3 LIBRARY OF FISH FUNCTIONS

Transcription

1 LIBRARY OF FISH FUNCTIONS LIBRARY OF FISH FUNCTIONS This section contains a library of FISH functions that have been written for general application in UDEC analysis. The functions can be used for various aspects of model generation and solution, including generation, plotting, assigning material properties and solution control. The FISH function files are contained in file FISH.ZIP in compressed format in the FISHBOWL directory. Individual files can be decompressed by typing the command pkunzip fish filename.fis in which filename corresponds to one of the file names described on the following pages. The general procedure to implement these FISH functions is performed in three steps: 1. The FISH file is first called by the UDEC data file with the command call filename.fis 2. Next, FISH variables, if given at the top of the function file, must be set in the data file with the command set var1 = value var2 = value... where var1, var2, etc. are the variable names that must be set to specified values. 3. Finally, the FISH function is invoked by entering the function name that follows directly after the DEFINE command in the FISH file. The name does not have to be given if a fishcall is used within the file. The FISH functions interact with UDEC in various ways. Turn to Section for a description of the different types of linkages between FISH and UDEC.

2 3-2 FISH in UDEC Table 3.1 General utility FISH functions Filename (.FIS) Command Purpose Variables SET before use Other.150 Function Required (.FIS) BOUNB bounb finds boundary blocks BOUNG boung finds boundary gridpoints BOUNZ bounz finds boundary zones PRSTRUC pr_struc prints selected structural element variables b_space PS3D ps3d computes 3D principal stresses SPEED set_zero speed calculates speed of calculations on your computer

3 LIBRARY OF FISH FUNCTIONS 3-3 Table 3.2 Plotting FISH functions Filename (.FIS) Command Purpose Variables SET before use Other.150 Function Required (.FIS) DISPMAG disp_mag calculates displacement magnitude at grid point to generate contour plot PQ history history qs ps calculates stress points p and q to generate a p-q diagram x_zone y_zone

4 3-4 FISH in UDEC Table 3.3 Solution control FISH functions Filename (.FIS) Command Purpose Variables SET before use Other.150 Function Required (.FIS) SERVO servo control to minimize inertial response to applied conditions high_unbal low_unbal high_val ZONK zonk relax gradually extracts region of zones to simulate excavation

5 LIBRARY OF FISH FUNCTIONS 3-5 Table 3.4 Special purpose FISH functions Filename (.FIS) DER Command Purpose Variables SET before use finds the derivative of a table derivative der_in der_out of values Other.150 Function Required (.FIS) ERFC erf erfc finds the error function of e_val complimentary error e_val e_val EXPINT exp_int finds the exponential integral of e_val e_val FFT finds the fast Fourier transform fftransform fft_in fft_out power spectrum of a table of values FROOT froot finds the root of a function bracketed in an interval c_x1 c_x2 func val INT finds the integral value of a integrate int_in int_out table of values LUDA solves systems of equations using ludcmp int_in int_out LU-decomposition SPEC spectrum finds the response spectrum of an accelerogram acc_in sv_out pmin damp sd_out sa_out pmax n_point

6 3-6 FISH in UDEC

7 LIBRARY OF FISH FUNCTIONS BOUNB.FIS/BOUNG.FIS/BOUNZ.FIS - 1 Finding Boundary Blocks, Gridpoints and Zones It is often useful to identify which blocks, gridpoints or zones lie along the external boundary or internal boundaries of a model. This allows the user to perform operations on these gridpoints or zones directly, rather than search the entire grid whenever these entities must be identified. For example, it may be necessary to monitor tunnel closure, or calculate stresses and displacements at the outer boundary. Three FISH functions, bounb, boung and bounz, that identify gridpoints or zones that lie along external or internal boundaries, are available. When BOUNB.FIS, BOUNG.FIS or BOUNZ.FIS is called, the addresses, gridpoints or zones that are on a boundary are printed. Note: If blocks have only two zones, bounz may report extra zones on boundary in the corners. In the example data file BOUN.DAT, boundary blocks are identified using BOUNB.FIS. Data File BOUNG.DAT new block split split split split delete gen quad 10 ; force the generation of the boundary corner list bound stress 0,0,0 ; force the generation of the interior boundary corner list boun int stress 0,0,0 ; ; Find gridpoints on boundaries ;ca boung.fis ;boung ; ; Find zones on boundaries ;ca bounz.fis ;bounz ; ; Find blocks on boundaries ca bounb.fis bounb ret

8 BOUNB.FIS/BOUNG.FIS/BOUNZ.FIS - 2 FISH in UDEC

9 LIBRARY OF FISH FUNCTIONS DER.FIS - 1 Finding the Derivative of a UDEC Table The FISH file DER.FIS integrates the values of a table and returns another table. The input table is specified with the der in argument, and the output table is specified with the der out argument. The function calculates the slopes between points in the input table, and locates the value midway between the points in the output table. Therefore, there will be n-1 points in the resulting table if there are n points in the source table. Figure 1 shows the result of taking the derivative of a simple cosine wave. Table 1 is the input data, and table 2 is the output. The input data is read using the TABLE 1 read test01.his command to copy the data into a table. Data File DER.DAT new title Example of DERIVATIVE FISH function table 1 read test01.his ; ca der.fis ; set der in 1 der out 2 derivative ; plot hold table 1 2 JOB TITLE : Example of Derivative FISH function UDEC (Version 5.00) LEGEND 17-Sep :24:27 cycle 0 time 0.000E+00 sec table plot -1.00E+00<tab 1> 1.00E E+00<tab 2> 1.00E+00 Vs. 0.00E+00<X value> 1.88E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+001) Figure 1 Derivative of a UDEC table (table 1 is a cosine wave; table 2 is the derivative)

