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               0, $CellContext`H$$ - $CellContext`tf1$$/
               2 - $CellContext`tf2$$/2}] + 
            Integrate[$CellContext`y4[$CellContext`s4]^2 \
$CellContext`\[Delta]4, {$CellContext`s4, 0, $CellContext`B2$$/2}] + 
            Integrate[$CellContext`y5[$CellContext`s5]^2 \
$CellContext`\[Delta]5, {$CellContext`s5, 0, $CellContext`B2$$/
               2}]]]; $CellContext`Jy = Simplify[
           N[
           Integrate[$CellContext`x1[$CellContext`s1]^2 \
$CellContext`\[Delta]1, {$CellContext`s1, 0, $CellContext`B1$$/2}] + 
            Integrate[$CellContext`x2[$CellContext`s2]^2 \
$CellContext`\[Delta]2, {$CellContext`s2, 0, $CellContext`B1$$/2}] + 
            Integrate[$CellContext`x3[$CellContext`s3]^2 \
$CellContext`\[Delta]3, {$CellContext`s3, 
               0, $CellContext`H$$ - $CellContext`tf1$$/
               2 - $CellContext`tf2$$/2}] + 
            Integrate[$CellContext`x4[$CellContext`s4]^2 \
$CellContext`\[Delta]4, {$CellContext`s4, 0, $CellContext`B2$$/2}] + 
            Integrate[$CellContext`x5[$CellContext`s5]^2 \
$CellContext`\[Delta]5, {$CellContext`s5, 0, $CellContext`B2$$/
               2}]]]; $CellContext`Jt = 
         N[$CellContext`\[Delta]1^3 ($CellContext`B1$$ (2^(-1)/
              3)) + $CellContext`\[Delta]2^3 ($CellContext`B1$$ (2^(-1)/
              3)) + $CellContext`\[Delta]3^3 (($CellContext`H$$ - \
$CellContext`tf1$$/2 - $CellContext`tf2$$/2)/
             3) + $CellContext`\[Delta]4^3 ($CellContext`B2$$ (2^(-1)/
              3)) + $CellContext`\[Delta]5^3 ($CellContext`B2$$ (2^(-1)/
              3))]; $CellContext`d = 
         Abs[$CellContext`xG]; $CellContext`h1 = ($CellContext`H$$ - \
$CellContext`tf1$$)/
           2 - $CellContext`yG; $CellContext`h2 = ($CellContext`H$$ - \
$CellContext`tf1$$)/
           2 - $CellContext`yG; $CellContext`h3 = $CellContext`d; \
$CellContext`h4 = ($CellContext`H$$ - $CellContext`tf2$$)/
           2 + $CellContext`yG; $CellContext`h5 = -(($CellContext`H$$ - \
$CellContext`tf2$$)/2 + $CellContext`yG); 
        Unset[$CellContext`c]; $CellContext`\[Omega]1[
           Pattern[$CellContext`s1, 
            Blank[]]] = $CellContext`c - 
          Integrate[$CellContext`h1, $CellContext`s1]; $CellContext`\[Omega]2[
          
           Pattern[$CellContext`s2$, 
            Blank[]]] = $CellContext`\[Omega]1[$CellContext`B1$$/2] - 
          Integrate[$CellContext`h2, $CellContext`s2$]; $CellContext`\[Omega]3[
           Pattern[$CellContext`s3$, 
            Blank[]]] = $CellContext`\[Omega]1[$CellContext`B1$$/2] - 
          Integrate[$CellContext`h3, $CellContext`s3$]; $CellContext`\[Omega]4[
           Pattern[$CellContext`s4$, 
            
            Blank[]]] = $CellContext`\[Omega]3[$CellContext`H$$ - \
$CellContext`tf1$$/2 - $CellContext`tf2$$/2] - 
          Integrate[$CellContext`h4, $CellContext`s4$]; $CellContext`\[Omega]5[
           Pattern[$CellContext`s5$, 
            
            Blank[]]] = $CellContext`\[Omega]3[$CellContext`H$$ - \
$CellContext`tf1$$/2 - $CellContext`tf2$$/2] - 
          Integrate[$CellContext`h5, $CellContext`s5$]; 
        Unset[$CellContext`c]; $CellContext`c = First[
           ReplaceAll[$CellContext`c, 
            Solve[Integrate[
                $CellContext`\[Omega]1[$CellContext`s1], {$CellContext`s1, 
                 0, $CellContext`B1$$/2}] + Integrate[
