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By Sunil S. Basarkar, Dilip M. Dewaikar, Rajesh H. Gajjar

"A load transfer program AXPILE was developed for simulation of the loadmovement characteristics of socketed piles of the Mumbai reg ion. This program used empirical relationships for ultimate shaft resistance at pile-rock interface recommended by Zhang and Einstein (1998). According to their recommendation, ultimate shaft resistance at pile-rock interface is related to the square-root of uniaxial compressive strength of intact socket material by a parameter C, which ranges from 0.4 for smooth sockets to 0.8 for rough sockets. Results of three instrumented pile load tests were compared with the AXPILE generated results. The comparison of analytical and field estimates of the unit shaft resistance confirm the recommendations of Zhang and Einstein (1998) on the unit side shear parameter c,, which was found to be in the range of 0.35 to 0.67.INTRODUCTIONInstrumented pile load tests were carried out on the test piles at the Bandra-Worli sea link project in Mumbai city of western India. High capacity socketed piles were required to take up loads from the superstructure. Hence, in addition to the available toe resistance, an accurate measurement of the shaft resistance in socket portion was desired. The instrumented load tests incorporated Osterbergcell (0-cell) assisted by strain gauge instrumentation to estimate the shaft resistance mobilization in the socket portion.AXPILE load transfer program was developed for load-movement simulation of socketed piles (Basarkar, 2004). The load transfer algorithm based on Coyle and Reese (1966) was incorporated in this program, which used empirical correlations and functions to account for load transfer in piles through shaft and toe resistance.This paper compares the computed unit shaft resistance from the AXPILE program with the measured values recorded from the instrumented pile load test."

Jan 1, 2005

By A. Haque

"This paper describes an improved formulation of an analytical shear strength model developed by Haque & Kodikara (2012) by incorporating the failure of regular concrete/rock joints subjected to CNS condition. Sokolovskii’s analytical solutions developed for failure stress of slopes have been found to be effective in this regard. The analytical models have been validated for predicting the shear behaviour of concrete/rock joints subjected to various CNS loading conditions. The model has been found to accurately predict the dilation, normal stress and shear stress behaviour for a regular triangular saw-tooth joint up to the peak shear stress. Moreover, the peak shear stresses of concrete-rock joints with asperity angles ranging from 5 to 27.5o compared reasonably well with the available laboratory test results.INTRODUCTIONA new shear strength model for elastic regular triangular concrete/rock joints loaded under Constant Normal Stiffness (CNS) boundary condition has been developed by Haque & Kodikara (2012). The salient feature of this model is its capability of accurately estimating the normal and shear stresses of joints with shear displacements under CNS condition by incorporating the elastic deformation of asperities. However, the shear strength model of Haque & Kodikara (2012) for the elastic asperities cannot be directly employed to predict the peak shear stress of a joint experiencing plasticity under applied loads. Therefore, the model inevitability requires articulation of an appropriate shear failure criterion simulating the boundary conditions of the loaded asperity to predict the peak shear stress and corresponding normal stress and dilation behaviour of concrete/rock joints. Sokolovskii (1960) developed a set of analytical solutions for failure stress of slopes comprising c-??soils, which can be exploited effectively to simulate the failure condition as encountered for loaded regular triangular concrete/rock joints (Seidel & Haberfield, 2002)."

Jan 1, 2015

By N. El-Kaddah, A. D. Patel, T. T. Natarajan

The phenomenon of particle migration in a conducting fluid under an applied electromagnetic force field finds its application in selective sorting of minerals and more recently in the removal of inclusion from molten metal. This paper deals with the analysis of the expulsion force and the flow around a spherical particle in crossed uniform electric and magnetic fields in .infinite and in confined medium. Analytical solutions will be presented for the electrk and magnetic fields as well as the induced flow at the limit of zero Hartmann number. The flow around the particle was investigated numerically at low and high values of Hartmann number. It will be shown the magnetic field has almost no effect on the velocity field when Hartmann number is less than 1.0. At higher Hartmann numbers the damping effect of the magnetic field was pronounced in the direction of the applied current. It is concluded that Hartmann number has little effect on the net force on the sphere particle.

