3 Cylindrical Hole in an Infinite Mohr-Coulomb Medium

Transcription

1 Cylindrical Hole in an Infinite Mohr-Coulomb Medium Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3.1 Problem Statement The problem concerns the determination of stresses and displacements for the case of a cylindrical hole in an infinite elasto-plastic medium subjected to in-situ stresses. The medium is assumed to be linearly elastic, perfectly plastic, with a failure surface defined by the Mohr-Coulomb criterion with both the associated (dilatancy = friction angle) and non-associated (dilatancy = 0) flow rules. This problem tests the Mohr-Coulomb plasticity model, the plane-strain condition and axisymmetric geometry in FLAC. The Mohr-Coulomb material is assigned the following properties: density (ρ) 2500 kg/m 3 shear modulus (G) 2.8 GPa bulk modulus (K) 3.9 GPa cohesion (c) 3.45 MPa friction angle (φ) 30 dilation angle (ψ) 0 and 30 An isotropic in-situ stress state exists with stresses equal to 30 MPa (tension positive). It is assumed that the problem is symmetric about both the horizontal and vertical axes. The radius of the hole is 1 m, and is assumed to be small compared to the length of the cylinder. This permits the use of the plane-strain condition.

2 3-2 Verification Problems 3.2 Closed-Form Solution The yield zone radius, R o, is given analytically by a theoretical model based on the solution of Salencon (1969): R o [ 2 = a K p + 1 P o + P i + q ] 1/(Kp 1) K p 1 q K p 1 (3.1) where: a = radius of hole K p = 1+sin φ 1 sin φ q =2 c tan (45 + φ/2) P o = initial in-situ stress magnitude and P i = internal pressure. The radial stress at the elastic/plastic interface is σ re = 1 K p + 1 (2P o q) (3.2) The stresses in the plastic zone are: σ r = σ θ = q K p 1 (P i + q K p 1 K p(p i + q K p 1 ) ( r a )K p 1 q K p 1 ) ( r a )K p 1 (3.3) where r = distance to the center of the hole. The stresses in the elastic zone are: σ r = P o + (P o σ re ) ( R o r )2 (3.4) σ θ = P o (P o σ re ) ( R o r )2

3 Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3-3 The displacements in the elastic and plastic regions are given by Salencon (1969). For the elastic region, ( u r = P o ( 2P o q K p + 1 ) ) ( R o 2G )(R o ) r (3.5) and for the plastic region: u r = r 2G χ χ = (2ν 1) ( P o + ) q K p 1 ( ) (1 ν)(k 2 ( )( ) p 1) q (Kp 1) ( Ro Ro + P i + K p + K ps K p 1 a r ( + (1 ν) (K )( ) pk ps + 1) (K p + K ps ) ν q ( r ) (Kp 1) P i + K p 1 a ) (Kps +1) (3.6) where: K ps = 1+sin ψ 1 sin ψ ψ = dilation angle ν = Poisson s ratio and G = shear modulus.

4 3-4 Verification Problems 3.3 FLAC Model The problem is first modeled as a two-dimensional plane-strain calculation using quarter-symmetry. The boundary conditions applied to the model are shown in Figure 3.1. The area representing the problem is divided into finite-difference zones, as shown in Figures 3.2 and 3.3, using the FISH data file HOLE.FIS (see Section 3 in the FISH volume). The outer boundary is located 10 m (five hole diameters) from the hole center. The model contains 900 rectangular zones oriented in a radial pattern, as indicated in Figures 3.2 and 3.3. This pattern minimizes the influence of the grid on localization effects (as discussed in Section in the User s Guide). P o P o Figure 3.1 Model for FLAC analysis quarter-symmetry

5 Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3-5 JOB TITLE :. (*10^1) May-08 9:46 step E+00 <x< 1.167E E+00 <y< 1.167E Grid plot 0 2E (*10^1) Figure 3.2 FLAC grid quarter-symmetry JOB TITLE :. 16-May-08 9:46 step E-01 <x< 2.000E E-01 <y< 2.000E+00 Grid plot E Figure 3.3 Zone geometry in region around hole

6 3-6 Verification Problems The problem is also modeled using axisymmetric geometry. Figure 3.4 shows the boundary conditions, and Figure 3.5 shows the zone geometry created for this calculation. A height of two zones is used to facilitate stress contour plots. Both models are subjected to an isotropic compressive stress of 30 MPa. The initial stress state is applied throughout each model first then the hole is removed. The data file MHOLE.DAT in Section 3.6 contains the FLAC commands for both the quartersymmetry grid model and the axisymmetric model. The comparison of FLAC results to the analytical solution is performed using FISH functions also contained in this data file. The theoretical relations are solved in FISH function theor. Then, the FLAC values for stress and displacement are compared to the solution values. The error distribution in radial and tangential stress is calculated in the function evals, and the error distribution in displacement is calculated in the function evald. Axis of Symmetry P o Figure 3.4 Model for FLAC analysis axisymmetric geometry

