(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 294154, 5624] NotebookOptionsPosition[ 287308, 5494] NotebookOutlinePosition[ 287898, 5514] CellTagsIndexPosition[ 287855, 5511] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Beam equations 2023", "Section", InitializationCell->True, CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4754623120573997`*^9, 3.4754623242566*^9}, {3.600410324619398*^9, 3.6004103283166046`*^9}, {3.600571616524475*^9, 3.600571617101676*^9}, 3.6005720003623533`*^9, {3.6971731591379333`*^9, 3.697173161237936*^9}, { 3.6972526005256643`*^9, 3.697252602065666*^9}, 3.7289615239416647`*^9, 3.760683546211379*^9, {3.7929365664229565`*^9, 3.792936566541724*^9}, 3.8251520926927586`*^9, 3.85665761838736*^9, {3.888370231650601*^9, 3.888370233021748*^9}},ExpressionUUID->"064353de-d88c-490a-aabe-\ 7b34514beb63"], Cell[CellGroupData[{ Cell["xyz-system", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6972526642557535`*^9, 3.697252670705762*^9}, { 3.697252700885804*^9, 3.697252701855806*^9}},ExpressionUUID->"ab002cd9-568c-424a-9781-\ 52ea039cdee4"], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{"limit", " ", "\[Alpha]"}], " ", "\[Rule]", " ", RowBox[{"Infinity", " ", "gives", " ", "Bernoulli", " ", "solution"}]}], "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"equilibrium", " ", "equations"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"equilibrium", "[", "xyz", "]"}], ":=", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Nx", "'"}], "[", "x", "]"}], "+", "bx"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Qy", "'"}], "[", "x", "]"}], "+", "by"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Qz", "'"}], "[", "x", "]"}], "+", "bz"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Tx", "'"}], "[", "x", "]"}], "+", "cx"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"My", "'"}], "[", "x", "]"}], "-", RowBox[{"Qz", "[", "x", "]"}], "+", "cy"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Mz", "'"}], "[", "x", "]"}], "+", RowBox[{"Qy", "[", "x", "]"}], "+", "cz"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"constitutive", " ", "equations"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"constitutive", "[", "xyz", "]"}], ":=", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Nx", "[", "x", "]"}], "-", RowBox[{"\[CapitalEpsilon]", " ", RowBox[{"(", RowBox[{ RowBox[{"A", " ", RowBox[{ RowBox[{"u", "'"}], "[", "x", "]"}]}], "-", RowBox[{"Sz", " ", RowBox[{ RowBox[{"\[Psi]", "'"}], "[", "x", "]"}]}], "+", RowBox[{"Sy", " ", RowBox[{ RowBox[{"\[Theta]", "'"}], "[", "x", "]"}]}]}], ")"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Qy", "[", "x", "]"}], "-", RowBox[{"G", " ", RowBox[{"(", RowBox[{ RowBox[{"A", " ", "\[Alpha]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"v", "'"}], "[", "x", "]"}], "-", RowBox[{"\[Psi]", "[", "x", "]"}]}], ")"}]}], "-", RowBox[{"Sy", " ", RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "x", "]"}]}]}], ")"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Qz", "[", "x", "]"}], "-", RowBox[{"G", " ", RowBox[{"(", RowBox[{ RowBox[{"A", " ", "\[Alpha]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"w", "'"}], "[", "x", "]"}], "+", RowBox[{"\[Theta]", "[", "x", "]"}]}], ")"}]}], "+", RowBox[{"Sz", " ", RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "x", "]"}]}]}], ")"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Tx", "[", "x", "]"}], "-", RowBox[{"G", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "Sy"}], " ", "\[Alpha]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"v", "'"}], "[", "x", "]"}], "-", RowBox[{"\[Psi]", "[", "x", "]"}]}], ")"}]}], "+", RowBox[{"Sz", " ", "\[Alpha]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"w", "'"}], "[", "x", "]"}], "+", RowBox[{"\[Theta]", "[", "x", "]"}]}], ")"}]}], "+", RowBox[{"Irr", " ", RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "x", "]"}]}]}], ")"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"My", "[", "x", "]"}], "-", RowBox[{"\[CapitalEpsilon]", " ", RowBox[{"(", RowBox[{ RowBox[{"Sy", " ", RowBox[{ RowBox[{"u", "'"}], "[", "x", "]"}]}], "-", RowBox[{"Iyz", " ", RowBox[{ RowBox[{"\[Psi]", "'"}], "[", "x", "]"}]}], "+", RowBox[{"Iyy", " ", RowBox[{ RowBox[{"\[Theta]", "'"}], "[", "x", "]"}]}]}], ")"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Mz", "[", "x", "]"}], "-", RowBox[{"\[CapitalEpsilon]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "Sz"}], " ", RowBox[{ RowBox[{"u", "'"}], "[", "x", "]"}]}], "+", RowBox[{"Izz", " ", RowBox[{ RowBox[{"\[Psi]", "'"}], "[", "x", "]"}]}], "-", RowBox[{"Izy", " ", RowBox[{ RowBox[{"\[Theta]", "'"}], "[", "x", "]"}]}]}], ")"}]}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", "quantities", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"displacements", "[", "xyz", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"u", "[", "x", "]"}], ",", RowBox[{"v", "[", "x", "]"}], ",", RowBox[{"w", "[", "x", "]"}], ",", RowBox[{"\[Phi]", "[", "x", "]"}], ",", RowBox[{"\[Theta]", "[", "x", "]"}], ",", RowBox[{"\[Psi]", "[", "x", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"forces", "[", "xyz", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"Nx", "[", "x", "]"}], ",", RowBox[{"Qy", "[", "x", "]"}], ",", RowBox[{"Qz", "[", "x", "]"}], ",", RowBox[{"Tx", "[", "x", "]"}], ",", RowBox[{"My", "[", "x", "]"}], ",", RowBox[{"Mz", "[", "x", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"functions", "[", "xyz", "]"}], ":=", RowBox[{"Join", "[", RowBox[{ RowBox[{"displacements", "[", "xyz", "]"}], ",", RowBox[{"forces", "[", "xyz", "]"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"some", " ", "common", " ", "boundary", " ", "conditions"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"clamped", "[", "xyz", "]"}], "[", "x_", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"u", "[", "x", "]"}], ",", RowBox[{"v", "[", "x", "]"}], ",", RowBox[{"w", "[", "x", "]"}], ",", RowBox[{"\[Phi]", "[", "x", "]"}], ",", RowBox[{"\[Theta]", "[", "x", "]"}], ",", RowBox[{"\[Psi]", "[", "x", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"free", "[", "xyz", "]"}], "[", "x_", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"Nx", "[", "x", "]"}], ",", RowBox[{"Qy", "[", "x", "]"}], ",", RowBox[{"Qz", "[", "x", "]"}], ",", RowBox[{"Tx", "[", "x", "]"}], ",", RowBox[{"My", "[", "x", "]"}], ",", RowBox[{"Mz", "[", "x", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"simply", "[", "xyz", "]"}], "[", "x_", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"u", "[", "x", "]"}], ",", RowBox[{"v", "[", "x", "]"}], ",", RowBox[{"w", "[", "x", "]"}], ",", RowBox[{"Tx", "[", "x", "]"}], ",", RowBox[{"My", "[", "x", "]"}], ",", RowBox[{"Mz", "[", "x", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"slide", "[", "xyz", "]"}], "[", "x_", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"Nx", "[", "x", "]"}], ",", RowBox[{"v", "[", "x", "]"}], ",", RowBox[{"w", "[", "x", "]"}], ",", RowBox[{"\[Phi]", "[", "x", "]"}], ",", RowBox[{"My", "[", "x", "]"}], ",", RowBox[{"Mz", "[", "x", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]"}]}]], "Input", InitializationCell->True, CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, 3.537622776495909*^9, { 3.537622908932484*^9, 3.537622917335965*^9}, {3.5376238837162385`*^9, 3.537624004391141*^9}, {3.5376240458145103`*^9, 3.5376242437438307`*^9}, { 3.537624281650999*^9, 3.5376244166457205`*^9}, {3.5376244546488943`*^9, 3.5376248198917847`*^9}, {3.5397503703948455`*^9, 3.5397504308449516`*^9}, 3.5627674253055143`*^9, {3.5781422724253883`*^9, 3.578142305481847*^9}, { 3.5828109510514917`*^9, 3.582810961924711*^9}, {3.6000023977095795`*^9, 3.6000024056343937`*^9}, {3.600002438425651*^9, 3.6000024578788853`*^9}, { 3.600002497565355*^9, 3.600002598559932*^9}, {3.6000027265893574`*^9, 3.6000028178183174`*^9}, {3.600004251273635*^9, 3.6000042647676587`*^9}, { 3.600070082152542*^9, 3.6000701428834486`*^9}, {3.6000701834435205`*^9, 3.600070190151532*^9}, {3.6000702259223948`*^9, 3.6000702268583965`*^9}, { 3.6000702793212886`*^9, 3.600070364949839*^9}, {3.600070434744362*^9, 3.6000704744932313`*^9}, {3.600070516816106*^9, 3.6000705186257086`*^9}, 3.600071039151823*^9, {3.6000711295227823`*^9, 3.600071143297606*^9}, { 3.6005715147810965`*^9, 3.600571515717098*^9}, {3.663135382391988*^9, 3.663135470097742*^9}, {3.6631360823052173`*^9, 3.66313609528444*^9}, { 3.663136161288156*^9, 3.663136161787357*^9}, {3.663136351093689*^9, 3.6631364825551205`*^9}, {3.6631365162355795`*^9, 3.6631365249247947`*^9}, {3.663220430524748*^9, 3.663220430758748*^9}, { 3.6972527130658216`*^9, 3.697252849546013*^9}, {3.6972530958863573`*^9, 3.697253118446389*^9}, {3.697253453686858*^9, 3.697253477746892*^9}, 3.697253957317563*^9, {3.69727811863721*^9, 3.697278131467228*^9}, { 3.760961617600359*^9, 3.760961641162077*^9}, 3.7609616930163608`*^9}, CellLabel->"In[1]:=",ExpressionUUID->"cc72c849-5d5a-40cb-ada4-785af365f33c"] }, Closed]], Cell[CellGroupData[{ Cell["snb-system", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6972526642557535`*^9, 3.697252670705762*^9}},ExpressionUUID->"05a2f398-c16e-4461-a17d-\ 36fd6f9726ee"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{"limit", " ", "\[Alpha]"}], " ", "\[Rule]", " ", RowBox[{"Infinity", " ", "gives", " ", "Bernoulli", " ", "solution"}]}], "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"equilibrium", "[", "snb", "]"}], ":=", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Ns", "'"}], "[", "s", "]"}], "-", RowBox[{ RowBox[{"Qn", "[", "s", "]"}], " ", "\[Kappa]"}], "+", "bs"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Qn", "'"}], "[", "s", "]"}], "+", RowBox[{ RowBox[{"Ns", "[", "s", "]"}], " ", "\[Kappa]"}], "-", RowBox[{ RowBox[{"Qb", "[", "s", "]"}], " ", "\[Tau]"}], "+", "bn"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Qb", "'"}], "[", "s", "]"}], "+", RowBox[{ RowBox[{"Qn", "[", "s", "]"}], " ", "\[Tau]"}], "+", "bb"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Ts", "'"}], "[", "s", "]"}], "-", RowBox[{ RowBox[{"Mn", "[", "s", "]"}], " ", "\[Kappa]"}], "+", "cs"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Mn", "'"}], "[", "s", "]"}], "+", RowBox[{ RowBox[{"Ts", "[", "s", "]"}], " ", "\[Kappa]"}], "-", RowBox[{ RowBox[{"Mb", "[", "s", "]"}], "*", "\[Tau]"}], "-", RowBox[{"Qb", "[", "s", "]"}], "+", "cn"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Mb", "'"}], "[", "s", "]"}], "+", RowBox[{ RowBox[{"Mn", "[", "s", "]"}], " ", "\[Tau]"}], "+", RowBox[{"Qn", "[", "s", "]"}], "+", "cb"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"constitutive", "[", "snb", "]"}], ":=", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Ns", "[", "s", "]"}], "-", RowBox[{"\[CapitalEpsilon]", " ", "A", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"u", "'"}], "[", "s", "]"}], "-", RowBox[{ RowBox[{"v", "[", "s", "]"}], " ", "\[Kappa]"}]}], ")"}]}], "+", RowBox[{"\[CapitalEpsilon]", " ", "Sb", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[Psi]", "'"}], "[", "s", "]"}], "+", RowBox[{ RowBox[{"\[Theta]", "[", "s", "]"}], " ", "\[Tau]"}]}], ")"}]}], "-", RowBox[{"\[CapitalEpsilon]", " ", "Sn", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[Theta]", "'"}], "[", "s", "]"}], "+", RowBox[{ RowBox[{"\[Phi]", "[", "s", "]"}], " ", "\[Kappa]"}], "-", RowBox[{ RowBox[{"\[Psi]", "[", "s", "]"}], " ", "\[Tau]"}]}], ")"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Qn", "[", "s", "]"}], "-", RowBox[{"G", " ", "A", " ", "\[Alpha]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"v", "'"}], "[", "s", "]"}], "+", RowBox[{ RowBox[{"u", "[", "s", "]"}], " ", "\[Kappa]"}], "-", RowBox[{ RowBox[{"w", "[", "s", "]"}], " ", "\[Tau]"}], "-", RowBox[{"\[Psi]", "[", "s", "]"}]}], ")"}]}], "+", RowBox[{"G", " ", "Sn", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "s", "]"}], "-", RowBox[{ RowBox[{"\[Theta]", "[", "s", "]"}], " ", "\[Kappa]"}]}], ")"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Qb", "[", "s", "]"}], "-", RowBox[{"G", " ", "A", " ", "\[Alpha]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"w", "'"}], "[", "s", "]"}], "+", RowBox[{ RowBox[{"v", "[", "s", "]"}], " ", "\[Tau]"}], "+", RowBox[{"\[Theta]", "[", "s", "]"}]}], ")"}]}], "-", RowBox[{"G", " ", "Sb", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "s", "]"}], "-", RowBox[{ RowBox[{"\[Theta]", "[", "s", "]"}], " ", "\[Kappa]"}]}], ")"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Ts", "[", "s", "]"}], "-", RowBox[{"G", " ", "Irr", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "s", "]"}], "-", RowBox[{ RowBox[{"\[Theta]", "[", "s", "]"}], " ", "\[Kappa]"}]}], ")"}]}], "-", RowBox[{"G", " ", "Sb", " ", "\[Alpha]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"w", "'"}], "[", "s", "]"}], "+", RowBox[{ RowBox[{"v", "[", "s", "]"}], " ", "\[Tau]"}], "+", RowBox[{"\[Theta]", "[", "s", "]"}]}], ")"}]}], "+", RowBox[{"G", " ", "Sn", " ", "\[Alpha]", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"v", "'"}], "[", "s", "]"}], "+", RowBox[{ RowBox[{"u", "[", "s", "]"}], " ", "\[Kappa]"}], "-", RowBox[{ RowBox[{"w", "[", "s", "]"}], " ", "\[Tau]"}], "-", RowBox[{"\[Psi]", "[", "s", "]"}]}], ")"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Mn", "[", "s", "]"}], "-", RowBox[{"\[CapitalEpsilon]", " ", "Inn", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[Theta]", "'"}], "[", "s", "]"}], "+", RowBox[{ RowBox[{"\[Phi]", "[", "s", "]"}], " ", "\[Kappa]"}], "-", RowBox[{ RowBox[{"\[Psi]", "[", "s", "]"}], " ", "\[Tau]"}]}], ")"}]}], "-", RowBox[{"\[CapitalEpsilon]", " ", "Sn", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"u", "'"}], "[", "s", "]"}], "-", RowBox[{ RowBox[{"v", "[", "s", "]"}], " ", "\[Kappa]"}]}], ")"}]}], "+", RowBox[{"\[CapitalEpsilon]", " ", "Ibn", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[Psi]", "'"}], "[", "s", "]"}], "+", RowBox[{ RowBox[{"\[Theta]", "[", "s", "]"}], " ", "\[Tau]"}]}], ")"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Mb", "[", "s", "]"}], "-", RowBox[{"\[CapitalEpsilon]", " ", "Ibb", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[Psi]", "'"}], "[", "s", "]"}], "+", RowBox[{ RowBox[{"\[Theta]", "[", "s", "]"}], " ", "\[Tau]"}]}], ")"}]}], "+", RowBox[{"\[CapitalEpsilon]", " ", "Sb", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"u", "'"}], "[", "s", "]"}], "-", RowBox[{ RowBox[{"v", "[", "s", "]"}], " ", "\[Kappa]"}]}], ")"}]}], "+", RowBox[{"\[CapitalEpsilon]", " ", "Inb", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[Theta]", "'"}], "[", "s", "]"}], "+", RowBox[{ RowBox[{"\[Phi]", "[", "s", "]"}], " ", "\[Kappa]"}], "-", RowBox[{ RowBox[{"\[Psi]", "[", "s", "]"}], " ", "\[Tau]"}]}], ")"}]}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"clamped", "[", "snb", "]"}], "[", "s_", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"u", "[", "s", "]"}], ",", RowBox[{"v", "[", "s", "]"}], ",", RowBox[{"w", "[", "s", "]"}], ",", RowBox[{"\[Phi]", "[", "s", "]"}], ",", RowBox[{"\[Theta]", "[", "s", "]"}], ",", RowBox[{"\[Psi]", "[", "s", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"free", "[", "snb", "]"}], "[", "s_", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"Ns", "[", "s", "]"}], ",", RowBox[{"Qn", "[", "s", "]"}], ",", RowBox[{"Qb", "[", "s", "]"}], ",", RowBox[{"Ts", "[", "s", "]"}], ",", RowBox[{"Mn", "[", "s", "]"}], ",", RowBox[{"Mb", "[", "s", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"simply", "[", "snb", "]"}], "[", "s_", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"u", "[", "s", "]"}], ",", RowBox[{"v", "[", "s", "]"}], ",", RowBox[{"w", "[", "s", "]"}], ",", RowBox[{"Ts", "[", "s", "]"}], ",", RowBox[{"Mn", "[", "s", "]"}], ",", RowBox[{"Mb", "[", "s", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"displacements", "[", "snb", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"u", "[", "s", "]"}], ",", RowBox[{"v", "[", "s", "]"}], ",", RowBox[{"w", "[", "s", "]"}], ",", RowBox[{"\[Phi]", "[", "s", "]"}], ",", RowBox[{"\[Theta]", "[", "s", "]"}], ",", RowBox[{"\[Psi]", "[", "s", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"forces", "[", "snb", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"Ns", "[", "s", "]"}], ",", RowBox[{"Qn", "[", "s", "]"}], ",", RowBox[{"Qb", "[", "s", "]"}], ",", RowBox[{"Ts", "[", "s", "]"}], ",", RowBox[{"Mn", "[", "s", "]"}], ",", RowBox[{"Mb", "[", "s", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"functions", "[", "snb", "]"}], ":=", RowBox[{"Join", "[", RowBox[{ RowBox[{"displacements", "[", "snb", "]"}], ",", RowBox[{"forces", "[", "snb", "]"}]}], "]"}]}], ";"}], "\[IndentingNewLine]"}]}]], "Input", InitializationCell->True, CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4754625674138002`*^9, 3.4754625790358*^9}, 3.4754707481998*^9, { 3.6005716753365784`*^9, 3.6005716876918*^9}, 3.6005717324482784`*^9, { 3.600593664589115*^9, 3.6005936722175283`*^9}, {3.6005937537744713`*^9, 3.6005938318994083`*^9}, {3.600593886733505*^9, 3.6005940857430544`*^9}, 3.6005941782044168`*^9, {3.6005942701041784`*^9, 3.600594408960022*^9}, { 3.6005944423128805`*^9, 3.600594462982917*^9}, {3.600594515149409*^9, 3.600595127372484*^9}, {3.6005952285698614`*^9, 3.6005955218971767`*^9}, { 3.600595556591638*^9, 3.6005957993592644`*^9}, {3.60059584254014*^9, 3.6005961280986414`*^9}, {3.6005963577778454`*^9, 3.6005963693530655`*^9}, {3.663672823546813*^9, 3.6636730038831296`*^9}, { 3.6636733305321035`*^9, 3.663673361030157*^9}, {3.69717157823372*^9, 3.6971719533642445`*^9}, {3.6972523864153643`*^9, 3.697252508205535*^9}, { 3.6972531687764597`*^9, 3.697253227016541*^9}, 3.6972534236468163`*^9, { 3.697253459656867*^9, 3.69725346177687*^9}}, CellLabel->"In[10]:=",ExpressionUUID->"b123c695-f0a8-49c9-8b5a-d535a1a4e2ed"] }, Open ]], Cell[CellGroupData[{ Cell["zxy-system (shear and torsion corrections and symmetry)", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6972526642557535`*^9, 3.697252670705762*^9}, { 3.697252700885804*^9, 3.697252701855806*^9}, {3.731830075930358*^9, 3.7318300954252224`*^9}, {3.7318301298139157`*^9, 3.7318301345137186`*^9}, {3.731831715855138*^9, 3.731831717355487*^9}},ExpressionUUID->"a7bd322b-bcff-446f-9164-\ d14fef465862"], Cell[BoxData[ RowBox[{ RowBox[{"(*", RowBox[{"equilibrium", " ", "equations"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"equilibrium", "[", "zxy", "]"}], ":=", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Fz", "'"}], "[", "z", "]"}], "+", "bz"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Fx", "'"}], "[", "z", "]"}], "+", "bx"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Fy", "'"}], "[", "z", "]"}], "+", "by"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Mz", "'"}], "[", "z", "]"}], "+", "cz"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Mx", "'"}], "[", "z", "]"}], "-", RowBox[{"Fy", "[", "z", "]"}], "+", "cx"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"My", "'"}], "[", "z", "]"}], "+", RowBox[{"Fx", "[", "z", "]"}], "+", "cy"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"constitutive", " ", "equations"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"constitutive", "[", "zxy", "]"}], ":=", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Fz", "[", "z", "]"}], "-", RowBox[{"\[CapitalEpsilon]", " ", "A", " ", RowBox[{ RowBox[{"uz", "'"}], "[", "z", "]"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Fx", "[", "z", "]"}], "-", RowBox[{"\[Kappa]1", " ", "G", " ", "A", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"ux", "'"}], "[", "z", "]"}], "-", RowBox[{"\[Theta]y", "[", "z", "]"}]}], ")"}]}], "-", RowBox[{"\[Kappa]4", " ", "G", " ", RowBox[{"Sqrt", "[", RowBox[{"A", " ", "Irr"}], "]"}], " ", RowBox[{ RowBox[{"\[Theta]y", "'"}], "[", "z", "]"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Fy", "[", "z", "]"}], "-", RowBox[{"\[Kappa]2", " ", "G", " ", "A", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"uy", "'"}], "[", "z", "]"}], "+", RowBox[{"\[Theta]x", "[", "z", "]"}]}], ")"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Mz", "[", "z", "]"}], "-", RowBox[{"\[Kappa]3", " ", "G", " ", "Irr", " ", RowBox[{ RowBox[{"\[Theta]z", "'"}], "[", "z", "]"}]}], "-", " ", RowBox[{"\[Kappa]4", " ", "G", " ", RowBox[{"Sqrt", "[", RowBox[{"A", " ", "Irr"}], "]"}], RowBox[{"(", " ", RowBox[{ RowBox[{ RowBox[{"ux", "'"}], "[", "z", "]"}], "-", " ", RowBox[{"\[Theta]y", "[", "z", "]"}]}], ")"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Mx", "[", "z", "]"}], "-", RowBox[{"\[CapitalEpsilon]", " ", "Iyy", " ", RowBox[{ RowBox[{"\[Theta]x", "'"}], "[", "z", "]"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"My", "[", "z", "]"}], "-", RowBox[{"\[CapitalEpsilon]", " ", "Ixx", " ", RowBox[{ RowBox[{"\[Theta]y", "'"}], "[", "z", "]"}]}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", "quantities", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"displacements", "[", "zxy", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"uz", "[", "z", "]"}], ",", RowBox[{"ux", "[", "z", "]"}], ",", RowBox[{"uy", "[", "z", "]"}], ",", RowBox[{"\[Theta]z", "[", "z", "]"}], ",", RowBox[{"\[Theta]x", "[", "z", "]"}], ",", RowBox[{"\[Theta]y", "[", "z", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"forces", "[", "zxy", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"Fz", "[", "z", "]"}], ",", RowBox[{"Fx", "[", "z", "]"}], ",", RowBox[{"Fy", "[", "z", "]"}], ",", RowBox[{"Mz", "[", "z", "]"}], ",", RowBox[{"Mx", "[", "z", "]"}], ",", RowBox[{"My", "[", "z", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"functions", "[", "zxy", "]"}], ":=", RowBox[{"Join", "[", RowBox[{ RowBox[{"displacements", "[", "zxy", "]"}], ",", RowBox[{"forces", "[", "zxy", "]"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"some", " ", "common", " ", "boundary", " ", "conditions"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"clamped", "[", "zxy", "]"}], "[", "z_", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"uz", "[", "z", "]"}], ",", RowBox[{"ux", "[", "z", "]"}], ",", RowBox[{"uy", "[", "z", "]"}], ",", RowBox[{"\[Theta]z", "[", "z", "]"}], ",", RowBox[{"\[Theta]x", "[", "z", "]"}], ",", RowBox[{"\[Theta]y", "[", "z", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"free", "[", "zxy", "]"}], "[", "z_", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"Fz", "[", "z", "]"}], ",", RowBox[{"Fx", "[", "z", "]"}], ",", RowBox[{"Fy", "[", "z", "]"}], ",", RowBox[{"Mz", "[", "z", "]"}], ",", RowBox[{"Mx", "[", "z", "]"}], ",", RowBox[{"My", "[", "z", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]"}]}]], "Input", InitializationCell->True, CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, 3.537622776495909*^9, { 3.537622908932484*^9, 3.537622917335965*^9}, {3.5376238837162385`*^9, 3.537624004391141*^9}, {3.5376240458145103`*^9, 3.5376242437438307`*^9}, { 3.537624281650999*^9, 3.5376244166457205`*^9}, {3.5376244546488943`*^9, 3.5376248198917847`*^9}, {3.5397503703948455`*^9, 3.5397504308449516`*^9}, 3.5627674253055143`*^9, {3.5781422724253883`*^9, 3.578142305481847*^9}, { 3.5828109510514917`*^9, 3.582810961924711*^9}, {3.6000023977095795`*^9, 3.6000024056343937`*^9}, {3.600002438425651*^9, 3.6000024578788853`*^9}, { 3.600002497565355*^9, 3.600002598559932*^9}, {3.6000027265893574`*^9, 3.6000028178183174`*^9}, {3.600004251273635*^9, 3.6000042647676587`*^9}, { 3.600070082152542*^9, 3.6000701428834486`*^9}, {3.6000701834435205`*^9, 3.600070190151532*^9}, {3.6000702259223948`*^9, 3.6000702268583965`*^9}, { 3.6000702793212886`*^9, 3.600070364949839*^9}, {3.600070434744362*^9, 3.6000704744932313`*^9}, {3.600070516816106*^9, 3.6000705186257086`*^9}, 3.600071039151823*^9, {3.6000711295227823`*^9, 3.600071143297606*^9}, { 3.6005715147810965`*^9, 3.600571515717098*^9}, {3.663135382391988*^9, 3.663135470097742*^9}, {3.6631360823052173`*^9, 3.66313609528444*^9}, { 3.663136161288156*^9, 3.663136161787357*^9}, {3.663136351093689*^9, 3.6631364825551205`*^9}, {3.6631365162355795`*^9, 3.6631365249247947`*^9}, {3.663220430524748*^9, 3.663220430758748*^9}, { 3.6972527130658216`*^9, 3.697252849546013*^9}, {3.6972530958863573`*^9, 3.697253118446389*^9}, {3.697253453686858*^9, 3.697253477746892*^9}, 3.697253957317563*^9, {3.69727811863721*^9, 3.697278131467228*^9}, { 3.7318301867962832`*^9, 3.7318301899137673`*^9}, 3.73183022017206*^9, { 3.731830252333604*^9, 3.731830358163036*^9}, {3.731830424418888*^9, 3.731830511661916*^9}, {3.7318305522481194`*^9, 3.7318305602836328`*^9}, { 3.7318306653961177`*^9, 3.7318307803698378`*^9}, {3.7318308194316225`*^9, 3.7318308509848747`*^9}, {3.731831067996001*^9, 3.731831074918455*^9}, { 3.731831494209321*^9, 3.7318314998648615`*^9}, {3.7318316637382765`*^9, 3.7318318217902308`*^9}, {3.731831866292759*^9, 3.731831931121459*^9}, { 3.7318320241150503`*^9, 3.731832056350881*^9}, {3.7318321036469564`*^9, 3.731832113067351*^9}, {3.731832151606052*^9, 3.7318321718691797`*^9}, { 3.7318322084165397`*^9, 3.7318322295740705`*^9}, {3.731832283007102*^9, 3.731832364435964*^9}, {3.731832394598748*^9, 3.731832401741023*^9}, 3.731832462948888*^9, {3.7318325897272468`*^9, 3.7318326678535776`*^9}, { 3.7318328922504473`*^9, 3.731832980998818*^9}, {3.7318376011543665`*^9, 3.731837610959304*^9}, {3.7318376526512194`*^9, 3.731837943323455*^9}, { 3.731845575007739*^9, 3.7318455758647804`*^9}, 3.7318456408032427`*^9, 3.731845710724454*^9, {3.732348568594884*^9, 3.7323485745174074`*^9}}, CellLabel->"In[18]:=",ExpressionUUID->"6f304dbc-7ab0-46e7-9178-4e87cfcdc12b"] }, Closed]], Cell[CellGroupData[{ Cell["xyz-system (torsion corrections)", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6972526642557535`*^9, 3.697252670705762*^9}, { 3.697252700885804*^9, 3.697252701855806*^9}, {3.731830075930358*^9, 3.7318300954252224`*^9}, {3.7318301298139157`*^9, 3.7318301345137186`*^9}, {3.731831715855138*^9, 3.731831717355487*^9}, { 3.860915692817583*^9, 3.8609157009158773`*^9}, 3.861165840437029*^9},ExpressionUUID->"1d403ced-785f-4cab-bf9b-\ 4393845ec623"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", RowBox[{"equilibrium", " ", "equations", " ", "Bernoulli", " ", "model"}], "*)"}], "\[IndentingNewLine]", RowBox[{"(*", RowBox[{ RowBox[{"area", " ", "centroid", " ", "origin"}], ",", " ", RowBox[{"z", "-", "down"}], ",", RowBox[{"x", "-", "axis"}], ",", " ", RowBox[{"y", "-", "to_me"}]}], "*)"}], "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"ey", ",", RowBox[{"ez", " ", "shear", " ", "centroid", " ", "position"}]}], "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"equilibrium", "[", "xyz", "]"}], ":=", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Fx", "'"}], "[", "x", "]"}], "+", "fx"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Fy", "'"}], "[", "x", "]"}], "+", "fy"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Fz", "'"}], "[", "x", "]"}], "+", "fz"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Mx", "'"}], "[", "x", "]"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"My", "'"}], "[", "x", "]"}], "-", RowBox[{"Fz", "[", "x", "]"}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Mz", "'"}], "[", "x", "]"}], "+", RowBox[{"Fy", "[", "x", "]"}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"constitutive", " ", "equations"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"constitutive", "[", "xyz", "]"}], ":=", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Fx", "[", "x", "]"}], "-", RowBox[{"EE", " ", "AA", " ", RowBox[{ RowBox[{"ux", "'"}], "[", "x", "]"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"uy", "'"}], "[", "x", "]"}], "-", RowBox[{"\[Theta]z", "[", "x", "]"}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"uz", "'"}], "[", "x", "]"}], "+", RowBox[{"\[Theta]y", "[", "x", "]"}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Mx", "[", "x", "]"}], "-", RowBox[{"GG", " ", "J", " ", RowBox[{ RowBox[{"\[Theta]x", "'"}], "[", "x", "]"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"My", "[", "x", "]"}], "-", RowBox[{"EE", " ", "Iyy", " ", RowBox[{ RowBox[{"\[Theta]y", "'"}], "[", "x", "]"}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Mz", "[", "x", "]"}], "-", RowBox[{"EE", " ", "Izz", " ", RowBox[{ RowBox[{"\[Theta]z", "'"}], "[", "x", "]"}]}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"(*", "quantities", "*)"}], "\n", RowBox[{ RowBox[{ RowBox[{"displacements", "[", "xyz", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"ux", "[", "x", "]"}], ",", RowBox[{"uy", "[", "x", "]"}], ",", RowBox[{"uz", "[", "x", "]"}], ",", RowBox[{"\[Theta]x", "[", "x", "]"}], ",", RowBox[{"\[Theta]y", "[", "x", "]"}], ",", RowBox[{"\[Theta]z", "[", "x", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"forces", "[", "xyz", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"Fx", "[", "x", "]"}], ",", RowBox[{"Fy", "[", "x", "]"}], ",", RowBox[{"Fz", "[", "x", "]"}], ",", RowBox[{"Mx", "[", "x", "]"}], ",", RowBox[{"My", "[", "x", "]"}], ",", RowBox[{"Mz", "[", "x", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"functions", "[", "xyz", "]"}], ":=", RowBox[{"Join", "[", RowBox[{ RowBox[{"displacements", "[", "xyz", "]"}], ",", RowBox[{"forces", "[", "xyz", "]"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"some", " ", "common", " ", "boundary", " ", "conditions"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"clamped", "[", "xyz", "]"}], "[", "x_", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"ux", "[", "x", "]"}], ",", RowBox[{"uy", "[", "x", "]"}], ",", RowBox[{"uz", "[", "x", "]"}], ",", RowBox[{"\[Theta]x", "[", "x", "]"}], ",", RowBox[{"\[Theta]y", "[", "x", "]"}], ",", RowBox[{"\[Theta]z", "[", "x", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"free", "[", "xyz", "]"}], "[", "x_", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"Fx", "[", "x", "]"}], ",", RowBox[{"Fy", "[", "x", "]"}], ",", RowBox[{"Fz", "[", "x", "]"}], ",", RowBox[{"Mx", "[", "x", "]"}], ",", RowBox[{"My", "[", "x", "]"}], ",", RowBox[{"Mz", "[", "x", "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "xyz", "]"}], ",", RowBox[{"constitutive", "[", "xyz", "]"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"clamped", "[", "xyz", "]"}], "[", "0", "]"}], "-", RowBox[{"{", RowBox[{ "ux1", ",", "uy1", ",", "uz1", ",", "\[Theta]x1", ",", "\[Theta]y1", ",", "\[Theta]z1"}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"clamped", "[", "xyz", "]"}], "[", "h0", "]"}], "-", RowBox[{"{", RowBox[{ "ux2", ",", "uy2", ",", "uz2", ",", "\[Theta]x2", ",", "\[Theta]y2", ",", "\[Theta]z2"}], "}"}]}]}], "]"}], "==", "0"}], "]"}], ",", RowBox[{"functions", "[", "xyz", "]"}], ",", "x"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"for", "=", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Fx", "[", "x", "]"}], ",", RowBox[{"Fy", "[", "x", "]"}], ",", RowBox[{"Fz", "[", "x", "]"}], ",", RowBox[{"Mx", "[", "x", "]"}], ",", RowBox[{"My", "[", "x", "]"}], ",", RowBox[{"Mz", "[", "x", "]"}]}], "}"}], "/.", "sol"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"FF", "=", RowBox[{"Simplify", "[", RowBox[{"Join", "[", RowBox[{ RowBox[{ RowBox[{"-", "for"}], "/.", RowBox[{"x", "->", "0"}]}], ",", RowBox[{"for", "/.", RowBox[{"x", "->", "h0"}]}]}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"MatrixForm", "[", RowBox[{"D", "[", RowBox[{"FF", ",", RowBox[{"{", RowBox[{"Join", "[", RowBox[{ RowBox[{"{", RowBox[{ "ux1", ",", "uy1", ",", "uz1", ",", "\[Theta]x1", ",", "\[Theta]y1", ",", "\[Theta]z1"}], "}"}], ",", RowBox[{"{", RowBox[{ "ux2", ",", "uy2", ",", "uz2", ",", "\[Theta]x2", ",", "\[Theta]y2", ",", "\[Theta]z2"}], "}"}]}], "]"}], "}"}]}], "]"}], "]"}], "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{ RowBox[{"Simplify", "[", RowBox[{"-", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"\[Delta]ux", "[", RowBox[{"1", ",", "0"}], "]"}], ",", RowBox[{"\[Delta]uy", "[", RowBox[{"1", ",", "0"}], "]"}], ",", RowBox[{"\[Delta]uz", "[", RowBox[{"1", ",", "0"}], "]"}], ",", RowBox[{"\[Delta]\[Theta]x", "[", RowBox[{"1", ",", "0"}], "]"}], ",", RowBox[{"\[Delta]\[Theta]y", "[", RowBox[{"1", ",", "0"}], "]"}], ",", RowBox[{"\[Delta]\[Theta]z", "[", RowBox[{"1", ",", "0"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"\[Delta]ux", "[", RowBox[{"2", ",", "0"}], "]"}], ",", RowBox[{"\[Delta]uy", "[", RowBox[{"2", ",", "0"}], "]"}], ",", RowBox[{"\[Delta]uz", "[", RowBox[{"2", ",", "0"}], "]"}], ",", RowBox[{"\[Delta]\[Theta]x", "[", RowBox[{"2", ",", "0"}], "]"}], ",", RowBox[{"\[Delta]\[Theta]y", "[", RowBox[{"2", ",", "0"}], "]"}], ",", RowBox[{"\[Delta]\[Theta]z", "[", RowBox[{"2", ",", "0"}], "]"}]}], "}"}]}], "]"}], ".", RowBox[{"(", RowBox[{ RowBox[{"0", "*", "FF"}], "+", RowBox[{"(", RowBox[{"FF", " ", "/.", RowBox[{"{", RowBox[{ RowBox[{"fx", "->", "0"}], ",", RowBox[{"fy", "->", "0"}], ",", RowBox[{"fz", "->", "0"}]}], "}"}]}], ")"}]}], ")"}]}]}], "]"}], "/.", "\[IndentingNewLine]", RowBox[{"Thread", "[", RowBox[{ RowBox[{"{", RowBox[{ "ux1", ",", "uy1", ",", "uz1", ",", "\[Theta]x1", ",", "\[Theta]y1", ",", "\[Theta]z1"}], "}"}], "->", RowBox[{"{", RowBox[{ RowBox[{"ux", "[", RowBox[{"1", ",", "0"}], "]"}], ",", RowBox[{"uy", "[", RowBox[{"1", ",", "0"}], "]"}], ",", RowBox[{"uz", "[", RowBox[{"1", ",", "0"}], "]"}], ",", RowBox[{"\[Theta]x", "[", RowBox[{"1", ",", "0"}], "]"}], ",", RowBox[{"\[Theta]y", "[", RowBox[{"1", ",", "0"}], "]"}], ",", RowBox[{"\[Theta]z", "[", RowBox[{"1", ",", "0"}], "]"}]}], "}"}]}], "]"}]}], "/.", "\[IndentingNewLine]", RowBox[{"Thread", "[", RowBox[{ RowBox[{"{", RowBox[{ "ux2", ",", "uy2", ",", "uz2", ",", "\[Theta]x2", ",", "\[Theta]y2", ",", "\[Theta]z2"}], "}"}], "->", RowBox[{"{", RowBox[{ RowBox[{"ux", "[", RowBox[{"2", ",", "0"}], "]"}], ",", RowBox[{"uy", "[", RowBox[{"2", ",", "0"}], "]"}], ",", RowBox[{"uz", "[", RowBox[{"2", ",", "0"}], "]"}], ",", RowBox[{"\[Theta]x", "[", RowBox[{"2", ",", "0"}], "]"}], ",", RowBox[{"\[Theta]y", "[", RowBox[{"2", ",", "0"}], "]"}], ",", RowBox[{"\[Theta]z", "[", RowBox[{"2", ",", "0"}], "]"}]}], "}"}]}], "]"}]}], "]"}], "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]"}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, 3.537622776495909*^9, { 3.537622908932484*^9, 3.537622917335965*^9}, {3.5376238837162385`*^9, 3.537624004391141*^9}, {3.5376240458145103`*^9, 3.5376242437438307`*^9}, { 3.537624281650999*^9, 3.5376244166457205`*^9}, {3.5376244546488943`*^9, 3.5376248198917847`*^9}, {3.5397503703948455`*^9, 3.5397504308449516`*^9}, 3.5627674253055143`*^9, {3.5781422724253883`*^9, 3.578142305481847*^9}, { 3.5828109510514917`*^9, 3.582810961924711*^9}, {3.6000023977095795`*^9, 3.6000024056343937`*^9}, {3.600002438425651*^9, 3.6000024578788853`*^9}, { 3.600002497565355*^9, 3.600002598559932*^9}, {3.6000027265893574`*^9, 3.6000028178183174`*^9}, {3.600004251273635*^9, 3.6000042647676587`*^9}, { 3.600070082152542*^9, 3.6000701428834486`*^9}, {3.6000701834435205`*^9, 3.600070190151532*^9}, {3.6000702259223948`*^9, 3.6000702268583965`*^9}, { 3.6000702793212886`*^9, 3.600070364949839*^9}, {3.600070434744362*^9, 3.6000704744932313`*^9}, {3.600070516816106*^9, 3.6000705186257086`*^9}, 3.600071039151823*^9, {3.6000711295227823`*^9, 3.600071143297606*^9}, { 3.6005715147810965`*^9, 3.600571515717098*^9}, {3.663135382391988*^9, 3.663135470097742*^9}, {3.6631360823052173`*^9, 3.66313609528444*^9}, { 3.663136161288156*^9, 3.663136161787357*^9}, {3.663136351093689*^9, 3.6631364825551205`*^9}, {3.6631365162355795`*^9, 3.6631365249247947`*^9}, {3.663220430524748*^9, 3.663220430758748*^9}, { 3.6972527130658216`*^9, 3.697252849546013*^9}, {3.6972530958863573`*^9, 3.697253118446389*^9}, {3.697253453686858*^9, 3.697253477746892*^9}, 3.697253957317563*^9, {3.69727811863721*^9, 3.697278131467228*^9}, { 3.7318301867962832`*^9, 3.7318301899137673`*^9}, 3.73183022017206*^9, { 3.731830252333604*^9, 3.731830358163036*^9}, {3.731830424418888*^9, 3.731830511661916*^9}, {3.7318305522481194`*^9, 3.7318305602836328`*^9}, { 3.7318306653961177`*^9, 3.7318307803698378`*^9}, {3.7318308194316225`*^9, 3.7318308509848747`*^9}, {3.731831067996001*^9, 3.731831074918455*^9}, { 3.731831494209321*^9, 3.7318314998648615`*^9}, {3.7318316637382765`*^9, 3.7318318217902308`*^9}, {3.731831866292759*^9, 3.731831931121459*^9}, { 3.7318320241150503`*^9, 3.731832056350881*^9}, {3.7318321036469564`*^9, 3.731832113067351*^9}, {3.731832151606052*^9, 3.7318321718691797`*^9}, { 3.7318322084165397`*^9, 3.7318322295740705`*^9}, {3.731832283007102*^9, 3.731832364435964*^9}, {3.731832394598748*^9, 3.731832401741023*^9}, 3.731832462948888*^9, {3.7318325897272468`*^9, 3.7318326678535776`*^9}, { 3.7318328922504473`*^9, 3.731832980998818*^9}, {3.7318376011543665`*^9, 3.731837610959304*^9}, {3.7318376526512194`*^9, 3.731837943323455*^9}, { 3.731845575007739*^9, 3.7318455758647804`*^9}, 3.7318456408032427`*^9, 3.731845710724454*^9, {3.732348568594884*^9, 3.7323485745174074`*^9}, { 3.860915722585951*^9, 3.860916091175351*^9}, 3.8609161834965935`*^9, { 3.8609162615491257`*^9, 3.860916455279315*^9}, {3.8609165534841857`*^9, 3.8609165822331924`*^9}, {3.8609166199986167`*^9, 3.860916727512003*^9}, { 3.8609168176293273`*^9, 3.8609168480202847`*^9}, {3.8609169528047533`*^9, 3.860916982055163*^9}, 3.860922780356757*^9, {3.860922904774488*^9, 3.8609229110197268`*^9}, {3.860922983294303*^9, 3.860922984350952*^9}, { 3.8609230189007025`*^9, 3.860923063948068*^9}, {3.860923101052407*^9, 3.8609234221918964`*^9}, {3.8609234582491074`*^9, 3.8609237621408215`*^9}, {3.8609238453446655`*^9, 3.860923957976555*^9}, { 3.861157300801337*^9, 3.8611574137566657`*^9}, {3.86116397799658*^9, 3.861164088901005*^9}, 3.8611641542845716`*^9, {3.8611641943305817`*^9, 3.8611645307746716`*^9}, {3.8611646323956256`*^9, 3.861164751984124*^9}, { 3.861165698358881*^9, 3.8611657864276686`*^9}, {3.8611658203573523`*^9, 3.8611658270840273`*^9}, {3.8611659116328144`*^9, 3.861165923203288*^9}, { 3.861165969859317*^9, 3.8611659701419663`*^9}, {3.86116667809354*^9, 3.861166679278653*^9}, {3.861166711286221*^9, 3.8611667121310463`*^9}, { 3.861166757710123*^9, 3.86116675796766*^9}, {3.8611668502467265`*^9, 3.8611668756045094`*^9}, {3.861166970038337*^9, 3.8611669702670755`*^9}, { 3.8612641169037857`*^9, 3.8612641408827114`*^9}, {3.8612641748386493`*^9, 3.8612641904609947`*^9}}, CellLabel->"In[36]:=",ExpressionUUID->"3e6fdcf2-47a5-4d38-b6fe-44f08499ac09"], Cell[BoxData[ RowBox[{"-", RowBox[{ FractionBox["1", SuperscriptBox["h0", "3"]], RowBox[{"(", RowBox[{ RowBox[{"AA", " ", "EE", " ", SuperscriptBox["h0", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"ux", "[", RowBox[{"1", ",", "0"}], "]"}], "-", RowBox[{"ux", "[", RowBox[{"2", ",", "0"}], "]"}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"\[Delta]ux", "[", RowBox[{"1", ",", "0"}], "]"}], "-", RowBox[{"\[Delta]ux", "[", RowBox[{"2", ",", "0"}], "]"}]}], ")"}]}], "+", RowBox[{"GG", " ", SuperscriptBox["h0", "2"], " ", "J", " ", RowBox[{"(", RowBox[{ RowBox[{"\[Delta]\[Theta]x", "[", RowBox[{"1", ",", "0"}], "]"}], "-", RowBox[{"\[Delta]\[Theta]x", "[", RowBox[{"2", ",", "0"}], "]"}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"\[Theta]x", "[", RowBox[{"1", ",", "0"}], "]"}], "-", RowBox[{"\[Theta]x", "[", RowBox[{"2", ",", "0"}], "]"}]}], ")"}]}], "+", RowBox[{"2", " ", "EE", " ", RowBox[{"(", RowBox[{ RowBox[{"3", " ", "Iyy", " ", RowBox[{"(", RowBox[{ RowBox[{"uz", "[", RowBox[{"1", ",", "0"}], "]"}], "-", RowBox[{"uz", "[", RowBox[{"2", ",", "0"}], "]"}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", RowBox[{"\[Delta]uz", "[", RowBox[{"1", ",", "0"}], "]"}]}], "-", RowBox[{"2", " ", RowBox[{"\[Delta]uz", "[", RowBox[{"2", ",", "0"}], "]"}]}], "-", RowBox[{"h0", " ", RowBox[{"(", RowBox[{ RowBox[{"\[Delta]\[Theta]y", "[", RowBox[{"1", ",", "0"}], "]"}], "+", RowBox[{"\[Delta]\[Theta]y", "[", RowBox[{"2", ",", "0"}], "]"}]}], ")"}]}]}], ")"}]}], "+", RowBox[{"h0", " ", "Iyy", " ", RowBox[{"(", RowBox[{ RowBox[{"h0", " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", RowBox[{"\[Delta]\[Theta]y", "[", RowBox[{"1", ",", "0"}], "]"}]}], "+", RowBox[{"\[Delta]\[Theta]y", "[", RowBox[{"2", ",", "0"}], "]"}]}], ")"}], " ", RowBox[{"\[Theta]y", "[", RowBox[{"1", ",", "0"}], "]"}]}], "+", RowBox[{"h0", " ", RowBox[{"(", RowBox[{ RowBox[{"\[Delta]\[Theta]y", "[", RowBox[{"1", ",", "0"}], "]"}], "+", RowBox[{"2", " ", RowBox[{"\[Delta]\[Theta]y", "[", RowBox[{"2", ",", "0"}], "]"}]}]}], ")"}], " ", RowBox[{"\[Theta]y", "[", RowBox[{"2", ",", "0"}], "]"}]}], "-", RowBox[{"3", " ", RowBox[{"\[Delta]uz", "[", RowBox[{"1", ",", "0"}], "]"}], " ", RowBox[{"(", RowBox[{ RowBox[{"\[Theta]y", "[", RowBox[{"1", ",", "0"}], "]"}], "+", RowBox[{"\[Theta]y", "[", RowBox[{"2", ",", "0"}], "]"}]}], ")"}]}], "+", RowBox[{"3", " ", RowBox[{"\[Delta]uz", "[", RowBox[{"2", ",", "0"}], "]"}], " ", RowBox[{"(", RowBox[{ RowBox[{"\[Theta]y", "[", RowBox[{"1", ",", "0"}], "]"}], "+", RowBox[{"\[Theta]y", "[", RowBox[{"2", ",", "0"}], "]"}]}], ")"}]}]}], ")"}]}], "+", RowBox[{"Izz", " ", RowBox[{"(", RowBox[{ RowBox[{"3", " ", RowBox[{"uy", "[", RowBox[{"1", ",", "0"}], "]"}], " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", RowBox[{"\[Delta]uy", "[", RowBox[{"1", ",", "0"}], "]"}]}], "-", RowBox[{"2", " ", RowBox[{"\[Delta]uy", "[", RowBox[{"2", ",", "0"}], "]"}]}], "+", RowBox[{"h0", " ", RowBox[{"(", RowBox[{ RowBox[{"\[Delta]\[Theta]z", "[", RowBox[{"1", ",", "0"}], "]"}], "+", RowBox[{"\[Delta]\[Theta]z", "[", RowBox[{"2", ",", "0"}], "]"}]}], ")"}]}]}], ")"}]}], "-", RowBox[{"3", " ", RowBox[{"uy", "[", RowBox[{"2", ",", "0"}], "]"}], " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", RowBox[{"\[Delta]uy", "[", RowBox[{"1", ",", "0"}], "]"}]}], "-", RowBox[{"2", " ", RowBox[{"\[Delta]uy", "[", RowBox[{"2", ",", "0"}], "]"}]}], "+", RowBox[{"h0", " ", RowBox[{"(", RowBox[{ RowBox[{"\[Delta]\[Theta]z", "[", RowBox[{"1", ",", "0"}], "]"}], "+", RowBox[{"\[Delta]\[Theta]z", "[", RowBox[{"2", ",", "0"}], "]"}]}], ")"}]}]}], ")"}]}], "+", RowBox[{"h0", " ", RowBox[{"(", RowBox[{ RowBox[{"3", " ", RowBox[{"\[Delta]uy", "[", RowBox[{"1", ",", "0"}], "]"}]}], "-", RowBox[{"3", " ", RowBox[{"\[Delta]uy", "[", RowBox[{"2", ",", "0"}], "]"}]}], "+", RowBox[{"2", " ", "h0", " ", RowBox[{"\[Delta]\[Theta]z", "[", RowBox[{"1", ",", "0"}], "]"}]}], "+", RowBox[{"h0", " ", RowBox[{"\[Delta]\[Theta]z", "[", RowBox[{"2", ",", "0"}], "]"}]}]}], ")"}], " ", RowBox[{"\[Theta]z", "[", RowBox[{"1", ",", "0"}], "]"}]}], "+", RowBox[{"h0", " ", RowBox[{"(", RowBox[{ RowBox[{"3", " ", RowBox[{"\[Delta]uy", "[", RowBox[{"1", ",", "0"}], "]"}]}], "-", RowBox[{"3", " ", RowBox[{"\[Delta]uy", "[", RowBox[{"2", ",", "0"}], "]"}]}], "+", RowBox[{"h0", " ", RowBox[{"\[Delta]\[Theta]z", "[", RowBox[{"1", ",", "0"}], "]"}]}], "+", RowBox[{"2", " ", "h0", " ", RowBox[{"\[Delta]\[Theta]z", "[", RowBox[{"2", ",", "0"}], "]"}]}]}], ")"}], " ", RowBox[{"\[Theta]z", "[", RowBox[{"2", ",", "0"}], "]"}]}]}], ")"}]}]}], ")"}]}]}], ")"}]}]}]], "Output", CellChangeTimes->{{3.8609233444466724`*^9, 3.860923363548772*^9}, { 3.860923547872297*^9, 3.86092355626967*^9}, {3.860923658155687*^9, 3.8609236773507147`*^9}, {3.8609237474866333`*^9, 3.8609237638096213`*^9}, 3.860923960474923*^9, 3.861157437679527*^9, {3.861164253501075*^9, 3.86116428324804*^9}, 3.8611643314282627`*^9, 3.8611643830605435`*^9, 3.861164532448831*^9, {3.8611647276783466`*^9, 3.8611647537614164`*^9}, 3.8611657933231907`*^9, 3.861165926293292*^9, 3.86116597240394*^9, { 3.861166692819927*^9, 3.861166715233869*^9}, 3.861166759195815*^9, 3.861166878128951*^9, 3.861166971456128*^9, 3.8612641306062827`*^9, 3.861264205061757*^9}, CellLabel->"Out[46]=",ExpressionUUID->"f5b9213b-2bb7-49e5-963d-3f7db3da5e22"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Examples", "Section", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622997047524*^9, {3.5376238658302155`*^9, 3.5376238672312956`*^9}, { 3.600002366135124*^9, 3.6000023692707295`*^9}, {3.6004096425861998`*^9, 3.6004096564234247`*^9}, {3.6004097207111373`*^9, 3.6004097213507385`*^9}, {3.600583614831902*^9, 3.6005836171180325`*^9}},ExpressionUUID->"05f91828-714a-44d3-8284-\ 0a01a3d0257d"], Cell[CellGroupData[{ Cell["Lecture problem 4.1 (cantilever xyz)", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6972528859460635`*^9, 3.697252899226082*^9}, 3.697691252966739*^9, {3.7289617343781796`*^9, 3.728961755357093*^9}, { 3.729161066723563*^9, 3.729161066957983*^9}, {3.789956342760171*^9, 3.789956347753665*^9}, {3.7899564437921624`*^9, 3.7899564457919617`*^9}, 3.825152135298736*^9},ExpressionUUID->"006aeb29-bd62-422a-b8bd-\ b4677696f61c"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bx", "\[Rule]", "0"}], ",", RowBox[{"by", " ", "\[Rule]", "b"}], ",", " ", RowBox[{"bz", "\[Rule]", "0"}], ",", RowBox[{"cx", "\[Rule]", "0"}], ",", RowBox[{"cy", "\[Rule]", "0"}], ",", RowBox[{"cz", "\[Rule]", "0"}], ",", RowBox[{"Sy", "\[Rule]", "0"}], ",", RowBox[{"Sz", "\[Rule]", "0"}], ",", " ", RowBox[{"Iyz", "\[Rule]", "0"}], ",", RowBox[{"Izy", "\[Rule]", "0"}], ",", RowBox[{"Iyy", "\[Rule]", "\[CapitalIota]"}], ",", RowBox[{"Izz", "\[Rule]", "\[CapitalIota]"}], ",", RowBox[{"Irr", "\[Rule]", RowBox[{"2", " ", "\[CapitalIota]"}]}], ",", " ", RowBox[{"\[Alpha]", "\[Rule]", "1"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "xyz", "]"}], ",", RowBox[{"constitutive", "[", "xyz", "]"}], ",", RowBox[{ RowBox[{"clamped", "[", "xyz", "]"}], "[", "0", "]"}], ",", RowBox[{ RowBox[{"free", "[", "xyz", "]"}], "[", "L", "]"}]}], "]"}], " ", "/.", "rule"}]}], ";"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"functions", "[", "xyz", "]"}], ",", "x"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"u", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", "/.", RowBox[{"x", "\[Rule]", "L"}]}], ")"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"v", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", "/.", RowBox[{"x", "\[Rule]", "L"}]}], ")"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"w", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", "/.", RowBox[{"x", "\[Rule]", "L"}]}], ")"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\<\[Psi][L] = \>\"", ",", RowBox[{"Simplify", "[", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"\[Psi]", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", "/.", RowBox[{"x", "\[Rule]", "L"}]}], ")"}], "]"}]}], " ", "]"}], "\[IndentingNewLine]"}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.6972529515461555`*^9, 3.697253084716342*^9}, {3.697253246136568*^9, 3.6972532878466263`*^9}, {3.697253341536701*^9, 3.697253385806763*^9}, { 3.697254029617665*^9, 3.6972541087377753`*^9}, {3.6972541537278385`*^9, 3.697254192057892*^9}, {3.6972542390879583`*^9, 3.6972542801280155`*^9}, { 3.697254317298067*^9, 3.69725432622808*^9}, {3.6972543601881275`*^9, 3.6972544415582414`*^9}, {3.6972550776891317`*^9, 3.6972551527992373`*^9}, {3.697690411465561*^9, 3.6976904555756226`*^9}, { 3.6976912961467996`*^9, 3.697691321896836*^9}, 3.7289617606979027`*^9, { 3.729161125292254*^9, 3.729161138512396*^9}, {3.729161169421898*^9, 3.729161340564952*^9}, {3.7292449011918864`*^9, 3.729244917872322*^9}, { 3.7292450116027665`*^9, 3.729245017369953*^9}},ExpressionUUID->"0986619f-9eaf-4026-a2ca-\ aa521b1c62f7"], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"u[L] = \"\>", "\[InvisibleSpace]", "0"}], SequenceForm["u[L] = ", 0], Editable->False]], "Print", CellChangeTimes->{ 3.729245022557714*^9},ExpressionUUID->"e18949b8-c613-4b8d-88a6-\ 8b69fa74cd72"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"v[L] = \"\>", "\[InvisibleSpace]", RowBox[{ FractionBox["1", "8"], " ", "b", " ", SuperscriptBox["L", "2"], " ", RowBox[{"(", RowBox[{ FractionBox["4", RowBox[{"A", " ", "G"}]], "+", FractionBox[ SuperscriptBox["L", "2"], RowBox[{"\[CapitalEpsilon]", " ", "\[CapitalIota]"}]]}], ")"}]}]}], SequenceForm[ "v[L] = ", ((Rational[1, 8] $CellContext`b) $CellContext`L^2) ( 4 ($CellContext`A^(-1)/$CellContext`G) + $CellContext`L^2 ($CellContext`\ \[CapitalEpsilon]^(-1)/$CellContext`\[CapitalIota]))], Editable->False]], "Print", CellChangeTimes->{ 3.729245022557714*^9},ExpressionUUID->"b66cc749-8959-43d2-aa26-\ bd67ceed2817"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"w[L] = \"\>", "\[InvisibleSpace]", "0"}], SequenceForm["w[L] = ", 0], Editable->False]], "Print", CellChangeTimes->{ 3.729245022557714*^9},ExpressionUUID->"c3a3a8e2-ff09-47ed-9be1-\ 9351b5099b47"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"\[Psi][L] = \"\>", "\[InvisibleSpace]", FractionBox[ RowBox[{"b", " ", SuperscriptBox["L", "3"]}], RowBox[{"6", " ", "\[CapitalEpsilon]", " ", "\[CapitalIota]"}]]}], SequenceForm[ "\[Psi][L] = ", (( Rational[1, 6] $CellContext`b) $CellContext`L^3) \ ($CellContext`\[CapitalEpsilon]^(-1)/$CellContext`\[CapitalIota])], Editable->False]], "Print", CellChangeTimes->{ 3.729245022573373*^9},ExpressionUUID->"8bb8d1c5-e098-4759-b455-\ 2a0e83238c52"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Lecture problem 4.2 (cantilever xyz)", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6972528859460635`*^9, 3.697252899226082*^9}, 3.697691252966739*^9, {3.7289617343781796`*^9, 3.728961755357093*^9}, { 3.729161066723563*^9, 3.729161066957983*^9}, {3.7291614169479938`*^9, 3.7291614171042624`*^9}, 3.7899563559148884`*^9, 3.789956456926772*^9, 3.8251521371990843`*^9},ExpressionUUID->"1d5fe2a2-f2a8-4c00-acb2-\ 0cfe7d7827b0"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bx", "\[Rule]", "0"}], ",", RowBox[{"by", " ", "\[Rule]", "b"}], ",", " ", RowBox[{"bz", "\[Rule]", "0"}], ",", RowBox[{"cx", "\[Rule]", "0"}], ",", RowBox[{"cy", "\[Rule]", "0"}], ",", RowBox[{"cz", "\[Rule]", "0"}], ",", RowBox[{"Sy", "\[Rule]", "0"}], ",", RowBox[{"Sz", "\[Rule]", "0"}], ",", " ", RowBox[{"Iyz", "\[Rule]", "0"}], ",", RowBox[{"Izy", "\[Rule]", "0"}], ",", RowBox[{"Iyy", "\[Rule]", "\[CapitalIota]"}], ",", RowBox[{"Izz", "\[Rule]", "\[CapitalIota]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "xyz", "]"}], ",", RowBox[{"constitutive", "[", "xyz", "]"}], ",", RowBox[{ RowBox[{"clamped", "[", "xyz", "]"}], "[", "0", "]"}], ",", RowBox[{ RowBox[{"clamped", "[", "xyz", "]"}], "[", "L", "]"}]}], "]"}], " ", "/.", "rule"}]}], ";"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"functions", "[", "xyz", "]"}], ",", "x"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{ RowBox[{"u", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ",", " ", RowBox[{"\[Alpha]", "\[Rule]", "1"}]}], "]"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{ RowBox[{"v", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ",", RowBox[{"\[Alpha]", "\[Rule]", "1"}]}], "]"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\<\[Psi][L] = \>\"", ",", RowBox[{"Simplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{ RowBox[{"\[Psi]", "[", "x", "]"}], " ", "/.", " ", "sol"}], ",", " ", RowBox[{"\[Alpha]", "\[Rule]", "1"}]}], "]"}], "]"}]}], " ", "]"}], "\[IndentingNewLine]"}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.6972529515461555`*^9, 3.697253084716342*^9}, {3.697253246136568*^9, 3.6972532878466263`*^9}, {3.697253341536701*^9, 3.697253385806763*^9}, { 3.697254029617665*^9, 3.6972541087377753`*^9}, {3.6972541537278385`*^9, 3.697254192057892*^9}, {3.6972542390879583`*^9, 3.6972542801280155`*^9}, { 3.697254317298067*^9, 3.69725432622808*^9}, {3.6972543601881275`*^9, 3.6972544415582414`*^9}, {3.6972550776891317`*^9, 3.6972551527992373`*^9}, {3.697690411465561*^9, 3.6976904555756226`*^9}, { 3.6976912961467996`*^9, 3.697691321896836*^9}, 3.7289617606979027`*^9, { 3.729161125292254*^9, 3.729161138512396*^9}, {3.729161169421898*^9, 3.729161340564952*^9}, {3.7291614229170876`*^9, 3.729161567291951*^9}, { 3.7292451693636384`*^9, 3.7292451757055583`*^9}, 3.761469815063783*^9, { 3.7614699958806887`*^9, 3.7614700508837376`*^9}, {3.7614700865001106`*^9, 3.761470095478616*^9}, {3.7614741030292683`*^9, 3.7614741148233995`*^9}},ExpressionUUID->"bca76ba3-440a-4727-9426-\ 831715647e88"] }, Closed]], Cell[CellGroupData[{ Cell["Lecture problem 4.3 (cantilever xyz)", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6972528859460635`*^9, 3.697252899226082*^9}, 3.697691252966739*^9, {3.7289617343781796`*^9, 3.728961755357093*^9}, { 3.729161066723563*^9, 3.729161066957983*^9}, {3.7291614169479938`*^9, 3.7291614171042624`*^9}, 3.729161757784644*^9, 3.7899563590871553`*^9, 3.7899564595958834`*^9, 3.8251521389209538`*^9},ExpressionUUID->"65b3d199-96d1-449a-8732-\ 3b5053c6f489"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bx", "\[Rule]", "0"}], ",", RowBox[{"by", " ", "\[Rule]", "0"}], ",", " ", RowBox[{"bz", "\[Rule]", "\[Rho]gA"}], ",", RowBox[{"cx", "\[Rule]", "0"}], ",", RowBox[{"cy", "\[Rule]", "0"}], ",", RowBox[{"cz", "\[Rule]", "0"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "xyz", "]"}], ",", RowBox[{ RowBox[{"free", "[", "xyz", "]"}], "[", "L", "]"}]}], "]"}], " ", "/.", "rule"}]}], ";"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"forces", "[", "xyz", "]"}], ",", "x"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Nx", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Qy", "[", "x", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Qz", "[", "x", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Tx", "[", "x", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"My", "[", "x", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Mz", "[", "x", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]"}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.6972529515461555`*^9, 3.697253084716342*^9}, {3.697253246136568*^9, 3.6972532878466263`*^9}, {3.697253341536701*^9, 3.697253385806763*^9}, { 3.697254029617665*^9, 3.6972541087377753`*^9}, {3.6972541537278385`*^9, 3.697254192057892*^9}, {3.6972542390879583`*^9, 3.6972542801280155`*^9}, { 3.697254317298067*^9, 3.69725432622808*^9}, {3.6972543601881275`*^9, 3.6972544415582414`*^9}, {3.6972550776891317`*^9, 3.6972551527992373`*^9}, {3.697690411465561*^9, 3.6976904555756226`*^9}, { 3.6976912961467996`*^9, 3.697691321896836*^9}, 3.7289617606979027`*^9, { 3.729161125292254*^9, 3.729161138512396*^9}, {3.729161169421898*^9, 3.729161340564952*^9}, {3.7291614229170876`*^9, 3.729161567291951*^9}, { 3.7291617820998154`*^9, 3.72916179094451*^9}, {3.7291618418090873`*^9, 3.7291619627126083`*^9}},ExpressionUUID->"bb1a09a3-51ff-4bef-a59d-\ 299fb02a4b72"] }, Closed]], Cell[CellGroupData[{ Cell["Lecture problem 4.4 (cantilever xyz offset)", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6972528859460635`*^9, 3.697252899226082*^9}, 3.697691252966739*^9, {3.7289617343781796`*^9, 3.728961755357093*^9}, { 3.729161066723563*^9, 3.729161066957983*^9}, {3.7291614169479938`*^9, 3.7291614171042624`*^9}, 3.7291628285924234`*^9, {3.729162942526467*^9, 3.729162945151785*^9}, 3.7899563620515733`*^9, 3.7899564620196486`*^9, 3.8251521408049912`*^9},ExpressionUUID->"47c6dfc5-81f4-4d13-b8df-\ fbf830b3fc7d"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bx", "\[Rule]", "0"}], ",", RowBox[{"by", " ", "\[Rule]", "b"}], ",", " ", RowBox[{"bz", "\[Rule]", "0"}], ",", RowBox[{"cx", "\[Rule]", "0"}], ",", RowBox[{"cy", "\[Rule]", "0"}], ",", RowBox[{"cz", "\[Rule]", "0"}], ",", RowBox[{"Sy", "\[Rule]", "0"}], ",", RowBox[{"Sz", "\[Rule]", RowBox[{ RowBox[{"-", "r"}], " ", "A"}]}], ",", " ", RowBox[{"Iyz", "\[Rule]", "0"}], ",", RowBox[{"Izy", "\[Rule]", "0"}], ",", RowBox[{"Iyy", "\[Rule]", "\[CapitalIota]"}], ",", RowBox[{"Izz", "\[Rule]", RowBox[{"\[CapitalIota]", "+", RowBox[{"A", " ", RowBox[{"r", "^", "2"}]}]}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "xyz", "]"}], ",", RowBox[{"constitutive", "[", "xyz", "]"}], ",", RowBox[{ RowBox[{"clamped", "[", "xyz", "]"}], "[", "0", "]"}], ",", RowBox[{ RowBox[{"free", "[", "xyz", "]"}], "[", "L", "]"}]}], "]"}], " ", "/.", "rule"}]}], ";"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"functions", "[", "xyz", "]"}], ",", "x"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"u", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", "/.", RowBox[{"x", "\[Rule]", "L"}]}], ",", " ", RowBox[{"\[Alpha]", "\[Rule]", "Infinity"}]}], "]"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"v", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", "/.", RowBox[{"x", "\[Rule]", "L"}]}], " ", ",", RowBox[{"\[Alpha]", "\[Rule]", "Infinity"}]}], "]"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\<\[Psi][L] = \>\"", ",", RowBox[{"Simplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[Psi]", "[", "x", "]"}], " ", "/.", " ", "sol"}], "/.", RowBox[{"x", "\[Rule]", "L"}]}], ",", " ", RowBox[{"\[Alpha]", "\[Rule]", "Infinity"}]}], "]"}], "]"}]}], " ", "]"}], "\[IndentingNewLine]"}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.6972529515461555`*^9, 3.697253084716342*^9}, {3.697253246136568*^9, 3.6972532878466263`*^9}, {3.697253341536701*^9, 3.697253385806763*^9}, { 3.697254029617665*^9, 3.6972541087377753`*^9}, {3.6972541537278385`*^9, 3.697254192057892*^9}, {3.6972542390879583`*^9, 3.6972542801280155`*^9}, { 3.697254317298067*^9, 3.69725432622808*^9}, {3.6972543601881275`*^9, 3.6972544415582414`*^9}, {3.6972550776891317`*^9, 3.6972551527992373`*^9}, {3.697690411465561*^9, 3.6976904555756226`*^9}, { 3.6976912961467996`*^9, 3.697691321896836*^9}, 3.7289617606979027`*^9, { 3.729161125292254*^9, 3.729161138512396*^9}, {3.729161169421898*^9, 3.729161340564952*^9}, {3.7291614229170876`*^9, 3.729161567291951*^9}, { 3.7291628552514014`*^9, 3.7291628840513134`*^9}},ExpressionUUID->"852413a6-712c-4a2f-b8de-\ cd95ebf6f13e"] }, Closed]], Cell[CellGroupData[{ Cell["Lecture problem 4.5 (half-circle snb)", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.7289615946007776`*^9, 3.7289616467307887`*^9}, { 3.7291631200139813`*^9, 3.729163149204567*^9}, 3.7291644717611895`*^9, 3.789956365054686*^9, 3.7899564651324177`*^9, 3.8251521426478596`*^9},ExpressionUUID->"bce8176b-3520-4462-b0af-\ a7a3aeda299e"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bs", "\[Rule]", "0"}], ",", RowBox[{"bn", "\[Rule]", "0"}], ",", RowBox[{"bb", "\[Rule]", "0"}], ",", RowBox[{"cs", "\[Rule]", "0"}], ",", RowBox[{"cn", "\[Rule]", "0"}], ",", RowBox[{"cb", "\[Rule]", "0"}], ",", RowBox[{"\[Tau]", "\[Rule]", "0"}], ",", RowBox[{"\[Kappa]", "->", RowBox[{"1", "/", "R"}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "snb", "]"}], ",", RowBox[{ RowBox[{ RowBox[{"free", "[", "snb", "]"}], "[", RowBox[{"Pi", " ", "R"}], "]"}], "-", RowBox[{"{", RowBox[{"0", ",", "0", ",", "P", ",", "0", ",", "0", ",", "0"}], "}"}]}]}], "]"}], " ", "/.", "rule"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"forces", "[", "snb", "]"}], ",", "s"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Ns", "[", "s", "]"}], " ", "/.", " ", "sol"}], " ", "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Qn", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Qb", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Ts", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], " ", "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Mn", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Mb", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.697364372157997*^9, 3.697364411619254*^9}, {3.697690490155671*^9, 3.6976905424457445`*^9}, {3.7291631839269705`*^9, 3.7291631880836554`*^9}, {3.7291632270408883`*^9, 3.729163251137182*^9}, { 3.729163313346614*^9, 3.729163318019148*^9}}, CellLabel->"In[25]:=",ExpressionUUID->"9032ada9-f181-4057-b774-73aae42b3638"] }, Closed]], Cell[CellGroupData[{ Cell["Lecture problem 4.6 (quarter-circle snb)", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.7289615946007776`*^9, 3.7289616467307887`*^9}, { 3.729138788591694*^9, 3.7291388008694696`*^9}, {3.7291634578308454`*^9, 3.729163478270488*^9}, 3.7291644744958477`*^9, 3.789956369138775*^9, 3.789956467572611*^9, 3.8251521447794733`*^9},ExpressionUUID->"7613c0b2-bbf0-4f69-b2cf-\ 04c894d5c781"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bs", "\[Rule]", "0"}], ",", RowBox[{"bn", "\[Rule]", "0"}], ",", RowBox[{"bb", "\[Rule]", "0"}], ",", RowBox[{"cs", "\[Rule]", "0"}], ",", RowBox[{"cn", "\[Rule]", "0"}], ",", RowBox[{"cb", "\[Rule]", "0"}], ",", RowBox[{"\[Tau]", "\[Rule]", "0"}], ",", RowBox[{"\[Kappa]", "->", RowBox[{"1", "/", "R"}]}], ",", "\[IndentingNewLine]", RowBox[{"Sn", "\[Rule]", "0"}], ",", RowBox[{"Sb", "\[Rule]", "0"}], ",", RowBox[{"Inn", "\[Rule]", "\[CapitalIota]"}], ",", RowBox[{"Ibb", "\[Rule]", "\[CapitalIota]"}], ",", RowBox[{"Ibn", "\[Rule]", "0"}], ",", RowBox[{"Inb", "\[Rule]", "0"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "snb", "]"}], ",", RowBox[{"constitutive", "[", "snb", "]"}], ",", RowBox[{ RowBox[{"clamped", "[", "snb", "]"}], "[", "0", "]"}], ",", RowBox[{ RowBox[{ RowBox[{"free", "[", "snb", "]"}], "[", RowBox[{"Pi", " ", RowBox[{"R", "/", "2"}]}], "]"}], "-", RowBox[{"{", RowBox[{"P", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}]}]}], "]"}], " ", "/.", "rule"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"functions", "[", "snb", "]"}], ",", "s"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"u", "[", "s", "]"}], " ", "/.", " ", "sol"}], " ", "/.", RowBox[{"s", "\[Rule]", RowBox[{"Pi", " ", RowBox[{"R", "/", "2"}]}]}]}], ",", RowBox[{"\[Alpha]", "\[Rule]", "Infinity"}]}], "]"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"v", "[", "s", "]"}], " ", "/.", " ", "sol"}], "/.", RowBox[{"s", "\[Rule]", RowBox[{"Pi", " ", RowBox[{"R", "/", "2"}]}]}]}], ",", RowBox[{"\[Alpha]", "\[Rule]", "Infinity"}]}], "]"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\<\[Psi](s)= \>\"", ",", RowBox[{"FullSimplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[Psi]", "[", "s", "]"}], " ", "/.", " ", "sol"}], "/.", RowBox[{"s", "\[Rule]", RowBox[{"Pi", " ", RowBox[{"R", "/", "2"}]}]}]}], ",", RowBox[{"\[Alpha]", "\[Rule]", "Infinity"}]}], "]"}], " ", "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.697364372157997*^9, 3.697364411619254*^9}, {3.697690490155671*^9, 3.6976905424457445`*^9}, {3.729138832200755*^9, 3.729139077882655*^9}, { 3.7291391324355974`*^9, 3.7291391703926563`*^9}, {3.7291392363965936`*^9, 3.7291392582268443`*^9}, {3.7291634930218964`*^9, 3.729163495631676*^9}, { 3.729163545261688*^9, 3.7291636375841603`*^9}, {3.7291636776977625`*^9, 3.7291636910898285`*^9}, {3.729163724089055*^9, 3.7291637452631474`*^9}}, CellLabel->"In[34]:=",ExpressionUUID->"794ed346-a549-41ed-9654-ec9fccb1373a"] }, Closed]], Cell[CellGroupData[{ Cell["Lecture problem 4.7 (half-circle snb)", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.7289615946007776`*^9, 3.7289616467307887`*^9}, { 3.729138788591694*^9, 3.7291388008694696`*^9}, {3.7291634578308454`*^9, 3.729163478270488*^9}, {3.7291638711514363`*^9, 3.729163876292599*^9}, 3.72916447696486*^9, 3.789956374232876*^9, 3.789956473967645*^9, 3.8251521505687885`*^9},ExpressionUUID->"5626b671-a555-4d64-8c94-\ 8b6559ba2a7f"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bs", "\[Rule]", "0"}], ",", RowBox[{"bn", "\[Rule]", "0"}], ",", RowBox[{"bb", "\[Rule]", "\[Rho]gA"}], ",", RowBox[{"cs", "\[Rule]", "0"}], ",", RowBox[{"cn", "\[Rule]", "0"}], ",", RowBox[{"cb", "\[Rule]", "0"}], ",", RowBox[{"\[Tau]", "\[Rule]", "0"}], ",", RowBox[{"\[Kappa]", "->", RowBox[{"1", "/", "R"}]}], ",", "\[IndentingNewLine]", RowBox[{"Sn", "\[Rule]", "0"}], ",", RowBox[{"Sb", "\[Rule]", "0"}], ",", RowBox[{"Inn", "\[Rule]", "\[CapitalIota]"}], ",", RowBox[{"Ibb", "\[Rule]", "\[CapitalIota]"}], ",", RowBox[{"Ibn", "\[Rule]", "0"}], ",", RowBox[{"Inb", "\[Rule]", "0"}], ",", RowBox[{"Irr", "\[Rule]", RowBox[{"2", "\[CapitalIota]"}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "snb", "]"}], ",", RowBox[{"constitutive", "[", "snb", "]"}], ",", RowBox[{ RowBox[{"clamped", "[", "snb", "]"}], "[", "0", "]"}], ",", RowBox[{ RowBox[{"clamped", "[", "snb", "]"}], "[", RowBox[{"Pi", "*", "R"}], "]"}]}], "]"}], " ", "/.", "rule"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"functions", "[", "snb", "]"}], ",", "s"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\<\[Phi](L/2)= \>\"", ",", RowBox[{"FullSimplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[Phi]", "[", "s", "]"}], " ", "/.", " ", "sol"}], " ", "/.", RowBox[{"s", "\[Rule]", RowBox[{"Pi", " ", RowBox[{"R", "/", "2"}]}]}]}], ",", RowBox[{"\[Alpha]", "\[Rule]", "Infinity"}]}], "]"}], "]"}]}], "]"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"w", "[", "s", "]"}], " ", "/.", " ", "sol"}], " ", "/.", RowBox[{"s", "\[Rule]", RowBox[{"Pi", " ", RowBox[{"R", "/", "2"}]}]}]}], ",", RowBox[{"\[Alpha]", "\[Rule]", "Infinity"}]}], "]"}], "]"}]}], "]"}], "\[IndentingNewLine]", "\[IndentingNewLine]"}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.697364372157997*^9, 3.697364411619254*^9}, {3.697690490155671*^9, 3.6976905424457445`*^9}, {3.729138832200755*^9, 3.729139077882655*^9}, { 3.7291391324355974`*^9, 3.7291391703926563`*^9}, {3.7291392363965936`*^9, 3.7291392582268443`*^9}, {3.7291634930218964`*^9, 3.729163495631676*^9}, { 3.729163545261688*^9, 3.7291636375841603`*^9}, {3.7291636776977625`*^9, 3.7291636910898285`*^9}, {3.729163724089055*^9, 3.7291637452631474`*^9}, { 3.729163916140255*^9, 3.7291640045714884`*^9}, {3.7291640411848593`*^9, 3.729164050591931*^9}, {3.7614651998977733`*^9, 3.761465201038422*^9}, { 3.7614652697252607`*^9, 3.7614652780826902`*^9}, {3.7614653164018235`*^9, 3.7614653329604597`*^9}, 3.761470868702693*^9, {3.825152228130761*^9, 3.8251522468621235`*^9}, {3.825152369125183*^9, 3.82515241499487*^9}, { 3.8251524473901963`*^9, 3.8251524478386555`*^9}, {3.82515253381339*^9, 3.825152645638444*^9}, {3.825152691250884*^9, 3.8251527091195803`*^9}}, CellLabel->"In[71]:=",ExpressionUUID->"0356b5d1-5a75-4b1f-98d3-ffdd89b5eded"] }, Closed]], Cell[CellGroupData[{ Cell["Example problem 2", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6972528859460635`*^9, 3.697252899226082*^9}, { 3.697464827703618*^9, 3.6974648360436296`*^9}, {3.7291344563518124`*^9, 3.7291344655087395`*^9}, {3.7291527409070597`*^9, 3.729152744031857*^9}, { 3.7609600370332837`*^9, 3.7609600421383896`*^9}, {3.7899564783083982`*^9, 3.7899564815906363`*^9}},ExpressionUUID->"d1d11315-3d04-4ac8-9c35-\ 7a72b66c5535"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bx", "\[Rule]", "0"}], ",", RowBox[{"by", " ", "\[Rule]", "0"}], ",", " ", RowBox[{"bz", "\[Rule]", RowBox[{"b", RowBox[{"(", RowBox[{"1", "-", RowBox[{"x", "/", "L"}]}], ")"}]}]}], ",", RowBox[{"cx", "\[Rule]", "0"}], ",", RowBox[{"cy", "\[Rule]", "0"}], ",", RowBox[{"cz", "\[Rule]", "0"}], ",", RowBox[{"Sy", "\[Rule]", "0"}], ",", RowBox[{"Sz", "\[Rule]", "0"}], ",", " ", RowBox[{"Iyz", "\[Rule]", "0"}], ",", RowBox[{"Izy", "\[Rule]", "0"}], ",", RowBox[{"Iyy", "\[Rule]", "\[CapitalIota]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "xyz", "]"}], ",", RowBox[{"constitutive", "[", "xyz", "]"}], ",", RowBox[{ RowBox[{"clamped", "[", "xyz", "]"}], "[", "0", "]"}], ",", RowBox[{ RowBox[{"free", "[", "xyz", "]"}], "[", "L", "]"}]}], "]"}], " ", "/.", "rule"}]}], ";"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"functions", "[", "xyz", "]"}], ",", "x"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Qz", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], ",", RowBox[{"\[Alpha]", "\[Rule]", "Infinity"}]}], "]"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"My", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], ",", RowBox[{"\[Alpha]", "\[Rule]", "Infinity"}]}], "]"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"w", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], ",", RowBox[{"\[Alpha]", "\[Rule]", "Infinity"}]}], "]"}], "]"}]}], "]"}], "\[IndentingNewLine]"}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.6972529515461555`*^9, 3.697253084716342*^9}, {3.697253246136568*^9, 3.6972532878466263`*^9}, {3.697253341536701*^9, 3.697253385806763*^9}, { 3.697254029617665*^9, 3.6972541087377753`*^9}, {3.6972541537278385`*^9, 3.697254192057892*^9}, {3.6972542390879583`*^9, 3.6972542801280155`*^9}, { 3.697254317298067*^9, 3.69725432622808*^9}, {3.6972543601881275`*^9, 3.6972544415582414`*^9}, {3.6972550776891317`*^9, 3.6972551527992373`*^9}, {3.697464845793644*^9, 3.6974649348237686`*^9}, { 3.6974649717638197`*^9, 3.697465031933904*^9}, {3.697465065083951*^9, 3.6974650864839807`*^9}, {3.697465175614105*^9, 3.6974652364441905`*^9}, 3.6974653757643857`*^9, {3.697465587824682*^9, 3.6974656235847325`*^9}, { 3.697465692394829*^9, 3.6974657187048655`*^9}, {3.69769096026633*^9, 3.697690962576333*^9}, {3.7291527576563945`*^9, 3.7291527822028446`*^9}, { 3.729152852802981*^9, 3.72915291736015*^9}, {3.729153391870802*^9, 3.7291534436087537`*^9}, {3.7291541748035603`*^9, 3.7291541852445717`*^9}, {3.760960090317384*^9, 3.760960254304179*^9}, { 3.7609603144243307`*^9, 