Verification Problems : 7 : Uniaxial Compressive …web.mst.edu/~norbert/ge5471/Assignments/Assign 1 - FLAC I...Uniaxial Compressive Strength of a Jointed Rock Sample 7-1 7 Uniaxial

Uniaxial Compressive Strength of a Jointed Rock Sample 7 - 1

7 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 themajor principal stress and the joints. In FLAC, this behavior of a jointed sample can be modeledusing two different approaches:

1. The sample can be considered as a continuum with a plastic anisotropy in thedirection 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 waysto 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/m3

shear modulus (G) 70 MPabulk modulus (K) 100 MPacohesion (c) 2 kPafriction angle (φ) 40◦dilation angle (ψ) 0◦

The joint properties are:

normal stiffness (kn) 1 GPa/mshear stiffness (ks) 1 GPa/mcohesion (cj ) 1 kPafriction angle (φj ) 30◦dilation angle (ψj ) 0◦

The calculations are performed under plane-strain conditions, so the test sample is equivalent toa long pillar. It is also assumed that the rock matrix and the joints have elastic, perfectly plasticbehavior, with no strain-softening.

FLAC Version 6.0

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 φj tan β) > 0, for

σ1 = σ3 − 2 (cj + |σ3| tan φj )

(1 − tan φj tan β) sin 2β(7.1)

where β is the angle formed by σ1 and the joint (see Figure 7.1).

β

σ 1

σ 1

10

5

Figure 7.1 Problem geometry

For those combinations of cj , φj , σ3 and β for which Eq. (7.1) is not satisfied, slip in the jointcannot occur, and the only alternative is the failure of the rock matrix, which, according to theMohr-Coulomb failure criterion, will occur for

σ1 = Nφσ3 − 2c√Nφ (7.2)

where:Nφ = 1+sin φ

1−sin φ ;

c = intact material cohesion; and

φ = intact material angle of internal friction.

FLAC Version 6.0


In the uniaxial compression test, σ3 = 0, so Eqs. (7.1) and (7.2) can be rewritten as

σ1 = −2 cj(1 − tan φj tan β) sin 2β

(7.3)

and

σ1 = −2c√Nφ (7.4)

The maximum pressure for a uniaxial compressive test (σc) of a jointed sample will then be

σc =

⎧⎪⎨⎪⎩

min{2c√Nφ, 2cj(1 − tan φj tan β) sin 2β } if (1 − tan φj tan β) > 0

(7.5)2c

√Nφ if (1 − tan φj tan β) < 0

7.3 FLAC Model

Two different types of mesh are used in this analysis: one for the ubiquitous-joint model, andanother for the model with an interface. Each model is loaded until failure occurs, and then thefailure stress and type of failure mode are noted. Constant velocity boundary conditions are appliedto the top and bottom of each model for a specified number of steps, to reach the failure state. Notethat combined damping is used in both models, because velocity vectors are all nonzero in the finalstate (see Section 1.3.4 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 ofβ, because the inclination of the joints in this model is controlledby 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 hsol,contained in “JROCK.DAT,” a MODEL null command is issued prior to the calculations for eachvalue of β. This command resets displacements, velocities, stresses and properties to zero. Thevertical 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 onefile: “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 appliedvelocity loading condition. This occurs for failure either along the ubiquitous-joint plane or withinthe intact material. The FLAC solution at each value of β is then determined at the end of each3000 step increment.

FLAC Version 6.0


FLAC (Version 6.00)

LEGEND

15-May-08 9:24 step 57000 -3.954E+00 <x< 8.954E+00 -1.454E+00 <y< 1.145E+01

Grid plot

0 2E 0

Boundary plot

0 2E 0

0.000

0.200

0.400

0.600

0.800

1.000

(*10^1)

-2.000 0.000 2.000 4.000 6.000 8.000

JOB TITLE : Compressive Strength of a Jointed Sample (UBI)

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. Inthis case, the joint is explicitly modeled, which requires that a different grid be generated for eachvalue of β. The input file for each value of β is “JROCKB.DAT” (see Section 7.7). The mesh isnow created using two GENERATE commands, keeping a strip of null zones (j = 6) between thetwo sides of the joint. In order to make the grid generation a parametric process, the coordinatesof the corners and the ranges of the GENERATE commands are calculated by FISH. As shown inFigure 7.3, the order in which the corners are numbered depends on β. When tan β < 0.5, the jointwill intersect the top and the bottom of the sample; the numbers used in the GENERATE commandsappear in Figure 7.3(a). For tan β > 0.5, the joint will intersect the sides of the sample; the numbersused 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 NEWcommand must be issued after each analysis. The NEW command will reset the histories and theFISH functions, so each case must be saved in a separate file. In order to make the interpretation ofthe results simple, the values ofbeta andsigmav for each case are written to a file “M7RES.BIN”using FISH I/O routines (see Section 2.8.4 in the FISH volume), and retrieved at the completionof all cases.

