(************** Content-type: application/mathematica **************
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Cell[TextData[
"FormTangentStiff2DTwoNodeBar[XY1_,XY2_,uXY1_,uXY2_,Em_,A0_,s0_]:=\n  \
Module[{X1,Y1,X2,Y2,X21,Y21,uX1,uY1,uX2,uY2,\n     \
uX21,uY21,L0,L,e,s,ax,ay,Ke}, \n    {X1,Y1}=XY1; {X2,Y2}=XY2; X21=X2-X1; \
Y21=Y2-Y1; \n    {uX1,uY1}=uXY1; {uX2,uY2}=uXY2; uX21=uX2-uX1; uY21=uY2-uY1;\n\
    L0=Sqrt[X21^2+Y21^2];     L=Sqrt[(X21+uX21)^2+(Y21+uY21)^2];\n    \
e=(L-L0)*(L+L0)/(2*L0^2); s=s0+Em*e; \n    ax=(X21+uX21)/L0;         \
ay=(Y21+uY21)/L0;\n    Ke=(Em*A0/L0)*{{ ax*ax, ax*ay,-ax*ax,-ax*ay},\n        \
           { ax*ay, ay*ay,-ay*ax,-ay*ay},\n                   {-ax*ax,-ay*ax, \
ax*ax, ay*ax},\n                   {-ax*ay,-ay*ay, ay*ax, ay*ay}} +\n      \
(A0*s/L0)*{{1,0,-1,0},{0,1,0,-1},{-1,0,1,0},{0,-1,0,1}};\n      \
Return[Simplify[Ke]]\n  ];\n         \n\
Ke=FormTangentStiff2DTwoNodeBar[{0,-S+H},{0,H},{0,0},{0,uY},Em,A0,0];\n\
Ke=Simplify[Ke/.{Sqrt[S^2]->S,(1/Sqrt[S^2])->1/S}];\n\
Print[\"Ke=\",Ke//InputForm];      \n\
Ke=FormTangentStiff2DTwoNodeBar[{-4,0},{0,3},{0,0},{0,0}, Em,A0,s0];\n\
(*Print[\"Ke=\",Ke];*)\nPrint[Eigenvalues[Ke]];"], "Input",
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"MergeElemIntoMasterStiff[Ke_,eftab_,Km_]:=   \n  \
Module[{i,j,ii,jj,neldof,K}, K=Km;\n       neldof=Dimensions[Ke][[1]]; \n     \
  For[i=1, i<=neldof, i++, ii=eftab[[i]];\n         For[j=i, j<=neldof, j++, \
jj=eftab[[j]];\n           If [ii>0 && jj>0, \n           \
K[[jj,ii]]=K[[ii,jj]]+=Ke[[i,j]]] \n            ]\n          ]; Return[K]\n   \
     ];\n        \nAssembleMasterStiffOfShallowArch[S_,H_,Em_,A0_,uX_,uY_]:=\n\
  Module[{K1,K2,K},\n    K=Table[0,{2},{2}]; \n    \
K1=FormTangentStiff2DTwoNodeBar[{-S/2,0},{0,H},\n             \
{0,0},{uX,uY},Em,A0,0]; Print[\"K1=\",K1//MatrixForm];\n    \
K=MergeElemIntoMasterStiff[K1,{0,0,1,2},K];\n    \
K2=FormTangentStiff2DTwoNodeBar[{0,H},{S/2,0},\n             \
{uX,uY},{0,0},Em,A0,0]; Print[\"K2=\",K2//MatrixForm];\n    \
K=MergeElemIntoMasterStiff[K2,{1,2,0,0},K];\n    Return[Simplify[K]]\n        \
];\n        \nClearAll[S,H,Em,A0,uX,uY]; S=2; H=1; Em=2; A0=Sqrt[2]; uX=0; \
uY=0; \nK=Simplify[AssembleMasterStiffOfShallowArch[S,H,Em,A0,uX,uY]];\n\
Print[\"Master stiffness matrix = \",K//InputForm];\n(*detK=Simplify[Det[K]]; \
Print[\"det of K=\",detK//InputForm];*) "], "Input",
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Cell[TextData[
"\nK = {{(4*A0*Em*(S^2 + 6*uX^2 + 4*H*uY + 2*uY^2))/(4*H^2 + S^2)^(3/2), \n   \
 (16*A0*Em*uX*(H + uY))/(4*H^2 + S^2)^(3/2)}, \n   {(16*A0*Em*uX*(H + \
uY))/(4*H^2 + S^2)^(3/2), \n    (8*A0*Em*(2*H^2 + uX^2 + 6*H*uY + \
3*uY^2))/(4*H^2 + S^2)^(3/2)}};\nPrint[K//FortranForm];"], "Input",
  AspectRatioFixed->True],

