(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 4.2'

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(*NotebookOptionsPosition[    179228,       7962]*)
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Notebook[{
Cell[TextData[{
  StyleBox["Here are the instructions:\n\n1.  Complete the missing code in \
Cells 1 and 3  where you see blue text enclosed \n     in (* ... *).  Red \
statements are for testing.  Once these cells work go to step 2.\n2.  Execute \
Cells 1 through 5 in any order (twice each if you want to  get rid of  \n     \
annoying warning messages) to initialize all necessary modules. \n     \
Shortcut for ",
    FontFamily->"Palatino"],
  StyleBox["Mathematica 4.0 ",
    FontFamily->"Palatino",
    FontSlant->"Italic"],
  StyleBox["  users: select Kernel ->Evaluation->Evaluate\n     \
Initialization  (In ",
    FontFamily->"Palatino"],
  StyleBox["Mathematica",
    FontFamily->"Palatino",
    FontSlant->"Italic"],
  StyleBox[" 2.2, Action->Evaluate Initialization).\n3.  Cell 6 contains the \
driver program to run the plane frame of Ex 22.3.  Run by\n     selecting the \
cell and executing.\n4.  Execute Cell 6A to generate deformed shape plot \
frames.   Animate by double\n     clicking one of the frames.  Animation may \
be slowed down by the playback\n     button controls on the left side of the \
bottom window bar.\n5.  If the plots produced by  running a main program \
appear too small,  click \n     on the top  one  (only)  with the mouse,  \
grap a corner \"handle\" and enlarge it.  \n     Then rerun  the program.",
    FontFamily->"Palatino"]
}], "Text",
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  Background->RGBColor[0.463401, 1, 0.300023]],

Cell[TextData[{
  "Cell 1.  Assembly module for plane frame structure.  The element stiffness \
module \n",
  StyleBox["PlaneBeamColumn2Stiffness ",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  "that supports the assembler has been \ndescribed in Chapter 21."
}], "Text",
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  CellMargins->{{12, 51}, {Inherited, Inherited}},
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  Background->RGBColor[0, 1, 1]],

Cell[CellGroupData[{

Cell[TextData[{
  "PlaneBeamColumn2Stiffness[ncoor_,mprop_,fprop_,opt_]:= Module[\n \
{x1,x2,y1,y2,x21,y21,Em,Gm,rho,alpha,A,Izz,num,c,s,L,LL,\n   \
LLL,ra,rb,T,Kebar,Ke}, \n  {{x1,y1},{x2,y2}}=ncoor; {x21,y21}={x2-x1,y2-y1}; \
\n  {Em,Gm,rho,alpha}=mprop; {A,Izz}=fprop; {num}=opt;\n   \
LL=Simplify[x21^2+y21^2]; L=PowerExpand[Sqrt[LL]]; LLL=L*LL; \n   c=x21/L; \
s=y21/L; ra=Em*A/L; rb= 2*Em*Izz/LLL; \n   Kebar= ra*{\n  { \
1,0,0,-1,0,0},{0,0,0,0,0,0},{0,0,0,0,0,0},\n  {-1,0,0, \
1,0,0},{0,0,0,0,0,0},{0,0,0,0,0,0}} +\n          rb*{\n  { 0,0,0,0,0,0},{0, \
6, 3*L,0,-6, 3*L},{0,3*L,2*LL,0,-3*L,  LL},\n  { 0,0,0,0,0,0},{0,-6,-3*L,0, \
6,-3*L},{0,3*L,  LL,0,-3*L,2*LL}};\n   \
T={{c,s,0,0,0,0},{-s,c,0,0,0,0},{0,0,1,0,0,0},\n      \
{0,0,0,c,s,0},{0,0,0,-s,c,0},{0,0,0,0,0,1}};\n   Ke=Transpose[T].Kebar.T; If \
[num,Ke=N[Ke]];\n   Return[Ke]\n ];\n\
PlaneFrameMasterStiffness[nodcoor_,elenod_,\n  \
elemat_,elefab_,eleopt_]:=Module[\n  \
{numele=Length[elenod],numnod=Length[nodcoor],\n",
  "  e,eNL,eftab,ni,nj,i,j,ncoor,mprop,fprop,opt,Ke,K},\n  \
K=Table[0,{3*numnod},{3*numnod}];\n  For [e=1, e<=numele, e++,\n",
  StyleBox["      eNL=elenod[[e]]; {ni,nj}=eNL; \n      \
eftab={3*ni-2,3*ni-1,3*ni,3*nj-2,3*nj-1,3*nj}; \n      \
ncoor={nodcoor[[ni]],nodcoor[[nj]]};           \n      mprop=elemat[[e]]; \
fprop=elefab[[e]]; opt=eleopt;  \n      \
Ke=PlaneBeamColumn2Stiffness[ncoor,mprop,fprop,opt];\n      Print[\"Ke for \
element \",e,\" is \",Ke//MatrixForm];\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        ];",
    FontColor->RGBColor[0, 0, 1]],
  "\n      ];\n  Return[K]",
  "\n  ];\n",
  StyleBox["nodcoor={{0,0},{10,0},{10,10}};\nelenod= {{1,2},{2,3},{1,3}}; \
elemat= Table[{100,0,0,0},{3}];\nelefab= {{1,10},{1/2,10},{2*Sqrt[2],10}}; \
eleopt= {True};\n\
K=PlaneFrameMasterStiffness[nodcoor,elenod,elemat,elefab,eleopt];\n\
Print[\"Master Stiffness of Example Frame:\"];\nPrint[K//MatrixForm]; \
Print[Chop[Eigenvalues[K]]]; ",
    FontColor->RGBColor[1, 0, 0]]
}], "Input",
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          0.0, 0.0, 10.0, 0.0, 0.0}, {0.0, -12.0, -60.0, 0.0, 12.0, -60.0}, {
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                {\(-60.`\), "0.`", "400.`", "60.`", "0.`", "200.`"},
                {\(-12.`\), "0.`", "60.`", "12.`", "0.`", "60.`"},
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                  "12.121320343559642`", \(-21.213203435596427`\)},
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                }], "\[NoBreak]", ")"}],
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        MatrixForm[ {{12.121320343559642, 
          7.8786796564403581, -21.213203435596427, -12.121320343559642, \
-7.8786796564403581, -21.213203435596427}, {7.8786796564403581, 
          12.121320343559642, 
          21.213203435596427, -7.8786796564403581, -12.121320343559642, 
          21.213203435596427}, {-21.213203435596427, 21.213203435596427, 
          282.84271247461902, 21.213203435596427, -21.213203435596427, 
          141.42135623730951}, {-12.121320343559642, -7.8786796564403581, 
          21.213203435596427, 12.121320343559642, 7.8786796564403581, 
          
