key to close the plot window and return to the 3dec> prompt. If a Windows-compatible printer is installed, type call test3.dat
and the plot shown in Figure 2.1 should be sent to your printer. If you do not have a printer connected, type quit
to stop the installation testing.
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User’s Guide
Example 2.1 3DEC output from “TEST1.DAT” >pri max No. No. No. No. No.
Cycles = 2000 MFREE of blocks (total) of blocks (visible) of vertices of zones
block vol. block mass zone vol. zone mass zone stress s11 s22 s33 s12 s13 s23 grid-point x-vel y-vel z-vel fx fy fz x-dis y-dis z-dis No. of contacts No. of sub-contacts
= 27571 2 2 154 399
min 2.887E+02 5.774E+05 6.659E-01 1.332E+03 min -2.210E+04 -1.273E+05 -2.104E+04 -4.905E+03 -1.111E+04 -4.386E+03
MTOP =
max 7.113E+02 1.423E+06 6.841E+00 1.368E+04 max 2.365E+04 0.000E+00 1.896E+04 3.180E+04 8.867E+03 5.525E+03 5.424E-01 3.139E-01 1.488E-03 1.613E+05 1.302E+06 2.333E+03 2.862E-01 1.753E-01 8.237E-04
1250000
ISMAX = 249999
average 5.000E+02 1.000E+06 2.506E+00 5.013E+03 average -1.479E+03 -4.358E+04 -5.028E+00 8.059E+03 1.665E+01 1.671E+01 3.478E-01 2.011E-01 3.512E-04 2.956E+04 1.299E+05 2.786E+02 1.833E-01 1.118E-01 1.929E-04
total 1.000E+03 2.000E+06 1.000E+03 2.000E+06 s.dev. 7.536E+03 3.975E+04 6.906E+03 8.208E+03 1.704E+03 1.449E+03
1 58
If you are not able to reproduce the results of any or all of these three tests, you should review the system requirements and installation steps in Sections 2.1 through 2.1.7. If you are still having difficulty, we recommend that you contact Itasca and describe the problem you have encountered and the type of computer you are using (see Section 6.2 for error-reporting procedures).
3DEC Version 3.0
GETTING STARTED
2-9
3DEC (Version 3.00) 12-Aug-02
16:01
dip= 70.00 above dd = 200.00 center 5.000E+00 5.000E+00 5.000E+00 cut-pl. 0.000E+00 mag = 1.00 cycle 2000
Itasca Consulting Group, Inc.
Figure 2.1
PostScript plot from “TEST3.DAT”
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User’s Guide
2.2 A Simple Tutorial — Use of Common Commands This section is provided for the new user who wishes to begin experimenting with 3DEC right away. A simple example is presented to help you learn some of the basic aspects of solving problems with 3DEC. The example is a three-dimensional model of a sedimentary rock slope. This is a cut slope in rock with steeply dipping foliation planes and is based on an actual problem described by Starfield and Cundall, 1988. A rotational failure was found to occur with simultaneous sliding along both the foliation planes and shallow-dipping fracture planes. The rotational failure mode was identified by two-dimensional distinct element analysis as the principal mechanism for the slope collapse. The three-dimensional model contains two intersecting discontinuities in the slope, forming a wedge. We will evaluate the stability of the slope for different values of joint friction. (The data file, “TUT.DAT,” included in the “\Tutorial\Beginner” directory, contains all the commands we are about to enter interactively.) We run this problem interactively (i.e., by typing the commands from the keyboard, pressingat the end of each command line, and seeing the results directly). To begin, load 3DEC by double-clicking on “3DEC.BAT” in the “\Tutorial\Beginner” directory. Your computer will load the program and display the initial heading followed by the interactive prompt 3dec>. We begin by specifying a single polyhedral block using the POLY brick command.* Type poly brick
(0,80)
(0,50)
(-30,80)
and pressto continue. This command creates a brick-shaped polyhedron which extends from coordinates 0 to 80 units in the x-direction, from 0 to 50 units in the y-direction, and from -30 units to 80 units in the z-direction. To see the polyhedron, type plot
A perspective view of the polyhedron will appear on the screen. The model is viewed from a viewing plane which is defined as being oriented parallel to and coincident with the graphics screen. The model view is defined in terms of the position of the viewing plane relative to the model reference axes. The model axes are a left-handed set (x,y,z) oriented, by default, as x (east), y (vertically up) and z (north). The default view of the model is from the viewing plane oriented parallel to the xy-plane of the model, with the centroid of the model positioned at the center of the screen. The model can be moved and rotated by pressing selected keys on the keyboard. For example, to rotate the model about the x- or y-axes of the viewing plane, press the <3> key and then the arrow keys on the numeric keypad (up/down arrow keys cause rotation about an axis pointing to the right in the viewing plane, left/right arrow keys cause rotation about an axis pointing upward in the viewing plane). The user should turn to Section 5 for a full description of the facilities available in the graphical interface. * See the command reference list in Section 1.2 in the Command Reference for further details. Note that command words can be abbreviated (see Section 2.5).
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To continue the problem and return to the 3dec> prompt, strike thekey. The polyhedron is now split into separate polyhedra by using the JSET command. First, we create boundary blocks that will confine the slope blocks. Enter the commands jset jset
dip 90 dip 90
dd 180 dd 180
origin 0,0,0 origin 0,0,50
These commands create two joint planes through the model at locations defined by a dip angle (dip), a dip direction (dd), and a location on the plane (origin). The dip angle and dip direction are oriented relative to the model axes. (See Section 3.2.2 for further information on locating joint planes in the model.) The bounding blocks are then hidden from view before we introduce joint planes that represent the actual joint structure in the slope. (Note that blocks hidden from view will not be cut by the JSET command.) To hide the bounding blocks, type hide 0,80 0,50 hide 0,80 0,50 mark region 1
-30,0 50,80
Blocks located in the range 0
dip 2.5 dip 2.5
dd 235 dd 315
or 30,12.5,0 or 35,30,0
and the high angle foliation planes with the command jset dip 76
dd 270
spacing 4
num 5
or 38,12.5,0
The last command contains two additional keywords that allow us to generate a set of joints automatically. The spacing keyword specifies an average spacing between joint planes, and the num keyword defines the number of joints in the joint set. We now hide the slope blocks and create a horizontal joint plane that is the base of the slope excavation. hide 30,80 0,50 0,50 jset dip 0 dd 0 or 0,10,0 hide 0,80 0,10 -30,80 mark region 2
We assign region number 2 to the blocks within the excavation region. Finally, we hide the blocks surrounding the slope blocks and create the joint planes that define the wedge in the slope. seek hide region 0 hide 0,80 0,10 hide 55,80 0,50 hide 0,30 0,50
0,50 0,50 0,50
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jset jset
User’s Guide
dip 70 dip 60
dd 200 dd 330
or 0,0,35 or 50,50,15
We can view the slope and joint planes by hiding the boundary blocks and the blocks representing the excavation. seek hide region 0 2
We view the slope oriented at a selected perspective view defined by a dip angle and dip direction relative to the viewing plane. We also magnify the view by a factor of 2. plot dip 70
dd
210
mag 2
axes
color
material
Figure 2.2 shows the model at this view.
3DEC (Version 3.00) 27-Aug-02
9:25
dip= 70.00 above dd = 210.00 center 4.000E+01 2.500E+01 2.500E+01 cut-pl. 0.000E+00 mag = 2.00 cycle 0
Y z
x
Itasca Consulting Group, Inc.
Figure 2.2
3DEC Version 3.0
3DEC model of a rock slope
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If you wish to make a hardcopy of a plot, enter the command COPY after returning to the 3dec> prompt, and the plot will be sent (by default) to a Windows printer.* Alternatively, you can send the plot to a file for printing at some later time. For example, the commands set plot po bw copy slope.ps
will create a monochrome PostScript file, “SLOPE.PS,” of the last-viewed plot. The file can be sent to a PostScript printer. The default size and orientation of a 3DEC plot is 8.5 in. × 11 in. landscape.† You can print this file without exiting 3DEC, if you wish. Type sys
dos
to spawn a DOS command process. You can then send the “SLOPE.PS” file to your PostScript printer by using the DOS COPY command: copy slope.ps Lp 1
Type the DOS command exit
to return to 3DEC and the 3dec> prompt.
* The printer type can be changed with the SET plot command, and the output port can be changed or a filename can be specified with the SET output command — see Section 1 in the Command Reference. † The size and orientation can be changed via the SET command. For example, to fit two 3DEC PostScript plots on the same page for an 8.5 in. × 11 in. portrait plot, use the following command to orient the top figure. set plot post 72 396 0.6 0.6
For the bottom figure, use: set plot post 72 36 0.6 0.6
Each SET command should be given prior to issuing the COPY command.
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User’s Guide
Next, the boundary blocks are immobilized and gravity is activated by typing seek fix 0 80 0 10 0 50 fix 55 80 0 50 0 50 fix region 0 hide region 0 delete region 2 gravity 0 -10 0 seek
The FIX commands fix the current velocity (i.e., zero) of all blocks within the specified ranges. The GRAVITY command assigns a gravitational acceleration in the negative y-direction. In this case we specify a value of 10 m/sec2 . Material properties are assigned to a property number for the blocks and joints by typing prop mat=1 dens=2000 prop jmat=1 kn=1e9 ks=1e9 f=89. prop jmat=2 kn=1e9 ks=1e9 f=0.0
For this problem, the mass density of all blocks is specified to be 2,000 units (kg/m3 , in this case). Note that the mass density is assigned, not the unit weight of the block material. For this exercise, the blocks are assumed to be rigid; block deformability is neglected. Two different material numbers are assigned to joints in the model. Both material numbers have the same contact normal (kn) and shear (ks) stiffness equal to 1.0 × 109 (here, Pa/m). Joint material 1 has a friction angle equal to 89◦ and joint material 2 has a friction angle equal to 0◦ . Joint material 2 is assigned to the joint contacts between the slope blocks and the boundary blocks, with the command change dip 90 dd 180 jmat=2
This provides a frictionless boundary along the vertical joint planes of the boundary blocks. At this point, the problem is ready to be executed. As will be seen later, it is often helpful to judge behavior (i.e., equilibrium, stability, instability) by observing the motion of specified points in the rock mass. In this problem, we monitor the y-velocity of a point at the location x = 30, y = 30, z = 30. The command used to record this motion is hist
yvel
(30,30,30) type 1
Following execution of this command, the program returns information about the selected monitoring point (30,30,30). The keyword type instructs the program to print the value (in this case, the y-velocity of point (30,30,30)) on the screen at specified intervals. Five hundred calculation cycles are executed by typing step
500
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During execution, the current cycle count, the calculation time, the maximum out-of-balance force, the y-velocity of the block vertex closest to point (30,30,30) and the clock time are printed on the screen every 10 cycles. Inspection of these values indicates that equilibrium has been obtained. (The velocity and out-of-balance force approach zero.) A graphical representation of this behavior is obtained by typing plot
hist 1
To give hardcopy plots a heading, type title new title>ROCK SLOPE STABILITY
Next, type plot pen hist 1
to create a hardcopy plot of the y-velocity history (see Figure 2.3). Rock Slope Stability
3DEC (Version 3.00)
(E-002) 1.0
HISTORY PLOT 27-Aug-02 9:26 cycle 500 0.5 Hist. no. 1 -2.775E-02 to 1.248E-09 VS Time
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5 0.0
0.5
1.0
1.5
2.0
2.5
3.0 (E-001)
Figure 2.3
3.5
4.0
4.5
5.0
5.5
6.0 Itasca Consulting Group, Inc.
History of y-velocity for initial rock slope
It is often helpful to save this initial state so that it can be restarted at any time — for example, to perform parameter studies. To save the current state (in a file called “SLOPE.SAV”), type save
slope.sav
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User’s Guide
The behavior of the slope can be studied by reducing the friction of the joints. We reduce the friction angle to 6◦ with the following command. prop jmat=1 f=6.0
Next, the calculation process continues; the problem state after 2000 additional cycles (2500 cycles total) is shown in Figure 2.4. This figure was obtained following execution of the following commands. cycle 2000 hide reg 0 title new title> ROCK SLOPE STABILITY -- WEDGE FAILURE plot dip 70 dd 210 mag 2
The figure shows the failure mode that develops in the slope. The failure mode combines rotational failure along the foliation planes and rotational failure of the wedge. The wedge failure dominates the failure, as shown by the block plot in Figure 2.4. The rotational mechanism contributes to the collapse. This can be seen in a vertical cross-section plot taken through the model. Enter the command plot xsec dip 90 dd 180 mag 4 wire disp blue
to view a vertical section through the wedge (see Figure 2.5). Note that cross-sectional plots can be oriented at any angle through the model, and various parameters can be presented on these sections. From this point, you may wish to play with the various features of 3DEC in an attempt to stabilize the slope. Try restarting the previous file you created by entering rest slope.sav
Try using the structural element logic described in Section 4 in Theory and Background to model rock anchors or tiebacks to support the slope. (An example illustrating support for this slope is given in Section 4.2.1.7 in Theory and Background.) To exit 3DEC, type quit
This ends the initial tutorial. In the following sections, we will present other features of 3DEC. We recommend that you read the rest of Getting Started for a beginner’s guide to the mechanics of using 3DEC. As you become more familiar with the code, turn to Section 3 for additional details on problem solving with 3DEC.
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ROCK SLOPE STABILTY -- WEDGE FAILURE
3DEC (Version 3.00) 27-Aug-02
9:26
dip= 70.00 above dd = 210.00 center 4.000E+01 2.500E+01 2.500E+01 cut-pl. 0.000E+00 mag = 2.00 cycle 2500
Y z
x
Itasca Consulting Group, Inc.
Figure 2.4
Rock slope failure in progress
ROCK SLOPE STABILTY -- WEDGE FAILURE
3DEC (Version 3.00) Cross section plot: 27-Aug-02 9:26 geometric scale 0
2E+01 vector scale
0
2E+01
dip= 90.00 above dd = 190.00 center 4.000E+01 2.500E+01 2.500E+01 cut-pl. 0.000E+00 mag = 4.00 cycle 2500
Max disp
in plane = 3.648E+00
Itasca Consulting Group, Inc.
Figure 2.5
Vertical cross-section through wedge showing displacement vectors
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User’s Guide
2.3 Nomenclature The nomenclature used in 3DEC is similar, for the most part, to that used in continuum stress analysis programs. In addition though, special terminology is used to describe the discontinuum features in a 3DEC model. The basic definitions are given here for clarification. Figure 2.6 is provided to illustrate 3DEC terminology.
fault discontinuity
joint discontinuity
cable
block
in-situ horizontal boundary stress zone
interior boundary
gridpoint
(excavation)
roller bottom boundary
Figure 2.6
Example of a 3DEC model (not to scale)
3DEC MODEL — The 3DEC model is created by the user to simulate a physical problem. When referring to a 3DEC model, we imply a sequence of 3DEC commands (see Section 1 in the Command Reference) which define the problem conditions for numerical solution. BLOCK — The block is the fundamental geometric entity for the distinct element calculation. The 3DEC model is created by either “cutting” a single block into many smaller blocks, or creating separate blocks and joining them together. Each block is an independent entity that may be detached from other blocks or may interact with other blocks via surface forces. Another term for block is polyhedron.
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CONTACT — Each block is connected to adjacent blocks via point contacts. A contact may be considered a boundary condition that applies external forces to each block. SUB-CONTACT — Each contact is divided into sub-contacts for both rigid and deformable blocks. Interaction forces between blocks are applied at sub-contacts. DISCONTINUITY — A discontinuity is a geologic feature that separates a physical mass into distinct parts. Discontinuities, for example, include joints, faults and fractures and other discontinuous features in a rock mass. To be represented in 3DEC, a discontinuity must have a trace length scale that is approximately of the same order as the engineering structure being analyzed. A discontinuity in 3DEC is defined by at least one contact between blocks. ZONE — Deformable blocks are composed of tetrahedral finite-difference zones. Mechanical changes (e.g., stress/strain) are calculated within each zone. Mixed-discretization (m-d) zones are special zones that are composed of two overlays of five tetrahedral sub-zones. m-d zones provide accurate solutions for block plasticity analysis. GRIDPOINT — Gridpoints are associated with the corners of the tetrahedral finite-difference zones (or sub-zones of m-d zones). There are always four gridpoints associated with each zone. A set of x-, y-, z-coordinates is assigned to each gridpoint, thus specifying the exact location of the finite-difference zones. Other terms for gridpoint are nodal point and node. MODEL BOUNDARY — The model boundary is the periphery of the 3DEC model. Internal boundaries (i.e., holes within the model) are also model boundaries. BOUNDARY CONDITION — A boundary condition is the prescription of a constraint or controlled condition along a model boundary (e.g., a fixed displacement or force for mechanical problems). INITIAL CONDITIONS — This is the state of all variables in the model (e.g., stresses) prior to any loading change or disturbance (e.g., excavation). NULL BLOCK — Null blocks are blocks that represent voids (i.e., no material present) within the model. Null blocks can be made “real” later in an analysis — for example, to simulate backfilling. (Once a block is deleted from a model it cannot be restored.) BLOCK CONSTITUTIVE MODEL — The block constitutive (or material) model represents the deformation and strength behavior prescribed to the zones of deformable blocks in a 3DEC model. Several constitutive models are available in 3DEC to simulate different types of behavior commonly associated with geologic materials. JOINT CONSTITUTIVE MODEL — The joint constitutive model represents the normal and shear interaction between blocks at their contact (sub-contact) points. The joint model includes a normal
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User’s Guide
and shear elastic stiffness component and a limiting shear and tensile strength component. The basic joint model is the Coulomb-slip model. STRUCTURAL ELEMENT — Structural elements are one-dimensional elements that represent the interaction of structures (such as rock bolts or cable bolts) with a rock mass. Material nonlinearity is possible with structural elements. Geometric nonlinearity occurs as a result of the large-strain formulation. STEP — Because 3DEC is an explicit code, the solution to a problem requires a number of computational steps. During computational stepping, the information associated with the phenomenon under investigation is propagated across the blocks in the model. A certain number of steps is required to arrive at an equilibrium (or steady-flow) state for a static solution. Typical problems are solved within 2000 to 4000 steps, although large complex problems can require tens of thousands of steps to reach a steady state. When using the dynamic analysis option, STEP or CYCLE refers to the actual timestep for the dynamic problem. Other terms for step are timestep and cycle. STATIC SOLUTION — A static or quasi-static solution is reached in 3DEC when the rate of change of kinetic energy in a model approaches a negligible value. This is accomplished by damping the equations of motion. At the static solution stage, the model will be either at a state of force equilibrium or at a state of steady flow of material if a portion (or all) of the model is unstable (i.e., fails) under the applied loading conditions. This is the default calculation mode in 3DEC and can also be invoked with the DAMP auto or DAMP local command. UNBALANCED FORCE — The unbalanced force indicates when a mechanical equilibrium state (or the onset of joint slip or plastic flow) is reached for a static analysis. A model is in exact equilibrium if the net nodal force vector at each block centroid or gridpoint is zero. The maximum nodal force vector is monitored in 3DEC and printed to the screen when the STEP or CYCLE command is invoked. The maximum nodal force vector is also called the “unbalanced” or “outof-balance” force. The maximum unbalanced force will never exactly reach zero for a numerical analysis. The model is considered to be in equilibrium when the maximum unbalanced force is small compared to the representative forces in the problem. If the unbalanced force approaches a constant nonzero value, this probably indicates that joint slip or block failure and plastic flow are occurring within the model. DYNAMIC SOLUTION — For a dynamic solution, the full dynamic equations of motion (including inertial terms) are solved; the generation and dissipation of kinetic energy directly affect the solution. Dynamic solutions are required for problems involving high frequency and short duration loads — e.g., seismic or explosive loading. The dynamic calculation is an optional module to 3DEC (see Section 2 in Optional Features).
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2.4 The 3DEC Model For most geomechanics analyses, the creation of a 3DEC model begins with a single block of a size that spans the physical region being analyzed.* The model features are then introduced by cutting this block into smaller blocks whose boundaries represent both the geologic structure (e.g., faults, bedding planes, joint structure) and engineered structures such as underground excavations and tunnels. All blocks in the model are defined by the x-, y-, z-coordinates of their vertices and centroid. Contacts between blocks, as well as gridpoints within deformable blocks, are also defined by their coordinate position. Model generation involves cutting the model block along planes whose positions are defined by an orientation (dip and dip direction) and one location on the plane. All entities of the 3DEC model (blocks, vertices, contacts, gridpoints and zones) are identified uniquely by an address number in the main data array, allocated automatically by 3DEC. These numbers may also be used to refer to a particular entity. The numbering system is not sequential for each entity, so the user must identify the number via a plot or printout.† For example, Figure 2.7 illustrates a 3DEC model block of the following dimensions: 10 units (say, meters) in the x-direction, 10 units in the y-direction and 10 units in the z-direction. The model block is divided into two blocks separated by a horizontal discontinuity located through the center of the block. The model shown in Figure 2.7 was created with the commands listed in Example 2.2, shown below. Example 2.2 3DEC model block divided into two blocks poly brick 0,10 0,10 0,10 jset origin 5 5 5 prop jmat 1 kn 1.33e7 ks 1.33e7 fric 20.0 prop mat 1 dens 2000 plot hold dip 70 dd 210 color mat cyc 1 ret
* The 3DEC model can also be generated by creating separate blocks and joining them together. This can be useful for building multiple blocky structures such as a masonry wall or arch bridge — e.g., see Section 3 in the Examples volume. † In general, address numbers should be avoided if possible when referring to particular entities in limiting the range of application of a command. Address numbers will likely change with different versions of 3DEC. Other optional range phases, as listed in Section 1.1.3 in the Command Reference, should be used whenever possible.
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User’s Guide
3DEC (Version 3.00) 27-Aug-02
9:42
dip= 70.00 above dd = 210.00 center 5.000E+00 5.000E+00 5.000E+00 cut-pl. 0.000E+00 mag = 1.00 cycle 0
Itasca Consulting Group, Inc.
Figure 2.7
3DEC model block divided into two blocks
The two blocks have block numbers 218 and 1078. The blocks are connected by one contact located at the center of the adjacent faces of the two blocks. The contact number is 1739. This information can be obtained with the following commands. print block print contact
Upon cycling, contacts are automatically decomposed into sub-contacts at which mechanical interactions between blocks are calculated. Sub-contacts are created by triangulating interacting block faces. For rigid blocks, each triangular section of the face is associated with a vertex on the face. In this example, we issue the command cycle 1
to create the sub-contacts. Eight sub-contacts are created; associated with each of the four vertices defining the two contacting faces. The sub-contact numbers can be viewed with the command print contact location
The two blocks may be made deformable by creating finite-difference zones in each block. The blocks in Example 2.2 are made deformable by adding the command gen edge 20
The two blocks are each subdivided into six zones with each zone defined by four gridpoints. The zone and gridpoint numbers and the gridpoint coordinates are printed with the command
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print zone location
Note that if we take one cycle, eight sub-contacts are created as before for the rigid blocks, but the sub-contact numbers are different because their addresses are created after those for the zones and gridpoints. The address numbers also act as pointers to storage locations of all state variables in the model. Data associated with each entity in the model are stored with that entity number. For example, block forces, velocities and displacements for rigid blocks are stored with each block number. For deformable blocks, vector quantities (e.g., forces, velocities, displacements) for a block are stored with gridpoint numbers, while scalar and tensor quantities (e.g., stresses, material property numbers) are stored with zone numbers. Contact data such as contact force, velocity and flow rates are stored at sub-contact numbers. FISH can be used to access 3DEC data via the address numbers. See Section 4 in the FISH volume for lists of the variables that can be accessed.
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User’s Guide
2.5 Command Syntax All input commands* to 3DEC are word-oriented and consist of a primary command word followed by one or more keywords and values, as required. Some commands accept switches — that is, keywords that modify the action of the command. Each command has the following format:
COMMAND keyword value . . .. . . Here, optional parameters are denoted by < >, while the ellipses ( . . . ) indicate that an arbitrary number of such parameters may be given. The commands are typed literally on the command line. You will note that only the first few letters are in bold type. The program requires these letters, at a minimum, to be typed to recognize the command; command input is not case-sensitive. The entire word for commands and keywords may be entered if the user so desires. Many of the keywords are followed by a series of values which provide the numeric input required by the keyword. The decimal point may be omitted from a real value, but may not appear in an integer value. Commands, keywords and numeric values may be separated by any number of spaces or by any of the following delimiters: ( ) , = A semicolon ( ; ) may be used to precede comments; anything that follows a semicolon in an input line is ignored. It is useful, and strongly recommended, to include comments in data files. Not only is the input documented in this way, the comments are echoed to the output as well, providing the opportunity for quality assurance in your analysis. A single input line, including comments, may contain up to 80 characters. If more than 80 characters are required to describe a particular command sequence, then an ampersand (&) can be given at the end of an input line to denote that the next line will be a continuation of that line. A total of 1024 characters per command sequence are allowed. Please note that the typographical conventions listed in Table 2.2 are used throughout this manual. * The commands and their meanings are presented in Section 1 in the Command Reference; a summary is given in Section 1 in the Command and FISH Reference Summary.
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Table 2.2
2 - 25
Typographical conventions
Type style
Used for
BOLD bold bold
3DEC commands and FISH statements 3DEC keywords and FISH internal variables and functions user-defined FISH variables and functions menu items and buttons with the hot-keys underlined place-holders for variables button with the hot-key underlined type the key between < > (here, ) on the keyboard hold down the first key while pressing the second (here,and the key)
Initial Caps
var Press Me
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User’s Guide
2.6 Mechanics of Using 3DEC 3DEC is based upon a command-driven format. Word commands control the operation of the program. This section provides an introduction to the basic commands a new user needs to perform simple 3DEC calculations. If you have not done so already, run the tutorial problem in Section 2.2 for an example of command-driven analysis with 3DEC. In order to set up a model to run a simulation with 3DEC, three fundamental components of a problem must be specified: (1) a distinct-element model that matches the problem geometry; (2) constitutive behavior and material properties; and (3) boundary and initial conditions. The model block defines the geometry of the problem. The constitutive behavior and associated material properties dictate the type of response the model will display upon disturbance (e.g., deformational response due to excavation). Boundary and initial conditions define the in-situ state (i.e., the condition before a change or disturbance in problem state is introduced). After these conditions are defined in 3DEC, an alteration is made (e.g., excavate material or change boundary conditions), and the resulting response of the model is calculated. The actual solution of the problem is different for an explicit-solution program like 3DEC than it is for conventional implicit-solution programs. (See the background discussion in Section 1.2.2 in Theory and Background.) 3DEC uses an explicit time-marching method to solve the algebraic equations. The solution is reached after a series of computational steps. In 3DEC, the number of steps required to reach a solution is controlled manually by the user. The user ultimately must determine if the number of steps is sufficient to reach the solved state. See Section 2.6.4 for ways in which this is done. The general solution procedure for an explicit static* analysis with 3DEC is illustrated in Figure 2.8. This procedure is convenient because it represents the sequence of processes that occur in the physical environment. The basic 3DEC commands needed to perform simple analyses with this solution procedure are described below.
* Dynamic analysis with 3DEC is discussed in Section 2 in Optional Features.
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MODEL SETUP (1) Generate model block, cut block to create problem geometry (2) Define constitutive behavior and material properties (3) Specify boundary and initial conditions
Step to equilibrium state
Examine the Model Response
PERFORM ALTERATIONS For Example: Excavate material Change boundary conditions
Step to solution
Examine the Model Response
REPEAT FOR ADDITIONAL ALTERATIONS
Figure 2.8
General solution procedure for static analysis in geomechanics
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User’s Guide
2.6.1 Model Generation The 3DEC model is usually created by cutting the original 3DEC block into smaller blocks that represent boundaries of physical features in the problem. The simplest block to create is brickshaped and is generated with the command poly brick
xl, xu
yl, yu
zl, zu
where (xl, xu), (yl, yu) and (zl, zu) are the lower- and upper-coordinate limits of the brick in the x-, y- and z-directions. The primary command used to create geologic structure (e.g., joints) is jset
The JSET command can either create individual joints or invoke an automatic joint set generator to create a set of joints defined by characteristic parameters — i.e., dip angle, dip direction, spacing, spatial location and persistence. The following example illustrates block cutting with the JSET command. The complete description for this command is given in Section 1 in the Command Reference. Joint generation is explained in more detail in Section 3.2.2. Example 2.3 Block model with three intersecting joint planes poly jset jset jset plot ret
brick -1 1 -1 1 -1 1 dd 270 dip 65 origin 0.3,0,0 dd 230 dip 40 origin 0,0,-0.3 dd 320 dip 50 origin 0,0,0.3 hold dip 70 dd 200 color mat
The three JSET commands define three joint planes through the model. The joints are located by their dip direction, dd, dip angle, dip, and a single point on the plane, origin. See Figure 3.5 in Section 3.2.2 for the definition of the orientations for dip direction and dip angle relative to the 3DEC model axes. By typing the command plot dip 70 dd 200
a plot of the model blocks oriented relative to the model reference axes is shown in the graphics mode (see Figure 2.9).
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3DEC (Version 3.00) 27-Aug-02
9:43
dip= 70.00 above dd = 200.00 center -2.980E-08 -2.980E-08 -2.980E-08 cut-pl. 0.000E+00 mag = 1.00 cycle 0
Itasca Consulting Group, Inc.
Figure 2.9
Block model with three intersecting joint planes
Shapes of engineered structures must also be cut in the 3DEC block, and these must be created before model execution begins. The JSET command can also be used to create shapes in a model. Boundaries of excavations are created as joint planes. An additional command, TUNNEL, is provided specifically to create tunnel shapes. The TUNNEL command creates a tunnel whose boundary is formed by planar segments that connect the two end faces of the tunnel, designated face A and face B. For example, a square-shaped tunnel can be created in the Example 2.3 model by adding the commands in Example 2.4. Example 2.4 Tunnel in jointed rock tunnel
a (-.3,-.3,-1.5) (-.3,.3,-1.5) (.3,.3,-1.5) (.3,-.3,-1.5) b (-.3,-.3,1.5) (-.3,.3,1.5) (.3,.3,1.5) (.3,-.3,1.5) remove -0.3,0.3 -0.3,0.3 -1.5,1.5 plot hold dip 70 dd 200 color mat ; plot excavation and joint structure only plot hold exc joint ret
&
Four vertices define each tunnel face; the vertices for each face must be entered in the same order to form connecting planes between the faces. The REMOVE command is used to delete the blocks
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User’s Guide
within the tunnel region. The resulting model is shown in Figure 2.10. The excavation blocks and the joint structure can also be plotted separately with the command (see Figure 2.11): plot exc
joint
Note that only the joints created by the JSET command are plotted in Figure 2.11. The “fictitious” joints created when the tunnel excavation was made with the TUNNEL command are not shown. These joints lock the adjoining blocks together so that they behave as one block. Joining blocks via fictitious joints can also be accomplished with the JOIN command. See Section 3.2.2 for details.
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dip= 70.00 above dd = 200.00 center -2.980E-08 -2.980E-08 -2.980E-08 cut-pl. 0.000E+00 mag = 1.00 cycle 0
Itasca Consulting Group, Inc.
Figure 2.10 Tunnel in jointed rock
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3DEC (Version 3.00) 27-Aug-02
9:43
dip= 70.00 above dd = 200.00 center -2.980E-08 -2.980E-08 -2.980E-08 cut-pl. 0.000E+00 mag = 1.00 cycle 0
Itasca Consulting Group, Inc.
Figure 2.11 Tunnel in jointed rock — excavation and joint structure
2.6.2 Assigning Material Models 2.6.2.1 Block Models Once all block cutting is complete, material behavior models must be assigned for all the blocks and discontinuities in the model. By default, all blocks are rigid. In most analyses, blocks should be made deformable. Only for cases in which stress levels are very low or the intact material possesses high strength and low deformability can the rigid block assumption be applied (for example, see the slope failure tutorial in Section 2.2). Blocks are made deformable via the command gen edge v
or gen quad ndiv i1
i2
i3
The GEN (or GENERATE) command invokes an automatic mesh generator that fills each block with tetrahedral-shaped finite difference zones. The command GEN edge v will work for blocks of any arbitrary shape. The value v defines the average edge length of the tetrahedral zones — i.e., the smaller the value for v, the higher the density of zones in a block. Care should be taken, though, to not create zones that have a high aspect ratio; a practical limit on aspect ratio is approximately 1:5 for reasonable solution accuracy. Type
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User’s Guide
plot zol
to check the zoning in a model. The command GEN quad ndiv i1 i2 i3 should be used if blocks are prescribed a plastic material model. This type of zoning provides a more accurate solution for plasticity problems (see Section 1.2.2.5 in Theory and Background for a description of this type of zoning). The GEN quad command, however, may not work for all block shapes; if not, the GEN edge command should be used for the remaining blocks. There are five built-in material models for deformable blocks in 3DEC; these are described in Section 2 in Theory and Background. Three models are sufficient for most analyses the new user will make. These are assigned by the following commands excavate ; null model change cons=1; elastic model change cons=2; Mohr-Coulomb model
The EXCAVATE command simulates the excavation or removal of material that will be replaced at a later stage in the analysis. Blocks within the region that is excavated can be changed back into elastic or elastic-plastic material with the FILL command. For example, if the excavation for the tunnel model of Example 2.4 is to be filled at a later stage, the EXCAVATE command should be used in place of the REMOVE command. If a block is deleted via the REMOVE or DELETE command, it cannot be restored at a later stage. The CHANGE command changes the material model assigned to a deformable block. Blocks changed to cons=1 are assigned isotropic elastic material behavior, while blocks changed to cons=2 are assigned Mohr-Coulomb plasticity behavior. By default, all deformable blocks are assigned cons=1. The models are described briefly in Table 3.2 in Section 3.7.1. The blocks changed to cons=1 and cons=2 must have material properties assigned via the PROPERTY mat command. Note that properties are not assigned to specific blocks, but rather to a material number. Properties may be assigned to as many as 50 material numbers. The material numbers are then assigned to blocks with the CHANGE mat command. For the elastic model, the required properties are (1) density; (2) bulk modulus; and (3) shear modulus.
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NOTE: Bulk modulus, K, and shear modulus, G, are related to Young’s modulus, E, and Poisson’s ratio, ν, by:
K =
E 3(1 − 2ν)
(2.1)
G =
E 2(1 + ν)
(2.2)
or E =
9KG 3K + G
(2.3)
ν =
3K − 2G 2(3K + G)
(2.4)
For the Mohr-Coulomb plasticity model, the required properties are: (1) density; (2) bulk modulus; (3) shear modulus; (4) friction angle; (5) cohesion; (6) dilation angle; and (7) tensile strength. If any of these properties are not assigned, their values are set to zero by default. For both the elastic and Mohr-Coulomb models, density, bulk modulus and shear modulus must be assigned positive values for 3DEC to execute.
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User’s Guide
2.6.2.2 Joint Models In addition to block material models, a material model must also be assigned to all discontinuities (i.e., contacts) in the model. There are two built-in constitutive models for discontinuities (summarized in Table 3.3 in Section 3.7.2). The model sufficient for most analyses is the (elastic-perfectly plastic) Coulomb slip model, which is assigned to discontinuities with the command change jcons=1
By default, all discontinuities are assigned jcons=1. The material models for discontinuities also have material properties assigned with the PROPERTY jmat command. As with blocks, properties are not assigned directly to the discontinuities but to material numbers. The material numbers are then assigned to the discontinuities with the CHANGE jmat command. For the Coulomb slip model, the required properties are: (1) normal stiffness; (2) shear stiffness; (3) friction angle; (4) cohesion; (5) dilation angle; and (6) tensile strength. If any of these properties are not assigned, their values are set to zero by default. Normal and shear stiffnesses must be assigned positive values for 3DEC to execute. Example 2.5 demonstrates the application of block and joint material models to the tunnel example. Example 2.5 Assigning material models and properties gen edge 1.0 prop mat=1 dens 2000 bulk 1.5e9 g .6e9 prop jmat=1 kn 1e9 ks 1e9 coh 1e9 ten 1e9 ret
The commands in Example 2.5 are entered to assign a mass density of 2000 kg/m3 , bulk and shear moduli of 1.5 GPa and 0.6 GPa to the deformable blocks, normal and shear stiffnesses of 1.0 GPa/m, and cohesion and tensile strength of 1.0 GPa to the joints. Note that we assign a high cohesion and tensile strength to the joints to prevent any slip or separation from occurring when we bring the model to an initial force-equilibrium state.
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2.6.3 Applying Boundary and Initial Conditions Boundary and initial conditions must not be applied until after all block cutting is complete and the mesh for deformable blocks is generated. Mechanical boundary conditions are generally applied with the BOUNDARY command. This command is used to specify force, stress and velocity (displacement) boundary conditions. Boundary forces and stresses can be applied to both rigid and deformable blocks, but boundary velocities can only be applied to deformable blocks. (See the commands FIX, FREE and APPLY to apply boundary conditions to rigid blocks.) Table 2.3 provides a summary of the boundary condition commands and their effect. Refer to Section 1.3 in the Command Reference for a complete listing of keywords to these commands. Table 2.3 Boundary condition command summary Command BOUNDARY
Effect stress xload yload zload xvel yvel zvel
FIX FREE APPLY
total stress applied to rigid or deformable blocks load applied in x-direction to rigid or deformable blocks load applied in y-direction to rigid or deformable blocks load applied in z-direction to rigid or deformable blocks x-velocity applied to deformable blocks y-velocity applied to deformable blocks z-velocity applied to deformable blocks velocities fixed for rigid or deformable blocks velocities freed for rigid or deformable blocks
xvel yvel
x-velocity applied to rigid blocks y-velocity applied to rigid blocks
zvel
z-velocity applied to rigid blocks
The commands BOUNDARY xload, yload and zload apply x-, y- and z-components of force at boundary vertices. The command BOUNDARY stress specifies components of the total stress tensor applied at the boundary. The commands BOUNDARY xvel, yvel and zvel fix the x-, y- and zcomponents of velocity at selected boundary gridpoints. Note that by using the BOUNDARY command, a condition or constraint is imposed that will not change (unless specifically changed by the user). Initial stress conditions can be specified for all zone stresses in deformable blocks and all normal and shear stresses along joints between rigid blocks or deformable blocks. The INSITU command is used to initialize stresses. By using this command, initial values are assigned to stresses; these can change while the computation proceeds. The initial stress state can also include the effect of gravity. This is invoked with the following command.
