4 Cylindrical Hole in an Infinite Hoek-Brown Medium
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1 Cylindrical Hole in an Infinite Hoek-Brown Medium Cylindrical Hole in an Infinite Hoek-Brown Medium 4.1 Problem Statement Stresses and displacements are calculated for the case of a cylindrical hole in an infinite Hoek-Brown medium subjected to an in-situ stress field and supported by an internal pressure. We consider a cylindrical hole with a radius b = 2 m, created within an infinite body under a uniform compressive stress of magnitude σ o = 30 MPa. Support is provided by an internal pressure of p i =5MPa. The body has the following material properties: shear modulus (G) 2.20 GPa bulk modulus (K) GPa density (ρ) 2000 kg/m 3 Values of the Hoek-Brown strength properties for the rock are: m b 1.7 s a 0.5 uniaxial compressive strength of the intact rock (σ ci ) 30MPa 4.2 Closed-Form Solution The analytical solution for this problem is provided by Carranza-Torres and Fairhurst (1999) for both associated plastic flow and non-associated plastic flow (with zero dilation). In this verification exercise, we only present the case for non-associated flow. The scaled far-field stresses, S o, and scaled internal pressure, P i, are determined by Eqs. (4.1) and (4.2): S o = σ o m b σ ci + s m 2 b = (4.1) P i = p i m b σ ci + s m 2 b = (4.2) The scaled critical internal pressure, Pi cr, at which the elastic limit of the stress state is reached, is calculated as
2 4-2 Verification Problems The critical internal pressure, p cr i, is then Pi cr = 1 [ 1 ] S o = (4.3) 16 p cr i = [ P cr i ] s m 2 m b σ ci = 15.8 MPa (4.4) b A plastic region develops uniformly around the hole because p i <p cr i. The extent of the failure zone is [ ( b pl = b exp 2 Pi cr )] P i = 1.62b = 3.2 m (4.5) The solution for the radial stress, σ r, and tangential stress, σ θ, in the plastic region, r b pl, is: S r (r) = [ P cr i + 1 ( )] r 2 2 ln (4.6) b pl S θ (r) = S r (r) + S r (r) (4.7) [ ] σ r (r) = S r (r) s m 2 m b σ ci (4.8) b [ ] σ θ (r) = S θ (r) s m 2 m b σ ci (4.9) b The solution for the stress state in the elastic region, r>b pl, is: σ r (r) = σ o (σ o p cr i ) ( bpl r ) 2 (4.10) σ θ (r) = σ o + (σ o p cr i ) ( bpl r ) 2 (4.11)
3 Cylindrical Hole in an Infinite Hoek-Brown Medium 4-3 For the case of non-associated flow with zero dilation, the radial displacement in the plastic region is u r b 2G = (1 pcr i σ o σ ci 2 ( r A 1 1 b pl + ) bpl A r D + b A 1 1 b pl 2(S o Pi cr )(1 A 1 ) 3 ) A1 ( C r + 4(S o Pi cr ln r )(1 A 1 ) b pl ) D 2(S o P cr i )(1 A 1 ) 3 r b pl ( (1 A 1 ) ln r b pl 1 where: A 1 = K ψ ; A 2 =1 ν νk ψ ; A 3 = ν (1 ν)k ψ ; C = A 2 A 3 ; [ D = A 2 2(1 A1 ) Pi cr 1 ] [ A 3 2(1 A1 ) Pi cr ] A 1 ; and K ψ = 1+sin ψ 1 sin ψ. ψ is the dilation angle, G is shear modulus and ν is Poisson s ratio. b pl ( r b pl ) A1 ) 2 (4.12) 4.3 FLAC Model The FLAC model created for this problem is a plane-strain model with the plane of analysis oriented normal to the axis of the hole. Because of symmetry, only a quarter of the problem needs to be analyzed. The model and boundary conditions are shown in Figure 4.1. A special FISH function was used to create the finite-difference zoning in the FLAC model. The function is contained in the file QDONUT.FIS (see Section 3 in the FISH volume), which is called from HOEKHOLE.DAT (see Section 4.6). The grid is shown in Figure 4.2. As the figure indicates, it is a radially symmetric mesh with increasing zone size away from the hole. The grid contains 3600 zones, and the boundary is located at 20 radii from the center of the hole.