          21.213203435596427}, {-7.8786796564403581, -12.121320343559642, \
-21.213203435596427, 7.8786796564403581, 
          12.121320343559642, -21.213203435596427}, {-21.213203435596427, 
          21.213203435596427, 141.42135623730951, 
          21.213203435596427, -21.213203435596427, 282.84271247461902}}]],
      Editable->False]], "Print"],

Cell[BoxData[
    \("Master Stiffness of Example Frame:"\)], "Print"],

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\(-7.878679656440358`\), \(-21.213203435596427`\)},
            {"7.878679656440358`", "24.121320343559642`", 
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            {\(-21.213203435596427`\), "81.21320343559643`", 
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            {\(-10.`\), "0.`", "0.`", "22.`", "0.`", \(-60.`\), \(-12.`\), 
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            {"0.`", \(-12.`\), \(-60.`\), "0.`", "17.`", \(-60.`\), 
              "0.`", \(-5.`\), "0.`"},
            {"0.`", "60.`", "200.`", \(-60.`\), \(-60.`\), "800.`", "60.`", 
              "0.`", "200.`"},
            {\(-12.121320343559642`\), \(-7.878679656440358`\), 
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Cell["\<\
Cell 2.  These modules apply the displacement boundary conditions \
by modifying the force
vector and the master stiffness matrix.  This implementation of \
ModifyNodeForcesFor DBC
cannot handle nonzero prescribed displacements.\
\>", "Text",
  CellFrame->True,
  CellMargins->{{12, 51}, {Inherited, Inherited}},
  CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 1]],

Cell[CellGroupData[{

Cell[TextData[{
  "ModifiedMasterStiffness[pdof_,K_] := Module[\n  \
{i,j,k,n=Length[K],np=Length[pdof],Kii,Kmod}, Kmod=K; \n  For [k=1,k<=np,k++, \
i=pdof[[k]]; Kii=Kmod[[i,i]]; \n      For [j=1,j<=n,j++, \
Kmod[[i,j]]=Kmod[[j,i]]=0];\n      If [Kii==0, Kii=1]; Kmod[[i,i]]=Kii\n      \
]; \n  Return[Kmod]\n];\n\nModifyNodeForces[pdof_,f_] := Module[\n  \
{i,k,np=Length[pdof],fmod}, fmod=f; \n     For [k=1,k<=np,k++, i=pdof[[k]]; \
fmod[[i]]=0];\n  Return[fmod]\n];\n\n",
  StyleBox["K=Array[Kij,{6,6}];\nPrint[\"Assembled Master \
Stiffness:\"];Print[K//MatrixForm];\nK=ModifiedMasterStiffness[{1,2,4},K];\n\
Print[\"Master Stiffness Modified For Displacement B.C.:\"];\n\
Print[K//MatrixForm];\nf=Array[fi,{6}];\nPrint[\"Node Force Vector:\"]; \
Print[f];\nf=ModifyNodeForces[{1,2,4},f];\nPrint[\"Node Force Vector Modified \
For Displacement B.C.:\"];\nPrint[f];",
    FontColor->RGBColor[1, 0, 0]]
}], "Input",
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Cell[BoxData[
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Cell[BoxData[
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            {\(Kij[1, 1]\), \(Kij[1, 2]\), \(Kij[1, 3]\), \(Kij[1, 
                4]\), \(Kij[1, 5]\), \(Kij[1, 6]\)},
            {\(Kij[2, 1]\), \(Kij[2, 2]\), \(Kij[2, 3]\), \(Kij[2, 
                4]\), \(Kij[2, 5]\), \(Kij[2, 6]\)},
            {\(Kij[3, 1]\), \(Kij[3, 2]\), \(Kij[3, 3]\), \(Kij[3, 
                4]\), \(Kij[3, 5]\), \(Kij[3, 6]\)},
            {\(Kij[4, 1]\), \(Kij[4, 2]\), \(Kij[4, 3]\), \(Kij[4, 
                4]\), \(Kij[4, 5]\), \(Kij[4, 6]\)},
            {\(Kij[5, 1]\), \(Kij[5, 2]\), \(Kij[5, 3]\), \(Kij[5, 
                4]\), \(Kij[5, 5]\), \(Kij[5, 6]\)},
            {\(Kij[6, 1]\), \(Kij[6, 2]\), \(Kij[6, 3]\), \(Kij[6, 
                4]\), \(Kij[6, 5]\), \(Kij[6, 6]\)}
            }], "\[NoBreak]", ")"}],
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Cell[BoxData[
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Cell[BoxData[
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            {"0", "0", \(Kij[3, 3]\), "0", \(Kij[3, 5]\), \(Kij[3, 6]\)},
            {"0", "0", "0", \(Kij[4, 4]\), "0", "0"},
            {"0", "0", \(Kij[5, 3]\), "0", \(Kij[5, 5]\), \(Kij[5, 6]\)},
            {"0", "0", \(Kij[6, 3]\), "0", \(Kij[6, 5]\), \(Kij[6, 6]\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Print"],