Jan 1, 1999

By Ebelia Manda, Peter RK Chileshe, Sydney Chinyanta, Abdul Samson, Victor Mwango Bowa

Buttressing is a widely employed control measure for mitigating slope failure in various settings such as openpit mines, road sections, dams, and hilly or rocky terrains. This paper presents a study on the effects of buttressing, specifically concerning slide-head toppling failure. Drawing insights from a case study of slide-head toppling in Zambia, our investigation employs both analytical (limit equilibrium) and numerical (3DEC software) modeling techniques. The analytical and numerical models exhibit notable similarities in their schematics and overall results, revealing a subtle effect of buttressing in controlling slide-head toppling failure. This study contributes valuable insights into the nuanced role of buttressing in slope stability, with implications for optimizing buttress design parameters. While the overall findings suggest a modest influence of buttressing on slide-head toppling failure, further discussions delve into the specific implications and potential refinements for practical applications.

Jan 1, 2025

By H. P. Wang, E. M. Perry

"Analytical solutions for investigating the solidification of casting problems with initial superheat and natural boundary conditions have been derived using an integral method. There are two moving interfaces in the solution field, one for the phase change line and the other for the superheat line. Two coupled nonlinear differential equations, representing two moving layers, were calculated by the Runge-Kutta numerical integration scheme. The solutions are in terms of six dimensionless parameters: phase change number, two Biot numbers for interfaces, diffusivity ratio, and dimensionless superheat temperature and time. Parametric studies on these parameters have been conducted to provide generic information about this phase change problem. This analytical information is extremely useful for validating numerical solutions. A simplified I-D finite element model was constructed for the purpose of comparison. The excellent agreement between the analytical and finite element solutions has proven that they are mutually validated.This validated finite element modeling tool was then used for modeling an investment casting process of a super-alloy with a simple geometry: a two-step plate. The nonlinear latent heat release, a common property of alloy materials, was approximated by several piecewise linear segments. The transient temperature response of this 2-D finite element solution, including a metal/mold gap resistance, shows excellent agreement with experimental data.This fundamental work provides a solid basis for further work on the development of a process modeling tool for the complex investment casting process.IntroductionAt present, the investment casting process is used in a variety of industries. In the aircraft engine industry, in particular, investment casting is used to produce parts ranging from turbine blades to engine frames. Because a large single casting can replace many component parts that require machining, fabrication, and assembly, this one-piece casting approach becomes a very cost-effective manufacturing process.The main constraint in using this process for mass production is the long design-to-production lead time. The standard rigging design is still an art, based on the experience of casting experts, rather than on scientific methods. Many trials are often required before making a sound final casting. In order to .reduce the number of expensive casting trials, the development of computer-aided tooling technology is needed.The fundamental physics describing this manufacturing process can be simply described. They reduce to a solid/liquid phase change problem, with two parts - first, the thermal balance of the initial superheat temperature of the molten metal and the preheat temperature of the ceramic shell through heat conduction, and second, the energy release from the mold surface to the ambient by radiation."

