THE UNIVERSITY OF MI CHIGAN COLLEGE OF ENGINEERING Department of Engineering Mechanics Technical Report AXIALLY SYMMETRIC IDENTATION OF COHESIONLESS SOILS L.,A. Larkin ORA Projec-t O403: under contract with: DEPARTMENT OF THE ARMY ORDNANCE TANK-AUTOMOTIVE COMMAND DETROIT ORDNANCE DISTRICT CONTRACT NO. DA-20-018-ORD-23276 DETROIT, MICHIGAN administered through OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR September 1963

TABLE OF CONTENTS Page LIST OF FIGURES v SUMMARY 1 I. INTRODUCTION 2 II. GOVERNING EQUATIONS 3 IIIo ANALYSIS OF THE PROBLEM 6 IV. SOLUTION OF THE PROBLEM 9 REFERENCES 19 APPENDIX 20 iii

LIST OF FIGURES Figure Page 1. Section of Coulomb yield surface with plane v; = constant, 4 2. Principal stress directions and limiting shear stress directions 4 35 Mohr's circle representation of the limiting stress state, 4 4. Field of stress characteristics. 8 5. Illustration of typical calculation situation. 10 6. Stress characteristics and punch pressure distribution for = 200. 13 7 Stress characteristics and punch pressure distribution for 7 = 3.. 1 8. Stress characteristics and punch pressure distribution for $ = 30~. 15 9. Stress characteristics and punch pressure distribution for 3 550 16 10. Stress characteristics and punch pressure distribution for = 40~ 17 11. Variation of OB/OA with angle of friction. 18 12. Variation of average punch pressure with angle of friction. 18 v

SUMMARY The stress distribution in a cohesionless soil, with weight, indented by a smooth, rigid punch is discussed~ The governing equations are hyperbolic, and computational procedures based on the method of characteristics are developed. Detailed numerical solutions obtained on the IBM 7090 computer at The University of Michigan are presented for angles of friction of 200, 250, 300, 350, and 40. 1

I INTRODUCTION This investigation is concerned with the determination of the stress distribution in a granular material when the material is indented by a smooth, flat-ended circular puncho The material is assumed to be cohesionless and to obey the Coulomb yield criterion. Shield' has investigated the axially symmetric behavior of an ideally plastic material which obeys the Tresca yield criterion and he has given a detailed discussion of the indentation problem for this materialo Shield's analysis has been extended to materials which obey the Coulomb criterion by Cox, Eason, and Hopkins for the case of weightless material and by Cox3 for the material with weight. The stress fields computed by Cox3 were for the particular case where the atmospheric pressure acts as a surcharge. This case is not normally met in practice, and..in this paper the numerical work is extended to the more usual circumstance of zero. surcharge. 2

IIo GOVERNING EQUATIONS It will be assumed that the material yields under a general state of stress according to the Coulomb criteriono The cross-section} a = const., of the resulting yield surface plotted in principal stress space (see Shield4) is shown in Figure 1.o Taking advantage of the symmetry of this yield surface the discussion will be limited to the sides and corners where,1 > c2 C Cox, Eason, and Hopkins2 have investigated the type of axially symmetric solution associated with each side and corner, called regimes. Only the kinematically determinate groups regimes AB and EF, and the statically determinate group, regimes A and F, give non-trivial stress or velocity solutions. The latter group appears to be the more promising for the solution of the indentation problem, For the corners A and F the circumferential stress ag is equal to one of the principal stresses in the r-z plane and the limiting Coulomb shear stresses act in this plane Introducing the variables n and ao, where T is the inclination from the r-axis of the clockwise rotating limiting shear stress (see Figure 2) and ao is the radius of the Mohr's circle for stress, then, from Figure 3, stress components satisfying the Coulomb yield criterion are~ Or (c cotan. - o cosec ) - ao sin ($ + 2) (la) az = (c cotan - 0o cosec $) + ao sin (4 + 2r) (lb) Tzr = (c cotan $ - ca cosec $) cos ($ + 2~) (lc) =Q - (c cotan $ - ao cosec +) +w ao (ld) where $- angle of friction c - cohesion + 1 for regime F I 1 fr r e 1 for regime A 3

/E F Figure 1. Section of Coulomb yield surface with plane AQ = constant. z + (/T a2 2 Tlimit 1^ /IT limit Figure 2. Principal stress directions and limiting shear stress directions. Figure 3. Mohr's circle representation of the limiting stress state. 4

