7 Uniaxial Compressive Strength of a Jointed Rock Sample

Transcription

1 Uniaxial Compressive Strength of a Jointed Rock Sample Uniaxial Compressive Strength of a Jointed Rock Sample 7.1 Problem Statement The uniaxial compressive strength of a jointed rock sample is a function of the angle formed by the major principal stress and the joints. In FLAC, this behavior of a jointed sample can be modeled using two different approaches: 1. The sample can be considered as a continuum with a plastic anisotropy in the direction of the joint. In this case, the ubiquitous-joint model can be used. 2. The joints can be individually modeled using interfaces. Both approaches are verified with this test problem. This test also demonstrates two different ways to perform parametric analysis with FLAC, based on each approach. The rock sample has a height/width ratio of 2. The rock mass has the following material properties: density 2000 kg/m 3 shear modulus (G) 70MPa bulk modulus (K) 100 MPa cohesion (c) 2 kpa friction angle (φ) 40 dilation angle (ψ) 0 The joint properties are: normal stiffness (k n ) shear stiffness (k s ) cohesion (c j ) friction angle (φ j ) dilation angle (ψ j ) 1 GPa/m 1 GPa/m 1 kpa 30 0 The calculations are performed under plane-strain conditions, so the test sample is equivalent to a long pillar. It is also assumed that the rock matrix and the joints have elastic, perfectly plastic behavior, with no strain-softening.

2 7-2 Verification Problems 7.2 Analytic Solution The plane-of-weakness model (Jaeger and Cook 1979) predicts that slip will occur in a triaxial test, provided (1 tan φ j tan β) > 0, for σ 1 = σ 3 2 (c j + σ 3 tan φ j ) (1 tan φ j tan β)sin 2β (7.1) where β is the angle formed by σ 1 and the joint (see Figure 7.1). σ 1 β 10 Figure 7.1 Problem geometry σ 1 5 For those combinations of c j, φ j, σ 3 and β for which Eq. (7.1) is not satisfied, slip in the joint cannot occur, and the only alternative is the failure of the rock matrix, which, according to the Mohr-Coulomb failure criterion, will occur for σ 1 = N φ σ 3 2c N φ (7.2) where: N φ = 1+sin φ 1 sin φ ; c = intact material cohesion; and φ = intact material angle of internal friction.

3 Uniaxial Compressive Strength of a Jointed Rock Sample 7-3 In the uniaxial compression test, σ 3 =0,soEqs. (7.1) and (7.2) can be rewritten as σ 1 = 2 c j (1 tan φ j tan β)sin 2β (7.3) and σ 1 = 2c N φ (7.4) The maximum pressure for a uniaxial compressive test (σ c ) of a jointed sample will then be min{2c 2c N φ, j (1 tan φ j tan β) sin 2β } if (1 tan φ j tan β) > 0 σ c = 2c N φ if (1 tan φ j tan β) < 0 (7.5) 7.3 FLAC Model Two different types of mesh are used in this analysis: one for the ubiquitous-joint model, and another for the model with an interface. Each model is loaded until failure occurs, and then the failure stress and type of failure mode are noted. Constant velocity boundary conditions are applied to the top and bottom of each model for a specified number of steps, to reach the failure state. Note that combined damping is used in both models, because velocity vectors are all nonzero in the final state (see Section in Theory and Background). Both models are contained in the data file JROCK.DAT (see Section 7.6). Ubiquitous-joint model Figure 7.2 shows the zone geometry used for the ubiquitous-joint model. The grid is the same for all values of β, because the inclination of the joints in this model is controlled by the material property jangle. Fairly accurate results are obtained with only 50 elements. The effect of the variation of β is studied every 5 from 90 to 0. In the FISH function, contained in JROCK.DAT, a MODEL null command is issued prior to the calculations for each value of β. This command resets displacements, velocities, stresses and properties to zero. The vertical stress (sigmav), analytical solution (anal), the value of β (beta) and vertical strain (ve) are tracked in histories. This approach allows us to save the entire parametric analysis in only one file: M7A. SAV. The results can be printed or plotted with the aid of the begin and skip switches. For this test, the failure state is found to be reached within 3000 calculational steps for the applied velocity loading condition. This occurs for failure either along the ubiquitous-joint plane or within the intact material. The FLAC solution at each value of β is then determined at the of each 3000 step increment.

