University of Florida/Eml4507/Team 7 Report 6

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Problem 1[edit]

On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.

Find[edit]

Solve the general eigenvalue problem by first converting it to the standard eigenvalue problem for the spring-mass-damper system given on p.53-13 of 'Fead.s13.sec.53b'.

Given[edit]



Solution[edit]

Converting the general eigenvalue problem to the standard eigenvalue problem and solving it

General eigenvalue form:



Standard eigenvalue form:



Where,




















Calculating the determinant



Therefore, the Eigenvalues are:

The next step is finding the Eigenvectors



Letting and calculating

  





Letting and calculating

  




These results compare favorably with those from problem 5.6 using CALFEM.

Mass Orthogonality:
The eigenvectors are therefore proven to be mass-orthogonal.

Problem 2[edit]

On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.

Find[edit]

The Lowest 3 eigenpairs by transforming the generalized eigenvalue problem into a standard eigenvalue problem and plot the 3 lowest mode shapes in a gif image.

Given[edit]

Figure 6.1: 25 member truss with L=0.3 m

Solution[edit]

 
clear
clc

%GIVENS
L=0.3;
E=100e9;
A=1e-4;
rho=5000;
ep=[E A];

%ELEMENT DOF
Edof=zeros(25,5);
for i=1:6
    j=2*(i-1);
    Edof(i,:)=[i,1+j,2+j,3+j,4+j];
end
for i=7:12
    j=2*(i+1);
    Edof(i,:)=[i,j-1,j,j+1,j+2];
end
for i=13:19
    j=2*(i-13);
    Edof(i,:)=[i,1+j,2+j,15+j,16+j];
end
for i=20:25
    j=2*(i-20);
    Edof(i,:)=[i,1+j,2+j,17+j,18+j];
end

%COORDINATES
ex=zeros(14,2);
ey=zeros(14,2);
for i=1:25
    if i<7
        ex(i,:)=[L*(i-1),L*i];
        ey(i,:)=[0,0];
    elseif i>6&&i<13
        ex(i,:)=[L*(i-7),L*(i-6)];
        ey(i,:)=[L,L];
    elseif i>12&&i<20
        ex(i,:)=[L*(i-13),L*(i-13)];
        ey(i,:)=[0,L];
    else
        ex(i,:)=[L*(i-20),L*(i-19)];
        ey(i,:)=[0,L];
    end
end

%STIFFNESS AND MASS MATRIX
K=zeros(28);
M=zeros(28);
for i=1:25
    E=ep(1);A=ep(2);
    xt=ex(i,2)-ex(i,1);
    yt=ey(i,2)-ey(i,1);
    L=sqrt(xt^2+yt^2);
    l=xt/L; m=yt/L;
    ke=E*A/L*[l^2 l*m -l^2 -l*m;
              l*m m^2 -l*m -m^2;
              -l^2 -l*m l^2 l*m;
              -l*m -m^2 l*m m^2;];
    m=L*A*rho;
    me=[m/2 0 0 0;
        0 m/2 0 0;
        0 0 m/2 0;
        0 0 0 m/2];
    edoft=Edof(i,2:5);
    K(edoft,edoft)=K(edoft,edoft)+ke;
    M(edoft,edoft)=M(edoft,edoft)+me;
end

%FIND THE EIGEN VALUES AND VECTORS
ndof=size(K,1);
freedof=[1:ndof]';
fixdof=[1 15 16]';
freedof(fixdof(:))=[];
Kred=K(freedof,freedof);
Mred=M(freedof,freedof);

Ks=Mred^(-1/2)*Kred*Mred^(-1/2);
[X1,D1]=eig(Ks);
X2=Mred^(-1/2)*X1;
for j=1:size(X2)
    d=sqrt(X2(:,j)'*X2(:,j));
    X3(:,j)=X2(:,j)/d;
end
[D1,i]=sort(diag(D1));
X4=X3(:,i);
eigenval=D1;
eigenvec=zeros(size(K,1),size(X4,1));
eigenvec(freedof,:)=X4;

%PLOT THE GIF
for kk=1:3
    eigenvalx=eigenval(kk);
    eigenvecx=eigenvec(:,kk);
    w=sqrt(eigenvalx);
    T=2*pi/w
    dt=T/100;
    t=0:dt:T;
    figure(kk)
    
    if kk==1;
        filename = 'mode1 r5p2.gif';
    elseif kk==2;
        filename = 'mode2 r5p2.gif';
    elseif kk==3;
        filename = 'mode3 r5p2.gif';
    end
    
    for jj = 1:100
        tt=t(jj);
        d=eigenvecx*sin(w*tt);
        d=d';
        rr=Edof(:,2:5);
        ed=zeros(25,4);
        for i=1:25;
            ed(i,1:4)=d(rr(i,:));
        end
        xd=(ex+ed(:,[1 3]))';
        yd=(ey+ed(:,[2 4]))';
        s1=['-' , 'k'];
        axis manual
        plot(xd,yd,s1)
        set(gca, 'ylim', [-1.5 1.5], 'xlim', [0 3]);
        drawnow
        frame = getframe(1);
        im = frame2im(frame);
        [imind,cm] = rgb2ind(im,256);
        if jj == 1;
            imwrite(imind,cm,filename,'gif', 'Loopcount',inf);
        else
            imwrite(imind,cm,filename,'gif','WriteMode','append');
        end
    end
end

The resulting egienpairs are:

The mode shapes for the previous eigenpairs are"

Figure 4.2: First eigen shape
Figure 4.3: Second eigen shape
Figure 4.4: Third eigen shape

Problem 3[edit]

On our honor, the solution to this problem was generated by referring to the solution to Problem 3.1 at these locations: Eml4507 Team7 Report3, 
Team Negative Damping (3): Report 3, Eml4507.s13.team2/Report3, and EML4507.s13.team4ever.R3. Their solutions were not copied, but consulted.

Find[edit]

Using FE matlab code, with non-zero prescribed displacement degrees-of-freedom, using the same "connectivity array" given in the same problem statement, provide all of the unknown global displacement degrees-of-freedom, force components(reactions), and member forces. Plot the deformed shape superposed on the undeformed shape. Verify the results with CALFEM.

Compare the reactions in the case of non-zero prescribed disp dofs with the reactions in the case of zero prescribed disp dofs.

Given[edit]

Figure 1.1: KS 2008 p.81 Example 2.5 [1]

In the non-zero prescribed case:








In the zero prescribed case:









Solution[edit]

MATLAB code:

%Displacements
d3= 0.02; %in m
d4= -0.01; 
d5= -0.03;
d6= 0.05;
d7 = 0;
d8 = 0;
ph1= 0;
ph2 = 0;
dsmall = [d3; d4; d5; d6;]; %ph =placeholder, displacement unknown
theta=45;
%Property Matrix= [element number; EA/L; angle;];
P = [1:3; 10^5*206 10^5*206 10^5*206; -pi/6 pi/2 -5*pi/6;];
%angles refer to 1->3, 1->2, 1->4
%Global Coordinate System

GNC = [ 0, 0;
        0, 1;
     .866, -.5;
     -.866, -.5;];

lm = [.866, -.5;
      0   ,    1;
      -.866, -.5;];

%populate global force matrix
dof = 8; %4 nodes, 2 directions reach
F = zeros(dof,1);
F(1) = 20000*cos(pi/4);
F(2) = 20000*sin(pi/4);

%populate stiffness matrices
    lsq1 = (lm(1,1))^2;
    msq1 = (lm(1,2))^2;
    ltimesm1 = lm(1,1)*lm(1,2);
    Ke1 = [ lsq1, ltimesm1, 0, 0, -lsq1, -ltimesm1, 0, 0;
             ltimesm1, msq1, 0, 0 -ltimesm1, -msq1, 0, 0;
             0, 0, 0, 0, 0, 0, 0, 0;
             0, 0, 0, 0, 0, 0, 0, 0;
             -lsq1, -ltimesm1, 0, 0, lsq1, ltimesm1, 0, 0;
             -ltimesm1, -msq1, 0, 0, ltimesm1, msq1, 0, 0;
             0, 0, 0, 0, 0, 0, 0, 0;
             0, 0, 0, 0, 0, 0, 0, 0;]; 
         
    lsq2 = (lm(2,1))^2;
    msq2 = (lm(2,2))^2;
    ltimesm2 = lm(2,1)*lm(2,2);
    Ke2 = [ lsq2, ltimesm2, -lsq2, -ltimesm2, 0, 0, 0, 0;
             ltimesm2, msq2, -ltimesm2, -msq2, 0, 0, 0, 0;
             -lsq2, -ltimesm2, lsq2, ltimesm2, 0, 0, 0, 0;
             -ltimesm2, -msq2, ltimesm2, msq2, 0, 0, 0, 0;
             0, 0, 0, 0, 0, 0, 0, 0;
             0, 0, 0, 0, 0, 0, 0, 0;
             0, 0, 0, 0, 0, 0, 0, 0;
             0, 0, 0, 0, 0, 0, 0, 0;];
         
    lsq3 = (lm(3,1))^2;
    msq3 = (lm(3,2))^2;
    ltimesm3 = lm(3,1)*lm(3,2);
    Ke3 = [ lsq3, ltimesm3, 0, 0, 0, 0, -lsq3, -ltimesm3;
             ltimesm3, msq3, 0, 0, 0, 0, -ltimesm3, -msq3;
             0, 0, 0, 0, 0, 0, 0, 0;
             0, 0, 0, 0, 0, 0, 0, 0;
             0, 0, 0, 0, 0, 0, 0, 0;
             0, 0, 0, 0, 0, 0, 0, 0;
             -lsq3, -ltimesm3, 0, 0, 0, 0, lsq3, ltimesm3;
             -ltimesm3, -msq3, 0, 0, 0, 0, ltimesm3, msq3;]; 

Kglobal = P(2,1)*(Ke1+Ke2+Ke3)

