B. Table of Contents

Transcription

1 A. Abstract In this report the results of three years of research, conducted simultaneously at the department of geological and environmental sciences at Ben-Gurion University (BGU), department of civil engineering and the Technion (Technion) and the department of civil and environmental engineering at the University of California at Berkeley (UCB) are summarized. The goal of this research was to study the deformation of a horizontally layered and vertically jointed roof of underground openings in rocks using documented case histories, analytical and numerical solutions, and physical modeling tests. For case histories the 3000 years old underground water storage facility at Tel Beer Sheva was used as a model. The numerical solutions utilized the Discontinuous Deformation Analysis (DDA) method. Physical modeling tests were centered on scale models of the analyzed problem using centrifuge modeling. Much effort by both the BGU and UCB teams was dedicated to validation of the numeric DDA method. As a result several papers have already been published and some are being submitted for publications at present. We used both analytical solutions and shaking table experiments in the DDA validation study and came up with a set of guidelines concerning numerical control parameters, primarily the accuracy and sensitivity of the selected contact spring stiffness, time step size, and energy dissipation parameter. We believe that we have brought DDA to a level that it can be applied to solve complex problems in discontinuous media with reasonable certainty regarding the accuracy of the numeric solution. Centrifuge model tests were performed by the Technion group and the exact same beam configurations were then modeled by DDA. The set of centrifuge model tests represents a breakthrough in physical modeling technology and in our ability to measure deformation characteristics of a discontinuous beam. The comparison with DDA simulations seems promising yet some DDA limitations become clear as a result of this exercise, first and foremost the inability of DDA to calculate stress distributions through the body of individual blocks. For discontinuous beam problems a small number of blocks this becomes a significant disadvantage. Future research in discontinuous deformation modeling is required in order to resolve this issue. 1

2 B. Table of Contents 1 Introduction Stability analysis in stratified and jointed rock mass Rock mass classification The voussoir Beam Analogy Numerical Modeling Numerical modeling of jointed beams Physical Modeling of the Voussoir beam Current Research Motivation Objectives of research Organization of Research Centrifuge modeling Introduction Shear similitude problem Experimental set up Materials tested and specimen preparation Testing procedure Test Results Discussion and Conclusions Voussoir beam model Experimental set up Results Validation of DDA DDA Basics Previous validation of DDA Rotation and impact verification Rotation verification Impact verification Conclusions Validation of DDA using analytical solution and a shaking table model Comparison of DDA and analytical solutions

3 Comparison between DDA and a shaking table model Discussion Conclusions Validation of DDA using centrifuge modeling A simple two-block system Centrifuge model DDA centrifuge model Selection of numeric control parameters DDA model results Comparison of centrifuge model and DDA - Discussion Validation of DDA using a case study Tel Beer-Sheva water reservoir Rock mass structure Rock mass mechanical properties Stability analysis using Voussoir beam model DDA analysis material properties, boundary, and initial conditions DDA analysis of a single multi-jointed layer DDA analysis of a sequence of multi-jointed layers Limitations of DDA analysis Comparison of DDA and Voussoir models Stability of underground openings in bedded and jointed rock mass Model geometry and material properties Discussion and Conclusions References cited

4 C. Stability of Underground Openings in Bedded and Jointed Rock 1 Introduction Most rock masses are discontinuous over a wide range of scales, from macroscopic to microscopic. In sedimentary rocks the two major sources of discontinuities are: 1) bedding planes, and 2) joints. The intersection of bedding planes and joints forms a blocky rock mass. Before the excavation of an underground opening the blocky rock mass is assumed to be in a state of static equilibrium and in optimum packing arrangement. Excavation of an opening disturbs the initial equilibrium, and the stresses in the rock mass tend to readjust until new equilibrium is attained. During the readjustment of the internal stresses, and hence the rearrangement of load resisting forces, some displacements of the rock blocks occurs. Failure occurs when the stresses can no longer readjust to form a stable, load resisting structure. This may occur either when the material strength is exceeded at some locations or when movements of the rock blocks preclude stable geometric configuration, without strength failure. Joints and beddings are sources of weakness in an otherwise competent rock mass, therefore large displacements and rotations are only possible across these discontinuities. The displacements and rotations of the rock block along and across the joints is the source of the volume change. The interaction forces between blocks result in: 1) an increase in formation stresses, due to volume expansion in restricted volume, tending to create stable conditions; and 2) application of forces that can cause an increased displacement, tending to cause rock mass failure. The interaction between the stabilizing and destabilizing factors shape the overall behavior of the blocky rock mass Stability analysis in stratified and jointed rock mass Investigation of immediate roof stability commenced more than a century ago when Fayol (1885) conducted experiments on a stack of wood beams spanning a simple support, simulating the bedded sequence of roof span. By noting the deflection of the lowest beam as 4

5 successive beams where loaded onto the stack, Fayol demonstrated that at a certain stage none of the added load of an upper beam was carried by the lowest member. The load of the upper beams was transmitted laterally to the supports, rather than vertically as transverse loads to the lower members. For such a configuration beam theory can be employed to assess deflection, shear stresses, and maximum stresses in the immediate roof as a function of elastic parameters, rock density, and beam geometry (Obert and Duvall, 1967). Goodman (1989) incorporated inter-bedding friction into the beam analysis, thus extending the capabilities of this method. These analyses however are limited to continuous, clamped beams only. The analysis of a stratified and jointed roof is complicated by the fact that there is no closedformed analytical solution for the interaction of these blocks. In absence of closed form solution, the practicing engineer/geologist should relay on other methods for assessing the stability of the roof. Three different methods are currently in practice: 1) empiric rock mass classification; 2) semi-analytical methods; 3) numerical methods. These methods are widely used today, either stand alone or in an integrated manner, in all areas of geological and civil engineering Rock mass classification Standard engineering design both in continuous and structurally discontinuous rock is largely based on empirical rock mass classification methods, mostly assessing the expected stand-up time and the required support loads. Terzaghi (1946) formulated the first rational method of classification by evaluating the rock loads appropriate to the design of steel sets. Lauffer (1956) introduced the concept of Stand Up Time, which estimates the time to failure for any active unsupported span as a function of rock structure. Development of new support techniques, i.e. the use of rock bolts and shotcrete, gave rise to new Rock Mass Classification Methods encompassing all aspects of support design. Two of the most prominent methods are the Geomechanics Classification (a.k.a. RMR) of Bieniawski (1973) and the Quality system of Barton et al., (1974) both based on extensive database of case studies. These empirical methods are widely used to day by practitioners world wide, mostly as checkup on their design. Two major drawbacks of the empirical classification methods are to be noted: 1) rock mass classifications are a very general and a rather coarse approach in that it caters for all possible rock masses and type of excavation; 2) absence of mechanistic basis. Recent study in Israel (Polishook and Flexer, 1998; Tsesarsky and Hatzor, 2000) show that these methods are in some cases over conservative, even when a simple rock mass is encountered (homogenous 5

