| Author | Message | |||
     johroedel Member Username: johroedel Post Number: 4 Registered: 12-2009 |
Dear all, I am concerning a very simple and basic problem: the bending of a cantilever beam. I have written a model similar to those in Deformation and Vibration book and compared the result with analytic solutions. Although, the overall impression is good (as in the Backstrom book), I am not very satisfied with the details. I have the impression that my FlexPDE-Model is generally too stiff. The difference between analytic solution and FEM solution of the displacement at the loading point is in the order of 5%. If we consider the fact, that the analytic (Bernoulli) solution does not include shear deformations, we can conclude that the real displacement is actually larger, so that the error would be larger than 5%. In the particular case it is almost 10%! I have compared this finding with results published in a book ( B. Klein, FEM: Grundlagen und Anwendungen der Finite-element-methode, p. 119). Klein has published a convergence analysis which shows that with 16x32 triangular quadratic (6-node) elements the deflection of the beam converges to a value larger as the simple bending solution (but slighly smaller than the Timoshenko solution). This shows that FEM with triangular elements (as FlexPDE) should be generally be able to generate a result better than the analytic solution. My question is: what could be the reason that my result is not better? Is there a difference in the solution algorithms between different FEM programsm which could be responsible for that. If the analytical solution (without shear) is quadratic in x direction, shouldn't the FEM with quadratic elements obtain the exact solution? I assume that the mesh density is not the problem as Kleins mesh has half the density of the attached model. Has anyone an idea what is the reason for the strange behavior of the solution at the fixed point on the left side of the problem? Best regards | |||
| johroedel Member Username: johroedel Post Number: 5 Registered: 12-2009 |
here is the pde file
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| rgnelson Moderator Username: rgnelson Post Number: 1396 Registered: 06-2003 |
Since we were able to reproduce Timoshenko's solution exactly in our Bentbar.pde example (q.v.), I have to assume that the deviation you see is due to discrepancies between the analytic and FEM solutions as regards the boundary conditions applied at the fixed end.
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| jerrybrown11743 Member Username: jerrybrown11743 Post Number: 62 Registered: 03-2004 |
The problem is apparently shear. If you reduce the dimension, B, the analytic solution agrees with the FEM model. You might want to consult Timoshenko's Strength of Materials, Part I. In it, he develops an analytic model that includes shear. You've also got a problem with your constants C11 and C12. They are not the correct values for plane stress.
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| jerrybrown11743 Member Username: jerrybrown11743 Post Number: 63 Registered: 03-2004 |
Note: I also included an alternative for the boundary condition on V at the anchored end. This isn't mandatory. But, I believe it is more realistic than a single point anchor. |
bending cantilever beam and comparison with analytic solution