Study on the simulation of bridge deformation in a mining subsidence area

Abstract

Bridges in mining areas deform primarily because of surface subsidence caused by underground mining. Analysis of these deformations should consider the synergistic effects between the foundation soil and the bridge superstructure. The geology of mining areas, which is inherently complex, significantly effects the selection of soil mechanics parameters, potentially leading to errors in model calculations. This study focuses on the bridge in the Fengfeng mining area of Handan, Hebei, and uses the probabilistic integral method to predict subsidence at the bottom section below the surface over a specified period. A finite element model of the bridge and foundation soil is constructed, utilising these predicted subsidence contour lines as boundary conditions for simulating the surrounding surface subsidence. Additionally, monitoring data of surface and bridge subsidence from the same period are collected to validate the simulation accuracy, analyse error types and causes, and propose model improvements. After the feasibility of the improvement schemes is validated, the revised model is employed to predict the bridge failure trend. The model parameters can be adjusted based on surface subsidence monitoring data, and by integrating these adjustments with the predicted subsidence results at the bottom section during various mining stages, the bridge failure trend can be accurately predicted.

Introduction

An important method for analysing the deformation of buildings and structures in mining areas is to investigate the synergistic effect between building foundations and structures in mining areas by establishing a synergistic mechanics model of foundations and structures in mining areas and analysing the change rule of additional internal forces of buildings in mining areas¹. Owing to the many buildings and structures in China’s coal mines and mining areas, deformation monitoring and early warning of surface buildings and structures in the mining and subsequent management of mining areas are crucial. Numerical simulation is a commonly used method for surface settlement simulation that can be used to analyse the impact of mining on the surface deformation of a mining area and to plan mining by simulation in subsequent coal mining work, which can help reduce the damage caused by coal mining to the rock layer and the ground surface and provide an effective guide for coal mining projects^2–5. To simulate the deformation of bridges and other buildings and structures in mining areas, obtaining the subsidence state of the ground surface in their vicinity is essential⁶. Researchers usually simplify the surface or building model and carry out simulations in a targeted manner according to the degree to which the information of the surface settlement state and specific engineering needs are obtained. For example, Liu et al.⁷ established a mathematical model of the cooperative deformation of building structures under surface curvature deformation under the premise of a known surface deformation trend, analysed the stress distribution of local building structures, and derived the causes of building damage. Zhu et al.⁸ simplified the structural model of surface bridges by constructing a large-scale mechanical model extending from the surface to the underground coal face under the condition that the surface subsidence state in the vicinity of bridges is unknown, simulating the overall displacement and deformation of the surface bridge caused by mining, and analysing the main factors causing damage to the bridge structure, as well as possible further damage. Owing to the complex geological structure of the mining area, underground excavation of a large coal face has a complex effect on the ground surface, among which uneven settlement of the ground surface leads to displacement of the bridge in both the vertical direction and horizontal direction, which in leads to tilting and torsion of the bridge, threatening its safety. Eventually, the deformation of the bridge is reflected mainly in the deformation of the bridge abutment, cracking of the bridge deck and rotational twisting of the bridge span^9,10. Simultaneously, the shear stress further causes the expansion of the cracks in the bridge foundation, abutment and bridge deck over time, which affects the bearing capacity and durability of the bridge. On the other hand, the inclination of bridges can affect vehicle operation, and for most old bridges with deteriorated pavement conditions due to ageing, corrosion, and increased gross vehicle weights, various effects can accelerate bridge damage¹¹.

In the study of ground subsidence caused by underground mining, mine subsidence prediction is the core problem. Among many subsidence prediction methods, the stochastic medium model proposed by Polish scholar J. Litwinisyn is a mature methods. Liu Baochen and Liao Guohua introduced this theory into China, and based on it, they solved the simplified solution and created the probability integral method (PIM). The domestic “three below” coal mining environment is rich, and many application scenarios are suitable for the PIM; thus, the method has been rapidly promoted, and many production practices have confirmed that for the calculation of the maximum subsidence area, the probability integral prediction results have good accuracy^12–14. With a variety of new monitoring methods being applied to the monitoring of coal mining areas, domestic observation data have been accumulated, and the parameter inversion model has been updated. Improving the parameter inversion model for the local conditions of each mining area can accurately anticipate the surface subsidence caused by mining^15–18. The Fengfeng mining area involved in this study has a large amount of measured data and has accumulated many inversion parameters through a long period of observation, which is a sufficiently convincing value of the expected surface subsidence. Therefore, this paper uses the results of ground subsidence as the boundary conditions of the finite element model to simulate the deformation of the bridge subsidence on the surface.