These equationq, together with the equilibrium equations r. aTzr + ar - a 0 (2a) z + +1 +.0 (2b) 6r 6z r ar az r where y-weight density of the materials form a hyperbolic set of equations The characteristics are described by tan n (3a) dz dr tan ( + 2 + ) (3b) The characteristics given by Eq. (3a) will be denoted first characteristics and those described by Eq~ (3b) will be called second characteristics Expressing each length in terms of a fundamental length R and introducing the dimensionless stress F = ao/Ry, the variation of F and r along the characteristic directions is given by dF + 2 tan $ F dr + F tan [w cos $ dr + (l-w sin $) dz] (4a) - tan. $ (sin ( dr + cos 4 dz) = 0 on the first characteristics and by dF - 2 tan $ F d + F tan [w cos $ dr - (1-w sin %) dz] r (4b) + tan $ (sin p dr - cos $ dz) = 0 on the second characteristicso 5

IIIo ANALYSIS OF THE PROBLEM Let the material occupy the region z > 0 with the origin of coordinates located at the center of the indenter, and set the radius of the indenter as the fundamental length R. Assume that the cylindrical indenter is rigid, flat ended, and perfectly smooth. (The assumption of perfect smoothness is sufficient to render the problem statically determinates) The stress boundary conditions under the indenter are then T = 0 for < r < R; z = 0 (5) zr The regimes A and F are distinguished by comparing the relative magnitude of the unequal principal stress with that of the two equal ones. At A the unequal principal stress is larger and at F it is smaller than the two equal stresses. It seems reasonable to suppose that the vertical stress under the indenter is smaller (larger negatively) than the radial or circumferential stresses. At or near the free surface, the radial stress is probably smaller than the vertical stress, which is zero at the surface, and it might well be smaller than the circumferential stress. This suggests the use of regime F throughout and in consequence, w =+ 1 will be used in the equations along the characteristics. The flat surface of the granular material outside the punch surface is usually stress free. (The atmosphere will not provide a confining pressure unless the surface is sealed off and the air evacuated from the material.) The boundary conditions on the free surface are then Trz = =0 on r>R; z = (6) Expressing these boundary conditions in terms of the dependent variables q and F Tn =~ - on r > R.. z 0 4j 2 F = - cotan ( - 6

For a cohesionless material F vanishes and the direction of -the characteristics is undetermined, but if a cohesionless material is considered as the limiting state as the cohesion, c, approaches zero, then it is seen that - ^-_; F = 0 on r > R; (8) z = 0 Similarly, expressing Eqo (5), the boundary conditions under the punch, in terms of the variables F and n yields 3 * 4 2 on 0 < r < R; (9) z 0 =- [F (1 + cosec $) - cotan c] yR yR From the boundary conditions Eqs. (8) and (9), it can be seen that the characteristics must change their angle rapidly in going from the stress free boundary to the flat indenter boundary. This suggests a field of characteristics as shown in Figure 4 in which a fan of characteristics centeredat the edge.of the punch accomplishes the required change in angle. Such a type of field is to be expected from the solution of the analogous plane strain problem and from the circular indentation solutions by Cox and Shield" The fan center, point A, is the point of intersection for all of the first characteristics in the fan, hence the inclination of the first characteristics takes on many values here. Just to the right of A the boundary conditions are given by Eq. (8) and so the characteristic at the edge of the fan, AC, is initially inclined at i/4 - $/2 to the r-axis. The conditions just to the left of A are given by Eqo (9) and hence the characteristic, AD, starts out at 3/4T - $/2 to the r-axis. This limits the included angle of the fan to J/2. The variation of the stress variable, F, around the singular point A can be found by considering the point to be a degenerate second characteristic of zero length. Then Eq. (4b) becomes d (cotan ~ In F) - 2d = 0 7

Integrating and imposing the known conditions at the free boundary, Eq.(7), the variation of the stress parameter around the singularity becomes ~n ~ = 2 ( -~ 5)1tano. (10) c cos o 2 - 1 - sin For the case of a cohesionless material, the logarithmic term becomes indefinitely large and the only way to preserve the equality (10) is to set F = O. Hence all the stress components are zero if the singularity is approached from any direction. z 2nd Char. D O ~ f 1st Char. A B ^ R~ R Indenter Figure 4. Field of stress characteristics. 8

