(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 4.2'

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Notebook[{
Cell["\<\
FormIntForce2DTwoNodeBar[XY1_,XY2_,uXY1_,uXY2_,Em_,A0_,s0_]:=
  Module[{X1,Y1,X2,Y2,X21,Y21,uX1,uY1,uX2,uY2,uX21,uY21,L0,L,
     e,s,ax,ay,ifv}, 
    {X1,Y1}=XY1; {X2,Y2}=XY2; X21=X2-X1; Y21=Y2-Y1; 
    {uX1,uY1}=uXY1; {uX2,uY2}=uXY2; uX21=uX2-uX1; uY21=uY2-uY1;
    L0=Sqrt[X21^2+Y21^2];     L=Sqrt[(X21+uX21)^2+(Y21+uY21)^2];
    e=(L-L0)*(L+L0)/(2*L0^2); s=s0+Em*e; 
    ax=(X21+uX21)/L0;         ay=(Y21+uY21)/L0;
    ifv=A0*s*{{-ax}, {-ay}, {ax}, {ay}};
    Return[Simplify[ifv]]
        ];       
pe=FormIntForce2DTwoNodeBar[{2,3},{5,7},{1,0},{1,0}, 20,12, 0];
Print[\"Internal elem force pe=\",pe];
pe=FormIntForce2DTwoNodeBar[{2,3},{5,7},{1,0},{1,0}, 20,12, s0];
Print[\"Internal elem force pe=\",pe];\
\>", "Input",
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Cell["\<\
MergeElemIntoMasterIntForce[pe_,eftab_,p_]:=   
  Module[{i,ii,neldof,fmaster}, fmaster=p;
       neldof=Length[eftab];
       For[i=1, i<=neldof, i++, ii=eftab[[i]];
           If [ii>0,fmaster[[ii,1]]+=pe[[i,1]]] 
          ]; Return[fmaster]
  ];
        
AssembleMasterIntForceOfTwoBarArch[span_,height_,Em_,A0_,uX_,uY_]:=
  Module[{f1,f2,fmaster},
    fmaster=Table[0,{2},{1}];
    f1=FormIntForce2DTwoNodeBar[{-span/2,0},{0,height},
             {0,0},{uX,uY},Em,A0,0]; 
    fmaster=MergeElemIntoMasterIntForce[f1,{0,0,1,2},fmaster];
    f2=FormIntForce2DTwoNodeBar[{0,height},{span/2,0},
             {uX,uY},{0,0},Em,A0,0];
    fmaster=MergeElemIntoMasterIntForce[f2,{1,2,0,0},fmaster];
    Return[Simplify[fmaster]]
  ]; 
              
ClearAll[Em,A0,S,H,uX,uY,f];
p=AssembleMasterIntForceOfTwoBarArch[2,2.5,10,.75,-0.4,0.25];
Print[\"Master Int Force p= \",p//InputForm];
p=AssembleMasterIntForceOfTwoBarArch[S,H,Em,A0,uX,uY];
Print[\"Master Int Force p= \",p//InputForm];\
\>", "Input",
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Cell["\<\
ClearAll[Em,A0,S,H,uX,uY,pX,pY,lambda];
pX = (4*A0*Em*uX*(S^2+2*uX^2+4*H*uY+2*uY^2))/(4*H^2+ S^2)^(3/2);
pY = (8*A0*Em*(H+uY)*(uX^2+2*H*uY+uY^2))/(4*H^2+S^2)^(3/2); 
roots=Solve[pX==0, uX];
Print[\"roots of pX=0 are \",roots//InputForm];
Print[\"lambda=\",Simplify[pY/.roots]//InputForm];\
\>", "Input",
  CellFrame->True,
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  Background->RGBColor[1, 0.647715, 0.521981]],

Cell[BoxData[
    \(General::"spell1" \(\(:\)\(\ \)\) 
      "Possible spelling error: new symbol name \"\!\(roots\)\" is similar to \
existing symbol \"\!\(Roots\)\"."\)], "Message"]
}, Open  ]],