In recent years, surface bridge structure monitoring has developed rapidly; it is an important part of bridge deformation monitoring for obtaining bridge structure displacement measurement data through various monitoring methods, analysing bridge movement and deformation, warning of damage early, and proposing safety management plans^19,20. Among them, terrestrial 3D laser scanning technology has been proven to meet the general structural deformation monitoring accuracy requirements and has abundant bridge monitoring application cases^21–24. Terrestrial 3D laser scanning technology also performs well in the field of mining surface monitoring because it can adapt to the complex monitoring environment in mining areas^25,26. However, miners usually do not have complete information about the effect of mining deformation on bridge objects, such as monthly deformation indicators or continuous settlement functions, but only the final value of the indicator at the end of a particular cycle. This is a drawback of 3D laser scanning technology applied to mine monitoring, and this lack of information flow directly contributes to the difficulty of proper monitoring of bridge health²⁷.

Conventional numerical modelling can be used to predict the results of surface settlement in the vicinity of a bridge and to obtain the safe state of the bridge for any period by updating the model parameters. However, to analyse the synergistic effect between the surface and the bridge in the mining area via numerical simulation, i.e., the displacement response of the bridge caused by the transmission of surface subsidence, it is necessary to obtain geotechnical samples through field drilling holes and simulate the changes in the rock body via the mechanical parameters of the samples. Due to the differences in the scales of the geotechnical samples and the natural rock body and the complex changes in the mechanical parameters of the rock layer on a large scale, the mechanical properties of the geotechnical samples cannot completely represent those of the rock body. The simulated strain and actual deformation values are bound to deviate^28,29. Currently, most of the numerical simulations of structural deformation of bridges and other structures in mining areas are limited to qualitative analysis, i.e., confirming the causes of deformation and the deformation trend of the structures during the mining process. Owing to the limitations of the mechanical model, obtaining the simulated variables close to the measured results is usually difficult, and the simulated strain results are not available. Thus, the simulation method cannot further extend and effectively integrate with the actual measured results in the quantitative analysis of deformation of structures in the mining area. As a result, the simulation method cannot be further extended in the field of quantitative analysis of the deformation of mine buildings and structures, and combining effectively with actual measurements is difficult. To address this problem, the authors calculate the difference between the surface simulation results and the measured results from actual projects, analyse the laws contained therein, propose solutions to solve the simulation errors of surface bridges in coal mine areas, and use the corrected model to predict the damage trend of bridges. First, for the railway bridge above the coal face in the Fengfeng mine area, a surface-bridge cooperative finite element model is constructed to simulate the simulated strains of the settlement of the surface and the bridge structure. Moreover, the surface structure monitoring data from the same period are obtained via ground-based, three-dimensional laser scanning technology. The measured and simulated results of the surface settlement are compared to calculate the simulation error of the settlement of the sampling points on the surface, analyse the reasons for the simulation errors as well as the main parameters affecting the simulation results, and propose model improvement and parameter conversion solutions. This study proposes a model improvement and parameter conversion scheme. After the surface settlement simulation results of the converted model match the actual measurement and verification results, the simulation effect of the bridge superstructure is further verified, which proves that the conversion scheme is feasible. The study shows that after reasonable adjustment, the model can control the relative error of the surface bridge settlement simulation within 10%, and in the subsequent prediction work, the feedback situation in the mining area is consistent with the results, which proves that the model can accurately predict the settlement trend of the bridge in the next stage.

Study area

Within the surveillance zone of the railway bridge, three coal mining faces are identified, as depicted in Fig. 1. The 08201 face is characterised by a surface elevation ranging from + 140 to + 150 m and a coal face elevation between − 260 and − 310 m, with an average dip extending approximately 120 m. Mining activities here spanned from January to October 2015. For the 08203 face, the surface elevation is similarly between + 140 and + 150 m, whereas the coal face descends to elevations of -260 to -310 m, boasting an average strike length of 290 m and a dip length of 120 m. Excavation began in January 2016 and ended in September 2018. The follow-up mining endeavor, designated 08202, presents a surface elevation from + 145 to + 150 m, with the coal face elevation plunging to -230 to -300 m.

Fig. 1.

Scanning area and coal faces.

The railway bridge under observation was located atop the 08201 and 08202 coal faces. Orienting north‒south, the bridge extends 48.61 m in length and 5 m in width and is equipped with railroad tracks, a pedestrian pathway, and stone parapets on both sides, with certain sections buried beneath ground level. The upper segments of the bridge piers and the principal structure of the bridge were fabricated via a method that integrates casting and splicing techniques. The bridge architecture is segmented into five distinct parts, which include the abutments on either side and three arches situated centrally, as shown in Fig. 2. Each segment is constructed from cast concrete blocks and encased with masonry bricks along the periphery, utilising identical materials for both the blocks and the external wall bricks. The external walls boast a thickness of 0.6 m. Subsequent to the assembly of the bridge segments, they undergo a final casting and splicing process to integrate seamlessly with the bridge piers.

Fig. 2.

Reinforced railway bridge.

To preserve the integrity of the bridge during mining operations, in 2015, the construction team reinforced each of its three arches with four sets of arch-shaped steel trusses. These trusses, constructed from double-spliced equal-angle steels, measured 20 cm in width and 22 mm in thickness. In the course of further observations, it was noted that the reinforced steel structures within each arch tunnel remained intact, showing no substantial damage or separation from the arch walls. Concurrently, through-cracks emerged at the splicing joints of the bridge’s external walls.