4 4-4 Verification Problems JOB TITLE :. (*10^1) FLAC (Version 6.00) LEGEND 16-May-08 9:33 step E+00 <x< 4.637E E+00 <y< 4.637E+01 Boundary plot 0 1E 1 Fixed Gridpoints -direction Y Y-direction Net Applied Forces max vector = 3.141E E 8 YY YYYY YYY Y Y Y Y Y Y Y Y Y Y Y Y Y Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure (*10^1) Model for FLAC analysis of a cylindrical hole in an infinite medium JOB TITLE :. (*10^1) FLAC (Version 6.00) LEGEND May-08 9:34 step E+00 <x< 4.637E E+00 <y< 4.637E Grid plot 0 1E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA (*10^1) Figure 4.2 FLAC zone geometry for the plane-strain model
5 Cylindrical Hole in an Infinite Hoek-Brown Medium Results and Discussion The analytical solution for radial and tangential stresses and radial displacement provided in Eqs. (4.1) through (4.12) are programmed in the FISH functions in HB SOLUTION.FIS, listed in Section 4.7. The analytical results and FLAC results are then copied into tables for comparison. Figure 4.3 shows the input required for the FISH functions to produce the analytical solutions and compare to FLAC results. The stresses and displacement are compared along a radial path measured from the center of the hole. The FISH variable radius tunnel is the radius of the tunnel, rb max defines the length of the normalized path ( number of measurement points along the path. r radius tunnel ), and npts defines the Figure 4.3 Fish input dialog Figure 4.4 shows the radial and tangential stresses calculated by FLAC, compared to the analytical solution for σ r, and σ θ. Figure 4.5 compares radial displacement, u r. The agreement is very good in both comparisons. A plot of the Hoek-Brown failure envelope is shown in Figure 4.6. Zone stresses are shown on this plot and indicate the extent of the failed zone. Figure 4.7 displays the zone plasticity indicators, which also indicate the extent of the failure region. This corresponds to the analytical solution of 3.2m(Eq. (4.5)).
6 4-6 Verification Problems JOB TITLE :. FLAC (Version 6.00) LEGEND 16-May-08 9:37 step 5668 Table Plot sigr/sigci (analytical) sigt/sigci (analytical) sigt/sigci (FLAC) sigr/sigci (FLAC) Itasca Consulting Group, Inc. Minneapolis, Minnesota USA -01 (10 ) Figure 4.4 Comparison of σ r and σ θ for the cylindrical hole in an infinite Hoek-Brown medium (along the normalized path 1<r/radius tunnel < 4) JOB TITLE :. FLAC (Version 6.00) LEGEND 16-May-08 9:38 step 5668 Table Plot ur/r [%] (analytical) ur/r [%] (FLAC) Itasca Consulting Group, Inc. Minneapolis, Minnesota USA -01 (10 ) Figure 4.5 Comparison of u r for the cylindrical hole in an infinite Hoek-Brown medium (along the normalized path 1<r/radius tunnel < 4)
7 Cylindrical Hole in an Infinite Hoek-Brown Medium 4-7 JOB TITLE :. FLAC (Version 6.00) LEGEND 16-May-08 9:39 step 5668 Failure Surface Plot Major Prin. Stress vs. Minor Prin. Stress Zone Stress States Hoek-Brown Failure Surf. s = E-03 UCS = E+07 m = E+00 Tension = E (10 ) Itasca Consulting Group, Inc. Minneapolis, Minnesota USA 06 (10 ) Figure 4.6 Hoek-Brown failure envelope JOB TITLE :. FLAC (Version 6.00) LEGEND 16-May-08 9:40 step E+00 <x< 5.769E E+00 <y< 5.453E+00 Plasticity Indicator shear-p shear-n shear-p Boundary plot E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 4.7 Plasticity indicators for Hoek-Brown model
8 4-8 Verification Problems 4.5 References Carranza-Torres, C., and C. Fairhurst. The Elasto-plastic Response of Underground Excavations in Rock Masses that Satisfy the Hoek-Brown Failure Criterion, Int. J. Rock Mech. Min. Sci., 36, (1999). Hoek, E., and E. T. Brown. Empirical Strength Criterion for Rock Masses, J. Geotech. Engng. Div. ASCE, 106, (1980). Hoek, E., and E. T. Brown. Underground Excavations in Rock. London: IMM, Hoek E., and E. T. Brown. The Hoek-Brown Failure Criterion A 1988 Update, in Rock Engineering For Underground Excavations (Proceedings of the 15th Canadian Rock Mechanics Symposium, October 1988), pp Toronto: University of Toronto, Department of Civil Engineering, 1988.