6 massive rock with widely spaced joints). Riedmuller and Schubert (1999), based on extensive tunneling practice in the Austrian Alps, show that rock mass classification is inadequate for support design and stability evaluation in complex geological conditions. The absence of true understanding of deformation mechanisms and the over conservative nature of the empirical methods will eventually lead to over conservative support design and unnecessary inflated project costs The voussoir Beam Analogy As noted by Fayol (1885) underground strata tend to separate upon deflection such that each laminated beam transfers its own weight to the abutments rather than loading the beam beneath. Stability of the excavation in this situation can be determined by analyzing the stability of a single beam deflecting under its own weight. Evans (1941) in his fundamental work established the relationship between vertical deflection, lateral thrust and stability of natural or artificially jointed roof. This work coined the term Voussoir Beam spanning an excavation, using the analogy of the masonry voussoir arch. The basic voussoir concept accepts that the beam may not carry longitudinal tensile stresses and it is confined between the abutments, i.e. lateral constrains are applied. The geometry and the forces acting in the voussoir beam are shown in Figure 1.1a. The overturning gravitational-reaction couple is equilibrated by the lateral thrust couple formed by beam deflection, where W is the weight of the beam, S is the beam span, T is the axial thrust and Z is the lever arm. The structure presented in Figure 1.1a is statically indeterminate since the lever arm Z is not known. In order to treat the posed problem analytically Evans assumed that a parabolic compressive arch structure of constant thickness is formed within the beam (Fig 1.1b). He also assumed an identical thickness of the arch at the abutments and at mid-span equal to half of the beam thickness. Three modes of failure are considered: 1) crushing of the rock at the abutments or at mid-span; 2) buckling (snap through) failure of the beam; 3) sliding between the blocks and the abutments. Beer and Meek (1982) reformulated and extended Evans s approach, introduced a coherent system of static equations, and evaluated the thickness of the compressive arch at the abutments and mid-span. Brady and Brown (1985) introduced an iterative algorithm for the evaluation of voussoir beam stability. The iterative approach assumes initial load distribution and line of action, i.e. assuming initial n and Z. The analysis provides the compressive zone thickness, and the maximum axial thrust. The factor of safety against the previously 6

7 mentioned failure modes can be calculated provided that the compressive strength of the rock, and the shear strength of the discontinuities are known. Diederichs and Kaiser (1999) further improved the classic iterative approach by introducing improved assumptions for lateral stress distribution and arch compression, and by providing a numerical buckling limit. The major advantages of the voussoir beam technique are the ability to assess previously ignored failure by shear along the abutments, and providing static (although undetermined) formulation of the discussed problem. Two main disadvantages of this method regarding the actual geometry of the problem should be mentioned. a t Voussoir block Abutment Mid-Span Crack S f c b t nt W 2 S 4 Z V T Figure 1.1. Evans s Voussoir beam: a) conceptual model and definition; b) compressive stress distribution in the beam. First, the voussoir beam analogue overlooks the geometrical and mechanical properties of the transverse joints, i.e. joint friction and joints spacing. Second, only a single layer of the roof is considered in this analogue. It is not unreasonable to assume that the mutual interaction of the individual blocks and the layers in laminated and stratified rock masses will differ from those described by Fayol. In absence of analytical solution, the stability of underground openings in laminated and jointed rock masses should be sought by means of numerical and physical modeling. 7

8 Numerical Modeling In rock mechanics, numerical methods are widely used to analyze the behavior of rock masses. For discontinuous rock masses the Finite Element Method (FEM) and the Distinct Element Method (DEM) are the most popular. The FEM is probably the most popular method in civil and rock engineering. Implementation of discontinuities into FEM has been motivated by rock mechanics need since the late 1960s. With the introduction of the joint element, first by Goodman et al., (1968), discontinuous rock mass can be analyzed by FEM. The FEM suffers from the fact that the global stiffness matrix tends to be ill conditioned when many joint elements are incorporated. Block rotations, complete detachment and large-scale fracture opening cannot be treated because the general continuum assumption in FEM formulations requires that fracture elements cannot be torn apart (Jing, 2002). Discrete Element Method (DEM) represents a different approach. The key feature of DEM is that the domain of interest is treated as an assembly of rigid or deformable blocks or particles. The contacts between the blocks are recognized and updated during the entire motion/deformation process, and represented by appropriate constitutive models. Thus, DEM allows finite displacements and rotations of discrete bodies, including complete detachment. The foundation of the method is the formulation and solution of equations of motion of rigid and/or deformable bodies using implicit (DDA - based on FEM discretization Shi, 1988) or explicit (UDEC - based on FDM discretization Cundall, 1971). The basic difference between the discontinuous and the continuum-based models is that the contact patterns between components of the system is continuously changing with the deformation process for the former, but are fixed for later Numerical modeling of jointed beams Wrigth (1972) conducted linear analysis of the voussoir beam by FEM, and supported the failure modes as proposed by Evans. Two models of voussoir beams were compared, one with a single mid-span joint and the other with 19 joints and concluded that the voussoir with a single mid-span joint is the worst case. Chugh (1977) studied the stability of a jointed beam by using the stiffness matrix for beam elements. Pender (1985) demonstrated the effect of joint dilation in the stability analysis of voussoir beam using a simplified model. Sepehr and Stimpson (1988) numerically studied the jointed roof in horizontally bedded strata with emphasis on developing the relationship between the roof deflection and joint spacing rather than analyzing failure modes. Passaris et al., (1993) have shown that crushing in high stresses 8

9 area and shear sliding are the most common failure modes encountered in mining environment, and showed that Wright s conclusion is erroneous, i.e. the multi jointed beam being the worst case. In their study however, the crushing failure was studied under the precondition that there was no shear sliding along the joints. Ran et al., (1994) studied the behavior of the jointed beam using non-linear FEM, and showed that the no shear preconditioning may result in over conservative estimate of roof strength. Both Passaris et al., (1993) and Ran et al., (1994) extended the analysis to multiple joints of variable spacing, however friction along joints was not modeled. FEM have limited applicability to the analysis of jointed rock masses since only small displacements/rotations are allowed, discontinuities are modeled as artificial-numerical interfaces, and new contacts are not automatically detected. Sofianos and Kapenis (1998) studied the stability of the classic mid-span jointed voussoir beam using UDEC. The mid-span joint model considers friction and cohession along joints, although prescribing values rarely encountered in rocks: zero friction at the mid span and = 89 0, c = 10GPa at the abutment. Thus, elastic displacement at mid-span is prevented, and only separation without shear is allowed at the abutments, i.e. crushing will occur before slip commences. Nomikos et al., (2002) investigated the influence of joint frequency and compliance under similar boundary condition, thus precluding shear along abutments and offcenter joints Physical Modeling of the Voussoir beam Physical modeling of laminated rock masses began with Fayol s experiment in Bucky (1931) studied the integrity of mine roof structures in rock, using a centrifuge (the first mention of anyone actually undertaking centrifuge modeling). Small preformed rock structures were subjected to increasing accelerations until they ruptured. There was little or no instrumentation on the models and their significance is largely historical. This work was pioneering, but saw little continuation or development. Evans (1941) studied the amount of deflection of brick beams. The deflection was analyzed as function of lateral thrust (amount and eccentricity) and beam geometry. Evans summarized the experiments in the following the tests on brick beams have served to show that, provided the end reactions are adequate, a voussoir beam can be quite stable under it s own weight even when traversed by numerous breaks and incapable of taking tensile forces. Sterling (1981) performed a series of experiments on single and multi-layered rock beams simulating the behavior of continuous rock beams from initial structural integrity to incipient 9

10 cracking and up to voussoir beam geometry. The experiment design provided data on the applied transverse load, induced beam deflection, induced lateral thrust and eccentricity of the lateral thrust. Sterling drew the following conclusions 1. Roof beds cannot be simulated by continuous, elastic beams or plates, since their behavior is dominated by the blocks generated by natural joints or induced transverse cracks. 2. Roof bed behavior is determined by the lateral thrust generated by deflection under gravity loading, of the voussoir beam against the confinement of the abutting rock. 3. A voussoir beam behaves elastically over the range of it satisfactory range. In addition, failure mode has been ascribed to various span to depth ratios and beam rock strength. Although pioneering, Sterling s experiments overlooked physical and geometrical properties of rock joints, and again concentrated on crushing strength and buckling limits of the threehinged voussoir beam. Passaris et al., (1993) and Ran et al., (1994) performed physical modeling of the voussoir beam using blocks of lightweight (and low strength) concrete, complemented with numerical (FEM) modeling. This research addressed the mechanical properties of joints, i.e. shear stiffness, and to a lesser extent geometrical properties. It has been shown, both numerically and experimentally, that the strength and stability of the voussoir beam is decreased with increasing number of blocks Current Research Motivation From the background material described above it is clear that the classical notation and solution of the Voussoir beam (the three hinged beam) is inadequate if a stability analysis of underground opening roof in laminated and jointed rock mass is attempted. The application of numerical methods is therefore inevitable. In order to closely simulate the deformation characteristic of a laminated voussoir beam the numerical method should allow rigid body displacement and deformation to occur simultaneously. The model must incorporate the influence of joint friction on block displacement, stress transfer, and arching mechanism that develop with ongoing beam deformation. The Discontinuous Deformation Analysis (DDA) was developed specifically to meet such requirements. The scope of this research project is to investigate the deformation characteristics of the laminated voussoir beam using DDA. 11