7 Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3-7 JOB TITLE : May-08 9:47 step E-01 <x< 1.050E E+00 <y< 5.500E Grid plot 0 2E (*10^1) Figure 3.5 FLAC grid axisymmetric geometry 3.4 Results and Discussion Figures 3.6, 3.7 and 3.8 show a direct comparison between FLAC results and the analytical solution along a radial line for the quarter-symmetry case. Normalized stresses, σ r /P o and σ θ /P o, are plotted versus normalized radius, r/a, infigure 3.6 normalized displacement, u r /a, is plotted versus normalized radius in Figure 3.7 for the associated flow case, and in Figure 3.8 for the nonassociated flow case. (Note that the stress states are identical for both plastic flow rules.) The error distribution throughout the entire FLAC grid is also presented. Figures 3.9 through 3.11 show contour plots of tangential stress error, radial stress error and radial displacement error, respectively, for the non-associated flow case.

8 3-8 Verification Problems JOB TITLE :. 16-May-08 9:50 step 3262 Table Plot Table 10 Table 11 Table 20 Table Figure 3.6 Stress solution comparison tangential stress: analytical (Table 10) vs FLAC (Table 11) radial stress: analytical (Table 20) vs FLAC (Table 21) JOB TITLE :. 16-May-08 9:50 step 3262 Table Plot Table 30 Table (10 ) Figure 3.7 Radial displacement solution comparison associated flow: analytical (Table 30) vs FLAC (Table 31)

9 Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3-9 JOB TITLE :. -02 (10 ) 16-May-08 9:51 step Table Plot Table 30 Table Figure 3.8 Radial displacement solution comparison non-associated flow: analytical (Table 30) vs FLAC (Table 31) JOB TITLE :. (*10^1) May-08 10:59 step E+00 <x< 1.167E E+00 <y< 1.167E Boundary plot 0 2E 0 EX_ 6 Contours -3.50E E E E E E E E E E Contour interval= 5.00E-01 Extrap. by averaging (*10^1) Figure 3.9 Error distributions for tangential stress

10 3-10 Verification Problems JOB TITLE :. (*10^1) May-08 11:01 step E+00 <x< 1.167E E+00 <y< 1.167E Boundary plot 0 2E 0 EX_ 7 Contours -2.00E E E E E E E E E Contour interval= 2.50E-01 Extrap. by averaging (*10^1) Figure 3.10 Error distributions for radial stress JOB TITLE :. (*10^1) May-08 9:53 step E+00 <x< 1.167E E+00 <y< 1.167E Boundary plot 0 2E 0 EX_ 8 Contours -5.00E E E E E E E E E Contour interval= 5.00E (*10^1) Figure 3.11 Error distributions for displacements

11 Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3-11 For the axisymmetric geometry, Figures 3.12, 3.13 and 3.14 show a direct comparison between FLAC results and the analytical solution. Figures 3.15, 3.16 and 3.17 show the error distributions for tangential stress, radial stress and radial displacement. JOB TITLE : Hole in Mohr-Coulomb Medium, Axisymmetric Case 16-May-08 9:56 step 1902 Table Plot Table 10 Table 11 Table 20 Table Figure 3.12 Stress solution comparison axisymmetric case tangential stress: analytical (Table 10) vs FLAC (Table 11) radial stress: analytical (Table 20) vs FLAC (Table 21)

12 3-12 Verification Problems JOB TITLE : Hole in Mohr-Coulomb Medium, Axisymmetric Case 16-May-08 9:56 step 1902 Table Plot Table 30 Table (10 ) Figure 3.13 Radial displacement solution comparison axisymmetric case (associated flow): analytical (Table 30) vs FLAC (Table 31) JOB TITLE : Hole in Mohr-Coulomb Medium, Axisymmetric Case 16-May-08 9:57 step 2127 Table Plot Table 30 Table (10 ) Figure 3.14 Radial displacement solution comparison axisymmetric case (non-associated flow): analytical (Table 30) vs FLAC (Table 31)

13 Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3-13 JOB TITLE : Hole in Mohr-Coulomb medium, Axisymmetric Case May-08 11:04 step E-01 <x< 1.050E E+00 <y< 5.500E Boundary plot 0 2E 0 EX_ 6 Contours -3.00E E E E E E E E E+00 Contour interval= 5.00E-01 Extrap. by averaging (*10^1) Figure 3.15 Error distributions in tangential stress axisymmetric case JOB TITLE : Hole in Mohr-Coulomb medium, Axisymmetric Case May-08 11:04 step E-01 <x< 1.050E E+00 <y< 5.500E Boundary plot 0 2E 0 EX_ 7 Contours -1.75E E E E E E E E+00 Contour interval= 2.50E-01 Extrap. by averaging (*10^1) Figure 3.16 Error distributions in radial stress axisymmetric case

