2.6 Initial parameter vector for an element

Consider again the assembled system of equations (2.1) of the boundary value problem. It is necessary but not sufficient for a well posed boundary problem that the rank of equations (2.1) be equal to the number of linear equations: $N_{eqs} = rank\left(spA\right)$ (see excerpt 2.10).

Computing diary excerpt 2.10 ( spESTframe93LaheWFI.m )  

#==================================================================== 
spA_rank =  67
spA =
Compressed Column Sparse (rows = 67, cols = 67, nnz = 181 [4%])
#====================================================================

Solving the sparse system of equations. To solve the sparse system of equations (2.1), the command

$$\mathbf{Z}=\mathbf{spA}\backslash \mathbf{B} \qquad$$ (2.52)

is made use of (see excerpt 2.11).

Computing diary excerpt 2.11 ( spESTframe93LaheWFI.m )  

#==================================================================== 
spA =
Compressed Column Sparse (rows = 67, cols = 67, nnz = 181 [4%])
#==================================================================== 
 Solving the sparse system of equations  
 Z=spA\B; 
Support_reactions = Support reactions begin from row: 61
#==================================================================== 
 No     Z

  1   -9.857e-11
  2   -1.706e+02
  3    6.483e+00  
  4   -2.267e+01
  5   -1.285e+01
  6   -7.196e+01
  7   -0.000e+00
  8    0.000e+00  
  9    4.246e+01  
 10    2.267e+01
 11    1.285e+01
 12    0.000e+00
 13   -1.686e+02
 14   -2.623e+01
 15    8.975e+00 
 16   -1.372e+01
 17    4.616e+00
 18    4.209e+01
 19   -1.686e+02
 20   -2.623e+01
 21    6.483e+00 
 22    1.618e+01
 23   -2.043e+01
 24    7.196e+01
 ...   ...  

 ...   ... 
 61    1.285e+01
 62   -2.267e+01 
 63    2.728e+01
 64    4.870e+00
 65   -1.026e+02
 66    4.948e+01
 67    1.800e+00

#====================================================================

Support reactions are in global coordinates.

It is necessary to verify the structure for a static equilibrium:

• the sum of the external forces (load forces and support reactions) is zero,
• the sum of the moments of all the external forces about any point is zero.

Dividing now each of the displacements and rotations by the scaling multiplier, $i_{0}= EI_{basic}/l_{basic}$, we obtain an unscaled initial parameter vector (Sign Convention 2) in local coordinates (see excerpt 2.12).

Computing diary excerpt 2.12 ( spESTframe93LaheWFI.m )  

============================================================================
 Unscaled initial parameter vector 
Element    u          w          fi               N          Q          M 
----------------------------------------------------------------------------
   1   -0.000e+00    0.000e+00   1.189e-02      22.670     12.850      0.000
   2   -4.721e-02   -7.343e-03   1.815e-03      16.182    -20.426     71.959
   3   -0.000e+00    0.000e+00   0.000e+00      -4.870     27.275   -102.642
   4   -4.721e-02    7.343e-03   2.513e-03      39.371     -7.946     46.197
   5   -2.192e-15    4.778e-02   2.340e-04      -1.800    -40.125    -26.181
----------------------------------------------------------------------------