Kunmeng Li, Kaiyuan Jiang, Yuanhui Li,c,*, Xin Wang, Kai Liu, Shuai Xu

a Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, School of Resources and Civil Engineering, Northeastern University, Shenyang 110819, China

b Key Laboratory of Liaoning Province on Deep Engineering and Intelligent Technology, School of Resources and Civil Engineering, Northeastern University, Shenyang 110819, China

c China-Canada Centre of Deep Mining Innovation, Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, School of Resources and Civil Engineering, Northeastern University, Shenyang 110819, China

d Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, United Kingdom

Keywords:Pillar stability Load bearing capacity Expandable prop Slenderness ratio Eccentric distance

A B S T R A C T This paper aims to determine the load bearing capacity of pre-stressed expandable props with different geometries and load eccentricities for flexible support in underground mining or excavation.It is deduced that the expandable device could have much higher strength(>89 MPa)by laboratory tests,and the load bearing capacity of the expandable prop may depend on the stability of the supporting steel pipe structure.A good agreement was found between the laboratory test and numerical results in terms of the load bearing capacity and the final macro-bending failure pattern for expandable props with heights of 1.5 and 2.7 m,and the theoretical calculation for the strength of traditional steel structures is not directly suitable for the expandable props.Moreover, additional numerical simulations were performed for the expandable props with different normalized slenderness ratios λn and loading eccentric distances e.The variation of stability coefficient of the expandable prop is in line with the Perry-Robertson equation and its correlation coefficients are fitted as a of 0.979 and b of 0.314.For estimating the load bearing capacity of the expandable props, the strength equation for traditional steel structures is improved by introducing a bending magnification factor and by modifying the normalized slenderness ratio to a converted slenderness ratio.Based on the underground field monitoring for the strength of expandable props with different heights,the empirical eccentric distances were back calculated,and a safety factor is introduced to obtain the designed strength of the expandable prop.In addition, a four-step design procedure is proposed for the expandable prop.

1.Introduction

In underground mines, where non-caving methods are used to extract tabular deposits, natural ore pillars are commonly left behind to support the stope roof for local and regional stability,thus providing a safe working environment for personnel [1].The failure to recover the natural ore pillars in the post-mining period may reduce the profitability for mining operations.It is a common practice to recover the pillars partially or completely using methods such as direct drilling and blasting [2–4] and replacing them with artificial props [5–8].Compared with the direct drilling and blasting approach, the installation of artificial props is a safer method for personnel working in the mining area as the back is supported by the artificial structures or systems prior to the recovery of natural pillars.

Based on the load bearing and deformation characteristics,artificial props can be divided into four categories, namely, (1) brittle prop,such as sand props[9]and traditional concrete prop[10–13];(2)perfectly plastic props,such as can supports[14],cluster props[15,16], rocprops [17], spider props [18], ball buster and quick sticks [9]; (3) strain-softening props, such as pumpables [19–21]and (4) strain-hardening props, such as omni props and Big Johns[22] and Little John Extremes [15].The advantages and disadvantages of the support systems have been discussed in detail in previous studies by Barczak [9] and Li et al.[23].In general, the first three types of artificial props have been widely used in the mining industry,while the high pre-stressed and strain-hardening or softening props, as one of the ‘yielding support’ systems [24,25], are still in the research and development stage.

Li et al.[23]developed a new type of standing artificial support system called the pre-stressed expandable prop,which can provide an active support force through chemical expansion towards the roof.Additionally, the pre-stressed expandable prop can maintain its stability under large deformations if it is correctly designed and installed based on the roof and support system properties.It generates a much higher pre-stress force[23]compared with other pre-stressed methods,such as wooden wedges[26,27],mechanical devices [28–30], air-filled rubber bladders [15], grout-filled bags[28] and water-inflated steel diaphragms [31].In addition to the high preload capacity, the strain-hardening or strain-softening property is another advantage of the pre-stressed expandable prop[32].Li.et al.[23]studied an expandable device that fits into a laboratory testing machine to determine the post-yield characteristics of the expandable prop, and the testing results showed that the load bearing process of the pre-stressed expandable prop was divided into three stages, namely, pre-stressing by expansion, linear or non-linear deformation under loading and strain-hardening or strain-softening after reaching the yield strength.However, the load and displacement characteristics of the expandable device were only explored in a low range of loading (up to 1000 kN)due to limitations in the test setup, which could not fully reveal the load bearing and instability characteristics of the expandable structure for the expandable props.