Cell[TextData[
"\nsol=Solve[S^2 + 4*H*uY + 2*uY^2==0,uY];\nPrint[sol];"], "Input",
  AspectRatioFixed->True],

Cell[TextData[
"MergeElemIntoMasterStiff[K_,eftab_,Km_]:=   \n  Module[{i,j,ii,jj,neldof,K}, \
K=Km;\n       neldof=Dimensions[K][[1]];\n       For[i=1, i<=neldof, i++, \
ii=eftab[[i]];\n         For[j=i, j<=neldof, j++, jj=eftab[[j]];\n           \
If [ii>0 && jj>0, \n           K[[jj,ii]]=K[[ii,jj]]+=K[[i,j]]] \n            \
]\n          ]; Return[K]\n        ];\n        \n\
AssembleMasterStiffOfShallowArch[S_,H_,Em_,A0_,uX_,uY_]:=\n  \
Module[{K1,K2,K3,K},\n    K=Table[0,{2},{2}]; \n    \
K1=FormTangentStiff2DTwoNodeBar[{-S/2,0},{0,H},\n             \
{0,0},{uX,uY},Em,A0,0]; (*Print[K1//TableForm];*)\n    \
K=MergeElemIntoMasterStiff[K1,{0,0,1,2},K];\n    \
K2=FormTangentStiff2DTwoNodeBar[{0,H},{S/2,0},\n             \
{uX,uY},{0,0},Em,A0,0]; (*Print[K2//TableForm];*)\n    \
K=MergeElemIntoMasterStiff[K2,{1,2,0,0},K];\n    \
K3=FormTangentStiff2DTwoNodeBar[{0,-S+H},{0,H},\n             \
{0,0},{uX,uY},Em,A0,0]; (*Print[K2//TableForm];*)\n    \
K=MergeElemIntoMasterStiff[K3,{0,0,1,2},K];\n    Return[Simplify[K]]\n        \
];\n        \nClearAll[S,H,Em,A0,uX,uY]; \n\
K=AssembleMasterStiffOfShallowArch[2,H,1,1,0,uY];\nK=Simplify[Expand[K]];\n\
Print[\"Master stiffness matrix = \",K//InputForm];"], "Input",
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uY^2/(1 + H^2)^(3/2);\na=Coefficient[Kxx,uY,0]; b=Coefficient[Kxx,uY,1]; \
c=Coefficient[Kxx,uY,2];\nd=Simplify[b^2-4*a*c]; Print[\"d=\",d//InputForm]; \
\nPlot[d,{H,0,2},PlotRange->{0,1}];\nPrint[FindRoot[d==0,{H,1.2}]];"], "Input",\

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  ImageRangeCache->{{{0, 476}, {292, 0}} -> {-0.174391, -0.0611163, \
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}, Open  ]],

