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M.R. Yeung

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Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 45th U.S. Rock Mechanics / Geomechanics Symposium, June 26–29, 2011

Paper Number: ARMA-11-203

Abstract

ABSTRACT In the present high-order 3-D DDA method, block contact constraints are enforced using the penalty method. This approach is quite simple, but may lead to inaccuracies that may be large for small values of the penalty number. The penalty method also creates block contact overlap, which violates the physical constraints of the problem. These limitations are overcome by using the augmented Lagrangian method that is used for normal contacts in this research. In this paper, contact constraints are enforced in high-order 3-D DDA using the augmented Lagrangian method and the formulations are presented. Moreover, a code has been programmed by Visual C++ and an illustrative example is used to validate the new formulations and the code. Using the augmented Lagrangian method to enforce contact restraints retains the simplicity of the Penalty method and reduces the disadvantages of it. 1. INTRODUCTION The discontinuous deformation analysis (DDA), which is an energy-based method, is an alternative to the distinct element method (DEM) for discontinuity-based problems [1]. This method can be used to solve problems involving discontinuous media. Original DDA formulation utilizes first order displacement functions to describe the block movement and deformation. Therefore, stress or strain is assumed constant through the block and the capability of block deformation is limited. This may yield unreasonable results when the block deformation is large and geometry of the block is irregular. There are some published papers on deformable blocks in 3-D DDA. Beyabanaki et al. [2-4] implemented Trilinear and Serendipity hexahedron FEM Meshes into 3-D DDA. Beyabanaki et al. [5-7] presented 3-D DDA with second-and third-order displacement functions. Beyabanaki et al. [8] presented 3-D DDA with n th -order displacement functions. Recently, contact theory of n th -order 3-D DDA is presented by Beyabanaki et al. [9]. The penalty method was originally used by the abovementioned 3-D DDA researchers to enforce contact constraints at the block interface. The accuracy of the contact solution depends highly on the choice of the penalty number and the optimal number cannot be explicitly found beforehand. Obviously, the penalty number should be very large to achieve zero interpenetration distance. However, a very high penalty number leads to progressive ill-conditioning of the resulting system and thus one cannot hope to achieve high-accuracy solutions with this approach. A well-known method to overcome these problems for equality constrained problems is the augmented Lagrangian method [10]. The augmented Lagrangian method has been advocated by Lin et al. [11] in two dimensional discontinuous deformation analysis. In this research, the same method has been implemented in high-order three-dimensional discontinuous deformation analysis and an illustrative example is presented for demonstrating this new approach. 2. HIGH-ORDER 3-D DDA In the original 3-D DDA, the block displacements function is equivalent to the complete first-order displacement approximation; constant strains and constant stresses are assumed within each block. When displacement functions are taken as n th -order functions: The high-order function is necessary in most engineering analyses since it can represent stress concentrations within one block. In two dimensions, the contact types between blocks include corner-to-corner, corner-to-edge and edge-toedge;

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 45th U.S. Rock Mechanics / Geomechanics Symposium, June 26–29, 2011