In field applications, the expandable prop has two key components, namely, the expandable base or structure, which is an enlarged version of the expandable device, and the support pipes[23].Hereafter, the support structure consists of multiple steel posts or pipes, which are relatively convenient for transportation and construction in underground mines to achieve fast support,and it is flexible to obtain the length of steel posts or pipes according to the height of the support location conditions.The steel posts or pipes have a high compressive resistance ability to meet the support requirements for underground excavation spaces by optimising the structural parameters.Presently,the most used expandable props in the field consist of five pipes.The pre-stress is applied by the expandable base to the roof through the post or pipe structure.As a system, the prop reaches its bearing capacity when one of its components yields.The height of the pre-stressed expandable prop varies accordingly to fit the actual mining situations.It is common that the load eccentricity is generated in the process of welding and fixing the steel posts or pipes to the top platen of the expandable base, and it is also affected by the roof-contacted structure and irregularities at contact surfaces, resulting in the non-uniform loading on the expandable prop.All these factors affect the overall bearing capacity and stability of the prestressed expandable props.

The support structure of the expandable prop is structurally similar to the traditional steel structures, including type a, type b, type c and type d [33–36], which are all composed of multiple steel tubes, bars or plate.Compared with traditional steel structures, steel strips or braces are not added to connect the support component of the expandable props to quickly support and reduce the failure caused by welding.Traditional steel structures have no expandable structure and therefore no active support properties.The theoretical equation for the load bearing capacity of traditional steel structures has been well developed with different slenderness ratios and loading eccentric distances,which is applied tentatively to obtain the strength of some support bodies that are similar to the expandable prop, and provides a basis for reference for determining the load bearing capacity of the expandable props.Shi et al.[37] evaluated the mechanical properties of high-strength steel columns under different sections by axial loading based on theoretical calculation of traditional steel structures.Kang et al.[38] analysed the global buckling behaviour of welded Q460GJ steel box columns under axial compression according to the steel structure theoretical equation.

In this study,compressive tests were conducted for the expandable device and the entire prop system to experimentally determine the load bearing capacity of the pre-stressed expandable props.The failure process and the weakness portion of the prop system are discussed,and the load bearing mechanism is analyzed in Section 2.Meanwhile,the load bearing performance and capacity of the two expandable props were further investigated and evaluated using numerical simulation and theoretical analysis.Based on the strength calculations for traditional steel structures,the theoretical equation for the load bearing capacity of the expandable props was developed in Section 3 by considering both the influence of normalized slenderness ratios λnand loading eccentric distances e.The load bearing characteristic of the expandable props was monitored and analysed in Section 4 for mine applications with different λnvalues,and the empirical loading eccentric distances are back calculated.The designed strength of the expandable props for on-site application was determined.Finally,the design standard for the pre-stressed expandable props was established in Section 5.The results from laboratory experiments, theoretical analysis, numerical simulation and field tests of the pre-stressed expandable props are summarised and presented in this study.The results will further optimize the support parameters of the expandable props for improved ore recovery and mine safety.

2.Evaluation of the load bearing capacity of pre-stressed expandable props

The developed pre-stressed expandable props can provide high bearing loads to stabilise mine roofs at relatively small deformation.Li et al.[23] developed an expandable prop that consists of two main components,namely,an expandable device and one post structure.Here, the designed test programme aims to assess the stability of the whole expandable prop structure in the laboratory.Two main test schemes were designed.Firstly, the expandable device is evaluated at a high load scenario, for instance, the loads can reach up to 2800 kN,which is higher than that of the previous study(i.e.,1000 kN from Li et al.[23]).In addition to the high loading test for expandable devices, the expandable props were also tested with two heights of 1.5 and 2.7 m for thin deposits.Moreover,the load bearing behaviour and capacity of expandable props are also analysed using numerical simulation and calculated using theoretical analysis.

2.1.Load bearing performance of the expandable prop in laboratory test

2.1.1.Laboratory testing of the expandable devices

The developed expandable prop system depends on the expansion of the confined water-sensitive agent, which can provide a high pre-stress.The expandable devices were tested under high loads to further investigate the load bearing characteristic of the expandable prop.The expandable device is a scaled-down version of the expandable component of the expandable prop for the laboratory scale tests[23],i.e.,the expandable base.The device is made of steel with top and bottom covers and an inner tube, and the expanding material was pre-packed in cartridges,and the columns of these cartridges were placed inside the device.When the expanding material reacts with water and expands in volume,the top and bottom covers move up and down, respectively, and both ends of the expandable device generate pre-stresses if the boundaries are restrained.Fig.1a and b shows the expandable device dimensions and the setup of the expansion and loading,respectively.The test arrangement follows the same procedure as stated by Li et al.[23], and the load and displacement of expandable devices can be recorded by the test machine.The obvious difference between the two tests lies in the extremely high loading to the expandable devices, and this study performs a higher load.By overloading the expandable devices,the maximum bearing capacity and failure pattern are explored for the expandable structure of pre-stressed expandable props.Four tests were conducted with different reserved gaps(defined as the distances between the loading head and the top of the expandable device) of 0, 10, 30 and 50 mm, respectively.The reverse loading tests for the expandable devices were also performed after the formation of an initial prestress.The tests were terminated when reaching approximately 2800 kN because the maximum loading capacity of the test machine is 3000 kN.

