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(*NotebookFileLineBreakTestNotebookFileLineBreakTest*)(*NotebookOptionsPosition[ 118175, 2375]*)(*NotebookOutlinePosition[ 119274, 2410]*)(* CellTagsIndexPosition[ 119230, 2406]*)(*WindowFrame->Normal*)

Notebook[{Cell["\<\To use this program, here are the instructions:1. Execute Cells 1 through 10, in that order (twice each to get rid of annoying warning messages) to initialize all necessary modules. Shortcut for Mathematica savvy users: select Kernel \->Evaluation->Evaluate

Initialization (In Mathematica 2.2,Action->Evaluate Initialization).2. Cells 11 through 13 contain driver programs for doing test problems Run them and check that the answers (displacements and stresses) are \correct.3. It is recommended you place the main programs for the problems in \following Cells. Those in Cells 11 -- 13 may be used as \"templates\".4. If the plots produced by running a driver program appear too small, click on the top one (only) with the mouse, grab a corner \"handle\" and enlarge it. Then rerun the program. \\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 0.727611, 0.140017]],

Cell[TextData[{ StyleBox[" ", FontFamily->"Palatino"], "Cell 0. Mathematica version-dependent settings (to get plots displayed \correctly)."}], "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell["\<\DisplayChannel=$DisplayFunction;If [$VersionNumber>=6.0, DisplayChannel=Print]; (* fix for Mathematica 6 & later *) \\>", "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell[TextData[{ " ", StyleBox["Off[General::spell1]; Off[General::spell];", FontColor->RGBColor[0, 0, 1]]}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 1. Modules QuadGaussRuleInfo, TrigGaussRuleInfo and \LineGaussRuleInfo provide quadrature information for iso-P quadrilaterals, triangles, and lines, \respectively. For quadrilaterals the rules implemented are 1x1 through 5x5. These are \identified by the number of points in each direction, p=1, ... 5. See IFEM Notes Ch 23 for \argument rules. For triangles the rules are p=1,3,-3,6,-6,7; see IFEM Ch 24.

This code is used by all iso-P stiffness and stress modules. Hence it must be \initialized before their appearance.\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell["\<\QuadGaussRuleInfo[{rule_,numer_},point_]:= Module[ {\[Xi],\[Eta],p1,p2,i,j,w1,w2,m,info={{Null,Null},0}}, If [Length[rule]==2, {p1,p2}=rule, p1=p2=rule]; If [p1<0, Return[QuadNonProductGaussRuleInfo[ {-p1,numer},point]]]; If [Length[point]==2, {i,j}=point, m=point; j=Floor[(m-1)/p1]+1; i=m-p1*(j-1) ]; {\[Xi],w1}= LineGaussRuleInfo[{p1,numer},i]; {\[Eta],w2}= LineGaussRuleInfo[{p2,numer},j]; info={{\[Xi],\[Eta]},w1*w2}; If [numer, Return[N[info]], Return[Simplify[info]]];];

TrigGaussRuleInfo[{rule_,numer_},point_]:= Module[ {zeta,p=rule,i=point,g1,g2,info={{Null,Null,Null},0} }, If [p== 1, info={{1/3,1/3,1/3},1}]; If [p== 3, info={{1,1,1}/6,1/3}; info[[1,i]]=2/3]; If [p==-3, info={{1,1,1}/2,1/3}; info[[1,i]]=0 ]; If [p== 6, g1=(8-Sqrt[10]+Sqrt[38-44*Sqrt[2/5]])/18; g2=(8-Sqrt[10]-Sqrt[38-44*Sqrt[2/5]])/18; If [i<4, info={{g1,g1,g1},(620+Sqrt[213125- 53320*Sqrt[10]])/3720}; info[[1,i]]=1-2*g1]; If [i>3, info={{g2,g2,g2},(620-Sqrt[213125- 53320*Sqrt[10]])/3720}; info[[1,i-3]]=1-2*g2]]; If [p== -6, If [i<4, info={{1,1,1}/6,3/10}; info[[1,i]]=2/3]; If [i>3, info={{1,1,1}/2,1/30}; info[[1,i-3]]=0]]; If [p== 7, g1=(6-Sqrt[15])/21; g2=(6+Sqrt[15])/21; If [i<4, info={{g1,g1,g1},(155-Sqrt[15])/1200}; info[[1,i]]= 1-2*g1];

If [i>3&&i<7, info={{g2,g2,g2},(155+Sqrt[15])/1200}; info[[1,i-3]]=1-2*g2]; If [i==7, info={{1/3,1/3,1/3},9/40} ]]; If [numer, Return[N[info]], Return[Simplify[info]]];]; LineGaussRuleInfo[{rule_,numer_},point_]:= Module[ {g2={-1,1}/Sqrt[3],w3={5/9,8/9,5/9}, g3={-Sqrt[3/5],0,Sqrt[3/5]}, w4={(1/2)-Sqrt[5/6]/6, (1/2)+Sqrt[5/6]/6, (1/2)+Sqrt[5/6]/6, (1/2)-Sqrt[5/6]/6}, g4={-Sqrt[(3+2*Sqrt[6/5])/7],-Sqrt[(3-2*Sqrt[6/5])/7], Sqrt[(3-2*Sqrt[6/5])/7], Sqrt[(3+2*Sqrt[6/5])/7]}, g5={-Sqrt[5+2*Sqrt[10/7]],-Sqrt[5-2*Sqrt[10/7]],0, Sqrt[5-2*Sqrt[10/7]], Sqrt[5+2*Sqrt[10/7]]}/3, w5={322-13*Sqrt[70],322+13*Sqrt[70],512, 322+13*Sqrt[70],322-13*Sqrt[70]}/900, i=point,p=rule,info={Null,0}}, If [p==1, info={0,2}]; If [p==2, info={g2[[i]],1}]; If [p==3, info={g3[[i]],w3[[i]]}]; If [p==4, info={g4[[i]],w4[[i]]}]; If [p==5, info={g5[[i]],w5[[i]]}]; If [numer, Return[N[info]], Return[Simplify[info]]];]; \\>", "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 2A: Modules for 4-node iso-P bilinear quadrilateral ring \element for axisymmetric solids. These are documented in AFEM Chapter 12.\\>", "Text",\

CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[TextData[{ "Quad4IsoPRingStiffness[ncoor_,Emat_,options_]:=Module[\n \{p=2,numer=False,Jcons=False,Kfac=1,qcoor,k,\n \r1,r2,r3,r4,z1,z2,z3,z4,Nf,N1,N2,N3,N4,A0,Jdet,Be,\n \dNr1,dNr2,dNr3,dNr4,dNz1,dNz2,dNz3,dNz4,rk,w,c,Ke,\n \Ke0=Table[0,{8},{8}],modname=\"Quad4IsoPRingStiffness: \"},\n If \[Length[options]==1,{numer}=options];\n If \[Length[options]==2,{numer,p}=options];\n If \[Length[options]==3,{numer,p,Jcons}=options];\n If \[Length[options]==4,{numer,p,Jcons,Kfac}=options];\n If [p<1||p>5, \Print[modname,\"illegal p:\",p]; Return[Ke0]];\n \{{r1,z1},{r2,z2},{r3,z3},{r4,z4}}=ncoor;\n \

A0=((r3-r1)*(z4-z2)-(r4-r2)*(z3-z1))/2;\n If [numer&&(A0<=0), Print[modname,\\"Neg or zero area\"]; \n Return[Ke0]]; Ke=Ke0;\n For [k=1,k<=p*p,k++, \\n {qcoor,w}= QuadGaussRuleInfo[{p,numer},k];\n \{Nf,{dNr1,dNr2,dNr3,dNr4},{dNz1,dNz2,dNz3,dNz4},\n \Jdet}=Quad4IsoPRingShapeFunDer[ncoor,qcoor,Jcons];\n If \[numer&&(Jdet<=0), Print[modname,\"Neg or zero\",\n \" Gauss point \Jacobian at k=\",k]; Return[Ke0]];\n {N1,N2,N3,N4}=Nf; \rk=r1*N1+r2*N2+r3*N3+r4*N4; \n Be={{ dNr1, 0, dNr2, 0, dNr3, 0, \dNr4, 0},\n { 0,dNz1, 0,dNz2, 0,dNz3, 0,dNz4},\n \ {N1/rk, 0,N2/rk, 0,N3/rk, 0,N4/rk, 0},\n { dNz1,dNr1, \dNz2,dNr2, dNz3,dNr3, dNz4,dNr4}};\n c=Kfac*w*rk*Jdet; If \[numer,Be=N[Be]; c=N[c]]; \n Ke+=c*Transpose[Be].(Emat.Be); \n ]; \ClearAll[Ke0,Be]; Return[Ke] ];\n\n\Quad4IsoPRingShapeFunDer[ncoor_,qcoor_,Jcons_]:= Module[\n { \r1,r2,r3,r4,z1,z2,z3,z4,\[Xi],\[Eta],Nf,dNr,dNz,A0,A1,A2,Jdet},\n \{{r1,z1},{r2,z2},{r3,z3},{r4,z4}}=ncoor; {\[Xi],\[Eta]}=qcoor;\n Nf={(1-\\[Xi])*(1-\[Eta]),(1+\[Xi])*(1-\[Eta]),(1+\[Xi])*(1+\[Eta]),(1-\[Xi])*(1+\\[Eta])}/4; \n A0=((r3-r1)*(z4-z2)-(r4-r2)*(z3-z1))/2;\n \A1=((r3-r4)*(z1-z2)-(r1-r2)*(z3-z4))/2;\n \A2=((r2-r3)*(z1-z4)-(r1-r4)*(z2-z3))/2;\n Jdet=(A0+A1*\[Xi]+A2*\[Eta])/4; \If [Jcons,Jdet=A0/4];\n \dNr={z2-z4+(z4-z3)*\[Xi]+(z3-z2)*\[Eta],z3-z1+(z3-z4)*\[Xi]+(z1-z4)*\[Eta],\n \ z4-z2+(z1-z2)*\[Xi]+(z4-z1)*\[Eta],z1-z3+(z2-z1)*\[Xi]+(z2-z3)*\[Eta]}/(8*\Jdet);\n \dNz={r4-r2+(r3-r4)*\[Xi]+(r2-r3)*\[Eta],r1-r3+(r4-r3)*\[Xi]+(r4-r1)*\[Eta],\n \ r2-r4+(r2-r1)*\[Xi]+(r1-r4)*\[Eta],r3-r1+(r1-r2)*\[Xi]+(r3-r2)*\[Eta]}/(8*\Jdet);\n Return[{Nf,dNr,dNz,Jdet}]\n];\n\n\Quad4IsoPRingBodyForces[ncoor_,bfor_,options_]:=Module[\n \{p=2,numer=False,Jcons=False,Kfac=1,qcoor,k,m,\n \r1,r2,r3,r4,z1,z2,z3,z4,N1,N2,N3,N4,dNr,dNz,Jdet,Be,\n \br1,bz1,br2,bz2,br3,bz3,br4,bz4,brc,bzc,bk,rk,w,c,fe,\n \fe0=Table[0,{8}],modname=\"Quad4IsoPRingBodyForces: \"},\n If \[Length[options]==1,{numer}=options];\n If \[Length[options]==2,{numer,p}=options];\n If \[Length[options]==3,{numer,p,Jcons}=options];\n If \[Length[options]==4,{numer,p,Jcons,Kfac}=options];\n If [p<1||p>5, \Print[modname,\"p out of range\"]; Return[fe0]];\n \{{r1,z1},{r2,z2},{r3,z3},{r4,z4}}=ncoor;\n \A0=((r3-r1)*(z4-z2)-(r4-r2)*(z3-z1))/2;\n If [numer&&(A0<=0), Print[modname,\\"Neg or zero area\"]; \n Return[fe0]]; fe=fe0; m=Length[bfor]; \n If \

[m!=2&&m!=4, Print[modname,\" Illegal bfor\"]; Return[fe0]]; \n If [m==2, \br1=br2=br3=br4=bfor[[1]];bz1=bz2=bz3=bz4=bfor[[2]]];\n If \[m==4,{{br1,bz1},{br2,bz2},{br3,bz3},{br4,bz4}}=bfor];\n For \[k=1,k<=p*p,k++, \n {qcoor,w}= QuadGaussRuleInfo[{p,numer},k];", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "\n {{N1,N2,N3,N4},dNr,dNz,Jdet}=\n \Quad4IsoPRingShapeFunDer[ncoor,qcoor,Jcons];", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "\n If [numer&&(Jdet<=0), Print[modname,\"Neg or zero\",\n \" \Gauss point Jacobian at k=\",k]; Return[fe0]];\n \rk=r1*N1+r2*N2+r3*N3+r4*N4; c=Kfac*w*Jdet*rk; ", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "\n brk=br1*N1+br2*N2+br3*N3+br4*N4;\n \bzk=bz1*N1+bz2*N2+bz3*N3+bz4*N4;\n bk={N1*brk,N1*bzk,N2*brk,N2*bzk,\n \ N3*brk,N3*bzk,N4*brk,N4*bzk};\n If [numer,bk=N[bk]]; fe+=c*bk; \n\ ]; If[!numer, fe=Simplify[fe]];\n Return[fe] ];\n", StyleBox[" ", FontColor->RGBColor[0, 0, 1]], " \nQuad4IsoPRingStresses[ncoor_,Emat_,ue_,options_]:= \n \Module[{numer=False,g=1/Sqrt[3],Jcons=False,w0=0,\n \eps=10.^(-9),r1,r2,r3,r4,z1,z2,z3,z4,Nf,N1,N2,N3,N4,\n \dNr1,dNr2,dNr3,dNr4,dNz1,dNz2,dNz3,dNz4,\n \T1,T2,T3,T4,Td,Tg4,Jdet,qcoor,w,c,Be,\n \gctab={{0,0}},k,kg,rk,sigg,sige,udis=ue,\n \modname=\"Quad4IsoPRingStresses: \"},\n If \[Length[options]==1,{numer}=options];\n If \[Length[options]==2,{numer,g}=options];\n If \[Length[options]==3,{numer,g,w0}=options];\n If [Head[g]==Symbol||g>0, \Td=4*g^2*(4+w0); \n T1=4*g^2*w0; T2=4+4*g^2+w0+2*g*(4+w0); \n \T3=-4+4*g^2-w0; T4=4+4*g^2+w0-2*g*(4+w0);\n \Tg4={{T1,T2,T3,T4,T3},{T1,T3,T2,T3,T4},\n \{T1,T4,T3,T2,T3},{T1,T3,T4,T3,T2}}/Td;", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], " \n gctab={{0,0},{-1,-1},{1,-1},{1,1},{-1,1}}*g];\n \kg=Length[gctab]; sigg=Table[{0,0,0,0},{kg}];\n If [numer, gctab=N[gctab]; \Tg4=N[Tg4]; udis=N[ue] ]; \n {{r1,z1},{r2,z2},{r3,z3},{r4,z4}}=ncoor;\n For \[k=1,k<=kg,k++, qcoor=gctab[[k]]; \n \{Nf,{dNr1,dNr2,dNr3,dNr4},{dNz1,dNz2,dNz3,dNz4},\n \Jdet}=Quad4IsoPRingShapeFunDer[ncoor,qcoor,Jcons];\n {N1,N2,N3,N4}=Nf; \rk=r1*N1+r2*N2+r3*N3+r4*N4; \n Be={{ dNr1, 0, dNr2, 0, dNr3, 0, \dNr4, 0},\n { 0,dNz1, 0, dNz2, 0,dNz3, 0,dNz4},\n \ {N1/rk, 0,N2/rk, 0,N3/rk, 0,N4/rk, 0},\n { dNz1,dNr1, \dNz2,dNr2, dNz3,dNr3, dNz4,dNr4}};\n If [numer,Be=N[Be]]; \

sigg[[k]]=Emat.(Be.udis);\n ] ; \n If [kg==1, \sige=Table[sigg[[1]],{4}], sige=Tg4.sigg];\n If [numer, \sige=Chop[sige,eps]]; \n If [!numer,sige=Simplify[sige]]; Return[sige] ]; \n \ \n", StyleBox["(* ClearAll[Em,\[Nu],a,b,d,h,p,numer]; \nEm=96; \[Nu]=1/3; a=4; \b=2; d=1; e=3; Kfac=1;\nncoor={{d,0},{a+d,0},{a+d,b+e},{d,b}}; numer=True; \Jcons=True;\n", FontColor->RGBColor[0, 0, 1]], "\[Sigma]rr", StyleBox["\nPrint[\"Emat=\",Emat//MatrixForm];\nFor [p=1,p<=5,p++, \Print[\"Gauss rule p=\",p]; \n \Ke=Quad4IsoPRingStiffness[ncoor,Emat,{numer,p,Jcons,Kfac}];\n \Ke=Simplify[Ke]; Print[\"Ke=\",Ke//MatrixForm]; \n Print[\"Eigenvalues of \Ke=\",Chop[Eigenvalues[N[Ke]]]];\n ]; *)\n \n(* \ClearAll[a,b,d,h,p,numer]; \na=6; b=2; d=1; br=3; bz=1;\nJcons=False; \numer=True;\nncoor={{d,0},{a+d,0},{a+d,b},{d,b}}; \nFor [p=1,p<=2,p++", FontColor->RGBColor[0, 0, 1]], ",", StyleBox[" Print[\"Gauss rule p=\",p]; \n \fe=Quad4IsoPRingBodyForces[ncoor,{3,-1},{numer,p}];\n Print[\"fe \=\",Transpose[Partition[fe,2]]//MatrixForm]; \n \bfor={{1,0},{6,0},{6,0},{1,0}};\n \fe=Quad4IsoPRingBodyForces[ncoor,bfor,{numer,p}];\n Print[\"fe \=\",Transpose[Partition[fe,2]]//MatrixForm];\n ]; *)\n \n(* \ClearAll[Em,\[Nu],a,b,d,err,ezz,grz,ur,uz,r,z]; \nEm=2500; \[Nu]=1/4; \n\{err,ezz,err,grz}={3/80,-1/40,3/80,4/50};\n\ncoor={{d,0},{a+d,0},{a+d,b},{d,b}}; num=False;\n\Emat=Em/((1+\[Nu])*(1-2*\[Nu]))*{{1+\[Nu],\[Nu],\[Nu],0},{\[Nu],1+\[Nu],\[Nu],\0},\n {\[Nu],\[Nu],1+\[Nu],0},{0,0,0,1/2-\[Nu]}};\n\{err,ezz,err,grz}={3/80,-1/40,3/80,4/50};\nur[r_,z_]:=err*r; \uz[r_,z_]:=ezz*z+grz*r;\nue=Table[{0,0},{4}];\nFor \[n=1,n<=4,n++,{rn,zn}=ncoor[[n]]; \n ue[[n]]={ur[rn,zn],uz[rn,zn]} ];\n\ue=Flatten[ue]; Print[\"ue=\",ue]; \n\sige=Quad4IsoPRingStresses[ncoor,Emat,ue,{}];\nPrint[\"Corner \stresses=\",sige//MatrixForm]; *)", FontColor->RGBColor[0, 0, 1]]}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 2B: Modules for 8-node iso-P serendipity quadrilateral ring \element for axisymmetric solids. These are documented in AFEM Chapter 12.\\>", \"Text", CellFrame->True,

CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[TextData[{ "Quad8IsoPRingStiffness[ncoor_,Emat_,options_]:=Module[\n \{p=2,numer=False,Jcons=False,Kfac=1,qcoor,k,\n \r1,r2,r3,r4,r5,r6,r7,r8,z1,z2,z3,z4,z5,z6,z7,z8,\n \Nf,N1,N2,N3,N4,N5,N6,N7,N8,\n dNr1,dNr2,dNr3,dNr4,dNr5,dNr6,dNr7,dNr8,\n \dNz1,dNz2,dNz3,dNz4,dNz5,dNz6,dNz7,dNz8,\n \rk,w,c,A0,Jdet,Be,Ke,Ke0=Table[0,{16},{16}],\n \modname=\"Quad8IsoPRingStiffness: \"}, Ke=Ke0;\n If \[Length[options]==1,{numer}=options];\n If \[Length[options]==2,{numer,p}=options];\n If \[Length[options]==3,{numer,p,Jcons}=options];\n If \[Length[options]==4,{numer,p,Jcons,Kfac}=options];\n If [p<1||p>5, \Print[modname,\"illegal p:,p\"]; Return[Ke0]];\n \{{r1,z1},{r2,z2},{r3,z3},{r4,z4},\n {r5,z5},{r6,y6},{r7,z7},{r8,z8}}=ncoor;\\n A0=((r5-r7)*(z6-z8)-(r6-r8)*(z5-z7))/4;\n If [numer&&(A0<=0), \Print[modname,\"Neg or zero area\"]; \n Return[Ke0]]; \n For \[k=1,k<=p*p,k++, \n {qcoor,w}= QuadGaussRuleInfo[{p,numer},k];\n \{{N1,N2,N3,N4,N5,N6,N7,N8},\n {dNr1,dNr2,dNr3,dNr4,dNr5,dNr6,dNr7,dNr8},\\n {dNz1,dNz2,dNz3,dNz4,dNz5,dNz6,dNz7,dNz8},\n \Jdet}=Quad8IsoPRingShapeFunDer[ncoor,qcoor,Jcons];\n If \[numer&&(Jdet<=0), Print[modname,\"Neg or zero\",\n \" Gauss point \Jacobian at k=\",k]; Return[Ke0]];\n \rk=r1*N1+r2*N2+r3*N3+r4*N4+r5*N5+r6*N6+r7*N7+r8*N8; \n Be={{ dNr1, 0, \dNr2, 0, dNr3, 0, dNr4, 0,\n dNr5, 0, dNr6, 0, dNr7, \0, dNr8, 0},\n { 0,dNz1, 0,dNz2, 0,dNz3, 0,dNz4,\n \ 0,dNz5, 0,dNz6, 0,dNz7, 0,dNz8},\n {N1/rk, \0,N2/rk, 0,N3/rk, 0,N4/rk, 0,\n N5/rk, 0,N6/rk, 0,N7/rk, \ 0,N8/rk, 0},\n { dNz1,dNr1, dNz2,dNr2, dNz3,dNr3, dNz4,dNr4,\n \ dNz5,dNr5, dNz6,dNr6, dNz7,dNr7, dNz8,dNr8}};\n \c=Kfac*w*rk*Jdet; If[!numer, Be=Simplify[Be]];\n If [numer,Be=N[Be]; \c=N[c]]; \n Ke+=c*Transpose[Be].(Emat.Be); \n ]; Return[Ke] ];\n\n\Quad8IsoPRingShapeFunDer[ncoor_,qcoor_,Jcons_]:=Module[\n \{r1,r2,r3,r4,r5,r6,r7,r8,z1,z2,z3,z4,z5,z6,z7,z8,\n \\[Xi],\[Eta],rv,zv,A0,dN\[Xi],dN\[Eta],N1B,N2B,N3B,N4B,J11,J12,J21,J22,\n \Nf,dNr,dNz,Jdet}, {\[Xi],\[Eta]}=qcoor; \n \{{r1,z1},{r2,z2},{r3,z3},{r4,z4},{r5,z5},{r6,z6},{r7,z7},\n {r8,z8}}=ncoor; \

A0=((r5-r7)*(z6-z8)-(r6-r8)*(z5-z7))/4; \n N1B=(1-\[Xi])*(1-\[Eta])/4; \N2B=(1+\[Xi])*(1-\[Eta])/4; \n N3B=(1+\[Xi])*(1+\[Eta])/4; \N4B=(1-\[Xi])*(1+\[Eta])/4;\n Nf={-N1B*(1+\[Xi]+\[Eta]), -N2B*(1-\[Xi]+\\[Eta]),-N3B*(1-\[Xi]-\[Eta]),\n -N4B*(1+\[Xi]-\[Eta]), \2*N1B*(1+\[Xi]), 2*N3B*(1-\[Eta]),\n 2*N3B*(1-\[Xi]), \2*N4B*(1-\[Eta])};\n dN\[Xi]={(1-\[Eta])*(2*\[Xi]+\[Eta]),(1-\[Eta])*(2*\\[Xi]-\[Eta]),(1+\[Eta])*(2*\[Xi]+\[Eta]),\n (1+\[Eta])*(2*\[Xi]-\\[Eta]), 4*\[Xi]*(\[Eta]-1),2*(1-\[Eta]^2),\n \-4*\[Xi]*(1+\[Eta]),-2*(1-\[Eta]^2)}/4;\n dN\[Eta]={(1-\[Xi])*(\[Xi]+2*\\[Eta]),-(1+\[Xi])*(\[Xi]-2*\[Eta]),(1+\[Xi])*(\[Xi]+2*\[Eta]),\n \-(1-\[Xi])*(\[Xi]-2*\[Eta]), -2*(1-\[Xi]^2),-4*(1+\[Xi])*\[Eta],\n \2*(1-\[Xi]^2),-4*(1-\[Xi])*\[Eta]}/4;\n rv={r1,r2,r3,r4,r5,r6,r7,r8}; \zv={z1,z2,z3,z4,z5,z6,z7,z8}; \n J11=dN\[Xi].rv; J12=dN\[Xi].zv; J21=dN\\[Eta].rv; J22=dN\[Eta].zv;\n Jdet=Simplify[J11*J22-J12*J21]; If \[Jcons,Jdet=A0];\n dNr=( J22*dN\[Xi]-J12*dN\[Eta])/Jdet; \n dNz=(-J21*dN\\[Xi]+J11*dN\[Eta])/Jdet;\n Return[{Nf,dNr,dNz,Jdet}] ];\n \n\Quad8IsoPRingBodyForces[ncoor_,bfor_,options_]:=Module[\n \{p=2,numer=False,Jcons=False,Kfac=1,qcoor,k,m,mOK,\n \r1,r2,r3,r4,r5,r6,r7,r8,z1,z2,z3,z4,z5,z6,z7,z8,\n \Nf,N1,N2,N3,N4,N5,N6,N7,N8,dNr,dNz,\n br1,br2,br3,br4,br5,br6,br7,br8,\n \bz1,bz2,bz3,bz4,bz5,bz6,bz7,bz8,\n rk,w,c,A0,Jdet,fe,fe0=Table[0,{16}],\n \modname=\"Quad8IsoPRingBodyForces: \"}, fe=fe0;\n If \[Length[options]==1,{numer}=options];\n If \[Length[options]==2,{numer,p}=options];\n If \[Length[options]==3,{numer,p,Jcons}=options];\n If \[Length[options]==4,{numer,p,Jcons,Kfac}=options];\n If [p<1||p>5, \Print[modname,\"illegal p:\",p]; Return[fe0]];\n \{{r1,z1},{r2,z2},{r3,z3},{r4,z4},\n {r5,z5},{r6,z6},{r7,z7},{r8,z8}}=ncoor;\\n A0=((r5-r7)*(z6-z8)-(r6-r8)*(z5-z7))/4;\n If [numer&&(A0<=0), \Print[modname,\"Neg or zero area\"]; \n Return[fe0]]; m=Length[bfor]; \mOK=MemberQ[{2,4,8},m];\n If [!mOK, Print[modname,\" Illegal bfor\"]; \Return[fe0]]; \n If [m==2, br1=br2=br3=br4=br5=br6=br7=br8=bfor[[1]]; \n \ bz1=bz2=bz3=bz4=bz5=bz6=bz7=bz8=bfor[[2]]];\n If \[m==4,{{br1,bz1},{br2,bz2},{br3,bz3},{br4,bz4}}=bfor;\n \{br5,bz5,br6,bz6,br7,bz7,br8,bz8}={br1+br2,\n \bz1+bz2,br2+br3,bz2+bz3,br3+br4,bz3+bz4,\n br4+br1,bz4+bz1}/2];\n \ If [m==8,{{br1,bz1},{br2,bz2},{br3,bz3},{br4,bz4},\n \{br5,bz5},{br6,bz6},{br7,bz7},{br8,bz8}}=bfor]; \n For [k=1,k<=p*p,k++, \n \ {qcoor,w}= QuadGaussRuleInfo[{p,numer},k];", StyleBox[" ", FontColor->RGBColor[1, 0, 0]],

"\n {{N1,N2,N3,N4,N5,N6,N7,N8},dNr,dNz,Jdet}=\n \Quad8IsoPRingShapeFunDer[ncoor,qcoor,Jcons];", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "\n If [numer&&(Jdet<=0), Print[modname,\"Neg or zero\",\n \" \Gauss point Jacobian at k=\",k]; Return[fe0]];\n \rk=r1*N1+r2*N2+r3*N3+r4*N4+r5*N5+r6*N6+r7*N7+r8*N8;", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "\n brk=br1*N1+br2*N2+br3*N3+br4*N4+br5*N5+br6*N6+br7*N7+br8*N8;\n \ bzk=bz1*N1+bz2*N2+bz3*N3+bz4*N4+bz5*N5+bz6*N6+bz7*N7+bz8*N8;\n \bk={N1*brk,N1*bzk,N2*brk,N2*bzk,N3*brk,N3*bzk,N4*brk,N4*bzk,\n \N5*brk,N5*bzk,N6*brk,N6*bzk,N7*brk,N7*bzk,N8*brk,N8*bzk};\n \c=Kfac*w*Jdet*rk; If [numer,bk=N[bk];c=N[c]]; fe+=c*bk; \n ]; \Return[fe] ];\n \nQuad8IsoPRingStresses[ncoor_,Emat_,ue_,options_]:= \n \Module[{numer=False,g=1/Sqrt[3],Jcons=False,w0=0,\n \eps=10.^(-9),r1,r2,r3,r4,r5,r6,r7,r8,z1,z2,z3,z4,\n \z5,z6,z7,z8,Nf,N1,N2,N3,N4,N5,N6,N7,N8,\n \dNr1,dNr2,dNr3,dNr4,dNr5,dNr6,dNr7,dNr8,\n \dNz1,dNz2,dNz3,dNz4,dNz5,dNz6,dNz7,dNz8,\n \T1,T2,T3,T4,T5,T6,Td,Tg8,Jdet,qcoor,w,c,Be,\n \gctab={{0,0}},k,kg,rk,sigg,sige,udis=ue,\n modname=\"Quad8IsoPRingStresses: \\"},\n If [Length[options]==1,{numer}=options];\n If \[Length[options]==2,{numer,g}=options];\n If \[Length[options]==3,{numer,g,w0}=options];\n If [Head[g]==Symbol||g>0, \Td=4*g^2*(4+w0); \n T1=4*g^2*w0; T2=4+4*g^2+w0+2*g*(4+w0); \n \T3=-4+4*g^2-w0; T4=4+4*g^2+w0-2*g*(4+w0);\n T5=g*(4+4*g+w0); \T6=g*(-4+4*g-w0);\n Tg8={{T1,T2,T3,T4,T3},{T1,T3,T2,T3,T4},\n \{T1,T4,T3,T2,T3},{T1,T3,T4,T3,T2},\n \{T1,T5,T5,T6,T6},{T1,T6,T5,T5,T6},\n \{T1,T6,T6,T5,T5},{T1,T5,T6,T6,T5}}/Td;", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], " \n gctab={{0,0},{-1,-1},{1,-1},{1,1},{-1,1}}*g];\n \kg=Length[gctab]; sigg=Table[{0,0,0,0},{kg}];\n If [numer, gctab=N[gctab]; \Tg8=N[Tg8]; udis=N[ue]]; \n {{r1,z1},{r2,z2},{r3,z3},{r4,z4},\n \{r5,z5},{r6,z6},{r7,z7},{r8,z8}}=ncoor;\n For [k=1,k<=kg,k++, \qcoor=gctab[[k]];\n {{N1,N2,N3,N4,N5,N6,N7,N8},\n \{dNr1,dNr2,dNr3,dNr4,dNr5,dNr6,dNr7,dNr8},\n \{dNz1,dNz2,dNz3,dNz4,dNz5,dNz6,dNz7,dNz8},\n \Jdet}=Quad8IsoPRingShapeFunDer[ncoor,qcoor,Jcons];\n \rk=r1*N1+r2*N2+r3*N3+r4*N4+r5*N5+r6*N6+r7*N7+r8*N8; \n Be={{ dNr1, 0, \dNr2, 0, dNr3, 0, dNr4, 0,\n dNr5, 0, dNr6, 0, dNr7, \0, dNr8, 0},\n { 0,dNz1, 0,dNz2, 0,dNz3, 0,dNz4,\n \ 0,dNz5, 0,dNz6, 0,dNz7, 0,dNz8},\n {N1/rk, \0,N2/rk, 0,N3/rk, 0,N4/rk, 0,\n N5/rk, 0,N6/rk, 0,N7/rk, \

0,N8/rk, 0},\n { dNz1,dNr1, dNz2,dNr2, dNz3,dNr3, dNz4,dNr4,\n \ dNz5,dNr5, dNz6,dNr6, dNz7,dNr7, dNz8,dNr8}};\n If \[numer,Be=N[Be]]; sigg[[k]]=Emat.(Be.udis)\n ] ;\n If [kg==1, \sige=Table[sigg[[1]],{4}], sige=Tg8.sigg];\n If [numer, \sige=Chop[sige,eps]]; \n If [!numer,sige=Simplify[sige]]; Return[sige] ];", StyleBox["\n ", FontColor->RGBColor[0, 0, 1]], " \n(*", StyleBox["ClearAll[Em,\[Nu],a,b,d,p,numer];\na=4; b=2; d=0; Em=96; \\[Nu]=1/3; \nncoor={{d,0},{a+d,0},{a+d,b},{d,b},{a/2+d,0},{a+d,b/2},\n \{a/2+d,b},{d,b/2}};\n\Emat=Em/((1+\[Nu])*(1-2*\[Nu]))*{{1-\[Nu],\[Nu],\[Nu],0},{\[Nu],1-\[Nu],\[Nu],\0},\n {\[Nu],\[Nu],1-\[Nu],0},{0,0,0,1/2-\[Nu]}};\n\Print[\"Emat=\",Emat//MatrixForm];\nFor [p=1,p<=4,p++, \n \Ke=Quad8IsoPRingStiffness[ncoor,Emat,{True,p}];\n Ke=Simplify[Chop[Ke]]; \Print[\"Ke=\",Ke//MatrixForm]; \n Print[\"Eigenvalues of \Ke=\",Chop[Eigenvalues[N[Ke]],.000001]]; \n ]; *)\n \n\(*ClearAll[a,b,d,p];\na=3; b=2; d=1;\n\ncoor={{d,0},{a+d,0},{a+d,b},{d,b},{a/2+d,0},{a+d,b/2},\n \{a/2+d,b},{d,b/2}};\nFor [p=1,p<=3,p++, For [case=1,case<=2,case++,\n If \[case==1, bfor={36,-18}];\n If [case==2, \bfor=Table[{60*ncoor[[i,1]],0},{i,8}]];\n \fe=Quad8IsoPRingBodyForces[ncoor,bfor,{True,p}];\n fe=Simplify[Chop[fe]]; \ \n Print[\"fe=\",Transpose[Partition[fe,2]]//MatrixForm];\n \frsum=Sum[fe[[2*i-1]],{i,8}]; fzsum=Sum[fe[[2*i]],{i,8}];\n \Print[\"frsum=\",frsum,\" fzsum=\",fzsum];\n ]]; *)\n \n\(*ClearAll[Em,\[Nu],a,b,d,err,ezz,grz,ur,uz,r,z]; \nEm=2500; \[Nu]=1/4; d=1; \a=3; b=2;\n{err,ezz,err,grz}={3/80,-1/40,3/80,4/50};\n\ncoor={{d,0},{a+d,0},{a+d,b},{d,b},{a/2+d,0},{a+d,b/2},{a/2+d,b},{d,b/2}}; \n\Emat=Em/((1+\[Nu])*(1-2*\[Nu]))*{{1+\[Nu],\[Nu],\[Nu],0},{\[Nu],1+\[Nu],\[Nu],\0},\n {\[Nu],\[Nu],1+\[Nu],0},{0,0,0,1/2-\[Nu]}};\n\{err,ezz,err,grz}={3/80,-1/40,3/80,4/50};\nur[r_,z_]:=err*r; \uz[r_,z_]:=ezz*z+grz*r;\nue=Table[{0,0},{8}];\nFor [n=1,n<=8,n++, \{rn,zn}=ncoor[[n]]; \n ue[[n]]={ur[rn,zn],uz[rn,zn]} ];\nue=Flatten[ue]; \Print[\"ue=\",ue]; \nsige=Quad8IsoPRingStresses[ncoor,Emat,ue,{True}];\n\Print[\"Corner stresses=\",sige//MatrixForm];*)", FontColor->RGBColor[0, 0, 1]], "\n"}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 3: Assembly of master stiffness as a full symmetric matrix. \Axisymmetric,2 dof per node. Two iso-P quads implemented: 4-Node and 8-Node.Uses element stiffness modules,which must be initialized at this \point.\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[TextData[{ "RingMasterStiffness[nodrzc_,eletyp_,elenod_,elemat_,defopt_]:=\n \Module[{numele=Length[elenod],numnod=Length[nodrzc],\n \numer=False,p=2,Jcons=False,Kfac=1,e,type,etype,enl,eftab,\n \i,ii,j,jj,n,nn,Em,\[Nu],Emat,ncoor,Ke,neldof,K,\n \modname=\"RingMasterStiffness: \"},\n K=Table[0,{2*numnod},{2*numnod}];\n \If [Length[defopt]==1,{numer}=defopt];\n If \[Length[defopt]==2,{numer,p}=defopt];\n If \[Length[defopt]==3,{numer,p,Jcons}=defopt];\n If \[Length[defopt]==4,{numer,p,Jcons,Kfac}=defopt];\n For [e=1,e<=numele,e++,\n\ type=eletyp[[e]]; etype=StringTake[type,5]; \n If \[etype!=\"Quad4\"&&etype!=\"Quad8\", Print[modname,\n \"Illegal \element type, e=\",e]; Return[Null]];\n enl=elenod[[e]]; \nn=Length[enl];\n \eftab=Flatten[Table[{2*enl[[i]]-1,2*enl[[i]]},{i,nn}]];\n \ncoor=Table[nodrzc[[enl[[i]]]],{i,nn}]; \n \{Em,\[Nu]}=elemat[[e]]; \n Emat=Em/((1+\[Nu])*(1-2*\[Nu]))*{{1-\[Nu],\\[Nu],\[Nu],0},{\[Nu],1-\[Nu],\[Nu],0},\n \{\[Nu],\[Nu],1-\[Nu],0},{0,0,0,1/2-\[Nu]}};\n If \[numer,ncoor=N[ncoor];Emat=N[Emat]];\n options={numer,p,Jcons,Kfac}; \n \ If [etype==\"Quad4\", \n If [type==\"Quad4.1\", \options={numer,1,Jcons,Kfac}]; \n \Ke=Quad4IsoPRingStiffness[ncoor,Emat,options] ];\n If \[etype==\"Quad8\",\n If [type==\"Quad8.3\", \options={numer,3,Jcons,Kfac}]; \n \Ke=Quad8IsoPRingStiffness[ncoor,Emat,options] ];\n If \[numer,Ke=Chop[Ke]]; If [!numer,Ke=Simplify[Ke]];\n ", StyleBox[" (*Print[\"Ke=\",Ke//MatrixForm];*)", FontColor->RGBColor[1, 0, 0]], "\n neldof=Length[Ke];\n For [i=1,i<=neldof,i++,ii=eftab[[i]];\n\ For [j=i,j<=neldof,j++,jj=eftab[[j]];\n \K[[jj,ii]]=K[[ii,jj]]+=Ke[[i,j]] ]];\n ]; \n Return[K]; \n];\n\n(*", StyleBox["ClearAll[Em,\[Nu],a,b,e,p,numer,exx,eyy,gxy]; \na=2; b=1; Em=12; \\[Nu]=1/3; \nnodrzc={{0,b},{0,0},{a/2,b},{a/2,0},{a,b},{a,0}};\neletyp=Table[\\"Quad4\",{2}]; \nelenod={{1,2,4,3},{3,4,6,5}};\n\