Cell[BoxData[
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Cell[BoxData[
    \({fi[1], fi[2], fi[3], fi[4], fi[5], fi[6]}\)], "Print"],

Cell[BoxData[
    \("Node Force Vector Modified For Displacement B.C.:"\)], "Print"],

Cell[BoxData[
    \({0, 0, fi[3], 0, fi[5], fi[6]}\)], "Print"]
}, Open  ]],

Cell["\<\
Cell 3.  Computation of internal forces (axial forces) for all \
elements of a plane truss,
from the computed nodal displacements.\
\>", "Text",
  CellFrame->True,
  CellMargins->{{12, 51}, {Inherited, Inherited}},
  CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}},
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Cell[CellGroupData[{

Cell[TextData[{
  "PlaneFrameIntForces[nodcoor_,elenod_,elemat_,elefab_,\n  eleopt_,u_]:= \
Module[{numele=Length[elenod],\n  numnod=Length[nodcoor],e,eNL,eftab,ni,nj,i,\
\n  ncoor,mprop,fprop,opt,ue,p},\n  p=Table[0,{numele}]; ue=Table[0,{6}];\n  \
For [e=1, e<=numele, e++, \n    eNL=elenod[[e]]; {ni,nj}=eNL;\n    \
ncoor={nodcoor[[ni]],nodcoor[[nj]]};  ",
  StyleBox["   ",
    FontColor->RGBColor[1, 0, 0]],
  "      \n    mprop=elemat[[e]]; fprop=elefab[[e]]; opt=eleopt;\n    \
eftab={3*ni-2,3*ni-1,3*ni,3*nj-2,3*nj-1,3*nj}; \n    For [i=1,i<=6,i++, \
ii=eftab[[i]]; ue[[i]]=u[[ii]]];\n    \
p[[e]]=PlaneBeamColumn2IntForces[ncoor,mprop,fprop,opt,ue]\n    ]; \n  \
Return[p]\n];\nPlaneBeamColumn2IntForces[ncoor_,mprop_,fprop_,opt_,ue_]:=\n \
Module[{x1,x2,y1,y2,x21,y21,Em,Gm,rho,alpha,A,Izz,num,LL,L,\n  \
dvy,cv1,cv2,pe=Table[0,{3}]},\n",
  StyleBox["  {{x1,y1},{x2,y2}}=ncoor; {x21,y21}={x2-x1,y2-y1};\n  \
{Em,Gm,rho,alpha}=mprop; {A,Izz}=fprop; {num}=opt;\n  If \
[num,{x21,y21,Em,A,Izz}=N[{x21,y21,Em,A,Izz}]]; \n  LL=x21^2+y21^2; \
L=PowerExpand[Sqrt[LL]];\n  \
pe[[1]]=Em*A*(x21*(ue[[4]]-ue[[1]])+y21*(ue[[5]]-ue[[2]]))/LL;\n  \
duy=(x21*(ue[[5]]-ue[[2]])-y21*(ue[[4]]-ue[[1]]))/L;\n  cv1= \
6*duy/LL+2*(-2*ue[[3]]-ue[[6]]  )/L;\n  cv2=-6*duy/LL+2*(   \
ue[[3]]+2*ue[[6]])/L;\n  pe[[2]]=Em*Izz*cv1; pe[[3]]=Em*Izz*cv2;",
    FontColor->RGBColor[0, 0, 1]],
  "\n  Return[pe]\n];  \n",
  StyleBox["ClearAll[L,Em,A1,A2,Izz1,Izz2,Izz3,uz1,uz2,uz2,uz3]; \n\
nodcoor={{0,0},{L,0},{L,L}}; elenod= {{1,2},{2,3},{1,3}};\nelemat= \
Table[{Em,0,0,0},{3}];\nelefab= {{A1,Izz1},{A2,Izz2},{A3,Izz3}}; eleopt= \
{False};\nu={0,uy1,0, 0,uy2,0, 0,uy3,0};\n\
p=PlaneFrameIntForces[nodcoor,elenod,elemat,elefab,eleopt,u];\nPrint[\"Int \
Forces of Example Frame:\"]; Print[p//MatrixForm]; ",
    FontColor->RGBColor[1, 0, 0]]
}], "Input",
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  CellMargins->{{12, 51}, {Inherited, Inherited}},
  InitializationCell->True,
  CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0.699214, 1, 0.0500191]],

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    \("Int Forces of Example Frame:"\)], "Print"],

