A new method is introduced to test the tensile strength of concrete: the cylinder transverse splitting test. Compared with the cylinder splitting (or Brazilian) test, the cylinder transverse splitting test involves different load positions and offers the advantages of doubling the data volume with the same number of specimens and improving the detection accuracy.

 Methods

Finite element analysis software was used to simulate the concrete cylinder transverse splitting test and the stress distribution on the failure surface was determined.

 Results and Discussion

The results show that the fracture of a cylinder is mainly determined by tensile stress. The splitting strength of normal concrete and crumb rubber concrete, a new environmentally friendly concrete material that has gathered considerable attention in recent years, was obtained by cylinder transverse and cube splitting tests. The cylinder transverse splitting test data show a stable correlation with the cube splitting test data and well characterize the concrete strength. The relationship between the cylinder nominal transverse splitting strength (f 2) and cube splitting strength (f 1) was established on the basis of linear analysis of the test data: f 1 = 0.51f 2. The calculated cube splitting strength is in good agreement with the test values.

 cylinder transverse splitting method nominal transverse splitting strength cube splitting strength crumb rubber concrete rubber content section-at-acceptance Structural Materials 1 Introduction

The tensile strength of concrete is substantially less than its compressive strength. Cracks in concrete easily propagate under tension, which can affect its service life and durability (Bhanja and Sengupta, 2005; Ge et al., 2022). A large number of laboratory studies have evaluated the tensile strength of concrete using three main methods: direct tensile tests; beam flexural tests; and splitting tensile tests (Rashid, et al., 2002; Lu and Li, 2011). Direct tension is theoretically considered to yield a tensile strength similar to the true strength of the concrete under pure uniaxial tension (Wu et al., 2012), but is often difficult to apply owing to the lack of control over the initial crack position, local stress concentrations, and eccentricity strain (Swaddiwudhipong et al., 2003; Zi et al., 2008; Sarfarazi et al., 2018). Furthermore, tensile strength values derived from beam flexural tests tend to be higher than those obtained from direct tensile tests (Raphael, 1984). There are two types of splitting tensile tests listed in GB/T50081-2002. (2002): the cube splitting test (Figure 1A) and the cylinder splitting (or Brazilian) test (Figure 1B). Previous studies have shown that the tensile strength obtained from cylinder splitting tests are closer to the true tensile strength than flexural tests (Mindess et al., 2002; Wang et al., 2022). The results of cylinder splitting tests have also been shown to provide better tensile strength predications than cube splitting tests (Nilsson, 1961).

Two types of concrete specimen splitting tests: the cube splitting test (A) and the cylinder splitting test (B).

The measurement of concrete strength is important for evaluating the construction quality of completed concrete projects. The drilled-core method is commonly used to detect the concrete structure strength because of its immediacy and reliability (Ge et al., 2020). In this approach, cylindrical specimens with a diameter of 100 mm are drilled from a concrete structure and the compressive strength and splitting strength are tested. The Brazilian test is used to measure the tensile strength of concrete within the core-drilling method. The drilling position and number of samples is stipulated to ensure minimum damage to the concrete structure within the detection range (JGJ/T384-2016, 2016), which can be difficult. The drilled-core method has strict requirements on the height-diameter ratio, perpendicularity, and flatness of the end face of the cylindrical specimen, which increases the difficulty of drilling and specimen processing.

The following limitations are often encountered when using cylinder splitting to measure the tensile strength of concrete following the drilled-core method (JGJ/T384-2016, 2016). 1) Differing from cylindrical specimens made in the laboratory, cylindrical core specimens that are drilled on concrete structures often have variable lengths with uneven end faces. Cylinder core specimens must thus be processed prior to testing with a height-to-diameter ratio of 2. 2) Tensile strength data are often limited because each cylindrical core specimen has only one tensile strength value and concrete structures have a limited amount of available specimen owing to drill core damage (Wang et al., 2023). 3) Specialized skills, tools, and working conditions are required to obtain a complete concrete cylinder core. To improve this situation, Yuan et al. (2006) presented the cylinder transverse splitting method to test the splitting strength of concrete cylinder specimens. Differing from the Brazilian test, the load in the cylinder transverse splitting method is applied across the diameter and perpendicular to the central axis. The direction of the specimen bus bar is thus pulled, which results in splitting tensile failure of the cross-section (Figure 2). The transverse splitting method has the following advantages: 1) increased amount of tensile strength data (a concrete cylinder specimen can be split multiple times); 2) reduced concrete core specimen processing procedure (the end of the concrete core specimen does not need to be cut and leveled); and 3) increased use ratio of the concrete core specimen (a specimen that is shorter than the minimum length specified in JGJ/T384-2016. (2016) can be tested).