Paper Number: ARMA-11-535

Abstract

ABSTRACT This paper reports results from a study of sliding and toppling failures in jointed rock slopes using the base friction modeling technique. The base friction modeling technique is a commonly used physical modeling technique in rock mechanics to study behavior of rock masses. In a base friction model test, a two-dimensional model is prepared and laid horizontally on the belt of the base friction machine. The model is then dragged by the belt against a fixed barrier. The frictional drag between the base of model and the belt simulates gravity in the direction of the belt movement. Results from base friction model tests on model rock slopes consisting of rectangular wooden blocks show that the factors affecting model behavior are 1) the geometry of the block system that comprises the rock slope, and 2) the friction angle of the block-to-block contact surfaces. Base friction results are compared with analytical results, and there is good agreement between our experimental results and analytical results. All the experimental results show that as the friction angle increases the initial behavior changes from sliding to toppling and sliding and then to toppling. A special case of a “cross-jointed” rock mass with an amount of “offset,” was also studied. Base friction model test results show that as the amount of offset increases, the initial behavior of the slope changes from toppling to toppling and sliding and then to sliding. INTRODUCTION The stability of rock slopes is of practical significance in the field of rock mechanics. A slope may be permanently stable, locally unstable but globally stable, or locally as well as globally unstable depending upon the geologic conditions and the geometry of the slope [1]. Toppling failure has attracted particular interest in the field of rock mechanics because the speed of the failure can range from low to extremely high [2]. Past research has analyzed rock toppling using 1-g physical and two-dimensional analytical models. The purpose of this research was to study toppling as a two-dimensional problem using the base friction modeling technique. The base friction modeling technique is a common technique for analyzing problems of complex movement patterns in a rock mass. It was first suggested by Goodman in 1969 [3]. He simulated the gravitational force by pulling out a piece of paper underneath a physical model, and demonstrated the movement pattern of the model. The base friction technique is not only a qualitative but also a quantitative tool for rock mechanics problems, with an accurate measurement system, proper scaling and interpretation. A two-dimensional model is prepared and laid horizontally on the belt of the base friction machine. The model is then dragged by the belt against a fixed barrier. The frictional drag between the base of model and the belt simulates gravity in the direction of the belt movement. Factors affecting toppling were studied by changing the geometry of the rock block system that comprised the rock slope. Also, the effect of the friction angle on toppling was observed by using two different friction angles.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 45th U.S. Rock Mechanics / Geomechanics Symposium, June 26–29, 2011

Paper Number: ARMA-11-489

Abstract

ABSTRACT Large deformations and rotations of rock blocks may occur under high stresses around deep underground rock engineering works such as deep mines and deep tunnels. The original discontinuous deformation analysis (DDA) was developed by Shi and Goodman to analyze large deformations, rotations and displacements of rock blocks by accumulating small components of these quantities in a time-marching scheme. The small rotation angle approximation adopted in the original DDA may induce block expansion with rotation (free expansion). Some methods have been used to study and reduce the rotation errors, including the Taylor series method and trigonometric method. Based on mechanics, the free expansion is caused by the error due to the approximation of the real behavior using a displacement function, and the theory used to describe the geometrical relationship between movement (displacement and rotation) and deformation. The original DDA uses the geometrical relationship that is based on small strain theory. Small strain theory gives a linear relationship between displacement and strain that does not consider the high order components and the decomposition of rotation and displacement. The finite deformation theory, however, decomposes the displacement and rotation and gives a nonlinear relationship between displacement and strain. Therefore the finite deformation theory can handle the block rotation problem more correctly. In our previous work, we compared the displacement field around a circle tunnel obtained from the finite deformation theory and small strain theory. The results show that the difference increases as the deformation increases. The rotation error due to the small strain assumption and the validity of the finite deformation theory were also studied by Chen using an analytical method. In this paper, we use the finite deformation theory to adjust the displacement and strain components in DDA to study the free expansion of blocks in a model in which a rock block falls down a slope. The area of the block is monitored during the fall. The expansion of the block computed by DDA modified by the finite deformation theory, and the original DDA will be compared. The result shows that the DDA modified with finite deformation theory can eliminate the free expansion. INTRODUCTION The discontinuous deformation analysis (DDA) developed by Shi and Goodman was a 2-D numerical method for the statics and dynamics of discontinuous block systems. Shi adopted the small angle approximations (sin r 0 ˜0, cos r 0 ˜1) in the original DDA formulation for the displacements of a point within the block. This simplification is convenient for formula derivation and computationally efficient and fast, particularly for small rotation of the blocks. However, the linearization of the rotational displacements causes the blocks to expand with every increment of rotation and results in distortion of stress and velocity fields. Ke proposed to use a postadjustment with a maximum rotation limit, 0.1 radian, set up to reduce the errors in computing contact forces.This approach is easy to implement and can prevent the blocks from expanding. Koo and Chern also proposed to use linear displacement function and post-correction.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 44th U.S. Rock Mechanics Symposium and 5th U.S.-Canada Rock Mechanics Symposium, June 27–30, 2010

