Introduction

As coal resources are progressively depleted at shallower depths, extraction from deeper coal seams has become increasingly prevalent. Following the mining of such deep seams1,2,3, significant stress redistribution occurs within the roof and floor strata surrounding the working face, resulting in substantial differences compared to the pre-mining stress conditions4,5,6,7. Understanding the stress distribution (both vertical and shear stresses) and mining-induced failure morphology within the floor of these deep-seam working faces is therefore critical8,9,10,11. This understanding is crucial for assessing potential impacts on underlying coal seams and pinpointing stress concentration zones. Crucially, as mining depth increases, the influence of key factors becomes amplified12,13,14,15,16. The stress distribution within the floor of steeply inclined seams and its consequent failure characteristics diverge markedly from those observed in near-horizontal seams at moderate depths. Furthermore, the stability of deep seams becomes increasingly sensitive to variables such as depth, seam dip angle, and panel width17,18,19.

Recent years have witnessed extensive research on the mechanical challenges in coal seam mining. Liu et al20,21.employed a combined theoretical analysis, numerical simulation, and field measurements to examine strata movement along the strike in inclined seams, identifying a fundamental relationship between coal pillar stress and seam dip angle. Lu and Zhou et al22,23,24,25.applied limit equilibrium theory to characterize the abutment stress distribution across coal pillars of varying widths, providing a theoretical foundation for stress patterns in the underlying strata. Meng and Lu et al26,27,28,29,30.analyzed stress distribution within the floor strata beneath coal pillars by simplifying the concentrated pillar load and applying elastic theory. Wei and Wang et al31,32,33,34,35. developed a more realistic coal pillar loading model to investigate stress transfer and distribution patterns in the floor beneath pillars of different widths. Li and Zhang et al36,37,38,39.used physical simulations incorporating the characteristic ‘inverted trapezoid’ stress field formed by pillars after upper seam extraction to define the affected stress zone in the floor. Zhang and Zhao et al40,41,42,43.derived analytical formulas to calculate stress distributions in floor strata under mining influence, specifically evaluating the effects of pillar size on the extent of stress distribution. Wang et al44,45,46.utilized numerical simulations to quantify the disturbance range of floor stress during protective seam mining. Zhu et al47,48.employed analog materials modeling and numerical simulations to study stress evolution in the floor during close-distance coal seam extraction, revealing a periodic fluctuation pattern in stress distribution. Li et al49,50,51. investigated the optimization of drawing parameters for top coal caving, the fracture evolution law of overburden strata controlled by multiple key strata, and the rational layout of gate roads in sectional coal pillars under the mining conditions of deep, hard coal seams, employing an integrated methodology that incorporates PFC and FLAC3D numerical simulations, physical similarity modeling, and in-situ monitoring.

The transmission of concentrated stress from extracted upper seams to underlying seams constitutes a critical input for mine design and hazard prevention in multi-seam coal mining. While substantial research exists, investigations addressing how coal seam dip angle governs the distribution of pillar-induced concentrated stress at floor strata—particularly stress propagation mechanisms beneath remnant pillars in deeper, inclined seams—remain scarce.

This research investigates typical deep, inclined coal seams utilizing a tripartite methodology, which includes physical similarity simulation experiments replicating mining conditions, FLAC3D numerical modeling of field-calibrated, dip-specific mining scenarios, and theoretical analysis based on elastic mechanics. This integrated approach facilitates a systematic examination of dip angle-dependent stress variation patterns within the deep coal seam floor strata.

Materials and methods

Experimental approach and workflow

To investigate dip-angle dependent stress propagation within floor strata of deep coal mining faces, a series of laboratory experiments was implemented (Fig. 1). The methodology integrates three complementary approaches: physical simulations replicating deep inclined coal seams, FLAC3D numerical modeling calibrated with experimental data and theoretical analysis based on semi-infinite elastic space mechanics. This integrated approach enables a comprehensive examination of dip-angle-governed stress distribution patterns and damage initiation mechanisms in deep coal seam floors.

