Development Of Plastic Zone Research Articles

Simulation and field application study of roadway remodeling load-bearing arch technology

This paper presents a numerical simulation study on the impact of material strength, thickness, and lateral pressure coefficient on the load-bearing arches used in tunnel reconstruction, as well as industrial experiments based on these arches. The study compares three different cross-sectional shapes—straight-wall semicircular arches, flat-bottom straight-wall semicircular arches, and inverted straight-wall semicircular arches—through multi-dimensional numerical simulations investigating various parameters such as material strength, thickness, and lateral pressure coefficient. We analyze the maximum load-bearing capacity of the reconstructed arches and the evolution characteristics of the plastic zones in the tunnel support structures with different cross-sectional shapes. The results indicate that among the factors affecting the ultimate load-bearing capacity of the arches, the ranking is as follows: tensile strength > thickness > lateral pressure coefficient. Additionally, the shape of the cross-section significantly influences the evolution of plastic zones during tunnel failure. The technology of reconstructing load-bearing structures in coal mine tunnels effectively addresses the deformation issues caused by the superimposition of multiple stresses in the tunnels of Daniu Coal Mine. This provides valuable insights for understanding and managing significant deformations resulting from concentrated stresses from in-situ rock, tectonic forces, and mining-induced stresses.

Stability Analysis of Borehole Walls When Drilling with Normal-Temperature Drilling Fluids in Permafrost Strata

Permafrost is a temperature-sensitive geological formation characterized by low elasticity and high plasticity. Inappropriate engineering design during borehole drilling in permafrost can result in the collapse of surrounding strata. To evaluate the stability of borehole walls, a finite element model was developed based on the inherent physical properties of permafrost. This model was utilized to investigate the thermal, stress, and plastic yield zone evolution around the borehole during drilling with normal-temperature fluids. The borehole expansion rate was employed as a quantitative measure to assess wall stability. The analysis reveals that the strata adjacent to the borehole, when drilled with normal-temperature fluids, experience thawing and yielding, with secondary stress concentrations in unthawed strata driving the progressive expansion of the plastic zone. The degree of plastic deformation diminishes with increasing distance from the borehole. Consequently, the borehole expansion rate was utilized to evaluate collapse risk under varying conditions, including permafrost thickness, depth, plastic strain thresholds, and drilling fluid densities. The findings suggest that normal-temperature drilling fluids are appropriate for thin permafrost layers, whereas for thicker permafrost, adjustments in drilling fluid density are required to ensure the stability of borehole walls due to the elevated temperatures and geostress at greater depths.

Effects of drilling parameters on stress relief performance during mining

Abstract This study investigated the effects of borehole diameter and borehole interval on the stress relief performance of coal seams during mining. A discrete element numerical model was established based on the Voronoi and block models, and the stress relief performance was investigated in terms of axial stress, vertical displacement, and plastic zone development using the control variates method. The results showed that a decrease in the distance from the workface was accompanied by an increase in the effects of mining, the stress concentration, and the vertical displacement of the roof. Coal mining had positive effects on fracture propagation in front of the workface. As the borehole interval increased, the density of boreholes in the coal seams decreased, reducing the stress concentration and fracture propagation in front of the workface. The stress relief performance of the boreholes in the coal seams was improved at an borehole interval of 8 m. However, as the borehole diameter increased, the stress concentration near the boreholes decreased, reducing the effects of the workface mining on stress relief through the boreholes. Meanwhile, increasing the borehole diameter decreased the vertical displacement of the roof but exacerbated fracture propagation in front of the workface, which improved the stress relief performance of the boreholes in the coal seams. This study provides a reference for arranging boreholes in coal seams during mining.

Predicting the dynamic response of elastic-plastic beams by dimensional analysis

Elastic-plastic beams subjected to dynamic loading are widely found in engineering. However, due to the complex coupling of geometric and material nonlinearity, there is no complete analytical solution available. In this paper, based on energy conservation and dimensional analysis, a dominant dimensionless number, the loading intensity ξ, is proposed. Including both the effects of geometry, material and load, ξ could be used to predict the significant responses of an elastic-plastic beam under uniformly pressure loading, such as energy ratio, deflection ratio and deformation mechanism. With combination of loading intensity ξ and dimensionless stiffness β, the dimensionless maximum and final deflections of elastic-plastic beams under pressure loading can be predicted directly. Dimensionless numbers of beams with different cross-sections, or under different loading forms are also analyzed and confirmed.Then, elastic effect over dynamic response of elastic-plastic beams is analyzed. For final deflection, it is found ξ>5 is the applicability of theoretical solution based on rigid, perfectly plastic (R-PP) material simplification. However, R-PP results can't be used in initial plastic hinge location and subsequent plastic zone evolution, even for very large ξ.Using these dimensionless numbers, experimental or simulation data could be expressed in a normalized and comparable form, and the physical mechanisms will be more intuitive and clearer, which is valuable for scaled dynamic tests.

