Evaluation Model on Activation Classification of Coal Mine Goaf Ground Considering High-Speed Railway Loads

Abstract

The construction and expansion of high-speed railway networks in China has been occurring at a fast pace, resulting in some lines crossing through coal mine goaf sites. However, the embankment and train loads may trigger the activation of the coal mine goaf ground, posing a threat to traffic safety. To ensure the safety of construction and railway lines, an evaluation model on the activation classification of coal mine goaf ground, taking into account the high-speed railway load, is proposed, which is mainly applicable for middle–deep and level goaf areas using a longwall mining method. Firstly, 12 influencing factors are selected as the underlying evaluation indexes, and the catastrophe progression method model for evaluating the coal mine goaf ground stability is constructed. The findings of the evaluation were found to align with the actual results, indicating the reliability of the model. Then, the additional stress calculation model for high-speed railway ground with different embankment heights, train speeds, and axle loads was established, and the train load disturbance depth with a 5% criterion was determined. The influence degree of load on high-speed railway was divided, and the weight of each factor was determined. Finally, the extension comprehensive evaluation method was used to unite the stability grade of the coal mine goaf site and the influence degree of the train, so the evaluation model on activation classification was proposed. The accurateness and reliability of the proposed model was verified using the Taijiao high-speed railway cases and the model test.

1. Introduction

Land waste from mining activities has become a major constraint to sustainable development [1,2]. In the past few years, there has been a significant increase in the implementation of engineering construction projects within coal mine goaf sites, and numerous scholars have performed many studies on the suitability of coal mine goaf site construction. As China continues to enhance its high-speed railway facilities, it is unavoidable that certain railways will need to traverse coal mine goaf sites. For example, in the Qingshui Coalfield of Shanxi Province, the completed Taijiao section, Shitai section, and Taixi section, as well as the planned Qingyin high-speed railway and Taiyan high-speed railway, totaling five sections of lines, pass through a coal mine goaf site. Therefore, a study on the suitability of high-speed railway construction in coal mine goaf sites has important practical value and practical significance.

Many studies have been carried out on the stability analysis and evaluation of coal mine goaf construction sites. In view of the fuzziness and randomness of the stability classification of goaf, Ma et al. [3] introduced a novel approach called the entropy weight–normal cloud model. This model effectively combines entropy weight to calculate the comprehensive certainty, resulting in evaluation results. To enhance the applicability of coal mine goaf stability evaluation outcomes, based on the fuzzy characteristics of coal mine goaf stability evaluation and treatment measure decision-making, He et al. [4] employed a double-layer method of fuzzy comprehensive evaluation to construct a model for assessing the coal mine goaf stability. In their analysis of stability problems, Swift and Reddish [5] considered a probabilistic approach. A method of Bayes discriminant analysis was presented by Hu and Li [6] to detect the risk of complicated goafs in mines. Through the use of a newly developed viscoelastic model, Wang et al. [7] examined the creep failure of a roof stratum supported by pillars in goaf. To analyze and evaluate the stability of coal mine goafs, Sun et al. [8] put forward a three-zone model. According to the research conducted by Helm et al. [9], a total of 1320 numerical simulation models were established in order to investigate the impact of surface traffic facilities due to shallow-buried room-and-pillar mine goaf ground. They highlighted the significant influence of seasonal water-level fluctuations on the stability of the overlying strata and the surface within the coal mine goaf ground. In a different study, Qin et al. [10] put forth an enhanced TrAdaBoost algorithm by incorporating the notion of a dynamic factor. This modification aimed to enhance the algorithm’s generalizability across various samples of coal mine goaf ground. Using the Reissner theory as a foundation for thick plates, Guo et al. [11] devised a model for evaluating the potential risks associated with instability at expressway construction sites. This model incorporated seven key factors for assessment and leveraged fuzzy theory. Taking into account the impact of roof material characteristics, María-Belén et al. [12] conducted an analysis on roof stability using a calculation approach rooted in material resistance. Deb and Choi [13] analyzed the weight relationships of the influencing factors of the stability of the old goaf and predicted the occurrence probability of the surface collapse pit through the fuzzy set theory. Long et al. [14] introduced an innovative ANN model along with its accompanying “optimal” hyperparameters. This model proves to be a valuable and convenient tool for forecasting land subsidence caused by underground mining activities. Adam Smoliński [15] studied the influence of selectively leaving rocks underground on the geological environment after coal mining, and these behaviors may change and affect the engineering geology, which is one of the stability factors of goaf.

Although there are some studies on the stabilization of coal mine goaf sites and the influence of external loads on coal mine goaf ground, the exploration of the “activation” of coal mine goaf ground in relation to high-speed railways is relatively scarce. The evaluation of high-speed railway coal mine goaf sites is a complex problem. It requires consideration not only of goaf stability but also of the coupling effect between the high-speed railway engineering load and the coal mine goaf site. In order to be able to establish a reliable evaluation method, the stability evaluation of high-speed railway coal mine goaf site is categorized into two subfactors: coal mine goaf stability grade evaluation and high-speed railway engineering impact evaluation. The coal mine goaf site stability grade is calculated by using the catastrophe progression method. Through the theoretical derivation and the establishment of the calculation model of the additional stress of the high-speed railway foundation, the influence grades of various factors in the high-speed railway project are divided. Finally, the extension comprehensive evaluation method is employed to combine the two influential factors to ascertain the activation grade of the ground in coal mine goaf.

2. Stability Evaluation of Coal Mine Goaf Site

2.1. Principle of Catastrophe Progression Method

The catastrophe progression method [16] evolved on the basis of the catastrophe theory [17,18]. Compared with the analytic hierarchy process [19], fuzzy mathematics [20], and other evaluation methods, the primary benefit of utilizing the catastrophe progression approach lies in the fact that it does not need to weigh the indicators at all grades, and the relative importance of each indicator is considered in the evaluation process. It not only reduces the intervention of human subjectivity but also has a relatively simple calculation process. The main evaluation steps are as follows:

(1)

Establishing a hierarchical structure

Firstly, the evaluation system’s overarching objective is decomposed into multiple grades according to the internal mechanism of action. This decomposition further forms a multi-layered system, consisting of multiple indicators. The evaluation system’s overarching objective is decomposed into the first layer of evaluation indicators and gradually decomposed into subindicators that can be quantitatively or qualitatively described.

(2)

Establishing the catastrophe model of each grade

Table 1. Functions of each catastrophe model.

Name	Dimension of Control Variables	Model Functions	Structural Representation
folding	1	x3 + ax
cusp	2	x4 + ax2 + bx
swallowtail	3	x5 + ax3 + bx2 + cx
butterfly	4	x6 + ax4 + cx3 + bx2 + dx

illustrates the mathematical model and model functions of the seven primary catastrophe system models utilized in catastrophe theory. These models include the folding catastrophe model, cusp catastrophe model, swallowtail catastrophe model, butterfly catastrophe model, elliptic umbilical point catastrophe model, hyperbolic umbilical point catastrophe model, and parabolic umbilical point catastrophe model. Due to the consideration of the dimension of state variables, the most commonly used models in the catastrophe progression method are the first four.