Jan 1, 1986

By D Oliveira

The use of an equivalent continuum for a rock mass is not always suitable for situations where the failure is structurally controlled by discontinuities as in the case of wedges in the tunnel roof. In these instances, discontinuum approaches are usually preferred. Rock joints that are filled with soft infill are likely to be the weakest planes in a rock mass, having a dominant influence on its overall shear behaviour. In this case, the joint material model adopted for the discontinuities should be able to describe important mechanisms, such as asperity sliding and shearing, post-peak behaviour, asperity deformation, and the effect of the soft infilling. The latest version of a soil-infilled joint model is discussed here. It describes more comprehensively than previous models the occurrence of dilation and compression with lateral displacements, and also represents the hardening mechanism related to asperity interference as observed in the laboratory that cannot be readily captured by the existing joint models. An analytical approach for the analysis of rock wedges structurally controlled by soil infilled joints is presented.The analytical approach presented in this paper for assessment of symmetric rock wedges in the roof of a tunnel demonstrates, that the peak shear strength of the joint may not be fully mobilised before detachment of the wedge. The analysis verifi ed that an overestimation of the factor of safety may be observed when an elastic-perfectly-plastic joint model was assumed, whereas the computed maximum factor of safety was considerably less when adopting an appropriate soil-infilled joint model.

Mar 8, 2011

By Andrew P. Gize

The analytical approaches used to solve a specific problem are dependent upon the information required, coupled with limitations imposed by available facilities, costs, and time. The organic matter associated with ore deposits has the potential to yield information for two problems. Information can be obtained to provide guidelines for prospecting, and, secondly, to solve genetic problems. A major potential of organic matter in providing prospecting guides lies in the high thermal sensitivity of organic matter in sediments which are metamorphosed at grades lower than greenschist facies. The organic matter can then be used to map thermal anomalies potentially associated with mineralization. The analytical approach is to determine the thermal maturity of the total organic matter, or of specific fractions, using techniques which are relatively quick and inexpensive. Consideration of genetic models (for example, reduction, complexing, involvement of oil field brines) generally requires more specific information than the overall thermal maturity of the organic matter, necessitating infor¬mation on molecular composition and structure. The facilities, time, and cost of this information are greater than for mapping thermal anomalies. The analytical techniques discussed include elemental analyses, spectroscopy (infra-red, visible and ultra-violet), Fourier Transform infra-red, nuclear magnetic resonance, extraction, pyrolysis techniques, chromatography and mass spectrometry.

Jan 1, 2013

By Dean Newman, Michael Behrens, Mark Trim, Justin Arifin

Road tunnels in recent infrastructure projects in Sydney, Australia, are designed to have a fire resistance level of either a 4-hour cellulosic fire or a 2-hour hydrocarbon fire. Where permanent ground support for tunnels consists of systematic rock bolts and relatively thin layer of shotcrete, it is important to ensure that during and after the fire event the ground support is designed/predicted not to pose risks to the tunnel occupant. The global behavior of the ground support can be predicted using finite element analysis, and its resistance is derived with consideration of the material strength degradation due to the elevated temperature.

Dec 1, 2022

By V. A. Scovazzo

The analytical pillar design procedure employing the Wilson equation is based on the site-specific material strength of the floor, pillar, and roof rock. Stable pillars so designed will sometimes be larger than those designed using empirical methods thus providing more stability or at other times smaller than empirically designed pillars thus conserving natural resources. Presented here is a detailed procedure for the analytical design of soft rock pillars using the Wilson Equation. An example of this procedure is presented that uses data from a mine in the Owl Seam in Southern Appalachia. These Owl Seam pillars are now in use and are performing as expected. Also addressed is an assessment of the Dr. Scovazzo?s analytically designed pillars based on pillar performance data gathered by Dr. Mark in the Southern Appalachian Coal Field.