11 1.3. Objectives of research There were four main goals in this research: 1. Investigation of shear stress displacement similitude of rock joint interfaces in centrifuge modeling. 2. Investigation of multi jointed Voussoir beam kinematics and deformation using centrifuge modeling. 3. Validation of Discontinuous Deformation Analysis (DDA) using analytical solutions, centrifuge and physical models, and case studies. 4. Development of a series of design charts and tables for different rock mass properties and Voussoir geometries Organization of Research The research were performed trough a collaboration of four Principal Investigators: 1. Dr. Yossef H. Hatzor Department of Geological and Environmental Sciences, Ben Gurion University of the Negev (BGU). Functioned as research program coordinator. 2. Dr. Mark L. Talesnick Faculty of Civil Engineering, The Technion. 3. Prof. Nicholas Sitar Department of Civil and Environmental Engineering, University of California, Berkley (UCB). 4. Dr. Gen-hua Shi Independent consultant. Author and developer of DDA. Each of the PI s was in charge of a research group dealing with different aspects of the research. BGU group: (i) Validation of DDA using physical models and case studies. (ii) Rock mechanics testing (iii) Development of design charts and tables. Technion group: (i) Development of centrifuge modeling technology. (ii) Centrifuge modeling of single layered, multi jointed Voussoir beam. UCB group: (i) Software development and referencing. (ii) Validation of DDA using analytical solutions. Dr. Shi: 11

12 (i) (ii) (iii) DDA Software development and improvement. Implementation of rock bolts into the DDA code. Implementation of direct dynamic loading function in to DDA code. 2 Centrifuge modeling 2.1. Introduction Use of physical models to study the response of underground opening in jointed rock masses is not a new concept, however their use in recent years has been eclipsed or replaced by numerical modeling. This is unfortunate since the appropriate application of any numerical model should result in the mirroring of actual physical conditions, which is not always apparent in the outcome of numerical techniques. Physical models can be divided into three different groups; full scale models where stresses, deformations and displacements are in proper similitude with one another. Small scale models where stresses, strains and displacements are not necessarily in similitude with the prototype full-scale conditions. Finally, centrifuge modeling is an attempt to bring stresses and strains that develop in a small-scale model into proper similitude with the full-scale prototype (Ko, 1988). Use of centrifuge modeling has become mainstream in soil mechanics research. Use of centrifuge modeling in rock mechanics research has been reported upon as early as the 1930 s (Clark 1988). The majority of the earlier works focused on the response of intact rock beams, one of the more interesting studies was reported upon by Stephansson (1971). A common assumption is that centrifuge modeling allows complete similitude between the model and the true to life prototype. Meaning that stresses and strains which develop in a true to life prototype can be reproduced in a model scaled down n times when placed in a centrifuge and accelerated to a gravity field of ng, (Schofield, 1988). This situation may be true for continuous media, however it s validity in the case of discontinuous media such as a jointed rock mass has never really been claimed or considered. The advancements in numerical and computational capabilities over the past 20 years has led to the development of numerous computerized frameworks aimed at determining the stability and kinematics of rock masses and their interaction with engineering structures. Most important is the family of discrete element methods (DEM, DDA), which model the development of inelastic deformations in discontinuous rock masses. 12

13 The overall aim of this part of the study is to perform a set of physical tests on scale models in a centrifuge. The geometry that was chosen to model was that of a Voussoir beam, as shown in Figure 2.1. The reason for performing the tests in the centrifuge is based on the capability of applying stresses similar in magnitude to those expected in actual field situations. Furthermore, testing in the centrifuge allows uniform conditions of load to be applied along the entire extent of the beam. Application of such conditions in the laboratory or field is very difficult and extremely expensive. The predictive capability of different distinct element methods will be assessed based on their ability to properly predict the measured response of the centrifuge model. The centrifuge model need not be an actual representation of a true field situation. This apparent drawback is not detrimental to the validation process. The model test must be properly designed, with distinct boundary conditions and degrees of freedom. The experiment must include accurate and meaningful measurement of those variables to be predicted by the computational schemes. Accurate definition of material and interface properties is essential if the comparison between the actual model response and the computed predictions are to be of any value. As is often the case, in either numerical or analytical modeling of boundary value problems involving geomaterials, the selection of the required material and interface parameters is a challenging task. The data presented herein is an attempt to properly and rationally define the material parameters required to model the behavior of rock block interfaces in the centrifuge models. Blocks making up the Voussoir Beam Block Interfaces Voussoir Beam abutments Figure 2.1. Schematic drawing of the Voussoir beam modeled in the centrifuge. Three aspects of interface response may be important: The ultimate shear capacity of the interface, the shear stiffness (or a functional describing the shear stiffness) of the interface, 13

14 and the dilative/contractive reaction of the interface in response to applied shear and normal stresses. It is of interest to consider the different aspects of interface response within the light of centrifuge testing in general. The use of centrifuge modeling is considered mainstream in soil mechanics and foundation engineering research. A major advantage in centrifuge modeling of soil is the assumed similitude between stresses and strains in the model and the prototype. Based on such an assumption the deformations within the model scale to those expected to develop in the field prototype. However, in situations that include structural members within the soil mass, such as piles or reinforcing elements, there is no assurance that the level of similitude assumed between stress and strain in the geomaterials, will be compatible to that between shear stresses and displacements at interface boundaries. Kim et al. (1982) discussed such a discrepancy in the determination of pile skin resistance and the critical relative displacement in shaft pull-out tests. This being said, the effect of non-similitude on the results of centrifuge model testing is clearly neglected in the published literature. The question as to the scalability of shear stress-shear displacement response at the boundary of continuous elements remains unanswered. Research into the effect of block size on the shear response of rock block interfaces has been dealt with comprehensively by Barton and Bandis (1982) and Bandis et al. (1983). They interpreted data from a wide range of sources in an attempt to develop a tool to extrapolate shear stiffness values determined from laboratory tests, to parameters required in the analysis of jointed rock masses. Data and interpretation published by Barton and Bandis (1982) and Bandis et al (1983) have been used here in order to consider the scaling of the shear stressshear displacement response of rock block interfaces in centrifuge models. Physical modeling of the Voussoir beam geometry has been performed by Sterling and Nelson (1978) and Passaris et al. (1993). In both these studies scale models were tested in the laboratory and model geometries were confined to the simple case of a two block (threehinged) beam Shear similitude problem Experimental set up The thrust of the experimental program was aimed at defining the ultimate shear resistance and the development of shear resistance as a function of shear displacement imposed along a 14