                $CellContext`\[Omega]2[$CellContext`s2], {$CellContext`s2, 
                 0, $CellContext`B1$$/2}] + Integrate[
                $CellContext`\[Omega]3[$CellContext`s3], {$CellContext`s3, 
                 0, $CellContext`H$$ - $CellContext`tf1$$/
                 2 - $CellContext`tf2$$/2}] + Integrate[
                $CellContext`\[Omega]4[$CellContext`s4], {$CellContext`s4, 
                 0, $CellContext`B2$$/2}] + Integrate[
                $CellContext`\[Omega]5[$CellContext`s5], {$CellContext`s5, 
                 0, $CellContext`B2$$/2}] == 
             0, $CellContext`c]]]; $CellContext`check = Chop[Integrate[
             $CellContext`\[Omega]1[$CellContext`s1], {$CellContext`s1, 
              0, $CellContext`B1$$/2}] + Integrate[
             $CellContext`\[Omega]2[$CellContext`s2], {$CellContext`s2, 
              0, $CellContext`B1$$/2}] + Integrate[
             $CellContext`\[Omega]3[$CellContext`s3], {$CellContext`s3, 
              0, $CellContext`H$$ - $CellContext`tf1$$/2 - $CellContext`tf2$$/
              2}] + Integrate[
             $CellContext`\[Omega]4[$CellContext`s4], {$CellContext`s4, 
              0, $CellContext`B2$$/2}] + Integrate[
             $CellContext`\[Omega]5[$CellContext`s5], {$CellContext`s5, 
              0, $CellContext`B2$$/2}]]; $CellContext`xC = 
         Chop[((-1)/$CellContext`Jx) (
            Integrate[($CellContext`\[Omega]1[$CellContext`s1] $CellContext`\
\[Delta]1) $CellContext`y1[$CellContext`s1], {$CellContext`s1, 
               0, $CellContext`B1$$/2}] + 
            Integrate[($CellContext`\[Omega]2[$CellContext`s2] $CellContext`\
\[Delta]2) $CellContext`y2[$CellContext`s2], {$CellContext`s2, 
               0, $CellContext`B1$$/2}] + 
            Integrate[($CellContext`\[Omega]3[$CellContext`s3] $CellContext`\
\[Delta]3) $CellContext`y3[$CellContext`s3], {$CellContext`s3, 
               0, $CellContext`H$$ - $CellContext`tf1$$/
               2 - $CellContext`tf2$$/2}] + 
            Integrate[($CellContext`\[Omega]4[$CellContext`s4] $CellContext`\
\[Delta]4) $CellContext`y4[$CellContext`s4], {$CellContext`s4, 
               0, $CellContext`B2$$/2}] + 
            Integrate[($CellContext`\[Omega]5[$CellContext`s5] $CellContext`\
\[Delta]5) $CellContext`y5[$CellContext`s5], {$CellContext`s5, 
               0, $CellContext`B2$$/2}])]; $CellContext`yC = 
         Chop[(1/$CellContext`Jy) (
            Integrate[($CellContext`\[Omega]1[$CellContext`s1] $CellContext`\
\[Delta]1) $CellContext`x1[$CellContext`s1], {$CellContext`s1, 
               0, $CellContext`B1$$/2}] + 
            Integrate[($CellContext`\[Omega]2[$CellContext`s2] $CellContext`\
\[Delta]2) $CellContext`x2[$CellContext`s2], {$CellContext`s2, 
               0, $CellContext`B1$$/2}] + 
            Integrate[($CellContext`\[Omega]3[$CellContext`s3] $CellContext`\
\[Delta]3) $CellContext`x3[$CellContext`s3], {$CellContext`s3, 
               0, $CellContext`H$$ - $CellContext`tf1$$/
               2 - $CellContext`tf2$$/2}] + 
            Integrate[($CellContext`\[Omega]4[$CellContext`s4] $CellContext`\
\[Delta]4) $CellContext`x4[$CellContext`s4], {$CellContext`s4, 
               0, $CellContext`B2$$/2}] + 
            
            Integrate[($CellContext`\[Omega]5[$CellContext`s5] $CellContext`\
\[Delta]5) $CellContext`x5[$CellContext`s5], {$CellContext`s5, 
               0, $CellContext`B2$$/
               2}])]; $CellContext`PC = {$CellContext`xC, $CellContext`yC}; \