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    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {
              "0", \(\(6\ Em\ Izz1\ \((\(-uy1\) + 
                        uy2)\)\)\/L\^2\), \(-\(\(6\ Em\ Izz1\ \((\(-uy1\) + 
                          uy2)\)\)\/L\^2\)\)},
            {\(\(A2\ Em\ \((\(-uy2\) + uy3)\)\)\/L\), "0", "0"},
            {\(\(A3\ Em\ \((\(-uy1\) + 
                        uy3)\)\)\/\(2\ L\)\), \(\(3\ Em\ Izz3\ \((\(-uy1\) + 
                        uy3)\)\)\/\(\@2\ L\^2\)\), \(-\(\(3\ Em\ Izz3\ \
\((\(-uy1\) + uy3)\)\)\/\(\@2\ L\^2\)\)\)}
            }], "\[NoBreak]", ")"}],
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        MatrixForm[ BoxForm`e$]]]], "Print"],

Cell["\<\
Cell 4 collectively groups plot modules that support preprocessing, \
that is, draw pictures of the FEM 
model and BCs.
It is broken down inro three cells:  4A, 4B, 4C because of the length of \
these modules.\
\>", "Text",
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  CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 1]],

Cell["\<\
Cell 4A.  PlotLineElements draws only the element (bars) without \
any labels.\
\>", "Text",
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  CellMargins->{{12, 51}, {Inherited, Inherited}},
  CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 1]],