Jan 1, 2010

By M. Hesham El Naggar, Mehdi Heidari

"Equivalent fixed-base cantilever model is a common approach for performance base seismic design of bridge structures supported by extended pile-shaft. In this method, global displacement ductility factor is related to local curvature ductility demand by assuming a concentrated plastic hinge at location of maximum bending moment. Equivalent plastic hinge length is one of the key parameters that should be assumed accurately in kinematic expression of the cantilever model. Although, some expressions have proposed to estimate the length of the in-ground plastic hinge; however, these relations yield differing results. In this paper, a general expression is derived based on the concept of distributed plasticity to predict the plastic region of yielded piles in clay and sand. A length modification factor is defined to relate the equivalent length of plastic hinge to the real plastic zone. Then, results from nonlinear pushover analyses are used to propose an equation for the modification factor as a function of displacement ductility factor.IntroductionDuring severe earthquake events, development of the plastic hinge in pile foundation below ground level is inevitable for bridge structures supported on extended pile-shafts. A common approach for seismic design of the yielding pile-shafts is to replace the pile-soil system by an equivalent fixed-base cantilever assuming the equivalent system has the same global response as that of the pile-shaft (Budek et al. 2000; Chai 2002). In this method, plastic curvature is concentrated at the center of the equivalent plastic hinge below ground surface. Prior knowledge of plastic-hinge length is essential for proper assessment of assess plastic displacement at the top of the pile-shaft. Unfortunately, existing recommendations for estimating plastichinge length are largely incomplete and/or inadequate.During the last two decades, several relationships have proposed for evaluating the equivalent plastic hinge length, Lp, based on experimental and numerical studies. Budek et al. (2000) conducted numerical modeling to calculate the plastic-hinge length of piles embedded in cohesionless soil. They found that the plastichinge length varies linearly with both soil stiffness and aboveground height, e. Chai and Hutchinson (2002) conducted full-scale experimental tests on reinforced concrete piles in sand. They reported that Lp depends primarily on the aboveground height of the pile, neither sensitive to the soil stiffness nor to the value of displacement ductility factor. They stated that the equivalent plastic hinge length is relatively independent of the displacement ductility factor and is not very sensitive to the change in soil density. On the contrary, Goel (2014) and Liu et al. (2016) conducted nonlinear numerical analyses which revealed that Lp changes with the displacement ductility factor and even suggested that using different values of Lp for different seismic design levels is essential for a reliable estimation of the ductility capacity."

Jan 1, 2017

By A Gerdes, H BrStz, E GSbler, D Davis, F Henjes-Kunst, S Dewaele, T Graupner, T Oberthnr, M Sitnikova

Following the United Nations initiative to fingerprint the origin of conflict materials, the German Federal Ministry for Economic Cooperation and Development has decided to fund a pilot study on coltan ores. Since 2006, our working group has been investigating columbite- tantalite (coltan) mineralisation, especially in Africa, also within the wider framework of establishing certified trade chains. More than 300 samples were obtained from the worldÆs major coltan producing areas. Special attention is, however, directed to samples and concentrates from Ta-Nb-Sn provinces in Africa: Democratic Republic of the Congo, Rwanda, Mozambique, Ethiopia and Namibia. Using state of the art analytical tools, we investigate mineralogical and chemical parameters obtained from columbite-tantalite ores and concentrates in order to distinguish between ore provinces, likely even down to the deposit scale. Methods employed include fully automated electron microscopy (mineral liberation analysis), electron microprobe analysis (major and minor elements), laser ablation plasma-source mass spectrometry (trace elements and U-Pb dating), X-ray fluorescence spectroscopy (bulk major and trace elements), X-ray diffraction analysis (mineralogy and structure) and thermal-ionisation mass-spectrometry (U-Pb dating). Major and trace element concentration patterns, mineral assemblages in the ore concentrates, and zoning characteristics in the different pegmatites from Africa distinctly differ from each other. Furthermore, the following age populations are evident:Archaean (>2.6 Ga);Palaeoproterozoic (1.9 - 2.1 Ga);early Neoproterozoic (æKibaranÆ; 0.98 - 0.93 Ga); andlate Neoproterozoic to early Palaeozoic (ca 0.5 Ga). Currently, we focus on the resolution of the fingerprinting system from region via ore province down to deposit scale. Our preliminary analytical results indicate that a certification scheme including fingerprinting of sources of coltan ores is feasible.

