1. Introduction
The shafts of coal mines are very significant to the safety of production in coal mines. Once the shaft lining breaks, it will not only cause great economic losses, but also pose a serious threat to the safety of underground miners. The artificial freeze method is usually used to construct vertical shafts passing through extra-thick alluvial layers, which is also called freezing shaft sinking. However, water gushing, or leakage, often occurs after the frozen soil thaws. For mass concrete pouring projects, such as freezing shafts, the internal temperature of the concrete can reach 60 to 80 °C [
1,
2], and at this time, the temperature inside the freezing wall is about −10 to −5 °C, and the air temperature inside the shaft is about 0 °C. The huge temperature difference will cause the shaft lining concrete to undergo additional temperature shrinkage, and the shrinkage of the shaft lining concrete is bound to be constrained by the freezing wall or the outside shaft lining, which will produce large temperature stress in the shaft lining concrete [
3,
4]. However, the early tensile strength of high-strength concrete is low. When the tensile stress caused by temperature stress is greater than its tensile strength, the shaft lining will produce temperature cracks. When the frozen wall is completely thawed, under the action of high-pressure water in the extra-thick alluvium, the temperature cracks in the shaft lining concrete will continue to expand and even penetrate, and then water leakage will occur [
5,
6], which will seriously threaten the mine safety production. Because of the freezing shaft lining of freezing shaft sinking in extra-thick alluvium, it is urgent to apply high-performance shaft lining concrete with good crack resistance and strong toughness.
Certain amounts of polypropylene plastic steel fiber (PPSF) and polyvinyl alcohol fiber (PVAF) can be added to concrete, which will greatly enhance its crack resistance and toughness [
7,
8,
9]. At the same time, the synergistic effect and superposition effect resulting from multi-scale and multi-element mixing of fibers can delay the formation of cracks in the hardening stage of concrete, enhance the impermeability and crack resistance of concrete [
10,
11,
12,
13] and improve the mechanical properties of hybrid-fiber-reinforced concrete (HFRC) compared with those of concrete mixed with single fibers [
14,
15]. Because of its good anti-cracking and impermeability properties, many scholars have applied fiber concrete to shaft lining structure model tests and studied the force characteristics and failure mechanism of the fiber concrete shaft lining structure model. Yao et al. [
16] carried out a model failure test of HFRC shaft lining based on similarity theory and verified that, as a shaft lining construction material, HFRC is superior to ordinary concrete. Qin et al. [
17] studied the compressive strength and failure characteristics of ordinary high-strength concrete and HFRC shaft lining models through indoor model tests. The mixed-use of steel fiber and PPSF restricts the expansion of cracks, improves the stress performance of the shaft lining, and mitigates the problems of hoop cracking, water permeability, and uneven deformation during use. However, when mixing shaft lining concrete, the dispersion of steel fibers is difficult to control, and steel fibers will greatly reduce the fluidity of concrete, which is not convenient for engineering applications. Taking into account the workability of concrete during the construction of the project, Yao et al. [
18] studied the mechanical properties of concrete with hybrid PPSF and PVAF and then carried out a similar model test on the shaft lining structure based on similarity theory. By comparing it with the ordinary concrete group, it was found that hybrid-fiber had little effect on improving the uniaxial compressive strength of concrete. However, it can greatly reduce the cracking of concrete and improve the ultimate capacity of the shaft lining.
With the rapid development of computer technology, many scholars have carried out numerical simulations of fiber-reinforced concrete, thereby enriching the research theory and research methods of fiber-reinforced concrete. Qureshi et al. [
19] calibrated the mechanical parameters of concrete and steel bars by basic mechanical property tests and carried out finite element modeling. It was found that the FEM was in good agreement with the experimental results. This research provided a reference for the numerical simulation of concrete structures. The interaction between the fiber and concrete interface is also a major difficulty in the process of modeling. Carozzi et al. [
20] used the tangential stress of the non-linear interface to characterize the interaction between the fibers and the surrounding mortar. The meshes were divided by non-linear truss elements. This research provided a reference for defining the interaction between the fiber and concrete interface. Radtke et al. [
21] modeled fibers by applying discrete force to the meshes. The background meshes represent the matrix, while the discrete force represents the interaction between the fibers and the matrix. This is a novel calculation method for describing fiber-reinforced concrete.
