TALK=f;RUN(1,1) DISPLAY This problem is the same as that of s104; but nz is given a value greater than 1 as a test of capability, while nx retains its unity value. The stresses, strains and displacements should be independent of z. A long, hollow, thick-walled cylinder, immersed in an outer fluid, contains a second fluid having a different pressure. Since conditions vary with radius alone, nx and nz are set = 1 . Two constraints on its length are considered, namely: 1. the cylinder length may extend freely; and 2. the cylinder length may not extend at all. To enable the computed stresses and strains to be compared with theoretical (text-book) values, the latter have been computed. So as to make comparison easy, variables named P1/T etc have also been computed. They represent the calculated values divided by text-book ones. They should all be, and indeed are, close to 1.0; but zero is printed when the text-book value is zero Parameters which may be changed by editing this file include: (a) Material properties: Young's modulus and Poisson's ratio (b) Geometric quantities: inner and outer radius, and length. (c) Mechanical loadings: inner and outer pressures (d) The grid fineness (via NYCYL) (e) The grid-fineness distributions in x and y (via POWERX and POWERY) The text-book solution may be found in section 14.4 of Benham, Crawford and Armstrong, Mechanics of Engineering Materials; and also in Love's Mathematical Theory of Elasticity p144 . ENDDIS ************************************************************ Group 1. Run Title and Number ************************************************************ READQ1=T begin_readq1 TEXT(Pressurised long cylinder [R,Z] Declarations and settings REAL(PI,PO,RI,RO,POISSON,YOUNG,POWERY) PI=1.e7 ! inner pressure PO=1.e6 ! outer pressure RI=0.2 ! inner radius RO=0.8 ! outer radius YOUNG = 1/(0.5E-11) ! Young's modulus POISSON=0.3 ! Poisson's ratio POWERY = 1.25 integer(caseno,nycyl) ! ncyl is y-direction cell number in solid mesg(:title: caseno = 2 Grid settings nx=1 nycyl=10 nz=10 **** !!! *** ISG59 = 1 ************************************************************ ************************************************************ Group 2. Time dependence STEADY = T ************************************************************ Group 3. X-Direction Grid Spacing CARTES = F XULAST = 3.1416 GRDPWR(X,NX,XULAST,1) ************************************************************ Group 4. Y-Direction Grid Spacing NREGY=3 ! 3 regions IREGY=1;GRDPWR(Y,1,RI,1.0) ! single inner fluid cell IREGY=2;GRDPWR(Y,NYCYL,RO-RI,POWERY) ! ncyl solid cells IREGY=3;GRDPWR(Y,1,0.01*RO,1.0) ! single outer fluid cell ************************************************************ Group 5. Z-Direction Grid Spacing ZWLAST = 1.000000E+00 GRDPWR(Z,NZ,1.0,1.0) ! uniform z grid ************************************************************ Group 7. Variables: STOREd,SOLVEd,NAMEd ONEPHS = T SOLVE(P1,V1,W1) STORE(P1TH,PRPS,DEN1,ENUL,DVO1,DRH1) ! TH means theoretical STORE(STRX,SXTH,EPSX,EXTH,STRY,SYTH) ! provide storage for STORE(EPSY,EYTH,STRZ,SZTH,EPSZ,EZTH) ! text-book values ************************************************************ GROUP 8. ITERATION NUMBERS ETC RESFAC=1.e-7 RESREF(V1)=0.0 ! to prevent premature exit LITER(V1) = 100 ! from solver RESREF(u1)=0.0; LITER(u1) = 100 ************************************************************ GROUP 9. PROPERTIES CSG10='Q1' ! materials with various POISSON ratios MATFLG=T;NMAT=1 160 7800.0 0.3 473.0 43.0 1.0e-5 0.5E-11 ************************************************************ GROUP 11. INITIAL VALUES FIINIT(PRPS)=0 PATCH(INNER,INIVAL,1,1,1,1,1,NZ-1,1,1) ! inner fluid COVAL(INNER,P1,FIXVAL,PI) PATCH(INNER,INIVAL,1,1,NY,NY,1,NZ-1,1,1) ! outer fluid COVAL(INNER,P1,FIXVAL,PO) PATCH(CYLINDER,INIVAL,1,1,2,NY-1,2,NZ-1,1,1) ! VIA INIT(CYLINDER,PRPS,FIXVAL,160) ! material. Insert other ************************************************************ GROUP 13. BOUNDARY & SPECIAL SOURCES