Geometric stiffness matrix for a beam element 1

[kiso]

Hello all!

Can anyone tell me where I can find the geometric stiffness matrix for a 12-dof beam element.

ps. I allready posted this question last week, but it somehow disappeared ....

 

[rlnorton]

The best published writeup I've seen is in "Theory of Matrix Structural Analysis" by J.S. Przemieniecki, pp 388-391, McGraw-Hill, 1968. His derivation is for a 2D beam (6 DOF), but is easily extended to 3D (12 DOF). I've successfully used this to determine beam buckling as well as the effect of end-loading on the lateral frequencies.

I believe that the matrix is also laid out in the COSMIC NASTRAN Theoretical Manual, although I don't have my copy at hand.

If you still need more info, let me know.

Bob

 

[kiso]

Yes, I mean the beam column element. I just didn't know that was the right name for it. Thanks for both of you for your interest.

I'll try to derive it. If I think of it quickly, I would just place the same terms (as in the 4 dof element) for the rest of the four dofs. This would give me the needed 8 dofs. The rest of the dofs would be zero (axial and the torsion terms). I hope this isn't a problem in the buckling analysis, where it needs to be positive definite. I guess the standard stiffness matrix have to be suppressed before the buckling analysis.

Regards
Kim

 

[vonlueke]

Here is the standard three-dimensional, 12-dof beam element stiffness matrix (without moment amplification effect of axial load, cited by rajbeer, above, which might be a fairly complex derivation in 3-D), with usual nomenclature and usual sign conventions (i.e., all end displacements and end forces, and all double arrowheads of end rotations and end moments, depicted positive along positive local x, y, z axes). It's listed here in ANSI C, though easy to discern the meaning and convert to any other language desired.

framel3d - Compute three-dimensional, frame-element, local stiffness matrix as defined in V. James Meyers, Matrix Analysis of Structures, 1983, p. 409.

double E,area,Iy,Iz,J,G,k[12][12];
int i,j;

/*

Compute member local stiffness matrix. (Meyers, p. 409, Eq. 8.18.)

 */

k[ 1][ 1] =  E*area/L;
k[ 1][ 7] = -k[1][1];
k[ 2][ 2] =  12.0*E*Iz/(L*L*L);
k[ 2][ 6] =  6.0*E*Iz/(L*L);
k[ 2][ 8] = -k[2][2];
k[ 2][12] =  k[2][6];
k[ 3][ 3] =  12.0*E*Iy/(L*L*L);
k[ 3][ 5] = -6.0*E*Iy/(L*L);
k[ 3][ 9] = -k[3][3];
k[ 3][11] =  k[3][5];
k[ 4][ 4] =  G*J/L;
k[ 4][10] = -k[4][4];
k[ 5][ 5] =  4.0*E*Iy/L;
k[ 5][ 9] =  6.0*E*Iy/(L*L);
k[ 5][11] =  2.0*E*Iy/L;
k[ 6][ 6] =  4.0*E*Iz/L;
k[ 6][ 8] = -6.0*E*Iz/(L*L);
k[ 6][12] =  2.0*E*Iz/L;
k[ 7][ 7] =  k[1][1];
k[ 8][ 8] =  k[2][2];
k[ 8][12] = -k[2][6];
k[ 9][ 9] =  k[3][3];
k[ 9][11] =  k[5][9];
k[10][10] =  k[4][4];
k[11][11] =  k[5][5];
k[12][12] =  k[6][6];

/*

Fill in lower part of symmetric member local stiffness matrix.

 */

for(i=2;i<=12;i++)
   for(j=1;j<ii;j++)
      k[i][j] = k[j][i];

Good luck.

 

[vonlueke]

Forgot to mention, each nonlisted matrix element in my above post has a value of zero.

 

[fahmad9]

check following reference
Matrix Structural Analysis By William Weaver and James Gere
Second Edition page 182