10 DER.FIS - 2 FISH in UDEC

11 LIBRARY OF FISH FUNCTIONS DISPMAG.FIS - 1 Plotting Displacement Magnitude Contours The user may write a function to calculate a special grid variable for plotting. For example, when the function disp mag is invoked, the displacement magnitudes are calculated at all gridpoints in the model, and stored in the FISH grid variable gp free. This array can then be plotted with the PLOT command. By typing plot gp free fill alias displacement magnitude a filled contour plot will be generated. Note that this function uses an extra grid variable accessed by gp extra (gi). The alias keyword is added to rename ex 1 to displacement magnitude in the plot legend. Figure 1 shows the plot. Data File DISPMAG.DAT block crack gen quad 4 bound yvel 0.0 range prop mat 1 dens 1000 bulk 2e8 shear 1e8 prop jmat 1 jkn 1.33e7 jks 1.33e7 jfric 30.0 set grav 0-10 damp auto cyc 100 call dispmag.fis disp mag plot hold boun gp extr inv fill alias displacement magnitude ret

12 DISPMAG.FIS - 2 FISH in UDEC JOB TITLE : Example of Grid Point Extra function (*10^1) UDEC (Version 5.00) LEGEND Sep :27:20 cycle 100 time 1.433E-01 sec boundary plot displacement magnitude contour interval= 1.000E E-03 to 9.000E E E E E E E E E E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 1 Contours of displacement magnitude

13 LIBRARY OF FISH FUNCTIONS ERFC.FIS - 1 Error Function and Complementary Error Function The file ERFC.FIS contains two FISH functions that calculate the error function, and complementary error function, erf(x) = 2 π x erfc(x) = 2 π 0 x e t2 dt e t2 dt of a real variable x using the rational approximation in section in Abramowitz and Stegun (1970). The error magnitude is less than The value of x is defined by e val; the functions erf and erfc return the corresponding function value. The following data file plots functions erf and erfc in the interval [0, 1.5]. Data File ERFC.DAT title Error and Complementary Error Functions ca erfc.fis suppress def plot erf dx = 1.5/20. e val = -dx loop ii (1,21) e val = e val + dx xtable(1,ii) = e val ytable(1,ii) = erf xtable(2,ii) = e val ytable(2,ii) = erfc end loop end plot erf plot hold table 1 2 ret

14 ERFC.FIS - 2 FISH in UDEC Reference Abramowitz, M., and I. A. Stegun. Handbook of Mathematical Functions. New York: Dover Publications Inc. (1970). JOB TITLE : Error and Complimentary Error Functions UDEC (Version 5.00) 1.20 LEGEND 17-Sep :34:48 cycle 0 time 0.000E+00 sec table plot 0.00E+00<tab 1> 9.66E E-02<tab 2> 1.00E+00 Vs. 0.00E+00<X value> 1.50E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1 Error and complementary error functions

15 LIBRARY OF FISH FUNCTIONS EXPINT.FIS - 1 Exponential Integral Function The file EXPINT.FIS contains a FISH function that calculates the exponential integral function, E 1 (x) = x e t t dt of a real and positive variable x, using polynomial approximations in sections and in Abramowitz and Stegun (1970). The error magnitude is less than for x 1, and less than for x>1. The value of x is defined by e val, and the function exp int returns the corresponding value of E 1. The following data file plots function E 1 in the interval [0,1.6]. Data File EXPINT.DAT title Exponential Integral Function ca expint.fis suppress def plot e1 dx = 1.6/20. e val = 0. loop ii (1,20) e val = e val + dx xtable(1,ii) = e val ytable(1,ii) = exp int end loop end plot e1 plot hold table 1 ret Reference Abramowitz, M., and I. A. Stegun. Handbook of Mathematical Functions. New York: Dover Publications Inc. (1970).

16 EXPINT.FIS - 2 FISH in UDEC JOB TITLE : Exponential Integral Function UDEC (Version 5.00) 2.20 LEGEND 17-Sep :36:36 cycle 0 time 0.000E+00 sec table plot 8.63E-02<tab 1> 2.03E+00 Vs. 8.00E-02<X value> 1.60E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1 Exponential integral function