15 rock block interface. The dilative/compressive nature of the interface has not been addressed in this stage of the centrifuge testing. The testing configuration was specially designed for the purpose of testing rock interfaces in the centrifuge. The specimen is comprised of two rock blocks; one larger block fixed to the base of the centrifuge arm, and a second smaller block (cube of 50 mm a side) sliding on the larger block. The interface between the two blocks is oriented perpendicular to the radial direction of the centrifuge. The system is depicted in the photograph of Figure 2.2 and includes integration of two sub-systems. Sub-system A was designed to apply controlled rates of relative shear displacement between the two rock blocks. The heart of the sub-system is an AC servo motor (1) 1, mounted with a planetary gear (1:40) and a linear ball screw actuator (2) with lead of 10mm per revolution. The rate of revolution of the motor axis is controlled using a servo amplifier, mounted on the centrifuge axis and receives a feedback signal from a resolver mounted on the motor. All testing was performed at a rate of 0.1 mm/min. Actual shear displacements were monitored with a LVDT (3). Two limit switches (4) were installed to protect the system from damage that could occur due to over travel. Figure 2.2 View of the centrifuge test set up. 1 the numbers relate to those shown in Figure 2.2 and Figure

16 Figure 2.3. Load cell. Sub-system B is responsible for shear load transfer and measurement. The heart of this subsystem is a specially manufactured load cell (5). Measurement of shear load in the centrifuge is difficult due to the fact that any type of load measuring device has mass and is subjected to forces due to changes in the acceleration field. It was, therefore, of prime importance to minimize the interaction between the actual shear load and parasitic signals generated due to changes in the centrifugal acceleration. The design incorporated the load cell as the method of transfer of shear displacement between the two blocks of the specimen. The concept of the load cell was based on the drag element of sting balances used in the testing of aeronautical models in wind tunnels (Ringel, 2002). The load cell (Figure 2.3) is made of four slender beams which connect two rigid end plates forming a cube with external dimensions 90mm a side. High resistance strain gages (4500W) were bonded to the slender beams close to the rigid connection in a moment bridge configuration. Two separate load bridges were constructed in this manner. This configuration insures extremely high sensitivity to loads causing shear deformation to the cubical frame, while normal loads due to inertial effects are theoretically eliminated. Parasitic shear loads due to centrifugal accelerations were calibrated and found to be, on average, 0.1 N/g. The duplication of the load bridges was originally conceived to be a backup to the measuring system; however both bridges performed excellently, even at an acceleration of 70 g, and showed only minimal differences. The load cell is capable of reliably resolving loads of magnitude 0.3 N with a full scale of 500 N. The load cell frame is mounted on the mobile block of the ball screw actuator. The mobile portion of the specimen (6) is placed within the load cell frame. Relative shear displacement between the two blocks is imposed as the linear actuator brings the load cell frame into contact with the mobile block. The line of contact between the rigid end of the load cell and 16

17 the specimen side is made at a distance of 0.5 mm above the plane of the stationary block (7) therefore the development of moments is largely eliminated. Shear resistance due to friction between the specimen and the actuator face (due to earth s gravity) was minimized by inserting a teflon sheet (8) between the two. Signal conditioning for the load cell and the LVDT was fixed on the centrifuge axis. Relative shear displacement measured at the LVDT was corrected for the shear flexibility of the load cell as a function of shear load. The entire testing setup was controlled by software developed in LabView (National Instruments); data was sampled and stored to a desktop computer by way of a 16 bit resolution A/D data acquisition card (National Instruments). All control and measurement signals were fed back and forth between the controlling computer and the model through a series of gold plated slip rings mounted on the centrifuge axis Materials tested and specimen preparation All the specimens were prepared from several blocks of chalk sampled from the Tel Be er Sheva where archeological studies have uncovered an ancient underground water reservoir, whose roof is comprised of horizontally layered and vertically jointed chalk. The roof acts as a set of adjoining Voussoir beams with joint spacing of between 25 cm to 75 cm and thickness of 50 cm (Hatzor and Benari, 1998). Specimens were prepared by sawing sections of chalk under dry conditions, and then casting them in plaster to standard dimensions. The larger, stationary blocks were cast to dimensions of 195mm*150mm and 55mm in height. The smaller, mobile blocks were cast to dimensions of 50mm*50mm*50mm. In order to allow for higher normal stresses than possible due to the density of the chalk and plaster alone, the mass of the mobile specimens was adjusted by encapsulating lead shot into the plaster at the time of casting. The mass of the specimens ranged from 169 grams to 392 grams. All of the specimens were of the same nominal dimensions. This artificial adjustment is of value in the study of scaling effects on interface response. The interface plane of both the mobile and stationary specimens was either ground, using a surface grinder (G series) or saw-cut (SC series) flush prior to casting in the plaster. The roughness of the ground surfaces was less then 0.01 mm. The asperity height of the saw cut surfaces was approximately 0.5 mm. The cast was poured with the prepared surface, face down on a glass plate and a thin layer of dry fine sand spread between the specimen and the form sides. After a period of approximately 6 hours the form was removed and the sand layer 17

18 blown off the bottom of the specimen sides. The purpose of the sand layer was to prevent development of friction between the stationary rock block and the edges of the mobile block, which otherwise would have been filled with plaster. Two weeks after preparation, the mobile specimens were re-massed and the dimensions of the prepared interface surfaces measured Testing procedure The stationary block is fixed in the test system such that the prepared face stands vertically, oriented perpendicular to the radius of the centrifuge, with the prepared surface facing the centrifuge axis. The mobile block was then positioned within the load cell frame with the prepared face placed in contact with the prepared surface of the stationary block. The centrifuge was activated and brought to a revolution rate inducing a gravity field of 5 g at the radial position of the interface between the two blocks. The displacement control system was then enabled and the load cell/frame advanced at a rate of 0.1mm/min in a direction towards the mobile block. The block was displaced up to 0.5 mm after which the displacement direction was reversed and the shear load removed. The entire procedure was repeated for acceleration levels of 10, 20 and 40 g s. In one testing run, an acceleration level of 70 g s was applied. Prior to imposing relative displacement at any specific g level, the outputs of the load cell bridges were zeroed, such that any parasitic readings, due to changes in the acceleration field, were completely removed Test Results Figure 2.4 and Figure 2.5 present the development of shear stress as a function of the imposed shear displacement as recorded in 5 sets of centrifuge tests. Results of tests performed on ground surfaces are shown in Figure 2.4a and b. Note that the plots of shear stress do not all originate from the origin, however, all computations have been based on the proper relative shear displacement. The two sets of results illustrate very consistent behavior. In both sets the shear stresses increase continuously with relative shear displacement, reaching a maximum following relative shear displacements of between 0.05 mm and 0.1 mm. Subsequent shear displacements result in negligible changes in shear stress. This ultimate shear stress represents the frictional strength of the interface under the particular normal stress developed due to the centrifugal field. The shear loading curve illustrates a nearly linear increase with relative shear displacement up to between 70% -90% of the ultimate frictional strength, after which the shear stress increases non linearly towards the stable maximum. It is interesting 18

19 and important to note that the unloading curves in all cases are completely vertical, indicating that no energy is stored in the system. The slope of the linear portion of the loading curve is representative of the initial shear stiffness of the interface plane. For the data shown in 2. 4a and b the shear stiffnesses are tabulated as a function of normal stress in the upper portion of Table 1. The values range from 617 kpa/mm to 983 kpa/mm, and do not indicate any trend in shear stiffness as a function of normal stress. Table 1 also includes the relative shear displacement at which ultimate shear stresses were achieved, and the ultimate shear stress developed in each loading cycle. Figures 2.5a,b and c show shear stress-shear displacement records measured for saw-cut interfaces. Three important, however subtle, differences are noteworthy, in comparison to the test results of ground interfaces: 1. The initial shear stiffnesses are somewhat higher than those determined from the ground interface data. The shear stiffnesses range from 932 kpa/mm to 1580 kpa/mm and are tabulated as a function of normal stress in the lower portion of Table 1. As in the case of the data recorded from the ground interface tests, no trend in the magnitude of the shear stiffness is noted as a function of the normal stress. 2. The unload portions of the shear stress-shear displacement curve are not, in all cases, vertical. In several of the tests, notably those shown in Figure 2.5a and c, the unload segments of the curves illustrate a small degree of stored energy which might be attributed to the elastic rebound of the interface asperities. 3. The shear stress-shear displacement plots do not reach a flat maximum as is the obvious case in the shear of the ground interfaces. In the case of the sawcut interfaces, the ultimate shear stress is obtained following a gradual increase in shear stress over significant shear displacement. This response is most obvious in tests performed at normal stresses greater than 18 kpa. This behavior is typical of the development of shear resistance along rough surfaces in first time loading (Jaeger and Cook 1976). 19