(3)

Dimensionless treatment of evaluation index

The lowest-level indicators obtained through decomposition usually have different dimensions, so the direct comparison of the relative importance of each indicator will produce significant errors. To prevent issues arising from different dimensions, the range transformation method is usually used to non-dimensionalize the index. In the process of conversion, there are generally two types of indicators, namely forward and reverse indicators, where the larger value is the better type and the smaller value is the better type.

$$y_{ij}=\frac{x_{ij}-x_{\min (j)}}{x_{\max (j)}-x_{\min (j)}}$$

(1)

$$y_{ij}=\frac{x_{\max (j)}-x_{ij}}{x_{\max (j)}-x_{\min (j)}}$$

(2)

where y_ij is the dimensionless index, x_ij is the index value, x_max(j) is the maximum value in the index sample, and x_min(j) is the minimum value in the index sample.

(4)

Control variable normalization formula

The normalization formula of the catastrophe model is acquired by deriving the bifurcation point set equation from the catastrophe system. The potential function of the catastrophe system can be represented as f(x). Based on the catastrophe theory, the equation for the critical point set on the equilibrium surface is obtained by using the first derivative of the potential function f(x), that is, f′(x) = 0. The second derivative of the potential function is used to obtain the singular point set equation, that is, f″(x) = 0. By combining the equation of the critical point set and the singular point set and eliminating x, the bifurcation point set equation of the catastrophe system is obtained. When the control variables satisfy the bifurcation point set equation, the system mutates. The bifurcation point set equation is decomposed and its normalization formula is derived. The function of the normalization formula is to transform the different mass states of the control variables into the same mass state.

Based on the established research [21], the cusp catastrophe system’s normalization formula is:

$$x_a=\sqrt{\left|a\right|}, x_b=\sqrt[3]{\left|b\right|}$$

(3)

For the normalization of the swallowtail mutation system, the formula is:

$$x_a=\sqrt{\left|a\right|}, x_b=\sqrt[3]{\left|b\right|}, x_c=\sqrt[4]{\left|c\right|}$$

(4)

Similarly, the normalization formula for the butterfly mutation system can be expressed as:

$$x_a=\sqrt{\left|a\right|}, x_b=\sqrt[3]{\left|b\right|}, x_c=\sqrt[4]{\left|c\right|}, x_d=\sqrt[5]{\left|d\right|}$$

(5)

(5)

Using normalization formula for comprehensive evaluation

There are three principles of comprehensive evaluation:

(a)

When the control variables in a system can mutually complement each other, the mean value of the sum of each variable is taken according to the “average principle”.

(b)

When the control variables in a system are not complementary and insufficient, the minimum value is selected according to “large take small”.

(c)

When the control variables in a system exceed a certain threshold to be complementary, according to the “average principle”, the threshold conditions are not met, according to the “large take small”.

2.2. Stability Evaluation System of Coal Mine Goaf Site

The coal mine goaf site undergoes a highly intricate deformation process. To establish a scientific system to evaluate the stability of the coal mine goaf site, it is crucial to thoroughly account for the impact brought about by a diverse array of factors on said site. By referencing the works of He [4], Guo [22], and Ren [23], this paper classifies the control variables affecting the stability of coal mine goaf into four categories: coal mine goaf parameters, engineering geology, hydrological conditions, and external disturbance. The construction of the stability evaluation system of the coal mine goaf site is shown in Figure 1, where the parameters of the goaf are: span (C1), mining depth-to-thickness ratio (C2), area (C3), and specification (C4) (ratio of span to height); engineering geology: rock mass structure (C5), rock quality index (C6), and geological structure (C7); hydrological conditions: underground visible water (C8), and groundwater body (C9); external effects: surrounding mining (C10), adjacent coal mine goaf conditions (C11), and engineering layout (C12).

Of these, rock mass structure, geological structure, underground visible water, groundwater body, surrounding mining, adjacent coal mine goaf working conditions, and engineering layout are qualitative indicators which need to be quantified before the calculation of catastrophe progression method. The quantitative method is as follows.

Rock mass structure: 1 is a complete block structure, 2 is a layered structure, 3 is a cataclastic structure, and 4 is a loose structure.

Geological structure: 1 is no fault and fold, 2 is the small influence of fault and fold, 3 is a partial cutting of the fault, 4 is a fault penetrating surrounding rock.

Underground visible water: 1 is no water trace, 2 is visible water trace during rainfall, 3 is water trace during heavy rainfall, and 4 is water trace during rainy season.

Groundwater body: 1 is no water body around the surrounding rock, 2 is the surrounding rock being little influenced by the water body, 3 is the surrounding rock being influenced by the water body, and 4 is the surrounding rock being greatly influenced by the water body.

Surrounding mining: 1 is no other operation influence in the mining area, 2 is the general influence of other operations in the mining area, 3 is the great influence of other operations in the mining area, and 4 is the great influence of other operations in the mining area.

Working conditions in the adjacent coal mine goaf: 1 is no other mining area within the influence area; 2 is a small area and small number of mining areas within the influence area; 3 is a large area and a large number of mining areas within the influence area but the distribution is dispersed; and 4 is a large area and a large number of mining areas within the influence area and the distribution is centralized.

Engineering layout: 1 is reasonable, 2 is relatively reasonable, 3 is partially reasonable, and 4 is extremely unreasonable.

2.3. Determination of Stability Evaluation Factors

Thirty coal mine goaf site samples in Reference [22] was selected as the learning samples of the catastrophe progression method of coal mine goaf site stability. Each qualitative index was converted according to the quantitative method given in Section 2.2.

From Table 1 and

Table 2. Example data of the coal mine goaf site.

NO.	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12
1	85	17.27	4330	2	3	39	1	2	2	4	4	2
2	63	27.78	1300	2	2	49	2	2	2	4	4	1
3	65	23.08	2480	3	3	37	3	2	2	4	4	2
4	69	13.50	1680	4	3	50	2	2	2	4	4	3
5	65	14.50	1890	3	3	41	2	2	2	4	4	2
6	170	12.29	5740	4	3	52	3	2	2	4	4	3
7	31	19.06	2910	1	2	53	1	1	1	1	1	1
8	89	16.00	2180	2	2	59	1	2	2	4	4	1
9	59	33.50	1200	1	1	66	1	2	2	1	1	1
10	90	42.24	4010	2	1	50	2	3	3	4	4	2
11	39	20.33	2260	2	1	59	2	1	1	1	1	1
12	37	22.31	1450	1	1	61	1	2	2	1	1	1
13	36	28.71	2590	1	1	52	1	2	2	3	3	1
14	65	17.33	2430	1	1	55	1	1	1	3	3	1
15	83	46.22	2350	1	3	56	3	2	2	3	3	2
16	68	20.80	1800	2	1	54	1	1	1	3	3	1
17	55	25.69	1600	3	1	57	2	2	2	4	4	2
18	120	14.28	4580	3	1	42	2	2	2	4	4	4
19	59	8.74	1950	2	1	55	1	1	1	3	3	1
20	110	26.00	5000	1	1	52	1	1	1	3	3	1
21	85	13.85	3270	2	1	51	1	1	1	3	3	1
22	65	24.00	1300	2	1	54	2	1	1	4	3	2
23	73	29.03	1730	2	1	55	2	1	1	4	4	1
24	74	38.33	1870	1	1	53	1	1	1	3	3	1
25	68	16.43	1170	2	1	53	1	1	1	3	3	1
26	65	14.38	2490	2	1	54	1	1	1	1	1	1
27	78	17.69	2440	2	1	53	1	1	1	1	1	1
28	80	10.00	3480	3	1	43	1	2	2	4	4	3
29	85	28.75	1410	2	1	55	1	1	1	1	1	1
30	65	20.59	1750	1	1	44	1	3	3	3	3	2

, it can be seen that C1, C2, C3, and C4 utilize the butterfly mutation model; C5, C6, and C7 utilize the swallowtail mutation model; C8 and C9 utilize the cusp mutation model; C10, C11, and C12 utilize the swallowtail mutation model; and self-parameter factor B1, engineering geological factor B2, hydrological condition factor B3, and external disturbance factor B4 utilize the butterfly mutation model. Using the normalization formula of each catastrophe model, the utility function of each index in each sample and the catastrophe progression of the overall goal of evaluation are calculated. The calculation results can be found in

Table 3. Utility function value of the indicator factors.