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Schematic diagram showing the experimental workflow and the experimental program design.

Design of physical similarity simulation experiment

Physical similarity simulation

A generalized model of Shanghai temple coal mine was employed in this study, as plotted in Fig. 2, this model has a depth of 800 m, which is a typical deep coal mine. Therefore, in order to effectively simulate the mechanical properties of the prototype, a load was applied to the upper part of the similar model experiment. The dimensions of the studied physical model are 1300 mm× 900 mm× 100 mm with a geometric scale of 1:200.

As Fig. 2 illustrates, the stratigraphic architecture comprises ten discrete layers systematically grouped into five lithological units: S1 mudstone, S2 coarse-grained sandstone, S3 coal seam, S4 fine-grained sandstone, and S5 siltstone, all of which maintain a uniform 22° dip angle.

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Similar simulation tests and geological stratification map.

Similar materials and similarity ratios

The model maintains strict adherence to geometric, static, and temporal similitude principles relative to the prototype. Defining length (L), mass (M), and time (T) as fundamental dimensions, scaling ratios were established with dimensional ratio = 1:200 and density ratio = 1:1.5. Given the invariant scaling of gravitational acceleration (1:1), temporal scaling derives as 14.14:1 according to similitude laws and dimensional analysis (Table 1). Layer-specific geomechanical parameters are documented in Table 2. Simulated strata were formulated from sand-calcium carbonate-gypsum composites, with mixture proportions for five stratigraphic types optimized through orthogonal experimentation to satisfy scaling constraints for elastic modulus and compressive strength. Weight ratios (sand: CaCO3: gypsum) for units S#1–S#5 are: 5:0.6:0.4, 7:0.5:0.5, 6:0.2:0.8, 9:0.8:0.2 and 3:0.5:0.5 respectively.

Table 1 Similarity scaling ratios.
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Table 2 Geomechanical parameters of coal-bearing strata.
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Real-time stress monitoring system

Stress monitoring in the similarity simulation experiment was conducted in real time using a system comprising flexible membrane pressure sensors (range: 0–0.1 MPa) and a DH3816N static strain data acquisition instrument, as illustrated in Fig. 5. During the experiment, the sensors were installed horizontally across 29 monitoring points within the deep coal seam strata. As shown in Fig. 2, the monitoring points were primarily located in the S#3 coal seam and the adjacent S#2 coarse sandstone layer, exhibiting a non-uniform distribution. Most sensors were positioned in the middle and lower sections of the coal seam, as well as near the interfaces between the coal seam and the surrounding sandstone roof and floor. Excavation will result in a concentrated area of principal stress distribution in the lower right corner of the inclined coal seam, and further estimation of the impact range of stress fluctuations may occur along the direction of coal seam inclination. The layout aims to effectively capture the dynamic stress redistribution within the coal seam and surrounding rock during mining, with particular focus on elucidating the evolution mechanisms of asymmetric stress fields and potential failure zones under conditions of inclined coal seams.

Numerical model setup

Numerical simulations were implemented by using the FLAC3D, with the model geometry and mechanical parameters defined by Fig. 2. Figure 3 depicts the final stratigraphic configuration and mesh discretization.

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Numerical simulation modeling and hierarchical diagram.

The numerical model enforced boundary conditions consistent with the physical simulation: usual displacement constraints on lateral boundaries, horizontal displacement constraints at the base, and exclusive application of overburden stress at the upper boundary without supplementary loading.

The mechanical properties of all stratigraphic units in the numerical model maintained strict parametric congruence with the physical similarity simulation (Table 2) to compute elastoplastic states. During simulation, the sole perturbation implemented was progressive excavation along both lateral flanks of the Seam #3, advancing 120 m from the working face. All other parameters remained invariant, with persistent usual displacement constraints enforced on lateral boundaries.