Paper Number: ARMA-10-316

Abstract

ABSTRACT: In this study, contact theory (including contact detection and mechanics) of deformable blocks in nth-order Three- Dimensional Discontinuous Deformation Analysis (3-D DDA) is presented. When high-order 3-D DDA is employed, block faces may deform and not remain planar. In this case, conventional contact models cannot be used. To deal with this difficulty, the authors propose a simple technique. In this research, formulations of stiffness and force matrices in nth-order 3-D DDA due to normal and shear contact forces are presented, as well. One illustrative example is used to validate the new formulations and codes for high orders of displacement functions. 1. INTRODUCTION The Discontinuous Deformation Analysis (DDA) method is a recently developed technique that is a member of the family of DEM methods. It is a pioneering method used to analyze the mechanical behavior of discrete blocks, In contrast, DDA as a complete block kinematics - a key component in dealing with interacting discrete blocks, guarantees the system equilibrium at any time, and is a real-time analysis. Both static and dynamic analyses can be conducted with the DDA method [1]. Original DDA formulation utilizes first order displacement functions to describe the block movement and deformation. Therefore, stress or strain is assumed constant through the block and the capability of block deformation is limited. This may yield unreasonable results when the block deformation is large and geometry of the block is irregular. In 2-D, to overcome these limitations, some approaches have been attempted. An approach to resolve this problem was to glue small blocks together to form a larger block using artificial joints [2] and sub-blocks [3]. Some researchers added finite element meshes in the blocks so that stress variations within the blocks can be accounted for [4-6]. An alternative approach is to include more polynomial terms in displacement function. Chern et al. [7] and Koo et al. [8] added the second-order to DDA. Ma et al. [9] and Koo and Chern [10] implemented the third-order displacement function in the 2-D DDA method. Hsiung [11] developed a more general formulation of the 2-D DDA. There are some published papers on deformable blocks in 3-D DDA. Beyabanaki et al. [12-14] implemented Trilinear and Serendipity hexahedron FEM Meshes into 3-D DDA. Beyabanaki et al. [15-17] presented 3-D DDA with second- and third-order displacement functions. In this paper, contact theory of nth-order 3-D DDA is presented. In this research, formulation of normal and shear contact forces are presented and applied to two examples. 2. APPROXIMATION OF GENERAL HIGHORDER DISPLACEMENT FUNCTIONS IN 3-D DDA In 3-D DDA, the large displacements are an accumulation of the small displacements and deformations in a time step. Within each time step, the X ,Y and Z displacements, (u,v,w) , at any point (x, y, z) in a block can be represented using the approximation of a polynomial displacement function. In the original 3-D DDA, the block displacements function is equivalent to the complete first-order displacement approximation; constant strains and constant stresses are assumed within each block.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 44th U.S. Rock Mechanics Symposium and 5th U.S.-Canada Rock Mechanics Symposium, June 27–30, 2010