In December 2015, our research successfully captured an inaugural set of three-dimensional laser scanning data, detailing the surface and bridge structures within the designated monitoring zone. Preceding this achievement, mining activities at the 08201 site ended by October 2015, in contrast to the 08202 and 08203 sites, which remained unexploited. In April 2018, we acquired the second set of data concerning the same structures, employing an innovative, target-free subsidence monitoring technique rooted in terrestrial laser scanning (TLS), as delineated by Gu et al.³⁰. By this juncture, more than half of the 08203 site, distantly situated from the bridge, had been excavated. Conversely, the 08202 site, positioned proximally to the north side of the bridge, awaited commencement of mining operations. The cross-section of the bridge in addition to its coordinate system is shown in Fig. 3.

Fig. 3.

Cross section of the bridge and independent coordinate system.

Methodology

PIM parameters

In this work, the values of surface deformation after coal face mining expected by the PIM serve as subsequent model boundary conditions. The probabilistic integral method evolves from the random medium model, which has two basic assumptions:

1. Mining-induced movements in all directions are direction independent;

2. The surface deformation caused by underground mining of large coal faces is a linear superposition of the surface deformation caused by mining of multiple small faces. Under these two assumptions, the subsidence value W_e(x) of the surface point x induced by unit mining in a two-dimensional plane is expressed as:

1

In the formula, r = H₀/tannβ is a constant, which is referred to as the main influence radius, where H₀ is the average mining depth and tanβ indicates the main influence angle tangent.

Equation (1) shows that the surface subsidence influence function caused by unit mining obeys an N(0,r/ suitable distribution.

Surface movement is a three-dimensional problem; i.e., the subsidence of the surface point is due to the mining of the coal seam in two directions: along the strike direction (x-axis direction) and the inclination direction (y-axis direction), and the three-dimensional spatial relationship is shown in Fig. 4. Under three-dimensional conditions, the subsidence value W(x, y)_A of a surface point A(x, y) caused by mining in a certain mining area is expressed as:

2

Fig. 4.

3D coordinate system based on the stochastic medium model³¹.

where is the spatial probability density function.

Before the model boundary conditions are set, the results of the probability integral dynamic prediction of the surface subsidence above the three coal faces in the peak mining area, namely, 08201, 08202 and 08203, were obtained. According to the observation results of subsidence values in the peak mine area in previous years and a comprehensive analysis of the actual situation in the area, the main inversion parameters involved in this analysis are determined, as shown in Table 1:

Table 1.

Main inversion parameters of the PIM.

Mining thickness	Subsidence factor q	Mainly affects angular tangent tanβ	Maximum subsidence angle θ	Horizontal movement factor b	Aberration distance S0
5.5 m	0.78	1.7	83	0.3	0 m

Modeling

In the process of mine surface subsidence, the foundation soil, bridge pier and bridge superstructure form a static equilibrium system, and the state of the bridge structure is determined by the magnitude of the deformation resistance stiffness of the three factors and their relative relationships³². Therefore, to simulate the bridge force state accurately, this paper considers the synergistic effects of the three factors during the deformation analysis of the stone arch bridge and constructs a complete bridge–soil model:

1. A bridge model with an axial length of 50 m and a transverse width of 5 m is established on the basis of the bridge section structure shown in Fig. 3, and the direction of the bridge axis is 14° north by west. Four groups of double-pieced equilateral corners with widths of 20 cm and thicknesses of 22 mm were installed in each archway.

2. According to the stratigraphic situation revealed by the rock layer, the surface layer of the rock layer in the mining area is loess sandwiched with pebbles, with an average thickness of 65 m. The main radius of influence of underground coal face mining calculated by the dynamically predicted parameter is approximately 250–270 m. To ensure that the size of the model covers the actual monitoring range and to avoid wasting computational resources, the overall model is determined as follows: the length of the bridge is 550 m, the width of the bridge is the thickness of the soil model is 65 m and the bridge axis direction is 5 m. and the thickness of the soil model is 65 m; and the bridge axis direction is 5 m. To improve the calculation accuracy, the river part under the bridge was refined according to Fig. 3.

In order to accurately predict the response of the bridge under actual geologic conditions, the authors employed the contact units CONTA174 and TARGE170 of ANSYS to establish the contact pairs between the bridge and the geologic entity, thereby simulating the complex interactions between the two. Furthermore, the displacement consistency was ensured through the implementation of The coupling between the nodes was achieved by merging consistent nodes on the contact surfaces of the steel structure-masonry and the bridge-entity, thus creating cemented contacts between the bridge and the geological model. This enabled the transfer of forces and displacements while maintaining node merging, thereby simulating the mechanical coupling between the bridge and the geological foundation. To prevent the contact surface grid size gap from being too large to affect the calculation results, the model adopts a gradient grid, the bridge grid spacing is 1 m, the river channel grid is 2.5 m, and the loess layer adopts an 8 m spacing grid. The model has a total of 188,288 cells and 138,390 nodes. The overall model and the local model of the bridge and the river below it are shown in Fig. 5:

Fig. 5.