%find unknown displacements, after reducing
Fsmall = [F(1) F(2)]';
Ksmall = zeros([2,6]);

for i=1:2
        for j=1:6
        Ksmall (i,j) = Kglobal((i+1),j);
       end
end

unknownds = Ksmall\Fsmall
d1 = unknownds(1);
d2 = unknownds(2);

Ksmall2 = zeros([4,4]);

for i=1:4
        for j=1:4
        Ksmall2 (i,j) = Kglobal((i+1),j);
       end
end

unknownFs = Ksmall2* dsmall

P1 = 10^5*206*.866*(d5-d1)-.5*(d6-d2)
P2 = 10^10*206*0*(d3-d1)-1*(d4-d2)
P3 = 10^10*206*-.866*(d7-d1)-.5*(d8-d2)

Resulting in:

Kglobal =

 Columns 1 through 7
  3.0898e+07            0            0            0  -1.5449e+07   8.9198e+06  -1.5449e+07
           0     3.09e+07            0    -2.06e+07   8.9198e+06    -5.15e+06  -8.9198e+06
           0            0            0            0            0            0            0
           0    -2.06e+07            0     2.06e+07            0            0            0
 -1.5449e+07   8.9198e+06            0            0   1.5449e+07  -8.9198e+06            0
  8.9198e+06    -5.15e+06            0            0  -8.9198e+06     5.15e+06            0
 -1.5449e+07  -8.9198e+06            0            0            0            0   1.5449e+07
 -8.9198e+06    -5.15e+06            0            0            0            0   8.9198e+06
 Column 8
 -8.9198e+06
   -5.15e+06
           0
           0
           0
           0
  8.9198e+06
    5.15e+06

unknownds =

  0.00045767
  0.00045767
unknownFs =
  -1.339e+06
           0
   1.236e+06
 -3.9818e+05


P1 =

 -5.3519e+05


P2 =

    1045.8

P3 =

    22.884


In the zero case: In the zero prescribed case:








Simply strike the rows and columns, resulting in:

unknownds =

  0.00045767
  0.00045767

All the outer node reactions are zero, since their displacements are zero.

P1 =

 -0.34500e+05


P2 =

 -0.94400e+05

P3 =

 1.29000e+05

Problem 4[edit]

On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.

Find[edit]

Solve the initial truss structure using truss FEs

Given[edit]

3dtruss.JPG

Solution[edit]

 
function FEAr6p4

nodelocation=[-1.5 0 8;
     1.5 0 8;
    -1.5 1.5 4;
    1.5 1.5 4;
    1.5 -1.5 4;
    -1.5 -1.5 4;
    -4 4 0;
    4 4 0;
    4 -4 0;
    -4 -4 0];

%INPUT:   ex = [-1.5 1.5]
%         ey = [0 0]         element node coordinates
%         ez = [z1 z2]
%    
%         ep = [E A]           E: Young's modulus
%                              A: cross section area
%
   
ex=[nodelocation(1,1) nodelocation(2,1);
    nodelocation(4,1) nodelocation(1,1);
    nodelocation(3,1) nodelocation(2,1);
    nodelocation(5,1) nodelocation(1,1);
    nodelocation(6,1) nodelocation(2,1);
    nodelocation(4,1) nodelocation(2,1);
    nodelocation(5,1) nodelocation(2,1);
    nodelocation(3,1) nodelocation(1,1);
    nodelocation(6,1) nodelocation(1,1);
    nodelocation(6,1) nodelocation(3,1);
    nodelocation(5,1) nodelocation(4,1);
    nodelocation(4,1) nodelocation(3,1);
    nodelocation(6,1) nodelocation(5,1);
    nodelocation(10,1) nodelocation(3,1);
    nodelocation(7,1) nodelocation(6,1);
    nodelocation(9,1) nodelocation(4,1);
    nodelocation(8,1) nodelocation(5,1);
    nodelocation(7,1) nodelocation(4,1);
    nodelocation(8,1) nodelocation(3,1);
    nodelocation(10,1) nodelocation(5,1);
    nodelocation(9,1) nodelocation(6,1);
    nodelocation(10,1) nodelocation(6,1);
    nodelocation(7,1) nodelocation(3,1);
    nodelocation(8,1) nodelocation(4,1);
    nodelocation(9,1) nodelocation(5,1);];

ey=[nodelocation(1,2) nodelocation(2,2);
    nodelocation(4,2) nodelocation(1,2);
    nodelocation(3,2) nodelocation(2,2);
    nodelocation(5,2) nodelocation(1,2);
    nodelocation(6,2) nodelocation(2,2);
    nodelocation(4,2) nodelocation(2,2);
    nodelocation(5,2) nodelocation(2,2);
    nodelocation(3,2) nodelocation(1,2);
    nodelocation(6,2) nodelocation(1,2);
    nodelocation(6,2) nodelocation(3,2);
    nodelocation(5,2) nodelocation(4,2);
    nodelocation(4,2) nodelocation(3,2);
    nodelocation(6,2) nodelocation(5,2);
    nodelocation(10,2) nodelocation(3,2);
    nodelocation(7,2) nodelocation(6,2);
    nodelocation(9,2) nodelocation(4,2);
    nodelocation(8,2) nodelocation(5,2);
    nodelocation(7,2) nodelocation(4,2);
    nodelocation(8,2) nodelocation(3,2);
    nodelocation(10,2) nodelocation(5,2);
    nodelocation(9,2) nodelocation(6,2);
    nodelocation(10,2) nodelocation(6,2);
    nodelocation(7,2) nodelocation(3,2);
    nodelocation(8,2) nodelocation(4,2);
    nodelocation(9,2) nodelocation(5,2);];

ez=[nodelocation(1,3) nodelocation(2,3);
    nodelocation(4,3) nodelocation(1,3);
    nodelocation(3,3) nodelocation(2,3);
    nodelocation(5,3) nodelocation(1,3);
    nodelocation(6,3) nodelocation(2,3);
    nodelocation(4,3) nodelocation(2,3);
    nodelocation(5,3) nodelocation(2,3);
    nodelocation(3,3) nodelocation(1,3);
    nodelocation(6,3) nodelocation(1,3);
    nodelocation(6,3) nodelocation(3,3);
    nodelocation(5,3) nodelocation(4,3);
    nodelocation(4,3) nodelocation(3,3);
    nodelocation(6,3) nodelocation(5,3);
    nodelocation(10,3) nodelocation(3,3);
    nodelocation(7,3) nodelocation(6,3);
    nodelocation(9,3) nodelocation(4,3);
    nodelocation(8,3) nodelocation(5,3);
    nodelocation(7,3) nodelocation(4,3);
    nodelocation(8,3) nodelocation(3,3);
    nodelocation(10,3) nodelocation(5,3);
    nodelocation(9,3) nodelocation(6,3);
    nodelocation(10,3) nodelocation(6,3);
    nodelocation(7,3) nodelocation(3,3);
    nodelocation(8,3) nodelocation(4,3);
    nodelocation(9,3) nodelocation(5,3);];


E=3*10^7;  A=0.0218; 
 
 B=zeros(3, 25);
 Kefinal=zeros(25,1);
 
 
 for i=1:25
  
     
  B(1,i)=ex(i,2)-ex(i,1) ;
  B(2,i)= ey(i,2)-ey(i,1);
  B(3,i)=ez(i,2)-ez(i,1) ;
  
  for i=1:25
 l=B(:,i);
 L=sqrt(l'*l);
Kle=E*A/L*[1 -1;
            -1  1];
n=l'/L   
  G=[   n   zeros(size(n));
    zeros(size(n))   n   ];

 Ke=G'*Kle*G
 
 end
 end
 
%  b=B*B'
%  L=sqrt(b)
% Kle=E*A/L*[1 -1;
%             -1  1];
% 
%   n=B'/L;   
%   G=[   n   zeros(size(n));
%     zeros(size(n))   n   ]
% 
% Ke=G'*Kle*G;

OUTPUT:

  Columns 1 through 4

    1.0745   -2.3640    1.7192   -1.0745
   -2.3640    5.2007   -3.7823    2.3640
    1.7192   -3.7823    2.7508   -1.7192
   -1.0745    2.3640   -1.7192    1.0745
    2.3640   -5.2007    3.7823   -2.3640
   -1.7192    3.7823   -2.7508    1.7192

  Columns 5 through 6

    2.3640   -1.7192
   -5.2007    3.7823
    3.7823   -2.7508
   -2.3640    1.7192
    5.2007   -3.7823
   -3.7823    2.7508


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
           0           0           0
     -218000           0           0
           0           0           0
           0           0           0

  Columns 4 through 6

     -218000           0           0
           0           0           0
           0           0           0
      218000           0           0
           0           0           0
           0           0           0


Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689   -5.5171   -4.1378
    2.0689    1.0345   -2.7585   -2.0689
   -5.5171   -2.7585    7.3561    5.5171
   -4.1378   -2.0689    5.5171    4.1378
   -2.0689   -1.0345    2.7585    2.0689
    5.5171    2.7585   -7.3561   -5.5171

  Columns 5 through 6

   -2.0689    5.5171
   -1.0345    2.7585
    2.7585   -7.3561
    2.0689   -5.5171
    1.0345   -2.7585
   -2.7585    7.3561


Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689    5.5171   -4.1378
   -2.0689    1.0345   -2.7585    2.0689
    5.5171   -2.7585    7.3561   -5.5171
   -4.1378    2.0689   -5.5171    4.1378
    2.0689   -1.0345    2.7585   -2.0689
   -5.5171    2.7585   -7.3561    5.5171

  Columns 5 through 6

    2.0689   -5.5171
   -1.0345    2.7585
    2.7585   -7.3561
   -2.0689    5.5171
    1.0345   -2.7585
   -2.7585    7.3561


Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689   -5.5171   -4.1378
   -2.0689    1.0345    2.7585    2.0689
   -5.5171    2.7585    7.3561    5.5171
   -4.1378    2.0689    5.5171    4.1378
    2.0689   -1.0345   -2.7585   -2.0689
    5.5171   -2.7585   -7.3561   -5.5171

  Columns 5 through 6

    2.0689    5.5171
   -1.0345   -2.7585
   -2.7585   -7.3561
   -2.0689   -5.5171
    1.0345    2.7585
    2.7585    7.3561


Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689    5.5171   -4.1378
    2.0689    1.0345    2.7585   -2.0689
    5.5171    2.7585    7.3561   -5.5171
   -4.1378   -2.0689   -5.5171    4.1378
   -2.0689   -1.0345   -2.7585    2.0689
   -5.5171   -2.7585   -7.3561    5.5171

  Columns 5 through 6

   -2.0689   -5.5171
   -1.0345   -2.7585
   -2.7585   -7.3561
    2.0689    5.5171
    1.0345    2.7585
    2.7585    7.3561


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887   -0.5033         0
         0   -0.5033    1.3422         0
         0         0         0         0
         0   -0.1887    0.5033         0
         0    0.5033   -1.3422         0

  Columns 5 through 6

         0         0
   -0.1887    0.5033
    0.5033   -1.3422
         0         0
    0.1887   -0.5033
   -0.5033    1.3422


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887    0.5033         0
         0    0.5033    1.3422         0
         0         0         0         0
         0   -0.1887   -0.5033         0
         0   -0.5033   -1.3422         0

  Columns 5 through 6

         0         0
   -0.1887   -0.5033
   -0.5033   -1.3422
         0         0
    0.1887    0.5033
    0.5033    1.3422


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887   -0.5033         0
         0   -0.5033    1.3422         0
         0         0         0         0
         0   -0.1887    0.5033         0
         0    0.5033   -1.3422         0

  Columns 5 through 6

         0         0
   -0.1887    0.5033
    0.5033   -1.3422
         0         0
    0.1887   -0.5033
   -0.5033    1.3422


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887    0.5033         0
         0    0.5033    1.3422         0
         0         0         0         0
         0   -0.1887   -0.5033         0
         0   -0.5033   -1.3422         0

  Columns 5 through 6

         0         0
   -0.1887   -0.5033
   -0.5033   -1.3422
         0         0
    0.1887    0.5033
    0.5033    1.3422


Ke =

  Columns 1 through 3

           0           0           0
           0      218000           0
           0           0           0
           0           0           0
           0     -218000           0
           0           0           0

  Columns 4 through 6

           0           0           0
           0     -218000           0
           0           0           0
           0           0           0
           0      218000           0
           0           0           0


Ke =

  Columns 1 through 3

           0           0           0
           0      218000           0
           0           0           0
           0           0           0
           0     -218000           0
           0           0           0

  Columns 4 through 6

           0           0           0
           0     -218000           0
           0           0           0
           0           0           0
           0      218000           0
           0           0           0


Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
           0           0           0
     -218000           0           0
           0           0           0
           0           0           0

  Columns 4 through 6

     -218000           0           0
           0           0           0
           0           0           0
      218000           0           0
           0           0           0
           0           0           0


Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
           0           0           0
     -218000           0           0
           0           0           0
           0           0           0

  Columns 4 through 6

     -218000           0           0
           0           0           0
           0           0           0
      218000           0           0
           0           0           0
           0           0           0


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640    1.7192   -1.0745
    2.3640    5.2007    3.7823   -2.3640
    1.7192    3.7823    2.7508   -1.7192
   -1.0745   -2.3640   -1.7192    1.0745
   -2.3640   -5.2007   -3.7823    2.3640
   -1.7192   -3.7823   -2.7508    1.7192

  Columns 5 through 6

   -2.3640   -1.7192
   -5.2007   -3.7823
   -3.7823   -2.7508
    2.3640    1.7192
    5.2007    3.7823
    3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640    1.7192   -1.0745
   -2.3640    5.2007   -3.7823    2.3640
    1.7192   -3.7823    2.7508   -1.7192
   -1.0745    2.3640   -1.7192    1.0745
    2.3640   -5.2007    3.7823   -2.3640
   -1.7192    3.7823   -2.7508    1.7192

  Columns 5 through 6

    2.3640   -1.7192
   -5.2007    3.7823
    3.7823   -2.7508
   -2.3640    1.7192
    5.2007   -3.7823
   -3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640   -1.7192   -1.0745
   -2.3640    5.2007    3.7823    2.3640
   -1.7192    3.7823    2.7508    1.7192
   -1.0745    2.3640    1.7192    1.0745
    2.3640   -5.2007   -3.7823   -2.3640
    1.7192   -3.7823   -2.7508   -1.7192

  Columns 5 through 6

    2.3640    1.7192
   -5.2007   -3.7823
   -3.7823   -2.7508
   -2.3640   -1.7192
    5.2007    3.7823
    3.7823    2.7508


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
           0           0           0
     -218000           0           0
           0           0           0
           0           0           0

  Columns 4 through 6

     -218000           0           0
           0           0           0
           0           0           0
      218000           0           0
           0           0           0
           0           0           0


Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689   -5.5171   -4.1378
    2.0689    1.0345   -2.7585   -2.0689
   -5.5171   -2.7585    7.3561    5.5171
   -4.1378   -2.0689    5.5171    4.1378
   -2.0689   -1.0345    2.7585    2.0689
    5.5171    2.7585   -7.3561   -5.5171

  Columns 5 through 6

   -2.0689    5.5171
   -1.0345    2.7585
    2.7585   -7.3561
    2.0689   -5.5171
    1.0345   -2.7585
   -2.7585    7.3561


Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689    5.5171   -4.1378
   -2.0689    1.0345   -2.7585    2.0689
    5.5171   -2.7585    7.3561   -5.5171
   -4.1378    2.0689   -5.5171    4.1378
    2.0689   -1.0345    2.7585   -2.0689
   -5.5171    2.7585   -7.3561    5.5171

  Columns 5 through 6

    2.0689   -5.5171
   -1.0345    2.7585
    2.7585   -7.3561
   -2.0689    5.5171
    1.0345   -2.7585
   -2.7585    7.3561


Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689   -5.5171   -4.1378
   -2.0689    1.0345    2.7585    2.0689
   -5.5171    2.7585    7.3561    5.5171
   -4.1378    2.0689    5.5171    4.1378
    2.0689   -1.0345   -2.7585   -2.0689
    5.5171   -2.7585   -7.3561   -5.5171

  Columns 5 through 6

    2.0689    5.5171
   -1.0345   -2.7585
   -2.7585   -7.3561
   -2.0689   -5.5171
    1.0345    2.7585
    2.7585    7.3561


Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689    5.5171   -4.1378
    2.0689    1.0345    2.7585   -2.0689
    5.5171    2.7585    7.3561   -5.5171
   -4.1378   -2.0689   -5.5171    4.1378
   -2.0689   -1.0345   -2.7585    2.0689
   -5.5171   -2.7585   -7.3561    5.5171

  Columns 5 through 6

   -2.0689   -5.5171
   -1.0345   -2.7585
   -2.7585   -7.3561
    2.0689    5.5171
    1.0345    2.7585
    2.7585    7.3561


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887   -0.5033         0
         0   -0.5033    1.3422         0
         0         0         0         0
         0   -0.1887    0.5033         0
         0    0.5033   -1.3422         0

  Columns 5 through 6

         0         0
   -0.1887    0.5033
    0.5033   -1.3422
         0         0
    0.1887   -0.5033
   -0.5033    1.3422


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887    0.5033         0
         0    0.5033    1.3422         0
         0         0         0         0
         0   -0.1887   -0.5033         0
         0   -0.5033   -1.3422         0

  Columns 5 through 6

         0         0
   -0.1887   -0.5033
   -0.5033   -1.3422
         0         0
    0.1887    0.5033
    0.5033    1.3422


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887   -0.5033         0
         0   -0.5033    1.3422         0
         0         0         0         0
         0   -0.1887    0.5033         0
         0    0.5033   -1.3422         0

  Columns 5 through 6

         0         0
   -0.1887    0.5033
    0.5033   -1.3422
         0         0
    0.1887   -0.5033
   -0.5033    1.3422


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887    0.5033         0
         0    0.5033    1.3422         0
         0         0         0         0
         0   -0.1887   -0.5033         0
         0   -0.5033   -1.3422         0

  Columns 5 through 6

         0         0
   -0.1887   -0.5033
   -0.5033   -1.3422
         0         0
    0.1887    0.5033
    0.5033    1.3422


Ke =

  Columns 1 through 3

           0           0           0
           0      218000           0
           0           0           0
           0           0           0
           0     -218000           0
           0           0           0

  Columns 4 through 6

           0           0           0
           0     -218000           0
           0           0           0
           0           0           0
           0      218000           0
           0           0           0


Ke =

  Columns 1 through 3

           0           0           0
           0      218000           0
           0           0           0
           0           0           0
           0     -218000           0
           0           0           0

  Columns 4 through 6

           0           0           0
           0     -218000           0
           0           0           0
           0           0           0
           0      218000           0
           0           0           0


Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
           0           0           0
     -218000           0           0
           0           0           0
           0           0           0

  Columns 4 through 6

     -218000           0           0
           0           0           0
           0           0           0
      218000           0           0
           0           0           0
           0           0           0


Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
           0           0           0
     -218000           0           0
           0           0           0
           0           0           0

  Columns 4 through 6

     -218000           0           0
           0           0           0
           0           0           0
      218000           0           0
           0           0           0
           0           0           0


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640    1.7192   -1.0745
    2.3640    5.2007    3.7823   -2.3640
    1.7192    3.7823    2.7508   -1.7192
   -1.0745   -2.3640   -1.7192    1.0745
   -2.3640   -5.2007   -3.7823    2.3640
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  Columns 5 through 6