Jan 1, 2008

By O. Acaroglu

Selection of the most suitable roadheaders (boom type tunnelling machines) for the galleries/tunnels to be excavated in certain rock and under known project conditions is important for avoiding problems during application. This research utilized a multiple attribute decision-making method and studied how this will be applied for the selection of boom-type excavation machines. The Cayirhan Coal Basin was used for selecting roadheaders using the analytical hierarchy process, which is one of the multiple attribute decision-making methods. To conclude, this study reveals that decision-making methods can be used in the process of selecting roadheaders for the excavation of galleries. Keywords: roadheaders, multiple attribute decision-making, analytical hierarchy process.

Jan 1, 2006

By Mark E. Hammele

A number of new thoughts have recently emerged concerning the role of air decks, stemming, gas confinement and gross motion in specific blasting environments and applications. Field research in full-scale production environments is based on the theoret ical work f i rst proposed at the Universi ty of Maryland’s Fracture Mechanics Laboratory.

Jan 1, 1989

By Tim Ross

Several shear failures were observed in Shaft #1 at the Mountain Coal Company, LLC, West Elk Mine, after mining longwall Panel 23, 1,100 ft to the east of Shaft #1. It was speculated that this shear damage could be related to differential ground movement caused by in situ stress relief from the ?stress shadow? of the caved zone above longwall Panel 23. A numerical study was conducted to assess the possibility of the shaft shear damage being caused by in situ stress relief and the potential for additional damage to Shaft #1 and two other nearby shafts, due to mining nearby longwall Panel 24. Three-dimensional (3D) models were built in FLAC3D to simulate past and future mining near shafts, the estimated local anisotropic and directional horizontal stresses, and the overlying variable surface topography. The numerical analyses indicated that stress relief due to mining Panel 23 caused the shear damage to Shaft #1and that additional damage to Shaft #1 and the other two shafts, would likely result from mining longwall Panel 24. Additional shear damage was documented in Shaft #1 when longwall Panel 24 was mined, confirming the results of the numerical analyses.

Jan 1, 2006

By Charles R. Barnhart

The Nitrogen Dioxide Cloud: After one year of testing it can be stated that we do know and understand a great deal about why the nitrogen dioxide formation is exacerbated in cast blasting. Picture 1 is an example of how concentrated the NOx can be from a cast blast. It has long been felt that 300 or more milliseconds of timing needs to be put between each row of detonations to allow the rock a maximum time of flight so the cast percentage is high. Our experiments have found that higher cast percentages can make with timing between 150 to 200 milliseconds between rows. Picture 2 show a definite improvement in nitrogen dioxide concentration and most of that was the result of shooting a previously shot end of the cast pattern that had not been dug, thus no confinement of the powder columns.

Jan 1, 2004

Obtaining correct assays for gold determinations relies on the correct methods being chosen and the assayer being made aware of the mineralogy of the sample and its general origin. Usually when a geologist or metallurgist obtains "incorrect" gold assays the problem can be traced to poor communication between the geologist or metallurgist and the assayer.