For each value of β, the file “JROCK.DAT” calls the file “JROCKB.DAT,” sets the appropriatevalue of β, calls hsol, saves the results, and issues a NEW command. hsol performs 10000calculational steps to reach the failure state in both the solid material and on the interface, and thenexecutes a SOLVE command to ensure that steady-state flow is obtained. On completion of all of thecases, the values from the file “JROCKRES.BIN” are written to tables for comparison of results.

FLAC Version 6.0


1 2

34

5

61

2

3

4

5 6

34 6(a) (b)

Figure 7.3 Corner numbers for the interface model: (a) for tan β < 0.5; (b)for tan β > 0.5

FLAC (Version 6.00)

LEGEND

15-May-08 9:26 step 8030 -4.167E+00 <x< 9.167E+00 -1.667E+00 <y< 1.167E+01

Grid plot

0 2E 0

Boundary plot

0 2E 0

0.000

0.200

0.400

0.600

0.800

1.000

(*10^1)

-3.000 -1.000 1.000 3.000 5.000 7.000 9.000

JOB TITLE : Compressive Strength of a Jointed Sample (INT)


Figure 7.4 Grid used for the interface model (β = 45◦)

FLAC Version 6.0


7.4 Results and Discussion

Figure 7.5 compares FLAC ’s ubiquitous-joint model and the analytical solution. This figure iscreated 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 issuingthe 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.

FLAC (Version 6.00)

LEGEND

16-May-08 11:28 step 57000 HISTORY PLOT Y-axis : 2 sigmav (FISH)

3 anal (FISH)

X-axis : 4 beta (FISH)

1 2 3 4 5 6 7 8 9

(10 ) 01

4.000

5.000

6.000

7.000

8.000

(10 ) 03

JOB TITLE : Compressive Strength of a Jointed Sample (UBI)


Figure 7.5 Comparison of uniaxial compressive strength values –ubiquitous-joint model (cross) versus analytical solution (line)

FLAC Version 6.0


Figure 7.6 presents the results obtained using the interface model. Three different modes of failureare observed:

1. No slip (β = 0◦, 5◦, and from 55◦ to 90◦) – This mode involves plastic failureof the rock matrix and no slip in the interface. In this case, results closelymatch 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 samplefor β = 50◦ using a magnification factor of 200. The stress-strain curve for thisvalue of β appears in Figure 7.8. The compressive strength oscillates aboutthe value predicted from Eq. (7.5). (Note that this oscillation can be reducedby decreasing the magnitude of the applied velocity.) No failure of the rockmatrix is involved in this mode.

3. Slip at tan β < 0.5 (β = 10◦ to 25◦) – For these values of β, the interfacetouches the platens, and both slipping and rock matrix failure occur, as shownin Figures 7.9 and 7.10 for β = 20◦. The compressive strength obtained for thisrange 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 modelappears to produce a more representative behavior for the applied test conditions.

FLAC (Version 6.00)

LEGEND

9-Jul-08 11:02 step 10000 Table PlotFLAC - interface model

analytical solution

1 2 3 4 5 6 7 8 9

(10 ) 01

4.000

5.000

6.000

7.000

8.000

(10 ) 03


Figure 7.6 Comparison of uniaxial compressive strength values – interfacemodel versus analytical solution

FLAC Version 6.0


FLAC (Version 6.00)

LEGEND

9-Jul-08 11:03 step 10000 -4.167E+00 <x< 9.167E+00 -1.667E+00 <y< 1.167E+01

Boundary plot

0 2E 0

Exaggerated Boundary Disp.

Magnification = 2.000E+02Max Disp = 1.408E-03

0.000

0.200

0.400

0.600

0.800

1.000

(*10^1)

-3.000 -1.000 1.000 3.000 5.000 7.000 9.000


Figure 7.7 Deformed sample for β = 50◦

FLAC (Version 6.00)

LEGEND

9-Jul-08 11:04 step 10000 HISTORY PLOT Y-axis : 2 sigmav (FISH)

3 anal (FISH)

X-axis : 5 ve (FISH)

2 4 6 8 10 12 14 16 18 20

(10 )-05

1.000

2.000

3.000

4.000

5.000

6.000

(10 ) 03


Figure 7.8 Stress-strain curve for β = 50◦

FLAC Version 6.0


FLAC (Version 6.00)

LEGEND

9-Jul-08 11:05 step 10000 -4.167E+00 <x< 9.167E+00 -1.667E+00 <y< 1.167E+01

Boundary plot

0 2E 0

Exaggerated Boundary Disp.