14 3-14 Verification Problems JOB TITLE : Hole in Mohr-Coulomb medium, Axisymmetric Case May-08 11:05 step E-01 <x< 1.050E E+00 <y< 5.500E Boundary plot 0 2E 0 EX_ 8 Contours 2.50E E E E E E E E E-01 Contour interval= 2.50E (*10^1) Figure 3.17 Error distribution in radial displacements axisymmetric case In both the quarter-symmetry model and the axisymmetric model, the errors are small the average stress error is less than 2%, while the average displacement error is less than 3%. Additional plots indicating the accuracy of the FLAC results are presented in Figures 3.18 through 3.20.

15 Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3-15 JOB TITLE :. (*10^1) May-08 10:05 step E+00 <x< 1.167E E+00 <y< 1.167E Plasticity Indicator * at yield in shear or vol. X elastic, at yield in past Boundary plot E (*10^1) Figure 3.18 Plasticity state indicators quarter-symmetry JOB TITLE :. 07 (10 ) 16-May-08 10:04 step E+00 <x< 1.167E E+00 <y< 1.167E Linear Profile Y-axis : YY-stress X-axis : Distance From ( 0.00E+00, 0.00E+00) To ( 1.00E+01, 0.00E+00) Figure 3.19 Linear profile of tangential stress quarter-symmetry

16 3-16 Verification Problems JOB TITLE :. 16-May-08 10:05 step 3262 Failure Surface Plot Major Prin. Stress vs. Minor Prin. Stress Zone Stress States Mohr-Coulomb Fail. Surf. Friction = E+01 Cohesion = E+06 Tension = E (10 ) (10 ) Figure 3.20 Zone stress states quarter-symmetry 3.5 Reference Salencon, J. Contraction Quasi-Statique D une Cavité a Symétrie Sphérique Ou Cylindrique Dans Un Milieu Elastoplastique, Annales Des Ponts Et Chaussées, 4, (1969).

17 Cylindrical Hole in an Infinite Mohr-Coulomb Medium Data File MHOLE.DAT Project Record Tree export *** Branch: Plane strain **** new... State: m3a.sav... title HOLE IN MOHR-COULOMB MEDIUM (ASSOCIATED FLOW) config extra=8 g mo mo call hole.fis set rmin=1 rmul=10 gratio=1.1 hole prop shear=2.8e9 bulk=3.9e9 dens=2500 coh=3.45e6 fric=30 ten=1e10 prop dil 0.0 non-associated flow prop dil 30.0 associated flow ini sxx=-30e6 syy=-30e6 szz=-30e6 fix y j 1 fix x j 31 app sxx=-30e6 syy=-30e6 i 31 hist unbal hist xd i 1 j 1 hist xv i 1 j 1 hist sxx i 1 j 1 hist syy i 1 j 1 hist szz i 1 j 1 solve save m3a.sav... State: m3a v.sav... rest m3aa.sav ****************** define the constants ********************* def parm p0=30e6 p1=0.0 rmin=1.0 s=cohesion(1,1) fi=friction(1,1)*degrad dil=dilation(1,1)*degrad bm=bulk mod(1,1) sm=shear mod(1,1) nu=(3.0*bm-2.0*sm)/(6.0*bm+2.0*sm)

18 3-18 Verification Problems kp=(1.0+sin(fi))/(1.0-sin(fi)) kps=(1.0+sin(dil))/(1.0-sin(dil)) q=2*s*cos(fi)/(1.0-sin(fi)) rp=rmin*(2.0/(kp+1.0)*(p0+q/(kp-1.0))/(p1+q/(kp-1.0)))ˆ(1.0/(kp-1)) sre=(2.0*p0-q)/(kp+1.0) parm *********** calculate the theoretical results ************** the theoretical results are stored in the following arrays tangential stress... EX 1 radial stress... EX 2 x displacements... EX 3 y displacements... EX 4 displacements magnitude... EX 5 def theor loop i (1,izones) loop j (1,jzones) xc=.25*(x(i,j)+x(i,j+1)+x(i+1,j+1)+x(i+1,j)) yc=.25*(y(i,j)+y(i,j+1)+y(i+1,j+1)+y(i+1,j)) rz=sqrt(xcˆ2+ycˆ2) if rz<=rp then ex 1(i,j)=q/(kp-1)-kp*(p1+q/(kp-1.0))*(rz/rmin)ˆ(kp-1.0) ex 2(i,j)=q/(kp-1)-(p1+q/(kp-1.0))*(rz/rmin)ˆ(kp-1.0) else if rz=0.0 then ex 1(i,j) = 0 ex 2(i,j) = 0 else ex 1(i,j)=-p0-(p0-sre)*(rp/rz)ˆ2 ex 2(i,j)=-p0+(p0-sre)*(rp/rz)ˆ2 if if loop i (1,igp) loop j (1,jgp) ro=sqrt(x(i,j)ˆ2+y(i,j)ˆ2) if ro#0 then if ro<=rp then d1=(2.0*nu-1.0)*(p0+q/(kp-1.0)) d2a=(1.0-nu)*(kpˆ2-1.0)/(kp+kps) d2b=(p1+q/(kp-1.0))*(rp/rmin)ˆ(kp-1.0)*(rp/ro)ˆ(kps+1.0) d3a=(1.0-nu)*(kp*kps+1.0)/(kp+kps)-nu