20 Shear Stress (kpa) Shear Stress (kpa) (a) GA-G2 mass gr g level - Normal Stress (kpa) 5 g g g g Shear Displacement (mm) (b) GB-G3 mass gr g level - Normal Stress (kpa) 5g g g g g Shear Displacement (mm) Figure 4 - Shear stress versus shear displacement for ground surface interfaces. (a) GA-G2 (b) GB-G3. Figure 2.4. Shear stress versus shear displacement for ground surface interfaces:(a) GA-G2 (b) GB-G3. 21

21 Shear Stress (kpa) Shear Stress (kpa) Shear Stress (kpa) Figure 2.6 shows the ultimate shear stress versus developed normal stress for each shear loading cycle of all tests performed, for both the ground and saw cut interfaces. All the data fall along a well defined singular linear function which passes directly through the origin. This plot represents the frictional strength of the interface. For the ground and saw cut interfaces tested there is no significant difference in the value of the angle of friction which varies between 34 and 36. The angle of friction is independent of the acceleration level at which the test was performed. Zimmie et al. (1994) presented data which illustrated that the dynamic angle of friction between a geotextile and a geomembrane liner was also independent of the applied acceleration. They did not consider the effect of the acceleration on the stiffness of the interface SCA-SC1 mass gr g level - Normal Stress (kpa) 5g g g G SCA-SC2 mass grams g level - Normal Stress (kpa) 5 g g g g (a) 10 0 (b) Shear Displacement (mm) Shear Displacement (mm) SCB-SC3 mass grams g level - Normal Stress (kpa) 5 g g g g (c) Shear Displacement (mm) Figure 5 - Shear stress versus shear displacement for saw-cut surface interfaces. (a) SCA-SC1 (b) SCA-SC2 (c) SCB-SC3. Figure 2.6. Shear stress versus shear displacement for saw-cut surface interfaces: a) SCA- SC1; b) SCA-SC2; c) SCB-SC3. 21

22 Shear Stress (kpa) Discussion and Conclusions 1. The shear stress shear displacement plots are surprisingly linear. 2. The fact that no post peak behaviour is noted in the shear stressshear displacement plots for either the ground, or saw cut interfaces indicates that no asperity locking is occurring and that the asperities of one block are simply riding on top of the asperities of the lower block. 3. The shear stiffness of the saw-cut interfaces is approximately 30% higher than those of their ground counterparts. The shear stiffness was found to be independent of the applied acceleration. 4. The failure envelope is extremely linear passing directly through the origin, for both the interfaces with ground surfaces and the saw-cut surfaces. One particular aspect of the data requires additional discussion. What is the effect of the magnitude of the acceleration on the shear response along block interfaces? For instance; when placing a block of dimensions 50 mm by 50 mm by 50 mm in a centrifuge and inducing an acceleration field of n times gravity, does this imply that the size of the block scales accordingly? Will the shear stiffness of the interface be affected by the application of the acceleration field? GA-G2 GB-G3 SCA-SC1 SCA-SC2 SCB-SC Normal Stress (kpa) Figure 6 - Frictional strength criterion. 22

23 Figure 2.7. Frictional strength criterion. Consider the shear stress-shear displacement response of blocks GA-G2 tested under an acceleration of 40 g, to that of blocks GB-G3 tested under an acceleration of 70 g. In both cases the normal stress was close to 48 kpa, this due to the different masses of the specimens. Figure 2.7 compares the shear stress-shear displacement plots recorded in the two tests. The data illustrate that the shear stress- shear displacement response is almost identical, and unaffected by the difference in acceleration fields under which the tests were performed. The fact that the shear stiffnesses are the same, and the relative shear displacements required to reach maximum shear resistance are similar suggests that no dimensional scaling of the relative shear displacement is warranted, despite the difference in the applied gravity fields. Barton and Bandis (1982) considered different aspects of joint response through the analysis of laboratory and field tests on interface surfaces. Based on results of numerous shear tests on rock joints, they came to the conclusion that in the case of laboratory sized specimens (~ mm in length) the peak shear stress is reached after a relative shear displacement of approximately 1% of the block length. They further noted that smoother interfaces required less relative displacement to reach peak shear stresses than do rough interface surfaces. Figure 2.8a and b (after Barton and Bandis, 1982) illustrate that for a given normal stress the shear stiffness decreases with increased block length, and that for a given block dimension the shear stiffness increases as the applied normal stress is increased. The data shown in Figure 2.8a were collected from a wide variety of sources and relate to the entire spectrum of interface conditions, therefore the scatter is significant. 2. 8b presents data pertinent to a specific interface condition; therefore the results are significantly more focused. Based on their interpretations, for any specific block size a tenfold increase in normal stress leads to a tenfold increase in shear stiffness. Likewise, a tenfold increase in block length (at any specific normal stress) results in a tenfold reduction in shear stiffness. 23

24 Shear Stress (kpa) GA-G2, 40g GB-G3, 70 g Shear Displacement (mm) Figure 7 - Comparison of shear stress shear displacement response for two different ground interfaces. Figure 2.7. Comparison of shear stress shear displacement response for two different ground interfaces It is interesting to consider how the data from the current set of centrifuge tests compares to the trends presented by Barton and Bandis (1982). The shear stiffness of the interfaces tested in the centrifuge ranged between 0.6 MPa/mm and 1.4 MPa/mm. This range is somewhat higher than the range of MPa/mm that could be interpreted from the graphs of Figure 2.8, for a block size of 50mm sheared under a normal stress varying between 0.01MPa and 0.1 MPa. The shear displacement required to reach maximum shear resistance was found to be significantly smaller than the 1% suggested by Barton and Bandis. The measured displacements were never greater than 0.25% of the specimen length. This departure from Barton and Bandis is most likely due to the relatively smooth interface geometries that were tested in the current centrifuge study. The main discrepancy of the current centrifuge data, compared to the plots provided by Barton and Bandis is that as the normal stress in the centrifuge was increased there was no corresponding increase seen in the shear stiffness of the interface. From the data presented above it seems that the concept of dimensional scaling in the centrifuge, for situations including rock block interfaces, is not warranted. This conclusion is 24

25 almost intuitive; however it is essential to consider when attempting to physically model a situation which includes a discontinuous media. The effect of such a conclusion on the analysis of centrifuge models and their usefulness in modeling actual geo-engineering situations must be addressed more fully by all researchers in this field. Future testing will include blocks of different sizes and interfaces of different roughnesses so that both scaling, and the issue of normal stress dependence, may be considered. Figure 2.8. a) Shear stiffness as a function of block size and normal stress, compilation of data from the literature (top); b)shear stiffness as a function of block size and normal stress extrapolated from laboratory data (bottom). After Barton and Bandis (1982). 25