IV. SOLUTION OF THE PROBLEM The field of characteristics and the stress field can be obtained by the integration of the differential equations involving F and ri along the characteristics simultaneously with the differential equations of the characteristics themselves. The solution of these equations usually involves a numerical procedure because of the difficulties in obtaining an analytical solution. The numerical procedure is based on the approximation of Eqso (3) and (4) by finite difference equations. These finite difference equations are in turn used to extend the solution from the boundary values. In the usual situation, F, n, r, and z are known at two intersections P, Q of the characteristics. It is desired to extend the solution to point 1, (see Figure 5). The unknown coordinates (ra, z1) can be initially approximated by finding the intersection of the two characteristics extended at their initial inclination. However, subsequent approximations for ri and zl can be made more accurately by extending each characteristic one-half its projection at its initial angle and the second half at the approximated angle of that characteristic at the point being determined. Therefore, if half the vertical projection is used then rl and zl are found by solving -~_ = + (cotan cotan otan ) for first characteristic and zi-Zp Zp1a) r r -_ — = [cotan (n + </2 + d) + cotan (rz + T/2 + $)] for the(llb z - ZQ 2 Q second characteristic. The equations on the dependent variables F and T along the characteristics, Eqs (4), will be replaced by the finite difference equations 9

nd Char. 1st Char. rl, Z1) / p 2 (rp, zp) rQ, ZQ) r Figure 5. Illustration of typical calculation situation. Fi-Fp+2tan~ F F1 - Fp + 2 tan 1 P (1 - p) 1 + ( /() tn [cos 1 (rl - rp) + (1 - sin () (zi - zp)] (12a) tan $ [sin j (ri - rp) + cos $ (zl - zp)] = 0 on a first characteristic. F1 - FQ -( 2 + rl+ tan ( [cos d (rl -rQ) - (1- sin $) (zi- zQ)] (12b) tan $ [sin 4 (ri - rQ) - cos $ (zl - zQ)] = 0 on a second characteristic. 10

Solving for F1 and ni F1 = 1 Fp + FQ - tan. [Fp (Tl - qp) - FQ (Ti - Q) + cos F( ( -) + F ( )) + (1 - K sin) F pr -F) - F r 2* )(13a) P +i Q 1 + r/ P r1 + Q 1 + r - sin (rQ - rp) - cos (2z -zp - zQ)]] where D =2 + tan [ Q - p + cos, rp + r, - r+ (1 - sin c1) - Zp" Z_ -) = FP + - cos - r - r - (1L sin ) ( - ZL + " ZQ 2 P Q 1 ri + rp r+ r + rp + - - sin r r + r + r- + F1 + Fp F+ + F + F +F (13b) cotan -Fp - F F1+ Fp F1 + After the initial approximations for rl and zl are calculated., Ti is estimated and a new F1 and. 1 are calculated from Eqsc (13)o Subsequently a new rl and zl can be found from Eqso (11) and the iteration performed until successive values become sufficiently unchanged. In this manner the characteristics and stresses in the region ABC are uniquely determined from the known boundary AB and the boundary values of F and n along it. Following the same procedure the stresses and char11

acteristics are uniquely determined in the fan CAD from the calculated values of F and'l along the characteristic AC and from their boundary values at the singularity A. Finally the region ADO is uniquely calculated starting from the previously determined values of the independent variables on the characteristic AD and knowing that the second characteristics terminate on the line z 0 at an inclination of it/4 + (/2 to the r-axis (see Eq. (9)). The radius at which this occurs can be found from Eq. (llb) and the corresponding value of F under the indenter can be found from Eq. (12b) and subsequently substituted into Eq. (9) to find the normal pressure under the indenter. The above procedure was programmed on the IBM 7090 digital computer at The University of Michigan. The iteration for each point was terminated when the change in every variable (r, z, F-and ) fell below 10-5 of its previous value. The number of iterations required to attain this accuracy was usually five and it was never more than nine. Whenever a characteristic experienced a change in angle of more than 60 from point to point, the mesh size was decreased locally until this requirement was' satisfied. As a result, the mesh size became very small near the singularity point. The largest field calculated required just under six minutes total computer time; each of the others needed less than three minutes total. The equations in the fan region ACD and consequently in the region ADO were poorly conditioned. Furthermore, it was found that the solution could be improved by the use of a small surface pressure along AB coupled with a very fine mesh at the singularity. This is not an unreasonable step, because in the physical problem the boundary conditions given by Eq. (8) will be perturbed by the initial pre-flow sinkage of the indenter and the slight overburden then added along AB will provide a source for the pressure necessary to make the solution well behaved. A pressure of the order of 10 Ry was assumed which corresponds to an initial sinkage of the order of 10' R where the thin overburden has no strength. The resulting fields of characteristics and pressure distributions under the punch for angles of friction varying from 20o-40~ are shown in Figures 6-10. Figure 11 is a graph of the ratio OB/OA and Figure 12 is a plot of the average indenter pressure. The appendix contains the computer program as written in the Michigan Algorithm Decoder (MAD) language. An overall equilibrium check equating the resisting forces acting on the outer second characteristic to the combined forces from the soil weight and from the indenter was made for the $ = 300 case. The error in vertical equilibrium amounted to less than 0.4% of the punch load. 12