elemat=Table[{Em,\[Nu]},{2}];\ndefopt={True,2,False};\n\K=RingMasterStiffness[nodrzc,eletyp,elenod,elemat,defopt];\n\Print[\"K=\",Chop[K]//MatrixForm];\nPrint[\"diag of \K=\",Table[K[[i,i]],{i,12}]]; \nPrint[\"eigs of \K=\",Chop[Eigenvalues[N[K]]]];", FontColor->RGBColor[0, 0, 1]], "\n", StyleBox["\nClearAll[Em,\[Nu],a,b,e,p,numer,exx,eyy,gxy]; \na=2; b=1; \Em=12; \[Nu]=1/3; \n\nodrzc={{0,b},{0,b/2},{0,0},{a/2,b},{a/2,0},{a,b},{a,b/2},{a,0}};\n\eletyp=Table[\"Quad8\",{1}]; \nelenod={{1,3,8,6,2,5,7,4}};\nelemat=Table[{Em,\\[Nu]},{1}];\ndefopt={True,2,False};\n\K=RingMasterStiffness[nodrzc,eletyp,elenod,elemat,defopt];\n\Print[\"K=\",Chop[K]//MatrixForm];\nPrint[\"diag of \K=\",Table[K[[i,i]],{i,16}]]; \nPrint[\"eigs of \K=\",Chop[Eigenvalues[N[K]]]];*)", FontColor->RGBColor[0, 0, 1]]}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 4: Application of displacement boundary conditions on master \stiffness equations. Code only takes single freedom constraints,but accepts nonzero prescribed \displacements.\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[TextData[{ "ModifiedMasterStiffness[pdof_,Km_]:= Module[{i,j,k,K},K=Km; \n For \[k=1,k<=Length[pdof],k++,i=pdof[[k]];\n For [j=1,j<=Length[K],j++,\n \ K[[i,j]]=K[[j,i]]=0]; K[[i,i]]=1;\n ]; Return[K];\n];\n\ModifiedNodeForces[pdof_,pdofv_,Km_,nfv_,bfor_]:= Module[\n \{i,j,k,l,d,kk=Length[pdof],n=Length[Km],fixed,rhs},\n rhs=nfv+bfor; \d=pdofv; fixed=Table[False,{n}]; \n Do [i=pdof[[k]]; \fixed[[i]]=True,{k,1,kk}];\n For [k=1,k<=kk,k++,i=pdof[[k]]; \n For \[j=1,j<=n,j++,If [fixed[[j]],Continue[]]; \n \rhs[[j]]=rhs[[j]]-Km[[i,j]]*d[[k]]; \n ]; rhs[[i]]=d[[k]]; \n \ ]; \n Return[rhs];\n];\n \n", StyleBox["(*K=Array[Ks,{6,6}];\nPrint[\"Master stiffness \

matrix:\"];Print[K//TableForm];\np=Array[ps,{6}];\nPrint[\"Node force vector:\\"];Print[p];\np=ModifiedNodeForces[{1,2,4},{d1,d2,d3},K,p];\nPrint[\"Node \Force vector modified for displacement B.C.:\"];\nPrint[p];\n\K=ModifiedMasterStiffness[{1,2,4},K];\nPrint[\"Master stiffness modified for \displacement B.C.:\"];\nPrint[K//TableForm];*)", FontColor->RGBColor[0, 0, 1]]}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 5: Assembly of consistent force vector associated with body \forces. Axisymmetric, 2 dof per node. Both 4-node and 8-node iso-P quads \implemented.Uses element body force modules,which must be initialized at this \point.\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[TextData[{ "RingBodyForces[nodrzc_,eletyp_,elenod_,elebfor_,defopt_]:=\n \Module[{numele=Length[elenod],numnod=Length[nodrzc],\n \numer=False,p=2,Jcons=False,Kfac=1,e,type,etype,enl,eftab,\n i,ii,n,nn,Em,\\[Nu],Emat,ncoor,neldof,fe,fe0,f,\n modname=\"RingBodyForces: \"},\n \f=Table[0,{2*numnod}]; fe0=Table[0,{16}];\n If \[Length[defopt]==1,{numer}=defopt];\n If \[Length[defopt]==2,{numer,p}=defopt];\n If \[Length[defopt]==3,{numer,p,Jcons}=defopt];\n If \[Length[defopt]==4,{numer,p,Jcons,Kfac}=defopt];\n \options={numer,p,Jcons,Kfac}; \n For [e=1,e<=numele,e++,\n \type=eletyp[[e]]; etype=StringTake[type,5]; \n If \[etype!=\"Quad4\"&&etype!=\"Quad8\", Print[modname,\n \"Illegal \element type,e=\",e]; Return[Null]];\n enl=elenod[[e]]; nn=Length[enl];\\n eftab=Flatten[Table[{2*enl[[i]]-1,2*enl[[i]]},{i,nn}]];\n \ncoor=Table[nodrzc[[enl[[i]]]],{i,nn}]; \n bfor=elebfor[[e]];\n \If [numer,ncoor=N[ncoor]; bfor=N[bfor]];\n If [etype==\"Quad4\", \n \ fe=Quad4IsoPRingBodyForces[ncoor,bfor,options] ];\n If \

[etype==\"Quad8\", \n \fe=Quad8IsoPRingBodyForces[ncoor,bfor,options] ];\n neldof=Length[fe];\n\ For [i=1,i<=neldof,i++,ii=eftab[[i]];\n f[[ii]]+=fe[[i]] ];\\n ];\n Return[f]; \n];\n\n(*", StyleBox["ClearAll[a,b,e,p]; a=2; b=1; \n\nodrzc={{0,b},{0,0},{a/2,b},{a/2,0},{a,b},{a,0}};\nelenod= \{{1,2,4,3},{3,4,6,5}};\n(*Plot2DElementsAndNodes[nodrzc,elenod,1/2.5,\n \\"Mesh for body forces\",True,True];*)\neletyp= {\"Quad4\",\"Quad4\"};\n\elebfor={{1,-2},{1,-2}}; defopt={True};\n\f=RingBodyForces[nodrzc,eletyp,elenod,elebfor,defopt];\nPrint[\"Body force \vector:\",Transpose[Partition[f,2]]//MatrixForm]; \n\nClearAll[a,b,e,p]; a=2; \b=1; \nnodrzc={{0,b},{0,b/2},{0,0},{a/2,b},{a/2,0},{a,b},{a,b/2},{a,0}};\n\eletyp=Table[\"Quad8\",{1}]; \nelenod={{1,3,8,6,2,5,7,4}};\nelebfor={{1,-2}}; \defopt={True,2};\nf=RingBodyForces[nodrzc,eletyp,elenod,elebfor,defopt];\n\Print[\"Body force vector:\",Transpose[Partition[f,2]]//MatrixForm];*)", FontColor->RGBColor[0, 0, 1]]}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 6: Computation of nodal stresses (averaged across all elements \meeting at a node) in Axisymmetric structure. Quad 4 and Quad8 implemented. Stress modules are \expected to be initialized.\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[TextData[{ "RingNodalStresses[nodrzc_,eletyp_,elenod_,elemat_,\n \noddis_,defopt_]:=Module[{numele=Length[elenod],\n \numnod=Length[nodrzc],numer=False,g=1/Sqrt[3],\n \w0=0,options,e,type,etype,enl,eftab,i,j,k,\n \n,nn,Em,\[Nu],Emat,ncoor,ue,ncount,esig,sige,sig,\n \modname=\"RingNodalStresses: \"}, \n sige=Table[{0,0,0,0},{numele}]; \n \sig= Table[{0,0,0,0},{numnod}]; \n If [Length[defopt]==1,{numer}=defopt];\n \If [Length[defopt]==2,{numer,g}=defopt];\n If \

[Length[defopt]==3,{numer,g,w0}=defopt];\n options={numer,g,w0}; \ncount=Table[0,{numnod}]; ", StyleBox["\n ", FontColor->RGBColor[1, 0, 0]], "For [e=1,e<=numele,e++, \n type=eletyp[[e]]; \etype=StringTake[type,5]; \n enl=elenod[[e]]; nn=Length[enl]; \n \eftab=Flatten[Table[{2*enl[[i]]-1,2*enl[[i]]},{i,nn}]];\n \ncoor=Table[nodrzc[[enl[[i]]]],{i,nn}];\n \ue=Flatten[Table[noddis[[enl[[i]]]],{i,nn}]];\n {Em,\[Nu]}=elemat[[e]]; \\n Emat=Em/((1+\[Nu])*(1-2*\[Nu]))*{{1-\[Nu],\[Nu],\[Nu],0},{\[Nu],1-\\[Nu],\[Nu],0},\n {\[Nu],\[Nu],1-\[Nu],0},{0,0,0,1/2-\[Nu]}}; \\n If [numer,ncoor=N[ncoor];ue=N[ue];Emat=N[Emat]];\n If \[etype==\"Quad4\",\n \esig=Quad4IsoPRingStresses[ncoor,Emat,ue,options]", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "];\n If [etype==\"Quad8\",\n \esig=Quad8IsoPRingStresses[ncoor,Emat,ue,options]", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "];\n For [i=1,i<=nn,i++, n=enl[[i]]; ncount[[n]]++; \n \sig[[n]]+=esig[[i]] ]; \n ];\n For [n=1,n<=numnod,n++,k=ncount[[n]];If \[k>1,sig[[n]]/=k]];\n Return[sig]];\n \n(*", StyleBox["ClearAll[Em,\[Nu],a,b,exx,eyy,gxy]; a=2; b=1;\nEm=120; \[Nu]=1/3; \exx=0.15; eyy=-0.125; gxy=-0.070;\n\nodrzc={{0,b},{0,0},{a/2,b},{a/2,0},{a,b},{a,0}};\nelenod= \{{1,2,4,3},{3,4,6,5}};\nnumnod=Length[nodrzc]; numele=Length[elenod];\n\eletyp=Table[\"Quad4\",{numele}]; \nelemat=Table[{Em,\[Nu]},{numele}]; \n\defopt={True,2,1,1};\nr=Table[nodrzc[[i,1]],{i,numnod}]; \z=Table[nodrzc[[i,2]],{i,numnod}];\n\noddis=Table[{r[[i]]*exx,r[[i]]*gxy+z[[i]]*eyy},{i,numnod}];\nPrint[\"noddis=\\",Transpose[noddis]//MatrixForm];\n\sig=RingNodalStresses[nodrzc,eletyp,elenod,elemat,noddis,defopt];\n\Print[\"Computed node stresses=\",Transpose[Chop[N[sig]]]//MatrixForm]; *)\n\n\(*ClearAll[Em,\[Nu],a,b,e,p,numer,exx,eyy,gxy]; a=2; b=1;\nEm=120; \[Nu]=1/3; \exx=0.15; eyy=-0.125; gxy=-0.070;\n\nodrzc={{0,b},{0,b/2},{0,0},{a/2,b},{a/2,0},{a,b},{a,b/2},{a,0}};\n\elenod={{1,3,8,6,2,5,7,4}};\nnumnod=Length[nodrzc]; numele=Length[elenod];\n\eletyp=Table[\"Quad8\",{numele}]; elemat=Table[{Em,\[Nu]},{numele}];\n\defopt={False};\nr=Table[nodrzc[[i,1]],{i,numnod}]; \z=Table[nodrzc[[i,2]],{i,numnod}];\n\noddis=Table[{r[[i]]*exx,r[[i]]*gxy+z[[i]]*eyy},{i,numnod}]; \n\Print[\"noddis=\",Transpose[noddis]//MatrixForm];\n\sig=RingNodalStresses[nodrzc,eletyp,elenod,elemat,noddis,defopt];\n\Print[\"Computed node stresses=\",Transpose[Chop[sig]]//MatrixForm]; *)",

FontColor->RGBColor[0, 0, 1]]}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 7: Mesh plot modules. Written by C. A. Felippa,1997/98. Mathematica adaptation of ancient (1966) Fortran code.\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[TextData[{ "Plot2DElements[xycoor_,elenod_,aspect_,label_]:= \n\Module[{enl,n,nc,xyc,poly=sides={},numele=Length[elenod]},\n Do \[enl=elenod[[ne]]; nc=Length[enl]; \n If [nc!=3 && nc!=4,Continue[]]; \xyc=Table[0,{nc+1}];\n enl=AppendTo[enl,enl[[1]]]; \n Do \[n=enl[[i]]; xyc[[i]]=xycoor[[n]],{i,1,nc+1}];\n poly= \AppendTo[poly,Graphics[Polygon[Take[xyc,nc]]]];\n \sides=AppendTo[sides,Graphics[Line[xyc]]],\n {ne,1,numele}];\n \Show[Graphics[RGBColor[1,1,0]],poly,\n \Graphics[Thickness[0.008]],Graphics[RGBColor[0,0,0]],\n \sides,AspectRatio->aspect,PlotLabel->label ];\n ];\n \n\Plot2DElementsAndNodes[xycoor_,elenod_,aspect_,label_,\n \elabels_,nlabels_]:= Module[{i,j,k,e,enl,elab,n,nlab,\n \x,y,delx,dely,xmin,xmax,ymin,ymax,xy0,eps,ex,ey,b,nb,\n \numele=Length[elenod],rade,gside,gpoly,\n \numnod=Length[xycoor],radn,pnod=pnlab={},\n \pefill=pesid=pelab=pecirc=pedisk={}},\n \x=Table[xycoor[[i,1]],{i,1,numnod}]; \n \y=Table[xycoor[[i,2]],{i,1,numnod}];\n \{xmin,xmax,ymin,ymax}=N[{Min[x],Max[x],Min[y],Max[y]}];\n \{delx,dely}={xmax-xmin,ymax-ymin}/Sqrt[N[numnod]];\n \radn=0.07*Min[delx,dely]; {ex,ey}={-2.3*radn,1.25*radn}; \n \rade=0.16*Min[delx,dely]; \n \(*Print[\"delx,dely,rade=\",{delx,dely,rade}];*) \n Do [enl=elenod[[e]]; \n=Length[enl];\n If [n==3,b={1,2,3};nb=3];\n If \[n==4,b={1,2,3,4};nb=4];\n If [n==6,b={1,4,2,5,3,6};nb=6];\n If \[n==8,b={1,5,2,6,3,7,4,8};nb=8];\n If [n==9,b={1,5,2,6,3,7,4,8};nb=8];\n\ gside=gpoly=Table[xycoor[[enl[[b[[i]]]]]],{i,nb}]; \n \AppendTo[gside,gpoly[[1]]];\n xy0={0,0};Do \[xy0+=gside[[i]],{i,nb}];xy0=xy0/nb; \n \pefill=AppendTo[pefill,Graphics[Polygon[gpoly]]];\n pesid= \AppendTo[pesid, Graphics[Line[gside]]];\n If [n==9,xy0+={0,-2*rade}];\n \ If [elabels,pedisk= AppendTo[pedisk,\n \

Graphics[Disk[xy0,rade]]];\n \pecirc=AppendTo[pecirc,Graphics[Circle[xy0,rade]]];\n \elab=FontForm[e,{\"Times\",11}];\n pelab= \AppendTo[pelab,Graphics[Text[elab,xy0]]]],\n {e,1,numele}];\n Do \[nlab=FontForm[n,{\"Times\",11}];\n If [nlabels,\n \pnlab=AppendTo[pnlab,\n \Graphics[Text[nlab,xycoor[[n]]+{ex,ey}]]]];\n pnod= AppendTo[pnod, \n \ Graphics[Disk[xycoor[[n]],radn]]],\n {n,1,numnod}]; \eps=0.1*(ymax-ymin);\n Show[Graphics[RGBColor[1,1,0]],pefill,\n \Graphics[GrayLevel[0]],\n Graphics[AbsoluteThickness[1.5]],pesid,\n \ Graphics[GrayLevel[0]],pnod,\n Graphics[AbsoluteThickness[0.8]],\n \ Graphics[GrayLevel[1]],pedisk,\n \Graphics[GrayLevel[0]],pecirc,pelab,\n Graphics[GrayLevel[0]],pnlab,\n \ PlotLabel->label,\n \PlotRange->{{xmin-eps,xmax+eps},{ymin-eps,ymax+eps}},\n \Axes->False,AspectRatio->aspect];\n];\n\n", StyleBox["(*xycoor=N[{{0,10},{-8.66,5},{-8.66,-5},{0,-10},{8.66,-5},{8.66,5}\,\n {-4.33,2.5},{-4.33,-2.5},{4.33,-2.5},{4.33,2.5},{0,0}}];\n\elenod={{1,2,7},{2,3,8,7},{3,4,8},{1,7,11},{7,8,11},{8,4,11},\n \{1,11,10},{11,9,10},{11,4,9},{1,10,6},{10,9,5,6},{4,5,9}};\n \n\aspect=10/8.66;\nPlot2DElements[xycoor,elenod,aspect,\"hexagon test mesh\"];\n\Plot2DElementsAndNodes[xycoor,elenod,aspect,\n \"hexagon test mesh with \elem & node labels\",True,True];*)", FontColor->RGBColor[0, 0, 1]], "\n "}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 8: Contour plot modules. Written by C. A. Felippa, 1997. Redone 2008. Mathematica adaptation of ancient (1966) Fortran code.\\>", \"Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[TextData[{ "ContourBandColor[f_,fmin_,fmax_]:= Module[{fs,h,s,b=1},\n If [fmax==0 \||fmin>=fmax, (* White if wrong inputs *)\n \

