(************** Content-type: application/mathematica **************

                    Mathematica-Compatible Notebook

This notebook can be used with any Mathematica-compatible
application, such as Mathematica, MathReader or Publicon. The data
for the notebook starts with the line containing stars above.

To get the notebook into a Mathematica-compatible application, do
one of the following:

* Save the data starting with the line of stars above into a file
  with a name ending in .nb, then open the file inside the
  application;

* Copy the data starting with the line of stars above to the
  clipboard, then use the Paste menu command inside the application.

Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode.  Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).

NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing
the word CacheID, otherwise Mathematica-compatible applications may
try to use invalid cache data.

For more information on notebooks and Mathematica-compatible 
applications, contact Wolfram Research:
  web: http://www.wolfram.com
  email: info@wolfram.com
  phone: +1-217-398-0700 (U.S.)

Notebook reader applications are available free of charge from 
Wolfram Research.
*******************************************************************)

(*CacheID: 232*)


(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[     96352,       2505]*)
(*NotebookOutlinePosition[     99244,       2587]*)
(*  CellTagsIndexPosition[     98682,       2570]*)
(*WindowFrame->Normal*)



Notebook[{
Cell[BoxData[
    \(ClearAll["\<`*\>"]\)], "Input",
  AspectRatioFixed->True],

Cell["\<\
Finite Element Analysis of
Two and Three Dimensional Truss Structures\
\>", "Title",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[{
  StyleBox[" by  Zafer Gurdal ", "Author"],
  StyleBox["\[Copyright]", "Author",
    FontFamily->"Symbol",
    FontSlant->"Plain"],
  StyleBox[
  " Copyright 1993\n Departments of Aerospace and Ocean Engineering \n    and \
Engineering Science and Mechanics - 0219\n Virginia Polytehnic Institute and \
State University\n Blacksburg, VA 24061", "Author"]
}], "Subsubtitle",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[{
  "This notebook is intended to be used as a tutorial to demonstrate the use \
of ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " and ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " programing for displacement and stresses analysis of 2-D and 3-D truss \
structures."
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[TextData["Initializations"], "Section",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[{
  "Following lines establish the environment for the tutorial and load the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "  package that has the various modules needed for the calculations.  ",
  "\n",
  "The first line in the next cell is to display two dimensional lists as \
matrices, and the next one sets the default font for the displayed text.   "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[BoxData[{
    \(\($PrePrint = If[MatrixQ[#1], MatrixForm[#1], #1] &;\)\), "\n", 
    \(\($DefaultFont = {"\<Times-Italic\>", 12};\)\)}], "Input",
  AspectRatioFixed->True],

Cell[TextData[{
  "Next we need to load the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "  package  ",
  StyleBox["FEM.m",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  "  from the appropriate directory.   This Program is provided in the link \
\"FEM.m\" under the PROGRAMS on the course web page. First. download the \
FEM.m file as a text file to one of your directories so that you can load it \
into your ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " notebook session.  \nTo find the current directory ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "  expects to load from use"
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(Directory[]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \("C:\\Program Files\\Wolfram Research\\Mathematica\\4.1"\)], "Output"]
}, Open  ]],

Cell["\<\
Following line may be used to set the current directory to be the one that \
includes the program that you would like to load.\
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(SetDirectory["\<C:\Documents and Settings\zgurdal\My \
Documents\VATECH\COURSES\PROGRAM_M_FILES\>"]\)], "Input"],

Cell[BoxData[
    \("C:\\Documents and Settings\\zgurdal\\My \
Documents\\VATECH\\COURSES\\PROGRAM_M_FILES"\)], "Output"]
}, Open  ]],

Cell["To find the names of the files in the current directory  use", "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(FileNames[]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \({"FEM.m"}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "To load a program file use   ",
  StyleBox["<<",
    FontColor->RGBColor[1, 0, 0]],
  "   with appropriate path leading from the  current directory to the \
directory in which the file you want to load resides  "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[BoxData[
    \(<< "\<FEM.m\>"\)], "Input",
  AspectRatioFixed->True],

Cell[TextData[{
  "In order to find the operations that are currently supported by the ",
  StyleBox["FEM.m",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  "  package use;  "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?SymbolicFEM\)\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \("Following modules are available in the truss Finite   Element Program, \
       showNodes,         elementLength, dirCos,          eStiffMat, \
globalMatrix,          assembleGlobalMatrix, reducedGlobalMatrix,          \
showStructure.        There will also be plotting utilities in the near \
future"\)], "Print",
  CellTags->"Info3216495278-8722696"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Input ", "Section",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["\<\
The elements of the truss are defined by three lists which specify  a) the \
location of the node points,  b) nodal connectivity of the truss elements,   \
and c)  cross-sectional properties of the elements.   \
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell["a) Node Points:  ", "Subsection",
  FontSize->18],

Cell[TextData[{
  "The first list is the list of  x- y- and z-coordinates of the node points \
along with the node numbers.   For a 2-D truss structure with nn nodes, the \
form of the list is \n",
  Cell[BoxData[
      FormBox[
        RowBox[{
          FormBox[
            RowBox[{
              RowBox[{
                FormBox[
                  RowBox[{
                    FormBox[\({{x\_1\),
                      "TraditionalForm"], ",", \(y\_1\)}],
                  "TraditionalForm"], "}"}], ",", \({x\_2\)}],
            "TraditionalForm"], ",", \(y\_2\)}], TraditionalForm]]],
  "}, ......., ",
  Cell[BoxData[
      \(TraditionalForm\`{x\_nn, \ y\_nn\)]],
  "}.  \nFor 3-D trusses, the z-coordimates are added for each node.  Note \
that the numbering of the nodes is arbitrary, and may not follow a sequential \
numbering scheme.  "
}], "Text",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(nodeLocations = {\ {1, {0, \ 0}}, \ {2, {h, \ 0}}, \ {3, \ {h, \ 
            h}}, \ {4, \ {2  h, \ 0}}}\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \({{1, {0, 0}}, {2, {h, 0}}, {3, {h, h}}, {4, {2\ h, 0}}}\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["b) Element Connectivity:  ", "Subsection",
  FontSize->18],

Cell[TextData[{
  "Each element is connected to two nodes.  The nodal connectivity is the \
list of two nodes numbers that define each element.  For a truss with ",
  StyleBox["ne",
    FontColor->RGBColor[1, 0, 0]],
  " elements, we have \n{{1, ",
  Cell[BoxData[
      \(TraditionalForm\`{nd\_1\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`nd\_2\)]],
  "}}, {2, ",
  Cell[BoxData[
      \(TraditionalForm\`{nd\_1\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`nd\_2\)]],
  "}},..., {",
  StyleBox["ne, ",
    FontColor->RGBColor[1, 0, 0]],
  Cell[BoxData[
      \(TraditionalForm\`{nd\_1\)]],
  ",",
  StyleBox[" ",
    FontColor->RGBColor[1, 0, 0]],
  Cell[BoxData[
      \(TraditionalForm\`nd\_2\)]],
  "}}}\nNumbering of the elements is also arbitrary, and does not have to \
follow a sequential scheme.  The node numbers, however, must appear in the \
node point list discussed above.  "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(elementNodes = {{1, {1, 3}}, {2, {1, 2}}, {3, {2, 3}}, {4, {2, 
            4}}, {5, {3, 4}}\ }\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \({{1, {1, 3}}, {2, {1, 2}}, {3, {2, 3}}, {4, {2, 4}}, {5, {3, 
          4}}}\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["c) Element Properties:  ", "Subsection",
  FontSize->18],

Cell[TextData[{
  "Cross-sectional properties, and the modulus of each element are rovided in \
a list of length ",
  StyleBox["ne",
    FontColor->RGBColor[1, 0, 0]],
  " in which the first entry is the area of the member and the second is the \
Young's modulus of the material."
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(elementProperties = {\ {1, {Ar, YE}}, {2, {Ar, YE}}, {3, {Ar, 
            YE}}, {4, {Ar, YE}}, {5, {Ar, YE}}\ \ }\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \({{1, {Ar, YE}}, {2, {Ar, YE}}, {3, {Ar, YE}}, {4, {Ar, YE}}, {5, {Ar, 
          YE}}}\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
d) Graphical Representation of the Input (Numeric Data Only):  \
\>", "Subsection",
  FontSize->18],

Cell[TextData[{
  "If the numerical values of the node point coordinates and the element \
properties are provided, it is possible to graphically verify the model.  Two \
utility modules, ",
  StyleBox["showNodes",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " and ",
  StyleBox["showElements",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " may be used for this purpose.   "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?showNodes\)\)], "Input"],

Cell[BoxData[
    \("showNodes[nodes, Options]       shows the nodal points in either 2- or \
3-D space depending  on the input. The argument nodes is a list of node point \
coordinates provided in the following form, {NodeNo, {x-coord, y-coord, \
z-coord}}. Default options are pointColor -> RGBColor[1,0,0] pointSize  -> \
PointSize[0.01]"\)], "Print",
  CellTags->"Info3216495292-5915026"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?showElements\)\)], "Input"],

Cell[BoxData[
    \("showElements[nodes, locs, Options]       shows the configuration of \
the elements. The argument nodes is a list of nodal connectivity for the \
elements.  The form of  the list for a truss element is {ElementNo,{Node1, \
Node2}}.   For a triangular element, for example, the form is \
{ElementNo,{Node1, Node2, Node3}}.  The argument locs is the list of nodal \
coordinates.  Its form, for example, for a 2-D problem is  {NodeNo, {x-coord, \
y-coord}}. Type Options[showElements]  to see the optional arguments."\)], \
"Print",
  CellTags->"Info3216495292-1617452"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(nodeLocations\)], "Input"],

Cell[BoxData[
    \({{1, {0, 0}}, {2, {h, 0}}, {3, {h, h}}, {4, {2\ h, 0}}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(nodes\  = \ nodeLocations\  /. \ h\  -> \ 100.0\)], "Input"],

Cell[BoxData[
    \({{1, {0, 0}}, {2, {100.`, 0}}, {3, {100.`, 100.`}}, {4, {200.`, 
          0}}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(g1\  = \ 
      showNodes[\ nodes, \ pointSize -> PointSize[0.035], \ 
        pointColor -> RGBColor[0, 1, 0]]\)], "Input"],

Cell[GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
%%IncludeResource: font Times-Italic
%%IncludeFont: Times-Italic
/Times-Italic findfont 12  scalefont  setfont
% Scaling calculations
0.0238095 0.0047619 0.0147151 0.00588604 [
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
0 1 0 r
.035 w
.02381 .01472 Mdot
.5 .01472 Mdot
.5 .60332 Mdot
.97619 .01472 Mdot
% End of Graphics
MathPictureEnd
\
\>"], "Graphics",
  ImageSize->{288, 177.938},
  ImageMargins->{{0, 0}, {0, 0}},
  ImageRegion->{{0, 1}, {0, 1}},
  ImageCache->GraphicsData["Bitmap", "\<\
CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40006`00012Q000`40O003h00O1Woo2@?Pa7oo2@?P
a7oo2@?P1goo0005Ool;0n32Ool;0n32Ool;0n06Ool000Aoo`d3h<1oo`d3h<1oo`d3h0Eoo`0017oo
3@?P`7oo3@?P`7oo3@?P1Goo0003Ool?0n2nOol?0n2nOol?0n04Ool000=oo`l3h;ioo`l3h;ioo`l3
h0Aoo`000goo3`?P_Woo3`?P_Woo3`?P17oo0003Ool?0n2nOol?0n2nOol?0n04Ool000=oo`l3h;io
o`l3h;ioo`l3h0Aoo`0017oo3@?P`7oo3@?P`7oo3@?P1Goo0004Ool=0n30Ool=0n30Ool=0n05Ool0
00Eoo`/3h<9oo`/3h<9oo`/3h0Ioo`001Woo2@?Pa7oo2@?Pa7oo2@?P1goo0008Ool50n38Ool50n38
Ool50n09Ool00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mo
ok5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo003oOonaOol00?mook5oo`00ogoo/Goo
003EOol50n3FOol00==oo`T3h=Aoo`00dWoo2`?Pdgoo003AOol=0n3BOol00=5oo`d3h=9oo`00d7oo
3`?PdGoo003@Ool?0n3AOol00=1oo`l3h=5oo`00d7oo3`?PdGoo003@Ool?0n3AOol00=5oo`d3h=9o
o`00dGoo3@?PdWoo003BOol;0n3COol00001\
\>"],
  ImageRangeCache->{{{0, 431}, {265.875, 0}} -> {-5.19691, -2.50052, 0.73223, \
0.592386}}],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(elements\  = \ 
      elementProperties\  /. \ {Ar -> 1.0, \ YE -> 1.0\ 10^7}\)], "Input"],

Cell[BoxData[
    \({{1, {1.`, 1.`*^7}}, {2, {1.`, 1.`*^7}}, {3, {1.`, 1.`*^7}}, {4, {1.`, 
          1.`*^7}}, {5, {1.`, 1.`*^7}}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(g2\  = \ 
      showElements[\ \ elementNodes, \ nodes, \ 
        AspectRatio -> \ Automatic]\)], "Input"],

