Although it is well known that Simcenter supports linear buckling solutions (SOL 105), some engineers are reluctant to use this method for composite laminate, based on the perception that the linear buckling analysis does not adequately account for the low transverse shear stiffness of the laminate and is thus not-conservative. This paper demonstrates that Simcenter linear buckling analyses are, in fact, adequate for composite laminates, even for sandwich structures.
Buckling of plates inherently involves large transverse deflections and out-of-plane bending, for which transverse shear becomes significant. Since composite materials have relatively low transverse shear stiffness compared to their in-plane properties, standard plate buckling equations will overestimate the buckling strength of composite laminates. This error will be larger for thicker laminates. This behavior is well known to composite structural analysts and nothing new has been presented thus far.
There is this persistent belief among some engineers that the linear buckling solution implemented in engineering simulation software suffers from the same deficiency. They will instead prefer to use a non-linear finite-element solution to determine the buckling strength of a laminate. The idea is that the gradually applied load during the non-linear solution will cause it to stop converging when the buckling strength has been exceeded.
This non-linear approach is sound, but does have some caveats:
- The onset of instability must be artificially introduced for flat surfaces;
- The model could stop converging for reasons other than buckling of the structure, requiring additional interpretation;
- The non-linear solution takes more time to solve (Many load steps are needed for an accurate prediction);
The question is therefore whether or not the non-linear approach is worth this extra effort (i.e. Are the results from the linear buckling approach actually not conservative?).
We will now investigate a buckling problem with the two approaches previously discussed using Simcenter. Let us begin with a simple 18in by 10in flat plate simply supported on three edges as shown below:
In this first example, we will define an 8-ply laminate [45/0/-45/90]S of T300/5208 (Properties used shown below).
We set-up the boundary conditions as:
At this point, we can readily perform a linear buckling analysis (SOL 105), where we predict a first buckling mode at 181.3lbf (200lbf applied * 0.906538).
We will now verify whether or not we get similar results with a non-linear analysis. First, because we have a perfectly flat plate, we must artificially introduce the onset of instability by slightly moving a node out of the plane of the plate. We thus translate a node near the center of the plate as shown below:
If this step was not done, the plate would most likely simply compress but would not bend out of its initial plane.
We now set-up the non-linear solution (SOL 106), selecting the global iterative solver.
Then we set a large number of increments (100 here) and change the convergence criteria as Displacement and Work.
Finally, we change the parameters to account for large displacements:
In order to achieve buckling, the applied load must exceed the critical load. If the applied load is insufficient, we will not know how close to buckling we are. We set the applied load to 250lbf, well above the critical load predicted by the linear buckling solution.
The last increment of the solution to converge is the 72nd. As a result, we calculate the critical load as 250lbf*72/100=180lbf. The next increment would have corresponded to a load of 182.5lbf. We can see that the critical load predicted by the linear buckling analysis fits right between these two values. At first glance, it seems that linear buckling solution is no less conservative than the non-linear solution.
Let us change the lay-up to [(-45/90/45/0)
S_2]S, increasing its thickness to 32 plies as to generate a fair amount of transverse shear stresses. The linear buckling solution now predicts a critical load of 10,746.6lbf. Applying 11,000lbf to the non-linear model, we obtain convergence up to increment 97, corresponding to a load of 10,670lbf. This is again just below the SOL 105 prediction.
If we further add a honeycomb core (5056, 3/16” cells, 3.1PCF, 1/8” thick) to the center of this laminate, we increase the buckling critical load to 30,202.8 (according to the SOL 105). Applying 31,000lbf to the non-linear model, we obtain convergence up to increment 97, corresponding to a load of 30,070lbf.
With these 3 tests, we have shown that the only conservatism coming from using a non-linear solution is due to the incremental nature of the load application. Indeed the last load for which convergence is observed was found to be below the critical buckling load. Note that this may not always be the case, depending on the load applied and the number of increments selected to run the non-linear model. Nevertheless, this exercise has shown a clear relation between the results of a linear buckling analysis (SOL 105) and a non-linear analysis (SOL 106), both performed with Simcenter.
The results of a Simcenter linear buckling analysis (SOL 105) can therefore be considered as sound for composite laminates and sandwich panels. One should however be mindful that this may not be the case for other simulation suites. Also, analysts should also consider that other failure modes may, in fact, be the critical ones in certain situations (e.g. delamination, core failure…). These failure modes would not necessarily be exposed in either a SOL 105 or a SOL 106.