NO.	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12
1	0.612	0.228	0.309	0.667	0.333	0.390	1.000	0.667	0.667	0.000	0.000	0.667
2	0.770	0.508	0.972	0.667	0.667	0.490	0.667	0.667	0.667	0.000	0.000	1.000
3	0.755	0.383	0.713	0.333	0.333	0.370	0.333	0.667	0.667	0.000	0.000	0.667
4	0.727	0.127	0.888	0.000	0.333	0.500	0.667	0.667	0.667	0.000	0.000	0.333
5	0.755	0.154	0.842	0.333	0.333	0.410	0.667	0.667	0.667	0.000	0.000	0.667
6	0.000	0.095	0.000	0.000	0.333	0.520	0.333	0.667	0.667	0.000	0.000	0.333
7	1.000	0.275	0.619	1.000	0.667	0.530	1.000	1.000	1.000	1.000	1.000	1.000
8	0.583	0.194	0.779	0.667	0.667	0.590	1.000	0.667	0.667	0.000	0.000	1.000
9	0.799	0.661	0.993	1.000	1.000	0.660	1.000	0.667	0.667	1.000	1.000	1.000
10	0.576	0.894	0.379	0.667	1.000	0.500	0.667	0.333	0.333	0.000	0.000	0.667
11	0.942	0.309	0.761	0.667	1.000	0.590	0.667	1.000	1.000	1.000	1.000	1.000
12	0.957	0.362	0.939	1.000	1.000	0.610	1.000	0.667	0.667	1.000	1.000	1.000
13	0.964	0.533	0.689	1.000	1.000	0.520	1.000	0.667	0.667	0.333	0.333	1.000
14	0.755	0.229	0.724	1.000	1.000	0.55	1.000	1.000	1.000	0.333	0.333	1.000
15	0.626	1.000	0.742	1.000	0.333	0.560	0.333	0.667	0.667	0.333	0.333	0.667
16	0.734	0.322	0.862	0.667	1.000	0.540	1.000	1.000	1.000	0.333	0.333	1.000
17	0.827	0.452	0.906	0.333	1.000	0.570	0.667	0.667	0.667	0.000	0.000	0.667
18	0.360	0.148	0.254	0.333	1.000	0.420	0.667	0.667	0.667	0.000	0.000	0.000
19	0.799	0.000	0.829	0.667	1.000	0.550	1.000	1.000	1.000	0.333	0.333	1.000
20	0.432	0.461	0.162	1.000	1.000	0.520	1.000	1.000	1.000	0.333	0.333	1.000
21	0.612	0.136	0.540	0.667	1.000	0.510	1.000	1.000	1.000	0.333	0.333	1.000
22	0.755	0.407	0.972	0.667	1.000	0.540	0.667	1.000	1.000	0.000	0.333	0.667
23	0.698	0.541	0.877	0.667	1.000	0.550	0.667	1.000	1.000	0.000	0.000	1.000
24	0.691	0.789	0.847	1.000	1.000	0.530	1.000	1.000	1.000	0.333	0.333	1.000
25	0.734	0.205	1.000	0.667	1.000	0.530	1.000	1.000	1.000	0.333	0.333	1.000
26	0.755	0.150	0.711	0.667	1.000	0.540	1.000	1.000	1.000	1.000	1.000	1.000
27	0.662	0.239	0.722	0.667	1.000	0.530	1.000	1.000	1.000	1.000	1.000	1.000
28	0.647	0.034	0.495	0.333	1.000	0.430	1.000	0.667	0.667	0.000	0.000	0.333
29	0.612	0.534	0.947	0.667	1.000	0.550	1.000	1.000	1.000	1.000	1.000	1.000
30	0.755	0.316	0.873	1.000	1.000	0.440	1.000	0.333	0.333	0.333	0.333	0.667

and Figure 2.

Figure 2 shows a high accuracy of the catastrophe progression approach in assessing the coal mine goaf stability, only 1 of the 30 samples has an error in the prediction result, and the accuracy rate reaches 96.7%. The stability evaluation criteria of different coal mine goaf sites are divided by using the calculation results of catastrophe series:

• Stable: 0.961 < A ≤ 1;
• Basically stable: 0.893 < A ≤ 0.961;
• Less stable: 0.737 < A ≤ 0.893;
• Unstable: 0 ≤ A ≤ 0.737.

3. Influence Degree Division of High-Speed Railway

The influence of railway construction on the ground can generally be viewed from three aspects [24]: train axle load, train speed, and subgrade height. The biggest difference between high-speed railway and traditional railway is the high speed of the train operation, which puts forward strict control requirements for the vibration and settlement of railway subgrade. By conducting calculations and analyzing the additional stress impact on the base, caused by the three variables mentioned above, the influence is combined with the stability grade of the goaf. This analysis is crucial for evaluating the overall construction safety of coal mine goafs when affected by high-speed railways. The research also examines the depth of load disturbance under various factors and categorizes their influence on the coal mine goaf site into different grades. This study is divided into four grades:

S = [S1, S2, S3, S4] = [No influence, Low influence, Medium influence, High influence]

Two carriage links are selected for analysis, depicted in Figure 3, assuming that the weight load on the axle is equally distributed to each wheel, and the forces acting on the rails are regarded as concentrated forces, which are distributed at the wheel–rail contact area. The spacing of the two wheels on a single bogie measures 2.5 m, whereas the distance between the two neighboring bogies at the carriage connection is 7.8 m.

(1)

Wheel–rail contact area

S. Kumar et al. [25] proposed a wheel–rail contact area calculation method, with reference to the classical solution of Hertz in 1881 for the contact stress between two smooth surface elastomers, considering the contact area as an ellipse, whose long axis a and short axis b can be derived from Equation (6):

$$a=\alpha \sqrt[3]{\frac{Nm}{n}},\begin{array}{cc}& b=\beta \end{array}\sqrt[3]{\frac{Nm}{n}}$$

(6)

$$m=\frac{4}{\frac{1}{r_1}+\frac{1}{r_1^{\prime }}+\frac{1}{r_2}+\frac{1}{r_2^{\prime }}}$$

(7)

$$n=\frac{4E}{3(1-v^2)}$$

(8)

where N is the positive pressure between wheels and rails; m, n, α, and β are dimensionless parameters; r₁ and r₁′ are the radii of curvature of the rail surface; r₂ and r₂′ are the radii of curvature of the wheel surface; and E and v are the elastic modulus and Poisson coefficient of the wheel and rail materials (assuming the same wheel and rail materials). Here r₁ = 25.4 cm, r₂ = 41.91 cm, r₁′ = r₂′ = $\infty$, E = 2.1 × 10⁵ MPa, and v = 0.3.

The values of α and β are shown in

Table 4. Parameters of α and β.

θ	20	30	35	40	45	50	55	60	65
α	3.778	2.731	2.397	2.136	1.926	1.754	1.611	1.486	1.378
β	0.408	0.493	0.530	0.567	0.604	0.641	0.678	0.717	0.759

.

