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The Simply Supported Beam Problem:
Concentrated Load Case\
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"Determine the reactions at each of the supports, the shear and bending \
moment, the deflection along the span, and the slope at the endpoints."], 
  "Special1",
  Evaluatable->False,
  AspectRatioFixed->True]
}, Open  ]],

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Cell[TextData[StyleBox["Variables",
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Cell[TextData[
"\tL  = beam length\n\tP  = the concentrated force\n\ta  = x coordinate of \
applied force\n\tRa = the reaction at support A\n\tRb = the reaction at \
support A\n\ty(x)  = deflection as a function of x\n\tV(x)  = shear as a \
function of x\n\tM(x)  = bending moment as a function of x\n\n\tA subscript \
of '1', refers to interval 0 \[LessEqual] x \[LessEqual] a\n\tA subscript of \
'2', refers to interval a \[LessEqual] x \[LessEqual] L\n\nNote that the \
point x = a is shared by intervals 1 & 2."], "Special1",
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    \(Rb =. \), 
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    \(shear2 =. \), 
    \(bending1 =. \), 
    \(bending2 =. \), 
    \(yA =. \), 
    \(yB =. \), 
    \(y =. \), 
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Cell[TextData[{
  StyleBox["Sum Forces & Moments",
    Evaluatable->False,
    AspectRatioFixed->True,
    FontVariations->{"Underline"->True}],
  StyleBox["",
    Evaluatable->False,
    AspectRatioFixed->True]
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Cell[CellGroupData[{

Cell[TextData[
"First we sum forces and moments to deduce the values of the reactions at \
supports A and B:"], "Special1",
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Cell[BoxData[{
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repetitive evaluation under varying conditions:"], "Special1",
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Cell[BoxData[{
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Cell[TextData[
"We now have the reactions at each support in terms of the load, placement \
and beam length.  Now we need to compute the shear, bending moment, beam \
deflection and end slopes.  Before we get into that however, we need to \
establish some sign conventions."], "Special1",
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Cell[TextData[StyleBox["Shear and Bending",
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Cell[TextData[
"Consider the following beam with imaginary cutting sections labeled A' and \
B':"], "Special1",
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"The maximum magnitude of M(x) is obtained by substituting 'a' for x in the \
moment expressions above.  This is left as an exercise for the reader.  We \
now will repeat this argument using the symbolic manipulation facilities of \
Mathematica\[Trademark]:"], "Special1",
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"We now assign y deflection coordinates to two intervals along the beam.  The \
y1 coordinate refers to the interval 0 \[LessEqual] x \[LessEqual] a, and the \
y2 coordinate refers to the interval a < x \[LessEqual] L. To compute \
spanwise deflection and slope we refer to linear bending theory which (under \
the assumption of constant stiffness) states that:\n"], "Text",
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  "\nApplying this to the y1 interval  (0 \[LessEqual] x \[LessEqual] a) and \
integrating:",
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We have four equations, and four unknown constants C1, C2, C1' and C2'.  The \
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  AspectRatioFixed->True,
  AnimationDisplayTime->0]
}, Open  ]],

Cell[BoxData[{
    \(\(bc1 = 
      enSconce[Reverse[\(displacement1 /. x \[Rule] 0\) /. y1[0] \[Rule] 0]]; 
    \)\), 
    \(\(bc2 = 
      enSconce[Reverse[\(displacement2 /. x \[Rule] L\) /. y2[L] \[Rule] 0]]; 
    \)\), 
    \(\(bc3 = 
      \((Last[displacement1] /. x \[Rule] a)\) - 
          \((Last[displacement2] /. x \[Rule] a)\) == 0; \)\), 
    \(\(bc4 = 
      \((Last[slope1] /. x \[Rule] a)\) - 
          \((Last[slope2] /. x \[Rule] a)\) == 0; \)\), 
    \(\(constants = 
      First[Simplify[Solve[{bc1, bc2, bc3, bc4}, {C1, C2, C1p, C2p}]]]; 
    \)\)}], "Input",
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Cell[BoxData[{
    \(\(collector[u_] := 
      First[u] \[Rule] 
        Simplify[Collect[Collect[Last[u], {EI, Ra, Rb, P}], EI]]; \)\), 
    \(outPut[
      foo1 = collector[constants\[LeftDoubleBracket]1\[RightDoubleBracket]]]
      \), 
    \(outPut[
      foo2 = collector[constants\[LeftDoubleBracket]2\[RightDoubleBracket]]]
      \), 
    \(outPut[
      foo3 = collector[constants\[LeftDoubleBracket]3\[RightDoubleBracket]]]
      \), 
    \(outPut[
      foo4 = collector[constants\[LeftDoubleBracket]4\[RightDoubleBracket]]]
      \)}], "Input",
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  AnimationDisplayTime->0],

Cell[BoxData[
    \("C1 -> (6*a*L^2*Rb - 2*L^3*Rb + 2*a^3*(Ra + Rb) - 3*a^2*L*(Ra + \
2*Rb))/(6*(E*I)*L)"\)], "Print"],

Cell[BoxData[
    \("C2p -> (a^2*(3*L*Rb - 2*a*(Ra + Rb)))/(6*(E*I))"\)], "Print"],

Cell[BoxData[
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  "Print"],

Cell[BoxData[
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Cell[TextData[
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Cell[BoxData[
    \(Simplify[Collect[constants /. reactions, {P, EI, L}]]\)], "Input",
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  AnimationDisplayTime->0],