z 4 -.5.2 0 kr 1.2...5.7.8. ~1 1.2. 1.4 1.5 1.6 1.7 1.0A 2.0 3.0 4.0_ 6.0 7.0 Figure 6. Stress characteristics and punch pressure distribution for $ = 200.

z.6.2 |.1.2,, 4,,6,7,8.oA. 1. 12 1.3 1.4 1.5 1.6 1.7 1. 8 1.9 2.0 ts r~~~~~~A 10.0 10.0 - 15.0 Figure 7. Stress characteristics and punch pressure distribution for 2 = 25~.

z.7.6 D.5.4 C.3.2.1 Ol.1.2.3.4..6.7.8 1.0 1.1 1.2 1.3 14 1.5 1.6 1.7 1-8 1.9 2.0 2.1 2.2 2.3 2.4 \31 10.0 20.0 N 30.040.0 Figure 8. Stress characteristics and punch pressure distribution for $ = 300.

i z.8.6.4.2 o B.2.4.6. 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 ~~-~H /20 0 / 40 60 100 120 Figure 9. Stress characteristics and punch pressure distribution for $ = 350

1.0.8.6 -.4.2 t0: B.2.4.6 I l..0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 50o A 100 150 b 200 250 300 Figure 10. Stress characteristics and punch pressure distribution for $ = 40".

4.0 0 3 5.0 O O cl 2.0 1.0 I I I i 20 25 30 35 40 Angle of Friction Figure 11. Variation of OB/OA with angle of friction. 60.~ 50 at 40 30 / a) 20 10 _ I I 20 25 30 35 40 Angle of Friction Figure 12. Variation of average punch pressure with angle of friction. 18

REFERENCES 1. R. T. Shield, "On the Plastic Flow of Metals Under Conditions of Axial Symmetry," Proc. Royal. Soc. A, vol. 233, 1955, ppo 267-2860 2. A. Do Cox, G. Eason, and H. G. Hopkins, "Axially Symmetric Plastic Deformation in Soils"'PhiL Transo Royal Soc., vol. 254, Ao 1036^ 1961, ppo 1-45. 3. A. D. Cox, "Axially Symmetric Plastic Deformation in Soils —IIo Indentation of Ponderable Soils," International Journ. of Mecho Sci., vol. 4, 1962, pp. 371-380. 4. Ro T. Shield, "On Coulomb's Law of Failure in Soils," Journo Mech, and Phys. of Solids, volo 4, 1955^ ppo 10-16. 19