Jan 1, 1986

By Abhijit Saha

"Two analytical closed-form elastic-rigid-plastic solution methods to predict the rigid foundation behaviour on stone-column reinforced soil are compared, both of which takes into account the stone material yield within the host soil according to stress-dilatancy theory. These are the mechanistic methods developed by the author (2004, 2005) [that presently incorporates the dilatancy angle as per Schanz and Vermeer (1996)] and that developed by Pulko and Majes (2006), that takes into account the column yield as per Rowe dilatancy theory. Since optimal design of stone column necessitates optimum stress concentration on column, which is generally composed of dense gravel with/or without sand, the vertical stress on granular column is often close to its peak strength and the material dilates. To arrive at closed-form analytic solution to this complex soil-stone column interaction problem, some common assumptions like axi-symmetric soilstone column ‘unit-cell’, elastic host soil and rigid-plastic Mohr-Coulomb stone column material are combined with equilibrium and kinematic conditions. The results reflect the beneficial effect of dilatancy within the optimal range or techno-economic domain of area-ratio. The settlement predictions are compared with other analytical methods available in literature as well as with measured settlement reduction of stonecolumn reinforced ground at different subsoil conditions around the globe reported in literature.INTRODUCTIONStone columns or granular piles are frequently used in stabilization of soft clays, silts and loose silty-sands with large amount of fines. The majority of analytical methods assume infinitely wide loaded area with end-bearing columns having constant diameter and spacing that approximate to axi-symmetric conditions, and the approach is known as ‘unit-cell concept’. Several analytical solutions are available, and many of them (Aboshi et al; 1979, Balaam & Booker; 1981) are based on elastic approach considering the stonecolumn and surrounding soil as elastic materials. However, elastic methods give the ratio between the vertical stress in the column to soil (stress concentration factor) approximately equal to the ratio of constrained modulus of both materials. This ratio was found to be considerably higher than measured in the field and it is believed that elastic methods may overestimate the effects of stone columns on settlement reduction (Barksdale & Bachus, 1983). The elastic and elasto-plastic solutions presented by Balaam and Booker (1981, 1985) indicate that the problem can be idealized by assuming that the stone column is in a triaxial state and possibly yielding, such that there is no shear stress at the stone-soil interface and no yielding in the soil. Similar assumptions of stone column to be in a state of limiting equilibrium and under a triaxial stress state are considered in other methods also (Priebe; 1976, Van Impe & De Beer; 1983, Van Impe & Madhav; 1992, Saha & De; 1994, Saha; 2004, 2005, Pulko & Majes; 2006). In majority of the methods it is assumed that under load the stone-column yields at constant volume. However, the last three cases cited consider another parameter; the dilation of stone column. In Van Impe and Madhav’s (1992) and author’s (2004, 2005) method, the dilation is taken into account through the final value of the volumetric strain of column, whereas Pulko & Majes (2006) considered the stone column as Mohr-Coulomb rigid-plastic; dilating at yield in line with Rowe (1962) dilatancy theory."

Jan 1, 2015

"6.1. Mineralogical Techniques ApplicableTo the Platinum-Group Elements6.1.1. IntroductionThe named platinum-group minerals (PGM) usually occur as grains smaller than about 200 microns and, even when present as larger grains as in nuggets, these are often complex multimineral intergrowths. Therefore, the PGM must be examined by microscopic techniques and, because all named PGM are opaque, they are suitable for microscopic study by reflected light. Special techniques for concentrating PGM and preparing suitable polished sections are detailed in Chapter 5. Another characteristic of the PGM is that the majority have very high reflectance at 589 nm, i.e., over 450Jo, but they range from about 38 to 74%. In contrast, micro-indentation hardness has a wider range from - 60 to - 3,000 kg I mm, and the PGM range from brittle to malleable.The small grain-size of PGM imposes certain limitations, or complications, on the usual mineralogical determinative methods of reflectance, micro-indentation hardness and X-ray diffraction. In addition, certain combinations of elements in some PGM render electron-probe analyses difficult. The usefulness and shortcoming of these methods, as applied specifically to the PGM, are discussed in the following sections. These are not, however, meant to be exhaustive descriptions of the techniques, which are fully described in other texts.6.1.2. Micro-Indentation HardnessDetermination of the Vickers Hardness Number (VHN) is discussed in detail in recent publications, such as Bowie and Simpson (1977) and Craig and Vaughan (1981). In addition, the recommendations of the Commission on Ore Microscopy (COM) of the International Mineralogical Association (IMA) are given by Criddle (1980). The usual small grain-size of the PGM, however, precludes the widespread use of this technique and very few PGM have, in fact, had their VHN determined at the COM-recommended standard load of 100 gf (cf. Cabri, 1981b, in Chapter 7). Examination of VHN data reported in the literature reveals that the few PGM whose microindentation hardness has been measured have incomplete VHN data, measured with loads other than 100 gf (Chapter 7). The small grain-size may also result in unreliable VHN data because, even though the correct load may have been applied, a thin grain will give erroneous data due to the matrix, or the grain may even shatter or pop out of the section. On the other hand, micro-indentations may give useful qualitative diagnostic clues. Moncheite, for example, shows fractures near the indentation (Chapter 7, Plate 1). In other cases, intergrown PGM produce different-sized indentations: e.g., froodite-michenerite-gold in Cabri (1972); isoferroplatinumtulameenite in Cabri et al. (1973); michenerite-moncheite in Cabri and Laflamme (1976) and Chapter 7, Plate 1; altaitemichenerite- testibiopalladite in Hudson eta/. (1978) and Plate 1, Chapter 7."