Cell[CellGroupData[{

Cell[TextData[{
  "PlotLineElements[nodcoor_,elenod_,aspect_,title_]:= Module[\n \
{eNL,e,n,ni,nj,nc,xyc,sides={},numele=Length[elenod],\n  \
numnod=Length[nodcoor],x,y,xmin,xmax,ymin,ymax,aratio},\n  \
x=Table[nodcoor[[i,1]],{i,1,numnod}]; \n  \
y=Table[nodcoor[[i,2]],{i,1,numnod}];\n \
{xmin,xmax,ymin,ymax}=N[{Min[x],Max[x],Min[y],Max[y]}];\n For \
[e=1,e<=numele,e++, eNL=elenod[[e]]; nc=Length[eNL]; \n     If [nc!=2, \
Continue[]]; {ni,nj}=eNL; \n     xyc={{x[[ni]],y[[ni]]},{x[[nj]],y[[nj]]}};   \
   \n     sides=AppendTo[sides,Graphics[Line[xyc]]]\n     ];\n If [aspect>0, \
aratio=aspect, aratio=(ymax-ymin)/(xmax-xmin)];\n If [aspect<0, \
aratio=Automatic];\n Show[Graphics[AbsoluteThickness[2]],\n      \
Graphics[RGBColor[0,0,0]],\n      sides, Background->RGBColor[1,1,0],\n      \
AspectRatio->aratio,PlotLabel->title ];\n ];\n",
  StyleBox[" \nClearAll[L,H];\nL=10000; H=6000; \n\
NodeCoordinates={{-L,0},{-L,H/2},{-L,H},{-L/2,H},{0,H},{L/2,H},\n         \
{L,H},{L,H/2},{L,0},{0,H/2},{0,0}};\nElemNodeLists= \
{{1,2},{2,3},{3,4},{4,5},{5,6},{6,7},{7,8},\n                \
{8,9},{5,10},{10,11}};\nnumnod=Length[nodcoor]; numele=Length[elenod];\n\
aspect=0; elabinfo={True,0.12}; nlabinfo={True,0.06};\n\n\
PlotLineElements[NodeCoordinates,ElemNodeLists,\n    aspect,\"test mesh\"];",
    FontColor->RGBColor[1, 0, 0]]
}], "Input",
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  CellMargins->{{12, 51}, {Inherited, Inherited}},
  InitializationCell->True,
  CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}},
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Cell["\<\
Cell 4B. PlotLineElementsAndNodes drwas element and nodes, and \
optionally labels them.\
\>", "Text",
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  CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}},
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Cell[TextData[{
  "PlotLineElementsAndNodes[nodcoor_,elenod_,aspect_,title_,\n  \
elabinfo_,nlabinfo_]:= Module[{i,j,k,e,eNL,n,ni,nj,x,y,\n   \
delx,dely,xmin,xmax,ymin,ymax,xy0,epsx,epsy,ex,ey,\n   \
numele=Length[elenod],elabels,nlabels,frade,fradn,\n   \
numnod=Length[nodcoor],rade,radn,elab,nlab,pnod=pnlab={},\n   \
pesid=pelab=pecirc=pedisk={},aratio},\n   \
x=Table[nodcoor[[i,1]],{i,1,numnod}]; \n   \
y=Table[nodcoor[[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};\n   {elabels,frade}=elabinfo; \
{nlabels,fradn}=nlabinfo;\n   radn=fradn*Min[delx,dely]; \
{ex,ey}={2*radn,1.6*radn}; \n   rade=frade*Min[delx,dely]; \n   For \
[e=1,e<=numele,e++, eNL=elenod[[e]]; n=Length[eNL];\n       If [n!=2, \
Continue[]]; {ni,nj}=eNL;\n       gside={nodcoor[[ni]],nodcoor[[nj]]}; \n     \
  xy0=(gside[[1]]+gside[[2]])/2; \n       pesid= AppendTo[pesid, \
Graphics[Line[gside]]];\n       If [elabels, \n          \
pedisk=AppendTo[pedisk,Graphics[Disk[xy0,rade]]]; \n          \
pecirc=AppendTo[pecirc, Graphics[Circle[xy0,rade]]];\n          \
elab=FontForm[e,{\"Times\",11}];\n          pelab= AppendTo[pelab, \
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pnlab=AppendTo[pnlab, \n             \
Graphics[Text[nlab,nodcoor[[n]]+{ex,ey}]]]];\n       pnod= AppendTo[pnod,  \n \
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aratio=aspect, aratio=(ymax-ymin)/(xmax-xmin)];\n   If [aspect<0, \
aratio=Automatic];\n   Show[Graphics[GrayLevel[0]],\n        \
Graphics[AbsoluteThickness[3]],pesid,\n        Graphics[GrayLevel[0]],pnod,\n \
       Graphics[AbsoluteThickness[1]],\n        \
Graphics[RGBColor[.99,.99,.99]],pedisk,\n        \
Graphics[GrayLevel[0]],pecirc,pelab,\n        Graphics[GrayLevel[0]],pnlab,\n \
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PlotRange->{{xmin-epsx,xmax+epsx},{ymin-epsy,ymax+epsy}},\n        \
Axes->False,AspectRatio->aratio];\n];\n\n",
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{L,H},{L,H/2},{L,0},{0,H/2},{0,0}};\nElemNodeLists= \
{{1,2},{2,3},{3,4},{4,5},{5,6},{6,7},{7,8},\n                \
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aspect=-1; elabinfo={True,0.10}; nlabinfo={True,0.04};\n\n\
PlotLineElementsAndNodes[NodeCoordinates,ElemNodeLists,aspect,\n   \"test \
mesh with elem & node labels\",elabinfo,nlabinfo];",
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}], "Input",
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applies forces and\nsupport conditions.  Not very successful yet, because ",
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  " plotting capabilities are not\nexactly state of the art."
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Cell 5.  Plot modules that support postprocessing.   As in the case \
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Cells 5A, 5B and 5C to keep them short.\
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x=Table[nodcoor[[i,1]],{i,1,numnod}]; \n \
y=Table[nodcoor[[i,2]],{i,1,numnod}];\n \
{xmin,xmax,ymin,ymax}=N[{Min[x],Max[x],Min[y],Max[y]}];\n For \
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Cell["\<\
Cell 5B. Plot Axial Stress level displays stress level and sign \
using color: red for tension,
blue for compression, white for zero stress.  A black background is \
used.\
\>", "Text",
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  "PlotAxialStressLevel[nodcoor_,elenod_,s_,sfac_,aspect_,title_]:= \n \
Module[{eNL,e,n,ni,nj,nc,xyc,sides={},numele=Length[elenod],\n  \
numnod=Length[nodcoor],x,y,xmin,xmax,ymin,ymax,\n  \
smin,smax,fmax,sval,c1,c2,c3,aratio},\n  \
x=Table[nodcoor[[i,1]],{i,1,numnod}]; \n  \
y=Table[nodcoor[[i,2]],{i,1,numnod}];\n \
{xmin,xmax,ymin,ymax}=N[{Min[x],Max[x],Min[y],Max[y]}];\n smin=Min[s]; \
smax=Max[s]; fmax=Max[Abs[smax],Abs[smin]];\n For [e=1,e<=numele,e++, \
eNL=elenod[[e]]; nc=Length[eNL];\n     sval=N[s[[e]]*sfac];\n     \
{c1,c2,c3}=LineColor[sval,fmax];",
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xyc={{x[[ni]],y[[ni]]},{x[[nj]],y[[nj]]}};    \n     \
sides=AppendTo[sides,Graphics[RGBColor[c1,c2,c3]]];\n     \
sides=AppendTo[sides,Graphics[Line[xyc]]]\n     ];\n If [aspect>0, \
aratio=aspect, aratio=(ymax-ymin)/(xmax-xmin)];\n \
Show[Graphics[AbsoluteThickness[3]],\n      Graphics[RGBColor[0,0,0]], sides, \
\n      Background->GrayLevel[0],\n      AspectRatio->aratio, \
PlotLabel->title ];\n ];\n\nLineColor[f_,fmax_]:= Module[{r,RGBmax={1,0,0},  \
\n   RGBmin={0,0,1}, RGBzero={1,1,1}, RGBout={0,0,0}},\n   If [f==0 || \
fmax==0,   \n       Return[RGBzero]];  (* White if f=0 *)\n   If [f>fmax || \
f<-fmax, \n       Return[RGBout ]];  (* Black if outside range *)\n   If \
[f>0, r= N[f/fmax]; \n       Return[r*RGBmax+(1-r)*RGBzero]]; (* positive *)\n\
   If [f<0, r=-N[f/fmax]; \n       Return[r*RGBmin+(1-r)*RGBzero]]; (* \
negative *)\n];\n",
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    {40,8},{40,0},{50,5},{50,0},{60,0}};\nelenod= \
{{1,3},{3,5},{5,7},{7,9},{9,11},{11,12},\n         \
{1,2},{2,4},{4,6},{6,8},{8,10},{10,12},\n         \
{2,3},{4,5},{6,7},{8,9},{10,11},\n         {2,5},{4,7},{7,8},{9,10}};\n\
numele=Length[elenod];\nu={0,0, 0,-0.10, 0,-0.10, 0,-0.15, 0,-0.15, 0,-0.20, \
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sigma=Table[ne-10,{ne,1,numele}];\naspect=0;\n\
PlotAxialStressLevel[nodcoor,elenod,sigma,1.0,aspect,\n   \"test axial stress \
level plot\"];\n   \n(* \nu={0,0, 0,-0.10, 0,-0.10, 0,-0.15, 0,-0.15, \
0,-0.20, \n    0,-0.20, 0,-0.15, 0,-0.15, 0,-0.10, 0,-0.10, 0,0};\nFor \
[t=0.,t<=N[Pi],t=t+N[Pi/6],  amp=10*Sin[t];\n    \
PlotDeformedShape[nodcoor,elenod,u,amp,aspect,\n    \"deformed shape\"]]; *)",
    
