(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' 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). 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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[ 50538, 2573]*) (*NotebookOutlinePosition[ 51204, 2596]*) (* CellTagsIndexPosition[ 51160, 2592]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Formulation of finite element matrices for a 2d elastic \ isoparametric 4-node quadrilateral element - Eigenvalue analysis of stiffness \ matrix (energy modes)\ \>", "Title", FontSize->36], Cell[BoxData[ \(Clear["\"]; Off[General::spell, General::spell1];\)], "Input",\ FontSize->36], Cell[CellGroupData[{ Cell["Shape functions and derivatives", "Subtitle", FontSize->36], Cell[BoxData[{ \(\(\[Phi]\_1\ = \ \(1\/4\) \((1 + \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\);\)\), \ "\[IndentingNewLine]", \(\(\[Phi]\_2\ = \ \(1\/4\) \((1 - \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\);\)\), \ "\[IndentingNewLine]", \(\(\[Phi]\_3\ = \ \(1\/4\) \((1 - \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\);\)\), \ "\[IndentingNewLine]", \(\(\[Phi]\_4\ = \ \(1\/4\) \((1 + \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\);\)\), \ "\[IndentingNewLine]", \(\(d\[Phi]1\[Xi]1\ = \ D[\[Phi]\_1, \[Xi]\_1];\)\), "\[IndentingNewLine]", \(\(d\[Phi]1\[Xi]2\ = \ D[\[Phi]\_1, \[Xi]\_2];\)\), "\[IndentingNewLine]", \(\(d\[Phi]2\[Xi]1\ = \ D[\[Phi]\_2, \[Xi]\_1];\)\), "\[IndentingNewLine]", \(\(d\[Phi]2\[Xi]2\ = \ D[\[Phi]\_2, \[Xi]\_2];\)\), "\[IndentingNewLine]", \(\(d\[Phi]3\[Xi]1\ = \ D[\[Phi]\_3, \[Xi]\_1];\)\), "\[IndentingNewLine]", \(\(d\[Phi]3\[Xi]2\ = \ D[\[Phi]\_3, \[Xi]\_2];\)\), "\[IndentingNewLine]", \(\(d\[Phi]4\[Xi]1\ = \ D[\[Phi]\_4, \[Xi]\_1];\)\), "\[IndentingNewLine]", \(\(d\[Phi]4\[Xi]2\ = \ D[\[Phi]\_4, \[Xi]\_2];\)\)}], "Input", FontSize->36] }, Open ]], Cell[CellGroupData[{ Cell["Element displacement interpolation matrix", "Subtitle", FontSize->36], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"H", " ", "=", " ", RowBox[{"(", GridBox[{ {\(\[Phi]\_1\), "0", \(\[Phi]\_2\), "0", \(\[Phi]\_3\), "0", \(\[Phi]\_4\), "0"}, {"0", \(\[Phi]\_1\), "0", \(\[Phi]\_2\), "0", \(\[Phi]\_3\), "0", \(\[Phi]\_4\)} }], ")"}]}], ";"}], "\[IndentingNewLine]", \(MatrixForm[ Transpose[H]]\)}], "Input", FontSize->36], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/4\ \((1 + \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\), "0"}, {"0", \(1\/4\ \((1 + \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\)}, {\(1\/4\ \((1 - \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\), "0"}, {"0", \(1\/4\ \((1 - \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\)}, {\(1\/4\ \((1 - \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\)\), "0"}, {"0", \(1\/4\ \((1 - \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\)\)}, {\(1\/4\ \((1 + \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\)\), "0"}, {"0", \(1\/4\ \((1 + \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Element nodal displacement vector and dispacement \ interpolation\ \>", "Subtitle", FontSize->36], Cell[CellGroupData[{ Cell[BoxData[{ \(Un\ = \ Table[U\_\(a, i\), {a, 4}, {i, 2}] // Flatten\), "\n", \(ue\ = \ H . 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