Cell[GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
%%IncludeResource: font Times-Italic
%%IncludeFont: Times-Italic
/Times-Italic findfont 12  scalefont  setfont
% Scaling calculations
0.0238095 0.0047619 0.0147151 0.00588604 [
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
1 0 0 r
.02 w
[ ] 0 setdash
.02381 .01472 m
.5 .60332 L
s
.02381 .01472 m
.5 .01472 L
s
.5 .01472 m
.5 .60332 L
s
.5 .01472 m
.97619 .01472 L
s
.5 .60332 m
.97619 .01472 L
s
% End of Graphics
MathPictureEnd
\
\>"], "Graphics",
  ImageSize->{288, 177.938},
  ImageMargins->{{0, 0}, {0, 0}},
  ImageRegion->{{0, 1}, {0, 1}},
  ImageCache->GraphicsData["Bitmap", "\<\
CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40006`00012Q000`40O003h00Oogoo/Goo000:Ool2
O03oOonHOol2O00;Ool000Uoo`=l0?mooiQoo`=l00Yoo`001Wooog`0Xg`027oo0006OoooO02SO008
Ool000Eoooml0:Il00Ioo`001Wooog`0Y7`01goo0006OoooO02TO007Ool000Ioooml0:=l00Qoo`00
1Wooog`0Xg`027oo0006OoooO02SO008Ool000Yoo`Yl0;moo`Ql0<1oo`Yl00]oo`002Woo2W`0_goo
27`0`7oo2W`02goo000;Ool:O02nOol8O02oOol:O00<Ool000aoo`Yl0;eoo`Ql0;ioo`Yl00eoo`00
3Goo2W`0_7oo27`0_Goo2W`03Woo000>Ool:O02kOol8O02lOol:O00?Ool000ioo`Yl0;]oo`Ql0;ao
o`Yl00moo`003goo2W`0^Woo27`0^goo2W`047oo000@Ool:O02iOol8O02jOol:O00AOol0015oo`Yl
0;Qoo`Ql0;Uoo`Yl019oo`004Woo2W`0]goo27`0^7oo2W`04goo000BOol;O02fOol8O02gOol;O00C
Ool001=oo`Yl0;Ioo`Ql0;Moo`Yl01Aoo`0057oo2W`0]Goo27`0]Woo2W`05Goo000EOol:O02dOol8
O02eOol:O00FOol001Ioo`Yl0;=oo`Ql0;Aoo`Yl01Moo`005goo2W`0/Woo27`0/goo2W`067oo000G
Ool:O02bOol8O02cOol:O00HOol001Qoo`Yl0;5oo`Ql0;9oo`Yl01Uoo`006Goo2W`0/7oo27`0/Goo
2W`06Woo000JOol:O02_Ool8O02`Ool:O00KOol001]oo`Yl0:ioo`Ql0:moo`Yl01aoo`006goo2W`0
[Woo27`0[goo2W`077oo000LOol:O02]Ool8O02^Ool:O00MOol001eoo`Yl0:aoo`Ql0:eoo`Yl01io
o`007Woo2W`0Zgoo27`0[7oo2W`07goo000OOol:O02ZOol8O02[Ool:O00POol001moo`Yl0:Yoo`Ql
0:]oo`Yl021oo`0087oo2W`0ZGoo27`0ZWoo2W`08Goo000QOol:O02XOol8O02YOol:O00ROol0029o
o`Yl0:Moo`Ql0:Qoo`Yl02=oo`008goo2W`0YWoo27`0Ygoo2W`097oo000SOol;O02UOol8O02VOol;
O00TOol002Aoo`Yl0:Eoo`Ql0:Ioo`Yl02Eoo`009Goo2W`0Y7oo27`0YGoo2W`09Woo000VOol:O02S
Ool8O02TOol:O00WOol002Moo`Yl0:9oo`Ql0:=oo`Yl02Qoo`00:7oo2W`0XGoo27`0XWoo2W`0:Goo
000XOol:O02QOol8O02ROol:O00YOol002Uoo`Yl0:1oo`Ql0:5oo`Yl02Yoo`00:Woo2W`0Wgoo27`0
X7oo2W`0:goo000[Ool:O02NOol8O02OOol:O00/Ool002aoo`Yl09eoo`Ql09ioo`Yl02eoo`00;7oo
2W`0WGoo27`0WWoo2W`0;Goo000]Ool:O02LOol8O02MOol:O00^Ool002ioo`Yl09]oo`Ql09aoo`Yl
02moo`00;goo2W`0VWoo27`0Vgoo2W`0<7oo000`Ool:O02IOol8O02JOol:O00aOol0031oo`Yl09Uo
o`Ql09Yoo`Yl035oo`00<Goo2W`0V7oo27`0VGoo2W`0<Woo000bOol:O02GOol8O02HOol:O00cOol0
03=oo`Yl09Ioo`Ql09Moo`Yl03Aoo`00=7oo2W`0UGoo27`0UWoo2W`0=Goo000dOol;O02DOol8O02E
Ool;O00eOol003Eoo`Yl09Aoo`Ql09Eoo`Yl03Ioo`00=Woo2W`0Tgoo27`0U7oo2W`0=goo000gOol:
O02BOol8O02COol:O00hOol003Qoo`Yl095oo`Ql099oo`Yl03Uoo`00>Goo2W`0T7oo27`0TGoo2W`0
>Woo000iOol:O02@Ool8O02AOol:O00jOol003Yoo`Yl08moo`Ql091oo`Yl03]oo`00>goo2W`0SWoo
27`0Sgoo2W`0?7oo000lOol:O02=Ool8O02>Ool:O00mOol003eoo`Yl08aoo`Ql08eoo`Yl03ioo`00
?Goo2W`0S7oo27`0SGoo2W`0?Woo000nOol:O02;Ool8O02<Ool:O00oOol003moo`Yl08Yoo`Ql08]o
o`Yl041oo`00@7oo2W`0RGoo27`0RWoo2W`0@Goo0011Ool:O028Ool8O029Ool:O012Ool0045oo`Yl
08Qoo`Ql08Uoo`Yl049oo`00@Woo2W`0Qgoo27`0R7oo2W`0@goo0013Ool:O026Ool8O027Ool:O014
Ool004Aoo`Yl08Eoo`Ql08Ioo`Yl04Eoo`00AGoo2W`0Q7oo27`0QGoo2W`0AWoo0015Ool;O023Ool8
O024Ool;O016Ool004Ioo`Yl08=oo`Ql08Aoo`Yl04Moo`00Agoo2W`0PWoo27`0Pgoo2W`0B7oo0018
Ool:O021Ool8O022Ool:O019Ool004Uoo`Yl081oo`Ql085oo`Yl04Yoo`00BWoo2W`0Ogoo27`0P7oo
2W`0Bgoo001:Ool:O01oOol8O020Ool:O01;Ool004]oo`Yl07ioo`Ql07moo`Yl04aoo`00C7oo2W`0
OGoo27`0OWoo2W`0CGoo001=Ool:O01lOol8O01mOol:O01>Ool004ioo`Yl07]oo`Ql07aoo`Yl04mo
o`00CWoo2W`0Ngoo27`0O7oo2W`0Cgoo001?Ool:O01jOol8O01kOol:O01@Ool0051oo`Yl07Uoo`Ql
07Yoo`Yl055oo`00DGoo2W`0N7oo27`0NGoo2W`0DWoo001BOol:O01gOol8O01hOol:O01COol0059o
o`]l07Ioo`Ql07Moo`]l05=oo`00Dgoo2W`0MWoo27`0Mgoo2W`0E7oo001DOol:O01eOol8O01fOol:
O01EOol005Eoo`Yl07Aoo`Ql07Eoo`Yl05Ioo`00EWoo2W`0Lgoo27`0M7oo2W`0Egoo001GOol:O01b
Ool8O01cOol:O01HOol005Moo`Yl079oo`Ql07=oo`Yl05Qoo`00F7oo2W`0LGoo27`0LWoo2W`0FGoo
001IOol:O01`Ool8O01aOol:O01JOol005Yoo`Yl06moo`Ql071oo`Yl05]oo`00Fgoo2W`0KWoo27`0
Kgoo2W`0G7oo001KOol:O01^Ool8O01_Ool:O01LOol005aoo`Yl06eoo`Ql06ioo`Yl05eoo`00GGoo
2W`0K7oo27`0KGoo2W`0GWoo001NOol:O01[Ool8O01/Ool:O01OOol005moo`Yl06Yoo`Ql06]oo`Yl
061oo`00Ggoo2W`0JWoo27`0Jgoo2W`0H7oo001POol:O01YOol8O01ZOol:O01QOol0065oo`Yl06Qo
o`Ql06Uoo`Yl069oo`00HWoo2W`0Igoo27`0J7oo2W`0Hgoo001SOol:O01VOol8O01WOol:O01TOol0
06=oo`]l06Eoo`Ql06Ioo`]l06Aoo`00I7oo2W`0IGoo27`0IWoo2W`0IGoo001UOol:O01TOol8O01U
Ool:O01VOol006Ioo`Yl06=oo`Ql06Aoo`Yl06Moo`00Igoo2W`0HWoo27`0Hgoo2W`0J7oo001XOol:
O01QOol8O01ROol:O01YOol006Qoo`Yl065oo`Ql069oo`Yl06Uoo`00JGoo2W`0H7oo27`0HGoo2W`0
JWoo001ZOol:O01OOol8O01POol:O01[Ool006]oo`Yl05ioo`Ql05moo`Yl06aoo`00K7oo2W`0GGoo
27`0GWoo2W`0KGoo001/Ool:O01MOol8O01NOol:O01]Ool006eoo`Yl05aoo`Ql05eoo`Yl06ioo`00
KWoo2W`0Fgoo27`0G7oo2W`0Kgoo001_Ool:O01JOol8O01KOol:O01`Ool0071oo`Yl05Uoo`Ql05Yo
o`Yl075oo`00L7oo2W`0FGoo27`0FWoo2W`0LGoo001aOol:O01HOol8O01IOol:O01bOol0079oo`Yl
05Moo`Ql05Qoo`Yl07=oo`00Lgoo2W`0EWoo27`0Egoo2W`0M7oo001dOol:O01EOol8O01FOol:O01e
Ool007Aoo`]l05Aoo`Ql05Eoo`]l07Eoo`00MGoo2W`0E7oo27`0EGoo2W`0MWoo001fOol:O01COol8
O01DOol:O01gOol007Moo`Yl059oo`Ql05=oo`Yl07Qoo`00N7oo2W`0DGoo27`0DWoo2W`0NGoo001i
Ool:O01@Ool8O01AOol:O01jOol007Uoo`Yl051oo`Ql055oo`Yl07Yoo`00NWoo2W`0Cgoo27`0D7oo
2W`0Ngoo001kOol:O01>Ool8O01?Ool:O01lOol007aoo`Yl04eoo`Ql04ioo`Yl07eoo`00OGoo2W`0
C7oo27`0CGoo2W`0OWoo001mOol:O01<Ool8O01=Ool:O01nOol007ioo`Yl04]oo`Ql04aoo`Yl07mo
o`00Ogoo2W`0BWoo27`0Bgoo2W`0P7oo0020Ool:O019Ool8O01:Ool:O021Ool0085oo`Yl04Qoo`Ql
04Uoo`Yl089oo`00PGoo2W`0B7oo27`0BGoo2W`0PWoo0022Ool:O017Ool8O018Ool:O023Ool008=o
o`Yl04Ioo`Ql04Moo`Yl08Aoo`00Q7oo2W`0AGoo27`0AWoo2W`0QGoo0025Ool:O014Ool8O015Ool:
O026Ool008Eoo`]l04=oo`Ql04Aoo`]l08Ioo`00QWoo2W`0@goo27`0A7oo2W`0Qgoo0027Ool:O012
Ool8O013Ool:O028Ool008Qoo`Yl045oo`Ql049oo`Yl08Uoo`00RGoo2W`0@7oo27`0@Goo2W`0RWoo
002:Ool:O00oOol8O010Ool:O02;Ool008Yoo`Yl03moo`Ql041oo`Yl08]oo`00Rgoo2W`0?Woo27`0
?goo2W`0S7oo002<Ool:O00mOol8O00nOol:O02=Ool008eoo`Yl03aoo`Ql03eoo`Yl08ioo`00SWoo
2W`0>goo27`0?7oo2W`0Sgoo002>Ool:O00kOol8O00lOol:O02?Ool008moo`Yl03Yoo`Ql03]oo`Yl
091oo`00T7oo2W`0>Goo27`0>Woo2W`0TGoo002AOol:O00hOol8O00iOol:O02BOol0099oo`Yl03Mo
o`Ql03Qoo`Yl09=oo`00TWoo2g`0=Woo27`0=goo2g`0Tgoo002COol:O00fOol8O00gOol:O02DOol0
09Aoo`Yl03Eoo`Ql03Ioo`Yl09Eoo`00UGoo2W`0=7oo27`0=Goo2W`0UWoo002FOol:O00cOol8O00d
Ool:O02GOol009Moo`Yl039oo`Ql03=oo`Yl09Qoo`00Ugoo2W`0<Woo27`0<goo2W`0V7oo002HOol:
O00aOol8O00bOol:O02IOol009Uoo`Yl031oo`Ql035oo`Yl09Yoo`00VWoo2W`0;goo27`0<7oo2W`0
Vgoo002KOol:O00^Ool8O00_Ool:O02LOol009]oo`Yl02ioo`Ql02moo`Yl09aoo`00W7oo2W`0;Goo
27`0;Woo2W`0WGoo002MOol:O00/Ool8O00]Ool:O02NOol009ioo`Yl02]oo`Ql02aoo`Yl09moo`00
Wgoo2W`0:Woo27`0:goo2W`0X7oo002OOol:O00ZOol8O00[Ool:O02POol00:1oo`Yl02Uoo`Ql02Yo
o`Yl0:5oo`00XGoo2W`0:7oo27`0:Goo2W`0XWoo002ROol:O00WOol8O00XOol:O02SOol00:=oo`Yl
02Ioo`Ql02Moo`Yl0:Aoo`00Xgoo2g`09Goo27`09Woo2g`0Y7oo002TOol:O00UOol8O00VOol:O02U
Ool00:Eoo`Yl02Aoo`Ql02Eoo`Yl0:Ioo`00YWoo2W`08goo27`097oo2W`0Ygoo002WOol:O00ROol8
O00SOol:O02XOol00:Qoo`Yl025oo`Ql029oo`Yl0:Uoo`00Z7oo2W`08Goo27`08Woo2W`0ZGoo002Y
Ool:O00POol8O00QOol:O02ZOol00:Yoo`Yl01moo`Ql021oo`Yl0:]oo`00Zgoo2W`07Woo27`07goo
2W`0[7oo002/Ool:O00MOol8O00NOol:O02]Ool00:aoo`Yl01eoo`Ql01ioo`Yl0:eoo`00[Goo2W`0
77oo27`07Goo2W`0[Woo002^Ool:O00KOol8O00LOol:O02_Ool00:moo`Yl01Yoo`Ql01]oo`Yl0;1o
o`00/7oo2W`06Goo27`06Woo2W`0/Goo002`Ool:O00IOol8O00JOol:O02aOol00;5oo`Yl01Qoo`Ql
01Uoo`Yl0;9oo`00/Woo2W`05goo27`067oo2W`0/goo002cOol:O00FOol8O00GOol:O02dOol00;Ao
o`Yl01Eoo`Ql01Ioo`Yl0;Eoo`00]7oo2g`057oo27`05Goo2g`0]Goo002eOol:O00DOol8O00EOol:
O02fOol00;Ioo`Yl01=oo`Ql01Aoo`Yl0;Moo`00]goo2W`04Woo27`04goo2W`0^7oo002hOol:O00A
Ool8O00BOol:O02iOol00;Uoo`Yl011oo`Ql015oo`Yl0;Yoo`00^Goo2W`047oo27`04Goo2W`0^Woo
002jOol:O00?Ool8O00@Ool:O02kOol00;]oo`Yl00ioo`Ql00moo`Yl0;aoo`00_7oo2W`03Goo27`0
3Woo2W`0_Goo002mOol:O00<Ool8O00=Ool:O02nOol00;eoo`Yl00aoo`Ql00eoo`Yl0;ioo`00_Woo
2W`02goo27`037oo2W`0_goo002oOol:O00:Ool8O00;Ool:O030Ool00<1oo`Yl00Uoo`Ql00Yoo`Yl
0<5oo`00`Goo2W`027oo27`02Goo2W`0`Woo0031Ool:O008Ool8O009Ool:O032Ool00<9oo`Yl00Mo
o`Ql00Qoo`Yl0<=oo`00`goo2W`01Woo27`01goo2W`0a7oo0034Ool:O005Ool8O006Ool:O035Ool0
0<Eoo`Yl00Aoo`Ql00Eoo`Yl0<Ioo`00aGoo2g`00goo27`017oo2g`0aWoo0036Ool:O003Ool8O004
Ool:O037Ool00<Moo`Yl009oo`Ql00=oo`Yl0<Qoo`00b7oo2W`000=oog`0O0001W`00Woo2W`0bGoo
0039OolBO0000gooO01l0008O03:Ool00<Yooa]l0<]oo`00bWoo6g`0bgoo003;OolIO03<Ool00<ao
oaMl0<eoo`00cGoo5G`0cWoo003>OolCO03?Ool00<iooa=l0<moo`00cgoo4G`0d7oo003@Ool?O03A
Ool00=5oo`el0=9oo`00dWoo2g`0dgoo003BOol;O03COol00=9oo`]l0==oo`00dgoo27`0eGoo003C
Ool8O03EOol00==oo`Ql0=Eoo`00eWoo0g`0egoo003oOonaOol00001\
\>"],
  ImageRangeCache->{{{0, 431}, {265.875, 0}} -> {-5.19691, -2.50052, 0.73223, \
0.592386}}],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Show[\ g2, \ g1, \ AspectRatio\  -> \ Automatic\ ]\)], "Input"],