   -2.3640   -1.7192
   -5.2007   -3.7823
   -3.7823   -2.7508
    2.3640    1.7192
    5.2007    3.7823
    3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640    1.7192   -1.0745
   -2.3640    5.2007   -3.7823    2.3640
    1.7192   -3.7823    2.7508   -1.7192
   -1.0745    2.3640   -1.7192    1.0745
    2.3640   -5.2007    3.7823   -2.3640
   -1.7192    3.7823   -2.7508    1.7192

  Columns 5 through 6

    2.3640   -1.7192
   -5.2007    3.7823
    3.7823   -2.7508
   -2.3640    1.7192
    5.2007   -3.7823
   -3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640   -1.7192   -1.0745
   -2.3640    5.2007    3.7823    2.3640
   -1.7192    3.7823    2.7508    1.7192
   -1.0745    2.3640    1.7192    1.0745
    2.3640   -5.2007   -3.7823   -2.3640
    1.7192   -3.7823   -2.7508   -1.7192

  Columns 5 through 6

    2.3640    1.7192
   -5.2007   -3.7823
   -3.7823   -2.7508
   -2.3640   -1.7192
    5.2007    3.7823
    3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640   -1.7192   -1.0745
    2.3640    5.2007   -3.7823   -2.3640
   -1.7192   -3.7823    2.7508    1.7192
   -1.0745   -2.3640    1.7192    1.0745
   -2.3640   -5.2007    3.7823    2.3640
    1.7192    3.7823   -2.7508   -1.7192

  Columns 5 through 6

   -2.3640    1.7192
   -5.2007    3.7823
    3.7823   -2.7508
    2.3640   -1.7192
    5.2007   -3.7823
   -3.7823    2.7508


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
           0           0           0
     -218000           0           0
           0           0           0
           0           0           0

  Columns 4 through 6

     -218000           0           0
           0           0           0
           0           0           0
      218000           0           0
           0           0           0
           0           0           0


Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689   -5.5171   -4.1378
    2.0689    1.0345   -2.7585   -2.0689
   -5.5171   -2.7585    7.3561    5.5171
   -4.1378   -2.0689    5.5171    4.1378
   -2.0689   -1.0345    2.7585    2.0689
    5.5171    2.7585   -7.3561   -5.5171

  Columns 5 through 6

   -2.0689    5.5171
   -1.0345    2.7585
    2.7585   -7.3561
    2.0689   -5.5171
    1.0345   -2.7585
   -2.7585    7.3561


Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689    5.5171   -4.1378
   -2.0689    1.0345   -2.7585    2.0689
    5.5171   -2.7585    7.3561   -5.5171
   -4.1378    2.0689   -5.5171    4.1378
    2.0689   -1.0345    2.7585   -2.0689
   -5.5171    2.7585   -7.3561    5.5171

  Columns 5 through 6

    2.0689   -5.5171
   -1.0345    2.7585
    2.7585   -7.3561
   -2.0689    5.5171
    1.0345   -2.7585
   -2.7585    7.3561


Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689   -5.5171   -4.1378
   -2.0689    1.0345    2.7585    2.0689
   -5.5171    2.7585    7.3561    5.5171
   -4.1378    2.0689    5.5171    4.1378
    2.0689   -1.0345   -2.7585   -2.0689
    5.5171   -2.7585   -7.3561   -5.5171

  Columns 5 through 6

    2.0689    5.5171
   -1.0345   -2.7585
   -2.7585   -7.3561
   -2.0689   -5.5171
    1.0345    2.7585
    2.7585    7.3561


Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689    5.5171   -4.1378
    2.0689    1.0345    2.7585   -2.0689
    5.5171    2.7585    7.3561   -5.5171
   -4.1378   -2.0689   -5.5171    4.1378
   -2.0689   -1.0345   -2.7585    2.0689
   -5.5171   -2.7585   -7.3561    5.5171

  Columns 5 through 6

   -2.0689   -5.5171
   -1.0345   -2.7585
   -2.7585   -7.3561
    2.0689    5.5171
    1.0345    2.7585
    2.7585    7.3561


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887   -0.5033         0
         0   -0.5033    1.3422         0
         0         0         0         0
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         0         0
   -0.1887    0.5033
    0.5033   -1.3422
         0         0
    0.1887   -0.5033
   -0.5033    1.3422


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887    0.5033         0
         0    0.5033    1.3422         0
         0         0         0         0
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         0         0
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         0         0
    0.1887    0.5033
    0.5033    1.3422


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887   -0.5033         0
         0   -0.5033    1.3422         0
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         0         0
   -0.1887    0.5033
    0.5033   -1.3422
         0         0
    0.1887   -0.5033
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Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887    0.5033         0
         0    0.5033    1.3422         0
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  Columns 5 through 6

         0         0
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    0.1887    0.5033
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Ke =

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           0      218000           0
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  Columns 4 through 6

           0           0           0
           0     -218000           0
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Ke =

  Columns 1 through 3

           0           0           0
           0      218000           0
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  Columns 4 through 6

           0           0           0
           0     -218000           0
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Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
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  Columns 4 through 6

     -218000           0           0
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Ke =

  Columns 1 through 3

      218000           0           0
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  Columns 4 through 6

     -218000           0           0
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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640    1.7192   -1.0745
    2.3640    5.2007    3.7823   -2.3640
    1.7192    3.7823    2.7508   -1.7192
   -1.0745   -2.3640   -1.7192    1.0745
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  Columns 5 through 6

   -2.3640   -1.7192
   -5.2007   -3.7823
   -3.7823   -2.7508
    2.3640    1.7192
    5.2007    3.7823
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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640    1.7192   -1.0745
   -2.3640    5.2007   -3.7823    2.3640
    1.7192   -3.7823    2.7508   -1.7192
   -1.0745    2.3640   -1.7192    1.0745
    2.3640   -5.2007    3.7823   -2.3640
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  Columns 5 through 6

    2.3640   -1.7192
   -5.2007    3.7823
    3.7823   -2.7508
   -2.3640    1.7192
    5.2007   -3.7823
   -3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640   -1.7192   -1.0745
   -2.3640    5.2007    3.7823    2.3640
   -1.7192    3.7823    2.7508    1.7192
   -1.0745    2.3640    1.7192    1.0745
    2.3640   -5.2007   -3.7823   -2.3640
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  Columns 5 through 6

    2.3640    1.7192
   -5.2007   -3.7823
   -3.7823   -2.7508
   -2.3640   -1.7192
    5.2007    3.7823
    3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640   -1.7192   -1.0745
    2.3640    5.2007   -3.7823   -2.3640
   -1.7192   -3.7823    2.7508    1.7192
   -1.0745   -2.3640    1.7192    1.0745
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  Columns 5 through 6

   -2.3640    1.7192
   -5.2007    3.7823
    3.7823   -2.7508
    2.3640   -1.7192
    5.2007   -3.7823
   -3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007   -2.3640    3.7823   -5.2007
   -2.3640    1.0745   -1.7192    2.3640
    3.7823   -1.7192    2.7508   -3.7823
   -5.2007    2.3640   -3.7823    5.2007
    2.3640   -1.0745    1.7192   -2.3640
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  Columns 5 through 6

    2.3640   -3.7823
   -1.0745    1.7192
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   -2.3640    3.7823
    1.0745   -1.7192
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Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
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  Columns 4 through 6

     -218000           0           0
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      218000           0           0
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689   -5.5171   -4.1378
    2.0689    1.0345   -2.7585   -2.0689
   -5.5171   -2.7585    7.3561    5.5171
   -4.1378   -2.0689    5.5171    4.1378
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  Columns 5 through 6

   -2.0689    5.5171
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689    5.5171   -4.1378
   -2.0689    1.0345   -2.7585    2.0689
    5.5171   -2.7585    7.3561   -5.5171
   -4.1378    2.0689   -5.5171    4.1378
    2.0689   -1.0345    2.7585   -2.0689
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  Columns 5 through 6

    2.0689   -5.5171
   -1.0345    2.7585
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   -2.0689    5.5171
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689   -5.5171   -4.1378
   -2.0689    1.0345    2.7585    2.0689
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   -4.1378    2.0689    5.5171    4.1378
    2.0689   -1.0345   -2.7585   -2.0689
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  Columns 5 through 6

    2.0689    5.5171
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    1.0345    2.7585
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689    5.5171   -4.1378
    2.0689    1.0345    2.7585   -2.0689
    5.5171    2.7585    7.3561   -5.5171
   -4.1378   -2.0689   -5.5171    4.1378
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  Columns 5 through 6

   -2.0689   -5.5171
   -1.0345   -2.7585
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Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887   -0.5033         0
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         0         0
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Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887    0.5033         0
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  Columns 5 through 6

         0         0
   -0.1887   -0.5033
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Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887   -0.5033         0
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  Columns 5 through 6

         0         0
   -0.1887    0.5033
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    0.1887   -0.5033
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Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887    0.5033         0
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  Columns 5 through 6

         0         0
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    0.1887    0.5033
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Ke =

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  Columns 4 through 6

           0           0           0
           0     -218000           0
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           0      218000           0
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Ke =

  Columns 1 through 3

           0           0           0
           0      218000           0
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  Columns 4 through 6

           0           0           0
           0     -218000           0
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Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
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  Columns 4 through 6

     -218000           0           0
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Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
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  Columns 4 through 6

     -218000           0           0
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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640    1.7192   -1.0745
    2.3640    5.2007    3.7823   -2.3640
    1.7192    3.7823    2.7508   -1.7192
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  Columns 5 through 6

   -2.3640   -1.7192
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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640    1.7192   -1.0745
   -2.3640    5.2007   -3.7823    2.3640
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  Columns 5 through 6

    2.3640   -1.7192
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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640   -1.7192   -1.0745
   -2.3640    5.2007    3.7823    2.3640
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  Columns 5 through 6