4 7-4 Verification Problems JOB TITLE : Compressive Strength of a Jointed Sample (UBI) (*10^1) FLAC (Version 6.00) LEGEND 15-May-08 9:24 step E+00 <x< 8.954E E+00 <y< 1.145E Grid plot 0 2E 0 Boundary plot 0 2E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 7.2 Grid used for the ubiquitous-joint model Interface model When the interface logic is used, a different approach must be followed. In this case, the joint is explicitly modeled, which requires that a different grid be generated for each value of β. The input file for each value of β is JROCKB.DAT (see Section 7.7). The mesh is now created using two GENERATE commands, keeping a strip of null zones (j = 6) between the two sides of the joint. In order to make the grid generation a parametric process, the coordinates of the corners and the ranges of the GENERATE commands are calculated by FISH. As shown in Figure 7.3, the order in which the corners are numbered deps on β. When tan β<0.5, the joint will intersect the top and the bottom of the sample; the numbers used in the GENERATE commands appear in Figure 7.3(a). For tan β>0.5, the joint will intersect the sides of the sample; the numbers used appear in Figure 7.3(b). Figure 7.4 shows the grid obtained using this method for β =45. The MODEL null command will not reset the stresses in the interface for this case, so a NEW command must be issued after each analysis. The NEW command will reset the histories and the FISH functions, so each case must be saved in a separate file. In order to make the interpretation of the results simple, the values of beta and sigmav for each case are written to a file M7RES.BIN using FISH I/O routines (see Section in the FISH volume), and retrieved at the completion of all cases. For each value of β, the file JROCK.DAT calls the file JROCKB.DAT, sets the appropriate value of β, calls, saves the results, and issues a NEW command. performs calculational steps to reach the failure state in both the solid material and on the interface, and then executes a SOLVE command to ensure that steady-state flow is obtained. On completion of all of the cases, the values from the file JROCKRES.BIN are written to tables for comparison of results.

5 Uniaxial Compressive Strength of a Jointed Rock Sample Figure (a) (b) Corner numbers for the interface model: (a) for tan β<0.5; (b) for tan β>0.5 4 JOB TITLE : Compressive Strength of a Jointed Sample (INT) (*10^1) FLAC (Version 6.00) LEGEND May-08 9:26 step E+00 <x< 9.167E E+00 <y< 1.167E Grid plot 0 2E Boundary plot 0 2E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 7.4 Grid used for the interface model (β = 45 )

6 7-6 Verification Problems 7.4 Results and Discussion Figure 7.5 compares FLAC s ubiquitous-joint model and the analytical solution. This figure is created with the command plot history 2 3 cross vs 4 begin 3000 skip 30 A record of numerical data can also be written in ASCII form to the file FLAC.HIS by issuing the command hist write 2 3 vs 4 begin 3000 skip 30 The match is excellent, with the error below 1% for all values of β. JOB TITLE : Compressive Strength of a Jointed Sample (UBI) FLAC (Version 6.00) LEGEND 16-May-08 11:28 step HISTORY PLOT Y-axis : 2 sigmav (FISH) 3 anal (FISH) X-axis : 4 beta (FISH) 03 (10 ) Itasca Consulting Group, Inc. Minneapolis, Minnesota USA 01 (10 ) Figure 7.5 Comparison of uniaxial compressive strength values ubiquitous-joint model (cross) versus analytical solution (line)