17 LIBRARY OF FISH FUNCTIONS FFT.FIS - 1 Finding the Fast Fourier Transform Power Spectrum of a UDEC Table The FISH file FFT.FIS performs a fast Fourier transform on a table of data, resulting in a power spectrum that is output to another table. The input table is specified by the first argument, and the output table is specified by the second argument. There are several definitions for a power spectrum. The one used here is adapted from Press et al. (1992). The power spectrum is a set of N/2 real numbers defined as P 0 = 1 N 2 ( f 0 ) 2 (1) P k = 1 N 2 [( f k ) 2 + ( f N k ) 2 ] (2) P N = 1 2 N 2 ( f N ) 2 (3) 2 where N is half the number of points in the original data field; P is the power spectrum output; f is the result of the fast Fourier transform of the original data; and k varies from 0 to N 2. Note that an array, worka, is used to manipulate the table data. The array dimension (n point) is defined from two conditions: (1) to be greater than the number of elements in the input table; and (2) to be a power of 2. (The array dimension need not be declared manually.) The fft algorithm requires input data with a constant timestep. So, a timestep is calculated, and the data are interpolated from the table and stored in the array for processing. The following example verifies the fft FISH function. The history input is the sum of a sine wave at 1 Hz and an amplitude of 1, a cosine wave at 5 Hz and an amplitude of 2, and a sine wave at 10 Hz and an amplitude of 3. The combined history input is calculated by the FISH function cr tab. The input is plotted in Figure 1. The power spectrum shown in Figure 2 consists of three sharp peaks at 1, 5 and 10 Hz, with increasing peak values. Reference Press, W. H., et al. Numerical Recipes in C. Cambridge: Cambridge University Press (1992).

18 FFT.FIS - 2 FISH in UDEC Data File FFT.DAT def cr tab i = 1 p2 = 2.*pi loop while i <= num point xx = end time*float(i)/float(num point) i = i + 1 yy = sin(xx*p2/per1)+2.*cos(5.*xx*p2/per1)+3.*sin(10.*xx*p2/per1) table(1,xx) = yy end loop end set num point 1024 end time 12.0 set per1 1.0 cr tab suppress def tab ind fft in = 1 fft out = 2 end tab ind ; ATTENTION: fft.fis uses a temporary setup to erase table ca fft.fis suppress fftransform plot hold table 1 plot hold table 2 ret

19 LIBRARY OF FISH FUNCTIONS FFT.FIS - 3 JOB TITLE :. UDEC (Version 5.00) 6.00 LEGEND 17-Sep :38:24 cycle 0 time 0.000E+00 sec table plot -5.38E+00<tab 1> 5.47E+00 Vs. 1.17E-02<X value> 1.20E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+001) Figure 1 Sum of three input waves JOB TITLE :. UDEC (Version 5.00) (e+001) 1.00 LEGEND 17-Sep :39:16 cycle 0 time 0.000E+00 sec table plot 1.14E-31<tab 2> 9.00E+00 Vs. 0.00E+00<X value> 2.13E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+001) Figure 2 Power spectrum; power versus frequency in Hz

20 FFT.FIS - 4 FISH in UDEC

21 LIBRARY OF FISH FUNCTIONS FROOT.FIS - 1 Root of a Function in an Interval The file FROOT.FIS contains a FISH function that calculates the root of a function f(x), known to lie in the interval [a,b], using Brent s method algorithm (Press et al. 1986). The FISH function func must be specified. It returns the value of f for the argument x defined by c x. The interval bounds a and b are assigned using c x1 and c x2, respectively. The root, returned as froot, is refined until its accuracy is tol. The following data file plots the function f(x)= tan x x and marks its root located in the interval [ π 2, 3π 2 ]. Reference Press, W. H., et al. Numerical Recipes: The Art of Scientific Computing (FORTRAN Version). Cambridge: Cambridge University Press (1986). Data File FROOT.DAT title Root of a function in an interval ca froot.fis def func func = tan(c x0) - cx0 end def itis root ; find the root of function func ; in the interval ] pi/2, 3pi/2 [ ; with accuracy tol c x1 = 1.01 * pi/2. c x2 = 0.99 * 3.*pi/2. tol = 1.e-4 plot func root = froot c x0 = root xtable(2,1) = root ytable(2,1) = func end def plot func ; calculate func at 20 points in the interval and ; store the values in table 1 dx = (c x2 - cx1)/20. c x0 = c x1 loop ii (1,20) c x0 = c x0 + dx xtable(1,ii) = c x0

22 FROOT.FIS - 2 FISH in UDEC ytable(1,ii) = func end loop end itis root plot hold table 1 line 2 cross ret JOB TITLE : Root of a function in an interval UDEC (Version 5.00) (e+001) 2.00 LEGEND 17-Sep :44:46 cycle 0 time 0.000E+00 sec table plot -7.58E+00<tab 1> 1.65E E-04<tab 2> 9.68E-04 X X X Vs. 1.74E+00<X value> 4.67E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1 Function and its root in an interval