Analysis of test results

Reliability verification of numerical simulation

The experimental protocol involves bidirectional excavation sequences: initial extraction from the upper section of the remnant coal pillar, followed by lower-section extraction. Each excavation step is carried out in increments of 5 centimeters and is systematically executed in six stages to simulate the continuous mining process. Data acquisition followed model stabilization. Comparative results from physical similarity simulations and numerical modeling are presented in Fig. 4. Figure 4(a) documents failure morphology in the physical model, while Fig. 4(b) illustrates the post-excavation plastic failure zone in the Seam #3 numerical simulation. Cross-validation reveals a phenomenological correspondence—namely, the arc-shaped roof collapse observed during physical excavation (Fig. 4(a)) aligns mechanistically with the computational dome-shaped plastic failure (Fig. 4(b)).

Figure 5 further details the stress distribution patterns following the excavation of Seam #3. Both methodologies identify consistent mechanical responses in the floor strata: principal stress redistribution exhibits (i) central concentration with peripheral dispersion, and (ii) depth-dependent stress migration. Specifically, beneath the protective pillar, with maximum principal stress decreases with depth and migrates from the right margin toward the central axis with increasing depth.

Quantitative validation was performed across 29 monitoring points (physical model). Measured stress values demonstrate < 4.2% deviation from computational results (Fig. 5), with potential minor discrepancies attributable to sensor orientation variables during physical testing. This inter-methodological consistency confirms simulation reliability.

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Diagrams of the failure zone in similar simulation tests (a) and the plastic failure zone in numerical simulation (b).

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Comparison chart of stress changes between similar and numerical simulation.

The stress transmission law of the base plate under different inclination angles

Figure 6 shows the vertical stress distribution and stress profile below the residual coal pillar under multiple numerical simulation experiments at inclinations of 15 °, 22 °, 30 °, 45 °, and 60 °. Comparative analysis of Fig. 6(a), (b), (e) demonstrate that for shallow-dip coal seams (≤ 30°), stress peaks consistently occur at the down-dip end. With increasing depth, the stress concentration zone progressively migrates from the down-dip side toward the central up-dip region, exhibiting quadratic decay kinetics.

Conversely, Fig. 6c and d, and 6f reveal distinct characteristics in steeply inclined seams (≥ 45°): stress peaks remain persistently localized at the down-dip end without spatial migration despite depth progression. These peaks exhibit systematically lower magnitudes than those in shallow-dip seams, with stress attenuation following uniform-gradient decay profiles.

Fig. 6
Fig. 6The alternative text for this image may have been generated using AI.Fig. 6The alternative text for this image may have been generated using AI.
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Stress cloud maps of the coal seam floor at inclinations of 15°(a), 30°(b), 45°(c) and 60°(d) and stress lateral line maps at inclinations of 15–30°(e) and 45–60°(f).

Stress distribution solution of the bottom plate in semi-wireless space

Mechanical model of floor strata beneath deep inclined remnant coal pillars

Post-extraction in deep inclined coal seams induces bilateral abutment pressures laterally adjacent to isolated pillars along the working face. Crucially, unlike conventional mining scenarios where overburden stresses redistribute to surrounding strata, these stresses concentrate predominantly within remnant pillars, resulting in significant stress concentrations within the underlying floor strata. Figure 7a quantifies this phenomenon through the inclined abutment pressure distribution along a 22°-dipping remnant pillar. The inherent differential overburden depths across steeply inclined workings—distinct from near-horizontal seams—produce substantial variations in peak magnitude and spatial distribution of lateral support stresses within floor strata.

To mechanistically analyze this behavior, floor strata along the dip direction were idealized as a semi-infinite elastic continuum. Within this framework, stress distribution patterns and failure characteristics beneath deep inclined remnant pillars were systematically investigated. The lateral abutment pressures (Fig. 7a) were resolved into transverse and longitudinal components relative to the floor plane, then modeled as linear distributed loads applied to the semi-infinite continuum. This formulation establishes a rigorous mechanical model for floor stress beneath deep inclined remnant pillars, with Cartesian coordinates (Fig. 7b) defining key parameters: seam dip angle (β), excavation depth at panel intersection (H), and unit weight of floor strata (γ).