Fig.1.Schematic of the laboratory testing setup (from the previous study of Li et al.[23]).

Fig.2a shows the axial load-displacement curves for the four tests with different reserved gaps.It was found that the initial load decreases as the reserved gap increases,and the initial load reaches roughly 500 kN for the reserved gap of 0 mm.It is safe to say that the prop modulus or stiffness and the initial load from prestressing decrease with an increase of the reserved gap.As shown in Fig.1b, the reserved gap determines the free allowable vertical movement of the expandable device, that is, the gap controls the expandable volume of the device.In other words, the density and initial stiffness of the expanding material inside the device increase as the reserved gap decreases.This observation demonstrates the importance of prop installation quality in the underground ground support project in terms of conditions at the roof or floor contacts such as the effective gap.

Fig.2.Load and displacement characteristics of the expandable device.

All tests were terminated when the axial load reaches 2800 kN,as it approached the maximum loading capacity of the test machine.The strength of the expandable device is calculated roughly to be 89 MPa, corresponding to a loading of 2800 kN.At the end of the tests, the expandable devices were still functioning without large deformation or failure.As shown in Fig.2b, only a slight extrusion deformation was observed near the connection area between the upper and lower cylinders.Although a slight extrusion deformation is found at the body of the expandable device at the end of test, it is still reasonable to note that the expandable device could have a strength of 89 MPa or an even higher ultimate maximum strength based on the applied load and device dimensions.

2.1.2.Laboratory testing of the pre-stressed expandable props

Considering the high strength of the expandable device, the support structure could be the weakest part of the expandable prop.The expandable props were overloaded in the laboratory to understand the failure pattern and the maximum bearing capacity of the support pipes.The pre-stressed expandable props are generally used for the recovery of residual pillars in room and pillar mining or roof support in cut-and-fill stopping.The majority of mines that use the above two mining cases normally have an average stope height of 3.0 m.The height of the steel pipe structure is 2.7 m,allowing 0.3 m for the expandable base.An expandable prop with a much lower height of 1.8 m(1.5 m with the support pipes in this case)was also tested to further study the influence of the slenderness ratio λ on the load bearing characteristic of the prestressed expandable prop.As shown in Fig.3, the loading system consisted of a servo-controlled compression machine with a loading capacity of 5000 kN for the entire prop system.The support structure with five pipes is made of A3 steel Q295.The outer and inner diameters of the pipe are 0.108 and 0.096 m, respectively.

Fig.3.Loading testing setup of a pre-stressed expandable prop.

The loading process was applied in two steps:(1)the preload or initial load is generated by lowering the press head to touch the prop top and by activating the expanding agent, and (2) the load is increased gradually.During the loading process, the axial loadtime and axial load-displacement curves were monitored, and the failure patterns of the expandable props were also recorded.To simulate the quasi-static deformation process of the stope roof caused by the increase of excavation size and the time lapse, the loading deformation rate of the compression machine was set to 1 mm/min,and the displacement was 2 mm for each loading increment.After the preload was generated, each loading increment was maintained for 0.15 days to observe the time effects on the prop stability.

Fig.4 shows the axial load-time curves for the support pipes with heights of 1.5 and 2.7 m.As shown in Fig.4,as the activation of the expanding agent, both the two props generate a preload or initial load rapidly, and the load can reach up to 1725 kN, which remains steady until 0.6 days later when the first load increment is applied.For each loading cycle, the axial load increases rapidly,followed by a decreasing trend, and then shows a relatively constant load although associated with certain fluctuations.This phenomenon is referred to as a‘deformation lag’as reported by Li et al.[23].For the 1.5 m-high pipes, the load increases steadily until reaching a prop capacity of 2689 kN (9.6 MPa in the expandable device) and then decreases gradually at a low rate.As shown in Fig.5a, the pipe support structure of the expandable prop begins to deform once one of the steel pipes starts to bend close to the middle section.As shown in Fig.5b, the deformation of the pipe support structure increases as the loading continues, where more steel pipes start bending,indicating the gradual loss of prop capacity.All six loading increments were maintained for 0.15 days, and the load is approximately constant during each cycle.For the 2.7 m-high pipes, the initial load is almost the same as that of the 1.5 m-high pipes, i.e., 1725 kN, as shown in Fig.4.However,once the pre-stress is applied, the steel pipe structure starts to deform in the form of bending(Fig.6a).This observation indicates that the load bearing capacity has been reached for the prop.As the bearing load on the expandable prop continues to increase, the load bearing capacity decreases, and the extent of reduction increases.As shown in Fig.6b, more steel pipes seem to be unstable at final stages of loading.When the load is 1335 kN,most of the pipes were bent in the support structure,and the bending locations were near the middle sections.