Return[Graphics[RGBColor[1,1,1]] ]];\n If [f>fmax || f<fmin, (* Black if \outside range *)\n Return[Graphics[RGBColor[0,0,0]] ]]; \n \fs=(fmax-f)/(fmax-fmin); h=0.70*fs; s=1-1/(1+100*(fs-1/2)^2);", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "\n Return[Graphics[Hue[h,s,b]]]];\n \n\TrigBandSegment[xs_,ys_,fs_,fv_,eps_]:=Module[{x1,x2,x3,\n \y1,y2,y3,f1,f2,f3,x21,y21,x32,y32,x13,y13,f21,f32,f31,\n \\[Zeta]211,\[Zeta]212,\[Zeta]311,\[Zeta]313,\[Zeta]322,\[Zeta]323,P1,P3,P21,\P32,P13}, \n {x1,x2,x3}=xs; {y1,y2,y3}=ys; {f1,f2,f3}=fs; P1={x1,y1}; \P3={x3,y3};\n If [fv<f1, Return[{{P1,P1},-1}]]; If [fv>f3, \Return[{{P3,P3},0}]];\n f21=f2-f1; If [f21<eps,f1=f1-eps;f21=eps];\n \f32=f3-f2; If [f32<eps,f3=f2+eps;f32=eps]; f31=f3-f1; \n \\[Zeta]212=(fv-f1)/f21; \[Zeta]212=Max[Min[\[Zeta]212,1],0]; \[Zeta]211=1-\\[Zeta]212;\n \[Zeta]313=(fv-f1)/f31; \[Zeta]313=Max[Min[\[Zeta]313,1],0]; \\[Zeta]311=1-\[Zeta]313;", StyleBox["\n", FontColor->RGBColor[1, 0, 0]], " P21={x2*\[Zeta]212+x1*\[Zeta]211,y2*\[Zeta]212+y1*\[Zeta]211};\n \P13={x1*\[Zeta]311+x3*\[Zeta]313,y1*\[Zeta]311+y3*\[Zeta]313};\n If \[fv<=f2,Return[{{P21,P13},1}]];\n \[Zeta]323=(fv-f2)/f32; \\[Zeta]323=Max[Min[\[Zeta]323,1],0]; \[Zeta]322=1-\[Zeta]323; \n P32={x3*\\[Zeta]323+x2*\[Zeta]322,y3*\[Zeta]323+y2*\[Zeta]322};\n \Return[{{P32,P13},2}]];\n \n\ContourBandPlotFunctionOnTrig3[xyc_,fc_,{fmin_,fmax_,finc_,eps_},\n \lines_]:=Module[{nv,iv,xc1,yc1,xc2,yc2,xc3,yc3,fc1,fc2,fc3,\n \xs,ys,fs,x1,x2,x3,y1,y2,y3,f1,f2,f3,firsthit,Pc1,Pc2,Pc3,\n \ftop,fbot,favg,Pb,Pt,tb,tt,P1,P2,P3,P4,Q1,Q2,Q3,Q4,ptab,\n \flast,Plast,tlast,kb,kl,p,pb,pl}, \n If [fmax<fmin||finc<=0, \Return[{{},{}}]];\n nv=Floor[(fmax-fmin+eps)/finc]; nv=Min[nv,1000];", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "\n pb=Table[{},{nv}]; pl=Table[{},{nv+1}]; \n \{{xc1,yc1},{xc2,yc2},{xc3,yc3}}=N[xyc]; {fc1,fc2,fc3}=N[fc];\n \{{x1,y1,f1},{x2,y2,f2},{x3,y3,f3}}=Sort[{{xc1,yc1,fc1},\n \{xc2,yc2,fc2},{xc3,yc3,fc3}},#1[[3]]<=#2[[3]]&]; \n xs={x1,x2,x3}; \ys={y1,y2,y3}; fs={f1,f2,f3}; \n Pc1={x1,y1}; Pc2={x2,y2}; Pc3={x3,y3};\n \ptab={{{},{Pc1,Pc2,Pc3},{Pc1,Q4,Q3},{Pc1,Pc2,Q3,Q4}},\n \{{Null},{},{Null},{Null}},\n \{{Null},{Pc3,Pc2,Q1,Q2},{Q1,Q2,Q4,Q3},{Pc2,Q1,Q2,Q4,Q3}},\n \{{Null},{Pc3,Q1,Q2},{Null},{Q1,Q2,Q4,Q3}}};\n flast=fmin-1000; \firsthit=True; kb=kl=0;\n For [iv=1,iv<=nv,iv++, \n \fbot=N[fmin+(iv-1)*finc]; ftop=N[fmin+iv*finc];\n If [ftop<=f1 || \fbot>=f3, Continue[]]; favg=(fbot+ftop)/2;\n If [fbot==flast, \{fbot,Pb,tb}={flast,Plast,tlast},\n \{Pb,tb}=TrigBandSegment[xs,ys,fs,fbot,eps],\n \{Pb,tb}=TrigBandSegment[xs,ys,fs,fbot,eps]]; \n \{Pt,tt}=TrigBandSegment[xs,ys,fs,ftop,eps];\n {P1,P2}=Pb; {P3,P4}=Pt; \{flast,Plast,tlast}={ftop,Pt,tt};\n If [lines,\n If \[firsthit&&(tb>0), ",

StyleBox[" \n ", FontColor->RGBColor[1, 0, 0]], "pl[[++kl]]=Graphics[Line[Pb]]; firsthit=False];\n\t If [tt>0, \pl[[++kl]]=Graphics[Line[Pt]]]\n\t ];\n \p=ptab[[tb+2,tt+2]]/.{Q1->P1,Q2->P2,Q3->P3,Q4->P4};\n \pb[[++kb]]={ContourBandColor[favg,fmin,fmax],Graphics[Polygon[p]]};\n\t ]; \", StyleBox["\n ", FontColor->RGBColor[1, 0, 0]], "If [kb>0,pb=Take[pb,kb],{},{}]; If [kl>0,pl=Take[pl,kl],{},{}];\n \ClearAll[ptab]; Return[{pb,pl}];\n ];\n \n\ContourBandPlotFunctionOnQuad4[xyc_,fc_,{fmin_,fmax_,finc_,eps_},\n \lines_]:=Module[{x1,x2,x3,x4,x5,y1,y2,y3,y4,y5,f1,f2,f3,f4,f5,\n \e,xytab,ftab,xyt,ft, pbq,plq,pbt,plt },\n pbq=plq=Table[{},{4}];\n \{{x1,y1},{x2,y2},{x3,y3},{x4,y4}}=xyc; {f1,f2,f3,f4}=fc;\n \x5=(x1+x2+x3+x4)/4; y5=(y1+y2+y3+y4)/4; f5=(f1+f2+f3+f4)/4;\n \xytab={{{x1,y1},{x2,y2},{x5,y5}},{{x2,y2},{x3,y3},{x5,y5}},\n \{{x3,y3},{x4,y4},{x5,y5}},{{x4,y4},{x1,y1},{x5,y5}}};\n \ftab={{f1,f2,f5},{f2,f3,f5},{f3,f4,f5},{f4,f1,f5}};\n For [e=1,e<=4,e++, \xyt=xytab[[e]]; ft=ftab[[e]];\n \{pbq[[e]],plq[[e]]}=ContourBandPlotFunctionOnTrig3[\n \xyt,ft,{fmin,fmax,finc,eps},lines];\n ]; \n Return[{pbq,plq}];\\n\t];\n\t\nContourBandPlotFunctionOnQuad9[xyc_,fc_,{fmin_,fmax_,finc_,eps_},\\n lines_]:=Module[{x1,x2,x3,x4,x5,x6,x7,x8,x9,y1,y2,y3,y4,\n \y5,y6,y7,y8,y9,f1,f2,f3,f4,f5,f6,f7,f8,f9,\n e,xytab,ftab,xyt,ft, \pbq,plq,pbt,plt },\n pbq=plq=Table[{},{4}];\n \{{x1,y1},{x2,y2},{x3,y3},{x4,y4},\n \{x5,y5},{x6,y6},{x7,y7},{x8,y8},{x9,y9}}=xyc; \n \{f1,f2,f3,f4,f5,f6,f7,f8,f9}=fc;\n \xytab={{{x1,y1},{x5,y5},{x9,y9},{x8,y8}},\n \{{x2,y2},{x6,y6},{x9,y9},{x5,y5}},\n \{{x3,y3},{x7,y7},{x9,y9},{x6,y6}},\n \{{x4,y4},{x8,y8},{x9,y9},{x7,y7}}};\n \ftab={{f1,f5,f9,f8},{f2,f6,f9,f5},{f3,f7,f9,f6},{f4,f8,f9,f7}};\n For \[e=1,e<=4,e++, xye=xytab[[e]]; fe=ftab[[e]];\n \{pbq[[e]],plq[[e]]}=ContourBandPlotFunctionOnQuad4[\n \xye,fe,{fmin,fmax,finc,eps},lines];\n ]; \n Return[{pbq,plq}];\\n\t];\n\t\nContourBandPlotFunctionOnQuad8[xyc_,fc_,{fmin_,fmax_,finc_,eps_},\\n lines_]:=Module[{x1,x2,x3,x4,x5,x6,x7,x8,x9,y1,y2,y3,y4,\n \y5,y6,y7,y8,y9,f1,f2,f3,f4,f5,f6,f7,f8,f9,\n e,xytab,ftab,xyt,ft, \pbq,plq,pbt,plt },\n pbq=plq=Table[{},{4}];\n \{{x1,y1},{x2,y2},{x3,y3},{x4,y4},\n {x5,y5},{x6,y6},{x7,y7},{x8,y8}}=xyc; \\n {f1,f2,f3,f4,f5,f6,f7,f8}=fc;\n x9=(2*(x5+x6+x7+x8)-(x1+x2+x3+x4))/4;\\n y9=(2*(y5+y6+y7+y8)-(y1+y2+y3+y4))/4;\n \f9=(2*(f5+f6+f7+f8)-(f1+f2+f3+f4))/4;\n \xytab={{{x1,y1},{x5,y5},{x9,y9},{x8,y8}},\n \{{x2,y2},{x6,y6},{x9,y9},{x5,y5}},\n \{{x3,y3},{x7,y7},{x9,y9},{x6,y6}},\n \{{x4,y4},{x8,y8},{x9,y9},{x7,y7}}};\n \

ftab={{f1,f5,f9,f8},{f2,f6,f9,f5},{f3,f7,f9,f6},{f4,f8,f9,f7}};\n For \[e=1,e<=4,e++, xye=xytab[[e]]; fe=ftab[[e]];\n \{pbq[[e]],plq[[e]]}=ContourBandPlotFunctionOnQuad4[\n \xye,fe,{fmin,fmax,finc,eps},lines];\n ]; \n Return[{pbq,plq}];\\n\t];\n\t\n\ContourBandPlotFunctionOnQuad16[xyc_,fc_,{fmin_,fmax_,finc_,eps_},\n \lines_]:=Module[{x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,\n \x11,x12,x13,x14,x15,x16,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,\n \y11,y12,y13,y14,y15,y16,f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,\n \f11,f12,f13,f14,f15,f16,\n e,xytab,ftab,xyt,ft, pbq,plq,pbt,plt },\n \pbq=plq=Table[{},{9}];\n {{x1,y1},{x2,y2},{x3,y3},{x4,y4},\n \{x5,y5},{x6,y6},{x7,y7},{x8,y8},\n {x9,y9},{x10,y10},{x11,y11},{x12,y12},\n\ {x13,y13},{x14,y14},{x15,y15},{x16,y16}}=xyc; \n \{f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12,\n f13,f14,f15,f16}=fc;\n \xytab={{{x1,y1}, {x5,y5}, {x13,y13},{x12,y12}},\n {{x2,y2}, \{x7,y7}, {x14,y14},{x6,y6}},\n {{x3,y3}, {x9,y9}, \{x15,y15},{x8,y8}},\n {{x4,y4}, {x11,y11},{x16,y16},{x10,y10}},\n \ {{x5,y5}, {x6,y6}, {x14,y14},{x13,y13}},\n {{x7,y7}, \{x8,y8}, {x15,y15},{x14,y14}},\n {{x9,y9}, \{x10,y10},{x16,y16},{x15,y15}},\n \{{x11,y11},{x12,y12},{x13,y13},{x16,y16}},\n \{{x13,y13},{x14,y14},{x15,y15},{x16,y16}}};\n \ftab={{f1,f5,f13,f12},{f2,f7,f14,f6},{f3,f9,f15,f8},\n \{f4,f11,f16,f10},{f5,f6,f14,f13},{f7,f8,f15,f14},\n \{f9,f10,f16,f15},{f11,f12,f13,f16},{f13,f14,f15,f16}};\n For [e=1,e<=9,e++, \xye=xytab[[e]]; fe=ftab[[e]];\n \{pbq[[e]],plq[[e]]}=ContourBandPlotFunctionOnQuad4[\n \xye,fe,{fmin,fmax,finc,eps},lines];\n ]; \n Return[{pbq,plq}];\\n\t];\n\t\nContourBandPlotFunctionOnTrig6[xyc_,fc_,{fmin_,fmax_,finc_,eps_},\\n lines_]:=Module[{x1,x2,x3,x4,x5,x6,y1,y2,y3,y4,y5,y6,\n \f1,f2,f3,f4,f5,f6,e,xytab,ftab,xyt,ft,pbt,plt,pbe,ple},\n \pbt=plt=Table[{},{4}];\n \{{x1,y1},{x2,y2},{x3,y3},{x4,y4},{x5,y5},{x6,y6}}=xyc; \n \{f1,f2,f3,f4,f5,f6}=fc;\n \xytab={{{x1,y1},{x4,y4},{x6,y6}},{{x2,y2},{x5,y5},{x4,y4}},\n \{{x3,y3},{x6,y6},{x5,y5}},{{x4,y4},{x5,y5},{x6,y6}}};\n \ftab={{f1,f4,f6},{f2,f5,f4},{f3,f6,f5},{f4,f5,f6}};\n For [e=1,e<=4,e++, \xyt=xytab[[e]]; ft=ftab[[e]];\n \{pbt[[e]],plt[[e]]}=ContourBandPlotFunctionOnTrig3[\n \xyt,ft,{fmin,fmax,finc,eps},lines];\n ]; \n Return[{pbt,plt}];\\n\t];\n\t\n\ContourBandPlotFunctionOnTrig10[xyc_,fc_,{fmin_,fmax_,finc_,eps_},\n \lines_]:=Module[{x1,x2,x3,x4,x5,x6,x7,x8,x9,x0,y1,y2,y3,\n \y4,y5,y6,y7,y8,y9,y0,f1,f2,f3,f4,f5,f6,f7,f8,f9,f0,\n \e,xytab,ftab,xyt,ft,pbt,plt},\n pbt=plt=Table[{},{9}];", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "\n {{x1,y1},{x2,y2},{x3,y3},{x4,y4},{x5,y5},{x6,y6},\n \

{x7,y7},{x8,y8},{x9,y9},{x0,y0}}=xyc; \n \{f1,f2,f3,f4,f5,f6,f7,f8,f9,f0}=fc;\n \xytab={{{x1,y1},{x4,y4},{x9,y9}},{{x2,y2},{x6,y6},{x5,y5}},\n \{{x3,y3},{x8,y8},{x7,y7}},{{x9,y9},{x4,y4},{x0,y0}},\n \{{x4,y4},{x5,y5},{x0,y0}},{{x5,y5},{x6,y6},{x0,y0}},\n \{{x6,y6},{x7,y7},{x0,y0}},{{x7,y7},{x8,y8},{x0,y0}},\n \{{x8,y8},{x9,y9},{x0,y0}}};\n \ftab={{f1,f4,f9},{f2,f6,f5},{f3,f8,f7},{f9,f4,f0},\n \{f4,f5,f0},{f5,f6,f0},{f6,f7,f0},{f7,f8,f0},\n {f8,f9,f0}};\n For \[e=1,e<=9,e++, xyt=xytab[[e]]; ft=ftab[[e]];\n \{pbt[[e]],plt[[e]]}=ContourBandPlotFunctionOnTrig3[\n \xyt,ft,{fmin,fmax,finc,eps},lines];\n ]; \n Return[{pbt,plt}];\n\\t];\n\t\nContourPlotValueLegend[fmin_,fmax_,xyleg_,mv_]:=Module[\n \{f,fs,finc,fa=Abs[fmin],fb=Abs[fmax],xylinL,xylinR,xytext,\n \black=Graphics[RGBColor[0,0,0]],baselineskip=10,\n \white=Graphics[RGBColor[1,1,1]],preleg,plab,iv,\n \thick=Graphics[AbsoluteThickness[10]],xf1,xf2,xf3,xf4,\n thin= \Graphics[AbsoluteThickness[1]],color,padOK},\n xf1=Offset[{-30, \10},xyleg]; xf2=Offset[{85,10},xyleg];\n xf3=Offset[{ \85,-10-baselineskip*(mv+1)},xyleg];\n \xf4=Offset[{-30,-10-baselineskip*(mv+1)},xyleg];\n preleg={thin,white,\n \ Graphics[Polygon[{xf1,xf2,xf3,xf4}]], black,\n \Graphics[Line[{xf1,xf2,xf3,xf4,xf1}]],thick,\n \Graphics[Text[\"LEGEND\",Offset[{5,0},xyleg],{-1,0}]]};\n \finc=(fmax-fmin)/mv; plab=Table[{},{mv+1}];", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "\n padOK=(Max[fa,fb]<=1000)&&(Min[fa,fb]>=1/100);\n For \[iv=1,iv<=mv+1,iv++, f=Chop[N[fmin+(iv-1)*finc]]; \n \xylinL=Offset[{-20,-baselineskip*iv},xyleg];\n xylinR=Offset[{ \5,-baselineskip*iv},xyleg]; \n xytext=Offset[{ \80,-baselineskip*iv},xyleg];\n If [padOK, \fs=PaddedForm[f,4,NumberSigns->{\"-\",\"+\"}],\n \fs=ScientificForm[f,4,NumberSigns->{\"-\",\"+\"}]];\n \color=ContourBandColor[f,fmin,fmax]; \n \plab[[iv]]={color,Graphics[Line[{xylinL,xylinR}]],\n \black,Graphics[Text[fs,xytext,{1,0}]]}]; \n Return[{preleg,plab}]];\n \n \ContourBandPlotNodeFuncOver2DMesh[nodxyz_,elenod_,fn_,{fmin_,fmax_,finc_},\n \{colors_,mesh_,nodes_,bacg_,lines_,bound_},xyleg_,aspect_,label_]:= Module[\n \ {enl,i,k,n,ne,xyc,fc,c,nn,pb,pl,eps,prebacg,pbacg,preband,pband,preline,\n \bnt,bn,nbn,enb,xyb,bseg,ebs,seg,rseg,nbseg,prebound,pbound,\n \pline,premesh,pmesh,prenode,preleg,pleg,pnode,numele=Length[elenod]},\n \bnt={{},{},{1,2,3,1},{1,2,3,4,1},{},{1,4,2,5,3,6,1},\n \{},{1,5,2,6,3,7,4,8,1},{1,5,2,6,3,7,4,8,1},\n \{1,4,5,2,6,7,3,8,9,1},{},{},{},{},{},{1,5,6,2,7,8,3,9,10,4,11,12,1}}; \n \eps=10.^(-8)*Max[Abs[fmax],Abs[fmin]]; pband=pmesh=pnode=pline=pbacg={}; \n \

If [colors,pband=Table[{},{numele}]]; If [mesh, pmesh=Table[{},{numele}]]; \n \ If [nodes, pnode=Table[{},{numele}]]; If [lines,pline=Table[{},{numele}]]; \n\ If [bacg, pbacg=Table[{},{numele}]]; preleg=pleg=bseg=pbound={};\n \prebacg= {Graphics[RGBColor[.7,.7,.7]]}; preband={};\n preline= \{Graphics[RGBColor[0,0,0]],Graphics[AbsoluteThickness[1]]};\n premesh= \{Graphics[RGBColor[1,0,0]],Graphics[AbsoluteThickness[2]],\n \Graphics[AbsoluteDashing[{1,10}]]};\n prenode= \{Graphics[RGBColor[0,0,0]],Graphics[AbsoluteDashing[{}]],\n \Graphics[AbsolutePointSize[5]]};\n \prebound={Graphics[RGBColor[0,0,1]],Graphics[AbsoluteDashing[{}]],\n \ Graphics[AbsoluteThickness[2]]};\n For [e=1,e<=Length[elenod],e++, \n \enl=elenod[[e]]; ne=Length[enl]; \n If [!MemberQ[{3,4,6,8,9,10,16},ne], \Print[\"ContourBandPlot:\",\n \"Elem \",e,\" with \",ne,\" nodes \skipped\"]; Continue[]];\n fc=Table[fn[[enl[[i]]]],{i,ne}]; \xyc=Table[nodxyz[[enl[[i]]]],{i,ne}];\n bn=bnt[[ne]]; nbn=Length[bn]; \xyb=Table[xyc[[bn[[i]]]],{i,nbn}];\n If [bound, \enb=Table[enl[[bn[[i]]]],{i,nbn}]; \n \ebs=Table[{enb[[i]],enb[[i+1]]},{i,1,nbn-1}]; ", StyleBox["\n", FontColor->RGBColor[1, 0, 0]], " For [i=1,i<=nbn-1,i++, seg=ebs[[i]]; rseg=Reverse[seg]; \n \ If [MemberQ[bseg,rseg], {{k}}=Position[bseg,rseg,1,1]; \n \bseg=Drop[bseg,{k}]; Continue[]]; \n AppendTo[bseg,seg]];\n \ ];\n If [ne==3, {pb,pl}=ContourBandPlotFunctionOnTrig3[\n \ xyc,fc,{fmin,fmax,finc,eps},lines]];\n If [ne==4, \{pb,pl}=ContourBandPlotFunctionOnQuad4[\n \xyc,fc,{fmin,fmax,finc,eps},lines]];\n If [ne==6, \{pb,pl}=ContourBandPlotFunctionOnTrig6[\n \xyc,fc,{fmin,fmax,finc,eps},lines]];\n If [ne==8, \{pb,pl}=ContourBandPlotFunctionOnQuad8[\n \xyc,fc,{fmin,fmax,finc,eps},lines]];\n If [ne==9, \{pb,pl}=ContourBandPlotFunctionOnQuad9[\n \xyc,fc,{fmin,fmax,finc,eps},lines]];\n If \[ne==10,{pb,pl}=ContourBandPlotFunctionOnTrig10[\n \xyc,fc,{fmin,fmax,finc,eps},lines]];\n If \[ne==16,{pb,pl}=ContourBandPlotFunctionOnQuad16[\n \xyc,fc,{fmin,fmax,finc,eps},lines]]; \n If [colors, pband[[e]]=pb]; If \[lines, pline[[e]]=pl];\n If [mesh, pmesh[[e]]=Graphics[Line[xyb]]]; \n\ If [bacg, pbacg[[e]]=Graphics[Polygon[xyb]]]; \n If [nodes, \pnode[[e]]=Table[Graphics[Point[xyb[[i]] ]],{i,ne}]]; ", StyleBox[" ", FontColor->RGBColor[1, 0, 0]],