Cell[GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: .5 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
%%IncludeResource: font Times-Italic
%%IncludeFont: Times-Italic
/Times-Italic findfont 12  scalefont  setfont
% Scaling calculations
0.0238095 0.0047619 0.0119048 0.0047619 [
[ 0 0 0 0 ]
[ 1 .5 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 0 m
1 0 L
1 .5 L
0 .5 L
closepath
clip
newpath
1 0 0 r
.02 w
[ ] 0 setdash
.02381 .0119 m
.5 .4881 L
s
.02381 .0119 m
.5 .0119 L
s
.5 .0119 m
.5 .4881 L
s
.5 .0119 m
.97619 .0119 L
s
.5 .4881 m
.97619 .0119 L
s
0 1 0 r
.035 w
.02381 .0119 Mdot
.5 .0119 Mdot
.5 .4881 Mdot
.97619 .0119 Mdot
% End of Graphics
MathPictureEnd
\
\>"], "Graphics",
  ImageSize->{288, 144},
  ImageMargins->{{0, 0}, {0, 0}},
  ImageRegion->{{0, 1}, {0, 1}},
  ImageCache->GraphicsData["Bitmap", "\<\
CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40006`0000f1000`40O003h00O1Woo2@?Pa7oo2@?P
a7oo2@?P1goo0005Ool;0n32Ool;0n32Ool;0n06Ool000Aoo`d3h<1oo`d3h<1oo`d3h0Eoo`0017oo
3@?P`7`03@?P`7`03@?P1Goo0003Ool?0n2nO00?0n2nO00?0n04Ool000=oo`l3h;il00l3h;il00l3
h0Aoo`000goo3`?P_W`03`?P_W`03`?P17oo0003Ool?0n2nO00?0n2nO00?0n04Ool000=oo`l3h;il
00l3h;il00l3h0Aoo`0017oo3@?P`7`03@?P`7`03@?P1Goo0004Ool=0n30O00=0n30O00=0n05Ool0
00Eoo`/3h0El0;eoo`/3h;aoo`Il00/3h0Ioo`001Woo2@?P1g`0_Goo2@?P_7oo27`02@?P1goo0008
Ool50n0:O02lOol2O0050n000g`0Oomoo`2jOol;O0050n09Ool000eoo`]l0;]oo`Ql0;]oo`al00io
o`003Woo2g`0^Woo27`0^Woo37`03goo000?Ool;O02iOol8O02iOol<O00@Ool0011oo`]l0;Qoo`Ql
0;Qoo`al015oo`004Goo2g`0]goo27`0]goo37`04Woo000BOol;O02fOol8O02fOol<O00COol001=o
o`]l0;Eoo`Ql0;Eoo`al01Aoo`0057oo2g`0]7oo27`0]7oo37`05Goo000EOol;O02cOol8O02cOol<
O00FOol001Ioo`]l0;9oo`Ql0;9oo`al01Moo`005goo2g`0/Goo27`0/Goo37`067oo000HOol;O02`
Ool8O02`Ool<O00IOol001Uoo`]l0:moo`Ql0:moo`al01Yoo`006Woo2g`0[Woo27`0[Woo37`06goo
000KOol;O02]Ool8O02]Ool<O00LOol001aoo`]l0:aoo`Ql0:aoo`al01eoo`007Goo2g`0Zgoo27`0
Zgoo37`07Woo000NOol;O02ZOol8O02ZOol<O00OOol001moo`]l0:Uoo`Ql0:Uoo`al021oo`0087oo
2g`0Z7oo27`0Z7oo37`08Goo000QOol;O02WOol8O02WOol<O00ROol0029oo`]l0:Ioo`Ql0:Ioo`al
02=oo`008goo2g`0YGoo27`0YGoo37`097oo000TOol;O02TOol8O02TOol<O00UOol002Eoo`]l0:=o
o`Ql0:=oo`al02Ioo`009Woo2g`0XWoo27`0XWoo37`09goo000WOol;O02QOol8O02QOol<O00XOol0
02Qoo`]l0:1oo`Ql0:1oo`al02Uoo`00:Goo2g`0Wgoo27`0Wgoo37`0:Woo000ZOol;O02NOol8O02N
Ool<O00[Ool002]oo`]l09eoo`Ql09eoo`al02aoo`00;7oo2g`0W7oo27`0W7oo37`0;Goo000]Ool;
O02KOol8O02KOol<O00^Ool002ioo`]l09Yoo`Ql09Yoo`al02moo`00;goo2g`0VGoo27`0VGoo37`0
<7oo000`Ool;O02HOol8O02HOol<O00aOol0035oo`]l09Moo`Ql09Moo`al039oo`00<Woo2g`0UWoo
27`0UWoo37`0<goo000cOol;O02EOol8O02EOol<O00dOol003Aoo`]l09Aoo`Ql09Aoo`al03Eoo`00
=Goo2g`0Tgoo27`0Tgoo37`0=Woo000fOol;O02BOol8O02BOol<O00gOol003Moo`]l095oo`Ql095o
o`al03Qoo`00>7oo2g`0T7oo27`0T7oo37`0>Goo000iOol;O02?Ool8O02?Ool<O00jOol003Yoo`]l
08ioo`Ql08ioo`al03]oo`00>goo2g`0SGoo27`0SGoo37`0?7oo000lOol;O02<Ool8O02<Ool<O00m
Ool003eoo`]l08]oo`Ql08]oo`al03ioo`00?Woo2g`0RWoo27`0RWoo37`0?goo000oOol;O029Ool8
O029Ool<O010Ool0041oo`]l08Qoo`Ql08Qoo`al045oo`00@Goo2g`0Qgoo27`0Qgoo37`0@Woo0012
Ool;O026Ool8O026Ool<O013Ool004=oo`]l08Eoo`Ql08Eoo`al04Aoo`00A7oo2g`0Q7oo27`0Q7oo
37`0AGoo0015Ool;O023Ool8O023Ool<O016Ool004Ioo`]l089oo`Ql089oo`al04Moo`00Agoo2g`0
PGoo27`0PGoo37`0B7oo0018Ool;O020Ool8O020Ool<O019Ool004Uoo`]l07moo`Ql07moo`al04Yo
o`00BWoo2g`0OWoo27`0OWoo37`0Bgoo001;Ool;O01mOol8O01mOol<O01<Ool004aoo`]l07aoo`Ql
07aoo`al04eoo`00CGoo2g`0Ngoo27`0Ngoo37`0CWoo001>Ool;O01jOol8O01jOol<O01?Ool004mo
o`]l07Uoo`Ql07Uoo`al051oo`00D7oo2g`0N7oo27`0N7oo37`0DGoo001AOol;O01gOol8O01gOol<
O01BOol0059oo`]l07Ioo`Ql07Ioo`al05=oo`00Dgoo2g`0MGoo27`0MGoo37`0E7oo001DOol;O01d
Ool8O01dOol<O01EOol005Eoo`]l07=oo`Ql07=oo`al05Ioo`00EWoo2g`0LWoo27`0LWoo37`0Egoo
001GOol;O01aOol8O01aOol<O01HOol005Qoo`]l071oo`Ql071oo`al05Uoo`00FGoo2g`0Kgoo27`0
Kgoo37`0FWoo001JOol;O01^Ool8O01^Ool<O01KOol005]oo`]l06eoo`Ql06eoo`al05aoo`00G7oo
2g`0K7oo27`0K7oo37`0GGoo001MOol;O01[Ool8O01[Ool<O01NOol005ioo`]l06Yoo`Ql06Yoo`al
05moo`00Ggoo2g`0JGoo27`0JGoo37`0H7oo001POol;O01XOol8O01XOol<O01QOol0065oo`]l06Mo
o`Ql06Moo`al069oo`00HWoo2g`0IWoo27`0IWoo37`0Hgoo001SOol;O01UOol8O01UOol<O01TOol0
06Aoo`]l06Aoo`Ql06Aoo`al06Eoo`00IGoo2g`0Hgoo27`0Hgoo37`0IWoo001VOol;O01ROol8O01R
Ool<O01WOol006Moo`]l065oo`Ql065oo`al06Qoo`00J7oo2g`0H7oo27`0H7oo37`0JGoo001YOol<
O01NOol8O01OOol<O01ZOol006Yoo`al05eoo`Ql05ioo`al06]oo`00Jgoo37`0G7oo27`0GGoo37`0
K7oo001/Ool<O01KOol8O01LOol<O01]Ool006eoo`al05Yoo`Ql05]oo`al06ioo`00KWoo37`0FGoo
27`0FWoo37`0Kgoo001`Ool;O01HOol8O01IOol<O01`Ool0075oo`]l05Moo`Ql05Qoo`al075oo`00
LWoo2g`0EWoo27`0Egoo37`0LWoo001cOol;O01EOol8O01FOol<O01cOol007Aoo`]l05Aoo`Ql05Eo
o`al07Aoo`00MGoo2g`0Dgoo27`0E7oo37`0MGoo001fOol;O01BOol8O01COol<O01fOol007Moo`]l
055oo`Ql059oo`al07Moo`00N7oo2g`0D7oo27`0DGoo37`0N7oo001iOol;O01?Ool8O01@Ool<O01i
Ool007Yoo`]l04ioo`Ql04moo`al07Yoo`00Ngoo2g`0CGoo27`0CWoo37`0Ngoo001lOol;O01<Ool8
O01=Ool<O01lOol007eoo`]l04]oo`Ql04aoo`al07eoo`00OWoo2g`0BWoo27`0Bgoo37`0OWoo001o
Ool;O019Ool8O01:Ool<O01oOol0081oo`]l04Qoo`Ql04Uoo`al081oo`00PGoo2g`0Agoo27`0B7oo
37`0PGoo0022Ool;O016Ool8O017Ool<O022Ool008=oo`]l04Eoo`Ql04Ioo`al08=oo`00Q7oo2g`0
A7oo27`0AGoo37`0Q7oo0025Ool;O013Ool8O014Ool<O025Ool008Ioo`]l049oo`Ql04=oo`al08Io
o`00Qgoo2g`0@Goo27`0@Woo37`0Qgoo0028Ool;O010Ool8O011Ool<O028Ool008Uoo`]l03moo`Ql
041oo`al08Uoo`00RWoo2g`0?Woo27`0?goo37`0RWoo002;Ool;O00mOol8O00nOol<O02;Ool008ao
o`]l03aoo`Ql03eoo`al08aoo`00SGoo2g`0>goo27`0?7oo37`0SGoo002>Ool;O00jOol8O00kOol<
O02>Ool008moo`]l03Uoo`Ql03Yoo`al08moo`00T7oo2g`0>7oo27`0>Goo37`0T7oo002AOol;O00g
Ool8O00hOol<O02AOol0099oo`]l03Ioo`Ql03Moo`al099oo`00Tgoo2g`0=Goo27`0=Woo37`0Tgoo
002DOol;O00dOol8O00eOol<O02DOol009Eoo`]l03=oo`Ql03Aoo`al09Eoo`00UWoo2g`0<Woo27`0
<goo37`0UWoo002GOol;O00aOol8O00bOol<O02GOol009Qoo`]l031oo`Ql035oo`al09Qoo`00VGoo
2g`0;goo27`0<7oo37`0VGoo002JOol;O00^Ool8O00_Ool<O02JOol009]oo`]l02eoo`Ql02ioo`al
09]oo`00W7oo2g`0;7oo27`0;Goo37`0W7oo002MOol;O00[Ool8O00/Ool<O02MOol009ioo`]l02Yo
o`Ql02]oo`al09ioo`00Wgoo2g`0:Goo27`0:Woo37`0Wgoo002POol;O00XOol8O00YOol<O02POol0
0:5oo`]l02Moo`Ql02Qoo`al0:5oo`00XWoo2g`09Woo27`09goo37`0XWoo002SOol;O00UOol8O00V
Ool<O02SOol00:Aoo`]l02Aoo`Ql02Eoo`al0:Aoo`00YGoo2g`08goo27`097oo37`0YGoo002VOol;
O00ROol8O00SOol<O02VOol00:Moo`]l025oo`Ql029oo`al0:Moo`00Z7oo2g`087oo27`08Goo37`0
Z7oo002YOol;O00OOol8O00POol<O02YOol00:Yoo`]l01ioo`Ql01moo`al0:Yoo`00Zgoo2g`07Goo
27`07Woo37`0Zgoo002/Ool;O00LOol8O00MOol<O02/Ool00:eoo`]l01]oo`Ql01aoo`al0:eoo`00
[Woo2g`06Woo27`06goo37`0[Woo002_Ool;O00IOol8O00JOol<O02_Ool00;1oo`]l01Qoo`Ql01Uo
o`al0;1oo`00/Goo2g`05goo27`067oo37`0/Goo002bOol;O00FOol8O00GOol<O02bOol00;=oo`]l
01Eoo`Ql01Ioo`al0;=oo`00]7oo2g`057oo27`05Goo37`0]7oo002eOol;O00COol8O00DOol<O02e
Ool00;Ioo`]l019oo`Ql01=oo`al0;Ioo`00]goo2g`04Goo27`04Woo37`0]goo002hOol;O00@Ool8
O00AOol<O02hOol00;Uoo`]l00moo`Ql011oo`al0;Uoo`00^Woo2g`03Woo27`03goo37`0^Woo002k
Ool;O00=Ool8O00>Ool<O02kOol00;aoo`]l00aoo`Ql00eoo`al0;aoo`00_Goo2g`02goo27`037oo
37`0_Goo002nOol;O00:Ool8O00;Ool<O02nOol00;moo`]l00Uoo`Ql00Yoo`al0;moo`00`7oo2g`0
27oo27`02Goo37`0`7oo0031Ool;O007Ool8O008Ool<O031Ool00<9oo`]l00Ioo`Ql00Moo`al0<9o
o`00`goo2g`01Goo27`01Woo37`0`goo0034Ool;O004Ool8O005Ool<O034Ool00<Eoo`]l00=oo`Ql
00Aoo`al0<Eoo`00aWoo2g`00Woo27`00goo37`0aWoo0037Ool;O0000gooO01l0006O002Ool<O037
Ool00<Qooa=l0003Ooml07`000Yl0<Qoo`00bGoo7W`0bGoo003:OolLO03:Ool00<]ooaYl0<]oo`00
c7oo2G`01@?P2W`0c7oo003=Ool6O0090n07O03=Ool00<ioo`Al00/3h0El0<ioo`00cgoo0W`03@?P
0g`0cgoo003@Ool00g`00n03h00;0n02O03@Ool00=1oo`l3h=5oo`00d7oo3`?PdGoo003@Ool?0n3A
Ool00=1oo`l3h=5oo`00d7oo3`?PdGoo003AOol=0n3BOol00=5oo`d3h=9oo`00\
\>"],
  ImageRangeCache->{{{0, 431}, {215, 0}} -> {-5.24524, -2.50054, 0.732566, \
0.732566}}],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell["\<\
There are other graphics utility modules that can be used to represent the \
cross-sectional areas of truss members.  Those will be introduced later in \
representing the structures.   \
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Element Stiffness Matrices and Global Stiffness Matrix", "Section",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[{
  "In addition to the input element and nodal data,  certain properties of \
the elements are needed in order to construct the element stiffness matrix of \
the truss elements.  These properties are the length of the elements and \
their position in space, which are described by the direction cosines of the \
angles each member make from the coordinate axes. The following modules, ",
  StyleBox["elementLengths",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " and ",
  StyleBox["dirCos",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " may be used to compute those properties.  "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell["a) Direction Cosines:  ", "Subsection",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?elementLength\)\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \("elementLength[enodes, nlocs]          computes the length of every \
truss element using the element         nodal connectivity matrix enodes and \
nodal location         matrix nlocs."\)], "Print",
  CellTags->"Info3216495390-5511677"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(lengths = elementLength[elementNodes, nodeLocations]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {"1", \(\@2\ \@h\^2\)},
            {"2", \(\@h\^2\)},
            {"3", \(\@h\^2\)},
            {"4", \(\@h\^2\)},
            {"5", \(\@2\ \@h\^2\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(lengths\  /. \ h -> 100\)], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {"1", \(100\ \@2\)},
            {"2", "100"},
            {"3", "100"},
            {"4", "100"},
            {"5", \(100\ \@2\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell["Direction cosines of the elements are obtained by using .   ", "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?dirCos\)\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \("dirCos[enodes, nlocs_List, elens] computes the          directional \
cosines of elements, enodes, whose nodal          coordinates are given in \
nlocs and lengths given in elens.   The module may be used for either a \
single element, or for the   entire list of elements."\)], "Print",
  CellTags->"Info3216495390-3443889"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(cos = dirCos[elementNodes, nodeLocations, lengths]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \({{1, {h\/\(\@2\ \@h\^2\), h\/\(\@2\ \@h\^2\)}}, {2, {h\/\@h\^2, 
          0}}, {3, {0, h\/\@h\^2}}, {4, {h\/\@h\^2, 
          0}}, {5, {h\/\(\@2\ \@h\^2\), \(-\(h\/\(\@2\ \@h\^2\)\)\)}}}\)], \
"Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(cos\  /. \ h -> 100\)], "Input"],

Cell[BoxData[
    \({{1, {1\/\@2, 1\/\@2}}, {2, {1, 0}}, {3, {0, 1}}, {4, {1, 
          0}}, {5, {1\/\@2, \(-\(1\/\@2\)\)}}}\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["b) Element Stiffness Matrices:  ", "Subsection",
  FontSize->18],

Cell["\<\
Element stiffness matrices may be computed directly using the element \
properties, and lengths and directional cosines.   \
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?eStiffMat\)\)], "Input"],

Cell[BoxData[
    \("eStiffMat[eprops,elens,cs]         Computes the element stiffness \
matrices of a list of          elements with properties eprops, lengths \
elens, and direction          cosines cs. The element stiffness matrix of a \
single element may   also be computed by providing only a single eprops, \
elens, and cs."\)], "Print",
  CellTags->"Info3216495391-2149240"]
}, Open  ]],

Cell["\<\
Make sure that the element directional cosines are defined (either \
symbolically or numerically) before executing the next line.   \
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(locals = eStiffMat[elementProperties, lengths\ , \ cos\ ]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \({{{\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\), \(Ar\ YE\)\/\(2\ \@2\ \@h\^2\), \
\(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\)\)\)}, {\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\), \(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\)\)\)}, {\(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\
\ \@2\ \@h\^2\)\)\), \(Ar\ YE\)\/\(2\ \@2\ \@h\^2\), \(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\)}, {\(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \
\@2\ \@h\^2\)\)\), \(Ar\ YE\)\/\(2\ \@2\ \@h\^2\), \(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\)}}, {{\(Ar\ YE\)\/\@h\^2, 0, \(-\(\(Ar\ YE\)\/\@h\^2\)\), 0}, {0, 0, 
          0, 0}, {\(-\(\(Ar\ YE\)\/\@h\^2\)\), 0, \(Ar\ YE\)\/\@h\^2, 0}, {0, 
          0, 0, 0}}, {{0, 0, 0, 0}, {0, \(Ar\ YE\)\/\@h\^2, 
          0, \(-\(\(Ar\ YE\)\/\@h\^2\)\)}, {0, 0, 0, 
          0}, {0, \(-\(\(Ar\ YE\)\/\@h\^2\)\), 
          0, \(Ar\ YE\)\/\@h\^2}}, {{\(Ar\ YE\)\/\@h\^2, 
          0, \(-\(\(Ar\ YE\)\/\@h\^2\)\), 0}, {0, 0, 0, 
          0}, {\(-\(\(Ar\ YE\)\/\@h\^2\)\), 0, \(Ar\ YE\)\/\@h\^2, 0}, {0, 0, 
          0, 0}}, {{\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\), \(-\(\(Ar\ YE\)\/\(2\ \@2\
\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(Ar\ YE\)\/\(2\ \@2\
\ \@h\^2\)}, {\(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\), \(Ar\ YE\)\/\(2\ \@2\ \@h\^2\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\
\)\)}, {\(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(Ar\ YE\)\/\(2\ \@2\ \@h\^2\
\), \(Ar\ YE\)\/\(2\ \@2\ \@h\^2\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\)}, \
{\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\
\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)}}}\)], \
"Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(locals[\([1]\)]\)], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(\(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\)\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \
\@2\ \@h\^2\)\)\)},
            {\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(\(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\)\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \
\@2\ \@h\^2\)\)\)},
            {\(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \
\@2\ \@h\^2\)\)\), \(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(\(Ar\ YE\)\/\(2\ \@2\
\ \@h\^2\)\)},
            {\(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \
\@2\ \@h\^2\)\)\), \(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(\(Ar\ YE\)\/\(2\ \@2\
\ \@h\^2\)\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(locals[\([2]\)]\)], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(\(Ar\ YE\)\/\@h\^2\), "0", \(-\(\(Ar\ YE\)\/\@h\^2\)\), "0"},
            {"0", "0", "0", "0"},
            {\(-\(\(Ar\ YE\)\/\@h\^2\)\), "0", \(\(Ar\ YE\)\/\@h\^2\), "0"},
            {"0", "0", "0", "0"}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(locals[\([3]\)]\)], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {"0", "0", "0", "0"},
            {"0", \(\(Ar\ YE\)\/\@h\^2\), "0", \(-\(\(Ar\ YE\)\/\@h\^2\)\)},
            {"0", "0", "0", "0"},
            {"0", \(-\(\(Ar\ YE\)\/\@h\^2\)\), "0", \(\(Ar\ YE\)\/\@h\^2\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(locals[\([4]\)]\)], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(\(Ar\ YE\)\/\@h\^2\), "0", \(-\(\(Ar\ YE\)\/\@h\^2\)\), "0"},
            {"0", "0", "0", "0"},
            {\(-\(\(Ar\ YE\)\/\@h\^2\)\), "0", \(\(Ar\ YE\)\/\@h\^2\), "0"},
            {"0", "0", "0", "0"}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(locals[\([5]\)]\)], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(\(Ar\ YE\)\/\(2\ \@2\
\ \@h\^2\)\)},
            {\(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(\(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\)\), \(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\)\)\)},
            {\(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(\(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\)\), \(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\)\)\)},
            {\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(\(Ar\ YE\)\/\(2\ \@2\
\ \@h\^2\)\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["c) Global Stiffness Matrix:  ", "Subsection",
  FontSize->18],