9 Cylindrical Hole in an Infinite Hoek-Brown Medium Data File HOEKHOLE.DAT ;Project Record Tree export ;Title:Hole in Hoek-Brown Medium ;... State: hb1.sav... config grid 60,60 model elastic set echo off call qdonut.fis set rmin=2.0 rmul=20 gratio=1.05 qdonut model hoekbrown notnull prop bulk= e9 shear=2.2e9 hbs= hbmb=1.7 hbsigci=3.0e7 & hba=0.5 hbs3cv=0.0 notnull prop density 2000 notnull initial sxx -3.0E7 initial syy -3.0E7 initial szz -3.0E7 apply nstress -3.0E7 from 61,61 to 61,1 apply nstress from 1,61 to 1,1 fix y j 1 fix x j 61 history 999 unbalanced solve save hb1.sav ;... State: hb2.sav... set echo off call hb solution.fis set sig0=30e6 pi=5e6 sigci=30e6 mb=1.7 s=3.9e-3 G=2.2e9 set nu=0.25 npts=100 rb max=4.0 radius tunnel=2.0 hb solution save hb2.sav ;*** plot commands **** ;plot name: Unbalanced force plot hold history 999 ;plot name: radial and hoop stresses label table 11 sigt/sigci (analytical) label table 10 sigr/sigci (analytical) label table 21 sigr/sigci (FLAC)
10 4-10 Verification Problems label table 20 sigt/sigci (FLAC) label table 12 ur/r [label table 22 ur/r [plot hold table 21 cross 20 cross 11 line 10 line ;plot name: radial displacement label table 22 ur/r [label table 12 ur/r [plot hold table 22 cross 12 line ;plot name: fail Hoek set hbm 1.7 hbs ucs 3.0E7 plot hold fail hoek principal ;plot name: plasticity plot plot hold plasticity bound
11 Cylindrical Hole in an Infinite Hoek-Brown Medium Data File HB SOLUTION.FIS ;Name:hb solution ;Diagram: ;Input: sig0/float/30e6/applied pressure on outer boundary ;Input: pi/float/5e6/applied pressure on inner boundary ;Input: sigci/float/30e6/hoek-brown constant ;Input: mb/float/1.7/hoek-brown constant ;Input: s/float/3.9e-3/hoek-brown constant ;Input: G/float/2.2e9/shear modulus ;Input: nu/float/0.25/poisson s ratio ;Input: npts/int/100/number of points along measurement path ;Input: rb max/float/4.0/length of measurement path ;Input: radius tunnel/float/2.0/tunnel radius def sigrhb ; input - rb (radial distance over tunnel radius) ; output - sigrhb (transformed radial stress) sigrhb = (sqrt( picrhb)+0.5*ln( rb/ xi))ˆ2 end ; def sigthb ; input - rb (radial distance over tunnel radius) ; output - sigthb (transformed hoop stress) sigthb = sigrhb + sqrt( sigrhb) end ; def urb2gsig0 ; input - rb (radial distance over tunnel radius) ; output - urb2gsig0 (transformed radial convergence) temp1 = ( A1+1)/( A1-1)* rb/ xi temp2 = ( D/2.0/( sig0hb- picrhb)/(1- A1)ˆ3-2.0/( A1-1))*( rb/ xi)ˆ A1 temp3 = C/4.0/( sig0hb- picrhb)/(1- A1)* rb/ xi*(ln( rb/ xi))ˆ2 temp4 = D/2.0/( sig0hb- picrhb)/(1- A1)ˆ3* rb/ xi*((1- A1)*ln( rb/ xi)-1) urb2gsig0 = xi*(1- picrsigci)*( temp1+ temp2+ temp3+ temp4) end ; def compute analytical solution pihb = pi/( sigci* mb) + s/ mbˆ2 sig0hb = sig0/( sigci* mb) + s/ mbˆ2 picrhb = 1.0/16.0*(1-sqrt(1+16* sig0hb))ˆ2 picrsigci = ( picrhb- s/ mbˆ2)* mb xi = exp(2*(sqrt( picrhb)-sqrt( pihb))) A1 = -1.0 A2 = 1 - nu - nu A3 = nu - (1- nu) C = A2 - A3