Finite element model of the railway bridge and soil.

Model parameters

The soil strength parameters used in this paper were averaged from the experimental results of the in situ soil samples, which were based on borehole data collected near the mine. Since the stone masonry constituting the main structure of the bridge is a type of nonhomogeneous and anisotropic material³³, experiments have shown that, under the conditions of the same production and moulding, curing and compression test methods, the stress and deformation laws and mechanical properties of the stone masonry are essentially similar to those of the concrete^34,35. Therefore, this simulation uses the most commonly used solid65 solid unit to simulate the concrete to define the stone masonry, and the relevant parameters are selected in conjunction with the actual situation of the bridge and the relevant design codes of the masonry structure. The relevant parameters are selected in conjunction with the actual situation of the bridge and the relevant design codes for masonry structures³⁶. The Beam188 unit is used to define the arch steel structure. The relevant parameters are shown in Table 2:

Table 2.

Material properties of the model.

		Bulk modulus E (GPa)	Shear modulus G (MPa)	Cohesion c (MPa)	Internal friction angle φ (°)	Tensile strength (MPa)	Density ρ (kg/m3)
Soil	Loess layer	30	10	0.5	20	3	1700

		Bulk modulus E (GPa)	Poisson’s ratio µ	Yield strength (MPa)	Compressive strength (MPa)	Tensile strength (MPa)	Density ρ (kg/m3)
Bridge	Masonry	12	0.255	345	3.71	0.11	2500
	Reinforced steel	206	0.3				7850

Boundary condition setting

In this study, the configuration of the model boundary conditions represents a crucial phase in the numerical simulation of bridge analysis, as it serves to guarantee the stability of the model and the dependability of the ensuing results³⁷. The configuration of model boundary conditions comprises the application of normal, symmetric boundary constraints around the geological model, which serve to limit the horizontal movement of the constrained area and prevent any horizontal displacement of the model edges. Additionally, the bridge abutments and piers are fully fixed, while the geological solid components and ancillary parts are completely fixed, thus ensuring the stability of the geological model throughout the simulation process. The global boundary conditions of the model are illustrated in Fig. 6, while the boundary conditions imposed locally on the geology are shown in Fig. 7.

Fig. 6.

Boundary conditions of the model.

Fig. 7.

Boundary conditions for localized models.

The authors obtained the results of the predicted ground cross-section settlement caused by the working face mining through the probabilistic integration method. These results are now applied to the bottom of the ground model as a vertical constraint, and the surface settlement and the settlement and deformation state of the bridge on the surface are calculated through the results of the predicted ground cross-section settlement. When the displacement is applied as the boundary condition of the model, the boundary constraint is in close proximity to the surface of the model, which results in excessive stress on the bridge structure and ultimately leads to the failure of the calculation. In light of the aforementioned considerations, the settlement contour of the bottom cross-section, situated at a vertical distance of 65 m below the surface, as proposed by the authors, is as follows: The predicted results of the settlement of the bottom cross-section of the loess layer were applied to the bottom of the model as the boundary constraint^38–40. The coordinates of the bridge were used to ensure that the settlement and deformation state of the bridge on the surface were calculated through the coordinates of the bottom cross-section of the loess layer. The spatial positioning of the bridge was adjusted to ensure that the relative position between the bridge model and the contours could be aligned with the actual bridge position after the contours were applied to the model. The boundary condition imposed at the bottom of the geology in the model is illustrated in Fig. 7a, while the normal symmetry constraint imposed around the address is shown in Fig. 7b.

The results of the boundary condition imposition at the bottom of the model and the spatial relationship with the bridge are shown in Fig. 8a. To show the boundary constraints near the bridge more clearly and facilitate the subsequent comparison, the contour lines near the bridge are enlarged and refined, and the bridge steel model is used to display the bridge location to avoid occlusion. The results are shown in Fig. 8b.

Fig. 8.

Settlement at the bottom of the model.

Load setting

In this study, the model is subjected to four distinct types of loads:

1. Gravity load: The model globally experiences a gravitational acceleration of 9.8 N/kg.

2. Ballast on the bridge deck: Comprising crushed stone, the ballast has a unit weight of 20 kN/m³. Given its height of 0.5 m, the resultant force from the ballast (F(ballast)) is computed as 20*0.5*5*48.63 = 2431.5 KN.

3. Sleepers and Rails: The concrete sleepers on the bridge deck account for 39.2 KN/m, and the rails account for 0.5 KN/m. Consequently, the cumulative force on the bridge deck (F(deck)) is (39.2 + 0.5) × 48.63 = 1903.611 KN.