The above research is mainly focused on mechanical property tests of fiber-reinforced concrete specimens or shaft lining structure models, but numerical simulation research on the HFRC shaft lining structure is less involved. Therefore, using the ANSYS finite element analysis software, this research first studies the influence of the thickness-diameter ratio, concrete design strength, PVAF content, and PPSF content on the HFRC shaft lining structure and explores the mechanical characteristics of the shaft lining structure. Next, according to the simulation results, the empirical calculation formula for the ultimate capacity of this new type of shaft lining structure is obtained by fitting. Then, the rationality of the empirical calculation formula is verified through a shaft lining structure model test. Finally, through a range analysis of the ultimate capacity of the shaft lining structure, the order of influence on the ultimate capacity of the shaft lining structure is analyzed. The research results are expected to provide a certain reference for designing this kind of shaft lining structure.
2. Establishment of the Numerical Model of the HFRC Shaft Lining Structure
In the analysis of reinforced concrete structures, ANSYS can not only provide data on the basic mechanical characteristics analysis, including the displacement, strain, and stress caused by the structure under load, but also record and analyze the concrete compression yield, plastic creep of steel bars and bond-slip between steel bars and concrete. Therefore, it is a feasible method to simulate the freezing shaft lining structures by ANSYS [
20,
21,
22,
23,
24].
2.1. Element Type
For modeling the shaft lining structure, separate models were adopted. In this process, HFRC was simulated by SOLID65 element, which can be used to simulate reinforced composite materials (such as steel bars and fibers), concrete cracking (three orthogonal directions), crushing, plastic deformation, etc. The steel bars were simulated by LINK180 element, which is a spatial rod element with functions, such as plasticity, creep, large deformation, large strain, etc. It has a wide range of engineering applications and can be used to simulate steel bars, trusses, springs, etc. It was assumed that there was no relative slip between the two types of elements, and the coordination of the displacement of the concrete elements and the steel bar elements was realized by sharing the nodes [
25,
26].
2.2. Material Constitutive Model and Parameters
In this numerical simulation, the uniaxial compression constitutive relation of HFRC was selected according to Formula (1), which can better reflect the rising and falling parts of the stress–strain relationship curve [
27].
In these formulas,
and
stand for the stress and strain of concrete;
and
stand for the parameter values of the rising and falling parts of the concrete stress–strain curve and were calculated by Formulas (3) and (4);
stands for the axial compressive strength of concrete;
stands for the peak strain of concrete under compression.
In these formulas, stands for the cubic compressive strength.
On the falling part of the concrete uniaxial compressive stress–strain curve, when the stress was reduced to 0.5
, the corresponding compressive strain was
. When calculating and analyzing the concrete structures, the uniaxial compressive strain should not exceed
, which is given by Formula (5):
Before the numerical simulation, a uniaxial compression test and an axial compression test of HFRC were carried out. The obtained mechanical parameters of concrete are shown in
Table 1. In this table, C-1 to C-9 represents the number of concretes selected for the nine shaft linings in this numerical simulation. The elastic strain modulus is the ratio between a load of 40% of the axial compressive strength and the corresponding strain. In addition, Poisson’s ratio was uniformly taken as 0.2.
The axial compressive strength of C-1 concrete was 62.16 MPa, the cubic compressive strength was 80.6 MPa, and the peak compressive strain of concrete was 2.124 × 10
3 . The uniaxial compressive strength of concrete can be obtained from Formula (1) to Formula (5). The pressure constitutive relationship curve is shown in
Figure 1. Similarly, the uniaxial compression constitutive relationship curves from C-2 to C-9 can be obtained.