19 Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3-19 d3b=(p1+q/(kp-1.0))*(ro/rmin)ˆ(kp-1.0) dd=ro*(d1+d2a*d2b+d3a*d3b)/(2.0*sm) else dd=(p0-sre)*rp/(2.0*sm)*(rp/ro) if ex 3(i,j)=-dd*x(i,j)/ro ex 4(i,j)=-dd*y(i,j)/ro ex 5(i,j)=-dd if theor ************** evaluate the error in stresses ********************** the errors in stress calculations are evaluated for each zone and are stored in the following arrays: the total average errors are calculated and stored in: average average def evals ert=0 err=0 loop i (1,izones) loop j (1,jzones) temp1=.5*(sxx(i,j)+syy(i,j)) temp2=sqrt(sxy(i,j)ˆ2+.25*(sxx(i,j)-syy(i,j))ˆ2) stm=temp1-temp2 ex 6(i,j)=100*(stm-ex 1(i,j))/p0 ert=ert+ex 6(i,j)ˆ2 srm=temp1+temp2 ex 7(i,j)=100*(srm-ex 2(i,j))/p0 err=err+ex 7(i,j)ˆ2 ert=sqrt(ert/(izones*jzones)) err=sqrt(err/(izones*jzones)) evals ************** evaluate the error in displacements **************** the errors in displacement calculations are evaluated for each zone and are stored in the following array: the total average error is calculated and stored in: average

20 3-20 Verification Problems def evald erd=0 loop i (1,igp) loop j (1,jgp) temp3=100*sqrt((xdisp(i,j)-ex 3(i,j))ˆ2+(ydisp(i,j)-ex 4(i,j))ˆ2) if ex 5(1,1)#0 then ex 8(i,j)=temp3/ex 5(1,1) else ex 8(i,j)=0.0 if erd=erd+ex 8(i,j) erd=erd/(igp*jgp) evald ************** compare results in tables **************** def tabm The theoretical and numerical results are stored in terms of normalized radius in the following tables: table 10: tangential stress theoretical table 11: tangential stress numerical table 20: radial stress theoretical table 21: radial stress numerical table 30: radial displacement theoretical table 31: radial displacement numerical loop i(1,izones) j = 1 xc=.25*(x(i,j)+x(i,j+1)+x(i+1,j+1)+x(i+1,j)) yc=.25*(y(i,j)+y(i,j+1)+y(i+1,j+1)+y(i+1,j)) rz=sqrt(xcˆ2+ycˆ2) temp1=.5*(sxx(i,j)+syy(i,j)) temp2=sqrt(sxy(i,j)ˆ2+.25*(sxx(i,j)-syy(i,j))ˆ2) stm=temp1-temp2 srm=temp1+temp2 xtable(10,i) = rz ytable(10,i) = -ex 1(i,1)/p0 xtable(11,i) = rz ytable(11,i) = -stm/p0 xtable(20,i) = rz ytable(20,i) = -ex 2(i,1)/p0 xtable(21,i) = rz ytable(21,i) = -srm/p0

21 Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3-21 loop i (1,igp) j = 1 rg = sqrt(x(i,j)ˆ2+y(i,j)ˆ2)/rmin temp1=sqrt(xdisp(i,j)ˆ2+ydisp(i,j)ˆ2)/rmin xtable(30,i) = rg ytable(30,i) = -ex 5(i,j) / rmin xtable(31,i) = rg ytable(31,i) = temp1 tabm ****************** create plots ******************************* scline scline save m3a v.sav *** Branch: Plane strain-non **** new... State: m3a-non.sav... title HOLE IN MOHR-COULOMB MEDIUM (ASSOCIATED FLOW) config extra=8 g mo mo call hole.fis set rmin=1 rmul=10 gratio=1.1 hole prop shear=2.8e9 bulk=3.9e9 dens=2500 coh=3.45e6 fric=30 ten=1e10 prop dil 0.0 non-associated flow prop dil 30.0 associated flow ini sxx=-30e6 syy=-30e6 szz=-30e6 fix y j 1 fix x j 31 app sxx=-30e6 syy=-30e6 i 31 hist unbal hist xd i 1 j 1 hist xv i 1 j 1 hist sxx i 1 j 1 hist syy i 1 j 1 hist szz i 1 j 1 solve save m3a-non.sav... State: m3a-non v.sav...