Figure 7b illustrates a linearly distributed load with inflection points projecting vertically onto the x-axis at positions o, a, b, c, d, e, f, and g, where the distances s₁–s₇ denote the intervals between adjacent projections. Concentration coefficients k₁, k₂, k₃, and k₄ for transverse abutment pressures satisfy k₁> k₂> 1 > k₃> k₄. Load segments ①②③④⑤⑥⑦ represent transverse components longitudinal to the floor plane, inducing compressive failure in strata, while segments ⑧⑨⑩⑪⑫⑬⑭ correspond to perpendicular components parallel to the floor, generating shear failure through slippage. The caving zone imposes transverse load \(\:\gamma\:{H}_{m}\text{cos}\beta\:\)(Segment ④) and longitudinal load \(\:\gamma\:{H}_{m}\text{sin}\beta\:\) (Segment ⑪). Stress evolution follows defined trajectories: transverse load increases linearly from \(\:{K}_{4}\gamma\:H\text{cos}\beta\:\:\)at 0 to \(\:{K}_{2}\gamma\:\left(H+{x}_{a}\text{sin}\beta\:\right)\text{cos}\beta\:\) at a, then decreases to in-situ stress \(\:\gamma\:\left(H+{x}_{b}\text{sin}\beta\:\right)\text{cos}\beta\:\) at b. Symmetric behavior occurs along the lower edge transverse load increases from \(\:{K}_{3}\gamma\:\left(H+{x}_{d}\text{sin}\beta\:\right)\text{cos}\beta\:\) at d to\(\:{K}_{1}\gamma\:\left(H+{x}_{e}\text{sin}\beta\:\right)\text{cos}\beta\:\) at e, then decreases to \(\:\gamma\:\left(H+{x}_{f}\text{sin}\beta\:\right)\text{cos}\beta\:\) at f. The longitudinal load exhibits a piecewise-linear trajectory along the excavation periphery: initiating at the upper edge as \(\:{K}_{4}\gamma\:H\text{sin}\beta\:\) at point o, it ascends to \(\:{k}_{2}\gamma\:\left(H+{x}_{a}\text{sin}\beta\:\right)\text{sin}\beta\:\) at point a, thereafter diminishing to the pristine lithostatic value \(\:\gamma\:\left(H+{x}_{b}\text{sin}\beta\:\right)\text{sin}\beta\:\) at point b; analogously, at the lower edge the stress climbs from \(\:{k}_{3}\gamma\:\left(H+{x}_{d}\text{sin}\beta\:\right)\text{sin}\beta\:\) at point d to \(\:{k}_{1}\gamma\:\left(H+{x}_{e}\text{sin}\beta\:\right)\text{sin}\beta\:\) at point e, before relaxing once more to the unperturbed geostatic state \(\:\gamma\:\left(H+{x}_{f}\text{sin}\beta\:\right)\text{sin}\beta\:\) at point f.

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The stress distribution of the lateral support of the deeply inclined remaining coal pillar (a) and the stress mechanical model of the base plate (b).

The stress distribution of the bottom plate of the deeply inclined remaining coal pillar

Based on the elastic theory of homogeneous isotropic semi-infinite continua (Timoshenko & Woinowsky-Krieger,1959), analytical expressions for stresses \(\:{\sigma\:}_{y}\) at arbitrary points within inclined working face floor strata along the coal seam dip direction are derived as given by Eq. (1), respectively.