Cylinder transverse splitting method.

Yu et al. (2010) theoretically verified the rationality of multiple transverse cleavage rafting and concluded that higher specimen quantity improves the detection accuracy and reduces the confidence interval difference and upper limit of the variation coefficient. An experimental study by Xiao et al. (2011) showed that the splitting strength (detected by the cylinder transverse splitting method) of a concrete cylinder correlates well with the concrete compressive strength, and that the cylinder transverse splitting method can be used to more accurately and conveniently determine the concrete materiality than traditional core axial compression methods. Yuan et al. (2016) used a finite element model to analyze the rationality of the stress distribution on the transection of cylindrical concrete and found that the tensile failure of the transection depends on the cylinder’s axial tensile stress.

Although the cylinder transverse splitting method has been shown to be feasible for testing the concrete tensile strength in many aspects (e.g., testing, mathematical statistics, finite element modeling), the cylinder transverse splitting strength is difficult to determine using a reasonable calculation formula. The cylinder transverse splitting strength was previously expressed by the nominal transverse splitting strength, P/A, where P is the failure load and A is the surface area cleavage. In this study, a strong correlation is found between the nominal transverse splitting strength of cylindrical concrete obtained by the cylinder transverse splitting method and the splitting strength of cubic concrete. A calculation formula to test the concrete tensile strength by the cylindrical transverse splitting method is established using the regression of a large number of experimental data. On the basis of previous tests, this study developed an improved specimen clamp for the cylinder transverse splitting method. In addition to normal concrete (NC), crumb rubber concrete (CRC) was also investigated to increase the amount of data and explore a wider range of concrete composition.

CRC is a new environmentally friendly concrete material made by the addition of rubber particles (e.g., crushed automobile tire waste) into ordinary concrete and has broad engineering application prospects (Lin et al., 2023). The rubber particles have a good elasticity, which can alleviate the stress concentration inside the concrete and reduce the probability of primary crack formation and concrete structural damage caused by an external load (Youssf et al., 2023). The addition of rubber particles to NC improves its crack resistance (Li et al., 2021; Elsayed et al., 2022; Feng et al., 2022), impact resistance (Topcu and Avcular, 1997), freeze-thaw durability (Savas et al., 1997; Paine and Dhir, 2002; Khan et al., 2021), abrasion resistance (Hang and Fan, 2011; Sun et al., 2021; Zhang et al., 2023), and reduces its elasticity modulus (Hernandez et al., 2002; Benazzouk et al., 2003; Haldar and Karmakar, 2021). Rubber particles can also reduce the concrete strength (Xu et al., 2020; Xu et al., 2021; Han et al., 2023). Segre et al. (2002) pointed out that zinc stearate tends to reduce the binding force between rubber particles and cement by infrared rays and chemical titration. Although Segre and Jocke (2000) found that rubber particles soaked in saturated NaOH solution for 20 min prior to mixing with the cement slurry improved the compressive strength, splitting strength, and bending strength of rubber cement mortar, they were unable to reach the strength level without the addition of rubber particles. The CRC tensile strength testing method must be thoroughly investigated to ensure engineering application safety (Adeboje et al., 2020).

 2 Materials and methods 2.1 Materials

Ordinary Portland Cement 42.5 was used in this study and its properties are listed in Table 1. The coarse aggregates were limestone gravel and the fine aggregates were natural river sand and rubber particles of 1–3 and 3–6 mm, the latter of which were obtained by crushing scrap tires. The properties of the coarse and fine aggregates are listed in Table 2. The apparent density of the rubber particles is 1119 kg/m3. The grading curve of the fine aggregates and rubber particles is shown in Figure 3. The indexes of the fine and coarse aggregates meet the requirements of GB/T14684-2011. (2011) and GB/T14685-2011. (2011).

Cement properties.

Setting time Compressive strength Flexural strength
Initial Final 3 days 28 days 3d ays 28 days
169 min 278 min 21.5 MPa 48.3 MPa 5.4 MPa 9.4 MPa

Aggregate properties.