Study on significance of pillar dimensions on stability of caverns in rocks with high overburden

The present study aims to examine the influence of pillar widths on the stability of caverns. Case study considering two caverns viz. Powerhouse Cavern (PHC) and Transformerhall Cavern (TC), in a major hydro-electric project in the eastern Himalayas is considered. 2D and 3D numerical analysis was carried out for w/B ratios 1.5 and 3.0 respectively. Primary aspects like major principal stress and development of plastic zone were investigated for the two pillar widths. An optimum pillar width was observed that resulted in reduced stress acting along the cavern periphery, a better stress distribution, and no overlap of plastic zones between the caverns. Further, the optimum pillar width resulted in a better stress-redistribution with progress of excavation and the in-situ stress became constant at an earlier stage of excavation. Observations from comparative analysis revealed that a pillar width nearly equal to the largest dimension or twice the width of the larger of the caverns in the group resulted in a better stability and hence can be considered as the optimum width. Furthermore, the analysis suggests that along the pillar width, maximum stress was observed at mid-height, and it is more in the vicinity of the face of the caverns.

Mechanical analysis of frost heaving of weakly cemented soft rock tunnels in cold regions

To reveal the mechanical response of non-uniform frost heave of surrounding rock in soft rock tunnel in cold region, the stress solution of frozen surrounding rock considering the temperature field was deduced by applying elastic mechanics' displacement potential theory and the superposition principle. Then the plastic zone evolution of frozen surrounding rock was clarified by using the limit equilibrium approach. Furthermore, the analytical solution is verified with the existing solutions, and proved to be reasonable. The calculation results showed that as the temperature drops, the growth rate of the frozen zone radius gradually slows down while the stress increment increases. Frost heaving effect lead to a decrease in the critical state value of the lateral pressure coefficient (λ) for tension–compression. Circumferential stresses show a significant stress drop at the frozen boundary, and radial stresses show extreme values. Frost heaving enlarges the circumferential range of the plastic zone. Under low initial vertical stress conditions (p), when λ or 1/λ decreases from 1 to 0, the plastic zone gradually increases, while its range is always in the frozen zone. However, with an increase in “p” and a decrease in temperature to a certain extent, both inner and outer plastic zones emerged.

Research on Multi-Physical Field Characteristics of Deep Coal Seam Mining Based on the Rock-Coal-Rock Model

In order to disclose the multi-physical field characteristics of the deep coal seam mining process and their dynamic evolution legislation, based on the “rock-coal-rock” model, during the mining process, the stress field, displacement field, energy field, and plastic zone evolution process are all simulated using FLAC3D6.0. The findings show that stress in the original rock is redistributed as a result of coal seam mining, creating a pressure relief zone in the middle of the goaf and advanced support pressure in the front part of the working face. The roof falls following the termination of coal seam mining. The collapsed blocks fill the middle of the goaf, playing a supporting role. The floor bulges as a new supporting pressure zone forms and builds up high elasticity. The stress reduction zone shifts from a rectangular to an inner circular distribution and an outer square as the working face’s mining distance increases and the range of the fracture field expands accordingly. In addition, a complete model was constructed to verify the correctness of the “rock-coal-rock” model. The stress, displacement, and energy curves of the overlying strata at a distance of 12 m from the bottom of the coal seam in the middle of the goaf obtained by the two methods were basically consistent. Ultimately, the findings of the numerical simulation were compared with the advanced support pressure data that were acquired on-site and they were good. This work can provide a reference for the safe mining of deep coal seams.

