(************** 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[ 49245, 1099]*) (*NotebookOutlinePosition[ 50235, 1129]*) (* CellTagsIndexPosition[ 50191, 1125]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ \(\(Clear["\"];\)\)], "Input", FontSize->24], Cell[TextData[{ "Beam bending using Ritz method\n", StyleBox["Consider a simply supported beam with a distributed load of the \ form: q(x) = q0 ", "Subtitle"], StyleBox[Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(x1(L - x1)\), StyleBox[ RowBox[{" ", StyleBox[" ", "Subtitle"]}]], StyleBox["and", "Subtitle"], StyleBox[" ", "Subtitle"], StyleBox["a", "Subtitle"], StyleBox[" ", "Subtitle"], StyleBox["concentrated", "Subtitle"], StyleBox[" ", "Subtitle"], StyleBox["load", "Subtitle"], StyleBox[" ", "Subtitle"], StyleBox["acting", "Subtitle"], StyleBox[" ", "Subtitle"], StyleBox["at", "Subtitle"], StyleBox[" ", "Subtitle"], StyleBox["x1", "Subtitle"]}], StyleBox[" ", "Subtitle"], StyleBox["=", "Subtitle"], StyleBox[" ", "Subtitle"], FractionBox[ StyleBox["L", "Subtitle"], "2"]}], TraditionalForm]], "Subtitle"], "Subtitle"], StyleBox[". Compute an approximate solution. The properties of the beam are \ as follows:", "Subtitle"] }], "Title", FontSize->24], Cell[CellGroupData[{ Cell["1) Seek approximate solutions of the form:", "Subtitle", FontSize->24], Cell[BoxData[{ \(Ck[n_] := \ Table[ck[i], {i, n}]\), "\[IndentingNewLine]", \(\[Phi]k[n_]\ := \ Table[Sin[\((2\ i\ - 1)\)\ \[Pi]\ x1\/L], {i, n}]\), "\[IndentingNewLine]", \(U3[n_]\ := \ Ck[n] . \[Phi]k[n]\)}], "Input", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(U3[1]\)], "Input", FontSize->24], Cell[BoxData[ \(ck[1]\ Sin[\(\[Pi]\ x1\)\/L]\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 2) Define the approximate potential \[CapitalPi](Ck) for this \ problem\ \>", "Subtitle", FontSize->24], Cell[BoxData[{\(d2U3[n_]\ := \ D[U3[n], {x1, 2}]\ // Simplify\), "\[IndentingNewLine]", \(U3qt[ n_]\ := \ U3[n]\ /. \ x1\ \[Rule] \ L\/4\), "\[IndentingNewLine]", \(q\ = \ q0\ x1\ \((L - x1)\);\), "\[IndentingNewLine]", RowBox[{ RowBox[{ StyleBox["\[CapitalPi]", FontSize->24], "[", "n_", "]"}], " ", ":=", \(\[Integral]\_0\%L\((\ \(\(ym\ im\)\/2\) \((d2U3[n])\)^2 - U3[n]\ q)\) \[DifferentialD]x1 - P\ U3qt[n] // Expand\)}]}], "Input", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(\[CapitalPi][1]\)], "Input", FontSize->24], Cell[BoxData[ \(\(-\(\(P\ ck[ 1]\)\/\@2\)\) - \(4\ L\^3\ q0\ ck[1]\)\/\[Pi]\^3 + \(im\ \ \[Pi]\^4\ ym\ ck[1]\^2\)\/\(4\ L\^3\)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 3) Minimize the potential to find the value of the coefficients \ that best approximate this problem\ \>", "Subtitle", FontSize->24], Cell[BoxData[ \(grad\[CapitalPi][n_]\ := \ Table[D[\[CapitalPi][n], ck[i]], {i, n}]\)], "Input", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(grad\[CapitalPi][1]\)], "Input", FontSize->24], Cell[BoxData[ \({\(-\(P\/\@2\)\) - \(4\ L\^3\ q0\)\/\[Pi]\^3 + \(im\ \[Pi]\^4\ ym\ \ ck[1]\)\/\(2\ L\^3\)}\)], "Output"] }, Open ]], Cell[BoxData[ \(solution[n_]\ := \ Solve[grad\[CapitalPi][n]\ \[Equal] \ 0\ , \ Ck[n]] // Flatten\)], "Input", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(solution[2]\)], "Input", FontSize->24], Cell[BoxData[ \({ck[ 1] \[Rule] \(2\ L\^3\ \((P\/\@2 + \(4\ L\^3\ \ q0\)\/\[Pi]\^3)\)\)\/\(im\ \[Pi]\^4\ ym\), ck[2] \[Rule] \(2\ L\^3\ \((P\/\@2 + \(4\ L\^3\ q0\)\/\(27\ \[Pi]\^3\))\ \)\)\/\(81\ im\ \[Pi]\^4\ ym\)}\)], "Output"] }, Open ]], Cell[BoxData[ \(U3final[n_]\ := \ U3[n]\ /. \ solution[n]\)], "Input", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(U3final[2]\)], "Input", FontSize->24], Cell[BoxData[ \(\(2\ L\^3\ \((P\/\@2 + \(4\ L\^3\ q0\)\/\[Pi]\^3)\)\ Sin[\(\[Pi]\ \ x1\)\/L]\)\/\(im\ \[Pi]\^4\ ym\) + \(2\ L\^3\ \((P\/\@2 + \(4\ L\^3\ \ q0\)\/\(27\ \[Pi]\^3\))\)\ Sin[\(3\ \[Pi]\ x1\)\/L]\)\/\(81\ im\ \[Pi]\^4\ ym\ \)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Plot the solution", "Subtitle", Evaluatable->False, FontSize->24], Cell[CellGroupData[{ Cell[BoxData[{ \(\(values\ = \ {ym\ \[Rule] \ 70\ 10\^9, L \[Rule] \ 3, im \[Rule] \ \(0.05\ 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