    2.3640    1.7192
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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640   -1.7192   -1.0745
    2.3640    5.2007   -3.7823   -2.3640
   -1.7192   -3.7823    2.7508    1.7192
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  Columns 5 through 6

   -2.3640    1.7192
   -5.2007    3.7823
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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007   -2.3640    3.7823   -5.2007
   -2.3640    1.0745   -1.7192    2.3640
    3.7823   -1.7192    2.7508   -3.7823
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  Columns 5 through 6

    2.3640   -3.7823
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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007    2.3640   -3.7823   -5.2007
    2.3640    1.0745   -1.7192   -2.3640
   -3.7823   -1.7192    2.7508    3.7823
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  Columns 5 through 6

   -2.3640    3.7823
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Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
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     -218000           0           0
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  Columns 4 through 6

     -218000           0           0
           0           0           0
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      218000           0           0
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689   -5.5171   -4.1378
    2.0689    1.0345   -2.7585   -2.0689
   -5.5171   -2.7585    7.3561    5.5171
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  Columns 5 through 6

   -2.0689    5.5171
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    2.0689   -5.5171
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689    5.5171   -4.1378
   -2.0689    1.0345   -2.7585    2.0689
    5.5171   -2.7585    7.3561   -5.5171
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  Columns 5 through 6

    2.0689   -5.5171
   -1.0345    2.7585
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   -2.0689    5.5171
    1.0345   -2.7585
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689   -5.5171   -4.1378
   -2.0689    1.0345    2.7585    2.0689
   -5.5171    2.7585    7.3561    5.5171
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    2.0689   -1.0345   -2.7585   -2.0689
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  Columns 5 through 6

    2.0689    5.5171
   -1.0345   -2.7585
   -2.7585   -7.3561
   -2.0689   -5.5171
    1.0345    2.7585
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689    5.5171   -4.1378
    2.0689    1.0345    2.7585   -2.0689
    5.5171    2.7585    7.3561   -5.5171
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Ke =

  1.0e+005 *

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Ke =

  1.0e+005 *

  Columns 1 through 4

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Ke =

  1.0e+005 *

  Columns 1 through 4

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         0    0.1887   -0.5033         0
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         0         0
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    0.1887   -0.5033
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Ke =

  1.0e+005 *

  Columns 1 through 4

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         0    0.1887    0.5033         0
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         0         0
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Ke =

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           0           0           0
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Ke =

  Columns 1 through 3

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Ke =

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Ke =

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      218000           0           0
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  Columns 4 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640    1.7192   -1.0745
    2.3640    5.2007    3.7823   -2.3640
    1.7192    3.7823    2.7508   -1.7192
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640    1.7192   -1.0745
   -2.3640    5.2007   -3.7823    2.3640
    1.7192   -3.7823    2.7508   -1.7192
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  Columns 5 through 6

    2.3640   -1.7192
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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640   -1.7192   -1.0745
   -2.3640    5.2007    3.7823    2.3640
   -1.7192    3.7823    2.7508    1.7192
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  Columns 5 through 6

    2.3640    1.7192
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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640   -1.7192   -1.0745
    2.3640    5.2007   -3.7823   -2.3640
   -1.7192   -3.7823    2.7508    1.7192
   -1.0745   -2.3640    1.7192    1.0745
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  Columns 5 through 6

   -2.3640    1.7192
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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007   -2.3640    3.7823   -5.2007
   -2.3640    1.0745   -1.7192    2.3640
    3.7823   -1.7192    2.7508   -3.7823
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  Columns 5 through 6

    2.3640   -3.7823
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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007    2.3640   -3.7823   -5.2007
    2.3640    1.0745   -1.7192   -2.3640
   -3.7823   -1.7192    2.7508    3.7823
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  Columns 5 through 6

   -2.3640    3.7823
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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007    2.3640    3.7823   -5.2007
    2.3640    1.0745    1.7192   -2.3640
    3.7823    1.7192    2.7508   -3.7823
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  Columns 5 through 6

   -2.3640   -3.7823
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Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

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      218000           0           0
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  Columns 4 through 6

     -218000           0           0
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Ke =

  1.0e+004 *

  Columns 1 through 4

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    2.0689    1.0345   -2.7585   -2.0689
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689    5.5171   -4.1378
   -2.0689    1.0345   -2.7585    2.0689
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  Columns 5 through 6

    2.0689   -5.5171
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689   -5.5171   -4.1378
   -2.0689    1.0345    2.7585    2.0689
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689    5.5171   -4.1378
    2.0689    1.0345    2.7585   -2.0689
    5.5171    2.7585    7.3561   -5.5171
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  Columns 5 through 6

   -2.0689   -5.5171
   -1.0345   -2.7585
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Ke =

  1.0e+005 *

  Columns 1 through 4

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         0    0.1887   -0.5033         0
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Ke =

  1.0e+005 *

  Columns 1 through 4

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Ke =

  1.0e+005 *

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Ke =

  1.0e+005 *

  Columns 1 through 4

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Ke =

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  Columns 4 through 6

           0           0           0
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Ke =

  Columns 1 through 3

           0           0           0
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  Columns 4 through 6

           0           0           0
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Ke =

  Columns 1 through 3

      218000           0           0
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  Columns 4 through 6

     -218000           0           0
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Ke =

  Columns 1 through 3

      218000           0           0
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  Columns 4 through 6

     -218000           0           0
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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640    1.7192   -1.0745
    2.3640    5.2007    3.7823   -2.3640
    1.7192    3.7823    2.7508   -1.7192
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640    1.7192   -1.0745
   -2.3640    5.2007   -3.7823    2.3640
    1.7192   -3.7823    2.7508   -1.7192
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  Columns 5 through 6

    2.3640   -1.7192
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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640   -1.7192   -1.0745
   -2.3640    5.2007    3.7823    2.3640
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640   -1.7192   -1.0745
    2.3640    5.2007   -3.7823   -2.3640
   -1.7192   -3.7823    2.7508    1.7192
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  Columns 5 through 6

   -2.3640    1.7192
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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007   -2.3640    3.7823   -5.2007
   -2.3640    1.0745   -1.7192    2.3640
    3.7823   -1.7192    2.7508   -3.7823
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  Columns 5 through 6

    2.3640   -3.7823
   -1.0745    1.7192
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    1.0745   -1.7192
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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007    2.3640   -3.7823   -5.2007
    2.3640    1.0745   -1.7192   -2.3640
   -3.7823   -1.7192    2.7508    3.7823
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  Columns 5 through 6

   -2.3640    3.7823
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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007    2.3640    3.7823   -5.2007
    2.3640    1.0745    1.7192   -2.3640
    3.7823    1.7192    2.7508   -3.7823
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  Columns 5 through 6

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    2.3640    3.7823
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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007   -2.3640   -3.7823   -5.2007
   -2.3640    1.0745    1.7192    2.3640
   -3.7823    1.7192    2.7508    3.7823
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    2.3640   -1.0745   -1.7192   -2.3640
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  Columns 5 through 6

    2.3640    3.7823
   -1.0745   -1.7192
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   -2.3640   -3.7823
    1.0745    1.7192
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Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

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Ke =

  1.0e+004 *

  Columns 1 through 4

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    2.0689    1.0345   -2.7585   -2.0689
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689    5.5171   -4.1378
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689   -5.5171   -4.1378
   -2.0689    1.0345    2.7585    2.0689
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689    5.5171   -4.1378
    2.0689    1.0345    2.7585   -2.0689
    5.5171    2.7585    7.3561   -5.5171
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  Columns 5 through 6

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Ke =

  1.0e+005 *

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Ke =

  1.0e+005 *

  Columns 1 through 4

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         0    0.1887    0.5033         0
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Ke =

  1.0e+005 *

  Columns 1 through 4

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Ke =

  1.0e+005 *

  Columns 1 through 4

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Ke =

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  Columns 4 through 6

           0           0           0
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Ke =

  Columns 1 through 3

           0           0           0
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  Columns 4 through 6

           0           0           0
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Ke =

  Columns 1 through 3

      218000           0           0
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  Columns 4 through 6

     -218000           0           0
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Ke =

  Columns 1 through 3

      218000           0           0
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  Columns 4 through 6

     -218000           0           0
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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640    1.7192   -1.0745
    2.3640    5.2007    3.7823   -2.3640
    1.7192    3.7823    2.7508   -1.7192
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640    1.7192   -1.0745
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    1.7192   -3.7823    2.7508   -1.7192
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640   -1.7192   -1.0745
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640   -1.7192   -1.0745
    2.3640    5.2007   -3.7823   -2.3640
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007   -2.3640    3.7823   -5.2007
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007    2.3640   -3.7823   -5.2007
    2.3640    1.0745   -1.7192   -2.3640
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007    2.3640    3.7823   -5.2007
    2.3640    1.0745    1.7192   -2.3640
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007   -2.3640   -3.7823   -5.2007
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    2.6865    2.6865    4.2984   -2.6865
    2.6865    2.6865    4.2984   -2.6865
    4.2984    4.2984    6.8775   -4.2984
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  Columns 5 through 6

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Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
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Ke =

  Columns 1 through 3

      218000           0           0
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  Columns 4 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689    5.5171   -4.1378
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  Columns 5 through 6

    2.0689   -5.5171
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689   -5.5171   -4.1378
   -2.0689    1.0345    2.7585    2.0689
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689    5.5171   -4.1378
    2.0689    1.0345    2.7585   -2.0689
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Ke =

  1.0e+005 *

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         0         0
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Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
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Ke =

  1.0e+005 *

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Ke =

  1.0e+005 *

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         0         0
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Ke =

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  Columns 4 through 6

           0           0           0
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Ke =

  Columns 1 through 3

           0           0           0
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  Columns 4 through 6

           0           0           0
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Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
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  Columns 4 through 6

     -218000           0           0
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Ke =

  Columns 1 through 3

      218000           0           0
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  Columns 4 through 6