$CellContext`e = 
         Abs[$CellContext`xC - $CellContext`d]; $CellContext`\[Omega]c1[
           Pattern[$CellContext`s1, 
            Blank[]]] = 
         Simplify[$CellContext`\[Omega]1[$CellContext`s1] + $CellContext`xC \
$CellContext`y1[$CellContext`s1] - $CellContext`yC \
$CellContext`x1[$CellContext`s1]]; $CellContext`\[Omega]c2[
           Pattern[$CellContext`s2, 
            Blank[]]] = 
         Simplify[$CellContext`\[Omega]2[$CellContext`s2] + $CellContext`xC \
$CellContext`y2[$CellContext`s2] - $CellContext`yC \
$CellContext`x2[$CellContext`s2]]; $CellContext`\[Omega]c3[
           Pattern[$CellContext`s3, 
            Blank[]]] = 
         Simplify[$CellContext`\[Omega]3[$CellContext`s3] + $CellContext`xC \
$CellContext`y3[$CellContext`s3] - $CellContext`yC \
$CellContext`x3[$CellContext`s3]]; $CellContext`\[Omega]c4[
           Pattern[$CellContext`s4, 
            Blank[]]] = 
         Simplify[$CellContext`\[Omega]4[$CellContext`s4] + $CellContext`xC \
$CellContext`y4[$CellContext`s4] - $CellContext`yC \
$CellContext`x4[$CellContext`s4]]; $CellContext`\[Omega]c5[
           Pattern[$CellContext`s5, 
            Blank[]]] = 
         Simplify[$CellContext`\[Omega]5[$CellContext`s5] + $CellContext`xC \
$CellContext`y5[$CellContext`s5] - $CellContext`yC \
$CellContext`x5[$CellContext`s5]]; $CellContext`check2 = Chop[Integrate[
             $CellContext`\[Omega]c1[$CellContext`s1], {$CellContext`s1, 
              0, $CellContext`B1$$/2}] + Integrate[
             $CellContext`\[Omega]c2[$CellContext`s2], {$CellContext`s2, 
              0, $CellContext`B1$$/2}] + Integrate[
             $CellContext`\[Omega]c3[$CellContext`s3], {$CellContext`s3, 
              0, $CellContext`H$$ - $CellContext`tf1$$/2 - $CellContext`tf2$$/
              2}] + Integrate[
             $CellContext`\[Omega]c4[$CellContext`s4], {$CellContext`s4, 
              0, $CellContext`B2$$/2}] + Integrate[
             $CellContext`\[Omega]c5[$CellContext`s5], {$CellContext`s5, 
              0, $CellContext`B2$$/2}]]; $CellContext`J\[Omega] = N[
           Chop[
           Integrate[$CellContext`\[Omega]c1[$CellContext`s1]^2 $CellContext`\
\[Delta]1, {$CellContext`s1, 0, $CellContext`B1$$/2}] + 
            Integrate[$CellContext`\[Omega]c2[$CellContext`s2]^2 \
$CellContext`\[Delta]2, {$CellContext`s2, 0, $CellContext`B1$$/2}] + 
            Integrate[$CellContext`\[Omega]c3[$CellContext`s3]^2 \
$CellContext`\[Delta]3, {$CellContext`s3, 
               0, $CellContext`H$$ - $CellContext`tf1$$/
               2 - $CellContext`tf2$$/2}] + 
            Integrate[$CellContext`\[Omega]c4[$CellContext`s4]^2 \
$CellContext`\[Delta]4, {$CellContext`s4, 0, $CellContext`B2$$/2}] + 
            Integrate[$CellContext`\[Omega]c5[$CellContext`s5]^2 \
$CellContext`\[Delta]5, {$CellContext`s5, 0, $CellContext`B2$$/
               2}]]]; $CellContext`c1 = $CellContext`G$$ $CellContext`Jt; \
$CellContext`c2 = $CellContext`Ee$$ $CellContext`J\[Omega]; $CellContext`\
\[Theta]usol[
           Pattern[$CellContext`z$, 
            Blank[]]] = First[
           ReplaceAll[
            $CellContext`\[Theta][$CellContext`z$], 
            
            DSolve[{$CellContext`c1 
               Derivative[2][$CellContext`\[Theta]][$CellContext`z$] == 0, 
              Derivative[2][$CellContext`\[Theta]][$CellContext`L$$] == 
              0, $CellContext`\[Theta][0] == 
              0, $CellContext`c1 