Cell["\<\
Print[Simplify[Solve[-S^2 - 4*H*uY - \
2*uY^2==0,uY]]//InputForm];\
\>", "Input",
  CellFrame->True,
  CellMargins->{{12, 34}, {Inherited, Inherited}},
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Cell["\<\
ClearAll[Em,A0,S,H,uX,uY,pX,pY,lambda];
lambdaP=(8*A0*Em*uY*(H + uY)*(2*H + uY))/(4*H^2 + S^2)^(3/2); 
lambdaS=(-4*A0*Em*S^2*(H + uY))/(4*H^2 + S^2)^(3/2); 
case1={H->Sqrt[3]/3,S->2,A0->1,Em->1};
case2={H->Sqrt[2],  S->2,A0->1,Em->1};
case3={H->Sqrt[3],  S->2,A0->1,Em->1};
case4={H->3,        S->2,A0->1,Em->1};
Plot[{lambdaP/.case1,lambdaS/.case1},{uY,0,-2*H/.case1},
     AxesLabel->{\"uY\",\"lambda\"},PlotLabel->\"Case 1: H=Sqrt[3]/3\"];
Plot[{lambdaP/.case2,lambdaS/.case2},{uY,0,-2*H/.case2},
     AxesLabel->{\"uY\",\"lambda\"},PlotLabel->\"Case 2: H=Sqrt[2]\"];
Plot[{lambdaP/.case3,lambdaS/.case3},{uY,0,-2*H/.case3},
     AxesLabel->{\"uY\",\"lambda\"},PlotLabel->\"Case 3: H=Sqrt[3]\"];
Plot[{lambdaP/.case4,lambdaS/.case4},{uY,0,-2*H/.case4},
     AxesLabel->{\"uY\",\"lambda\"},PlotLabel->\"Case 4: H=3\"];\
\>", "Input",\

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}, Open  ]],

Cell["\<\
ClearAll[S,H,Em,A0,uX,uY];  
lambdaP=N[8*A0*Em*uY*(H+uY)*(2*H+uY)/(4*H^2+S^2)^(3/2)];
sol=Solve[D[lambdaP,uY]==0,uY];
Print[Simplify[lambdaP/.sol]//InputForm];\
\>", "Input",
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Cell["\<\
ClearAll[S,H,Em,A0,uX,uY]; H=Sqrt[3.];

lambdamax[S_,H_,Em_,A0_]:=3.0792*A0*Em*H^3/(4*H^2+S^2)^(3/2);
primarypath[S_,H_,Em_,A0_,uY_]:=
  {0,uY,8*A0*Em*uY*(H+uY)*(2*H+uY)/(4*H^2+S^2)^(3/2)};
secondarypath1[S_,H_,Em_,A0_,uY_]:=Module[{c,uYB1,uYB2},
  uYB1=Re[N[-H+Sqrt[H^2-S^2/2]]]; uYB2=Re[N[-H-Sqrt[H^2-S^2/2]]];
  c=4*A0*Em*S^2/(4*H^2+S^2)^(3/2);
  If [N[uY]>=uYB1, Return[{0,uYB1,-c*(H+uYB1)}]];
  If [N[uY]<=uYB2, Return[{0,uYB2,-c*(H+uYB2)}]]; 
  Return[{Sqrt[-S^2/2-2*H*uY-uY^2],uY,-c*(H+uY)}]
        ];
secondarypath2[S_,H_,Em_,A0_,uY_]:=Module[{c,uYB1,uYB2},
  uYB1=Re[N[-H+Sqrt[H^2-S^2/2]]]; uYB2=Re[N[-H-Sqrt[H^2-S^2/2]]];
  c=4*A0*Em*S^2/(4*H^2+S^2)^(3/2);
  If [N[uY]>=uYB1, Return[{0,uYB1,-c*(H+uYB1)}]];
  If [N[uY]<=uYB2, Return[{0,uYB2,-c*(H+uYB2)}]]; 
  Return[{-Sqrt[-S^2/2-2*H*uY-uY^2],uY,-c*(H+uY)}]
        ];
lambdarange=1.1*{-lambdamax[2,H,1,1],lambdamax[2,H,1,1]};
pp=ParametricPlot3D[primarypath[2,H,1,1,uY], {uY,0,-2*H},  
   PlotPoints->201, PlotRange->{{-H,H},{0,-2*H},lambdarange},
   BoxRatios->{1,2,1},AxesLabel->{\"uX\",\"uY\",\"lambda\"},
   DisplayFunction->Identity];
ps1=ParametricPlot3D[secondarypath1[2,H,1,1,uY], {uY,0,-2*H},  
   PlotPoints->201, PlotRange->{{-H,H},{0,-2*H},lambdarange},
   BoxRatios->{1,2,1},AxesLabel->{\"uX\",\"uY\",\"lambda\"},
   DisplayFunction->Identity];
ps2=ParametricPlot3D[secondarypath2[2,H,1,1,uY], {uY,0,-2*H},  
   PlotPoints->201, PlotRange->{{-H,H},{0,-2*H},lambdarange},
   BoxRatios->{1,2,1},AxesLabel->{\"uX\",\"uY\",\"lambda\"},
   DisplayFunction->Identity];
Show[pp,ps1,ps2,ViewPoint->{3,1,2},DisplayFunction->$DisplayFunction];\
\>", \
"Input",
  CellFrame->True,
  CellMargins->{{12, 34}, {Inherited, Inherited}},
  CellLabelMargins->{{10, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[1, 0.647715, 0.521981]],