22 3-22 Verification Problems rest m3aa.sav ****************** define the constants ********************* def parm p0=30e6 p1=0.0 rmin=1.0 s=cohesion(1,1) fi=friction(1,1)*degrad dil=dilation(1,1)*degrad bm=bulk mod(1,1) sm=shear mod(1,1) nu=(3.0*bm-2.0*sm)/(6.0*bm+2.0*sm) kp=(1.0+sin(fi))/(1.0-sin(fi)) kps=(1.0+sin(dil))/(1.0-sin(dil)) q=2*s*cos(fi)/(1.0-sin(fi)) rp=rmin*(2.0/(kp+1.0)*(p0+q/(kp-1.0))/(p1+q/(kp-1.0)))ˆ(1.0/(kp-1)) sre=(2.0*p0-q)/(kp+1.0) parm *********** calculate the theoretical results ************** the theoretical results are stored in the following arrays tangential stress... EX 1 radial stress... EX 2 x displacements... EX 3 y displacements... EX 4 displacements magnitude... EX 5 def theor loop i (1,izones) loop j (1,jzones) xc=.25*(x(i,j)+x(i,j+1)+x(i+1,j+1)+x(i+1,j)) yc=.25*(y(i,j)+y(i,j+1)+y(i+1,j+1)+y(i+1,j)) rz=sqrt(xcˆ2+ycˆ2) if rz<=rp then ex 1(i,j)=q/(kp-1)-kp*(p1+q/(kp-1.0))*(rz/rmin)ˆ(kp-1.0) ex 2(i,j)=q/(kp-1)-(p1+q/(kp-1.0))*(rz/rmin)ˆ(kp-1.0) else if rz=0.0 then ex 1(i,j) = 0 ex 2(i,j) = 0 else ex 1(i,j)=-p0-(p0-sre)*(rp/rz)ˆ2 ex 2(i,j)=-p0+(p0-sre)*(rp/rz)ˆ2 if

23 Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3-23 if loop i (1,igp) loop j (1,jgp) ro=sqrt(x(i,j)ˆ2+y(i,j)ˆ2) if ro#0 then if ro<=rp then d1=(2.0*nu-1.0)*(p0+q/(kp-1.0)) d2a=(1.0-nu)*(kpˆ2-1.0)/(kp+kps) d2b=(p1+q/(kp-1.0))*(rp/rmin)ˆ(kp-1.0)*(rp/ro)ˆ(kps+1.0) d3a=(1.0-nu)*(kp*kps+1.0)/(kp+kps)-nu d3b=(p1+q/(kp-1.0))*(ro/rmin)ˆ(kp-1.0) dd=ro*(d1+d2a*d2b+d3a*d3b)/(2.0*sm) else dd=(p0-sre)*rp/(2.0*sm)*(rp/ro) if ex 3(i,j)=-dd*x(i,j)/ro ex 4(i,j)=-dd*y(i,j)/ro ex 5(i,j)=-dd if theor ************** evaluate the error in stresses ********************** the errors in stress calculations are evaluated for each zone and are stored in the following arrays: the total average errors are calculated and stored in: average average def evals ert=0 err=0 loop i (1,izones) loop j (1,jzones) temp1=.5*(sxx(i,j)+syy(i,j)) temp2=sqrt(sxy(i,j)ˆ2+.25*(sxx(i,j)-syy(i,j))ˆ2) stm=temp1-temp2 ex 6(i,j)=100*(stm-ex 1(i,j))/p0 ert=ert+ex 6(i,j)ˆ2 srm=temp1+temp2 ex 7(i,j)=100*(srm-ex 2(i,j))/p0 err=err+ex 7(i,j)ˆ2