23 LIBRARY OF FISH FUNCTIONS HOEK.FIS - 1 Adapting the Mohr-Coulomb Model to a Hoek-Brown Failure Surface The FISH routines in HOEK.FIS adapt the Mohr-Coulomb model in UDEC to approximate the nonlinear failure surface for a Hoek-Brown material (Hoek and Brown 1982). The Hoek-Brown failure criterion is based on a nonlinear relation between major and minor principal stresses, σ 1 and σ 3 : σ 1 = σ 3 σ 3 σ c m + σ 2 c s (1) where σ c is the unconfined compressive strength of the intact rock, m and s are material constants of the rock mass, and compressive stresses are negative (UDEC convention). For a given value of σ 3, a tangent to the function (Eq. (1)) will represent an equivalent Mohr-Coulomb yield criterion in the form σ 1 = N φ σ 3 σ M c (2) where N φ = 1+sin φ 1 sin φ = tan2 ( φ ). By substitution, σ M c is σ M c = σ 1 + σ 3 N φ = σ 3 + σ 3 σ c m + σc 2s σ 3 N φ = σ 3 (1 N φ ) + σ 3 σ c m + σc 2s (3) σ M c is the apparent uniaxial compressive strength of the rock mass for that value of σ 3. The tangent to the function (1) is defined by N φ (σ 3 ) = σ 1 σ 3 = 1 + σ c m 2 σ 3 σ c m + sσ 2 c (4) The cohesion (c) and friction angle (φ) can then be obtained from N φ and σ M c :

24 HOEK.FIS - 2 FISH in UDEC φ = 2 tan 1 N φ 90 (5) c = σ M c 2 N φ (6) The comparison of the Mohr-Coulomb linear approximation to the Hoek-Brown yield surface is shown in the figure. These equivalent c and φ are a good approximation of the nonlinear yield surface for values of the minor principal stress that are close to the given σ 3. The FISH function cfi calculates the value of c and φ for each zone every ns steps. Thus, as σ 3 changes, the values of c and φ will also change. Note that the instantaneous values of c and φ calculated in this way closely match those calculated using Hoek s (1990) expressions based on normal and shear stress. Hoek and Brown (1982) also define constants m r and s r for properties of a broken rock mass. If failure occurs, m and s are changed to m r and s r to represent sudden post-failure response. A progressive strain-softening behavior could be modeled by replacing the Mohr-Coulomb model with the strain-softening model. The Hoek-Brown parameters σ c, m, s, m r and s r are set in HOEK.FIS via the variables hb sc, hb mmi, hb ssi, hb mmr and hb ssr, respectively, through the SET command. The FISH function cfi is called to update cohesion, friction and tension variables in the Mohr-Coulomb model. The dilation angle may be specified using the variable hoek psi (use hoek psi = fi for an associated flow rule see example below). Note that, if σ 3 becomes tensile, the yield surface remains linear with the slope N φ (σ 3 ) defined at σ 3 = 0. The user controls the update process by specifying ns and nsup through the SET command. ns defines the number of steps taken before cfi is called to update properties. nsup defines the total number of times cfi is to be called. ns nsup corresponds to the total number of steps in the UDEC run. If not specified, the default for ns is 5. This may require variation depending on the nonlinearity of the failure surface. A triaxial compression test on a Hoek-Brown material sample is provided below as an example application of this routine. The test is strain-controlled, and an associated flow rule is selected, for the numerical simulation. References Hoek, E. Estimating Mohr-Coulomb Friction and Cohesion Values from the Hoek-Brown Failure Criterion, Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 27(3), (1990). Hoek, E., and E. T. Brown. Underground Excavations in Rock. London: IMM (1982).

25 LIBRARY OF FISH FUNCTIONS HOEK.FIS - 3 Figure 1 Linear approximation of the Hoek-Brown failure criterion Data File HOEK.DAT new title Triaxial test on a Hoek-Brown material round =.001 block gen quad.2 zone model mohr bound stress range bound xvel 0.0 range bound yvel.05 range bound yvel -.05 range zone bulk shear 400 tension 1e10 prop mat 1 dens 2e-3 prop jmat 1 jkn 133 jks 133 insitu stress szz -5 hist syy label hist 1 YY - Stress hist sxx label hist 2

26 HOEK.FIS - 4 FISH in UDEC XX - Stress hist szz hist ydisp label hist 4 Y - Displacement hist xdisp call block.fin call zmat.fin call hoek.fis def hoek psi hoek psi = fi end ;associated flow rule set hb mmi=1.0 hb mmr=1.0 set hb ssi= hb ssr= set hb sc=50.0 set nsup= 1200 ns=10 ; note, UDEC will cycle nsup*ns times supsolve plot hold his 1 2 vs -4 plot hold bou disp ret JOB TITLE : Triaxial test on a Hoek-Brown material UDEC (Version 5.00) (e+001) LEGEND 17-Sep :46:35 cycle time 6.762E-01 sec history plot Y-axis: YY - Stress XX - Stress X-axis: Time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-001) Figure 2 Stresses versus top vertical displacement

27 LIBRARY OF FISH FUNCTIONS HOEK.FIS - 5 JOB TITLE : Triaxial test on a Hoek-Brown material UDEC (Version 5.00) LEGEND 17-Sep :46:35 cycle time 6.762E-01 sec boundary plot displacement vectors maximum = 4.510E E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 3 Displacement vectors at end of simulation