Material Size (mm) Apparent density (kg/m3) Mud content (%) Crushed index (%) Fineness modulus
coarse aggregates 5∼20 2710 0.42 9.3 --
aggregates 0∼4.75 2650 1.5 -- 2.7

Grading curves of sand and rubber particles.

 2.2 Mixtures

Six water-cement ratios (W/C) were used, as listed in Table 3. The CRC was produced by partially replacing the fine aggregates with rubber particles in volumetric proportions of 1%, 3%, 5%, 10%, and 15% (Table 4). The samples covered a relevant concrete strength range that is commonly used in concrete engineering.

Mixture proportions of NC (kg/m3).

Mixtures Water–cement ratio Water Cement Crushed stones Sand
NC0.74 0.74 195 262.4 1197.3 733.8
NC0.62 0.62 195 312.2 1170.6 717.5
NC0.54 0.54 195 362.1 1143.8 701.1
NC0.42 0.42 195 461.9 1161.6 598.4
NC0.38 0.38 195 511.7 1133.1 583.7
NC0.35 0.35 195 557.14 1107.48 570.52

Mixture proportions of CRC (kg/m3).

Mixtures Rubber content (%) Water–cement ratio Water Cement Crushed stones Sand Rubber
CRC0.74-1 1 0.74 195 262.4 1197.3 726.5 3.00
CRC0.54-1 0.54 195 362.1 1143.8 694.1 2.90
CRC0.42-1 0.42 195 461.9 1161.6 592.4 2.48
CRC0.38-1 0.38 195 511.7 1133.1 577.9 2.42
CRC0.35-1 0.35 195 557.14 1107.48 564.8 2.36
CRC0.74-3 3 0.74 195 262.4 1197.3 711.8 9.00
CRC0.54-3 0.54 195 362.1 1143.8 680 8.71
CRC0.42-3 0.42 195 461.9 1161.6 580 7.43
CRC0.38-3 0.38 195 511.7 1133.1 566.2 7.25
CRC0.35-3 0.35 195 557.14 1107.48 553.4 7.09
CRC0.74-5 5 0.74 195 262.4 1197.3 679.3 15.00
CRC0.54-5 0.54 195 362.1 1143.8 666.1 14.50
CRC0.42-5 0.42 195 461.9 1161.6 568.4 12.40
CRC0.38-5 0.38 195 511.7 1133.1 554.7 12.10
CRC0.35-5 0.35 195 557.14 1107.48 541.92 11.80
CRC0.74-10 10 0.74 195 262.4 1197.3 660.8 30.00
CRC0.54-10 0.54 195 362.1 1143.8 631.1 29.00
CRC0.42-10 0.42 195 461.9 1161.6 538.4 24.80
CRC0.38-10 0.38 195 511.7 1133.1 525.7 24.20
CRC0.35-10 0.35 195 557.14 1107.48 513.3 23.60
CRC0.74-15 15 0.74 195 262.4 1197.3 624.3 45.00
CRC0.54-15 0.54 195 362.1 1143.8 596.1 43.50
CRC0.42-15 0.42 195 461.9 1161.6 508.4 37.20
CRC0.38-15 0.38 195 511.7 1133.1 496.7 36.30
CRC0.35-15 0.35 195 557.14 1107.48 484.7 35.40
2.3 Specimen preparation

Each mixture proportion contained cubic and cylindrical transverse splitting specimens. The specimens for testing the mechanical properties were prepared based on SL352-2006. (2006) and the number and size are listed in Table 5.

Specimen number and size for each group.

Test Number Size
Standard cube splitting strength 3 150 mm × 150 mm × 150 mm
Non-standard cylinder transverse splitting method 3 ∅ 100 mm × 200 mm
2.4 Test method 2.4.1 Finite element simulations