"\n ];\n If [!colors,pband={}]; If [!mesh, pmesh={}]; If [!nodes, \pnode={}]; \n If [!lines, pline={}]; If [!bacg, pbacg={}]; pleg={};", StyleBox["\n ", FontColor->RGBColor[1, 0, 0]], "nbseg=Length[bseg];", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "\n If [nbseg>0, pbound=Table[0,{nbseg}]; \n For [k=1,k<=nbseg,k++, \{i,j}=bseg[[k]];\n \pbound[[k]]=Graphics[Line[{nodxyz[[i]],nodxyz[[j]]}]] ]];\n If \[Length[xyleg]==0||!colors, Show[prebacg,pbacg,\n \preband,pband,preline,pline,premesh,pmesh,prenode,\n \pnode,prebound,pbound,TextStyle->{FontFamily->\"Courier\",\n \FontSlant->\"Plain\", FontSize->10},\n \AspectRatio->aspect,PlotLabel->label, \n \DisplayFunction->DisplayChannel ]];\n If [Length[xyleg]>0&&colors,\n \{preleg,pleg}=ContourPlotValueLegend[fmin,fmax,xyleg,6];\n \Show[prebacg,pbacg,preband,pband,preline,pline,\n \premesh,pmesh,prenode,pnode,prebound,pbound,preleg,pleg, \n \TextStyle->{FontFamily->\"Courier\",\n FontSlant->\"Plain\", \FontSize->9},\n AspectRatio->aspect,PlotLabel->label,\n \DisplayFunction->DisplayChannel ]];\n \ClearAll[pband,pline,pmesh,pnode,pbacg,pbound,pleg]; \n ];\n \n", StyleBox["(*nodxyz=N[{{0,6},{0,3},{0,0},{5/2,6},{5/2,3},\n \{5/2,0},{5,6},{5,3},{5,0}}]; \nelenod= {{1,3,9,7,2,6,8,4,5}};\n\\[Sigma]rr=Table[8*nodxyz[[n,1]]/5-4,{n,1,9}]; \[Sigma]rrmax=4; \\[Sigma]rrmin=-4;\n \[Sigma]rrinc=.5; aspect=6/5; Print[\"\[Sigma]rr=\",\\[Sigma]rr];\nContourBandPlotNodeFuncOver2DMesh[nodxyz,elenod,\[Sigma]rr,{\\[Sigma]rrmin,\[Sigma]rrmax,\[Sigma]rr\[Sigma]rrinc},\n \{True,True,True,True,True,True},{2,2.5},aspect,\"Sigma-xx (white: zero)\"];\n\\[Sigma]zz=Table[4*nodxyz[[n,2]]/6,{n,1,9}]; \[Sigma]zzmax=4; \\[Sigma]zzmin=-4;\n\[Sigma]zzinc=.5; aspect=6/5; Print[\"\[Sigma]zz=\",\\[Sigma]zz];\nContourBandPlotNodeFuncOver2DMesh[nodxyz,elenod,\[Sigma]zz,{\\[Sigma]zzmin,\[Sigma]zzmax,\[Sigma]zzinc},\n \{True,True,True,True,True,True},{2,2.5},aspect,\"Sigma-yy (white: \zero)\"];*)", FontColor->RGBColor[0, 0, 1]]}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["Cell 9A: Print utility modules", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}},

ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell["\<\PrintRingNodeCoordinates[nodrzc_,title_,digits_]:= Module[ {numnod=Length[nodrzc],n,x,y,z,d=2,f=4,tab}, tab=Table[0,{numnod}]; If [Length[digits]==2,{d,f}=digits]; For [n=1,n<=numnod,n++,{x,y}=nodrzc[[n]]; tab[[n]]={ToString[n],PaddedForm[x,{d,f}],PaddedForm[y,{d,f}]}]; If [StringLength[title]>0, Print[title]]; Print[TableForm[tab, TableAlignments->{Right}, TableDirections->{Column,Row},TableSpacing->{0,2}, TableHeadings ->{None,{\"node\", \"r-coor\", \"z-coor\"}}]]; ClearAll[tab]; ]; PrintRingElementTypeNodes[eletyp_,elenod_,title_,digits_]:= Module[{numele=Length[elenod],e,d=6,f=3,tab}, tab=Table[0,{numele}]; If [Length[digits]==2,{d,f}=digits]; For [e=1,e<=numele,e++, tab[[e]]={ToString[e],eletyp[[e]],ToString[elenod[[e]]]}]; If [StringLength[title]>0, Print[title]]; Print[TableForm[tab, TableAlignments->Right, TableDirections->{Column,Row},TableSpacing->{0,2}, TableHeadings->{None,{\"elem\", \"type\", \"node-list\"}}]]; ClearAll[tab]; ]; PrintRingElementMaterials[elemat_,title_,digits_]:= Module[{numele=Length[elemat],e,mat,d=6,f=3,tab}, tab=Table[0,{numele}]; If [Length[digits]==2,{d,f}=digits]; For [e=1,e<=numele,e++, mat=elemat[[e]]; tab[[e]]={ToString[e],ToString[mat]}]; If [StringLength[title]>0, Print[title]]; Print[TableForm[tab, TableAlignments->Right, TableDirections->{Column,Row},TableSpacing->{0,2}, TableHeadings->{None,{\"elem\", \"material: {E,\[Nu]}\"}}]]; ClearAll[tab]; ]; PrintRingElemBodyForces[elebfor_,title_,digits_]:= Module[{numele=Length[elebfor],e,br,bz, br1,bz1,br2,bz2,br3,bz3,br4,bz4,tab,d=8,f=4}, If [numele<=0,Print[\"No body forces specified\"]; Return[]]; tab=Table[0,{numele}]; If [Length[digits]==2,{d,f}=digits]; If [StringLength[title]>0, Print[title]]; If [Length[elebfor[[1]]]==2, For [e=1,e<=numele,e++, {br,bz}=elebfor[[e]]; tab[[e]]={ToString[e],PaddedForm[br,{d,f}], PaddedForm[bz,{d,f}]}]; Print[TableForm[tab,TableAlignments->{Right}, TableDirections->{Column,Row},TableSpacing->{0,1}, TableHeadings ->{None,{\"elem\", \"r-bforce\", \"z-bforce\"}}]] ]; If [Length[elebfor[[1]]]==4, For [e=1,e<=numele,e++, {{br1,bz1},{br2,bz2},{br3,bz3},{br4,bz4}}=elebfor[[e]];

tab[[e]]={ToString[e], PaddedForm[br1,{d,f}],PaddedForm[bz1,{d,f}], PaddedForm[br2,{d,f}],PaddedForm[bz2,{d,f}], PaddedForm[br3,{d,f}],PaddedForm[bz3,{d,f}], PaddedForm[br4,{d,f}],PaddedForm[bz4,{d,f}]}]; Print[TableForm[tab,TableAlignments->{Right}, TableDirections->{Column,Row},TableSpacing->{0,1}, TableHeadings ->{None,{\"elem\", \"br1\",\"bz1\",\"br2\",\"bz2\", \"br3\",\"bz3\",\"br4\",\"bz4\"}}]]; ]; ClearAll[tab]; ]; PrintRingElemTracForces[eletfor_,title_,digits_]:= Module[{numele=Length[eletfor],e,tab, ne,side,tr1,tr2,tz1,tz2,d=8,f=4}, If [numele<=0,Print[\"No surface traction forces specified\"]; Return[]]; tab=Table[0,{numele}]; If [Length[digits]==2,{d,f}=digits]; For [e=1,e<=numele,e++, {ne,side,{tr1,tr2},{tz1,tz2}}=eletfor[[e]]; tab[[e]]={ToString[ne],ToString[side], PaddedForm[tr1,{d,f}],PaddedForm[tr2,{d,f}], PaddedForm[tz1,{d,f}],PaddedForm[tz2,{d,f}]}]; If [StringLength[title]>0, Print[title]]; Print[TableForm[tab,TableAlignments->{Right}, TableDirections->{Column,Row},TableSpacing->{0,1}, TableHeadings ->{None,{\"elem\", \"side\", \"r-tforce1\", \\"r-tforce2\", \"z-tforce1\", \"z-tforce2\"}}]]; ClearAll[tab]; ]; PrintRingFreedomActivity[nodtag_,nodval_,title_,digits_]:= Module[ {numnod=Length[nodtag],n,t,v,d=6,f=2,tab}, tab=Table[0,{numnod}]; If [Length[digits]==2,{d,f}=digits]; For [n=1,n<=numnod,n++, t=nodtag[[n]]; v=nodval[[n]]; tab[[n]]={ToString[n], PaddedForm[t[[1]],d],PaddedForm[t[[2]],d], PaddedForm[v[[1]],{d,f}],PaddedForm[v[[2]],{d,f}]}]; If [StringLength[title]>0, Print[title]]; Print[TableForm[tab,TableAlignments->{Right}, TableDirections->{Column,Row},TableSpacing->{0,1}, TableHeadings->{None,{\"node\", \"r-tag\", \"z-tag\", \"r-value\", \"z-value\"}} ]]; ClearAll[tab]; ];

PrintRingNodeDisplacements[noddis_,title_,digits_]:= Module[ {numnod=Length[noddis],n,ur,uz,d=8,f=6,tab}, tab=Table[0,{numnod}]; If [Length[digits]==2,{d,f}=digits]; For [n=1,n<=numnod,n++, {ur,uz}=noddis[[n]]; tab[[n]]={ToString[n],PaddedForm[ur,{d,f}], PaddedForm[uz,{d,f}]}]; If [StringLength[title]>0, Print[title]]; Print[TableForm[tab,TableAlignments->{Right}, TableDirections->{Column,Row},TableSpacing->{0,1},

TableHeadings->{None,{\"node\", \"r-disp\", \"z-disp\"}}]]; ClearAll[tab]; ]; PrintRingNodeForces[nodfor_,title_,digits_]:= Module[ {numnod=Length[nodfor],n,fr,fz,tab,d=8,f=4}, tab=Table[0,{numnod}]; If [Length[digits]==2,{d,f}=digits]; For [n=1,n<=numnod,n++, {fr,fz}=nodfor[[n]]; tab[[n]]={ToString[n], PaddedForm[fr,{d,f}],PaddedForm[fz,{d,f}]}]; If [StringLength[title]>0, Print[title]]; Print[TableForm[tab,TableAlignments->{Right}, TableDirections->{Column,Row},TableSpacing->{0,1}, TableHeadings ->{None,{\"node\", \"r-force\", \"z-force\"}}]]; \ ClearAll[tab]; ]; PrintRingNodeStresses[nodsig_,title_,digits_]:= Module[{numnod=Length[nodsig],n,\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\\[Theta],\[Sigma]rz,d=8,f=4,tab}, tab=Table[0,{numnod}]; If [Length[digits]==2,{d,f}=digits]; For [n=1,n<=numnod,n++, {\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\\[Sigma]rz}=nodsig[[n]]; tab[[n]]={ToString[n], PaddedForm[\[Sigma]rr,{d,f}], PaddedForm[\[Sigma]zz,{d,f}], PaddedForm[\[Sigma]\[Theta]\[Theta],{d,f}], \PaddedForm[\[Sigma]rz,{d,f}]} ]; If [StringLength[title]>0, Print[title]]; Print[TableForm[tab,TableAlignments->{Right}, TableDirections->{Column,Row},TableSpacing->{0,1}, TableHeadings ->{None,{\"node\", \"sigma-rr\", \"sigma-zz\", \"sigma-\[Theta]\[Theta]\", \"sigma-xy\"}}]]; ClearAll[tab]; ]; PrintRingAnalysisSolution[noddis_,nodfor_,nodsig_,title_,digits_]:= Module[{numnod=Length[noddis],n,ur,uz,fr,fz, \[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz,d=6,f=4,\tab}, tab=Table[0,{numnod}]; If [Length[digits]==2,{d,f}=digits]; For [n=1,n<=numnod,n++, {ur,uz}=noddis[[n]]; {fr,fz}=nodfor[[n]]; {\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz}=nodsig[[n]\]; tab[[n]]={ToString[n], PaddedForm[ur,{d,f}], PaddedForm[uz,{d,f}], PaddedForm[\[Sigma]rr,{d,f}], PaddedForm[\[Sigma]zz,{d,f}], PaddedForm[\[Sigma]\[Theta]\[Theta],{d,f}], \PaddedForm[\[Sigma]rz,{d,f}], PaddedForm[fr,{d,f}], PaddedForm[fz,{d,f}]} ]; If [StringLength[title]>0, Print[title]]; Print[TableForm[tab,TableAlignments->{Right}, TableDirections->{Column,Row},TableSpacing->{0,1},

TableHeadings ->{None,{\"node\",\"r-disp\",\"z-disp\",\"sigma-rr\", \"sigma-zz\",\"sigma-\[Theta]\[Theta]\",\"sigma-xy\",\"r-force\",\"z-\force\"}}]]; ClearAll[tab]; ]; PrintRingNodeDispStresses[noddis_,nodsig_,title_,digits_]:= Module[{numnod=Length[noddis],n,ur,uz, \[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz,d=6,f=4,\tab}, tab=Table[0,{numnod}]; If [Length[digits]==2,{d,f}=digits]; For [n=1,n<=numnod,n++, {ur,uz}=noddis[[n]]; {\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz}=nodsig[[n]\]; tab[[n]]={ToString[n], PaddedForm[ur,{d,f}], PaddedForm[uz,{d,f}], PaddedForm[\[Sigma]rr,{d,f}], PaddedForm[\[Sigma]zz,{d,f}], PaddedForm[\[Sigma]\[Theta]\[Theta],{d,f}], \PaddedForm[\[Sigma]rz,{d,f}]} ]; If [StringLength[title]>0, Print[title]]; Print[TableForm[tab,TableAlignments->{Right}, TableDirections->{Column,Row},TableSpacing->{0,1}, TableHeadings ->{None,{\"node\",\"r-disp\",\"z-disp\", \"sigma-rr\",\"sigma-zz\",\"sigma-\[Theta]\[Theta]\",\"sigma-xy\"}}]];\

ClearAll[tab]; ]; \\>", "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 9B: Mesh generation utility modules\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[TextData[{ "GenQuad4NodeCoordinates[rzcorn_,Ner_,Nez_]:= Module[\n \{k=0,i,j,dr,dz,numnod,N1,N2,N3,N4,\[Xi],\[Eta],\n \rc1,rc2,rc3,rc4,yc1,yc2,yc3,yc4,r,z,nodrzc}, \n numnod=(Ner+1)*(Nez+1); \nodrzc=Table[{0,0},{numnod}];\n \{{rc1,zc1},{rc2,zc2},{rc3,zc3},{rc4,zc4}}=rzcorn;\n For [i=1,i<=Ner+1,i++, \

For [j=1,j<=Nez+1,j++,\n \[Xi]=N[2*(i-1)/Ner-1]; \\[Eta]=N[2*(j-1)/Nez-1];\n N1=(1-\[Xi])*(1-\[Eta])/4; N2=(1+\[Xi])*(1-\\[Eta])/4;\n N3=(1+\[Xi])*(1+\[Eta])/4; N4=(1-\[Xi])*(1+\[Eta])/4;\n \ r=rc1*N1+rc2*N2+rc3*N3+rc4*N4;\n z=zc1*N1+zc2*N2+zc3*N3+zc4*N4;\n \nodrzc[[++k]]={r,z};\n ]];\n Return[nodrzc]];\n \n\GenQuad8NodeCoordinates[rzcorn_,Ner_,Nez_]:= Module[\n \{k=0,i,j,dr,dz,numnod,N1,N2,N3,N4,\[Xi],\[Eta],\n \rc1,rc2,rc3,rc4,yc1,yc2,yc3,yc4,r,z,nodrzc}, \n \numnod=(2*Ner+1)*(2*Nez+1)-Ner*Nez; \n nodrzc=Table[{0,0},{numnod}];\n \{{rc1,zc1},{rc2,zc2},{rc3,zc3},{rc4,zc4}}=rzcorn;\n For \[i=1,i<=2*Ner+1,i++, For [j=1,j<=2*Nez+1,j++,\n If \[Mod[i,2]==0&&Mod[j,2]==0,Continue[]];\n \[Xi]=N[(i-Ner-1)/Ner]; \\[Eta]=N[(j-Nez-1)/Nez];\n N1=(1-\[Xi])*(1-\[Eta])/4; N2=(1+\[Xi])*(1-\\[Eta])/4;\n N3=(1+\[Xi])*(1+\[Eta])/4; N4=(1-\[Xi])*(1+\[Eta])/4;\n \ r=rc1*N1+rc2*N2+rc3*N3+rc4*N4;\n z=zc1*N1+zc2*N2+zc3*N3+zc4*N4; \n \If [Mod[i,2]==0&&Mod[j,2]==0,Continue[]];\n nodrzc[[++k]]={r,z};\n \]];\n Return[nodrzc]]; \n \n\GenGradedQuad4NodeCoordinates[rzcorn_,Ner_,Nez_,sr_,sz_]:= \n\Module[{k=0,i,j,dr,dz,numnod,N1,N2,N3,N4,\[Xi],\[Eta],\n \rc1,rc2,rc3,rc4,yc1,yc2,yc3,yc4,r,z,elenod}, \n numnod=(Ner+1)*(Nez+1); \rzcoor=Table[0,{numnod}];\n \{{rc1,zc1},{rc2,zc2},{rc3,zc3},{rc4,zc4}}=rzcorn;\n For [i=1,i<=Ner+1,i++,\n\ If [i<=Length[sr],\[Xi]=sr[[i]],\[Xi]=2*(i-1)/Ner-1];\n For \[j=1,j<=Nez+1,j++,\n If \[j<=Length[sz],\[Eta]=sz[[j]],\[Eta]=2*(j-1)/Nez-1];\n N1=(1-\[Xi])*(1-\\[Eta])/4; N2=(1+\[Xi])*(1-\[Eta])/4;\n N3=(1+\[Xi])*(1+\[Eta])/4; \N4=(1-\[Xi])*(1+\[Eta])/4;\n r=rc1*N1+rc2*N2+rc3*N3+rc4*N4;\n \z=zc1*N1+zc2*N2+zc3*N3+zc4*N4;\n rzcoor[[++k]]={r,z};\n ]];\n \Return[rzcoor]];\n\nGenQuad4ElemNodes[Ner_,Nez_]:= Module[\n \{k,i,j,c1,c2,numele,elenod},numele=Ner*Nez; \n \elenod=Table[{0,0,0,0},{numele}]; k=0;\n For [i=1,i<=Ner,i++, For \[j=1,j<=Nez,j++,\n c1=(Nez+1)*(i-1)+j; c2=c1+Nez+1; \n \elenod[[++k]]={c1,c2,c2+1,c1+1}\n ]]; Return[elenod]];\n \n\GenQuad8ElemNodes[Ner_,Nez_]:= Module[\n {k,i,j,c1,c2,c3,numele,elenod}, \numele=Ner*Nez; \n elenod=Table[{0,0,0,0,0,0,0,0},{numele}]; k=0;\n For \[i=1,i<=Ner,i++, For [j=1,j<=Nez,j++,\n c1=(3*Nez+2)*(i-1)+2*j-1; \n \ c3=c1+2*Nez-j+2; c2=c3+Nez+j;\n \elenod[[++k]]={c1,c2,c2+2,c1+2,c3,c2+1,c3+1,c1+1}\n ]]; \Return[elenod]]; \n ", StyleBox[" \n(* Ner=2;Nez=2; rzcorn={{0,0},{1,0},{1,1},{0,1}};\n\

