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  StyleBox["*gm}};\n   If [numer,KGe=N[KGe]]; If [!numer,KGe=Simplify[KGe]];\n\
   Return[KGe] ];\n   \n",
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Print[Chop[Eigenvalues[N[KG]]]];",
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  StyleBox["Cell 4.  Module to form the tangent stiffness matrix of the \
geometrically nonlinear, C0 plane beam.",
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Cell[TextData[{
  "TLPlaneBeamTanStiff",
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Cell["\<\
Cell 4a.  This is a Finite Difference check (by Central FDs)  that \
K = partial p /partial u\
\>", "Text",
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Cell["\<\
ClearAll[EA,GA,EI,N0,V0,M0,eps,tol,numer]; tol=0.001; eps=0.001; \
numer=True;
 {X1,Y1}={-3,7}; {X2,Y2}={-8,11}; u1={-.6,-.9,0}; u2={1.5,2.1,0};
 XYcoor={{X1,Y1},{X2,Y2}}; X21=X2-X1; Y21=Y2-Y1; 
 L0=Sqrt[X21^2+Y21^2]; u1={0.1,0.25,-0.3}; u2={0.45,0.58,-0.6};  
 EA=100; GA=240; EI=300;  (* GA=12*EI/L0^2; RBF *)
 S={EA,GA,EI}; z0={10,-20,30};
 u0=Flatten[{u1,u2}]; KFD=Table[0,{6},{6}];
 (*p0=TLPlaneBeamIntForce[XYcoor,u1,u2,S,z0];*)
 For [i=1, i<=6, i++,
   u=u0; u[[i]]=u[[i]]+eps; uXY\[Theta]={Take[u,3],Take[u,-3]};
   pplus= TLPlaneBeamIntForce[XYcoor,S,z0,uXY\[Theta],numer];
   u=u0; u[[i]]=u[[i]]-eps; uXY\[Theta]={Take[u,3],Take[u,-3]};
   pminus=TLPlaneBeamIntForce[XYcoor,S,z0,uXY\[Theta],numer];
   KFD[[i]]=Flatten[Simplify[(pplus-pminus)/(2*eps)]] ];
 KFD=N[KFD];
 Print[\"K by finite differences:\"];Print[KFD//MatrixForm];
 u=u0; uXY\[Theta]={Take[u,3],Take[u,-3]};
 K=TLPlaneBeamTanStiff[XYcoor,S,z0,uXY\[Theta],numer];
 Print[\"K by stiffness modules:\"]; Print[K//MatrixForm];
 Print[\"This diff should be zero or small\",
       \" (Chop used with \",tol,\" tolerance)\"];
 Print[Chop[K-KFD,tol]//MatrixForm];
 Print[Chop[Eigenvalues[K]]];
 Print[Chop[Eigenvalues[KFD]]];\
\>", "Input",
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Cell[TextData[{
  StyleBox["Cell 5.  Module  ",
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  "CantBeamMasterIntForce",
  StyleBox["  assembles the internal force vector of\na compressed cantilever \
beam-column  of total length L, clamped on the left, and discretized\nwith Ne \
elements.  The structure is subjected to an applied axial end force P, \
positive\nto the right (so that P<0 means the beam-column is compressed).\n\
Support module: ",
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  " MergeElemIntoMasterIntForce"
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Cell[TextData[{
  "CantBeamMasterIntForce[{L_,P_,Ne_},{Em_,A0_,I0_},u_,numer_]:=\n  \
Module[{e,numnod,numdof,u1,u2,u3,Le,X1,X2,S0,z0,\n          \
XYcoor,eftab,pe,p}, numdof=3*numnod-3; Le=L/Ne;\n   numnod=Ne+1; \
p=Table[0,{numdof},{1}];\n   z0={P,0,0}; S0={Em*A0,12*Em*I0/Le^2,Em*I0}; (* \
RBF *)\n   For [e=1, e<=Ne, e++,\n     X1=(e-1)*Le; X2=e*Le; \
XYcoor={{X1,0},{X2,0}}; \n     If [e==1, u1={0,0,0}; u2=Take[u,{3*e-2,3*e}];\n\
         eftab={0,0,0,3*e-2,3*e-1,3*e}];\n     If [e>1, \
u1=Take[u,{3*e-5,3*e-3}]; u2=Take[u,{3*e-2,3*e}];\n         \
eftab={3*e-5,3*e-4,3*e-3,3*e-2,3*e-1,3*e}];\n     \
pe=TLPlaneBeamIntForce[XYcoor,S0,z0,{u1,u2},numer];\n",
  StyleBox["     (*Print[\"pe=\",pe];*)",
    FontColor->RGBColor[1, 0, 0]],
  "\n     p=MergeElemIntoMasterIntForce[pe,eftab,p];\n   ];\n   \
Return[Simplify[p]]\n ];\n MergeElemIntoMasterIntForce[pe_,eftab_,pm_]:=   \n \
 Module[{i,ii,nf=Length[eftab],p}, p=pm; \n         For [i=1, i<=nf, i++, \
ii=eftab[[i]];\n              If [ii<=0,Continue[]];\n              \
p[[ii,1]]+=pe[[i,1]] \n             ]; Return[p]\n        ];"
}], "Input",
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Cell[TextData[{