A 2D stability analysis of the rock surrounding underground liquified natural gas storage cavern based on COMSOL Multiphysics

Underground liquified natural gas (LNG) storage is essential in guaranteeing national energy strategic reserves, and its construction is being accelerated. The stability of surrounding rock of underground LNG storage caverns under stress-low temperature coupling effect is the key factor determining the feasibility of LNG storage. First, a mathematical model used for controlling the stress-low temperature coupling and the processes of rock damage evolution is given, followed by a 2-D numerical execution process of the mathematical model mentioned above described based on Comsol Multiphysics and Matlab code. Finally, a series of 2-D simulations are performed to study the influence of LNG storage cavern layout, burial depth, temperature and internal pressure on the stability of surrounding rocks of these underground storage caverns. The results indicate that all the factors mentioned above affect the evolution of deformation and plastic zone of surrounding rocks. The research results contribute to the engineering design of underground LNG storage caverns.

Influence of principal stress orientation on stress distribution and plastic zone evolution of rock surrounding tunnels

During the excavation of tunnels, the principal stress orientation changes, with a significant impact on the stress distribution, mechanical properties, and plastic zone evolution of rocks surrounding tunnels, causing severe deformation control, and monitoring problems in the stability of tunnels. Currently, biaxial compression tests were conducted to explore the influence of principal stress orientation on mechanical properties of rocks surrounding tunnels. The analytical solution of stress and the model of plastic zone of rocks considering the principal stress orientation and the distance from the excavation boundary were established to reveal the failure mechanism of surrounding rock under different principal stress orientations. With an increase in the angle between the principal stress orientation and the long axis of tunnel, the maximum tangential stress around the tunnel gradually changed to the minimum horizontal principal stress direction, and its value gradually increased, leading to quicker failure of surrounding rocks, and reducing the strength enhancement effect of the same section size of the tunnel. However, the increase in the angle reduced the damage range and the range of the plastic zone around the tunnel and caused the plastic zone to gradually approach the bottom and roof. The maximum depth of the plastic zone remained parallel to or nearly parallel to the minimum horizontal principal stress direction. When the principal stress orientation was kept constant, the maximum depth of the plastic zone shifted to the minimum horizontal principal stress direction with an increase in the vertical principal stress.

Instability mechanism and control technology of soft rock roadways affected by mining in karst mountain area

The deformation and damage of soft rock roadways in karst mountainous area is serious, which greatly restricts the safe and efficient mining of coal resources. To explore the instability mechanism of soft rock roadway under the influence of mining, the 226 rail cross-cut of Shanjiaoshu Coal Mine in China was taken as an engineering background. The methods of field monitoring and drilling imaging were adopted to analyze the roadway failure characteristics. In addition, the distributions of the plastic zone, surface displacement, and fracture fractal evolution characteristics of the surrounding rock for different stress wave peak and stress wave frequency are compared and analyzed using UDEC software. Then, the combined support scheme of “secondary strong bolting and grouting” is proposed to control the severe deformation of the roadway. The primary support mainly adopts short hollow grouting anchor cable to strengthen the shallow broken surrounding rock within 0–3 m. The secondary reinforced support adopts long hollow grouting anchor cable or constant resistance large deformation anchor cable to fully mobilize the bearing capacity of deep surrounding rock and to ensure the long-term stability of the roadway. On-site monitoring showed that the scheme could effectively control the severe deformation of the roadway, which also provided a reference for other similar roadway support.

Durability of an adhesively bonded joint between steel ship hull and sandwich superstructure pre-exposed to saline environment

ABSTRACT This paper outlines an experimental investigation into the durability of large-scale adhesively bonded joints with a thick layer of methyl methacrylate adhesive (MMA). Ageing has been performed by immersion in a 3.5 wt% NaCl solution for 10 weeks at 50°C. Two aged and one unaged specimen were subjected to tensile testing, and three aged and one unaged specimen were loaded up to ~ 3.5 million fatigue cycles followed by a residual tensile test. The ductility of the adhesive is affected by ageing and fatigue testing. Despite a decrease in ductility, the plastic zone development was adequate for the required strain redistribution without compromising the joint performance (strength and stiffness) demonstrating the fatigue tolerance of the joint. The shear, longitudinal, and peel strain values in the adhesive bulk are evaluated by digital image correlation. The shear strength values are significantly higher than the requirements following from the design. All specimens failed by sudden delamination of the composite plate. Post-mortem analysis showed no corrosion travel at the interface of steel and adhesive.