APPENDIX: MAD COMPUTER PROGRAM COMPILE MAODEXECUTE.DUMP. PRINT OBJECT PRINT RESULTS M, i, J, R(IJ), Z(I,J), ETA(I,J), F(I,J) ~~R ~ -~~IKANS-KER TU STAKI R AXIALLY SYMMETRIC PUNCH PROBLEM OTHERWISE -.~-~' ~! M=M+1 START READ FORMAT INPUT*ANGLE, W. MESH, N, QUIT. P. ALPHA END OF CONDITIONAL PRINT FORMAT HEAD WHENEVER BETA.E. 1. PRINT FORMAT OUTPUTANGLE. W.MESH, N. QUIT. P. ALPHA TANETA=SIN.(OLDETA)/COS.(OLDETA) REALD AND PRINT DATA IAETPH=SIN.(OLDETA+.5*PI+FHI)/CO.(UOLDETA+.5*PI+PHI)I PRINT FORMAT TOP DENOM=TANP+TANETA-TANQ-TAETPH INTEGER N. M, P, I1 J. QUiT, WQUITOl,.QUITO2 Rl=(RP*(TANP+TANETA)-RQ*(TANQ+TAETPH)+2.*(ZQ-ZP))/DENOM DIMENSION R(D500,MTRX), Z(5000,MTRX), ETA(5000,MTRX). Zl=(ZQ*(TANP+TANETA)-ZP*(TANQ+TAETPH)+.5*(RP-RQ)*(TANP+TANETA 1 F(U5000MTRX ) 1) (TANQ+TAETPH))/DENOM VECTOR VALUES MTRX= A. 2,0,45 OTHERWISE IN ILKNAL FUNCIIUN A.(Q,IJ)=IW+J-1)*(J-I)+1-N+J -IANEIA=CUO.(ULDtIA)/SIN. ULUDTAI PI=3..1415927 TAETPH=COS.(OLDETA+.5*PI+PHI)/SIN.(OLDETA+.5*PI+PHI) NUMBou~. AUENUM='1ANP+TANElA-TANU-TAElPH LAST-0. ZI=(ZP* TANP+TANETA)-ZQ*(TANQ+TAETPH)+2.*(RQ-RP) )/DENOM PHI=PI/lB~.*ANLE - ~ ~ l(MiKU*( IANP+TANETA)-KP*l IANU+IAEtTPH)+.5*tZP-ZU)*(TANP+TANETA SINPHI=SIN.IPHI) 1)*(TANQ+TAETPH))/DENOM tUbmHI=tLUSil (t'lI EI N Ut CUNDI'IUNAL TANPHI=SINPHI/COSPHI R(1,J)=RI ~ J'i~ ~/(IJ)=l1 M=0 F( 1.J)=(FP+FQ-TANPHI*(FP*(OLDETA-ETA,(I J-1))-FQ*(OLDETA-ETA(I ETA(NJ)=.25*PI-.5*PHI I-1.J) )+CUSPiH *(P*(K1-KP )/ I+KP)+R t*1KI-KU)/IK1+KQ) I+ R(N,J)=1.0 2(1.-SINPHI)*(FP*(Z1-ZP)/(Rl+RP)-FQ*<Z1-ZQ)/(RI+RQ) )-INPHI* ZCN~J)t=R. ~ ~-KHU-KP)E }-LU Hm 1*P Z. * I -LF-ZU) ) /I ( Z + I ANPHIT l rA J IJ- TA. (I F(NJ)=ALPHA 4J-1)+COSPHI* (R1-RP)/(Rl+RP)+(R1-RQ)/(RI+RQ))+(l.-SINPHI)*((Z PRINT FORMAT RESULTI.MN,J.R(NJ),Z(N,J),180./PI*ETA(N,J), 51-ZP)/IRI+RP)-(Z1-ZQ)/R+RQ) ) ) 1F(NJ) ETA (IJ)=.5* ETA(IJ-1)+ETA(I-1,J)-COSPHI*((RI-RP)/(RI+RP) THROUGH SAM, FOR I=N+1,11.E.N+W+1 1 -(RI-RO)/(R+KR) )- 1.-SINPHI )*( Z-ZP)/(KI+RP)+(Zl-ZC)/IKI+R ETA(I,J)=ETA(I-1,J)+.5*PI/'W 2Q))+COSPHI*((Z1-ZP)/(F(I,J)+FP)-(ZIl-ZQ)/(F(I,J)+FQ))+SINPHI* ( I. JI= 1 0 l K-K ) / ( IF +,J) + (KI -I I /-( I. / I ANPH I* Z.l J=0, 4((F(I,J)-FP)/(F(IJ)+FP)-(F(I,J)-FQ)/(F(I,J)+FQ))) F(I'J)C=ALPHA*EXP.(22*TANPHI*(ETA(IJ)-.25*PI+.*PrPn:I) UIV=ETA(IJ)-OLDETA SAM PRINT FORMAT RESULTM I,J.R(I,J),Z I,J )180./PI*ETA(I,J),F(I, WHENEVER.ABS.(Z(I,J)-OLDZ).G.I.E-5*OLDZ.OR. 1J) 1.ABS.R( I,J)-OLDR )...E-6*OLDR.OR. SUM=0. 2.ABS. ( F,J)OLDF).G.l.E-5*OLDF.OR. LOOP J=J+1 3.AJS.(ETA(I.J)-OLOETA).G.1.E-5*OLDETA WHENEVER J.E.QUITO1, MESH=MESHO1 WHENEVER DEV*OLDDEV.G.0.'WHENEVER J.E.QUITO2, MESH=MESHU2 OLDZ=Z(I,J) WHENEVER J.E.QUIT OLDR=R IJ) WHENEVER NUMB.E.O. OLDF=F(I,J) QUITO1=99 OLDETA=ETA(iJ) 5U1 U'=~~ - LULVtV=VtV MESH=MESHO3 TRANSFER TO ITERAT NUMB=1. OTHERWISE OTHERWISE OLDZ=.5*Z(I.J)+IOLDZ) MESH=MESHO3 OLDR=.5*(R(IJ)+OLDR) END OF CONDITIONAL OLDF=.5*(F(I J)+OLDF) Nu ur UNU) JIJUNAL ULUE I A=. D I A t?J +UL IA OVER' I=N-J+1 OLDDEV=DEV ETA(I,J).25*PI-.5*pHI lIANS-tR TO ITEKA~ Z(I,J)=0. END OF CONDITIONAL R(I1J)=R(I+ivJ-1)+1I/MESH NU Ut UNDIIIUNAL F(IPJ)= ALPHA WHENEVER.ABS.(ETA(I,J)-ETA(I-1,J) ).G.INCRE M=U - A~ t TSHrosmt~n TRANSFER TO TEST QUITO1=QUITO1+2 BEGIN 1=1+1 UIOZ=QU ITOO+ZT WHENEVER I.E.N+W+J-1 WHENEVER NUMB.E 1. WHENEVER LASTE.2. UIT=J+2 TRANSFER TO STAR TR I SE.END OF CONDITIONAL QUIT=QUIT+2 END OF CONDITIONAL END OF CONDITIONAL WHENEVER I.E.N+W+J-1 TRANSFER TO OVER OLDETA= 75*PI-.5 PHI N F CONDITONAL ETA(IJ)=OLDETA TEST PRINT FORMAT RESULTM,IJ,R(I,J),Z(I,J),180./PI*ETA(I,J),F(I, WHENEVER.ABS.(ETA(ItJ)-ETA(I-1,J))C.G.INCRE, TRANSFER TO GAK IJ)'TANQ=COS.(ETA(1-1,J)+.5*PI+PHI)/SIN.(ETA(I-1i,J)+.5*PI+PHI) TRANSFER TO BEGIN TAETPH=COS.(OLDETA+.5*PI+PHI)/SIN.(OLDETA+.5*PI+PHI) VEtCOR VALUES INPUT=P10S,., 110, F1 Ot 3110, E0l.5*$ R~(IJ)=R(I-1,J)-.5*Z(I-1,J)(ATANQ+TAETPH) VECTOR VALUES HEAD=$//18H ANGLE OF FRICTIONS5,12HFAN DIVISION Z(IJ)=O. 1510, 4HGRIDS10, IHNS10, 11HPRINT LIMITSSI, 15HITERATION LIMIT H=-F (I- J)/(R(ItJ)+RI1-1,J))*(SINPHI*(R I.J-R(I-1.J) )- 2S3.8HPRESSURE*$ 1ANPHI*(1.-SINPHI)*(ZIJ)-Z(I-TJ))) -TANPHI*SINPHI(R(I.) VtIU VALUES UUIPUI=/bZFtU.J3,SIU,11U 9M Ut bU.j,.I'13,SltZ 2-R(I-1,JI)+SINPHI*(Z(I,J)-Z(I-l,J) C 113,522,13,58,E10.2*$ F(IJ)=(F(I-1i,J)*(1+TANPHI*(OLDETA-ETAI-.1,J)))+H)/ -VECTOR VALUES TOP=$//S6,lHMS4,4HI. JSZO.lHRS24.1HZS23, l(1.-TANPHI*(OLDETA-ETA I-1,J ))+SINPHI*(R(I,J)-R(I _-1,j))/ 13HETAS24,1HF//*$ 2(R(IJ)+R(I-iJ))-TANPHI*( 1.SINPHI)*Z(I J)-Z(i-1 )/ - VECTOR VALUES RESULT=5,12,S2, 213,4E25.5*$ 3(R(IJ)+R(I-l.AT I VECTOR VALUES ANSWER=$S9,213,3E25.5,13H PRESSURE IS E12.5*$ PRINI FURMAT ANSWER. I.JN(RIJ),Z(IJ), IO,*ETA(IJ)/PI VELIUK VALUES KtMAKR=CI/H PUUR CUNVKbmtNCLEXN HNI F(I- J1 /SINPHI+. O VECTOR VALUES PRESSR=$/21H AVERAGE PRESSURE IS E12.5*$ WHENEVER (RII- IJ- ))-K )I,J)).G.R(IJ)ENOFPO M WHENEVER LAST.E.0. DATAO MESH=2.*MESH 30.00256.00 3- 15 50 E-2 LAST=LAST+ 1. bUMSUM+i.l(l./SINPHI)*lI,JC II I*,J)Rk(I,J),_ PRINT FORMAT PRESSR2 SUM END OF CONDITIONAL OTHERWISE ~ UM M+ IITI /S ll INPHlf )~II 1ITFIJ)ll IIj.J- l))i.S*KI*J)+l.-1,~~ END OF CONDITIONAL TRANSFER TO LOOP END OF CONDITIONAL OLDETA=ETA(I,J-1) OLDF=FTIT.J —1) OLDR=R(I.J-1).OLDZ=Z(I,J-1) ZP=Z(I,J-1 ) z e=z(l,J) RP=R(I J-1) KQ=REII-1 J) FP=F( I,J-1 FQFCI-i.J) OLDDEV=1. M=O WHENEVER.ABS.(.5*PI-PHI-OLDETA) -.L.175 1ANFICIN.tLIA(It.J-)I/LUS.blEIAII.J-i~C TANQ=SIN (ETA( 1-1,J)+.5*PI+PHI /COS.(ETA(I-l,J)+.5*PI+PHI) BETA=1. OTHERWISE TANP=COS.(ETA I,J-1))/SIN.(ETA(I,J-1)),TANQOCOS.(ETA( I-l,J)+.5*PI+PHI)/SIN.(ETAII-l,J)+.5.pl+PHI) END OF CONDITIONAL ITERAT WHENEVER M.E.P PAINT FORMAT REMARK" 20