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gravity
User’s Guide
gx
gy
gz
The first value is the gravitational acceleration component in the x-direction, the second value is that in the y-direction and the third is that in the z-direction. Gravity can be omitted from a model if the stress variation due to gravity is small across the model compared to the in-situ stresses. Gravity is often applied to help identify loose blocks around an opening. This is demonstrated below. If stresses due to gravity are the same magnitude as the in-situ stresses, then a stress gradient should be applied with the INSITU command to speed convergence to the initial equilibrium. Boundary and initial conditions can be applied to the tunnel model, for example, with the commands listed in Example 2.6, below. Example 2.6 Applying boundary and initial conditions bound -1,1 0.9,1.1 -1,1 stress 0.0,-1.0e6,0.0 bound -1.1,-0.9 -1,1 -1,1 xvel 0.0 bound 0.9,1.1 -1,1 -1,1 xvel 0.0 bound -1,1 -1,1 -1.1,-0.9 zvel 0.0 bound -1,1 -1,1 0.9,1.1 zvel 0.0 bound -1,1 -1.1,-0.9 -1,1 yvel 0.0 grav 0,-10,0 insitu stress -0.5e6 -1.0e6 -0.5e6 0.0 0.0 0.0 ret
0.0,0.0,0.0
A stress boundary of 1.0 MPa is applied in the vertical direction to the top boundary. Roller boundary conditions are assigned to the lateral boundaries, and the bottom boundary is fixed from movement in the y-direction. A gravitational acceleration of 10 m/sec2 acts in the negative y-direction. A zero-velocity boundary along the bottom boundary is particularly important when gravity is acting; this prevents the model from moving. Note that stress boundaries affect all degrees-of-freedom. Thus, stress boundary conditions should always be applied before velocity boundary conditions at the same boundary corners; otherwise, the prescribed velocity constraint will be lost. Also, note that x-, y- and z-coordinate ranges are specified for each of the four BOUND commands. Care should be taken to ensure that the boundary affected by the BOUND command falls completely within the range.
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Type print bound
to check boundary conditions. The INSITU command initializes all stresses in the y-direction to -1.0 MPa and in the x- and zdirections to -0.5 MPa. 2.6.4 Stepping to Initial Equilibrium The 3DEC model must be at an initial force-equilibrium state before alterations can be performed. The boundary conditions and initial conditions may be assigned such that the model is exactly at equilibrium initially. However, often it is necessary to calculate the initial equilibrium state under the given boundary and initial conditions, particularly for problems with complex geometries or multiple materials. This is done by using either the STEP (or CYCLE) command. With the STEP command, the user specifies a number of calculation steps to perform in order to bring the model to equilibrium. The model is in equilibrium when the net nodal force vector at each centroid of rigid blocks or gridpoint of deformable blocks is zero (see Section 1.2.2.5 in Theory and Background). The maximum nodal force vector (called the maximum “out-of-balance” or “unbalanced” force) is monitored in 3DEC and printed to the screen when the STEP command is invoked. In this way, the user can assess when equilibrium has been reached. For a numerical analysis, the out-of-balance force will never reach exactly zero. It is sufficient, though, to say that the model is in equilibrium when the maximum unbalanced force is small compared to the total applied forces in the problem. For example, if the maximum unbalanced force is initially 1 MN and drops to approximately 100 N, then the model can be considered at equilibrium, within 0.01% of the initial maximum unbalanced force. This is an important aspect of numerical problem-solving with 3DEC. The user must decide when the model has reached equilibrium. There are several features built into 3DEC to assist with this decision. The history of the maximum unbalanced force may be recorded with the following command: hist
unbal
Additionally, the history of selected variables (e.g., velocity or displacement at a gridpoint) may be recorded. The following commands are examples: hist hist
xvel 5,5,5 ydisp 0,11,0
The first history records x-velocity at a gridpoint location closest to (x = 5, y = 5, z = 5), while the second records y-displacement at a location closest to (x = 0, y = 11, z = 0) in the model. After running several hundred (or thousand) calculation steps, a history of these records may be plotted to indicate the equilibrium condition. By default, 3DEC performs a static analysis by applying a mechanical damping algorithm known as adaptive global (or auto) damping. This algorithm is described in Section 1.2.2.7 in Theory and
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Background. The data file in Example 2.7 illustrates the process to reach an initial equilibrium state. Example 2.7 Stepping to initial equilibrium hist hist hist step save ret
unbal ydis 0.3,0.3,0 ty 1 500 tun0.sav
The initial unbalanced force is approximately 0.2 MN. After 500 steps, this force has dropped to around 5 N. By plotting the two histories, it can be seen that the maximum unbalanced force has approached zero, while the displacement has approached a constant magnitude of approximately 4.3 × 10−4 m. Type plot hist 1 plot hist 2
to view these plots. The number following PLOT hist corresponds to the order in which the histories are entered in the data file. Figures 2.12 and 2.13 show the unbalanced force and displacement history plots, respectively.
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HISTORY PLOT 27-Aug-02 9:58 cycle 500 Hist. no. 1 5.457E+00 to 1.155E+05
1.2
VS Time 1.0
0.8
0.6
0.4
0.2
0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
(E-002)
-0.2
Figure 2.12 Maximum unbalanced force history
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3DEC (Version 3.00)
(E-004) 0.0
HISTORY PLOT 27-Aug-02 9:58 cycle 500 -0.5 Hist. no. 2 -4.313E-04 to -1.742E-05 -1.0
VS Time
-1.5
-2.0
-2.5
-3.0
-3.5
-4.0
-4.5
-5.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
(E-002)
Itasca Consulting Group, Inc.
Figure 2.13 y-displacement history at (.3, .3, 0) It is very important in an analysis that the model be at equilibrium before alterations are made. Several histories should be recorded throughout a model to ensure that a large force imbalance does not exist. It does not affect the analysis adversely if more steps are taken than are needed to reach equilibrium. However, it will affect the analysis if an insufficient number of steps is taken. A 3DEC calculation can be interrupted at any time during stepping by pressing. It often is convenient to use the STEP command with a high step number and periodically interrupt the stepping, check the histories, and resume stepping until the equilibrium condition is reached. 2.6.5 Performing Alterations 3DEC allows model conditions to be changed at any point in the solution process. These changes may be of the following forms: • excavation of material; • addition or deletion of boundary loads or stresses; • fix or free velocities of boundary corners; or • change of material model or properties for blocks or discontinuities.
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Excavation is performed with either the DELETE (or REMOVE) command or EXCAVATE command. Loads and stresses are applied with the BOUNDARY xload, yload, zload or stress command. Boundary vertices are fixed via the BOUNDARY xvel, yvel or zvel command. The constraint at boundaries is removed with BOUNDARY xfree, yfree and zfree. Material models for deformable blocks and discontinuities are changed with the CHANGE command, while properties are changed with the PROPERTY command. It should be evident that several commands can be repeated to perform various model alterations. For example, continue Example 2.7 from the initial equilibrium stage using the commands in Example 2.8. Example 2.8 Reduce the strength of the joints rest tun0.sav ; reduce friction along joints prop jmat 1 fric 6.0 coh 0.0 ten 0.0 ; reset time hist disp hist unbal hist ydisp 0.3,0.3,-0.1 hist ty 2 cycle 5000 save tun1.sav hide -.4 .5 .3 .8 -1 -.5 pl hold dip 90 dd 190 ret
The three joints and the excavation in this model form an isolated wedge in the roof of the excavation. The wedge is potentially unstable and can slide along the joint plane dipping at 65◦ . In Example 2.8 we reduce the strength of the joint structure in the model by setting the cohesion and tensile strength to zero and the friction angle to 6◦ .* The failure after an additional 5000 cycles is shown by the block plot in Figure 2.14. The wedge in the roof of the excavation has become detached from the surrounding blocks and is falling into the excavation. Blocks in front of the unstable wedge are hidden for better viewing of the wedge. (Hidden blocks are still present for mechanical calculations.) The instability is also indicated by the y-displacement history plot in Figure 2.15. The history at location (0.3,0.3,-0.1) corresponds to one vertex on the wedge. Note that we reset the time, history records and displacement in the model in Example 2.8, so that only the change in displacement due to the drop in joint strength is monitored.
* Alternatively, we could start the analysis with the tunnel blocks still in place and the joint strength set to a low value, and solve for an initial equilibrium state. Then, we could excavate the tunnel and monitor the response.
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3DEC (Version 3.00) 27-Aug-02
10:08
dip= 90.00 above dd = 190.00 center -2.980E-08 -2.980E-08 -2.980E-08 cut-pl. 0.000E+00 mag = 1.00 cycle 5500
Y z
x
Itasca Consulting Group, Inc.
Figure 2.14 Sliding wedge in tunnel
3DEC (Version 3.00)
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HISTORY PLOT 27-Aug-02 10:08 cycle 5500 -1.0 Hist. no. 2 -7.410E-02 to -2.787E-05 VS Time
-2.0
-3.0
-4.0
-5.0
-6.0
-7.0
-8.0
-9.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
(E-001)
3.5
4.0
4.5
5.0
5.5 Itasca Consulting Group, Inc.
Figure 2.15 y-displacement history at (.3, .3, -0.1)
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User’s Guide
2.6.6 Saving/Restoring Problem State Two other commands, SAVE and RESTORE, are helpful when performing analyses in stages. At the end of one stage (e.g., initial equilibrium), the model state can be saved by typing save
file.sav
where file.sav is a user-specified filename. The extension “.SAV” identifies this file as a saved file (see Section 2.9). This file can be restored at a later time by typing rest
file.sav
and the model state at the point at which the model was saved will be restored. It is not necessary to build the model from the beginning every time a change is made; merely save the model before the change and restore it whenever a new change is to be analyzed. For example, in the previous example, the state should be saved after the initial equilibrium stage. Then, different methods can be evaluated to stabilize the falling block. For example, we inserted the following save tun0.sav
after the STEP 500 command at the end of the data file for Example 2.7. Now we try stabilizing the block with cable reinforcement using the commands in Example 2.9. A single cable is installed through the wedge and into the surrounding rock. See Section 4 in Theory and Background and Section 1 in the Command Reference for descriptions of the STRUCT cable command and cable parameters assigned with the PROPERTY command. Example 2.9 Stabilize roof block with a cable bolt rest tun0.sav ; reduce friction along joints prop jmat 1 fric 6.0 coh 0.0 ten 0.0 ; ; add cable support struct cable 0.3 0.3 0.0 0.7 0.7 0.0 prop 1 seg 4 struct prop 1 area 5e-4 e 1e9 yield 1e6 kbond 15e8 sbond 1e9 ; reset time hist disp hist unbal hist ydisp 0.3,0.3,-0.1 hist ty 2 cycle 5000 save tun2.sav ret
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After the run is completed, the saved file, “TUN2.SAV,” can be restored and evaluated to study the effect of cable reinforcement. A history of y-displacement shows that the wedge has stopped moving after 4.3 × 10−3 m of vertical displacement. The file “TUN0.SAV” can be restored again and different cable locations, orientations and properties investigated. Several files can be linked together, with RESTORE tun0.sav beginning each section and a different filename saved after execution. Each save file can then be evaluated separately after the entire run is completed.
3DEC (Version 3.00)
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HISTORY PLOT 27-Aug-02 10:12 cycle 5500 -0.5 Hist. no. 2 -4.318E-03 to -2.692E-05 -1.0
VS Time
-1.5
-2.0
-2.5
-3.0
-3.5
-4.0
-4.5
-5.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
(E-001)
3.5
4.0
4.5
5.0
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Figure 2.16 y-displacement history at (.3, .3, -0.1) — wedge is stable
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User’s Guide
2.6.7 Summary of Commands for Simple Analyses The major command words described in this section are summarized in Table 2.4. These are all that are needed to begin performing simple analyses with 3DEC. Start by running simple tests with these commands (e.g., direct shear tests on single joints or simple excavation stability analyses). It may be helpful to review the detailed description of these commands in Section 1.3 in Theory and Background. Then try adding more complexity to the model. Before running very detailed simulations though, we recommend that you read Section 3, which provides guidance on problem solving in general. Table 2.4
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Basic commands for simple analyses
Function
Command
Block Model Creation Block Cutting
POLY JSET TUNNEL
Material Model & Properties for Blocks and Joints
GEN CHANGE PROPERTY
Boundary/Initial Conditions
BOUNDARY INSITU
Initial Equilibrium (with gravity)
GRAVITY STEP
Perform Alterations
DELETE CHANGE PROPERTY BOUNDARY STRUCTURE cable
Monitor Model Response
HISTORY PLOT
Save/Restore Problem State
SAVE RESTORE
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2.7 Sign Conventions The following sign conventions are used in 3DEC and must be kept in mind when entering input or evaluating results. BLOCK MOTION — Positive motion is in the positive coordinate axes directions. DIRECT STRESS — Positive stresses indicate tension; negative stresses indicate compression. SHEAR STRESS — With reference to Figure 2.17, a positive shear stress points in the positive direction of the coordinate axis of the second subscript if it acts on a surface with an outward normal in the positive direction. Conversely, if the outward normal of the surface is in the negative direction, then the positive shear stress points in the negative direction of the coordinate axis of the second subscript. The shear stresses shown in Figure 2.17 are all positive. z
σzz σzx
σzy
σxy σyy
σyx
σxz
σxz
σxx
σyz σyx
σyy
σxy
σyz σxx
y
σzx σzy
σzz
x
Figure 2.17 Sign convention for positive stress components
DIRECT STRAIN — Positive strain indicates extension; negative strain indicates compression. SHEAR STRAIN — Shear strain follows the convention of shear stress (see above). PORE PRESSURE — Fluid pore pressure is positive in compression.
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User’s Guide
DIP, DIP DIRECTION — Dip and dip direction assume that the x-direction corresponds to “East,” z-direction to “North” and y-direction to “Up.” The dip angle is measured in the negative ydirection from the global xz-plane. The dip direction angle is measured in the global xz-plane, clockwise from the positive z-axis. The x-, y- and z-components of vector quantities, such as forces, displacements and velocities, are positive when pointing in the directions of the positive x-, y- and z-coordinate space.
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2.8 Systems of Units 3DEC accepts any consistent set of engineering units. Examples of consistent sets of units for basic parameters are shown in Table 2.5. The user should apply great care when converting from one system of units to another. An excellent reference on the subject of units and conversion between the Imperial and SI systems can be found in the Journal of Petroleum Technology (December 1977). No conversions are performed in 3DEC except for friction and dilation angles, which are entered in degrees. Table 2.5
Systems of units — mechanical parameters SI
Length Density Force Stress Gravity
where
1 bar 1 atm 1 slug 1 snail 1 gravity
m kg / m3 N Pa m / sec2
= = = = =
m 103 kg / m3 kN kPa m / sec2
m 106 kg / m3 MN MPa m / sec2
Imperial cm 106 g / cm3 Mdynes bar cm / s2
ft slugs / ft3 lbf lbf / ft2 ft / sec2
in snails / in3 lbf psi in / sec2
106 dynes / cm2 = 105 N / m2 = 105 Pa; 1.013 bars = 14.7 psi = 2116 lbf / ft2 = 1.01325 × 105 Pa; 1 lbf - s2 / ft = 14.59 kg; 1 lbf -s2 / in; and 9.81 m / s2 = 981 cm / s2 = 32.17 ft / s2 .
When selecting a system of units, care should be taken to avoid calculations that approach the precision limits of the computer hardware. For Pentium-based computers, the range is approximately 10−35 to 1035 in single precision. If numbers exceed these limits, it is likely that the program will crash or, at least, produce artifacts in the model that may be difficult to identify or detect.
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2.9 Files There are several types of files that are either used or created by 3DEC. The files are distinguished by their extensions and are described below. INITIALIZATION FILE “3DEC.INI” — This is a formatted ASCII file, created by the user, that 3DEC will automatically access upon start-up or when a NEW command is issued. 3DEC searches for the file “3DEC.INI” in the directory in which the code is executed and, if not found, in the “\ITASCA\System” folder. The file may contain any valid 3DEC command(s) (see Section 1 in the Command Reference). Although this file does not need to exist (i.e., no errors will result if it is absent), it is normally used to change default options in 3DEC to those preferred by the individual user each time a new analysis is run (see Section 2.1.7). DATA FILES The user has a choice of running 3DEC interactively (i.e., entering 3DEC commands while in the 3DEC environment) or via a data file (also called a “batch file”). The data file is a formatted ASCII text file created by the user which contains the set of 3DEC commands that represents the problem being analyzed. In general, creating data files is the most efficient way to use 3DEC. To use data files with 3DEC, see the CALL command in Section 1 in the Command Reference. Data files can have any filename and any extension. It is recommended that a common extension (e.g., “.DAT”) for 3DEC input commands, and “.FIS” for FISH function statements) be used to distinguish these files from other types of files. Important note: The end of each line in a text file must be terminated by a carriage return. If not, the line will not be processed. It is a good idea to put in a “ret” line or comment as the last line of a data file in order to avoid this. SAVE FILES “3DEC.SAV” — This file is created by 3DEC at the user’s request when issuing the command SAVE. The default file name is “3DEC.SAV,” which will appear in the default directory when quitting 3DEC. The user may specify a different filename by issuing the command SAVE filename, where filename is a user-specified filename. “3DEC.SAV” is a binary file containing the values of all state variables and user-defined conditions. The primary reason for creating Save files is to allow one to investigate the effect of parameter variations without having to rerun a problem completely. A Save file can be restored and the analysis continued at a subsequent time (see the RESTORE command in Section 1 in the Command Reference). Normally, it is good practice to create several Save files during a 3DEC run. LOG FILES “3DEC.LOG” — This file is created by 3DEC at the user’s request when issuing the command SET log on. It is a formatted ASCII file. The default name of the file is “3DEC.LOG,” which will appear in the default directory after quitting 3DEC. The user may specify a different filename by issuing the command SET log filename, where filename is a user-supplied filename. The command
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may be issued interactively or be part of a data file. Subsequent to the SET log on command, all text appearing on the screen will be copied to the log file. The log file is useful in providing a record of the 3DEC work session; it also provides a document for quality-assurance purposes. HISTORY FILES “3DEC.HIS” — This file is created by 3DEC at the user’s request when issuing the command HISTORY write n, where n is a history number (see the HISTORY command, Section 1 in the Command Reference). It is a formatted ASCII file. The default name of the file is “3DEC.HIS,” which will appear in the default directory after quitting 3DEC. The user may specify a different filename by issuing the command SET hisfile filename. The user-supplied filename takes the place of “3DEC.HIS.” The command may be issued interactively or be part of a data file. A record of the history values is written to the file, which can be examined using any text editor that can access formatted ASCII files. Alternatively, the file may be processed by a commercial graph-plotting or spreadsheet package. TABLE FILES “3DEC.TAB” — This file is created by 3DEC at the user’s request when issuing the command TABLE n write dx, where n is the table number and dx specifies the abscissa spacing for the data points (see the TABLE command in Section 1.3 in the Command Reference). It is a formatted ASCII file. The default name of the file is “3DEC.TAB,” which will appear in the default directory after quitting 3DEC. The user may specify a different filename by adding the filename to the end of the TABLE n write dx command. The file will consist of a single column of y-data at an even spacing of dx. If dx = 0, the data will be the actual x,y pairs in table n. PLOT FILES Plot files are created at the user’s request by issuing the command COPY filename in the command mode, after first creating the plot. By default, a PostScript file will be created with the user-specified filename when COPY filename is issued. The output type can be changed with the SET plot command. PCX output can also be created by either setting this output mode on with the SET pcx on command before creating the plot, or by pressing thekey while in the graphics-screen mode. When PCX mode is turned on, or the key is pressed in the graphics-screen mode, a PCX screen dump will be written to a file named “3DEC.PCX.” Only one screen image can be written to a file. The user may specify a different title name with the command SET pcxfile filename where the user-specified filename takes the place of “3DEC.PCX.” PCX files consist of bitmaps of screen images; they are accepted by many image display and manipulation programs. MOVIE FILES “3DEC.DCX” — This file is created by 3DEC at the user’s request when issuing the command MOVIE on. Its purpose is to capture graphics images for playback on the computer monitor as a movie at a later time. Note that this feature will only work with VGA graphics. The default file name is “3DEC.DCX,” which will appear in the default directory when quitting 3DEC. The user
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may specify a different filename by issuing the command MOVIE file filename, where filename takes the place of “3DEC.DCX.” A DCX file format is used for the movie file. See the MOVIE command in Section 1.3 in the Command Reference. Note that the DCX format is limited to 1024 frames. 2.10 References Journal of Petroleum Technology. “The SI Metric System of Units and SPE’s Tentative Metric Standard,” 1575-1616 (December, 1977).
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3 PROBLEM SOLVING WITH 3DEC This section provides guidance in the use of 3DEC in problem solving for rock mechanics engineering.* In Section 3.1, an outline of the steps recommended for performing a geomechanics analysis is given, followed in Sections 3.2 through 3.10 by an examination of specific aspects that must be considered in any model creation and solution. These include: • model generation (Section 3.2); • choice of rigid or deformable block analysis (Section 3.3); • boundary and initial conditions (Sections 3.4 and 3.5); • loading and sequential modeling (Section 3.6); • choice of block and joint constitutive models and material properties (Sections 3.7 and 3.8); • ways to improve modeling efficiency (Section 3.9); and • interpretation of results (Section 3.10). Finally, the philosophy of modeling in the field of geomechanics is examined in Section 3.11; the novice modeler in this field may wish to consult this section first. The methodology of modeling in geomechanics can be significantly different from that in other engineering fields, such as structural engineering. It is important to keep this in mind when performing any geomechanics analysis.
* Problem solving for coupled mechanical-thermal analysis is described in Section 1 in Optional Features, and problem solving for dynamic analysis is discussed in Section 2 in Optional Features.
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3.1 General Approach The modeling of geo-engineering processes involves special considerations and a design philosophy different from that followed for design with fabricated materials. Analyses and designs for structures and excavations in or on rocks and soils must be achieved with relatively little site-specific data and an awareness that deformability and strength properties may vary considerably. It is impossible to obtain complete field data at a rock or soil site. For example, information on stresses, properties and discontinuities can only be partially known, at best. Since the input data necessary for design predictions are limited, a numerical model in geomechanics should be used primarily to understand the dominant mechanisms affecting the behavior of the system. Once the behavior of the system is understood, it is then appropriate to develop simple calculations for a design process. This approach is oriented toward geotechnical engineering, in which there is invariably a lack of good data; but in other applications, it may be possible to use 3DEC directly in design if sufficient data, as well as an understanding of material behavior, are available. The results produced in a 3DEC analysis will be accurate when the program is supplied with appropriate data. Modelers should recognize that there is a continuous spectrum of situations, as illustrated in Figure 3.1, below.
Typical situation
Data
Approach
Figure 3.1
Simple geology; $$$ spent on site investigation
Complicated geology; inaccessible; no testing budget
COMPLETE
NONE
Investigation of mechanisms
Bracket field behavior by parameter studies
Predictive (direct use in design)
Spectrum of modeling situations
3DEC may be used either in a fully predictive mode (right-hand side of Figure 3.1) or as a “numerical laboratory” to test ideas (left-hand side). It is the field situation (and budget), rather than the program, that determine the types of use. If enough data of a high quality are available, 3DEC can give good predictions. Since most 3DEC applications will be for situations in which little data are available, this section discusses the recommended approach for treating a numerical model as if it were a laboratory test. The model should never be considered as a “black box” that accepts data input at one end and produces a prediction of behavior at the other. The numerical “sample” must be prepared carefully, and several samples tested, to gain an understanding of the problem. Table 3.1 lists the steps recommended to perform a successful numerical experiment; each step is discussed separately.
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Table 3.1
3-3
Recommended steps for numerical analysis in geomechanics
Step 1
Define the objectives for the model analysis
Step 2
Create a conceptual picture of the physical system
Step 3
Construct and run simple idealized models
Step 4
Assemble problem-specific data
Step 5
Prepare a series of detailed model runs
Step 6
Perform the model calculations
Step 7
Present results for interpretation
3.1.1 Step 1: Define the Objectives for the Model Analysis The level of detail to be included in a model often depends on the purpose of the analysis. For example, if the objective is to decide between two conflicting mechanisms that are proposed to explain the behavior of a system, then a crude model may be constructed, provided that it allows the mechanisms to occur. It is tempting to include complexity in a model just because it exists in reality. However, complicating features should be omitted if they are likely to have little influence on the response of the model, or if they are irrelevant to the model’s purpose. Start with a global view and add refinement as (and if) necessary. 3.1.2 Step 2: Create a Conceptual Picture of the Physical System It is important to have a conceptual picture of the problem to provide an initial estimate of the expected behavior under the imposed conditions. Several questions should be asked when preparing this picture. For example, is it anticipated that the system could become unstable? Is the predominant mechanical response linear or nonlinear? Are there well-defined discontinuities that may affect the behavior, or does the material behave essentially as a continuum? Is there an influence from groundwater interaction? Is the system bounded by physical structures, or do its boundaries extend to infinity? Is there any geometric symmetry in the physical structure of the system? These considerations will dictate the gross characteristics of the numerical model, such as the design of the model geometry, the types of material models, the boundary conditions, and the initial equilibrium state for the analysis. They will determine whether a three-dimensional model is required or if a two-dimensional model can be used to take advantage of geometric conditions in the physical system.
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3.1.3 Step 3: Construct and Run Simple Idealized Models When idealizing a physical system for numerical analysis, it is more efficient to construct and run simple test models first, before building the detailed model. Simple models should be created at the earliest possible stage in a project to generate both data and understanding. The results can provide further insight into the conceptual picture of the system; Step 2 may need to be repeated after simple models are run. Simple models can reveal shortcomings that can be remedied before any significant effort is invested in the analysis. For example, do the selected material models sufficiently represent the expected behavior? Are the boundary conditions influencing the model response? The results from the simple models can also help guide the plan for data collection by identifying which parameters have the most influence on the analysis.
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3.1.4 Step 4: Assemble Problem-Specific Data The types of data required for a model analysis include: • details of the geometry (e.g., profile of underground openings, surface topography, dam profile, rock/soil structure); • locations of geologic structure (e.g., faults, bedding planes, joint sets); • material behavior (e.g., elastic/plastic properties, post-failure behavior); • initial conditions (e.g., in-situ state of stress, pore pressures, saturation); and • external loading (e.g., explosive loading, pressurized cavern). Since, typically, there are large uncertainties associated with specific conditions (in particular, state of stress, deformability and strength properties), a reasonable range of parameters must be selected for the investigation. The results from the simple model runs (in Step 3) can often prove helpful in determining this range and in providing insight for the design of laboratory and field experiments to collect the needed data. 3.1.5 Step 5: Prepare a Series of Detailed Model Runs Most often, the numerical analysis will involve a series of computer simulations that include the different mechanisms under investigation and span the range of parameters derived from the assembled database. When preparing a set of model runs for calculation, several aspects, such as those listed below, should be considered. 1. How much time is required to perform each model calculation? It can be difficult to obtain sufficient information to arrive at a useful conclusion if model runtimes are excessive. Consideration should be given to performing parameter variations on multiple computers to shorten the total computation time. 2. The state of the model should be saved at several intermediate stages so that the entire run does not have to be repeated for each parameter variation. For example, if the analysis involves several loading/unloading stages, the user should be able to return to any stage, change a parameter and continue the analysis from that stage. Consideration should be given to the amount of disk space required for Save files. 3. Are there a sufficient number of monitoring locations in the model to provide for a clear interpretation of model results and for comparison with physical data? It is helpful to locate several points in the model at which a record of the change of a parameter (such as displacement, velocity or stress) can be monitored during the calculation. Also, the maximum unbalanced force in the model should always be monitored to check the equilibrium or failure state at each stage of an analysis.
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3.1.6 Step 6: Perform the Model Calculations It is best to first make one or two model runs split into separate sections before launching a series of complete runs. The runs should be checked at each stage to ensure that the response is as expected. Once there is assurance that the model is performing correctly, several data files can be linked together to run a complete calculation sequence. At any time during a sequence of runs, it should be possible to interrupt the calculation, view the results, and then continue or modify the model as appropriate. 3.1.7 Step 7: Present Results for Interpretation The final stage of problem solving is the presentation of the results for a clear interpretation of the analysis. This is best accomplished by displaying the results graphically, either directly on the computer screen or as output to a hardcopy plotting device. The graphical output should be presented in a format that can be directly compared to field measurements and observations. Plots should clearly identify regions of interest from the analysis, such as locations of calculated stress concentrations, or areas of stable movement versus unstable movement in the model. The numeric values of any variable in the model should also be readily available for more-detailed interpretation by the modeler. We recommend that these seven steps be followed to solve geo-engineering problems efficiently. The following sections describe the application of 3DEC to meet the specific aspects of each of these steps in this modeling approach.
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3.2 Model Generation One of the main tasks when performing a 3DEC analysis is building a numerical representation of the real world situation. All 3DEC models will have an outer boundary to which boundary conditions will be applied. In addition, most models will have features such as joints, faults, cracks, excavations, or material property interfaces. Quite often the features to be modeled are not continuous through the entire model. The complex shapes which result from the intersections of noncontinuous planes in space are difficult to create and visualize. The purpose of this section is to give some guidance in using a systematic approach to model building. The first rule is: keep the model simple. Usually, a numerical model is being developed because the real life situation is too complex to understand. There is a tendency to attempt to include every possible feature in a numerical model. This results in a 3DEC model that is also too complex to understand. The goal of the modeler should be to understand the mechanisms, properties, and parameters that determine the model’s behavior. The time required to calculate a solution also increases with increasing complexity. Therefore, it is to the modeler’s advantage to start with a model that only includes the minimum of features. The complexity of the model can then be increased by adding features one at a time and noting the effect. By using this approach, the modeler has the best chance of gaining understanding of the parameters that are critical to the real life situation. 3DEC is different from conventional numerical programs in the way that the model geometry is created. A 3DEC model can be created in two ways: (1) by splitting a polyhedron into separate polyhedra; and (2) by creating separate polyhedra and joining them together. For most geomechanics analyses, a single block is created first, with a size that encompasses the physical region being analyzed. Then, this block is cut into smaller blocks whose boundaries represent both geologic features and engineered structures in the model. This cutting process is termed collectively as joint generation; however, “joints” represent both physically real geologic structures and boundaries of man-made structures or materials that will be removed or changed during the subsequent stages of the 3DEC analysis. In this latter case, the joints are fictitious entities and their presence should not influence model results. The representation of fictitious joints is discussed in Section 3.2.3. 3.2.1 Fitting the 3DEC Model to a Problem Region The 3DEC model geometry must represent the physical problem to a sufficient extent to capture the dominant mechanisms related to the geologic structure in the region of interest. The following aspects must be considered. 1. In what detail should the geologic structure (e.g., faults, joints, bedding planes, etc.) be represented? 2. How will the location of the model boundaries influence model results? 3. If deformable blocks are used, what density of zoning is required for accurate solution in the region of interest? All three aspects determine the size of the 3DEC model that is practical for analysis.
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As mentioned above, there are two different starting points in building 3DEC models. The first method is to describe a simple starting shape and slice it up to create the desired geometrical features. The second involves defining complex polyhedral shapes and putting them together to form the continuous mass. Both of these approaches make use of the POLY command. There are five forms of the POLY command available in 3DEC. POLY face POLY brick POLY cube POLY prism POLY tunnel By using the POLY face command, virtually any shape polyhedra can be defined. Each face is defined by a list of vertex coordinates. The list must be entered in counterclockwise order, looking at the face from outside the polyhedra. All points on a face must be coplanar, and the resulting polyhedra created by the face commands must be convex. Continuation lines are allowed, but the coordinates for each vertex may not be split between lines. All faces required to close the polyhedra must be specified. A simple example of using the POLY face command to generate a cube (1 unit on each side) is as follows: Example 3.1 A cube generated with the POLY face command new poly face face face face face face ret
& 0,0,0 0,0,0 0,0,0 1,1,1 1,1,1 1,1,1
1,0,0 0,0,1 0,1,0 1,1,0 1,0,1 0,1,1
1,1,0 1,0,1 0,1,1 1,0,0 0,0,1 0,1,0
0,1,0 1,0,0 0,0,1 1,0,1 0,1,1 1,1,0
& & & & &
The model created with the example is shown in Figure 3.2. In this case, a simple regular rightangled solid is produced. The command can also be used to create complex shapes. Because of the large amount of input required, the POLY face command is often best used in conjunction with the external pre-processor program PGEN — see Section 5 in Theory and Background for examples.
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x
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Figure 3.2
Cubic model created with the POLY face command
The POLY brick command provides a simpler alternative to the POLY face command when the problem region is a regular six-sided “brick-shaped” region. The parameters for the brick keyword are the x-, y-, and z-limits of the solid (i.e., the region extends from coordinates xl to xu in the x-direction, from coordinates yl to yu in the y-direction, and from coordinates zl to zu in the z-direction). For example, to create the same model generated in Figure 3.2, the command is: Example 3.2 A cube generated with the POLY brick command new poly brick 0,1 ret
0,1
0,1
POLY cube is a tool for generating irregularly-shaped boundaries. These boundaries may represent geologic contacts or the borders of excavations. This is intended as an alternative to the PGEN program. The advantage of POLY cube over the PGEN program is that the resulting shapes are easier to zone and can be zoned as mixed discretization zones for plasticity. The disadvantage is that the shapes can be complex in only two dimensions. Using PGEN, they can be zoned in three dimensions. See Section 3.2.3.2 for an example of the complex use of this tool. The POLY prism command is an extension of the POLY brick command to create prism-shaped polyhedra. The two parallel faces of the prism are defined by an arbitrary number of vertices. The opposing vertices on each face are then automatically connected to form the prism. The first face
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(face a) is defined by vertices entered in either a clockwise or counterclockwise order. The opposite face (face b) must have its vertices entered in the same order as the corresponding vertices for face a. Faces a and b must be planar and convex. The prism shown in Figure 3.3 is created by the commands listed in Example 3.3. Example 3.3 An octahedral-shaped prism generated with the POLY prism command new poly prism a (0,0,0) (-.5,.87,0) (-.5,1.87,0) (0,2.74,0) & (1,2.74,0) (1.5,1.87,0) (1.5,.87,0) (1,0,0) & b (0,0,4) (-.5,.87,4) (-.5,1.87,4) (0,2.74,4) & (1,2.74,4) (1.5,1.87,4) (1.5,.87,4) (1,0,4) ret
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Itasca Consulting Group, Inc.
Figure 3.3
An octahedral-shaped prism generated with the POLY prism command
The POLY tunnel command is specifically designed to generate a circular-shaped tunnel model. This command works by constructing the model with individual blocks. This is in contrast to the TUNNEL command, discussed in Section 3.2.3, which cuts an arbitrarily-shaped tunnel out of an existing block. The blocks created by the POLY tunnel command are all six-sided with low aspect ratios and are intended for use with the GEN quad command. (This command is recommended for deformable-block plasticity analysis — see Section 3.3.) The user needs only to specify the orientation, dimensions, and the number of blocks to be used for the tunnel. Additional jointing
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can be added with the JSET command, if desired. For example, to create a tunnel model with the following dimensions: radius
2.0 m
length
20.0 m
outside boundary
3.0 r
dip
horizontal
heading
south
blocks in each octant
1
annular blocks
2
blocks along the axis
3
use the POLY tunnel command as given in Example 3.4. Example 3.4 A tunnel model generated with the POLY tunnel command new poly tunnel rad=2 leng=-10,10 ratr=3.0 dip=0 dd=0 nr=2 nt=1 nx=3 delete -2 2, -2 2 -10 10 ret
The model with the tunnel deleted is shown in Figure 3.4.
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Figure 3.4
Tunnel model created with the POLY tunnel command
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3.2.2 Joint Generation The JSET command is used to make additional cuts in the solids, created with the POLY command, to define joints, faults, and holes or excavations. The JSET command can be used to make single cuts or multiple parallel cuts. Statistical parameters may be used to vary orientation, spacing and persistence to match logged jointing data. The JSET command is first demonstrated in this section for making single cuts. Some planning should be made to optimize the sequence in which joints are created. Joints which define the geometry of excavations are usually cut first (see Section 3.2.3), followed by the minor joints or joint sets. Through-going faults are usually defined last. The primary keywords for the JSET command are dip, dd (dip direction), and origin (origin). Unless other keywords are used, the JSET command will create a single plane cutting through the model in the orientation specified. The origin point may be any point on the plane. Figure 3.5 shows how the orientations for dip and dip direction relate to the coordinate axes in 3DEC. The dip range is from 0 to 90◦ . The dip direction range is from 0 to 360◦ . Up (y)
North (z)
= Dip direction
S
tr
ik
e
lin
e
= Dip
East (x)
Joint plane
Figure 3.5
Terms describing the attitude of an inclined plane: dip angle, α, is positive measured downward from the horizontal (xz) plane; dip direction, β, is positive measured clockwise from north (z)
Control of the continuity of cuts made using the JSET command is accomplished with the HIDE and SEEK commands. The JSET command will only cut blocks that are currently visible.
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Example 3.5 illustrates the creation of a noncontinuous joint. Example 3.5 Creation of a noncontinuous vertical joint new poly jset hide jset seek ret
brick 0,1 0,1 0,1 dip 0 dd 0 or 0,.5,0 dip 0 dd 0 or 0,.5,0 below dip 90 dd 90 or .5,0,0
; ; ; ; ;
create a block make a horizontal cut hide the bottom block vertical cut through top block only make all blocks visible
Figure 3.6 shows the full model and the joint structure plot for this example. Note that the vertical joint does not penetrate the bottom block. 3DEC automatically assigns a joint ID number = 2 to the horizontal joint and a joint ID number = 3 to the vertical joint. If desired, the joint ID number can be controlled with the JSET command. For example, jset dip 0 dd 0 or 0,.5,0
id = 1000
will create a horizontal joint with an ID number of 1000. The ID numbers for joint faces and contacts are given sequentially as they are created. Therefore, if two faults are defined that intersect, the edge to edge contacts at the line of intersection will have the joint IDs of the second fault defined. This order becomes important if different properties are to be assigned to the different faults. Property numbers of the face-to-face contacts that comprise most of the area of joints can be assigned using the CHANGE command, in which the particular joint to be changed is identified using either its joint ID number or orientation. The difficulty comes in assigning property numbers to the edge-to-edge contacts that are created at joint and fault intersections. It is easy to assign property numbers to edge-to-edge contacts by use of the joint ID number. It is difficult to assign property numbers to edge-to-edge contacts by orientation because they have a different orientation than either of the two intersecting planes that created them. There is a PLOT option that allows plotting of joint material properties (the joint material item in the options menu). This plot is useful in checking that the edge-to-edge contacts are assigned the correct properties.
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(a) full-solid view
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(b) joint-structure view Figure 3.6
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Concave blocks can be made by use of the JOIN command. The blocks that have been joined are still convex, but the join logic locks the interface between them. For example, add the following commands at the end of Example 3.5. hide (0.5,1.0) (0.5,1.0) (0,1) join on
Figure 3.7 shows the concave block that is created. Note that only visible blocks can be joined.
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Figure 3.7
Concave block created with the JOIN command
Joined blocks are plotted in the same color on the graphics screen. Also, contacts between joined blocks are identified as master-slave (m-s) contacts. Type PRINT contact to check the contact type. Note that “slaved” blocks will be automatically joined if they are connected to the same “master” block. For example, if block A and block B are joined, and block A and block C are joined, then block B will be joined automatically to block C. The JSET command can also be used to generate a set of joints automatically based upon physically measured parameters (i.e., joint dip, dip direction, spacing and persistence). By hiding selected blocks, a set of noncontinuous joints can be generated. In Example 3.6, a jointed rock slope is created containing both shallow and deeply dipping joint sets. Two noncontinuous fractures are also created to define a rock wedge in the slope; see Figure 3.8.