Cell[BoxData[
    \(ParametricPlot3D::"ppcom" \(\(:\)\(\ \)\) 
      "Function \!\(primarypath[\(\(2, H, 1, 1, uY\)\)]\) cannot be compiled; \
plotting will proceed with the uncompiled function."\)], "Message"],

Cell[BoxData[
    \(ParametricPlot3D::"ppcom" \(\(:\)\(\ \)\) 
      "Function \!\(secondarypath1[\(\(2, H, 1, 1, uY\)\)]\) cannot be \
compiled; plotting will proceed with the uncompiled function."\)], "Message"],

Cell[BoxData[
    \(ParametricPlot3D::"ppcom" \(\(:\)\(\ \)\) 
      "Function \!\(secondarypath2[\(\(2, H, 1, 1, uY\)\)]\) cannot be \
compiled; plotting will proceed with the uncompiled function."\)], "Message"],

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plotload =Table[0,{nmax+1}];plotarrow=Table[0,{nmax+1}];
Do [uY=-2.5*n*H/nmax;      d= N[-S^2/2-2*H*uY-uY^2];  
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\>"],
  ImageRangeCache->{{{0, 474}, {474, 0}} -> {-6.00006, -6.00006, 0.0253167, \
0.0253167}}]
}, Open  ]],

Cell["\<\
ClearAll[S,H,Em,A0,uX,uY,a,c,d];  S=2;H=.8;A0=1; Em=1;
nmax =20;   c= N[4*A0*Em/(4*H^2+S^2)^(3/2)];
Do [uY=-2.5*n*H/nmax;      d= N[-S^2/2-2*H*uY-uY^2];  
    If[d <=0., uX=0;       lambda= N[2*c*uY*(H+uY)*(2*H+uY)],
               uX=Sqrt[d]; lambda= N[-c*S^2*(H+uY)]];
    plotelem1=Graphics[Line[{{-S/2,0},{uX,H+uY}}]];
    plotelem2=Graphics[Line[{{ uX,H+uY},{S/2,0}}]];
    plotload =Graphics[Line[{{ uX,H+uY},{uX,H+uY-lambda}}]];
    a = Min[.5*Abs[lambda],0.2*H];
    If [lambda>0,    plotarrow=Graphics[{
        Line[{{ uX,H+uY},{uX-a/2,H+uY-a}}],
        Line[{{ uX,H+uY},{uX+a/2,H+uY-a}}]} ],
                     plotarrow=Graphics[{
        Line[{{ uX,H+uY},{uX-a/2,H+uY+a}}],
        Line[{{ uX,H+uY},{uX+a/2,H+uY+a}}]} ]  ];
    Show[Graphics[Thickness[.012]],plotelem1,plotelem2,
      Graphics[Thickness[.005]],Graphics[RGBColor[1,0,1]],plotload,plotarrow,
      PlotRange->{{-2*H,2*H},{-2*H,2*H}},AspectRatio->1],
      {n,0,nmax}];\
\>", "Input",
  CellFrame->True,
  CellMargins->{{12, 34}, {Inherited, Inherited}},
  CellLabelMargins->{{10, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[1, 0.647715, 0.521981]],

