It has been proved that by dispersing steel fiber into a concrete matrix evenly, a complex constrained network would be formed that can enhance the integrity of concrete and restrain the formation of macro-cracks, which provides a good synergistic effect on improving the concrete mechanical performance (Xu et al., 2011; Chi et al., 2014a; Chi et al., 2014b; Su et al., 2018; Meng et al., 2021). Based on previous studies (Chi et al., 2014a; Su et al., 2018), the volume fraction and aspect ratio of steel fiber (SF) are taken as the variables that would influence the axial strength of HFRC. In this study, the hooked-end steel fibers with the aspect ratio of 30, 60 and 80 respectively were adopted to enhance the mechanical performance of HFRC, as shown in Table 1, and the material properties of polypropylene fibers are also listed in Table 2.

Given the levels chosen for the fiber variables and confinements (5–20 MPa), 36 conventional triaxial test groups and 75 true axial test were designed (as shown in Tables 3, 4) based on partial single variable principle. In order to reduce the test error, three specimens have been tested in each group, and then, a total of 111 groups of specimens (108 cylinders for conventional triaxial test and 225 cubes for true triaxial test) were fabricated to investigate the influence of these variables on the axial strength of HFRC under different confinement levels. For convenience, the nomenclature of the specimens was taken as shown in Figure 2, and the fiber information of these specimens for conventional triaxial tests and true triaxial tests are summarized in Tables 3, 4, respectively.

To exclude the influence of concrete strength grade, the C50 and C60 grades of HFRC were selected respectively in the conventional triaxial tests and true triaxial tests. The river sand with fineness modulus of 2.8 and the gravel with particle size of 5–15 mm were adopted. The mixture proportions of the concrete matrix are listed in Table 5.

A detail mixing procedure is shown as follows:

(1) Dry cement particles and fine aggregates were added into a mixer and mixed for 120s.

The specimens were demoulded after 24 h, and then stored in a curing room with a constant humidity of 95% and temperature of 20°C for 28 days. After that, Φ50 mm × 150 mm cylinders were drilled out from 150 mm × 150 mm × 150 mm standard cubic specimens, and then the specimens were cut by 25 mm from the upper part and bottom of the cylinders, respectively.

In the conventional triaxial test, the MTS 815.3 (full-digitally servo-controlled stiffness testing system) was employed. In order to prevent the oil from dipping into the material due to the formation of a macro crack that may propagate to the surface of the specimen with the evolution of damage, the heat shrinkable tube (HST) was used and wrapped on the surface of the specimen. Furthermore, to reduce the friction resistance, a thin layer of grease was applied between the specimen and the steel column. The extensometers and the radial strain gauge were used to measure the axial strain and lateral strain respectively, as shown in Figure 3A, and a schematic diagram of the installation is shown in Figure 3B. Before the loading, a pre-pressure was applied and held at the fixed value of 0/5/10/20 MPa during the whole loading procedure.

In the true triaxial test, the lateral stress with a value pair of (5 MPa, 10 MPa), (4 MPa, 15 MPa), or (3 MPa, 20 MPa) was applied, respectively, to investigate the influence of lateral stress on the axial strength of HFRC. The true triaxial apparatus was employed and the installation of the cubic HFRC specimens is shown in Figure 4.

For each group, three identical specimens were tested to reduce the dispersion of test results, and the value of the axial stress at the peak point of the axial stress-axial strain curve was taken as the axial strength of the HFRC specimen, as shown in Figure 5. The axial strength of HFRC of each test group is determined as the average of the axial stress values of three specimens in one group, the results of which are summarized in Tables 3, 4 by follows.

In engineering situations, the passive constraint are the most representative stress states of concrete members, such as FRP confined concrete members. In this situation, with the increase of axial load, the lateral stress tends to increase proportionally to a certain range, and the axial peak stress are positively correlated with the ratio of lateral stress and axial stress. Therefore, the ratio of the principal stresses σ 1 / σ 3 and σ 2 / σ 3 would be set as independent variables that influence the axial strength, which is often determined by the confinement coefficient of confinement concrete in practice. Besides, since the calculation of passive constraints is complex and lacks mechanical basis, in 2020, Yang and Feng etc. (Yang and Feng, 2020) proposed a method that bridge the passive confinement and active confinement, which makes the active constraint model can be used as a means to calculate the passive constraint problem. Therefore, an active confinement model in which the lateral stresses σ 1 and σ 2 are selected as the independent variables is also studied. In the following sections, aiming at conventional triaxial compression (Section 3) and true axial compression (Section 4), the passive confinement model (Sections 3.1, 4.1) and active confinement model (Sections 3.2, 4.2) for axial strength calculation are respectively established, for the purpose of providing reference for engineering calculation.