Cell[TextData[{
  "Computation of the global stiffness matrix may be done either one of two \
ways.  If the element stiffness matrices are available, then ",
  StyleBox["assembleGlobalMatrix[...]",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " may be used.   If the element stiffness matrices are not computed, then \
one can directly compute the global stiffness matrix using ",
  StyleBox["globalMatrix[.....]",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " .   "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?assembleGlobalMatrix\)\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \("assembleGlobalMatrix[elmat,nlocs,enodes] computes the         global \
stiffness matrix from the element stiffness matrices         provided in \
elmat."\)], "Print",
  CellTags->"Info3216495391-7681631"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(globalK = 
      assembleGlobalMatrix[locals, nodeLocations, elementNodes]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(\(Ar\ YE\)\/\@h\^2 + \(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(\(Ar\ \
YE\)\/\(2\ \@2\ \@h\^2\)\), \(-\(\(Ar\ YE\)\/\@h\^2\)\), 
              "0", \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\
\(2\ \@2\ \@h\^2\)\)\), "0", "0"},
            {\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(\(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\)\), "0", 
              "0", \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\
\(2\ \@2\ \@h\^2\)\)\), "0", "0"},
            {\(-\(\(Ar\ YE\)\/\@h\^2\)\), "0", \(\(2\ Ar\ YE\)\/\@h\^2\), "0",
               "0", "0", \(-\(\(Ar\ YE\)\/\@h\^2\)\), "0"},
            {"0", "0", "0", \(\(Ar\ YE\)\/\@h\^2\), 
              "0", \(-\(\(Ar\ YE\)\/\@h\^2\)\), "0", "0"},
            {\(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \
\@2\ \@h\^2\)\)\), "0", "0", \(\(Ar\ YE\)\/\(\@2\ \@h\^2\)\), 
              "0", \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(\(Ar\ YE\)\/\(2\
\ \@2\ \@h\^2\)\)},
            {\(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \
\@2\ \@h\^2\)\)\), "0", \(-\(\(Ar\ YE\)\/\@h\^2\)\), 
              "0", \(\(Ar\ YE\)\/\@h\^2 + \(Ar\ YE\)\/\(\@2\ \@h\^2\)\), \
\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\)},
            {"0", "0", \(-\(\(Ar\ YE\)\/\@h\^2\)\), 
              "0", \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(\(Ar\ YE\)\/\(2\
\ \@2\ \@h\^2\)\), \(\(Ar\ YE\)\/\@h\^2 + \(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \
\(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\)},
            {"0", "0", "0", 
              "0", \(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(-\(\(Ar\ YE\)\/\(2\ \
\@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(\(Ar\ YE\)\/\(2\
\ \@2\ \@h\^2\)\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[TextData[{
  "In a commercial code, element stiffness matrices are transparent to the \
user.  Providing only the nodal connectivity, node point locations, and \
element properties are enough to compute the lobal stiffness matrix.  In this \
program the function             ",
  StyleBox["globalMatrix[.....]",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  "   computes the global stiffness matrix directly.  "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?globalMatrix\)\)], "Input"],

Cell[BoxData[
    \("globalMatrix[enodes,nlocs,eprops] assembles the          global \
stiffness matrix for the structure directly from          the list of element \
nodal connectivity enodes, the list of          nodal coordinates nlocs, and \
the list of element properties          Ei Ai provided in eprops. "\)], \
"Print",
  CellTags->"Info3216495391-4699456"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(globalKnew = 
      globalMatrix[elementNodes, nodeLocations, elementProperties]\)], "Input",\

  AspectRatioFixed->True],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(\(Ar\ YE\)\/\@h\^2 + \(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(\(Ar\ \
YE\)\/\(2\ \@2\ \@h\^2\)\), \(-\(\(Ar\ YE\)\/\@h\^2\)\), 
              "0", \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\
\(2\ \@2\ \@h\^2\)\)\), "0", "0"},
            {\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(\(Ar\ YE\)\/\(2\ \@2\ \
\@h\^2\)\), "0", 
              "0", \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\
\(2\ \@2\ \@h\^2\)\)\), "0", "0"},
            {\(-\(\(Ar\ YE\)\/\@h\^2\)\), "0", \(\(2\ Ar\ YE\)\/\@h\^2\), "0",
               "0", "0", \(-\(\(Ar\ YE\)\/\@h\^2\)\), "0"},
            {"0", "0", "0", \(\(Ar\ YE\)\/\@h\^2\), 
              "0", \(-\(\(Ar\ YE\)\/\@h\^2\)\), "0", "0"},
            {\(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \
\@2\ \@h\^2\)\)\), "0", "0", \(\(Ar\ YE\)\/\(\@2\ \@h\^2\)\), 
              "0", \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(\(Ar\ YE\)\/\(2\
\ \@2\ \@h\^2\)\)},
            {\(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \
\@2\ \@h\^2\)\)\), "0", \(-\(\(Ar\ YE\)\/\@h\^2\)\), 
              "0", \(\(Ar\ YE\)\/\@h\^2 + \(Ar\ YE\)\/\(\@2\ \@h\^2\)\), \
\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\)},
            {"0", "0", \(-\(\(Ar\ YE\)\/\@h\^2\)\), 
              "0", \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(\(Ar\ YE\)\/\(2\
\ \@2\ \@h\^2\)\), \(\(Ar\ YE\)\/\@h\^2 + \(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \
\(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\)},
            {"0", "0", "0", 
              "0", \(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\), \(-\(\(Ar\ YE\)\/\(2\ \
\@2\ \@h\^2\)\)\), \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(\(Ar\ YE\)\/\(2\
\ \@2\ \@h\^2\)\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(globalK \[Equal] globalKnew\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(True\)], "Output"]
}, Open  ]]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Nodal Restraints and Reduced Global Stiffness Matrix", "Section",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[{
  "Restraint on nodal degrees of freedom is specified by a list whose \
elements are the node number, ",
  StyleBox["no",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  ",  and its associated restraint state,  ",
  StyleBox["{no, {uR, vR, wR}}",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  " .   For the four bar truss (four nodes) structure with the two lefth hand \
nodes restrained (nodes 1 and 3) the following specifies that the  both the u \
and v displacements of those nodes are restrained to have zero displacements \
\"0\".   The free nodes have \"1\". "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(nodeRestraints = {{1, {0, 0}}, {2, {1, 1}}, {3, {1, 1}}, \ {4, {1, 
            0}}\ }\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \({{1, {0, 0}}, {2, {1, 1}}, {3, {1, 1}}, {4, {1, 0}}}\)], "Output"]
}, Open  ]],

Cell[TextData[
"Given the nodal restraint vector, the following module reduces the stiffness \
matrix.  "], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->14],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?reduceGlobalMatrix\)\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \("reduceGlobalMatrix[glb, rest] reduces         the global stiffness \
matrix glb with the nodal restraints         provided in rest."\)], "Print",
  CellTags->"Info3216495482-6674438"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Kstiff = reduceGlobalMatrix[globalK, nodeRestraints]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(\(2\ Ar\ YE\)\/\@h\^2\), "0", "0", 
              "0", \(-\(\(Ar\ YE\)\/\@h\^2\)\)},
            {"0", \(\(Ar\ YE\)\/\@h\^2\), "0", \(-\(\(Ar\ YE\)\/\@h\^2\)\), 
              "0"},
            {"0", "0", \(\(Ar\ YE\)\/\(\@2\ \@h\^2\)\), 
              "0", \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\)},
            {"0", \(-\(\(Ar\ YE\)\/\@h\^2\)\), 
              "0", \(\(Ar\ YE\)\/\@h\^2 + \(Ar\ YE\)\/\(\@2\ \@h\^2\)\), \
\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)},
            {\(-\(\(Ar\ YE\)\/\@h\^2\)\), 
              "0", \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(\(Ar\ YE\)\/\(2\
\ \@2\ \@h\^2\)\), \(\(Ar\ YE\)\/\@h\^2 + \(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell["\<\
Note that, for free nodes the \"1\" may be dropped from the restrained nodes \
list.  In a commercial code, only those nodes that are restrained are \
specified.  \
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(nodeRestraints = {{1, {0, 0}}, \ {4, {1, 0}}\ }\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \({{1, {0, 0}}, {4, {1, 0}}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Kstiff1\  = reduceGlobalMatrix[globalK, nodeRestraints]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(\(2\ Ar\ YE\)\/\@h\^2\), "0", "0", 
              "0", \(-\(\(Ar\ YE\)\/\@h\^2\)\)},
            {"0", \(\(Ar\ YE\)\/\@h\^2\), "0", \(-\(\(Ar\ YE\)\/\@h\^2\)\), 
              "0"},
            {"0", "0", \(\(Ar\ YE\)\/\(\@2\ \@h\^2\)\), 
              "0", \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\)},
            {"0", \(-\(\(Ar\ YE\)\/\@h\^2\)\), 
              "0", \(\(Ar\ YE\)\/\@h\^2 + \(Ar\ YE\)\/\(\@2\ \@h\^2\)\), \
\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)},
            {\(-\(\(Ar\ YE\)\/\@h\^2\)\), 
              "0", \(-\(\(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)\), \(\(Ar\ YE\)\/\(2\
\ \@2\ \@h\^2\)\), \(\(Ar\ YE\)\/\@h\^2 + \(Ar\ YE\)\/\(2\ \@2\ \@h\^2\)\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Kstiff\  == Kstiff1\)], "Input"],

Cell[BoxData[
    \(True\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData["Nodal Forces and Solution"], "Section",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["\<\
Nodal forces are specified similar to the nodal restraints with a list of \
node numbers at which the forces are specified.  The forces are the second \
elements of the elements of the list.  For the 4-bar truss with a force \
acting at node 2, vertically down, we have  \
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(nodeForces = {\ {3, \ {0, \(-Pload\)}}\ \ }\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \({{3, {0, \(-Pload\)}}}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "For a complete force vector (before reducing the size to account for the \
nodal restraints, one can use the module ",
  StyleBox["forceVector",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " to see the complete right-hand-side force vector.  "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?forceVector\ \)\)], "Input"],

Cell[BoxData[
    \("forceVector[forces_, nNodes_Integer]  needs a description written for \
it"\)], "Print",
  CellTags->"Info3216495494-6568000"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(forceVector[\ nodeForces, \ 4]\)], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {"0", "0"},
            {"0", "0"},
            {"0", \(-Pload\)},
            {"0", "0"}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[TextData[{
  "To reduce the nodal forces use ",
  StyleBox["reduceForceVector",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " .  This module constucts the right-hand-side vector that can be used to \
solve the displacement degrees of freedom."
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?reduceForceVector\)\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \("reduceForceVector[forc, rest, nNodes] reduces the force         vector \
forc using restraint definitions rest."\)], "Print",
  CellTags->"Info3216495494-2764126"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(RHS\  = \ 
      reduceForceVector[\ nodeForces, \ nodeRestraints, \ 4]\)], "Input"],

Cell[BoxData[
    \({0, 0, 0, \(-Pload\), 0}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "The unknowns of the problem are the displacements at those nodes that are \
free to move. To solve for the unknown displacements, ",
  Cell[BoxData[
      \(TraditionalForm\`{u\_1\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`v\_1, \ u\_2, \ ...\( .\(, \ etc\)\)\)]],
  "}, set up the system, "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(unknown = {u2, v2, u3, v3, \ u4}\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \({u2, v2, u3, v3, u4}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(system = Kstiff . unknown == RHS\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \({\(2\ Ar\ u2\ YE\)\/\@h\^2 - \(Ar\ u4\ YE\)\/\@h\^2, \(Ar\ v2\ \
YE\)\/\@h\^2 - \(Ar\ v3\ YE\)\/\@h\^2, \(Ar\ u3\ YE\)\/\(\@2\ \@h\^2\) - \(Ar\
\ u4\ YE\)\/\(2\ \@2\ \@h\^2\), \(Ar\ u4\ YE\)\/\(2\ \@2\ \@h\^2\) - \(Ar\ v2\
\ YE\)\/\@h\^2 + 
          v3\ \((\(Ar\ YE\)\/\@h\^2 + \(Ar\ YE\)\/\(\@2\ \@h\^2\))\), \
\(-\(\(Ar\ u2\ YE\)\/\@h\^2\)\) - \(Ar\ u3\ YE\)\/\(2\ \@2\ \@h\^2\) + \(Ar\ \
v3\ YE\)\/\(2\ \@2\ \@h\^2\) + 
          u4\ \((\(Ar\ YE\)\/\@h\^2 + \(Ar\ YE\)\/\(2\ \@2\ \@h\^2\))\)} == \
{0, 0, 0, \(-Pload\), 0}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(soln = Flatten[\ N[Solve[system, unknown]]\ ]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \({u2 \[Rule] \(0.5`\ \@h\^2\ Pload\)\/\(Ar\ YE\), 
      u3 \[Rule] \(0.5`\ \@h\^2\ Pload\)\/\(Ar\ YE\), 
      u4 \[Rule] \(\@h\^2\ Pload\)\/\(Ar\ YE\), 
      v2 \[Rule] \(-\(\(1.9142135623730951`\ \@h\^2\ Pload\)\/\(Ar\ YE\)\)\), 
      v3 \[Rule] \(-\(\(1.9142135623730951`\ \@h\^2\ Pload\)\/\(Ar\ YE\)\)\)}\
\)], "Output"]
}, Open  ]],

Cell["\<\
Rather than soling the system of equations, we could have inverted the matrix \
and multiply the force vector.  \
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(Kinverse\  = \ Inverse[Kstiff]\)], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(\@h\^2\/\(Ar\ YE\)\), \(-\(\@h\^2\/\(2\ Ar\ YE\)\)\), \(\@h\^2\
\/\(2\ Ar\ YE\)\), \(-\(\@h\^2\/\(2\ Ar\ YE\)\)\), \(\@h\^2\/\(Ar\ YE\)\)},
            {\(-\(\@h\^2\/\(2\ Ar\ YE\)\)\), \(\(2\ h\^6\ \((\(3\ Ar\^4\ \
YE\^4\)\/\(4\ h\^4\) + \(Ar\^4\ YE\^4\)\/\(\@2\ h\^4\))\)\)\/\(Ar\^5\ \@h\^2\ \
YE\^5\)\), \(-\(\@h\^2\/\(2\ Ar\ YE\)\)\), \(\(2\ h\^6\ \((\(Ar\^4\ YE\^4\)\/\
\(4\ h\^4\) + \(Ar\^4\ YE\^4\)\/\(\@2\ h\^4\))\)\)\/\(Ar\^5\ \@h\^2\ \
YE\^5\)\), \(-\(\@h\^2\/\(Ar\ YE\)\)\)},
            {\(\@h\^2\/\(2\ Ar\ YE\)\), \(-\(\@h\^2\/\(2\ Ar\ YE\)\)\), \(\(2\
\ h\^6\ \((\(Ar\^4\ YE\^4\)\/\(4\ h\^4\) + \(Ar\^4\ YE\^4\)\/\(\@2\ h\^4\))\)\
\)\/\(Ar\^5\ \@h\^2\ YE\^5\)\), \(-\(\@h\^2\/\(2\ Ar\ YE\)\)\), \
\(\@h\^2\/\(Ar\ YE\)\)},
            {\(-\(\@h\^2\/\(2\ Ar\ YE\)\)\), \(\(2\ h\^6\ \((\(Ar\^4\ YE\^4\)\
\/\(4\ h\^4\) + \(Ar\^4\ YE\^4\)\/\(\@2\ h\^4\))\)\)\/\(Ar\^5\ \@h\^2\ \
YE\^5\)\), \(-\(\@h\^2\/\(2\ Ar\ YE\)\)\), \(\(2\ h\^6\ \((\(Ar\^4\ YE\^4\)\/\
\(4\ h\^4\) + \(Ar\^4\ YE\^4\)\/\(\@2\ h\^4\))\)\)\/\(Ar\^5\ \@h\^2\ \
YE\^5\)\), \(-\(\@h\^2\/\(Ar\ YE\)\)\)},
            {\(\@h\^2\/\(Ar\ YE\)\), \(-\(\@h\^2\/\(Ar\ YE\)\)\), \
\(\@h\^2\/\(Ar\ YE\)\), \(-\(\@h\^2\/\(Ar\ YE\)\)\), \(\(2\ \@h\^2\)\/\(Ar\ \
YE\)\)}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Kinverse\  . \ RHS\)], "Input"],

Cell[BoxData[
    \({\(\@h\^2\ Pload\)\/\(2\ Ar\ YE\), \(-\(\(2\ h\^6\ Pload\ \((\(Ar\^4\ \
YE\^4\)\/\(4\ h\^4\) + \(Ar\^4\ YE\^4\)\/\(\@2\ h\^4\))\)\)\/\(Ar\^5\ \@h\^2\ \
YE\^5\)\)\), \(\@h\^2\ Pload\)\/\(2\ Ar\ YE\), \(-\(\(2\ h\^6\ Pload\ \((\(Ar\
\^4\ YE\^4\)\/\(4\ h\^4\) + \(Ar\^4\ YE\^4\)\/\(\@2\ h\^4\))\)\)\/\(Ar\^5\ \
\@h\^2\ YE\^5\)\)\), \(\@h\^2\ Pload\)\/\(Ar\ YE\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(soln\)], "Input"],