ANSYS software was used to simulate the transverse splitting tests of the concrete cylinders. The simulations included solid 65 as the concrete material, a sample size of Φ0.1 × 0.1 m, elastic modulus of 30,000 MPa, and Poisson’s ratio of 0.2. The constitutive relation adopts the Drucher-Prager model. The failure criterion is an improved William-Warnke five-parameter failure surface, which requires that the following parameters be defined: uniaxial tensile strength; uniaxial compression strength; biaxial compression strength; and uniaxial and biaxial compression strength under a certain confining pressure. Owing to the lack of multi-axis test parameters, ANSYS only requires uniaxial tensile strength and uniaxial compression strength input values, and the other parameters were calculated by the ANSYS default formula. The fracture opening shear transfer coefficient of concrete was 0.3, the fracture closing shear transfer coefficient was 0.9, and the uniaxial tensile strength was 3 MPa. Concrete crushing failure was not considered in the calculation. The remaining concrete parameters were set to the ANSYS default values. The parameters are selected according to the mechanical properties of the most common C30 concrete. The elastic modulus is about 30000 MPa, and the tensile strength is about 3 MPa (1/10 of the compressive strength). The shear stress transfer coefficient of open fissure in concrete is 0.3–0.5, which is 0.3 in this paper. The shear stress transfer coefficient of closed cracks is 0.9–1.0, which is 0.9 in this paper.

 2.4.2 Cube splitting tests

Cube splitting tests were performed in accordance with SL352-2006. (2006). The test instrument was a hydraulic universal testing machine with a maximum test force of 2000 kN. The experimental device is shown in Figure 4, which includes a steel square filler strip with a cross-section of 5 × 5 mm and length of 200 mm. To apply a linear uniform load on the concrete surface, the filler strip was placed between the concrete surface and pressure plate (Tang, 1994; Rocco et al., 2000; Olesen et al., 2006). The cube splitting strength of concrete was calculated according to: f 1 = 2 P 1 π A 1 = 0.637 P 1 A 1 where f 1 is the cube splitting strength (MPa), P 1 is the failure load (N), and A 1 is the cross-sectional area (mm2).

Cube splitting test.

 2.4.3 Cylinder transverse splitting method

The setup used in the cylinder transverse splitting test is shown in Figure 5. One of each of the upper and lower pressure plates of the hydraulic servo universal testing machine was used (Figure 6). The test instrument was a hydraulic universal testing machine with a maximum test force of 1000 kN. The contact position with the concrete specimen was an anti-arc pressure cutter (equivalent of a filler strip) and the contact surface was 5-mm wide.

Schematic diagram of the test fixture: ① pressing plate; ② stiffening rib; ③ pressure cutter; and ④ connecting bolt.

Setup installation.

The cylinder transverse splitting test process was as follows. The cylinder specimens were cured in a standard curing room for 28 days and immediately tested after removing the surface moisture. A pencil was used to mark the circumference parallel to the specimen cross-section to determine the splitting position (Figure 7A). The displacement boundary conditions of the cylinder specimens are related closely to the cylinder shape (Jia, 1997; Kanos et al., 2006; Yang and Li, 2002; Zhou et al., 2008; Wu et al., 2021) and the height-diameter ratio is the basic parameter that reflects the cylinder shape characteristics. Yuan et al. (2016) found that the transverse fracture failure load changed little with increasing specimen height when the cylinder height-diameter ratio was greater than or equal to 0.7 and is thus considered stable. The cylinder specimen could therefore be uniformly split three times. The height-diameter ratio was 2 in the first splitting and 1 in the second and third splittings (Figure 7B). A vernier caliper was used to measure the cylinder diameter along the predetermined splitting position. The mean value of three measurements was taken as the effective specimen diameter to calculate the cross-sectional area A 2 . The sample was placed between the aligned upper and lower pressure knives according to the predetermined splitting position. The failure load P 2 was recorded at 0.04–0.06 MPa/s.

Transverse splitting position of cylinder specimen (D = 100 mm, L = 200 mm, l = 50 mm) (A). The cylinder specimen could be uniformly split three times (B).

The cylinder nominal transverse strength of concrete is defined as: f 2 = P 2 A 2 where f 2 is the nominal cylinder transverse strength (MPa), P 2 is the failure load (N), and A 2 is the cross-sectional area (mm2).

 3 Results and discussion 3.1 Comparison of forces

In the cube splitting tests, an evenly distributed compression load was applied to the middle of the upper and lower surfaces of the concrete specimen (Figure 8A). The compression load caused compressive stress in the axial surface of the concrete specimen (same plane as the load) and uniform tensile stress in most of the middle area (Figure 8B). The fracture surface exhibited failure under tensile stress (Figure 8C). The cylinder transverse splitting tests differed in that the cylinder specimen was uniformly loaded on the cross-sectional curve along the circumference (Figure 9A). The load direction was the plumb direction, the same as in the cube splitting tests. The results from the ANSYS stress analysis show that the concrete specimen cross-section was under a state of tension and compression complex stress (Figure 9B), whereas most of the central area was under a state of tension stress, which influences the splitting failure. The fracture surface performance is consistent with the stress analysis results (Figure 9C), which indicates that the cylinder transverse splitting test can be used to effectively measure the concrete splitting strength.