Magnification = 2.000E+02Max Disp = 2.253E-03

0.000

0.200

0.400

0.600

0.800

1.000

(*10^1)

-3.000 -1.000 1.000 3.000 5.000 7.000 9.000


Figure 7.9 Deformed sample for β = 20◦

FLAC (Version 6.00)

LEGEND

9-Jul-08 11:05 step 10000 -4.167E+00 <x< 9.167E+00 -1.667E+00 <y< 1.167E+01

Boundary plot

0 2E 0

Plasticity Indicator* at yield in shear or vol.X elastic, at yield in past

0.000

0.200

0.400

0.600

0.800

1.000

(*10^1)

-3.000 -1.000 1.000 3.000 5.000 7.000 9.000


Figure 7.10 Failed zones for β = 20◦

FLAC Version 6.0


FLAC (Version 6.00)

LEGEND

9-Jul-08 11:06 step 10000 HISTORY PLOT Y-axis : 2 sigmav (FISH)

3 anal (FISH)

X-axis : 5 ve (FISH)

0 4 8 12 16 20

(10 )-05

0.000

1.000

2.000

3.000

4.000

5.000

6.000

(10 ) 03


Figure 7.11 Stress-strain curve for β = 20◦

7.5 Reference

Jaeger, J. C., and N. G. W. Cook. Fundamentals of Rock Mechanics, 3rd Ed. New York: Chapmanand Hall, 1979.

FLAC Version 6.0


7.6 Data File “JROCK.DAT”

;Project Record Tree export

;*** BRANCH: UBI ****new

;... STATE: JROCKA ....configg 5 10set mess offdef hsol

loop k (0,18)beta=90.0*(18.0-k)/18.0alfa=90-betacommand

mo nullmo ubipro den 2000 bulk 1e8 she 7e7 fric 40 co 2e3 ten 2400pro jco 1e3 jfric 30 jang alfa jten 2000fix y j 1fix y j 11ini yvel -1e-7 j 11ini yvel 1e-7 j 1set st damp combstep 3000print betaprint sigmavprint anal

end commandend loop

enddef sigmav

sum=0.0loop i (1,igp)

sum=sum+yforce(i,jgp)end loopsigmav=sum/(x(igp,jgp)-x(1,jgp))

enddef ve

ve=(ydisp(3,1)-ydisp(3,11))/(y(3,11)-y(3,1))enddef anal

mc=cohesion(1,1)mfi=friction(1,1)*degradjc=jcohesion(1,1)

FLAC Version 6.0


jfi=jfriction(1,1)*degradsm=2.0*mc*cos(mfi)/(1.0-sin(mfi))if beta=90*int(beta/90) then

sj=-1else

divsj=((1.0-tan(jfi)*tan(beta*degrad))*sin(2.0*beta*degrad))if divsj=0.0 then

sj=-1else

sj=2.0*jc/divsjend if

end ifif sj<0 then

anal=smelse

anal=min(sj,sm)end if

endhist nstep 100hist unbalhist sigmavhist analhist betahist vehist yv i 1 j 1hsolsave jrocka.sav

;*** BRANCH: INTERFACE - 0 ****new

;... STATE: JROCKB00 ....config;--- 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) = 0stat = write(zero,1)stat = close

endstartupcall jrockb.datset beta 00hsolsave jrockb00.sav

FLAC Version 6.0



;... STATE: JROCK05 ....configcall jrockb.datset beta 05hsolsave jrock05.sav


;... STATE: JROCKB10 ....configcall jrockb.datset beta 10hsolsave jrockb10.sav






;... STATE: JROCKB25 ....config

FLAC Version 6.0


call jrockb.datset beta 25hsolsave jrockb25.sav




;... STATE: JROCKB35 ....configca jrockb.datset beta 35hsolsave jrockb35.sav





;*** BRANCH: INTERFACE - 50 ****

FLAC Version 6.0


new









;... STATE: JROCKB70 ....configcall jrockb.datset beta 70

FLAC Version 6.0


hsolsave jrockb70.sav









;... STATE: JROCKBFINAL ....def put to table ; Put results & analytical solutions in tablesloop n (1,narr)