$$\:{\sigma\:}_{y}={{\upsigma\:}}_{y1}+{\sigma\:}_{y2}+{\sigma\:}_{y3}+{\sigma\:}_{y4}+{\sigma\:}_{y5}+{\sigma\:}_{y6}+{\sigma\:}_{y7}+{\sigma\:}_{y8}+{\sigma\:}_{y9}+{\sigma\:}_{y10}+{\sigma\:}_{y11}+{\sigma\:}_{y12}+{\sigma\:}_{y13}+{\sigma\:}_{y14}$$
$$\:=\frac{2}{\pi\:}{\int\:}_{-\left({s}_{1}+{s}_{2}+{s}_{3}\right)}^{-\left({s}_{1}+{s}_{2}\right)}\frac{\gamma\:{H}_{m1}cos\beta\:{y}^{3}}{{\left[{\left(x-\epsilon\:\right)}^{2}+{y}^{2}\right]}^{2}}{d}_{\epsilon\:}+\frac{2}{\pi\:}{\int\:}_{-\left({s}_{1}+{s}_{2}\right)}^{-{s}_{1}}\frac{\left(A+\epsilon\:B\right){y}^{3}}{{\left[{\left(x-\epsilon\:\right)}^{2}+{y}^{2}\right]}^{2}}{d}_{\epsilon\:}+\frac{2}{\pi\:}{\int\:}_{-{s}_{1}}^{0}\frac{\left(C+\epsilon\:D\right){y}^{3}}{{\left[{\left(x-\epsilon\:\right)}^{2}+{y}^{2}\right]}^{2}}{d}_{\epsilon\:}+\frac{2}{\pi\:}{\int\:}_{0}^{{s}_{4}}\frac{\left(q+\epsilon\:p\right){y}^{3}}{{\left[{\left(x-\epsilon\:\right)}^{2}+{y}^{2}\right]}^{2}}{d}_{\epsilon\:}$$
$$\:+\frac{2}{\pi\:}{\int\:}_{{s}_{4}}^{{s}_{4+}{s}_{5}}\frac{\left(E+\epsilon\:F\right){y}^{3}}{{\left[{\left(x-\epsilon\:\right)}^{2}+{y}^{2}\right]}^{2}}{d}_{\epsilon\:}+\frac{2}{\pi\:}{\int\:}_{{s}_{4}+{s}_{5}}^{{s}_{4+}{s}_{5}+{s}_{6}}\frac{\left(G+\epsilon\:I\right){y}^{3}}{{\left[{\left(x-\epsilon\:\right)}^{2}+{y}^{2}\right]}^{2}}{d}_{\epsilon\:}$$
$$+\frac{2}{\pi\:}{\int\:}_{{s}_{4}+{s}_{5}+{s}_{6}}^{{s}_{4}+{s}_{5+}{s}_{6}+{s}_{7}}\frac{\gamma\:{H}_{m2}cos\beta\:{y}^{3}}{{\left[{\left(x-\epsilon\:\right)}^{2}+{y}^{2}\right]}^{2}}{d}_{\epsilon\:}+\frac{2}{\pi\:}{\int\:}_{-\left({s}_{1}+{s}_{2}+{s}_{3}\right)}^{-\left({s}_{1}+{s}_{2}\right)}\frac{\gamma\:{H}_{m1}sin\beta\:\left(x-\epsilon\:\right){y}^{2}}{{\left[{\left(x-\epsilon\:\right)}^{2}+{y}^{2}\right]}^{2}}{d}_{\epsilon\:}$$
$$\:+\frac{2}{\pi\:}{\int\:}_{-\left({s}_{1}+{s}_{2}\right)}^{-{s}_{1}}\frac{\left(A+\epsilon\:B\right){tan\beta\:\left(x-\epsilon\:\right)y}^{2}}{{\left[{\left(x-\epsilon\:\right)}^{2}+{y}^{2}\right]}^{2}}{d}_{\epsilon\:}+\frac{2}{\pi\:}{\int\:}_{-{s}_{1}}^{0}\frac{\left(C+\epsilon\:D\right)tan\beta\:\left(x-\epsilon\:\right){y}^{2}}{{\left[{\left(x-\epsilon\:\right)}^{2}+{y}^{2}\right]}^{2}}{d}_{\epsilon\:}+\frac{2}{\pi\:}{\int\:}_{0}^{{s}_{4}}\frac{\left(q+\epsilon\:p\right){tan\beta\:\left(x-\epsilon\:\right)y}^{2}}{{\left[{\left(x-\epsilon\:\right)}^{2}+{y}^{2}\right]}^{2}}{d}_{\epsilon\:}$$
$$\:+\frac{2}{\pi\:}{\int\:}_{{s}_{4}}^{{s}_{4+}{s}_{5}}\frac{\left(E+\epsilon\:F\right){tan\beta\:\left(x-\epsilon\:\right)y}^{2}}{{\left[{\left(x-\epsilon\:\right)}^{2}+{y}^{2}\right]}^{2}}{d}_{\epsilon\:}+\frac{2}{\pi\:}{\int\:}_{{s}_{4}+{s}_{5}}^{{s}_{4+}{s}_{5}+{s}_{6}}\frac{\left(G+\epsilon\:I\right){tan\beta\:\left(x-\epsilon\:\right)y}^{2}}{{\left[{\left(x-\epsilon\:\right)}^{2}+{y}^{2}\right]}^{2}}{d}_{\epsilon\:}+\frac{2}{\pi\:}{\int\:}_{{s}_{4}+{s}_{5}+{s}_{6}}^{{s}_{4}+{s}_{5}+{s}_{6}+{s}_{7}}\frac{\gamma\:{H}_{m2}sin\beta\:\left(x-\epsilon\:\right){y}^{2}}{{\left[{\left(x-\epsilon\:\right)}^{2}+{y}^{2}\right]}^{2}}{d}_{\epsilon\:}$$
(1)