4. Vertical train load: According to the Chinese railway bridge design specifications and related research results^41,42, combined with the traditional train coal transport mode used in the peak mining area, the “medium–live load” shown in Fig. 9 is chosen to represent the vertical load generated when the train passes through the railway bridge:

Fig. 9.

Chinese railway standard live load.

to represent the vertical load exerted by trains traversing the railway bridge. This load is quantified as F(train) = 220 × 5 + 30 × 92 + 11.13 × 80 = 4750.4 KN.

Considering that the fill thickness of the bridge is less than 1 m, a dynamic coefficient is incorporated into the total load calculation:

3

Within α =4(1-h) ≤ 2, where h = 0.5 m and L = 48.63 m. This results in a dynamic coefficient of 1 + μ =1.15. Thus, the total load (F(total)) is calculated as (F(ballast) + F(deck) + F(train)) × 1.15 = 10448.338 KN. When the vertical load of the train is uniformly distributed across the bridge deck, it yields a uniform pressure (P(uniform)) of 42.971 KPa.

Analysis and improvement

Analysis of simulated results

Since this paper uses the PIM-predicted results of subsidence at the bottom of the loess layer as the model boundary constraints, during the simulation process, the bottom cross-section subsidence of the loess layer is transferred from the bottom of the model to the ground surface, which leads to the results of the ground surface subsidence. it is necessary to compare the simulated and measured results of surface settlement and the simulation results of the bridge can be proved to be valid only if the surface simulation results are verified to be correct. For the two periods of surface data acquired in December 2015 and April 2018, the measured surface settlement contour data are obtained via the kriging interpolation technique. Moreover, the simulated surface settlement results are extracted. Now, the settlement contours near the bridges in the two results are locally enlarged, and the bridge locations are marked in the measured contours (Fig. 10a) and compared with the simulated surface settlement results (Fig. 10b).

Fig. 10.

Comparison of the measured and modelled surface settlements.

The measured contours show that the surface settlement near the bridge is between − 0.1 m and − 0.5 m, that the surface settlement near the north abutment is greater, between − 0.3 m and − 0.5 m, and that a small part of the north side of the bridge is greater than − 0.5 m. In addition, in the location slightly farther north of the bridge, the surface settlement is close to the range of -0.5 m and − 0.9 m, whereas the simulation results show that the surface settlement near the bridge is greater than the measured value. The simulation results show that the surface settlement near the bridge is between − 0.14 m and − 0.41 m, that the surface settlement on the north side is greater, and that between − 0.38 m and − 0.41 m, the simulated settlement is obviously smaller than the measured settlement. Thus, a simulation error exists in the construction model in this paper, and extraction of the settlement results of the corresponding ground sampling points in the simulation and the measured results and their comparison is necessary to determine the type of error.

Since the river and the surrounding terrain below the bridge are restored during the model construction process, extracting the surface points near the bridge for validation can minimise the effect of model construction on the simulation results and can also reflect the settlement state of the bridge to a certain extent. Along the axis of the bridge, 20 surface points are extracted from the surface below and near the bridge, the simulated and measured results of the settlement at the sampling points are recorded, and the difference from the simulated settlement to the measured settlement results is used to determine the simulated true error of each sampling point. The above results are recorded in Table 3:

Table 3.

Settlement results and simulation errors of sampling points (m).

Sampling point number	1	2	3	4	5	6	7	8	9	10
Actual settlement	-0.499	-0.413	-0.444	-0.440	-0.411	-0.438	-0.517	-0.557	-0.531	-0.631
Simulated settlement	-0.381	-0.283	-0.286	-0.285	-0.323	-0.358	-0.342	-0.403	-0.353	-0.402
Simulation error	0.118	0.130	0.158	0.155	0.088	0.079	0.174	0.153	0.178	0.229
Relative error	-23.6	-31.4%	-35.5%	-35.2%	-21.3%	-18.1%	-21.5%	-20.8%	-24.2	-36.3%

Sampling point number	11	12	13	14	15	16	17	18	19	20
Actual settlement	-0.522	-0.583	-0.400	-0.431	-0.335	-0.572	-0.457	-0.403	-0.400	-0.402
Simulated settlement	-0.340	-0.405	-0.309	-0.294	-0.271	-0.386	-0.306	-0.303	-0.335	-0.300
Simulation error	0.183	0.178	0.091	0.136	0.065	0.186	0.151	0.100	0.065	0.102
Relative error	-26.9	-30.6%	-22.8	-31.7%	-43.2%	-21.6%	-33.0%	-24.9%	-28.8%	-25.3%

As shown in Table 3, most of the sampling point simulation errors are distributed between 0.1 m and 0.2 m, and the calculated simulation root mean square error (RMSE) is 0.132 m. All the sampling point simulation relative error values are more than 10% of the measured settlement. In the simulation and prediction work, the relative error value is generally controlled to be less than 10%^43,44, so the results cannot meet the application requirements. The error values of each sampling point indicates that a systematic error exists in the simulated settlement results of the model. Therefore, further analysis of the reasons for the error and corrections are necessary.