FLAC Version 6.0


beta = beta values(n)xtable(10,n) = betaytable(10,n) = load values(n)xtable(11,n) = betaytable(11,n) = anal

endLoopendput to tablesave jrockbfinal.sav

;*** plot commands ****;plot name: gridplot hold grid bound white;plot name: Uniaxial Strength - UBIplot hold history 2 line 3 cross begin 3000 skip 30 vs 4;plot name: Deformed sampleplot hold bound bound magnify 200.0 green;plot name: Stress-strainplot hold history 2 line 3 line vs 5;plot name: Failed zonesplot hold bound plasticity;plot name: Comparison of uniaxial strengthlabel table 10FLAC - interface modellabel table 11analytical solutionplot hold table 11 both 10 both

FLAC Version 6.0


7.7 Data File “JROCKB.DAT”

set mess=off echo offtitlecompressive strength of a jointed sample (INT)g 10 11def jrockio

; Read current number of recordsarray arrrec(1)stat = open(’jrockres.bin’,0,0)stat = read(arrrec,1)nrec = arrrec(1)narr = nrec + 1

endjrockiodef update the file

; Create arrays to hold old+new results ... read & write newarray beta values(narr) load values(narr)if nrec > 0

stat = read(beta values,nrec)stat = read(load values,nrec)

endifstat = closebeta values(narr) = betaload values(narr) = sigmavarrrec(1) = narrstat = open(’jrockres.bin’,1,0)stat = write(arrrec,1)stat = write(beta values,narr)stat = write(load values,narr)stat = close

enddef anal ; Analytical solution for joint ... infinite sample

mc = cohesion(1,1)mfi = friction(1,1) * degradjc = 1e3jfi = 30.0 * degradsm = 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 = -1else

sj = 2.0 * jc / sjdemend ifif sj < 0 then

FLAC Version 6.0


anal = smelse

anal = min(sj,sm)end if

enddef hsol

i1 = 1i2 = igpi3 = 1i4 = igpj1 = 1j2 = jgp / 2j3 = jgp / 2 + 1j4 = jgpif beta = 90.0 then

tb = 1e10else

tb = tan(beta*degrad)end ifif tb > 0.5 then

x1 = 0.0y1 = 0.0x2 = 0.0y2 = 5.0 - 2.5 / tbx3 = 5.0y3 = 5.0 + 2.5 / tbx4 = 5.0y4 = 0.0x5 = 0.0y5 = 10.0x6 = 5.0y6 = 10.0fi1 = 1fi2 = igpfi3 = 1fi4 = igpfj1 = 1fj2 = 1fj3 = jgpfj4 = jgp

elsex1 = 0.0y1 = 10.0x2 = 2.5 + 5.0 * tby2 = 10.0x3 = 2.5 - 5.0 * tb

FLAC Version 6.0


y3 = 0.0x4 = 0.0y4 = 0.0x5 = 5.0y5 = 10.0x6 = 5.0y6 = 0.0fi1 = igpfi2 = igpfi3 = 1fi4 = 1fj1 = 1fj2 = jgpfj3 = 1fj4 = jgp

end ifcommand

mo mo j 1 5mo mo j 7 11gen x1 y1 x2 y2 x3 y3 x4 y4 i i1 i2 j j1 j2gen x2 y2 x5 y5 x6 y6 x3 y3 i i3 i4 j j3 j4fix y i fi1 fi2 j fj1 fj2fix y i fi3 fi4 j fj3 fj4ini yvel 1e-7 i fi1 fi2 j fj1 fj2ini yvel -1e-7 i fi3 fi4 j fj3 fj4pro den 2000 bulk 1e8 she 7e7 fric 40 co 2e3 ten 2400 j 1 5pro den 2000 bulk 1e8 she 7e7 fric 40 co 2e3 ten 2400 j 7 11int 1 aside from 1 6 to 11 6 bside from 1 7 to 11 7int 1 kn 1e9 ks 1e9 fric 30 co 1e3set ncw=50 st damp comb step=4000step 10000solve force=0.5

end commands1 = string(beta)s2 = string(sigmav)s3 = string(anal)oo = out(’beta = ’+s1+’ sigmav = ’+s2+’ anal = ’+s3)update the file

enddef sigmav

sum = 0.0loop i (fi1,fi2)

loop j (fj1,fj2)sum = sum - yforce(i,j)

end loopend loop

FLAC Version 6.0


sigmav = sum / 5.0enddef ve

ve=(ydisp(fi1,fj1)-ydisp(fi3,fj3))/10.0endhist nstep 50hist unbalhist sigmavhist analhist betahist vehist yv i 1 j 1return

FLAC Version 6.0


FLAC Version 6.0

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Verification Problems : 7 : Uniaxial Compressive …web.mst.edu/~norbert/ge5471/Assignments/Assign 1 - FLAC I...Uniaxial Compressive Strength of a Jointed Rock Sample 7-1 7 Uniaxial