3.760960387237884*^9}}, CellLabel->"In[37]:=",ExpressionUUID->"c0786013-67e5-419c-af0a-3c65f07f5b77"] }, Closed]], Cell[CellGroupData[{ Cell["Example problem 3", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6972528859460635`*^9, 3.697252899226082*^9}, { 3.697464827703618*^9, 3.6974648360436296`*^9}, {3.7291344563518124`*^9, 3.7291344655087395`*^9}, 3.7899564911817603`*^9},ExpressionUUID->"b2f79b14-f462-42e7-9a06-\ c51278085057"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bx", "\[Rule]", "0"}], ",", RowBox[{"by", " ", "\[Rule]", "0"}], ",", " ", RowBox[{"bz", "\[Rule]", "0"}], ",", RowBox[{"cx", "\[Rule]", "0"}], ",", RowBox[{"cy", "\[Rule]", "0"}], ",", RowBox[{"cz", "\[Rule]", "0"}], ",", RowBox[{"Sy", "\[Rule]", "0"}], ",", RowBox[{"Sz", "\[Rule]", "0"}], ",", " ", RowBox[{"Iyz", "\[Rule]", "0"}], ",", RowBox[{"Izy", "\[Rule]", "0"}], ",", RowBox[{"Iyy", "\[Rule]", "\[CapitalIota]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "xyz", "]"}], ",", RowBox[{"constitutive", "[", "xyz", "]"}], ",", RowBox[{ RowBox[{"clamped", "[", "xyz", "]"}], "[", "0", "]"}], ",", RowBox[{ RowBox[{ RowBox[{"free", "[", "xyz", "]"}], "[", "L", "]"}], "-", RowBox[{"{", RowBox[{"0", ",", "0", ",", "P", ",", "0", ",", "0", ",", "0"}], "}"}]}]}], "]"}], " ", "/.", "rule"}]}], ";"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"functions", "[", "xyz", "]"}], ",", "x"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"w", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], ",", RowBox[{"\[Alpha]", "\[Rule]", "Infinity"}]}], "]"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\<\[Theta][x] = \>\"", ",", RowBox[{"Simplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"\[Theta]", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], ",", RowBox[{"\[Alpha]", "\[Rule]", "Infinity"}]}], "]"}], "]"}]}], "]"}], "\[IndentingNewLine]"}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.6972529515461555`*^9, 3.697253084716342*^9}, {3.697253246136568*^9, 3.6972532878466263`*^9}, {3.697253341536701*^9, 3.697253385806763*^9}, { 3.697254029617665*^9, 3.6972541087377753`*^9}, {3.6972541537278385`*^9, 3.697254192057892*^9}, {3.6972542390879583`*^9, 3.6972542801280155`*^9}, { 3.697254317298067*^9, 3.69725432622808*^9}, {3.6972543601881275`*^9, 3.6972544415582414`*^9}, {3.6972550776891317`*^9, 3.6972551527992373`*^9}, {3.697464845793644*^9, 3.6974649348237686`*^9}, { 3.6974649717638197`*^9, 3.697465031933904*^9}, {3.697465065083951*^9, 3.6974650864839807`*^9}, {3.697465175614105*^9, 3.6974652364441905`*^9}, 3.6974653757643857`*^9, {3.697465587824682*^9, 3.6974656235847325`*^9}, { 3.697465692394829*^9, 3.6974657187048655`*^9}, {3.69769096026633*^9, 3.697690962576333*^9}, {3.760960784647874*^9, 3.7609608099057693`*^9}}, CellLabel->"In[49]:=",ExpressionUUID->"f0b86a41-2f52-47eb-87d6-a3a579200d84"] }, Closed]], Cell[CellGroupData[{ Cell["Example problem 4", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6972528859460635`*^9, 3.697252899226082*^9}, { 3.697464827703618*^9, 3.6974648360436296`*^9}, {3.697465912365137*^9, 3.697465912595137*^9}, {3.7291345705356245`*^9, 3.729134577739562*^9}, 3.789956495930732*^9},ExpressionUUID->"985b8f42-9df1-4a85-8a2c-\ 0ed0937539a6"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bx", "\[Rule]", "0"}], ",", RowBox[{"by", " ", "\[Rule]", "0"}], ",", " ", RowBox[{"bz", "\[Rule]", "b"}], ",", RowBox[{"cx", "\[Rule]", "0"}], ",", RowBox[{"cy", "\[Rule]", "0"}], ",", RowBox[{"cz", "\[Rule]", "0"}], ",", RowBox[{"Sy", "\[Rule]", "0"}], ",", RowBox[{"Sz", "\[Rule]", "0"}], ",", " ", RowBox[{"Iyz", "\[Rule]", "0"}], ",", RowBox[{"Izy", "\[Rule]", "0"}], ",", RowBox[{"Iyy", "\[Rule]", "\[CapitalIota]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "xyz", "]"}], ",", RowBox[{"constitutive", "[", "xyz", "]"}], ",", RowBox[{ RowBox[{"simply", "[", "xyz", "]"}], "[", "0", "]"}], ",", RowBox[{ RowBox[{"slide", "[", "xyz", "]"}], "[", "L", "]"}]}], "]"}], " ", "/.", "rule"}]}], ";"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"functions", "[", "xyz", "]"}], ",", "x"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"w", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], " ", "/.", RowBox[{"{", " ", RowBox[{ RowBox[{"x", "\[Rule]", RowBox[{"L", "/", "2"}]}], ",", RowBox[{"\[Alpha]", "\[Rule]", "1"}]}], "}"}]}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\<\[Theta][x] = \>\"", ",", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"\[Theta]", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], "/.", " ", RowBox[{"{", " ", RowBox[{ RowBox[{"x", "\[Rule]", RowBox[{"L", "/", "2"}]}], ",", RowBox[{"\[Alpha]", "\[Rule]", "1"}]}], "}"}]}], "]"}]}], " ", "]"}], "\[IndentingNewLine]"}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.6972529515461555`*^9, 3.697253084716342*^9}, {3.697253246136568*^9, 3.6972532878466263`*^9}, {3.697253341536701*^9, 3.697253385806763*^9}, { 3.697254029617665*^9, 3.6972541087377753`*^9}, {3.6972541537278385`*^9, 3.697254192057892*^9}, {3.6972542390879583`*^9, 3.6972542801280155`*^9}, { 3.697254317298067*^9, 3.69725432622808*^9}, {3.6972543601881275`*^9, 3.6972544415582414`*^9}, {3.6972550776891317`*^9, 3.6972551527992373`*^9}, {3.697464845793644*^9, 3.6974649348237686`*^9}, { 3.6974649717638197`*^9, 3.697465031933904*^9}, {3.697465065083951*^9, 3.6974650864839807`*^9}, {3.697465175614105*^9, 3.6974652364441905`*^9}, 3.6974653757643857`*^9, {3.697465587824682*^9, 3.6974656235847325`*^9}, { 3.697465692394829*^9, 3.6974657187048655`*^9}, {3.697465932475165*^9, 3.69746597904523*^9}, {3.697466038865314*^9, 3.697466039565315*^9}, { 3.697466076935367*^9, 3.6974660936553907`*^9}, {3.6976909038662505`*^9, 3.6976909062262535`*^9}, {3.729134586302946*^9, 3.7291345965540867`*^9}, { 3.7291346614059453`*^9, 3.729134696003601*^9}}, CellLabel->"In[54]:=",ExpressionUUID->"abd606e0-ad5b-489b-b7b3-63f20e950cb5"] }, Closed]], Cell[CellGroupData[{ Cell["Example problem 5", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6972528859460635`*^9, 3.697252899226082*^9}, { 3.697464827703618*^9, 3.6974648360436296`*^9}, {3.697465912365137*^9, 3.697465912595137*^9}, {3.7291345705356245`*^9, 3.729134577739562*^9}, 3.7291349817094*^9, 3.789956498723392*^9},ExpressionUUID->"ac29112e-37bb-4091-82d1-\ 87b34b95efcd"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bx", "\[Rule]", "0"}], ",", RowBox[{"by", " ", "\[Rule]", "b"}], ",", " ", RowBox[{"bz", "\[Rule]", "b"}], ",", RowBox[{"cx", "\[Rule]", "0"}], ",", RowBox[{"cy", "\[Rule]", "0"}], ",", RowBox[{"cz", "\[Rule]", "0"}], ",", RowBox[{"Sy", "\[Rule]", "0"}], ",", RowBox[{"Sz", "\[Rule]", "0"}], ",", " ", RowBox[{"Iyz", "\[Rule]", "0"}], ",", RowBox[{"Izy", "\[Rule]", "0"}], ",", RowBox[{"Iyy", "\[Rule]", "\[CapitalIota]"}], ",", RowBox[{"Izz", "\[Rule]", "\[CapitalIota]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "xyz", "]"}], ",", RowBox[{"constitutive", "[", "xyz", "]"}], ",", RowBox[{ RowBox[{"simply", "[", "xyz", "]"}], "[", "0", "]"}], ",", RowBox[{ RowBox[{"slide", "[", "xyz", "]"}], "[", "L", "]"}]}], "]"}], " ", "/.", "rule"}]}], ";"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"functions", "[", "xyz", "]"}], ",", "x"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"Limit", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"v", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], ",", " ", RowBox[{"\[Alpha]", "\[Rule]", "Infinity"}]}], "]"}], "]"}]}], "]"}], "\[IndentingNewLine]"}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.6972529515461555`*^9, 3.697253084716342*^9}, {3.697253246136568*^9, 3.6972532878466263`*^9}, {3.697253341536701*^9, 3.697253385806763*^9}, { 3.697254029617665*^9, 3.6972541087377753`*^9}, {3.6972541537278385`*^9, 3.697254192057892*^9}, {3.6972542390879583`*^9, 3.6972542801280155`*^9}, { 3.697254317298067*^9, 3.69725432622808*^9}, {3.6972543601881275`*^9, 3.6972544415582414`*^9}, {3.6972550776891317`*^9, 3.6972551527992373`*^9}, {3.697464845793644*^9, 3.6974649348237686`*^9}, { 3.6974649717638197`*^9, 3.697465031933904*^9}, {3.697465065083951*^9, 3.6974650864839807`*^9}, {3.697465175614105*^9, 3.6974652364441905`*^9}, 3.6974653757643857`*^9, {3.697465587824682*^9, 3.6974656235847325`*^9}, { 3.697465692394829*^9, 3.6974657187048655`*^9}, {3.697465932475165*^9, 3.69746597904523*^9}, {3.697466038865314*^9, 3.697466039565315*^9}, { 3.697466076935367*^9, 3.6974660936553907`*^9}, {3.6976909038662505`*^9, 3.6976909062262535`*^9}, {3.729134586302946*^9, 3.7291345965540867`*^9}, { 3.7291346614059453`*^9, 3.729134696003601*^9}, {3.7291349933984165`*^9, 3.729135038715684*^9}, 3.729137489348261*^9, {3.760960918479903*^9, 3.760960928474247*^9}, {3.760961010985484*^9, 3.760961026301402*^9}, { 3.7609610758570623`*^9, 3.7609610840317755`*^9}, {3.760961120658888*^9, 3.760961123587323*^9}, {3.7609612641276474`*^9, 3.7609613476967983`*^9}, { 3.7609615128658776`*^9, 3.7609615151468267`*^9}, {3.7609617057924185`*^9, 3.760961709252054*^9}}, CellLabel-> "In[143]:=",ExpressionUUID->"c751fb3c-c274-4a37-8c7e-f6c8efd3142b"] }, Closed]], Cell[CellGroupData[{ Cell["Example problem 6", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6974344862207527`*^9, 3.6974344863907523`*^9}, 3.6976910842865033`*^9, {3.7291375217579207`*^9, 3.7291375239143925`*^9}, 3.7899565011402254`*^9},ExpressionUUID->"ad58db85-6e4e-4cf9-aea2-\ fe550af5a594"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bs", "\[Rule]", "0"}], ",", RowBox[{"bn", "\[Rule]", "0"}], ",", RowBox[{"bb", "\[Rule]", "0"}], ",", RowBox[{"cs", "\[Rule]", "0"}], ",", RowBox[{"cn", "\[Rule]", "0"}], ",", RowBox[{"cb", "\[Rule]", "0"}], ",", RowBox[{"\[Tau]", "\[Rule]", RowBox[{"4", " ", RowBox[{"Pi", "/", "L"}]}]}], ",", RowBox[{"\[Kappa]", "\[Rule]", "0"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "snb", "]"}], ",", RowBox[{ RowBox[{ RowBox[{"free", "[", "snb", "]"}], "[", "L", "]"}], "-", RowBox[{"{", RowBox[{"0", ",", "0", ",", "P", ",", "0", ",", "0", ",", "0"}], "}"}]}]}], "]"}], "/.", "rule"}]}], " ", ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"forces", "[", "snb", "]"}], ",", "s"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Ns", "[", "s", "]"}], " ", "/.", " ", "sol"}], " ", "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Qn", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Qb", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Ts", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Mn", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Mb", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.697364372157997*^9, 3.697364411619254*^9}, {3.697434495460765*^9, 3.697434521790802*^9}, {3.697690355685483*^9, 3.6976903875455275`*^9}}, CellLabel-> "In[147]:=",ExpressionUUID->"40e59b0f-6f51-4a96-8ca3-c5d0c7af77cd"] }, Closed]], Cell[CellGroupData[{ Cell["Example problem 7", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6974344862207527`*^9, 3.6974344863907523`*^9}, 3.697449470857856*^9, {3.729137677330374*^9, 3.7291376793774137`*^9}, 3.789956503511906*^9},ExpressionUUID->"6c9b4103-896a-4dc6-b865-\ 26c876ff4c2e"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bs", "\[Rule]", "0"}], ",", RowBox[{"bn", "\[Rule]", "0"}], ",", RowBox[{"bb", "\[Rule]", "0"}], ",", RowBox[{"cs", "\[Rule]", "0"}], ",", RowBox[{"cn", "\[Rule]", "0"}], ",", RowBox[{"cb", "\[Rule]", "0"}], ",", RowBox[{"\[Tau]", "\[Rule]", "0"}], ",", RowBox[{"\[Kappa]", "\[Rule]", RowBox[{"1", "/", "R"}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "snb", "]"}], ",", RowBox[{ RowBox[{ RowBox[{"free", "[", "snb", "]"}], "[", RowBox[{"Pi", " ", RowBox[{"R", "/", "2"}]}], "]"}], "-", RowBox[{"{", RowBox[{"0", ",", "0", ",", RowBox[{"-", "P"}], ",", "0", ",", "0", ",", "0"}], "}"}]}]}], "]"}], " ", "/.", "rule"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], " ", ",", RowBox[{"forces", "[", "snb", "]"}], ",", "s"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Ns", "[", "s", "]"}], " ", "/.", " ", "sol"}], " ", "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Qn", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Qb", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Ts", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Mn", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Mb", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.697364372157997*^9, 3.697364411619254*^9}, {3.697434495460765*^9, 3.697434521790802*^9}, {3.6974494800178685`*^9, 3.697449649338105*^9}, { 3.6974497365782275`*^9, 3.697449764068266*^9}, {3.697449849648386*^9, 3.697449854128392*^9}, {3.697449891648445*^9, 3.6974500004785967`*^9}, { 3.697450033448643*^9, 3.6974500679186916`*^9}, {3.697450157838817*^9, 3.697450161958823*^9}, {3.697455511860576*^9, 3.697455514010579*^9}, { 3.6976906128058434`*^9, 3.6976906522458982`*^9}, {3.697690766386058*^9, 3.697690779346076*^9}, {3.6976911466565905`*^9, 3.6976911642166147`*^9}, { 3.729137740227687*^9, 3.72913778338817*^9}, 3.729137820714054*^9, { 3.7291378672501726`*^9, 3.729137889064778*^9}, {3.7291379747615857`*^9, 3.729137975902319*^9}, {3.7291380173322287`*^9, 3.729138024708011*^9}}, CellLabel-> "In[156]:=",ExpressionUUID->"f18bf471-d153-4197-9a8a-80ac235a29e3"] }, Closed]], Cell[CellGroupData[{ Cell["Example problem 8", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6974344862207527`*^9, 3.6974344863907523`*^9}, 3.697449470857856*^9, {3.729137677330374*^9, 3.7291376793774137`*^9}, 3.729138074728773*^9, 3.789956506331716*^9},ExpressionUUID->"4d9190d6-4237-4017-8f74-\ 1fd24cbdab08"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bs", "\[Rule]", "0"}], ",", RowBox[{"bn", "\[Rule]", "0"}], ",", RowBox[{"bb", "\[Rule]", "0"}], ",", RowBox[{"cs", "\[Rule]", "0"}], ",", RowBox[{"cn", "\[Rule]", "0"}], ",", RowBox[{"cb", "\[Rule]", "0"}], ",", RowBox[{"\[Tau]", "\[Rule]", "0"}], ",", RowBox[{"\[Kappa]", "\[Rule]", RowBox[{"1", "/", "R"}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "snb", "]"}], ",", RowBox[{ RowBox[{ RowBox[{"free", "[", "snb", "]"}], "[", "0", "]"}], "-", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "P", ",", "0", ",", "0"}], "}"}]}]}], "]"}], " ", "/.", "rule"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], " ", ",", RowBox[{"forces", "[", "snb", "]"}], ",", "s"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Ns", "[", "s", "]"}], " ", "/.", " ", "sol"}], " ", "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Qn", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Qb", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Ts", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Mn", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Mb", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.697364372157997*^9, 3.697364411619254*^9}, {3.697434495460765*^9, 3.697434521790802*^9}, {3.6974494800178685`*^9, 3.697449649338105*^9}, { 3.6974497365782275`*^9, 3.697449764068266*^9}, {3.697449849648386*^9, 3.697449854128392*^9}, {3.697449891648445*^9, 3.6974500004785967`*^9}, { 3.697450033448643*^9, 3.6974500679186916`*^9}, {3.697450157838817*^9, 3.697450161958823*^9}, {3.697455511860576*^9, 3.697455514010579*^9}, { 3.6976906128058434`*^9, 3.6976906522458982`*^9}, {3.697690766386058*^9, 3.697690779346076*^9}, {3.6976911466565905`*^9, 3.6976911642166147`*^9}, { 3.7291381254999723`*^9, 3.7291381265938854`*^9}, {3.729138170926679*^9, 3.7291382193064594`*^9}, {3.729138282485258*^9, 3.729138285282423*^9}}, CellLabel-> "In[165]:=",ExpressionUUID->"8d300df9-089f-4959-94d0-66831d92ae9d"] }, Closed]], Cell[CellGroupData[{ Cell["Example problem 9", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6974344862207527`*^9, 3.6974344863907523`*^9}, 3.697449470857856*^9, {3.729137677330374*^9, 3.7291376793774137`*^9}, { 3.729155492572383*^9, 3.729155497025456*^9}, {3.760961983523259*^9, 3.7609619886968155`*^9}, 3.7899565108321805`*^9},ExpressionUUID->"edf4395a-fc4f-496b-b821-\ ce7ded231efe"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bs", "\[Rule]", "0"}], ",", RowBox[{"bn", "\[Rule]", "0"}], ",", RowBox[{"bb", "\[Rule]", "b"}], ",", RowBox[{"cs", "\[Rule]", "0"}], ",", RowBox[{"cn", "\[Rule]", "0"}], ",", RowBox[{"cb", "\[Rule]", "0"}], ",", RowBox[{"\[Tau]", "\[Rule]", "0"}], ",", RowBox[{"\[Kappa]", "\[Rule]", RowBox[{"1", "/", "R"}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "snb", "]"}], ",", RowBox[{ RowBox[{"free", "[", "snb", "]"}], "[", RowBox[{"Pi", " ", RowBox[{"R", "/", "2"}]}], "]"}]}], "]"}], " ", "/.", "rule"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], " ", ",", RowBox[{"forces", "[", "snb", "]"}], ",", "s"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Ns", "[", "s", "]"}], " ", "/.", " ", "sol"}], " ", "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Qn", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Qb", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Ts", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Mn", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Mb", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.697364372157997*^9, 3.697364411619254*^9}, {3.697434495460765*^9, 3.697434521790802*^9}, {3.6974494800178685`*^9, 3.697449649338105*^9}, { 3.6974497365782275`*^9, 3.697449764068266*^9}, {3.697449849648386*^9, 3.697449854128392*^9}, {3.697449891648445*^9, 3.6974500004785967`*^9}, { 3.697450033448643*^9, 3.6974500679186916`*^9}, {3.697450157838817*^9, 3.697450161958823*^9}, {3.697455511860576*^9, 3.697455514010579*^9}, { 3.6976906128058434`*^9, 3.6976906522458982`*^9}, {3.697690766386058*^9, 3.697690779346076*^9}, {3.6976911466565905`*^9, 3.6976911642166147`*^9}, { 3.729137740227687*^9, 3.72913778338817*^9}, 3.729137820714054*^9, { 3.7291378672501726`*^9, 3.729137889064778*^9}, {3.7291379747615857`*^9, 3.729137975902319*^9}, {3.7291380173322287`*^9, 3.729138024708011*^9}, { 3.7291555280177145`*^9, 3.7291555414107866`*^9}, 3.7291573568380594`*^9, 3.7291586056263833`*^9, {3.729158656870234*^9, 3.7291586690265427`*^9}, 3.72922244579767*^9}, CellLabel-> "In[174]:=",ExpressionUUID->"fb7686e1-ed73-4549-8da0-67eaccdee858"] }, Closed]], Cell[CellGroupData[{ Cell["Example problem 10", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.7289615946007776`*^9, 3.7289616467307887`*^9}, { 3.729138788591694*^9, 3.7291388008694696`*^9}, 3.789956514036558*^9},ExpressionUUID->"54d696cb-6e14-4e2a-a400-\ 9dd418b4e2f7"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bs", "\[Rule]", "0"}], ",", RowBox[{"bn", "\[Rule]", "0"}], ",", RowBox[{"bb", "\[Rule]", "0"}], ",", RowBox[{"cs", "\[Rule]", "0"}], ",", RowBox[{"cn", "\[Rule]", "0"}], ",", RowBox[{"cb", "\[Rule]", "0"}], ",", RowBox[{"\[Tau]", "\[Rule]", "0"}], ",", RowBox[{"\[Kappa]", "->", RowBox[{"1", "/", "R"}]}], ",", "\[IndentingNewLine]", RowBox[{"Sn", "\[Rule]", "0"}], ",", RowBox[{"Sb", "\[Rule]", "0"}], ",", RowBox[{"Inn", "\[Rule]", "\[CapitalIota]"}], ",", RowBox[{"Ibb", "\[Rule]", "\[CapitalIota]"}], ",", RowBox[{"Ibn", "\[Rule]", "0"}], ",", RowBox[{"Inb", "\[Rule]", "0"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "snb", "]"}], ",", RowBox[{"constitutive", "[", "snb", "]"}], ",", RowBox[{ RowBox[{"clamped", "[", "snb", "]"}], "[", "0", "]"}], ",", RowBox[{ RowBox[{ RowBox[{"free", "[", "snb", "]"}], "[", RowBox[{"Pi", " ", RowBox[{"R", "/", "2"}]}], "]"}], "-", RowBox[{"{", RowBox[{"0", ",", "P", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}]}]}], "]"}], " ", "/.", "rule"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"functions", "[", "snb", "]"}], ",", "s"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{ RowBox[{"u", "[", "s", "]"}], " ", "/.", " ", "sol"}], " ", "/.", RowBox[{"\[Alpha]", "\[Rule]", "1"}]}], " ", "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{ RowBox[{"v", "[", "s", "]"}], " ", "/.", " ", "sol"}], "/.", RowBox[{"\[Alpha]", "\[Rule]", "1"}]}], " ", "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\<\[Psi](s)= \>\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{ RowBox[{"\[Psi]", "[", "s", "]"}], " ", "/.", " ", "sol"}], "/.", RowBox[{"\[Alpha]", "\[Rule]", "1"}]}], " ", "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.697364372157997*^9, 3.697364411619254*^9}, {3.697690490155671*^9, 3.6976905424457445`*^9}, {3.729138832200755*^9, 3.729139077882655*^9}, { 3.7291391324355974`*^9, 3.7291391703926563`*^9}, {3.7291392363965936`*^9, 3.7291392582268443`*^9}}, CellLabel-> "In[183]:=",ExpressionUUID->"03250248-2eb6-4657-97dc-3a855f663467"] }, Closed]], Cell[CellGroupData[{ Cell["Assignment 1", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6972528859460635`*^9, 3.697252899226082*^9}, 3.697691252966739*^9, {3.7289617343781796`*^9, 3.728961755357093*^9}, { 3.729161066723563*^9, 3.729161066957983*^9}, {3.7292140394406013`*^9, 3.729214048003995*^9}, {3.8883702885369663`*^9, 3.8883702911067038`*^9}},ExpressionUUID->"49f66745-cab2-42f5-88b3-\ 0675bcc5c206"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bx", "\[Rule]", "0"}], ",", RowBox[{"by", " ", "\[Rule]", "b"}], ",", " ", RowBox[{"bz", "\[Rule]", "0"}], ",", RowBox[{"cx", "\[Rule]", "0"}], ",", RowBox[{"cy", "\[Rule]", "0"}], ",", RowBox[{"cz", "\[Rule]", "0"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "xyz", "]"}], ",", RowBox[{ RowBox[{ RowBox[{"free", "[", "xyz", "]"}], "[", "L", "]"}], "-", RowBox[{"{", RowBox[{"P", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}]}]}], "]"}], "/.", "rule"}]}], ";"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"forces", "[", "xyz", "]"}], ",", "x"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"(", RowBox[{ RowBox[{"Nx", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"(", RowBox[{ RowBox[{"Qy", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"(", RowBox[{ RowBox[{"Mz", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], "]"}]}], "]"}], "\[IndentingNewLine]"}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.6972529515461555`*^9, 3.697253084716342*^9}, {3.697253246136568*^9, 3.6972532878466263`*^9}, {3.697253341536701*^9, 3.697253385806763*^9}, { 3.697254029617665*^9, 3.6972541087377753`*^9}, {3.6972541537278385`*^9, 3.697254192057892*^9}, {3.6972542390879583`*^9, 3.6972542801280155`*^9}, { 3.697254317298067*^9, 3.69725432622808*^9}, {3.6972543601881275`*^9, 3.6972544415582414`*^9}, {3.6972550776891317`*^9, 3.6972551527992373`*^9}, {3.697690411465561*^9, 3.6976904555756226`*^9}, { 3.6976912961467996`*^9, 3.697691321896836*^9}, 3.7289617606979027`*^9, { 3.729161125292254*^9, 3.729161138512396*^9}, {3.729161169421898*^9, 3.729161340564952*^9}, {3.729214060145916*^9, 3.729214094962048*^9}, { 3.729214128559123*^9, 3.729214181049144*^9}, {3.729214219490755*^9, 3.729214226960372*^9}}, CellLabel->"In[40]:=",ExpressionUUID->"d9ac3563-69a1-4b8d-9966-ba5107446602"] }, Closed]], Cell[CellGroupData[{ Cell["Assignment 2", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6974344862207527`*^9, 3.6974344863907523`*^9}, 3.6976910842865033`*^9, {3.7291375217579207`*^9, 3.7291375239143925`*^9}, { 3.7292151542939053`*^9, 3.7292151572314253`*^9}, {3.8883702931772676`*^9, 3.8883702957883005`*^9}},ExpressionUUID->"4134aba9-323b-4b4b-b5d2-\ 2f5253e5c448"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bs", "\[Rule]", "0"}], ",", RowBox[{"bn", "\[Rule]", "0"}], ",", RowBox[{"bb", "\[Rule]", "0"}], ",", RowBox[{"cs", "\[Rule]", "0"}], ",", RowBox[{"cn", "\[Rule]", "0"}], ",", RowBox[{"cb", "\[Rule]", "0"}], ",", RowBox[{"\[Tau]", "\[Rule]", "0"}], ",", RowBox[{"\[Kappa]", "\[Rule]", RowBox[{"1", "/", "R"}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "snb", "]"}], ",", RowBox[{ RowBox[{ RowBox[{"free", "[", "snb", "]"}], "[", RowBox[{"Pi", " ", "R"}], "]"}], "-", RowBox[{"{", RowBox[{ RowBox[{"-", "P"}], ",", "0", ",", "0", ",", "0", ",", "0", ",", RowBox[{"-", "H"}]}], "}"}]}]}], "]"}], "/.", "rule"}]}], " ", ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"forces", "[", "snb", "]"}], ",", "s"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Ns", "[", "s", "]"}], " ", "/.", " ", "sol"}], " ", "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Qn", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Mb", "[", "s", "]"}], " ", "/.", " ", "sol"}], "]"}]}], "]"}]}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.697364372157997*^9, 3.697364411619254*^9}, {3.697434495460765*^9, 3.697434521790802*^9}, {3.697690355685483*^9, 3.6976903875455275`*^9}, { 3.7292151624822364`*^9, 3.72921524266241*^9}}, CellLabel->"In[46]:=",ExpressionUUID->"7d721f21-2378-4717-936b-724118e92229"] }, Closed]], Cell[CellGroupData[{ Cell["Assignment 4", "Subsection", CellChangeTimes->{ 3.4710857577642*^9, {3.4710858112098*^9, 3.4710858239082003`*^9}, { 3.4710859049969997`*^9, 3.4710859061046*^9}, {3.4710892678536*^9, 3.4710892701*^9}, 3.4710893817648*^9, {3.4710894793115997`*^9, 3.4710895039596*^9}, {3.4710905118050003`*^9, 3.471090512741*^9}, { 3.4725252603414*^9, 3.472525264101*^9}, {3.4725253064862003`*^9, 3.4725253370934*^9}, {3.4747629998582*^9, 3.4747630055678*^9}, { 3.4752273878873997`*^9, 3.4752273904926*^9}, {3.4753663741484003`*^9, 3.4753664116352*^9}, {3.4753773567911997`*^9, 3.475377356916*^9}, 3.4753775870472*^9, {3.4754620951394*^9, 3.4754621031266003`*^9}, { 3.4779795207265997`*^9, 3.4779795250576*^9}, 3.537621939388029*^9, 3.537622934627954*^9, {3.5376230222229643`*^9, 3.537623029743394*^9}, { 3.537623460058007*^9, 3.537623470980632*^9}, {3.537624955779557*^9, 3.5376249594157653`*^9}, {3.6000028730424147`*^9, 3.6000028738224154`*^9}, 3.6000042319920015`*^9, {3.6005962565960674`*^9, 3.6005962590140715`*^9}, 3.6006031322879376`*^9, 3.6636738373457937`*^9, {3.6971725349450593`*^9, 3.69717255003508*^9}, {3.6972528859460635`*^9, 3.697252899226082*^9}, { 3.697464827703618*^9, 3.6974648360436296`*^9}, {3.7291344563518124`*^9, 3.7291344655087395`*^9}, {3.7292162281744947`*^9, 3.729216231393631*^9}, 3.760859874061782*^9, {3.8883702988384905`*^9, 3.8883703049523797`*^9}},ExpressionUUID->"1cc1bc5b-b244-47a7-a86b-\ ad614c5cad7e"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"rule", "=", RowBox[{"{", RowBox[{ RowBox[{"bx", "\[Rule]", "0"}], ",", RowBox[{"by", " ", "\[Rule]", "0"}], ",", " ", RowBox[{"bz", "\[Rule]", "b"}], ",", RowBox[{"cx", "\[Rule]", "0"}], ",", RowBox[{"cy", "\[Rule]", "0"}], ",", RowBox[{"cz", "\[Rule]", "0"}], ",", RowBox[{"Sy", "\[Rule]", "0"}], ",", RowBox[{"Sz", "\[Rule]", "0"}], ",", " ", RowBox[{"Iyz", "\[Rule]", "0"}], ",", RowBox[{"Izy", "\[Rule]", "0"}], ",", RowBox[{"Iyy", "\[Rule]", "\[CapitalIota]"}], ",", RowBox[{"Izz", "\[Rule]", "\[CapitalIota]"}], ",", RowBox[{"Irr", "\[Rule]", RowBox[{"2", "\[CapitalIota]"}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"equs", "=", RowBox[{ RowBox[{"Join", "[", RowBox[{ RowBox[{"equilibrium", "[", "xyz", "]"}], ",", RowBox[{"constitutive", "[", "xyz", "]"}], ",", RowBox[{ RowBox[{"clamped", "[", "xyz", "]"}], "[", "0", "]"}], ",", RowBox[{ RowBox[{"free", "[", "xyz", "]"}], "[", "L", "]"}]}], "]"}], " ", "/.", "rule"}]}], ";"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"Thread", "[", RowBox[{"equs", "\[Equal]", "0"}], "]"}], ",", RowBox[{"functions", "[", "xyz", "]"}], ",", "x"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"(", RowBox[{ RowBox[{"Nx", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"(", RowBox[{ RowBox[{"Qz", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"(", RowBox[{ RowBox[{"My", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"(", RowBox[{ RowBox[{"u", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Simplify", "[", RowBox[{"(", RowBox[{ RowBox[{"w", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\<\[Theta][x] = \>\"", ",", RowBox[{"Simplify", "[", RowBox[{"(", RowBox[{ RowBox[{"\[Theta]", "[", "x", "]"}], " ", "/.", " ", "sol"}], " ", ")"}], "]"}]}], "]"}], "\[IndentingNewLine]"}]}]], "Input", CellChangeTimes->{ 3.4747630249118*^9, {3.4747631189174*^9, 3.4747632157778*^9}, { 3.4747632855098*^9, 3.4747632963361998`*^9}, {3.4747634024006*^9, 3.474763402697*^9}, {3.474763495517*^9, 3.4747635001814003`*^9}, { 3.4747637730566*^9, 3.474763803851*^9}, {3.4747642087646*^9, 3.4747642600573997`*^9}, {3.4747643726426*^9, 3.4747644737774*^9}, { 3.4752269899314003`*^9, 3.4752270948882*^9}, {3.4752271345902*^9, 3.4752272632434*^9}, {3.475227295551*^9, 3.4752275983314*^9}, { 3.475227632591*^9, 3.4752276649932003`*^9}, {3.4752276971146*^9, 3.4752277780172*^9}, {3.4752278086712*^9, 3.4752279362948*^9}, { 3.4752279825643997`*^9, 3.4752280182416*^9}, {3.4752280573196*^9, 3.4752280750411997`*^9}, {3.4752281205931997`*^9, 3.4752281779231997`*^9}, {3.4752283129412003`*^9, 3.4752284125678*^9}, { 3.4752286421373997`*^9, 3.4752286429174*^9}, {3.4752290214202003`*^9, 3.4752292420810003`*^9}, {3.4752292747942*^9, 3.4752293745874*^9}, { 3.4752294050854*^9, 3.4752295128190002`*^9}, {3.475230037291*^9, 3.4752300676486*^9}, {3.4752302150998*^9, 3.475230248749*^9}, { 3.4753664935702*^9, 3.4753668085810003`*^9}, {3.4753668484702*^9, 3.4753670665114*^9}, {3.4753671011902*^9, 3.4753674723454*^9}, { 3.4753676229322*^9, 3.4753677863733997`*^9}, {3.475367825233*^9, 3.475367848243*^9}, {3.475368029983*^9, 3.4753680589522*^9}, { 3.475369411392*^9, 3.4753695644904003`*^9}, {3.4753696954256*^9, 3.475369726766*^9}, {3.4753698525643997`*^9, 3.4753698555128*^9}, { 3.4753773724536*^9, 3.4753774958652*^9}, 3.4753775502468*^9, { 3.4753776430044003`*^9, 3.4753776659364*^9}, {3.4753777121748*^9, 3.4753777832483997`*^9}, {3.4753778188164*^9, 3.475377922336*^9}, { 3.4753779601348*^9, 3.4753784001952*^9}, {3.4753785090208*^9, 3.475378568722*^9}, {3.4753786124955997`*^9, 3.4753786164112*^9}, { 3.4753786468156*^9, 3.4753786657384*^9}, {3.4753787045355997`*^9, 3.4753787267032003`*^9}, 3.4753910727106*^9, {3.4753931910636*^9, 3.4753932441756*^9}, {3.4754621543882*^9, 3.4754622450397997`*^9}, { 3.4778876497897997`*^9, 3.4778877494737997`*^9}, {3.477887825399*^9, 3.4778878298606*^9}, {3.4778878701085997`*^9, 3.477887883041*^9}, { 3.4778879526482*^9, 3.477887957843*^9}, {3.4778879979974003`*^9, 3.4778880009614*^9}, {3.4778880335654*^9, 3.4778880462326*^9}, { 3.4778880897098*^9, 3.4778881201142*^9}, {3.4778882394908*^9, 3.477888243734*^9}, {3.4778884935368*^9, 3.477888539042*^9}, { 3.477889451995*^9, 3.4778895118416*^9}, {3.4779795395165997`*^9, 3.4779795402136*^9}, {3.4779797634996*^9, 3.4779797999356003`*^9}, 3.4779798372766*^9, {3.4779799820666*^9, 3.4779801217206*^9}, { 3.4779801745425997`*^9, 3.4779801845926*^9}, {3.4779804353096*^9, 3.4779804588506002`*^9}, {3.4779804893566*^9, 3.4779806203556004`*^9}, { 3.4779807300538*^9, 3.4779807467418003`*^9}, {3.4779808817128*^9, 3.4779809449638*^9}, {3.4781573512264*^9, 3.4781573649856*^9}, { 3.478157397808*^9, 3.4781574763227997`*^9}, {3.4781575335436*^9, 3.4781576406532*^9}, {3.4781576880615997`*^9, 3.478157739292*^9}, { 3.478157788042*^9, 3.4781577938452*^9}, {3.4781578276036*^9, 3.4781579446191998`*^9}, {3.4781579974564*^9, 3.4781579980959997`*^9}, 3.47815804396*^9, {3.4781582919532003`*^9, 3.47815834816*^9}, { 3.4781583838694*^9, 3.4781583926366*^9}, 3.4781585049566*^9, { 3.478162407691*^9, 3.4781624349442*^9}, {3.4781624661286*^9, 3.4781624682033997`*^9}, 3.4781626733746*^9, {3.5376219474134884`*^9, 3.537621988198821*^9}, {3.537622048951296*^9, 3.537622120225373*^9}, { 3.5376222446004868`*^9, 3.537622312108348*^9}, {3.537622795498996*^9, 3.537622894561662*^9}, {3.537623044440235*^9, 3.5376230936910515`*^9}, { 3.537623228199745*^9, 3.537623231323924*^9}, {3.5376233146726913`*^9, 3.537623328831501*^9}, {3.537623482339281*^9, 3.5376234825212917`*^9}, { 3.537623521274508*^9, 3.5376235261817884`*^9}, {3.537624917437364*^9, 3.537625128320426*^9}, {3.5376251730519843`*^9, 3.5376251803164*^9}, { 3.537671339414649*^9, 3.537671356705638*^9}, {3.5376809528035088`*^9, 3.53768095299552*^9}, {3.6000029089848776`*^9, 3.6000029208252983`*^9}, { 3.6000029756593943`*^9, 3.6000030098078547`*^9}, {3.600003158522916*^9, 3.60000318349856*^9}, 3.6000032324514456`*^9, {3.6000033076123776`*^9, 3.60000340005814*^9}, {3.600003431055394*^9, 3.6000034794310794`*^9}, { 3.6000035325647726`*^9, 3.6000035614872236`*^9}, {3.6000036318433475`*^9, 3.600003647334174*^9}, 3.600003682933437*^9, 3.6000042191843786`*^9, { 3.6000042874812984`*^9, 3.600004390035879*^9}, {3.6000045146644974`*^9, 3.6000045412001443`*^9}, {3.600070569247798*^9, 3.600070571525402*^9}, { 3.600596261572476*^9, 3.6005963099949613`*^9}, {3.6005963772466793`*^9, 3.6005963778550806`*^9}, {3.6005964615336275`*^9, 3.6005964934512835`*^9}, {3.6005965263517413`*^9, 3.6005965785806327`*^9}, {3.6005966096402874`*^9, 3.600596735236108*^9}, { 3.6005968172922525`*^9, 3.6005968222686605`*^9}, {3.6005975946152177`*^9, 3.6005976378428936`*^9}, {3.6005977203514385`*^9, 3.60059777807154*^9}, { 3.600597821283616*^9, 3.600597939235423*^9}, {3.600597981823498*^9, 3.6005980087335453`*^9}, {3.600603138793149*^9, 3.600603177684017*^9}, { 3.600603426223654*^9, 3.6006034698569307`*^9}, {3.6006035079365973`*^9, 3.600603558699086*^9}, {3.600603598447956*^9, 3.6006036328304167`*^9}, { 3.600603663812071*^9, 3.600603671144084*^9}, {3.6006092873029766`*^9, 3.6006092888161793`*^9}, {3.600701212913769*^9, 3.6007012407286177`*^9}, { 3.6636736831863227`*^9, 3.663673688209532*^9}, {3.6640795480320005`*^9, 3.664079619454086*^9}, {3.6640796991846457`*^9, 3.664079717333684*^9}, { 3.6971720519443827`*^9, 3.69717222127462*^9}, {3.697172261514676*^9, 3.6971723572348104`*^9}, {3.697172562075097*^9, 3.6971725624350977`*^9}, 3.6971726063951592`*^9, {3.6972525155055447`*^9, 3.697252531465567*^9}, { 3.6972529515461555`*^9, 3.697253084716342*^9}, {3.697253246136568*^9, 3.6972532878466263`*^9}, {3.697253341536701*^9, 3.697253385806763*^9}, { 3.697254029617665*^9, 3.6972541087377753`*^9}, {3.6972541537278385`*^9, 3.697254192057892*^9}, {3.6972542390879583`*^9, 3.6972542801280155`*^9}, { 3.697254317298067*^9, 3.69725432622808*^9}, {3.6972543601881275`*^9, 3.6972544415582414`*^9}, {3.6972550776891317`*^9, 3.6972551527992373`*^9}, {3.697464845793644*^9, 3.6974649348237686`*^9}, { 3.6974649717638197`*^9, 3.697465031933904*^9}, {3.697465065083951*^9, 3.6974650864839807`*^9}, {3.697465175614105*^9, 3.6974652364441905`*^9}, 3.6974653757643857`*^9, {3.697465587824682*^9, 3.6974656235847325`*^9}, { 3.697465692394829*^9, 3.6974657187048655`*^9}, {3.69769096026633*^9, 3.697690962576333*^9}, {3.729216248879525*^9, 3.729216272491686*^9}, { 3.7292163444521027`*^9, 3.729216447228859*^9}, {3.7608597693364944`*^9, 3.7608597776627054`*^9}}, CellLabel->"In[31]:=",ExpressionUUID->"7713952d-10be-4b56-9090-67fea8947e12"] }, Closed]] }, Open ]] }, AutoGeneratedPackage->Automatic, WindowSize->{869.5, 422.5}, WindowMargins->{{-19, Automatic}, {-13.75, Automatic}}, PrivateNotebookOptions->{"VersionedStylesheet"->{"Default.nb"[8.] -> False}}, TaggingRules-><|"TryRealOnly" -> False|>, Magnification:>1.5 Inherited, FrontEndVersion->"13.1 for Microsoft Windows (64-bit) (June 16, 2022)", StyleDefinitions->"Default.nb", ExpressionUUID->"2f6610a0-22c5-4378-a20e-3960633be76e" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[567, 22, 1244, 19, 100, "Section",ExpressionUUID->"064353de-d88c-490a-aabe-7b34514beb63", InitializationCell->True], Cell[CellGroupData[{ Cell[1836, 45, 1319, 20, 81, "Subsection",ExpressionUUID->"ab002cd9-568c-424a-9781-52ea039cdee4"], Cell[3158, 67, 13789, 309, 915, "Input",ExpressionUUID->"cc72c849-5d5a-40cb-ada4-785af365f33c", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell[16984, 381, 1269, 19, 55, "Subsection",ExpressionUUID->"05a2f398-c16e-4461-a17d-36fd6f9726ee"], Cell[18256, 402, 13443, 323, 952, "Input",ExpressionUUID->"b123c695-f0a8-49c9-8b5a-d535a1a4e2ed", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell[31736, 730, 1516, 22, 81, "Subsection",ExpressionUUID->"a7bd322b-bcff-446f-9164-d14fef465862"], Cell[33255, 754, 12436, 252, 809, "Input",ExpressionUUID->"6f304dbc-7ab0-46e7-9178-4e87cfcdc12b", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell[45728, 1011, 1567, 23, 55, "Subsection",ExpressionUUID->"1d403ced-785f-4cab-bf9b-4393845ec623"], Cell[CellGroupData[{ Cell[47320, 1038, 19331, 416, 1584, "Input",ExpressionUUID->"3e6fdcf2-47a5-4d38-b6fe-44f08499ac09"], Cell[66654, 1456, 7549, 185, 463, "Output",ExpressionUUID->"f5b9213b-2bb7-49e5-963d-3f7db3da5e22"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[74264, 1648, 1093, 17, 77, "Section",ExpressionUUID->"05f91828-714a-44d3-8284-0a01a3d0257d"], Cell[CellGroupData[{ Cell[75382, 1669, 1541, 22, 81, "Subsection",ExpressionUUID->"006aeb29-bd62-422a-b8bd-b4677696f61c"], Cell[CellGroupData[{ Cell[76948, 1695, 9881, 178, 414, "Input",ExpressionUUID->"0986619f-9eaf-4026-a2ca-aa521b1c62f7"], Cell[CellGroupData[{ Cell[86854, 1877, 252, 7, 5390609, "Print",ExpressionUUID->"e18949b8-c613-4b8d-88a6-8b69fa74cd72"], Cell[87109, 1886, 745, 20, 11119328, "Print",ExpressionUUID->"b66cc749-8959-43d2-aa26-bd67ceed2817"], Cell[87857, 1908, 252, 7, 5390609, "Print",ExpressionUUID->"c3a3a8e2-ff09-47ed-9be1-9351b5099b47"], Cell[88112, 1917, 534, 15, 10637082, "Print",ExpressionUUID->"8bb8d1c5-e098-4759-b455-2a0e83238c52"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[88707, 1939, 1543, 22, 55, "Subsection",ExpressionUUID->"1d5fe2a2-f2a8-4c00-acb2-0cfe7d7827b0"], Cell[90253, 1963, 9659, 169, 385, "Input",ExpressionUUID->"bca76ba3-440a-4727-9426-831715647e88"] }, Closed]], Cell[CellGroupData[{ Cell[99949, 2137, 1571, 23, 55, "Subsection",ExpressionUUID->"65b3d199-96d1-449a-8732-3b5053c6f489"], Cell[101523, 2162, 9390, 166, 414, "Input",ExpressionUUID->"bb1a09a3-51ff-4bef-a59d-299fb02a4b72"] }, Closed]], Cell[CellGroupData[{ Cell[110950, 2333, 1626, 23, 55, "Subsection",ExpressionUUID->"47c6dfc5-81f4-4d13-b8df-fbf830b3fc7d"], Cell[112579, 2358, 9848, 177, 471, "Input",ExpressionUUID->"852413a6-712c-4a2f-b8de-cd95ebf6f13e"] }, Closed]], Cell[CellGroupData[{ Cell[122464, 2540, 1448, 21, 55, "Subsection",ExpressionUUID->"bce8176b-3520-4462-b0af-a7a3aeda299e"], Cell[123915, 2563, 9088, 166, 414, "Input",ExpressionUUID->"9032ada9-f181-4057-b774-73aae42b3638"] }, Closed]], Cell[CellGroupData[{ Cell[133040, 2734, 1501, 22, 55, "Subsection",ExpressionUUID->"7613c0b2-bbf0-4f69-b2cf-04c894d5c781"], Cell[134544, 2758, 9881, 181, 442, "Input",ExpressionUUID->"794ed346-a549-41ed-9654-ec9fccb1373a"] }, Closed]], Cell[CellGroupData[{ Cell[144462, 2944, 1543, 22, 55, "Subsection",ExpressionUUID->"5626b671-a555-4d64-8c94-8b6559ba2a7f"], Cell[146008, 2968, 9934, 172, 471, "Input",ExpressionUUID->"0356b5d1-5a75-4b1f-98d3-ffdd89b5eded"] }, Closed]], Cell[CellGroupData[{ Cell[155979, 3145, 1534, 22, 55, "Subsection",ExpressionUUID->"d1d11315-3d04-4ac8-9c35-7a72b66c5535"], Cell[157516, 3169, 10028, 179, 385, "Input",ExpressionUUID->"c0786013-67e5-419c-af0a-3c65f07f5b77"] }, Closed]], Cell[CellGroupData[{ Cell[167581, 3353, 1406, 21, 55, "Subsection",ExpressionUUID->"b2f79b14-f462-42e7-9a06-c51278085057"], Cell[168990, 3376, 9430, 164, 357, "Input",ExpressionUUID->"f0b86a41-2f52-47eb-87d6-a3a579200d84"] }, Closed]], Cell[CellGroupData[{ Cell[178457, 3545, 1448, 21, 55, "Subsection",ExpressionUUID->"985b8f42-9df1-4a85-8a2c-0ed0937539a6"], Cell[179908, 3568, 9675, 169, 357, "Input",ExpressionUUID->"abd606e0-ad5b-489b-b7b3-63f20e950cb5"] }, Closed]], Cell[CellGroupData[{ Cell[189620, 3742, 1472, 22, 55, "Subsection",ExpressionUUID->"ac29112e-37bb-4091-82d1-87b34b95efcd"], Cell[191095, 3766, 9600, 160, 328, "Input",ExpressionUUID->"c751fb3c-c274-4a37-8c7e-f6c8efd3142b"] }, Closed]], Cell[CellGroupData[{ Cell[200732, 3931, 1380, 20, 55, "Subsection",ExpressionUUID->"ad58db85-6e4e-4cf9-aea2-fe550af5a594"], Cell[202115, 3953, 8981, 165, 414, "Input",ExpressionUUID->"40e59b0f-6f51-4a96-8ca3-c5d0c7af77cd"] }, Closed]], Cell[CellGroupData[{ Cell[211133, 4123, 1374, 20, 55, "Subsection",ExpressionUUID->"6c9b4103-896a-4dc6-b865-26c876ff4c2e"], Cell[212510, 4145, 9718, 176, 442, "Input",ExpressionUUID->"f18bf471-d153-4197-9a8a-80ac235a29e3"] }, Closed]], Cell[CellGroupData[{ Cell[222265, 4326, 1400, 21, 55, "Subsection",ExpressionUUID->"4d9190d6-4237-4017-8f74-1fd24cbdab08"], Cell[223668, 4349, 9559, 172, 414, "Input",ExpressionUUID->"8d300df9-089f-4959-94d0-66831d92ae9d"] }, Closed]], Cell[CellGroupData[{ Cell[233264, 4526, 1478, 22, 55, "Subsection",ExpressionUUID->"edf4395a-fc4f-496b-b821-ce7ded231efe"], Cell[234745, 4550, 9729, 174, 385, "Input",ExpressionUUID->"fb7686e1-ed73-4549-8da0-67eaccdee858"] }, Closed]], Cell[CellGroupData[{ Cell[244511, 4729, 1353, 20, 55, "Subsection",ExpressionUUID->"54d696cb-6e14-4e2a-a400-9dd418b4e2f7"], Cell[245867, 4751, 9159, 165, 357, "Input",ExpressionUUID->"03250248-2eb6-4657-97dc-3a855f663467"] }, Closed]], Cell[CellGroupData[{ Cell[255063, 4921, 1497, 22, 55, "Subsection",ExpressionUUID->"49f66745-cab2-42f5-88b3-0675bcc5c206"], Cell[256563, 4945, 8969, 155, 357, "Input",ExpressionUUID->"d9ac3563-69a1-4b8d-9966-ba5107446602"] }, Closed]], Cell[CellGroupData[{ Cell[265569, 5105, 1455, 21, 55, "Subsection",ExpressionUUID->"4134aba9-323b-4b4b-b5d2-2f5253e5c448"], Cell[267027, 5128, 8413, 148, 357, "Input",ExpressionUUID->"7d721f21-2378-4717-936b-724118e92229"] }, Closed]], Cell[CellGroupData[{ Cell[275477, 5281, 1501, 22, 55, "Subsection",ExpressionUUID->"1cc1bc5b-b244-47a7-a86b-ad614c5cad7e"], Cell[276981, 5305, 10299, 185, 471, "Input",ExpressionUUID->"7713952d-10be-4b56-9090-67fea8947e12"] }, Closed]] }, Open ]] } ] *)