ElemNodes=GenQuad4ElemNodes[Ner,Nez];\n\NodeCoordinates=GenQuad4NodeCoordinates[rzcorn,Ner,Nez];\n\Plot2DElementsAndNodes[NodeCoordinates,ElemNodes,1,\n \"Quad4 \mesh\",True,True];\nElemNodes=GenQuad8ElemNodes[Ner,Nez];\n\NodeCoordinates=GenQuad8NodeCoordinates[rzcorn,Ner,Nez];\n\Plot2DElementsAndNodes[NodeCoordinates,ElemNodes,1,\n \"Quad8 \mesh\",True,True]; *)", FontColor->RGBColor[0, 0, 1]]}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["Cell 9C. Stress contour plot driver", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[TextData[{ "ContourPlotStresses[nodrzc_,elenod_,nodsig_,whichones_,\n \showrange_,plotopt_,legend_,aspect_]:=Module[\n {sigeps=10.^(-3),nbands=10,\\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz,n,\n \numnod=Length[nodrzc],\n options={True,True,True,False,True,True},\n \\[Sigma]rrmax,\[Sigma]zzmax,\[Sigma]\[Theta]\[Theta]max,\[Sigma]rzmax,\[Sigma]\rrmin,\[Sigma]zzmin,\[Sigma]\[Theta]\[Theta]min,\[Sigma]rzmin,\n \\[Sigma]rrinc,\[Sigma]zzinc,\[Sigma]\[Theta]\[Theta]inc,\[Sigma]rzinc,plotrr,\plotzz,plot\[Theta]\[Theta],plotrz},\n \\[Sigma]rr=Table[nodsig[[n,1]],{n,numnod}]; \n \\[Sigma]zz=Table[nodsig[[n,2]],{n,numnod}];\n \\[Sigma]\[Theta]\[Theta]=Table[nodsig[[n,3]],{n,numnod}];\n \\[Sigma]rz=Table[nodsig[[n,4]],{n,numnod}];\n {\[Sigma]rrmax,\[Sigma]zzmax,\\[Sigma]\[Theta]\[Theta]max,\[Sigma]rzmax}=Abs[{Max[\[Sigma]rr],\n Max[\\[Sigma]zz],Max[\[Sigma]\[Theta]\[Theta]],Max[\[Sigma]rz]}]+sigeps; \n {\\[Sigma]rrmin,\[Sigma]zzmin,\[Sigma]\[Theta]\[Theta]min,\[Sigma]rzmin}=Abs[{\Min[\[Sigma]rr],\n Min[\[Sigma]zz],Min[\[Sigma]\[Theta]\[Theta]],Min[\\[Sigma]rz]}]+sigeps;\n \[Sigma]rrmax=Max[\[Sigma]rrmax,\[Sigma]rrmin]; \\[Sigma]rrmin=-\[Sigma]rrmax;\n \\[Sigma]zzmax=Max[\[Sigma]zzmax,\[Sigma]zzmin]; \\[Sigma]zzmin=-\[Sigma]zzmax;\n \[Sigma]\[Theta]\[Theta]max=Max[\[Sigma]\\[Theta]\[Theta]max,\[Sigma]\[Theta]\[Theta]min]; \\[Sigma]\[Theta]\[Theta]min=-\[Sigma]\[Theta]\[Theta]max;\n \

\[Sigma]rzmax=Max[\[Sigma]rzmax,\[Sigma]rzmin]; \\[Sigma]rzmin=-\[Sigma]rzmax;\n \{\[Sigma]rrinc,\[Sigma]zzinc,\[Sigma]\[Theta]\[Theta]inc,\[Sigma]rzinc}={\\[Sigma]rrmax-\[Sigma]rrmin,\[Sigma]zzmax-\[Sigma]zzmin,\n \[Sigma]\\[Theta]\[Theta]max-\[Sigma]\[Theta]\[Theta]min,\[Sigma]rzmax-\[Sigma]rzmin}/\nbands;\n {plotrr,plotzz,plot\[Theta]\[Theta],plotrz}=whichones;\n If \[Length[plotopt]>0,options=plotopt];\n If [plotrr, If [showrange, Print[\"\\[Sigma]rr min,max,inc=\",\n {\[Sigma]rrmin,\[Sigma]rrmax,\\[Sigma]rrinc}]];\n ContourBandPlotNodeFuncOver2DMesh[nodrzc,\n \elenod,\[Sigma]rr,{\[Sigma]rrmin,\[Sigma]rrmax,\[Sigma]rrinc},options,\n \ legend,aspect,\"Radial stress \[Sigma]rr\"]];\n If [plotzz, If [showrange, \Print[\"\[Sigma]zz min,max,inc=\",\n \{\[Sigma]zzmin,\[Sigma]zzmax,\[Sigma]zzinc}]];\n \ContourBandPlotNodeFuncOver2DMesh[nodrzc,\n \elenod,\[Sigma]zz,{\[Sigma]zzmin,\[Sigma]zzmax,\[Sigma]zzinc},options,\n \ legend,aspect,\"Axial stress \[Sigma]zz\"]];\n If [plot\[Theta]\[Theta], If \[showrange, Print[\"\[Sigma]\[Theta]\[Theta] min,max,inc=\",\n \ {\[Sigma]\[Theta]\[Theta]min,\[Sigma]\[Theta]\[Theta]max,\[Sigma]\[Theta]\\[Theta]inc}]];\n ContourBandPlotNodeFuncOver2DMesh[nodrzc,\n \elenod,\[Sigma]\[Theta]\[Theta],{\[Sigma]\[Theta]\[Theta]min,\[Sigma]\[Theta]\\[Theta]max,\[Sigma]\[Theta]\[Theta]inc},options,\n \legend,aspect,\"Hoop stress \[Sigma]\[Theta]\[Theta]\"]];\n If [plotrz, If \[showrange, Print[\"\[Sigma]rz min,max,inc=\",\n \{\[Sigma]rzmin,\[Sigma]rzmax,\[Sigma]rzinc}]];\n \ContourBandPlotNodeFuncOver2DMesh[nodrzc,\n \elenod,\[Sigma]rz,{\[Sigma]rzmin,\[Sigma]rzmax,\[Sigma]rzinc},options,\n \ legend,aspect,\"Shear stress \[Sigma]rz\"]]; \n \ClearAll[\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz]; \Return[]];", StyleBox[" ", FontColor->RGBColor[0, 0, 1]]}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 9D: Analytical (exact) solutions for some benchmark problems, \also

modules to extract tables and do FEM vs exact plots. Source: Timoshenko'sTheory of Elasticity Ch 2 and Timoshenko's Theory of Plates and Shells Ch \4.\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[TextData[{ "PressThickCylinder[{a_,b_},{Em_,\[Nu]_},p_,r_,numer_]:=\n Module[{ur,uz,\\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz,err,ezz,e\[Theta]\\[Theta],grz,ueg}, \n \[Sigma]rr=-p*a^2(b^2/r^2-1)/(b^2-a^2);\n \\[Sigma]\[Theta]\[Theta]= p*a^2(b^2/r^2+1)/(b^2-a^2);\n \\[Sigma]zz=2*\[Nu]*a^2*p/(b^2-a^2); \[Sigma]rz=0; ezz=grz=0;\n err= \a^2*p*(1+\[Nu])*(b^2-r^2*(1-2*\[Nu]))/((a^2-b^2)*Em*r^2); \n e\[Theta]\\[Theta]=-a^2*p*(1+\[Nu])*(b^2+r^2*(1-2*\[Nu]))/((a^2-b^2)*Em*r^2); \n \ur=((1+\[Nu])/Em)*a^2*p/(b^2-a^2)((1-2*\[Nu])*r +b^2/r); uz=0;\n \ueg={ur,uz,\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz};\n \If [!numer, ueg=Simplify[ueg]]; If [numer, ueg=N[ueg]]; \n \Return[ueg]];", StyleBox[" \n \n", FontColor->RGBColor[0, 0, 1]], "RotatingThinDisk[{a_,b_,h_},{Em_,\[Nu]_,\[Rho]_},\[Omega]_,{r_,z_},numer_]:\=\n Module[{ur,uz,\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz,\err,ezz,e\[Theta]\[Theta],grz,ueg}, \n \ur=((a^2*(3+\[Nu])*(-(r^2*(-1+\[Nu]))+b^2*(1+\[Nu]))-r^2*(-1+\[Nu])*\n \ (-(r^2*(1+\[Nu]))+b^2*(3+\[Nu])))*\[Rho]*\[Omega]^2)/(8*Em*r);\n uz=\\[Rho]*\[Omega]^2*z*\[Nu]*(1+\[Nu])*(2*\[Nu]-1)*(-2*r^2*(1+\[Nu])+a^2*(3+\[Nu]\)+\n b^2*(3+\[Nu]))/(4*Em*(1+3*\[Nu]));\n \[Sigma]rr=\[Rho]*\\[Omega]^2*(3+\[Nu])/8*(b^2+a^2-a^2*b^2/r^2-r^2);\n \[Sigma]\[Theta]\\[Theta]=\[Rho]*\[Omega]^2*(3+\[Nu])/8*(b^2+a^2+a^2*b^2/r^2-(1+3*\[Nu])*r^2/(\3+\[Nu]));\n \[Sigma]zz=\[Sigma]rz=0;\n \ueg={ur,uz,\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz};\n \If [!numer, ueg=Simplify[ueg]]; If [numer, ueg=N[ueg]]; \n \Return[ueg]];", StyleBox[" \n\n", FontColor->RGBColor[0, 0, 1]], "PointLoadCircPlate[{R_,h_},{Em_,\[Nu]_},P_,{rr_,z_},numer_]:=\n \Module[{Dm,Mrr,M\[Theta]\[Theta],c,ur,uz,\[Sigma]rr,\[Sigma]zz,\[Sigma]\\[Theta]\[Theta],\[Sigma]rz,duzdr,r,usig},", StyleBox["\n",

FontColor->RGBColor[1, 0, 0]], " r=Max[R/1000,rr];", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "If [Head[rr]===Symbol,r=rr];\n Dm=Em*h^3/(12*(1-\[Nu]^2)); \c=(3+\[Nu])/(1+\[Nu]);\n uz=P/(16*Pi*Dm)*(c*(R^2-r^2)+2*r^2*Log[r/R]);\n \ ur=\[InvisibleSpace](P*r*z*(c-1-2*Log[r/R]))/(8*Dm*Pi);\n If [rr==0, \uz=c*P*R^2/(16*Pi*Dm); ur=0];\n Mrr=-(P/(4*Pi))*(1+\[Nu])*Log[R/r];\n M\\[Theta]\[Theta]=-(P/(4*Pi))*((1+\[Nu])*Log[R/r]+1-\[Nu]);\n \\[Sigma]rr=-12*Mrr*z/h^3; \\[Sigma]\[Theta]\[Theta]=-12*M\[Theta]\[Theta]*z/h^3; \\[Sigma]zz=\[Sigma]rz=0;\n usig=Simplify[{ur,uz,\[Sigma]rr,\[Sigma]zz,\\[Sigma]\[Theta]\[Theta],\[Sigma]rz}];\n If [!numer, usig=Simplify[usig]]; \If [numer,usig=N[usig]];\n Return[usig]];", StyleBox[" ", FontColor->RGBColor[0, 0, 1]], "\n", StyleBox["\n", FontColor->RGBColor[0, 0, 1]], "ExactSolution[nodrzc_,dimen_,{Em_,\[Nu]_,\[Rho]_},load_,\n \solution_,numer_]:=Module[{n,r,z,p,P,R,\n ur,uz,\[Sigma]rr,\[Sigma]zz,\\[Sigma]\[Theta]\[Theta],\[Sigma]rz,numnod=Length[nodrzc],\n \exactnoddis,exactnodsig,modname=\"ExactSolution: \"},\n \exactnoddis=Table[{0,0},{numnod}];\n exactnodsig=Table[{0,0,0,0},{numnod}]; \\n If [solution==\"PressThickCylinder\", p=load; {a,b}=dimen;\n For \[n=1,n<=numnod,n++, {r,z}=nodrzc[[n]];\n \{ur,uz,\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz}=\n \PressThickCylinder[{a,b},{Em,\[Nu]},p,r,numer];\n \exactnoddis[[n]]={ur,uz};\n exactnodsig[[n]]={\[Sigma]rr,\[Sigma]zz,\\[Sigma]\[Theta]\[Theta],\[Sigma]rz}];\n \Return[{exactnoddis,exactnodsig}]];\n If [solution==\"RotatingThinDisk\", \\[Omega]=load; {a,b,h}=dimen;\n For [n=1,n<=numnod,n++, \{r,z}=nodrzc[[n]];\n {ur,uz,\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\\[Theta],\[Sigma]rz}=\n RotatingThinDisk[{a,b,h},{Em,\[Nu],\[Rho]},\\[Omega],{r,z},numer];\n exactnoddis[[n]]={ur,uz};\n \exactnodsig[[n]]={\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz}];\\n Return[{exactnoddis,exactnodsig}]];\n If \[solution==\"PointLoadCircPlate\", P=load; {R,h}=dimen;\n For \[n=1,n<=numnod,n++, {r,z}=nodrzc[[n]];\n \{ur,uz,\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz}=\n \PointLoadCircPlate[{R,h},{Em,\[Nu]},-P,{r,z},numer];\n \exactnoddis[[n]]={ur,uz};\n exactnodsig[[n]]={\[Sigma]rr,\[Sigma]zz,\

\[Sigma]\[Theta]\[Theta],\[Sigma]rz}];\n \Return[{exactnoddis,exactnodsig}]]; \n Print [modname,solution,\" not \available\"];\n Return[{exactnoddis,exactnodsig}]]; "}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 9E. Driver to get FEM result vs exact solution r-plots for \benchmark problems of Chapter 13.\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[TextData[{ "FillUpFEMResultPlotArray[etype_,nodrzc_,noddis_,nodsig_,\n \Ner_,Nez_,iwhat_,ilayer_]:=Module[{n,nn,n1,n2,n3,n4,\n \er,ez,k,ir,nr,cn,ncn,ntab,rtab,vtab,vFEM,m=ilayer,\n \numnod=Length[nodrzc],num9nod},\n", StyleBox[" (*Print[\"enter FillUpFEMResultPlotArray\"]; \Print[\"numnod=\",numnod];*)", FontColor->RGBColor[1, 0, 0]], "\n If [etype==\"Quad4\",\n nr=Ner+1; \ntab=Table[(Nez+1)*n-Nez+m,{n,Ner+1}]];\n If [etype==\"Quad8\", \num9nod=(2*Ner+1)*(2*Nez+1);\n nr=2*Ner+1; \ntab=Table[(2*Nez+1)*(ir-1)+1+m,{ir,1,nr}];\n \cn=Table[2*Nez+3+2*(ez-1)+(4*Nez+2)*(er-1),{er,Ner},{ez,Nez}];\n ", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], " cn=Flatten[cn];", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "ncn=Table[0,{num9nod}]; k=0;\n For [n=1,n<=num9nod,n++, If \[MemberQ[cn,n],k++];\n ncn[[n]]=k];", StyleBox["\n (*Print[\"cn=\",cn]; Print[\"ncn=\",ncn];", FontColor->RGBColor[1, 0, 0]], "\n ", StyleBox["Print[\"original ntab=\",ntab];*)", FontColor->RGBColor[1, 0, 0]], "\n For [n=1,n<=nr,n++, nn=ntab[[n]]; k=ncn[[nn]];\n If \[MemberQ[cn,nn],ntab[[n]]=-nn; Continue[]];\n \ntab[[n]]=ntab[[n]]-k]]; ", StyleBox[" \n (*Print[\"updated ntab=\",ntab];*)\n", FontColor->RGBColor[1, 0, 0]], " If [Mod[m,2]==1, ", StyleBox["\n ", FontColor->RGBColor[1, 0, 0]], "For [n=1,n<=nr,n++, nn=ntab[[n]]; If [nn>0, Continue[]]; \n \

nn=Abs[nn]; ntab[[n]]={ntab[[n-1]],ntab[[n+1]],\n \nn-1-ncn[[nn-1]],nn+1-ncn[[nn+1]]}; ", StyleBox["\n ", FontColor->RGBColor[1, 0, 0]], " ]]; \n vFEM=Table[0,{nr}]; \n rtab=Table[nodrzc[[n,1]],{n,numnod}];\n \ If [iwhat<=2, vtab=Table[noddis[[n,iwhat]], {n,numnod}]];\n If \[iwhat>=3, vtab=Table[nodsig[[n,iwhat-2]],{n,numnod}]];\n", StyleBox[" (*Print[\"rtab=\",rtab]; Print[\"vtab=\",vtab];*)\n", FontColor->RGBColor[1, 0, 0]], " For [ir=1,ir<=nr,ir++, n=ntab[[ir]];\n If [Length[n]==2, \{n1,n2}=n; \n vFEM[[ir]]={rtab[[n1]]+rtab[[n2]],\n \vtab[[n1]]+vtab[[n2]]}/2; Continue[]];\n If [Length[n]==4, \{n1,n2,n3,n4}=n; \n vFEM[[ir]]={rtab[[n1]]+rtab[[n2]]+\n \ rtab[[n3]]+rtab[[n4]],vtab[[n1]]+vtab[[n2]]+\n \vtab[[n3]]+vtab[[n4]]}/4; Continue[]]; \n \vFEM[[ir]]={rtab[[n]],vtab[[n]]}; \n ]; \ClearAll[cn,ncn,ntab,rtab,vtab];\n", StyleBox[" (*Print[\"vFEM=\",vFEM//MatrixForm];*)", FontColor->RGBColor[1, 0, 0]], "\n Return[vFEM]];\n", StyleBox[" ", FontColor->RGBColor[0, 0, 1]], " \nRadialPlotFEMvsExact[etype_,nodrzc_,noddis_,nodsig_,\n dimen_,{Em_,\\[Nu]_,\[Rho]_},load_,Nerz_,prob_,what_,ilayer_,\n \numer_]:=Module[{n,i,nr,plotwhat,rbeg,rend,iwhat,r,z,\n Ner,Nez=1,ur,uz,\\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz,Q4layer,Q8layer,\n \ a,b,h,hh,R,p,P,\[Omega],vFEM,vexact,probOK,whatOK,name,\n \zlayer,modname=\"RadialPlotFEMvsExact: \"}, \n ClearAll[r,z];\n \probOK={\"PressThickCylinder\",\"PointLoadCircPlate\",\n \\"RotatingThinDisk\"};\n \whatOK={\"ur\",\"uz\",\"\[Sigma]rr\",\"\[Sigma]zz\",\"\[Sigma]\[Theta]\[Theta]\\",\"\[Sigma]rz\"};\n Q4zlayer=(hh/2)*{{-1,1},{-1,0,1},{-1,-1/3,1/3,1},\n \ {-1,-1/2,0,1/2,1},{-1,-3/5,-1/5,1/5,3/5,1},\n \{-1,-2/3,-1/3,0,1/3,2/3,1}};\n Q8zlayer=(hh/2)*{{-1,0,1},{-1,-1/2,0,1/2,1},\\n {-1,-2/3,-1/3,0,1/3,2/3,1}}; \n Ner=Nerz; If \[Length[Nerz]==2,{Ner,Nez}=Nerz];\n If [!MemberQ[probOK,prob], \Print[modname,\n \"exact sol of problem \",prob,\n \" not \implemented\"]; Return[]];\n If [!MemberQ[whatOK,what], Print[modname,\n \ \"plot of component \",what,\n \" not implemented\"]; Return[]]; \n \ If [prob==\"PressThickCylinder\", p=load; \n {a,b}=Take[dimen,2]; \{rbeg,rend}={a,b};\n {ur,uz,\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\\[Theta],\[Sigma]rz}=PressThickCylinder[\n \{a,b},{Em,\[Nu]},p,r,numer];\n title=\"Pressurized thick cyl: \"];\n \If [prob==\"RotatingThinDisk\", \[Omega]=load; \n {a,b,h}=dimen; \