12 4-12 Verification Problems D = A2*(2*sqrt( picrhb)*(1- A1)-1) - A3*(2*sqrt( picrhb)*(1- A1)- A1) loop i (1, npts) rb = 1 + float(i-1)/float( npts-1)*( rb max-1) if pi/ sigci > picrsigci ; then solution is elastic... sigrsigcisol = sig0/ sigci-( sig0/ sigci- pi/ sigci)*(1/ rb)ˆ2 sigtsigcisol = sig0/ sigci+( sig0/ sigci- pi/ sigci)*(1/ rb)ˆ2 urbsol = sig0/2.0/ G*(1- picrsigci* sigci/ sig0)*(1/ rb)*100 xtable(10,i) = rb ytable(10,i) = sigrsigcisol xtable(11,i) = rb ytable(11,i) = sigtsigcisol xtable(12,i) = rb ytable(12,i) = urbsol else ; then solution is elasto-plastic... if rb > xi sigrsigcisol = sig0/ sigci-( sig0/ sigci- picrsigci)*( xi/ rb)ˆ2 sigtsigcisol = sig0/ sigci+( sig0/ sigci- picrsigci)*( xi/ rb)ˆ2 urbsol = sig0/2.0/ G*(1- picrsigci* sigci/ sig0)* xiˆ2/ rb*100.0 else sigrsigcisol = ( sigrhb- s/ mbˆ2)* mb sigtsigcisol = ( sigthb- s/ mbˆ2)* mb urbsol = sig0/2.0/ G* urb2gsig0*100.0 end if xtable(10,i) = rb ytable(10,i) = sigrsigcisol xtable(11,i) = rb ytable(11,i) = sigtsigcisol xtable(12,i) = rb ytable(12,i) = urbsol end if end loop end def extract flac results rad max = rb max* radius tunnel ; Stresses counter = 1 jh = int(60/2) loop i (1,60) xcoord =.25*(x(i, jh)+x(i, jh+1)+x(i+1, jh+1)+x(i+1, jh)) ycoord =.25*(y(i, jh)+y(i, jh+1)+y(i+1, jh+1)+y(i+1, jh)) sxx = sxx(i, jh) syy = syy(i, jh) sxy = sxy(i, jh) smin = 0.5*( sxx+ syy)+sqrt(( sxx- syy)ˆ2/4.0+ sxyˆ2) smax = 0.5*( sxx+ syy)-sqrt(( sxx- syy)ˆ2/4.0+ sxyˆ2) sigt = smax
13 Cylindrical Hole in an Infinite Hoek-Brown Medium 4-13 sigr = smin rad = sqrt( xcoord* xcoord+ ycoord* ycoord) if rad <= rad max xtable(20, counter) = rad/ radius tunnel ytable(20, counter) = - sigr/ sigci xtable(21, counter) = rad/ radius tunnel ytable(21, counter) = - sigt/ sigci counter = counter + 1 end if end loop ; ; Displacements counter = 1 loop i (1,igp) xcoord = x(i, jh) ycoord = y(i, jh) ux = xdisp(i, jh) uy = ydisp(i, jh) ur = sqrt( ux* ux+ uy* uy) rad = sqrt( xcoord* xcoord+ ycoord* ycoord) if rad <= rad max xtable(22, counter) = rad/ radius tunnel ytable(22, counter) = ur/ radius tunnel*100 counter = counter + 1 end if end loop END ; def hb solution compute analytical solution extract flac results end
14 4-14 Verification Problems
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