The angles in the table are calculated from the following equations:

$$\cos \theta =\frac{B}{A}$$

(9)

$$A=\frac{2}{m}$$

(10)

$$B=\frac{1}{2}{\left[{\left(\frac{1}{r_1}-\frac{1}{r_1^{\prime }}\right)}^2+{\left(\frac{1}{r_2}-\frac{1}{r_2^{\prime }}\right)}^2+2\left(\frac{1}{r_1}-\frac{1}{r_1^{\prime }}\right)\left(\frac{1}{r_2}-\frac{1}{r_2^{\prime }}\right)\cos 2\psi \right]}^{\frac{1}{2}}$$

(11)

where ψ is the angle between the wheel and the main curvature plane of the rail, take 90° for calculation.

From Equation (6), it can be seen that:

a/b = α/β

(12)

The contact area can be obtained as:

$$S=\pi (ab)=\pi (a^2\frac{\beta }{\alpha })=\pi \alpha \beta {(N\frac{m}{n})}^{\frac{2}{3}}$$

(13)

(2)

Wheel–rail contact positive pressure

Liang et al. [26] proposed that the static load superimposed on a series of sinusoidal functions can be used to combine into an excitation force function to simulate the train load, and the formula can effectively respond to the additional dynamic load caused by track unevenness and rail surface wave wear. The load calculation equation is:

$$F(t)=P_0+{\displaystyle \sum _{i=1}^3{P}_i}\sin ({\omega }_{i}t)$$

(14)

where $P_i=m_0{a}_i{\omega }_i^2$, ${\omega }_i=2\pi V/L_i$, and P₀ are the static wheel loads and m₀ is the underspring mass; a_i represents the vector height; moreover, the circular frequency under uneven conditions is denoted by ${\omega }_i$, the train speed is represented by V, and L_i is the corresponding wavelength. The unevenness management values are shown in

Table 5. Irregularity selection.

Control Conditions	Wavelength (m)	Positive Vector (mm)
According to the smoothness of travel (I)	50	16
	20	9
	10	5
Dynamic additional loads acting on the line (II)	5	2.5
	2	0.6
	1	0.3
Waveform wear (III)	0.5	0.1
	0.05	0.005

.

(3)

Calculation of the depth of influence of additional stress

In order to facilitate the calculation and for the purpose of safety design, to study the additional stress impact depth when the two vehicles meet, first use Equations (6)–(13) to calculate the wheel–rail contact area, then select the peak value of Equation (14) as the wheel–rail positive pressure value for the additional stress impact depth calculation in order to obtain the additional stress magnitude of the base train load; the calculation model is shown in Figure 4.

Assuming that the ground is a uniform, continuous, and isotropic semi-infinite space elastomer, the additional stress at any point within the ground soil is obtained by combining Boussinesq’s solution:

$${\sigma }_{\theta }=\frac{3P{z}^3}{2\pi R^5}$$

(15)

When the wheel–rail contact surface is calculated in accordance with Equations (6)–(13), it can be observed that the shape is elliptical, which is simplified to rectangle for the convenience of calculation, and the length of one side of the rectangle is fixed and equal to the width of the rail’s top surface width, i.e., 70 mm.

After obtaining the value of the additional stress in the subgrade caused by the train, the stress value generated by the static load on the embankment is calculated, and the superimposed stress value of the two is taken as the force acting uniformly on the subgrade, the subgrade is considered a strip ground, and the depth of the additional stress influence is calculated. As a high-speed railroad project has very strict settlement requirements, a depth when the additional stress is less than 5% of the soil’s self-weight is used as the influence depth calculation standard.

3.1. Calculation of Additional Stress

To analyze the influence of different influencing factors on the influence depth of additional stress, the calculation groups are presented as shown in

Table 6. Grouping of additional stress calculation models for high-speed railway ground.

Group	Axle Load (t)	Speed (km/h)	Subgrade Height (m)	Additional Stress in the Base of Subgrade Load (kPa)	Additional Stress in the Base of Train Load (kPa)
Axis recombination	14	250	4	56.16	9.62
	18	250	4	56.16	11.04
	22	250	4	56.16	12.51
	26	250	4	56.16	14.02
	30	250	4	56.16	15.02
Vehicle speed group	22	150	4	56.16	9.38
	22	200	4	56.16	11.01
	22	250	4	56.16	12.51
	22	300	4	56.16	14.14
Subgrade height group	22	250	3.5	50.04	11.68
	22	250	4	56.16	12.51
	22	250	4.5	62.13	12.89
	22	250	5	67.96	12.94
	22	250	5.5	73.68	12.75

, and the additional stress values of the ground caused by subgrade load and train load are calculated.

During the calculation process, the bed’s top surface showcases a width of 13.4 m. The subgrade bed features a slope ratio of 1:1.5, while the ballast bed displays a top surface width of 8.2 m. The height of the ballast bed measures 0.35 m, and its slope ratio stands at 1:1.75. When calculating the self-weight of the rock and soil, in the study area, the density of loess is 1600 kg/m³ at 15 m below the surface, and the density of mudstone is 1860 kg/m³ at 15~50 m depth.

3.2. Influence Degree Analysis of Train Axle Load

Different types of trains have different axle loads. The force exerted on the ground of the coal mine goaf increases as the axle load of the train increases, leading to a greater influence depth. By constructing the model for the working conditions of axle loads, as presented in Table 6, it becomes possible to calculate the disturbance depth caused by train loads under different axle load conditions. As illustrated in Figure 5, the influence depth of load tends to increase as the axle load increases, and there is a good linear relationship between them. The axle load ranges from 14 to 30 tons, and the load influence depth ranges from 31.61 m to 32.99 m.

According to the relationship between the calculation results and Figure 5,

Table 7. Classification of train axle load influence degree.

Grade	Axle Load Interval
Ⅰ	14t < P ≤ 18t
Ⅱ	18t < P ≤ 22t
Ⅲ	22t < P ≤ 26t
Ⅳ	26t < P ≤ 30t

illustrates the extent of impact for this categorization and the associated range of axle load.

3.3. Influence Degree Analysis of Train Speed

Train running speed has a significant impact on the dynamic load within the subgrade of high-speed trains. This dynamic load plays a crucial role in determining whether there will be ground “activation” deformation in coal mine goafs. A study by Hu and Li [27], which is based on field measured data, has found that the dynamic stress of the subgrade no longer changes with the speed when it is lower than 150 km/h or higher than 300 km/h. However, when the train speed falls within the range of 150 km/h to 300 km/h, the dynamic stress of the subgrade changes linearly with the speed. These findings align with the results obtained from the presented calculation model, as shown in Figure 6. The model confirms that the load influence depth is linearly related to the train speed when it is between 150 km/h and 300 km/h. When the velocity changes from 150 km/h to 300 km/h, the influence depth ranges from 31.55 m to 32.77 m.

Table 8. Classification of train speed influence degree.

Grade	Speed Interval
Ⅰ	120 < V ≤ 150
Ⅱ	150 < V ≤ 200
Ⅲ	200 < V ≤ 300
Ⅳ	300 < V ≤ 350

presents the impact degree and the corresponding range of speeds for this classification based on the calculation results, Figure 6, and the research conducted by Hu [27] and Bian [28].

3.4. Influence Degree Analysis of Subgrade Height

The main impact of the track lies in selecting the type of track [24] and the height of the roadbed [28]. Chen et al. [24] introduced a formula for the decay of dynamic stresses along the depth of the subgrade:

$$\eta =1-\frac{z}{a+b\times z}$$

(16)

where the attenuation coefficient is represented by η and the depth of roadbed is denoted as z. The fitting coefficients are a and b, for the ballastless track, and the values of a and b are 2.12 and 1.18, respectively. On the other hand, for the ballast track, the corresponding values are 0.64 for a and 0.86 for b.