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Cell 6.  The main driver program.  Defines the frame problem and \
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ClearAll[L,H,a,Em,P];
L=10000; H=6000;  Em=35000; a=500; A=a^2; Izz=a^4/12;
P=4800;
NodeCoordinates={{-L,0},{-L,H/2},{-L,H},{-L/2,H},{0,H},{L/2,H},
         {L,H},{L,H/2},{L,0},{0,H/2},{0,0}};
ElemNodeLists= {{1,2},{2,3},{3,4},{4,5},{5,6},{6,7},{7,8},
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numnod=Length[NodeCoordinates]; 
numele=Length[ElemNodeLists]; numdof=3*numnod;
ElemMaterial= Table[{Em,0,0,0},{numele}];
ElemFabrication= Table[{A,Izz},{numele}];
ProcessOptions= {True};
aspect=0;
PlotLineElements[NodeCoordinates,ElemNodeLists,aspect,
   \"frame mesh\"];
PlotLineElementsAndNodes[NodeCoordinates,ElemNodeLists,aspect,
   \"frame mesh with elem & node labels\",{True,0.10},{True,0.04}];
  
FreedomTag=FreedomValue=Table[{0,0,0},{numnod}]; 
FreedomValue[[4]]={0,-P,0}; 
FreedomValue[[3]]={P/2,0,0};   
Print[\"Applied node forces=\"]; Print[FreedomValue];
FreedomTag[[1]]=FreedomTag[[9]]=FreedomTag[[11]]={1,1,1}; (* fixed *)

f=Flatten[FreedomValue];
K=PlaneFrameMasterStiffness[NodeCoordinates,
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pdof={}; 
For[n=1,n<=numnod,n++, For[j=1,j<=3,j++,
    If [FreedomTag[[n,j]]>0, AppendTo[pdof,3*(n-1)+j]]]];
Print[\"pdof=\",pdof];  
Kmod=ModifiedMasterStiffness[pdof,K];
fmod=ModifyNodeForces [pdof,f];
u=LinearSolve[Kmod,fmod];  u=Chop[u,.000001];

Print[\"Computed Nodal Displacements:\"]; 
NodeForces=NodeDisplacements=Table[{0,0,0},{numnod}];
For [n=1,n<=numnod,n++, For[j=1,j<=3,j++,
    NodeDisplacements[[n,j]]=u[[3*(n-1)+j]] ]];
Print[NodeDisplacements//MatrixForm] 
f=Simplify[K.u];           f=Chop[f,.000001];
Print[\"External Node Forces Including Reactions:\"]; 
For [n=1,n<=numnod,n++, For[j=1,j<=3,j++,
    NodeForces[[n,j]]=f[[3*(n-1)+j]] ]];
Print[NodeForces//MatrixForm];

p=PlaneFrameIntForces[NodeCoordinates,ElemNodeLists,
    ElemMaterial,ElemFabrication,eleopt,u];
p=Chop[p,.000001]; 
Print[\"Internal Member Forces:\"];
Print[p//MatrixForm];

amp=10000.;
PlotPlaneFrameDeformedShape[NodeCoordinates,ElemNodeLists,u,
   {amp,amp,0,16},aspect,\"Deformed shape (magnified x10000)\"];
    
H=500;  
ElemFabrication= Table[{A,Izz,H},{numele}];   
fmax=0.25;  Nsub{16,8}; aspect=-1; 
title=\"Axial stress in frame members\";
PlotPlaneFrameStress[NodeCoordinates,ElemNodeLists,
    ElemFabrication,p,fmax,Nsub,aspect,title]; 
    \
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Cell["\<\
Cell 6A.  Animation of deflections.  Uses the displacements u \
computed by driver program to
plot a sequence of deformed shapes with amplitude varying as sine of \
pseudotime t.
To animate, click twice on a plot, then use the VCR buttons  that appear on \
the bottom of
the window to regulate frame speed.\
\>", "Text",
  CellFrame->True,
  CellMargins->{{12, 51}, {Inherited, Inherited}},
  CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 1]],

Cell["\<\
ampmax=10000.; ampmin=0;
For [t=0.,t<=N[Pi],t=t+N[Pi/6],  amp=ampmax*Sin[t];
    PlotPlaneFrameDeformedShape[NodeCoordinates,ElemNodeLists,
    u,{amp,ampmax,ampmin,16},-1,\"deformed shape\"]];
   \
\>", "Input",
  CellFrame->True,
  CellMargins->{{12, 51}, {Inherited, Inherited}},
  CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 0]],