28 HOEK.FIS - 6 FISH in UDEC

29 LIBRARY OF FISH FUNCTIONS INT.FIS - 1 Finding the Integral of a UDEC Table The FISH file INT.FIS integrates the values of a table, and returns another table. The ID number of the input table is the first argument, and the output table is the second argument. If the new output table already exists, all data points in it will be deleted and overwritten with the new values. If the output table does not exist, one is created. The function integrates using the trapezoidal rule, with the same number of points in the result tables as are in the source tables. The function assumes an integration constant of zero. The figure shows the result of a simple cosine wave integration. Table 1 is the input data, and table 2 is the output data. The input data are read using the cr tab function to create the data and copy it into a table. Data File INT.DAT new title Example of INTEGRATE FISH function def cr tab val = pi * 6.e-3 loop ii (1,1000) xx = float(ii-1) * val xtable(1,ii) = xx ytable(1,ii) = cos(xx) end loop end cr tab suppress ; ca int.fis suppress ; set int in 1 int out 2 integrate plot hold table 1 2

30 INT.FIS - 2 FISH in UDEC JOB TITLE : Example of INTEGRATE FISH function UDEC (Version 5.00) 1.20 LEGEND 17-Sep :53:54 cycle 0 time 0.000E+00 sec table plot -1.00E+00<tab 1> 1.00E E+00<tab 2> 1.00E+00 Vs. 0.00E+00<X value> 1.88E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+001) Figure 1 Simple cosine wave integration

31 LIBRARY OF FISH FUNCTIONS LUDA.FIS - 1 Matrix Inversion via LU-Decomposition A pair of FISH functions, ludcmp and lubksb, can be used to solve the system of equations Ax = b (1) where A is a given square matrix of size n, b is a given vector of size n, and x is the desired solution vector of size n. The FISH function ludcmp performs an LU-decomposition on the matrix A. The FISH function lubksb performs the back-substitution operation to produce the solution vector x. Both of these functions implement the algorithm described in Press et al. (1986). An example problem demonstrating the use of the two FISH functions is provided in the data file below. The size of the matrix is passed as an argument to make random data. Reference Press, W. H., et al. Numerical Recipes: The Art of Scientific Computing (FORTRAN Version). Cambridge: Cambridge University Press (1986). Data File LUDA.DAT new ; Test LU-decomposition FISH functions on random matrices. ; new def luda setup lu nn = 8 ; define dimensions of matrices end ; luda setup ; def make random data array aa(lu nn,lu nn) indx(lu nn) bb(lu nn) ; needed by LUDA functions array xx(lu nn) ; just used locally ;--- fill aa for test loop i (1,lu nn) loop j (1,lu nn) aa(i,j) = urand end loop end loop ;--- set "unknowns" loop i (1,lu nn) xx(i) = float(i) xtable(1,i) = float(i) ytable(1,i) =xx(i)

32 LUDA.FIS - 2 FISH in UDEC end loop ;--- multiply by matrix to get r.h.s. loop i (1,lu nn) sum = 0.0 loop j (1,lu nn) sum = sum + aa(i,j) * xx(j) end loop bb(i) = sum ii = out( b( +string(i)+ ) = +string(sum)) end loop end;--- look at results --- def look loop i (1,lu nn) xtable(2,i) = float(i) ytable(2,i) = bb(i) ii = out( x( +string(i)+ ) = +string(bb(i))) end loop end ; call luda.fis ; support functions for LU-decomposition ; set log on make random data ; Right-hand sides... ludcmp lubksb look ; Following list should correspond to original vector of x values ; =================================================================== label table 1 Set values label table 2 Calculated values pl tab 1 2 cross return

33 LIBRARY OF FISH FUNCTIONS LUDA.FIS - 3 JOB TITLE : Example of LU-Decomposition FISH function UDEC (Version 5.00) 9.00 LEGEND 17-Sep :56:16 cycle 0 time 0.000E+00 sec table plot Set values Calculated values X X X Vs. 1.00E+00<X value> 8.00E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1 Comparison of unknowns

34 LUDA.FIS - 4 FISH in UDEC

35 LIBRARY OF FISH FUNCTIONS PQ.FIS - 1 P-Q Stress Diagram Often, the user may wish to print or plot problem variables that are not directly accessible through the UDEC HISTORY command. It is quite simple for the user to write a FISH function that will calculate the desired variable directly in UDEC. The data file PQ.DAT illustrates the use of FISH to calculate the stress point p,q, and to plot a p-q diagram via the HISTORY and PLOT commands. The generalized stress components p and q are expressed in terms of principal stresses: p = 1 3 (σ 1 + σ 2 + σ 3 ) q = 1 2 (σ 1 σ 2 ) 2 + (σ 2 σ 3 ) 2 + (σ 1 σ 3 ) 2 (1) Note that p is an effective pressure, defined in terms of the effective principal stresses. Data File PQ.DAT block gen quad 4 call pq.fis set x zon = 2.5 set y zon = 10.0 bound xvel 0.0 range bound yvel 0.01 range bound yvel range prop mat 1 dens 2000 bulk 2e8 shear 1e8 prop jmat 1 jkn 1.33e7 jks 1.33e7 jfric 30.0 damp auto hist qs hist ps cyc 100 bound yvel 0.0 range bound yvel 0.0 range bound xvel -.01 range cyc 300 plot hold his 1 vs his 2 ret