AD Accession No. UNCLASSIFIED AD Accession No. UNCLASSIFIED The University of Michigan, Office The University of Michigan, Office of Research Administration, Ann of Research Administration, Ann Arbor, Mich. Axially Symmetric In- Arbor, Mich. Axially Symmetric Indentation of Cohesionless Soils, by L. dentation of Cohesionless Soils, by L. A. Larkin. Report No. 04403-10-T, Sept. A. Larkin. Report No. 04403-10-T, Sept. 1963, 23. incl. illus., Project 04403 1963, 23 p. incl. illus., Project 04403 (Contract No. DA-20-018-ORD-23276) (Contract No. DA-20-018-ORD-23276) UNCLASSIFIED UNCLASSIFIED The stress distribution in a cohesion- The stress distribution in a cohesionless soil, with weight, indented by a less soil, with weight, indented by a smooth, rigid punch is discussed. The smooth, rigid punch is discussed. The governing equations are hyperbolic UNCLASSIFIED governing equations are hyperbolic UNCLASSIFIED [~_____ (over), (over) UNCLASSIFIED UNCLASSIFIED and computational procedures based on and computational procedures based on the method of characteristics are de- the method of characteristics are developed. Detailed numerical solu- veloped. Detailed numerical solutions obtained on the IBM 7090 computer tions obtained on the IBM 7090 computer at The University of Michigan are pre- at The University of Michigan are presented for angles of friction of 200, sented for angles of friction of 20~, 250, 300, 350, and 40 250, 300, 35, and 40~ UNCLASSIFIED UNCLASSIFIED... ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I