where \(\:\epsilon\:\) is the variable of integration in Eq. (1), and the other integration parameters are as follows:

$$\:q=\gamma\:Hcos\beta\:$$
$$\:p=\gamma\:sin\beta\:cos\beta\:$$
$$\:A=\frac{\left({k}_{2}{s}_{1}+{k}_{2}{s}_{2}-{s}_{1}\right)q-\left({k}_{2}{s}_{1}-{s}_{1}\right)\left({s}_{1}+{s}_{2}\right)p}{{s}_{2}}$$
$$\:B=\frac{\left({k}_{2}-1\right)q-\left({k}_{2}{s}_{1}-{s}_{1}-{s}_{2}\right)p}{{s}_{2}}$$
$$\:C={k}_{4q}$$
$$\:D=\frac{\left({k}_{4}-{k}_{2}\right)q+{k}_{2}{s}_{1}p}{{s}_{1}}$$
$$\:E=\frac{\left({k}_{3}{s}_{5}-{k}_{1}{s}_{4}+{k}_{3}{s}_{4}\right)q+\left({k}_{3}{s}_{4}{s}_{5}-{k}_{1}{s}_{4}{s}_{4}-{k}_{1}{s}_{4}{s}_{5}+{k}_{3}{s}_{4}{s}_{4}\right)p}{{s}_{5}}$$
$$\:F=\frac{\left({k}_{1}-{k}_{3}\right)q+\left({k}_{1}{s}_{4}{+{k}_{1}s}_{5}-{k}_{3}{s}_{4}\right)p}{{s}_{5}}$$
$$\:G={k}_{1}q+{k}_{1}\left({s}_{4}+{s}_{5}\right)p-\frac{\left({1-k}_{1}\right)q+\left({s}_{4}+{s}_{5}+{s}_{6}-{k}_{1}{s}_{4}-{k}_{1}{s}_{5}\right)p}{{s}_{6}}\left({s}_{4}+{s}_{5}\right)$$
$$\:I=\frac{\left({1-k}_{1}\right)q+\left({s}_{4}+{s}_{5}+{s}_{6}-{k}_{1}{s}_{4}-{k}_{1}{s}_{5}\right)p}{{s}_{6}}$$