Fig.4.Axial load history for the pre-stressed expandable props.

Fig.5.Observed instability of the 1.5 m-high expandable prop.

Fig.6.Observed instability of the 2.7 m-high expandable prop.

Fig.7 shows the axial load-displacement curves for both tests.For the 1.5 m-high pipes, the loading process shows three phases,namely, pre-stress generation, load carrying or bearing and yielding.However,for the 2.7 m-high pipes,the bearing load is only stabilised for a very short time after the pre-stress is applied and then decreases as the deformation increases.This behaviour reflects a strain-softening characteristic in the prop yielding phase.The load bearing phase is shortened or skipped due to the high preload.To improve the load bearing capacity for high expandable props,several methods can be adopted,such as reducing the prop expansion by modifying the expanding agent or introducing gaps and strengthening the steel pipes.As also shown in Fig.7, the axial load-displacement curves are smooth when the expandable prop is in a stable state.The curves become wavy or rough once the peak load is reached because the axial bearing load starts to decrease once one or more pipes become unstable as a result of uneven loading.The remaining pipes carry more load with further deformation,which may result in a temporary increase in prop strength.

Fig.7.Axial load-displacement curves of the expandable props.

2.2.Load bearing performance of the expandable props using numerical modelling

To further analyse the load bearing characteristic of the expandable props, a finite element method numerical modelling with the Abaqus software was conducted for the support structure of the expandable props with the heights of 1.5 and 2.7 m.As shown in Fig.8a, the numerical model consists of solid elements with eight nodes and 6 degrees of freedom.The ends of the five pipes are welded to rigid steel plates with sufficient stiffness, and the tie connection is adopted for numerical modelling of the Abaqus software to represent the actual welding.The upper and lower ends of the expandable props were articulated,i.e.,the displacement of the centre point of the lower limiting plate is restricted, and the rotation of the lower limiting plate in space is free.At the same time,the displacement of the centre point of the upper limiting plate in the horizontal direction is restricted, and the displacement of the centre point of the upper limiting plate in the vertical direction and the rotation of the upper limiting plate in space is free.Meanwhile, the numerical model of the expandable props has an initial defect with one-thousandth of the height, and the ideal uniaxial loading tests without load eccentricity were performed.Pipes are simulated as A3 steel of Q295.Young’s modulus, Poisson’s ratio and the yield strength of steel material are 206 GPa, 0.3 and 295.0 MPa,respectively.Displacement loading and statics analysis were adopted for the purpose of achieving the full-period load bearing curve for the expandable props.The total time step, the maximum time step and the minimum time step are 1, 0.1,0.001, respectively, and the adaptive analysis step identification was accepted.Non-linear finite element analysis was executed,including a large deflection analysis and a bilinear material model.The vertical load and displacement of the expandable props are monitored during the loading process.

Fig.8.Load bearing performance of the expandable props with λn of 0.2717 and 0.473 from numerical simulation.

The support structure for the expandable prop is a semi-integral component, which is between the rigid frame and lattice column.The slenderness ratio λ of the expandable prop is small if the calculation method for the lattice column is adopted.Therefore, in this study, the slenderness ratio λ of the expandable props is obtained by back calculation of λ=(π2EA/NEx)0.5[33,39], where NExis the elastic buckling strength of the expandable props obtained by numerical simulation, E is the elastic modulus and A is the crosssectional area of expandable props.The normalized slenderness ratio is calculated as λn=(λ/π)×(f/E)0.5[33,39], where f is the yield strength of steel material.NEx, λ and λnare 37,714 and 12,213 kN, 22.74 and 39.97 kN, and 0.2717 and 0.473 kN for the expandable props of 1.5 and 2.7 m, respectively.

As shown in Fig.8b, the load–displacement curves of the expandable props with λnof 0.2717 and 0.473 show a similar evolution trend.The bearing load increases firstly on expandable props and then decreases after reaching the ultimate strength.The load bearing capacity for the expandable prop with λnof 0.2717(height of 1.5 m) would be about 2616 kN, while the bearing capacity for the expandable prop with a height of 2.7 m is 2432 kN.When the load reaches the maximum value, the deformation of the support pipes of 2.7 m is larger than that of the pipes of 1.5 m.

As shown in Fig.9, the buckling instability is the main failure form of the expandable props for the two normalized slenderness ratio λncases.Compared with case with λnof 0.473, a relatively greater proportion of plastic yielding zone is observed for the expandable prop with λnof 0.2717.For both two cases, an elastoplastic buckling and a C-shaped bending pattern are found at the central portion.