In the cube splitting tests, an evenly distributed compression load was applied to the middle of the upper and lower surfaces of the concrete specimen (A). The compression load caused compressive stress in the axial surface of the concrete specimen and uniform tensile stress in most of the middle area (B). The fracture surface exhibited failure under tensile stress (C).

In the cylinder transverse splitting tests, the cylinder specimen was uniformly loaded on the cross-sectional curve along the circumference (A). The concrete specimen cross-section was under a state of tension and compression complex stress (B), whereas most of the central area was under a state of tension stress, which influences the splitting failure. The fracture surface performance is consistent with the stress analysis results (C).

 3.2 Comparison of standard deviations

Three batches of cubic and cylindrical specimens of NC were made at different times (in 2011, 2014, and 2017) for the cube and cylinder transverse splitting tests. The standard deviation indicates the degree of data dispersion; smaller standard deviations reflect more concentrated data distributions. The relative precision of the test data can thus be explained by comparing the standard deviations obtained during the cylinder transverse splitting versus the cube splitting tensile failure loads. Each group of the second test (in 2014) consisted of three cubic and two cylindrical specimens and thus involved three cubic splitting tensile failure tests and six cylinder transverse splitting failure tests, as shown in Table 6. The standard deviation of the cylinder transverse splitting test results are generally smaller than those obtained from the cube splitting test results. The precision of the concrete tensile strength values obtained by the cylinder transverse splitting test is slightly higher than that of the cube splitting tests.

Test result.

W/C 0.75 0.54
Particle size and mixing amount of rubber Cube splitting tensile failure load Standard deviation Cylinder transverse splitting tensile failure load Standard deviation Cube splitting tensile failure load Standard deviation Cylinder transverse splitting tensile failure load Standard deviation
NC 48.07 10.10 25.24 2.15 90.46 3.48 13.19 9.97
70.74 31.46 88.42 41.92
68.00 27.20 96.61 34.92
27.77 41.07
27.83 37.75
30.89 40.35
3–6mm, 5% 63.07 6.10 28.65 3.85 93.13 3.67 32.95 4.24
59.44 37.66 88.00 45.44
73.80 27.40 84.18 39.23
28.20 33.07
25.89 36.49
27.86 38.16
3–6mm, 10% 53.59 6.80 17.53 6.35 85.70 1.05 33.26 4.3
61.15 38.63 83.17 43.89
70.23 25.57 84.81 39.21
23.10 31.87
26.09 36.86
27.95 32.33
3–6mm, 15% 72.00 8.04 20.79 2.39 82.74 4.31 31.98 2.98
52.42 28.07 72.78 39.89
60.47 23.29 80.80 34.59
21.69 32.59
24.90 30.68
22.61 32.78
1–3mm, 5% 59.61 4.04 21.08 2.49 83.15 4.46 24.08 4.81
67.23 28.31 83.15 38.16
57.95 27.26 92.61 36.28
27.15 37.55
27.81 36.57
24.81 34.15
1–3mm, 10% 62.33 4.11 17.15 2.49 78.59 2.30 28.77 2.28
56.88 24.09 80.16 35.93
52.28 24.01 84.05 32.92
23.08 35.14
24.37 32.56
22.50 33.16
1–3mm, 15% 53.11 1.04 14.29 2.58 83.23 4.43 28.10 1.93
50.70 18.51 79.14 33.48
51.19 22.12 72.48 29.63
21.60 32.15
19.61 33.28
20.34 31.30
W/C 0.42 0.38
Particle size and mixing amount of rubber Split tensile failure load of cube standard deviation transverse Split tensile failure load of cylinder standard deviation Split tensile failure load of cube standard deviation transverse Split tensile failure load of cylinder standard deviation
NC 102.06 4.92 45.96 3.18 133.99 7.01 48.44 3.87
108.71 54.59 116.91 59.16
114.1 45.65 124.01 52.81
47.27 56.8
45.42 55.18
46.92 49.25
3–6mm, 5% 107.49 4.41 37.58 5.81 122.75 3.22 49.11 2.46
99.8 56.35 123.21 57
110.23 45.94 116.17 52.06
43.16 50.41
42.02 52.96
47.62 51.95
3–6mm, 10% 106.14 5.73 46.02 5.67 85.70 3.06 43.96 7.39
101.56 56.98 83.17 66.39
92.37 43.34 84.81 50.03
44.27 46.49
40.51 48.34
39.98 47.16
3–6mm, 15% 80.46 8.04 40.94 6.23 110.43 5.96 38.59 7.63
96.78 57.3 96.26 61.51
98.18 40.64 100.34 45.97
41.83 41.26
40.65 44.26
39.43 40.4
1–3mm, 5% 109.46 5.42 39.34 5.92 123.14 9.26 51.71 2.71
106.56 58.04 102.16 58.77
96.78 45.68 120.09 51.63
42.4 50.61
43.09 52.69
46.15 51.81
1–3mm, 10% 98.51 1.91 34.98 4.44 118.06 7.58 42.66 6.21
100.94 47.76 110.24 61.37
103.18 43.88 99.56 47.6
45.47 49.68
39.7 45.11
38.03 44.24
1–3mm, 15% 102.32 4.92 26.73 5.07 103.98 7.01 34.53 4.56
90.36 39.35 107.27 47.76
95.02 38.2 91.04 40.21
40.24 39.05
40.42 42.09
41.83 46.9
W/C 0.35
Particle size and mixing amount of rubber Split tensile failure load of cube standard deviation transverse Split tensile failure load of cylinder standard deviation
NC 133 3.22 51.01 5.03
140.19 65.18
133.78 62.96
58.16
60.09
53.13
3–6mm, 5% 137.69 4.55 44.15 6.39
129.67 62.93
126.98 56.98
57.08
61.87
51.47
3–6mm, 10% 127.91 20.47 44.13 5.64
93.47 62.58
142.24 51.13
54.33
48.66
52.04
3–6mm, 15% 111.45 13.31 42.92 7.71
104.12 65.74
80.27 46.26
44.31
46.03
50.25
1–3mm, 5% 126.67 4.88 46.88 3.42
131.63 56.21
119.72 55.92
54.38
53.93
57.3
1–3mm, 10% 97.12 12.39 42.91 5.73
115.53 57.48
127.21 43.65
48.19
53.14
56.14
1–3mm, 15% 105.45 1.34 40.72 5.01
108.37 53.52
105.63 44.65
44.95
40.71
51.94
3.3 NC splitting strength