               Derivative[
                1][$CellContext`\[Theta]][$CellContext`L$$] == \
$CellContext`Mt$$}, 
             $CellContext`\[Theta][$CellContext`z$], $CellContext`z$]]]; 
        Unset[$CellContext`\[Theta]]; $CellContext`\[Theta]sol[
           Pattern[$CellContext`z$, 
            Blank[]]] = Quiet[
           ReplaceAll[
            $CellContext`\[Theta][$CellContext`z$], 
            
            NDSolve[{$CellContext`c2 
                Derivative[
                 4][$CellContext`\[Theta]][$CellContext`z$] - $CellContext`c1 
               Derivative[2][$CellContext`\[Theta]][$CellContext`z$] == 0, 
              Derivative[2][$CellContext`\[Theta]][$CellContext`L$$] == 
              0, $CellContext`\[Theta][0] == 0, 
              Derivative[1][$CellContext`\[Theta]][0] == 
              0, $CellContext`c1 
                Derivative[
                 1][$CellContext`\[Theta]][$CellContext`L$$] - \
$CellContext`c2 
               Derivative[
                3][$CellContext`\[Theta]][$CellContext`L$$] == \
$CellContext`Mt$$}, 
             $CellContext`\[Theta][$CellContext`z$], {$CellContext`z$, 
              0, $CellContext`L$$}]]]; Grid[{{
            Show[
             RegionPlot[$CellContext`sec], 
             ListLinePlot[{
               Table[
                Callout[
                 Part[$CellContext`points, $CellContext`i], 
                 ToString[$CellContext`i]], {$CellContext`i, 1, 6}]}, 
              PlotStyle -> Black, PlotMarkers -> {Automatic, 5}], 
             ListPlot[{
               Callout[$CellContext`PG, 
                ToString[
                 N[$CellContext`PG]]]}, PlotStyle -> Red, 
              PlotMarkers -> {Automatic, 5}], 
             ListPlot[{
               Callout[$CellContext`PG, "G", Left]}, PlotStyle -> Red, 
              PlotMarkers -> {Automatic, 5}], 
             ListPlot[{
               Callout[$CellContext`PC, 
                ToString[
                 N[$CellContext`PC]]]}, PlotStyle -> Magenta, 
              PlotMarkers -> {Automatic, 5}], 
             ListPlot[{
               Callout[$CellContext`PC, "C", Left]}, PlotStyle -> Magenta, 
              PlotMarkers -> {Automatic, 5}], AspectRatio -> Automatic, Axes -> 
             True, AxesLabel -> {"x", "y"}, 
             PlotRange -> {{(-1.5) (Max[$CellContext`B1$$, $CellContext`B2$$]/
                 2), 1.5 (Max[$CellContext`B1$$, $CellContext`B2$$]/
                 2)}, {(-0.6) $CellContext`H$$, 0.6 $CellContext`H$$}}, 
             ImageSize -> 300], 
            Show[{
              ParametricPlot[{
                $CellContext`x1[$CellContext`s1], 
                $CellContext`y1[$CellContext`s1]}, {$CellContext`s1, 
                0, $CellContext`B1$$/2}], 
              ParametricPlot[{
                $CellContext`x2[$CellContext`s2], 
                $CellContext`y2[$CellContext`s2]}, {$CellContext`s2, 
                0, $CellContext`B1$$/2}], 
              ParametricPlot[{
                $CellContext`x3[$CellContext`s3], 
                $CellContext`y3[$CellContext`s3]}, {$CellContext`s3, 
                0, $CellContext`H$$ - $CellContext`tf1$$/
                2 - $CellContext`tf2$$/2}], 
              ParametricPlot[{
                $CellContext`x4[$CellContext`s4], 
                $CellContext`y4[$CellContext`s4]}, {$CellContext`s4, 
                0, $CellContext`B2$$/2}], 
              ParametricPlot[{
