6.2. Validación experimental del modelo numérico
6.2.1. Validación experimental del campo de velocidades en el canal
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, pressing
<Enter> at 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 prompt3dec>.
We begin by specifying a single polyhedral block using the POLY brick command.* Type
poly brick (0,80) (0,50) (-30,80)
and press<Enter> to continue. This command creates a brick-shaped polyhedron which extends from coordinates 0 to 80 units in thex-direction, from 0 to 50 units in the y-direction, and from -30 units to 80 units in thez-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) andz (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 thex- 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 inSection 1.2in the Command Reference for further details. Note that command words can be abbreviated (seeSection 2.5).
To continue the problem and return to the3dec> prompt, strike the <Q> key. 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 dip 90 dd 180 origin 0,0,0 jset dip 90 dd 180 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. (SeeSection 3.2.2for 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 -30,0 hide 0,80 0,50 50,80 mark region 1
Blocks located in the range 0<x<80, 0<y<50, -30<z<0 and in the range 0<x<80, 0<y<50, 50<z<80 will be hidden from view. The visible blocks are assigned a region number. Region numbers facilitate the application of commands to a group of blocks within a specific region.
We now create the shallow-dipping fracture planes with the commands
jset dip 2.5 dd 235 or 30,12.5,0 jset dip 2.5 dd 315 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 auto-matically. 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 0,50 hide 55,80 0,50 0,50 hide 0,30 0,50 0,50
jset dip 70 dd 200 or 0,0,35 jset dip 60 dd 330 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 2.2shows the model at this view.
3DEC (Version 3.00)
Itasca Consulting Group, Inc.
x Y z
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
27-Aug-02 9:25
Figure 2.2 3DEC model of a rock slope
If you wish to make a hardcopy of a plot, enter the command COPY after returning to the3dec>
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 the3dec> 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 1in 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.
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 negativey-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 they-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 moni-toring 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
During execution, the current cycle count, the calculation time, the maximum out-of-balance force, they-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 they-velocity history (seeFigure 2.3).
3DEC (Version 3.00)
Figure 2.3 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
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 (seeFigure 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 inSection 4in Theory and Background to model rock anchors or tiebacks to support the slope. (An example illustrating support for this slope is given inSection 4.2.1.7in 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 toSection 3for additional details on problem solving with 3DEC.
3DEC (Version 3.00)
ROCK SLOPE STABILTY -- WEDGE FAILURE
Itasca Consulting Group, Inc.
x Y z
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
27-Aug-02 9:26
Figure 2.4 Rock slope failure in progress
3DEC (Version 3.00)
ROCK SLOPE STABILTY -- WEDGE FAILURE
Itasca Consulting Group, Inc.
Max disp in plane = 3.648E+00 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
27-Aug-02 9:26
geometric scale Cross section plot:
0 2E+01
Figure 2.5 Vertical cross-section through wedge showing displacement vec-tors
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
block
zone gridpoint cable
interior boundary (excavation)
roller bottom boundary in-situ
horizontal boundary stress
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 re-ferring to a 3DEC model, we imply a sequence of 3DEC commands (seeSection 1in 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
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 dis-tinct 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
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 com-putational steps. During comcom-putational 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 “out-of-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 2in Optional Features).