24 3-24 Verification Problems ert=sqrt(ert/(izones*jzones)) err=sqrt(err/(izones*jzones)) evals ************** evaluate the error in displacements **************** the errors in displacement calculations are evaluated for each zone and are stored in the following array: the total average error is calculated and stored in: average def evald erd=0 loop i (1,igp) loop j (1,jgp) temp3=100*sqrt((xdisp(i,j)-ex 3(i,j))ˆ2+(ydisp(i,j)-ex 4(i,j))ˆ2) if ex 5(1,1)#0 then ex 8(i,j)=temp3/ex 5(1,1) else ex 8(i,j)=0.0 if erd=erd+ex 8(i,j) erd=erd/(igp*jgp) evald ************** compare results in tables **************** def tabm The theoretical and numerical results are stored in terms of normalized radius in the following tables: table 10: tangential stress theoretical table 11: tangential stress numerical table 20: radial stress theoretical table 21: radial stress numerical table 30: radial displacement theoretical table 31: radial displacement numerical loop i(1,izones) j = 1 xc=.25*(x(i,j)+x(i,j+1)+x(i+1,j+1)+x(i+1,j)) yc=.25*(y(i,j)+y(i,j+1)+y(i+1,j+1)+y(i+1,j)) rz=sqrt(xcˆ2+ycˆ2) temp1=.5*(sxx(i,j)+syy(i,j))

25 Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3-25 temp2=sqrt(sxy(i,j)ˆ2+.25*(sxx(i,j)-syy(i,j))ˆ2) stm=temp1-temp2 srm=temp1+temp2 xtable(10,i) = rz ytable(10,i) = -ex 1(i,1)/p0 xtable(11,i) = rz ytable(11,i) = -stm/p0 xtable(20,i) = rz ytable(20,i) = -ex 2(i,1)/p0 xtable(21,i) = rz ytable(21,i) = -srm/p0 loop i (1,igp) j = 1 rg = sqrt(x(i,j)ˆ2+y(i,j)ˆ2)/rmin temp1=sqrt(xdisp(i,j)ˆ2+ydisp(i,j)ˆ2)/rmin xtable(30,i) = rg ytable(30,i) = -ex 5(i,j) / rmin xtable(31,i) = rg ytable(31,i) = temp1 tabm ****************** create plots ******************************* scline scline save m3a-non v.sav *** Branch: axisymmetric **** new... State: m3b.sav... title HOLE IN MOHR-COULOMB MEDIUM (AXISYMMETRY, ASSOCIATED FLOW RULE) config ax extra=8 g 31 2 mo mo gen rat prop shear=2.8e9 bulk=3.9e9 dens=2500 coh=3.45e6 fric=30 ten=6e6 prop dil 0.0 non-associated flow prop dil 30.0 associated flow ini sxx=-30e6 syy=-30e6 szz=-30e6 fix y j 1 fix y j 3 app press 30e6 i 32 hist unbal

26 3-26 Verification Problems hist xd i 1 j 1 hist xv i 1 j 1 hist sxx i 1 j 1 hist syy i 1 j 1 hist sig1 i 1 j 1 hist sig2 i 1 j 1 hist szz i 1 j 1 solve save m3b.sav... State: m3b v.sav... rest m3bb.sav ****************** define the constants ********************* def parm p0=30e6 p1=0.0 rmin=1.0 s=cohesion(1,1) fi=friction(1,1)*degrad dil=dilation(1,1)*degrad bm=bulk mod(1,1) sm=shear mod(1,1) nu=(3.0*bm-2.0*sm)/(6.0*bm+2.0*sm) kp=(1.0+sin(fi))/(1.0-sin(fi)) kps=(1.0+sin(dil))/(1.0-sin(dil)) q=2*s*cos(fi)/(1.0-sin(fi)) rp=rmin*(2.0/(kp+1.0)*(p0+q/(kp-1.0))/(p1+q/(kp-1.0)))ˆ(1.0/(kp-1)) sre=(2.0*p0-q)/(kp+1.0) parm *********** calculate the theoretical results ************** the theoretical results are stored in the following arrays tangential stress... EX 1 radial stress... EX 2 x displacements... EX 3 y displacements... EX 4 displacements magnitude... EX 5 def theor loop i (1,izones) loop j (1,jzones) rz=.25*(x(i,j)+x(i,j+1)+x(i+1,j+1)+x(i+1,j)) if rz<=rp then ex 1(i,j)=q/(kp-1)-kp*(p1+q/(kp-1.0))*(rz/rmin)ˆ(kp-1.0)