26 2.3. Voussoir beam model Experimental set up Figures 2.9a and b show a schematic and a photograph of the experimental set-up while inflight in the centrifuge. The set-up included a Voussoir beam comprised of six gypsum blocks 50mm*50mm*50mm aligned between two end abutments. 180 grit abrasive paper was glued to the interface plane of each block and abutment. Abutment A (Figure 2.9a) was brought into contact with the beam such that a nominal load of 50 N was developed and then locked in place. No lateral movement of either abutment was allowed and the alignment of blocks remained supported by the shear stresses at the abutment interfaces alone. The entire test setup was slowly accelerated in a 4 m diameter centrifuge up to radial accelerations of 40 times gravity. Throughout each centrifuge run the following variables were monitored; (i) The compressive thrust developed perpendicular to the abutment face. The thrust was monitored by a pancake load cell placed behind abutment A. (ii) The deflection of each block was monitored at both lateral edges by an LVDT. This allowed block rotation to be identified if undergone. (iii) Each block was tagged with four dots, one at each corner of each block. The tags were monitored by a high resolution digital camera positioned above the test rig and rotated together with the model. The camera was controlled by remote control and capable of operating at g levels of up to 50 g. Image analysis was employed to track the tag movement relative to tags mounted on a stationary reference beam (see Figure 4b) rendering a 2D picture of displacements in the plane of the beam. (iv) A single block was instrumented with six 120 Ohm foil strain gauges (Figure 2.9b), three each on the front a back faces of the block. A quarter bridge configuration was constructed with each leg containing two gauges. The gauges were positioned as to monitor the strain developed in the top middle and bottom portions of the block as acceleration and associated thrust developed. The set up was placed in flight three times, prior to each run the location of the blocks were altered such that the instrumented block filled each of the different positions 1 through 3 (Figure 2.9a). Additional runs were performed for a frictional interface comprised of 360 grit 26

27 abrasive paper. The results of these runs were used for DDA validation, and they are described in section 3.4. Figure 2.9. Voussoir beam experimental set-up tested in the centrifuge. a) Schematic drawing (top); and b) In flight photograph at 18 g (bottom Results Figure 2.10 shows the development of thrust as a function of the applied centrifugal acceleration for each of three different runs performed (180C, 180D and 180E). The plots are essentially the same indicating that in terms of the developed thrust results are consistent and unique. Figures 6a,b and c show the development of beam deflection based on LVDT measurements as a function of position along the beam for accelerations of 10g, 24g, and 40g 27

28 respectively. Based on the uniformity of the results shown in Figure 2.10 it might be expected that the deflection curves would be unique for each of the three tests for each specific g level. As is seen in Figure 2.11 this is not the case, however the measured deflections are relatively tightly bunched with maximum variations on the order of 0.2 mm. It is noted that the deflections are not symmetric about the center of the beam between block positions 3 and 4. Rather the maximum deflections were measured at the interface between blocks 2 and 3. This is a result of the fact that the radial ray of the centrifuge arm was not placed in alignment with the centerline of the beam model, rather with the interface between blocks 2 and 3. This condition must be resolved in order for the data to be a viable benchmark test. The form and presentation of the results are interesting since they illustrate how the blocks slide along interfaces and rotate individually within the beam while maintaining overall stability. Some of the most exciting data stems from monitoring the development of strain within individual blocks along the beam axis as the model was accelerated and the beam underwent deflection. Each graph in Figure 2.12 plots the development of axial strain at a specific height along the beam, either top middle or bottom as a function of the position along the beam (defined as block position). For instance, Figure 2.12a plots the strain measured along the top of the beam. In each of the three block positions it may be seen that compressive strain (and stress) develop indicating that the entire top layer of the beam plays a role in supporting the beam. The maximum strain ( ) was measured in test 180C where the measuring block was located at the beam center. On the other hand, consider the plots of Figure 2.12c which illustrates the response at the bottom portion of the beam. In this case it may be seen that only block position 1, next to the abutment undergoes compression. The bottom beam fiber in block positions 2 and 3 actually illustrate relative tension. Since actual tension can not be imposed across an interface the measured tensile strains are in fact due to unloading of initially applied compressive stress with the development of eccentric loading over the block height. The data shown in Figure 2.12b represents the transition of stress along the middle of the beam. The beam center (block position 3) undergoes almost no relative axial deformation, only close to the abutment position does the block center participate in picking up the compressive thrust developed due to deflection of the beam. 28

29 Figure Development of thrust as a function of acceleration for three centrifuge runs. Figure Beam deflection at acceleration levels of 10g, 24g, 32g and 40g as a function of position along the beam. 29

30 Figure Measured strain as a function of acceleration and block position. 31

31 3 Validation of DDA 3.1. DDA Basics DDA models a discontinuous material as a system of individually deformable blocks that move independently without interpenetration. In the DDA method the formulation of the blocks is very similar to the definition of a finite element mesh. A finite element type of problem is solved in which all elements are physically isolated blocks bounded by preexisting discontinuities. The discontinuities can in general be located anywhere with any direction, and length. Therefore, elements of any shape are expected. Both FEM and DDA require integration of polynomial functions over a general polygon area. In FEM, integrations are preformed using the Gaussian quadrature, which is only suitable for integration in triangular and rectangular elements. In DDA integrations are preformed using the analytical Simplex solution (Shi, 1993), thus the elements can assume any given topology. The displacements (u, v) at any point (x, y) in a block, can be related in two dimensions to six displacement variables D u v r ε ε T i (3.1) x y γ xy where (u 0, v 0 ) are the rigid body translations of a specific point (x 0, y 0 ) within a block, (r 0 ) is the rotation angle of the block with a rotation center at (x 0, y 0 ), and x, y and xy are the normal and shear strains of the block. For a two-dimensional formulation of DDA, the center of rotation (x 0, y 0 ) coincides with block centroid (x c, y c ). Shi (1988) showed that the complete first order approximation of block displacement takes the following form u v T D D (3.2) i i ( y y ) ( x x ) 0 ( x x ) 0 0 ( y y ) 0 ( y y ) / 2 ( x x ) / 2 0 This equation enables the calculation of displacements at any point (x, y) of the block when the displacements are given at the center of rotation and when the strains are known. By adopting first order displacement approximation, each block is a constant strain/stress element. The local equations of equilibrium are derived using FEM style potential energy minimization. In DDA individual blocks form a system of blocks through contacts between blocks and displacement constrains which are imposed on a single block. For a block system defined by n blocks the simultaneous equilibrium equations are i 31

32 K K K K n1 K K K K n2 K K K K n3 K K K K 1n 2n 3n nn D D D D n F1 F2 F3 Fn or D F K (3.3) where K ij are 6 6 sub-matrices defined by the interactions of blocks i and j, D i is a 6 1 displacement variables sub-matrix, and F i is a 6 1 loading sub-matrix. In total the number of displacement unknowns is the sum of the degrees of freedom of all the blocks. The diagonal sub-matrices K ij (i = j) represent the sum of contributing sub-matrices for the i-th block, namely block inertia and elastic strain energy. The off diagonal sub-matrices K ij (i j) represent the sum of contributing sub-matrices of contacts between blocks i and j and other inter-element actions like bolting. The i-th row of (3) consists of six linear equations where d ri are the deformation variables of block i. 0, r 1,,6 (3.4) d ri The solution to the system of equations (3) is constrained by inequalities associated with block kinematics, as well as the no penetration and no tension condition between blocks. The kinematic constrains on the system are imposed using the penalty method. Contact detection is performed in order to determine possible contacts between blocks. Numerical penalties analogous to stiff springs are applied at the contacts to prevent penetration. Tension or penetration at the contacts results in expansion or contraction of the springs, a process that adds energy to the block system. Thus, the minimum energy solution is one with no tension or penetration. When the system converges to an equilibrium state the energy of the contact forces is balanced by the penetration energy, resulting in inevitable very small penetrations. The energy of the penetrations is used to calculate the contact forces, which are in turn used to calculate the frictional forces along the interfaces between blocks. Shear displacement along the interfaces is modeled using Coulomb - Mohr failure criterion. Fixed boundary conditions are enforced in a manner consistent with the penalty method formulation. Stiff springs are applied at fixed points. Displacement of the fixed points adds considerable energy to the block system. Thus, a minimum energy solution satisfies the no displacement condition of the fixed points. The solution of the system of equations is iterative. First, the solution is checked to see how well the constrains are satisfied. If tension or penetration are found at contacts the constrains are adjusted by selecting new position for the contact springs and a modified 32