Under some special concrete members, such as confinement concrete members, there would be a certain proportional relationship between the lateral stress and the axial stress, and the ratio of the axial stress and lateral stress is always determined by the confinement coefficient, which can be valued by the member design. Assume that the ratio of lateral stress to axial stress σ 1 / σ 3 is known as the independent variable, and the material is subjected to low confinement (e.g., σ 1 / σ 3 ≤ 0.3 in general). Based on our previous study (Li et al., 2017), the addition of SF would enhance the compressive strength of HFRC significantly while the effect of PF is much smaller. Therefore, a fiber correction factor is introduced in the formula and a power function as shown in the following is adopted to capture the main features of the test results, f 3 f c u = 1 + a λ S F 1 + b σ 1 σ 3 c (1) where f c u is the cube compressive strength of the plain concrete matrix of HFRC; λ S F is defined as the characteristic parameter of SF, which is equal to the product of volume fraction and aspect ratio of SF.

The regression analysis on the experimental results in Table 3 indicates that the parameters in Eq. 1 are a = 0.1735, b = 1.6417 and c = 0.3012, i.e., f 3 f c u = 1 + 0.1735 λ S F 1 + 1.6417 σ 1 σ 3 0.3012 (2)

The correlation coefficient R ² between the predictions of the model and the test values is 0.8015.

Taking λ S F and σ 1 / σ 3 as the X and Y-axis, respectively, and f 3 / f c u as the Z-axis, the fitting result diagram of fitting surface including the test results is shown in Figure 6A.

The diagram shows that some discreteness between the fitting results and the test values can be observed. This is mainly attributed to the lack of mechanics or physics basis reflecting the causal relationship between the stress ratio and the axial strength. In fact, the mechanism of concrete material yield behavior is the propagation of inner cracks. In this damage process, all of the three principal stresses play a complex role on the cracks evolution, such as the tensile cracks introduced by the tensile stress on some sections and the shear cracks induced by the maximum shear stress on others sections. Mathematically, these principal stresses should usually satisfy some complicated equations, such as W-W yield model (Willam and Warner, 1974; Yin et al., 2014), Barcelona model (Lubliner et al., 1989; Faria et al., 1998; Wu et al., 2006) and so on, rather than the explicit equation between the axial stress and the ratio of lateral stresses, when the material yield. Therefore, due to the lack of the mechanical mechanism of Eq. 2, there would be some intrinsic deviation that hardly can be eliminated. However, even so, the increase of axial strength with the increase of confinement is reasonably reflected by the statistical formula Eq. 2, which can provide a beneficial reference to the engineering construction.

In practice, the HFRC members are usually under passive constraints. However, because of the uncertainty of lateral principal stress in the passive confinement problem, the axial peak stress is hardly mathematically calculated. A functional relation between the axial stress and the lateral stresses can help the calculation of passive confinement problem calculation, and therefore an establishment of active confinement model is valuable. In such cases, a known confining pressure would be the independent variable and the axial strength is a function of confining pressure. For the situation that σ 1 / σ 3 and σ 2 / σ 3 are less than 0.3, a power function is adopted to reflect the evolution laws of the strengths as follows, f 3 f c u = 1 + a λ S F 1 + b σ 1 f c u c (3) where σ 1 = σ 2 is the lateral stress that is equal to the known confinement pressure P. Through regression analysis, the parameters are determined as a = 0.1510, b = 2.9390, and c = 0.8432, respectively, and the correlation coefficient R 2 = 0.9496 . Hence, the formula can be rewritten as, f 3 f c u = 1 + 0.151 λ S F 1 + 2.939 σ 1 f c u 0.8432 (4)