For representative parameters (H = 800 m, Hₘ= 15 m, γ = 24 kN/m³, k₁= 2.8, k₂= 2.3, k₃= 0.32, k₄= 0.1, s₁= s₂= s3 = 3 m, s4 = 12 m, s5 = s6 = s7 =5 m), Figs. 8(a), 8(b), 8(c), 8(d) and 8(e) present contour maps of vertical stress \(\:{\sigma\:}_{y}\) distribution along the dip direction in inclined working face floor strata at seam dip angles of 15°, 22°, 30°, 45°, and 60°, with contour values denoting magnitudes relative to in-situ stress. Analysis reveals distinct distribution patterns: vertical stress contours beneath deep, inclined remnant pillars exhibit symmetric configurations at lower dip angles (≤ 30°) and transition to asymmetric patterns at steeper inclinations (≥ 45°). Notably, these contours adopt a “ladle-shaped” morphology characterized by downward-increasing magnitude gradients relative to the working face. Contour deflection angles intensify with increasing seam dip, demonstrating greater prominence and angular magnitude on the down-dip flank compared to the up-dip side.

Fig. 8
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Isoclines of vertical stress of the tilted workface floor along the tilted direction for different dip angles (a)15°, (b)22°, (c)30°, (d)45°, (e)60°.

Discussion

This study identifies two distinct stress transmission modes in floor strata beneath deep remnant coal pillars, governed fundamentally by seam dip angle. In shallow-dip seams (≤ 30°), the stress peak migrates from the down-dip side towards the central up-dip region with increasing depth, following quadratic decay kinetics. Conversely, in steep-dip seams (≥ 45°), the stress peak remains anchored at the down-dip end, exhibiting uniform-gradient decay without significant migration. This divergence is attributed to the dominant role of gravity-induced asymmetric loading in steep dips, which overwhelms the force vector inclination effect prevalent in shallower seams.

The high concordance (< 4.2% deviation) between physical simulations and numerical model robustly validates the adopted methodology. The theoretical model, derived from elastic half-space theory, successfully replicates the observed “ladle-shaped” stress contours and provides a mechanistic explanation for the dip-dependent phenomena, advancing beyond previous models that simplified pillar loading or ignored force resolution in inclined systems.

These findings address a recognized knowledge gap by elucidating the stress propagation mechanism beneath remnant pillars in deep, inclined seams. While prior research established general stress concentration concepts, this work demonstrates that dip angle is a fundamental control variable dictating the very mode of stress transmission. The results provide a critical framework for optimizing pillar design and assessing stability in deep mining, enabling more targeted hazard assessment and strategic planning for multi-seam operations. Future work should focus on incorporating layered elastoplasticity and time-dependent effects to enhance the model’s realism.

Conclusions

This study investigates stress distribution patterns in floor strata beneath deep inclined remnant coal pillars through integrated physical similarity simulations and FLAC3D numerical model, systematically analyzed stress transmission mechanisms across varying dip angles. Key findings demonstrate that:

(1) Post-extraction in deep inclined seams induces stress redistribution, resulting in arched failure patterns within roof strata bilaterally, with remnant pillars becoming peak stress zones due to overburden loading; concurrently, vertical stress within pillar floor strata exhibits gradient attenuation with increasing depth;

(2) Two characteristic stress distribution modes emerge: for shallow-dip seams (15–30°), stress peaks initially localize at the down-dip end and progressively migrate toward the central up-dip region with depth, whereas in steep-dip seams (30–60°), peaks persist at the down-dip end without spatial migration;

(3) Mechanical model reveals distinct stress decay behaviors—shallow-dip seams exhibit quadratic decay with near-floor stresses around 70 MPa and far-floor stresses between 25 and 30 MPa, while steep-dip seams show uniform decay with near-floor stresses around 45–50 MPa and far-floor stresses between 15 and 20 MPa.

This study has obtained the stress distribution characteristics in the floor strata beneath remnant coal pillars in deep inclined coal seams, providing a quantitative basis and theoretical support for the design of pillar arrangement, stability control, and floor disaster prevention. Future research will focus on the elastoplastic response in layered anisotropic strata, the spatiotemporal evolution of mining-induced stress, and hydro-mechanical coupling effects, aiming to develop high-precision prediction models and enhance forecasting and control capabilities for safe mining under complex deep geological conditions.