Jan 1, 1979

By A. G. Sharkey

THE PURPOSE of this investigation was to develop methods for analyzing coal derivatives and products from the Fischer-Tropsch synthesis. As part of this investigation it was necessary to study the mass spectral characteristics of many classes of compounds. The mass spectra of alcohols, the trimethylsilyl ether derivatives of alcohols, acetal-type compounds, ketones, esters, and naphthenes were correlated with structure, and analytical methods were formulated. A method was devised to determine the ratio of branched to normal hydrocarbons up to CIS in the Fischer-Tropsch product. Low-ionizing-voltage mass spectrometry was combined with type-analyses to analyze tar acids and neutral oils from coal. The necessary sensitivity correlations at low-ionizing voltage were developed, particularly for phenolic compounds. Special analytical techniques were developed for the mass spectral analyses of specific compounds and compound types including (1) oxides of nitrogen, (2) hydrogen sulfide, (3) isomers of butenes and pentenes, and (4) hydrogen deuteride. An investigation was made of the effects of various hydrocarbons and oxygenated compounds on tungsten filaments in the mass spectrometer. The operating characteristics of rhenium filaments under similar conditions were studied. Several improved sample handling techniques were developed, including an automatic manometer and a self-filling micropipet. Studies were made of anomalous and negative peaks in the mass spectra of certain gases.

Jan 1, 1967

By Debasis Deb, Ankur Narang, G. P. Raja Sekhar, Kamal C. Das

"The stability of underground structures made (especially) in jointed rock mass is of the utmost important to designers, engineers and operators. Rock bolting is generally being practised to reinforce excavation walls and roofs by minimizing the movement of rock joints. This study proposes a new analytical model for the prediction of displacements, axial load and shear stress of a fully grouted rock bolt intersected by single or multiple rock joint(s). The governing equations for rock bolt are derived based on the assumption that shear force generated by slippage along the bolt-grout interface has a linear relationship with respect to the relative displacement at the interface. The proposed model is novel compared to the existing literature in the sense that it accounts for rock kinematics as an enriched sum of a continuous part (elastic) and a discontunous (jump) part due to the presence of joint planes. The complete solution of the bolt displacement, axial load and shear stress is derived as a function of the rock displacements, and, it comprises of Heaviside and/or Dirac delta functions in the solution space due to the jump at the intersection point(s) of bolt(s) and joint(s).Our novel solution is able to predict the elastic behavior of a grouted bolt, subjected to opening of the joint planes in pull-out test and also in excavated tunnel. These two cases also serve as benchmarks for grouted rock bolts with multiple joints and can be used for validation of numerical and experimental models. Further, the solution is applied to analyze the behaviour of grouted rock bolt as well as cable bolt installed in circular tunnel, with variations in rock joint(s) opening displacement, shear stiffness in the boltgrout interface and locations of the joint planes. Results indicate that for a specified grout stiffness, the maximum axial load developed in the bolt for discontinuous distribution of rock mass displacement is significantly higher than that of the continuous distribution of rock displacement. This paper provides an analytical interpretation of bolt beaviour considering multiple joint planes with variable joint opening displacements. The results obtained via the analytical method are in line with the corresponding investigation done by other authors. Thus, our analytical treatment can be used to understand bolt behaviour in realistic situations."

Jan 1, 2015