Investigation of frequency-dependent fatigue crack growth behavior in perfluorosulfonic-acid membrane: In-situ experiment, constitutive modeling and crack growth prediction

Fatigue crack growth behavior of perfluorosulfonic-acid membrane is investigated under distinct loading frequencies. The evolution of cyclic plastic zone (CPZ) is traced based on in-situ optical microscopy testing and digital image correlation (DIC) technique. The crack tip deformation process is directly observed by in-situ scanning electron microscope (SEM). The experimental results show that fatigue cracks experiencing a higher loading frequency display a comparatively reduced propagation rate. The CPZ can be solely used to interpret the fatigue crack growth rate variation under different loading frequencies. The crack-front microspores are easily formed under lower loading frequency due to the enhancement of near-tip plasticity deformation. Then, a cyclic elastic-viscoplastic constitutive model is developed, which successfully capture the mechanical behavior of the membrane. Finally, based on the plastically dissipated energy, a numerical method incorporated with the developed material’s constitutive relationship is established, which gives reasonable prediction of fatigue crack growth under various loading frequencies.

Laboratory Experimental Study on the Pressure Relief Effect of Boreholes in Sandstone under High-Stress Conditions

To study the effects of deep rock drilling pressure relief under high stress conditions in enhanced geothermal systems, two kinds of drilling pressure relief experiments were conducted on sandstone—the staged drilling of pressure relief holes before peak stress and one-time drilling. Pressure relief experiments were carried out on sandstone with two borehole methods of the stage-by-stage drilling and one-time drilling of pressure relief boreholes ahead of the experiments. FLAC3D was used to analyze the plastic zone evolution during drilling and the relationship between stress and plastic zone volume. The results reveal the pre-peak stress change characteristics and pressure relief features of non-prefabricated boreholes under high stress. The experiments show that the staged drilling of pressurized samples involves stages of rapid and gradual decreases in stress, with total relief amplitudes increasing but single-borehole relief decreasing with more holes. Under the same conditions, staged drilling has better relief effects and results in greater energy dissipation, indicating that incremental pre-peak pressure relief is beneficial for reducing the surrounding rock’s impact tendency and improving stability. The research results can provide good guidance and reference for the long-term stability analysis of borehole-containing rock and rock burst hazard control.

Evolution Characteristics of Plastic Zone in Jointed Rock Mass of High-Temperature Hydraulic Tunnel

The evolution characteristics and the extent of the plastic zone in rock mass can reflect the failure characteristics and destruction degree of a hydraulic tunnel. In this research, we derived the equation of a plastic zone range based on a gateway-like structure high-temperature tunnel through a theoretical analysis. On-site monitoring and discrete element model simulation were combined to analyze the temperature field law and plastic zone evolution characteristics of the jointed rock mass at high temperatures. The results show that the joints affect the temperature field variation in rock mass, and the vertical and horizontal joint groups pose significantly greater influence than the inclined joint group on temperature field. The sensitivity of the joint internal friction angle and thermal expansion coefficient to the range of the plastic zone is relatively small. Under various joint spacings, the influence of horizontal and inclined joint groups on the plastic zone morphology decreases, while the vertical joint group exhibits an incremental influence on the plastic zone morphology. Similar to the influence of the temperature field, the vertical and horizontal joint groups have a significantly greater influence on the plastic zone range than the inclined joint group. Under various rock mass temperatures, the plastic zone in rock mass results in the occurrence of uneven expansions along the direction of the joint dip angle, which changes the potential failure direction of rock mass and increases the potential destruction degree of rock mass, whereas it exhibits a smaller uniform expansion perpendicular to the joint dip angle, and the boundary of the plastic zone coincides with the joint surface, relatively hindering the potential destruction degree of rock mass. The research results of this study have certain reference values for the stability control of jointed rock mass in high-temperature hydraulic tunnels.