7 Uniaxial Compressive Strength of a Jointed Rock Sample 7-7 Figure 7.6 presents the results obtained using the interface model. Three different modes of failure are observed: 1. No slip (β =0,5, and from 55 to 90 ) This mode involves plastic failure of the rock matrix and no slip in the interface. In this case, results closely match those predicted by Eq. (7.4), with a maximum error of less than 0.5%. 2. Slip at tan β>0.5(β = 30 to 50 ) Figure 7.7 shows the deformed sample for β =50 using a magnification factor of 200. The stress-strain curve for this value of β appears in Figure 7.8. The compressive strength oscillates about the value predicted from Eq. (7.5). (Note that this oscillation can be reduced by decreasing the magnitude of the applied velocity.) No failure of the rock matrix is involved in this mode. 3. Slip at tan β<0.5 (β = 10 to 25 ) For these values of β, the interface touches the platens, and both slipping and rock matrix failure occur, as shown in Figures 7.9 and 7.10 for β =20. The compressive strength obtained for this range of β lies between that predicted by Eqs. (7.3) and (7.4) (see Figure 7.11). While the ubiquitous-joint model precisely reproduces the analytical model, the interface model appears to produce a more representative behavior for the applied test conditions. JOB TITLE : Compressive Strength of a Jointed Sample (INT) FLAC (Version 6.00) 9-Jul-08 11:02 step LEGEND Table Plot FLAC - interface model analytical solution 03 (10 ) (10 ) Figure 7.6 Comparison of uniaxial compressive strength values interface model versus analytical solution

8 7-8 Verification Problems JOB TITLE : Compressive Strength of a Jointed Sample (INT) (*10^1) FLAC (Version 6.00) LEGEND Jul-08 11:03 step E+00 <x< 9.167E E+00 <y< 1.167E Boundary plot 0 2E 0 Exaggerated Boundary Disp. Magnification = 2.000E+02 Max Disp = 1.408E Figure 7.7 Deformed sample for β = 50 JOB TITLE : Compressive Strength of a Jointed Sample (INT) FLAC (Version 6.00) 9-Jul-08 11:04 step LEGEND HISTORY PLOT Y-axis : 2 sigmav (FISH) 3 anal (FISH) X-axis : 5 ve (FISH) 03 (10 ) (10 ) Figure 7.8 Stress-strain curve for β = 50

9 Uniaxial Compressive Strength of a Jointed Rock Sample 7-9 JOB TITLE : Compressive Strength of a Jointed Sample (INT) (*10^1) FLAC (Version 6.00) LEGEND Jul-08 11:05 step E+00 <x< 9.167E E+00 <y< 1.167E Boundary plot 0 2E 0 Exaggerated Boundary Disp. Magnification = 2.000E+02 Max Disp = 2.253E Figure 7.9 Deformed sample for β = 20 JOB TITLE : Compressive Strength of a Jointed Sample (INT) (*10^1) FLAC (Version 6.00) LEGEND Jul-08 11:05 step E+00 <x< 9.167E E+00 <y< 1.167E Boundary plot 0 2E 0 Plasticity Indicator * at yield in shear or vol. X elastic, at yield in past Figure 7.10 Failed zones for β = 20

10 7-10 Verification Problems JOB TITLE : Compressive Strength of a Jointed Sample (INT) FLAC (Version 6.00) 9-Jul-08 11:06 step LEGEND HISTORY PLOT Y-axis : 2 sigmav (FISH) 3 anal (FISH) X-axis : 5 ve (FISH) 03 (10 ) (10 ) Figure 7.11 Stress-strain curve for β = Reference Jaeger, J. C., and N. G. W. Cook. Fundamentals of Rock Mechanics, 3rd Ed. New York: Chapman and Hall, 1979.

11 Uniaxial Compressive Strength of a Jointed Rock Sample Data File JROCK.DAT ;Project Record Tree export ;*** BRANCH: UBI **** ;... STATE: JROCKA... g 5 10 set mess off def loop k (0,18) beta=90.0*(18.0-k)/18.0 alfa=90-beta command mo null mo ubi pro den 2000 bulk 1e8 she 7e7 fric 40 co 2e3 ten 2400 pro jco 1e3 jfric 30 jang alfa jten 2000 fix y j 1 fix y j 11 ini yvel -1e-7 j 11 ini yvel 1e-7 j 1 set st damp comb step 3000 print beta print sigmav print anal command loop def sigmav sum=0.0 loop i (1,igp) sum=sum+yforce(i,jgp) loop sigmav=sum/(x(igp,jgp)-x(1,jgp)) def ve ve=(ydisp(3,1)-ydisp(3,11))/(y(3,11)-y(3,1)) def anal mc=cohesion(1,1) mfi=friction(1,1)*degrad jc=jcohesion(1,1)