The cube splitting strength and cylinder nominal transverse splitting strength of the NC show similar negative dependencies with W/C (Figure 10), which is in agreement with the W/C law of Abrams. The cylinder nominal transverse splitting strength is found to be higher than the cube splitting strength for concrete with the same W/C. These results are in agreement with previous studies (Malhotra, 1970; Hang and Fan, 2011) mainly because of the size effect.

Relationship between splitting strength (NC) and W/C: First batch (A); Second batch (B); Third batch (C).

 3.4 CRC splitting strength

Similar to NC, the cube splitting strength and cylinder nominal transverse splitting strength of CRC decrease with increasing W/C (Figure 11) and show the same trend regardless of the rubber particle size (3–6 or 1–3 mm) or rubber particle content (5%, 10%, or 15%). This regularity shows no correlation with rubber particle size and content. The samples with 1% and 3% rubber particle contents show the same behavior and are not discussed further.

Relationship between splitting strength (CRC) and W/C: CRC with 5% rubber particles (3–6 mm) (A); CRC with 5% rubber particles (1–3 mm) (B); CRC with 10% rubber particles (3–6 mm) (C); CRC with 10% rubber particles (1–3 mm) (D); CRC with 15% rubber particles (3–6 mm) (E); CRC with 15% rubber particles (1–3 mm) (F).

As shown in Figure 12, regardless of rubber particle size (3–6 or 1–3 mm) and W/C (0.35, 0.42, or 0.74), the cube splitting strength and cylinder nominal transverse splitting strength both consistently decrease with increasing rubber particle content. This is mainly because the rubber particle strength is far less than the sand strength, and the bond strength between the rubber and cement is far less than the bond strength between the sand and cement. The samples with 0.38 and 0.54 W/C show the same behavior and are not discussed here. The influence of rubber particle content on concrete strength is similar to previous results (Eldin and Senouci, 1993; Topcu, 1995; Toutanji, 1996; Ali et al., 2000). For the same rubber particle content, the cylinder nominal transverse splitting strength is higher than the cube splitting strength.