Cell[TextData[
"ClearAll[S,H,Em,A0,uX,uY];\n\
eigval1[S_,H_,Em_,A0_,uX_,uY_]:=(2*(256*A0*Em*H^8 + 256*A0*Em*H^6*S^2 + \n    \
    96*A0*Em*H^4*S^4 + 16*A0*Em*H^2*S^6 + A0*Em*S^8 + \n        \
512*A0*Em*H^6*uX^2 + 384*A0*Em*H^4*S^2*uX^2 + \n        96*A0*Em*H^2*S^4*uX^2 \
+ 8*A0*Em*S^6*uX^2 + \n        1024*A0*Em*H^7*uY + 768*A0*Em*H^5*S^2*uY + \n  \
      192*A0*Em*H^3*S^4*uY + 16*A0*Em*H*S^6*uY + \n        512*A0*Em*H^6*uY^2 \
+ 384*A0*Em*H^4*S^2*uY^2 + \n        96*A0*Em*H^2*S^4*uY^2 + 8*A0*Em*S^6*uY^2 \
- \n        (A0^2*Em^2*(4*H^2 + S^2)^6*\n           (16*H^4 - 8*H^2*S^2 + S^4 \
+ 32*H^2*uX^2 + 8*S^2*uX^2 + \n             16*uX^4 + 64*H^3*uY - 16*H*S^2*uY \
+ 64*H*uX^2*uY + \n             96*H^2*uY^2 - 8*S^2*uY^2 + 32*uX^2*uY^2 + \n  \
           64*H*uY^3 + 16*uY^4))^(1/2)))/(4*H^2 + S^2)^(9/2); \n\
eigval2[S_,H_,Em_,A0_,uX_,uY_]:=(2*(256*A0*Em*H^8 + 256*A0*Em*H^6*S^2 + \n    \
    96*A0*Em*H^4*S^4 + 16*A0*Em*H^2*S^6 + A0*Em*S^8 + \n        \
512*A0*Em*H^6*uX^2 + 384*A0*Em*H^4*S^2*uX^2 + \n        96*A0*Em*H^2*S^4*uX^2 \
+ \n        8*A0*Em*S^6*uX^2 + 1024*A0*Em*H^7*uY + \n        \
768*A0*Em*H^5*S^2*uY + 192*A0*Em*H^3*S^4*uY + \n        16*A0*Em*H*S^6*uY + \
512*A0*Em*H^6*uY^2 + \n        384*A0*Em*H^4*S^2*uY^2 + 96*A0*Em*H^2*S^4*uY^2 \
+ \n        8*A0*Em*S^6*uY^2 + \n        (A0^2*Em^2*(4*H^2 + S^2)^6*\n        \
   (16*H^4 - 8*H^2*S^2 + S^4 + 32*H^2*uX^2 + 8*S^2*uX^2 + \n             \
16*uX^4 + 64*H^3*uY - 16*H*S^2*uY + 64*H*uX^2*uY + \n             96*H^2*uY^2 \
- 8*S^2*uY^2 + 32*uX^2*uY^2 + \n             64*H*uY^3 + \
16*uY^4))^(1/2)))/(4*H^2 + S^2)^(9/2);\n           \n \
eig1=eigval1[S,H,1,1,0,uY]; eig2=eigval2[S,H,1,1,0,uY];\n \
eig1=Simplify[eig1]; \n eig2=Simplify[eig2];\n \
Print[\"eig1P=\",eig1//InputForm]; Print[\"eig2P=\",eig2//InputForm];\n \
(*Plot[{eig1,eig2},{uY,0,-2*H},PlotRange->{{-2*H,0},{-0.5,0.5}}];*)\n \n \
uX=Sqrt[-S^2/2-2*H*uY-uY^2];\n eig1=eigval1[S,H,1,1,uX,uY]; \
eig2=eigval2[S,H,1,1,uX,uY];\n eig1=Simplify[eig1]; \n eig2=Simplify[eig2];\n \
Print[\"eig1B=\",eig1//InputForm]; Print[\"eig2B=\",eig2//InputForm];\n \
uYB1=Re[N[-H+Sqrt[H^2-S^2/2]]]; uYB2=Re[N[-H-Sqrt[H^2-S^2/2]]];\n \
(*Plot[{eig1,eig2},{uY,uYB1,uYB2},PlotRange->{{-2*H,0},{-0.5,0.5}}];*)"], 
  "Input",
  CellFrame->True,
  CellMargins->{{10, 60}, {Inherited, Inherited}},
  CellLabelMargins->{{14, Inherited}, {Inherited, Inherited}},
  AspectRatioFixed->True,
  Background->RGBColor[1, 0.647715, 0.521981]],

Cell[TextData[
"ClearAll[H,S,Em,A0,uY];\ndetK=(32*A0^2*Em^2*(S^2 + 4*H*uY + 2*uY^2)*\n     \
(2*H^2 + 6*H*uY + 3*uY^2))/(4*H^2 + S^2)^3;\nsol=Simplify[Solve[detK==0,uY]]; \
\nPrint[sol//InputForm];\nlambdaCP=(8*A0*Em*uY*(H + uY)*(2*H + uY))/(4*H^2 + \
S^2)^(3/2)/.sol;\nlambdaCP=Simplify[Expand[lambdaCP]]; \n\
Print[\"lambdaCP=\",lambdaCP//InputForm];"], "Input",
  CellFrame->True,
  CellMargins->{{10, 60}, {Inherited, Inherited}},
  CellLabelMargins->{{14, Inherited}, {Inherited, Inherited}},
  AspectRatioFixed->True,
  Background->RGBColor[1, 0.647715, 0.521981]],

