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kno_0080oooo00<0mOGe0?;blP3lo?`08@3oool10?Win@40oooo0@3oool00?l0fMWI7@3IfMT00001 \ \>"], ImageRangeCache->{{{0, 283}, {283, 0}} -> {-0.00141501, 0.998585, 1.00001, \ 1.00001}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output", FontSize->18] }, Open ]], Cell[TextData[{ "Question 1: Given the following displacement field:\n", Cell[BoxData[{ \(u\_1 = \ \[Phi]\_1[x\_2]\), "\n", \(u\_2 = \ 0\), "\n", \(u\_3 = \ 0\)}], FontSize->18], "\n\nfor the aluminum plate of the figure whose dimensions are: width = \ 0.1 m, height = 0.1 m, thickness = 3 mm and whose elastic properties are: \ E=70 GPa and \[Nu]=0.3:\na) Give an example for ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\_1\)]], ", draw and provide an interpretation of how the plate deforms\nb) Evaluate \ the internal strain energy and the complementary strain energy of the plate\n\ c) Obtain numerical values of these quantities for ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\_1 = \ \[Gamma]\ x\_2\)]], " with \[Gamma] = 0.005" }], "Subtitle", FontSize->18], Cell[CellGroupData[{ Cell[BoxData[{ \(u\_1 = \ \[Phi]\_1[x\_2]\), "\[IndentingNewLine]", \(u\_2 = \ 0\), "\[IndentingNewLine]", \(u\_3 = \ 0\), "\[IndentingNewLine]", \(Table[\[Epsilon]\_\(i, j\) = \(1\/2\) \((\[PartialD]\_\(x\_j\)u\_i + \ \ \[PartialD]\_\(x\_i\)u\_j)\), \ {i, 3}, {j, 3}]\)}], "Input", FontSize->18], Cell[BoxData[ \(\[Phi]\_1[x\_2]\)], "Output", FontSize->18], Cell[BoxData[ \(0\)], "Output", FontSize->18], Cell[BoxData[ \(0\)], "Output", FontSize->18], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{\(1\/2\), " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", "\[Prime]", MultilineFunction->None], "[", \(x\_2\), "]"}]}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{\(1\/2\), " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", "\[Prime]", MultilineFunction->None], "[", \(x\_2\), "]"}]}], ",", "0", ",", "0"}], "}"}], ",", \({0, 0, 0}\)}], "}"}]], "Output", FontSize->18] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[\[Sigma]\_\(i, j\) = \ \[Lambda]\ Sum[\[Epsilon]\_\(k, k\), {k, 3}]\ KroneckerDelta[i, j] + \ 2 \[Mu]\ \[Epsilon]\_\(i, j\), \ {i, 3}, {j, 3}]\)], "Input", FontSize->18], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{"\[Mu]", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", "\[Prime]", MultilineFunction->None], "[", \(x\_2\), "]"}]}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"\[Mu]", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", "\[Prime]", MultilineFunction->None], "[", \(x\_2\), "]"}]}], ",", "0", ",", "0"}], "}"}], ",", \({0, 0, 0}\)}], "}"}]], "Output", FontSize->18] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(U\_0 = \ \(1\/2\) Sum[\(\[Sigma]\_\(i, j\)\) \[Epsilon]\_\(i, j\), {j, 3}, {i, 3}]\)], "Input", FontSize->18], Cell[BoxData[ RowBox[{\(1\/2\), " ", "\[Mu]", " ", SuperscriptBox[ RowBox[{ SubsuperscriptBox["\[Phi]", "1", "\[Prime]", MultilineFunction->None], "[", 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"\[IndentingNewLine]", \(\(t\ = \ 0.003\ Meter;\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(EE\ = \ 70\ 10\^9\ Newton\/Meter\^2;\)\), "\[IndentingNewLine]", \(\(\[Nu]\ = \ 0.3;\)\), "\[IndentingNewLine]", \(\[Mu]\ = \ EE\/\(2\ \((1 + \[Nu])\)\)\), "\[IndentingNewLine]", \(\[Gamma]\ = \ 0.005\), "\[IndentingNewLine]", \(U\)}], "Input", FontSize->18], Cell[BoxData[ \(\(2.692307692307692`*^10\ Newton\)\/Meter\^2\)], "Output"], Cell[BoxData[ \(0.005`\)], "Output"], Cell[BoxData[ \(20.19230769230769`\ Meter\ Newton\)], "Output"] }, Open ]], Cell[TextData[{ "Question 2: Use the PVD to obtain equilibrium equations and natural \ boundary conditions for a cantilever beam loaded with a distributed load f(x) \ (owe you figure):\n\n", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(\[Epsilon]\ = \ -\), "TraditionalForm"], "z", " ", \(w''\), \((x)\), " "}], TraditionalForm]]], "\n\[Sigma] = -E