2.3.Theoretical load bearing capacity of expandable props

Similarities are observed between the support component of the pre-stressed expandable prop and the traditional steel structures used in construction engineering.Fig.10a is a schematic diagram of a traditional steel structure, which is similar to the traditional steel structure type c in terms of the overall structure.Differences also exist between the two types.For instance,the traditional steel structure belongs to a lattice column group, and its sub-columns are interconnected by steel strips or braces to restrict the deformation and to maintain the stability of the overall prop structure.However, no steel braces are designed to be included between the support pipes of the expandable props to quickly support and reduce the failure caused by welding, which can be regarded as a semi-integral lattice column, as shown in Fig.10b.As the cross-sectional shape of the support structure is circular and the cross section is symmetric, the expandable prop is most akin to the traditional steel structure type a [33–36].Moreover,the theoretical strength calculation equation for the steel structure type a by axial loading is adopted to approximately evaluate the load bearing capacity of the expandable props.The equation is N=Sc×A×f,where A is the cross-sectional area of expandable props,f is the yield strength of steel material and Sc is the stability coefficient.Sc is 0.971 and 0.952 for the steel structure type a with λnof 0.2717 and 0.473,respectively[33–36].The load bearing capacities of the expandable props N are then calculated theoretically as 2708 and 2655 kN with the λnof 0.2717 and 0.473, respectively.

Fig.10.Difference between traditional steel structures and the expandable prop.

2.4.Evaluation for the load bearing capacity of the expandable props

As presented in Table 1, the strength of the expandable prop with λnof 0.2717 from numerical simulation is slightly lower than that obtained in the laboratory experiment, which may be caused by deviations such as the yield strength of support pipes in the laboratory being slightly greater than295.0 MPa.However, the simulated load bearing capacity of the expandable prop with λnof 0.473 is much larger than that from the experiment.Two possibilities are considered.The first is that the greater the normalized slenderness ratio λnof the expandable props, the larger the loading eccentric distance e in the laboratory.The second is that the load eccentricity of the expandable props is the same with λnof 0.2717 and 0.473,but the influence of the loading eccentric distance is much more pronounced for the load bearing capacity of the expandable props with large λn.A good agreement is found between laboratory tests and numerical simulations in terms of the load bearing performance and the final failure pattern of the expandable props.The expandable props begin to lose stability after reaching the maximum bearing load and ultimately produce a bending yielding failure.The deformation of the expandable props on one side is significant.In general,the load bearing capacity of the expandable prop with λnof 0.2717 is larger than that for the case with λnof 0.473, illustrating that the larger the normalized slenderness ratio λn, the smaller the load bearing capacity of the expandable prop.Besides, the load eccentricity has a certain effect on the strength of the expandable prop.The load bearing capacity of the expandable prop by the traditional theoretical calculation for the steel structure type a is larger than that by numerical simulation, indicating that the former is not particularly suitable for strength calculation of the expandable props.

Table 1 Load bearing capacity of expandable props with λn of 0.2717 and 0.473.

3.Determination of the load bearing capacity of expandable props

In the field application, as shown in Fig.11, the normalized slenderness ratio of expandable props λnis varied with different factors,such as the support location height,heterogeneous ground stress distribution and roof roughness.Moreover, the expandable props generally bear a non-axial (eccentric) load.Therefore, it is necessary to derive the theoretical calculation equation for load bearing capacity of the expandable props, considering both the influence of normalized slenderness ratio and loading eccentric distance.

Fig.11.Overview diagram of conditions that result in different heights and eccentricities of the expandable props.

3.1.Normalized slenderness ratio λn

The pre-stressed expandable props with different normalized slenderness ratios λnwere designed to perform uniaxial loading tests without load eccentricity using numerical simulation.Loading design and loading speed are the same as the numerical experiments for the support structure of the expandable props for the heights of 1.5 and 2.7 m.Displacement loading and statics analysis were adopted.As presented in Table 2, the heights of the support pipes were designed at 1.0, 1.8, 2.6, 3.4, 4.2, 5.0, 5.8, 6.6 and 7.4 m,and λnwas back calculated by the elastic buckling strengths as 0.189, 0.323, 0.46, 0.599, 0.738, 0.878, 1.018, 1.157 and 1.297,respectively.The load-displacement curve, instability pattern and load bearing capacity of expandable props were obtained and analyzed.As shown in Fig.12, the load on the expandable props first increases and then decreases with the increase in vertical loading displacement.Prior to the peak strength,the larger the normalized slenderness ratio of the expandable props λn, the lower the load bearing capacity and the more increased rate of the bearing load on the expandable props.No obvious ultimate strength of the expandable props with a small λnwas observed, such as λn=0.189.However, the extreme point instability pattern appears as the λnvalue increases.When the peak strength is reached, the vertical displacement of the expandable prop is larger with higher λn.As the loading progresses further, the bearing load on the expandable prop decreases, though there is still a certain amount of residual strength.