AD Accession No. UNCLASSIFIED AD Accession No. UNCLASSIFIED The University of Michigan, Office The University of Michigan, Office of Research Administration, Ann of Research Administration, Ann Arbor, Mich. Axially Symmetric In- Arbor, Mich. Axially Symmetric Indentation of Cohesionless Soils, by L. dentation of Cohesionless Soils, by L. A. Larkin. Report No. 04403-10-T, Sept. A. Larkin. Report No. 04403-10-T, Sept. 1963, 23 p. incl. illus., Project 04403 1963, 23 p. incl. illus., Project 04403 (Contract No. DA-20-018-ORD-23276) (Contract No. DA-20-018-ORD-23276) UNCLASSIFIED UNCLASSIFIED The stress distribution in a cohesion- The stress distribution in a cohesionless soil, with weight, indented by a less soil, with weight, indented by a smooth, rigid punch is discussed. The smooth, rigid punch is discussed. The governing equations are hyperbolic UNCLASSIFIED governing equations are hyperbolic, UNCLASSIFIED (ov) (over(or) UNCLASSIFIED UNCLASSIFIED and computational procedures based on and computational procedures based on the method of characteristics are de- the method of characteristics are developed. Detailed numerical solu- veloped. Detailed numerical solutions obtained on the IBM 7090 computer tions obtained on the IBM 7090 computer at The University of Michigan are pre- at The University of Michigan are presented for angles of friction of 20", sented for angles of friction of 20", 250, 30", 35~, and 40. 250, 30", 50,~ and 40. UNCLASSIFIED UNCLASSIFIED