Paper Number: ARMA-10-517

Abstract

ABSTRACT: The depth of tunnel excavations has increased in mining practice, as shallower mines have been exhausted. The excavation of a deep tunnel presents many challenges, including high stresses and high tunnel support costs. Therefore, it has become very important to accurately predict the displacement field in the rock surrounding the tunnel, which can then lead to a safer support scheme. In practice, the displacement field can be computed using classical small strain theory. In this paper, we use the finite deformation theory (SR additive decomposition) to compute the displacement field in the surrounding rock and then compare the results obtained with those from small strain theory. Furthermore, we computed using the finite deformation theory the deformation of the surrounding rock, assuming that the rock is viscoplastic. The results show that the difference in the solutions from the two theories is more significant near the tunnel wall, where a fractured and loosening zone can greatly affect the stability of the tunnel. The viscoplastic solution also shows that as time increases, the difference between the finite deformation solution and the small strain solution increases. 1. INTRODUCTION With the rise of exploitation depth, the deformation of deep tunnels under high in situ stress, the cost of tunnel support and the amount of renovation increase significantly [1, 2, 3, 4]. Therefore, the tasks of predicting the deformation of tunnels exactly and designing a reasonable support scheme have become more and more important. All of the above considerations should be based on the accurate description of the displacement field in the rock surrounding a tunnel. Although researchers tried to analyze and describe the deformation of tunnels [5, 6, 7, 8, 9], the finite deformation analytical solution of the surrounding rock displacement field has not been studied. The classical small strain theory has essential limitations [10, 11, 12, 13], whereas the finite deformation theory (additive decomposition), which is based on precise mathematical theory, has been validated to be a correct method to describe movement and deformation [14]. In this paper, the finite deformation theory is used to analyze the displacement field in the rock surrounding a tunnel which is located in a hydrostatic in situ stress field. The solution is compared with the solution from classical small strain theory. In addition, the deformation of the surrounding rock having viscoplastic properties is also analyzed with the finite deformation theory and the solution compared with that given by Lu and Mao (2008) [9]. 2. ANALYSIS OF DISPLACEMENT FIELD USING FINITE DEFORMATION THEORY2.1. Problem Definition The model in Fig. 1. is assumed as follows: (i) A circular tunnel is driven in a CHILE (Continuous, Homogeneous, Isotropic, and Linearly Elastic) rock mass with elastic modulus E and Poisson's ratio µ. (ii) The in situ stress is Pb while the supporting force is Pa 2.2. Derivation of Displacement Field in Surrounding Rock The finite deformation theory is a theory that decomposes movement into two parts, a deformation and a rotation [13, 14], called S-R decomposition.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 43rd U.S. Rock Mechanics Symposium & 4th U.S. - Canada Rock Mechanics Symposium, June 28–July 1, 2009

Paper Number: ARMA-09-005

Abstract

Abstract In the original 3-D DDA formulation, first-order displacement was assumed for block deformation, which precludes the application of it to problems with significant stress variations within blocks. This may yield unreasonable results when the block deformation is large and geometry of the block is irregular. Up to now, 3-D DDA with third-order displacement functions is developed. However, there are applications that may require using polynomials greater than the third-order to achieve better accuracy. This study presents the results of an effort to develop a more general approach in which the 3-D DDA is implemented with higher-order polynomial displacement functions. In this research, formulations of stiffness and force matrices in nth-order are presented and the codes have been programmed. An illustrative example is used to validate the new formulations and codes for different orders of displacement functions. By contrast, the results calculated for the same model by use of the third order 3-D DDA are far from the theoretical solution. 1. INTRODUCTION Discontinuous deformation analysis (DDA) is a discrete element method growing in popularity for geomechanical simulation. The DDA method is generally formulated using the principle of minimum potential energy [1,2]. In the original DDA formulation, a first-order displacement function was used to model block deformations, which does not allow for variable stress/strain distribution within a block. This approximation precludes the application of the first order polynomial function to problems with significant stress variations within blocks. This may yield unreasonable results when the block deformation is large or when the geometry of the block is irregular. In two dimensions, to overcome this shortcoming, several approaches have been attempted. One approach adopted was to glue small blocks together in an artificial manner to form a larger block. Ma et al. [4], Koo et al. [5] and Hsuing [6] implemented high-order displacement functions into the DDA algorithm. Shyu [7], Chang [8] and Grayeli & Mortazavi [9] implemented finite element mesh into the original DDA blocks to account for stress variations within the blocks. A major shortcoming of the original 3-D DDA formulation is that, similar to the 2-D DDA approach, the block deformation is modeled using a first-order polynomial displacement function, which does not allow for variable stress/strain distribution within a block. This approximation leads to large errors in problems in which stress variations within the blocks are expected to be significant. Practical examples would include cases where the sizes of rock blocks spanning the joints and discontinuities are large enough so that their deformability becomes important. Beyabanaki et al. [10, 11] has proposed the use of a third-order displacement function but this may still be inadequate for applications that require the use of polynomials greater than the thirdorder to attain better precision. In this paper, an effort is made to develop a more general formulation of the 3-D DDA and the codes have been developed and used to calculate deformation of a simply supported beam. 2. THE THEORY OF 3-D DDA Similar to 2-D, 3-D DDA analyzes a problem as an assembly of discrete blocks.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 42nd U.S. Rock Mechanics Symposium (USRMS), June 29–July 2, 2008