27 Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3-27 ex 2(i,j)=q/(kp-1)-(p1+q/(kp-1.0))*(rz/rmin)ˆ(kp-1.0) else ex 1(i,j)=-p0-(p0-sre)*(rp/rz)ˆ2 ex 2(i,j)=-p0+(p0-sre)*(rp/rz)ˆ2 if loop i (1,igp) loop j (1,jgp) ro=x(i,j) if ro<=rp then d1=(2.0*nu-1.0)*(p0+q/(kp-1.0)) d2a=(1.0-nu)*(kpˆ2-1.0)/(kp+kps) d2b=(p1+q/(kp-1.0))*(rp/rmin)ˆ(kp-1.0)*(rp/ro)ˆ(kps+1.0) d3a=(1.0-nu)*(kp*kps+1.0)/(kp+kps)-nu d3b=(p1+q/(kp-1.0))*(ro/rmin)ˆ(kp-1.0) dd=ro*(d1+d2a*d2b+d3a*d3b)/(2.0*sm) else dd=(p0-sre)*rp/(2.0*sm)*(rp/ro) if ex 3(i,j)=-dd ex 4(i,j)=0.0 ex 5(i,j)=-dd theor ************** evaluate the error in stresses ********************** the errors in stress calculations are evaluated for each zone and are stored in the following arrays: the total average errors are calculated and stored in: average average def evals ert=0 err=0 loop i (1,izones) loop j (1,jzones) ex 6(i,j)=100*(szz(i,j)-ex 1(i,j))/p0 ert=ert+ex 6(i,j)ˆ2 ex 7(i,j)=100*(sxx(i,j)-ex 2(i,j))/p0 err=err+ex 7(i,j)ˆ2 ert=sqrt(ert/(izones*jzones))

28 3-28 Verification Problems err=sqrt(err/(izones*jzones)) evals ************** evaluate the error in displacements **************** the errors in displacement calculations are evaluated for each zone and are stored in the following array: the total average error is calculated and stored in: average def evald erd=0 loop i (1,igp) loop j (1,jgp) temp3=100*sqrt((xdisp(i,j)-ex 3(i,j))ˆ2+(ydisp(i,j)-ex 4(i,j))ˆ2) if ex 8(i,j)#0.0 then ex 8(i,j)=temp3/ex 5(1,1) else ex 8(i,j)=temp3 if erd=erd+ex 8(i,j) erd=erd/(igp*jgp) evald ************** compare results in tables **************** def tabm The theoretical and numerical results are stored in terms of normalized radius in the following tables: table 10: tangential stress theoretical table 11: tangential stress numerical table 20: radial stress theoretical table 21: radial stress numerical table 30: radial displacement theoretical table 31: radial displacement numerical loop i(1,izones) j = 1 rz=.25*(x(i,j)+x(i,j+1)+x(i+1,j+1)+x(i+1,j)) xtable(10,i) = rz ytable(10,i) = -ex 1(i,1)/p0 xtable(11,i) = rz ytable(11,i) = -szz(i,j)/p0 xtable(20,i) = rz

29 Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3-29 ytable(20,i) = -ex 2(i,1)/p0 xtable(21,i) = rz ytable(21,i) = -sxx(i,j)/p0 loop i (1,igp) j = 1 rg = sqrt(x(i,j)ˆ2+y(i,j)ˆ2)/rmin temp1=sqrt(xdisp(i,j)ˆ2+ydisp(i,j)ˆ2)/rmin xtable(30,i) = rg ytable(30,i) = -ex 5(i,j) / rmin xtable(31,i) = rg ytable(31,i) = temp1 tabm ****************** create plots ******************************* scline save m3b v.sav *** Branch: axisymmetric-non **** new... State: m3b-non.sav... title HOLE IN MOHR-COULOMB MEDIUM (AXISYMMETRY, ASSOCIATED FLOW RULE) config ax extra=8 g 31 2 mo mo gen rat prop shear=2.8e9 bulk=3.9e9 dens=2500 coh=3.45e6 fric=30 ten=6e6 prop dil 0.0 non-associated flow prop dil 30.0 associated flow ini sxx=-30e6 syy=-30e6 szz=-30e6 fix y j 1 fix y j 3 app press 30e6 i 32 hist unbal hist xd i 1 j 1 hist xv i 1 j 1 hist sxx i 1 j 1 hist syy i 1 j 1 hist sig1 i 1 j 1 hist sig2 i 1 j 1 hist szz i 1 j 1 solve save m3b-non.sav

30 3-30 Verification Problems... State: m3b-non v.sav... rest m3bb.sav ****************** define the constants ********************* def parm p0=30e6 p1=0.0 rmin=1.0 s=cohesion(1,1) fi=friction(1,1)*degrad dil=dilation(1,1)*degrad bm=bulk mod(1,1) sm=shear mod(1,1) nu=(3.0*bm-2.0*sm)/(6.0*bm+2.0*sm) kp=(1.0+sin(fi))/(1.0-sin(fi)) kps=(1.0+sin(dil))/(1.0-sin(dil)) q=2*s*cos(fi)/(1.0-sin(fi)) rp=rmin*(2.0/(kp+1.0)*(p0+q/(kp-1.0))/(p1+q/(kp-1.0)))ˆ(1.0/(kp-1)) sre=(2.0*p0-q)/(kp+1.0) parm *********** calculate the theoretical results ************** the theoretical results are stored in the following arrays tangential stress... EX 1 radial stress... EX 2 x displacements... EX 3 y displacements... EX 4 displacements magnitude... EX 5 def theor loop i (1,izones) loop j (1,jzones) rz=.25*(x(i,j)+x(i,j+1)+x(i+1,j+1)+x(i+1,j)) if rz<=rp then ex 1(i,j)=q/(kp-1)-kp*(p1+q/(kp-1.0))*(rz/rmin)ˆ(kp-1.0) ex 2(i,j)=q/(kp-1)-(p1+q/(kp-1.0))*(rz/rmin)ˆ(kp-1.0) else ex 1(i,j)=-p0-(p0-sre)*(rp/rz)ˆ2 ex 2(i,j)=-p0+(p0-sre)*(rp/rz)ˆ2 if loop i (1,igp) loop j (1,jgp)