Fig.12.Load bearing performance of the expandable props with different λn under axis loading.

As shown in Fig.13, the expandable props show the bending instability, and the final buckling form is C-shaped.The degree of failure of support pipes is greater on one side of the expandable props.The middle part and both ends of support pipes yield, and the maximum deflection position is found in the middle of the expandable props.The proportion of the plastic zone that corresponds to the entire support pipes gradually decreases as the normalized slenderness ratio λnof the expandable props increases.In the meantime, the instability pattern is transformed from a fullsection buckling to an elastoplastic buckling and then to an elastic buckling as the volume ratio of the plastic zone becomes very small, such as for the expandable prop with λnof 1.297.

Fig.13.Instability pattern of the expandable props with different λn.

As presented in Table 2,the simulated stability coefficient of the expandable props can be calculated as Scs-Ep=N/Af according to the load bearing capacity N of expandable props with different λnvalues [33,39].As shown in Fig.14, the stability coefficient Scs-Epdecreases with the increase of λn.Moreover, Scs-Epshows a slower downward trend when λnis less than 0.5, because the ultimate strength of the expandable props is only related to the yield strength and is less affected by other factors such as the height of support pipes.When λnis between 0.5 and 1.2, Scs-Epdecreases relatively rapidly as the yielding strength and the height of support pipes jointly control the load bearing capacity of the expandable props.When λnis over 1.2, Scs-Epdecreases slowly again because the strength of the expandable props is only dominated by the height of support pipes.

Fig.14.Relationship between Scs-Ep and λn under axis loading.

As shown in Fig.14, the variation tendency and the magnitude of stability coefficient of the expandable prop are similar to the traditional steel structures type b,which is inconsistent with the previous analysis.Although the expandable prop is similar to the traditional steel structure type a in terms of the cross-sectional shape and cross-sectional symmetry of the support structure, the load bearing performance nearly agrees with the steel structure type b.Therefore,the Perry-Robertson equation[39]for traditional steel structures,i.e.,column buckling curves[33–36],is adopted to obtain the stability coefficient for expandable props ScEp:

where a and b are the correlation coefficients of ScEp.A non-linear fitting method is used to determine a=0.979 and b=0.314 (type b:a=0.965, b=0.300).The load bearing capacity of the expandable props without load eccentricity can be calculated by N=ScEpAf.

3.2.Different loading eccentric distances

According to the theoretical strength calculation equation of traditional steel structures [33,39], the load bearing capacity of the expandable props considering both the loading eccentric distance e and normalized slenderness ratio λnis:

where MXis the bending moment,i.e.,MX=N×e,and Wx1the section modulus.

To verify the suitability of Eq.(2), uniaxial loading tests were executed numerically on the expandable props with different normalized slenderness ratios λnand loading eccentric distances e.λnwas also 0.189, 0.323, 0.46, 0.599, 0.738, 0.878, 1.018, 1.157 and 1.297, which corresponded to the heights of support pipes as 1.0,1.8, 2.6, 3.4, 4.2, 5.0, 5.8, 6.6 and 7.4 m.e was designed as 0, 10,25, 40, 55, 70 and 85 mm, which is for hard rock.

As shown in Fig.15, the load bearing performances of the expandable props are analysed representatively with three kinds of typical normalized slenderness ratios λnof 0.189 (small), 0.738(medium) and 1.297 (large).As the loading eccentric distance e increases, the bearing load on the expandable prop has a lower increasing rate(small initial stiffness)before reaching the ultimate strength, and the load bearing capacity decreases.The load on theexpandable props begins to decrease as the loading progresses,but a certain residual strength is maintained and nearly equal for large λn,as shown in Fig.15b and c.After reaching the ultimate strength,the residual stiffness of the expandable props becomes larger as the loading eccentric distance e increases.As shown in Fig.15a,when λnis 0.189 (small), the ultimate load on expandable props remains approximatively unchanged and even increases slightly for the large loading eccentric distance e.

Fig.15.Load bearing performance of the expandable props with different e.

Table 2 Load bearing capacity of expandable props with different λn from numerical modelling results.

As shown in Fig.16, when λnis 0.189, as the loading eccentric distance e increases, the yielding zone of the expandable prop moves from the centre to the top and bottom of support pipes,and the buckling form transits from a C to S shape.When λnis 0.738 and 1.279, the yielding zone moves from the centre to the top of support pipes, and the stress of support pipes away from the compression side gradually increases.

Fig.16.Failure pattern of the expandable props with different e.

