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The\ required\ function\ for\ B\ is\ R^{\(-2\)} P_ 1^1 \((cos \((beta)\))\) sin \((theta)\), but\ this\ form\ would\ give\ m > n\ for\ A, E, so\ we\ need\ to\ use\ the\ log \((R + z)\)\ form\ of\ equations\ \((22.37)\) . To\ get\ the\ appropriate\ non - axisymmetric\ function, take\ log \((R + z)\)\ and\ differentiate\ with\ respect\ to\ y, obtaining\ \ *) \)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\[Phi] = C[1]*Sin[\[Beta]]* Sin[\[Theta]]/\((R*\((1 + Cos[\[Beta]])\))\)\), "\[IndentingNewLine]", \(\[Omega] = C[2]*Sin[\[Beta]]*Sin[\[Theta]]/R^2\), "\[IndentingNewLine]", \(\[Psi] = C[3]*Sin[\[Beta]]* Cos[\[Theta]]/\((R*\((1 + Cos[\[Beta]])\))\)\)}], "Input"], Cell[BoxData[ \(\(C[1]\ Sin[\[Beta]]\ Sin[\[Theta]]\)\/\(R\ \((1 + \ Cos[\[Beta]])\)\)\)], "Output"], Cell[BoxData[ \(\(C[2]\ Sin[\[Beta]]\ Sin[\[Theta]]\)\/R\^2\)], "Output"], Cell[BoxData[ \(\(C[3]\ Cos[\[Theta]]\ Sin[\[Beta]]\)\/\(R\ \((1 + \ Cos[\[Beta]])\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\[Sigma]RR = Simplify[D[\[Phi], R, R] + R*Cos[\[Beta]]*D[\[Omega], R, R] - 2*\((1 - \[Nu])\)*Cos[\[Beta]]*D[\[Omega], R] + \((2*\[Nu]/R)\)* Sin[\[Beta]]*D[\[Omega], \[Beta]] + \((2/R)\)* D[\[Psi], R, \[Theta]] - \((2/R^2)\)* D[\[Psi], \[Theta]]]\), "\[IndentingNewLine]", \(\[Sigma]\[Theta]\[Theta] = Simplify[\((1/R)\)*D[\[Phi], R] + \((Cot[\[Beta]]/R^2)\)* D[\[Phi], \[Beta]] + \((1/\((R*Sin[\[Beta]])\)^2)\)* D[\[Phi], \[Theta], \[Theta]] + \((1 - 2*\[Nu])\)*Cos[\[Beta]]* D[\[Omega], R] + \((2*\[Nu]/R)\)*Sin[\[Beta]]* D[\[Omega], \[Beta]] + \((Cos[\[Beta]]^2/\((R*Sin[\[Beta]])\))\)* D[\[Omega], \[Beta]] + \((Cot[\[Beta]]/\((R*Sin[\[Beta]])\))\)* D[\[Omega], \[Theta], \[Theta]] + \((2/\((R*Sin[\[Beta]])\)^2)\)* D[\[Psi], \[Theta]] - \((2/R)\)* D[\[Psi], R, \[Theta]] - \((2*Cot[\[Beta]]/R^2)\)* D[\[Psi], \[Beta], \[Theta]]]\), "\[IndentingNewLine]", \(\[Sigma]\[Beta]\[Beta] = Simplify[\((1/R)\)*D[\[Phi], R] + \((1/R^2)\)* D[\[Phi], \[Beta], \[Beta]] + \((Cos[\[Beta]]/R)\)* D[\[Omega], \[Beta], \[Beta]] + \((1 - 2*\[Nu])\)*Cos[\[Beta]]* D[\[Omega], R] + \((2*\((1 - \[Nu])\)*Sin[\[Beta]]/R)\)* D[\[Omega], \[Beta]] + \((2*Cot[\[Beta]]/R^2)\)* D[\[Psi], \[Theta], \[Beta]] - 2*\((Cot[\[Beta]]/R)\)^2* D[\[Psi], \[Theta]]]\), "\[IndentingNewLine]", \(\[Sigma]\[Theta]\[Beta] = \((1/\((R^2*Sin[\[Beta]])\))\)* D[\[Phi], \[Theta], \[Beta]] - \((Cot[\[Beta]]/\((R^2* Sin[\[Beta]])\))\)* D[\[Phi], \[Theta]] + \((Cot[\[Beta]]/R)\)* D[\[Omega], \[Beta], \[Theta]] + \((2*\((1 - \[Nu])\)/R)\)* D[\[Omega], \[Theta]] - \((1/\((R*Sin[\[Beta]]^2)\))\)* D[\[Omega], \[Theta]] + \((Cot[\[Beta]]/\((R^2*Sin[\[Beta]])\))\)* D[\[Psi], \[Theta], \[Theta]] - \((Sin[\[Beta]]/R)\)* D[\[Psi], R, \[Beta]] - \((Cos[\[Beta]]/R^2)\)* D[\[Psi], \[Beta], \[Beta]] + \((1/\((R^2*Sin[\[Beta]])\))\)* D[\[Psi], \[Beta]]\), "\[IndentingNewLine]", \(\[Sigma]\[Beta]R = \((1/R)\)*D[\[Phi], \[Beta], R] - \((1/R^2)\)* D[\[Phi], \[Beta]] + \((1 - 2*\[Nu])\)*Sin[\[Beta]]* D[\[Omega], R] + Cos[\[Beta]]* D[\[Omega], \[Beta], R] - \((2*\((1 - \[Nu])\)*Cos[\[Beta]]/R)\)* D[\[Omega], \[Beta]] + \((1/R^2)\)* D[\[Psi], \[Theta], \[Beta]] + \((Cot[\[Beta]]/R)\)* D[\[Psi], \[Theta], R] - \((2*Cot[\[Beta]]/R^2)\)* D[\[Psi], \[Theta]]\), "\[IndentingNewLine]", \(\[Sigma]R\[Theta] = \((1/\((R*Sin[\[Beta]])\))\)* D[\[Phi], R, \[Theta]] - \((1/\((R^2*Sin[\[Beta]])\))\)* D[\[Phi], \[Theta]] + Cot[\[Beta]]* D[\[Omega], R, \[Theta]] - \((2*\((1 - \[Nu])\)*Cot[\[Beta]]/R)\)* D[\[Omega], \[Theta]] + \((1/\((R^2*Sin[\[Beta]])\))\)* D[\[Psi], \[Theta], \[Theta]] - Sin[\[Beta]]*D[\[Psi], R, R] - \((Cos[\[Beta]]/R)\)* D[\[Psi], R, \[Beta]] + \((2*Cos[\[Beta]]/R^2)\)* D[\[Psi], \[Beta]] + \((Sin[\[Beta]]/R)\)* D[\[Psi], R]\), "\[IndentingNewLine]", \(\)}], "Input"], Cell[BoxData[ \(\(\(1\/\(R\^3\ \((1 + Cos[\[Beta]])\)\)\)\((\((2\ C[1] + 5\ C[2] - \[Nu]\ C[2] + 4\ C[3] - 2\ \((\(-5\) + \[Nu])\)\ C[ 2]\ Cos[\[Beta]] - \((\(-5\) + \[Nu])\)\ C[2]\ Cos[ 2\ \[Beta]])\)\ Sin[\[Beta]]\ Sin[\[Theta]])\)\)\)], "Output"], Cell[BoxData[ \(\(-\(\((\((8\ C[1] + 12\ C[2] - 24\ \[Nu]\ C[2] + 16\ C[3] + \((4\ C[1] + \((21 - 42\ \[Nu])\)\ C[2] + 8\ C[3])\)\ Cos[\[Beta]] - 12\ \((\(-1\) + 2\ \[Nu])\)\ C[2]\ Cos[2\ \[Beta]] + 3\ C[2]\ Cos[3\ \[Beta]] - 6\ \[Nu]\ C[2]\ Cos[ 3\ \[Beta]])\)\ Sin[\[Beta]]\ Sin[\[Theta]])\)/\((4\ R\^3\ \ \((1 + Cos[\[Beta]])\)\^2)\)\)\)\)], "Output"], Cell[BoxData[ \(\(\(1\/\(R\^3\ \((1 + Cos[\[Beta]])\)\^3\)\)\((2\ Cos[\[Beta]\/2]\^3\ \ Cos[\[Beta]]\ \((\(-2\)\ C[1] - 3\ C[2] + 6\ \[Nu]\ C[2] - 4\ C[3] + 4\ \((\(-1\) + 2\ \[Nu])\)\ C[ 2]\ Cos[\[Beta]] + \((\(-1\) + 2\ \[Nu])\)\ C[2]\ Cos[ 2\ \[Beta]])\)\ Sin[\[Beta]\/2]\ Sin[\[Theta]])\)\)\)], \ "Output"], Cell[BoxData[ \(General::"spell" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(\[Sigma]\[Theta]\[Beta]\ \)\" is similar to existing symbols \!\({\[Sigma]\[Beta]\[Beta], \[Sigma]\ \[Theta]\[Theta]}\)."\)], "Message"], Cell[BoxData[ \(\(C[2]\ Cos[\[Beta]]\ Cos[\[Theta]]\ Cot[\[Beta]]\)\/R\^3 - \(C[1]\ \ Cos[\[Theta]]\ Cot[\[Beta]]\)\/\(R\^3\ \((1 + Cos[\[Beta]])\)\) - \(C[3]\ \ Cos[\[Theta]]\ Cot[\[Beta]]\)\/\(R\^3\ \((1 + Cos[\[Beta]])\)\) - \(C[2]\ \ Cos[\[Theta]]\ Csc[\[Beta]]\)\/R\^3 + \(2\ \((1 - \[Nu])\)\ C[2]\ \ Cos[\[Theta]]\ Sin[\[Beta]]\)\/R\^3 + \(Csc[\[Beta]]\ \((\(C[1]\ Cos[\[Beta]]\ \ Cos[\[Theta]]\)\/\(R\ \((1 + Cos[\[Beta]])\)\) + \(C[1]\ Cos[\[Theta]]\ \ Sin[\[Beta]]\^2\)\/\(R\ \((1 + Cos[\[Beta]])\)\^2\))\)\)\/R\^2 - \ \(Sin[\[Beta]]\ \((\(-\(\(C[3]\ Cos[\[Beta]]\ Cos[\[Theta]]\)\/\(R\^2\ \((1 + \ Cos[\[Beta]])\)\)\)\) - \(C[3]\ Cos[\[Theta]]\ Sin[\[Beta]]\^2\)\/\(R\^2\ \ \((1 + Cos[\[Beta]])\)\^2\))\)\)\/R + \(Csc[\[Beta]]\ \((\(C[3]\ Cos[\[Beta]]\ \ Cos[\[Theta]]\)\/\(R\ \((1 + Cos[\[Beta]])\)\) + \(C[3]\ Cos[\[Theta]]\ \ Sin[\[Beta]]\^2\)\/\(R\ \((1 + Cos[\[Beta]])\)\^2\))\)\)\/R\^2 - \ \(\(1\/R\^2\)\((Cos[\[Beta]]\ \((\(3\ C[3]\ Cos[\[Beta]]\ Cos[\[Theta]]\ Sin[\ \[Beta]]\)\/\(R\ \((1 + Cos[\[Beta]])\)\^2\) - \(C[3]\ Cos[\[Theta]]\ Sin[\ \[Beta]]\)\/\(R\ \((1 + Cos[\[Beta]])\)\) + \(2\ C[3]\ Cos[\[Theta]]\ Sin[\ \[Beta]]\^3\)\/\(R\ \((1 + Cos[\[Beta]])\)\^3\))\))\)\)\)], "Output"], Cell[BoxData[ \(\(-\(\(2\ C[ 2]\ Cos[\[Beta]]\^2\ Sin[\[Theta]]\)\/R\^3\)\) - \(2\ \((1 - \ \[Nu])\)\ C[2]\ Cos[\[Beta]]\^2\ Sin[\[Theta]]\)\/R\^3 + \(3\ C[3]\ Cos[\ \[Beta]]\ Sin[\[Theta]]\)\/\(R\^3\ \((1 + Cos[\[Beta]])\)\) - \(2\ \((1 - 2\ \ \[Nu])\)\ C[2]\ Sin[\[Beta]]\^2\ Sin[\[Theta]]\)\/R\^3 + \(\(-\(\(C[1]\ Cos[\ \[Beta]]\ Sin[\[Theta]]\)\/\(R\^2\ \((1 + Cos[\[Beta]])\)\)\)\) - \(C[1]\ \ Sin[\[Beta]]\^2\ Sin[\[Theta]]\)\/\(R\^2\ \((1 + Cos[\[Beta]])\)\^2\)\)\/R - \ \(\(C[1]\ Cos[\[Beta]]\ Sin[\[Theta]]\)\/\(R\ \((1 + Cos[\[Beta]])\)\) + \ \(C[1]\ Sin[\[Beta]]\^2\ Sin[\[Theta]]\)\/\(R\ \((1 + \ Cos[\[Beta]])\)\^2\)\)\/R\^2 + \(\(-\(\(C[3]\ Cos[\[Beta]]\ Sin[\[Theta]]\)\/\ \(R\ \((1 + Cos[\[Beta]])\)\)\)\) - \(C[3]\ Sin[\[Beta]]\^2\ \ Sin[\[Theta]]\)\/\(R\ \((1 + Cos[\[Beta]])\)\^2\)\)\/R\^2\)], "Output"], Cell[BoxData[ \(\(-\(\(2\ C[ 2]\ Cos[\[Beta]]\ Cos[\[Theta]]\)\/R\^3\)\) - \(2\ \((1 - \ \[Nu])\)\ C[2]\ Cos[\[Beta]]\ Cos[\[Theta]]\)\/R\^3 - \(2\ C[1]\ \ Cos[\[Theta]]\)\/\(R\^3\ \((1 + Cos[\[Beta]])\)\) - \(C[3]\ Cos[\[Theta]]\)\/\ \(R\^3\ \((1 + Cos[\[Beta]])\)\) - \(3\ C[3]\ Cos[\[Theta]]\ \ Sin[\[Beta]]\^2\)\/\(R\^3\ \((1 + Cos[\[Beta]])\)\) - \(Cos[\[Beta]]\ \ \((\(-\(\(C[3]\ Cos[\[Beta]]\ Cos[\[Theta]]\)\/\(R\^2\ \((1 + Cos[\[Beta]])\)\ \)\)\) - \(C[3]\ Cos[\[Theta]]\ Sin[\[Beta]]\^2\)\/\(R\^2\ \((1 + \ Cos[\[Beta]])\)\^2\))\)\)\/R + \(2\ Cos[\[Beta]]\ \((\(C[3]\ Cos[\[Beta]]\ \ Cos[\[Theta]]\)\/\(R\ \((1 + Cos[\[Beta]])\)\) + \(C[3]\ Cos[\[Theta]]\ Sin[\ \[Beta]]\^2\)\/\(R\ \((1 + Cos[\[Beta]])\)\^2\))\)\)\/R\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(t1 = \[Sigma]\[Beta]R /. \[Beta] \[Rule] \[Beta]0\), "\ \[IndentingNewLine]", \(t2 = \[Sigma]\[Beta]\[Beta] /. \[Beta] \[Rule] \[Beta]0\), "\ \[IndentingNewLine]", \(t3 = Expand[\[Sigma]\[Theta]\[Beta] /. \[Beta] \[Rule] \[Beta]0]\)}], "Input"], Cell[BoxData[ \(\(-\(\(2\ C[ 2]\ Cos[\[Beta]0]\^2\ Sin[\[Theta]]\)\/R\^3\)\) - \(2\ \((1 - \ \[Nu])\)\ C[2]\ Cos[\[Beta]0]\^2\ Sin[\[Theta]]\)\/R\^3 + \(3\ C[3]\ Cos[\ \[Beta]0]\ Sin[\[Theta]]\)\/\(R\^3\ \((1 + Cos[\[Beta]0])\)\) - \(2\ \((1 - 2\ \ \[Nu])\)\ C[2]\ Sin[\[Beta]0]\^2\ Sin[\[Theta]]\)\/R\^3 + \(\(-\(\(C[1]\ \ Cos[\[Beta]0]\ Sin[\[Theta]]\)\/\(R\^2\ \((1 + Cos[\[Beta]0])\)\)\)\) - \ \(C[1]\ Sin[\[Beta]0]\^2\ Sin[\[Theta]]\)\/\(R\^2\ \((1 + Cos[\[Beta]0])\)\^2\ \)\)\/R - \(\(C[1]\ Cos[\[Beta]0]\ Sin[\[Theta]]\)\/\(R\ \((1 + \ Cos[\[Beta]0])\)\) + \(C[1]\ Sin[\[Beta]0]\^2\ Sin[\[Theta]]\)\/\(R\ \((1 + \ Cos[\[Beta]0])\)\^2\)\)\/R\^2 + \(\(-\(\(C[3]\ Cos[\[Beta]0]\ Sin[\[Theta]]\)\ \/\(R\ \((1 + Cos[\[Beta]0])\)\)\)\) - \(C[3]\ Sin[\[Beta]0]\^2\ \ Sin[\[Theta]]\)\/\(R\ \((1 + Cos[\[Beta]0])\)\^2\)\)\/R\^2\)], "Output"], Cell[BoxData[ \(\((2\ Cos[\[Beta]0\/2]\^3\ Cos[\[Beta]0]\ \((\(-2\)\ C[1] - 3\ C[2] + 6\ \[Nu]\ C[2] - 4\ C[3] + 4\ \((\(-1\) + 2\ \[Nu])\)\ C[ 2]\ Cos[\[Beta]0] + \((\(-1\) + 2\ \[Nu])\)\ C[2]\ Cos[ 2\ \[Beta]0])\)\ Sin[\[Beta]0\/2]\ Sin[\[Theta]])\)/\((R\^3\ \ \((1 + Cos[\[Beta]0])\)\^3)\)\)], "Output"], Cell[BoxData[ \(\(C[2]\ Cos[\[Beta]0]\ Cos[\[Theta]]\ Cot[\[Beta]0]\)\/R\^3 - \(C[2]\ \ Cos[\[Theta]]\ Csc[\[Beta]0]\)\/R\^3 + \(2\ C[2]\ Cos[\[Theta]]\ \ Sin[\[Beta]0]\)\/R\^3 - \(2\ \[Nu]\ C[2]\ Cos[\[Theta]]\ \ Sin[\[Beta]0]\)\/R\^3 + \(C[1]\ Cos[\[Theta]]\ Sin[\[Beta]0]\)\/\(R\^3\ \((1 \ + Cos[\[Beta]0])\)\^2\) + \(C[3]\ Cos[\[Theta]]\ Sin[\[Beta]0]\)\/\(R\^3\ \ \((1 + Cos[\[Beta]0])\)\^2\) - \(3\ C[3]\ Cos[\[Beta]0]\^2\ Cos[\[Theta]]\ \ Sin[\[Beta]0]\)\/\(R\^3\ \((1 + Cos[\[Beta]0])\)\^2\) + \(2\ C[3]\ \ Cos[\[Beta]0]\ Cos[\[Theta]]\ Sin[\[Beta]0]\)\/\(R\^3\ \((1 + \ Cos[\[Beta]0])\)\) - \(2\ C[3]\ Cos[\[Beta]0]\ Cos[\[Theta]]\ \ Sin[\[Beta]0]\^3\)\/\(R\^3\ \((1 + Cos[\[Beta]0])\)\^3\) + \(C[3]\ Cos[\ \[Theta]]\ Sin[\[Beta]0]\^3\)\/\(R\^3\ \((1 + Cos[\[Beta]0])\)\^2\)\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(c1 = Simplify[Expand[t1/Sin[\[Theta]]]]\), "\[IndentingNewLine]", \(c2 = Simplify[t2/Sin[\[Theta]]]\), "\[IndentingNewLine]", \(c3 = Simplify[t3/Cos[\[Theta]]]\)}], "Input"], Cell[BoxData[ \(\(-\(\((Cos[\[Beta]0\/2]\^2\ \((4\ C[1] + 6\ C[2] - 6\ \[Nu]\ C[2] + 2\ C[3] + \((\((7 - 5\ \[Nu])\)\ C[2] - 6\ C[3])\)\ Cos[\[Beta]0] + 2\ \((1 + \[Nu])\)\ C[2]\ Cos[2\ \[Beta]0] + C[2]\ Cos[3\ \[Beta]0] + \[Nu]\ C[2]\ Cos[ 3\ \[Beta]0])\))\)/\((R\^3\ \((1 + Cos[\[Beta]0])\)\^2)\)\ \)\)\)], "Output"], Cell[BoxData[ \(\((2\ Cos[\[Beta]0\/2]\^3\ Cos[\[Beta]0]\ \((\(-2\)\ C[1] - 3\ C[2] + 6\ \[Nu]\ C[2] - 4\ C[3] + 4\ \((\(-1\) + 2\ \[Nu])\)\ C[ 2]\ Cos[\[Beta]0] + \((\(-1\) + 2\ \[Nu])\)\ C[2]\ Cos[ 2\ \[Beta]0])\)\ Sin[\[Beta]0\/2])\)/\((R\^3\ \((1 + Cos[\ \[Beta]0])\)\^3)\)\)], "Output"], Cell[BoxData[ \(\((2\ Cos[\[Beta]0\/2]\^3\ \((2\ C[1] + 3\ C[2] - 6\ \[Nu]\ C[2] + 4\ C[3] - 4\ \((\(-1\) + 2\ \[Nu])\)\ C[ 2]\ Cos[\[Beta]0] + \((1 - 2\ \[Nu])\)\ C[2]\ Cos[ 2\ \[Beta]0])\)\ Sin[\[Beta]0\/2])\)/\((R\^3\ \((1 + Cos[\ \[Beta]0])\)\^3)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(t = Simplify[\[Sigma]\[Beta]R*Sin[\[Theta]] + \[Sigma]R\[Theta]* Cos[\[Theta]]*Cos[\[Beta]]]\), "\[IndentingNewLine]", \(c4 = Integrate[t, \ {\[Theta], \ 0, 2*\[Pi]}]\), "\[IndentingNewLine]", \(c5 = Simplify[Integrate[ c4*R^3*Sin[\[Beta]], \ {\[Beta], \ 0, \[Beta]0}]]\)}], "Input"], Cell[BoxData[ \(\((Cos[\[Beta]\/2]\^2\ \((\(-16\)\ C[1] - 40\ C[2] + 32\ \[Nu]\ C[2] + 4\ C[3] - 2\ \((8\ C[1] + \((26 - 16\ \[Nu])\)\ C[2] - 5\ C[3])\)\ Cos[\[Beta]] + 12\ \((\(-2\)\ C[2] + C[3])\)\ Cos[2\ \[Beta]] - 12\ C[2]\ Cos[3\ \[Beta]] + 6\ C[3]\ Cos[3\ \[Beta]] - 8\ C[1]\ Cos[\[Beta] - 2\ \[Theta]] + 2\ C[2]\ Cos[\[Beta] - 2\ \[Theta]] - 4\ \[Nu]\ C[2]\ Cos[\[Beta] - 2\ \[Theta]] - 19\ C[3]\ Cos[\[Beta] - 2\ \[Theta]] - 2\ C[2]\ Cos[3\ \[Beta] - 2\ \[Theta]] + 4\ \[Nu]\ C[2]\ Cos[3\ \[Beta] - 2\ \[Theta]] + 3\ C[3]\ Cos[3\ \[Beta] - 2\ \[Theta]] - 4\ C[2]\ Cos[2\ \((\[Beta] - \[Theta])\)] + 8\ \[Nu]\ C[2]\ Cos[2\ \((\[Beta] - \[Theta])\)] + 6\ C[3]\ Cos[2\ \((\[Beta] - \[Theta])\)] + 16\ C[1]\ Cos[2\ \[Theta]] + 8\ C[2]\ Cos[2\ \[Theta]] - 16\ \[Nu]\ C[2]\ Cos[2\ \[Theta]] + 20\ C[3]\ Cos[2\ \[Theta]] - 4\ C[2]\ Cos[2\ \((\[Beta] + \[Theta])\)] + 8\ \[Nu]\ C[2]\ Cos[2\ \((\[Beta] + \[Theta])\)] + 6\ C[3]\ Cos[2\ \((\[Beta] + \[Theta])\)] - 8\ C[1]\ Cos[\[Beta] + 2\ \[Theta]] + 2\ C[2]\ Cos[\[Beta] + 2\ \[Theta]] - 4\ \[Nu]\ C[2]\ Cos[\[Beta] + 2\ \[Theta]] - 19\ C[3]\ Cos[\[Beta] + 2\ \[Theta]] - 2\ C[2]\ Cos[3\ \[Beta] + 2\ \[Theta]] + 4\ \[Nu]\ C[2]\ Cos[3\ \[Beta] + 2\ \[Theta]] + 3\ C[3]\ Cos[ 3\ \[Beta] + 2\ \[Theta]])\))\)/\((8\ R\^3\ \((1 + Cos[\[Beta]])\)\^2)\ \)\)], "Output"], Cell[BoxData[ \(\(-\(\(1\/\(R\^3\ \((1 + Cos[\[Beta]])\)\^2\)\)\((2\ \[Pi]\ Cos[\[Beta]\ \/2]\^4\ \((4\ C[1] + 10\ C[2] - 8\ \[Nu]\ C[2] - C[3] + 6\ C[2]\ Cos[2\ \[Beta]] - 3\ C[3]\ Cos[2\ \[Beta]])\))\)\)\)\)], "Output"], Cell[BoxData[ \(\(-\[Pi]\)\ \((4\ C[1] + 10\ C[2] - 8\ \[Nu]\ C[2] - C[3] + \((4\ C[2] - 2\ C[3])\)\ Cos[\[Beta]0] + \((2\ C[2] - C[3])\)\ Cos[2\ \[Beta]0])\)\ Sin[\[Beta]0\/2]\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Solve[{c1 \[Equal] 0, c2 \[Equal] 0, c3 \[Equal] 0, c5 \[Equal] M}, {C[1], C[2], C[3]}]]\)], "Input"], Cell[BoxData[ \({{C[ 1] \[Rule] \(-\(\((M\ \((\(-13\) + 14\ \[Nu] + \((\(-4\) + 8\ \[Nu])\)\ Cos[\[Beta]0] + \((\(-7\) + 2\ \[Nu])\)\ Cos[ 2\ \[Beta]0])\)\ Csc[\[Beta]0\/2]\^4)\)/\((4\ \[Pi]\ \ \((14 + 8\ \[Nu] + 19\ \((1 + \[Nu])\)\ Cos[\[Beta]0] + \((2 + 8\ \[Nu])\)\ Cos[2\ \[Beta]0] + Cos[3\ \[Beta]0] + \[Nu]\ Cos[3\ \[Beta]0])\))\)\)\), C[2] \[Rule] \(-\(\((3\ M\ Csc[\[Beta]0\/2]\^4)\)/\((2\ \[Pi]\ \((14 \ + 8\ \[Nu] + 19\ \((1 + \[Nu])\)\ Cos[\[Beta]0] + \((2 + 8\ \[Nu])\)\ Cos[ 2\ \[Beta]0] + Cos[3\ \[Beta]0] + \[Nu]\ Cos[3\ \[Beta]0])\))\)\)\), C[3] \[Rule] \(-\(\((M\ Cos[\[Beta]0]\ \((\(-1\) + 2\ \[Nu] + \((1 + \[Nu])\)\ Cos[\[Beta]0])\)\ Csc[\ \[Beta]0\/2]\^4)\)/\((\[Pi]\ \((14 + 8\ \[Nu] + 19\ \((1 + \[Nu])\)\ Cos[\[Beta]0] + \((2 + 8\ \[Nu])\)\ Cos[2\ \[Beta]0] + Cos[3\ \[Beta]0] + \[Nu]\ Cos[ 3\ \[Beta]0])\))\)\)\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Sigma]RR = Simplify[\[Sigma]RR /. %20]\)], "Input"], Cell[BoxData[ \({\((3\ M\ \((\(-2\) - 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4\ Cos[\[Beta]0] - Cos[2\ \[Beta]0])\)\ Csc[\[Beta]0\/2]\^4\ \((Sin[\[Beta]\/2] \ - Sin[\(3\ \[Beta]\)\/2])\)\ Sin[\[Theta]])\)/\((2\ \[Pi]\ R\^3\ \((1 + Cos[\ \[Beta]])\)\^3\ \((14 + 8\ \[Nu] + 19\ \((1 + \[Nu])\)\ Cos[\[Beta]0] + \((2 + 8\ \[Nu])\)\ Cos[ 2\ \[Beta]0] + Cos[3\ \[Beta]0] + \[Nu]\ Cos[3\ \[Beta]0])\))\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Sigma]\[Theta]\[Beta] = Simplify[\[Sigma]\[Theta]\[Beta] /. %20]\)], "Input"], Cell[BoxData[ \({\((3\ M\ \((\(-1\) + 2\ \[Nu])\)\ Cos[\[Beta]\/2]\^3\ \((4\ Cos[\[Beta]] + Cos[2\ \[Beta]] - 4\ Cos[\[Beta]0] - Cos[2\ \[Beta]0])\)\ Cos[\[Theta]]\ Csc[\[Beta]0\/2]\^4\ Sin[\ \[Beta]\/2])\)/\((\[Pi]\ R\^3\ \((1 + Cos[\[Beta]])\)\^3\ \((14 + 8\ \[Nu] + 19\ \((1 + \[Nu])\)\ Cos[\[Beta]0] + \((2 + 8\ \[Nu])\)\ Cos[ 2\ \[Beta]0] + Cos[3\ \[Beta]0] + \[Nu]\ Cos[3\ \[Beta]0])\))\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Sigma]\[Beta]R = Simplify[\[Sigma]\[Beta]R /. %20]\)], "Input"], Cell[BoxData[ \({\((3\ M\ Cos[\[Beta]\/2]\^2\ \((Cos[\[Beta]] - 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