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Example 3.6 Rock slope containing continuous and noncontinuous joints new poly brick 0 80 0 50 -30 80 ; shallow-dipping fracture planes (continuous) jset dip 2.45 dd 235 org 30 12.5 0 jset dip 2.45 dd 315 org 35 30 0 ; high angle foliation planes (continuous) jset dip 76 dd 270 spac 16 num 3 org 30,12.5,0 ; intersecting discontinuities (non-continuous) hide 0 80 0 10 0 50 hide 55 80 0 50 0 50 jset dip 70 dd 200 org 0 0 35 jset dip 60 dd 330 org 50 50 15 seek hide 0,30 13,50 -30,80 ret
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Figure 3.8
Rock slope containing continuous and noncontinuous joints
Bear in mind that joints are displayed as straight-line segments in the 3DEC model; many segments may be required to fit an irregular joint structure. The modeler must decide the level at which the 3DEC joint geometry will match the physical jointing pattern. The effect of geometric irregularity
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on the response of a joint can also be taken into account via the joint material model — e.g., by varying properties along the joint. One final point is made concerning joint generation. When using continuum programs, it is usually appropriate to take advantage of symmetry conditions with excavation shapes in order to reduce the size of the model. Symmetry conditions cannot be imposed as easily with discontinuum programs because the presence of discontinuous features precludes symmetry except for special cases. For example, it is not possible to impose a vertical plane of symmetry through the model shown in Figure 3.8 because the joints in the model are not aligned with the vertical axis. 3.2.3 Creating Internal Boundary Shapes When fitting the 3DEC model to the problem region, polyhedral boundaries must also be defined to coincide with boundary shapes of the physical problem. These may be internal boundaries representing excavations or holes or external boundaries representing, for example, man-made structures such as earth dams or natural features such as mountain slopes. If the physical problem has a complicated boundary, it is important to assess whether simplification will have any effect on the questions that need to be answered (i.e., whether a simpler geometry will be sufficient to reproduce the important mechanisms). All physical boundaries to be represented in the model simulation (including regions that will be added or excavations created at a later stage in the simulation) must be defined before the solution process begins. Shapes of structures that will be added later in a sequential analysis must be defined and then “removed” (via the EXCAVATE command). Excavated blocks are added with the FILL command. Note that only deformable blocks can be excavated and filled. The creation of boundary shapes is performed with the following commands: JSET TUNNEL POLY cube Each command cuts the polyhedra into one or more segments that are fitted together in the desired shape. The JSET commands create planar joint segments as discussed above in Section 3.2.2. The TUNNEL command cuts a shape into the blocks. The POLY cube command creates a volume of cubed blocks and cuts a boundary.
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User’s Guide
3.2.3.1 Tunnel Command The TUNNEL command creates a tunnel whose boundary is formed by planar segments that connect two faces designated as face A and face B. The shape of the tunnel face is prescribed by an arbitrary number of vertices with coordinates (x1,y1,z1), (x2,y2,z2) (x3,y3,z3), etc. The same number of vertices must exist on both faces; the faces may be positioned either inside or outside the model. Example 3.7 presents a simple example for the creation of a horseshoe-shaped tunnel. The command REMOVE is used to delete the tunnel region (defined as region 1). The DELETE command may also be used; with the REMOVE command, the deleted region can still be viewed in plot mode, if desired. The resulting tunnel is shown in Figure 3.9. Example 3.7 Tunnel created with the TUNNEL command new poly brick -1.5,1.5 -1.5,1.5 -1.5,1.5 tunnel region 1 & a (-.3,0,-1.5) (-.3,.4,-1.5) (-.25,.47,-1.5) (-.15,.52,-1.5) (0,.55,-1.5) (.15,.52,-1.5) (.25,.47,-1.5) (.3,.4,-1.5) (.3,0,-1.5) b (-.3,0,1.5) (-.3,.4,1.5) (-.25,.47,1.5) (-.15,.52,1.5) (0,.55,1.5) (.15,.52,1.5) (.25,.47,1.5) (.3,.4,1.5) (.3,0,1.5) remove region 1 ret
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3DEC (Version 3.00) 27-Aug-02
10:51
dip= 70.00 above dd = 200.00 center 1.192E-07 1.192E-07 1.192E-07 cut-pl. 0.000E+00 mag = 1.00 cycle 0
Y z
x
Itasca Consulting Group, Inc.
Figure 3.9
Tunnel created with TUNNEL command
Usually, the model will be brought to an equilibrium state before the tunnel is excavated. The user must be careful that the fictitious joints along the tunnel boundary do not influence model response during the initial equilibrium calculation. If the TUNNEL command is used, the blocks are automatically joined at the fictitious joints. If JSET commands are used to create fictitious joints, then the JOIN command is recommended to join the blocks separated by the fictitious joint. Use the PRINT contact command to identify if the contact is a master-slave (m-s) contact between joined blocks. 3.2.3.2 POLY cube POLY cube is a tool for generating irregularly-shaped boundaries. These boundaries may represent geologic contacts or the borders of excavations. This is intended as an alternative to the PGEN program. The advantage of POLY cube over the PGEN program is that the resulting shapes are easier to zone and can be zoned as mixed discretization zones for plasticity. The disadvantage is that the shapes can be complex in only two dimensions. Using PGEN, the shapes can be complex in three dimensions. The POLY cube process is relatively simple. The user specifies information about the size and the orientation of the shape to be created. 3DEC then generates an area of six-sided polyhedra (cubes) which occupies this area. The orientation of the region can be along any line in space and can be rotated about that line. After the cubes are created, 3DEC uses the coordinates specified in a data file (normally “overlay.txt”) to cut the geometry out of the cubes. Each cube is cut only once by the boundary so the resolution of the shape is defined by the size of the cubes.
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User’s Guide
The orientation of the cube area is defined by the dip, ddirection, top and rotate keywords which define a control plane (see Figure 3.10). All blocks are created behind this control plane. The size and shape of the cubes are defined by the number and spacing keywords. An outer box can be created as a zoning transition by the box keyword. The IDs of the jointing are controlled by the id keyword. The region numbers can be set by the inside and outside keywords. The file which controls the cutting of the boundary shape is a simple text file of x,y,z coordinates. A DXF file may also be used, but the coordinates in the DXF file must define a contiguous polygon. The PGEN program may be used to edit the DXF file to connect discontinuous segments.
top point rotate
nz
ny y x z
nx spacing control plane defined by dip and dip direction
Figure 3.10 Elements of the POLY cube command This simple example demonstrates the construction of a 500 cube area which is cut by the polygon stored in “overlay.txt.”
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Example 3.8 Data file which generates a model using POLY cube command ; ; example of use of poly cube command ; poly br 0 100 0 100 0 40 poly cube dip 90 dd 180 num 10 10 5 spac 4 4 8 top seek hide reg 1 pl hold dip 100 dd 180 mag 2 color reg ret
50 50 0
The cutting geometry is controlled by the data points in the file “overlay.txt.” 31 31 31 41 37 51 41 61 51 69 63 61 63 51 47 43 ; end
0 0 0 0 0 0 0 0 of segment
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dip= 80.00 below dd = 0.00 center 5.000E+01 5.000E+01 2.000E+01 cut-pl. 0.000E+00 mag = 2.00 cycle 0
Itasca Consulting Group, Inc.
Figure 3.11 Resultant geometry from example
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User’s Guide
3.2.4 Selecting the Coordinate System As shown in Figure 3.5, the reference axes for 3DEC are a left-handed set (x,y,z) oriented, by default, as x (east), y (vertically up), and z (north). In generating a 3DEC model of a given region, the following conditions may require reorientation for input into the model: (1) geometry of the problem structures (i.e., mine layout, tunnel locations, etc.); (2) geometry of the geologic features (e.g., faults and joint sets); and (3) orientation of the in-situ stress field. In the ideal situation, the problem geometries would align with the 3DEC reference axes. In general though, this is not the case, and one or two of the problem geometries will require transformation to the 3DEC model reference frame. Typically, it is best to orient the 3DEC model axes to align with the geometry of the problem structures (e.g., the mine grid or centerline of a tunnel). For graphic presentation, it is best to position the origin of the model axes at the center of the structure for which the analysis is intended. In this case, the geologic features may require reorientation from global to local problem axes (see Section 3.2.5, below), and the field principal stresses may require transformation for application to the model boundary. The recommended procedure to execute this transformation is discussed in Section 3.5.8. For reasons of numerical precision, it is best to truncate coordinates to some convenient number. For example, if the units for a model are from x = 15,423 to 15,443, it is best to truncate the coordinates by 15,400 so that the model coordinates range from 23 to 43. 3.2.5 Orientation of Geologic Features to the Model Axes If the problem axes do not align with the model axes (i.e., positive z-axis (north), x-axis (east) and y-axis (up)), then the orientation of geologic features will have to be transformed from the problem axes to the model axes. This can be accomplished by making use of a stereonet. The reorientation of structural features to the model reference axes is demonstrated for the case of a single fault that crosses a mine raise. The raise is oriented with an axis dip of 84◦ and dip direction of 125◦ , and the fault is oriented 11◦ / 148◦ . The 3DEC model axes are located with the z-axis directed down the raise, the y-axis lying in the vertical plane containing the raise dip vector, and the x-axis lying in the horizontal plane. The origin of the model axes is located at the center of the raise (see Figure 3.12). The dip and dip direction of the fault must be redefined relative to the x,y,z-model axes as oriented in the figure.
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UP N ventilation raise
Raise Orientation b = 125º a = 84º
b
E
a N35º x
Model Axes
y z
Figure 3.12 Orientation of 3DEC model axes (x,y,z) relative to north-east-up reference axes The fault pole is plotted on the lower hemisphere stereonet shown in Figure 3.14. The relation between the fault and the model axes is found by plotting the positive x-, y- and z-axes of the model on the stereonet. The dihedral angle between the fault pole and each axis is then read from the stereonet. In this example, the dihedral angles are 86◦ for x-axis to fault pole, 73◦ for y-axis to fault pole, and 17◦ for z-axis to fault pole. On a second stereonet, Figure 3.13, the model axes are oriented to align with the axes of the stereonet: z (north), x (east) and y (up). The intersection of the three dihedral angles on this plot gives the pole of the fault (plotted on the upper hemisphere). The orientation of the fault relative to the model axes is thus 74◦ / 2◦ . 3.2.6 Choice of Model Scale Analysis of rock mass response involves several different scales. It is impossible and undesirable to include all features, details or rock mass response mechanisms in one model. It is also wellrecognized that location of the far-field boundary in the model can have a significant influence on results obtained for underground excavations. However, in three-dimensional analysis, it is not always feasible to place boundaries sufficiently far from the excavations to avoid adversely affecting the results. These two observations taken together suggest that a reasonable modeling approach involves starting with a large global model and proceeding through reasonable models to the smallest size required, with increasing complexity and detail added at each stage. 3DEC has an automatic method of recording stresses at specified locations so that they can be applied as boundary
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User’s Guide
tractions on smaller problems. This unique feature ensures stress compatibility between larger and smaller models. See Section 3.4.4.2 for additional information on applying this technique. 3.2.7 Incorporation of Discontinuities Selection of joint geometry for input to a model is a crucial step in distinct element analysis. Typically, only a very small percentage of joints can actually be included in the model in order to create models of reasonable size and execution speed for practical analysis. Thus, the modeler must filter joint geometry data and select only those joints that are most critical to the mechanical response by identifying those which are most susceptible to slip for the prescribed loading conditions. This may involve, for example, determining whether sufficient kinematic freedom is provided (e.g., through the use of block theory) or comparing in-situ observations and records (e.g., microseismic records to identify key joints). Once a consistent set of joints is selected, the geometric parameters (e.g., strike, dip, location) for these features are input into the model, and the practicality of the analysis in terms of required memory and runtime are assessed. If the model size is too large, the number of joints must be reduced, and it is necessary to further filter the input to bring the model to a practical size. This dilemma of balancing model size and critical joint structures is addressed by Hart (1993). In some cases it may not be clear exactly how important certain discontinuities are, or whether discontinuities should be incorporated into the model at all. The guiding philosophy in these cases is to first run the models with either no discontinuities or with the discontinuities in a welded state. Comparisons should then be made with observations. If adequate calibration of the model behavior can be obtained without discontinuities, then discontinuities need not be modeled. By rerunning the model and allowing certain discontinuities to slip, the modeler can determine exactly how the discontinuities affect the system behavior and whether that behavior agrees more closely with observations.
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N
x-axis
86º 73º
pole to fault 17º
E
z-axis
fault plane
Figure 3.13 Stereonet plot of fault relative to model axes
z
17º 73º
fault
86º
pole on
upper hemisphere fault plane 74º/2º
x y (up)
fault pole on lower hemisphere
Figure 3.14 Stereonet plot of pole to fault and model reference axes relative to problem north-east axes
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3.3 Selection of Deformable versus Rigid Blocks An important aspect of a discontinuum analysis is the decision to use rigid blocks or deformable blocks to represent the behavior of intact material. The considerations for rigid versus deformable blocks are discussed in this section. If a deformable block analysis is required, there are several different models available to simulate block deformability; these are discussed in Section 3.7. As mentioned in Section 1.1 in Theory and Background, early distinct element codes assumed that blocks were rigid. However, the importance of including block deformability has become recognized, particularly for stability analyses of underground openings and studies of seismic response of buried structures. One of the most obvious reasons to include block deformability in a distinct element analysis is the requirement to represent the “Poisson’s ratio effect” of a confined rock mass. 3.3.1 Poisson’s Effect Rock mechanics problems are usually very sensitive to the Poisson’s ratio chosen for a rock mass. This is because joints and intact rock are pressure-sensitive; their failure criteria are functions of the confining stress (e.g., the Mohr-Coulomb criterion). Capturing the true Poisson behavior of a jointed rock mass is critical for meaningful numerical modeling. The effective Poisson’s ratio of a rock mass is comprised of two parts: (1) a component due to the jointing; and (2) a component due to the elastic properties of the intact rock. Except at shallow depths or low confining stress levels, the compressibility of the intact rock makes a large contribution to the compressibility of a rock mass as a whole. Thus, the Poisson’s ratio of the intact rock has a significant effect on the Poisson’s ratio of a jointed rock mass. A single Poisson’s ratio, ν, is, strictly speaking, defined only for isotropic elastic materials. However, there are only a few jointing patterns which lead to isotropic elastic properties for a rock mass. Therefore, it is convenient to define a “Poisson effect” that can be used for discussion of anisotropic materials. The Poisson effect will be defined as the ratio of horizontal-to-vertical stress when a load is applied in the vertical direction and no strain is allowed in the horizontal direction; plane-strain conditions are assumed. As an example, the Poisson effect for an isotropic elastic material is σxx ν = σyy 1−ν
(3.1)
Consider the Poisson effect produced by the vertical jointing pattern shown in Figure 3.15. If this jointing were modeled with rigid blocks, applying a vertical stress would produce no horizontal stress at all. This is clearly unrealistic because the horizontal stress produced by the Poisson’s ratio of the intact rock is ignored.
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yy
yy
Figure 3.15 Model for Poisson’s effect in rock with vertical and horizontal jointing The joints and intact rock act in series. In other words, the stresses acting on the joints and on the rock are identical. The total strain of the jointed rock mass is the sum of the strain due to the jointing and the strain due to the compressibility of the rock. The elastic properties of the rock mass as a whole can be derived by adding the compliances of the jointing and the intact rock:
xx σxx rock jointing = C +C yy σyy
(3.2)
If the intact rock were modeled as an isotropic elastic material, its compliance matrix would be 1+ν C rock = E
1−ν −ν
−ν 1−ν
(3.3)
The compliance matrix due to the jointing is C jointing =
1 Skn
0
0
1 Skn
(3.4)
where S is the joint spacing, and kn is the normal stiffness of the joints.
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User’s Guide
If xx = 0 in Eq. (3.2), then (total)
C σxx = − 12 (total) σyy C
(3.5)
11
where C (total) = C (rock) + C (j ointing) . Thus, the Poisson effect for the rock mass as a whole is ν (1 + ν) σxx = σyy E/(Skn ) + (1 + ν)(1 − ν)
(3.6)
Eq. (3.6) is graphed as a function of the ratio E/(Skn ) in Figure 3.16. Also graphed are the results of several two-dimensional UDEC simulations run to verify the formula. The ratio E/(Skn ) is a measure of the stiffness of the intact rock in relation to the stiffness of the joints. For low values of E/(Skn ), the Poisson effect for the rock mass is dominated by the elastic properties of the intact rock. For high values of E/(Skn ), the Poisson effect is dominated by the jointing. Now consider the Poisson effect produced by joints dipping at various angles. The Poisson effect is a function of the orientation and elastic properties of the joints. Consider the special case shown in Figure 3.17. A rock mass contains two sets of equally spaced joints dipping at an angle, θ , from the horizontal. The elastic properties of the joints consist of a normal stiffness, kn , and a shear stiffness, ks . The blocks of intact rock are assumed to be completely rigid.
UDEC Simulations
0.4
Analytic Solution
xx
yy
0.3
0.2
0.1
0 0
0.5
1
1.5
E
2
SK n
Figure 3.16 Poisson’s effect for vertically-jointed rock (ν = 0.3 for intact rock)
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S
yy
O
yy
Figure 3.17 Model for Poisson’s effect in rock with joints dipping at angle θ from the horizontal and with spacing S The Poisson effect for this jointing pattern is cos2 θ [(kn / ks ) − 1] σxx = σyy sin2 θ + cos2 θ (kn / ks )
(3.7)
This formula is illustrated graphically for several values of θ in Figure 3.18. Also shown are the results of numerical simulations using UDEC. The UDEC simulations agree closely with Eq. (3.7).
0.8
xx
yy
0.6
0.4 Analytic Solution UDEC Simulation o O = 20 o O = 45
0.2
o O = 60
0 2
4
6
Kn
8
10
Ks
Figure 3.18 Poisson’s effect for jointed rock at various joint angles (blocks are rigid)
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User’s Guide
Eq. (3.7) demonstrates the importance of using realistic values for joint shear stiffness in numerical models. The ratio of shear stiffness to normal stiffness dramatically affects the Poisson response of a rock mass. If shear stiffness is equal to normal stiffness, the Poisson effect is zero. For more reasonable values of kn /ks , from 2.0 to 10.0, the Poisson effect is quite high, up to 0.9. Next, the contribution of the elastic properties of the intact rock will be examined for the case of θ = 45◦ . Following the analysis for the vertical jointing case, the intact rock will be treated as an isotropic elastic material. The elastic properties of the rock mass as a whole will be derived by adding the compliances of the jointing and the intact rock. The compliance matrix due to the two equally spaced sets of joints dipping at 45◦ is C
(j ointing)
1 = 2S kn ks
ks + kn ks − kn
ks − kn ks + kn
Thus, the Poisson effect for the rock mass as a whole is [ν(1 + ν)] / E + (kn − ks ) / (2S kn ks ) σxx = σyy [(1 + ν)(1 − ν)] / E + (kn + ks ) / (2S kn ks )
(3.8)
Eq. (3.8) is graphed for several values of the ratio E/(Skn ) in Figure 3.19 for the case of ν = 0.2. Also plotted are the results of UDEC simulations. For low values of E/(Skn ), the Poisson effect of a rock mass is dominated by the elastic properties of the intact rock. For high values of E/(Skn ), the Poisson effect is dominated by the jointing.
0.8 Rigid
locks
B
(
E SK n
=
)
xx
yy
0.6 0.5 E SK n =
0.4 Rock with No Joints
0.2
( E SK
n
= 0
)
UDEC Simulation
0 2
4
6
Kn
8
10
Ks
Figure 3.19 Poisson’s effect for rock with two equally spaced joint sets with θ = 45◦ (blocks are deformable with ν = 0.2)
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3.3.2 Zoning for Deformable Blocks If a deformable block analysis is required, models with different densities of block zoning should be evaluated, once the block cutting and boundary location have been established. The GENERATE edge and GENERATE quad commands are used to specify the zoning density. The highest density of zoning should be in regions of high stress or strain gradients (e.g., in the vicinity of excavations). For greatest accuracy, the aspect ratio of zone dimensions (i.e., tetrahedron base length to height ratio) should also be as near unity as possible; anything above 5:1 is potentially inaccurate. It is also not advisable to have large jumps in zone size between adjacent polyhedra. The ratio between zone volumes in adjacent polyhedra should not exceed roughly 4:1 for reasonable accuracy. Use the PRINT max command to find the maximum and minimum zone volumes. The GENERATE edge command will automatically create tetrahedral zones within an arbitrarilyshaped concave polyhedron. It is recommended that if block cutting results in blocks that are long and thin, that these blocks be further cut and joined before generating zones. By doing this, zones with an aspect ratio closer to unity can be generated. The GENERATE quad command will only generate zones within six-sided polyhedra. This command creates mixed-discretization (m-d) zones (two overlays of five tetrahedral zones) that provide better accuracy for problems involving failure and collapse of the intact blocks. (See Section 1.2.2.5 in Theory and Background for details.) It is recommended that the GENERATE quad command be used for analyses involving plastic failure of the intact material. The GENERATE edge command can provide reasonable accuracy for certain failure modes (e.g., confined compression loading); however, this type of zoning does not produce an accurate prediction for collapse loads in bearing capacity problems. Comparisons of results using GENERATE quad versus GENERATE edge for plasticity analysis are given in Sections 5 and 6 in the Verifications volume. The GENERATE edge command produces zoning that is more computationally efficient, and is recommended for blocks in regions where extensive intact material failure is not anticipated. In order to improve the calculational efficiency for models involving intact material failure, the GENERATE quad command can be used to generate m-d zones around an excavation, and the GENERATE edge command can then be used to generate zones in blocks at greater distance from the excavation (or for surrounding blocks that are not six-sided). The GENERATE hotetra command produces high order tetrahedral zones which can be used in blocks which cannot be zoned with m-d zoning. These zones have additional gridpoint nodes and are more accurate than the standard tetrahedral zoning. Note that high order tetrahedral zones are not compatible with the far field dynamic boundary or the extended zone models (cppudm). The analysis of large models can also be aided by using the GENERATE center command, which is a variation of the GENERATE edge command. This command allows the sizes of the tetrahedral zones to be increased gradually outward from a central point.
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User’s Guide
3.4 Boundary Conditions The boundary conditions in a numerical model consist of the values of field variables (e.g., stress, displacement) that are prescribed at the boundary of the model. Boundaries are of two categories: real and artificial. Real boundaries exist in the physical object being modeled — e.g., a tunnel surface or the ground surface. Artificial boundaries do not exist in reality, but they must be introduced in order to enclose the chosen number of elements (e.g., blocks). The conditions that can be imposed on each type are similar; these conditions are discussed first. Then (in Section 3.4.4), some suggestions are made concerning the location and choice of artificial boundaries and the effect they have on the solution. Mechanical boundaries are of two main types: prescribed displacement or prescribed stress. A free surface is a special case of the prescribed-stress boundary. The two types of mechanical boundary are described in Sections 3.4.1 and 3.4.2. Viscous boundaries, which are used for dynamic analysis, are described in Section 2 in Optional Features. 3.4.1 Stress Boundary By default, the boundaries of a 3DEC model are free of stress and any constraint. Forces or stresses may be applied to any boundary, or part of a boundary, by means of the BOUNDARY command. Note that forces and stresses can be applied to either rigid or deformable blocks. Individual components of the stress tensor (σxx , σyy , σzz , σxy , σxz and σyz ) are specified with the stress keyword. For example, the command boundary (0,10) (0,10) (-1,1) stress 0,-1e6,0 0,0,0
would apply σxx = 0, σyy = −106 and σzz = 0 and zero shear stresses to a model boundary lying within the coordinate window 0 < x < 10, 0 < y < 10, -1 < z < 1. The user should always make sure that the window encompasses all the boundary vertices designated for the assigned boundary condition. This can be done using the command print boundary state
Each exterior boundary vertex will be listed with its assigned boundary code. (See the PRINT boundary command in Section 1.3 in the Command Reference.) The boundary can move during a model calculation, so the user must make sure that the coordinate window is large enough to include the appropriate boundary vertices at the time the BOUNDARY command is executed. Alternatively, boundary conditions can be specified along a boundary defined by the orientation of the boundary face. For example, the same boundary condition above can be applied with the command boundary dip 90 dd 180 or 0,0,1 above stress 0,-1e6,0
0,0,0
This will apply the boundary condition along the boundary face located within the range defined by a plane with a dip angle of 90◦ , a dip direction of 270◦ and above the position x = 0, y = 0, z = 1.
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Compressive stresses have a negative sign, in accordance with the general sign convention for internal stresses in 3DEC. Also, 3DEC actually applies stress components as forces, or tractions, which result from a stress tensor acting on the given boundary plane. The tractions are divided into two components, permanent and transient. Permanent tractions are constant loads and transient tractions are time-varying loads applied for dynamic analysis (see Section 2 in Optional Features) by using the history keyword on the same command line as the stress keyword. Various forms of time-varying histories can be applied, including linear-varying, sine and cosine wave, and usersupplied functions; these are described in Section 1.3 in the Command Reference (see BOUNDARY history). Individual forces can be applied to the model boundary of rigid or deformable blocks by using the xload, yload and zload keywords that specify x-, y- and z-components of an applied force vector. 3.4.1.1 Applied Stress Gradient The BOUNDARY command may take additional keywords xgrad, ygrad and zgrad, which allow the applied stresses or forces to vary linearly over the specified range. Six parameters follow each of these keywords and describe the variation of the stress components in either the x-, y- or z-direction: xgrad sxxx syyx szzx sxyx sxzx syzx ygrad sxxy syyy szzy sxyy sxzy syzy zgrad sxxz syyz szzz sxyz sxzz syzz The stresses vary linearly with distance from the global coordinate origin of x = 0, y = 0, z = 0: ◦ σxx = σxx + (sxxx)x + (sxxy)y + (sxxz)z ◦ σyy = σyy + (syyx)x + (syyy)y + (syyz)z ◦ σzz = σzz + (szzx)x + (szzy)y + (szzz)z ◦ σxy = σxy + (sxyx)x + (sxyy)y + (sxyz)z
(3.9)
◦ σxz = σxz + (sxzx)x + (sxzy)y + (sxzz)z ◦ σyz = σyz + (syzx)x + (syzy)y + (syzz)z ◦ , σ ◦ , σ ◦ , σ ◦ , σ ◦ and σ ◦ are the stress components at the origin. where σxx yy zz xy xz yz
The operation of this feature is best explained by an example: boundary -.1,.1 -100,0 0,10 stress 0,-10e6,0 0,0,0
ygrad 0,1e5,0 0,0,0
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The stresses at the origin (x = 0, y = 0, z = 0) are ◦ = 0 σxx ◦ = −10 × 106 σyy ◦ σzz = 0
◦ = 0 σxy ◦ σxz = 0 ◦ = 0 σyz
The equation for the y-variation in stress component σyy is σyy = −10 × 106 + (105 )y The value for σyy at y = -100 is then −20 × 106 . At points in between, the y-variation is linearly scaled to the relative y-distance from the origin. Typically, applied stress gradients are used to reproduce the effects of increasing stress with depth caused by gravity. It is important to make sure that the applied gradient is compatible with the gradient specified with the INSITU command and the value of gravitational acceleration (GRAVITY command). Section 3.5 provides more details on this matter. 3.4.1.2 Changing Boundary Stresses As discussed above, transient loading can be performed with the history keyword for dynamic analysis. For static analysis, it may also be necessary to alter the values of applied stresses during the course of a 3DEC simulation. For example, the load on a footing may change. To effect a sudden change in an existing applied stress or load, a new BOUNDARY command is given with the range that encompasses the same boundary vertices as in the original command but with a change in stress value or variation. In this case, the new value will be added to the existing value.* If the stress is to be removed, the current value should be given with an opposite sign. If a transient load is changed (i.e., a load assigned with the history keyword), any new load with the same history type is added to the existing load; however, a new transient load with a different history type replaces the old transient load. * The user should be aware that this approach is different than that used in the Itasca code FLAC, in which the stresses are updated to the new values when a new boundary condition is specified.
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3.4.1.3 Checking the Boundary Condition The boundary stresses and loads may be verified with the command PRINT bound. The PRINT bound command lists the boundary vertex addresses along with current values and conditions assigned to each vertex. Once a BOUNDARY command is issued, a boundary vertex list is created for the model face that the boundary condition is assigned. Optional keywords can be used with the PRINT bound command to check the different conditions along the boundary. For example, print bound force
lists the permanent forces (fx,fy,fz) and incremental forces (fxi,fyi,fzi) added during the current loading stage. If transient loads are applied (with the BOUND. . . hist command), the total forces refer to the permanent plus transient loads at the current cycle number. The command print bound state
identifies the type of boundary condition assigned to a boundary vertex. 3.4.1.4 Cautions and Advice In this section, some miscellaneous difficulties with stress boundaries are described. With 3DEC, it is possible to apply stresses to the boundary of a body that has no displacement constraints (unlike many finite element programs, which require some constraints). The body will react in exactly the same way as a real body would — i.e., if the boundary stresses are not in equilibrium, then the whole body will start moving. A similar, but more subtle, effect arises when material is excavated from a body that is supported by a stress boundary condition: the body is initially in equilibrium under gravity, but the removal of material reduces the weight. The whole body then starts moving upward, as demonstrated in Example 3.9 and illustrated in Figure 3.20. Example 3.9 Uplift when material is removed new poly brick 0 10 0,10 0,10 jset dip 0 dd 180 or 0,5,0 gen quad ndiv 4 4 4 change cons 1 prop mat=1 dens 1000 bulk 8e9 g prop jmat=1 kn 1e10 ks 1e10 gravity 0,-10,0 bound 0,10 -0.1,.1 0,10 stress bound -.1,.1 0,10 0,10 xvel = bound 9.9,10.0 0,10 0,10 xvel = bound 0,10 0,10 -.1,.1 xvel = bound 0,10 0,10 9.9,10.1 xvel =
5e9
0,-1e5,0 0,0,0 0.0 0.0 0.0 0.0
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insitu stress 0,-1e5,0 0,0,0 ygrad 0,1e4,0 0,0,0 hist ydisp 5,2.5,5 step 300 excavate 0,10 5,10 0,10 step 100 plot hold dip 70 dd 150 axes zol color mat vel blue ret
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dip= 70.00 above dd = 150.00 center 5.000E+00 5.000E+00 5.000E+00 cut-pl. 0.000E+00 mag = 1.00 cycle 400
Max Velocity = 5.307E-03
Y z X Itasca Consulting Group, Inc.
Figure 3.20 Uplift when material is removed The difficulty encountered in running this data file can be eliminated by fixing the bottom boundary, rather than supporting it with stresses. Section 3.4.4 contains information relating to the location of such artificial boundaries. Finally, the stress boundary affects all degrees-of-freedom. Velocity boundary conditions must, therefore, be prescribed after stress boundary conditions affecting the same boundary corners. If the stress boundary is applied after the velocity boundary, the effect of the prescribed velocity will be lost. Example 3.10 demonstrates this problem: Example 3.10 Mixing stress and velocity boundary conditions new poly brick 0 10 0,10 gen quad ndiv 4 4 4
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prop mat=1 dens 1000 bulk 8e9 g bound 0,10 -0.1,.1 0,10 yvel = bound -0.1,.1 0,10 0,10 stress bound 9.9,10.1 0,10 0,10 stress bound 0,10 0,10 -0.1,.1 stress bound 0,10 0,10 9.9,10.1 stress bound 0,10 9.9,10.1 0,10 stress hist ydisp 0,0,0 step 100 plot hold zol axes color mat vel ret
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5e9 0.0 -1e5,0,0 -1e5,0,0 0,0,-1e5 0,0,-1e5 0,-2e5,0
0,0,0 0,0,0 0,0,0 0,0,0 0,0,0
blue
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dip= 90.00 above dd = 180.00 center 5.000E+00 5.000E+00 5.000E+00 cut-pl. 0.000E+00 mag = 1.00 cycle 100
Max Velocity = 6.545E-03
Y z
X
Itasca Consulting Group, Inc.
Figure 3.21 Mixing stress and velocity boundary conditions The fixed y-velocity boundary condition along the bottom boundary of the model is removed along the bottom edges of the block when the stress boundaries are applied. These points move downward, as indicated by the velocity vector plot in Figure 3.21, when the model is loaded.
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3.4.2 Displacement Boundary Displacements cannot be controlled directly in 3DEC; in fact, they play no part in the calculation process, as explained in Section 1.2.2 in Theory and Background. In order to apply a given displacement to a boundary of a deformable block model, it is necessary to fix the boundary and prescribe the boundary’s velocity for a given number of steps (using the BOUNDARY command). If the desired displacement is D, a velocity, V , is applied for a time increment, T (e.g., D = V T ), where T = !tN , !t is the timestep, and N is the number of steps (or cycles). In practice, V should be kept small and N large, in order to minimize shocks to the system being modeled. The BOUNDARY command is used to fix the velocity of gridpoints of deformable blocks in the x-, yor z-direction (BOUND xvel yvel or zvel) or in the normal direction (BOUND nvel) along boundaries not aligned with the x-, y- and z-axes. The velocity of rigid or deformable blocks can be fixed with the FIX command (at the current velocity). Use the APPLY command to specify a velocity other than the current value for rigid blocks. The velocity can also be altered with a FISH function. Time-varying velocity histories can be applied via the BOUND . . . hist command for deformable blocks or the APPLY . . . hist command for rigid blocks. This history keyword must appear on the same line as BOUND xvel, BOUND yvel or BOUND zvel to prescribe a velocity history. Histories can also be applied as FISH functions. As discussed above in Section 3.4.1.4, velocity boundaries should always be assigned after stress boundaries. Fixed velocity conditions can be removed for deformable blocks with the BOUND xfree, BOUND yfree or BOUND zfree command, and for rigid blocks with the FREE command. 3.4.3 Real Boundaries — Choosing the Right Type It is sometimes difficult to know the type of boundary condition to apply to a particular surface on the body being modeled. For example, in modeling a laboratory triaxial test, should the load applied by the platen be regarded as a stress boundary, or should the platen be treated as a rigid, displacement boundary? Of course, the whole testing machine, including the platen, could be modeled, but that might be very time-consuming. Remember that 3DEC takes a long time to converge if there is a large contrast in stiffnesses. In general, if the object applying the load is very stiff compared with the sample (say, more than 20 times stiffer), then it may be treated as a rigid boundary. If it is soft compared with the sample (say, 20 times softer), then it may be modeled as a stress-controlled boundary. Clearly, a fluid pressure acting on the surface of a body is of the latter category. Footings on jointed rock can often be represented as rigid boundaries that move with constant velocity for the purposes of finding the collapse load of the rock. This approach has another advantage — it is much easier to control the test and obtain a good load/displacement graph. It is well-known that stiff testing machines are more stable than soft testing machines.
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3.4.4 Artificial Boundaries Artificial boundaries fall into two categories: planes of symmetry and planes of truncation. Symmetry planes take advantage of symmetry conditions in a physical system; truncation planes are needed when modeling an infinite or very large system. 3.4.4.1 Symmetry Planes Sometimes it is possible to take advantage of the fact that the geometry and loading in a system are symmetrical about one or more planes. For example, if everything is symmetrical about a vertical (yz) plane, then the horizontal displacements on that plane will be zero. Therefore, we can make that plane a boundary and fix all gridpoints in the horizontal direction, using the command BOUND xvel=0. If velocities on the plane of symmetry are not already zero, they will be set to zero with this command. In the case considered, the y-component and z-component of velocity on the vertical plane of symmetry are not affected; they should not be fixed. Similar considerations apply to a horizontal plane of symmetry. The command BOUND nvel=0 can be used to set planes of symmetry that lie at angles to the coordinate axes. As discussed in Section 3.2.3, the presence of discontinuities makes the application of symmetry planes more difficult. When using symmetry planes in 3DEC, the modeler should always be careful to consider the effect of joint orientation. 3.4.4.2 Boundary Truncation — Location of the Far-Field Boundary Analysis of rock mass response involves several different scales. It is impossible and undesirable to include all features, details or rock mass response mechanisms in one model. It is also wellrecognized that location of the far-field boundary in the model can have a significant influence on results obtained for underground excavations. However, in three-dimensional analysis, it is not always feasible to place boundaries sufficiently far from the excavations to avoid adversely affecting the results. These two observations, taken together, suggest that a reasonable modeling approach involves starting with a large global model and proceeding through reasonable models to the smallest size required, with increasing complexity and detail added at each stage. 3DEC has an automatic method of recording stresses at specified locations so that they can be applied as boundary tractions on smaller problems. This unique feature ensures stress compatibility between larger and smaller models. An example of the application of this technique is a mine in which the local topographic relief is significant relative to the size of the mine, and in which a model incorporating both the surrounding topography and the mine geometry is prohibitively large. A coarse regional model incorporating a relatively large sample of the surrounding topography is first created, as shown in Figure 3.22. Following this, a smaller central portion of the regional model is generated, incorporating details of the mining geometry, which is more finely zoned. Boundary tractions applied to the detailed model are transferred from the regional model at points corresponding to the locations of the detailed model boundaries by the following procedure.
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1. Run Model A. 2. Enter the command SET log on. 3. Enter the command PRINT brick xl xu yl yu zl zu where (xl, xu), (yl, yu) and (zl, zu) correspond to the boundaries of Model B. 4. The loads printed to the log file for the PRINT brick command can be applied to Model B by using the BOUND xtraction, BOUND ytraction and BOUND ztraction commands. An important modeling decision when using this procedure is how extensive a volume should be incorporated into each of the regional and detailed models. In the regional model, the topographic relief should be small relative to the external dimensions of the model. A criterion recommended for selecting the depth of the model is that stresses near the bottom of the model should be relatively uniform — i.e., it should not be possible to determine if a point is beneath a mountain peak or a valley; otherwise, the finite depth of the model will influence the stress field near the surface. Analyses performed using two-dimensional models indicate that to satisfy this criterion, the thickness of the model beneath the valleys should be approximately three times the topographic amplitude.
Model B
Model A
Figure 3.22 Models used to transfer stress boundary conditions
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Boundary locations for the detailed model are chosen using the conventional criterion that induced stresses at the boundary location caused by internal changes (such as mining) should not be significant. An estimate of this distance can be obtained using the elastic solution for stress around a t , at a distance sphere in a hydrostatic stress field. The induced tangential component of stress, σind R from a cavity of radius a in a hydrostatic stress field po is: t σind a3 = po 2 R3
(3.10)
Because the decay of induced stress away from the excavation is more rapid than for a circular hole, the two-dimensional equivalent, boundaries around three-dimensional excavations do not have to be particularly far. For example, at a distance of approximately 2a, the induced stress is only 5% of the hydrostatic level and only 1% at a distance of 3.7a. Solutions for other shaped openings can also be used depending on the actual excavation shape. An important aspect of the point-wise boundary traction transfer to the smaller model is that a more complex boundary stress distribution can be generated than by conventional linear variations. The dimensions of boundaries in the detailed model can, therefore, be small relative to the surrounding topographic relief. This would be impossible to achieve using standard linearly-varying boundary stress distributions. An alternative to the stress transfer technique can be used when the entire model is not too big to fit in memory but runs too slowly to allow investigation of such things as strength variations or extraction sequences. In this case, the COUPLE command may be used. The COUPLE command allows the decoupling of an inner and outer region. After reaching internal equilibrium, the large mass shown in Figure 3.22 (Model A) is defined as a single region — for example, region 1 (see the MARK command). The outer area can be removed from calculation cycles by the COUPLE 1 off command. From this point, 3DEC does not include the blocks that lie in region 1 in the cycle calculations. The model forces at the interface between region 1 and all other regions are held constant. Changes in the inner model that affect the interface will not cause a change in the forces at the boundaries unless an outer region is turned back on (COUPLE 1 on). The status of any region can be determined by the PRINT region command.