Paper Number: ARMA-08-349

Abstract

ABSTRACT: In the original 3-D DDA formulation proposed by Shi, a first-order displacement function was utilized to describe the block movement and deformation. Therefore, stresses and strains throughout the block are assumed constant and the capability of block deformation is limited. In this paper, a third order displacement function is proposed for analysis of problems using 3-D DDA. The third-order displacement function allows nonlinear distribution of stress and strain within a discrete block, which significantly enhances the ability of 3-D DDA as an analysis technique. An illustrative example is presented to show the improvement achieved by this model. The calculated results show close agreement with the theoretical solutions. 1. INTRODUCTION The Discontinuous Deformation Analysis (DDA) is well-suited for investigating fractured rock mass behavior important to many geotechnical and structural problems [1, 2]. The method has the following major characteristics [2]: The principle of minimum total potential energy is used to calculate an approximate solution similar to FEM. Dynamic and static problems can be solved by applying the same formulations. Any constitutive law can be incorporated. Any contact criterion (i.e., Mohr-Coulomb criterion), boundary conditions (i.e., constraint displacement), and loading conditions (i.e., initial stress, inertia force, volume force, etc.) can be modeled. In the original DDA formulation, a first-order displacement function was used to model block deformations, which does not allow for variable stress/strain distribution within a block. This approximation precludes the application of the firstorder polynomial function to problems with significant stress variations within blocks. This may yield unreasonable results when the block deformation is large or when the geometry of the block is irregular. In two dimensions, to overcome this shortcoming, some approaches have been attempted. One approach adopted was to glue small blocks together in an artificial manner to form a larger block. Ma et al. [3], Koo et al. [4] and Hsuing [5] implemented high-order displacement functions into the DDA algorithm. Shyu [6], Chang [7] and Grayeli & Mortazavi [8] implemented finite element mesh into the original DDA blocks to account for stress variations within the blocks. In 3-D, there have been some published works but they use a linear displacement function as in the original 2-D DDA, so the stresses and strains within each block are constant. In this paper, the 3-D DDA with third-order displacement functions is derived. The Visual C++ .Net code for the third-order 3-D DDA has been developed and used to calculate a cantilever beam deformation under a force. 2. FUNDAMENTALS OF 3-D DDA In the DDA method, the formulation of blocks is very similar to the definition of a finite element mesh. A problem of the finite element type is solved in which all elements are physically isolated blocks bounded by preexisting discontinuities. When the blocks are in contact, Coulomb?s law is applied to the contact interfaces, and the simultaneous equilibrium equations are formed and solved at each loading or time increment. The large displacements are the accumulation of incremental displacements and deformations at successive time steps.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 42nd U.S. Rock Mechanics Symposium (USRMS), June 29–July 2, 2008