Cell[BoxData[
    \({u2 \[Rule] \(0.5`\ \@h\^2\ Pload\)\/\(Ar\ YE\), 
      u3 \[Rule] \(0.5`\ \@h\^2\ Pload\)\/\(Ar\ YE\), 
      u4 \[Rule] \(\@h\^2\ Pload\)\/\(Ar\ YE\), 
      v2 \[Rule] \(-\(\(1.9142135623730951`\ \@h\^2\ Pload\)\/\(Ar\ YE\)\)\), 
      v3 \[Rule] \(-\(\(1.9142135623730951`\ \@h\^2\ Pload\)\/\(Ar\ YE\)\)\)}\
\)], "Output"]
}, Open  ]],

Cell["\<\
The displacements of all the nodes  are (including the zero displacements \
specified through nodeRestraints) \
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(nodeRestraints\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \({{1, {0, 0}}, {4, {1, 0}}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(displacements = {0, 0, u2, v2, u3, v3, u4, 0} /. soln\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \({0, 
      0, \(0.5`\ \@h\^2\ Pload\)\/\(Ar\ YE\), \(-\(\(1.9142135623730951`\ \@h\
\^2\ Pload\)\/\(Ar\ YE\)\)\), \(0.5`\ \@h\^2\ Pload\)\/\(Ar\ YE\), \
\(-\(\(1.9142135623730951`\ \@h\^2\ Pload\)\/\(Ar\ YE\)\)\), \(\@h\^2\ \
Pload\)\/\(Ar\ YE\), 0}\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Structural Configuration", "Section",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(Information["\<showStructure\>", LongForm \[Rule] False]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \("showStructure[nodes,locs,areas_Optional]         shows the \
configuration of the truss by plotting the nodal          points and the \
truss members.  If the optional          cross-sectional areas of the members \
are provided in a list         areas, then the thicknesses of the members are \
shown           proportional to the areas of the members."\)], "Print",
  CellTags->"Info3216495516-7563905"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(undeforemed\  = \ showStructure[elementNodes, nodes]\)], "Input",
  AspectRatioFixed->True],

Cell[GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: .5 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
%%IncludeResource: font Times-Italic
%%IncludeFont: Times-Italic
/Times-Italic findfont 12  scalefont  setfont
% Scaling calculations
0.0238095 0.0047619 0.0119048 0.0047619 [
[ 0 0 0 0 ]
[ 1 .5 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 0 m
1 0 L
1 .5 L
0 .5 L
closepath
clip
newpath
0 g
2 Mabswid
[ ] 0 setdash
.02381 .0119 m
.5 .4881 L
s
.02381 .0119 m
.5 .0119 L
s
.5 .0119 m
.5 .4881 L
s
.5 .0119 m
.97619 .0119 L
s
.5 .4881 m
.97619 .0119 L
s
.03 w
.02381 .0119 Mdot
.5 .0119 Mdot
.5 .4881 Mdot
.97619 .0119 Mdot
% End of Graphics
MathPictureEnd
\
\>"], "Graphics",
  ImageSize->{288, 144},
  ImageMargins->{{0, 0}, {0, 0}},
  ImageRegion->{{0, 1}, {0, 1}},
  ImageCache->GraphicsData["Bitmap", "\<\
CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40006`0000f1000`40O003h00Oogoo/Goo0009Ool2
003;Ool2003;Ool2000;Ool000Ioo`P00<Eoo`P00<Eoo`P000Qoo`001Goo2P00`goo2P00`goo2P00
1goo0005Ool:0033Ool:0033Ool:0007Ool000Eoool00:D000Moo`0017ooo`00Y`001Woo0004Oooo
002W0006Ool000Eoo`X00<=oo`X00<=oo`X000Moo`001Goo2`00`Woo2P00`goo2P001goo0005Ool<
0031Ool:0032Ool;0007Ool000Ioo``00<5oo`P00<9oo`/000Qoo`002Goo0P0017oo1000`goo0`00
`goo10000goo0P002goo000@Ool40032Ool30032Ool4000AOol0015oo`@00<5oo`<00<5oo`@0019o
o`004Woo1000`7oo0`00`7oo10004goo000COol4002oOol3002oOol4000DOol001Aoo`@00;ioo`<0
0;ioo`@001Eoo`005Goo1000_Goo0`00_Goo10005Woo000FOol4002lOol3002lOol4000GOol001Mo
o`@00;]oo`<00;]oo`@001Qoo`0067oo1000^Woo0`00^Woo10006Goo000IOol4002iOol3002iOol4
000JOol001Yoo`@00;Qoo`<00;Qoo`@001]oo`006goo1000]goo0`00]goo100077oo000LOol4002f
Ool3002fOol4000MOol001eoo`@00;Eoo`<00;Eoo`@001ioo`007Woo1000]7oo0`00]7oo10007goo
000OOol4002cOol3002cOol4000POol0021oo`@00;9oo`<00;9oo`@0025oo`008Goo1000/Goo0`00
/Goo10008Woo000ROol4002`Ool3002`Ool4000SOol002=oo`@00:moo`<00:moo`@002Aoo`0097oo
1000[Woo0`00[Woo10009Goo000UOol4002]Ool3002]Ool4000VOol002Ioo`@00:aoo`<00:aoo`@0
02Moo`009goo1000Zgoo0`00Zgoo1000:7oo000XOol4002ZOol3002ZOol4000YOol002Uoo`@00:Uo
o`<00:Uoo`@002Yoo`00:Woo1000Z7oo0`00Z7oo1000:goo000[Ool4002WOol3002WOol4000/Ool0
02aoo`@00:Ioo`<00:Ioo`@002eoo`00;Goo1000YGoo0`00YGoo1000;Woo000^Ool4002TOol3002T
Ool4000_Ool002moo`@00:=oo`<00:=oo`@0031oo`00<7oo1000XWoo0`00XWoo1000<Goo000aOol4
002QOol3002QOol4000bOol0039oo`@00:1oo`<00:1oo`@003=oo`00<goo1000Wgoo0`00Wgoo1000
=7oo000dOol4002NOol3002NOol4000eOol003Eoo`@009eoo`<009eoo`@003Ioo`00=Woo1000W7oo
0`00W7oo1000=goo000gOol4002KOol3002KOol4000hOol003Qoo`@009Yoo`<009Yoo`@003Uoo`00
>Goo1000VGoo0`00VGoo1000>Woo000jOol4002HOol3002HOol4000kOol003]oo`@009Moo`<009Mo
o`@003aoo`00?7oo1000UWoo0`00UWoo1000?Goo000mOol4002EOol3002EOol4000nOol003ioo`@0
09Aoo`<009Aoo`@003moo`00?goo1000Tgoo0`00Tgoo1000@7oo0010Ool4002BOol3002BOol40011
Ool0045oo`@0095oo`<0095oo`@0049oo`00@Woo1000T7oo0`00T7oo1000@goo0013Ool4002?Ool3
002?Ool40014Ool004Aoo`@008ioo`<008ioo`@004Eoo`00AGoo1000SGoo0`00SGoo1000AWoo0016
Ool4002<Ool3002<Ool40017Ool004Moo`@008]oo`<008]oo`@004Qoo`00B7oo1000RWoo0`00RWoo
1000BGoo0019Ool40029Ool30029Ool4001:Ool004Yoo`@008Qoo`<008Qoo`@004]oo`00Bgoo1000
Qgoo0`00Qgoo1000C7oo001<Ool40026Ool30026Ool4001=Ool004eoo`@008Eoo`<008Eoo`@004io
o`00CWoo1000Q7oo0`00Q7oo1000Cgoo001?Ool40023Ool30023Ool4001@Ool0051oo`@0089oo`<0
089oo`@0055oo`00DGoo1000PGoo0`00PGoo1000DWoo001BOol40020Ool30020Ool4001COol005=o
o`@007moo`<007moo`@005Aoo`00E7oo1000OWoo0`00OWoo1000EGoo001EOol4001mOol3001mOol4
001FOol005Ioo`@007aoo`<007aoo`@005Moo`00Egoo1000Ngoo0`00Ngoo1000F7oo001HOol4001j
Ool3001jOol4001IOol005Uoo`@007Uoo`<007Uoo`@005Yoo`00FWoo1000N7oo0`00N7oo1000Fgoo
001KOol4001gOol3001gOol4001LOol005aoo`@007Ioo`<007Ioo`@005eoo`00GGoo1000MGoo0`00
MGoo1000GWoo001NOol4001dOol3001dOol4001OOol005moo`@007=oo`<007=oo`@0061oo`00H7oo
1000LWoo0`00LWoo1000HGoo001QOol4001aOol3001aOol4001ROol0069oo`@0071oo`<0071oo`@0
06=oo`00Hgoo1000Kgoo0`00Kgoo1000I7oo001TOol4001^Ool3001^Ool4001UOol006Eoo`@006eo
o`<006eoo`@006Ioo`00IWoo1000K7oo0`00K7oo1000Igoo001WOol4001[Ool3001[Ool4001XOol0
06Qoo`@006Yoo`<006Yoo`@006Uoo`00JGoo1000JGoo0`00JGoo1000JWoo001ZOol4001XOol3001X
Ool4001[Ool006]oo`@006Moo`<006Moo`@006aoo`00K7oo1000IWoo0`00IWoo1000KGoo001]Ool4
001UOol3001UOol4001^Ool006ioo`@006Aoo`<006Aoo`@006moo`00Kgoo1000Hgoo0`00Hgoo1000
L7oo001`Ool4001ROol3001ROol4001aOol0075oo`@0065oo`<0065oo`@0079oo`00LWoo1000H7oo
0`00H7oo1000Lgoo001cOol4001OOol3001OOol4001dOol007Aoo`@005ioo`<005ioo`@007Eoo`00
MGoo1000GGoo0`00GGoo1000MWoo001fOol4001LOol3001LOol4001gOol007Moo`@005]oo`<005]o
o`@007Qoo`00N7oo1000FWoo0`00FWoo1000NGoo001iOol4001IOol3001IOol4001jOol007Yoo`@0
05Qoo`<005Qoo`@007]oo`00Ngoo1000Egoo0`00Egoo1000O7oo001lOol4001FOol3001FOol4001m
Ool007eoo`@005Eoo`<005Eoo`@007ioo`00OWoo1000E7oo0`00E7oo1000Ogoo001oOol4001COol3
001COol40020Ool0081oo`@0059oo`<0059oo`@0085oo`00PGoo1000DGoo0`00DGoo1000PWoo0022
Ool4001@Ool3001@Ool40023Ool008=oo`@004moo`<004moo`@008Aoo`00Q7oo1000CWoo0`00CWoo
1000QGoo0025Ool4001=Ool3001=Ool40026Ool008Ioo`@004aoo`<004aoo`@008Moo`00Qgoo1000
Bgoo0`00Bgoo1000R7oo0028Ool4001:Ool3001:Ool40029Ool008Uoo`@004Uoo`<004Uoo`@008Yo
o`00RWoo1000B7oo0`00B7oo1000Rgoo002;Ool40017Ool30017Ool4002<Ool008aoo`@004Ioo`<0
04Ioo`@008eoo`00SGoo1000AGoo0`00AGoo1000SWoo002>Ool40014Ool30014Ool4002?Ool008mo
o`@004=oo`<004=oo`@0091oo`00T7oo1000@Woo0`00@Woo1000TGoo002AOol40011Ool30011Ool4
002BOol0099oo`@0041oo`<0041oo`@009=oo`00Tgoo1000?goo0`00?goo1000U7oo002DOol4000n
Ool3000nOol4002EOol009Eoo`@003eoo`<003eoo`@009Ioo`00UWoo1000?7oo0`00?7oo1000Ugoo
002GOol4000kOol3000kOol4002HOol009Qoo`@003Yoo`<003Yoo`@009Uoo`00VGoo1000>Goo0`00
>Goo1000VWoo002JOol4000hOol3000hOol4002KOol009]oo`@003Moo`<003Moo`@009aoo`00W7oo
1000=Woo0`00=Woo1000WGoo002MOol4000eOol3000eOol4002NOol009ioo`@003Aoo`<003Aoo`@0
09moo`00Wgoo1000<goo0`00<goo1000X7oo002POol4000bOol3000bOol4002QOol00:5oo`@0035o
o`<0035oo`@00:9oo`00XWoo1000<7oo0`00<7oo1000Xgoo002SOol4000_Ool3000_Ool4002TOol0
0:Aoo`@002ioo`<002ioo`@00:Eoo`00YGoo1000;Goo0`00;Goo1000YWoo002VOol4000/Ool3000/
Ool4002WOol00:Moo`@002]oo`<002]oo`@00:Qoo`00Z7oo1000:Woo0`00:Woo1000ZGoo002YOol4
000YOol3000YOol4002ZOol00:Yoo`@002Qoo`<002Qoo`@00:]oo`00Zgoo10009goo0`009goo1000
[7oo002/Ool4000VOol3000VOol4002]Ool00:eoo`@002Eoo`<002Eoo`@00:ioo`00[Woo100097oo
0`0097oo1000[goo002_Ool4000SOol3000SOol4002`Ool00;1oo`@0029oo`<0029oo`@00;5oo`00
/Goo10008Goo0`008Goo1000/Woo002bOol4000POol3000POol4002cOol00;=oo`@001moo`<001mo
o`@00;Aoo`00]7oo10007Woo0`007Woo1000]Goo002eOol4000MOol3000MOol4002fOol00;Ioo`@0
01aoo`<001aoo`@00;Moo`00]goo10006goo0`006goo1000^7oo002hOol4000JOol3000JOol4002i
Ool00;Uoo`@001Uoo`<001Uoo`@00;Yoo`00^Woo100067oo0`0067oo1000^goo002kOol4000GOol3
000GOol4002lOol00;aoo`@001Ioo`<001Ioo`@00;eoo`00_Goo10005Goo0`005Goo1000_Woo002n
Ool4000DOol3000DOol4002oOol00;moo`@001=oo`<001=oo`@00<1oo`00`7oo10004Woo0`004Woo
1000`Goo0031Ool4000AOol3000AOol40032Ool00<9oo`@0011oo`<0011oo`@00<=oo`00`goo1000
3goo0`003goo1000a7oo0034Ool4000>Ool3000>Ool40035Ool00<Eoo`@000eoo`<000eoo`@00<Io
o`00aWoo100037oo0`0037oo1000agoo0037Ool4000;Ool3000;Ool40038Ool00<Qoo`@000Yoo`<0
00Yoo`@00<Uoo`00bGoo10002Goo0`002Goo1000bWoo003:Ool40008Ool30008Ool4003;Ool00<]o
o`@000Moo`<000Moo`@00<aoo`00c7oo10001Woo0`001Woo1000cGoo003=Ool40005Ool30005Ool4
003>Ool00<ioo`@000Aoo`<000Aoo`@00<moo`00cgoo10000goo0`000goo1000d7oo003@Ool40002
Ool30002Ool4003AOol00=5oo`@00005Ool00000001oo`04003BOol00=9oo`/00==oo`00dWoo2P00
e7oo003BOol:003DOol00=9oo`X00=Aoo`00dGoo3000dgoo003AOol<003COol00=9oo`X00=Aoo`00
dWoo2P00e7oo003BOol:003DOol00001\
\>"],
  ImageRangeCache->{{{0, 431}, {215, 0}} -> {-5.24524, -2.50054, 0.732566, \
0.732566}}],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell["\<\
It is also possible to show the deformed shape of the structure by adding the \
nodal displacements to the node point locations. \
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(displacements\)], "Input"],

Cell[BoxData[
    \({0, 
      0, \(0.5`\ \@h\^2\ Pload\)\/\(Ar\ YE\), \(-\(\(1.9142135623730951`\ \@h\
\^2\ Pload\)\/\(Ar\ YE\)\)\), \(0.5`\ \@h\^2\ Pload\)\/\(Ar\ YE\), \
\(-\(\(1.9142135623730951`\ \@h\^2\ Pload\)\/\(Ar\ YE\)\)\), \(\@h\^2\ \
Pload\)\/\(Ar\ YE\), 0}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(UVdisplacements\  = \ 
      Partition[\ displacements, \ 2\ ]\  /. \ {h \[Rule] \ 100, \ 
          Pload\  \[Rule] 100\ 10^3, \ Ar \[Rule] 1, \ 
          YE \[Rule] \ 1.0\ 10^7}\)], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {"0", "0"},
            {"0.49999999999999994`", \(-1.9142135623730951`\)},
            {"0.49999999999999994`", \(-1.9142135623730951`\)},
            {"1.`", "0"}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(magFactor\  = \ 10\)], "Input"],

Cell[BoxData[
    \(10\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(nodesDeformed\  = \ 
      MapThread[\ {#1[\([1]\)], \ {#1[\([2, 1]\)] + 
                magFactor*#2[\([1]\)], \ #1[\([2, 2]\)] + 
                magFactor*#2[\([2]\)]}}\  &, \ {nodes, \ 
          UVdisplacements}]\)], "Input"],

Cell[BoxData[
    \({{1, {0, 0}}, {2, {105.`, \(-19.14213562373095`\)}}, {3, {105.`, 
          80.85786437626905`}}, {4, {210.`, 0}}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(deformed\  = \ showStructure[elementNodes, nodesDeformed]\)], "Input",
  AspectRatioFixed->True],