From Equation (16), it is evident that with the increase in subgrade height, the dynamic stress decays rapidly in the subgrade. However, in the coal mine goaf site, if the height of the roadbed is blindly expanded, the dynamic stress generated by the train load can be attenuated, but the increase in the weight of the roadbed may also make the site unstable [29]. Based on the findings presented in Table 6, it is apparent that the additional stress of the base caused by the train load increases first and then decreases with the change in the subgrade height. This is because the wheel–rail contact area is too small, leading to additional stress when subjected to vertical concentrated force. With the increase in subgrade height, the growth trend of base additional stress caused by the static load of subgrade is much larger than the attenuation trend of base additional stress caused by the train load. As shown in Figure 7, when the subgrade height changes from 3.5 m to 5.5 m, the load influence depth exhibits a linear correlation with the variation in subgrade height, and the range of load influence depth is 30.52~36.64 m.

According to the relationship between the calculation results and Figure 7,

Table 9. Classification of subgrade height influence degree.

Grade	Embankment Height Below Subgrade Bed
Ⅰ	H ≤ 3.75
Ⅱ	3.75 < H ≤ 4.0
Ⅲ	4.0 < H ≤ 4.5
Ⅳ	4.5 < H ≤ 5.5

depicts the influence degree of this classification and the associated subgrade height range.

4. Comprehensive Evaluation Model of Coal Mine Goaf Ground Considering High-Speed Railway Load

4.1. Combination of Catastrophe Progression Method and Extension Theory

The catastrophe progression method requires a large number of samples when predicting the target of an event. However, there are few data related to the development of high-speed railway within the coal mine subsidence area. Most of the routes are still in the planning stage, and it is impossible to predict the activation grade target of high-speed railway construction across the whole coal mine goaf site. With the increase in subfactor indicators, it is possible to cause the phenomenon of “attend to one thing and lost sight of another”, resulting in errors. Therefore, the total prediction target is divided into two parts. The stability grade of the coal mine goaf is forecasted using the catastrophe progression method, and then the extension theory is used to connect it with the train influence to complete the evaluation of the total target.

Extension theory is a technique aimed at solving intricate problems through the integration of the matter-element theory and the extension set theory. The comprehensive evaluation model based on extension theory has garnered widespread adoption across diverse domains [30,31,32]. This paper applies the catastrophe progression method to simplify the influential factors of coal mine goaf. Subsequently, the stability grade of the coal mine goaf site is utilized to evaluate the impact of the coal mine goaf. To construct the evaluation system of activation in coal mine goaf ground under high-speed railways, refer to Figure 8, and the coal mine goaf ground activation grade is classified into four grades:

V = [V₁, V₂, V₃, V₄] = [Not activated, Not easily activated, Easily activated, Activated].

4.2. Comprehensive Evaluation Model

(1)

Determination of classical and nodal domains

To clarify the evaluation system of the catastrophe progression method-extension comprehensive evaluation model, firstly, the classical domain and the joint domain are determined, and let

$$R_{0j}=(N_{0j},C,V_{0j})=\left[\begin{array}{ccc}{N}_{0j}& c_1& v_{0j1}\\ & c_2& v_{0j2}\\ & \dots & \dots \\ & c_n& v_{0jn}\end{array}\right]$$

The N_0j level is expressed as the j level, where c_n denotes each model’s evaluation index, and the V_0ji reflects the value range of each evaluation index at the j level.

Based on the classification value of the catastrophe progression method and the division of Table 7, Table 8 and Table 9, the classical domain is constructed.

The classical domain matter-elements of each grade affected by the coal mine goaf are as follows:

$$\begin{array}{l}{R}_{01}=\left[\begin{array}{ccc}{N}_{01}& \mathrm{stability} \mathrm{grade} \mathrm{of} \mathrm{coal} \mathrm{mine} \mathrm{goaf}& (0.961,1)\end{array}\right]\\ R_{02}=\left[\begin{array}{ccc}{N}_{02}& \mathrm{stability} \mathrm{grade} \mathrm{of} \mathrm{coal} \mathrm{mine} \mathrm{goaf}& (0.893,0.961)\end{array}\right]\\ R_{03}=\left[\begin{array}{ccc}{N}_{03}& \mathrm{stability} \mathrm{grade} \mathrm{of} \mathrm{coal} \mathrm{mine} \mathrm{goaf}& (0.737,0.893)\end{array}\right]\\ R_{04}=\left[\begin{array}{ccc}{N}_{04}& \mathrm{stability} \mathrm{grade} \mathrm{of} \mathrm{coal} \mathrm{mine} \mathrm{goaf}& (0,0.737)\end{array}\right]\end{array}$$

The nodal domain matter-element is:

$$R_D=\left[\begin{array}{ccc}D& \mathrm{stability} \mathrm{grade} \mathrm{of} \mathrm{coal} \mathrm{mine} \mathrm{goaf}& (0,1)\end{array}\right]$$

The classical domain matter-elements of each grade affected by the train are as follows:

$$R_{01}=\left[\begin{array}{ccc}{N}_{01}& \mathrm{train} \mathrm{axle}& (14,18)\\ & \mathrm{train} \mathrm{speed}& (120,150)\\ & \mathrm{subgrade} \mathrm{height}& (3.35,3.75)\end{array}\right]$$

$$R_{02}=\left[\begin{array}{ccc}{N}_{02}& \mathrm{train} \mathrm{axle}& (18,22)\\ & \mathrm{train} \mathrm{speed}& (150,200)\\ & \mathrm{subgrade} \mathrm{height}& (3.75,4.0)\end{array}\right]$$

$$R_{03}=\left[\begin{array}{ccc}{N}_{03}& \mathrm{train} \mathrm{axle}& (22,26)\\ & \mathrm{train} \mathrm{speed}& (200,300)\\ & \mathrm{subgrade} \mathrm{height}& (4.0,4.5)\end{array}\right]$$

$$R_{04}=\left[\begin{array}{ccc}{N}_{04}& \mathrm{train} \mathrm{axle}& (26,30)\\ & \mathrm{train} \mathrm{speed}& (300,350)\\ & \mathrm{subgrade} \mathrm{height}& (4.5,5.5)\end{array}\right]$$

The nodal domain matter-element is:

$$R_D=\left[\begin{array}{ccc}D& \mathrm{train} \mathrm{axle}& (14,30)\\ & \mathrm{train} \mathrm{speed}& (120,350)\\ & \mathrm{subgrade} \mathrm{height}& (3.35,5.5)\end{array}\right]$$

(2)

Determination of matter-element to be evaluated

The data of the object being evaluated, a_i, are represented using matter-elements. Through this representation, the matter-element R_i, which is the object that needs to be evaluated, is derived.

$$R_i=(p_i,C,V_i)=\left[\begin{array}{ccc}{p}_i& c_1& v_{i1}\\ & c_2& v_{i2}\\ & \dots & \dots \\ & c_n& v_{in}\end{array}\right]$$

P_i is represented as the i-th evaluation object, and v_ij is the value of p_i on the characteristic index c_j.