Cell["\<\

ClearAll[S,H,Em,A0,uX,uY,a,c,d];  S=2.;H=3;A0=1; Em=1;
nmax =20;   c= N[4*A0*Em/(4*H^2+S^2)^(3/2)];
Do [uY=-2.5*n*H/nmax;      d= N[-S^2/2-2*H*uY-uY^2];  
    If[d <=0., uX=0;       lambda= N[2*c*uY*(H+uY)*(2*H+uY)],
               uX=Sqrt[d]; lambda= N[-c*S^2*(H+uY)]];
    plotelem1=Graphics[Line[{{-S/2,0},{uX,H+uY}}]];
    plotelem2=Graphics[Line[{{ uX,H+uY},{S/2,0}}]];
    plotload =Graphics[Line[{{ uX,H+uY},{uX,H+uY-lambda}}]];
    a = Min[.5*Abs[lambda],0.2*H];
    If [lambda>0,    plotarrow=Graphics[{
        Line[{{ uX,H+uY},{uX-a/2,H+uY-a}}],
        Line[{{ uX,H+uY},{uX+a/2,H+uY-a}}]} ],
                     plotarrow=Graphics[{
        Line[{{ uX,H+uY},{uX-a/2,H+uY+a}}],
        Line[{{ uX,H+uY},{uX+a/2,H+uY+a}}]} ]  ];
    Show[Graphics[Thickness[.004]],plotelem1,plotelem2,
      Graphics[Thickness[.005]],Graphics[RGBColor[1,0,1]],plotload,plotarrow,
      PlotRange->{{-2*H,2*H},{-2*H,2*H}},AspectRatio->1],
      {n,0,nmax}];\
\>", "Input",
  CellFrame->True,
  CellMargins->{{11, 34}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}}],

Cell[CellGroupData[{

Cell["\<\
FormIntForce2DTwoNodeBar[XY1_,XY2_,uXY1_,uXY2_,Em_,A0_,s0_]:=
  Module[{X1,Y1,X2,Y2,X21,Y21,uX1,uY1,uX2,uY2,uX21,uY21,L0,L,e,s,ax,ay,ifv}, 
    {X1,Y1}=XY1; {X2,Y2}=XY2; X21=X2-X1; Y21=Y2-Y1; 
    {uX1,uY1}=uXY1; {uX2,uY2}=uXY2; uX21=uX2-uX1; uY21=uY2-uY1;
    L0=Sqrt[X21^2+Y21^2];     L=Sqrt[(X21+uX21)^2+(Y21+uY21)^2];
    e=(L-L0)*(L+L0)/(2*L0^2); s=s0+Em*e; 
    ax=(X21+uX21)/L0;         ay=(Y21+uY21)/L0;
    ifv=A0*s*{{-ax}, {-ay}, {ax}, {ay}};
    Return[Simplify[ifv]]
        ];
FormTangentStiff2DTwoNodeBar[XY1_,XY2_,uXY1_,uXY2_,Em_,A0_,s0_]:=
  Module[{X1,Y1,X2,Y2,X21,Y21,uX1,uY1,uX2,uY2,uX21,uY21,L0,L,e,s,ax,ay,esm}, 
    {X1,Y1}=XY1; {X2,Y2}=XY2; X21=X2-X1; Y21=Y2-Y1; 
    {uX1,uY1}=uXY1; {uX2,uY2}=uXY2; uX21=uX2-uX1; uY21=uY2-uY1;
    L0=Sqrt[X21^2+Y21^2];     L=Sqrt[(X21+uX21)^2+(Y21+uY21)^2];
    e=(L-L0)*(L+L0)/(2*L0^2); s=s0+Em*e; 
    ax=(X21+uX21)/L0;         ay=(Y21+uY21)/L0;
    esm=(Em*A0/L0)*{{ ax*ax, ax*ay,-ax*ax,-ax*ay},
                         { ax*ay, ay*ay,-ay*ax,-ay*ay},
                         {-ax*ax,-ay*ax, ax*ax, ay*ax},
                         {-ax*ay,-ay*ay, ay*ax, ay*ay}} +
        (A0*s/L0)*{{1,0,-1,0},{0,1,0,-1},{-1,0,1,0},{0,-1,0,1}};
        Return[Simplify[esm/.{Sqrt[S^2]->S,(1/Sqrt[S^2])->1/S}]]
        ];   