Cell[GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: .47619 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
%%IncludeResource: font Times-Italic
%%IncludeFont: Times-Italic
/Times-Italic findfont 12  scalefont  setfont
% Scaling calculations
0.0238095 0.00453515 0.0981503 0.00453515 [
[ 0 0 0 0 ]
[ 1 .47619 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 0 m
1 0 L
1 .47619 L
0 .47619 L
closepath
clip
newpath
0 g
2 Mabswid
[ ] 0 setdash
.02381 .09815 m
.5 .46485 L
s
.02381 .09815 m
.5 .01134 L
s
.5 .01134 m
.5 .46485 L
s
.5 .01134 m
.97619 .09815 L
s
.5 .46485 m
.97619 .09815 L
s
.03 w
.02381 .09815 Mdot
.5 .01134 Mdot
.5 .46485 Mdot
.97619 .09815 Mdot
% End of Graphics
MathPictureEnd
\
\>"], "Graphics",
  ImageSize->{288, 137.125},
  ImageMargins->{{0, 0}, {0, 0}},
  ImageRegion->{{0, 1}, {0, 1}},
  ImageCache->GraphicsData["Bitmap", "\<\
CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40006`0000cA000`40O003h00OeWoo0P00f7oo003C
Ool8003EOol00=9oo`X00=Aoo`00dWoo2P00e7oo003BOol:003DOol00<mooa400=1oo`00bWoo6`00
bgoo0034OolW0035Ool00;mooa0000=oo`X000Aooa000<1oo`00^Goo4@0027oo2P002Goo4@00^Woo
002dOol@000?Ool8000@Ool@002eOol00:iooa4001Moo`<001Mooa400:moo`00Z7oo4@007Goo0`00
7Goo4@00ZGoo002SOolA000ROol3000ROolA002TOol009eooa4002Qoo`<002Qooa4009ioo`00V7oo
4000;Woo0`00;Woo4000VGoo002BOolA000cOol3000cOolA002COol008eooa0003Uoo`<003Uooa00
08ioo`00Qgoo4@00?Woo0`00?Woo4@00R7oo0022Ool@0014Ool30014Ool@0023Ool007aooa4004Uo
o`<004Uooa4007eoo`00Mgoo4000Cgoo0`00Cgoo4000N7oo001aOolA001DOol3001DOolA001bOol0
06]ooa4005Yoo`<005Yooa4006aoo`00IWoo4@00Ggoo0`00Ggoo4@00Igoo001POolA001UOol3001U
OolA001QOol005]ooa0006]oo`<006]ooa0005aoo`00EGoo4@00L7oo0`00L7oo4@00EWoo001@Ool@
001fOol3001fOol@001AOol004Yooa4007]oo`<007]ooa4004]oo`00AGoo4000PGoo0`00PGoo4000
AWoo000oOolA0026Ool30026OolA0010Ool003Yooa0008aoo`<008aooa0003]oo`00=7oo4@00TGoo
0`00TGoo4@00=Goo000^OolA002GOol3002GOolA000_Ool002Uooa4009aoo`<009aooa4002Yoo`00
8goo4@00XWoo0`00XWoo4@0097oo0009Ool2000COol@002XOol3002XOol@000BOol2000;Ool000Io
o`P000Yooa400:eoo`<00:eooa4000Uoo`P000Qoo`001Goo2P0017oo4000/goo0`00/goo40000goo
2P001goo0005OolI002hOol3002hOolH0007Ool000Eooa<00;ioo`<00;iooa8000Moo`0017oo3`00
`goo0`00`goo3P001Woo0004Ool<0036Ool30035Ool<0006Ool000Eoo`/00<Ioo`<00<Ioo`X000Mo
o`001Goo3000aGoo0`00aGoo2`001goo0005Ool=0034Ool30034Ool<0007Ool000Ioo`P00003Ool0
000000800<=oo`<00<=oo``000Qoo`002Goo0P001Goo1@00`Goo0`00`Goo1@0017oo0P002goo000A
Ool50030Ool30030Ool5000BOol0019oo`D00;moo`<00;moo`D001=oo`0057oo1@00_Goo0`00_Goo
1@005Goo000EOol5002lOol3002lOol5000FOol001Ioo`D00;]oo`<00;]oo`D001Moo`0067oo1@00
^Goo0`00^Goo1@006Goo000IOol5002hOol3002hOol5000JOol001Yoo`D00;Moo`<00;Moo`D001]o
o`0077oo1000]Woo0`00]Woo10007Goo000MOol5002dOol3002dOol5000NOol001ioo`D00;=oo`<0
0;=oo`D001moo`007goo1@00/Woo0`00/Woo1@0087oo000QOol5002`Ool3002`Ool5000ROol0029o
o`D00:moo`<00:moo`D002=oo`008goo1@00[Woo0`00[Woo1@0097oo000UOol5002/Ool3002/Ool5
000VOol002Ioo`D00:]oo`<00:]oo`D002Moo`009goo1@00ZWoo0`00ZWoo1@00:7oo000YOol4002Y
Ool3002YOol4000ZOol002Yoo`D00:Moo`<00:Moo`D002]oo`00:goo1@00YWoo0`00YWoo1@00;7oo
000/Ool5002UOol3002UOol5000]Ool002ioo`D00:=oo`<00:=oo`D002moo`00;goo1@00XWoo0`00
XWoo1@00<7oo000`Ool5002QOol3002QOol5000aOol0039oo`@00:1oo`<00:1oo`@003=oo`00<goo
1@00WWoo0`00WWoo1@00=7oo000dOol5002MOol3002MOol5000eOol003Eoo`D009aoo`<009aoo`D0
03Ioo`00=goo1@00VWoo0`00VWoo1@00>7oo000hOol5002IOol3002IOol5000iOol003Uoo`D009Qo
o`<009Qoo`D003Yoo`00>goo1@00UWoo0`00UWoo1@00?7oo000lOol5002EOol3002EOol5000mOol0
03eoo`D009Aoo`<009Aoo`D003ioo`00?goo1000Tgoo0`00Tgoo1000@7oo0010Ool5002AOol3002A
Ool50011Ool0045oo`D0091oo`<0091oo`D0049oo`00@Woo1@00Sgoo0`00Sgoo1@00@goo0014Ool5
002=Ool3002=Ool50015Ool004Eoo`D008aoo`<008aoo`D004Ioo`00AWoo1@00Rgoo0`00Rgoo1@00
Agoo0018Ool50029Ool30029Ool50019Ool004Uoo`D008Qoo`<008Qoo`D004Yoo`00BWoo1@00Qgoo
0`00Qgoo1@00Bgoo001<Ool40026Ool30026Ool4001=Ool004eoo`D008Aoo`<008Aoo`D004ioo`00
CWoo1@00Pgoo0`00Pgoo1@00Cgoo001?Ool50022Ool30022Ool5001@Ool0055oo`D0081oo`<0081o
o`D0059oo`00DWoo1@00Ogoo0`00Ogoo1@00Dgoo001COol5001nOol3001nOol5001DOol005Eoo`D0
07aoo`<007aoo`D005Ioo`00EWoo1@00Ngoo0`00Ngoo1@00Egoo001GOol5001jOol3001jOol5001H
Ool005Uoo`@007Uoo`<007Uoo`@005Yoo`00FWoo1@00Mgoo0`00Mgoo1@00Fgoo001KOol5001fOol3
001fOol5001LOol005aoo`D007Eoo`<007Eoo`D005eoo`00GWoo1@00Lgoo0`00Lgoo1@00Ggoo001O
Ool5001bOol3001bOol5001POol0061oo`D0075oo`<0075oo`D0065oo`00HWoo1000L7oo0`00L7oo
1000Hgoo001SOol5001^Ool3001^Ool5001TOol006Aoo`D006eoo`<006eoo`D006Eoo`00IGoo1@00
K7oo0`00K7oo1@00IWoo001WOol5001ZOol3001ZOol5001XOol006Qoo`D006Uoo`<006Uoo`D006Uo
o`00JGoo1@00J7oo0`00J7oo1@00JWoo001[Ool5001VOol3001VOol5001/Ool006aoo`D006Eoo`<0
06Eoo`D006eoo`00KGoo1@00I7oo0`00I7oo1@00KWoo001_Ool4001SOol3001SOol4001`Ool0071o
o`D0065oo`<0065oo`D0075oo`00LGoo1@00H7oo0`00H7oo1@00LWoo001bOol5001OOol3001OOol5
001cOol007Aoo`D005eoo`<005eoo`D007Eoo`00MGoo1@00G7oo0`00G7oo1@00MWoo001fOol5001K
Ool3001KOol5001gOol007Qoo`D005Uoo`<005Uoo`D007Uoo`00NGoo1@00F7oo0`00F7oo1@00NWoo
001jOol5001GOol3001GOol5001kOol007aoo`@005Ioo`<005Ioo`@007eoo`00OGoo1@00E7oo0`00
E7oo1@00OWoo001nOol5001COol3001COol5001oOol007moo`D0059oo`<0059oo`D0081oo`00PGoo
1@00D7oo0`00D7oo1@00PWoo0022Ool5001?Ool3001?Ool50023Ool008=oo`D004ioo`<004ioo`D0
08Aoo`00QGoo1000CGoo0`00CGoo1000QWoo0026Ool5001;Ool3001;Ool50027Ool008Moo`D004Yo
o`<004Yoo`D008Qoo`00R7oo1@00BGoo0`00BGoo1@00RGoo002:Ool50017Ool30017Ool5002;Ool0
08]oo`D004Ioo`<004Ioo`D008aoo`00S7oo1@00AGoo0`00AGoo1@00SGoo002>Ool50013Ool30013
Ool5002?Ool008moo`D0049oo`<0049oo`D0091oo`00T7oo1@00@Goo0`00@Goo1@00TGoo002BOol4
0010Ool30010Ool4002COol009=oo`D003ioo`<003ioo`D009Aoo`00U7oo1@00?Goo0`00?Goo1@00
UGoo002EOol5000lOol3000lOol5002FOol009Moo`D003Yoo`<003Yoo`D009Qoo`00V7oo1@00>Goo
0`00>Goo1@00VGoo002IOol5000hOol3000hOol5002JOol009]oo`D003Ioo`<003Ioo`D009aoo`00
W7oo1@00=Goo0`00=Goo1@00WGoo002MOol5000dOol3000dOol5002NOol009moo`@003=oo`<003=o
o`@00:1oo`00X7oo1@00<Goo0`00<Goo1@00XGoo002QOol5000`Ool3000`Ool5002ROol00:9oo`D0
02moo`<002moo`D00:=oo`00Y7oo1@00;Goo0`00;Goo1@00YGoo002UOol5000/Ool3000/Ool5002V
Ool00:Ioo`D002]oo`<002]oo`D00:Moo`00Z7oo1@00:Goo0`00:Goo1@00ZGoo002YOol5000XOol3
000XOol5002ZOol00:Yoo`D002Moo`<002Moo`D00:]oo`00[7oo10009Woo0`009Woo1000[Goo002]
Ool5000TOol3000TOol5002^Ool00:ioo`D002=oo`<002=oo`D00:moo`00[goo1@008Woo0`008Woo
1@00/7oo002aOol5000POol3000POol5002bOol00;9oo`D001moo`<001moo`D00;=oo`00/goo1@00
7Woo0`007Woo1@00]7oo002eOol4000MOol3000MOol4002fOol00;Ioo`D001]oo`<001]oo`D00;Mo
o`00]goo1@006Woo0`006Woo1@00^7oo002hOol5000IOol3000IOol5002iOol00;Yoo`D001Moo`<0
01Moo`D00;]oo`00^goo1@005Woo0`005Woo1@00_7oo002lOol5000EOol3000EOol5002mOol00;io
o`D001=oo`<001=oo`D00;moo`00_goo1@004Woo0`004Woo1@00`7oo0030Ool5000AOol3000AOol5
0031Ool00<9oo`@0011oo`<0011oo`@00<=oo`00`goo1@003Woo0`003Woo1@00a7oo0034Ool5000=
Ool3000=Ool50035Ool00<Eoo`D000aoo`<000aoo`D00<Ioo`00agoo1@002Woo0`002Woo1@00b7oo
0038Ool50009Ool30009Ool50039Ool00<Uoo`D000Qoo`<000Qoo`D00<Yoo`00bgoo1@001Woo0`00
1Woo1@00c7oo003<Ool50005Ool30005Ool5003=Ool00<eoo`D000Aoo`<000Aoo`D00<ioo`00cgoo
10000goo0`000goo1000d7oo003@Ool500001Goo00000000Ool01@00dGoo003AOol=003BOol00=9o
o`/00==oo`00dWoo2P00e7oo003BOol:003DOol00=5oo``00==oo`00dGoo3000dgoo003BOol:003D
Ool00=9oo`X00=Aoo`00\
\>"],
  ImageRangeCache->{{{0, 431}, {204.688, 0}} -> {-5.54755, -21.6427, \
0.769472, 0.769472}}],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Show[\ undeforemed, \ deformed\ ]\)], "Input"],