Relationship between splitting strength (CRC) and content of rubber particles: CRC with 0.35W/C and rubber particles 3∼6 mm (A); CRC with 0.35W/C and rubber particles 1∼3 mm (B); CRC with 0.42W/C and rubber particles 3∼6 mm (C); CRC with 0.42W/C and rubber particles 1∼3 mm (D); CRC with 0.74W/C and rubber particles 3∼6 mm (E); CRC with 0.74W/C and rubber particles 1∼3 mm (F).

On the basis of the above analysis, the cylinder nominal transverse splitting strength can be used to represent the concrete tensile strength, similar to the cube splitting strength.

 3.5 Relationship between cube and cylinder nominal transverse splitting strength

The cube splitting strength is given in SL352-2006. (2006) as Eq. 1. The concrete splitting surface has a similar stress distribution under the condition of cylinder transverse and cube splitting (Figure 8B; Figure 9B). The cylinder transverse splitting strength is assumed to follow a formula in the same form as the cube splitting strength, namely: f 2 ′ = γ P 2 A 2 where f 2 ′ is the cylinder transverse splitting strength (MPa) and γ is the coefficient of stress distribution. The results show that an internal relationship exists between the cube and cylinder transverse splitting strengths for concrete: f 1 = β f 2 ′ where β is the coefficient related to the specimen size. Combined with Eq. 1 and Eq. 2, Eq. 3, Eq. 4, we obtain: f 1 = β γ P 2 A 2 = β γ f 2

For ψ = β γ , the relational expression between the cube and cylinder nominal transverse splitting strength is: f 1 = ψ f 2

The linear fitting results (that pass through the origin) of Eq. 6 are shown in Figure 13. The ψ of NC (0.50; Figure 13A) differs slightly from that of the CRC (0.50 in Figure 13B, 0.53 in Figure 13C), which indicates that the incorporation of rubber particles into NC has only a small effect on ψ . The slight difference in ψ between CRC with 3–6 mm (0.50, Figure 13B) and 1–3 mm rubber particles (0.53, Figure 13C) shows that the effect of rubber particle size on ψ is small. When all of the data are combined, ψ = 0.53 with an R2 (correlation coefficient) of 0.994 (Figure 13D).

Linear fitting of Ep. (6): NC (A); CRC (3–6 mm rubber particles) (B); CRC (1–3 mm rubber particles) (C); All concrete (D).

An insertion of the value of ψ into Eq. 6 yields: f 1 = 0.53 f 2

The test value is compared to the calculated value.

The calculated f 1 values from Eq. 7 and test value f 1 ′ are shown in Table 7. There are 17 data sets for NC and 54 data sets for CRC (27 for 3–6 mm CRC and 27 for 1–3 mm CRC) for a total of 71 data sets. Among the 71 data sets, the ratios of the calculated values f 1 to test value f 1 ′ gives a minimum of 0.80, maximum of 1.34, mean of 0.99, and mean square error of 0.087. The ratios are relatively centralized and concentrated around 1.00, as shown in Figure 14. The cube splitting strength calculated by the cylinder nominal transverse splitting strength via Eq. 7 is similar to that obtained by the cube splitting test.

Calculated and test splitting strengths of concrete.