As shown in Fig.17a and b, when λnis 0.189 (small) and 0.738(medium), the load bearing capacity of the expandable props obtained by theoretical calculation and numerical simulation is almost the same for a small load eccentricity.The deviation gradually increases as the loading eccentric distance e increases, leading to an inconsistent variation trend between theoretical calculation and numerical simulation.The traditional lattice columns synergistically yield due to the mutual restraint of steel stripes.However, the expandable props without braces will bend on one side and are in the elastic state on the other side according to the numerical simulations, resulting in an obviously heterogeneous distribution of the loading eccentric bending moment.For the small load eccentricity, the expandable props tend to be an integral structure, the bending moment distribution is relatively uniform,and the load bearing capacity deviation of the expandable props is relatively small between the numerical simulation and theoretical calculation.When λnis 1.297, the simulated strength of the expandable props is greater than that from theoretical calculation,but the load evolution trend is almost the same.The reason is that the yielding of expandable props tends to elastic buckling,and the height of pipes is the dominant factor that influences the strength of the expandable props.In this manner, the uneven bending moment distribution determined by the load eccentricity has a little effect on the load bearing capacity of the expandable props.

Fig.17.Load bearing capacity of the expandable props with different e.

Fig.18.Relationship between λk and λn.

Fig.19.Relationship between k and λn.

To obtain the theoretical equation for the load bearing capacity of the expandable props,the bending moment amplification factor k is added to Eq.(2) to amend the evolution trend of N, i.e., the curve slope, and the normalized slenderness ratio λnis modified to a converted slenderness ratio λkto optimise the magnitude of load bearing capacity.The revised theoretical equation for load bearing capacity of the expandable props is:

where NEx′is the elastic buckling strength related to the converted slenderness ratio λk,i.e.,NEx′=π2EA/.The modified stability coefficient ScEp′of the expandable prop is related to λk, i.e.ScEp′=[(0.979+0.314λk+)-((0.979+0.314λk+)-)0.5]/.The load bearing capacity of the expandable props with different λnand e can be fitted by an implicit function fitting method, as shown in the blue dotted curves in Fig.17 for three typical normalized slenderness ratios λn.The relations between λk,k and λnare obtained in the black solid lines of Figs.18 and 19,which can be fitted by a linear fitting method in the red dotted lines and Eqs.(4)–(6):

4.Designed load bearing capacity of the expandable props in the situation field

In engineering applications,the strength and failure patterns for expandable props with different heights were obtained by visual inspection and pressure sensors, and the monitoring process has been described in detail in the previous study[40].The load bearing performance of the expandable props is measured by setting the pressure sensors.The sampling frequency was set at 5 min for pressure sensors, and the monitoring data is transmitted to PCs for data analysis through an infrared data acquisition system.In the field application, the support pipes of the expandable props are made of A3 steel Q295, and the outer and inner diameters are 0.108 and 0.096 m, respectively.

Fig.20a shows the load bearing performance of the expandable props with three typical heights of support pipes of 1.0, 3.4 and 5.8 m, which correspond to the normalized slenderness ratios λnas 0.189, 0.599 and 1.018, respectively.The maximum height of support pipes considered was 7.2 m, and the applicable equation for determining the load bearing capacity of expandable props was found using numerical simulation.However, in the field, the height of the expandable prop is lower than 5.8 m, because the construction becomes difficult, and safety issues may preclude it if the support height of the expandable props is too large.After the bending begins, the bearing load on 1.0 m-high expandable prop remains basically constant but decreases for the expandable props with the heights of 3.4 and 5.8 m, which is consistent with the simulated results.Thus, it can be concluded that the larger the slenderness ratio of the expandable prop,the more likely there is to be instability at the extreme bearing load point, as also confirmed in Fig.15.

Fig.20.Load bearing performance and e′ of the expandable props with different H in the field application.

It is well known that uneven stope roofs, construction proficiency and other factors will lead to an increase of loading eccentric distance e, which would result in a decrease of the load bearing capacity of the expandable props and is difficult to measure.The empirical loading eccentric distances e′of the expandable props with different λnon site are back calculated by Eq.(3)based on the monitored strength of the expandable props NM, as presented in Table 3.It is worth mentioning that the height of support pipes presented in Table 3 only refers to an imprecise value(±100 mm)as the support height corresponding to the actual mining situations cannot be an accurate integer, so that the empirical loading eccentric distances e′are an average value for the expandable props with multiple approximate heights.As the height of the expandable props increases by 0.8 m from 1.0 m, the e′increases approximately 7–9 mm at a roughly the same increasing rate, as shown in Fig.20b and Table 3, and can be linearly fitted as e′=0.01003H+24.9804, where H is the height of support pipes.