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3.5 Initial Conditions In all civil or mining engineering projects, there is an in-situ state of stress in the ground, before any excavation or construction is started. By setting initial conditions in the 3DEC model, an attempt is made to reproduce this in-situ state, because it can influence the subsequent behavior of the model. Ideally, information about the initial state comes from field measurements but, when these are not available, the model can be run for a range of possible conditions. Although the range is potentially infinite, there are a number of constraining factors (e.g., the system must be in equilibrium, and the chosen yield and slip criteria must not be violated anywhere). In a uniform layer of soil or rock with a free surface, the vertical stresses are usually equal to gρz, where g is the gravitational acceleration, ρ is the mass density of the material, and z is the depth below surface. However, the in-situ horizontal stresses are more difficult to estimate. There is a common — but erroneous — belief that there is some “natural” ratio between horizontal and vertical stress, given by ν/(1 − ν), where ν is the Poisson’s ratio. This formula is derived from the assumption that gravity is suddenly applied to an elastic mass of material in which lateral movement is prevented. This condition hardly ever applies in practice due to repeated tectonic movements, material failure, overburden removal and locked-in stresses due to faulting and localization (see Section 3.11.3). Of course, if we had enough knowledge of the history of a particular volume of material, we might simulate the whole process numerically, so as to arrive at the initial conditions for our planned engineering works. This approach is not usually feasible. Typically, we compromise: a set of stresses is installed in the model, and then 3DEC is run until an equilibrium state is obtained. It is important to realize that there is an infinite number of equilibrium states for any given system. In the following sections, we examine progressively more complicated situations and the way in which the initial conditions may be specified. The user is encouraged to experiment with the various data files that are presented. 3.5.1 Uniform Stresses in an Unjointed Medium: No Gravity For an excavation deep underground, the gravitational variation of stress from top to bottom of the excavation may be neglected because the variation is small in comparison with the magnitude of stress acting on the volume of rock to be modeled. The GRAVITY command may be omitted, causing the gravitational acceleration to default to zero. The initial stresses are installed with the INSITU command — e.g., insitu stress -5e6 -1e7 -5e6 0.0 0.0 0.0
The components σ11 (or σxx ), σ22 (or σyy ) and σ33 (or σzz ) are set to compressive stresses of 5×106 , 107 and 5 × 106 , respectively, throughout the model. Range parameters may be added if the stresses are to be restricted to a subregion of the model. The INSITU command sets all stresses to the given values, but there is no guarantee that the stresses will be in equilibrium. There are at least three possible problems. First, the stresses may violate the yield criterion of a nonlinear constitutive model assigned to deformable blocks. In this case, plastic flow of zones in the blocks will occur immediately after the STEP command is given, and the stresses will readjust; this possibility should be checked by doing one trial step and examining the response (e.g., PLOT plas). Second, the stress
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state may result in slip or separation along joints within the model. The command PLOT vel should indicate locations where joint movement is occurring. Third, the prescribed stresses at the model boundary may not equal the given initial stresses. In this case, the boundary gridpoints will start to move as soon as a STEP command is given; again, output should be checked (e.g., PLOT vel) for this possibility. The commands in Example 3.11 produce a single block with initial stresses that are in equilibrium with prescribed boundary stresses. Example 3.11 Initial and boundary stresses in equilibrium new poly brick 0 10 0,10 0,10 gen edge 2.0 prop mat=1 dens 1000 bulk 8e9 g 5e9 bound (-0.1,0.1) (0,10) (0,10) bound (9.9,10.1) (0,10) (0,10) bound (0,10) (0,10) (-0.1,0.1) bound (0,10) (0,10) (9.9,10.1) bound (0,10) (-0.1,0.1) (0,10) bound (0,10) (9.9,10.1) (0,10) insitu stress -5e6 -1e7 -5e6 0,0,0 step 1 ret
stress stress stress stress stress stress
-5e6, 0, 0 -5e6, 0, 0 0, 0,-5e6 0, 0,-5e6 0,-1e7, 0 0,-1e7, 0
0,0,0 0,0,0 0,0,0 0,0,0 0,0,0 0,0,0
3.5.2 Stresses with Gradients in an Unjointed Medium: Uniform Material Variation in stress with depth cannot be ignored near the ground surface — the GRAVITY command is used to inform 3DEC that gravitational acceleration operates on the model. It is important to understand that the GRAVITY command does not directly cause stresses to appear in the model; it simply causes body forces to act on all gridpoints of deformable blocks (or centroids of rigid blocks). These body forces correspond to the weight of material surrounding each gridpoint. If no initial stresses are present, the forces will cause the material to move (during stepping) in the direction of the forces until equal and opposite forces are generated by zone stresses. Given the appropriate boundary conditions (e.g., fixed bottom, roller side boundaries), the model will, in fact, generate its own gravitational stresses compatible with the applied gravity. However, this process is inefficient, since many hundreds of steps may be necessary for equilibrium. It is better to initialize the internal stresses such that they satisfy both equilibrium and the gravitational gradient. The INSITU command must include the xgrad, ygrad and zgrad parameters so that the stress gradient matches the gravitational gradient gρ. The internal stresses must also match boundary stresses at stress boundaries. Even though the boundary and in-situ stresses are specified to produce a force balance, some cycling of the model is normally required. This is because the boundary forces are only applied at the end
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of a cycle; a small force imbalance is produced by the in-situ stresses. Usually, this imbalance is reduced within a few hundred cycles. Consider, for example, a 20 m × 20 m × 20 m box of homogeneous unjointed material at a depth of 200 m underground, with fixed base and stress boundaries on the other five sides. Example 3.12 produces an equilibrium system for this problem condition. Example 3.12 Initial stress state with gravitational gradient new poly brick 0 20 0,20 0,20 gen edge 4.0 prop mat=1 dens 2500 bulk 5e9 g 3e9 phi 35 change cons 2 gravity 0 -10 0 bound (-0.1,0.1) (0,20) (0,20) stress -2.75e6 -5.5e6 ygrad 1.25e4 2.5e4 1.25e4 0 0 0 bound (19.9,20.1) (0,20) (0,20) stress -2.75e6 -5.5e6 ygrad 1.25e4 2.5e4 1.25e4 0 0 0 bound (0,20) (0,20) (-0.1,0.1) stress -2.75e6 -5.5e6 ygrad 1.25e4 2.5e4 1.25e4 0 0 0 bound (0,20) (0,20) (19.9,20.1) stress -2.75e6 -5.5e6 ygrad 1.25e4 2.5e4 1.25e4 0 0 0 bound (0,20) (19.9,20.1) (0,20) stress 0 -5.0e6, insitu stress -2.75e6 -5.5e6 -2.75e6 0,0,0 & ygrad 1.25e4 2.5e4 1.25e4 0 0 0 bound (0,20) (-0.1,0.1) (0,20) yvel = 0.0 step 500 ret
-2.75e6
0,0,0 &
-2.75e6
0,0,0 &
-2.75e6
0,0,0 &
-2.75e6
0,0,0 &
0
0,0,0
In this example, horizontal stresses and gradients are equal to half the vertical stresses and gradients, but they may be set at any value that does not violate the yield criterion (Mohr-Coulomb, in this case). After preparing a data file such as the one above, the model should be cycled to check that an equilibrium state is reached. If material failure does occur (e.g., reduce phi = 10◦ ), this will show as an unbalanced force magnitude roughly the same order of magnitude as the applied loading. 3.5.3 Stresses with Gradients in a Nonuniform Material It is more difficult to give the initial stresses when materials of different densities are present. Consider a layered system with a free surface, enclosed in a box with roller side boundaries and fixed base. Suppose that the material has the following density distribution:
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1600 kg/m3 from 0 to 10 m depth 2000 kg/m3 from 10 to 15 m 2200 kg/m3 from 15 to 25 m An equilibrium state is produced by the data file in Example 3.13. Example 3.13 Initial stress gradient in a nonuniform material new poly brick 0 20 0,25 0,20 jset dip 0.0 or 0,10,0 jset dip 0.0 or 0,15,0 gen edge 5.0 change 0,20 0,10 0,20 mat 1 change 0,20 10,15 0,20 mat 2 change 0,20 15,25 0,20 mat 3 prop mat=1 dens 1600 bulk 5e9 g 3e9 prop mat=2 dens 2000 bulk 5e9 g 3e9 prop mat=3 dens 2200 bulk 5e9 g 3e9 change jmat 1 prop jmat 1 kn 1e10 ks 1e10 coh 1e10 gravity 0 -10 0 insitu 0 20 0 10 0 20 stress 0.0 -4.8e5 0.0 0,0,0 ygrad 0.0 2.2e4 0.0 0,0,0 insitu 0 20 10 15 0 20 stress 0.0 -4.6e5 0.0 0,0,0 ygrad 0.0 2.0e4 0.0 0,0,0 insitu 0 20 15 25 0 20 stress 0.0 -4.0e5 0.0 0,0,0 ygrad 0.0 1.6e4 0.0 0,0,0 bound (0,20) (-0.1,0.1) (0,20) yvel = 0.0 bound (-0.1,0.1) (0,20) (0,20) xvel = 0.0 bound (19.9,20.1) (0,20) (0,20) xvel = 0.0 bound (0,20) (0,20) (-0.1,0.1) zvel = 0.0 bound (0,20) (0,20) (19.9,20.1) zvel = 0.0 hist unbal step 500 ret
& & &
An individual block is created for each material density; fictitious joints separate each block. The internal stress profile is calculated manually for each block from the known overburden above it. Note that the example is simplified — in a real case, the elastic moduli would vary, and there would be horizontal stresses. If high horizontal stresses exist in a layer, these may also be installed with the INSITU command.
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This example is not in equilibrium at one calculation step; approximately 500 steps are required. The presence of the fictitious joints also prevents the model from being in equilibrium when the initial stresses match the boundary stresses. A jointed model will often require more steps to equilibrate than an unjointed model. 3.5.4 Compaction within a Model with Nonuniform Zoning Puzzling results are sometimes observed when a model with nonuniform zoning is allowed to come to equilibrium under gravity. A model that is composed of deformable blocks of different sizes will usually have nonuniform zoning. When a Mohr-Coulomb, or other nonlinear constitutive, model is assigned to the blocks, the final stress state and displacement pattern are not uniform, even though the boundaries are straight and the free surface is flat. The data file in Example 3.14 illustrates the effect — see Figure 3.23 for the generated plot showing vertical stress contours. Example 3.14 Nonuniform stress initialized in a model with nonuniform zoning new poly brick 0,10 0,10 0,10 jset dip 90.0 dd 90 or 3,0,0 gen edge 2.0 change cons 2 prop mat=1 dens 2000 bulk 2e8 g 1e8 phi 30 prop jmat 1 kn 1e10 ks 1e10 coh 1e10 ten 1e10 gravity 0 -10 0 bound (0,10) (-0.1,0.1) (0,10) yvel = 0.0 bound (-0.1,0.1) (0,10) (0,10) xvel = 0.0 bound (9.9,10.1) (0,10) (0,10) xvel = 0.0 bound (0,10) (0,10) (-0.1,0.1) zvel = 0.0 bound (0,10) (0,10) (9.9,10.1) zvel = 0.0 hist unbal step 1000 ; optional method 1 ; insitu stress -1.5e5,-2.0e5,-1.5e5 0,0,0 & ; ygrad 1.5e4, 2.0e4, 1.5e4 0,0,0 ; step 400 ; ; optional method 2 ; prop mat=1 bcoh 1e10 bten 1e10 ; step 750 ; prop mat=1 bcoh 0.0 bten 0.0 ; step 250 plot hold x smin mag 2 ret
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3DEC (Version 3.00) Cross section plot: 27-Aug-02 12:03 geometric scale 0
2E 00
dip= 90.00 above dd = 180.00 center 5.000E+00 5.000E+00 5.000E+00 cut-pl. 0.000E+00 mag = 2.00 cycle 1000 min. p.s. contours interval = 1.500E+04 from to -8.500E+04 -7.000E+04 -7.000E+04 -5.500E+04 -5.500E+04 -4.000E+04 -4.000E+04 -2.500E+04 -2.500E+04 -1.000E+04 -1.000E+04 5.000E+03
Itasca Consulting Group, Inc.
Figure 3.23 Nonuniform stresses Since we have roller boundaries on the four sides, we might expect the material to move down equally on all sides. However, the zones are not the same size in the blocks. For static analysis, 3DEC tries to keep the timestep equal for all zones, so it increases the inertial mass for the gridpoints of the smaller zones to compensate for their size. These gridpoints then accelerate more slowly than those for the larger zones. This would have no effect on the final state of a linear material, but it causes nonuniformity in a material that is path-dependent. For a Mohr-Coulomb material without cohesion, the situation is similar to dropping sand from some height into a container and expecting the final state to be uniform. In reality, a large amount of plastic flow would occur because the confining stress does not build up immediately. Even with a uniformly-zoned model, this approach is not a good one because the horizontal stresses depend on the dynamics of the process. The best solution is to use the INSITU stress command to set initial stresses to conform to the desired Ko value (ratio of horizontal to vertical stress). For example, the STEP 1000 command in the previous data file could be replaced by the following lines: insitu stress (-1.5e5,-2.0e5,-1.5e5,0,0,0) ygrad (1.5e4,2.0e4,1.5e4,0,0,0) step 400
&
A stable state is achieved with Ko = 0.75; fewer steps are needed to reach equilibrium and the stress state is uniform (see Figure 3.24). Note that there is a slight nonuniformity, but this is related to the contouring routine and the coarseness of the zoning.
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Alternatively, the model can be run with an elastic behavior for the initial equilibrium calculation and then changed to the nonlinear behavior model for the final state. Replace the STEP 1000 command with the following lines: prop step prop step
mat=1 btens=1e10 bcoh=1e10 750 mat=1 btens=0 bcoh=0 250
The result is the same as that shown in Figure 3.24. The material is prevented from yielding during the compaction process but the original properties are restored when equilibrium is achieved.
3DEC (Version 3.00) Cross section plot: 27-Aug-02 12:08 geometric scale 0
2E 00
dip= 90.00 above dd = 180.00 center 5.000E+00 5.000E+00 5.000E+00 cut-pl. 0.000E+00 mag = 2.00 cycle 400 min. p.s. contours interval = 3.000E+04 from to -1.600E+05 -1.300E+05 -1.300E+05 -1.000E+05 -1.000E+05 -7.000E+04 -7.000E+04 -4.000E+04 -4.000E+04 -1.000E+04 -1.000E+04 2.000E+04
Itasca Consulting Group, Inc.
Figure 3.24 Uniform stresses
3.5.5 Initial Stresses following a Model Change There may be situations in which one material model for deformable blocks is used in the process of reaching a desired stress distribution, but another model is used for the subsequent simulation. Models can be changed for entire blocks (via the CHANGE command). If one model is replaced by another non-null model, the stresses in the affected zones are preserved, as in Example 3.15.
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Example 3.15 Initial stresses following a model change new poly brick 0,5 0,5 0,5 gen edge 1.0 prop mat=1 dens 2000 bulk 3e8 g 2e8 gravity 0 -10 0 bound (0,10) (-0.1,0.1) (0,10) xvel = 0.0 hist unbal step 250 pause change cons 2 prop mat=1 dens 2000 bulk 3e8 g 2e8 phi 34 ret
yvel = 0.0
At this point in the run, the stresses generated by the initial elastic model still exist and act as initial stresses for the region containing the new Mohr-Coulomb model. Two points should be remembered. First, if a null block is created (via the EXCAVATE command) in any part of the model (even if it is subsequently replaced by another non-null block), all stresses are removed from the null block. Second, if one material model is replaced by another and the stresses should physically be zero in the new model, then an INSITU command must be used to reset the stresses to zero in this region. This situation would occur if rock is mined out and replaced by backfill; the backfill should start its life without stress. 3.5.6 Stresses in a Jointed Medium A spatial heterogeneity in an initial stress state can develop in a jointed and fractured medium. This results from the stress path followed during the geologic history of the medium and the physical processes, related to fracturing and slip and separation along discontinuities, which may have occurred at different stages in the history. Spatial heterogeneity of the stress state can be an important factor in the design of underground excavations, particularly if the resulting stress concentrations adversely influence the excavation stability. It is very difficult to determine whether the stress state installed in a jointed model is representative of the in-situ state of stress. As discussed later in Section 3.11.2, statistical analyses may provide a means to develop confidence in the model representation. One such study using UDEC is reported by Brady et al. (1986). There are certain modeling aspects that should be considered when bringing a jointed model to an equilibrated state. First, the INSITU command should be invoked after all joints are generated in the model. Then the normal and shear stresses along joints will be initialized, corresponding to the initial stress values resolved along the plane of each joint.
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User’s Guide
As mentioned previously, a jointed model will not be in equilibrium initially, even when internal stresses are set to match boundary stresses. Some calculation steps are required and the unbalanced force should be monitored. In addition, histories of velocities or displacements should be recorded at various locations in the model. These are good indicators of the calculation step at which motion is negligible. The user should always ensure that motion in the model has essentially stopped for the equilibrium stress state before beginning the next stage of an analysis. It is possible that, for the specified initial stress state and joint strength properties, some joints will slip or separate when the model is brought to an equilibrated state. Joint slip which is confined within the model is acceptable; “locked-in” stresses at the joint ends will result. However, the user should avoid conditions for which joint failure extends to the model boundary. This indicates that the model conditions are not well-posed. It may be necessary to reevaluate the assigned stress state, joint properties and joint orientations and locations. If conditions are such that joint failure still extends to a boundary, then a fixed boundary condition should be considered. This implies that the joint is truncated at the boundary. The data file in Example 3.16 demonstrates the case of a joint dipping at 60◦ confined between two joints dipping at 20◦ . The 60◦ joint slips for the prescribed initial stress while the 20◦ joints do not. The friction angle for all joints is 30◦ . Example 3.16 Slip of a confined joint new poly brick -10,10 -20,0 -10,10 jset dip 60 dd 90 or 0,-10,0 jset dip 20 dd 90 or 0,-8,0 jset dip 20 dd 90 or 0,-12,0 join -10 10 -20 -14 -10 10 on join -10 10 -7 0 -10 10 on gen edge 2.0 prop mat=1 dens 2000 bulk 8e9 g 5e9 prop jmat=1 kn 5e11 ks 2.5e11 fric 30 insitu stress -2.5e7,-1e7 -2.5e6 0,0,0 bound -10.1,-9.9 -20,0 -10,10 stress -2.5e6 0 0 bound 9.1,10.1 -20,0 -10,10 stress -2.5e6 0 0 bound -10,10 -20,0 -10.1,-9.9 stress 0 0 -2.5e6 bound -10,10 -20,0 9.9,10.1 stress 0 0 -2.5e6 bound -10,10 -.1,.1 -10,10 stress 0 -1e7 0 bound -10,10 -20.1,-19.9 -10,10 yvel 0.0 hist unbal hist ydis 0,-10,0 step 1000 plot hold x w mag 2 sxy jshear blue ret
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0 0 0 0 0
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Figure 3.25 shows a block plot on which the region of joint slip is indicated by joint shear vectors. Contours of σxy are also plotted and show the areas of locked-in stresses near the ends of the 60◦ joint.
3DEC (Version 3.00) Cross section plot: 27-Aug-02 12:34 geometric scale 0
5E 00 vector scale
0
1E-03
dip= 90.00 above dd = 180.00 center 0.000E+00 -1.000E+01 0.000E+00 cut-pl. 0.000E+00 mag = 2.00 cycle 1000 xy-stress contours interval = 5.000E+04 from to -1.500E+05 -5.000E+04 0.000E+00 1.000E+05 1.500E+05 2.500E+05 3.000E+05 4.000E+05 4.500E+05 5.500E+05
Max shear in plane = 3.890E-04
Itasca Consulting Group, Inc.
Figure 3.25 Slip of a confined joint; plot shows shear stress contours
3.5.7 Determination of the In-situ Stress State Knowledge of the virgin stress field is required in order to establish appropriate boundary and initial conditions for models of underground excavations. However, often it is not possible to perform in-situ stress measurements sufficiently far from underground excavations or topographic features to determine virgin stresses. This is particularly true for mines using massive mining methods. Boundary stresses may be approximated from regional stress compilations if available — e.g., Müller et al. (1992), Lindner and Halpern (1978), but three-dimensional modeling can be used to quantify the various forms of induced stress, such as those generated by topography, excavations, or material property variations. Subtracting the induced stress components from the total or measured stress enables the virgin stress field to be computed. The latter method has the advantage of being able to utilize stress measurements which are known to include induced stresses from various sources (or to check if the measurements are free of induced stress). It also enables a computation of the local stress field to be made, which is not necessarily represented by the regional stress field. The following procedure has been used in three-dimensional stress analyses to estimate virgin stresses from in-situ measurements that were influenced by various forms of induced stress.
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User’s Guide
The total stress field, σtot , at any point is the sum of virgin stress plus any induced stress components, σind . Virgin stress, in turn, is composed of gravitational stress, σgrav , plus an as-yet-undetermined additional horizontal component that will be referred to as a tectonic component, σtec . There are various geological reasons for why this additional horizontal component of stress should be incorporated into the total stress tensor. Eq. (3.11) relates these components: σtot = σgrav + σtec + σind
(3.11)
The induced stress, in turn, is composed of gravitational and tectonic components: σind = σ¯ grav + σ¯ tec
(3.12)
Substituting Eq. (3.12) into Eq. (3.11) produces: σtot = σgrav + σ¯ grav + σtec + σ¯ tec
(3.13)
which, upon regrouping, becomes: σtec + σ¯ tec = σtot − (σgrav + σ¯ grav )
(3.14)
Terms on the right-hand side of Eq. (3.14) are either known (stress measurements are representative of the total stress field) or can be computed using a model with only gravitational loading. To start the computation process, it should be assumed that the problem geometry (i.e., topography, nearby excavations) is the primary factor generating the induced stress field and that material property variations produce only second-order effects. (Experience has shown this to be a reasonable assumption.) First, a model is constructed, taking into account the topography and excavation geometry. This is run with gravitational loading only. The resulting stress field at the stress measurement points accounts for gravity, plus the gravitational component of induced stress caused by the problem geometry. The resultant calculated vertical stresses are compared to the corresponding measured vertical stresses, and the measured stresses (all components) are adjusted to bring the measured vertical stress into agreement with the calculated vertical stress. This latter step essentially scales the measured stress to the model. If the computed and measured vertical components of stress are found to differ by a large amount, then either the model is incorrect (e.g., incorrect densities), there are other unknown sources of induced stress (e.g., locked-in stresses from geological processes), or there are significant errors in the measurements. In these situations, it is wise to further investigate to determine the reason for the stress anomaly, because confidence in the stress field is a critical design requirement. The unknown tectonic components can be solved by applying unit normal or shear stress boundary conditions to the model and computing the resultant stress level at the stress measurement point.
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The correct “far-field” tectonic boundary stress is computed by scaling the unit stress results to match the magnitudes of components obtained using Eq. (3.14). The total stress field is specified by the combination of horizontal tectonic stresses applied at the boundary of the model, as well as gravitational stresses. 3.5.8 Transferring Field Stresses to Model Stresses A utility program, “TRANS.EXE,” is provided in the “\Tutorial\Solving” directory to transform field stresses into a set of stress components referenced to the local problem axes defined for the 3DEC model. The orientation of the local (model) axes is defined by the dip and dip direction of the local z-axis. The local y-axis lies in the vertical plane containing the z-axis dip vector, and the x-axis lies in the horizontal plane. The program “TRANS.EXE” calculates local stress components on the basis of the following input data. field principal stress 1 (σ1 ): magnitude, dip and dip direction field principal stress 2 (σ2 ): magnitude, dip and dip direction field principal stress 3 (σ3 ): magnitude, dip and dip direction local z-axis: dip and bearing The user must ensure that the directions of σ1 , σ2 and σ3 are orthogonal. “TRANS.EXE” computes, first, a set of stress components referenced to a left-handed set X, Y, Z of global axes which are oriented X (north), Y (east) and Z (vertically up). Then, the set of stress components referenced to the model axes are calculated. The output stress components are recorded on a file named “TRANS.REC.” The following example illustrates the transformation of field stresses to boundary stresses for the 3DEC model. A tunnel ventilation raise is oriented with an axis dip of 84◦ and dip direction of 125◦ , as shown previously in Figure 3.12. The 3DEC model axes are oriented as shown in that figure. The z-axis is directed down the raise, the y-axis lies in the vertical plane containing the raise dip vector, and the x-axis lies in the horizontal plane, directed N35◦ E. The field stresses for this problem are listed below. σ1 = 30 MPa directed 24◦ / 231◦ (dip / dip direction) σ2 = 15 MPa directed 5◦ / 138◦ (dip / dip direction) σ3 = 12 MPa directed 66◦ / 36◦ (dip / dip direction) For a z-axis orientation of dip = 84◦ and dip direction = 125◦ , program “TRANS.EXE” computes the following stress components relative to the local axis (in 3DEC, tensile stresses are considered positive).
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User’s Guide
Stress Data for 3DEC (left-handed axes) Field Stress: Principal Stress 1
Magnitude -30.0
Dip 24.0
Bearing 231.0
Principal Stress 2
Magnitude -15.0
Dip
Bearing 138.0
Principal Stress 3
Magnitude -12.0
Dip 66.0
5.0
Bearing
36.0
Stresses Relative to Global Axes (X (north), Y (east), Z (vertically up)): FXX -19.44
FYY -22.47
FZZ -15.09
FXY - 5.79
FYZ - 5.17
FZX - 4.38
Model z-Axis Orientation: Dip 84.0
Bearing 125.0
Stresses Relative to Model Axes: SXX -25.85
SYY -16.38
SZZ -14.74
SXY
SYZ
SXZ - 6.16
4.07
1.59
This information is contained in “TRANS.REC.” The boundary stresses applied to the model are then σxx = -25.8 MPa, σyy = -16.38 MPa, σzz = -14.74 MPa σxy = 4.07 MPa, σyz = 1.59 MPa, σzx = -6.16 MPa 3.5.9 Topographical Stresses The command INSITU topograph automatically calculates gravity loading in models which have an irregular top surface. The stresses are calculated based on the density of the overlying materials and specifies Ko values.
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3.6 Loading and Sequential Modeling By applying different model loading conditions at different stages of an analysis, it is possible to simulate changes in physical loading, such as sequences of excavation and construction. Changes in loading may be specified in a number of ways — e.g., by applying new stress or displacement boundaries, by changing the material model in blocks to either a null material or to a different material model, or by changing material properties. It is important to recognize that sequential modeling follows the stages of an engineering work. In most analyses, each work stage corresponds to a different static solution following a loading change — i.e., physical time is not a parameter. 3DEC can perform calculations for heat transfer and dynamic mechanical analysis as well (see Sections 1 and 2 in Optional Features). In these cases, a static solution for an equilibrium stress state may be followed, for example, by a dynamic calculation for an applied explosive excitation or a transient calculation for flow through joints. Time-dependent behavior, on the other hand, cannot be simulated directly. Some engineering judgment must be used to estimate the effects of time. For example, a model parameter may be changed after a pre-determined amount of displacement or strain has occurred. This displacement may be estimated to have occurred over a given period of time. A loading change must cause unbalanced forces to develop in order to effect a change in model response. Therefore, changing the elastic properties will have no effect, whereas changing strength properties will if the change causes the current stress state to exceed the failure limit. The recommended approach to sequential modeling is demonstrated by the following example. This problem involves the stability analysis of an underground opening in jointed rock and includes the evaluation of different types of support measures. The stages to be analyzed are: (1) equilibration at the in-situ stress state; (2) excavation of the tunnel; and (3) application of the tunnel support. The objective is to investigate the stability of the excavation under in-situ conditions and assess the effect of the support measures. Three types of support are evaluated: local reinforcement rock bolts, cable bolts and a continuous concrete liner. Note that this model is greatly simplified for rapid execution, but it still illustrates the recommended steps for loading and sequential modeling. The tunnel is located in rock containing three major faults: one dipping at 65◦ with a dip direction of 40◦ ; the second dipping at 70◦ with a dip direction of 270◦ ; and the third dipping at 60◦ with a dip direction of 130◦ . The tunnel is horseshoe-shaped and is centered along the z-axis of the model. The tunnel is created with the TUNNEL command. A second TUNNEL command is also used to define the location of the concrete liner. Note that this must be done before any cycling is performed. The model is created by the following series of commands beginning with Example 3.17:
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User’s Guide
Example 3.17 Stability analysis of an underground excavation — initial model new poly brick -10 10 -10 10 -10 10 ; --- tunz: FISH function to define tunnel geometry parameters -------; ... tunnel along z axis, from ZZA to ZZB ; ... semi-circular roof, centered at (TXC,TYC) ; def tunz ; zza = -10.0 zzb = 10.0 ; ; --- outer surface --txb1 = -4.0 tyb1 = -4.0 txb2 = 4.0 tyb2 = -4.0 ; txc = 0.0 tyc = 0.0 tr = 4.0 tx1 = txc + tr * cos(180*degrad) ty1 = tyc + tr * sin(180*degrad) tx2 = txc + tr * cos(135*degrad) ty2 = tyc + tr * sin(135*degrad) tx3 = txc + tr * cos(90*degrad) ty3 = tyc + tr * sin(90*degrad) tx4 = txc + tr * cos(45*degrad) ty4 = tyc + tr * sin(45*degrad) tx5 = txc + tr * cos(0*degrad) ty5 = tyc + tr * sin(0*degrad) ; ; --- inner surface --; thickness th th = 0.5 txb1i = -4.0 + th tyb1i = -4.0 + th txb2i = 4.0 - th tyb2i = -4.0 + th ; txc = 0.0 tyc = 0.0 tri = tr - th tx1i = txc + tri * cos(180*degrad)
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ty1i tx2i ty2i tx3i ty3i tx4i ty4i tx5i ty5i
= = = = = = = = =
tyc txc tyc txc tyc txc tyc txc tyc
+ + + + + + + + +
tri tri tri tri tri tri tri tri tri
* * * * * * * * *
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sin(180*degrad) cos(135*degrad) sin(135*degrad) cos(90*degrad) sin(90*degrad) cos(45*degrad) sin(45*degrad) cos(0*degrad) sin(0*degrad)
; end ; ; --------------------------------------------------------------------; (execute function) tunz ; ; create outer surface tunnel a txb1 tyb1 zza tx1 ty1 zza tx2 ty2 zza tx3 ty3 zza & tx4 ty4 zza tx5 ty5 zza txb2 tyb2 zza & b txb1 tyb1 zzb tx1 ty1 zzb tx2 ty2 zzb tx3 ty3 zzb & tx4 ty4 zzb tx5 ty5 zzb txb2 tyb2 zzb & reg 5 ; ; create inner surface tunnel a txb1i tyb1i zza tx1i ty1i zza tx2i ty2i zza tx3i ty3i zza & tx4i ty4i zza tx5i ty5i zza txb2i tyb2i zza & b txb1i tyb1i zzb tx1i ty1i zzb tx2i ty2i zzb tx3i ty3i zzb & tx4i ty4i zzb tx5i ty5i zzb txb2i tyb2i zzb & reg 7 ; ; --- NOTE: region inside inner surface is REG 7 ; region between surface (to be liner) is REG 5 ; save tun_a.sav ; ; --- joints --- 3 joints to form a wedge in the roof ; jset dd 270 dip 70 or 0,5.7 0 id 10 jset dd 40 dip 65 or 0,5.7 0 id 10 jset dd 130 dip 60 or 0,5.7 0 id 10 ; save tun_b.sav ; ; --- mesh generation --; rock blocks hide reg 5 7
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User’s Guide
gen ed 5 ; ; liner find reg 5 gen ed 2 ; find reg 7 gen ed 5 ; save tun_z.sav pl hold dip 70 dd 210 color mat ret
Figure 3.26 shows the resulting model configuration. The tunnel geometry parameters are defined in the FISH function tunz. The inner region of the tunnel is assigned region number 7, and the region corresponding to the liner is assigned region number 5. Note that the tunnel is created first and then the physical joint set is generated. Blocks are joined automatically with the TUNNEL commands. Zone generation is performed separately for the rock blocks, the liner blocks and the interior region of the tunnel.
3DEC (Version 3.00) 27-Aug-02
13:59
dip= 70.00 above dd = 210.00 center 0.000E+00 0.000E+00 0.000E+00 cut-pl. 0.000E+00 mag = 1.00 cycle 0
Y z
x
Itasca Consulting Group, Inc.
Figure 3.26 3DEC model of tunnel region Material properties are assigned to the rock blocks (mat 1), the concrete liner blocks (mat 5), the rock joints (jmat 1), the concrete-concrete joints (jmat 5) and the concrete-rock interface (jmat 6).
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The in-situ stress state and boundary conditions are applied assuming the tunnel is at a depth of 200 m and the ratio of horizontal to vertical stress is 0.5. Note that for a practical simulation, the boundaries are too close to the tunnel excavation and should be moved to a greater distance to minimize their influence on the model results — see Section 3.4.4.2. The commands to assign material properties and achieve the initial stress state are listed in Example 3.18. Example 3.18 Stability analysis of an underground excavation — initial equilibrium stress state rest tun_z.sav ; ; --- properties --; ; --- MAT 1 : rock --; density = 2700 kg/m3 = 0.0027e6 kg/m3 ; E=50 GPa, Poisson’s ratio=0.2 prop mat 1 dens 0.0027 k 27778 g 20833 ; ; --- MAT=5 : concrete liner --; density = 2400 kg/m3 = 0.0024e6 kg/m3 ; E=30 GPa, Poisson’s ratio=0.2 prop mat 5 dens 0.0025 k 16667 g 12500 ; ; --- JMAT=1 : rock joints --prop mat 1 kn 10000 ks 2000 fric 25 ; ; --- JMAT=5 : concrete-concrete joints (elastic) --prop mat 5 kn 30000 ks 12000 coh 1e6 tens 1e6 ; ; --- JMAT=6 : concrete-rock interface --prop mat 6 kn 10000 ks 2000 fric 0.001 ; ; --- assign material numbers --; initially all materials are rock change mat 1 change jmat 1 ; ; --- insitu stress state --; assume tunnel at 200 m depth ; vertical stress: syy=(0.0027*g)*(y-200) ; at y=0: syy=-5.4 ; y-gradient of syy: 0.027 ; (positive: less compression going up) ; horizontal sxx=szz=0.5*syy
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; insitu stress -2.7 -5.4 -2.7 0 0 0 & ygrad 0.0135 0.027 0.0135 0 0 0 ; ; gravity grav 0 -10 0 ; ; --- boundary conditions for insitu stress state --; top of model (y=10): syy=-0.027*190=-5.13 bound yr 9.9 10.1 stress 0 -5.13 0 0 0 0 ; bottom bound yr -10.1 -9.9 yvel 0 ; sides bound xr -10.1 -9.9 xvel 0 bound xr 9.9 10.1 xvel 0 bound zr -10.1 -9.9 zvel 0 bound zr 9.9 10.1 zvel 0 ; ; --- histories to monitor convergence --hist nc=1 unbal ; top of model hist xdis 0 10 0 ydis 0 10 0 zdis 0 10 0 ; save tun_c0.sav cycle 500 save tun_c.sav pl hold hist 2 3 4 ret
The maximum unbalanced force in the model and displacements at the top boundary are monitored to help make sure that an initial equilibrium stress state is reached within 1000 cycles. Figure 3.27 shows the x-, y- and z-displacement histories for the gridpoint (x = 0, y = 10, z = 0) at the top of the model.
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3DEC (Version 3.00)
(E-006) 0.4
HISTORY PLOT 27-Aug-02 14:07 cycle 500
0.2
Hist. no. 2 -9.059E-07 to 0.000E+00 0.0 Hist. no. 3 -1.517E-06 to -8.306E-08 Hist. no. 4 -1.335E-07 to 1.812E-07
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2.0
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(E-002)
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3.5
4.0
4.5 Itasca Consulting Group, Inc.
Figure 3.27 Displacement histories at top of model If the tunnel is excavated without support, a rock wedge detaches and falls from the roof. This is shown by running Example 3.19; the tunnel is excavated with the DELETE command, and the y-displacement at a location in the roof is monitored while the model is cycled. Figure 3.29 plots the y-displacement history and indicates that the position is moving downward. Figure 3.28 shows a close-up view of the detached wedge, with surrounding blocks hidden for better viewing. Example 3.19 Stability analysis of an underground excavation — unsupported tunnel rest tun_c.sav ; delete interior blocks remove reg 5 7 ; ; history point at tunnel roof reset disp time hist hist ydis 0 4 0 ; cycle 5000 ; save tun_x.sav pl hold hist 1 hide -.7 2.7 3.7 6 -10 -3 pl hold dip 90 dd 180 mag 2 ret
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User’s Guide
3DEC (Version 3.00)
(E-002) 0.0
HISTORY PLOT 27-Aug-02 14:10 cycle 5500 -0.4
Hist. no. 1 -2.227E-02 to -6.176E-05 VS Time
-0.8
-1.2
-1.6
-2.0
-2.4
-2.8 0.0
1.0
2.0
3.0 (E-001)
4.0
5.0
6.0
7.0 Itasca Consulting Group, Inc.
Figure 3.28 y-displacement history at tunnel roof
3DEC (Version 3.00) 27-Aug-02
14:10
dip= 90.00 above dd = 180.00 center 0.000E+00 0.000E+00 0.000E+00 cut-pl. 0.000E+00 mag = 2.00 cycle 5500
Itasca Consulting Group, Inc.
Figure 3.29 Close-up view of wedge in roof (surrounding blocks hidden)
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The effect of rock bolt support is evaluated first for local reinforcement elements (STRUCT axial) and then for fully-bonded cable elements (STRUCT cable). See Section 4 in Theory and Background for a detailed description of these two types of structural support. Example 3.20 lists the commands to excavate the tunnel and install the local reinforcement elements, and Example 3.21 lists those for cable element support. Note that we use the REMOVE command to excavate the tunnel this time. This has the same effect as the DELETE command, but now we can view the excavated region with the PLOT exc command. The reinforcement elements and cable elements are positioned in the same locations in the side walls and roof of the tunnel. Figure 3.30 shows the location of the cable elements around the tunnel excavation.
3DEC (Version 3.00) 27-Aug-02
14:07
dip= 80.00 above dd = 190.00 center 0.000E+00 0.000E+00 0.000E+00 cut-pl. 0.000E+00 mag = 2.00 cycle 500
Itasca Consulting Group, Inc.
