Discrete Model for Buckling Mode Shapes of a Clamped-Free Column

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The discussion focuses on deriving the buckling mode shape matrix for a clamped-free column using a discrete model. The buckling shape is defined by the equation v(x)=1-cos(n*pi/2*L) for odd integers n. To obtain a 3x3 matrix, participants suggest forming a stiffness matrix and solving the eigenvalue problem with given parameters L, E, and I. The conversation also addresses separating translation and rotation modes by performing a stress analysis and formulating the geometric stiffness matrix. The final step involves solving the eigensolution with both elastic and geometric stiffness matrices to determine buckling characteristics.
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the buckling shape for clamped-free column is v(x)=1-cos(n*pi/2*L), n=1,3,5 ...

how could i use the discrete model to get the buckling mode shape matrix?

for example 3x3 matrix

form a stiffness matrix and solve the eigenvalue problem?

L=1;E=1;I=1

l=L/3 % 3-element beam

k=(E*I/l^3)*[24 0 -12 6*l 0 0;
0 8*l^2 -6*l 2*l^2 0 0;
-12 -6*l 24 0 -12 6*l;
6*l 2*l^2 0 8*l^2 -6*l 2*l^2 ;
0 0 -12 -6*l 12 -6*l ;
0 0 6*l 2*l^2 -6*l 4*l^2]

[v,d]=eig(k)

v =
0.0524 0.1142 0.0947 0.3177 -0.7286 0.5860
-0.3090 -0.6433 -0.5420 0.4409 -0.0323 -0.0386
0.1941 0.2589 0.0603 0.5004 -0.3124 -0.7373
-0.5357 -0.1915 0.7575 0.2857 0.1437 -0.0133
0.3924 0.1830 -0.0086 0.6031 0.5834 0.3290
-0.6505 0.6602 -0.3460 0.0941 0.0970 0.0550
 
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how to separate the translation mode and rotation mode?
 
First you do a stress analysis for a unit applied load (i.e. the axial load on the column).

Then you formulate the geometric stiffness (or stress stiffness) matrix Kg for the stress distribution.

Then you do an eigensolution with the elastic stiffness Ke and Kg:

det(Ke + b Kg) = 0

where b is the "buckling factor", i.e. the scale factor from your unit load to the buckling load.

A textbook on the FE method will tell you how to set up Kg. (I had a quick look on the web but I couldn't find anything online)

If one of the eigenvectors of Ke is the same shape as the buckling mode, that is just a coincidence.
 
thank you~
 
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