The diagram of the fitting surface with the test results is shown in Figure 6B, from which it can be seen that compared to the illustration shown in Figure 6A, a satisfactory fitting result with a higher correlation coefficient of 0.9496 is obtained. This is due to the underlying physical mechanism that the concrete material will yield once the principal stresses exceed the critical values, hence, a causal relationship between the axial strength and the confinement pressure makes sense. In many previous achievements of yield model of concrete material, the yield behavior were described as an implicit mathematical form of F σ 1 , σ 2 , σ 3 = 0 (Willam and Warner, 1974; Ottosen and Krenk, 1979; Hsieh et al., 1982; Podgórski, 1985; Lubliner et al., 1989; Faria et al., 1998; Wu et al., 2006; Dede and Ayvaz, 2010a; Dede and Ayvaz, 2010b; Yin et al., 2014). These mathematical forms are verified to describe the yield behavior of concrete materials well in engineering practice. Although a more detailed micromechanical mechanism that how the principal stresses impact on the cracks is remains a puzzle, the macroscopic practice experience shows that there must be some objective causal relationship between the principal stresses and the yield behavior of concrete. Therefore, a simplified and explicit equation as Eq. 4 can be fitted to describe the yield behavior of HFRC well.

In the majority of practical situations, the lateral principal stresses are not equal to each other. But even so, a statistical relation between the lateral stresses and the axial stress can be mostly found when the concrete material yield.

When the stress ratios σ 1 / σ 3 and σ 2 / σ 3 are fixed, an empirical function in the following form is adopted to keep the consistency with the proposed function of Eq. 1, f 3 f c u = 1 + 0.1735 λ S F 1 + a σ 1 σ 3 0.3012 + 1.6417 − a σ 2 σ 3 0.3012 (5)

For the case of σ 1 = σ 2 , this formula can be completely reduced to the conventional triaxial regression formula Eq. 1 in previous section. Upon the regression analysis of the test data shown in Table 4, the parameters can be determined as a = 0.2916. Hence, Eq. 5 can be rewritten as f 3 f c u = 1 + 0.1735 λ S F 1 + 0.2961 σ 1 σ 3 0.3012 + 1.3456 σ 2 σ 3 0.3012 (6)

The correlation coefficient R 2 = 0.8063 . Taking the characteristic parameters of SF as 0.15 and 0.45 respectively, the spatial relationship between the fitting surfaces and the test points is shown in Figure 7.

In particular, when σ 1 = σ 2 , Eq. 6 can also be used to describe the axial strength of HFRC under conventional triaxial confinement. Taking the characteristic parameters of SF as 0, 0.3, 0.6, and 0.9 respectively, the spatial relationships between the fitting surfaces and the conventional triaxial test points in Table 3 are shown in Figure 8. It can be seen that the parameters fitted by the test results from the true triaxial testing shown in Table 4 can well match the results of the conventional triaxial tests (Table 3), which means that the empirical model Eq. 6 can be used to predict the axial strength of HFRC in both situations of true triaxial and conventional triaxial states. As discussed in Section 3.1, there is no objective causal relationship between the maximum yield principal stress (i.e., axial strength) and the stress ratios, and therefore, the mathematical form of F σ 3 , σ 1 / σ 3 , σ 2 / σ 3 = 0 is just a calculation formula in statistical sense, rather than an objective mechanical equation, of which the universality and accuracy is limited by this reason.

When the two lateral stresses are known as σ 1 and σ 2 , a quantitative relation between the lateral stresses and the axial stress can be founded statistically when the concrete material yields. For consistency, the equation in the following form is adopted to describe the relationship between the lateral stresses and the axial strength. f 3 f c u = 1 + 0.151 λ S F 1 + a σ 1 f c u 0.8432 + 2.939 − a σ 2 f c u 0.8432 (7) where σ 1 / f c u ≤ σ 2 / f c u . By regressing the test results shown in Table 4, the parameter can be determined as a = 0.5545, i.e., f 3 f c u = 1 + 0.151 λ S F 1 + 0.5545 σ 1 f c u 0.8432 + 2.3845 σ 2 f c u 0.8432 (8)

The correlation coefficient is R 2 = 0.9229 . Taking the characteristic parameters of SF as 0.15 and 0.45, respectively, the fitting results are shown in Figure 9.

In particular, when σ 1 = σ 2 , Eq. 8 can be reduced to the case of conventional triaxial confinement. Taking the characteristic parameters of SF as 0, 0.3, 0.6, and 0.9, respectively, the fitting results are shown in Figure 10. From the comparisons, it can be observed that the empirical model Eq. 8 can fit the conventional test data as well. Eq. 8 can be regarded as a simplified and statistical yield equation of HFRC, which can be used to describe the yield behavior of HFRC as well as plain concrete (set λ S F = 0 ) under both the true axial environment and the conventional triaxial environment with adequate accuracy. With the sacrifice of some physical mechanism and mechanical significance, the calculation simplification and practicability of the model have been greatly improved.