Model amendment

Because of the large deviation between the simulation results and the measured results, corrections to the model are necessary, and the reason for this deviation lies in the mechanical parameters. In geological exploration, local rock samples are usually obtained through drilling, and the mechanical parameters of the samples are obtained through experiments, which are used to define the mechanical properties of the rock formation in subsequent numerical simulations of the mine. However, unlike man-made components such as blocks and steel structures, rock formation is a naturally occurring and complex phenomenon, with different porosities and water contents in each region of the same rock formation. Large-scale rock formations do not have a single geotechnical property, and the underrepresentation of rock blocks is due to the above differences in scale. In actual simulation work, the use of rock mass parameters to simulate the rock mass will inevitably create differences between the simulation parameters and the actual situation, resulting in excessive attenuation of the displacement at the bottom of the model in the process of conduction to the ground, which ultimately leads to the simulation results in which the surface settlement is smaller than expected.

In this work, to define the mechanical properties of the soil body, a total of six parameters are listed in Table 1, namely, the bulk modulus E, shear modulus G, cohesion c, angle of internal friction φ, tensile strength, and density ρ. Each parameter reflects the deformation characteristics of the soil body when it is subjected to force to varying degrees, i.e., stiffness attributes, and to determine the adjustment scheme of the model parameters, a sensitivity analysis of the sensitivity of each parameter to the settlement of the soil body is necessary. In this work, control variable means are used to reduce the values of each parameter by a fixed proportion of 10%, and the trend of the ground simulation settlement relative to the value of the parameter is recorded. Since the established model mechanical parameters act on the global soil body, changing the parameters has a similar effect on the soil body at various points near the bridge. Any ground point near the bridge is chosen to record the changes in the settlement of the ground sampling points during the period of 10-50% (i.e., 100%*para-50%para) reduction in each parameter, and the sensitivity curves of each parameter are shown in Fig. 11:

Fig. 11.

Comparison of sensitivity analysis curves for each parameter.

Figure 11 shows that the bulk modulus parameter has a greater influence on soil settlement, and the other parameters have similar sensitivities, but none of them can have a significant effect on soil settlement. The bulk modulus describes the stiffness of the soil body to stress changes, reflecting the deformation characteristics of the soil body when it is subjected to force, and the value of the bulk modulus directly affects the stiffness of the soil body. A larger bulk modulus implies that the response of the soil body to external stress is stiffer, i.e., it has a stronger resistance to external forces and, relatively, a smaller deformation capacity. The effect of the bulk modulus on the soil stiffness is more pronounced in the simulation of extensive surface settlement. The other parameters affect the strength and stability of the soil body but have relatively small effects on the soil hardness. Therefore, this paper corrects the model by discounting the bulk modulus in Table 2.

In this study, the original volumetric modulus is scaled down by a fixed ratio of 10%, and the remaining parameters are kept unchanged. Thus, the results of the settlement of the sampling points in each simulation and the error in the simulation are recorded until the results meet the requirements of mining subsidence accuracy. In the trial calculation, the settlement of each sampling point shows a synchronous trend; that is, a reduction in the soil bulk modulus increases only the overall subsidence of the ground and does not change the original force and deformation characteristics of the soil, so new errors are not produced. When the volume modulus is reduced to 30% of the original volume modulus (30%*BM), the simulated settlement value is the closest to the measured result. The simulated surface settlement contour is shown in Fig. 12, and the coordinates of the sampling points and the simulation error are shown in Table 4.

Fig. 12.

Contour lines of surface subsidence at 30% bulk modulus.

Table 4.

Contour line of surface settlement at 0.3 times bulk modulus (m).

Sampling point number	1	2	3	4	5	6	7	8	9	10
Simulated settlement	-0.452	-0.423	-0.423	-0.463	-0.393	-0.379	-0.463	-0.466	-0.493	-0.502
Simulation error	0.047	-0.010	0.021	-0.023	0.018	0.058	0.024	0.031	0.038	0.129
Relative error	-9.5%	-2.5%	-4.7	-5.2%	-4.3	-13.3%	-4.8	-6.2%	-7.2	-20.5%

Sampling point number	11	12	13	14	15	16	17	18	19	20
Simulated settlement	-0.500	-0.495	-0.409	-0.394	-0.331	-0.476	-0.446	-0.413	-0.405	-0.400
Simulation error	-0.007	0.088	-0.009	0.037	0.005	0.016	0.011	-0.010	-0.005	0.002
Relative error	-1.5	-15.1%	-2.2	-8.5%	-1.4	-3.3%	-2.3	-2.4%	-1.2%	-0.4%

As shown in Fig. 10, when 30% of the original bulk modulus is used, the surface settlement around the bridge is between − 0.15 m and − 0.5 m, which is consistent with the measured surface settlement range shown in Fig. 8a. At this time, the simulation error of the sampling point is 0.043 m, with the exception of anomalies 6, 10 and 12, the simulation of the relative error of the sampling point is controlled to be less than 10% of the measured settlement, and the average value of the relative error is -5.8%, which is consistent with the application requirements of mine simulation and prediction. The above results prove that changing the bulk modulus can reduce the simulation error, that the correction scheme proposed in this paper significantly improves the simulation accuracy of surface settlement, and that the model can be used for further quantitative analysis.