36 PQ.FIS - 2 FISH in UDEC JOB TITLE :. UDEC (Version 5.00) (e+005) 1.20 LEGEND 17-Sep :57:33 cycle 400 time 8.165E-01 sec history plot Y-axis: 1 - Fish: qs X-axis: Time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-001) Figure 1 p-q plot

37 LIBRARY OF FISH FUNCTIONS PRSTRUC.FIS - 1 Printing Selected Structural-Element Variables The user can select specific variables to be printed from a UDEC model, even though they may not be directly available from the PRINT command. This can be done by accessing the data structure in UDEC directly, as described in Section 4. For example, the user may wish to print maximum and minimum stresses associated with structural elements for the case of structural elements installed on a regular spacing in the out-of-plane direction. The structural element variables stored by UDEC are scaled axial, shear forces and moments. In order to determine the actual forces and moments in the beams, the UDEC forces and moments must be multiplied by the spacing. Actual axial stresses are then derived using actual moment of inertia and area of the beam cross-section. (See Section in Special Features for further discussion on scaling a 2D UDEC model to simulate a 3D problem with regularly spaced structural elements.) The file PRSTRUC.FIS contains a FISH function pstruct that calculates the actual extreme values of axial stresses at the midpoint of beams, assuming a regular spacing defined by b space, and a cross-sectional height specified by b height. (Note that the beam formulation in UDEC assumes a linear variation of moment along the beam element.) The function calls the file STR.FIN to access values in the offsets associated with the structural-element data structure. The actual minimum and maximum axial stresses for each beam element are then printed in a list. The following data file illustrates the application of PRSTRUC.FIS : Data File PRSTRUC.DAT round 0.05 block crack crack crack crack del crack crack fix fix ; rock properties prop m=1 d= k=2e3 g=1e3 jkn=1e4 jks=1e4 jfric=10 jcoh=0 jtens=0 ; ; structural liner properties struct gen xc 0 yc 0 np 20 thick 0.1 mat 10 prop mat=10 st d= prop mat=10 st ymod= st prat=0.15 prop mat=10 st yield=40 st yresid=40 st ycomp 40 prop mat=10 if kn=1e4 if ks=1e4 if tens=0 if fric=50 if coh=0 ; ; gravity load

38 PRSTRUC.FIS - 2 FISH in UDEC set grav 0-10 ; hist unbal hist ydis 0 3 hist yvel 0 3 ; ; Case 1 : elastic analysis step 3000 set echo off call prstruc.fis set echo on set b space 5 b height=0.5 set log pr struc.log set log on pr struc set log off ret ; win save wedge1.sav plot block struct mom yrev fill disp ; ; Case 2 : yield strength = residual strength = 18 MPa prop mat=10 st yield=18 st ycomp=40 st yresid=18 step save wedge2.sav plot block struct mom yrev fill disp ; ; Case 3 : yield strength = 18 MPa residual strength = 16.7 MPa rest wedge1.sav prop mat=10 st yield=18 st ycomp=40 st yresid=16.7 step win plot block struct mom yrev fill disp save wedge3.sav ; ; Case 4 : elastic analysis with double layer of liner rest wedge0.sav ; structural liner properties struct gen xc 0 yc 0 np 100 thick 0.1 mat 10 struct gen xc 0 yc 0 np 100 thick 0.1 mat 10 prop mat=10 st d= prop mat=10 st ymod=21000 st prat=0.15 prop mat=10 st yield=40 st yresid=40 st ycomp 40 prop mat=10 if kn=1e4 if ks=1e4 if tens=0 if fric=50 if coh=0 ;

39 LIBRARY OF FISH FUNCTIONS PRSTRUC.FIS - 3 ; gravity load set grav 0-10 ; hist unbal hist ydis 0 3 hist yvel 0 3 ; step win save wedge4.sav plot hold block struct mom yrev fill disp The printout from this example is shown: ID F-axial Str-min Str-max e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-001

40 PRSTRUC.FIS - 4 FISH in UDEC

41 LIBRARY OF FISH FUNCTIONS PS3D.FIS - 1 Computing the 3D Principal Stresses Keywords sig1 and sig2 (used in printing and plotting) correspond to the two-dimensional major and minor principal stresses, respectively. As stated in Section 1 in the Command Reference, they refer to stresses in the xy-plane only. However, the out-of-plane stress szz may be the major or minor principal stress if the full three-dimensional stress tensor is considered. The FISH function ps3d computes the major stress, taking into account the out-of-plane stress, and places it in the z free zone variable. Data File PS3D.DAT block crack gen quad 4 bound yvel 0.0 range insitu szz -3e4 prop mat 1 dens 1000 bulk 2e8 shear 1e8 prop jmat 1 jkn 1.33e7 jks 1.33e7 jfric 30.0 set grav 0-10 damp auto cyc 100 call ps3d.fis ps3d plot hold boun z extra inv fill alias major principal stress

42 PS3D.FIS - 2 FISH in UDEC JOB TITLE : (*10^1) UDEC (Version 5.00) LEGEND Sep :31:38 cycle 100 time 1.433E-01 sec boundary plot major principal stress contour interval= 1.000E E+04 to E E E E E E E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 1 Major principal stress contours