Figure 3.30 Cable bolts positioned around tunnel excavation
Example 3.20 Stability analysis of an underground excavation — local reinforcement support rest tun_c.sav ; ; delete interior blocks remove region 7 ; delete liner blocks remove reg 5 ; --- install axial elements --struct axial -8 -2 -5 -3.9 -2 -5 struct axial -8 -2 0 -3.9 -2 0 struct axial -8 -2 5 -3.9 -2 5
prop 7 prop 7 prop 7
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struct axial -6.8 6.8 -5 -2.8 2.8 -5 prop 7 struct axial -6.8 6.8 0 -2.8 2.8 0 prop 7 struct axial -6.8 6.8 5 -2.8 2.8 5 prop 7 ; struct axial 8 -2 -5 3.9 -2 -5 prop 7 struct axial 8 -2 0 3.9 -2 0 prop 7 struct axial 8 -2 5 3.9 -2 5 prop 7 struct axial 6.8 6.8 -5 2.8 2.8 -5 prop 7 struct axial 6.8 6.8 0 2.8 2.8 0 prop 7 struct axial 6.8 6.8 5 2.8 2.8 5 prop 7 ; struct axial 0 4 -5 0 8 -5 prop 7 struct axial 0 4 0 0 8 0 prop 7 struct axial 0 4 5 0 8 5 prop 7 ; struct prop 7 rkax 250 rlen 0.10 rult 0.55 ; reset disp time hist ; history point at tunnel roof hist ydis 0 4 0 ; cy 2000 ; save tun_lr.sav pl hold hist 1 ret
Example 3.21 Stability analysis of an underground excavation — fully grouted cable support rest tun_c.sav ; ; delete interior blocks remove region 7 ; delete liner blocks remove reg 5 ; --- install cable elements --struct cable -8 -2 -5 -4.05 -2 -5 prop 8 seg 4 struct cable -8 -2 0 -4.05 -2 0 prop 8 seg 4 struct cable -8 -2 5 -4.05 -2 5 prop 8 seg 4 struct cable -6.8 6.8 -5 -2.85 2.85 -5 prop 8 seg 4 struct cable -6.8 6.8 0 -2.85 2.85 0 prop 8 seg 4 struct cable -6.8 6.8 5 -2.85 2.85 5 prop 8 seg 4 ; struct cable 8 -2 -5 4.05 -2 -5 prop 8 seg 4 struct cable 8 -2 0 4.05 -2 0 prop 8 seg 4
3DEC Version 3.0
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struct cable 8 -2 5 4.05 -2 5 prop 8 seg 4 struct cable 6.8 6.8 -5 2.85 2.85 -5 prop 8 struct cable 6.8 6.8 0 2.85 2.85 0 prop 8 struct cable 6.8 6.8 5 2.85 2.85 5 prop 8 ; struct cable 0 4.1 -5 0 8 -5 prop 8 seg 4 struct cable 0 4.1 0 0 8 0 prop 8 seg 4 struct cable 0 4.1 5 0 8 5 prop 8 seg 4 ; ; start with high SBOND struct prop 8 area 5e-4 e 100000 yield 0.55 kbond ; reset disp time hist ; history point at tunnel roof hist ydis 0 4 0 ; cycle 500 ; ; set real SBOND struct prop 8 sbond 0.8 ; cy 1500 ; save tun_cab.sav pl hold wire exc cable blue dip 80 dd 190 mag 2 pl hold hist 1 ret
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seg 4 seg 4 seg 4
15e4 sbond 1e6
The roof is stabilized for both types of reinforcement. The y-displacement history now indicates that the wedge movement stops at roughly 25 mm displacement for both the reinforcement elements and the cable elements (see Figures 3.31 and 3.32).
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3DEC (Version 3.00)
(E-003) 0.0
HISTORY PLOT 27-Aug-02 14:30 cycle 2500 -0.5
Hist. no. 1 -2.616E-03 to -6.176E-05 VS Time
-1.0
-1.5
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-2.5
-3.0
-3.5 0.0
0.4
0.8
1.2
1.6
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Itasca Consulting Group, Inc.
Figure 3.31 y-displacement history at tunnel roof — reinforcement element support
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HISTORY PLOT 27-Aug-02 14:28 cycle 2500 Hist. no. 1 -2.662E-03 to -6.071E-05
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2.0
2.4
2.8 Itasca Consulting Group, Inc.
Figure 3.32 y-displacement history at tunnel roof — cable support
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The axial forces that develop in the support are greatest in the roof elements. This is shown for both the reinforcement elements and the cable elements by the axial force plots in Figures 3.33 and 3.34.
3DEC (Version 3.00) 27-Aug-02
14:30
dip= 90.00 above dd = 180.00 center 0.000E+00 0.000E+00 0.000E+00 cut-pl. 0.000E+00 mag = 2.00 cycle 2500 interval = 5.000E-02 min max 2.500E-01 3.000E-01 2.000E-01 2.500E-01 1.500E-01 2.000E-01 1.000E-01 1.500E-01 5.000E-02 1.000E-01 0.000E+00 5.000E-02 Max Axial Rein Force= 2.861E-01
Itasca Consulting Group, Inc.
Figure 3.33 Axial forces in reinforcement elements
3DEC (Version 3.00) 27-Aug-02
14:28
dip= 90.00 above dd = 180.00 center 0.000E+00 0.000E+00 0.000E+00 cut-pl. 0.000E+00 mag = 2.00 cycle 2500
Max Axial Cable Force= 6.955E-02
Itasca Consulting Group, Inc.
Figure 3.34 Axial forces in cable elements
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User’s Guide
The model of a tunnel excavation and support sequence should simulate the change in stresses around the tunnel as the excavation advances, before the tunnel support is installed. This can be done in a 3DEC model by alternately excavating the tunnel in sections and installing support after each excavation section. This is the recommended approach to simulate support loading changes due to tunnel advancement. Alternatively, in this simplified model we simulate the effect of tunnel advancement by reducing the tractions at the tunnel periphery in increments and installing the liner before the tractions are completely removed. This demonstrates an approach for simulating a gradual excavation of a tunnel section. Example 3.22 shows the data file for this approach. Example 3.22 Stability analysis of an underground excavation — reduce tunnel tractions by 50% and install liner rest tun_c.sav ; delete interior blocks delete region 7 ; ; excavate liner blocks (not deleted) excavate reg 5 ; ; history point at tunnel roof hist xdis 0 4 0 ydis 0 4 0 zdis 0 4 0 ; ; simulate the removal of approximately 50% of insitu stress ; applying at liner-rock interface a stress state ; syy=-2.7 sxx=szz=-1.35 ; bound -4.1 -3.9 -4.1 0.1 -11 11 str -1.35 -2.7 -1.35 0 0 0 bound 3.9 4.1 -4.1 0.1 -11 11 str -1.35 -2.7 -1.35 0 0 0 bound -4.1 4.1 -4.1 -3.9 -11 11 str -1.35 -2.7 -1.35 0 0 0 ; note: need to include all faces on tunnel surface ; (inner radius must be a bit smaller than 4.0) bound yr -0.1 4.1 cyl 0 0 -11 0 0 11 3.5 4.1 & str -1.35 -2.7 -1.35 0 0 0 ; ; must again fix end-surfaces that were freed by BOU STRESS bou zr -10.1 -9.9 zvel 0 bou zr 9.9 10.1 zvel 0 ; ; check that sum of applied forces on tunnel surface is zero pr -5 5 -5 5 -11 11 bou for pause ; cycle 2000 ;
3DEC Version 3.0
PROBLEM SOLVING WITH 3DEC
save tun_l1.sav pause ; ; --- insert liner --; remove loads from tunnel surface ; bound -4.1 -3.9 -4.1 0.1 -11 11 xfree bound 3.9 4.1 -4.1 0.1 -11 11 xfree bound -4.1 4.1 -4.1 -3.9 -11 11 xfree bound yr -0.1 4.1 cyl 0 0 -11 0 0 11 ; ; must again fix end-surfaces bou zr -10.1 -9.9 zvel 0 bou zr 9.9 10.1 zvel 0 ; ; insert liner fill reg 5 mat 5 jmat 5 ; ; join liner blocks join reg 5 ; ; assign rock-liner interface material change rint 0 5 jmat 6 ; cy 2000 ; save tun_l2.sav hide seek reg 5 pl hold dip 80 dd 200 mag 2 pl hold hist 6 ret
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yfree zfree yfree zfree yfree zfree 3.5 4.1 xfree yfree zfree
The BOUND command is used to apply 50% of the in-situ stress state to the liner-rock interface, and the BOUND range covers all faces on the tunnel surface. The applied stresses at the tunnel surface should produce traction forces on the surface that sum to zero; this can be checked with the PRINT bound force command. The model is cycled to an equilibrium state with tunnel tractions reduced by 50%. Then, the tractions are removed completely, and the liner is installed (with the FILL region command). Figure 3.35 shows the liner blocks created for this model. The model is cycled to a new equilibrium state. The load that develops in the liner is due to the reduction of the tractions from 50% to zero. Note that the selection of a 50% reduction in tunnel tractions in this example is arbitrary and only for demonstration purposes. If it is necessary to simulate a gradual excavation, it may be necessary to reduce the tractions in smaller increments to minimize the effects of transient stress waves on the response of the model.
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User’s Guide
3DEC (Version 3.00) 27-Aug-02
14:45
dip= 80.00 above dd = 200.00 center 0.000E+00 0.000E+00 0.000E+00 cut-pl. 0.000E+00 mag = 2.00 cycle 4500
Itasca Consulting Group, Inc.
Figure 3.35 Thick concrete liner support — liner blocks The displacement of the roof is monitored in Figure 3.36. Roughly 1 mm of vertical displacement occurs when the tractions are reduced by 50% and an additional 1 mm displacement after the tunnel tractions are completely removed and the liner is installed.
3DEC (Version 3.00)
(E-003) 0.8
HISTORY PLOT 27-Aug-02 14:45 cycle 4500 0.4 Hist. no. 6 -2.113E-03 to 2.505E-06 VS Time
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4.0 Itasca Consulting Group, Inc.
Figure 3.36 y-displacement history at tunnel roof — tunnel liner added after tractions reduced by 50%
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If a more representative model of the liner behavior, including an elastic-plastic response, is required, then mixed-discretization zoning (see Section 3.3.2) should be used to define the liner with a minimum of five m-d zones across the liner thickness. The POLY prism command can be used to create liner blocks for the m-d zones. Example 3.23 presents a data file to create the liner with m-d zoning. Example 3.23 Stability analysis of an underground excavation — liner with m-d zoning rest tun_c0.sav ; delete interior blocks delete reg 5 7 ; --------------------------------------------------------------------; --- insert support with POLY prism commands --; poly prism a txb1 tyb1 zza tx1 ty1 zza & tx1i ty1i zza txb1i tyb1i zza & b txb1 tyb1 zzb tx1 ty1 zzb & tx1i ty1i zzb txb1i tyb1i zzb & reg 8 ; poly prism a tx1 ty1 zza tx2 ty2 zza & tx2i ty2i zza tx1i ty1i zza & b tx1 ty1 zzb tx2 ty2 zzb & tx2i ty2i zzb tx1i ty1i zzb & reg 8 ; poly prism a tx2 ty2 zza tx3 ty3 zza & tx3i ty3i zza tx2i ty2i zza & b tx2 ty2 zzb tx3 ty3 zzb & tx3i ty3i zzb tx2i ty2i zzb & reg 8 ; poly prism a tx3 ty3 zza tx4 ty4 zza & tx4i ty4i zza tx3i ty3i zza & b tx3 ty3 zzb tx4 ty4 zzb & tx4i ty4i zzb tx3i ty3i zzb & reg 8 ; poly prism a tx4 ty4 zza tx5 ty5 zza & tx5i ty5i zza tx4i ty4i zza & b tx4 ty4 zzb tx5 ty5 zzb & tx5i ty5i zzb tx4i ty4i zzb & reg 8 ; poly prism a tx5 ty5 zza txb2 tyb2 zza & txb2i tyb2i zza tx5i ty5i zza &
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User’s Guide
b tx5 ty5 zzb txb2 tyb2 zzb & txb2i tyb2i zzb tx5i ty5i zzb & reg 8 ; poly prism a txb2 tyb2 zza txb1 tyb1 zza & txb1i tyb1i zza txb2i tyb2i zza & b txb2 tyb2 zzb txb1 tyb1 zzb & txb1i tyb1i zzb txb2i tyb2i zzb & reg 8 ; gen quad ndiv 5 5 10 rmul 1.0 ; change reg 8 mat 5 change rint 8 8 jmat 5 change rint 0 8 jmat 6 ; ; --- MAT=5 : concrete liner --; density = 2400 kg/m3 = 0.0024e6 kg/m3 ; E=30 GPa, Poisson’s ratio=0.2 prop mat 5 dens 0.0025 k 16667 g 12500 ; ; --- JMAT=5 : concrete-concrete joints (elastic) --prop mat 5 kn 30000 ks 12000 coh 1e6 tens 1e6 ; ; --- JMAT=6 : concrete-rock interface --prop mat 6 kn 10000 ks 2000 fric 35 ; ; --------------------------------------------------------------------; ; history point at tunnel roof reset disp time hist hist ydis 0 4 0 ; cycle 2000 ; save tun_lin3.sav hide seek reg 8 pl hold dip 75 dd 188 color reg pl hold wire zol pl hold hist 1 pl dip 90 dd 180 x cent -1 2 2 mag 8 sscale 10 princ ccomp ret
Note that the prism-shaped blocks must be created before cycling is initiated. In this example, we delete the blocks in region 5 and insert prism-shaped blocks for the liner (defined now as region
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8). The m-d zoning is created with the GEN quad command and only elastic behavior is assigned to the liner material. If we wish to evaluate the elastic-plastic response, the bilinear material model (CHANGE cons 6 with the ubiquitous joint behavior suppressed) can be assigned to the liner material. The liner supports the entire load in this example. (We could also reduce the tractions as before in Example 3.22.) Figure 3.37 illustrates the liner blocks for this case, and Figure 3.38 shows the m-d zoning within the liner.
3DEC (Version 3.00) 27-Aug-02
14:52
dip= 75.00 above dd = 188.00 center 0.000E+00 0.000E+00 0.000E+00 cut-pl. 0.000E+00 mag = 2.00 cycle 2000
Itasca Consulting Group, Inc.
Figure 3.37 Thick concrete liner support — prism-shaped liner blocks
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User’s Guide
3DEC (Version 3.00) 27-Aug-02
14:52
dip= 75.00 above dd = 188.00 center 0.000E+00 0.000E+00 0.000E+00 cut-pl. 0.000E+00 mag = 2.00 cycle 2000
Y z
x
Itasca Consulting Group, Inc.
Figure 3.38 Thick concrete liner support — mixed-discretization zoning in liner blocks
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Figure 3.39 shows the plot of the y-displacement in the roof for this case. Approximately 1.6 mm displacement occurs when the liner supports the tunnel. The stresses in the liner are plotted in Figure 3.40.
3DEC (Version 3.00)
(E-003) 0.0
HISTORY PLOT 27-Aug-02 14:52 cycle 2000 -0.2 Hist. no. 1 -1.743E-03 to -3.421E-06 -0.4
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Figure 3.39 y-displacement history at tunnel roof — support by prism-shaped liner blocks
3DEC (Version 3.00) Cross section plot: 27-Aug-02 14:52 geometric scale 0
2E 00 vector scale
0
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dip= 90.00 above dd = 180.00 center -1.000E+00 2.000E+00 2.000E+00 cut-pl. 0.000E+00 mag = 8.00 cycle 2000
tension compression Max compress. stress -5.280E+00
Itasca Consulting Group, Inc.
Figure 3.40 Principal stress distribution in top section of liner
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User’s Guide
3.7 Choice of Constitutive Model This section provides an overview of the block and joint constitutive models in 3DEC as well as recommendations concerning when to use a model. Section 2 in Theory and Background presents background information on the block constitutive model formulations. Also see userdefined models and extended material models in Optional Features. The joint models are described in Sections 1.2.2.3 and 3 in Theory and Background. 3.7.1 Deformable-Block Material Models There are five built-in block material models in 3DEC: (1) null (EXCAVATE command); (2) elastic, isotropic (CHANGE cons = 1); (3) elastic, anisotropic (CHANGE cons = 3); (4) Mohr-Coulomb plasticity (CHANGE cons = 2); and (5) bilinear strain-hardening/softening, ubiquitous joint (CHANGE cons = 6). Note that the null model is assigned with the EXCAVATE command. The other four models are assigned with the CHANGE cons command. Model properties are then specified for the non-null models with the PROPERTY mat command for material property numbers, and the property numbers are assigned to the blocks with the CHANGE mat command. Each block model is designed to represent a specific type of constitutive behavior commonly associated with geologic materials. The null model is used to represent material which is removed from the model. The elastic, isotropic model is valid for homogeneous, isotropic, continuous materials which exhibit linear stress-strain behavior. The elastic, anisotropic model is appropriate for elastic materials that exhibit a well-defined elastic anisotropy. The Mohr-Coulomb plasticity model is used for materials that yield when subjected to shear loading, but the yield stress depends on the major and minor principal stresses only; the intermediate principal stress has no effect on yield. The bilinear strain-softening, ubiquitous joint model combines a strain-softening MohrCoulomb model for the matrix material with a strain-softening ubiquitous joint model to represent a well-defined strength anisotropy. This model includes a bilinear failure envelope for both the matrix and the ubiquitous joints. The material models in 3DEC are primarily intended for applications related to geotechnical engineering — e.g., underground construction, mining, slope stability, foundations, earth and rock-fill dams. When selecting a constitutive model for a particular engineering analysis, the following two considerations should be kept in mind: 1. What are the known characteristics of the material being modeled? 2. What is the intended application of the model analysis?
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Table 3.2 presents a summary of the 3DEC block models along with examples of representative materials and possible applications of the models. The elastic block model is generally applicable for cases in which slip along discontinuities is the predominant mechanism for failure. The MohrCoulomb model should be used when stress levels are such that failure of intact material is expected. Mohr-Coulomb parameters for cohesion and friction angle are usually available more often than other properties for geo-engineering materials.
Table 3.2
3DEC block constitutive models
Model
Representative Material
Example Application
null
void
holes, excavations, regions in which material will be added at later stage
elastic
homogeneous, isotropic continuum; linear stress-strain behavior
manufactured materials (e.g., steel) loaded below strength limit; factor-ofsafety calculation
Drucker-Prager plasticity
limited application; soft clays with low friction
common model for comparison to implicit finite-element programs
Mohr-Coulomb plasticity
loose and cemented granular materials; soils, rock, concrete
general soil or rock mechanics (e.g., slope stability and underground excavation)
strain-hardening / softening MohrCoulomb with ubiquitous-joint
granular materials that exhibit nonlinear material hardening or softening and/or thinly laminated material exhibiting strength anisotropy (e.g., slate)
studies in post-failure (e.g., progressive collapse, yielding pillar, caving) and excavation in closely bedded strata
The bilinear strain-softening, ubiquitous-joint model is actually a variation of the Mohr-Coulomb model. This model will produce identical results for shear failure to that for Mohr-Coulomb if the additional material parameters are set to high values. The only difference between the Mohr-Coulomb model and the bilinear model is the tensile failure criterion: In the Mohr-Coulomb model, a tension cutoff is specified. When any principal stress component in a zone exceeds the tension cutoff, all principal stress components in the zone are set to zero. In the bilinear model, tensile failure is defined by a tensile strength limit, and postfailure is governed by an associated plasticity flow rule. The value assigned for the tensile strength remains constant when tensile failure occurs. Tensile softening can be controlled with the strain-hardening/softening component of the bilinear model.
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User’s Guide
By default, the tensile strength is zero in both the Mohr-Coulomb and bilinear models. A comparison of the two tensile failure conditions is given in Example 3.25. The Mohr-Coulomb model is more computationally efficient than the bilinear model; the bilinear model requires increased memory and additional time for calculation. For example, plastic strain is not calculated directly in the Mohr-Coulomb model. If plastic strain is required, the bilinear model must be used. This model is primarily intended for applications in which the post-failure response is important — e.g., yielding pillars, caving or backfilling studies. 3.7.2 Joint Material Models There are two built-in models available to represent the material behavior of discontinuities: (1) joint area contact — Coulomb slip (CHANGE jcons = 1); and (2) continuously yielding (CHANGE jcons = 3). The joint models are assigned to one or more contacts by using the CHANGE jcons command. Joint model properties are then specified with the PROPERTY jmat command for material property numbers, and the property numbers are assigned to the contacts with the CHANGE jmat command. The joint constitutive models are designed to be representative of the physical response of rock joints. The joint area contact model is intended for closely packed blocks with area contact. The model provides a linear representation of joint stiffness and yield limit and is based upon elastic stiffness, frictional, cohesive and tensile strength properties and dilation characteristics common to rock joints. The model simulates displacement-weakening of the joint by loss of cohesive and tensile strength at the onset of shear or tensile failure. (A variation of the area contact model is also available (CHANGE jcons = 2) in which the cohesion and tensile strength are maintained following failure.) The continuously yielding joint model is a more complex model that simulates continuous weakening behavior as a function of accumulated plastic-shear displacement. Table 3.3 summarizes the 3DEC joint models and presents examples of representative materials and possible applications. The area contact Coulomb slip model is most applicable for general engineering studies. Coulomb friction and cohesion properties are usually available more often than other joint properties.
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Table 3.3
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3DEC joint constitutive models
Model
Representative Material
Example Application
area contact
joints, faults, bedding planes in rock
general rock mechanics (e.g., underground excavation)
continuously yielding
rock joints displaying progressive damage and hysteretic behavior
cyclic loading and load reversal with predominant hysteretic loop; dynamic analysis
The continuously yielding joint model is an empirical expression that requires more detailed knowledge of the joint behavior. The properties for the continuously yielding model are derived from laboratory test results relating joint shear stress to shear and normal displacement. It is always recommended that initial studies be based upon the Coulomb slip model first in order to develop a fundamental understanding of joint response before applying a more complex joint model. This is discussed further in the following section. A demonstration of the response of the continuously yielding model and the required properties are provided in Section 3 in Theory and Background. 3.7.3 Selection of an Appropriate Model A problem analysis should always start with simple block and joint material models; in most cases, an elastic block model (cons = 1) and a joint area contact Coulomb slip model (jcons = 1) should be used first. The elastic block model only requires three material parameters, mass density, bulk modulus and shear modulus (see Section 3.8.1.2). The Coulomb slip model requires six parameters: normal and shear stiffness, friction angle, cohesion, tensile strength and dilation angle. Estimates and references for these properties are given in Section 3.8.2. These material models provide a simple perspective of stress-deformation behavior in the 3DEC model; the results of these analyses can help the user assess if a more complex (or simpler) material model is needed to describe the block or joint behavior. For example, if the stresses and deformations in the blocks are low compared to the joint movements, then a simpler, rigid block model may be sufficient. It is often helpful to run simple tests of the selected material model before using it to solve the full-scale, boundary-value problem. This can provide insight into the expected response of the model compared to the known response of the physical material. The following example illustrates the use of a simple test model. The problem application is the analysis of joint slip around an underground excavation. A simple model is created to evaluate the adequacy of the Coulomb slip model to represent the response of a joint subjected to shear loading. The test is a simulation of a direct shear test, which consists of a single horizontal joint that is first subjected to a normal confining stress and then to a unidirectional shear displacement. Figure 3.41 shows the model; the joint is defined by one contact that is composed of 10 sub-contacts.
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User’s Guide
3DEC (Version 3.00) 28-Aug-02
9:28
dip= 70.00 above dd = 200.00 center 0.000E+00 -7.451E-09 -7.451E-09 cut-pl. 0.000E+00 mag = 1.00 cycle 15100
Max Velocity = 5.003E-03
Itasca Consulting Group, Inc.
Figure 3.41 Direct shear test model First, a normal stress of 20 MPa is applied that is representative of the confining stress acting on the joint. A horizontal velocity is then applied to the top block to produce a shear displacement that is also representative of the displacements expected in the problem application. For demonstration purposes, we only apply a small shear displacement of less than 1 mm to this model. The average normal and shear stresses and normal and shear displacements along the joint are measured with a FISH function (av str). With this information we can determine the peak and residual shear strengths and dilation that are produced with the different models. The data file for this test using the Coulomb slip model is contained in Example 3.24. Example 3.24 Direct shear test with Coulomb slip model new ; Coulomb slip joint model ; direct shear test ; poly brick -0.15,0.15 -0.10,0 -0.10,0.10 gen edge 1.0 poly brick -0.10,0.10 0,0.10 -0.10,0.10 gen edge 0.2 ; prop mat=1 d=0.0026 k=4000 g=3000 ;
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; Coulomb slip model change jcons=1 prop jmat=1 kn=100000 ks=100000 fric=30.0 dil 15 zdil 6e-4 ; prop jmat=1 coh 10 ; fix -0.15,0.15 -0.1,0 -0.10,0.10 ; ; normal load bound -0.10,0.10 0.09 0.11 -0.10,0.10 str 0 -50 0 0 0 0 ; step 100 ; ; function to calculate average joint stresses ; and average joint displacements ; def av_str whilestepping sstav = 0.0 nstav = 0.0 njdisp = 0.0 sjdisp = 0.0 ncon = 0 jarea = 0.04 ic = contact_head loop while ic # 0 icsub = c_cx(ic) loop while icsub # 0 ncon = ncon + 1 sstav = sstav + cx_xsforce(icsub) nstav = nstav + cx_nforce(icsub) njdisp = njdisp + cx_ndis(icsub) sjdisp = sjdisp + cx_xsdis(icsub) icsub = cx_next(icsub) endloop if ncon # 0 sstav = sstav / jarea nstav = nstav / jarea njdisp = njdisp / ncon sjdisp = - sjdisp / ncon endif ic = c_next{ic) endloop end ; reset jdisp ; shear load
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bound -0.11 0.11 -0.01 0.11 -0.11 0.11 bound -0.16 0.16 -0.11 0.11 -0.11 0.11 ; hist unbal ncyc 5 hist sstav nstav njdisp sjdisp ; hist sdis -1 0 -1 ndis -1 0 -1 hist sdis -1 0 1 ndis -1 0 1 hist sdis 0 0 0 ndis 0 0 0 hist sstr -1 0 -1 nstr -1 0 -1 hist sstr -1 0 1 nstr -1 0 1 hist sstr 0 0 0 nstr 0 0 0 hist sfor -1 0 -1 nfor -1 0 -1 ; cyc 15000 plot hold hist 2 vs 5 plot hold hist 4 vs 5 save cs_1.sav return
User’s Guide
xvel=0.005 zvel=0.0
The average shear stress versus shear displacement along the joint is plotted in Figure 3.42, and the average normal displacement versus shear displacement is plotted in Figure 3.43. These plots indicate that joint slip occurs for the prescribed model properties and conditions. The loading slope in Figure 3.42 is linear until a peak shear strength of approximately 2.9 MPa is reached. As indicated in Figure 3.43, the joint begins to dilate when the joint fails in shear, at roughly 0.3 mm shear displacement. Dilation occurs until the limiting shear displacement (zdilation = 0.6 mm) is reached for zero dilation. The maximum average dilation is approximately 0.077 mm.
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3DEC (Version 3.00)
(E+001) 3.5
HISTORY PLOT 28-Aug-02 9:28 cycle 15100 Hist. no. 2 2.948E-02 to 2.886E+01
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-0.5 Figure 3.42 Average shear stress versus shear displacement — Coulomb slip model
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(E-005) 9.0
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1.10 Itasca Consulting Group, Inc.
Figure 3.43 Average normal displacement versus shear displacement — Coulomb slip model
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User’s Guide
As these results indicate, the Coulomb slip model (jcons = 1) only defines a limiting shear strength value for the joint. The dilation that occurs after the joint begins to slip is approximated as a linear function of the dilation angle with a dilation limit that is a function of the shear displacement. (These functions are described in Section 1.2.2.3 in Theory and Background.) Other modifications to the joint behavior are also available for the Coulomb slip model. For example, a displacement-weakening behavior can be approximated by including a joint cohesion of 10 MPa (PROP jmat 1 coh = 10). At the onset of failure, the cohesion is set to zero. The results shown in Figure 3.44 illustrate the peak and residual strengths that develop when the effect of cohesion is included. Note that the drop in strength occurs abruptly. The maximum dilation, as shown in Figure 3.36, is lower than the previous case without cohesion for the same limiting shear displacement, because more shear displacement occurs before the joint fails initially. (Compare Figure 3.45 to Figure 3.43.)
3DEC (Version 3.00)
(E+001) 4.0
HISTORY PLOT 28-Aug-02 9:39 cycle 15100 3.5
Hist. no. 2 2.950E-02 to 3.257E+01 VS Hist. no. 5 2.629E-07 to 9.857E-04
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Figure 3.44 Average shear stress versus shear displacement — Coulomb slip model with peak and residual strength Note that if no weakening behavior is associated with a joint that has a cohesive strength, the command CHANGE jcons = 2 should be given in place of CHANGE jcons = 1. In this case there will be no change in the cohesion when the joint fails.
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3DEC (Version 3.00)
(E-005) 5.5
HISTORY PLOT 28-Aug-02 9:39 cycle 15100
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1.00
1.10 Itasca Consulting Group, Inc.
Figure 3.45 Average normal displacement versus shear displacement — Coulomb slip model with peak and residual strength The displacement-weakening behavior is produced automatically with the continuously yielding joint model (jcons = 3). This model simulates the progressive damage of the joint under shear. For details and an example direct shear test with the continuously yielding model, see Section 3 in Theory and Background. The material properties for the Coulomb slip model in these examples were selected to produce roughly the same response for joint shear strength and dilation for joints subjected to unidirectional shearing. For an actual application, properties should be selected (and adjusted as necessary) to simulate the response of the joints under the expected loading conditions. In most cases, the Coulomb model parameters are relatively easy to estimate (see Section 3.8.2), and the simple modifications that are available with the Coulomb model may be sufficient to approximate the joint behavior. For other joint models, such as the continuously yielding model, the determination of properties is more involved. In order to use the continuously yielding model it is necessary to run a series of joint shear tests to best-fit the model properties to physical test results. It is recommended that simple shear tests always be performed, regardless of the joint model selected, to ensure that the joint behaves as expected under the anticipated problem conditions. If it is necessary to simulate a complicated joint response, then a more complex joint model may be required. However, before going to a more complex model, it is usually helpful to apply a simple model first to establish a basis for evaluating the influence of the more complicated joint behavior.
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3.8 Material Properties 3DEC requires material properties for both the intact blocks and the discontinuities. This section provides an overview of typical properties used to represent the behavior of jointed rock and presents guidelines for selecting the appropriate properties for a given model. There are also special considerations such as the definition of post-failure properties and the extrapolation of laboratory-measured properties to the field scale. These topics are also discussed. The selection of properties is often the most difficult element in the generation of a model because of the high uncertainty in the property data base. It should be kept in mind when performing an analysis, especially in geomechanics, that the problem will always involve a data-limited system; the field data will never be known completely. However, with the appropriate selection of properties based upon the available data base, important insight to the physical problem can still be achieved. This approach to modeling is discussed further in Section 3.11. 3.8.1 Block Properties Properties assigned to blocks are generally derived from laboratory testing programs. The following four sections describe intrinsic (laboratory-scale) properties and list common values for various rocks. 3.8.1.1 Mass Density The mass density is required for every non-void material in a 3DEC model. This property has units of mass divided by volume and does not include the gravitational acceleration. In many cases, the unit weight of a material is prescribed. If the unit weight is given with units of force divided by volume, then this value must be divided by the gravitational acceleration before entering as 3DEC input for density. 3.8.1.2 Intrinsic Deformability Properties All material models for deformable blocks in 3DEC assume an isotropic material behavior in the elastic range described by two elastic constants, bulk modulus, K, and shear modulus, G. The elastic constants, K and G, are used in 3DEC rather than Young’s modulus, E, and Poisson’s ratio, ν, because it is believed that bulk and shear moduli correspond to more fundamental aspects of material behavior than do Young’s modulus and Poisson’s ratio. (See note 8 in Section 3.9 for justification for using (K,G) rather than (E,ν).) The equations to convert from (E,ν) to (K,G) are
K=
G=
3DEC Version 3.0
E 3(1 − 2ν) E 2(1 + ν)
(3.15)
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Eq. (3.15) should not be used blindly when ν is near 0.5, since the computed value of K will be unrealistically high and convergence to the solution will be very slow. It is better to fix the value of K at its known physical value (estimated from an isotropic compaction test or from the p-wave speed), and then compute G from K and ν. Some typical values for elastic constants are summarized in Table 3.4 for selected rocks.
Table 3.4 Selected elastic constants (laboratory-scale) for rocks (adapted from Goodman 1980) E (GPa)
ν
K (GPa)
G (GPa)
Berea sandstone
19.3
0.38
26.8
7.0
Hackensack siltstone
26.3
0.22
15.6
10.8
Bedford limestone
28.5
0.29
22.6
11.1
Micaceous shale
11.1
0.29
8.8
4.3
Cherokee marble
55.8
0.25
37.2
22.3
Nevada Test Site granite
73.8
0.22
43.9
30.2
3.8.1.3 Intrinsic Strength Properties The basic criterion for block material failure in 3DEC is the Mohr-Coulomb relation, which is a linear failure surface corresponding to shear failure: fs = σ1 − σ3 Nφ + 2c Nφ where Nφ σ1 σ3 φ c
(3.16)
= (1 + sin φ)/(1 − sin φ); = major principal stress (compressive stress is negative); = minor principal stress; = friction angle; and = cohesion.
Shear yield is detected if fs < 0. The two strength constants, φ and c, are conventionally derived from laboratory triaxial tests.
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User’s Guide
The Mohr-Coulomb criterion loses its physical validity when the normal stress becomes tensile but, for simplicity, the surface is extended into the tensile region to the point at which σ3 equals the uniaxial tensile strength, σ t . The minor principal stress can never exceed the tensile strength — i.e., ft = σ3 − σ t
(3.17)
Tensile yield is detected if ft > 0. Tensile strength for rock and concrete is usually derived from a Brazilian (or indirect tensile) test. Note that the tensile strength cannot exceed the value of σ3 corresponding to the apex limit for the Mohr-Coulomb relation. This maximum value is given by t = σmax
c tan φ
(3.18)
Typical values of cohesion, friction angle and tensile strength for a representative set of rock specimens are listed in Table 3.5.
Table 3.5 Selected strength properties (laboratory-scale) for rocks (adapted from Goodman 1980) friction angle (degrees)
cohesion (MPa)
tensile strength (MPa)
Berea sandstone
27.8
27.2
1.17
Repetto siltstone
32.1
34.7
—
Muddy shale
14.4
38.4
—
Sioux quartzite
48.0
70.6
—
Indiana limestone
42.0
6.72
1.58
Stone Mountain granite
51.0
55.1
—
Nevada Test Site basalt
31.0
66.2
13.1
The ubiquitous-joint component of the bilinear model also requires strength properties for the planes of weakness. Joint properties are discussed in Section 3.8.2, below. The properties for joint cohesion and friction angle also apply for the ubiquitous-joint model.
3DEC Version 3.0
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3.8.1.4 Post-Failure Properties In many instances, particularly in mining engineering, the response of a material after the onset of failure is an important factor in the engineering design. Consequently, the post-failure behavior must be simulated in the material model. In 3DEC, this is accomplished with properties which define three types of post-failure response: (1) shear dilatancy; (2) shear hardening/softening; (3) volumetric hardening/softening; and (4) tensile softening. These properties are only activated after the onset of failure, as defined by the Mohr-Coulomb relation. Shear dilatancy is assigned for the Mohr-Coulomb and bilinear strain-hardening/softening, ubiquitous joint model. Hardening/softening parameters are assigned for the bilinear model. Shear Dilatancy — Shear dilatancy, or dilatancy, is the change in volume that occurs with shear distortion of a material. Dilatancy is characterized by a dilation angle, ψ, which is related to the ratio of plastic volume change to plastic shear strain. This angle can be specified in the block plasticity models in 3DEC. The dilation angle is typically determined from triaxial tests or shear box tests. For example, the idealized relation for dilatancy, based upon the Mohr-Coulomb failure surface, is depicted for a triaxial test in Figure 3.46. The dilation angle is found from the plot of volumetric strain versus axial strain. Note that the initial slope for this plot corresponds to the elastic regime, while the slope used to measure the dilation angle corresponds to the plastic regime. |s1 -
s3| s1
1
2 c cos
E
elastic
f
- (s1 -
s3)
sin
f
e1
plastic
s2 = s3 ev s3
atan (1-2u)
atan
2 sin
y
1 - sin
y
e1
Figure 3.46 Idealized relation for dilation angle, ψ, from triaxial test results (Vermeer and de Borst 1984)
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User’s Guide
For soils, rocks, and concrete, the dilation angle is generally significantly smaller than the friction angle of the material. Vermeer and de Borst (1984) report the following typical values for ψ: dense sand loose sand normally consolidated clay granulated and intact marble concrete
15◦ < 10◦ 0◦ 12◦ − 20◦ 12◦
Vermeer and de Borst observe that values for the dilation angle are approximately between 0◦ and 20◦ , whether the material is soil, rock, or concrete. The default value for dilation angle is zero for all the constitutive models in 3DEC. Dilation angle can also be prescribed for the joints in the ubiquitous-joint component of the bilinear model. This property is typically determined from direct shear tests, and common values can be found in the references discussed in Section 3.8.2. Shear Hardening/Softening — The initiation of material hardening or softening is a gradual process once plastic yield begins. At failure, deformation becomes more and more inelastic as a result of micro-cracking in concrete and rock and particle sliding in soil. This also leads to degradation of strength in these materials and the initiation of shear bands. These phenomena, related to localization, are discussed further in Section 3.11. In 3DEC, shear hardening and softening are simulated by making Mohr-Coulomb properties (cohesion and friction, along with dilation) functions of plastic strain (see Section 2.3.5 in Theory and Background). These functions are accessed from the bilinear model, and can be specified by using the TABLE command. Hardening and softening parameters must be calibrated for each specific analysis with values that are generally back-calculated from results of laboratory triaxial tests. This is usually an iterative process. Investigators have developed expressions for hardening and softening; for example, Vermeer and de Borst (1984) propose the frictional hardening relation
sin φm
√ ep ef = 2 sin φ ep + ef
for ep ≤ ef (3.19)
sin φm = sin φ where φ φm ep ef
= ultimate friction angle; = mobilized friction angle; = plastic strain; and = hardening constant.