Cell[GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: .56734 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
%%IncludeResource: font Times-Italic
%%IncludeFont: Times-Italic
/Times-Italic findfont 12  scalefont  setfont
% Scaling calculations
0.0238095 0.00453515 0.100321 0.00453515 [
[ 0 0 0 0 ]
[ 1 .56734 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 0 m
1 0 L
1 .56734 L
0 .56734 L
closepath
clip
newpath
0 g
2 Mabswid
[ ] 0 setdash
.02381 .10032 m
.47732 .55384 L
s
.02381 .10032 m
.47732 .10032 L
s
.47732 .10032 m
.47732 .55384 L
s
.47732 .10032 m
.93084 .10032 L
s
.47732 .55384 m
.93084 .10032 L
s
.03 w
.02381 .10032 Mdot
.47732 .10032 Mdot
.47732 .55384 Mdot
.93084 .10032 Mdot
2 Mabswid
.02381 .10032 m
.5 .46702 L
s
.02381 .10032 m
.5 .01351 L
s
.5 .01351 m
.5 .46702 L
s
.5 .01351 m
.97619 .10032 L
s
.5 .46702 m
.97619 .10032 L
s
.03 w
.02381 .10032 Mdot
.5 .01351 Mdot
.5 .46702 Mdot
.97619 .10032 Mdot
% End of Graphics
MathPictureEnd
\
\>"], "Graphics",
  ImageSize->{288, 163.375},
  ImageMargins->{{0, 0}, {0, 0}},
  ImageRegion->{{0, 1}, {0, 1}},
  ImageCache->GraphicsData["Bitmap", "\<\
CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40006`0000mA000`40O003h00Oogoo/Goo003FOol2
003HOol00==oo`P00=Eoo`00dWoo2P00e7oo003BOol:003DOol00=9oo`X00=Aoo`00d7oo3`00dGoo
003:OolK003;Ool00<EoobD00<Ioo`00_goo4@000Woo2P000goo4@00`7oo002jOol@0008Ool:0009
Ool@002kOol00;Eooa0000ioo`P000mooa000;Ioo`00[goo40005goo0`005goo4000/7oo002ZOol@
000LOol3000LOol@002[Ool00:Aooa40025oo`<0025ooa400:Eoo`00Wgoo40009goo0`009goo4000
X7oo002JOol@000/Ool3000/Ool@002KOol009Aooa00039oo`<0039ooa0009Eoo`00Sgoo4000=goo
0`00=goo4000T7oo0029OolA000lOol3000lOolA002:Ool008Aooa00049oo`<0049ooa0008Eoo`00
Ogoo4000Agoo0`00Agoo4000P7oo001iOol@001=Ool3001=Ool@001jOol007Aooa00059oo`<0059o
oa0007Eoo`00KWoo4@00Egoo0`00Egoo4@00Kgoo001YOol@001MOol3001MOol@001ZOol006=ooa40
069oo`<0069ooa4006Aoo`00GWoo4000J7oo0`00J7oo4000Ggoo001IOol@001]Ool3001]Ool@001J
Ool005=ooa0007=oo`<007=ooa0005Aoo`00CWoo4000N7oo0`00N7oo4000Cgoo0018OolA001mOol3
001mOolA0019Ool004=ooa0008=oo`<008=ooa0004Aoo`00?Woo4000R7oo0`00R7oo4000?goo000h
Ool@002>Ool3002>Ool@000iOol003=ooa0009=oo`<009=ooa0003Aoo`00;Goo4@00V7oo0`00V7oo
4@00;Woo000XOol@002NOol3002NOol@000YOol002=ooa000:=oo`<00:=ooa0002Aoo`002Goo0P00
4Woo4000Wgoo0P0027oo0`00ZGoo40004Goo0P002goo0006Ool8000:Ool@002QOol80005Ool3002^
Ool@0009Ool80008Ool000Eoo`X000=ooa400:Eoo`X000Aoo`<00;9ooa80009oo`X000Moo`001Goo
6000Zgoo2P0017oo0`00/Woo7P001goo0005Oooo002A0002OolB0007Ool000Aoool009<000Moo`d0
00Ioo`0017ooo`00T`0027oo30001Woo0005Ool;002hOol:0004Ool3002bOol:000:Ool:0007Ool0
00Eoo``00;Moo`X000Aoo`<00;9oo`X000Uoo`/000Moo`001Goo3@00]Woo2P0017oo0`00/Goo2`00
27oo30001goo0006Ool>002eOol80005Ool3002`Ool;0007Ool=0008Ool000Uoo`8000=oo`L00;Mo
o`<000Moo`<00:moo`@000=oo`8000Uoo`D000Aoo`8000]oo`003goo1`00]Woo0`001goo0`00[Woo
10003Woo1@004Woo000@Ool7002eOol30007Ool3002]Ool4000>Ool4000DOol0015oo`P00;=oo`<0
00Moo`<00:aoo`@000eoo`D001Eoo`004Woo2000/Woo0`001goo0`00Zgoo10003Goo1@005Woo000C
Ool8002aOol30007Ool3002ZOol4000=Ool5000GOol001Aoo`T00:moo`<000Moo`<00:Uoo`@000ao
o`D001Uoo`005Goo2@00[Woo0`001goo0`00Z7oo100037oo1@006Woo000FOol9002]Ool30007Ool3
002WOol4000<Ool5000KOol001Moo`@00003Ool0000000<00:]oo`<000Moo`<00:Ioo`@000]oo`D0
01eoo`0067oo100000=oo`0000000`00ZWoo0`001goo0`00YGoo10002goo1@007Woo000IOol40000
0goo00000003002YOol30007Ool3002TOol4000;Ool5000OOol001Yoo`@0009oo`@00:Qoo`<000Mo
o`<00:=oo`@000]oo`@0025oo`006goo10000Woo1@00YWoo0`001goo0`00XWoo10002Woo1@008Woo
000LOol40002Ool5002UOol30007Ool3002QOol4000:Ool5000SOol001eoo`@0009oo`D00:Aoo`<0
00Moo`<00:1oo`@000Yoo`D002Aoo`007Woo10000goo1@00XWoo0`001goo0`00Wgoo10002Goo1@00
9Woo000OOol40003Ool5002QOol30007Ool3002NOol40009Ool5000WOol0021oo`@000=oo`D00:1o
o`<000Moo`<009eoo`@000Uoo`D002Qoo`008Goo100017oo1@00WWoo0`001goo0`00W7oo100027oo
1@00:Woo000ROol40004Ool5002MOol30007Ool3002KOol40008Ool5000[Ool002=oo`@000Aoo`D0
09aoo`<000Moo`<009Yoo`@000Qoo`D002aoo`0097oo10001Goo1@00VWoo0`001goo0`00VGoo1000
1goo1@00;Woo000UOol40005Ool5002IOol30007Ool3002HOol40007Ool5000_Ool002Ioo`@000Eo
o`D009Qoo`<000Moo`<009Moo`@000Moo`D0031oo`009goo10001Woo1000Ugoo0`001goo0`00UWoo
10001goo1000<Woo000XOol40006Ool5002EOol30007Ool3002EOol40006Ool5000cOol002Uoo`@0
00Ioo`D009Aoo`<000Moo`<009Aoo`@000Ioo`D003Aoo`00:Woo10001Woo1@00Tgoo0`001goo0`00
Tgoo10001Woo1@00=Goo000[Ool40007Ool5002AOol30007Ool3002BOol40005Ool5000gOol002ao
o`@000Moo`D0091oo`<000Moo`<0095oo`@000Eoo`D003Qoo`00;Goo10001goo1@00Sgoo0`001goo
0`00T7oo10001Goo1@00>Goo000^Ool40008Ool5002=Ool30007Ool3002?Ool40004Ool5000kOol0
02moo`@000Qoo`D008aoo`<000Moo`<008ioo`@000Aoo`D003aoo`00<7oo100027oo1@00Rgoo0`00
1goo0`00SGoo100017oo1@00?Goo000aOol40009Ool50029Ool30007Ool3002<Ool40003Ool5000o
Ool0039oo`@000Uoo`D008Qoo`<000Moo`<008]oo`@000=oo`D0041oo`00<goo10002Goo1@00Qgoo
0`001goo0`00RWoo10000goo1@00@Goo000dOol4000:Ool40026Ool30007Ool30029Ool40003Ool4
0013Ool003Eoo`@000Yoo`D008Aoo`<000Moo`<008Qoo`@0009oo`D004Aoo`00=Woo10002Woo1@00
Pgoo0`001goo0`00Qgoo10000Woo1@00AGoo000gOol4000:Ool50022Ool30007Ool30026Ool40002
Ool50016Ool003Qoo`@000]oo`D0081oo`<000Moo`<008Eoo`@00003Ool0000000<004Qoo`00>Goo
10002goo1@00Ogoo0`001goo0`00Q7oo100000=oo`0000000`00BGoo000jOol4000;Ool5001nOol3
0007Ool30023Ool400000goo00000003001:Ool003]oo`@000aoo`D007aoo`<000Moo`<0089oo`T0
04aoo`00?7oo100037oo1@00Ngoo0`001goo0`00PGoo2@00CGoo000mOol4000<Ool5001jOol30007
Ool30020Ool9001>Ool003ioo`@000eoo`@007Uoo`<000Moo`<007moo`P0051oo`00?goo10003Goo
1@00Mgoo0`001goo0`00OWoo2000DGoo0010Ool4000=Ool5001fOol30007Ool3001mOol8001BOol0
045oo`@000eoo`D007Eoo`<000Moo`<007aoo`P005=oo`00@Woo10003Woo1@00Lgoo0`001goo0`00
Ngoo1`00EGoo0013Ool4000>Ool5001bOol30007Ool3001jOol7001FOol004Aoo`@000ioo`D0075o
o`<000Moo`<007Uoo`L005Moo`00AGoo10003goo1@00Kgoo0`001goo0`00N7oo1P00FGoo0016Ool4
000?Ool5001^Ool30007Ool3001gOol6001JOol004Moo`@000moo`D006eoo`<000Moo`<007Ioo`H0
05]oo`00B7oo100047oo1@00Jgoo0`001goo0`00MGoo1@00GGoo0019Ool4000@Ool5001ZOol30007
Ool3001dOol5001NOol004Yoo`@0011oo`D006Uoo`<000Moo`<007=oo`D005moo`00Bgoo10004Goo
1000J7oo0`001goo0`00LWoo1000HGoo001<Ool4000AOol5001VOol30007Ool3001`Ool5001ROol0
04eoo`@0015oo`D006Eoo`<000Moo`<006moo`D006=oo`00CWoo10004Goo1@00I7oo0`001goo0`00
KWoo1@00I7oo001?Ool4000BOol5001ROol30007Ool3001/Ool6001UOol0051oo`@0019oo`D0065o
o`<000Moo`<006]oo`H006Ioo`00DGoo10004Woo1@00H7oo0`001goo0`00JWoo1P00Igoo001BOol4
000COol5001NOol30007Ool3001XOol7001XOol005=oo`@001=oo`D005eoo`<000Moo`<006Moo`L0
06Uoo`00E7oo10004goo1@00G7oo0`001goo0`00IWoo1`00JWoo001EOol4000DOol4001KOol30007
Ool3001TOol8001[Ool005Ioo`@001Aoo`D005Uoo`<000Moo`<006=oo`P006aoo`00Egoo100057oo
1@00F7oo0`001goo0`00HWoo2000KGoo001HOol4000DOol5001GOol30007Ool3001QOol8001^Ool0
05Uoo`@001Eoo`D005Eoo`<000Moo`<005moo`T006moo`00FWoo10005Goo1@00E7oo0`001goo0`00
GWoo2@00L7oo001KOol4000EOol5001COol30007Ool3001MOol9001aOol005aoo`@001Ioo`D0055o
o`<000Moo`<005]oo`D00003Ool000000080079oo`00GGoo10005Woo1@00D7oo0`001goo0`00FWoo
1@0000=oo`0000000P00Lgoo001NOol4000FOol5001?Ool30007Ool3001IOol500000goo00000002
001dOol005moo`@001Moo`D004eoo`<000Moo`<005Moo`D0009oo`@007Eoo`00H7oo10005goo1@00
C7oo0`001goo0`00EWoo1@000Woo1000MWoo001QOol4000GOol5001;Ool30007Ool3001EOol50002
Ool4001gOol0069oo`@001Qoo`@004Yoo`<000Moo`<005Aoo`@000=oo`@007Qoo`00Hgoo100067oo
1@00B7oo0`001goo0`00DWoo1@000goo1000NGoo001TOol4000HOol50017Ool30007Ool3001AOol5
0003Ool4001jOol006Eoo`@001Qoo`D004Ioo`<000Moo`<0051oo`D000=oo`@007]oo`00IWoo1000
6Goo1@00A7oo0`001goo0`00CWoo1@0017oo1000O7oo001WOol4000IOol50013Ool30007Ool3001=
Ool50004Ool4001mOol006Qoo`@001Uoo`D0049oo`<000Moo`<004aoo`D000Aoo`@007ioo`00JGoo
10006Woo1@00@7oo0`001goo0`00BWoo1@001Goo1000Ogoo001ZOol4000JOol5000oOol30007Ool3
0019Ool50005Ool40020Ool006]oo`@001Yoo`D003ioo`<000Moo`<004Qoo`D000Eoo`@0085oo`00
K7oo10006goo1@00?7oo0`001goo0`00AWoo1@001Woo1000PWoo001]Ool4000KOol5000kOol30007
Ool30015Ool50006Ool40023Ool006ioo`@001]oo`D003Yoo`<000Moo`<004Aoo`D000Ioo`@008Ao
o`00Kgoo100077oo1000>Goo0`001goo0`00@goo10001goo1000QGoo001`Ool4000LOol5000gOol3
0007Ool30011Ool50007Ool40026Ool0075oo`@001aoo`D003Ioo`<000Moo`<0041oo`D000Moo`@0
08Moo`00LWoo100077oo1@00=Goo0`001goo0`00?goo1@001goo1000R7oo001cOol4000MOol5000c
Ool30007Ool3000mOol50008Ool40029Ool007Aoo`@001eoo`D0039oo`<000Moo`<003aoo`D000Qo
o`@008Yoo`00MGoo10007Goo1@00<Goo0`001goo0`00>goo1@0027oo1000Rgoo001fOol4000NOol5
000_Ool30007Ool3000iOol50009Ool4002<Ool007Moo`@001ioo`D002ioo`<000Moo`<003Qoo`D0
00Uoo`@008eoo`00N7oo10007Woo1@00;Goo0`001goo0`00=goo1@002Goo1000SWoo001iOol4000O
Ool4000/Ool30007Ool3000fOol4000:Ool4002?Ool007Yoo`@001moo`D002Yoo`<000Moo`<003Ao
o`D000Yoo`@0091oo`00Ngoo10007goo1@00:Goo0`001goo0`00<goo1@002Woo1000TGoo001lOol4
000OOol5000XOol30007Ool3000bOol5000:Ool4002BOol007eoo`@0021oo`D002Ioo`<000Moo`<0
031oo`D000]oo`@009=oo`00OWoo100087oo1@009Goo0`001goo0`00;goo1@002goo1000U7oo001o
Ool4000POol5000TOol30007Ool3000^Ool5000;Ool4002EOol0081oo`@0025oo`D0029oo`<000Mo
o`<002aoo`D000aoo`@009Ioo`00PGoo10008Goo1@008Goo0`001goo0`00:goo1@0037oo1000Ugoo
0022Ool4000QOol5000POol30007Ool3000ZOol5000<Ool4002HOol008=oo`@0029oo`D001ioo`<0
00Moo`<002Qoo`D000eoo`@009Uoo`00Q7oo10008Woo1@007Goo0`001goo0`009goo1@003Goo1000
VWoo0025Ool4000ROol5000LOol30007Ool3000VOol5000=Ool4002KOol008Ioo`@002=oo`@001]o
o`<000Moo`<002Eoo`@000ioo`@009aoo`00Qgoo10008goo1@006Goo0`001goo0`008goo1@003Woo
1000WGoo0028Ool4000SOol5000HOol30007Ool3000ROol5000>Ool4002NOol008Uoo`@002=oo`D0
01Moo`<000Moo`<0025oo`D000ioo`@009moo`00RWoo100097oo1@005Goo0`001goo0`007goo1@00
3goo1000X7oo002;Ool4000TOol5000DOol30007Ool3000NOol5000?Ool4002QOol008aoo`@002Ao
o`D001=oo`<000Moo`<001eoo`D000moo`@00:9oo`00SGoo10009Goo1@004Goo0`001goo0`006goo
1@0047oo1000Xgoo002>Ool4000UOol5000@Ool30007Ool3000JOol5000@Ool4002TOol008moo`@0
02Eoo`D000moo`<000Moo`<001Uoo`D0011oo`@00:Eoo`00T7oo10009Woo1@003Goo0`001goo0`00
5goo1@004Goo1000YWoo002AOol4000VOol5000<Ool30007Ool3000FOol5000AOol4002WOol0099o
o`@002Ioo`D000]oo`<000Moo`<001Eoo`D0015oo`@00:Qoo`00Tgoo10009goo10002Woo0`001goo
0`0057oo10004Woo1000ZGoo002DOol4000WOol50008Ool30007Ool3000BOol5000BOol4002ZOol0
09Eoo`@002Moo`D000Moo`<000Moo`<0015oo`D0019oo`@00:]oo`00UWoo10009goo1@001Woo0`00
1goo0`0047oo1@004Woo1000[7oo002GOol4000XOol50004Ool30007Ool3000>Ool5000COol4002]
Ool009Qoo`@002Qoo`D000=oo`<000Moo`<000eoo`D001=oo`@00:ioo`00VGoo1000:7oo1@000Woo
0`001goo0`0037oo1@004goo1000[goo002JOol4000YOol80007Ool3000:Ool5000DOol4002`Ool0
09]oo`@002Uoo`L000Moo`<000Uoo`D001Aoo`@00;5oo`00W7oo1000:Goo1P001goo0`0027oo1@00
57oo1000/Woo002MOol4000ZOol40007Ool30007Ool4000EOol4002cOol009ioo`@002Yoo`D000Eo
o`<000Eoo`D001Eoo`@00;Aoo`00Wgoo1000:Goo1P0017oo0`0017oo1@005Goo1000]Goo002POol4
000XOol70003Ool30003Ool5000EOol4002fOol00:5oo`@002Moo`<00003Ool0000000d001Ioo`@0
0;Moo`00XWoo10009Woo0`000Woo3@005Woo1000^7oo002SOol4000UOol30003Ool;000FOol4002i
Ool00:Aoo`@002Aoo`<000=oo`X001Ioo`@00;Yoo`00YGoo10008goo0`000Woo300057oo1000^goo
002VOol4000ROol30002Ool<000COol4002lOol00:Moo`@0025oo`<000=oo`X001=oo`@00;eoo`00
Z7oo100087oo0`000goo2P004Woo1000_Woo002YOol4000OOol30003Ool:000AOol4002oOol00:Yo
o`@001ioo`<000Aoo`P0015oo`@00<1oo`00Zgoo10007Goo0`001goo0P004goo1000`Goo002/Ool4
000LOol3000KOol40032Ool00:eoo`@001]oo`<001Yoo`@00<=oo`00[Woo10006Woo0`006Goo1000
a7oo002_Ool4000IOol3000HOol40035Ool00;1oo`@001Qoo`<001Moo`@00<Ioo`00/Goo10005goo
0`005Woo1000agoo002bOol4000FOol3000EOol40038Ool00;=oo`@001Eoo`<001Aoo`@00<Uoo`00
]7oo100057oo0`004goo1000bWoo002eOol4000COol3000BOol4003;Ool00;Ioo`@0019oo`<0015o
o`@00<aoo`00]goo10004Goo0`0047oo1000cGoo002hOol4000@Ool3000?Ool4003>Ool00;Uoo`@0
00moo`<000ioo`@00<moo`00^Woo10003Woo0`003Goo1000d7oo002kOol4000=Ool3000<Ool4003A
Ool00;aoo`@000aoo`<000]oo`@00=9oo`00_Goo10002goo0`002Woo1000dgoo002nOol4000:Ool3
0009Ool4003DOol00;moo`@000Uoo`<000Qoo`@00=Eoo`00`7oo100027oo0`001goo1000eWoo0031
Ool40007Ool30006Ool4003GOol00<9oo`@000Ioo`<000Eoo`@00=Qoo`00`goo10001Goo0`0017oo
1000fGoo0034Ool40004Ool30003Ool4003JOol00<Eoo`@000=oo`<0009oo`@00=]oo`00aWoo1000
0Woo0`0000=oo`0000000P00g7oo0037Ool400000goo00000005003MOol00<Qoo`X00=ioo`00b7oo
2P00gWoo0038Ool:003NOol00<Qoo`X00=ioo`00agoo3000gGoo0037Ool<003MOol00<Qoo`X00=io
o`00b7oo2P00gWoo0038Ool:003NOol00001\
\>"],
  ImageRangeCache->{{{0, 431}, {244.063, 0}} -> {-5.45936, -22.1214, \
0.768858, 0.768858}}],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData["Element Strains"], "Section",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["\<\
In order to compute the strains in each element, nodal displacements need to \
be converted to the element coordinates and the displacement at the nodes of \
each element must be subtracted and divided by the length of the element.  \
The elements transformation matrix can be obtained if the directional cosines \
are available.  \
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontFamily->"Times",
  FontSize->18],

Cell[TextData[{
  "The module ",
  StyleBox["elementStrains",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " may be used to compute the strains.  Nodal displacements along with node \
point locations and element connectivity are needed for the strain \
computation.  "
}], "Text",
  FontSize->18],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(nodeLocations\)\(\ \)\)\)], "Input"],

Cell[BoxData[
    \({{1, {0, 0}}, {2, {h, 0}}, {3, {h, h}}, {4, {2\ h, 0}}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(elementNodes\)], "Input"],

Cell[BoxData[
    \({{1, {1, 3}}, {2, {1, 2}}, {3, {2, 3}}, {4, {2, 4}}, {5, {3, 
          4}}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(displacements\)], "Input"],