{rbeg,rend}={a,b};\n {ur,uz,\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\\[Theta],\[Sigma]rz}= RotatingThinDisk[\n {a,b,h},{Em,\[Nu],\\[Rho]},\[Omega],{r,z},numer];\n title=\"Rotating Thin Disk: \"];\n \If [prob==\"PointLoadCircPlate\", P=load; \n {R,h}=Take[dimen,2]; \{rbeg,rend}={R/1000,R};\n \{ur,uz,\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz}=\PointLoadCircPlate[\n \{R,h},{Em,\[Nu]},-P,{r,z},numer];\n If [etype==\"Quad4\",\n \zlayer=Q4zlayer[[Nez,ilayer+1]]/.hh->h];\n If [etype==\"Quad8\",\n \ zlayer=Q8zlayer[[Nez,ilayer+1]]/.hh->h];\n", StyleBox[" (*Print[\"ilayer=\",ilayer]; Print[\"zlayer=\",zlayer];*) \ ", FontColor->RGBColor[1, 0, 0]], "\n \{ur,\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz}={ur,\[Sigma]rr,\\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz}/.z->zlayer;\n \title=\"Point loaded circ plate: \"];\n {{iwhat}}=Position[whatOK,what];", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "name=whatOK[[iwhat]];\n If [iwhat<=2, plotwhat=\"disp \"<>name;\n \vexact={ur,uz}[[iwhat]] ]; \n If [iwhat>=3, plotwhat=\"stress \"<>name;\n \ vexact={\[Sigma]rr,\[Sigma]zz,\[Sigma]\[Theta]\[Theta],\[Sigma]rz}[[\iwhat-2]] ]; \n vFEM=FillUpFEMResultPlotArray[etype,nodrzc,noddis,nodsig,\n\ Ner,Nez,iwhat,ilayer]; \n If [numer, \{vFEM,rbeg,rend}=N[{vFEM,rbeg,rend}]];\n", StyleBox[" (*Print[\"vFEM=\",Transpose[vFEM]//MatrixForm];\n \Print[\"name=\",name,\" exact @ rbeg=\",N[vexact/.r->rbeg]];*)", FontColor->RGBColor[1, 0, 0]], "\n pFEM=ListPlot[vFEM,PlotJoined->True,PlotStyle->{RGBColor[1,0,0]},\n \ DisplayFunction->Identity];\n \pexact=Plot[vexact,{r,rbeg,rend},PlotStyle->{RGBColor[0,0,0]},\n \DisplayFunction->Identity];\n Show[Graphics[AbsoluteThickness[2]],pexact, \\n pFEM,TextStyle->{FontFamily->\"Times\",FontSize->9},\n \GridLines->Automatic,Axes->True,PlotRange->All,\n \PlotLabel->title<>plotwhat<>\" (black=exact,red=FEM)\",\n \AspectRatio->1/GoldenRatio,\n Frame->True, \DisplayFunction->DisplayChannel]; \n ClearAll[vFEM]; Return[]]; \n \n", StyleBox["(*etype=\"Quad8\"; Ner=2; Nez=2; iwhat=5; zlayer=3;\n\nodrzc=Table[{r[i],z[i]},{i,1,25}];\nnoddis=Table[{ur[i],uz[i]},{i,1,25}];\n\nodsig=Table[{\[Sigma]rr[i],\[Sigma]zz[i],\[Sigma]\[Theta]\[Theta][i],\[Sigma]\

rz[i]},{i,1,25}];\nvFEM=FillUpFEMResultPlotArray[etype,nodrzc,noddis,nodsig,\n\ Ner,Nez,zlayer,iwhat];\n\Print[\"vFEM=\",Transpose[vFEM]//MatrixForm];*)", FontColor->RGBColor[0, 0, 1]], " "}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 10: Driver for static analysis of axisymmetric solid problem.\\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[TextData[{ "RingAnalysisDriver[nodrzc_,eletyp_,elenod_,elemat_,elebfor_,\n \eletfor_,doftag_,dofval_,defopt_]:= Module[{n,ns,j,\n \numnod=Length[nodrzc],numele=Length[elenod],eps=10.^(-7),\n \f,fb,ft,K,Kmod,u,supdof,supval,noddis,nodfor,nodsig,\n \modname=\"RingAnalysisDriver: \"}, \n noddis=nodfor=nodsig={}; \fb=ft=Table[0,{2*numnod}];\n \K=RingMasterStiffness[nodrzc,eletyp,elenod,elemat,defopt];\n", StyleBox[" (*Print[\"Master Stiff K:\\n\",K//MatrixForm];\n Print[\"eigs \of K=\",Chop[Eigenvalues[N[K]]]];\n Print[\"doftag=\",doftag,\" \dofval=\",dofval]; *)\n ", FontColor->RGBColor[1, 0, 0]], "ns=0; Do [Do [If[doftag[[n,j]]>0,ns++],{j,1,2}],{n,1,numnod}];\n \supdof=supval=Table[0,{ns}];\n k=0; Do [Do \[If[doftag[[n,j]]>0,k++;supdof[[k]]=2*(n-1)+j;\n \supval[[k]]=dofval[[n,j]]],{j,1,2}],{n,numnod}];\n", StyleBox[" (*Print[\"supdof=\",supdof];*)\n ", FontColor->RGBColor[1, 0, 0]], "If [Length[elebfor]>0, ", StyleBox["\n ", FontColor->RGBColor[1, 0, 0]], "fb=RingBodyForces[nodrzc,eletyp,elenod,elebfor,defopt]];\n If \[Length[eletfor]>0, Print[modname,", StyleBox["\n ", FontColor->RGBColor[1, 0, 0]], "\" surface traction force lumping unavailable ", "- ", "sorry\"]];\n f=ModifiedNodeForces[supdof,supval,K,Flatten[dofval],fb+ft]; \ \n Kmod=ModifiedMasterStiffness[supdof,K];\n", StyleBox[" (*Print[\"Force vector f=\",f];*)\n (* Print[\"eigs of \Kmod=\",Eigenvalues[N[Kmod]]];*)",

FontColor->RGBColor[1, 0, 0]], "\n u=LinearSolve[Kmod,f]; u=Chop[u,eps]; noddis=Partition[u,2];\n \f=Simplify[K.u]; f=Chop[f,eps]; nodfor=Partition[f,2];\n \nodsig=RingNodalStresses[nodrzc,eletyp,elenod,elemat,\n noddis,defopt]; \nodsig=Chop[nodsig,eps]; \n ClearAll[K,Kmod,u,f,fb,ft]; \n \Return[{noddis,nodfor,nodsig}]; \n ]; "}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\ Cell 11: Pressurized thick cylinder with 2 Quad4 elements in theradial direction. Cell used for the driver script example of Chapter \13.\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, InitializationCell->True, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell["\<\ClearAll[Em,\[Nu],th,a,b,d,p,numer]; Em=1000.; \[Nu]=0; etype=\"Quad4\"; numer=True;Kfac=1; a=4; b=10; d=2; p=10; aspect=d/(b-a);

(* Define FEM model *)

NodeCoordinates=N[{{a,0},{a,d},{(a+b)/2,0}, {(a+b)/2,d},{b,0},{b,d}}]; ElemNodes={{1,3,4,2},{3,5,6,4}};numnod=Length[NodeCoordinates]; numele=Length[ElemNodes]; ElemType=Table[etype,{numele}]; ElemMaterial=Table[N[{Em,\[Nu]}],{numele}]; FreedomValues=Table[{0,0},{numnod}];FreedomTags=Table[{0,1},{numnod}]; pfor=N[Kfac*p*a*d];FreedomValues[[1]]=FreedomValues[[2]]={pfor/2,0}; ElemBodyForces={}; ElemTractionForces={}; DefaultOptions={numer};PrintRingNodeCoordinates[NodeCoordinates,\"Node Coordinates\",{2,4}];PrintRingElementTypeNodes[ElemType,ElemNodes,\"Element type & nodes\",{}];PrintRingElementMaterials[ElemMaterial,\"Material data\",{}]; PrintRingElemBodyForces[ElemBodyForces,\"Body forces\",{}];PrintRingElemTracForces[ElemTractionForces,\"Traction forces\",{}];PrintRingFreedomActivity[FreedomTags,FreedomValues,\"BC data\",{}];Plot2DElementsAndNodes[NodeCoordinates,ElemNodes,aspect, \"Pressurized thick cylinder mesh\",True,True];

(* Solve problem and print results *)

{NodeDisplacements,NodeForces,NodeStresses}=RingAnalysisDriver[ NodeCoordinates,ElemType,ElemNodes, ElemMaterial,ElemBodyForces,ElemTractionForces, FreedomTags,FreedomValues,DefaultOptions];PrintRingAnalysisSolution[NodeDisplacements,NodeForces, NodeStresses,\"Computed solution\",{6,6}]; {ExactNodeDisplacements,ExactNodeStresses}= ExactSolution[NodeCoordinates,{a,b},{Em,\[Nu],\[Rho]},p, \"PressThickCylinder\",numer];PrintRingNodeDispStresses[ExactNodeDisplacements, ExactNodeStresses,\"Exact (Lame) solution\",{}]; (* Plot Node Stress Distributions *)

legend={(a+2*b)/3,0.8*d};ContourPlotStresses[NodeCoordinates,ElemNodes,NodeStresses, {True,False,True,False},True,{},legend,aspect]; (* Through-wall plots comparing FEM vs exact solutions *)

pwhat={\"ur\",\"\[Sigma]rr\",\"\[Sigma]zz\",\"\[Sigma]\[Theta]\[Theta]\"}; If \[\[Nu]==0,pwhat={\"ur\",\"\[Sigma]rr\",\"\[Sigma]\[Theta]\[Theta]\"}];For [ip=1,ip<=Length[pwhat],ip++, what=pwhat[[ip]]; RadialPlotFEMvsExact[etype,NodeCoordinates,NodeDisplacements, NodeStresses, {a,b},{Em,\[Nu],\[Rho]},p,{2,1}, \"PressThickCylinder\",what,0,numer] ];\\>", "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 12: Pressurized thick cylinder with Quad4 elems: Ner >=1 \elements in radial direction, Nez=1 in the axial direction. a,b inner & outer radii, d \slice thickness\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell["\<\ClearAll[Em,\[Nu],th,a,b,d,p,Ner,Nez,numer]; Em=1000.; \[Nu]=0; etype=\"Quad4\"; numer=True;Kfac=1; a=4; b=10; d=2; p=10; Ner=4; Nez=1; aspect=d/(b-a);

(* Define FEM model using mesh generation *)

CornerNodeCoordinates=N[{{a,0},{b,0},{b,d},{a,d}}]; NodeCoordinates= GenQuad4NodeCoordinates[CornerNodeCoordinates,Ner,Nez];

ElemNodes=GenQuad4ElemNodes[Ner,Nez];numnod=Length[NodeCoordinates]; numele=Length[ElemNodes]; ElemType=Table[etype,{numele}]; ElemMaterial=Table[N[{Em,\[Nu]}],{numele}]; FreedomValues=Table[{0,0},{numnod}];FreedomTags=Table[{0,1},{numnod}]; pfor=N[Kfac*p*a*d];FreedomValues[[1]]=FreedomValues[[2]]={pfor/2,0}; ElemBodyForces={}; ElemTractionForces={}; DefaultOptions={numer};PrintRingNodeCoordinates[NodeCoordinates,\"Node Coordinates\",{2,4}];PrintRingElementTypeNodes[ElemType,ElemNodes,\"Element type & nodes\",{}];PrintRingElementMaterials[ElemMaterial,\"Material data\",{}]; PrintRingElemBodyForces[ElemBodyForces,\"Body forces\",{}];PrintRingElemTracForces[ElemTractionForces,\"Traction forces\",{}];PrintRingFreedomActivity[FreedomTags,FreedomValues,\"BC data\",{}];

Plot2DElementsAndNodes[NodeCoordinates,ElemNodes,aspect, \"Pressurized thick cylinder Quad4 mesh\",True,True];

(* Solve problem and print results *)

{NodeDisplacements,NodeForces,NodeStresses}=RingAnalysisDriver[ NodeCoordinates,ElemType,ElemNodes, ElemMaterial,ElemBodyForces,ElemTractionForces, FreedomTags,FreedomValues,DefaultOptions];PrintRingAnalysisSolution[NodeDisplacements,NodeForces, NodeStresses,\"Computed solution\",{6,6}]; {ExactNodeDisplacements,ExactNodeStresses}= ExactSolution[NodeCoordinates,{a,b},{Em,\[Nu],\[Rho]},p, \"PressThickCylinder\",numer];PrintRingNodeDispStresses[ExactNodeDisplacements, ExactNodeStresses,\"Exact (Lame) solution\",{}]; (* Contourplot Node Stress Distributions *)

legend={(a+b)/2,0.75*d}; whichones={True,True,True,False};If [\[Nu]==0, whichones={True,False,True,False}]; ContourPlotStresses[NodeCoordinates,ElemNodes,NodeStresses, whichones,True,{},legend,aspect]; (* Through-wall plots comparing FEM vs exact solutions *)

pwhat={\"ur\",\"\[Sigma]rr\",\"\[Sigma]zz\",\"\[Sigma]\[Theta]\[Theta]\"}; If \[\[Nu]==0,pwhat={\"ur\",\"\[Sigma]rr\",\"\[Sigma]\[Theta]\[Theta]\"}];For [ip=1,ip<=Length[pwhat],ip++, what=pwhat[[ip]]; RadialPlotFEMvsExact[etype,NodeCoordinates,NodeDisplacements, NodeStresses,{a,b},{Em,\[Nu],\[Rho]},p,{Ner,Nez}, \"PressThickCylinder\",what,0,numer] ];\\>", "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 13: Pressurized thick cylinder accepting both Quad4 or Quad8 \elems; Ner >=1 elements in radial direction, Nez (assumed 1) in the axial direction. a,b inner & outer \radii, d slice thickness\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[TextData[{ "ClearAll[Em,\[Nu],th,a,b,d,p,Ner,Nez]; \nEm=1000.; \[Nu]=0.0; \etype=\"Quad4\"; numer=True; Ner=4; Nez=1;\nKfac=1; a=4; b=10; d=2; \aspect=d/(b-a); p=10; \n\n(* Define FEM model *)\n\n\MeshCorners=N[{{a,0},{b,0},{b,d},{a,d}}]; \nIf [etype==\"Quad4\",\n \NodeCoordinates=GenQuad4NodeCoordinates[MeshCorners,Ner,Nez]; \n \ElemNodes=GenQuad4ElemNodes[Ner,Nez]];\nIf [etype==\"Quad8\",\n \NodeCoordinates=GenQuad8NodeCoordinates[MeshCorners,Ner,Nez]; \n \ElemNodes=GenQuad8ElemNodes[Ner,Nez]];\nnumnod=Length[NodeCoordinates]; \numele=Length[ElemNodes]; \nElemType=Table[etype,{numele}]; \n\ElemMaterial=Table[N[{Em,\[Nu]}],{numele}]; \n\FreedomTags=Table[{0,1},{numnod}];\nFreedomValues=Table[{0,0},{numnod}]; \pfor=N[Kfac*p*a*d];\nIf [etype==\"Quad4\",\n \FreedomValues[[1]]=FreedomValues[[2]]={pfor/2,0}]; \nIf [etype==\"Quad8\",\n \ FreedomValues[[1]]=FreedomValues[[3]]={pfor/6,0};\n FreedomValues[[2]] \={2*pfor/3,0}]; \nElemBodyForces= ElemTractionForces={}; \DefaultOptions={numer};\n(* Problem data print statements removed *)\n\Plot2DElementsAndNodes[NodeCoordinates,ElemNodes,aspect,\n \"Pressurized \thick cylinder\",True,True]; \n \n(* Solve problem and print results *)\\n\n{NodeDisplacements,NodeForces,NodeStresses}=RingAnalysisDriver[\n \NodeCoordinates,ElemType,ElemNodes,\n \ElemMaterial,ElemBodyForces,ElemTractionForces,\n \FreedomTags,FreedomValues,DefaultOptions];\n\PrintRingAnalysisSolution[NodeDisplacements,NodeForces,\n \NodeStresses,\"Computed solution\",{}]; \n\{ExactNodeDisplacements,ExactNodeStresses}=\n \ExactSolution[NodeCoordinates,{a,b},{Em,\[Nu],\[Rho]},p,\n \\"PressThickCylinder\",numer];\n\PrintRingNodeDispStresses[ExactNodeDisplacements,\n \ExactNodeStresses,\"Exact (Lame) solution\",{}]; \n\n(* Plot Node Stress \Distributions *)\n\nlegend={(a+b)/2,0.75*d}; \whichones={True,True,True,False}; \nIf [\[Nu]==0, \whichones={True,False,True,False}]; \n\ContourPlotStresses[NodeCoordinates,ElemNodes,NodeStresses,\n \

whichones,True,{},legend,aspect];\n \n(* Through-wall plots comparing FEM vs \exact solutions *)\n\npwhat={\"ur\",\"\[Sigma]rr\",\"\[Sigma]zz\",\"\[Sigma]\\[Theta]\[Theta]\"};", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "\nFor [ip=1,ip<=Length[pwhat],ip++, what=pwhat[[ip]];\n \RadialPlotFEMvsExact[etype,NodeCoordinates,NodeDisplacements,\n \NodeStresses,{a,b},{Em,\[Nu],\[Rho]},p,{Ner,Nez},\n \"PressThickCylinder\\",what,1,numer] ]; "}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 14. Benchmark #2: Rotating thin disk. Inner radius a, \exterior radius b, z-thickness h, massdensity \[Rho]. Disk rotates about z at \[Omega] rads/sec. Accepts both Quad4 \and Quad8, just change etype.Body force computed at element corner nodes. Ner arbitrary but Nez=1;\\>", \"Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[TextData[{ "ClearAll[Em,\[Nu],a,b,h,Kfac,\[Rho],\[Omega],Ner,Nez,numer]; \nEm=1000.; \\[Nu]=N[1/3]; Ner=2; Nez=1; etype=\"Quad8\";\nKfac=1; a=4; b=10; h=1; \aspect=h/(b-a); \[Rho]=3.0; \[Omega]=0.5;\nnumer=True;\n\n(* Define FEM \model *)\n\nMeshCorners=N[{{a,0},{b,0},{b,h},{a,h}}];\nIf [etype==\"Quad4\",\n\ NodeCoordinates=GenQuad4NodeCoordinates[MeshCorners,Ner,Nez]; \n \ ElemNodes= GenQuad4ElemNodes[Ner,Nez]];\nIf [etype==\"Quad8\",\n \NodeCoordinates=GenQuad8NodeCoordinates[MeshCorners,Ner,Nez]; \n \ElemNodes= GenQuad8ElemNodes[Ner,Nez]];\nnumnod=Length[NodeCoordinates]; \numele=Length[ElemNodes];\nElemType= Table[etype,{numele}]; \n\ElemMaterial= Table[{Em,\[Nu]},{numele}]; \n\ElemBodyForces=Table[{0,0},{numele}];\nFor [e=1,e<=numele,e++, \enl=ElemNodes[[e]];\n ncoor=Table[NodeCoordinates[[enl[[i]]]],{i,4}];\n \ {{r1,z1},{r2,z2},{r3,z3},{r4,z4}}=ncoor;\n ElemBodyForces[[e]]=\[Rho]*\\[Omega]^2*{{r1,0},{r2,0},{r3,0},{r4,0}}]; ",