33 versions of K and F are formed for which a new solution is attained. The process is repeated until each of the contacts converges to a constant state. The positions of the blocks are then updated according to the prescribed displacement variables. The large displacements and deformations are the accumulation of small displacements and deformations at each time step. Shear displacement along the interfaces is modeled using the Coulomb - Mohr failure criterion. The large displacements and deformations are the accumulation of small displacements and deformations at each time step. The time integration scheme in DDA adopts the Newmark (1959) approach, which for a single degree of freedom can be written in the following manner: u u i 1 i 1 u i u i tu i ( 0. 5 ) t ( 1 ) tu tu i 2 2 u t u where u, u, and u are acceleration, velocity, and displacement respectively, i 1 i i 1 (3.5) t is the time step, and are the collocation parameters. Unconditional stability of the scheme is assured for The DDA integration scheme uses and and, thus it is implicit and unconditionally stable. Computer implementation of DDA allows control over the analysis procedure through a set of user specified control parameters: 1 Dynamic control parameter (k01) defines the type of analysis required, from quasistatic to fully dynamic. For quasi-static analysis the velocity of each block at the beginning of each time step is set to zero, k01 = 0. In case of dynamic analysis the velocity of each block at the end of a time step is fully transferred to the next time step, k01 = 1. Different values from 0 to 1 correspond to different degrees of inter step velocity transfer, comparable to damping or energy dissipation. For example for an input value of k01 = 0.95 the velocity in the beginning of each time step is 5% lower then the terminal velocity at the previous time step. 2 Penalty value (g0) is the stiffness of the contact springs used to enforce contact constrains between blocks. 3 Upper limit of time step size (g1) the maximum time interval that can be used in a time step, should be chosen so that the assumption of infinitesimal displacement within the time step is satisfied. 4 Assumed maximum displacement ratio (g2) the calculated maximum displacement within a time step is limited to an assumed maximum displacement ( U max g2 ( y 2), 33

34 where h is the length of the analysis domain in the y-direction) in order to ensure infinitesimal displacements within a time step. g2 is also used to detect possible contacts between blocks Previous validation of DDA Yeung (1991) and MacLaughlin (1997) tested the accuracy of DDA for applications ranging from tunneling to slope stability, using problems for which analytical or semi-analytical solutions exist. Doolin and Sitar (2001) explored the kinematics of a block on an incline for sliding distances of up to 250 meters. Hatzor and Feintuch (2001) validated DDA using direct dynamic input. Analytical integration of sinusoidal functions of increasing complexity was compared to displacements prescribed by DDA for a single block on an incline subjected to the same acceleration functions as integrated analytically. The necessity for DDA validation using analytical solutions is evident if the method is to be adopted by the engineering profession. However, analytical solutions are only valid for the inherent underlying simplifying assumptions. This limitation can be overcome by comparison between DDA prediction and experimental results of carefully planned physical models. Up to date, such attempts have been limited, or practically non-existent for the dynamic problem. O sullivan and Bray (2001) simulated the behavior of hexagonally packed glass rods subjected to bi-axial compression, showing the advantages of DDA in the study of soil dynamics. McBride and Scheele (2001) validated DDA using a multi-block array on an incline subjected to gravitational loading, and a bearing capacity model. Validation of DDA using analytical solutions showed that DDA accurately predicts single block displacements, up to tens of meters. However, validation using physical models proved less successful. In particular, it was found that kinetic damping is required for reliable prediction of displacement (McBride and Scheele, 2001) Rotation and impact verification Rotation verification DDA is formulated using a linearized rotation, which works well for small cumulative rotation. Accumulating large rotations results in free expansion" (Ke 1993), where the 2D area of the block grows without bound. This increase in area is accompanied by energy growth due to increased block mass. Mitigating free expansion requires rotation post- 34

35 correction" (Ke 1993; MacLaughlin 1997), where higher order approximations or exact functions are used to update the rotations. For pure rotation, a single block 1 m 3 in volume (meter square, meter thick) was used, with plane strain boundary conditions. The block stiffness was set to 5x10 7, and gravitational acceleration was set to zero. The Poisson ratio was set at The time step was taken constant at 0.1s (Δt = 0.1) for 250 time steps, and the maximum displacement parameter was set at The condition number was computed and found to be roughly the same as the block stiffness, indicating that the block can be considered numerically rigid", and that no more than 9 significant digits could be possible in the solution vector of deformations. Although areal block growth is not physical, the linearized rotations require such behavior. Using an exact post-correction for rotations (Ke 1993; MacLaughlin 1997), the relative error was found to be approximately Two analyses are shown here for illustration using a rod (Fig. 1) of unit thickness, unit height and length 15 m. The gravitational force was set to 0, the Young's modulus was set sufficiently high (E = 5 x 10 7 ) and the initial angular velocity ω= 0.5 rad/sec. The rest of the parameters and properties were set as follows: Δt = 0.01, density ρ = 2.75, and Poisson's ratio ν = No viscous damping was used, and the analyses were performed using plane stress boundary conditions. Under these conditions the rod spins about its centroid, marked by measured point 2 in Fig. 3.1(a). The relative error for both the free expansion and the exact post-correction was below Figure 3.2(a) shows displacement with respect to time for a translating, rotating rod starting from a position similar to the position shown in Fig. 3.1(a). The computed and analytic values for point positions appear to match very well up to impact. The relative error is shown in Fig. 3.2(b) for the x and y positions at the corner point, and varies from 10-6 to 10-8 until impact, consistent with the numerical resolution of the problem Impact verification The impact of a rotating rod shown in Figure 1 was modeled with the following assumptions: 1. the rod and plate are rigid; 2. there is no adhesion or cohesion on plate; 3. Coulomb friction governs sliding behavior on the plate; and 4. there is no penetration of rod into plate. Figure 3.3 shows results from an experiment with the material properties of a 1 meter by 15 meter rod of unit thickness, and base properties, as follows: density ρ = 2.75 Mg/m 3, Young's modulus E = 5 x10 7, Poisson's ratio ν = 0.49, gravitational acceleration g = 9.81m/s 2, no viscous or contact damping, interface friction angle Φ = 0, and the rod with an initial angular velocity ω = 0.5 rad/s. The results show that the contact force increases with time step and 35

36 penalty (Fig. 3.3). At small time steps, the impact assumption is not valid because interpenetration is maintained for more than 1 successive time steps, which can be seen by examining the velocity of the interpenetrating point. The velocity of the interpenetrating point for these small time steps shows a deceleration phase before the velocity changes sign, and an acceleration phase before the contact is broken. For example, for the rotating rod with Δt = 0.001s, the contact was maintained for 5 time steps (3 deceleration and 2 acceleration) before the contact was opened. For a purely impact" break in the contact, the velocity immediately following the sign change should be the highest velocity, to reflect the downward gravitational force that acts on the mass after the contact is broken. At larger time steps, the impact is impulsive, with contact breaking in the time step immediately following contact initiation. Overall, small time steps appear to more accurately capture impacts Conclusions Our results to date show that the implementation of DDA accurately computes rotation for the analysis parameters and material properties stated above. However, assuming that accurate normal contact forces are necessary for computing accurate frictionally controlled shear forces, DDA contact algorithm needs to be improved to ensure accurate dynamic contact forces. Inaccurate impact forces may also induce inaccurate strains in colliding blocks, which would preclude implementing accurate block fracture algorithms. Figure 3.1. Rod impact and rotation geometry. 36

37 Figure 3.2. Combined rotation and translation relative error Figure 3.3. Impact force as a function of time step Validation of DDA using analytical solution and a shaking table model Comparison of DDA and analytical solutions A Fourier series composed of sine components represents the simplest form of harmonic oscillations: n a( t) a i sin( t) (3.6) i 1 where a i and are the amplitude (acceleration) and frequency respectively. The displacement of a mass subjected to dynamic loading is attained by double integration of the acceleration record (Eq. 2) from to t: i 37