     -218000           0           0
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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640    1.7192   -1.0745
    2.3640    5.2007    3.7823   -2.3640
    1.7192    3.7823    2.7508   -1.7192
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  Columns 5 through 6

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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640    1.7192   -1.0745
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  Columns 5 through 6

    2.3640   -1.7192
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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640   -1.7192   -1.0745
   -2.3640    5.2007    3.7823    2.3640
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  Columns 5 through 6

    2.3640    1.7192
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Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640   -1.7192   -1.0745
    2.3640    5.2007   -3.7823   -2.3640
   -1.7192   -3.7823    2.7508    1.7192
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  Columns 5 through 6

   -2.3640    1.7192
   -5.2007    3.7823
    3.7823   -2.7508
    2.3640   -1.7192
    5.2007   -3.7823
   -3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007   -2.3640    3.7823   -5.2007
   -2.3640    1.0745   -1.7192    2.3640
    3.7823   -1.7192    2.7508   -3.7823
   -5.2007    2.3640   -3.7823    5.2007
    2.3640   -1.0745    1.7192   -2.3640
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  Columns 5 through 6

    2.3640   -3.7823
   -1.0745    1.7192
    1.7192   -2.7508
   -2.3640    3.7823
    1.0745   -1.7192
   -1.7192    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007    2.3640   -3.7823   -5.2007
    2.3640    1.0745   -1.7192   -2.3640
   -3.7823   -1.7192    2.7508    3.7823
   -5.2007   -2.3640    3.7823    5.2007
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    3.7823    1.7192   -2.7508   -3.7823

  Columns 5 through 6

   -2.3640    3.7823
   -1.0745    1.7192
    1.7192   -2.7508
    2.3640   -3.7823
    1.0745   -1.7192
   -1.7192    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007    2.3640    3.7823   -5.2007
    2.3640    1.0745    1.7192   -2.3640
    3.7823    1.7192    2.7508   -3.7823
   -5.2007   -2.3640   -3.7823    5.2007
   -2.3640   -1.0745   -1.7192    2.3640
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  Columns 5 through 6

   -2.3640   -3.7823
   -1.0745   -1.7192
   -1.7192   -2.7508
    2.3640    3.7823
    1.0745    1.7192
    1.7192    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007   -2.3640   -3.7823   -5.2007
   -2.3640    1.0745    1.7192    2.3640
   -3.7823    1.7192    2.7508    3.7823
   -5.2007    2.3640    3.7823    5.2007
    2.3640   -1.0745   -1.7192   -2.3640
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  Columns 5 through 6

    2.3640    3.7823
   -1.0745   -1.7192
   -1.7192   -2.7508
   -2.3640   -3.7823
    1.0745    1.7192
    1.7192    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    2.6865    2.6865    4.2984   -2.6865
    2.6865    2.6865    4.2984   -2.6865
    4.2984    4.2984    6.8775   -4.2984
   -2.6865   -2.6865   -4.2984    2.6865
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  Columns 5 through 6

   -2.6865   -4.2984
   -2.6865   -4.2984
   -4.2984   -6.8775
    2.6865    4.2984
    2.6865    4.2984
    4.2984    6.8775


Ke =

  1.0e+004 *

  Columns 1 through 4

    2.6865   -2.6865    4.2984   -2.6865
   -2.6865    2.6865   -4.2984    2.6865
    4.2984   -4.2984    6.8775   -4.2984
   -2.6865    2.6865   -4.2984    2.6865
    2.6865   -2.6865    4.2984   -2.6865
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  Columns 5 through 6

    2.6865   -4.2984
   -2.6865    4.2984
    4.2984   -6.8775
   -2.6865    4.2984
    2.6865   -4.2984
   -4.2984    6.8775


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
           0           0           0
     -218000           0           0
           0           0           0
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  Columns 4 through 6

     -218000           0           0
           0           0           0
           0           0           0
      218000           0           0
           0           0           0
           0           0           0


Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689   -5.5171   -4.1378
    2.0689    1.0345   -2.7585   -2.0689
   -5.5171   -2.7585    7.3561    5.5171
   -4.1378   -2.0689    5.5171    4.1378
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    5.5171    2.7585   -7.3561   -5.5171

  Columns 5 through 6

   -2.0689    5.5171
   -1.0345    2.7585
    2.7585   -7.3561
    2.0689   -5.5171
    1.0345   -2.7585
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689    5.5171   -4.1378
   -2.0689    1.0345   -2.7585    2.0689
    5.5171   -2.7585    7.3561   -5.5171
   -4.1378    2.0689   -5.5171    4.1378
    2.0689   -1.0345    2.7585   -2.0689
   -5.5171    2.7585   -7.3561    5.5171

  Columns 5 through 6

    2.0689   -5.5171
   -1.0345    2.7585
    2.7585   -7.3561
   -2.0689    5.5171
    1.0345   -2.7585
   -2.7585    7.3561


Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689   -5.5171   -4.1378
   -2.0689    1.0345    2.7585    2.0689
   -5.5171    2.7585    7.3561    5.5171
   -4.1378    2.0689    5.5171    4.1378
    2.0689   -1.0345   -2.7585   -2.0689
    5.5171   -2.7585   -7.3561   -5.5171

  Columns 5 through 6

    2.0689    5.5171
   -1.0345   -2.7585
   -2.7585   -7.3561
   -2.0689   -5.5171
    1.0345    2.7585
    2.7585    7.3561


Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689    5.5171   -4.1378
    2.0689    1.0345    2.7585   -2.0689
    5.5171    2.7585    7.3561   -5.5171
   -4.1378   -2.0689   -5.5171    4.1378
   -2.0689   -1.0345   -2.7585    2.0689
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  Columns 5 through 6

   -2.0689   -5.5171
   -1.0345   -2.7585
   -2.7585   -7.3561
    2.0689    5.5171
    1.0345    2.7585
    2.7585    7.3561


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887   -0.5033         0
         0   -0.5033    1.3422         0
         0         0         0         0
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  Columns 5 through 6

         0         0
   -0.1887    0.5033
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         0         0
    0.1887   -0.5033
   -0.5033    1.3422


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887    0.5033         0
         0    0.5033    1.3422         0
         0         0         0         0
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  Columns 5 through 6

         0         0
   -0.1887   -0.5033
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         0         0
    0.1887    0.5033
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Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887   -0.5033         0
         0   -0.5033    1.3422         0
         0         0         0         0
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  Columns 5 through 6

         0         0
   -0.1887    0.5033
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         0         0
    0.1887   -0.5033
   -0.5033    1.3422


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887    0.5033         0
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         0         0         0         0
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  Columns 5 through 6

         0         0
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         0         0
    0.1887    0.5033
    0.5033    1.3422


Ke =

  Columns 1 through 3

           0           0           0
           0      218000           0
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  Columns 4 through 6

           0           0           0
           0     -218000           0
           0           0           0
           0           0           0
           0      218000           0
           0           0           0


Ke =

  Columns 1 through 3

           0           0           0
           0      218000           0
           0           0           0
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           0     -218000           0
           0           0           0

  Columns 4 through 6

           0           0           0
           0     -218000           0
           0           0           0
           0           0           0
           0      218000           0
           0           0           0


Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
           0           0           0
     -218000           0           0
           0           0           0
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  Columns 4 through 6

     -218000           0           0
           0           0           0
           0           0           0
      218000           0           0
           0           0           0
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Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
           0           0           0
     -218000           0           0
           0           0           0
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  Columns 4 through 6

     -218000           0           0
           0           0           0
           0           0           0
      218000           0           0
           0           0           0
           0           0           0


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640    1.7192   -1.0745
    2.3640    5.2007    3.7823   -2.3640
    1.7192    3.7823    2.7508   -1.7192
   -1.0745   -2.3640   -1.7192    1.0745
   -2.3640   -5.2007   -3.7823    2.3640
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  Columns 5 through 6

   -2.3640   -1.7192
   -5.2007   -3.7823
   -3.7823   -2.7508
    2.3640    1.7192
    5.2007    3.7823
    3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640    1.7192   -1.0745
   -2.3640    5.2007   -3.7823    2.3640
    1.7192   -3.7823    2.7508   -1.7192
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    2.3640   -5.2007    3.7823   -2.3640
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  Columns 5 through 6

    2.3640   -1.7192
   -5.2007    3.7823
    3.7823   -2.7508
   -2.3640    1.7192
    5.2007   -3.7823
   -3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640   -1.7192   -1.0745
   -2.3640    5.2007    3.7823    2.3640
   -1.7192    3.7823    2.7508    1.7192
   -1.0745    2.3640    1.7192    1.0745
    2.3640   -5.2007   -3.7823   -2.3640
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  Columns 5 through 6

    2.3640    1.7192
   -5.2007   -3.7823
   -3.7823   -2.7508
   -2.3640   -1.7192
    5.2007    3.7823
    3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640   -1.7192   -1.0745
    2.3640    5.2007   -3.7823   -2.3640
   -1.7192   -3.7823    2.7508    1.7192
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  Columns 5 through 6

   -2.3640    1.7192
   -5.2007    3.7823
    3.7823   -2.7508
    2.3640   -1.7192
    5.2007   -3.7823
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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007   -2.3640    3.7823   -5.2007
   -2.3640    1.0745   -1.7192    2.3640
    3.7823   -1.7192    2.7508   -3.7823
   -5.2007    2.3640   -3.7823    5.2007
    2.3640   -1.0745    1.7192   -2.3640
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  Columns 5 through 6

    2.3640   -3.7823
   -1.0745    1.7192
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   -2.3640    3.7823
    1.0745   -1.7192
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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007    2.3640   -3.7823   -5.2007
    2.3640    1.0745   -1.7192   -2.3640
   -3.7823   -1.7192    2.7508    3.7823
   -5.2007   -2.3640    3.7823    5.2007
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  Columns 5 through 6