12 7-12 Verification Problems jfi=jfriction(1,1)*degrad sm=2.0*mc*cos(mfi)/(1.0-sin(mfi)) if beta=90*int(beta/90) then sj=-1 else divsj=((1.0-tan(jfi)*tan(beta*degrad))*sin(2.0*beta*degrad)) if divsj=0.0 then sj=-1 else sj=2.0*jc/divsj if if if sj<0 then anal=sm else anal=min(sj,sm) if hist nstep 100 hist unbal hist sigmav hist anal hist beta hist ve hist yv i 1 j 1 save jrocka.sav ;*** BRANCH: INTERFACE - 0 **** ;... STATE: JROCKB00... ;--- Run several cases, and save results in a binary file --- def startup ; Initialize the results file with a zero array zero(1) stat = open( jrockres.bin,1,0) zero(1) = 0 stat = write(zero,1) stat = close startup set beta 00 save jrockb00.sav

13 Uniaxial Compressive Strength of a Jointed Rock Sample 7-13 ;*** BRANCH: INTERFACE - 5 **** ;... STATE: JROCK05... set beta 05 save jrock05.sav ;*** BRANCH: INTERFACE - 10 **** ;... STATE: JROCKB10... set beta 10 save jrockb10.sav ;*** BRANCH: INTERFACE - 15 **** ;... STATE: JROCKB15... set beta 15 save jrockb15.sav ;*** BRANCH: INTERFACE - 20 **** ;... STATE: JROCKB20... set beta 20 save jrockb20.sav ;*** BRANCH: INTERFACE - 25 **** ;... STATE: JROCKB25...

14 7-14 Verification Problems set beta 25 save jrockb25.sav ;*** BRANCH: INTERFACE - 30 **** ;... STATE: JROCKB30... set beta 30 save jrockb30.sav ;*** BRANCH: INTERFACE - 35 **** ;... STATE: JROCKB35... ca jrockb.dat set beta 35 save jrockb35.sav ;*** BRANCH: INTERFACE - 40 **** ;... STATE: JROCKB40... set beta 40 save jrockb40.sav ;*** BRANCH: INTERFACE - 45 **** ;... STATE: JROCKB45... set beta 45 save jrockb45.sav ;*** BRANCH: INTERFACE - 50 ****

15 Uniaxial Compressive Strength of a Jointed Rock Sample 7-15 ;... STATE: JROCKB50... set beta 50 save jrockb50.sav ;*** BRANCH: INTERFACE - 55 **** ;... STATE: JROCKB55... set beta 55 save jrockb55.sav ;*** BRANCH: INTERFACE - 60 **** ;... STATE: JROCKB60... set beta 60 save jrockb60.sav ;*** BRANCH: INTERFACE - 65 **** ;... STATE: JROCKB65... set beta 65 save jrockb65.sav ;*** BRANCH: INTERFACE - 70 **** ;... STATE: JROCKB70... set beta 70

16 7-16 Verification Problems save jrockb70.sav ;*** BRANCH: INTERFACE - 75 **** ;... STATE: JROCKB75... set beta 75 save jrockb75.sav ;*** BRANCH: INTERFACE - 80 **** ;... STATE: JROCKB80... set beta 80 save jrockb80.sav ;*** BRANCH: INTERFACE - 85 **** ;... STATE: JROCKB85... set beta 85 save jrockb85.sav ;*** BRANCH: INTERFACE - 90 **** ;... STATE: JROCKB90... set beta 90 save jrockb90.sav ;... STATE: JROCKBFINAL... def put to table ; Put results & analytical solutions in tables loop n (1,narr)

17 Uniaxial Compressive Strength of a Jointed Rock Sample 7-17 beta = beta values(n) xtable(10,n) = beta ytable(10,n) = load values(n) xtable(11,n) = beta ytable(11,n) = anal Loop put to table save jrockbfinal.sav ;*** plot commands **** ;plot name: grid plot hold grid bound white ;plot name: Uniaxial Strength - UBI plot hold history 2 line 3 cross begin 3000 skip 30 vs 4 ;plot name: Deformed sample plot hold bound bound magnify green ;plot name: Stress-strain plot hold history 2 line 3 line vs 5 ;plot name: Failed zones plot hold bound plasticity ;plot name: Comparison of uniaxial strength label table 10 FLAC - interface model label table 11 analytical solution plot hold table 11 both 10 both