Cell[BoxData[
    \({C1 \[Rule] 
        \(-\(\(a\ \((a\^2 - 3\ a\ L + 2\ L\^2)\)\ P\)\/\(6\ EI\ L\)\)\), 
      C2p \[Rule] \(a\^3\ P\)\/\(6\ EI\), 
      C1p \[Rule] \(-\(\(a\ \((a\^2 + 2\ L\^2)\)\ P\)\/\(6\ EI\ L\)\)\), 
      C2 \[Rule] 0}\)], "Output"]
}, Open  ]],

Cell[TextData[
"This requires little effort compared to checking by hand. We proceed to \
check our deflection and slope equations in a similar effortless manner:"], 
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  AspectRatioFixed->True,
  AnimationDisplayTime->0]
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Cell[CellGroupData[{

Cell[BoxData[{
    \(Simplify[
      Collect[\(displacement1 /. constants\) /. reactions, {P, EI, L}]]\), 
    \(Simplify[Collect[\(slope1 /. constants\) /. reactions, {P, EI, L}]]\), 
    \(Simplify[
      Collect[\(displacement2 /. constants\) /. reactions, {P, EI, L}]]\), 
    \(Simplify[Collect[\(slope2 /. constants\) /. reactions, {P, EI, L}]]
      \)}], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(y1[x] \[Rule] 
      \(-\(\(P\ x\ 
              \((a\^3 - 3\ a\^2\ L - L\ x\^2 + a\ \((2\ L\^2 + x\^2)\))
                \)\)\/\(6\ EI\ L\)\)\)\)], "Output"],

Cell[BoxData[
    \(dy1dx[x] \[Rule] 
      \(-\(\(P\ \((a\^3 - 3\ a\^2\ L + 2\ a\ L\^2 + 3\ a\ x\^2 - 3\ L\ x\^2)
                \)\)\/\(6\ EI\ L\)\)\)\)], "Output"],

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    \(y2[x] \[Rule] 
      \(a\ P\ \((L - x)\)\ \((a\^2 + x\ \((\(-2\)\ L + x)\))\)\)\/\(6\ EI\ L
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Cell[BoxData[
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      \(-\(\(a\ P\ \((a\^2 + 2\ L\^2 - 6\ L\ x + 3\ x\^2)\)\)\/\(6\ EI\ L
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Cell[CellGroupData[{

Cell[TextData["End Slopes"], "Section",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontWeight->"Bold",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->True,
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  "Shadow"->False}],

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"Our only remaining errand is to evaluate the slope expression at the \
endpoints and to remember that end slopes come out in radians:"], "Special1",
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  AspectRatioFixed->True],

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Cell[BoxData[{
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    \(\(riteEndSlope = Simplify[slope2 /. x \[Rule] L]; \)\), 
    \(outPut[leftEndSlope]\), 
    \(outPut[riteEndSlope]\)}], "Input",
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Cell[BoxData[
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}, Open  ]]
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Cell[CellGroupData[{

Cell[TextData["Output useable deflection expressions."], "Subsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[{
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      \(IF[AND[0 \[LessEqual] x, x \[LessEqual] a], A1, A2] /. 
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    \)\), 
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      fix[deflExpr, {"\<C\>" \[Rule] "\<a_C\>", 
          "\<EI\>" \[Rule] "\<(E*I_a)\>"}]]\), 
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          "\<EI\>" \[Rule] "\<(E*I_b)\>"}]]\), 
    \(outPut[
      fix[deflExpr, {"\<C\>" \[Rule] "\<c_C\>", 
          "\<EI\>" \[Rule] "\<(E*I_c)\>"}]]\)}], "Input",
  AspectRatioFixed->True],

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+ a_C1p*x + (Rb*((L*x^2)/2 - x^3/6))/(E*I_a))\""\)], "Print"],

Cell[BoxData[
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+ b_C1p*x + (Rb*((L*x^2)/2 - x^3/6))/(E*I_b))\""\)], "Print"],

Cell[BoxData[
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}, Open  ]]
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Cell[CellGroupData[{

Cell[TextData["Output useable constant expressions."], "Subsection",
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  AspectRatioFixed->True],

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Cell[BoxData[{
    \(outPut[
      fix[foo4, {"\<C\>" \[Rule] "\<a_C\>", "\<EI\>" \[Rule] "\<(E*I_a)\>"}]]
      \), 
    \(outPut[
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      \), 
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      \), 
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Cell[BoxData[
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Cell[BoxData[
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}, Open  ]]
}, Open  ]]
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Cell[CellGroupData[{

Cell[TextData[StyleBox["Testing",
  Evaluatable->False,
  AspectRatioFixed->True,
  FontVariations->{"Underline"->True}]], "Section",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[TextData["Convert displacement functions."], "Subsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[{
    \(ruleToFunction[displacement1]\), 
    \(ruleToFunction[displacement2]\)}], "Input",
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Information["\<y1\>", LongForm \[Rule] False]\), 
    \(Information["\<y2\>", LongForm \[Rule] False]\)}], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \("Global`y1"\)], "Print"],

Cell[BoxData[
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        ShowStringCharacters->True,
        NumberMarks->True],
      InputForm[ 
        Definition[ y1]],
      Editable->True,
      AutoDelete->True]], "Print"],

Cell[BoxData[
    \("Global`y2"\)], "Print"],

Cell[BoxData[
    InterpretationBox[
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        \(y2[x]\  := \ 
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        ShowStringCharacters->True,
        NumberMarks->True],
      InputForm[ 
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      Editable->True,
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Cell[CellGroupData[{

Cell[TextData["Input parameters for a realistic example.\n"], "Subsection",
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  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[{
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    \(\(tmp2 = \(\(y2[x] /. nconstants\) /. reactions\) /. inputs; \)\), 
    \(\(yA[x_] := tmp1; \)\), 
    \(\(yB[x_] := tmp2; \)\), 
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[TextData["Plot displacement functions.\n"], "Subsection",
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