MergeElemIntoMasterIntForce[pe_,eftab_,p_]:=   
  Module[{i,ii,neldof,fmaster}, fmaster=p;
       neldof=Dimensions[eftab][[1]];
       For[i=1, i<=neldof, i++, ii=eftab[[i]];
           If [ii>0,fmaster[[ii,1]]+=pe[[i,1]]] 
          ]; Return[fmaster]
        ];
MergeElemIntoMasterStiff[esm_,eftab_,msm_]:=   
  Module[{i,j,ii,jj,neldof,Kmaster}, Kmaster=msm;
       neldof=Dimensions[esm][[1]];
       For[i=1, i<=neldof, i++, ii=eftab[[i]];
         For[j=i, j<=neldof, j++, jj=eftab[[j]];
           If [ii>0 && jj>0, 
           Kmaster[[jj,ii]]=Kmaster[[ii,jj]]+=esm[[i,j]]] 
            ]
          ]; Return[Kmaster]
        ];
 AssembleMasterIntForceOfQuiz3Arch[span_,height_,Em_,A0_,
          uX2_,uY2_,uX3_,uY3_,uX4_,uY4_]:=
  Module[{K1,K2,K3,K4,K5,K6,fmaster,S,H},

    fmaster=Table[0,{6},{1}]; S=span; H=height;
    f1=FormIntForce2DTwoNodeBar[{-S/2,0},{-S/4,3*H/4},
             {0,0},{uX2,uY2},Em,4*A0,0]; 
    fmaster=MergeElemIntoMasterIntForce[f1,{0,0,1,2},fmaster];
    f2=FormIntForce2DTwoNodeBar[{-S/4,3*H/4},{0,H},
             {uX2,uY2},{uX3,uY3},Em,4*A0,0]; 
    fmaster=MergeElemIntoMasterIntForce[f2,{1,2,3,4},fmaster];
    f3=FormIntForce2DTwoNodeBar[{0,H},{S/4,3*H/4},
             {uX3,uY3},{uX4,uY4},Em,4*A0,0];
    fmaster=MergeElemIntoMasterIntForce[f3,{3,4,5,6},fmaster];
    f4=FormIntForce2DTwoNodeBar[{S/4,3*H/4},{S/2,0},
             {uX4,uY4},{0,0},Em,4*A0,0]; 
    fmaster=MergeElemIntoMasterIntForce[f4,{5,6,0,0},fmaster];
    f5=FormIntForce2DTwoNodeBar[{-S/2,0},{S/4,3*H/4},
             {0,0},{uX4,uY4},Em,A0,0]; 
    fmaster=MergeElemIntoMasterIntForce[f5,{0,0,5,6},fmaster];
    f6=FormIntForce2DTwoNodeBar[{-S/4,3*H/4},{S/2,0},
             {uX2,uY2},{0,0},Em,A0,0]; 
    fmaster=MergeElemIntoMasterIntForce[f6,{1,2,0,0},fmaster];
     (*Print[\"f1=\",f1];Print[\"f2=\",f2];Print[\"f3=\",f3];
     Print[\"f4=\",f4];Print[\"f5=\",f5];Print[\"f6=\",f6];*)
    Return[Simplify[fmaster]]
        ];
AssembleMasterStiffOfQuiz3Arch[span_,height_,Em_,A0_,
   uX2_,uY2_,uX3_,uY3_,uX4_,uY4_]:=
  Module[{K1,K2,K3,K4,K5,K6,Kmaster,S,H},
    Kmaster=Table[0,{6},{6}]; S=span; H=height;
    K1=FormTangentStiff2DTwoNodeBar[{-S/2,0},{-S/4,3*H/4},
             {0,0},{uX2,uY2},Em,4*A0,0]; (*Print[K1//TableForm];*)
    Kmaster=MergeElemIntoMasterStiff[K1,{0,0,1,2},Kmaster];
    K2=FormTangentStiff2DTwoNodeBar[{-S/4,3*H/4},{0,H},
             {uX2,uY2},{uX3,uY3},Em,4*A0,0]; (*Print[K2//TableForm];*)
    Kmaster=MergeElemIntoMasterStiff[K2,{1,2,3,4},Kmaster];
    K3=FormTangentStiff2DTwoNodeBar[{0,H},{S/4,3*H/4},
             {uX3,uY3},{uX4,uY4},Em,4*A0,0]; (*Print[K1//TableForm];*)
    Kmaster=MergeElemIntoMasterStiff[K3,{3,4,5,6},Kmaster];
    K4=FormTangentStiff2DTwoNodeBar[{S/4,3*H/4},{S/2,0},
             {uX4,uY4},{0,0},Em,4*A0,0]; (*Print[K2//TableForm];*)
    Kmaster=MergeElemIntoMasterStiff[K4,{5,6,0,0},Kmaster];
    K5=FormTangentStiff2DTwoNodeBar[{-S/2,0},{S/4,3*H/4},
             {0,0},{uX4,uY4},Em,A0,0]; (*Print[K1//TableForm];*)
    Kmaster=MergeElemIntoMasterStiff[K5,{0,0,5,6},Kmaster];
    K6=FormTangentStiff2DTwoNodeBar[{-S/4,3*H/4},{S/2,0},
             {uX2,uY2},{0,0},Em,A0,0]; (*Print[K2//TableForm];*)
    Kmaster=MergeElemIntoMasterStiff[K6,{1,2,0,0},Kmaster];
    Return[Simplify[Kmaster]]
        ];   
       