(3)

Determination of index weight

The three factors of axle load, vehicle speed, and subgrade height are independent of each other. At present, the main parameters of trains in China are as follows: the load is 14~30 tons, the speed is 120~350 km/h, and the subgrade height is 3.35~5.5 m. The influence of the upper and lower bounds of the factor interval is calculated via the fitting formula of each factor, and the influence of each factor on the disturbance depth is analyzed.

$$\left|y_i(x_{\max (i)}-x_{\min (i)})\right|=E_i$$

(17)

where E_i is the influence range of i factor, y_i is the fitting equation of i factor, x_max(i) is the minimum value of i factor, and x_min(i) is the maximum value of i factor.

Through the calculation, the influence degree of subgrade height is the highest, which is 6.38 m, and the influence degree of vehicle speed is basically the same as that of axle load, which is 1.22 m and 1.38 m, respectively. According to the proportion of influence degree, each factor index is determined. The weight of embankment height is 0.71, and the weights of vehicle speed and axle load are 0.14 and 0.15, respectively.

The determination of the initial layer index weight coefficient in the extension system depends on the evaluation system for ground stability in coal mine goaf conducted by previous researchers. The weight assigned to external load or traffic load falls within the range of 0.116 [33] to 0.232 [34]. For this study, the weight coefficients of 0.2 and 0.8 have been chosen to represent the train load and the stability of the coal mine goaf, respectively.

(4)

Establishment of correlation function

The calculation formula of correlation function is:

$$\{\begin{array}{c}{K}_j(v_{ki})=\frac{\rho (v_{ki},V_{0ji})}{\rho (v_{ki},V_{0Pi}) - \rho (v_{ki},V_{0ji})}\begin{array}{ccc}& & (v_{ki}\end{array}\notin V_{0ji})\\ K_j(v_{ki})=-\frac{\rho (v_{ki},V_{0ji})}{\left|b_{0ji} - a_{0ji}\right|}\begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}\begin{array}{cccc}& & & (v_{ki}\in V_{0ji})\end{array}\end{array}$$

(18)

The formula for the distance of the point from the interval in formula is:

$$\rho (v_{ki},V_{0ji})=\left|v_{ki}-\frac{a_{0ji}+b_{0ji}}{2}\right|-\frac{1(b_{0ji}-a_{0ji})}{2}$$

(19)

(5)

Normalization of correlation degree

The correlation degree is normalized by a specific equation (Equation (19)) to make analysis and comparison easier.

$$K_j^{\prime }(v_{ki})=\frac{K_j(v_{ki})}{\max \left|K_j(v_{ki})\right|}$$

(20)

(6)

Determination of “activation” grade in coal mine goaf ground

The weight coefficient and normalized correlation degree are calculated using Equation (20), resulting in the comprehensive correlation degree of the evaluation object.

$$K_j(p_k)={\displaystyle \sum _{i=1}^n{a}_1}{K}_j^{\prime }(v_{ki})$$

(21)

The “activation” grade of the ground in the coal mine goaf corresponds to the highest value observed in the comprehensive correlation degree.

5. Case Studies

An engineering example was chosen to illustrate the section of the Taijiao high-speed railway that passes over a coal mine goaf site. In the beginning, an evaluation model called the catastrophe progression method–extension comprehensive evaluation model was utilized to assess the level of activation in the coal mine goaf ground of the Taijiao high-speed railway. Subsequently, a model test was conducted based on the engineering geological conditions to validate the classification method for the “activation” of the coal mine goaf ground proposed in this study. The coal mine goaf beneath the Taijiao high-speed railway section measures approximately 110 m in length, 18 m in width, and has a mining thickness of 4 m. Moreover, the coal mine goaf is buried at an average depth of 56 m. The surrounding rock experienced prolonged water saturation, leading to a relatively loose rock mass structure and the development of joint fissures. Adjacent to the coal mine goaf, there are three other coal mine goafs with an inefficient engineering layout, and the dip angle of the ore layer is measured to 12°. The high-speed railway carries an axle load of 18 tons, achieving a design speed of 250 km/h, with a subgrade height standing at 4.15 m.

5.1. Case One

5.1.1. Calculation of Coal Mine Goaf Site Stability

The catastrophe series method was used to calculate the stability of coal mine goaf site, organizing the coal mine goaf parameters and normalizing them, and the results are shown in

Table 10. Engineering parameters and normalized results of the Taijiao high-speed railway.

		C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12
Taijiao high-speed railway	Actual parameters	110	14	1980	3	4	28	3	4	4	3	3	3
	Index after normalization	0.432	0.140	0.823	0.333	0.000	0.28	0.333	0.000	0.000	0.333	0.333	0.333

.

Through the normalization formula of the corresponding catastrophe model, the catastrophe progression was calculated, and the values of x_B1, x_B2, x_B3 and x_B4 in the coal mine goaf were 0.856, 0.778, 0 and 0.925, which met the threshold requirements. Therefore, the catastrophe progression A is selected as the “average principle”, so the value of A is 0.640, which belongs to the “unstable” coal mine goaf site.

5.1.2. Extension Theory Calculation of Activation Grade

(1)

Extension evaluation of stability at coal mine goaf site

The matter-element to be evaluated in this case is:

$$R_{01}=\left[\begin{array}{ccc}{p}_1& \mathrm{stability} \mathrm{grade} \mathrm{of} \mathrm{coal} \mathrm{mine} \mathrm{goaf}& 0.64\end{array}\right]$$

Through the calculation of Equations (18) and (19), the correlation matrix K₁ is obtained.

$$K_1=\left[\begin{array}{cccc}-0.471& -0.413& -0.212& 0.369\end{array}\right]$$

The normalization of the correlation matrix K′₁ is conducted based on Equation (20).

$$K_1^{\prime }=\left[\begin{array}{cccc}-1& -0.876& -0.451& 0.783\end{array}\right]$$

Equation (21) is utilized to compute the comprehensive correlation degree, resulting in the acquisition of the matrix K_p1. As this evaluation layer solely encompasses a single index, the weight coefficient is set to 1.

$$K_{p1}=A_1•K_1^{\prime }=\left[\begin{array}{cccc}-1& -0.876& -0.451& 0.783\end{array}\right]$$

(2)

Extension evaluation of train influence

Based on the overall analysis of the project, it can be seen that the matter-element to be evaluated for the Taijiao high-speed railway is:

$$R_1=\left[\begin{array}{ccc}{p}_1& \mathrm{train} \mathrm{axle}& 18\\ & \mathrm{train} \mathrm{speed}& 250\\ & \mathrm{subgrade} \mathrm{height}& 4.15\end{array}\right]$$

Through the calculation of Equations (18) and (19), the correlation matrix K₁ of the Taijiao high-speed railway is obtained.

$$K_1=\left[\begin{array}{cccc}0& 0& -0.333& -0.667\\ -0.5& -0.333& 1& -0.333\\ -0.448& -0.333& 1& -0.304\end{array}\right]$$

The correlation matrix K′₁ of the Taijiao high-speed railway is obtained via normalization according to Equation (20).

$$K_1^{\prime }=\left[\begin{array}{cccc}0& 0& -0.5& -1\\ -0.5& -0.333& 1& -0.333\\ -0.448& -0.333& 1& -0.304\end{array}\right]$$

The weight coefficient obtained from Section 4.2 is:

$$A_1=\left[\begin{array}{ccc}0.15& 0.14& 0.71\end{array}\right]$$

Calculate the comprehensive correlation degree using Equation (17) to obtain the Taijiao high-speed railway matrix K_p1:

$$K_{p1}=A_1•K_1^{\prime }=\left[\begin{array}{cccc}-0.388& -0.283& 0.775& -0.413\end{array}\right]$$

(3)

Extension pre-evaluation of “activation” grade in coal mine goaf ground

The current weighting coefficient A = [0.8 0.2] is obtained from Section 4.2, and the first-grade comprehensive evaluation of Taijiao high-speed railway is obtained according to Equation (17) as follows:

$$K=A•K_p=\left(\begin{array}{cc}0.8& 0.2\end{array}\right)\left[\begin{array}{cccc}-1& -0.876& -0.451& 0.783\\ -0.388& -0.283& 0.775& -0.413\end{array}\right]$$

The final pre-evaluation result is calculated as follows:

$$K=\left[\begin{array}{cccc}-0.878& -0.757& -0.206& 0.545\end{array}\right]$$

The final evaluation result is “activation”. In the actual project, the coal mine goaf was grouted during the construction of the regional section, and the corresponding anti-settlement design of the subgrade was carried out. The evaluation results align with the engineering evaluation report, demonstrating that this method possesses a particular point of reference and engineering significance in the construction of high-speed railway coal mine goaf site.