Cell["\<\
 ClearAll[L,H,a,Em,P];
R=10;  Em=10000; b=1; h=5/10; A=b*h; Izz=b*h^3/12;
P=4;  ne=32; nnod=ne+1;  NodeCoordinates=Table[{0,0},{nnod}];
For [n=1,n<=nnod,n++, arg=Pi/(2*ne)*(n-1);
    NodeCoordinates[[n]]=R*{Sin[arg],Cos[arg]}];
Print[\"NodeCoordinates=\",NodeCoordinates];
ElemNodeLists=Table[{0,0},{ne}];
For [e=1,e<=ne,e++,ElemNodeLists[[e]]={e,e+1}];
numnod=Length[NodeCoordinates]; 
numele=Length[ElemNodeLists]; numdof=3*numnod;
ElemMaterial= Table[{Em,0,0,0},{numele}];
ElemFabrication= Table[{A,Izz},{numele}];
ProcessOptions= {True};
aspect=0;
PlotLineElements[NodeCoordinates,ElemNodeLists,aspect,
   \"frame mesh\"];
PlotLineElementsAndNodes[NodeCoordinates,ElemNodeLists,aspect,
   \"frame mesh with elem & node labels\",{True,0.03},{True,0.02}];
  
FreedomTag=FreedomValue=Table[{0,0,0},{numnod}]; 
FreedomValue[[1]]={0,-P,0}; 
Print[\"Applied node forces=\"]; Print[FreedomValue];
FreedomTag[[1]]={1,0,1};
FreedomTag[[numnod]]={0,1,1};
f=Flatten[FreedomValue];
K=PlaneFrameMasterStiffness[NodeCoordinates,
  ElemNodeLists,ElemMaterial,ElemFabrication,ProcessOptions];
Print[\"K=\",K//MatrixForm];
pdof={}; 
For[n=1,n<=numnod,n++, For[j=1,j<=3,j++,
    If [FreedomTag[[n,j]]>0, AppendTo[pdof,3*(n-1)+j]]]];
Print[\"pdof=\",pdof];  
Kmod=ModifiedMasterStiffness[pdof,K];
Print[\"Kmod=\",Kmod//MatrixForm];
Print[\"eigs of Kmod=\",Chop[Eigenvalues[N[Kmod]]]];
fmod=ModifyNodeForces [pdof,f];
u=LinearSolve[Kmod,fmod];  u=Chop[u,.000001];

Print[\"Computed Nodal Displacements:\"]; 
NodeForces=NodeDisplacements=Table[{0,0,0},{numnod}];
For [n=1,n<=numnod,n++, For[j=1,j<=3,j++,
    NodeDisplacements[[n,j]]=u[[3*(n-1)+j]] ]];
Print[NodeDisplacements//MatrixForm]; 

f=Simplify[K.u]; f=Chop[f,.000001];
Print[\"External Node Forces Including Reactions:\"]; 
For [n=1,n<=numnod,n++, For[j=1,j<=3,j++,
    NodeForces[[n,j]]=f[[3*(n-1)+j]] ]];
Print[NodeForces//MatrixForm];

p=PlaneFrameIntForces[NodeCoordinates,ElemNodeLists,
    ElemMaterial,ElemFabrication,eleopt,u];
p=Chop[p,.000001]; 
Print[\"Internal Member Forces:\"];
Print[p//MatrixForm];

amp=1.;
PlotPlaneFrameDeformedShape[NodeCoordinates,ElemNodeLists,u,
   {amp,amp,0,16},aspect,\"Deformed shape (magnified x10000)\"];
   \
\>", "Input",
  CellFrame->True,
  CellMargins->{{12, 51}, {Inherited, Inherited}},
  CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 0]]
}, Open  ]],

Cell["\<\
ClearAll[L,H,a,Em,P];
NodeCoordinates={{0,a},{a,0}}; numnod=2; numele=1;
Print[\"NodeCoordinates=\",NodeCoordinates];
ElemNodeLists={{1,2}};
ElemMaterial= {{Em,0,0,0}};
ElemFabrication= {{A,\[Alpha]*a^2*A}};
ProcessOptions= {False};
(*aspect=0;
PlotLineElements[NodeCoordinates,ElemNodeLists,aspect,
   \"frame mesh\"];
PlotLineElementsAndNodes[NodeCoordinates,ElemNodeLists,aspect,
   \"frame mesh with elem & node labels\",{True,0.03},{True,0.02}];*)
   
FreedomTag={{1,0,1},{0,1,1}};
FreedomValue={{0,P,0},{0,0,0}};
f=Flatten[FreedomValue];
K=PlaneFrameMasterStiffness[NodeCoordinates,
  ElemNodeLists,ElemMaterial,ElemFabrication,ProcessOptions];
Print[\"K=\",K//MatrixForm];

pdof={}; 
For[n=1,n<=numnod,n++, For[j=1,j<=3,j++,
    If [FreedomTag[[n,j]]>0, AppendTo[pdof,3*(n-1)+j]]]];
Print[\"pdof=\",pdof];  
Kmod=ModifiedMasterStiffness[pdof,K];
Print[\"Kmod=\",Kmod//MatrixForm];
(*Print[\"eigs of Kmod=\",Chop[Eigenvalues[N[Kmod]]]];*)
fmod=ModifyNodeForces [pdof,f];
u=LinearSolve[Kmod,fmod];  u=Simplify[u]; 