Concrete NC 3–6 mm CRC 1–3 mm CRC f 1 / f 1 ′ Mean value Mean square error
Sequence Number f 2 f 1 f 1 ′ f 1 / f 1 ′ f 2 f 1 f 1 ′ f 1 / f 1 ′ f 2 f 1 f 1 ′ f 1 / f 1 ′
1 4.02 2.04 1.80 1.13 3.78 1.94 1.83 1.06 3.63 1.84 2.05 0.90 0.99 0.087
2 5.32 2.70 2.43 1.11 4.99 2.55 2.65 0.96 4.34 2.19 2.58 0.85
3 5.63 2.86 2.79 1.03 5.86 3.01 2.92 1.03 5.40 2.75 2.53 1.09
4 6.21 3.16 2.97 1.06 6.13 3.11 3.16 0.98 6.33 3.21 3.15 1.02
5 7.19 3.67 3.15 1.17 6.51 3.32 3.32 1.00 6.42 3.26 3.65 0.89
6 6.99 3.57 3.78 0.94 7.40 3.77 4.00 0.94 6.92 3.52 3.71 0.95
7 3.62 1.84 1.93 0.95 4.23 2.14 2.39 0.90 3.99 2.04 2.34 0.87
8 4.44 2.24 2.60 0.86 4.50 2.30 1.72 1.34 3.90 1.99 2.49 0.80
9 6.07 3.11 3.10 1.00 5.72 2.91 2.63 1.11 5.10 2.60 2.76 0.94
10 6.83 3.47 3.50 0.99 6.00 3.06 3.00 1.02 6.02 3.06 2.82 1.09
11 7.44 3.77 3.80 0.99 7.54 3.83 3.50 1.09 6.23 3.16 2.87 1.10
12 4.09 2.09 2.37 0.88 7.02 3.57 3.70 0.96 6.60 3.37 3.49 0.97
13 4.80 2.45 2.53 0.97 3.73 1.89 1.85 1.02 3.32 1.68 1.71 0.98
14 6.19 3.16 2.66 1.19 4.79 2.45 2.60 0.94 4.39 2.24 2.40 0.93
15 6.32 3.21 2.96 1.08 5.78 2.96 3.00 0.99 5.83 2.96 3.02 0.99
16 7.10 3.62 3.51 1.03 6.62 3.37 3.40 0.99 6.74 3.42 3.42 1.01
17 7.60 3.88 3.57 1.09 7.10 3.62 3.70 0.98 7.08 3.62 3.61 1.01
18 3.38 1.73 1.75 0.99 2.89 1.48 1.60 0.93
19 4.61 2.35 2.40 0.98 4.22 2.14 2.42 0.89
20 5.76 2.96 2.80 1.06 5.30 2.70 2.90 0.93
21 6.42 3.26 3.20 1.02 6.18 3.16 3.13 1.02
22 6.64 3.37 3.40 0.99 6.41 3.26 3.31 0.99
23 3.01 1.53 1.71 0.89 2.48 1.28 1.51 0.85
24 4.31 2.20 2.20 1.00 3.99 2.04 2.22 0.93
25 5.54 2.81 2.70 1.04 4.82 2.45 2.71 0.91
26 5.77 2.96 2.90 1.02 5.32 2.70 2.90 0.93
27 6.28 3.21 3.00 1.07 5.88 3.01 3.03 1.00

Distribution of ratio of calculated and tested cube splitting strength.

 4 Conclusion

The cube and cylinder nominal transverse splitting strengths of NC and CRC were determined and compared. On the basis of the results, the following conclusions can be drawn.

• The stress distribution in the tensile direction of concrete is obtained by simulating the cylinder transverse splitting test using finite element analysis software. The tensile stress distribution area of the splitting tensile failure surface is large, which is similar to the stress distribution of the cube splitting tensile failure surface. This indicates that failure during concrete cylinder transverse splitting tests is mainly determined by the tensile stress.

• The standard deviation of each group of concrete cylinder transverse splitting failure load tests is slightly lower than that obtained from the cube splitting failure load tests. This indicates a slightly higher precision of the cylinder tensile strength detected by the cylinder transverse splitting tests than that obtained by the cube splitting tensile tests.

• The cube and cylinder nominal transverse splitting strengths of NC and CRC regularly decrease with increasing water-cement ratio and rubber content, which indicates a certain internal relationship between the cube and cylinder nominal transverse splitting strengths.

• The relationship between the cube (150 × 150 × 150 mm) and cylinder (∅100 mm) nominal transverse splitting strength is established, f 1 = 0.53 f 2 , with a correlation coefficient R2 = 0.994. The influence of rubber particle content and particle size on ψ is small over the investigated range and can be ignored.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary material, further inquiries can be directed to the corresponding author.

 Author contributions

QC and SH performed material preparation, data collection, and modeling. FW performed result analysis and discussion. FW edited the manuscript. LF reviewed the manuscript. All authors contributed to the article and approved the submitted version.

 Funding

The research was supported by The National Key Research and Development Program of China (2022YFC3202300), Major Science and Technology Special Projects in Henan Province (201300311400), General Science Foundation Program of Henan Province (222300420491).

 Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

 Correction note

This article has been corrected with minor changes. These changes do not impact the scientific content of the article.

 Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

 Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fmats.2023.1216747/full#supplementary-material

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