(************** 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[ 39299, 918]*) (*NotebookOutlinePosition[ 39962, 941]*) (* CellTagsIndexPosition[ 39918, 937]*) (*WindowFrame->Normal*) Notebook[{ Cell["16.21 Techniques of structural analysis and design", "Title", TextAlignment->Center, FontSize->18], Cell["\<\ Home assignment 6 \ \>", "Subtitle", TextAlignment->Center, FontSize->18], Cell["\<\ Question 3: Problem 7.24 from textbook\ \>", "Subtitle"], Cell[TextData[{ StyleBox["The unknown displacement field for this problem is the deflection \ of the beam's neutral axis ", "Subsubtitle", FontSize->18], StyleBox[Cell[BoxData[ \(TraditionalForm\`\((w\_0)\) . \ We\ approximate\ this\ field\ using\ a\ Ritz\ approximation\ of\ \ the\ \(type : \[IndentingNewLine]w\_0\ ~\ c\_i \[Phi]\_i\)\)], "Subsubtitle", FontSize->18], "Subsubtitle"], StyleBox["The essential boundary conditions at the clamped end require:\n", "Subsubtitle", FontSize->18], StyleBox[Cell[BoxData[ \(TraditionalForm\`\(\[Phi]\_i\)(0)\)], "Subsubtitle", FontSize->18], "Subsubtitle"], StyleBox["=0, ", "Subsubtitle", FontSize->18], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[Phi]\_i'\)], "Subsubtitle", FontSize->18], "Subsubtitle"], StyleBox["(0) = 0\nA good candidate family of functions ", "Subsubtitle", FontSize->18], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[Phi]\_i\)], "Subsubtitle", FontSize->18], "Subsubtitle"], StyleBox["would be: ", "Subsubtitle", FontSize->18], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[Phi]\_i = \ x\^\(i + 1\), \ i = 1, n\)], "Subsubtitle", FontSize->18], "Subsubtitle"], StyleBox[". Note ", "Subsubtitle", FontSize->18], StyleBox[Cell[BoxData[ \(TraditionalForm\`x\^1\)], "Subsubtitle", FontSize->18], "Subsubtitle"], StyleBox[" does not satisfy the zero-slope boundary condition so it's been \ eliminated.\n\nIt is important to observe that the second half of the beam is \ going to rotate rigidly because the load is applied at x=L/2. Therefore, the \ chosen set of functions should be able to represent this deformation \ (completeness requirement). It can be observed that the chosen set of \ functions wil very slowly converge to the exact solution is this case since \ we are trying to represent a straight line (linear function) with a series of \ polynomials that do not contain the linear polynomial itself. This motivates \ the idea that it would be best if we could split the beam in two, choose \ independent approximations in each part and somehow link them together. This \ is the basis of the finite element method", "Subsubtitle", FontSize->18] }], "Subtitle", FontSize->168], Cell["\<\ Question 4: Problem 7.29 from textbook\ \>", "Subtitle"], Cell[BoxData[ \(\(\( (*\ clean\ all\ the\ variables\ \ *) \)\(\[IndentingNewLine]\)\(Clear["\"]; Off[General::spell, General::spell1];\)\)\)], "Input", FontSize->24], Cell["\<\ Clamped beam with concentrated load at x=L/2 \ \>", "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[\((x\/L)\)\^\(i + 1\), {i, n}]\), "\[IndentingNewLine]", \(w[n_]\ := \ Ck[n] . \[Phi]k[n]\), "\[IndentingNewLine]", \(w[2]\)}], "Input", FontSize->24], Cell[BoxData[ \(\(x\^2\ ck[1]\)\/L\^2 + \(x\^3\ ck[2]\)\/L\^3\)], "Output"] }, Open ]], Cell["\<\ 2) approximate potential \[CapitalPi](Ck) for this problem\ \>", \ "Subtitle", FontSize->24], Cell[BoxData[{ \(\[CapitalDelta][ n_]\ := \ \((w[n]\ /. \ x \[Rule] \ L\/2)\)\), "\[IndentingNewLine]", \(\[CapitalPi][ n_]\ := \(\(ym\ im\)\/2\) \(\[Integral]\_0\%L\ \ \(\((\[PartialD]\_\(x, x\)w[ n])\)\^2\) \[DifferentialD]x\)\ - \ \(F\_0\) \ \[CapitalDelta][n] // Expand\)}], "Input", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(\[CapitalPi][2]\)], "Input"], Cell[BoxData[ \(\(2\ im\ ym\ ck[1]\^2\)\/L\^3 + \(6\ im\ ym\ ck[1]\ ck[2]\)\/L\^3 + \(6\ \ im\ ym\ ck[2]\^2\)\/L\^3 - 1\/4\ ck[1]\ F\_0 - 1\/8\ ck[2]\ F\_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]\)], "Input", FontSize->24], Cell[BoxData[ \({\(4\ im\ ym\ ck[1]\)\/L\^3 + \(6\ im\ ym\ ck[2]\)\/L\^3 - F\_0\/4, \(6\ im\ ym\ ck[1]\)\/L\^3 + \(12\ im\ ym\ ck[2]\)\/L\^3 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