   -2.3640    3.7823
   -1.0745    1.7192
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    2.3640   -3.7823
    1.0745   -1.7192
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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007    2.3640    3.7823   -5.2007
    2.3640    1.0745    1.7192   -2.3640
    3.7823    1.7192    2.7508   -3.7823
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  Columns 5 through 6

   -2.3640   -3.7823
   -1.0745   -1.7192
   -1.7192   -2.7508
    2.3640    3.7823
    1.0745    1.7192
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Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007   -2.3640   -3.7823   -5.2007
   -2.3640    1.0745    1.7192    2.3640
   -3.7823    1.7192    2.7508    3.7823
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  Columns 5 through 6

    2.3640    3.7823
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    1.0745    1.7192
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Ke =

  1.0e+004 *

  Columns 1 through 4

    2.6865    2.6865    4.2984   -2.6865
    2.6865    2.6865    4.2984   -2.6865
    4.2984    4.2984    6.8775   -4.2984
   -2.6865   -2.6865   -4.2984    2.6865
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  Columns 5 through 6

   -2.6865   -4.2984
   -2.6865   -4.2984
   -4.2984   -6.8775
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Ke =

  1.0e+004 *

  Columns 1 through 4

    2.6865   -2.6865    4.2984   -2.6865
   -2.6865    2.6865   -4.2984    2.6865
    4.2984   -4.2984    6.8775   -4.2984
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  Columns 5 through 6

    2.6865   -4.2984
   -2.6865    4.2984
    4.2984   -6.8775
   -2.6865    4.2984
    2.6865   -4.2984
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Ke =

  1.0e+004 *

  Columns 1 through 4

    2.6865    2.6865   -4.2984   -2.6865
    2.6865    2.6865   -4.2984   -2.6865
   -4.2984   -4.2984    6.8775    4.2984
   -2.6865   -2.6865    4.2984    2.6865
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  Columns 5 through 6

   -2.6865    4.2984
   -2.6865    4.2984
    4.2984   -6.8775
    2.6865   -4.2984
    2.6865   -4.2984
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Ke =

   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN
   NaN   NaN   NaN   NaN   NaN   NaN


Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
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  Columns 4 through 6

     -218000           0           0
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      218000           0           0
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689   -5.5171   -4.1378
    2.0689    1.0345   -2.7585   -2.0689
   -5.5171   -2.7585    7.3561    5.5171
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  Columns 5 through 6

   -2.0689    5.5171
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    2.0689   -5.5171
    1.0345   -2.7585
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689    5.5171   -4.1378
   -2.0689    1.0345   -2.7585    2.0689
    5.5171   -2.7585    7.3561   -5.5171
   -4.1378    2.0689   -5.5171    4.1378
    2.0689   -1.0345    2.7585   -2.0689
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  Columns 5 through 6

    2.0689   -5.5171
   -1.0345    2.7585
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   -2.0689    5.5171
    1.0345   -2.7585
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378   -2.0689   -5.5171   -4.1378
   -2.0689    1.0345    2.7585    2.0689
   -5.5171    2.7585    7.3561    5.5171
   -4.1378    2.0689    5.5171    4.1378
    2.0689   -1.0345   -2.7585   -2.0689
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  Columns 5 through 6

    2.0689    5.5171
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   -2.0689   -5.5171
    1.0345    2.7585
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Ke =

  1.0e+004 *

  Columns 1 through 4

    4.1378    2.0689    5.5171   -4.1378
    2.0689    1.0345    2.7585   -2.0689
    5.5171    2.7585    7.3561   -5.5171
   -4.1378   -2.0689   -5.5171    4.1378
   -2.0689   -1.0345   -2.7585    2.0689
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  Columns 5 through 6

   -2.0689   -5.5171
   -1.0345   -2.7585
   -2.7585   -7.3561
    2.0689    5.5171
    1.0345    2.7585
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Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887   -0.5033         0
         0   -0.5033    1.3422         0
         0         0         0         0
         0   -0.1887    0.5033         0
         0    0.5033   -1.3422         0

  Columns 5 through 6

         0         0
   -0.1887    0.5033
    0.5033   -1.3422
         0         0
    0.1887   -0.5033
   -0.5033    1.3422


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887    0.5033         0
         0    0.5033    1.3422         0
         0         0         0         0
         0   -0.1887   -0.5033         0
         0   -0.5033   -1.3422         0

  Columns 5 through 6

         0         0
   -0.1887   -0.5033
   -0.5033   -1.3422
         0         0
    0.1887    0.5033
    0.5033    1.3422


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887   -0.5033         0
         0   -0.5033    1.3422         0
         0         0         0         0
         0   -0.1887    0.5033         0
         0    0.5033   -1.3422         0

  Columns 5 through 6

         0         0
   -0.1887    0.5033
    0.5033   -1.3422
         0         0
    0.1887   -0.5033
   -0.5033    1.3422


Ke =

  1.0e+005 *

  Columns 1 through 4

         0         0         0         0
         0    0.1887    0.5033         0
         0    0.5033    1.3422         0
         0         0         0         0
         0   -0.1887   -0.5033         0
         0   -0.5033   -1.3422         0

  Columns 5 through 6

         0         0
   -0.1887   -0.5033
   -0.5033   -1.3422
         0         0
    0.1887    0.5033
    0.5033    1.3422


Ke =

  Columns 1 through 3

           0           0           0
           0      218000           0
           0           0           0
           0           0           0
           0     -218000           0
           0           0           0

  Columns 4 through 6

           0           0           0
           0     -218000           0
           0           0           0
           0           0           0
           0      218000           0
           0           0           0


Ke =

  Columns 1 through 3

           0           0           0
           0      218000           0
           0           0           0
           0           0           0
           0     -218000           0
           0           0           0

  Columns 4 through 6

           0           0           0
           0     -218000           0
           0           0           0
           0           0           0
           0      218000           0
           0           0           0


Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
           0           0           0
     -218000           0           0
           0           0           0
           0           0           0

  Columns 4 through 6

     -218000           0           0
           0           0           0
           0           0           0
      218000           0           0
           0           0           0
           0           0           0


Ke =

  Columns 1 through 3

      218000           0           0
           0           0           0
           0           0           0
     -218000           0           0
           0           0           0
           0           0           0

  Columns 4 through 6

     -218000           0           0
           0           0           0
           0           0           0
      218000           0           0
           0           0           0
           0           0           0


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640    1.7192   -1.0745
    2.3640    5.2007    3.7823   -2.3640
    1.7192    3.7823    2.7508   -1.7192
   -1.0745   -2.3640   -1.7192    1.0745
   -2.3640   -5.2007   -3.7823    2.3640
   -1.7192   -3.7823   -2.7508    1.7192

  Columns 5 through 6

   -2.3640   -1.7192
   -5.2007   -3.7823
   -3.7823   -2.7508
    2.3640    1.7192
    5.2007    3.7823
    3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640    1.7192   -1.0745
   -2.3640    5.2007   -3.7823    2.3640
    1.7192   -3.7823    2.7508   -1.7192
   -1.0745    2.3640   -1.7192    1.0745
    2.3640   -5.2007    3.7823   -2.3640
   -1.7192    3.7823   -2.7508    1.7192

  Columns 5 through 6

    2.3640   -1.7192
   -5.2007    3.7823
    3.7823   -2.7508
   -2.3640    1.7192
    5.2007   -3.7823
   -3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745   -2.3640   -1.7192   -1.0745
   -2.3640    5.2007    3.7823    2.3640
   -1.7192    3.7823    2.7508    1.7192
   -1.0745    2.3640    1.7192    1.0745
    2.3640   -5.2007   -3.7823   -2.3640
    1.7192   -3.7823   -2.7508   -1.7192

  Columns 5 through 6

    2.3640    1.7192
   -5.2007   -3.7823
   -3.7823   -2.7508
   -2.3640   -1.7192
    5.2007    3.7823
    3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    1.0745    2.3640   -1.7192   -1.0745
    2.3640    5.2007   -3.7823   -2.3640
   -1.7192   -3.7823    2.7508    1.7192
   -1.0745   -2.3640    1.7192    1.0745
   -2.3640   -5.2007    3.7823    2.3640
    1.7192    3.7823   -2.7508   -1.7192

  Columns 5 through 6

   -2.3640    1.7192
   -5.2007    3.7823
    3.7823   -2.7508
    2.3640   -1.7192
    5.2007   -3.7823
   -3.7823    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007   -2.3640    3.7823   -5.2007
   -2.3640    1.0745   -1.7192    2.3640
    3.7823   -1.7192    2.7508   -3.7823
   -5.2007    2.3640   -3.7823    5.2007
    2.3640   -1.0745    1.7192   -2.3640
   -3.7823    1.7192   -2.7508    3.7823

  Columns 5 through 6

    2.3640   -3.7823
   -1.0745    1.7192
    1.7192   -2.7508
   -2.3640    3.7823
    1.0745   -1.7192
   -1.7192    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007    2.3640   -3.7823   -5.2007
    2.3640    1.0745   -1.7192   -2.3640
   -3.7823   -1.7192    2.7508    3.7823
   -5.2007   -2.3640    3.7823    5.2007
   -2.3640   -1.0745    1.7192    2.3640
    3.7823    1.7192   -2.7508   -3.7823

  Columns 5 through 6

   -2.3640    3.7823
   -1.0745    1.7192
    1.7192   -2.7508
    2.3640   -3.7823
    1.0745   -1.7192
   -1.7192    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007    2.3640    3.7823   -5.2007
    2.3640    1.0745    1.7192   -2.3640
    3.7823    1.7192    2.7508   -3.7823
   -5.2007   -2.3640   -3.7823    5.2007
   -2.3640   -1.0745   -1.7192    2.3640
   -3.7823   -1.7192   -2.7508    3.7823