The load bearing capacity of 1.5 m-high expandable prop is almost equal as evaluated in the laboratory and from numerical simulation 2689 and 2616 kN,respectively,indicating that the laboratory test is close to the axis loading (i.e., loading eccentric distance e≈0 mm).The strength of the 2.7 m-high expandable prop is 1725 kN, and the loading eccentric distance is back calculated by Eq.(3) as 44.67 mm, which is less than 52.06 mm in the field obtained by e′=0.01003H+24.9804.The reason could lie in the uneven stope roof, rough construction quality and other factors for the field tests.

To determine the designed load bearing capacity N′,a safety factor ζ is introduced, which is defined as the ratio of N′to the actual load bearing capacity N of an expandable prop in the field.One of the support parameters is N′, which depends on the stability of rock mass,stress state and other factors.The load bearing capacity of the expandable props N can be calculated from Eq.(3)according to empirical loading eccentric distance e′of the expandable props with different λnvalues.In the engineering application, the designed load bearing capacity N′is equal to the product of actual load bearing capacity N and safety factor ζ,i.e.N′=ζ×N.The support parameters, such as the quantity of support pipes and the yield strength, can be improved to meet support requirements, i.e., to satisfy the safety factor ζ.

5.Design standard of expandable props

Fig.21 displays a proposed design standard for the pre-stressed expandable props,and the design procedure consists of four steps.The first step is to determine the support demand.The second step is to design the expandable structure with a required preload.Li et al.[23] conducted a preliminary study on the relationship between the amount of expandable material, the reserved gap and the active preload magnitude.This paper mainly focused on the study of the third and fourth steps,and the theoretical calculation formula is established for the load bearing capacity of the expandable props, considering both the normalized slenderness ratio λnand the loading eccentric distance e.Meanwhile, the empirical loading eccentric distance e′of the expandable props with different λnis monitored and calculated in the field application as e′=0.01003H+24.9804.The safety factor ζ is introduced to determine the designed load bearing capacity of expandable props N’=ζ×N.

Fig.21.Proposed design standard for the pre-stressed expandable props.

6.Conclusions

The load bearing capacity of the pre-stressed expandable props was determined for different geometries and loading eccentricities by theoretical analysis,laboratory experiments,numerical simulation and on-site tests.

(1) The strength evaluation of expandable devices and the overall structure of the entire prop system with two heights of 1.5 and 2.7 m show that the load bearing capacity of the expandable props depends on the stability of supporting steel pipes and is affected by the normalized slenderness ratios λnand the loading eccentric distances e.Moreover,the load bearing capacity of the expandable props cannot be calculated directly by the strength formula for traditional steel structures.

(2) According to the load bearing performance of the expandable props with different λnand e values, the stability coefficient of the expandable props ScEpconforms to the Perry-Robertson formula with an axial loading (without load eccentricity), and its correlation coefficients were fitted as a=0.979 and b=0.314.To establish a theoretical formula for the load bearing capacity of the expandable props considering both λnand e, the strength calculation for traditional steel structures was analyzed to add a bending magnification factor k and modify λnto the converted slenderness ratio λk.The relations between λk, k and λnwere obtained as λk=1.026λn-0.088, k=0.6608λn-1.25514 (λn≤0.46) and k=-0.4766λn+1.795 (λn>0.46).

(3) The load bearing capacity of the expandable props with different λnvalues was monitored to back calculate the empirical loading eccentric distance e′in the field trials, which is linearly fitted as e′=0.01003H+24.9804.The safety factor ζ was introduced to determine the designed strength of the expandable props: N′=ζ×N.

(4) With the acquisitions from this study, a design standard for the expandable props is proposed and established,including four steps for on-site applications.

Table 3 Empirical e′ of expandable props with different λn in the field application.

The research outputs in this paper can provide theoretical and technical fundamentals for the design of expandable props.Consequently, based on the design standard of expandable props, the expected active support force can be obtained by adjusting the amount of the expandable material and the reserved gap between the support structure and stope roof.Moreover, the load bearing capacity can be adjusted by precisely designing support structures,such as the quantity of the support pipes.However, to accurately obtain the safety factor ζ, further work is needed to enrich the monitoring for the load bearing capacity of the expandable props corresponding to various on-site conditions and mines.It is also indispensable to strengthen the exploration of the expandable structure,to develop the full-scale design standard for expandable props.In the meantime,as the load eccentricity in industrial applications is caused by various factors,such as in the process of welding and fixing the steel posts or pipes to the top platen of the expandable base, the roof-contacted structure and irregularities at contact surfaces.Therefore, seeking the solutions to weaken the eccentric loading condition is an effective means to improve the load bearing capacity of the expandable prop.

Acknowledgments

This work was financially supported by the National Key ResearchandDevelopmentProgramofChina(No.2022YFC2903804), and the National Natural Science Foundation of China (Nos.52004054, 52274115, 51874068 and 52074062).The authors would like to thank Dr.Jerry Ran for providing comments and suggestions for improving the paper quality.