38 n ai d( t) sin it sin i i ( t ) cos 2 i i i 1 i (3.7) where is the time at which yield acceleration a y is attained. Hatzor and Feintuch (2001) showed that for an acceleration function consisting of sum of three sines (for arbitrary selected constants a 1 = a 2 = a 3 = DDA prediction is accurate within 15% of the analytic solution provided that the numeric control parameters g1, g2 are carefully selected. Moreover, they argued that the influence of higher order terms in a series of sines is negligible. These values produce a low frequency dynamic input assuring a nearly constant block velocity, which was attained at the beginning of the analysis (ca. 20% of elapsed time). In order to attain a more realistic simulation we have extended the analysis to higher frequencies, constraining the peak horizontal acceleration to 0.15g. The analysis was performed for a single block resting on a plane inclined = 15 0 to the horizontal. The block material properties were: density = 2700 kg/m 3, E = 5000 MPa, and = The friction angle of the sliding plane was set to = 15 0, thus the yield acceleration (a y = 0) was attained immediately at the beginning of analysis ( = 0 sec). Three different sets of frequencies were modeled (Table 3.1). Constant values of numeric spring stiffness (g0 = 1000 MN/m), assumed maximum displacement ratio (g2 = ), and dynamic control parameter (k01 = 1) were used. Each frequency set was modeled twice, first the time step was set to g1 = sec, then the time step was halved to g1 = sec. Input acceleration and comparison between the analytical solution and the numerical solution of the total displacement are presented in Figure 3.4, where excellent agreement between the analytical and DDA solutions are shown. The relative numeric error is defined by: where d and the norm operator. d d e (%) d N 100 (3.8) d N are the analytical and the numeric displacement vectors respectively. is The relative numeric error for g1 = sec simulations is within 4.5% for a numeric spring stiffness of MN/m. Halving the time step reduces the numeric error to 1.5%. We have further investigated the interrelationship between the numeric control parameters using the input function of set 2 (Table 3.1). Figure 2 shows the dependence of the numeric error on the choice of the numeric control parameters g1, g2, and the numeric spring stiffness 38

39 g0 (the penalty value). It is found that for an optimized set of g1and g2 (bold in Figure 3.5) the DDA solution is not sensitive to the penalty value, which can be changed over two orders of magnitude, from N/m to N/m, with no significant change in error. Within this range the numeric error never exceeds 10% and in most cases approaches the value of 1%. Departing from the optimal g1, g2 combination results in increased sensitivity of the DDA solution to the penalty value. The departure from the analytical solution occurs at lower penalty values with increasing time step size. 39

40 Displacement (m.) Acceleration (g) 0.2 a(t) sin t sin t sin t a Full Symbols - DDA output, g1 =0.0025sec Empty Symbols - DDA output, g1 =0.005sec Full Lines - Analytical Solution b Time (sec.) Figure 3.4. The loading function a(t) = a 1 sin( t) + a 2 sin( 2 t)+ a 3 sin( 3 t); b) Comparison between analytical and DDA solution for block displacement subjected to a sum of three sines loading function. All DDA simulations for:g0 = 1*10 9 N/m; g2 = ; block elastic modulus E = 5000*10 9 N/m. All input motions are Set a 1 (g) a 2 (g) a 3 (g) 1 8, 0.1 4, , , 0.1 5, , , , , Table 3.1. Frequency sets for sum of three sines dynamic input function. 41

41 10000 g1 = g2 = g1 = g2 = g1 = g2 = 0.01 g1 = g2 = EN Numeric spring stiffnes (10 6 N/m) Figure 3.5. Absolute numeric error of DDA ultimate displacement prediction as a function of spring stiffness, for a sum of three sines loading function Comparison between DDA and a shaking table model The physical modeling used in this part was performed by Wartman (1999) at the Earthquake Simulation Laboratory of the University of California at Berkley. The tests were performed on a large hydraulic driven shaking table, producing accurate, well controlled, and repeatable motions to frequencies up to 14 Hz. The table was driven by an MTS closed loop servo controlled hydraulic actuator. A Hewlett Packard 33120A arbitrary function generator produced the table command signal. A steel plane, inclined by , was fitted to the shaking table. The steel block was 2.54 cm thick, with area of 25.8 cm 2, and weight of 1.6 kg. Linear accelerometers were fitted on top of the sliding block and the inclined plane. Displacement transducers measured the relative displacement of the sliding block, and of the shaking table (Figure 3.6). In this study eight sinusoidal input motion tests were used for validation. The input frequencies, amplitudes, and block displacements are given in Table 3.2. A geotextile and a geomembrane were fitted to the face of the sliding block and the inclined plane respectively. The static friction angle ( ) of the interface was determined by Wartman using tilt tests and a value of was reported. Kim et al., (1999) found that the friction angle of the geotextile geomembrane interface exhibited pronounced strain rate 41

42 effects, and reported an increase by 20% over one log-cycle of strain rate. Wartman (1999) showed that the friction angle of the interface was controlled by two factors: 1) amount of displacement; and 2) sliding velocity. The back-calculated friction angle for the range of velocities and displacements measured was in the range of , Figure 3.7. The DDA version used in this research accepts a constant value of friction angle. Therefore, a single friction angle ( av ) value must be chosen for validation. The value of av was determined as follows. First, the measured displacement of the block was differentiated with respect to time and hence the velocity record was attained. Next, the velocity content for the duration of the test was computed. For an example, the 2.66 Hz input motion test showed that the upper bound velocity value was bellow 10 cm/sec (Figure 3.8a). This value was attained only for short periods during the test (Figure 3.8b). The velocity content chart shows that 80% of the velocities fall under 3 cm/sec (Figure 3.8b). Taking 3 cm/sec as the upper bound velocity, the corresponding friction angle should be av < 17 o. The 50% is 1 cm/sec value corresponding to friction angle of av = 16 o. In all DDA simulations a constant value of av = 16 o was used for the validation study. It should be noted that velocity dependence is a test artifact associated with particular geo-interface used, and is not expected in rock discontinuities (e.g. Crawford and Curran, 1981). For this part of the study the numeric control parameters are: penalty value g0 = 5*10 9 N/m, time step size g1 = sec, assumed maximum displacement g2 = The 2.66 Hz input motion is discussed here in detail and the results are shown in Figure 3.9. For k01 = 1 the numeric error is approximately 80%, but the ultimate displacement values are close, m measured displacement compared to m of calculated solution. Introducing some kinetic damping by reducing k01 improves the agreement between DDA and the physical test. Setting k01 = 0.98, corresponding to 2% velocity reduction, reduces the error to bellow 4%, and improves the tracking of the displacement history by DDA. Setting the k01 = 0.95 results in a highly un-conservative (under estimated) displacement prediction by DDA Plotting e d d d against the input motion frequency (Fig. 3.10) shows that in general N DDA accuracy increases with higher frequencies, and that for k01 = 1 the numeric error is always conservative (over prediction of displacement), with an exception at 6 Hz. Reducing k01 to 0.98 shows a similar effect for all frequencies, reducing the numeric error bellow 10%. 42

43 114.3 cm (45 in) 2 Rigid Block o Shaking Table Inclined Plane cm (48 in) No. Instrument Direction of Measurement 1 accelerometer parallel to plane 2 accelerometer parallel to plane 3 accelerometer horizontal 4 displacement transducer horizontal 5 displacement transducer parallel to plane Figure 3.6. a) General view of the inclined plane and the sliding block (top); b) Sliding block experimental setup and instrumentation location (bottom), from Wartman (1999). Test (Hz) d T (cm) d B (cm) a h (h) Table 3.2. Input motion summary for the shaking table model: is the input motion frequency, d T is the shaking table displacement, d B is relative block displacement, and a h is maximum horizontal table acceleration. 43