3DEC Version 3.0
for ep > ef
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Numerical testing conditions can influence the model response for shear hardening/softening behavior. The rate of loading can introduce inertial effects; this can be controlled by monitoring the unbalanced force and reducing the loading rate accordingly. A FISH function can be used to control the loading rate automatically. The results are also mesh-dependent; thus, it is important to evaluate the model behavior for differing zone sizes and mesh orientations whenever performing an analysis involving shear hardening or softening. Tensile Softening — At the initiation of tensile failure, the tensile strength of a material will generally drop to zero. In the Mohr-Coulomb model the tensile strength is set to zero when tensile failure occurs in a zone (instantaneous softening). The rate at which the tensile strength drops, or tensile softening occurs, can also be controlled by the plastic tensile strain in 3DEC. This function is accessed from the bilinear model, and can be specified by using the TABLE command. A simple tension test (Example 3.25) illustrates brittle tensile failure, as built into the Mohr-Coulomb model. The model is a tension test on a cubic block composed of Mohr-Coulomb material. The ends of the sample are pulled apart at a constant velocity. The test is performed with both the cons 2 and the cons 6 block models. Example 3.25 Tension test on tensile-softening material new poly brick 0 1 0 1 0 1 gen quad ndiv 1 1 1 rmul 1 ; Mohr-Coulomb (cons = 2) model change cons 2 prop mat=1 d=2500 k=1.19e10 g=1.1e10 bcoh 2.72e5 phi 44 bten 2e5 ; bilinear (cons = 6) model ; change cons 6 ; prop mat 1 dens 2500 k=1.19e10 g=1.1e10 bcoh 2.72e5 phi 44 bten 2e5 ; prop mat 1 jcubs 1e20 jtubs 1e20 jfubs 44 ; prop mat 1 ttab 1 ; table 1 0 2e5 9e-6 0 ; bound 0,1 -0.1,0.1 0,1 yvel -1e-5 bound 0,1 0.9,1.1 0,1 yvel 1e-5 def ax_str str = 0.0 ib = block_head ig = b_gp(ib) nx = 0 ny = 0 xbpos = 0.0 xbdis = 0.0 xtpos = 0.0 xtdis = 0.0 ybpos = 0.0
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User’s Guide
ybdis = 0.0 ytpos = 0.0 ytdis = 0.0 loop while ig # 0 if gp_y(ig) > ytop then str = str - gp_yforce{ig) ytpos = ytpos + gp_y(ig) ytdis = ytdis + gp_ydis(ig) end_if if gp_y(ig) < ybot then ny = ny + 1 ybpos = ybpos + gp_y(ig) ybdis = ybdis + gp_ydis(ig) end_if if gp_x(ig) < xbot then nx = nx + 1 xbpos = xbpos + gp_x(ig) xbdis = xbdis + gp_xdis(ig) end_if if gp_x(ig) > xtop then xtpos = xtpos + gp_x(ig) xtdis = xtdis + gp_xdis(ig) end_if ig = gp_next(ig) end_loop ax_str = str / area xbdis = xbdis / nx xbpos = xbpos / nx xtdis = xtdis / nx xtpos = xtpos / nx ybdis = ybdis / ny ybpos = ybpos / ny ytdis = ytdis / ny ytpos = ytpos / ny ex_str = (xtdis - xbdis) / (xtpos - xbpos) ey_str = (ytdis - ybdis) / (ytpos - ybpos) end set area = 1.0 ytop = 0.9 hist ax_str hist ex_str hist ey_str damp local step 20000 save mc.sav ; save bil.sav plot hold his 1 vs 3
3DEC Version 3.0
ybot = 0.1 xbot = 0.1 xtop = 0.9
PROBLEM SOLVING WITH 3DEC
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plot hold his 2 vs 3 ret
The plot of σyy stress versus yy-strain (Figure 3.47) shows that the average stress drops to zero for the Mohr-Coulomb model in cons 2. The stress will remain constant in the bilinear model without tensile softening. The brittleness of the tensile softening can be controlled by the plastic tensile-strain function. If Example 3.25 is repeated with cons 6 and a tensile softening table, an instantaneous softening response can be reduced, as shown in Figure 3.48.
3DEC (Version 3.00)
(E+005) 2.2
HISTORY PLOT 28-Aug-02 9:44 cycle 20000
2.0
Hist. no. 1 0.000E+00 to 1.999E+05 1.8 VS Hist. no. 3 1.002E-08 to 2.004E-05 1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0 0.0
0.4
0.8
1.2 (E-005)
-0.2
1.6
2.0
2.4
2.8 Itasca Consulting Group, Inc.
Figure 3.47 σyy stress versus yy-strain for tension test with cons 2 model
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User’s Guide
3DEC (Version 3.00)
(E+005) 2.2
HISTORY PLOT 28-Aug-02 9:54 cycle 20000
2.0
Hist. no. 1 0.000E+00 to 1.999E+05 1.8 VS Hist. no. 3 1.002E-08 to 2.004E-05 1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0 0.0
0.4
0.8
1.2 (E-005)
-0.2
1.6
2.0
2.4
2.8 Itasca Consulting Group, Inc.
Figure 3.48 σyy stress versus yy-strain for tension test with cons 6 model and tensile-softening table The average xx-strain and zz-strain across the model decreases until tensile failure initiates. With the cons 2 model, this strain is not controlled in the post-failure region; the model will continue to contract as indicated by the plot of xx-strain versus yy-strain in Figure 3.49. With the bilinear model, the strain is affected after the onset of tensile failure; the model expands in the x- and z-directions as tensile softening occurs, as indicated in Figure 3.50.
3DEC Version 3.0
PROBLEM SOLVING WITH 3DEC
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3DEC (Version 3.00)
(E-006) 0.0
HISTORY PLOT 28-Aug-02 9:44 cycle 20000 -0.5
Hist. no. 2 -2.935E-06 to -1.339E-09 VS Hist. no. 3 1.002E-08 to 2.004E-05
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5 0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
(E-005)
Itasca Consulting Group, Inc.
Figure 3.49 xx-strain versus yy-strain for tension test with cons 2 model
3DEC (Version 3.00)
(E-005) 2.4
HISTORY PLOT 28-Aug-02 9:54 cycle 20000 2.0
Hist. no. 2 -1.162E-06 to 1.956E-05 VS Hist. no. 3 1.002E-08 to 2.004E-05
1.6
1.2
0.8
0.4
0.0
-0.4
-0.8 0.0
0.4
0.8
1.2 (E-005)
1.6
2.0
2.4
2.8 Itasca Consulting Group, Inc.
Figure 3.50 xx-strain versus yy-strain for tension test with cons 6 model and tensile-softening table
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User’s Guide
Note that local damping (DAMP local) is used to minimize oscillations that can arise when the abrupt tensile failure occurs. Alternatively, if adaptive global damping is used by giving the DAMP auto command, oscillations are observed in the stress/strain plots. With adaptive global damping, the damping parameter is continually decreased as the model is stretched. When tensile failure occurs, the global damping parameter is low, and oscillations are produced that may affect the final solution state. With local damping, the amount of damping varies from gridpoint to gridpoint and is proportional to the unbalanced force. This damping minimizes the oscillations that are produced when the abrupt tensile failure occurs. (See Section 1.2.2.7 in Theory and Background for further discussion on damping.) The brittleness of the tensile softening can be controlled by the plastic tensile strain function, by using the bilinear model instead of the Mohr-Coulomb model. As with the shear-softening, the tensile-softening must be calibrated for each specific problem and mesh size, since the results will be mesh-dependent. 3.8.1.5 Extrapolation to Field-Scale Properties The material properties used in the 3DEC model should correspond as closely as possible to the actual values of the physical problem. Laboratory-measured properties generally should not be used directly in a 3DEC model for a full-scale problem. The presence of discontinuities in the model will account for a good portion of the scaling effect on properties. However, some adjustment of block properties will still probably be required to represent the influence of heterogeneities and micro-fractures, fissures and other small discontinuities on the rock mass response. Several empirical approaches have been proposed to derive field-scale properties. Some of the more-commonly-accepted methods are discussed. Deformability of a rock mass is generally defined by a modulus of deformation, Em . If the rock mass contains a set of relatively parallel, continuous joints with uniform spacing, the value for Em can be estimated by treating the rock mass as an equivalent transversely isotropic continuum. The relations in Section 3.8.2 can then be used to estimate Em in the direction normal to the joint set. Deformation moduli can also be estimated for cases involving more than one set of discontinuities. The references listed in Section 3.8.2 provide solutions for multiple joint sets. In practice, the rock mass structure is often much too irregular or sufficient data are not available to use the above approach. It is common to determine Em from a force-displacement curve obtained from an in-situ compression test. Such tests include plate bearing tests, flatjack tests, and dilatometer tests. Bieniawski (1978) developed an empirical relation for Em based upon field test results at sites throughout the world. The relation is based upon rock mass rating (RMR). For rocks with a rating higher than 55, the test data can be approximately fit to Em = 2(RMR) − 100 The units of Em are GPa.
3DEC Version 3.0
(3.20)
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For values of Em between 1 and 10 GPa, Serafim and Pereira (1983) found a better fit, given by Em = 10
RMR−10 40
(3.21)
References by Goodman (1980) and Brady and Brown (1985) provide additional discussion on these methods. The most-commonly-accepted approach to estimate rock mass strength is that proposed by Hoek and Brown (1980). They developed the empirical rock mass strength criterion σ1s = σ3 + (mσc σ3 + sσc2 )1/2
(3.22)
where σ1s = major principal stress at peak strength; σ3 = minor principal stress; m and s = constants that depend on the properties of the rock and the extent to which it has been broken before being subjected to failure stresses; and σc = uniaxial compressive strength of intact rock material. The unconfined compressive strength for a rock mass is given by qm = σc s 1/2
(3.23)
and the uniaxial tensile strength of a rock mass is σt =
1 σc [m − (m2 + 4s)1/2 ] 2
(3.24)
Table 3.6, from Hoek and Brown (1988), presents typical values for m and s for undisturbed and disturbed rock masses. It is possible to estimate Mohr-Coulomb friction angle and cohesion from the Hoek-Brown criterion (see, for example, Hoek 1990). For a given value of σ3 , a tangent to the function (Eq. (3.22)) will represent an equivalent MohrCoulomb yield criterion in the form σ1 = Nφ σ3 + σcM where Nφ =
1+sin φ 1−sin φ
(3.25)
= tan2 ( φ2 + 45◦ )
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User’s Guide
By substitution, σcM is: σcM
= σ1 − σ3 Nφ = σ3 +
σ3 σc m + σc2 s
− σ3 Nφ = σ3 (1 − Nφ ) +
σ3 σc m + σc2 s
σcM is the apparent uniaxial compressive strength of the rock mass for that value of σ3 . The tangent to the Eq. (3.22) is defined by: Nφ (σ3 ) =
∂σ1 σc m = 1 + ∂σ3 2 σ3 · σc m + sσc2
(3.26)
The cohesion (c) and friction angle (φ) can then be obtained from Nφ and σcM :
φ = 2 tan−1 σM c = c 2 Nφ
3DEC Version 3.0
Nφ − 90◦
(3.27) (3.28)
PROBLEM SOLVING WITH 3DEC
Typical values for Hoek-Brown rock-mass strength parameters (adapted from Hoek and Brown (1988))
Laboratory specimens free
m
7.00
10.00
15.00
17.00
from discontinuities
s
1.00
1.00
1.00
1.00
1.00
CSIR rating: RMR = 100
m
7.00
10.00
15.00
17.00
25.00
NGI rating: Q = 500
s
1.00
1.00
1.00
1.00
1.00
VERY GOOD QUALITY ROCK MASS Tightly interlocking undisturbed rock with unweathered joints at 1 to 3 m CSIR rating RMR = 85 NGI rating: Q = 100
m s m s
2.40 0.082 4.10 0.189
3.43 0.082 5.85 0.189
5.14 0.082 8.78 0.189
5.82 0.082 9.95 0.189
8.56 0.082 14.63 0.189
GOOD QUALITY ROCK MASS Fresh to slightly weathered rock, slightly disturbed with joints at 1 to 3 m CSIR rating: RMR = 65 NGI rating: Q = 10
m s m s
0.575 0.00293 2.006 0.0205
0.821 0.00293 2.865 0.0205
1.231 0.00293 4.298 0.0205
1.395 0.00293 4.871 0.0205
2.052 0.00293 7.163 0.0205
FAIR QUALITY ROCK MASS Several sets of moderately weathered joints spaced at 0.3 to 1 m CSIR rating: RMR = 44 NGI rating: Q = 1
m s m s
0.128 0.00009 0.947 0.00198
0.183 0.00009 1.353 0.00198
0.275 0.00009 2.03 0.00198
0.311 0.00009 2.301 0.00198
0.458 0.00009 3.383 0.00198
POOR QUALITY ROCK MASS Numerous weathered joints at 30-500 mm, some gouge; clean compacted waste rock CSIR rating: RMR = 23 NGI rating: Q = 0.1
m s m s
0.029 0.000003 0.447 0.00019
0.041 0.000003 0.639 0.00019
0.061 0.000003 0.959 0.00019
0.069 0.000003 1.087 0.00019
0.102 0.000003 1.598 0.00019
VERY POOR QUALITY ROCK MASS Numerous heavily weathered joints spaced <50 mm with gouge; waste rock with fines CSIR rating: RMR = 3 NGI rating: Q = 0.01
m s m s
0.007 0.0000001 0.219 0.00002
0.01 0.0000001 0.313 0.00002
0.015 0.0000001 0.469 0.00002
0.017 0.0000001 0.532 0.00002
0.025 0.0000001 0.782 0.00002
EMPIRICAL FAILURE CRITERION σ'1 = σ'3 ÷ √(mσcσ'3 ÷ sσ2c) σ'1 = major principal effective stress σ'3 = minor principaI effective stress σc = uniaxial compressive strength of intact rock, and m and s are empirical constants.
ARENACEOUS ROCKS WITH STRONG CRYSTALS AND POORLY DEVELOPED CRYSTAL CLEAVAGE — sandstone and quartzite
LITHIFIED ARGILLACEOUS ROCKS — mudstone, siltstone, shale and slate (normal to cleavage)
COARSE-GRAINED POLYMINERALLIC IGNEOUS & METAMORPHIC CRYSTALLINE ROCKS — amphibolite, gabbro gneiss, granite, norite, quartz-diorite
Undisturbed rock mass m and s values
CARBONATE ROCKS WITH WELLDEVELOPED CRYSTAL CLEAVAGE — dolomite, limestone and marble
Disturbed rock mass m and s values
FINE-GRAINED POLYMINERALLIC IGNEOUS CRYSTALLINE ROCKS — andesite, dolerite, diabase and thyolite
Table 3.6
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INTACT ROCK SAMPLES 25.00
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User’s Guide
3.8.2 Joint Properties Joint properties are conventionally derived from laboratory testing (e.g., triaxial and direct shear tests). These tests can produce physical properties for joint friction angle, cohesion, dilation angle, and tensile strength, as well as joint normal and shear stiffnesses. The joint cohesion and friction angle correspond to the parameters in the Coulomb strength criterion. Values for normal and shear stiffnesses for rock joints typically can range from roughly 10 to 100 MPa/m, for joints with soft clay in-filling, to over 100 GPa/m, for tight joints in granite and basalt. Published data on stiffness properties for rock joints are limited; summaries of data can be found in Kulhawy (1975), Rosso (1976), and Bandis et al. (1983). Approximate stiffness values can be back-calculated from information on the deformability and joint structure in the jointed rock mass and the deformability of the intact rock. If the jointed rock mass is assumed to have the same deformational response as an equivalent elastic continuum, then relations can be derived between jointed rock properties and equivalent continuum properties. For uniaxial loading of rock containing a single set of uniformly-spaced joints oriented normal to the direction of loading, the following relation applies: 1 1 1 = + Em Er kn s or kn =
Em Er s (Er − Em )
(3.29)
where Em = rock mass Young’s modulus; Er = intact rock Young’s modulus; kn
= joint normal stiffness; and
s
= joint spacing.
A similar expression can be derived for joint shear stiffness: ks = where Gm = rock mass shear modulus; Gr = intact rock shear modulus; and ks
= joint shear stiffness.
3DEC Version 3.0
Gm Gr s (Gr − Gm )
(3.30)
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The equivalent continuum assumption, when extended to three orthogonal joint sets, produces the following relations: Ei = Gij =
1 1 + Er si kni
−1
1 1 1 + + Gr si ksi sj ksj
−1
(i = 1, 2, 3)
(3.31)
(i , j = 1, 2, 3)
(3.32)
Several expressions have been derived for two- and three-dimensional characterizations and multiple joint sets. References for these derivations can be found in Singh (1973), Gerrard (1982(a) and (b)), and Fossum (1985). There is a limit to the maximum joint stiffnesses that are reasonable to use in a 3DEC model. If the physical normal and shear stiffnesses are less than ten times the equivalent stiffness of adjacent zones (see Eq. (3.33) in Section 3.9), then there is no problem in using physical values. If the ratio is more than ten, the solution time will be significantly longer than for the case in which the ratio is limited to ten, without much change in the behavior of the system. Serious consideration should be given to reducing supplied values of normal and shear stiffnesses to improve solution efficiency. There may also be problems with block interpenetration if the normal stiffness, kn , is very low. A rough estimate should be made of the joint normal displacement that would result from the application of typical stresses in the system (u = σ/kn ). This displacement should be small compared to a typical zone size. If it is greater than, say, 10% of an adjacent zone size, then either there is an error in one of the numbers or the stiffness should be increased. Published strength properties for joints are more readily available than stiffness properties. Summaries can be found, for example, in Jaeger and Cook (1969), Kulhawy (1975), and Barton (1976). Friction angles can vary from less than 10◦ for smooth joints in weak rock, such as tuff, to over 50◦ for rough joints in hard rock, such as granite. Joint cohesion can range from zero cohesion to values approaching the compressive strength of the surrounding rock. It is important to recognize that joint properties measured in the laboratory typically are not representative of those for real joints in the field. Scale dependence of joint properties is a major question in rock mechanics. Often, the only way to guide the choice of appropriate parameters is by comparison to similar joint properties derived from field tests; however, field test observations are extremely limited. Some results are reported by Kulhawy (1975).
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3.9 Tips and Advice When problem solving with 3DEC, it is often important to try to optimize the model for the most efficient analysis. This section provides several suggestions on ways to improve a model run. Also, some common pitfalls which should be avoided when preparing a 3DEC calculation are listed. 1. Designing the Model It is tempting to try to build as much detail of the geologic structure as possible into a 3DEC model. The main arguments against this approach are that (1) it is futile to ever expect to have sufficient data to model a jointed rock mass in every detail, (2) the computer hardware requirements for a detailed model quickly exceed that typically available for engineering projects, and (3) most importantly, a controlled engineering understanding of model results becomes less effective as more detail is added. Two considerations should be kept in mind when creating the 3DEC model. The first is whether or not a discontinuum analysis is actually required. This depends in large part on the ratio of the scale of the physical system under investigation to the average spacing of the joint structure. For example, if an excavation is made in a rock mass containing a single joint set with an average spacing of 1 m or less, and the minimum dimension of the excavation is 10 m, then a continuum analysis with a ubiquitous-joint material model may be a more reasonable approach. At this scale, the continuum analysis can produce a response that is broadly equivalent to that obtained when the joints are explicitly modeled. The analysis with 3DEC provides a more detailed analysis of the failure mechanism, but may require more computation time than the continuum analysis. In instances where the ratio of the physical system scale to the joint spacing exceeds approximately 10:1, a continuum analysis may be preferred. Analyses should be conducted with both continuum and discontinuum analyses when there is a question as to whether the continuum analysis is sufficient to represent the discontinuum response. See Board et al. (1996) for an illustration of the application of discontinuum and continuum analyses (using UDEC and FLAC) in a comparative study of toppling behavior of a jointed rock slope. The second consideration is the extent to which the detailed geologic structure should be included in the model. A detailed representation which includes the most critical joint structure (see Section 3.2.7) is usually only required within a limited region surrounding the area of interest (e.g., within a few tunnel radii surrounding a tunnel excavation). Generally, the greater jointing detail only needs to extend from the area of interest to a distance sufficient to encompass the region in which failure is anticipated. That is, the detailed geologic structure should extend beyond the distance to which joint slip and separation are calculated.
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2. Check Model Runtime The solution time for a 3DEC run is a function of both the number of rigid blocks or gridpoints in deformable blocks, and the number of contacts in a model. If there are very few contacts in the model, then the time is proportional to N 3/2 , where N is the number of rigid blocks or gridpoints in deformable blocks. This formula holds for elastic problems. The runtime will vary somewhat, but not substantially, for plasticity problems. The solution time will increase as more contacts are created in the model. It is important to check the speed of calculation on your computer for a specific model. An easy way to do this is to run the benchmark test described in Section 6. Then use this speed to estimate the speed of calculation for the specific model, based on interpolation from the number of gridpoints and contacts. 3. Effects on Runtime 3DEC will take a longer time to converge if: (a) there are large contrasts in stiffness in block materials or in joint materials or in block versus joint materials; or (b) there are large contrasts in block or zone sizes. The code becomes less efficient as these contrasts become greater. The effect of a contrast in stiffness should be investigated before performing a detailed analysis. For example, for mechanical-only calculations, joint normal and shear stiffnesses should be kept smaller than ten times the equivalent stiffness of the stiffest neighboring zone in blocks adjoining the joint — i.e.,
K + 4/3G kn and ks ≤ 10.0 max !zmin
(3.33)
where K and G are the bulk and shear moduli, respectively, of the block material, and !zmin is the smallest dimension of the zone adjoining the joint in the normal direction. If the joint stiffnesses are greater than 10 times the equivalent stiffness, the solution time of the model will be significantly longer than for the case in which the ratio is limited to ten, without a significant change in the behavior of the system. On the other hand, there may be problems if the normal stiffness, kn , is very low. A rough estimate should be made of the joint normal displacement that would result from the application of typical stresses in the system (u = σ/kn ). This displacement should be small compared to a typical zone size. If it is greater than roughly 10% of an adjacent zone size, then either there is an error in one of the numbers or the stiffness should be increased.
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4. Considerations for Density of Zoning 3DEC uses constant-strain elements in deformable blocks. If the gradient of stress/strain is high, many zones will be needed to represent the varying distribution. Run the same problem with different zoning densities to check the effect. Constant-strain zones are used in 3DEC because a better accuracy is achieved when modeling plastic flow with many low-order elements than with a few high-order elements. Try to keep the zoning as uniform as possible, particularly in the region of interest. Avoid long, thin zones with aspect ratios greater than 5:1. 5. Check Model Response 3DEC shows how a system behaves. Make frequent simple tests to check whether you are doing what you think you are doing. For example, if a loading condition and geometry are symmetrical, make sure that the response is symmetrical. After making a change in the model, execute a few calculation steps (say, 5 or 10) to verify that the initial response is of the correct sign and in the correct location. Do back-of-the-envelope estimates of the expected order of magnitude of stress or displacements and compare to the 3DEC output. If you apply a violent shock to the model, you will get a violent response. If you do physically unreasonable things to the model, you must expect strange results. If you get unexpected results at a given stage of an analysis, review the steps you followed up to this stage. Critically examine the output before proceeding with the model simulation. If, for example, everything appears reasonable except for large velocities in one corner block, do not go on until you understand the reason for this. In this case, you may have not fixed a boundary corner properly. 6. Use Bulk and Shear Moduli It is better to use bulk modulus, K, and shear modulus, G, than Young’s modulus, E, and Poisson’s ratio, ν, for elastic properties in 3DEC. The pair (K, G) makes sense for all elastic materials that do not violate thermodynamic principles. The pair (E, ν) does not make sense for certain admissible materials. At one extreme we have materials that resist volumetric change but not shear and at the other extreme materials that resist shear but not volumetric change. The first type of material corresponds to finite K and zero G, and the second to zero K and finite G. However, the pair (E, ν) is not able to characterize either the first or the second type of material. If we exclude the two limiting cases (conventionally, ν = 0.5 and ν = -1), the equations
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3K(1 − 2ν) = E (3.34) 2G(1 + ν) = E relate the two sets of constants. These equations hold however close we approach (but not reach) the limiting cases. We do not need to relate them to physical tests that may or may not be feasible; the equations are simply the consequence of two possible ways of defining coefficients of proportionality. Suppose we have a material in which the resistance to distortion progressively reduces, but in which the resistance to volume change remains constant. ν approaches 0.5 in this case. The equation 3K(1 − 2ν) = E must still be satisfied. There are two possibilities (argued on algebraic grounds, not physical): either E remains finite (and nonzero) and K tends to an arbitrarily large value; or K remains finite and E tends to zero. The first possibility we rule out because there is a limiting compressibility to all known materials (e.g., 2 GPa for water, which has a Poisson’s ratio of 0.5). This leaves the second, in which E is varying drastically, even though we supposed that the material’s principal mode of elastic resistance was unchanging. We deduce that the parameters (E, ν) are inadequate to express the material behavior. 7. Choice of Damping In most instances, it is recommended that DAMP local be used for static analyses. This is generally appropriate for static analysis for the reasons given in Section 1.2.2.7 in Theory and Background. Also, as demonstrated in Example 3.25, local damping is more suitable to minimize oscillations that may arise when abrupt failure occurs in the model. In some cases, particularly when calculating an initial equilibrium state, it may be more computationally efficient to use DAMP auto. As discussed in Section 1.2.2.7 in Theory and Background, local damping is most efficient when velocity components at gridpoints pass through zero periodically, because the mass-adjustment process depends on velocity sign changes. Adaptive global damping, on the other hand, applies a constant damping factor that is not affected by velocity sign-changes. If velocities act predominantly in one direction (e.g., due to gravity loading), then a system with local damping may take longer to converge than one with adaptive global damping. When in doubt, it is usually best to run the model with both DAMP local and DAMP auto and compare the calculation steps required to reach convergence.
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8. Avoiding Rounding Errors Most of the calculations in 3DEC are in single precision (e.g., coordinate updates and displacements). Usually, the significant figures that are available in single-precision, 32-bit computer arithmetic are sufficient to keep rounding errors from affecting the solution. However, there are circumstances in which numerical rounding errors can affect the results of an analysis. For example, rounding errors may become noticeable in problems that run for several hundred thousand cycles with a low applied velocity. To minimize rounding error problems, avoid large coordinates when creating a model. For example, it is tempting to adopt the same coordinate values as those used in mine plan and section drawings when creating a model for a mining application. This makes it easier to refer plots and results back to the mine drawings, but it may inadvertently magnify the rounding errors in the computation. 3DEC always updates coordinates progressively as deformations develop. If the initial coordinates are very large numerically (e.g., a coordinate range of 10000 to 10100), significant accuracy may be lost when small displacement increments are added to the coordinates. Also, searches for block contacts, which involve differences between coordinate values, may become unreliable. By changing the coordinates to range from 0 to 100, the problem can be avoided. 9. Determining Collapse Loads In order to determine a collapse load, it often is better to use “strain-controlled” boundary conditions rather than “stress-controlled” — i.e., apply a constant velocity and measure the boundary reaction forces rather than apply forces and measure displacements. A system that collapses becomes difficult to control as the applied load approaches the collapse load. This is true of a real system as well as a model system. 10. Determining Factor-of-Safety 3DEC does not calculate a “Factor-of-Safety” directly. If you need a safety factor, it can be determined for any selected parameter by calculating the ratio of the selected parameter value under given conditions to the parameter value that results in failure. For example,
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Note that the larger value is always divided by the smaller value (assuming that the system does not fail under the actual conditions). The definition of failure must be established by the user.
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3.10 Interpretation Because 3DEC models a nonlinear system as it evolves in time, the interpretation of results may be more difficult than with a conventional finite element program that produces a “solution” at the end of its calculation phase. There are several indicators that can be used to assess the state of the numerical model for a static analysis* — e.g., whether the system is stable, unstable, or is in steady-state plastic flow. The various indicators are described below. 3.10.1 Unbalanced Force Forces are accumulated at each centroid of rigid blocks and each gridpoint of deformable blocks. At equilibrium — or steady plastic flow in deformable blocks — the algebraic sum of these forces is almost zero (i.e., the forces acting on one side of the block centroid or gridpoint nearly balance those acting on the other). During timestepping, the maximum unbalanced force is determined for the whole model; this force is displayed continuously on the screen. It can also be saved as a history and viewed as a graph. The unbalanced force is important in assessing the state of the model for static analysis, but its magnitude must be compared with the magnitude of typical internal forces acting in the model; in other words, it is necessary to know what constitutes a “small” force. A representative internal gridpoint force for deformable blocks may be found by multiplying stress by zone area perpendicular to the force, using values that are typical in the area of interest in the model. Denoting R as the ratio of maximum unbalanced force to the representative internal force, expressed as a percentage, the value of R will never decrease to zero. However, a value of 1% or 0.1% may be acceptable as denoting equilibrium, depending on the degree of precision required (e.g., R = 1% may be good enough for an intermediate stage in a sequence of operations, while R = 0.1% may be used if a final stress or displacement distribution is required for inclusion in a report or paper). Note that a low value of R only indicates that forces balance at all gridpoints; however, steady plastic flow may be occurring, without acceleration. In order to distinguish between this condition and “true” equilibrium, other indicators, such as those described below, should be consulted. 3.10.2 Block/Gridpoint Velocities The velocities of rigid blocks and gridpoints of deformable blocks may be assessed by plotting the whole field of velocities (using the PLOT vel command) or by selecting certain key points in the model and tracking their velocities with histories (HIS xvel, HIS yvel or HIS zvel). Both types of plots are useful. Steady-state conditions are indicated if the velocity histories show horizontal traces in their final stages. If they have all converged to near-zero (in comparison to their starting values), then absolute equilibrium has occurred. If a history has converged to a nonzero value, then either the block is falling, or steady plastic flow is occurring at the block/gridpoint corresponding to that history. If one or more velocity history plots show fluctuating velocities, then the system is likely to be in a transient condition. * Interpretation of the state of a model for a dynamic analysis is discussed in Section 2 in Optional Features.
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The plot of the field of velocity vectors is more difficult to interpret, since both the magnitudes and the nature of the pattern are important. As with gridpoint forces, velocities never decrease precisely to zero. The magnitude of velocity should be viewed in relation to the displacement that would occur if a significant number of steps (e.g., 1000) were to be executed. For example, if current displacements in the system are of the order of 1 cm, the maximum velocity in the velocity plot is 10−3 m/sec and the timestep is 10−5 sec, then 1000 steps would produce an additional displacement of 10−5 m, or 10−3 cm, which is 0.1% of the current displacements. In this case, it can be said that the system is in equilibrium, even if the velocities all seem to be “flowing” in one direction. More often, the vectors appear to be random (or almost random) in direction and (possibly) in magnitude. This condition occurs when the changes in gridpoint force fall below the accuracy limit of the computer, which is around six decimal digits. A random velocity field of low amplitude is an infallible indicator of block stability and no plastic flow. If the vectors in the velocity field are coherent (i.e., there is some systematic pattern) and their magnitude is quite large (using the criterion described above), then either blocks are falling or slipping, plastic flow is occurring within blocks, or the system is still adjusting elastically (e.g., damped elastic oscillation is taking place). To confirm that continuing plastic flow is occurring, a plot of plasticity indicators should be consulted, as described below. If, however, the motion involves elastic oscillation, then the magnitude should be observed in order to indicate if such movement is significant. Seemingly meaningful patterns of oscillation may be seen; however, if amplitude is low, then the motion has no physical significance. 3.10.3 Plastic Indicators for Block Failure For most of the nonlinear block models in 3DEC, the command PLOT plas displays those zones in which the stresses satisfy the yield criterion. Such an indication usually denotes that plastic flow is occurring, but it is possible for a block zone simply to “sit” on the yield surface without any significant flow taking place. It is important to look at the whole pattern of plasticity indicators to see if a mechanism has developed. A failure mechanism is indicated if there is a contiguous line of active plastic zones that join two surfaces. The diagnosis is confirmed if the velocity plot also indicates motion corresponding to the same mechanism. Note that initial plastic flow often occurs at the beginning of a simulation, but subsequent stress redistribution unloads the yielding elements so that their stresses no longer satisfy the yield criterion (“yielded in past”). Only the actively yielding elements (“at yield surface”) are important to the detection of a failure mechanism. If there is no contiguous line or band of active plastic zones between boundaries, two patterns should be compared before and after the execution of, say, 500 steps. Is the region of active yield increasing or decreasing? If it is decreasing, then the system is probably heading for equilibrium; if it is increasing, then ultimate failure may be possible. If a condition of continuing plastic flow has been diagnosed, one further question should be asked: Does the active flow band(s) include zones adjacent to artificial boundaries? The term “artificial boundary” refers to a boundary that does not correspond to a physical entity, but exists simply to limit the size of the model that is used. If plastic flow occurs along such a boundary, then the solution is not realistic, because the mechanism of failure is influenced by a nonphysical entity. This comment only applies to the final steady-state solution; intermediate stages may exhibit flow along boundaries.
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3.10.4 Histories In any problem, there are certain variables that are of particular interest (e.g., displacements may be of concern in one problem, but stresses may be of concern in another). Liberal use should be made of the HIST command to track these important variables in the regions of interest. After some timestepping has taken place, the plots of these histories often provide the way to find out what the system is doing.
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3.11 Modeling Methodology 3.11.1 Modeling of Data-Limited Systems In a field such as geomechanics, where data are not always available, the methodology used in numerical modeling should be different from that used in a field such as mechanical engineering. Starfield and Cundall (1988) provide suggestions for an approach to modeling that is appropriate for a data-limited system. This paper should be consulted before any serious modeling with 3DEC is attempted. In essence, the approach recognizes that field data (such as in-situ stresses, material properties and geological features) will never be known completely. It is futile to expect the model to provide design data, such as expected displacements, when there is massive uncertainty in the input data. However, a numerical model is still useful in providing a picture of the mechanisms that may occur in particular physical systems. The model acts to educate the intuition of the design engineer by providing a series of cause-and-effect examples. The models may be simple, with assumed data that are consistent with known field data and engineering judgement. It is a waste of effort to construct a very large and complicated model that may be just as difficult to understand as the real case. Of course, if extensive field data are available, then these may be incorporated into a comprehensive model that can yield design information directly. More commonly, however, the data-limited model does not produce such information directly, but provides insight into mechanisms that may occur. The designer can then do simple calculations, based on these mechanisms, that estimate the parameters of interest or the stability conditions. 3.11.2 Modeling of Chaotic Systems In some calculations, especially in those involving discontinuous materials, the results can be extremely sensitive to very small changes in initial conditions or trivial changes in loading sequence. At first sight, this situation may seem unsatisfactory and may be taken as a reason to mistrust the computer simulations. However, the sensitivity exists in the physical system being modeled. There appear to be a least two sources for the seemingly erratic behavior. 1. There are certain geometric patterns of discontinuities that force the system to choose, apparently at random, between two alternative outcomes; the subsequent evolution depends on which choice is made. For example, Figure 3.51 illustrates a small portion of a jointed rock mass. If block A is forced to move down relative to B, it can either go to the left or to the right of B; the choice will depend on microscopic irregularities in geometry, properties, or kinetic energy.
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A
B
Figure 3.51 A small portion of a jointed rock mass 2. There are processes in the system that can be described as “softening” or, more generally, as cases of positive feedback. In a fairly uniform stress field, small perturbations are magnified in the subsequent evolution because a region that has more strain, softens more and thereby attracts more strain, and so on, in a cycle of positive feedback. Both phenomena give rise to behavior that is chaotic in its extreme form (Gleick (1987) and Thompson and Stewart (1986)). The study of chaotic systems reveals that the detailed evolution of such a system is not predictable, even in principle. The observed sensitivity of the computer model to small changes in initial conditions or numerical factors is simply a reflection of a similar sensitivity in the real world to small irregularities. There is no point in pursuing ever more “accurate” calculations, because the resulting model is unrepresentative of the real world, where conditions are not perfect. What should our modeling strategy be in the face of a chaotic system? It appears that the best we can expect from such a model is a finite spectrum of expected behavior; the statistics of a chaotic system are well-defined. We need to construct models that contain distributions of initial irregularities — e.g., by using 3DEC ’s adev parameter on the JSET command. Each model should be run several times, with different distributions of irregularities. Under these conditions, we may expect the fluctuations in behavior to be triggered by the imposed irregularities, rather than by artifacts of the numerical solution scheme. We can express the results in a statistical form.
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3.11.3 Localization, Physical Instability and Path-Dependence In many systems that can be modeled with 3DEC, there may be several paths that the solution may take, depending on rather small changes in initial conditions. This phenomenon is termed bifurcation. For example, a shear test on an elastic/plastic material may either deform uniformly, or it may exhibit shear bands, in which the shear strain is localized rather than being uniformly distributed. It appears that if a numerical model has enough degrees-of-freedom (i.e., enough elements), then localization is to be expected. Indeed, theoretical work on the bifurcation process (e.g., Rudnicki and Rice (1975) and Vardoulakis (1980)) shows that shear bands form even if the material does not strain-soften, provided that the dilation angle is lower than the friction angle. The “simple” Mohr-Coulomb material should always exhibit localization if enough elements exist to resolve one or more localized bands. A strain-softening material is more prone to produce bands. Some computer programs appear incapable of reproducing band formation, although the phenomenon is to be expected physically. However, 3DEC is able to allow bands to develop and evolve, partly because it models the dynamic equations of motion (i.e., the kinetic energy that accompanies band formation is released and dissipated in a physically realistic way). Several papers document the use of two-dimensional FLAC in modeling shear band formation (Cundall (1989), (1990), and (1991)). These should be consulted for details concerning the solution process. One aspect that is not treated well by 3DEC is the thickness of a shear band. In reality, the thickness of a band is determined by internal features of the material, such as grain size. These features are not built into 3DEC ’s constitutive models. Hence, the bands in 3DEC collapse down to the smallest width that can be resolved by the grid, which is one grid-width if the band is parallel to the grid, or about three grid-widths if the band cuts across the grid at an arbitrary angle. Although the overall physics of band formation is modeled correctly by 3DEC, band thickness and band spacing are grid-dependent. Furthermore, if the strain-softening model is used with a weakening material, the load/displacement relation generated by 3DEC for a simulated test is strongly grid-dependent. This is because the strain concentrated in a band depends on the width of the band (in length units), which depends on zone size, as we have seen. Hence, smaller zones lead to more softening, since we move out more rapidly on the strain axis of the given softening curve. To correct this grid dependence, some sort of length scale must be built into the constitutive model. There is controversy, at present, concerning the best way to do this. It is anticipated that future versions of 3DEC will include a length scale in the constitutive models — probably involving the use of a Cosserat material, in which internal spins and moments are taken into account. In the meantime, the processes of softening and localization may be modeled, but it must be recognized that the grid size and angle affect the results; models must be calibrated for each grid used. One topic that involves chaos, physical instability and bifurcation is path-dependence. In most nonlinear, inelastic systems, there are an infinite number of solutions that satisfy equilibrium, compatibility, and the constitutive relations. There is no “correct” solution to the physical problem unless the path is specified. If the path is not specified, all possible solutions are correct. This situation can cause endless debate among modelers and users, particularly if a seemingly irrelevant parameter in the solution process (e.g., damping) is seen to affect the final result. All the solutions are valid numerically. For example, a simulation done of a mining excavation with low damping may show a large overshoot and, hence, large final displacements, while high damping will eliminate the overshoot and give lower final displacements. Which one is more realistic? It depends on the path. If the excavation is done by explosion (i.e., suddenly), then the solution with overshoot may
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be the appropriate one; if the excavation is done by pick and shovel (i.e., gradually), then the second case may be more appropriate. For cases in which path-dependence is a factor, modeling should be done in a way that mimics the way the system evolves physically.