  StyleBox["Cell 6. Module ",
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  "CantBeamMasterTanStiff",
  StyleBox["    assembles the master stiffness matrix of\na compressed \
cantilever beam-column  of total length L, clamped on the left, and \
discretized\nwith Ne elements.  The structure is subjected to an applied \
axial end force P, positive\nto the right (so that P<0 means the beam-column \
is compressed).\nSupport module: ",
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  "MergeElemIntoMasterStiff"
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Cell[TextData[{
  "CantBeamMasterTanStiff[{L_,P_,Ne_},{Em_,A0_,I0_},u_,numer_]:=\n  \
Module[{e,numnod,numdof,u1,u2,u3,Le,X1,X2,S0,z0,\n         \
XYcoor,uXY\[Theta],eftab,pe,p}, numdof=3*numnod-3; Le=L/Ne;\n   numnod=Ne+1; \
K=Table[0,{numdof},{numdof}];\n   z0={P,0,0}; S0={Em*A0,12*Em*I0/Le^2,Em*I0}; \
(* RBF *)\n   For [e=1, e<=Ne, e++,\n     X1=(e-1)*Le; X2=e*Le; \
XYcoor={{X1,0},{X2,0}};\n     If [e==1,u1={0,0,0}; u2=Take[u,{3*e-2,3*e}];\n  \
       eftab={0,0,0,3*e-2,3*e-1,3*e}];\n     If [e>1, \
u1=Take[u,{3*e-5,3*e-3}]; u2=Take[u,{3*e-2,3*e}];\n         \
eftab={3*e-5,3*e-4,3*e-3,3*e-2,3*e-1,3*e}];\n     \
Ke=TLPlaneBeamTanStiff[XYcoor,S0,z0,{u1,u2},numer];\n",
  StyleBox["     (* Print[\"Ke=\",Ke//MatrixForm]; *)",
    FontColor->RGBColor[1, 0, 0]],
  "\n     K=MergeElemIntoMasterStiff[Ke,eftab,K];\n   ];\n   \
Return[Simplify[K]]\n ];\nMergeElemIntoMasterStiff[Ke_,eftab_,Km_]:=   \n  \
Module[{i,j,ii,jj,nf=Length[eftab],K}, K=Km; \n       For[i=1, i<=nf, i++, \
ii=eftab[[i]]; \n           If [ii<=0,Continue[]];\n           For [j=i, \
j<=nf, j++, jj=eftab[[j]];\n               If [jj>0, \n               \
K[[jj,ii]]=K[[ii,jj]]+=Ke[[i,j]]] \n               ]\n          ]; Return[K]\n\
        ];"
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Cell[TextData[{
  StyleBox[" Cell 7. The following script calculates the buckling load factor \
of the compressed beam-column for an\n increasing number of elements: 1,2, \
4,... etc, using several of the  foregoing modules in cells 1,2,4,5.   \n The \
calculation is brute force, looking for the roots of the exact characteristic \
polynomial of K\n with NSolve.  Method breaks  down at about 16 or 32 \
elements, depending on computational power, \n because the coefficients of \
det[K] grow very rapidly.    ",
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  "Needs modules in initialization cells.  ",
  StyleBox["\n This is the topic of Exercise 11.3",
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  " "
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Cell[TextData[{
  "ClearAll[L,P,Em,A0,I0]; Em=A0=I0=L=1; Ne=1; numer=False;\nFor [k=1, k<=2, \
k++, \n   Print[\"Number of elements=\",Ne];  \n   EAI={Em,A0,I0}; \
u=Table[0,{3*Ne+3}];  \n   K=CantBeamMasterTanStiff[{L,P,Ne},EAI,u,numer]; \n \
 ",
  StyleBox[" (*Print[\"K=\",K//MatrixForm];*)",
    FontColor->RGBColor[1, 0, 0]],
  "\n   detK=Det[K];  ",
  StyleBox["(*Print[\"det(K)=\",detK];*)",
    FontColor->RGBColor[1, 0, 0]],
  "\n   roots=NSolve[detK==0,P];\n   Print[\"roots of stability \
det=\",roots]; Pcr=P/.roots;\n   exact=-N[\[Pi]^2/4]; dmax=10^20; \
bestroot=Pcr[[1]]; \n   For [i=1,i<=Length[Pcr],i++, d=Abs[Pcr[[i]]-0];\n     \
   If [d>=dmax,Continue[]]; dmax=d; bestroot=Pcr[[i]]];       \n   \
Print[\"root closest to zero is \",bestroot,\n         \", error: \
\",(bestroot-exact)];   \n   Ne=2*Ne;\n ];\nPrint[\"exact buckling load coeff \
is \",-N[Pi^2/4]];"
}], "Input",
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Cell["\<\
Cell 8. This is another, quicker and numerically more stable \
method, to get the 
critical buckling load factor. Needs modules in initialization cells.   