StyleBox["\n", FontColor->RGBColor[1, 0, 0]], "FreedomTags=FreedomValues=Table[{0,0},{numnod}];\nIf [etype==\"Quad4\", \n\ For [n=1,n<=numnod-Nez,n=n+Nez+1, FreedomTags[[n]]={0,1}]];\nIf \[etype==\"Quad8\", \n For [n=1,n<=numnod-2*Nez,n=n+3*Nez+2, \FreedomTags[[n+1]]={0,1}]];\nElemTractionForces={}; DefaultOptions={True};\n\PrintRingNodeCoordinates[NodeCoordinates,\"Node Coordinates\",{4,2}];\n\PrintRingElementTypeNodes[ElemType,ElemNodes,\"Element type & nodes\",{}];\n\PrintRingElementMaterials[ElemMaterial,\"Material data\",{}]; \n\PrintRingElemBodyForces[ElemBodyForces,\"Body forces\",{5,3}];\n\PrintRingElemTracForces[ElemTractionForces,\"Traction forces\",{}];\n\PrintRingFreedomActivity[FreedomTags,FreedomValues,\"BC data\",{}];\n\Plot2DElementsAndNodes[NodeCoordinates,ElemNodes,aspect,\n \"Rotating disk \mesh\",True,True]; \n \n(* Solve problem and print results *)\n\n\{NodeDisplacements,NodeForces,NodeStresses}=RingAnalysisDriver[\n \NodeCoordinates,ElemType,ElemNodes,\n \ElemMaterial,ElemBodyForces,ElemTractionForces,\n \FreedomTags,FreedomValues,DefaultOptions];\n\PrintRingAnalysisSolution[NodeDisplacements,NodeForces,\n \NodeStresses,\"Computed solution\",{}]; \n\{ExactNodeDisplacements,ExactNodeStresses}=\n \ExactSolution[NodeCoordinates,{a,b,h},{Em,\[Nu],\[Rho]},\[Omega],\n \\"RotatingThinDisk\",numer];\n\PrintRingNodeDispStresses[ExactNodeDisplacements,\n \ExactNodeStresses,\"Exact solution\",{}]; \n \n(* Plot Node Stress \Distributions *)\n\nlegend={(a+b)/2,0.75*h}; \whichones={True,False,True,False}; \n\ContourPlotStresses[NodeCoordinates,ElemNodes,NodeStresses,\n \whichones,True,{},legend,aspect];\n \n(* Through-wall plots comparing FEM vs \exact solutions *)\n\npwhat={\"ur\",\"\[Sigma]rr\",\"\[Sigma]zz\",\"\[Sigma]\\[Theta]\[Theta]\"};\nFor [ip=1,ip<=Length[pwhat],ip++, what=pwhat[[ip]];\n \ RadialPlotFEMvsExact[etype,NodeCoordinates,NodeDisplacements,\n \NodeStresses,{a,b,h},{Em,\[Nu],\[Rho]},\[Omega],{Ner,Nez},\n \\"RotatingThinDisk\",what,0,numer] ]; "}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 15: Benchmark #3. Thin circular plate under central point \load. Exact results taken from Timoshenko's Theory of Plates and Shells, Chapter 4. Both Quad4 and Quad8 \done by same script; just change etypr. Ner elements along radius, Nez=2 (Quad4) or 1 \(Quad8) \

\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[CellGroupData[{

Cell[TextData[{ "ClearAll[Em,\[Nu],a,b,R,h,Kfac,P,Ner,Nez,numer]; \nEm=1000.; \\[Nu]=N[1/3]; Ner=4; etype=\"Quad4\";\nKfac=1; R=10; h=1; aspect=h/R; P=10.;\\nNez=2; If [etype==\"Quad8\", Nez=1]; numer=True;\n\n(* Define FEM model *)\\n\nMeshCorners=N[{{0,-h/2},{R,-h/2},{R,h/2},{0,h/2}}];\nIf \[etype==\"Quad4\",\n \NodeCoordinates=GenQuad4NodeCoordinates[MeshCorners,Ner,Nez]; \n \ElemNodes= GenQuad4ElemNodes[Ner,Nez]];\nIf [etype==\"Quad8\",\n \NodeCoordinates=GenQuad8NodeCoordinates[MeshCorners,Ner,Nez]; \n \ElemNodes= GenQuad8ElemNodes[Ner,Nez]];\nnumnod=Length[NodeCoordinates]; \numele=Length[ElemNodes];\nElemType= Table[etype,{numele}]; \n\ElemMaterial= Table[{Em,\[Nu]},{numele}]; \n\ElemBodyForces=ElemTractionForces={}; DefaultOptions={True};", StyleBox["\n", FontColor->RGBColor[1, 0, 0]], "FreedomTags=FreedomValues=Table[{0,0},{numnod}];\n\FreedomTags[[1]]=FreedomTags[[2]]=FreedomTags[[3]]={1,0};\n\FreedomTags[[numnod-1]]={0,1}; Ps=N[P*Kfac/(2*Pi)];\nIf [etype==\"Quad4\", \n\ FreedomValues[[1]]=FreedomValues[[3]]={0,-Ps/4}; \n \FreedomValues[[2]]={0,-Ps/2}]; \nIf [etype==\"Quad8\", \n \FreedomValues[[1]]=FreedomValues[[3]]={0,-Ps/6}; \n \FreedomValues[[2]]={0,-2*Ps/3}]; \n\PrintRingNodeCoordinates[NodeCoordinates,\"Node Coordinates\",{4,2}];\n\PrintRingElementTypeNodes[ElemType,ElemNodes,\"Element type & nodes\",{}];\n\PrintRingElementMaterials[ElemMaterial,\"Material data\",{}]; \n\PrintRingElemBodyForces[ElemBodyForces,\"Body forces\",{5,3}];\n\PrintRingElemTracForces[ElemTractionForces,\"Traction forces\",{}];\n\PrintRingFreedomActivity[FreedomTags,FreedomValues,\"BC data\",{}];\n\Plot2DElementsAndNodes[NodeCoordinates,ElemNodes,aspect,\n \"Point loaded \circular plate mesh\",True,True]; \n \n(* Solve problem and print results \*)\n\n{NodeDisplacements,NodeForces,NodeStresses}=RingAnalysisDriver[\n \NodeCoordinates,ElemType,ElemNodes,\n \ElemMaterial,ElemBodyForces,ElemTractionForces,\n \FreedomTags,FreedomValues,DefaultOptions];\n\PrintRingAnalysisSolution[NodeDisplacements,NodeForces,\n \NodeStresses,\"Computed solution\",{}]; \n\{ExactNodeDisplacements,ExactNodeStresses}=\n \ExactSolution[NodeCoordinates,{R,h},{Em,\[Nu],\[Rho]},P,\n \

\"PointLoadCircPlate\",numer];\n\PrintRingNodeDispStresses[ExactNodeDisplacements,\n \ExactNodeStresses,\"Exact solution\",{}]; \n \n(* Plot Node Stress \Distributions *)\n\nwhichones={True,False,True,False}; \n\ContourPlotStresses[NodeCoordinates,ElemNodes,NodeStresses,\n \whichones,True,{},{},aspect];\n \n(* Through-wall plots comparing FEM vs \exact solutions *)\n\npwhat={\"uz\",\"ur\",\"\[Sigma]rr\",\"\[Sigma]zz\",\"\\[Sigma]\[Theta]\[Theta]\"};\nFor [ip=1,ip<=Length[pwhat],ip++, \what=pwhat[[ip]];\n RadialPlotFEMvsExact[etype,NodeCoordinates,\n \NodeDisplacements,NodeStresses, {R,h},{Em,\[Nu],\[Rho]},\n P,{Ner,Nez},\\"PointLoadCircPlate\",what,0,numer] ];\n "}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 16: reserved for benchmark problem\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell["\<\

\\>", "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 17. Concrete pile embebded in elastic soil - very coarse \mesh.\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell["\<\ClearAll[Epile,\[Nu]pile,Esoil,\[Nu]soil,d,H,P,L,Kfac]; Epile=300000.; Esoil=Epile/50; \[Nu]pile=0.1; \[Nu]soil=0.4; P=500000; d=10; L=250; H=1.2*L; Ner=3; Nez=3; R=8*d; Kfac=1; etype=\"Quad4\"; aspect=H/(2*R);

(* Define FEM model *)

MeshCorners=N[{{0,-H},{R,-H},{R,0},{0,0}}];NodeCoordinates=GenGradedQuad4NodeCoordinates[MeshCorners, Ner,Nez,{-1,d/R-1,-0.4,1},{-1,1-2*L/H,0.1,1}]; ElemNodes=GenQuad4ElemNodes[Ner,Nez];numnod=Length[NodeCoordinates]; numele=Length[ElemNodes];ElemType= Table[etype,{numele}]; ElemMaterial= Table[{Esoil,\[Nu]soil},{numele}]; ElemMaterial[[2]]=ElemMaterial[[3]]={Epile,\[Nu]pile}; ElemBodyForces=ElemTractionForces={};FreedomTags= Table[{0,0},{numnod}]; For[n=2,n<=4,n++, FreedomTags[[n]]={1,0}];For[n=14,n<=16,n++, FreedomTags[[n]]={1,1}];For[n=1,n<=13,n=n+4,FreedomTags[[n]]={1,1}];FreedomValues= Table[{0,0},{numnod}];FreedomValues[[4]]={0,N[-P*Kfac/(2*Pi)]}; DefaultOptions={True};PrintRingNodeCoordinates[NodeCoordinates,\"Node Coordinates\",{4,2}];PrintRingElementTypeNodes[ElemType,ElemNodes,\"Element type & nodes\",{}];PrintRingElementMaterials[ElemMaterial,\"Material data\",{}]; PrintRingElemBodyForces[ElemBodyForces,\"Body forces\",{}];PrintRingElemTracForces[ElemTractionForces,\"Traction forces\",{}];PrintRingFreedomActivity[FreedomTags,FreedomValues,\"BC data\",{}];Plot2DElementsAndNodes[NodeCoordinates,ElemNodes,aspect, \"Pile in soil - very coarse mesh\",True,True]; (* Solve problem and print results *)

{NodeDisplacements,NodeForces,NodeStresses}=RingAnalysisDriver[ NodeCoordinates,ElemType,ElemNodes, ElemMaterial,ElemBodyForces,ElemTractionForces, FreedomTags,FreedomValues,DefaultOptions];PrintRingAnalysisSolution[NodeDisplacements,NodeForces, NodeStresses,\"Computed solution\",{6,3}]; (* Contour plot stress distributions *) ContourPlotStresses[NodeCoordinates,ElemNodes,NodeStresses, {True,True,True,True},True,{},{.4*R,-H/4},aspect];\\>", "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell["\<\Cell 18. A rotating sphere with the rough dimensions and \approximate properties of Mother Earth. Kg-force/m/sec unit system. The Earth spins with angular velocity \[Omega]=2\\[Pi] rad/(24hr)=2\[Pi]/86400 (rad/sec) about z. The planet radius is R= 6370Km=6370000m. For E take 1/3 of the rigidity of steel =(21000 Kgf/mm^2)/3=(7 10^9 Kgf/m^2, \

as recommendedin Love's Theory of Elasticity, Poisson's ratio \[Nu]=0.3 (my own guess), and \mass density \[Rho]=5.52 times the water density. Watch out for units: the centrifugal body force \\[Rho]\[Omega]^2r should come up in Kg/m^3. Very coarse mesh: 4 elements, constructed by eyeballing. \\\>", "Text", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[1, 1, 0]],

Cell[CellGroupData[{

Cell["\<\ClearAll[Em,\[Nu],a,b,h,Kfac,\[Rho],\[Gamma],g,\[Omega]]; Em=N[100]; \[Nu]=0; R=1.; \[Gamma]=5.52*1000; g=9.81; P=10000.\[Rho]=\[Gamma]/g; \[Omega]=0*N[2*Pi/(3600*24)]; Kfac=1; aspect=1; \etype=\"Quad4\";Print[\"Actual polar flattening of Earth =\",0.001*R/303,\" Km\"];

(* Define FEM model *)

NodeCoordinates={{0,0},{0.5,0},{1,0},{0,0.5},{0.4,0.4}, {0.916,0.4},{0.71,0.71},{0.4,0.916},{0,1}}*R; ElemNodes={{1,2,5,4},{2,3,6,5},{4,5,8,9},{5,6,7,8}};numnod=Length[NodeCoordinates]; numele=Length[ElemNodes];ElemType= Table[etype,{numele}]; ElemMaterial= Table[{Em,\[Nu]},{numele}]; ElemBodyForces= Table[{0,0},{numele}];FreedomTags= Table[{0,0},{numnod}]; FreedomValues= Table[{0,0},{numnod}]; FreedomTags[[1]]={1,1};FreedomTags[[2]]=FreedomTags[[3]]={0,1};FreedomTags[[4]]=FreedomTags[[9]]={1,0};FreedomValues[[3]]={-P,0};FreedomValues[[9]]={0,-P}/(2*N[Pi]);For [e=1,e<=numele,e++, enl=ElemNodes[[e]]; ncoor=Table[NodeCoordinates[[enl[[i]]]],{i,4}]; {{r1,z1},{r2,z2},{r3,z3},{r4,z4}}=ncoor; ElemBodyForces[[e]]=\[Rho]*\[Omega]^2*{{r1,0},{r2,0},{r3,0},{r4,0}}; ]; DefaultOptions={True};PrintRingNodeCoordinates[NodeCoordinates,\"Node Coordinates\",{2,4}];PrintRingElementTypeNodes[ElemType,ElemNodes,\"Element type & nodes\",{}];PrintRingElementMaterials[ElemMaterial,\"Material data\",{}]; PrintRingElemBodyForces[ElemBodyForces,\"Body forces\",{4,2}];PrintRingElemTracForces[ElemTractionForces,\"Traction forces\",{}];PrintRingFreedomActivity[FreedomTags,FreedomValues,\"BC data\",{}];Plot2DElementsAndNodes[NodeCoordinates,ElemNodes,aspect, \"Spinning Mother Earth - very coarse mesh\",True,True];

(* Solve problem and print results *)

{NodeDisplacements,NodeForces,NodeStresses}=RingAnalysisDriver[ NodeCoordinates,ElemType,ElemNodes, ElemMaterial,ElemBodyForces,ElemTractionForces, FreedomTags,FreedomValues,DefaultOptions];PrintRingAnalysisSolution[0.001*NodeDisplacements, 10.^(-6)*NodeForces, 10.^(-6)*NodeStresses, \"Solution: disp in Km, forces in MN, stresses in MPa\",{}]; (* Contour plot stress distributions *) ContourPlotStresses[NodeCoordinates,ElemNodes, 10.^(-6)*NodeStresses, {True,True,True,True},True,{},{R/3,R/2},aspect]; \\>", "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

Cell[BoxData[ \(10000.`\)], "Output"],

Cell[TextData[{ StyleBox["ClearAll[Em,\[Nu],a,b,h,Kfac,\[Rho],\[Gamma],g,\[Omega]]; \n\Em=10^7; \[Nu]=0.3; R=1.; P=100000;\n aspect=1; etype=\"Quad4\";\n \n(* \Define FEM model *)\n \n\NodeCoordinates={{0,0},{0,1}/5,{Cos[3*Pi/8],Sin[3*Pi/8]}/5,\n\{Cos[Pi/4],Sin[Pi/4]}/5,{Cos[Pi/8],Sin[Pi/8]}/5,{1,0}/5,{0,1}*2/5,\n\{Cos[3*Pi/8],Sin[3*Pi/8]}*2/5,{Cos[Pi/4],Sin[Pi/4]}*2/5,{Cos[Pi/8],Sin[Pi/8]}*\2/5,\n{1,0}*2/5,{0,1}*3/5,{Cos[3*Pi/8],Sin[3*Pi/8]}*3/5,{Cos[Pi/4],Sin[Pi/4]}*\3/5,\n{Cos[Pi/8],Sin[Pi/8]}*3/5,{1,0}*3/5,{0,1}*4/5,{Cos[3*Pi/8],Sin[3*Pi/8]}*\4/5,\n{Cos[Pi/4],Sin[Pi/4]}*4/5,{Cos[Pi/8],Sin[Pi/8]}*4/5,{1,0}*4/5,{0,1},\n\{Cos[3*Pi/8],Sin[3*Pi/8]},{Cos[Pi/4],Sin[Pi/4]},{Cos[Pi/8],Sin[Pi/8]},{1,0}}*\R; \nElemNodes=Table[{0,0,0,0},{n,1,18}];\nElemNodes[[1]]={1,4,3,2};\n\ElemNodes[[2]]={1,6,5,4};\nFor[n=3,n<=6,n++,ElemNodes[[n]]={n-1,n,n+4,n+5}];\n\For[n=7,n<=10,n++,ElemNodes[[n]]={n,n+5,n+6,n+1}];\n\For[n=11,n<=14,n++,ElemNodes[[n]]={n+1,n+6,n+7,n+2}];\n\For[n=15,n<=18,n++,ElemNodes[[n]]={n+2,n+7,n+8,n+3}];\n\numnod=Length[NodeCoordinates]; numele=Length[ElemNodes];\nElemType= \Table[etype,{numele}]; \nElemMaterial= Table[{Em,\[Nu]},{numele}]; \n\ElemBodyForces= Table[{0,0},{numele}];\nElemTractionForces={};\nFreedomTags= \

Table[{0,0},{numnod}]; \nFreedomValues= Table[{0,0},{numnod}]; \n\FreedomValues[[22]]={0,N[-P/(2*Pi)]}; \n\FreedomTags[[1]]={1,1};FreedomTags[[2]]=FreedomTags[[7]]=FreedomTags[[12]]=\FreedomTags[[17]]=FreedomTags[[22]]={1,0};\n\FreedomTags[[6]]=FreedomTags[[11]]=FreedomTags[[16]]=FreedomTags[[21]]=\FreedomTags[[26]]={0,1};\nFor [e=1,e<=numele,e++, enl=ElemNodes[[e]];\n \ncoor=Table[NodeCoordinates[[enl[[i]]]],{i,4}];\n \{{r1,z1},{r2,z2},{r3,z3},{r4,z4}}=ncoor;\n \ElemBodyForces[[e]]={{0,0},{0,0},{0,0},{0,0}};\n ]; \n\DefaultOptions={True};\nPrintRingNodeCoordinates[NodeCoordinates,\"Node \Coordinates\",{2,4}];\nPrintRingElementTypeNodes[ElemType,ElemNodes,\"Element \type & nodes\",{}];\nPrintRingElementMaterials[ElemMaterial,\"Material \data\",{}]; \nPrintRingElemBodyForces[ElemBodyForces,\"Body forces\",{4,2}];\n\PrintRingElemTracForces[ElemTractionForces,\"Traction forces\",{}];\n\PrintRingFreedomActivity[FreedomTags,FreedomValues,\"BC data\",{}];\n\Plot2DElementsAndNodes[NodeCoordinates,ElemNodes,aspect,\n \"Spinning \Mother Earth - very coarse mesh\",True,True];\n \n(* Solve problem and print \results *)\n \n\{NodeDisplacements,NodeForces,NodeStresses}=RingAnalysisDriver[\n \NodeCoordinates,ElemType,ElemNodes,\n \ElemMaterial,ElemBodyForces,ElemTractionForces,\n \FreedomTags,FreedomValues,DefaultOptions];\n\PrintRingAnalysisSolution[NodeDisplacements,\n NodeForces, NodeStresses,\n\ \"Solution: \",{}]; \n \n(* Contour plot stress distributions *)\n \n\ContourPlotStresses[NodeCoordinates,ElemNodes,\n 10.^(-6)*NodeStresses,\n \{True,True,True,True},True,{},{R/3,R/2},aspect]; ", FontSize->14], StyleBox["\n ", FontFamily->"Verdana", FontWeight->"Plain"], "\n \n "}], "Input", CellFrame->True, CellMargins->{{15, 80}, {Inherited, Inherited}}, CellLabelMargins->{{7, Inherited}, {Inherited, Inherited}}, ImageRegion->{{-0, 1}, {0, 1}}, Background->RGBColor[0, 1, 0]],

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