ClearAll[Em,A0,S,H,uX2,uX3,uY2,uY3,uX4,uY4,f,lambda]; Em=1;A0=1;S=4;H=5;

f=AssembleMasterIntForceOfQuiz3Arch[S,H,Em,A0,uX2,uY2,uX3,uY3,uX4,uY4];
K=AssembleMasterStiffOfQuiz3Arch[S,H,Em,A0,uX2,uY2,uX3,uY3,uX4,uY4];
(*Print[\"Int Force Vector f=\",f//InputForm];*)

lambda=0.; {uX2p,uY2p,uX3p,uY3p,uX4p,uY4p}={0.,0.,0.,0.,0.,0.};
nsteps=50;
plot1=Table[{0,0},{nsteps}];   plot2=Table[{0,0},{nsteps}]; 
Do [
     reqs={f[[1,1]]==0,f[[2,1]]==-lambda,f[[3,1]]==0,
       f[[4,1]]==-lambda,f[[5,1]]==0,f[[6,1]]==-lambda};
   sol=FindRoot[reqs, {uX2,uX2p},{uY2,uY2p},{uX3,uX3p},
        {uY3,uY3p},{uX4,uX4p},{uY4,uY4p}];
   sol=Chop[sol];
   {uX2p,uY2p,uX3p,uY3p,uX4p,uY4p}={uX2,uY2,uX3,uY3,uX4,uY4}/.sol;
   uX3p=0.;
   Print[\"lambda= \",lambda,\"  v=\",{uX2,uY2,uX3,uY3,uX4,uY4}/.sol];
   Kt=N[K/.sol]; eKt=Eigenvalues[Kt]; (*Print[\"K eigs=\",eKt];*)
   qz=Transpose[{{0},{-1},{0},{-1},{0},{-1}}].Eigenvectors[Kt];
   plot1[[i]]={lambda,eKt[[6]]};   plot2[[i]]={lambda,qz[[1,6]]};
   lambda=lambda+.002,
   {i,1,nsteps}];
 ListPlot[plot1,PlotJoined->True];
 ListPlot[plot2,PlotJoined->True];
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