Load capacity of penstock manifolds subjected to increasing settlements at the vertical supports

AbstractHydroelectric power plants usually have penstock manifolds near the powerhouse, including individual bifurcations to provide several turbines. These steel constructions have numerous vertical supports and are designed based on linear elastic behaviour, leading to very small allowable settlements at the supports.For a representative built manifold, with five bifurcations (six branch pipes), the elastic‐plastic load capacity was studied in detail, assuming dead load, internal pressure and increasing settlements at the supports. Different settlement patterns were studied. The development of plastic zones was analyzed, depending on the rate of settlement. Due to the high bending stiffness of the penstock manifold, the penstock detaches from the single supports due to the settlement, leading to a significant redistribution of the vertical reaction forces at the supports.The comprehensive calculations show adequate robustness of the penstock manifolds, even though in case of very high settlements at the supports. Due to the significant redistribution of the reaction forces at the supports, these vertical supports must have an increased bearing capacity, in comparison to the reaction forces due to dead load and internal pressure based on linear elastic analyses.

Numerical simulation study on the evolution characteristics of the stress induced by mining in deep adjacent working faces

BackgroundThe mining-induced stress in the surrounding rock after coal seam mining is the primary cause of damage and failure of the surrounding rock in the mining area. However, the magnitude and direction of the stress field induced by mining in the overburden strata during the excavation process of deep and adjacent coal seams are not yet clear, and it is difficult to determine how adjacent working faces interact with each other.ResultsIn this study, a large-scale numerical model was built using FLAC3D (Fast Lagrange Analysis Continua) to simulate the sequential mining process of three adjacent working faces (No.1, No.2, and No.3) in Liuzhuang Coal Mine located in southern China. The results showed that the maximum height of plastic zone development after mining in the No.1 working face was 41 m, and the maximum height of plastic zone development was 33.8 m away from the 13 coal seam. It did not affect the top and bottom of the No.3 working face. The development height of the plastic zone on the roof of the No.2 working face after mining was 52m, and the top and bottom plates of the No.3 working face remained intact. The plastic zone of the floor of the No.3 working face after mining was not communicated with the plastic zone of the roof of the No.1 and No.2 working faces. There was a complete rock layer between the two coal seams, and there was not the mutual influence of the mining activities. During the mining process of working faces No.1 and No.2, the range of dynamic pressure influence was extended up to 100m ahead of the working face. During the mining process of the No.3 working face, the range of dynamic pressure influence was 120 m ahead of the working face. The No.1 working face goaf and the mining of No.2 working face had not disturbance to the No.3 working face.ConclusionsAfter the mining of No.1 and No.2 working faces, the stress in the goaf significantly decreased, and the lateral support stress concentration area of No.1 and No.2 working faces had a relatively small impact on the 13 coal seam. Therefore, the No.1 goaf and mining of No.2 working faces had not disturbance to No.3 working face.

СТРІЧКОВИЙ ФУНДАМЕНТ З ПОВЗДОВЖНІМ ВИРІЗОМ ПО ПІДОШВІ МАСИВНОЇ ПІДПІРНОЇ СТІНИ

The paper analyzes the designs of traditional strip foundations with a flat bottom, the load from which causes the plane strain stress state of the soil base, and other variations of conventionally strip (continuous) foundations, which due to their shape (configuration) of the contact area with the base change its stress state, which enables designing more sustainable foundations for continuous buildings and structures. Proceeding from the solutions to the mixed problem of the theory of elasticity and plasticity using the Mohr-Coulomb criterion strength criterion, analytical studies of the development of plastic zones in the base of a strip foundation with a longitudinal cut-out have been conducted, which show that the limit state always occurs first in the foundation’s edge zones, that is, underneath the outer edges of the foundation. It is also noted, however, that the design resistance of soil decreases when there is no additional load in the area of the cut-out; therefore, a patented design has been proposed of a strip foundation with a longitudinal cut-out in the bottom, where the cut-out with the height is filled with low-modulus material to improve the design resistance of soil. Based on experimental and theoretical studies, methods have been proposed for calculating the soil base (design resistance , settlement and inclination ) for a strip foundation of a massive retaining wall with a longitudinal cut-out in the bottom. A real example shows that the total width of the strip foundation with a cut-out is shorter by 1.5 m in comparison with the continuous shape of the bottom of the foundation, which has a significant economic effect on every linear meter of the wall foundation. Overall, the proposed methods make it possible to reasonably design effective eccentrically loaded foundations with cut-outs in the bottom and to improve the permissible vertical pressure on the base in comparison with a continuous bottom, all other things being equal. Keywords: strip foundation, massive retaining wall, eccentricity, bottom, cut-out, soil base, calculation procedure.

