Buckling Mode of Cantilever Beam

Grasshopper Karamba3D
1

Hi

I have been trying to calculate the buckling load of cantilevered beams. With one end fully fixed and the other free, I get comparable results using the euler buckling equations.

When I then fixed the other end using either a pinned or fixed I get discrepancies as shown in the attached images

bucklingcomparison
bucklingcomparison1086×1022 143 KB

I tried the same analysis using 5 elements instead of 1 but this has made no difference.

I have also tried using the modify elements etc as shown in the example in the manual. This has made no difference to the comparison values.

I am attaching the gh file that I have been using. I am probably missing some important details.

Regards

William

Beambuckling.gh (77.7 KB)

2

Hi William,

thank you for this very interesting example!

These are the reasons why you do not get the exact values:

  • The I-beam under analysis has both a weak and a strong bending axis. In the fixed–hinged and hinged–hinged cases, the first two modes correspond to the weak axis, while the third mode relates to the strong axis. The buckling formula provides the first critical value for each axis. See:

    Beambuckling_cp.gh (105.3 KB)

  • The formulas do not account for shear stiffness. By setting Ay and Az to a very large value (as in the attached example), the Karamba3D results align more closely with the formula-based values.

  • The system must have sufficient degrees of freedom; the number of beam elements influences the outcome.

  • The input geometry was unevenly parametrized. Reparametrizing to produce beam segments of equal length improved the results.

  • There are unavoidable round-off errors during the eigenvalue calculation process.

– Clemens

3

Thank you for your swift response. I suspected that there was something that I had not incorporated/done in my script.