On my Mac PowerBook  it works up to 128 elements.  
This is the subject of Exercise 11.4\
\>", "Text",
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Cell[TextData[{
  StyleBox["ClearAll[L,P,Em,A0,I0]; Em=A0=I0=L=1; Ne=1; numer=False;\nFor \
[k=1, k<=2, k++, \n   Print[\"Number of elements=\",Ne];  \n   LPNe={L,P,Ne}; \
EAI={Em,A0,I0}; u=Table[0,{3*Ne+3}];  \n   K=",
    CellFrame->True,
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    CellLabelMargins->{{9, Inherited}, {Inherited, Inherited}},
    AspectRatioFixed->True],
  "CantBeamMasterTanStiff",
  StyleBox["[{L,P,Ne},EAI,u,numer]; \n   KM=Coefficient[K,P,0]; \
KG=Coefficient[K,P,1]; \n   A=LinearSolve[N[KM],N[KG]];\n ",
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  StyleBox["  \
(*Print[\"KM=\",KM//MatrixForm];Print[\"KG=\",KG//MatrixForm];*)\n   (*Print[\
\"A=\",A//MatrixForm];*) ",
    CellFrame->True,
    CellMargins->{{18, 43}, {Inherited, Inherited}},
    CellLabelMargins->{{9, Inherited}, {Inherited, Inherited}},
    AspectRatioFixed->True,
    FontColor->RGBColor[1, 0, 0]],
  StyleBox["\n   emax=-Max[Eigenvalues[A]]; Print[\"FEM lambda \
cr=\",(1/emax)];\n   Ne=2*Ne;\n ];\nPrint[\"exact buckling lambda coeff is \
\",-N[Pi^2/4]];",
    CellFrame->True,
    CellMargins->{{18, 43}, {Inherited, Inherited}},
    CellLabelMargins->{{9, Inherited}, {Inherited, Inherited}},
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  "       "
}], "Input",
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Cell["\<\
Cell 9.  Extrapolation of last cell results applying the Shanks \
transformation yields -\[Pi]^2/4 to 9 digits\
\>", "Text",
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Cell["\<\
Shanks[Z_]:= Module[{m,n,shankedZ},
  If [Length[Dimensions[Z]]<=1, n=Length[Z];  
      If [n<=2, Return[{}]]; shankedZ=Table[0,{n-2}];
      Do [
      shankedZ[[j]]=( Z[[j+2]]*  Z[[j]]  -Z[[j+1]]^2)/
                     (Z[[j+2]]-2*Z[[j+1]]+Z[[j]]),
      {j,1,n-2}]; Return[shankedZ];
     ];
  {m,n}=Dimensions[Z]; 
  If [n<=2, Return[{{}}]];
  shankedZ=Table[0,{m},{n-2}];
  Do [Do [
      shankedZ[[i,j]]=( Z[[i,j+2]]*  Z[[i,j]]  -Z[[i,j+1]]^2)/
                       (Z[[i,j+2]]-2*Z[[i,j+1]]+Z[[i,j]]),
   {j,1,n-2}], {i,1,m}];
  Return[shankedZ]];
  
Z={-3.165151389911680,-2.603908889389283,-2.499688345543354,
   -2.475364052205859,-2.469385123708907,-2.467896687873336,
   -2.467524972077594,-2.467432067643745};
Print[\"Shanked sequence=\",Shanks[Z]//InputForm,
     \" vs. exact: -2.467401100272339\"];
Print[\"Re-Shanked sequence=\",Shanks[Shanks[Z]]//InputForm,
     \" vs. exact: -2.467401100272339\"];
Print[\"Re-Re-Shanked sequence=\",Shanks[Shanks[Shanks[Z]]]//InputForm,
     \" vs. exact: -2.467401100272339\"];    \
\>", "Input",
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ShowCellLabel->True,
ShowCellTags->False,
RenderingOptions->{"ObjectDithering"->True,
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Magnification->2,
MacintoshSystemPageSetup->"\<\
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0000I00000400`<300000Gd000400@00000000000000060801T1T00000000000
00000000000000000000000000000000\>"
]

(*******************************************************************
Cached data follows.  If you edit this Notebook file directly, not
using Mathematica, you must remove the line containing CacheID at
the top of  the file.  The cache data will then be recreated when
you save this file from within Mathematica.
*******************************************************************)

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(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)