18 7-18 Verification Problems 7.7 Data File JROCKB.DAT set mess=off echo off title compressive strength of a jointed sample (INT) g def jrockio ; Read current number of records array arrrec(1) stat = open( jrockres.bin,0,0) stat = read(arrrec,1) nrec = arrrec(1) narr = nrec + 1 jrockio def update the file ; Create arrays to hold old+ results... read & write array beta values(narr) load values(narr) if nrec > 0 stat = read(beta values,nrec) stat = read(load values,nrec) if stat = close beta values(narr) = beta load values(narr) = sigmav arrrec(1) = narr stat = open( jrockres.bin,1,0) stat = write(arrrec,1) stat = write(beta values,narr) stat = write(load values,narr) stat = close def anal ; Analytical solution for joint... infinite sample mc = cohesion(1,1) mfi = friction(1,1) * degrad jc = 1e3 jfi = 30.0 * degrad sm = 2.0 * mc * cos(mfi) / (1.0-sin(mfi)) sjb = tan(jfi) * tan(beta*degrad) sjdem = (1.0-sjb) * sin(2.0*beta*degrad) if sjdem = 0.0 then sj = -1 else sj = 2.0 * jc / sjdem if if sj < 0 then

19 Uniaxial Compressive Strength of a Jointed Rock Sample 7-19 anal = sm else anal = min(sj,sm) if def i1 = 1 i2 = igp i3 = 1 i4 = igp j1 = 1 j2 = jgp / 2 j3 = jgp / j4 = jgp if beta = 90.0 then tb = 1e10 else tb = tan(beta*degrad) if if tb > 0.5 then x1 = 0.0 y1 = 0.0 x2 = 0.0 y2 = / tb x3 = 5.0 y3 = / tb x4 = 5.0 y4 = 0.0 x5 = 0.0 y5 = 10.0 x6 = 5.0 y6 = 10.0 fi1 = 1 fi2 = igp fi3 = 1 fi4 = igp fj1 = 1 fj2 = 1 fj3 = jgp fj4 = jgp else x1 = 0.0 y1 = 10.0 x2 = * tb y2 = 10.0 x3 = * tb

20 7-20 Verification Problems y3 = 0.0 x4 = 0.0 y4 = 0.0 x5 = 5.0 y5 = 10.0 x6 = 5.0 y6 = 0.0 fi1 = igp fi2 = igp fi3 = 1 fi4 = 1 fj1 = 1 fj2 = jgp fj3 = 1 fj4 = jgp if command mo mo j 1 5 mo mo j 7 11 gen x1 y1 x2 y2 x3 y3 x4 y4 i i1 i2 j j1 j2 gen x2 y2 x5 y5 x6 y6 x3 y3 i i3 i4 j j3 j4 fix y i fi1 fi2 j fj1 fj2 fix y i fi3 fi4 j fj3 fj4 ini yvel 1e-7 i fi1 fi2 j fj1 fj2 ini yvel -1e-7 i fi3 fi4 j fj3 fj4 pro den 2000 bulk 1e8 she 7e7 fric 40 co 2e3 ten 2400 j 1 5 pro den 2000 bulk 1e8 she 7e7 fric 40 co 2e3 ten 2400 j 7 11 int 1 aside from 1 6 to 11 6 bside from 1 7 to 11 7 int 1 kn 1e9 ks 1e9 fric 30 co 1e3 set ncw=50 st damp comb step=4000 step solve force=0.5 command s1 = string(beta) s2 = string(sigmav) s3 = string(anal) oo = out( beta = +s1+ sigmav = +s2+ anal = +s3) update the file def sigmav sum = 0.0 loop i (fi1,fi2) loop j (fj1,fj2) sum = sum - yforce(i,j) loop loop

21 Uniaxial Compressive Strength of a Jointed Rock Sample 7-21 sigmav = sum / 5.0 def ve ve=(ydisp(fi1,fj1)-ydisp(fi3,fj3))/10.0 hist nstep 50 hist unbal hist sigmav hist anal hist beta hist ve hist yv i 1 j 1 return

22 7-22 Verification Problems