z w''(x)\n\nVirtual displacement \[Delta]w, virtual \ strains:\n\[Delta]\[Epsilon] = -z \[Delta]w''(x)\n\nPVD reads in this case:\n\ \n", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[Integral]\_V \[Sigma]\ \ \[Delta]\[Epsilon]\ dx\)\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\^L\( f(x)\)\ \(\[Delta]w( x)\)\ dx\)]], "\n\nVirtual work of internal forces:\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[Integral]\_V \[Sigma]\ \ \[Delta]\[Epsilon]\ dx\ = \ \[Integral]\_V E\ \(z\^2\) w'' \((x)\)\ \[Delta]w'' \((x)\)\ dx\)\)\)]], " = \n\n= ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\^L E\ I\ w'' \((x)\)\ \[Delta]w'' \((x)\)\ dx\)]], "\n\nTo derive differential equations, want to transform \[Delta]w''(x) \ into something involving \[Delta]w(x) \[LongRightArrow] integration by parts\n\ \n", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\(\[Integral]\_0\^L\ \((EI\ w''\ \ \[Delta]w')\)'\ dx\ - \ \[Integral]\_0\^L\ \(\(\((EI\ w'')\)'\)\(\ \)\(\ \[Delta]w'\)\(\ \)\(dx\)\(\ \)\)\)\)\)]], "\n\n= ", Cell[BoxData[ \(TraditionalForm\`\((EI\ w''\ \[Delta]w')\)\( | \_L\)\(-\ \((EI\ w''\ \ \[Delta]w')\)\)\( | \_0\)\)]], " - \n\n", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\^L\ \(\(\((EI\ w'')\)'\)\(\ \)\(\ \[Delta]w'\)\(\ \)\(dx\)\(\ \)\)\)]], "\n\nThe second term is zero because ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Delta]w'\)\( | \_0\)\)\ = \ 0\)]], " (it has to satisfy the homogeneous essential boundary condition of zero \ slope at the clamp that the real solution satisfies). The third term needs to \ be integrated by parts one more time to obtain \[Delta]w inside integral:\n\n\ = ", Cell[BoxData[ \(TraditionalForm\`\((EI\ w''\ \[Delta]w')\)\( | \ \_L\)\(-\(\[Integral]\_0\^L\([\ \((EI\ w'')\)'\ \[Delta]w]\)'\ dx\)\)\ + \ \[Integral]\_0\^L\ \(\(\((EI\ w'')\)''\)\(\ \)\(\[Delta]w\)\(\ \)\(dx\)\(\ \)\ \)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(\((EI\ w''\ \[Delta]w')\)\( | \_L\)\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`\([\((EI\ w'')\)' \[Delta]w]\)\( | \_L\)\(-\ \ \([\((EI\ w'')\)' \[Delta]w]\)\)\( | \_0\)\(+\ \(\[Integral]\_0\^L\ \(\(\((EI\ \ w'')\)''\)\(\ \)\(\[Delta]w\)\(\ \)\(dx\)\(\ \ \)\)\)\)\)]], "\nThe second term in this expression also vanishes because ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Delta]w\)\( | \_0\)\)\ = \ 0\)]], " (it has to satisfy the homogeneous essential boundary condition of zero \ deflection at the clamp that the real solution satisfies). \n\nThe PVD then \ reads:\n", Cell[BoxData[ \(TraditionalForm\`\(\((EI\ w''\ \[Delta]w')\)\( | \_L\)\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`\(\(\((EI\ w'')\)' \[Delta]w\)\( | \_L\)\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[Integral]\_0\^L\ \(\(\((EI\ w'')\)''\)\(\ \ \)\(\[Delta]w\)\(\ \)\(dx\)\(\ \)\)\)\)\)]], "= - ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\^L\( f(x)\)\ \[Delta]w\ dx\)]], "\n\nThe only way this can be satisfied for all admissible \[Delta]w \ functions is if:\n\nEI w'' ", Cell[BoxData[ \(TraditionalForm\`\(\( | \_L\)\(\ \)\(\(=\)\(\ \)\(0\)\)\)\)]], " (no moment applied at cantilever tip)\n\n(EI w'')' ", Cell[BoxData[ \(TraditionalForm\`\(\( | \_L\)\(\(=\)\(\ \)\(0\)\)\)\)]], " (no shear force applied at cantilever tip)\n\n(EI w'')'' + f = 0 \ (differential equation of equilibrium of beam)" }], "Subtitle"], Cell["\<\ Question 3: Example 4.5 from textbook Question 4: Example 4.6 from textbook Question 5: Example 5.2 from textbook Question 5: Example 5.3 from textbook\ \>", "Subtitle"] }, FrontEndVersion->"4.2 for X", ScreenRectangle->{{0, 1600}, {0, 1200}}, WindowSize->{594, 600}, WindowMargins->{{150, Automatic}, {Automatic, 52}}, StyleDefinitions -> "Report.nb" ] (******************************************************************* Cached data follows. 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