Paper Number: ARMA-08-339

Abstract

ABSTRACT: A rock mass stability analysis method coupling block theory and three-dimensional discontinuous deformation analysis (3D DDA) has been developed. In this paper, two case histories are presented in which the coupled method developed was used to analyze the stability of underground openings. One case history involved a proposed new highway tunnel, and the other involved an underground chamber for a hydroelectric power plant. The stability analyses allowed the design of stabilization using rock reinforcement. It was found that for these two case histories, the stabilization designs from the coupled method gave stabilization designs that were less conservative than the actual designs. 1. INTRODUCTION Block theory, developed by Goodman and Shi [1], has some limitations. The mode and stability analyses consider only sliding modes and some special rotational modes and cannot handle general modes of simultaneous sliding and rotation. Furthermore, they do not consider dynamic equilibrium. To overcome these limitations, block theory?s removability analysis is coupled with three-dimensional discontinuous deformation analysis (3D DDA) single block analysis. 3D DDA single block analysis is used to perform the mode and stability analyses, because DDA considers dynamic equilibrium and can handle general modes of failure including simultaneous sliding and rotation. In this way, the advantages of both block theory and DDA can be combined. A design method coupling block theory and 3D DDA has been developed [2]. To test the coupled method, the coupled method has been applied to several case histories. Two of these case histories involved rock slopes, one involving a rock slope in the Three Gorges Reservoir area in China [2, 3], and one involving some rock slopes in Macau, China [3]. This paper presents two other case histories in which the coupled method was applied to analyze underground openings, one involving a tunnel and one involving an underground chamber. 2. CASE 1: THE PROPOSED ELK CREEK TUNNEL REPLACEMENT The Elk Creek Tunnel Project involved replacing an existing highway tunnel and two adjoining bridges near Elkton, Oregon [4]. The project was designed but has not been constructed. The new tunnel would be approximately 366 m long. The design called for a horseshoe-shaped tunnel, with a width of 13.1 m, a maximum height of 8.2 m, and a tunnel roof arch height of 4.5 m. The tunnel is horizontal and has a curved alignment in plan, with an average tunnel axis orientation of N56°E. 2.1. Rock Joint Properties The rock mass at the tunnel site consisted of sandstone and siltstone/shale. The structural features of the rock mass included bedding planes, joints, and a fault near the east portal of the existing tunnel. Four discontinuity sets were identified through statistical analysis of stereographic projections of joints and bedding planes measured in the field. The orientations of the discontinuity sets and the fault are given in Table 1 [4]. Based on laboratory direct shear and triaxial shear tests on discontinuities, the friction angles of joints in sandstone, joints in siltstone/shale, and bedding planes were estimated to be 42°, 35°, and 24°, respectively. The friction angle of the fault plane was estimated to be 35° [4].

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the Alaska Rocks 2005, The 40th U.S. Symposium on Rock Mechanics (USRMS), June 25–29, 2005

Paper Number: ARMA-05-751

Abstract

ABSTRACT: Acoustic emission monitoring has been used extensively in field and laboratory tests. Locating acoustic emissionsources has been proven to be a useful non-destructive analytic technique to study the progressive damage process of rock at itstrue stress states. In most cases, the P-wave velocity was treated as a known value. However, P-wave velocity is usually difficult toobtain since ultrasonic sensors are very sensitive to sudden shocks. In the present study, acoustic emission source location wasdeveloped by treating P-wave velocity as an unknown variable and then solving the problem by least square method. The accuracywas first validated by a benchmark test. Another test was conducted on granite in MTS 815 system with acoustic emissionmonitoring to demonstrate its application in study of damages within specimen under constant loading. The results indicated thatthe source location determined by the algorithm proposed in this paper can predict the damage zone fairly well. INTRODUCTION Acoustic emission (AE) is defined as the transientelastic wave generated by the rapid release ofenergy from a source within a material. Sinceacoustic emission monitoring is essentially passive,it provides an ideal non-destructive method forstudying failure mechanism of rock. One importantaspect is that counting the number of AE eventsprior to sample failure. Lockner & Byerlee [1]showed a correlation between AE rate and inelasticstrain rate. Lin et al [2] had studied the deformationcharacteristics, damage accumulation and strengthdegradation of granite under constant loading andproposed that cumulative AE count should be agood indicator of damage. Locating acoustic emission sources has been provento be a useful non-destructive analytical technique.Scholz [3] first located the hypocenters of microcracksfor Westerly granite under uniaxialcompression from acoustic emission and drew oneimportant conclusion in which the micro-cracks arequasi-uniformly distributed throughout thespecimen at lower load but nucleates in the fractureprocess zone at higher load. Mogi [4] confirmed thevalidity of this technique by a bending test. Furtherstudy of the nucleation and growth of fractures inbrittle rock was carried out by Lockner et al [5].The locations of acoustic emission sources duringdeformation of rock showed the nucleation andgrowth of macroscopic fault planes in granite. Linet al [6] studied progressive damage process ofgranite under constant loading by couting theacoustic emission and locating the acousticemission sources. Damage was quantified bycumulative acoustic count and the progressivedamage process was shown by spatial distributionof acoustic emission sources during three creepstages. Katsuyama [7] located the AE hypocentersby assuming that P-wave velocity was anisotropic. In the above studies, P-wave velocity was taken as aknown value. It is well known that ultrasonic sensoris very sensitive to sudden shock and damage. It isdifficult to obtain P-wave velocity during the wholetesting process. In this study, P-wave velocity willbe treated as an unknown variable in the sourcelocation algorithm.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the Alaska Rocks 2005, The 40th U.S. Symposium on Rock Mechanics (USRMS), June 25–29, 2005

