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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[ 163131, 3898]*) (*NotebookOutlinePosition[ 204240, 5236]*) (* CellTagsIndexPosition[ 204150, 5230]*) (*WindowFrame->Normal*) Notebook[{ Cell["16.21 Techniques of structural analysis and design", "Title", TextAlignment->Center, FontSize->18], Cell["\<\ Home assignment 4 \ \>", "Subtitle", TextAlignment->Center, FontSize->18], Cell[TextData[{ "Question 1: Note that we require the complete elasticity solution: ", Cell[BoxData[ \(TraditionalForm\`\(u\_\(\(i\)\(,\)\)\) \[Epsilon]\_ij, \ \[Sigma]\_ij\ \)]] }], "Subtitle", FontSize->18], Cell[CellGroupData[{ 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\(\[Sigma]\_12 = 0\)\);\)\)}], "Input",\ FontSize->18], Cell[TextData[{ "From equilibrium:\n", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\(x\_1\)\[Sigma]\_11 = 0\)]], " \[Implies] ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Sigma]\_11 = \ C\)\(,\)\)\)]], " b.c.: ", Cell[BoxData[ FormBox[ RowBox[{\(\(\[Sigma]\_11\)(\(-w\)\/2, x\_2, x\_3)\), " ", "=", " ", RowBox[{\(0\ \[Implies] C\), " ", "=", " ", RowBox[{"0", Cell[""]}]}]}], TraditionalForm]]], "\n\[DoubleLongRightArrow]" }], "Subtitle", FontSize->18], Cell[BoxData[ \(\(\[Sigma]\_11 = 0;\)\)], "Input", FontSize->18], Cell[" Similarly (from the plane stress condition)", "Subtitle", FontSize->18], Cell[BoxData[ \(\(\[Sigma]\_22 = 0;\)\)], "Input", FontSize->18], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\(x\_3\)\[Sigma]\_33 = 0\)]], " \n\nConstitutive:\n", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\_33 = \ \(1\/E\)[\[Sigma]\_33 - \[Nu]\ \ \((0 + 0)\)]\)]], "\n\nThen:\n", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\(x\_3\)\((E\ \[Epsilon]\_33)\)\ = \ \ \(0\ \[DoubleLongRightArrow]\ \[Epsilon]\_33 = \ \(\[PartialD]\_\(x\_3\)u\_3 \ = c\)\)\)]] }], "Subtitle", FontSize->18], Cell[BoxData[ \(\(u\_3 = c\ x\_3 + d;\)\)], "Input", FontSize->18], Cell[TextData[{ "\tThe bondary conditions are:\n\n", Cell[BoxData[ \(TraditionalForm\`u\_3[L\/2]\ = \ \(c\ L\/2 + d\ = \ 0\)\)]], "\n", Cell[BoxData[ \(TraditionalForm\`u\_3[\(-L\)\/2]\ = \ \(\(-c\)\ L\/2 + d\ = \ \[Delta]\)\)]] }], "Subtitle", FontSize->18], Cell[BoxData[{ \(\(rules\ = \ \ Solve[{c\ L\/2 + d\ \[Equal] 0, \ \(-c\)\ L\/2 + d\ \[Equal] \[Delta]}, {c, d}] // Flatten;\)\), "\[IndentingNewLine]", \(\(sol\ = \ u\_3 /. \ rules;\)\ (*\ this\ auxiliary\ step\ avoids\ infinite\ recursion\ *) \), "\ \[IndentingNewLine]", \(\(u\_3\ = \ sol;\)\)}], "Input", FontSize->18], Cell[BoxData[{ \(\(\[Epsilon]\_33 = \ \[PartialD]\_\(x\_3\)u\_3