Results and predictions

Bridge settlement analysis

After the correct results of the surface analysis are confirmed, to further verify the accuracy of the model for bridge analysis and compare the difference between the simulated settlements and measured settlements of the bridge structure, the simulation results of some characteristic points of the bridge abutment and superstructure were extracted at 0.3 times the bulk modulus and compared with the measured results.

For the two-phase ground scan data, the first-phase data collected in December 2015 are used as the baseline, the second-phase data collected in April 2018 are subjected to iterative ICP alignment with the first-phase data^45,46, the two-phase point clouds of the north side pier #I and the south side piers #II shown in Fig. 3 are extracted, and the centre of gravity settlement of the two piers is calculated after resampling with an interval of 0.01 m^47,48. Thereafter, the difference between the simulated settlement and the measured settlement result is used to obtain the difference between the measured deformation and the simulated strain difference, which is recorded as the simulation error of the bridge pier displacement. The above results are recorded in Table 5:

Table 5.

Bridge pier settlement and simulation error (m).

	#I pier	#II pier
Actual settlement measurement	-0.473	-0.464
Simulated settlement	-0.451	-0.442
Simulation error	0.032	0.32
Relative error	-0.07	-0.07

When scaled down to 0.3 times the original bulk modulus, the north abutment sinks 0.451 m, and the south abutment sinks 0.442 m. The simulation results are high, the measured results are 90%, and the relative error is controlled within 10%, which is consistent with the requirements of simulation and prediction accuracy for bridges in mining areas.

For the bridge superstructure, this study extracts the simulated settlement results of the arch steel structure and some characteristic points on the bridge facade and uses the measured results of the corresponding points for verification. The steel structure and façade feature points adopt the numbering rules shown in Fig. 13 below: a-f for the steel structure bottom edge end point number; A-F for the steel structure outside the arch ring bottom inflection point number; and each arch ring bottom feature point corresponds to the adjacent steel structure end point number in the arch ring. The west side arch steel structure points correspond to the same side points a1-f1, and the bridge deck points correspond to A1-F1; the east side arch steel structure points correspond to a2-f2, and the bridge deck points correspond to A2-F2.

Fig. 13.

Bridge superstructure and feature point numbering.

After the reduction to 0.3 times the original bulk modulus, the structural point settlements identified in Fig. 13 are extracted from the simulation results. The settlement simulation errors at the end points of the western arch steel structure and the inflection points along the arch are represented by solid dots and hollow dots, respectively, and the simulation errors of each point mode settlement are connected with a dotted line, as shown in Fig. 14a. The settlement simulation errors at the end points of the eastern arch steel structure and the inflection points along the arch are represented by solid triangular points and hollow triangular points, respectively, and the simulation errors of each point mode settlement are connected with a dotted line, as shown in Fig. 14b. Moreover, solid triangular points and hollow triangular points are used to represent the settlement simulation errors at the end points of the east arch steel structure and the inflection points along the arch, respectively, and the modal settlement simulation errors at each point are connected with the dotted lines in Fig. 14b.

Fig. 14.

Simulation error of steel structure endpoints and arch edge inflection points.

Since the model constructs a more detailed model of the bridge, the ground under the bridge and the river channel, the simulation results of the bridge structure are closer to the actual measurement results than some ground sampling points that are far from the bridge. The relative error of the simulation of the main structure of the bridge is controlled within 10%, which is consistent with the requirements for the application of the prediction in the mining area, and the average value of the relative error is close to 5%, which is highly accurate. Combined with the results of the surface analysis above, the modification scheme proposed in this paper significantly improves the accuracy of the model calculation and proves that the fineness of the model has a significant effect on the simulation results. In addition, the simulation error of the sampling point of arch No. 3 is larger than that of the other parts of the arch, and this phenomenon occurs in each simulation result. Thus, the safety status of the structure around arch No. 3 and the internal structure is the recommended focus in follow-up monitoring work.

Prediction of bridge damage

As shown in Fig. 1, the monitoring mine 08203 coal face was not fully mined in April 2018 and ended mining on 30 September of the same year. With the subsequent preparation for mining the 08202 coal face, as the coal face is located under the bridge, its mining work will have a greater impact on the bridge. Therefore, the damage state of the bridge is predicted after coal face 08202 is fully mined to provide a reference for the mining of the coal face and the operation of the bridge. The PIM is used to obtain the estimated settlement data of the bottom cross-section at 65 m below the ground surface after the full mining of the 08202 coal face, and the revised model is used to predict the bridge settlement after the full mining of the three coal faces. The settlement results of the sampling points are extracted from Fig. 13, the anomalous points e₁, e₂, f₂, and E₁, E₂, and F₂ are excluded, and the settlement results of the remaining sampling points are shown in Table 6.