31 Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3-31 ro=x(i,j) if ro<=rp then d1=(2.0*nu-1.0)*(p0+q/(kp-1.0)) d2a=(1.0-nu)*(kpˆ2-1.0)/(kp+kps) d2b=(p1+q/(kp-1.0))*(rp/rmin)ˆ(kp-1.0)*(rp/ro)ˆ(kps+1.0) d3a=(1.0-nu)*(kp*kps+1.0)/(kp+kps)-nu d3b=(p1+q/(kp-1.0))*(ro/rmin)ˆ(kp-1.0) dd=ro*(d1+d2a*d2b+d3a*d3b)/(2.0*sm) else dd=(p0-sre)*rp/(2.0*sm)*(rp/ro) if ex 3(i,j)=-dd ex 4(i,j)=0.0 ex 5(i,j)=-dd theor ************** evaluate the error in stresses ********************** the errors in stress calculations are evaluated for each zone and are stored in the following arrays: the total average errors are calculated and stored in: average average def evals ert=0 err=0 loop i (1,izones) loop j (1,jzones) ex 6(i,j)=100*(szz(i,j)-ex 1(i,j))/p0 ert=ert+ex 6(i,j)ˆ2 ex 7(i,j)=100*(sxx(i,j)-ex 2(i,j))/p0 err=err+ex 7(i,j)ˆ2 ert=sqrt(ert/(izones*jzones)) err=sqrt(err/(izones*jzones)) evals ************** evaluate the error in displacements **************** the errors in displacement calculations are evaluated for each zone and are stored in the following array: the total average error is calculated and stored in:

32 3-32 Verification Problems average def evald erd=0 loop i (1,igp) loop j (1,jgp) temp3=100*sqrt((xdisp(i,j)-ex 3(i,j))ˆ2+(ydisp(i,j)-ex 4(i,j))ˆ2) if ex 8(i,j)#0.0 then ex 8(i,j)=temp3/ex 5(1,1) else ex 8(i,j)=temp3 if erd=erd+ex 8(i,j) erd=erd/(igp*jgp) evald ************** compare results in tables **************** def tabm The theoretical and numerical results are stored in terms of normalized radius in the following tables: table 10: tangential stress theoretical table 11: tangential stress numerical table 20: radial stress theoretical table 21: radial stress numerical table 30: radial displacement theoretical table 31: radial displacement numerical loop i(1,izones) j = 1 rz=.25*(x(i,j)+x(i,j+1)+x(i+1,j+1)+x(i+1,j)) xtable(10,i) = rz ytable(10,i) = -ex 1(i,1)/p0 xtable(11,i) = rz ytable(11,i) = -szz(i,j)/p0 xtable(20,i) = rz ytable(20,i) = -ex 2(i,1)/p0 xtable(21,i) = rz ytable(21,i) = -sxx(i,j)/p0 loop i (1,igp) j = 1 rg = sqrt(x(i,j)ˆ2+y(i,j)ˆ2)/rmin temp1=sqrt(xdisp(i,j)ˆ2+ydisp(i,j)ˆ2)/rmin xtable(30,i) = rg

33 Cylindrical Hole in an Infinite Mohr-Coulomb Medium 3-33 ytable(30,i) = -ex 5(i,j) / rmin xtable(31,i) = rg ytable(31,i) = temp1 tabm ****************** create plots ******************************* scline save m3b-non v.sav *** plot commands **** plot name: grid plot hold grid plot name: Stresses comparison plot hold table 21 cross 20 line 11 cross 10 line plot name: Rad. disp. comp. plot hold table 31 cross 30 line plot name: Error - tang. stress plot hold bound ex 6 zone fill plot name: Error - radial stress plot hold bound ex 7 zone fill plot name: Error - disp plot hold bound ex 8 fill plot name: Plasticity indicator plot hold plasticity bound plot name: Zone stress state set pltc pltf 30.0 pltt 0.0 plot hold fail principal plot name: Profile of syy plot hold syy line (0.0,0.0) (10.0,0.0) 31

34 3-34 Verification Problems