                $CellContext`x5[$CellContext`s5], 
                $CellContext`y5[$CellContext`s5]}, {$CellContext`s5, 
                0, $CellContext`B2$$/2}], 
              ListPlot[{
                Table[
                 Callout[
                  Part[$CellContext`points, $CellContext`i], 
                  ToString[$CellContext`i]], {$CellContext`i, 1, 6}]}, 
               PlotStyle -> Black, PlotMarkers -> {Automatic, 5}], 
              ListPlot[{
                Callout[$CellContext`PG, 
                 ToString[
                  N[$CellContext`PG]]]}, PlotStyle -> Red, 
               PlotMarkers -> {Automatic, 5}], 
              ListPlot[{
                Callout[$CellContext`PG, "G", Left]}, PlotStyle -> Red, 
               PlotMarkers -> {Automatic, 5}], 
              ListPlot[{
                Callout[$CellContext`PC, 
                 ToString[
                  N[$CellContext`PC]]]}, PlotStyle -> Magenta, 
               PlotMarkers -> {Automatic, 5}], 
              ListPlot[{
                Callout[$CellContext`PC, "C", Left]}, PlotStyle -> Magenta, 
               PlotMarkers -> {Automatic, 5}]}, 
             PlotRange -> {{(-1.5) (Max[$CellContext`B1$$, $CellContext`B2$$]/
                 2), 1.5 (Max[$CellContext`B1$$, $CellContext`B2$$]/
                 2)}, {(-0.6) $CellContext`H$$, 0.6 $CellContext`H$$}}, 
             ImageSize -> 300], 
            Plot[{
              Evaluate[
               $CellContext`\[Theta]usol[$CellContext`z]], 
              Evaluate[
               $CellContext`\[Theta]sol[$CellContext`z]]}, {$CellContext`z, 
              0, $CellContext`L$$}, AxesLabel -> {"z", "\[Theta](z)"}, 
             PlotRange -> Automatic, AspectRatio -> 1/GoldenRatio, ImageSize -> 
             400, PlotLabels -> {"Uniform Torsion", "Non-uniform Torsion"}]}},
           Spacings -> {2, 0}]], 
      "Specifications" :> {{{$CellContext`B1$$, 200, 
          "\!\(\*SubscriptBox[\(B\), \(1\)]\) [mm]"}}, {{$CellContext`B2$$, 
          200, "\!\(\*SubscriptBox[\(B\), \(2\)]\) [mm]"}}, \
{{$CellContext`H$$, 400, "H [mm]"}}, {{$CellContext`tf1$$, 20, 
          "\!\(\*SubscriptBox[\(t\), SubscriptBox[\(f\), \(1\)]]\) [mm]"}}, \
{{$CellContext`tf2$$, 20, 
          "\!\(\*SubscriptBox[\(t\), SubscriptBox[\(f\), \(2\)]]\) [mm]"}}, \
{{$CellContext`tw$$, 20, 
          "\!\(\*SubscriptBox[\(t\), \(w\)]\) [mm]"}}, {{$CellContext`L$$, 
          1000, "L [mm]"}}, {{$CellContext`Ee$$, 210000, 
          "E [MPa]"}}, {{$CellContext`G$$, 80000, 
          "G [MPa]"}}, {{$CellContext`Mt$$, 10000, 
          "\!\(\*SubscriptBox[\(M\), \(t\)]\) [N mm]"}}}, "Options" :> {}, 
      "DefaultOptions" :> {}],
     ImageSizeCache->{1421., {206.8564453125, 213.1435546875}},
     SingleEvaluation->True],
    Deinitialization:>None,
    DynamicModuleValues:>{},
    SynchronousInitialization->True,
    UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$},
    UnsavedVariables:>{Typeset`initDone$$},
    UntrackedVariables:>{Typeset`size$$}], "Manipulate",
   Deployed->True,
   StripOnInput->False],
  Manipulate`InterpretManipulate[1]]], "Output",
 CellChangeTimes->{
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WindowSize->{1440., 758.25},
WindowMargins->{{0, Automatic}, {0, Automatic}},
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FrontEndVersion->"12.3 for Linux x86 (64-bit) (July 9, 2021)",
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