Table 6.

Prediction results of the settlement of the bridge structure points (m).

West sampling point	#1 arch				#2 arch			#3 arch
	a1	b1	A1	B1	c1	d1	C1	D1	f1	F1
Simulated settlement	-1.437	-1.347	-1.438	-1.350	-1.319	-1.217	-1.325	-1.222	-1.093	-1.095

East sampling point	#1 arch				#2 arch			#3 arch
	a1	b1	A1	B1	c1	d1	C1	D1	f1	F1
Simulated settlement	-1.412	-1.327	-1.410	-1.327	-1.300	-1.197	-1.301	-1.200	-1.068	-1.067

The simulation results show that up to full mining of the 08202 mine, the mine surface has already experienced a large degree of subsidence relative to December 2015 because of the excavation of the 08203 and 08202 underground coal faces. The maximum subsidence of the surface around the bridge is approximately 1.5 m, whereas the maximum settlement of the bridge structure is located at the north abutment, with a settlement of 1.52 m. difference in settlement between the north end and south end of the bridge has further increased. The bridge has not been damaged, and the structure of the bridge is more dangerous, with the bridge deck tilted and torn and a large amount of stress occurring at the south abutment and at the bottom of the No. 2 arch, the bridge may be damaged from the south side in the next stage. After the simulation results were submitted to the mine, the mine designed a corresponding maintenance programme in accordance with the predicted results. According to the follow-up inspections and feedback content, the maintenance programme ensured the normal operation of the railway, and the actual situation of bridge damage essentially coincided with the predicted results, proving that the programme provided by this study can be promoted and applied in similar mining work.

According to the above description, in large-scale surface deformation monitoring work involving mine buildings and structures, it is possible to use the PIM to predict the bottom cross-section settlement at a certain depth below the surface, simulate the surface settlement via the predicted results, obtain the measured surface settlement in the same period, and then use the measured surface data to calibrate the boundary conditions of the model and the mechanical parameters. Because the corrected model can be applied to the deformation analysis of buildings and structures in the next stage, there is no need to monitor the buildings and structures separately in the next stage, which can reduce monitoring costs. Moreover, the simulation results can also be used to check the anomalies in the measured data and correct the roughness in the measured data.

Conclusion

In this study, a prediction method based on numerical simulation is proposed for the deformation of bridges in mining areas under the influence of underground mining. Through the systematic analysis of the railway bridge in the Fengfeng mine area of Handan, Hebei, we established a finite element model of bridge‒soil interaction and used the settlement data of the 65 m bottom section below the ground surface predicted by the PIM as the boundary conditions of the model. We compared the measured results of surface settlement in the same period to verify the accuracy of the simulation results and reasonably adjusted the parameters of the model after determining that a systematic error existed in the ground simulation results relative to the actual measurements. After determining the systematic error of the ground simulation results relative to the actual measurement, the model parameters were reasonably adjusted so that the relative error of the ground settlement simulation was controlled within 10%, which improved the simulation accuracy of the model. The simulation results show that by precisely adjusting the bulk modulus of the soil model, we can significantly improve the accuracy of bridge deformation prediction so that the relative error between the simulation results and the measured data is controlled within 10%.

The study further confirms that the parameter-optimised model is able to simulate the settlement values of bridges accurately and control the relative error of the bridge settlement simulation within 10% The study is also able to predict the settlement trends of bridges effectively, which provides reliable technical support for safety assessment and maintenance decisions for bridges in mining areas. In addition, the methodology of this study is not only applicable to this case but also has the potential to be extended to the deformation analysis of buildings and structures in other mining areas. By predicting surface deformation and using it as a model boundary condition, the deformation and damage state of bridges during the mining process can be identified in advance, thus preventing potential structural hazards and risks.

Overall, this study improves the science and accuracy of deformation prediction for bridges in mining areas and provides new technical means for safety monitoring and risk management of surface buildings and structures in mining areas. However, this study also has several limitations. First, setting boundary conditions relies on probabilistic integral inversion parameters and projected results, and the parameter adjustment of the model relies on the availability of measured data, which may be difficult to obtain in some cases. Second, the generalisability of the model has not been fully validated for different types of geological conditions and bridge structures. Future work will explore the applicability of the model parameters under different geological conditions and further optimise the model in conjunction with more measured data for application in a wider range of engineering practices.

Author contributions

Chi Zhang (first author, chzhang@whu.edu.cn): Conceptualization, Methodology, Data acquisition, Visualization, Writing- Original draft preparation.Kan Wu (wukan6899@263.net): Resources, Project administration, ValidationShengxiang Huang (Corresponding author, shengxhuang@163.com): Data curation, Writing- Reviewing and EditingLingai Li(lla2019@whu.edu.cn): writing—review and editingXiaokang Rao(xiaokangrao@whu.edu.cn): writing—review and editing.

Data availability

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.