  Columns 5 through 6

   -2.3640   -3.7823
   -1.0745   -1.7192
   -1.7192   -2.7508
    2.3640    3.7823
    1.0745    1.7192
    1.7192    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    5.2007   -2.3640   -3.7823   -5.2007
   -2.3640    1.0745    1.7192    2.3640
   -3.7823    1.7192    2.7508    3.7823
   -5.2007    2.3640    3.7823    5.2007
    2.3640   -1.0745   -1.7192   -2.3640
    3.7823   -1.7192   -2.7508   -3.7823

  Columns 5 through 6

    2.3640    3.7823
   -1.0745   -1.7192
   -1.7192   -2.7508
   -2.3640   -3.7823
    1.0745    1.7192
    1.7192    2.7508


Ke =

  1.0e+004 *

  Columns 1 through 4

    2.6865    2.6865    4.2984   -2.6865
    2.6865    2.6865    4.2984   -2.6865
    4.2984    4.2984    6.8775   -4.2984
   -2.6865   -2.6865   -4.2984    2.6865
   -2.6865   -2.6865   -4.2984    2.6865
   -4.2984   -4.2984   -6.8775    4.2984

  Columns 5 through 6

   -2.6865   -4.2984
   -2.6865   -4.2984
   -4.2984   -6.8775
    2.6865    4.2984
    2.6865    4.2984
    4.2984    6.8775


Ke =

  1.0e+004 *

  Columns 1 through 4

    2.6865   -2.6865    4.2984   -2.6865
   -2.6865    2.6865   -4.2984    2.6865
    4.2984   -4.2984    6.8775   -4.2984
   -2.6865    2.6865   -4.2984    2.6865
    2.6865   -2.6865    4.2984   -2.6865
   -4.2984    4.2984   -6.8775    4.2984

  Columns 5 through 6

    2.6865   -4.2984
   -2.6865    4.2984
    4.2984   -6.8775
   -2.6865    4.2984
    2.6865   -4.2984
   -4.2984    6.8775


Ke =

  1.0e+004 *

  Columns 1 through 4

    2.6865    2.6865   -4.2984   -2.6865
    2.6865    2.6865   -4.2984   -2.6865
   -4.2984   -4.2984    6.8775    4.2984
   -2.6865   -2.6865    4.2984    2.6865
   -2.6865   -2.6865    4.2984    2.6865
    4.2984    4.2984   -6.8775   -4.2984

  Columns 5 through 6

   -2.6865    4.2984
   -2.6865    4.2984
    4.2984   -6.8775
    2.6865   -4.2984
    2.6865   -4.2984
   -4.2984    6.8775


Ke =

  1.0e+004 *

  Columns 1 through 4

    2.6865   -2.6865   -4.2984   -2.6865
   -2.6865    2.6865    4.2984    2.6865
   -4.2984    4.2984    6.8775    4.2984
   -2.6865    2.6865    4.2984    2.6865
    2.6865   -2.6865   -4.2984   -2.6865
    4.2984   -4.2984   -6.8775   -4.2984

  Columns 5 through 6

    2.6865    4.2984
   -2.6865   -4.2984
   -4.2984   -6.8775
   -2.6865   -4.2984
    2.6865    4.2984
    4.2984    6.8775

Problem 5[edit]

On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.

Find[edit]

Vibration analysis of 3-D truss system in R6.4

Given[edit]

3dtruss.JPG

Solution[edit]

 
clear
clc

E=3*10^7;  A=0.0218; 
rho= 7.3*10^(-4);
     
Edof=[1 1 2 3 4 5 6;
     2 1 2 3 10 11 12 ;
     3 4 5 6 7 8 9;
     4 1 2 3 13 14 15;
     5 4 5 6 16 17 18;
     6 4 5 6 10 11 12;
     7 4 5 6 13 14 15;
     8 1 2 3 7 8 9;
     9 1 2 3 16 17 18;
     10 7 8 9 16 17 18;
     11 10 11 12 13 14 15;
     12 7 8 9 10 11 12;
     13 13 14 15 16 17 18;
     14 7 8 9 28 29 30;
     15 16 17 18 19 20 21;
     16 10 11 12 25 26 27;
     17 13 14 15 22 23 24;
     18 10 11 12 19 20 21;
     19 7 8 9 22 23 24;
     20 13 14 15 28 29 30;
     21 16 17 18 25 26 27;
     22 16 17 18 28 29 30;
     23 7 8 9 19 20 21;
     24 10 11 12 22 23 24;
     25 13 14 15 25 26 27];
 
 coord=[-1.5   0   8;
        1.5    0   8;
        -1.5  1.5  4;
        1.5   1.5  4;
        1.5  -1.5  4;
        -1.5 -1.5  4;
        -4     4   0;
         4     4   0;
         4    -4   0;
        -4    -4   0];
    
 ex=zeros(25,2);
 ey=zeros(25,2);
 ez=zeros(25,2);
 for i=1:25
     ex(i,1)=coord((Edof(i,2)+2)/3,1);
     ex(i,2)=coord((Edof(i,5)+2)/3,1);
     ey(i,1)=coord((Edof(i,3)+1)/3,2);
     ey(i,2)=coord((Edof(i,6)+1)/3,2);
     ez(i,1)=coord(Edof(i,4)/3,3);
     ez(i,2)=coord(Edof(i,7)/3,3);
 end

K=zeros(75,75);
M=zeros(75);
for i=1:25
 
    
dx=ex(i,2)-ex(i,1) ;
dy= ey(i,2)-ey(i,1);
dz=ez(i,2)-ez(i,1) ;
 
 

L=sqrt(dx^2+dy^2+dz^2);
Kle=E*A/L*[1 -1;
           -1  1];
l=dx/L;
m=dy/L;
n=dz/L;
 G=[l m n 0 0 0;
    0 0 0 l m n];
mass=L*A*rho;
me=[mass/2 0 0 0 0 0;
   0 mass/2 0 0 0 0;
   0 0 mass/2 0 0 0;
   0 0 0 mass/2 0 0;
   0 0 0 0 mass/2 0;
   0 0 0 0 0 mass/2];
Ke=G'*Kle*G;
S=Edof(i,2:7);
K(S,S)=K(S,S)+Ke;
M(S,S)=M(S,S)+me;
 
end
nd=size(K,1);
fdof=[1:nd]';
bc=[25 26 22 24 19 21 28 30 ]';
fdof(bc(:))=[];
[X,D]=eig(K(fdof,fdof),M(fdof,fdof));
nfdof=size(X,1);
eigenval=diag(D);
eigenvec=zeros(nd,nfdof);
eigenvec(fdof,:)=X;
%top
for kk=1:3
   eigenvalx=eigenval(kk);
   eigenvecx=eigenvec(:,kk);
   w=sqrt(eigenvalx);
   T=2*pi/w;
   dt=T/100;
   t=0:dt:T;
   figure(kk)

   if kk==1;
       filename = 'mode1_top.gif';
   elseif kk==2;
       filename = 'mode2_top.gif';
   elseif kk==3;
       filename = 'mode3_top.gif';
   end

   for jj = 1:100
       tt=t(jj);
       d=eigenvecx*sin(w*tt);
       d=d';
       rr=Edof(:,2:7);
       ed=zeros(25,6);
       for i=1:25;
           ed(i,1:6)=d(rr(i,:));
       end
       xd=(ex+ed(:,[1 4]))';
       yd=(ey+ed(:,[2 5]))';
       zd=(ez+ed(:,[3 6]))';
       s1=['-' , 'k'];
       axis manual
       plot(xd,zd,s1)
       set(gca, 'ylim', [-12 12], 'xlim', [-12 12]);
       drawnow
       frame = getframe(1);
       im = frame2im(frame);
       [imind,cm] = rgb2ind(im,256);
       if jj == 1;
           imwrite(imind,cm,filename,'gif', 'Loopcount',inf);
       else
           imwrite(imind,cm,filename,'gif','WriteMode','append');
       end
   end
end
%side
 for kk=1:3
    eigenvalx=eigenval(kk);
    eigenvecx=eigenvec(:,kk)
    w=sqrt(eigenvalx)
    T=2*pi/w;
    dt=T/100
    t=0:dt:T;
    figure(kk)
 
    if kk==1;
        filename = 'mode1_side.gif'
    elseif kk==2;
        filename = 'mode2_side.gif'
    elseif kk==3;
        filename = 'mode3_side.gif'
    end
 
    for jj = 1:100
        tt=t(jj);
        d=eigenvecx*sin(w*tt);
        d=d';
        rr=Edof(:,2:7);
        ed=zeros(25,6);
        for i=1:25;
            ed(i,1:6)=d(rr(i,:));
        end
        xd=(ex+ed(:,[1 4]))';
        yd=(ey+ed(:,[2 5]))';
        zd=(ez+ed(:,[3 6]))';
        s1=['-' , 'k'];
        axis manual
        plot(xd,yd,s1)
        set(gca, 'ylim', [-12 12], 'xlim', [-12 12]);
        drawnow
        frame = getframe(1);
        im = frame2im(frame);
        [imind,cm] = rgb2ind(im,256);
        if jj == 1;
            imwrite(imind,cm,filename,'gif', 'Loopcount',inf);
        else
            imwrite(imind,cm,filename,'gif','WriteMode','append');
        end
    end
 end

Top and Side view for Mode 1:
Mode1 topfixed.gif Mode1 sidefixed.gif

Top and Side view for Mode 2:
Mode2 topfixed.gif Mode2 sidefixed.gif

Top and Side view for Mode 3:
Mode3 topfixed.gif Mode3 sidefixed.gif

References[edit]

  1. http://upload.wikimedia.org/wikiversity/en/2/22/Fead.s13.sec3.djvu

Contributing Team Members[edit]

Problem Assignments
Problem # Solved & Typed by Reviewed by
1 Zeyn Kermani All
2 Matthew Gidel All
3 Joshua Plicque All
4 Brandon Wright,Spencer Herran All
5 Kelvin Li,Matthew Gidel All

On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.