5.2. Case Two

5.2.1. Introduction of Instrument and Model

Based on the engineering geological conditions of the Taijiao high-speed railway coal mine goaf ground, the model test was carried out. The test instrument adopts the self-developed high-speed railway coal mine goaf ground pseudo-dynamic loading model test device, which is mainly composed of five parts: electric gantry, controller system, cooling system, hydraulic oil separator, and actuator, as shown in Figure 9.

The ballasted track served as the subgrade structure for the Taijiao high-speed railway. The subgrade bed’s bottom layer measured 2.3 m in height, while its surface layer stood at 0.7 m. Furthermore, the ballast layer maintained a thickness of 0.35 m. The strata of the road section in the study area were relatively gentle, showing a north–south overturning syncline distribution, and the coal rock strata were stable. According to the field drilling data, the formation lithology is mainly composed of old loess, silty clay, limestone, mudstone, sandstone and so on. The mining depth of the goaf selected in the model test was 28.1 m~48.8 m, and the mining height was 1.2~6.5 m. Most of the mined-out areas were semi-filled and loose in structure.

For the simulation, an area with a mining height of 4 m and a mining depth of 60 m was chosen to investigate the characteristics and mechanism of activation deformation in a coal mine goaf site under the load of a high-speed train. From the perspective of considering the operability of the test, the formation conditions were simplified from top to bottom into 15 m loess, 10 m limestone, 20 m sandstone, and 15 m mudstone. The similarity test adopted a scale model, the geometric similarity constant C_l is 100, and the heavy C_γ similarity constant was 1.5. Based on the theory of similarity, the similarity constant of the basic parameters in the physical model test of the coal mine goaf under the high-speed railway were obtained. The values are shown in

Table 11. Physical quantity similarity constant.

Physical Quantity	Similitude Parameter
Stress	150
Elastic modulus	150
Internal friction angle	1
Time	10
Displacement	100
Cohesive force	150

.

According to the similarity constants of the parameters in Table 11, the physical parameters of the model of each layer of rock and soil in the coal mine goaf were calculated. In this model test, river sand, gypsum and lime were selected as similar materials. Different ratios were selected for the production of simulated materials through similar material tests. The test results show that the ratio of loess is 573, the ratio of limestone is 437, the ratio of sandstone is 455, and the ratio of mudstone is 473. The first place is the ratio of sand to binder (the mass ratio of aggregate to binder material), the second and third places are the mass ratios of lime to gypsum, and the water consumption of each material is 8% of the total weight. Finally, a coal mine goaf model with a geometric similarity ratio of 1:100 was established, and the width, height, and thickness of the test model were 1.52 m × 0.7 m × 0.25 m, as shown in Figure 10.

After compiling the required test data, the control system will approach the required simulated waveform through iteration. As shown in Figure 11 and Figure 12, the “M” waveform of 350 km/h train load under the action of 1:40 similarity ratio, the error is less than 10%, which meets the test requirements.

5.2.2. Test Process and Analysis of Results

The self-developed high-speed railway coal mine goaf ground pseudo-dynamic loading model test device is used to apply the dynamic load of high-speed railway. Before the dynamic load is applied, a pre-pressure of 3 kN is applied to the subgrade, followed by simulating train travel through the test device, while observing the test phenomena such as crack development, stress time course changes, and surface settlement in the coal mine goaf. The monitoring equipment is shown in Figure 13, and the digital speckle technique is used to monitor the displacement, mainly by arranging speckles on the surface of the model, and then taking photos of the model with a high-speed camera and monitoring the deformation of the specified area by comparing the photos of different time periods.

(1)

Analysis of excavation settlement

Excavation was carried out after the model was demolded and air-dried, and the excavation scheme was divided into four excavations, each with a distance of 30 cm and a coal seam extraction height of 4 cm. During the excavation process, the model deformation was monitored by digital image correlation (DIC) equipment throughout, and each excavation was stopped for 30 min to continue the next excavation after the deformation of the coal mine goaf was stabilized. When the coal seam was mined to 47 m, the initial fracturing of the immediate roof occurred, accompanied by the emergence of numerous minuscule transverse fractures in the upper part of the new roof. When the coal seam was mined to 65 m, the second large-scale roof collapse phenomenon occurred, and the beam structure was separated from the seam at both ends of the coal mine goaf, with a break angle of about 64° at the cut-hole. When the coal seam was mined to 120 m, the third large-scale collapse occurred, and a large separation layer appeared around 22.5 m from the surface, which had a maximum void height of 2.3 m, and masonry beam structures and a large number of voids and voids also existed at the boundary of the coal mine goaf.

As shown in Figure 14, during the mining process, three large-scale collapses occurred in the coal mine goaf site, resulting in a maximum separation development height of 48.5 m. Through the treatment of DIC equipment, the surface subsidence curve during the mining process is shown in Figure 15. The maximum settlement of the coal mine goaf site is 1010.78 mm, and the surface position of the maximum subsidence value is biased to the cut side.

The stable state of goaf settlement before dynamic load loading after the excavation is shown in Figure 16. After completing the excavation, the coal mine goaf site is continuously observed. When the deformation of the coal mine goaf is stable, the maximum development height of the separation layer is 48.5 m from the mining base, and the maximum gap height of the separation layer is 52 cm. There is a large amount of activated space in the middle and edge of the coal mine goaf site. The coal mine goaf features a beam structure separation layer and cavity located in its middle section and at its boundary. This separation layer and cavity are susceptible to destabilization and activation due to external load disturbances.

(2)

Analysis of Loading settlement

After the settlement of the coal mine goaf is stable, soft soil and fine sand are used to backfill the surface at the position where the roadbed is placed to ensure the level of the roadbed. Due to the high strength of the roadbed, it is generally not damaged. This time, the concrete prefabricated roadbed is used. After the site becomes stable and the surface is filled horizontally, it is placed in the central position of the coal mine goaf. After the roadbed is placed and the secondary settlement of the coal mine goaf is stable, the dynamic load is applied by the actuator to observe the fracture development and surface settlement of the coal mine goaf site. The dynamic load is divided into a group according to 100 train load cycles. As depicted in Figure 17, during the application of the dynamic load, it can be observed that the separation position is continuously developed upward and some cracks are closed.

When the load is applied to the 900~1000th cycle, the coal mine goaf site mutates, as depicted in Figure 18, the boundary cracks of the coal mine goaf penetrate the surface, and the middle of the coal mine goaf site is separated and the cracks are closed. The subgrade is inclined as a whole, the inclination angle is about 12°, and the maximum settlement of the subgrade is 1230.74 mm.