PATCH(IN,CELL,1,1,1,1,2,NZ-1,1,1) ! pressurizing inner COVAL(IN,P1,FIXVAL,PI) ! fluid PATCH(OUT,CELL,1,1,NY,NY,2,NZ-1,1,1) ! pressurizing outer COVAL(OUT,P1,FIXVAL,PO) ! fluid PATCH(ZMINX,CELL,1,1,2,NY-1,1,1,1,1) ! of high-end air COVAL(ZMINX,W1,FIXVAL,0.0) COVAL(ZMINX,P1,FIXVAL,0.0) PATCH(ZMAXX,CELL,1,1,2,NY-1,NZ-1,NZ-1,1,1) ! of high-end air COVAL(ZMAXX,W1,FIXVAL,0.0) PATCH(ZMAXX1,CELL,1,1,1,NY,NZ,NZ,1,1) ! of high-end air COVAL(ZMAXX1,P1,FIXVAL,0.0) ! stating whether expansion is allowed in one-cell direcctions SPEDAT(BOUNDARY,XCONST,R,1.E20) ! because total angle is fixed ************************************************************ GROUP 15. TERMINATE SWEEPS LSWEEP = 400 ISG21 = LSWEEP ENDIT(V1) =0.0 RESREF(V1)=0.0 ENDIT(u1) =0.0 RESREF(u1)=0.0 ************************************************************ GROUP 17. RELAXATION #CONPROM ************************************************************ GROUP 19. DATA TRANSMITTED TO GROUND STRA = T ************************************************************ GROUP 23.FIELD PRINT-OUT & PLOT CONTROL TSTSWP = - 1 ! graphic-mode ISG52 = 3 ! probe & res output(den1,n,n,n,n,n,n) output(drh1,n,n,n,n,n,n) output(dvo1,n,n,n,n,n,n) output(enul,n,n,n,n,n,n) output(prps,n,n,n,n,n,n) NYPRIN=1 NZPRIN=1 IXMON = 1 IYMON = NY-1 IZMON = NZ/2 ISG52 = 3 ! probe & res Theoretical values according to Benham et al and Love (Mathematical Theory of Electricity) to be conveyed to earth by way of InForm statements. inform7begin ! Declarations first: REAL(RODRI, RISQ, ROSQ, RODRISQ, CONI, CONO, TERM) REAL( AA, BB, EE) REAL(LL,GG) ! Lame's constants ! deductions ! Lame's constants Lamda+LL, mu=GG LL = YOUNG*POISSON/((1+POISSON)*(1-2.0*POISSON)) ! in terms of YOUNG GG = YOUNG/(2.0*(1.0+POISSON)) ! and POISSON RISQ = RI * RI ROSQ = RO * RO RODRISQ = ROSQ / RISQ ! settings from formulae in the above text books ! all cases CONI = PI / (RODRISQ - 1) CONO = PO * RODRISQ / (RODRISQ - 1) (STORED VAR SYTH IS :CONI:*(1.0 - :ROSQ:/ Rg^2 ) - $ :CONO:*(1.0 - :RISQ:/RG^2) with imat>100) (STORED VAR SXTH IS :CONI:*(1.0 + :ROSQ:/ Rg^2 ) - $ :CONO:*(1.0 + :RISQ:/RG^2) with imat>100) BB = (PI-PO) * RISQ * ROSQ / ( 2.0*GG * (ROSQ-RISQ) ) ! Love's B AA = (PI*RISQ-PO*ROSQ) / ( 2.0*(LL+GG) * (ROSQ-RISQ) ) ! Love's A ! first part IF(CASENO.EQ.1) THEN ! ee and aa for no stress EE = - LL * (PI*RISQ-PO*ROSQ) / ( GG * (3.0*LL+2.0*GG) )! Love's e ee EE = EE / (ROSQ-RISQ) ! concluded ee AA = AA - EE*LL/( 2.0* (LL + GG)) ! Love's A concluded (STORED VAR SZTH IS 0.0 WITH IMAT>100) TERM = - LL / (GG * (3.0*LL + 2.0*GG) ) TERM = TERM * ( PI*RISQ - PO*ROSQ) TERM = TERM / ( ROSQ - RISQ ) (STORED VAR EZTH IS :TERM: WITH IMAT>100) ENDIF IF(CASENO.EQ.2) THEN ! fixed end (STORED VAR EZTH IS 0 WITH IMAT>100) TERM = LL/(LL+GG) TERM = TERM * ( PI*RISQ - PO*ROSQ) TERM = TERM / ( ROSQ - RISQ ) (STORED VAR SZTH IS :TERM: WITH IMAT>100) ENDIF IF(CASENO.EQ.3) THEN ! cylinder with end plates EE = (PI*RISQ - PO*ROSQ) / (ROSQ - RISQ) EE = EE / ( 3.0*LL + 2.0*GG ) AA = AA - EE*LL/( 2.0* (LL + GG)) ! Love's A TERM = (PI*RISQ - PO*ROSQ)/(ROSQ-RISQ) ! longitudinal tension (STORED VAR SZTH IS :TERM: WITH IMAT>100) PATCH(HIFORC,HIGH,1,NX,2,NY-1,NZ-1,NZ-1,1,1) ! apply to face in z COVAL(HIFORC,W1,FIXFLU,TERM) (STORED VAR EZTH IS :EE: WITH IMAT>100) TERM = TERM/(3.0*LL+2.0*GG) TERM = TERM* (ZWLAST-0.02) (STORED VAR W1TH IS :TERM:*IZ*0.5 WITH IMAT>100) ENDIF ! print to screen and satlog.txt caseno nz AA BB EE ! all cases (STORED VAR V1TH IS :AA:*RV + BB/RV) (STORED VAR EXTH IS (SXTH - :POISSON:*(SYTH+SZTH))/:YOUNG:) (STORED VAR EYTH IS (SYTH - :POISSON:*(SXTH+SZTH))/:YOUNG:) (STORED VAR P1TH IS (EXTH + EYTH + EZTH) WITH IMAT>100) (STORED VAR P1/T IS P1/P1TH) (STORED VAR V1/T IS V1/V1TH) (STORED VAR EX/T IS EPSX/EXTH) (STORED VAR EY/T IS EPSY/EYTH) IF(CASENO.EQ.1) THEN (STORED VAR EZ/T IS EPSZ/EZTH) ENDIF IF(CASENO.EQ.2) THEN ! cases 1 and 2 distinguished (STORED VAR SZ/T IS STRZ/SZTH) ENDIF IG(40)=0 spedat(rlxfac,rlxw1d,r,0.5) spedat(rlxfac,rlxv1d,r,0.5) inform7end STOP