Cell[BoxData[
    \({0, 
      0, \(0.5`\ \@h\^2\ Pload\)\/\(Ar\ YE\), \(-\(\(1.9142135623730951`\ \@h\
\^2\ Pload\)\/\(Ar\ YE\)\)\), \(0.5`\ \@h\^2\ Pload\)\/\(Ar\ YE\), \
\(-\(\(1.9142135623730951`\ \@h\^2\ Pload\)\/\(Ar\ YE\)\)\), \(\@h\^2\ \
Pload\)\/\(Ar\ YE\), 0}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(strains = 
      elementStrains[elementNodes, nodeLocations, displacements]\  // 
        N\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \({\(-\(\(0.7071067811865475`\ h\ Pload\)\/\(Ar\ \@h\^2\ YE\)\)\), \(0.5`\
\ h\ Pload\)\/\(Ar\ \@h\^2\ YE\), \(0.`\ h\ Pload\)\/\(Ar\ \@h\^2\ YE\), \
\(0.5`\ h\ Pload\)\/\(Ar\ \@h\^2\ YE\), \(-\(\(0.7071067811865475`\ h\ \
Pload\)\/\(Ar\ \@h\^2\ YE\)\)\)}\)], "Output"]
}, Open  ]]
}, Open  ]]
},
FrontEndVersion->"4.1 for Microsoft Windows",
ScreenRectangle->{{0, 1024}, {0, 723}},
WindowToolbars->{},
CellGrouping->Automatic,
WindowSize->{1016, 696},
WindowMargins->{{0, Automatic}, {Automatic, 0}},
PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}},
ShowCellLabel->True,
ShowCellTags->False,
RenderingOptions->{"ObjectDithering"->True,
"RasterDithering"->False},
CharacterEncoding->"MacintoshAutomaticEncoding",
Magnification->1.5,
StyleDefinitions -> "Classroom.nb"
]

(*******************************************************************
Cached data follows.  If you edit this Notebook file directly, not
using Mathematica, you must remove the line containing CacheID at
the top of  the file.  The cache data will then be recreated when
you save this file from within Mathematica.
*******************************************************************)

(*CellTagsOutline
CellTagsIndex->{
  "Info3216495278-8722696"->{
    Cell[6195, 221, 374, 6, 135, "Print",
      CellTags->"Info3216495278-8722696"]},
  "Info3216495292-5915026"->{
    Cell[11282, 406, 392, 6, 135, "Print",
      CellTags->"Info3216495292-5915026"]},
  "Info3216495292-1617452"->{
    Cell[11764, 420, 587, 9, 185, "Print",
      CellTags->"Info3216495292-1617452"]},
  "Info3216495390-5511677"->{
    Cell[36236, 954, 257, 4, 85, "Print",
      CellTags->"Info3216495390-5511677"]},
  "Info3216495390-3443889"->{
    Cell[37683, 1010, 347, 5, 110, "Print",
      CellTags->"Info3216495390-3443889"]},
  "Info3216495391-2149240"->{
    Cell[39065, 1061, 382, 6, 135, "Print",
      CellTags->"Info3216495391-2149240"]},
  "Info3216495391-7681631"->{
    Cell[45883, 1246, 225, 4, 60, "Print",
      CellTags->"Info3216495391-7681631"]},
  "Info3216495391-4699456"->{
    Cell[48804, 1317, 366, 6, 110, "Print",
      CellTags->"Info3216495391-4699456"]},
  "Info3216495482-6674438"->{
    Cell[52853, 1433, 204, 3, 60, "Print",
      CellTags->"Info3216495482-6674438"]},
  "Info3216495494-6568000"->{
    Cell[56902, 1567, 147, 3, 35, "Print",
      CellTags->"Info3216495494-6568000"]},
  "Info3216495494-2764126"->{
    Cell[57949, 1610, 184, 3, 60, "Print",
      CellTags->"Info3216495494-2764126"]},
  "Info3216495516-7563905"->{
    Cell[63902, 1798, 426, 6, 135, "Print",
      CellTags->"Info3216495516-7563905"]}
  }
*)

(*CellTagsIndex
CellTagsIndex->{
  {"Info3216495278-8722696", 97281, 2530},
  {"Info3216495292-5915026", 97397, 2533},
  {"Info3216495292-1617452", 97514, 2536},
  {"Info3216495390-5511677", 97631, 2539},
  {"Info3216495390-3443889", 97747, 2542},
  {"Info3216495391-2149240", 97865, 2545},
  {"Info3216495391-7681631", 97983, 2548},
  {"Info3216495391-4699456", 98100, 2551},
  {"Info3216495482-6674438", 98218, 2554},
  {"Info3216495494-6568000", 98335, 2557},
  {"Info3216495494-2764126", 98452, 2560},
  {"Info3216495516-7563905", 98569, 2563}
  }
*)

(*NotebookFileOutline
Notebook[{
Cell[1705, 50, 77, 2, 72, "Input"],
Cell[1785, 54, 142, 5, 174, "Title",
  Evaluatable->False],
Cell[1930, 61, 437, 11, 242, "Subsubtitle",
  Evaluatable->False],
Cell[2370, 74, 404, 14, 126, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[2799, 92, 92, 2, 93, "Section",
  Evaluatable->False],
Cell[2894, 96, 480, 12, 175, "Text",
  Evaluatable->False],
Cell[3377, 110, 176, 3, 98, "Input"],
Cell[3556, 115, 758, 23, 213, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[4339, 142, 70, 2, 72, "Input"],
Cell[4412, 146, 89, 1, 71, "Output"]
}, Open  ]],
Cell[4516, 150, 237, 7, 89, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[4778, 161, 132, 2, 124, "Input"],
Cell[4913, 165, 121, 2, 71, "Output"]
}, Open  ]],
Cell[5049, 170, 163, 4, 52, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[5237, 178, 70, 2, 72, "Input"],
Cell[5310, 182, 43, 1, 71, "Output"]
}, Open  ]],
Cell[5368, 186, 330, 10, 89, "Text",
  Evaluatable->False],
Cell[5701, 198, 73, 2, 72, "Input"],
Cell[5777, 202, 315, 11, 90, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[6117, 217, 75, 2, 72, "Input"],
Cell[6195, 221, 374, 6, 135, "Print",
  CellTags->"Info3216495278-8722696"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{
Cell[6618, 233, 73, 2, 93, "Section",
  Evaluatable->False],
Cell[6694, 237, 321, 8, 126, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[7040, 249, 55, 1, 86, "Subsection"],
Cell[7098, 252, 877, 23, 221, "Text"],

Cell[CellGroupData[{
Cell[8000, 279, 163, 3, 72, "Input"],
Cell[8166, 284, 89, 1, 71, "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{
Cell[8304, 291, 64, 1, 86, "Subsection"],
Cell[8371, 294, 1008, 34, 261, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[9404, 332, 154, 3, 72, "Input"],
Cell[9561, 337, 110, 2, 71, "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{
Cell[9720, 345, 62, 1, 86, "Subsection"],
Cell[9785, 348, 379, 11, 126, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[10189, 363, 173, 3, 98, "Input"],
Cell[10365, 368, 120, 2, 71, "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{
Cell[10534, 376, 109, 3, 86, "Subsection"],
Cell[10646, 381, 561, 18, 128, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[11232, 403, 47, 1, 72, "Input"],
Cell[11282, 406, 392, 6, 135, "Print",
  CellTags->"Info3216495292-5915026"]
}, Open  ]],

Cell[CellGroupData[{
Cell[11711, 417, 50, 1, 72, "Input"],
Cell[11764, 420, 587, 9, 185, "Print",
  CellTags->"Info3216495292-1617452"]
}, Open  ]],

Cell[CellGroupData[{
Cell[12388, 434, 46, 1, 72, "Input"],
Cell[12437, 437, 89, 1, 71, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[12563, 443, 80, 1, 72, "Input"],
Cell[12646, 446, 113, 2, 71, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[12796, 453, 144, 3, 98, "Input"],
Cell[12943, 458, 4213, 90, 279, 699, 42, "GraphicsData", "PostScript", \
"Graphics"],
Cell[17159, 550, 130, 3, 71, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[17326, 558, 110, 2, 72, "Input"],
Cell[17439, 562, 145, 2, 99, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[17621, 569, 127, 3, 72, "Input"],
Cell[17751, 574, 8941, 159, 279, 790, 54, "GraphicsData", "PostScript", \
"Graphics"],
Cell[26695, 735, 130, 3, 71, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[26862, 743, 83, 1, 72, "Input"],
Cell[26948, 746, 7698, 149, 228, 842, 60, "GraphicsData", "PostScript", \
"Graphics"],
Cell[34649, 897, 130, 3, 71, "Output"]
}, Open  ]],
Cell[34794, 903, 296, 8, 126, "Text",
  Evaluatable->False]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{
Cell[35139, 917, 121, 2, 93, "Section",
  Evaluatable->False],
Cell[35263, 921, 782, 20, 238, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[36070, 945, 61, 1, 86, "Subsection"],

Cell[CellGroupData[{
Cell[36156, 950, 77, 2, 72, "Input"],
Cell[36236, 954, 257, 4, 85, "Print",
  CellTags->"Info3216495390-5511677"]
}, Open  ]],

Cell[CellGroupData[{
Cell[36530, 963, 111, 2, 72, "Input"],
Cell[36644, 967, 344, 10, 198, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[37025, 982, 56, 1, 72, "Input"],
Cell[37084, 985, 323, 10, 168, "Output"]
}, Open  ]],
Cell[37422, 998, 163, 4, 52, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[37610, 1006, 70, 2, 72, "Input"],
Cell[37683, 1010, 347, 5, 110, "Print",
  CellTags->"Info3216495390-3443889"]
}, Open  ]],

Cell[CellGroupData[{
Cell[38067, 1020, 109, 2, 72, "Input"],
Cell[38179, 1024, 224, 4, 151, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[38440, 1033, 52, 1, 72, "Input"],
Cell[38495, 1036, 139, 2, 94, "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{
Cell[38683, 1044, 70, 1, 86, "Subsection"],
Cell[38756, 1047, 234, 7, 89, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[39015, 1058, 47, 1, 72, "Input"],
Cell[39065, 1061, 382, 6, 135, "Print",
  CellTags->"Info3216495391-2149240"]
}, Open  ]],
Cell[39462, 1070, 243, 7, 89, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[39730, 1081, 116, 2, 72, "Input"],
Cell[39849, 1085, 1774, 25, 629, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[41660, 1115, 48, 1, 72, "Input"],
Cell[41711, 1118, 867, 17, 224, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[42615, 1140, 48, 1, 72, "Input"],
Cell[42666, 1143, 401, 9, 178, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[43104, 1157, 48, 1, 72, "Input"],
Cell[43155, 1160, 401, 9, 178, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[43593, 1174, 48, 1, 72, "Input"],
Cell[43644, 1177, 401, 9, 178, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[44082, 1191, 48, 1, 72, "Input"],
Cell[44133, 1194, 867, 17, 224, "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{
Cell[45049, 1217, 67, 1, 86, "Subsection"],
Cell[45119, 1220, 652, 18, 165, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[45796, 1242, 84, 2, 72, "Input"],
Cell[45883, 1246, 225, 4, 60, "Print",
  CellTags->"Info3216495391-7681631"]
}, Open  ]],

Cell[CellGroupData[{
Cell[46145, 1255, 133, 3, 72, "Input"],
Cell[46281, 1260, 1873, 33, 404, "Output"]
}, Open  ]],
Cell[48169, 1296, 557, 14, 164, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[48751, 1314, 50, 1, 72, "Input"],
Cell[48804, 1317, 366, 6, 110, "Print",
  CellTags->"Info3216495391-4699456"]
}, Open  ]],

Cell[CellGroupData[{
Cell[49207, 1328, 141, 4, 72, "Input"],
Cell[49351, 1334, 1873, 33, 404, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[51261, 1372, 86, 2, 72, "Input"],
Cell[51350, 1376, 38, 1, 71, "Output"]
}, Open  ]]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{
Cell[51449, 1384, 119, 2, 93, "Section",
  Evaluatable->False],
Cell[51571, 1388, 696, 18, 201, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[52292, 1410, 145, 3, 72, "Input"],
Cell[52440, 1415, 86, 1, 71, "Output"]
}, Open  ]],
Cell[52541, 1419, 202, 6, 44, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[52768, 1429, 82, 2, 72, "Input"],
Cell[52853, 1433, 204, 3, 60, "Print",
  CellTags->"Info3216495482-6674438"]
}, Open  ]],

Cell[CellGroupData[{
Cell[53094, 1441, 111, 2, 72, "Input"],
Cell[53208, 1445, 853, 17, 270, "Output"]
}, Open  ]],
Cell[54076, 1465, 274, 8, 89, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[54375, 1477, 106, 2, 72, "Input"],
Cell[54484, 1481, 60, 1, 71, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[54581, 1487, 114, 2, 72, "Input"],
Cell[54698, 1491, 853, 17, 270, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[55588, 1513, 52, 1, 72, "Input"],
Cell[55643, 1516, 38, 1, 71, "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{
Cell[55730, 1523, 102, 2, 93, "Section",
  Evaluatable->False],
Cell[55835, 1527, 382, 9, 126, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[56242, 1540, 102, 2, 72, "Input"],
Cell[56347, 1544, 56, 1, 71, "Output"]
}, Open  ]],
Cell[56418, 1548, 405, 12, 127, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[56848, 1564, 51, 1, 72, "Input"],
Cell[56902, 1567, 147, 3, 35, "Print",
  CellTags->"Info3216495494-6568000"]
}, Open  ]],

Cell[CellGroupData[{
Cell[57086, 1575, 63, 1, 72, "Input"],
Cell[57152, 1578, 282, 9, 132, "Output"]
}, Open  ]],
Cell[57449, 1590, 391, 12, 90, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[57865, 1606, 81, 2, 72, "Input"],
Cell[57949, 1610, 184, 3, 60, "Print",
  CellTags->"Info3216495494-2764126"]
}, Open  ]],

Cell[CellGroupData[{
Cell[58170, 1618, 104, 2, 72, "Input"],
Cell[58277, 1622, 58, 1, 71, "Output"]
}, Open  ]],
Cell[58350, 1626, 420, 13, 126, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[58795, 1643, 91, 2, 72, "Input"],
Cell[58889, 1647, 54, 1, 71, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[58980, 1653, 91, 2, 72, "Input"],
Cell[59074, 1657, 560, 9, 221, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[59671, 1671, 104, 2, 72, "Input"],
Cell[59778, 1675, 346, 6, 150, "Output"]
}, Open  ]],
Cell[60139, 1684, 223, 7, 89, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[60387, 1695, 63, 1, 72, "Input"],
Cell[60453, 1698, 1427, 24, 350, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[61917, 1727, 51, 1, 72, "Input"],
Cell[61971, 1730, 393, 5, 192, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[62401, 1740, 37, 1, 72, "Input"],
Cell[62441, 1743, 346, 6, 150, "Output"]
}, Open  ]],
Cell[62802, 1752, 221, 7, 89, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[63048, 1763, 73, 2, 72, "Input"],
Cell[63124, 1767, 60, 1, 71, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[63221, 1773, 112, 2, 72, "Input"],
Cell[63336, 1777, 283, 5, 150, "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{
Cell[63668, 1788, 91, 2, 93, "Section",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[63784, 1794, 115, 2, 72, "Input"],
Cell[63902, 1798, 426, 6, 135, "Print",
  CellTags->"Info3216495516-7563905"]
}, Open  ]],

Cell[CellGroupData[{
Cell[64365, 1809, 111, 2, 72, "Input"],
Cell[64479, 1813, 7657, 148, 228, 833, 59, "GraphicsData", "PostScript", \
"Graphics"],
Cell[72139, 1963, 130, 3, 71, "Output"]
}, Open  ]],
Cell[72284, 1969, 217, 6, 89, "Text",
  Evaluatable->False],

Cell[CellGroupData[{
Cell[72526, 1979, 46, 1, 72, "Input"],
Cell[72575, 1982, 283, 5, 150, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[72895, 1992, 213, 4, 98, "Input"],
Cell[73111, 1998, 357, 9, 132, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[73505, 2012, 51, 1, 72, "Input"],
Cell[73559, 2015, 36, 1, 71, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[73632, 2021, 251, 5, 150, "Input"],
Cell[73886, 2028, 148, 2, 71, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[74071, 2035, 116, 2, 72, "Input"],
Cell[74190, 2039, 7361, 144, 218, 865, 59, "GraphicsData", "PostScript", \
"Graphics"],
Cell[81554, 2185, 130, 3, 71, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[81721, 2193, 66, 1, 72, "Input"],
Cell[81790, 2196, 12255, 222, 258, 1126, 80, "GraphicsData", "PostScript", \
"Graphics"],
Cell[94048, 2420, 130, 3, 71, "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{
Cell[94227, 2429, 92, 2, 93, "Section",
  Evaluatable->False],
Cell[94322, 2433, 442, 10, 163, "Text",
  Evaluatable->False],
Cell[94767, 2445, 339, 10, 127, "Text"],

Cell[CellGroupData[{
Cell[95131, 2459, 60, 1, 72, "Input"],
Cell[95194, 2462, 89, 1, 71, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[95320, 2468, 45, 1, 72, "Input"],
Cell[95368, 2471, 110, 2, 71, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[95515, 2478, 46, 1, 72, "Input"],
Cell[95564, 2481, 283, 5, 150, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[95884, 2491, 150, 4, 72, "Input"],
Cell[96037, 2497, 287, 4, 94, "Output"]
}, Open  ]]
}, Open  ]]
}
]
*)



(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)