Figure 19 and Figure 20 correspond to the overall settlement deformation cloud map under the action of “M” wave 250,000 times and 500,000 times, respectively. During the test, the subgrade has uneven settlement, with the deformation value being more prominent on the right side compared to the left side. As the number of cycles for the “M” wave increases, the uneven settlement experiences a substantial rise. In such instances, the influence depth of settlement deformation is also significantly greater than that of the left side. This is because the ground on the right side of the subgrade was broken during the loading process, which makes the dynamic stress more borne by the lower geotechnical body, thus making the accumulated settlement deformation of the right side of the ground significantly larger than the left side.

As shown in Figure 21 and Figure 22, the development of the separation layer and the trend of surface subsidence show three stages: slow development–rapid development–mutation. The slow development stage may demonstrate that under the impact of dynamic load, the upper space of the initial separation layer constitutes a simply supported beam structure serves the purpose of bearing the load and resisting deformation, so that the separation layer develops slowly. Due to an escalating number of cycles and the influence of vibration, the rock beam is layered, unable to bear the upper load, and its thickness decreases, which makes the development speed of the separation layer faster. When the thickness of the rock beam is reduced to a certain value, the upper load is much larger than the load it can bear, resulting in the bending or even breaking of the rock beam; thus, the surface of the coal mine goaf is unstable and the subgrade falls into the ground.

From the analysis of the above test results, it becomes evident that the main reason for the activation deformation of the coal mine goaf site of the high-speed railway is that the upper rock mass structure of the separation layer cannot bear the subgrade load and the dynamic load generated by the train. If the separation layer is grouted before the construction, the ground of the coal mine goaf site will be strengthened, the height of the separation layer will be reduced, and the top of the separation layer rock (soil) beam structure thickened, so that it is sufficient to bear the subgrade and the train load can effectively reduce the possibility of the activation deformation of the coal mine goaf site. The same can also be carried out before roadbed construction using methods such as the strong ramming on a pre-built site to compress the dense coal mine goaf site and closing the separation layer closer to the surface, thus reducing the activatable space of the coal mine goaf.

Through the test results, it becomes apparent that the complete roof cutting took place at the ultimate coal mine goaf location in the experimental model test of the Taijiao high-speed railway. Furthermore, a substantial deformation occurred in the high-speed railway embankment, aligning with the evaluation outcomes of the extension comprehensive evaluation method. It was proven that the activation classification evaluation method proposed in this paper has certain engineering guiding significance.

6. Discussions

The evaluation of activation classification in high-speed railway coal mine goaf ground is a relatively complex and novel research direction. There are not many achievements in this direction, and the methods used are different.

Most of the comprehensive evaluation methods are more subjective in the selection of weights, and the expert scoring method often employed for weights calculation, such as the research results of Wang [35] and Ren [20]. However, the expert scoring method can sometimes give full play to the engineering experience of experts, and also has a certain positive effect on the evaluation of the project.

In the study of Ren et al. [23], the neural network method was used to predict the stability of the coal mine goaf site. Subsequently, the activation grade of the coal mine goaf ground was assessed by combining the extension comprehensive evaluation method with the unascertained measure theory. However, a limitation exists in this evaluation method. Namely, the stability grade of the coal mine goaf site predicted by the neural network is a fixed value, specifically levels 1, 2, 3, and 4. This fixed value approach, wherein the stability grade falls within predetermined intervals ([0.5, 1.5], [1.5, 2.5], [2.5, 3.5], [3.5, 4.5]), presents a lack of alignment with the extension model fusion.

In this paper, an evaluation model on activation classification of coal mine goaf ground considering high-speed railway load is proposed, which is mainly applicable for middle–deep and level goaf areas using a longwall mining method. The prediction of the stability grade of the coal mine goaf via the catastrophe progression method not only has high accuracy but also has good fit with the extension comprehensive evaluation method. Because the output of the catastrophe progression method is an interval value of various stability grades, considering the influence of high-speed railway load on the activation of coal mine goaf ground, the influence degree of each influencing factor of the train is obtained by establishing the calculation model of the additional stress of the high-speed railway ground, and the weight factor is calculated by using the influence degree of each factor. The results obtained are scientific and objective. A comparison of the evaluation methods is shown in

Table 12. Comparisons of different methods.

Evaluation Methods	Weight Calculation Methods	Advantages	Disadvantages
ANP-entropy weight-fuzzy comprehensive evaluation method [35]	Expert scoring method	The use of entropy weighting reduces the error of human subjective factors	Too much reliance on questionnaires, questionable authenticity of questionnaires
AHP-fuzzy comprehensive evaluation method [20]	Expert scoring method	Detailed classification of each factor affiliation	Using the expert scoring method to determine the weight, the human subjective error is large
Neural network-extension comprehensive evaluation method [23]	Uncertainty measurement theory	Eliminate the error of human factors	The neural network method does not fit well with the topologically integrated evaluation method
Catastrophe progression method–extension comprehensive evaluation model	Bottom tier indicator weights are determined by the impact degree (few relevant studies), and first tier indicator weights refer to others’ results (many relevant studies and wide application)	The calculation of the bottom index weights is objective, and the first layer of index weights is determined by combining engineering experience, with a simple calculation process and accurate results	The application of catastrophe progression method requires a larger sample size

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7. Conclusions

To evaluate the safety of constructing coal mine goaf sites for high-speed railways, this study employs the catastrophe progression technique to forecast the stability grade of coal mine goaf site, considers the high-speed railway load, and analyzes the influence of subgrade height, vehicle speed, and axle load on construction safety. An evaluation model on activation classification of coal mine goaf ground is proposed. This model primarily applies to level goaf areas using a longwall mining method at middle–deep depths, and it serves the purpose of assessing the activation grade of goaf ground under high-speed railways. The key conclusions are summarized below:

• The assessment and evaluation of ground activation in coal mine goaf for high-speed railway projects are divided into the influence of coal mine goaf site stability grade and the influence of high-speed railway engineering. The catastrophe progression method is introduced to predict the stability grade of the coal mine goaf site. The theoretical analysis and the establishment of the additional stress calculation model are used to classify the influence degree of the influencing factors of the high-speed railway project. The influence grade of different factors on the coal mine goaf site is categorized into four grades: no influence, low influence, medium influence, and high influence.
• The influence of the stability grade of the coal mine goaf site as well as the influence of the high-speed railway project are comprehensively considered via the extension comprehensive evaluation method. This method is simple, accurate, and overcomes the shortcomings of the general algorithm weight selection with strong human subjectivity and subjective evaluation results.
• A constructed system for evaluating the stability of coal mine goaf sites incorporates the catastrophe progression technique to evaluate the grade of stability. The factors influencing the stability grade of coal mine goaf site are determined, and the stability grade classification includes stable, basically stable, less stable, and unstable. Through the verification of the example, the accuracy of this method can reach 96.7%.
• The analysis and evaluation results of engineering examples are ‘activation’, aligning with the conclusions presented in the engineering evaluation report. This shows that the evaluation results of activation in high-speed railway coal mine goaf ground based on catastrophe progression method–extension comprehensive evaluation method are accurate and reliable, which provides a new way for evaluating the activation of the foundation in high-speed railway goaf.
• The model test results show that the goaf site of the Taijiao high-speed railway model test experiences complete detachment of the entire roof. Additionally, significant deformations are detected in the high-speed railway subgrade, which align with the evaluation findings obtained through the extension comprehensive evaluation method. It was proven that the activation classification evaluation method proposed in this paper has certain engineering guiding significance.