43 LIBRARY OF FISH FUNCTIONS SERVO.FIS - 1 Servo Control A servo-control function is used to minimize the influence of inertial effects on the response of the model. The FISH file SERVO.FIS shows how the applied vertical velocities can be adjusted as a function of the maximum unbalanced force in the model. By preventing the unbalanced force from getting too high (i.e., controlling the inertial effects), the user has better control over model behavior. The control is specified by setting the upper limit for unbalanced force, high unbal, and lower limit, low unbal, with the SET command. The loading velocity is also controlled by specifying an upper limit (high vel). A command is not issued for this function because it is automatically invoked at every calculation step through the WHILE STEPPING FISH statement. This function is demonstrated for the problem of a triaxial compression test of a strain-softening material (data file SERVO.DAT ). The stress-strain response of the specimen indicates a weakening of the material after the peak strength is reached. The servo-control of the applied velocity allows for an analysis with minimal inertial effects. Note that FISH functions are built into the data file to calculate the average vertical stress, sigmav, and average vertical strain, ev, in order to generate the stress-strain plot shown in Figure 1. The servo-control function will need to be modified for different types of loading. Data File SERVO.DAT ; Triaxial test of strain-softening material ; with controlled velocity title Triaxial test of strain-softening material round.001 bl gen quad 1 zone model ss ca boucnr.fin call servo.fis bound xvel = 0 bound stress -1e6,0,-1e6 bound yvel -1e-2 bound yvel 1e-2 prop mat 1 den 2500 zone bulk 2e8 she 1e8 co 2e6 fric 45 ten 1e6 dil 10 zone ftab 1 ctab 2 dtab 3 table table 2 0 2e6.05 1e6.1 5e5 1 5e5 table insitu stress -1e6,0,-1e6 szz -1e6 def sigmav

44 SERVO.FIS - 2 FISH in UDEC sum=0.0 z count = 0 i b = block head loop while i b # 0 i z = b zone(i b) loop while i z # 0 sum = sum - z syy(i z) i z = z next(i z) z count = z count + 1 end loop i b = b next(i b) end loop sigmav = sum/z count end def ev ev = (gp ydis(i gb)-gp ydis(i gt))/(gp y(i gt)-gp y(i gb)) end def set lim i gt = gp near(0,10) i gb = gp near(0,0) end set lim hist sigmav label hist 1 Axial Stress hist ev label hist 2 Axial Strain hist yvel 0 0 label hist 3 Vertical Velocity hist unbal label hist 4 Maximum Unbalanced Force set high unbal=5e4 set low unbal=2e4 set high vel= 2 step 6000 save servo.sav plot hold his 1 vs 2 plot hold his 4 plot hold his 3 ret

45 LIBRARY OF FISH FUNCTIONS SERVO.FIS - 3 JOB TITLE : Triaxial test of strain softening material UDEC (Version 5.00) (e+007) 1.80 LEGEND 17-Sep :35:23 cycle 6000 time 3.569E+00 sec history plot Y-axis: Axial Stress X-axis: Axial Strain Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-002) Figure 1 Axial stress versus axial strain for a triaxial test with strainsoftening material with controlled velocity JOB TITLE : Triaxial test of strain softening material (e+005) UDEC (Version 5.00) 4.00 LEGEND 17-Sep :35:23 cycle 6000 time 3.569E+00 sec history plot Y-axis: 0 - maximum unbalanced f X-axis: Number of cycles Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e+003) Figure 2 Unbalanced force history for a triaxial test with strain-softening material with controlled velocity

46 SERVO.FIS - 4 FISH in UDEC JOB TITLE : Triaxial test of strain softening material (e-001) UDEC (Version 5.00) 3.50 LEGEND 17-Sep :35:23 cycle 6000 time 3.569E+00 sec history plot Y-axis: Vertical Velocity X-axis: Time Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 3 Vertical velocity history for a triaxial test with strain-softening material with controlled velocity

47 LIBRARY OF FISH FUNCTIONS SPEC.FIS - 1 Finding the Response Spectrum of an Acceleration History in a UDEC Table The FISH file SPEC.FIS finds the displacement response spectrum, the pseudo-velocity response spectrum and the pseudo-acceleration response spectrum of an input acceleration stored in a UDEC table. The ID number of the input table is defined by the acc in argument, and the three output tables are identified by the sd out, sv out and sa out arguments. If any of the output tables currently exists, it will be deleted and overwritten by the new results. The damping constant for the response analysis is specified by the dmp argument. The calculation is only approximate for damped responses; the higher dmp is, the less accurate the response. The range of periods over which the spectrum is calculated by the pmin and pmax arguments, and the number of points in the output tables, are defined by the n point argument. This routine can take considerable time to execute. If N i is the number of input points and N p is the number of points in the output, then the number of calculations increases as N p N i log(n i ). This formulation tends to give somewhat distorted results for periods approaching zero. However, improving the accuracy for small periods increases the calculation time. The algorithm was adapted from Craig (1981). As an example of its use, a simple sine wave was input into a UDEC table as an input acceleration. The function was then executed from a period of 0.5 to 2, with 50 points, in the output tables (see SPEC.DAT ). Figure 1 shows the input acceleration generated: a sine wave with a period of 1.0. Figures 2 through 4 show the various response spectrums generated, displaying the expected peaks at a period of 1.0. Reference Craig, Jr., R. R. Structural Dynamics An Introduction to Computer Methods. New York: John Wiley and Sons (1981).

