(************** 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[ 44027, 1016]*) (*NotebookOutlinePosition[ 44692, 1039]*) (* CellTagsIndexPosition[ 44648, 1035]*) (*WindowFrame->Normal*) Notebook[{ Cell["16.21 Techniques of structural analysis and design", "Title", TextAlignment->Center, FontSize->18], Cell["\<\ Home assignment 7 \ \>", "Subtitle", TextAlignment->Center, FontSize->18], Cell["Question 5: Problem 7.43 from textbook", "Subtitle"], Cell[BoxData[ \(\(\( (*\ clean\ all\ the\ variables\ \ *) \)\(\[IndentingNewLine]\)\(Clear["\"]; Off[General::spell, General::spell1];\)\)\)], "Input", FontSize->24], Cell["\<\ Cantilever beam loaded with a uniformly distributed load with a \ spring at free end. Algebraic basis functions\ \>", "Title", FontSize->24], Cell["1) approximation functions:", "Subtitle", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[{ \(Ck[n_] := \ Table[ck[i], {i, n}]\), "\[IndentingNewLine]", \(\[Phi]k[n_]\ := \ Table[x\^\(i + 1\), {i, n}]\), "\[IndentingNewLine]", \(w[n_]\ := \ Ck[n] . \[Phi]k[n]\), "\[IndentingNewLine]", \(w[2]\)}], "Input", FontSize->24], Cell[BoxData[ \(x\^2\ ck[1] + x\^3\ ck[2]\)], "Output"] }, Open ]], Cell["\<\ 2) approximate potential \[CapitalPi](Ck) for this problem\ \>", \ "Subtitle", FontSize->24], Cell[BoxData[{ \(\[CapitalDelta][n_]\ := \ w[n]\ /. x \[Rule] \ L\), "\[IndentingNewLine]", \(\[CapitalPi][ n_]\ := \(\(ym\ im\)\/2\) \(\[Integral]\_0\%L\ \ \(\((\[PartialD]\_\(x, x\)w[n])\)\^2\) \[DifferentialD]x\)\ + \ \(1\/2\) k\ \[CapitalDelta][n]\^2 - \[Integral]\_0\%L\( q\_0\) w[n] \[DifferentialD]x // Expand\)}], "Input", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(\[CapitalPi][1]\)], "Input"], Cell[BoxData[ \(1\/2\ k\ L\^4\ ck[1]\^2 + 2\ im\ L\ ym\ ck[1]\^2 - 1\/3\ L\^3\ ck[1]\ q\_0\)], "Output"] }, Open ]], 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][2] // Simplify\)], "Input", FontSize->24], Cell[BoxData[ \({L\ \((k\ L\^3\ \((ck[1] + L\ ck[2])\) + 2\ im\ ym\ \((2\ ck[1] + 3\ L\ ck[2])\))\) - \(L\^3\ q\_0\)\/3, L\^2\ \((k\ L\^3\ \((ck[1] + L\ ck[2])\) + 6\ im\ ym\ \((ck[1] + 2\ L\ ck[2])\))\) - \(L\^4\ q\_0\)\/4}\)], "Output"] }, Open ]], Cell[BoxData[ \(solution[n_]\ := \ Solve[grad\[CapitalPi][n]\ \[Equal] \ 0\ , \ Ck[n]] // Flatten\)], "Input", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(solution[1]\)], "Input", FontSize->24], Cell[BoxData[ \({ck[ 1] \[Rule] \(L\^3\ q\_0\)\/\(3\ \((k\ L\^4 + 4\ im\ L\ ym)\)\)}\)], \ "Output"] }, Open ]], Cell[BoxData[ \(wapprox[n_]\ := \ w[n]\ /. \ solution[n]\)], "Input", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(wapprox[1] // Simplify\)], "Input", FontSize->24], Cell[BoxData[ \(\(L\^2\ x\^2\ q\_0\)\/\(3\ k\ L\^3 + 12\ im\ ym\)\)], "Output"] }, Open ]], Cell["\<\ Plot the solution: it's very interesting to play with the values of \ k and plot the solution. 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