The selected failure criterion of concrete was Willam and Warnke’s five-parameter strength criterion, as shown in Formula (6), which takes into account the multiaxial stress state of concrete.
In the above formula, is the principal stress state function; is the failure surface function; , , , and are the uniaxial tensile strength, uniaxial compressive strength, biaxial compressive strength under hydrostatic pressure, and multiaxial compressive strength under hydrostatic pressure of concrete, respectively, and , , and .
The ideal elastic–plastic model was adopted for the steel bars, and the yield condition obeys the Mises criterion. In the numerical calculation, the elastic modulus was 2.1 × 105 MPa, Poisson’s ratio was 0.3, and the yield strength was 240 MPa.
2.3. Shaft Lining Simulation Scheme and Boundary Conditions
In the numerical simulation of this research, the only four factors considered were the thickness–diameter ratio, shaft concrete strength, PVAF content, and PPSF content, and it mainly analyzed the stress–strain characteristics of the HFRC shaft lining structure model under loads. The value of the thickness–diameter ratio was selected according to the thickness–diameter ratio of the shaft lining at the engineering site. The three levels of the thickness–diameter ratio were 0.2675, 0.2908, and 0.3140. The concrete of the shaft lining was selected from C-1 to C-9 in
Table 1. The PVAF content was 0.728 kg/m
3, 1.092 kg/m
3 and 1.456 kg/m
3. The PPSF content was 4 kg/m
3, 5 kg/m
3 and 6 kg/m
3. According to the four-factor three-level orthogonal test method, it was necessary to carry out the numerical simulation on the shaft lining structure with nine different parameters. The specific design parameters are shown in
Table 2.
In this simulation of the shaft lining, the hoop reinforcement ratio was 0.6%, the vertical reinforcement ratio was 0.3%, and the reinforcement diameter was 5 mm. According to the size of the test equipment, the outside diameter of the model was 925 mm, the height was 562.5 mm, and the thicknesses of the shaft lining corresponding to thickness–diameter ratios of 0.2675, 0.2908, and 0.3140 were 97.6 mm, 104.2 mm and 110.5 mm, respectively.
Considering the axial symmetry of the shaft lining structure in this numerical simulation, the ¼ 3D finite element calculation model was established according to the design parameters of the shaft lining [
28]. In the process of mesh generation, it is necessary to pay attention to the node sharing of steel bars and concrete. The surface load was simulated by applying a larger horizontal load. The boundary condition of the shaft lining model was that the upper and lower end faces were constrained by longitudinal displacement, two ¼ cross-sections were constrained by hoop displacement, and then a uniform surface load was applied on the outside surface of the model. The mesh division diagram and boundary conditions of the shaft lining model are shown in
Figure 2.
4. Range Analysis of the Ultimate Capacity of the Shaft Lining Structure
In order to study the influence of various factors on the ultimate capacity of the shaft lining, a range analysis of the ultimate bearing capacity was carried out. The analysis results are shown in
Table 6. The thickness–diameter ratio, shaft concrete design strength, PVAF content, and PPSF content are reported as factors A, B, C, and D, respectively.
From the comparison of the R value and
k value in
Table 6, it can be seen that the order of influence on the ultimate capacity of the shaft lining structure is the thickness–diameter ratio, the design strength of the concrete, the content of PPSF and the content of PVAF. The greater the thickness–diameter ratio and the design strength of the shaft lining concrete, the greater the
k value, that is, the greater the ultimate capacity of the shaft lining structure. In addition, the
k value of the hybrid-fiber reaches the maximum under the second-level combination.
In order to more intuitively observe the influence of each factor level on the ultimate capacity of the shaft lining structure model, the relationship curve between each factor and the ultimate capacity of the shaft lining structure is drawn, as shown in
Figure 14.