Paper Number: ARMA-05-752

Abstract

ABSTRACT: To overcome some limitations of the mode and stability analyses of Block Theory, Block Theory?s removabilityanalysis is coupled with three-dimensional Discontinuous Deformation Analysis (3D DDA) single block analysis. This results in adesign method for discontinuous rock masses: Block Theory?s removability analysis is used to identify all removable block types,and the 3D DDA single block analysis is used to determine the mode and stability of each removable block type. In this paper, acomputer program that implements the coupled method is introduced. Selecting a block size for design is also discussed. Thestability analyses will give the factors of safety and if necessary allow the design of stabilization using rock anchors. A case historyis used to test the method. The results show that the coupled method can be applied as a design method for rock slope engineering,with potential applications in the design of tunnels and underground chambers. INTRODUCTION Block Theory, developed by Goodman and Shi(1985), is a three-dimensional geometrical methodthat allows a rigorous inventory and analysis of rockblocks that can be formed by intersecting rock massdiscontinuities and free surfaces. A complete BlockTheory analysis consists of the removabilityanalysis, mode analysis and the stability analysis.The removability analysis gives a list of removableblocks that can be formed by intersecting rock massdiscontinuities and free surfaces. A removableblock is one that can move in some way into thefree space without pushing into neighbouring blocks.Whether it will move or not, however, also dependson whether it has a mode of failure, determined bythe mode analysis. Finally, a stability analysis isperformed on each removable block that has a modeof failure to arrive at the keyblocks, blocks thatwould fail without support. By the fundamentalaxiom of block theory, if all the keyblocks arestabilised, the entire rock mass will be stable.Therefore, one only needs to stabilise all keyblocksto assure stability of the entire rock mass. Thisrigorous narrowing down of the types of blocks thatneed to be considered, from the many possibilitiesthat exist in a discontinuous rock mass to just ahandful of keyblocks, is the beauty of Block Theory. At present, there are limitations to the mode andstability analyses of Block Theory. The mode andstability analyses consider only sliding modes andsome special rotational modes and cannot handlegeneral modes of simultaneous sliding and rotation.Furthermore, they do not consider dynamicequilibrium, which has been shown to give correctfailure modes of blocky systems. To overcomethese limitations, three-dimensional DiscontinuousDeformation Analysis (3D DDA) single blockanalysis, which has been developed from theoriginal two-dimensional DDA (2D DDA) (Shi andGoodman, 1988), is proposed to perform the modeand stability analyses because DDA considersdynamic equilibrium and can handle general modesof failure including simultaneous sliding androtation. In this way, the advantages of both BlockTheory and DDA can be realized.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the Alaska Rocks 2005, The 40th U.S. Symposium on Rock Mechanics (USRMS), June 25–29, 2005