Print[\"Computed Nodal Displacements:\"]; 
NodeForces=NodeDisplacements=Table[{0,0,0},{numnod}];
For [n=1,n<=numnod,n++, For[j=1,j<=3,j++,
    NodeDisplacements[[n,j]]=u[[3*(n-1)+j]] ]];
Print[NodeDisplacements//MatrixForm]; 
f=Simplify[K.u]; f=Chop[f,.000001];
Print[\"External Node Forces Including Reactions:\"]; 
For [n=1,n<=numnod,n++, For[j=1,j<=3,j++,
    NodeForces[[n,j]]=f[[3*(n-1)+j]] ]];
Print[NodeForces//MatrixForm];

p=PlaneFrameIntForces[NodeCoordinates,ElemNodeLists,
    ElemMaterial,ElemFabrication,eleopt,u];
p=Simplify[p];
Print[\"Internal Member Forces:\"];
Print[p//MatrixForm];\
\>", "Input",
  CellFrame->True,
  CellMargins->{{12, 51}, {Inherited, Inherited}},
  CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 0]],

Cell["UVA plane - 3-member plane frame model of wings", "Text",
  CellFrame->True,
  CellMargins->{{12, 51}, {Inherited, Inherited}},
  CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[1, 1, 0]],

Cell["\<\
ClearAll[L,H,a,b,Em,P,A,Izz]; Em=185648; a=30; b=9; A=0.7449; \
Izz=0.4434; 
P=2*4.4723; 
NodeCoordinates={{0,b},{a,b},{a,0},{0,0} }; 
numnod=Length[NodeCoordinates];
Print[\"NodeCoordinates=\",NodeCoordinates];
ElemNodeLists={{1,2},{2,3},{3,4}}; numele=Length[ElemNodeLists]; \
Print[numele];
ElemMaterial= Table[{Em,0,0,0},{numele}];
ElemFabrication= Table[{A,Izz},{numele}];
ProcessOptions= {False};
aspect=0;
PlotLineElements[NodeCoordinates,ElemNodeLists,aspect,
   \"UVA mesh\"];
PlotLineElementsAndNodes[NodeCoordinates,ElemNodeLists,aspect,
   \"UVA mesh with elem & node labels\",{True,0.12},{True,0.08}];

FreedomTag={{1,1,1},{0,0,0},{0,0,0},{1,1,1}};
FreedomValue={{0,0,0},{0,P/2,0},{0,P/2,0},{0,0,0}};
f=Flatten[FreedomValue];
K=PlaneFrameMasterStiffness[NodeCoordinates,
  ElemNodeLists,ElemMaterial,ElemFabrication,ProcessOptions];
K=N[K];
Print[\"K=\",K//MatrixForm]; 
Print[\"eigs of K=\",Chop[Eigenvalues[N[K]]]];
pdof={}; 
For[n=1,n<=numnod,n++, For[j=1,j<=3,j++,
    If [FreedomTag[[n,j]]>0, AppendTo[pdof,3*(n-1)+j]]]];
Print[\"pdof=\",pdof];  
Kmod=ModifiedMasterStiffness[pdof,K];
Print[\"Kmod=\",Kmod//MatrixForm];
Print[\"eigs of Kmod=\",Chop[Eigenvalues[N[Kmod]]]];
fmod=ModifyNodeForces [pdof,f];

u=LinearSolve[Kmod,fmod];  u=Simplify[u]; 

Print[\"Computed Nodal Displacements:\"]; 
NodeForces=NodeDisplacements=Table[{0,0,0},{numnod}];
For [n=1,n<=numnod,n++, For[j=1,j<=3,j++,
    NodeDisplacements[[n,j]]=u[[3*(n-1)+j]] ]];
Print[NodeDisplacements//MatrixForm]; 
f=Simplify[K.u]; f=Chop[f,.000001];
Print[\"External Node Forces Including Reactions:\"]; 
For [n=1,n<=numnod,n++, For[j=1,j<=3,j++,
    NodeForces[[n,j]]=f[[3*(n-1)+j]] ]];
Print[NodeForces//MatrixForm];

p=PlaneFrameIntForces[NodeCoordinates,ElemNodeLists,
    ElemMaterial,ElemFabrication,eleopt,u];
p=Simplify[p]; Print[\"Internal Member Forces:\"]; Print[p//MatrixForm];

amp=20.;
PlotPlaneFrameDeformedShape[NodeCoordinates,ElemNodeLists,u,
   {amp,amp,0,16},aspect,\"Deformed shape (magnified x1)\"];
   
H=2; 
ElemFabrication= Table[{A,Izz,H},{numele}];   
fmax=200;  Nsub={16,8}; aspect=-1; 
title=\"Axial stress in frame members\";
PlotPlaneFrameStress[NodeCoordinates,ElemNodeLists,
    ElemFabrication,p,fmax,Nsub,aspect,title]; \
\>", "Input",
  CellFrame->True,
  CellMargins->{{12, 51}, {Inherited, Inherited}},
  CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 0]]
},
FrontEndVersion->"4.2 for Macintosh",
ScreenRectangle->{{0, 1920}, {0, 1180}},
AutoGeneratedPackage->None,
WindowToolbars->{},
CellGrouping->Manual,
WindowSize->{1436, 1119},
WindowMargins->{{72, Automatic}, {Automatic, 5}},
PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}},
ShowCellLabel->True,
ShowCellTags->False,
RenderingOptions->{"ObjectDithering"->True,
"RasterDithering"->False},
Magnification->1.5,
MacintoshSystemPageSetup->"\<\
00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001
0000I00000400`<300000BL?00400@00000000000000060801T1T00000000000
00000000000000000000000000000000\>"
]

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*******************************************************************)