AD Accession No. UNCLASSIFIED AD Accession No. UNCLASSIFIED The University of Michigan, Office The University of Michigan, Office of Research Administration, Ann of Research Administration, Ann Arbor, Mich. Axially Symmetric In- Arbor, Mich. Axially Symmetric Indentation of Cohesionless Soils, by L. dentation of Cohesionless Soils, by L. A. Larkin. Report No. 04403-10-T, Sept. A. Larkin. Report No. 04403-10-T, Sept. 1963, 23 p. incl. illus., Project 04403 1963, 23 p. incl. illus., Project 04403 (Contract No. DA-20-018-ORD-23276) (Contract No. DA-20-018-ORD-23276) UNCLASSIFIED UNCLASSIFIED The stress distribution in a cohesion- The stress distribution in a cohesionless soil, with weight, indented by a less soil, with weight, indented by a smooth, rigid punch is discussed. The smooth, rigid punch is discussed. The governing equations are hyperbolic UNCLASSIFIED governing equations are hyperbolic UNCLASSIFIED (over).. (over) UNCLASSIFIED UNCLASSIFIED and computational procedures based on and computational procedures based on the method of characteristics are de- the method of characteristics are developed. Detailed numerical solu- veloped. Detailed numerical solutions obtained on the IBM 7090 computer tions obtained on the IBM 7090 computer at The University of Michigan are pre- at The University of Michigan are presented for angles of friction of 20~, sented for angles of friction of 20~, 250, 30", 35~, and 400. 250, 300, 35~, and 400~ UNCLASSIFIED UNCLASSIFIED. _..._..,.. ~~~~~~~~~~~~~~~~~~~~~~~I

AD Accession No. UNCLASSIFIED AD Accession No. UNCLASSIFIED The University of Michigan, Office The University of Michigan, Office of Research Administration, Ann of Research Administration, Ann Arbor, Mich. Axially Symmetric In- Arbor, Mich. Axially Symmetric Indentation of Cohesionless Soils, by L. dentation of Cohesionless Soils, by L. A. Larkin. Report No. 04403-10-T, Sept. A. Larkin. Report No. 04403-10-T, Sept. 1963, 23 p. incl. illus., Project 04403 1963, 23 p. incl. illus., Project 04403 (Contract No. DA-20-018-ORD-23276) (Contract No. DA-20-018-ORD-23276) UNCLASSIFIED UNCLASSIFIED The stress distribution in a cohesion- The stress distribution in a cohesionless soil, with weight, indented by a less soil, with weight, indented by a smooth, rigid punch is discussed. The smooth, rigid punch is discussed. The governing equations are hyperbolic UNCLASSIFIED governing equations are hyperbolic UNCLASSIFIED (over) (over) UNCLASSIFIED UNCLASSIFIED and computational procedures based on and computational procedures based on the method of characteristics are de- the method of characteristics are developed. Detailed numerical solu- veloped. Detailed numerical solutions obtained on the IBM 7090 computer tions obtained on the IBM 7090 computer at The University of Michigan are pre- at The University of Michigan are presented for angles of friction of 20", sented for angles of friction of 20~, 250, 30", 350, and 400. 250, 300, 35~, and 40. UNCLASSIFIED UNCLASSIFIED

UNIVERSITY OF MICHIGAN 3 901111111115 0366 22731111111 3 9015 03466 2273