Paper Number: ARMA-05-750

Abstract

ABSTRACT: Heat generated from the waste in the repository will result in an increase in rock temperature and substantially change the stress state of the host rock surrounding a nuclear waste repository. In this study, a series of uniaxial constant loading tests with acoustic emission monitoring has been conducted on wet granite in MTS 815 under different stress levels at room temperature, 50°C and 90°C. The results indicate that temperature under 50°C has no significant effect on the time dependent behavior. When the temperature is raised to 90°C, the crack damage stress, which is related to the long-term strength of granite, decreases to 90% of that at room temperature. The time to failure response decreases accordingly. Effect of temperature on the strain rate and acoustic emission rate under constant loading will also be discussed in the paper. Therefore, rock engineers need to consider the long-term in-situ strength of a rock at different temperatures in the design and construction of excavations in the rock for nuclear waste repositories. INTRODUCTION Geological disposal has been accepted as the best option for disposing high-level radioactive waste. Heat generated from the waste in the repository will result in an increase in rock temperature and substantially change the stress state in the host rock. It is also important to minimize potential pathways for the transport of radionuclides in the rock surrounding a repository. Preferred hydraulic pathways could be created if the rock around excavations is damaged due to stress redistribution or under sustained loading. The combined impact of temperature, stress and water on the time dependent behavior of granite is of particular interest and concern because it is in-situ loading a rock mass surrounding a nuclear waste repository will experience. Granite has been considered as an ideal site to construct nuclear waste repository due to its high structural capacity and low permeability. A number of studies had been conducted to study the time dependent behavior of rocks under constant loading at room temperature [1-5] and temperatures near melting [6-8]. However, there is only little information in the intermediate temperature range. Kranz et al [9] indicated that the effects of increasing temperatures were to weaken the rock appreciably and to reduce the time to failure by about two orders of magnitude from 24°C to 200°C. Lau et al [10] studied on the Lac du Bonnet pink granite and found that the higher the temperature (up to 125°C), the greater was the water-weakening effect. Failure of granite is brittle at low temperature and deform permanently by micro-fracturing at stresses greater than crack damage stress, at which the volumetric strain starts to reverse. This permanent strain is accompanied by a volume increase (dilatancy) with acoustic emission associated with micro-fracturing [11, 12]. Since acoustic emission monitoring is essentially passive, it provides an ideal non-destructive method for studying crack growth at its true stress state. Recently, Lin et al [13] studied the progressive damage process of granite under constant loading by acoustic emission source location technique and successfully mapped the crack propagation during the three creep stages.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the DC Rocks 2001, The 38th U.S. Symposium on Rock Mechanics (USRMS), July 7–10, 2001

Paper Number: ARMA-01-0451

Abstract

ABSTRACT: This paper presents the application of a Geographic Information System (GIS) to the assessment of karst collapse risk in Tangshart, China. It gives the main factors that influence the risk of karst collapse and the method for assessing it. Using the automated GIS-based risk assessment method developed, three zones with different risk levels are delineated in downtown Tangshan. Based on the risk assessment resuits, recommendations are given on the risk assessment method and development of downtown Tangshah. INTRODUCTION Tangshan is an important industrial city in China. In 1978 a catastrophic earthquake, with a magnitude of 7.8, rocked this city and took the lives of a quaaer of a million people. This earthquake also brought many unforeseen problems subsequently to Tangshah. Karst collapses are one of the unforeseen problems. From 1979 to 1994, 18 confirmed or possible karst collapses occurred with a total affected area of nearly 20 km 2. A karst collapse usually originates from the loss of stability of a hidden karst cave in a fragmented rock mass. Since karst caves are usually covered by Quaternary sediments, they are difficult to locate and their stability difficult to assess. An analysis of the factors influencing karst collapse leads to a regional risk assessment and a hazard zoning scheme for the re-construction of Tangsban, especially the new downtown.