Cell[CellGroupData[{

Cell[TextData[
"ClearAll[H,S,Em,A0,uY];\nlambdaL=(16*A0*Em*H^3)/(3*3^(1/2)*(4*H^2 + \
S^2)^(3/2));\nlambdaB=(2*2^(1/2)*A0*Em*S^2*(2*H^2 - S^2)^(1/2))/(4*H^2 + \
S^2)^(3/2);\nset={Em->1,A0->1,S->2};\n\
pL=Plot[lambdaL/.set,{H,0,5},DisplayFunction->Identity,\n   \
AxesLabel->{\"H\",\"lambda at CPs\"},\n   PlotLabel->\"lambdaCP vs H \
(S=2,E=A0=1)\"];\n\
pB=Plot[lambdaB/.set,{H,Sqrt[2.00001],5},DisplayFunction->Identity];\n\
Show[{pL,pB},DisplayFunction->$DisplayFunction];"], "Input",
  CellFrame->True,
  CellMargins->{{10, 60}, {Inherited, Inherited}},
  CellLabelMargins->{{14, Inherited}, {Inherited, Inherited}},
  AspectRatioFixed->True,
  Background->RGBColor[1, 0.647715, 0.521981]],

Cell[GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.190476 0.0147151 1.6219 [
[(1)] .21429 .01472 0 2 Msboxa
[(2)] .40476 .01472 0 2 Msboxa
[(3)] .59524 .01472 0 2 Msboxa
[(4)] .78571 .01472 0 2 Msboxa
[(5)] .97619 .01472 0 2 Msboxa
[(H)] 1.025 .01472 -1 0 Msboxa
[(lambdaCP vs H \\(S=2,E=A0=1\\))] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa
[(0.05)] .01131 .09581 1 0 Msboxa
[(0.1)] .01131 .17691 1 0 Msboxa
[(0.15)] .01131 .258 1 0 Msboxa
[(0.2)] .01131 .3391 1 0 Msboxa
[(0.25)] .01131 .42019 1 0 Msboxa
[(0.3)] .01131 .50129 1 0 Msboxa
[(0.35)] .01131 .58238 1 0 Msboxa
[(lambda at CPs)] .02381 .61803 0 -4 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
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\>"],
  ImageRangeCache->{{{0, 394}, {240, 0}} -> {-0.822088, -0.0313136, \
0.0161603, 0.00189786}}]
}, Open  ]],

Cell["\<\

ClearAll[H,S,Em,A0,uY];
lambdaL=(16*A0*Em*H^3)/(3*3^(1/2)*(4*H^2 + S^2)^(3/2));
lambdaB=(2*2^(1/2)*A0*Em*S^2*(2*H^2 - S^2)^(1/2))/(4*H^2 + S^2)^(3/2);
set={Em->1,A0->1,S->2};
dlambdaBH=Simplify[D[lambdaB/.set,H]];
sol=Solve[dlambdaBH==0,H];
Print[N[(lambdaL/.set)/.H->Sqrt[3]]];
Print[N[(lambdaB/.set)/.H->Sqrt[3]]];
dlambdaBH=Simplify[D[lambdaB,H]*Sqrt[S^2/4+H^2]/H /.set];
Print[\"dlambdaBH=\",dlambdaBH];
sol=Solve[dlambdaBH==0,H]; Print[\"sol=\",sol]
Print[N[(lambdaL/.set)/.sol]];
Print[N[(lambdaB/.set)/.sol]];
(*pL=Plot[lambdaL/.set,{H,0,5},DisplayFunction->Identity,
   AxesLabel->{\"H\",\"lambda at CPs\"},
   PlotLabel->\"lambdaCP vs H (S=2,E=A0=1)\"];
pB=Plot[lambdaB/.set,{H,Sqrt[2.00001],5},DisplayFunction->Identity];
Show[{pL,pB},DisplayFunction->$DisplayFunction];*)\
\>", "Input",
  CellFrame->True,
  AspectRatioFixed->True,
  Background->GrayLevel[0.833326]],