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3.12 References Bandis, S. C., A. C. Lumsden and N. R. Barton. “Fundamentals of Rock Joint Deformation,” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 20(6), 249-268 (1983). Barton, N. “The Shear Strength of Rock and Rock Joints,” Int. J. Rock Mech. Min. Sci. & Geotech. Abstr., 13, 255-279 (1976). Bieniawski, Z. T. “Determining Rock Mass Deformability: Experience from Case Histories,” Int. J. Rock Mech. Min. Sci., 15, 237-247 (1978). Board, M., E. Chacon, P. Varona and L. Lorig. “Comparative Analysis of Toppling Behaviour at Chuquicamata Open-Pit Mine, Chile,” Trans. Instn. Min. Metall., Sec. A, 105, A11-A21, 1996. Brady, B. H. G., and E. T. Brown. Rock Mechanics for Underground Mining. London: George Allen and Unwin., 1985. Brady, B. H. G., J. V. Lemos and P. A. Cundall. “Stress Measurement Schemes for Jointed and Fractured Rock,” in Rock Stress and Rock Stress Measurements, pp. 167-176. Luleå, Sweden: Centek Publishers, 1986. Clark, I. H. “The Cap Model for Stress Path Analysis of Mine Backfill Compaction Processes,” in Computer Methods and Advances in Geomechanics, Vol. 2, pp. 1293-1298. Rotterdam: A. A. Balkema, 1991. Cundall, P. A. “Numerical Experiments on Localization in Frictional Material,” Ingenieur-Archiv, 59, 148-159 (1988). Cundall, P. A. “Numerical Modelling of Jointed and Faulted Rock,” in Mechanics of Jointed and Faulted Rock, pp. 11-18. Rotterdam: A. A. Balkema, 1990. Cundall, P. A. “Shear Band Initiation and Evolution in Frictional Materials,” in Mechanics Computing in 1990s and Beyond (Proceedings of the Conference, Columbus, Ohio, May, 1991), Vol. 2: Structural and Material Mechanics, pp. 1279-1289. New York: ASME, 1991. Fossum, A. F. “Technical Note: Effective Elastic Properties for a Randomly Jointed Rock Mass,” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 22(6), 467-470 (1985). Gerrard, C. M. “Elastic Models of Rock Masses Having One, Two and Three Sets of Joints,” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 19, 15-23 (1982b). Gerrard, C. M. “Equivalent Elastic Moduli of a Rock Mass Consisting of Orthorhombic Layers,” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 19, 9-14 (1982a). Gleick, J. Chaos: Making a New Science. New York: Penguin Books, 1987. Goodman, R. E. Introduction to Rock Mechanics. New York: John Wiley and Sons, 1980. Hart, R. D. “An Introduction to Distinct Element Modeling for Rock Engineering,” in Comprehensive Rock Engineering, Vol. 2, pp. 245-261. Oxford: Pergamon Press, Ltd., 1993.
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Hoek, E. “Estimating Mohr-Coulomb Friction and Cohesion Values from the Hoek-Brown Failure Criterion,” in Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 27(3), 227-229 (1990). Hoek, E., and E. T. Brown. “The Hoek-Brown Failure Criterion — a 1988 Update,” in Rock Engineering for Underground Excavations, pp. 31-38. Toronto: University of Toronto, 1988. Hoek, E., and E. T. Brown. Underground Excavations in Rock. London: Instn. Min. Metall., 1980. Huang, X., B. C. Haimson, M. E. Plesha and X. Qiu. “An Investigation of the Mechanics of Rock Joints — Part I. Laboratory Investigation,” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 30, 257-269 (1993). Jaeger, J. C., and N. G. W. Cook. Fundamentals of Rock Mechanics, 2nd Ed. London: Chapman and Hall, 1969. Jing, L., E. Nordlund and O. Stephansson. “An Experimental Study on the Anisotropy an StressDependency of the Strength and Deformability of Rock Joints,” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 29, 535-542 (1992). Kulhawy, Fred H. “Stress Deformation Properties of Rock and Rock Discontinuities,” Engineering Geology, 9, 327-350 (1975). Lindner, E. N., and J. A. Halpern. “In-Situ Stress in North America: A Compilation,” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 15, 183-203 (1978). Lorig, L. J., and B. H. G. Brady. “An Improved Procedure for Excavation Design in Stratified Rock,” in Rock Mechanics — Theory-Experiment-Practice, pp. 577-586. New York: Association of Engineering Geologists, 1983. Müller, B., M. L. Zoback, K. Fuchs, L. Mastin, S. Gregersen, N. Pavoni, O. Stephansson and C. Ljunggren. “Regional Patterns of Tectonic Stress In Europe,” J. Geophys. Res., 97(B8), 1178311803 (1992). Rosso, R. S. “A Comparison of Joint Stiffness Measurements in Direct Shear, Triaxial Compression, and In-Situ,” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 13, 167-172 (1976). Rudnicki, J. W., and J. R. Rice. “Conditions for the Localization of the Deformation in PressureSensitive Dilatant Materials,” J. Mech. Phys. Solids, 23, 371-394 (1975). Serafim, J. L., and J. P. Pereira. “Considerations of the Geomechanical Classification of Bieniawski,” Proceedings of the International Symposium on Engineering Geology and Underground Construction Lisbon 1983, Vol. 1, pp. II.33-42. Lisbon: SPGILNEC, 1983. Singh, B. “Continuum Characterization of Jointed Rock Masses: Part I — The Constitutive Equations,” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 10, 311-335 (1973). Souley, M., F. Homand and B. Amadei. “An Extension to the Saeb and Amadei Constitutive Model for Rock Joints to Include Cyclic Load Paths,” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 32, 101-109 (1995).
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Starfield, A. M., and P. A. Cundall. “Towards a Methodology for Rock Mechanics Modelling,” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 25(3), 99-106 (1988). Thompson, J. M. T., and H. B. Stewart. Nonlinear Dynamics and Chaos. New York: John Wiley and Sons, 1986. Vardoulakis, I. “Shear Band Inclination and Shear Modulus of Sand in Biaxial Tests,” Int. J. Numer. Anal. Meth. in Geomechanics, 4, 103-119 (1980). Vermeer, P. A., and R. de Borst. “Non-Associated Plasticity for Soils, Concrete and Rock,” Heron, 29(3), 3-64 (1984).
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4 FISH BEGINNER’S GUIDE 4.1 Introduction and Overview FISH is a programming language embedded within 3DEC that enables the user to define new variables and functions. These functions may be used to extend 3DEC ’s usefulness or add userdefined features. For example, new variables may be plotted or printed, special model generators may be implemented, servo-control may be applied to a numerical test, unusual distributions of properties may be specified, and parameter studies may be automated. FISH was developed in response to requests from users who wanted to do things with Itasca software that were either difficult or impossible to do with existing program structures. Rather than incorporate many new and specialized features into the standard code, it was decided that an embedded language would be provided so that users could write their own functions. Some useful FISH functions have already been written; a library of these is provided with the 3DEC program. It is possible for someone without experience in programming to write simple FISH functions or to modify some of the simpler existing functions. Section 4.2 contains an introductory tutorial for non-programmers. However, FISH programs can also become very complicated (which is true of code in any programming language); for more details, refer to Section 2 in the FISH volume. As with all programming tasks, FISH functions should be constructed in an incremental fashion, checking operations at each level before moving on to more complicated code. FISH does less error-checking than most compilers, so all functions should be tested on simple data sets before using them for real applications. FISH programs are simply embedded in a normal 3DEC data file — lines following the word DEFINE are processed as a FISH function; the function terminates when the word END is encountered. Functions may invoke other functions, which may invoke others, and so on. The order in which functions are defined does not matter as long as they are all defined before they are used (e.g., invoked by a 3DEC command). Since the compiled form of a FISH function is stored in 3DEC ’s memory space, the SAVE command saves the function and the current values of associated variables. A complete definition of FISH language rules and intrinsic functions is provided in Section 2 in the FISH volume. This includes rules for syntax, data types, arithmetic, variables and functions. All FISH language names are described in Section 2 in the FISH volume, and a summary of the names is provided in Section 2 in the Command and FISH Reference Summary.
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4.2 Tutorial This section is intended for people who have run 3DEC (at least for simple problems) but have not used the FISH language; no programming experience is assumed. To get the maximum benefit from the examples given here, you should try them out with 3DEC running interactively. The short programs may be typed in directly. After running an example, give the 3DEC command NEW to “wipe the slate clean,” ready for the next example. Alternatively, the more lengthy programs may be created on file and CALLed when required. Type the lines in Example 4.1 after 3DEC ’s command prompt, pressingat the end of each line. Example 4.1 Defining a FISH function def abc abc = 22 * 3 + 5 end
Note that the command prompt changes to Def> after the first line has been typed in; then it changes back to the usual prompt when the command END is entered. This change in prompt lets you know if you are sending lines to 3DEC or to FISH. Normally, all lines following the DEFINE statement are taken as part of the definition of a FISH function (until the END statement is entered). However, if you type in a line that contains an error (e.g., you type the = sign instead of the + sign), then you will get the 3DEC prompt back again. In this case, you should give the NEW command and try again from the beginning. Since it is very easy to make mistakes, FISH programs are normally typed into a file using an editor. These are then CALLed into 3DEC just like a regular 3DEC data file. We will describe this process later; for now, we’ll continue to work interactively. Assuming that you typed in the above lines without error and that you now see the 3DEC prompt 3Dec>, you can “execute” the function abc,* defined earlier in Example 4.1, by typing the line print abc
The message abc =
71
should appear on the screen. By defining the symbol abc (using the DEFINE ... END construction, as in Example 4.1), we can now refer to it in many ways using 3DEC commands. For example, the PRINT command causes the value of a FISH symbol to be displayed; the value is computed by the series of arithmetic operations in the line abc = 22 * 3 + 5
* We will use courier boldface to identify user-defined FISH functions and declared variables in the text.
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This is an “assignment statement.” If an equal sign is present, the expression on the right-hand side of the equal sign is evaluated and given to the variable on the left-hand side. Note that arithmetic operations follow the usual conventions; addition, subtraction, multiplication and division are done with the signs +, -, * and /, respectively. The sign ˆ denotes “raised to the power of.” We now type in a slightly different program (using the command NEW to erase the old one): Example 4.2 Using a variable new def abc hh = 22 abc = hh * 3 + 5 end
Here we introduce a “variable,” hh, which is given the value of 22 and then used in the next line. If we give the command PRINT abc, then exactly the same output as in the previous case appears. However, we now have two FISH symbols; they both have values, but one (abc) is known as a “function” and the other (hh) as a “variable.” The distinction is as follows. When a FISH symbol name is mentioned (e.g., in a PRINT statement), the associated function is executed if the symbol corresponds to a function; however, if the symbol is not a function name, then the current value of the symbol is simply used. The following experiment may help to clarify the distinction between variables and functions. Before doing the experiment, note that 3DEC ’s SET command can be used to set the value of any user-defined FISH symbol, independent of the FISH program in which the symbol was introduced. Now type in the following lines without giving the command NEW, since we want to keep our previously-entered program in memory. Example 4.3 SETting variables set abc=0 hh=0 print hh print abc print hh
The SET command sets the values of both abc and hh to zero. Since hh is a variable, the first PRINT command simply displays the current value of hh, which is zero. The second PRINT command causes abc to be executed (since abc is the name of a function); the values of both hh and abc are thereby recalculated. Accordingly, the third PRINT statement shows that hh has indeed been reset to its original value. As a test of your understanding, you should type in the slightly modified sequence shown in Example 4.4 and figure out why the displayed answers are different.
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Example 4.4 Test your understanding of function and variable names new def abc abc = hh * 3 + 5 end set hh=22 print abc set abc=0 hh=0 print hh print abc print hh
At this stage, it may be useful to list the most important 3DEC commands that directly refer to simple FISH variables or functions. (In Table 4.1, below, var stands for the name of the variable or function.) Table 4.1 Commands that directly refer to FISH names PRINT SET HISTORY
var var = value var
We have already seen examples of the first two (refer to Examples 4.3 and 4.4); the third case is useful when histories are required of things that are not provided in the standard 3DEC list of history variables. Example 4.5 shows how this can be done. Example 4.5 Capturing the history of a FISH variable new poly brick 0,10 0,10 0,10 gen edge 10 prop mat=1 dens 1000 k 1e9 g 0.7e9 bound 0, 10 -0.01, 0.01 0,10 yvel 0.0 grav 0 -10 0 def stress_y zoneIdx = b_zone(block_head) stress_y = z_syy(zoneIdx) end hist stress_y cyc 200 pl his 1 hold
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In this example, a history of the vertical stress in one zone is recorded. The symbols b zone(), block head and z syy() are pre-defined names that permit access to 3DEC ’s data structures. We obtained the index of the first zone in the one block in our model. With that index we can access a number of parameters associated with that zone. In this case, we have accessed the vertical stress and monitored its change in a history. In addition to the above-mentioned pre-defined variable names, there are many other pre-defined objects available to a FISH program. These fall into several classes; one such class consists of scalar variables, which are single numbers — for example,
clock
clock time in hundredths of a second
unbal
maximum unbalanced force
pi
π
step
current step number
urand
random number drawn from uniform distribution between 0.0 and 1.0.
This is just a small selection; the full list is given in Section 2.5.2 in the FISH volume. Another useful class of built-in objects is the set of intrinsic functions, which enables things like sines and cosines to be calculated from within a FISH program. A complete list is provided in Section 2.5.4 in the FISH volume; a few are given below:
abs(a)
absolute value of a
cos(a)
cosine of a (a is in radians)
log(a)
base-ten logarithm of a
max(a,b)
returns maximum of a, b
sqrt(a)
square root of a
An example in the use of intrinsic functions will be presented later, but now we must discuss one further way in which a 3DEC data file can make use of user-defined FISH names. Wherever a number is expected in a 3DEC input line, you may substitute the name of a FISH variable or function. This simple statement is the key to a very powerful feature of FISH that allows such things as ranges, applied stresses, properties, etc. to be computed in a FISH function and used by 3DEC input in symbolic form. Hence, parameter changes can be made very easily, without the need to change many numbers in an input file. As an example, let us assume that we know the Young’s modulus and Poisson’s ratio of a material. Although properties may be specified using Young’s modulus and Poisson’s ratio, internally 3DEC
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uses the bulk and shear moduli, we may derive these with a FISH function, using Eqs. (4.1) and (4.2): G=
E 2(1 + ν)
(4.1)
K=
E 3(1 − 2ν)
(4.2)
Coding Eqs. (4.1) and (4.2) into a FISH function (called derive) can then be done as shown in Example 4.6, below. Example 4.6 FISH functions to calculate bulk and shear moduli new def derive s_mod = y_mod / (2.0 * (1.0 + p_ratio)) b_mod = y_mod / (3.0 * (1.0 - 2.0 * p_ratio)) end set y_mod = 5e8 p_ratio = 0.25 derive print b_mod s_mod
Note that, here, we execute the function derive by giving its name by itself on a line; we are not interested in its value, only what it does. If you run this example, you will see that values are computed for the bulk and shear moduli, b mod and s mod, respectively. These can then be used, in symbolic form, in 3DEC input as shown in Example 4.7. Example 4.7 Using symbolic variables in 3DEC input poly brick -1,1 -1,1 -1,1 jset gen edge 1 prop mat 1 density = 2000 k =b_mod g = s_mod print property block
The validity of this operation may be checked by printing the bulk and shear moduli with the PRINT property block command. In these examples, our property input is given via the SET command — i.e., to variables y mod and p ratio, which stand for Young’s modulus and Poisson’s ratio, respectively. In passing, note that there is great flexibility in choosing names for FISH variables and functions; the underline character ( ) may be included in a name. Names must begin with a non-number and must not contain any of the arithmetic operators (+, –, /, * or ˆ). A chosen name should not be
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the same as one of the built-in (or reserved) names; Section 2.2.2 in the FISH volume contains a complete list of names to be avoided, as well as some rules that should be followed. In the above examples, we checked the computed values of FISH variables by giving their names explicitly as arguments to a PRINT command. Alternatively, we can list all current variables and functions. A printout of all current values, sorted alphabetically by name, is produced by giving the command print fish
We now examine ways in which decisions can be made and repeated operations can be done in FISH programs. The following FISH statements allow specified sections of a program to be repeated many times:
LOOP
var (expr1, expr2)
ENDLOOP The words LOOP and ENDLOOP are FISH statements, the symbol var stands for the loop variable, and expr1 and expr2 stand for expressions (or single variables). Example 4.8 shows the use of a loop (or repeated sequence) to produce the sum and product of the first 10 integers. Example 4.8 Controlled loop in FISH new def xxx sum = 0 prod = 1 loop n (1,10) sum = sum + n prod = prod * n endloop end xxx print sum, prod
In this case, the loop variable n is given successive values from 1 to 10, and the statements inside the loop (between the LOOP and ENDLOOP statements) are executed for each value. As mentioned, variable names or an arithmetic expression could be substituted for the numbers 1 or 10. A practical use of the LOOP construct is to install a nonlinear initial distribution of elastic moduli in a 3DEC model. Suppose that the Young’s modulus at a site is given by Eq. (4.3). √ E = E◦ + c y
(4.3)
where y is the depth below surface, and c and E◦ are constants. We write a FISH function to install appropriate values of bulk and shear modulus in the model, as in Example 4.9.
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Example 4.9 Applying a nonlinear initial distribution of moduli new poly brick 0,30 -30,0 0,30 jset dip 36 dd 270 spac 6 jset dip -58 dd 270 spac 6
num 20 num 20
def install iprop = 0 loop while iprop < 7 iprop = iprop + 1 y_depth1 = (float(iprop) - 1.0) * 5.0 y_depth2 = y_depth1 + 5.0 y_mod = y_zero + cc * sqrt((y_depth1 + y_depth2) / 2.0) command prop mat = iprop ymod = y_mod prop mat = iprop prat = 0.25 dens = 2000 endcommand bi = block_head loop while bi # 0 y_depth = float(-b_y(bi)) if y_depth > y_depth1 then if y_depth <= y_depth2 then b_mat(bi) = iprop endif endif bi = b_next(bi) endloop endloop end set y_zero = 1e7 cc = 1e8 install plot color mat hold
Having seen several examples of FISH programs, let’s briefly examine the question of program syntax and style. A complete FISH statement must occupy one line; there are no continuation lines. If a formula is too long to fit on one line, then a temporary variable must be used to split the formula. Example 4.10 shows how this can be done.
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Example 4.10 Splitting lines new def long_sum ;example of a sum of many things temp1 = v1 + v2 + v3 + v4 + v5 + v6 + v7 + v8 + v9 + v10 long_sum = temp1 + v11 + v12 + v13 + v14 + v15 end
In this case, the sum of 15 variables is split into two parts. Note also the use of the semicolon in line 1 of Example 4.10 to indicate a comment. Any characters that follow a semicolon are ignored by the FISH compiler, but they are echoed to the log file. It is good programming practice to annotate programs with informative comments. Some of the programs have been shown with indentation — that is, space inserted at the beginning of some lines to denote a related group of statements. Any number of space characters may be inserted (optionally) between variable names and arithmetic operations to make the program more readable. Again, it is good programming practice to include indentation to indicate things like loops, conditional clauses and so on. Spaces in FISH are “significant” in the sense that space characters may not be inserted into a variable or function name. One other topic that should be addressed now is that of variable type. You may have noticed, when printing out variables from the various program examples, that numbers are either printed without decimal points or in “E-format” — that is, as a number with an exponent denoted by “E.” At any instant in time, a FISH variable or function name is classified as one of three types: integer, floating-point or string. These types may change dynamically, depending on context, but the casual user should not normally have to worry about the type of a variable, since it is set automatically. Consider Example 4.11. Example 4.11 Variable types new def haveone aa = 2 bb = 3.4 cc = ’Have a nice day’ dd = aa * bb ee = cc + ’, old chap’ end haveone print fish
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The resulting screen display looks like this: Value ----2 3.4000e+000 - string 6.8000e+000 - string 0
Name ---aa bb cc dd ee haveone
The variables aa, bb and cc are converted to integer, float and string, respectively, corresponding to the numbers (or strings) that were assigned to them. Integers are exact numbers (without decimal points) but are of limited range; floating-point numbers have limited precision (about six decimal places) but are of much greater range; string variables are arbitrary sequences of characters. There are various rules for conversion between the three types. For example, dd becomes a floating-point number because it is set to the product of a floating-point number and an integer; the variable ee becomes a string because it is the sum (concatenation) of two strings. The topic can get quite complicated, but it is fully explained in Sections 2.2.4 and 2.2.5 in the FISH volume. There is a further language element in FISH that is commonly used — the IF statement. The following three statements allow decisions to be made within a FISH program.
IF
expr1 test expr2 THEN
ELSE ENDIF These statements allow conditional execution of FISH program segments; ELSE and THEN are optional. The item test consists of one of the following symbols or symbol-pairs: =
#
>
<
>=
<=
The meanings are standard except for #, which means “not equal.” The items expr1 and expr2 are any valid expressions or single variables. If the test is true, then the statements immediately following IF are executed until ELSE or ENDIF is encountered. If the test is false, the statements between ELSE and ENDIF are executed if the ELSE statement exists; otherwise, the program jumps to the first line after ENDIF. The action of these statements is illustrated in Example 4.12.
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Example 4.12 Action of the IF ELSE ENDIF construct new def abc if xx > 0 then abc = 33 else abc = 11 end_if end set xx = 1 print abc set xx = -1 print abc
The displayed value of abc in Example 4.12 depends on the set value of xx. You should experiment with different test symbols (e.g., replace > with <). Until now, our FISH programs have been invoked from 3DEC either by using the PRINT command, or by giving the name of the function on a separate line of 3DEC input. It is also possible to do the reverse — that is, to give 3DEC commands from within a FISH function. Most valid 3DEC commands can be embedded between the following two FISH statements:
COMMAND ENDCOMMAND There are two main reasons for sending out 3DEC commands from a FISH program. First, it is possible to use a FISH function to perform operations that are not possible using the pre-defined variables that we already discussed. Second, we can control a complete 3DEC run with FISH.
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5 GRAPHICAL INTERFACE 3DEC contains a graphical interface to facilitate both model creation and presentation of results. When 3DEC is in the graphics screen mode, the user can enter input interactively in order to “move” the model for better viewing and select various types of output for graphical presentation (e.g., vectors, tensors, contours). The graphical interface is displayed when the command PLOT is given from the command mode. A graphical plot of the model appears on the screen along with a menu box containing a list of the active keystrokes that can be used to manipulate the model. The user can view the results of the keystroke (or combination of keystrokes) directly on the screen.* A mouse can also be used when the cursor is active. The elements of the graphical interface are illustrated in Figure 5.1. When in the graphics mode, the 3DEC model plot is viewed from a “viewing plane.” The viewing plane is always oriented parallel to and coincident with the graphics screen (see Figure 5.1). The model view is defined in terms of the position of the viewing plane relative to the model reference axes. The model axes are a left-hand set (x,y,z) oriented, by default, as x (east), y (vertically up) and z (north). (See Section 3.2.4 for further discussion on the model axes.) The default view of the model is from the viewing plane oriented parallel to the xy-plane of the model, with the centroid of the model positioned at the center of the screen. A “cut-plane” is also defined in the interface to permit the cutting of blocks (i.e., splitting a block into two blocks) while in the graphics mode (see Figure 5.1). The cut-plane, like the viewing plane, is always parallel to the screen, but it also can be moved into and out of the screen (i.e., along the normal to the viewing plane). viewing plane (screen)
model axes y z
x
cut plane
center of screen
3DEC menu box
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Figure 5.1
3DEC graphical interface (DOS version)
* Note that an hourglass will appear on the screen while the action of a keystroke is being performed. The time required to complete a keystroke action will depend upon the number of blocks (and zones) in the model.
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5.1 Overview Seven different colors are available with shading to create three-dimensional perspective plots and cross-section plots of the 3DEC model. A menu box also appears to the right of the graphical display on the screen. The main menu is visible in the menu box when first entering the screen mode, and all active keystrokes are listed. When a keystroke is selected from the main menu, the results are viewed directly. (It is not necessary to follow the keystroke with thekey.) For certain keystrokes, an action is taken (e.g., by pressing the key, the size of the plot is magnified). For other keystrokes, a new menu will appear in the menu box (e.g., by pressing the key, a ColorMode menu appears, and a listing of block coloring options is given). There are three levels of menus in the graphical interface. These are summarized in the menu guide shown in Figure 5.2. The required keystrokes to move from the main menu to a second-level menu, and from second-level to third-level menus, are shown in bold on the figure. When in a second-level or third-level menu, a new list of active keystrokes is displayed. The user has three options in a second-level or third-level menu: (1) pressing a key for a selected action and then pressing (or left mouse button) to invoke this action; (2) pressing the key to reset all actions to their default condition and then pressingto invoke the reset; or (3) pressing the key to escape from this menu and returning to the preceding menu. The keystroke actions are described in the following sections for the main menu and all secondand third-level menus. Pressing the arrow keys will rotate or translate the model in directions as specified in the main menu (menu items 1-5). Moving the mouse with the left button pressed is the same as repeatedly pressing the arrow keys. Clicking on a block with the right mouse button will center the view and rotation on the centroid of that block. The background color may be changed by SET back = color. The default background color is gray. The hardcopy background can be changed by SET plot background = color. The default background for hardcopies is iwhite.
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COLOR MODE INTERROGATE JOINT
COLOR
MODE
TARGET
ACTIVE
LINER/CABLE
MENU
BLOCK
(P)
MATERIAL NUMBER
(O)
CONSTITUTIVE NUMBER
JOIN 2 (X)
MAIN
BLOCK
BLOCKS
CROSS SECTION
SPECIAL OPTIONS RUNNING
STRESS
PLOT
VECTOR PLOT
(F10)
STRESSES/STATE
COLOR
VECTORS
HARD COPY
SECOND-LEVEL MENUS
Figure 5.2
COLOR
THIRD-LEVEL MENUS
3DEC menu guide
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5.2 Menus 5.2.1 Main Menu
C
Color Mode Blk changes the colors for blocks. When thekey is pressed, theColorMode menu appears (see Section 5.2.2).
J
Joint Structure displays the joint structure only. All blocks are hidden. When thekey is pressed, the JointMode menu appears with options for displaying joints (see Section 5.2.3).
K
“Knife” (cut blocks) All visible blocks intersected by the cut-plane are split into two. The cut-plane is parallel to the screen.
L
Struct Structural liner or cable information is displayed on 3D wireframe plots. Thekey only operates (and is visible in the menu in the place of the DisplayTarget menu item) after the (wireframe) key is pressed. When the key is pressed, the Liner/Cable menu appears (see Section 5.2.5).
T
Display Target A target or cursor is displayed (except in wireframe mode), and the TargetActive menu appears (see Section 5.2.4). The cursor can be moved with the arrow keys or with a mouse.
M
Magnify The size of the plot is magnified. Repeated typing of thekey increases the magnification.
O
Special Options Plotting options are available by pressing thekey. When this key is pressed, the SpecialOptions menu appears (see Section 5.2.6).
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Quit Plot Mode The program switches to command-line mode (see Section 1 in the Command Reference).
R
Run (do cycles) The model will begin cycling from the screen mode. When thekey is pressed, the Running menu appears. The cycle number and values of variables, if selected in the Stresses or Vectors menus, are printed in the menu box. The model plot is refreshed every 10 cycles. Press the key to stop the run.
S
Stresses Stresses are displayed on surfaces or two-dimensional cross sections through the model. This menu permits plotting principal stress tensors, planar tractions, stress contours and plasticity indicators. Thekey only operates (and is visible in the menu) after the(cross section) key is pressed. When the key is pressed, the Stress menu appears (see Section 5.2.7).
U
“Un-magnify” The size of the plot is diminished (un-magnified). Repeated typing of the key decreases the magnification.
V
Vec/Con Vectors are displayed in wireframe mode in 3D or cross-section plots. Contours are displayed on cross-section plots. If in 3D block model mode, thekey only operates (and is visible in the menu) after the (wireframe) key is pressed. When the key is pressed, the Vector(andContours) menu appears, and displacement and velocity vectors and contours can be plotted (see Section 5.2.8). If in JointStructure mode ( key and <1> key), then the key operates to plot joint stress and displacement contours (see Section 5.2.8).
W
Wire-Frame Display A wireframe plot of the model is displayed. The solid model plot is displayed when thekey is pressed again.
X
Cross Section A two-dimensional cross section is created through the model at the position of the cut-plane.
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The following seven keys in the Main Menu control the action of the arrow keys in the numeric keypad. The arrow keys are used to move objects, change the view, move the cursor, and so on. The particular action of the keys depends on the mode that is currently set. Moving the mouse with the left button pressed is the same as repeatedly pressing the arrow keys. There are 5 modes, set by typing the appropriate key (on the top row of the keyboard):
1
Eye Position In mode 1, the up and down arrows move the eye position closer to or farther from the object (i.e., the perspective view is changed).
2
left-right up-down In mode 2, the four arrow keys cause the displayed block system to move to the left or right, or up or down.
3
x-rotate y-rotate In mode 3, the up/down arrows cause the block system to rotate about an axis pointing to the right in the plane of the screen; the left/right keys cause a rotation about an axis pointing upward in the plane of the screen.
4
Move Cut-plane In mode 4, the up/down keys cause the cut-plane to move nearer to or farther from the screen. The cut-plane is always oriented parallel with the plane of the screen.
5
z-move z-rotate In mode 5, the up/down keys cause the block system to move nearer to or farther from the screen; the left/right keys cause a rotation about an axis normal to the plane of the screen.
+
Increase Movement This increases (by a factor of 5) the movement caused by the arrow keys.
–
Decrease Movement This decreases (by a factor of 5) the movement caused by the arrow keys.
3DEC Version 3.0
GRAPHICAL INTERFACE
(F2)
5-7
PCX File Each timeis pressed, a PCX image of the screen is created. Any previously existing file will be overwritten unless autoname is on.
(F9)
Movie Capture This captures screen plots to a movie file for later replay as a movie. The <(F9)> key only operates (and is visible in the menu) after the MOVIE on command is given in command-line mode (see Section 1.3 in the Command Reference).
(F10)
Copy PostScript/Bitmap/Printer This causes a hardcopy plot of the current screen plot to be made. This keystroke performs the same action as the COPY command (see Section 1.3 in the Command Reference). The file type is defined by the SET plot command. The file name is set by the SET out command.
The main menu also contains information on the current viewing position of the model plot, cutplane, magnification and cycle number. The model view is defined in terms of the position of the viewing plane relative to the model reference axes. The viewing plane is located by a dip angle and dip direction* relative to the model axes and by location of the center of the viewing plane (i.e., the screen center) relative to the origin of the model axes. (See Figure 5.3.)
* In 3DEC, the following definitions apply. Bearing is the horizontal angle measured clockwise between a line in the horizontal plane (i.e., the model xz-plane) and the z- (North) coordinate direction. Dip angle is the inclination (in degrees) of the line of greatest slope of an inclined plane measured relative to the horizontal plane (i.e., the model xz-plane). Dip direction is the bearing in the horizontal plane of the dip angle projected to the horizontal plane, measured in degrees from model North (+ z-axis). Strike is the bearing of the intersection between a given plane and a horizontal plane. Strike is perpendicular to the dip direction.
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User’s Guide
dip: Dip angle, in degrees, of the viewing plane is measured downward from the model xz-plane. The range for the dip angle is 0 ≤ dip ≤ 90◦ . If the dip direction is pointing from the model axes toward the viewing plane, the word “above” follows the dip angle. If the dip direction is pointing away from the viewing plane, the word “below” follows the dip angle. (default dip = 90◦ ) dd: Dip direction, in degrees, of the viewing plane is measured clockwise from the positive z-axis. The range for dip direction is 0 ≤ dd ≤ 360◦ . (default dd = 180◦ ) center: Location of the center of the viewing plane is measured relative to the origin of the model axes. (default center = x,y,z-coordinates of model centroid) cut-pl.: Location of the cut-plane is measured relative to the center of the viewing plane. The value is negative if the cut-plane is located inside the screen (away from the viewer) and is positive if located outside the screen (toward the viewer). The default location of the cut-plane is parallel and coincident with the viewing plane. mag: magnification factor (default mag = 1) cycle: current cycle number
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GRAPHICAL INTERFACE
5-9
z
model axes
y x DD
DIP
center distance
center
2.0 3DEC Version
viewing plane (screen)
Figure 5.3
Location of viewing plane in terms of dip, dip direction and center distance from model axes
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User’s Guide
5.2.2 Select Color Mode Menu
B
Block Sequence The blocks are plotted in different colors. Colors will change as the model is altered.
C
Constitutive Number The block colors are assigned according to constitutive model number:
M
0
(null model)
blue
1
(elastic model)
green
2
(Mohr Coulomb model)
cyan
3
(anisotropic elastic model)
red
6
(bilinear strain softening Mohr Coulomb matrix ubiquitous joint model)
blue
Material Type The block colors are assigned according to material type number specified for the block (colors are mod 6, so material 7 will be the same color as material 1):
R
1
green
6
blue
2
cyan
7
green
3
red
8
cyan
4
magenta
9
red
5
yellow
10 magenta
Region Number The block colors are assigned according to region number specified for the block (colors are mod 6):
Z
0
green
4
yellow
1
cyan
5
blue
2
red
6
green
3
magenta
colors repeat
Freeze Color The present block color state is frozen and will not change when blocks are hidden or deleted.
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5.2.3 Select Joint Mode Menu
0
Joint Mode Off Blocks are plotted.
1
Joint Structure All joints are plotted in perspective view. (Blocks are hidden from view.) Joints can be plotted individually by joint number n if the PLOT joint n command is issued first from the command mode.
2
Cons Near X Sect Only joints within a tolerance, ETOL, of the viewing plane are plotted. (Type PRINT state for current value of ETOL. Type SET etol to change tolerance.)
3
Flow Structure Shows fluid flow planes.
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User’s Guide
5.2.4 Target Active Menu In TargetActive mode, either a mouse or the arrow keys can be used to move the cursor. The left mouse button performs the same action as thekey. To return to the main menu from the TargetActive menu, press the key.
I
Interrogate Block When the key is pressed, the cursor can be moved to each block in the model; then, by pressing thekey or left mouse button, current information on the block will appear in the menu box. An example menu is shown in Figure 5.4. Press (or right mouse button) to return to the TargetActive menu.
C
Color Block The color of a block can be changed. When thekey is pressed, a select BlockColor menu will appear, and a different color can be chosen. The menu contains the following color choices: 0
black
4
red
1
blue
5
magenta
2
green
6
yellow
3
cyan
7
white
First, press the number corresponding to the selected color and(or left button) to change the block to this color; then, press to return to the TargetActive menu.
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GRAPHICAL INTERFACE
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Block 34527 Mat 1 Con 1 Reg 0 Rigid Block X 7.45E-01 Y 5.77E-01 Z -4.96E-01 Volume 2.64E-01 Mass 5.29E+02 X Y Z Tx Ty Tz
Disp 5.07E-07 -6.03E-06 -5.60E-07 0.00E+00 0.00E+00 0.00E+00
Block 217 Mat 2 Con 1 Reg 10 Zones 6 X 1.50E+00 Y 2.50E+00 Z 0.00E+00 Volume 3.50E+02 Mass 5.29E+02 Vel 1.25E-04 -8.55E-05 -1.35E-04 -4.26E-04 -5.25E-06 -4.41E-04
Force Sums X 0.00E+00 Y 0.00E+00 Z 0.00E+00 Tx -3.68E+02 Ty -4.53E+00 Tz -3.40E+02
(a) rigid block information Figure 5.4
X Y Z Tx Ty Tz
Disp -3.99E-07 -5.42E-06 0.00E+00 0.00E+00 0.00E+00 0.00E+00
Vel 1.19E-10 2.64E-10 0.00E+00 0.00E+00 0.00E+00 0.00E+00
Stress Avg S1 -6.61E+04 S2 -1.40E+04 S3 -1.08E+04 SS 2.77E+04
(b) deformable block information
Example interrogate block menu
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P
User’s Guide
Material # The material type number for a block can be changed. When thekey is pressed, a (P)SelectMaterialNumber menu will appear, and a different material type number can be chosen. This menu contains the following material number choices: 1 material 1
6
material 6
2 material 2
7
material 7
3 material 3
8
material 8
4 material 4
9
material 9
5 material 5
a
material 10
First, press the number or letter corresponding to the selected material type number and(or left button) to change the block to this number; then, press to return to the TargetActive menu.
O
Constitutive # The constitutive number for a block can be changed. When thekey is pressed, a (O)SelectConstitutiveNumber menu will appear, and a different constitutive number can be chosen. This menu contains the following constitutive number choices: 1 2
cons 1 cons 2
elastic Mohr-Coulomb
First, press the number corresponding to the selected constitutive model and(or left button) to change the block to this number; then, press to return to the TargetActive menu.
J
Join 2 Blocks Two blocks can be joined. (One block becomes the master block and the other the slave.) After thekey is pressed, the cursor is moved to the first block to be joined and the key (or left mouse button) pressed; then, the cursor is moved to the second block (it must be adjacent to the first), and the key (or left button) pressed again. The two blocks will then be joined. Several blocks can be joined by repeating this procedure.
3DEC Version 3.0
GRAPHICAL INTERFACE
F
5 - 15
Find Block Hidden blocks can be found. If thekey is pressed, the key (or left button) can then be pressed to restore (i.e., make visible) the last block hidden while in text mode. If the key (or left button) is pressed repeatedly, all hidden blocks will be restored.
H
Hide Block Blocks are hidden from view. After thekey is pressed, the cursor can be moved to a block that is to be hidden; then, by pressing the key (or left button), the block will be made invisible. It is put on a stack and can be recalled with the key in TargetActive mode. When blocks are invisible, they cannot be split by the key or JSET command; in this way, discontinuous joints can be made. However, the invisible blocks still interact normally with other blocks and are remembered on restart. Only visible blocks can be deleted, by the key or by the DELETE command. Only visible blocks are affected by the CHANGE command and have region numbers assigned or changed by the MARK command.
D
Delete Block Blocks are deleted. After thekey is pressed, the cursor can be moved to a block that is to be deleted; then, by pressing the key (or left button), the block will be deleted. Caution: Deleted blocks cannot be restored.
E
Face Generator This is a utility which allows the user to click on three scalar symbols on the screen and generate a POLY face command in the log file. Note that SET log on must be specified by the user.
G
Joint Set Generator This is a utility which allows the user to click on three scalar symbols and generate a JSET command which would cut the defined plane. Note that SET log on must be specified by the user.
L
Line Generator This is a utility which allows the user to draw on the surface of a block. The line segments are appended to the overlay plotting file. The default name for this file is “OVERLAY.TXT.”
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S
User’s Guide
Scalar Generator This is a utility which creates scalars for each click of the mouse. The type and magnitude are both equal to 1. If SET log on is specified, the SCALAR command will also be written to the log.
A
Automatic Refresh When the key is active, the plot will be redrawn (screen refreshed) each time a block is hidden or deleted.
M
Manual Refresh When thekey is active, the plot will not be redrawn (screen refreshed) each time a block is hidden or deleted. The screen will only be refreshed when the key is pressed.
R
Refresh The screen is refreshed. Thekey is only active (and visible in the menu) after the key is pressed. When is pressed, the plot is redrawn (screen refreshed).
X
3D View/Cross Section A switch from 3D perspective view to cross-section view can be made. Thekey is pressed to switch between 3D perspective view and cross-section view while the target is active.
Z
Zoom Zoom is used to window in on a specific area. A cross-hair cursor will appear. Move the cross-hair to one corner of the new window. Press and hold the left mouse button. Stretch the rubber-band box to the new dimension and release the left mouse button again. Use the (un-magnify) key in the main menu to zoom back out again.