48 SPEC.FIS - 2 FISH in UDEC Data File SPEC.DAT def cr tab i = 0 p2 = 2.*pi loop while i <= num point xx=end time*float(i)/float(num point) i = i+1 yy = sin(xx*p2/per1) table(1,xx) = yy end loop end set num point 250 end time 3.0 set per1 1.0 cr tab suppress ; ATTENTION: spec.fis uses a temporary set-up for erasing table ca spec.fis suppress set pmin = 0.5 set pmax = 2. set dmp = 0. set acc in = 1 set sd out = 2 set sv out = 3 set sa out = 4 set n point = 50 spectra ; label table 1 input acceleration label table 2 displacement response label table 3 pseudo-velocity response label table 4 pseudo-acceleration response plot hold table 1 plot hold table 2 plot hold table 3 plot hold table 4 ret

49 LIBRARY OF FISH FUNCTIONS SPEC.FIS - 3 JOB TITLE :. UDEC (Version 5.00) 1.00 LEGEND 17-Sep :39:30 cycle 0 time 0.000E+00 sec table plot input acceleration Vs. 0.00E+00<X value> 3.00E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1 Input acceleration JOB TITLE :. UDEC (Version 5.00) (e-001) 2.80 LEGEND 17-Sep :40:27 cycle 0 time 0.000E+00 sec table plot displacement response Vs. 5.00E-01<X value> 2.00E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 2 Displacement response spectrum

50 SPEC.FIS - 4 FISH in UDEC JOB TITLE :. UDEC (Version 5.00) 1.60 LEGEND 17-Sep :41:31 cycle 0 time 0.000E+00 sec table plot pseudo-velocity response Vs. 5.00E-01<X value> 2.00E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 3 Pseudo-velocity response spectrum JOB TITLE :. UDEC (Version 5.00) (e+001) 1.00 LEGEND 17-Sep :42:13 cycle 0 time 0.000E+00 sec table plot pseudo-acceleration respons Vs. 5.00E-01<X value> 2.00E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 4 Pseudo-acceleration response spectrum

51 LIBRARY OF FISH FUNCTIONS SPEED.FIS - 1 Runtime Test The calculation speed of UDEC can be determined with the FISH file SPEED.FIS. The clock is first set to zero with the function set zero. After stepping, the calculation speed is printed by typing print speed The speed is measured in zone-steps per second. The data file SPEED.DAT illustrates the use of these functions. Data File SPEED.DAT block crack gen quad 4 bound yvel 0.0 range prop mat 1 dens 1000 bulk 2e8 shear 1e8 prop jmat 1 jkn 1.33e7 jks 1.33e7 jfric 30.0 set grav 0-10 damp auto call speed.fis set num zones = 144 set zero cyc 100 print speed cyc 200 print speed cyc 400 print speed ret

52 SPEED.FIS - 2 FISH in UDEC

53 LIBRARY OF FISH FUNCTIONS ZONK.FIS - 1 Gradual Unloading of Void Regions The FISH function ZONK.FIS detects a void within a model, and slowly relaxes the forces around the void region. This facility is useful for simulating a gradual excavation in elasto-plastic material. The influence of transients on material failure is minimized; the solution is more static. The boundary of the extracted region is detected and modified by specifying the FISH function zonk. The forces are then relaxed by specifying the function relax. The example data file ZONK.DAT illustrates the use of ZONK.FIS to simulate a gradual excavation in Mohr-Coulomb material. Data File ZONK.DAT new round 0.01 block jset ; Generate cavern crack crack crack arc arc arc arc arc arc crack crack crack crack crack crack crack crack crack crack crack crack gen edge 15 change cons 3 Prop m=1 bulk=3.9e9 shear=2.8e9 dens=2500 coh=3.45e6 fric 30 dil 0 ten 1e10 Prop jmat=1 jkn=10e9 jks=1e9 jfric=40 jcoh=0.4e6 ; Crack 1 bound yr stress -3.95e e6 bound yr yvel 0 bound xr xvel 0

54 ZONK.FIS - 2 FISH in UDEC bound xr xvel 0 set grav 0-10 insitu stress -3.95e e6 ygrad insitu szz=-3.95e6 ; hist unbal solve ratio 1e-4 save stage0.sav call zonk.fis del xr yr ; del reg del reg del ; ; must fix points and cycle once to get reaction forces ; boun int xvel 0 yvel 0 cycle 1 ; hist sxx hist xdis zonk relax save stage1.sav label hist 2 XX - Stress label hist 3 X - Displacement pl hist -2 vs -3 ret

55 LIBRARY OF FISH FUNCTIONS ZONK.FIS - 3 JOB TITLE :. UDEC (Version 5.00) (e+006) LEGEND 17-Sep :43:48 cycle 4080 time 3.027E+00 sec history plot Y-axis: XX - Stress X-axis: X - Displacement Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (e-002) Figure 1 Horizontal stress vs displacement at history point

56 ZONK.FIS - 4 FISH in UDEC