Hi everyone,
I'm running shell buckling analysis with a shell with perfect geometry and consider geometric nonlinearities. The riks algorithm is set on automatic incrementation. In many cases, the solver gives out a warning message reporting negative eigenvalues, which means, that the bifurcation load may have been exceeded. However, the algorithm still increases the load proportionality factor and just 'runs over' the bifurcation load. This also happens if I decrease the initial and maximum increment. My solution so far is to check the message file for negative eigenvalues. However, this is inconvenient for automatisation of evaluation. Do you know of any other solution?
Dear Manuela, you should check your model's boundary conditions. Maybe your model is statically overdeterminated. What kind of constraints do you use? What does the warning massage contain?
Dear Manuela Böhm
In my opinion, negative eigenvalues indicate that the structure is unstable and the algorithm is predicting an instability buckling load that is lower than the actual buckling load.
One solution could be to adjust the solution parameters, such as reducing the load increments and increasing the number of load steps, to ensure that the algorithm converges to the correct buckling load. You can also try to change the solution method to a more robust one, such as the arc-length method. The arc-length method can handle problems where the buckling load is difficult to predict and it also helps in avoiding over-shooting the bifurcation load.
Thanks for your answers. I'm simulating very short cylindrical shells under negative pressure. The boundary conditions are okay, and it is exactly what Rut Su said, the bifurcation load is lower than the actual buckling load. By choosing very small increments and many load steps, the algorithm converges. I'm just very surprised, that this happens since I chose a robust arc-length method (riks is an arc-length solver) which should avoid over-shooting the bifurcation load. I would have expected that the solver decreases the increments enough if I choose automatic incrementation.
"Riks method" for the "perfect geometry" case often gives a false buckling loading. You have to count only on negative eigenvalues and compare with the value calculated by "Buckle method" to be sure. This problem is quite well known for Riks' method. This is why for the method of Riks, the solution often is to implement a small initial defect (very small) to have a right value of buckling load.
If you add a small imperfection, the solver should converge much faster. What is the loading, you mentioned negative pressure, do you mean external pressure which compresses the cylinder surface?
Just to expand on what Giovanni Zucco has said and to clarify a couple of common misconceptions. There is no "issue" in what Riks is doing; in fact, it is behaving exactly as it should. The Riks arc-length method is a Newton solver for tracing solutions to a system of equations with the ability of passing a limit point, i.e. a maximum or minimum, in the driving parameter. At a limit point the number of negative eigenvalues of the tangent stiffness matrix will always change, i.e. if a limit point corresponds to a loss of stability the change is from 0 to 1 negative eigenvalues. But a limit point isn't the only way a system can lose stability. The second possible instability for a single changing parameter is a bifurcation point, where a second equilibrium path intersects your current one. Depending on the nature of this bifurcation point (super- or supercritical) and depending on which path of the bifurcation manifold the solver is currently on, the number of negative eigenvalues may change or may not change. In structural applications when we start at zero load, the solver will generally be on some fundamental path with 0 negative eigenvalues and if it passes a bifurcation point before reaching a limit point will then show a change from zero to one negative eigenvalue. At this point you have reached an instability and the structure will buckle if modelled dynamically rather than quasi-statically. There is nothing "wrong" with Riks in continuing on the fundamental path, however, because its purpose is to follow an equilibrium branch in a direction tangential to the current path. To perturb the solver onto the bifurcated path one could (i) keep the structure perfect and push the Riks solver onto the bifurcated path (branch switching), e.g. using a secondary force or changing the predictor, or (ii) embed an imperfection which breaks the perfect bifurcation such that the imperfect prebuckling and imperfect postbuckling curves are connected. The second option is the more straightforward in commercial FE packages but has significant limitations when mutliple bifurcation points are nearly coincident as different imperfections will trigger different buckling modes. As such the Riks solver doesn't give a "false" buckling load, because one should always be looking at the number of negative eigenvalues while path-following.
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