(ins)
Change Speed The speed of the cursor movement with the arrow keys is changed. The <(ins)> key is a toggle to increase or decrease the movement of the cursor when an arrow key is pressed.
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5.2.5 Structure Menu
1
Axial Reinforcement Local reinforcement locations.
2
Axial Force — Axial Reinforcement Axial (local) reinforcement axial forces are plotted.
3
Cable Bolt Cable geometry is plotted.
4
Axial Force — Cable Axial cable forces are plotted at the center of each element.
5
Liner Liner plate elements are plotted.
6
Beam Beam element location.
M
Color by Magnitude The axial cable force vectors are colored by magnitude.
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User’s Guide
5.2.6 Special Options Menu
C
Freeze Colors Block colors will not change when blocks are deleted, excavated or hidden.
E
Show Excavations Only excavated blocks (i.e., blocks removed for calculation purposes by the EXCAVATE or REMOVE command; see Section 1.3 in the Command Reference) are plotted.
F
Freeze Scale The present scale for vector plots will remain constant for all vector plots. The scale can be “unfrozen” by returning to the SpecialOptions menu and pressing thekey again.
A
x-y-z Axis Labels x-, y-, z-coordinate axes are drawn in the lower-left corner of the screen to help the user orient the model. The axes can be removed by pressing the key again.
N
N-E-U Axis Labels N- (north), E- (east), U- (up) coordinate axes are drawn in the lower-left corner of the screen to help the user orient the model. The axes can be removed by pressing thekey again.
L
Hardcopy Legend A legend box replaces the menu box with the menu used for hardcopy screen dumps. When the legend box is visible, all screen-mode keys still operate. To return to the menu box, type thekey followed by the key and
.
J
Joint Material plots contacts associated with a joint plane. Contacts are displayed as diamonds if the contact type is face-to-face. All other contact types are displayed as arrows. The color of the symbol indicates the joint material type number. (For clarity, this menu item should only be used with one joint plane at a time.)
S
User-Defined Scalars plots scalar quantities defined by the SCALAR command (see Section 1.3 in the Command Reference).
3DEC Version 3.0
GRAPHICAL INTERFACE
V
5 - 19
User-Defined Vectors plots vector quantities defined by the VECTOR command (see Section 1.3 in the Command Reference).
Y
User-Defined Tensors plots tensor quantities defined by the TENSOR command (see Section 1.3 in the Command Reference).
Z
Plot Zone Outlines plots zone outlines on faces of deformable blocks.
H
Hide MS Construction does not plot lines between joined blocks.
P
Perspective Plot turns on/off perspective plotting.
D
DXF turns on DXF file overlay defined by the PLOT dxf or SET dxf command.
K
User-Defined Labels allows user-defined labels to be plotted.
1
X-Boundary Condition plots symbols which represent the boundary condition applied to gridpoints in the x-direction.
2
Y-Boundary Condition plots symbols which represent the boundary condition applied to gridpoints in the y-direction.
3
Z-Boundary Condition plots symbols which represent the boundary condition applied to gridpoints in the z-direction.
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User’s Guide
5.2.7 Stresses Menu
P
Principal Stresses plots principal stress tensors, indicating magnitudes and directions. By default, tensors are colored by magnitude of maximum compressive stress, PS1.
T
Planar Tractions plots tractions acting on cross section. Circles indicate relative magnitude of tractions, and an arrow indicates direction and relative magnitude of shear component of traction. If only a dot shows in the center of the circle, the traction is entirely a normal stress; if the arrow extends to the full radius of the circle, the traction is entirely a shear stress. By default, circles are colored by the magnitude of the ratio of shear-to-normal stress.
C
Select Color Mode The color mode for stress and planar traction plots can be changed. When thekey is pressed, a SelectColorMode menu appears, and a different color mode can be chosen. For stress plots, the menu contains the following menu choices:
0
compr/tens Stresses are colored by magnitude relative to maximum compressive stress.
1
sigma 1 Stresses are colored by magnitude relative to maximum major principal stress, PS1.
2
sigma 2 Stresses are colored by magnitude relative to intermediate principal stress, PS2.
3
sigma 3 Stresses are colored by magnitude relative to maximum minor principal stress, PS3.
4
shear Stresses are colored by magnitude relative to maximum shear stress.
3DEC Version 3.0
GRAPHICAL INTERFACE
5
5 - 21
slip Stresses are colored by slip condition. Parameters must be set by a SET pltphi command.
6
plastic state Stresses are colored by plastic state.
7
Mohr-C FOS Stresses are colored by Mohr-Coulomb strength/stress factor. Parameters must be defined by a SET pltphi, SET pltcoh or SET plttens command.
8
Hoek-Brown FOS Stresses are colored by Hoek-Brown strength/stress factor. Parameters are defined by a SET ucs, SET hbs or SET hbm command.
For planar traction plots, the menu contains the following menu choices:
0
all same color All tractions are plotted red.
1
magnitude Tractions are colored by magnitude relative to maximum compressive stress.
2
shear stress Tractions are colored by magnitude relative to maximum shear stress.
3
excess shear stress Tractions are colored by magnitude relative to maximum excess shear stress. Parameters must be set by a SET pltphi or SET pltcoh command.
X
Component Arrow Arrows are displayed on the component of principal stress which is being used to scale colors.
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Y
User’s Guide
Plastic Indicators Plasticity indicators are plotted, identifying zones that have failed based on the MohrCoulomb failure criterion. If the symbols are red, the stress state is currently at the yield limit. If the color is cyan, the stress state is below the yield limit. The symbols shown below identify the failure mode. If no symbol is shown, the zone is elastic and has never failed.
matrix shear matrix tension ubiquitous joint shear ubiquitous joint tension Figure 5.5
1
Symbols identifying failure mode
Max. p.s. Contours maximum principal stress contours*
2
Int. p.s. Contours intermediate principal stress contours∗
3
Min. p.s. Contours minimum principal stress contours∗
4
xx-stress Contours xx-stress contours∗
* Use the PLOT command to control contouring parameters.
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5
5 - 23
xy-stress Contours xy-stress contours∗
6
xz-stress Contours xz-stress contours∗
7
yy-stress Contours yy-stress contours∗
8
yz-stress Contours yz-stress contours∗
9
zz-stress Contours zz-stress contours∗
A
Use Zone Averaging The zone averaging technique for contouring stresses (see the PLOT command in Section 1.3 in the Command Reference) is turned on or off.
B
Use Block Fill Contouring Fills entire area of zone cross section with color representing the stress in that zone. This more accurately represents the stresses than zone averaging.
G
Show Contour Grid The grid used to interpolate values for stress contour plotting is displayed.∗
Z
Show Zone X Sect The intersections of zones with the cross-section plane are plotted.
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User’s Guide
5.2.8 Vectors (and Contours) Menu If in 3D block model mode, the following keys operate.
D
Displacement Vectors Displacement vectors are plotted at vertices.
N
Joint Normal Displ. Relative normal displacement vectors along joints are plotted at contacts.
S
Joint Shear Displ. Relative shear displacement vectors along joints are plotted at contacts.
V
Velocity Vectors Velocity vectors are plotted at vertices.
C
Color by Magnitude The color of vectors can be changed. When thekey is pressed, a SelectColor menu will appear, and a different color mode can be chosen. The menu contains the following color choices.
0
All Same Color All vectors are red.
1
magnitude Vectors are colored by magnitude relative to maximum vector value.
2
x-displacement/velocity Vectors are colored by magnitude relative to maximum x-displacement (velocity) value.
3
y-displacement/velocity Vectors are colored by magnitude relative to maximum y-displacement (velocity) value.
3DEC Version 3.0
GRAPHICAL INTERFACE
4
5 - 25
z-displacement/velocity Vectors are colored by magnitude relative to maximum z-displacement (velocity) value.
1
x-disp. Contours x-displacement contours*
2
y-disp. Contours y-displacement contours∗
3
z-disp. Contours z-displacement contours∗
4
x-velocity Contours x-velocity contours∗
5
y-velocity Contours y-velocity contours∗
6
z-velocity Contours z-velocity contours∗
T
Temperature Contours temperature contours (only active for thermal configuration)
Z
Show Zone X Sect The intersections of zones with the cross-section plane are plotted.
G
Show Contour Grid The grid is used to interpolate values for displacement, and velocity contour plotting is displayed.∗
* Use the PLOT command to control contouring parameters.
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User’s Guide
If in JointStructure mode, the following keys operate. These keys only apply for deformable blocks:
1
Normal Stress joint normal stress contours
2
Shear Stress joint shear stress contours
3
Joint Normal Displ. joint normal displacement contours
4
Joint Shear Displ. joint shear displacement contours
If in FlowStructure and Solid mode, the following keys operate.
1
Pore Pressure Contour
2
Aperture Contour
If in FlowStructure and Wireframe mode, the following key operates.
1
Discharge Vectors
3DEC Version 3.0
MISCELLANEOUS
6-1
6 MISCELLANEOUS 6.1 3DEC Runtime Benchmark 3DEC has been tested on a number of different computers. The calculation rates are compared here for the following benchmark problem: a cubic model that contains 125 blocks subject to applied pressure boundary conditions. The timing test is made for both a rigid block analysis (with 1000 vertices) and a deformable block analysis (with 750 zones and 1000 gridpoints). The model is run for 1000 steps, and the rate is calculated by a FISH function. The data file is given in Example 6.1; Table 6.1 summarizes the calculation rates for different computers. Table 6.1
3DEC runtime calculation rates
Computer
Deformable Blocks
Pentium Pro Pentium Pro Pentium II Pentium II AMD Athelon AMD Athelon AMD Athelon 1600+ Pentium 4 Pentium 4 Pentium 4
sec / gp / 1000 steps 0.131 0.101 0.088 0.061 0.026 0.021 0.019 0.013 0.012 0.010
200 MHz 266 MHz 300 MHz 450 MHz 1000 MHz 1200 MHz 1400 MHz 2000 MHz 2260 MHz 2800 MHz
3DEC Version 3.0
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Example 6.1 Benchmark data file — “TIMING.DAT” def rate rate = (t1 - t0) / (1000) end def time_0 t0 = clock / 100.0 end def time_1 t1 = clock / 100.0 end poly brick 0,10 0,10 0,10 jset dip 0 dd 180 spac 2 num 20 jset dip 90 dd 180 spac 2 num 20 jset dip 90 dd 90 spac 2 num 20 prop jmat 1 kn 1e9 ks 1e9 fric 45.0 ; ; for deformable block model gen ed 4.0 prop mat 1 dens 1000 prop mat 1 bulk 1e9 g 7e8 ; bound 0,10 9.9,10.0 0,10 stress 0 -2e6 0 0 0 0 bound 9.9,10.1 0,10 0,10 stress -2e6 0 0 0 0 0 bound 0,10 0,10 9.9,10.1 stress 0 0 -2e6 0 0 0 ; ; for deformable block model bound -.1,.1 0,10 0,10 xvel 0.0 bound 0,10 -.1,.1 0,10 yvel 0.0 bound 0,10 0,10 -.1,.1 zvel 0.0 ; ; for rigid block model ; apply 0 2 0 10 0 10 xvel 0.0 ; apply 0 10 0 2 0 10 yvel 0.0 ; apply 0 10 0 10 0 2 zvel 0.0 time_0 cyc 1000 time_1 print rate
3DEC Version 3.0
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MISCELLANEOUS
6-3
6.2 Error Reporting Although 3DEC has been tested extensively, it is almost impossible to test all available combinations of options in a code as complex as 3DEC. For this reason, some errors may have evaded our notice. If you discover a genuine bug, please let us know as soon as possible so that we may correct it. 6.2.1 Reporting via Internet Itasca’s current Internet e-mail address is [email protected] Please include the same information requested on the error notification form (in Section 6.2.2), followed by the contents of your data file. 6.2.2 Reporting via Fax A sample form for you to copy and mail or fax to us is given on the next page. Please fill out the form completely, as this is the minimum information we will need to find and correct the error. The sample file should, if possible, contain the minimum number of commands necessary to produce the error. We may have to contact you for further information if we are unable to duplicate the error. Be aware that it is always possible that the error is peculiar to your hardware, making it impossible for us to duplicate. 6.3 Technical Support Service Itasca and its offices and agents will provide telephone support, at no cost, to assist code owners in the installation of Itasca codes on their computer system. Additionally, general assistance may be provided in aiding the owner to understand the capabilities of the various features of the code. However, no-cost assistance is not provided for help in applying an Itasca code to specific userdefined problems. Questions should, in the first instance, be directed to the office or agent where 3DEC was purchased.
3DEC Version 3.0
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MISCELLANEOUS
6-5
ERROR NOTIFICATION FORM Found By:
Phone:
Fax:
Email: Computer:
RAM:
3DEC Data Serial No.:
Version*
Key No.:
Options
Description:
* Type PRINT version to report your complete version number Please attach a sample input file that produces the error. Itasca Consulting Group, Inc. Mill Place 111 Third Avenue South, Suite 450 Minneapolis, Minnesota 55401 USA
Phone: Fax: E-Mail: Web:
(1) 612-371-4711 (1) 612·371·4717 [email protected] www.itascacg.com
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BIBLIOGRAPHY
7-1
7 BIBLIOGRAPHY Adachi, T., Y. Ohnishi and K. Arai. “Investigation of Toppling Slope Failure at Route 305 in Japan,” in Proceedings of the 7th International Congress on Rock Mechanics (Aachen, Germany, September, 1991), Vol. 2, pp. 843-846. Rotterdam: A. A. Balkema, 1991. Al-Harthi, A., and S. Hencher. “Physical and Numerical Modelling of Underground Excavations in Dilational Rock Masses,” in Proceedings of the ISRM Regional Conference on Fractured and Jointed Rock Masses (Lake Tahoe, June, 1992), Vol. 3, pp. 608-615. Berkeley, California: Lawrence Berkeley Laboratory, 1992. Alfonsi, P., J. L. Durville and X. Rachez. “Modelisation Numerique d’une Fondation sur Versant Rocheux par la Methode des Elements Distincts: Comparaison 2-D/3-D,” in Proceedings of the 9th ISRM Congress on Rock Mechanics (Paris, 1999), Vol. 1, pp. 71-76. Rotterdam: A. A. Balkema, 1999. Antikainen, J., A. Simonen and I. Makinen. “3D Modelling of the Central Pillar in the Pyhasalmi Mine,” in Innovative Mine Design for the 21st Century (Proceedings of the International Congress on Mine Design, Kingston, Ontario, Canada, August, 1993), pp. 631-640. W. F. Bawden and J. F. Archibald, Eds. Rotterdam: A. A. Balkema, 1993. Bandis, S. C., N. R. Barton and M. Christianson. “Application of a New Numerical Model of Joint Behaviour to Rock Mechanics Problems,” in Fundamentals of Rock Joints (Proceedings of the International Symposium on Fundamentals of Rock Joints, Björkliden, September, 1985), pp. 345-356. Luleå, Sweden: Centek Publishers, 1985. Barla, G., M. Borri-Brunetto and G. Gerbaudo. “Physical and Mathematical Modelling of a Jointed Rock Mass for the Study of Block Toppling,” in Proceedings of the ISRM Regional Conference on Fractured and Jointed Rock Masses (Lake Tahoe, June, 1992), Vol. 3, pp. 616-623. Berkeley, California: Lawrence Berkeley Laboratory, 1992. Baroudi, H., J. P. Piguet, J. P. Josien, I. Arif and P. Lebon. “Modelling of Underground Cavity Storage and Consideration of Rock Mass Discontinuities,” in Proceedings of the 7th International Congress on Rock Mechanics (Aachen, Germany, September, 1991), Vol. 2, pp. 1073-1081. Rotterdam: A. A. Balkema, 1991. Barton, N. “Modelling Jointed Rock Behavior and Tunnel Performance,” World Tunnelling, 4(7), 414-416 (November, 1991). Barton, N., L. Harvik, M. Christianson, S. Bandis, A. Makurat, P. Chryssanthakis and G. Vik. “Numerical Analyses and Laboratory Tests to Investigate the Ekofisk Subsidence,” Fjellsprengningsteknikk Bergmekanikk/Geoteknikk, 21.1-21.23, 1985. Barton, N., L. Harvik, M. Christianson and G. Vik. “Estimation of Joint Deformations, Potential Leakage and Lining Stresses for a Planned Urban Road Tunnel,” in Large Rock Caverns (Proceedings of the Conference on Large Rock Caverns, Helsinki, 1986), pp. 1171-1182. Oxford: Pergamon Press, 1985.
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User’s Guide
Barton, N., R. Lien, F. Løset, T. Löken, E. Grimstad, H. Hansteen, L. Harvik and M. Christianson. “Methods for Selecting Support in Sub-Sea Rock Tunnels,” Proceedings of the International Symposium on Strait Crossings (Stavanger, Norway, October, 1986), Vol. 2, pp. 715-731. Trondheim, Norway: Tapir, 1986. Barton, N., F. Løset, A. Smallwood, G. Vik, C. Rawlings, P. Chryssanthakis, H. Hansteen and T. Ireland. “Radioactive Waste Repository Design Using Q and UDEC-BB,” in Proceedings of the ISRM Regional Conference on Fractured and Jointed Rock Masses (Lake Tahoe, June, 1992), Vol. 3, pp. 735-742. Berkeley, California: Lawrence Berkeley Laboratory, 1992. Barton, N., A. Makurat, M. Christianson and S. Bandis. “Modelling Rock Mass Conductivity Changes in Disturbed Zones,” in Rock Mechanics: Proceedings of the 28th U.S. Symposium (Tucson, June-July, 1987), pp. 563-574. Rotterdam: A. A. Balkema, 1987. Barton, N., K. Monsen, P. Chryssanthakis and O. Norheim. “Rock Mechanics Design for High Pressure Gas Storage in Shallow Lined Caverns,” in Storage of Gases in Rock Caverns, pp. 159176. Rotterdam: A. A. Balkema, 1989. Barton, N., L. Tunbridge, F. Løset, H. Westerdahl, J. Kristiansen, G. Vik and P. Chryssanthakis. “Norwegian Olympic Ice Hockey Cavern of 60 m Span,” in Proceedings of the 7th International Congress on Rock Mechanics (Aachen, Germany, September, 1991), Vol. 2, pp. 1073-1081. Rotterdam: A. A. Balkema, 1991. Bigarre, P., K. Ben Slimane and J. Tinucci. “3-Dimensional Modelling of Fault-Slip Rockbursting,” in Rockbursts and Seismicity in Mines 93 (Proceedings of the International Symposium, Kingston, Ontario, Canada, August, 1993), pp. 315-319. R. Paul Young, Ed. Rotterdam: A. A. Balkema, 1993. Blair, Stephen C., Steven R. Carlson and Jeffrey L. Wagoner. “Analysis of Geomechanical Behavior for the Drift Scale Test,” in Proceedings of the 9th International High-Level Radioactive Waste Management Conference (IHLRWM, Las Vegas, April-May 2001), Paper 08-3. La Grange Park, Illinois: American Nuclear Society, Inc., 2001. Blair, Stephen C., Steven R. Carlson and Jeffery L. Wagoner. “Distinct Element Modeling of the Drift Scale Test,” in Rock Mechanics in the National Interest (Proceedings of the 38th U. S. Rock Mechanics Symposium, Washington, D.C., July 2001), Vol.1, pp. 527-531. Lisse, The Netherlands: Swets & Zeitlinger B. V., 2001. Board, M. Examination of the Use of Continuum versus Discontinuum Models for Design and Performance Assessment for the Yucca Mountain Site. U.S. Nuclear Regulatory Commission, NUREG/CR-5426, August, 1989. Board, M. UDEC (Universal Distinct Element Code) Version ICG1.5, Vols. 1-3. U.S. Nuclear Regulatory Commission, NUREG/CR-5429, September 1989. Board, Mark, Richard Brummer and Shawn Seldon. “Use of Numerical Modeling for Mine Design and Evaluation,” in Underground Mining Methods: Engineering Fundamentals and International Case Studies, pp. 483-491. W. A. Hustrulid and R. L. Bullock, Eds. Littleton, Colorado: SME, 2001.
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Board, M., T. Rorke, G. Williams and N. Gay. “Fluid Injection for Rockburst Control in Deep Mining,” in Rock Mechanics (Proceedings of the 33rd U.S. Symposium on Rock Mechanics), pp. 111-120. J. R. Tillerson and W. R. Wawersik, Eds. Rotterdam: A. A. Balkema, 1993. Board, M., S. Seldon, R. Brummer and R. Pakalnis. “Analysis of the Failure of a Large Hangingwall Wedge: Kidd Mine Division, Falconbridge, Ltd.,” CIM Bull., 93(1043), 89-97 (September 2000). Borri-Brunetto, M. “A Direct Variational Approach to Static Analysis of Discontinua,” in Second European Speciality Conference on Numerical Methods in Geotechnical Engineering (Santander, Spain, September, 1990), pp. 33-44, 1990. Brady, B., and J. Lemos. “Dynamic Analysis of Surface Rock Structures,” in Proceedings of the 2nd Italian Conference on Rock Mechanics and Rock Engineering, 1988. Brady, B. H., S. H. Hsiung, A. H. Chowdhury and J. Philip. “Verification Studies on the UDEC Computational Model of Jointed Rock,” in Mechanics of Jointed and Faulted Rock, pp. 551-558. Rotterdam: A. A. Balkema, 1990. Brady, B. H. G., M. L. Cramer and R. D. Hart. “Preliminary Analysis of a Loading Test on a Large Basalt Block,” Int. J. Rock Mech., 22(5), 345-348 (1985). Chen, S. G., J. G. Cai, J. Zhao and Y. X. Zhou. “3DEC Modeling of a Small-Scale Field Explosion Test,” in Pacific Rocks 2000: Rock Around the Rim (Proceedings of the 4th North American Rock Mechanics Symposium, Seattle, July-August 2000), pp. 571-576. J. Girard et al., Eds. Rotterdam: A. A. Balkema 2000. Choi, S. K. “The Application of the Distinct Element Method for Rock Mechanics Problems,” in Proceedings of the 1st U.S. Conference on Discrete Element Methods (Golden, Colorado, 1989). G. G. W. Mustoe et al., Eds. Golden, Colorado: CSM Press, 1990. Choi, S. K., and M. A. Coulthard. “Modelling of Jointed Rock Masses Using the Distinct Element Method,” in Mechanics of Jointed and Faulted Rock, pp. 471-478. Rotterdam: A. A. Balkema, 1990. Christianson, M. Sensitivity of the Stability of a Waste Emplacement Drift to Variation in Assumed Rock Joint Parameters in Welded Tuff. U.S. Nuclear Regulatory Commission, NUREG/CR-5336, April, 1989. Christianson, M., J. Itoh and S. Nakaya. “Seismic Analysis of the 25 Stone Buddhas Group at Hakone, Japan,” in Rock Mechanics (Proceedings of the 35th U.S. Symposium, University of Nevada, Reno, June, 1995), pp. 107-112. J. J. K. Daemen and R. A. Schultz, Eds. Rotterdam: A. A. Balkema, 1995. Christianson, M. C., and B. Brady. Analysis of Alternative Waste Isolation Concepts. U.S. Nuclear Regulatory Commission, NUREG/CR-5389, June, 1989. Chryssanthakis, P., and N. Barton. “Predicting Performance of the 62 m Span Ice Hockey Cavern in Gjøvik, Norway,” in Proceedings of the ISRM Regional Conference on Fractured and Jointed Rock Masses (Lake Tahoe, June, 1992), Vol. 3, pp. 624-631. Berkeley, California: Lawrence Berkeley Laboratory, 1992.
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Chryssanthakis, P., K. Monsen and N. Barton. “Validation of UDEC-BB Against the CSM Block Test and Large Scale Application to Glacier Loading of Jointed Rock Masses,” in Proceedings of the 7th International Congress on Rock Mechanics (Aachen, Germany, September, 1991), Vol. 1, pp. 693-698. Rotterdam: A. A. Balkema, 1991. Contador, Nolberto V., and Marcelo F. Glavic. “Sublevel Open Stoping at El Soldado Mine: A Geomechanic Challenge,” in Underground Mining Methods: Engineering Fundamentals and International Case Studies, pp. 325-332. W. A. Hustrulid and R. L. Bullock, Eds. Littleton, Colorado: SME, 2001. Coulthard, M. A. “Distinct Element Modelling of Mining-Induced Subsidence — A Case Study,” in Proceedings of the ISRM Regional Conference on Fractured and Jointed Rock Masses (Lake Tahoe, June, 1992), Vol. 3, pp. 751-758. Berkeley, California: Lawrence Berkeley Laboratory, 1992. Coulthard, M. A., and I. H. Clark. “Computational Stress Analysis for Mine Excavation Design,” in Computer Applications in the Mineral Industry (Proceedings of the 2nd Australian Conference, The University of Wollongong, N.S.W., July, 1991), pp. 165-170. E. Y. Baafi, Ed. Wollongong: University of Wollongong, 1991. Coulthard, M. A., and A. J. Dutton. “Numerical Modeling of Subsidence Induced by Underground Coal Mining,” in Key Questions in Rock Mechanics: Proceedings of the 29th U.S. Symposium (University of Minnesota, June, 1988), pp. 529-536. Rotterdam: A. A. Balkema, 1988. Coulthard, M. A., N. C. Journet and C. F. Swindells. “Integration of Stress Analysis into Mine Excavation Design,” in Rock Mechanics (Proceedings of the 33rd U.S. Symposium, Santa Fe, June, 1992), pp. 451-460. J. R. Tillerson and W. R. Wawersik, Eds. Rotterdam: A. A. Balkema, 1992. Cundall, P. A. “Alternative User Interfaces for Programs that Model Nonlinear Systems,” in Applications of Computational Mechanics in Geotechnical Engineering, pp. 343-352. Vargas et al., Eds. Rotterdam: A. A. Balkema, 1994. Cundall, P. A. “Formulation of a Three-Dimensional Distinct Element Model — Part I: A Scheme to Detect and Represent Contacts in a System Composed of Many Polyhedral Blocks,” Int. J. Rock Mech., Min. Sci. & Geomech. Abstr., 25, 107-116 (1988). Cundall, P. A. “Numerical Modeling of Jointed and Faulted Rock,” in Mechanics of Jointed and Faulted Rock, pp. 11-18. Rotterdam: A. A. Balkema, 1990. Cundall, P. A., and R. D. Hart. “Development of Generalized 2-D and 3-D Distinct Element Programs for Modeling Jointed Rock,” Itasca Consulting Group Report to U.S. Army Engineering Waterways Experiment Station, May, 1983; published as Misc. Paper SL-85-1, U.S. Army Corps of Engineers, 1985. Cundall, P. A., and R. D. Hart. “Numerical Modeling of Discontinua,” in Comprehensive Rock Engineering, Vol. 2, pp. 231-243. J. A. Hudson, Sr. Ed. Oxford: Pergamon Press Ltd., 1993.
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Cundall, P. A., and J. V. Lemos. “Numerical Simulation of Fault Instabilities with the Continuously Yielding Joint Model,” in Rockbursts and Seismicity in Mines, pp. 147-152. C. Fairhurst, Ed. Rotterdam: A. A. Balkema, 1990. Damjanac, Branko, and Charles Fairhurst. “Ecoulement tri-dimensionnel d’eau sous pression dans les milieux fracturés," in La sécurité des grands ouvrages: Hommage á Pierre Londe (October 2000), pp. 5-19. Paris: Presses de l’école nationale des Ponts et chaussées, 2000. Damjanac, Branko, Charles Fairhurst and Terje Brandshaug. “Numerical Simulation of the Effects of Heating on the Permeability of a Jointed Rock Mass,” in Proceedings of the 9th ISRM Congress on Rock Mechanics (Paris, 1999), Vol. 2, pp. 881-885. Rotterdam: A. A. Balkema, 1999. Dasgupta, B. “Numerical Modeling of Large Underground Caverns for Hydro Power Projects,” in Trends in Rock Mechanics, Geotechnical Special Publication No. 102, Proceedings of Sessions of Geo-Denver 2000 (August 2000, Denver), pp. 50-64. J. F. Labuz et al., Eds. Reston, Virginia: ASCE, 2000. Dasgupta, B, R. Dham and L. J. Lorig. “Three-Dimensional Discontinuum Analysis of the Underground Power House for Sardar Sarovar Project, India,” in Proceedings of the Eighth International Congress on Rock Mechanics (Tokyo, September 1995), Vol. II, pp. 551-554. T. Fujii, Ed. Rotterdam: A. A. Balkema, 1995. Dasgupta, B., and L. J. Lorig. “Numerical Modelling of Underground Power Houses in India,” in Proceedings of the International Workshop on Observational Method of Construction of Large Underground Caverns in Difficult Ground Conditions, (8th ISRM International Congress on Rock Mechanics, Tokyo, September, 1995), pp. 65-74. S. Sakurai, Ed. Dasgupta, B., K. N. Reddy and S. Nayak. “Discontinuum Analysis of Rock Slopes,” in Proceedings of the Asian Regional Symposium on Rock Slopes (New Delhi, December, 1992), pp. 157-164. New Delhi: Oxford EIBH Publications, 1993. Dasgupta, B., and V. M. Sharma. “Numerical Modelling of Underground Power Houses in India,” in Distinct Element Modeling in Geomechanics, pp. 187-217. V. M. Sharma et al., Eds. New Delhi, Oxford & IBH Publishing Co., 1999. Dasgupta, B., M. K. V. Sharma, M. Verman and V. M. Sharma. “Design of Underground Caverns for Tehri Hydropower Project, India by Numerical Modelling,” in Proceedings of the 9th ISRM Congress on Rock Mechanics (Paris, 1999), Vol. 1, pp. 357-358. Rotterdam: A. A. Balkema, 1999. Dialer, C. “A Distinct Element Approach for the Deformation Behavior of Shear Stressed Masonry Panels,” in Proceedings of the 6th Canadian Masonry Symposium (University of Saskatchewan, June, 1992), pp. 765-776, 1992. Dialer, C., and M. Karaca. “Application of DEM to Problems in Rock Mechanics, Structural Engineering and Material Testing,” submitted to the 2nd International Conference on Discrete Element Methods (DEM), Massachusetts Institute of Technology, 1993.
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3DEC Version 3.0
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User’s Guide
Lorig, L. J., and P. A. Cundall. “Modeling of Reinforced Concrete Using the Distinct Element Method,” in Fracture of Concrete and Rock, pp. 459-471. S. P. Shah and S. E. Swartz, Eds. Bethel, Conn.: SEM, 1987. Lorig, L. J., and B. Dasgupta. Analysis of Emplacement Borehole Rock and Liner Behavior for a Repository at Yucca Mountain. U.S. Nuclear Regulatory Commission, NUREG/CR-5427, September, 1989. Lorig, L. J., R. D. Hart, M. P. Board and G. Swan. “Influence of Discontinuity Orientations and Strength on Cavability in a Confined Environment,” in Rock Mechanics As a Guide for Efficient Utilization of Natural Resources, pp. 167-174. A. Wahab Khair, Ed. Rotterdam: A. A. Balkema, 1989. Lorig, L. J., R. D. Hart and P. A. Cundall. “Slope Stability Analysis of Jointed Rock Using the Distinct Element Method,” Transportation Research Record, 1330, Soils, Geology, and Foundations (Behavior of Jointed Rock Masses and Reinforced Soil Structures), 1-9 (1991). Lorig, L. J., and B. E. Hobbs. “Numerical Modeling of Slip Instability Using the Distinct Element Method with State Variable Friction Laws,” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 27(6), 525-534 (1990). Mack, M. “Verification of Thermal Logic in the Three-Dimensional Code 3DEC,” ICG Report to U.S. NRC, Contract No. NRC-02-85-002, August, 1989. Mack, M. G., T. Brandshaug and B. Brady. “Rock Mass Modification Around a Nuclear Waste Repository in Welded Tuff,” Itasca Consulting Group Report to the U.S. Nuclear Regulatory Commission, Contract 02-85-002, Topical Report 006-01-T5, March, 1989; NUREG/CR-5390, August, 1989. Makurat, A., N. Barton, G. Vik and S. Bandis. “Investigation of Disturbed Zone Effects and Support Strategies for the Fjellinjen Road Tunnels Under Oslo,” in Proceedings of the International Congress on Progress and Innovation in Tunnelling (Toronto, September, 1989), pp. 125-134. Toronto: TAC/NRC/ITA, 1989. McKinnon, S. D. “Analysis of Stress Measurements Using a Numerical Model Methodology,” Int. J. Rock Mech. Min. Sci., 38, 699-709 (2001). McNearny, R. L., and J. F. Abel, Jr. “Large-Scale Two-Dimensional Block Caving Model Tests,” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 30(2), 93-109, 1993. Morrison, D. M., G. Swan and C. H. Scholz. “Chaotic Behavior and Mining-Induced Seismicity,” in Innovative Mine Design for the 21st Century (Proceedings of the International Congress on Mine Design, Kingston, Ontario, Canada, August, 1993), pp. 233-237. W. F. Bawden and J. F. Archibald, Eds. Rotterdam: A. A. Balkema, 1993. Ng, L. K. W., G. Swan and M. Board. “The Application of an Energy Approach in Fault Models for Support Design,” in Rockbursts and Seismicity in Mines 93 (Proceedings of the International Symposium, Kingston, Ontario, Canada, August 1993), pp. 387-391. R. Paul Young, Ed. Rotterdam: A. A. Balkema, 1993.
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Nordlund, E., and L. Jing. “Distinct Element Modelling of Jointed Rock Pillars,” in Proceedings of the 1st U.S. Conference on Discrete Element Methods (Golden, Colorado, October, 1990). G. G. W. Mustoe et al., Eds. Golden, Colorado: CSM Press, 1989. Nordlund, E., and G. Radberg. “Determination of Failure Modes in Jointed Pillars By Numerical Modeling,” in Proceedings of the ISRM Regional Conference on Fractured and Jointed Rock Masses (Lake Tahoe, June, 1992), Vol. 3, pp. 751-758. Berkeley, California: Lawrence Berkeley Laboratory, 1992. O’Connor, K. M., and C. H. Dowding. “Monitoring and Simulation of Mining-Induced Subsidence,” in Mechanics of Jointed and Faulted Rock, pp. 781-787. Rotterdam: A. A. Balkema, 1990. O’Hearn, B., D. Morrison, G. Allan, M. Board and R. Hart. “Use of Numerical Modeling in Mine Design at Falconbridge, Ltd.,” SME Annual Meeting (Phoenix, January, 1988), Geomechanics, Session II, 1988. O’Hearn, B., and G. Swan. “The Use of Models in Sill Mat Design at Falconbridge,” in Innovations in Mining Backfill Technology (Montreal, October, 1989), pp. 139-146. Rotterdam: A. A. Balkema, 1989. Papastamatiou, D., I. Psycharis, P. Carydis, C. Papantonopoulos, H. Mouzakis, J. V. Lemos and C. Zambas. Monuments Under Seismic Action: A Numerical and Experimental Approach, National Technical University of Athens, Laboratory for Earthquake Engineering, Report to E. U. Environment Programme, NTUA/LEE-97-01, May 1997. Pritchard, M. A., K. W. Savigny and S. G. Evans. “Toppling and Deep-Seated Landslides in Natural Slopes,” in Mechanics of Jointed and Faulted Rock, pp. 937-943. Rotterdam: A. A. Balkema, 1990. Roest, J. P. A., R. D. Hart and L. J. Lorig. “Modelling Fault-Slip in Underground Mining with the Distinct Element Method,” in Proceedings of the 6th International IAEG Congress (Amsterdam, August, 1990), pp. 105-110. Rotterdam: A. A. Balkema, 1990. Rosengren, L., M. Board, N. Krauland and S. Sandström. “Numerical Analysis of the Effectiveness of Reinforcement Methods at the Kristineberg Mine in Sweden,” in Rock Support in Mining and Underground Construction (Proceedings of the International Symposium on Rock Support, Sudbury, Ontario, Canada, June, 1992), pp. 507-514. P. K. Kaiser and D. R. McCreath, Eds. Rotterdam: A. A. Balkema, 1993. Salo, J., R. Riekkola, E. Johansson, M. Hakala, P. Särkka and H. Kuula. “Further Development of the Continuously-Yielding Joint Model for Studying Disposal of High-Level Nuclear Waste in Crystalline Rock (Yhteistyöprojekti II),” Saanio & Riekkola OY, Report T-2000-41/97, November 1997. Santarelli, F. J., D. Dahen, H. Baroudi and K. B. Slimane. “Mechanisms of Borehole Instability in Heavily Fractured Rock,” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 29(5), 457-467, 1992.
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Tinucci, J. P., and J. Israelsson. “Site Characterization and Validation Excavation Stress Effects Around the Validation Drift,” SKB Stripa Project Technical Report 91-20, August, 1991. Tinucci, J. P., A. R. Leach and A. J. S. Spearing. “Improved Seismic Ground Conditions with the Double-Cut Mining Method in Wide Tabular Reef Extraction,” in Rockbursts and Seismicity in Mines 93 (Proceedings of the International Symposium, Kingston, Ontario, Canada, August, 1993), pp. 429-434. R. Paul Young, Ed. Rotterdam: A. A. Balkema, 1993. Tolppanen, P. J., E. J. W. Johansson and R. Riekkola. “Comparison of Vertical and Horizontal Deposition Hole Concept for Disposal of Spent Fuel Based on the Rock Mechanical In-Situ Stress/Strength Analyses,” in Preprints of Contributions to the Workshop on Computational Methods in Engineering Geology (Lund, Sweden, October, 1996), pp. 230-237. R. Pusch and R. Adey, Eds. Lund: Clay Technology AB, 1996. Tolppanen, P. J., E. J. W. Johansson and J. P. Salo. “Rock Mechanical Analyses of In-Situ Stress/Strength Ratio at the Posiva Oy Investigation Sites, Kivetty, Olkiluoto and Romuvaara, in Finland,” in Prediction and Performance in Rock Mechanics & Rock Engineering (Proceedings of ISRM International Symposium EUROCK ’96, Turin, September, 1996), Vol. 1, pp. 435-442. G. Barla, Ed. Rotterdam: A. A. Balkema, 1996. Valdivia, C., and L. Lorig. “Slope Stability at Escondida Mine,” in Slope Stability in Surface Mining, Ch. 17, pp. 153-162. W. A. Hustrulid, M. K. McCarter and D. J. A. Van Zyl, Eds. Littleton, Colorado: SME, 2000. Vervoort, A. “Initial Roof Movement During Development of Room and Pillar Sections,” in Computer Methods and Advances in Geomechanics (Proceedings of the 7th International Conference, Cairns, Australia, May, 1991). Rotterdam: A. A. Balkema, 1991. Vonk, R. A., H. S. Rutten, J. G. M. van Mier and H. J. Fijneman. “Micromechanical Simulation of Concrete Softening,” in Fracture Processes in Concrete, Rock and Ceramics, Section 10. London: E. & F. N. Spon, 1991.
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User’s Guide
NOTE: NUREG reports may be ordered from: The National Technical Information Service 5285 Fort Royal Road Springfield, Virginia 22161 Telephone: (703)-487-4650
3DEC Version 3.0