Numerical Study on the Mechanism of Coal and Gas Outburst in the Coal Seam Thickening Area during Mining

Most coal and gas outbursts occur in the coal thickness variation zone. However, it is difficult to illustrate the mechanism of outbursts in coal thickening areas by physical simulation experiments. In this study, a coupled multi-field model, established by considering the stress–strain field, gas transport field and damage field, was used to investigate the evolution of stress, gas pressure and plastic failure zones under different variation gradients and amplitudes of coal thickness. The simulation results show that the stress peak at the coal thickening transition zone caused by mining is higher than that at the constant thickness coal seam. The stress peak at the coal thickening transition zone decreases from 18.8 MPa to 16.9 MPa with the increase in the transition zone from 0 m to 10 m under the constant coal thickness variation from 3 m to 7 m; while it increases from 16.2 MPa to 19.3 MPa with the increase in the transition zone from 2 m to 10 m under the constant coal thickness variation gradient of 45°. Similarly, the plastic deformation volume of the coal seam between the driving face and the coal thickening interface increases with the increase in the coal thickness variation gradient and amplitude. In addition, the gas pressure in the fracture declines slower in the coal thickness variation zone affected by the higher coal thickness variation gradients or amplitudes. The mechanism for outbursts occurring in the increasing coal thickness area was further discussed, and combined with the simulation results for the energy principle of outbursts. Compared with the constant thickness coal seam, the elastic energy increases from 1.85 MJ to 1.94 MJ, and the free gas expansion energy increases from 24.19 MJ to 50.57 MJ when the coal thickness varies from 3 m to 13 m within a 10 m transition zone. The variation of coal thickness causes higher stress, higher gas pressure and low coal strength, which triggers outbursts more easily. The research could provide the theoretical support to prevent and control outbursts in coal seam thickening areas during mining.

Constitutive modeling of a laumontite-rich tight rock and the application to poromechanical analysis of deeply drilled wells

Geological formations containing laumontite-rich rock are usually treated as problematic for geo-energy production projects because the presence of laumontite mineral can promote complex mechanical behaviors. Previous laboratory results indicate that rock formations with a higher laumontite content display severe stress sensitivity in poromechanical responses. With an increase in confining pressure, there is a transition from dilation to compression regime and the resulting localization styles range from shear dilation to compaction bands. In this study, we conduct finite element modeling of constitutive behaviors of rocks retrieved from the tight glutenite reservoir formation using a thermodynamic-consistent plasticity model. The shear dilation to compaction transition is well characterized. Poromechanical analysis is also conducted to analyze the plastic zone development around a borehole drilled in an over pressured reservoir. The simulated stress-paths of key points around the borehole are used to demonstrate the plastic strain development processes. The impact of in-situ stress on the wellbore stability is highlighted, and a comparison with the results from using the traditional plastic constitutive model is conducted.

Theoretical analysis of grouting reinforcement for surrounding rock of deep shaft based on stability and water inflow control

To address the challenge of designing grouting reinforcement in a deep shaft to control water, this study established an elastic-plastic analytical formula for the grouted rock surrounding a shaft under the combined action of thermal, hydraulic, and mechanical fields based on the Mohr–Coulomb yield criterion. The various rules and influencing factors of stability and impermeability of the grouted rock surrounding the shaft are calculated and analyzed. The analysis shows that damage to the surrounding rock is aggravated by the action of high ground temperature and high water pressure in deep shafts, and the influence of water pressure is particularly significant. The larger the radius and cohesion of the grouted surrounding rock, the smaller the radius of its plastic zone. With an increase in the elastic modulus ratio between the grouted surrounding rock and the original rock, the radius of its plastic zone increases, so the elastic modulus of the grouted surrounding rock design should consider the original surrounding rock. With a decrease in the permeability coefficient of the grouted surrounding rock, the radial stress decreases, the tangential stress increases, and the radius of the plastic zone increases. The development of plastic zone not only affects the stability but also causes an increase in the permeability coefficient. There is mutual restriction and influence between the shaft water inflow, the permeability coefficient, the radius of the grouted surrounding rock, and the radius of its plastic zone. In the design of grouting reinforcement, the stability of surrounding rock and the control requirements of shaft water inflow should be comprehensively considered and the optimal parameters should be selected based on theoretical calculation. This study provides a theoretical basis for the optimization of grouting reinforcement parameters in a deep shaft.