Cell["\<\
ClearAll[S,H,A0,Em,V0]; Em=1; S=2; A0=1;
L0=V0/(2*A0); H=Sqrt[L0^2-S^2/4]; 
lambdaL=(16*A0*Em*H^3)/(3*3^(1/2)*(4*H^2 + S^2)^(3/2));
lambdaL=Simplify[lambdaL]; Print[\"L0=\",L0];Print[\"H=\",H];
Print[lambdaL];Plot[lambdaL/V0,{V0,2,10}];
lambdaB=(2*2^(1/2)*A0*Em*S^2*(2*H^2 - S^2)^(1/2))/(4*H^2 + S^2)^(3/2);
lambdaB=Simplify[lambdaB];
Print[lambdaB];Plot[lambdaB/V0,{V0,2,10}];
Plot[lambdaB,{V0,2,10}];\
\>", "Input",
  CellFrame->True,
  AspectRatioFixed->True,
  Background->GrayLevel[0.833326]],

Cell["\<\

K=(EA/L)*{{0,0},{0,1}}+m*g/L*{{1,0},{0,1}};
Ms={{1/Sqrt[m],0},{0,1/Sqrt[m]}};
eig=Simplify[Eigenvalues[Ms.K.Ms]];
Print[\"squared frequencies=\",eig];
\
\>", "Input",
  CellFrame->True,
  AspectRatioFixed->True,
  Background->GrayLevel[0.833326]],

Cell[TextData[{
  StyleBox["Em=A0=1; S=2; s=Sin[a]; c=Cos[a]; H=(Sin[a]/Cos[a])*S/2;\n\
lambdaL=(16*A0*Em*H^3)/(3*3^(1/2)*(4*H^2 + S^2)^(3/2));\n\
Print[\"lambdaL=\",Simplify[lambdaL]//InputForm];\n",
    AspectRatioFixed->True],
  StyleBox["lambdaL=Simplify[(2*Tan[a]^3)/(3*3^(1/2)*(Sec[a]^3))];\n",
    AspectRatioFixed->True,
    FontWeight->"Plain"],
  StyleBox["Print[\"lambdaL=\",lambdaL];",
    AspectRatioFixed->True]
}], "Input",
  CellFrame->True,
  AspectRatioFixed->True,
  Background->GrayLevel[0.833326]],

Cell["\<\

AssembleMasterStiffOfQuiz3Arch[S_,H_,Em_,A0_,uX_,uY_]:=
  Module[{K1,K2,K3,K4,K5,K6,K,S,H},
    K=Table[0,{2},{2}]; S=S; H=H;
    K1=FormTangentStiff2DTwoNodeBar[{-S/2,0},{0,3*H/4},
             {0,0},{uX,uY},Em,4*A0,0]; (*Print[K1//TableForm];*)
    K=MergeElemIntoMasterStiff[K1,{0,0,1,2},K];
    K2=FormTangentStiff2DTwoNodeBar[{-S/2,S},{-S/2,H},
             {uX,uY},{0,0},Em,4*A0,0]; (*Print[K2//TableForm];*)
    K=MergeElemIntoMasterStiff[K2,{1,2,0,0},K];
    K3=FormTangentStiff2DTwoNodeBar[{0,H},{S/2,3*H/4},
             {0,0},{uX,uY},Em,4*A0,0]; (*Print[K1//TableForm];*)
    K=MergeElemIntoMasterStiff[K3,{0,0,1,2},K];
    K4=FormTangentStiff2DTwoNodeBar[{S/2,3*H/4},{S,0},
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    K5=FormTangentStiff2DTwoNodeBar[{-S/2,0},{0,H},
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Cell["\<\
ClearAll[V0,H,S,Em,A0];  (*S=2; Em=1; A0=1;*)
H=Sqrt[V0/(2*A0)-S^2/4];
(*L0=Sqrt[S^2/4+H^2]; A0=V0/(2*L0);*)
lambdaL=(16*A0*Em*H^3)/(3*3^(1/2)*(4*H^2 + S^2)^(3/2))/V0;
lambdaB=(2*2^(1/2)*A0*Em*S^2*(2*H^2 - S^2)^(1/2))/(4*H^2 + S^2)^(3/2)/V0;
(*Plot[{lambdaL,lambdaB},{H,0,3},PlotRange->All];*)
Plot[{lambdaL,lambdaB},{V0,0,10},PlotRange->All];
(*Plot[{lambdaL,lambdaB},{H,0,3},PlotRange->All];*)
V0best=5;Hbest=H/.V0->V0best;
Print[{Hbest,N[ArcTan[Hbest]*(180/Pi)]}];
Print[Simplify[D[lambdaL,V0]]];
Print[Simplify[lambdaL]];
\
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