Earlier I posted a white paper addressing some of the short-comings associated with analyzing the buckling behavior of composite parts in NX Nastran. Due to a few un-answered questions as well as feedback from the original paper, it felt prudent to update my findings, and clear up some unresolved results. It is assumed that the reader has already read the first paper, which can be found here
. In this follow up, I will provide an updated recommendation on when and how one can employ offsets on 2D elements, as well as how to handle short wavelength buckling in cored panels.
Buckling with offsets: On the Element or Property
One question that resulted from the original white paper was: “How did you apply your offsets? On the property or on the element?” As it turns out, this question matters a lot, though at the surface level, a NASTRAN user might think “well why should it matter? An offset shouldn’t be any different whether it is applied to the element or property”. To reduce complexity, the 6 test coupons from the first paper used CQUAD4/R/8 elements and a metallic plate PSHELL property with a thickness of 0.05”. The only way to apply offsets to elements using the PSHELL property is to apply the offset to the element card.
Buckling Test Coupons
The same 6 test coupons were updated to use a PCOMP laminate property containing 1 ply of the same Aluminum material with a ply thickness of 0.05” (the PSHELL thickness of the original plates). Having updated the coupons to use PCOMP as opposed to PSHELL properties, the right hand side coupons could also be updated to move the offset to the property card instead of on the element. The original 3 benchmarks were then repeated and the results of those benchmarks can be seen in Table 1 through Table 3.
PCOMP Offset Buckling Benchmark Results
From these results two things become clear. The first is yet another confirmation that the CQUAD8 elements are still unusable as they fail to accurately capture the correct buckling behavior in the shear simply supported benchmark. For those parties interested, this is using Simcenter Nastran 2019.1 (the successor to NX Nastran 12.2). The second more enlightening piece is that when offsets are applied to the PCOMP property rather than the element, the results produced between the CQUAD4 and CQUADR elements are fairly similar. So why does it matter where the offsets are applied? These updated findings show that as long as the user is only applying offsets to the PCOMP card and not the element card, using CQUAD4 elements is the most correct method for modeling composite structures as it both handles offsets correctly and has an accurate transverse shear stiffness.
Theoretical Discussion of Short Wavelength Buckles
In section 7.3 in the NX Nastran Theoretical guide (page 323 in the pdf that ships with Simcenter NASTRAN 2019.1), the author goes into detail about how the geometric (they use differential) stiffness matrix is formulated for any 2D plane stress element. Below is an infinitesimal section cored panel, simplified to a 1-dimensional shear deformable beam:
Cored Panel Infinitesimal Segment
With the above exaggerated shear strain, it is not hard to see that as the shear strain increases, the length of the beam does as well. Since the facesheets are typically stiff, rather than dramatically increasing in length the way the core would, they develop a large in-plane load. This load then serves to somewhat restore the de-stabilizing effect that would occur when the whole section is in compression, so the total potential energy of the beam from equation 1 then becomes:
When cored panels shear, in reality the facesheets develop an in-plane restorative tension fighting the buckling load in the structure, preventing it from buckling in a purely shear dominated behavior.
Test Model Setup
To explore the theory that the restorative facesheet tension and the importance of the transverse shear stiffness to the buckling load, four test coupons were generated in order to model structures with low transverse shear stiffness.
Short Wavelength Buckling Test Coupons
The coupons model cored panels, whose layups are: [0/452/0/0.125” core/0/452/0]. The light (1) and dark (3) orange coupons are modeled exclusively using CQUAD4 elements and a PCOMP property. The light (2) and dark (4) green coupons are modeled using CQUAD4 elements for the facesheets ([0/452/0]) and 4 CHEXA elements extruded through the thickness of the core. To differentiate between the two modeling approaches, we’ll call the orange coupons 2D, and the green coupons 2.5D. Below is a depiction of one of the 2.5D coupon up close:
2.5D Test Coupon
One might notice the red lines running through the thickness of this coupon. These are RBE2 elements that are used to ensure that at the boundary of the plates, the facesheets are forced to move together, and serves as a convenient place at which to load and constrain the plate.
Simply Supported Uniaxial Buckling Benchmark
Typically, the short wavelength buckles only appear in circumstances where the plate is restrained on all sides, such as in a simply supported case. As such, coupons 1 and 2 were pinned in the 2 and 3 (y an z) directions along both short edges, pinned in the 1 and 3 (x and z) direction on the bottom edge, and in the 3 direction on the upper edge:
Simply supported boundary conditions
The load was applied as an enforced displacement of 0.012893 in the -y direction. As expected, the first 393 buckle mode shapes are all seemingly random variations of short-wavelength buckles with the first eigenvalue at ~0.999902 and progressively increasing at least until 1.249227. Below is one example:
Coupon 1 simply supported mode shape 9
In contrast, the first eigenvalue and mode shape pair produced by the 2.5D model predict a mode shape that seems plausible, at an eigenvalue of 1.200:
Coupon 2 simply supported mode shape 1
It’s important to note that the critical eigenvalue produced by the 2.5D coupon (2) model is never predicted by coupon (1). What is concerning is that not only does the 2D modeling technique create many fictitious eigenvalues, but it also doesn’t correctly predict the correct eigenvalue/mode either.
Expanding on core instabilities
As previously stated, some have argued that the short wavelength buckles are shear crimping, or an indication that shear crimping might be occurring. Using the actual material properties, the shear crimping critical load can be calculated as:
The load at which coupon (1) exhibits short wavelength buckles is ~226 lb/in, well over the actual critical crimping load. After further investigation, my understanding is that shear crimping is a failure mode only possible in celled core, as it is the through-thickness buckling of the cell walls. Clearly the 2.5D coupon (2) model doesn’t have the adequate fidelity to model those stability failure modes. With that in mind, it seems inconsistent to imply that the 2D CQUAD4 model would then be capable of capturing that behavior at all.
Buckling Sensitivity to Transverse Shear Stiffness of Core
The NASTRAN documentation indicated that in order to mitigate the short wavelength buckling problem, one simply needs to set the value of the shear strain to zero. One way to achieve this via modeling is to set the core transverse shear stiffness to a very large number. To test this hypothesis’ validity, we can simply do a sensitivity study alluded to in 2.1.2 by repeating the simply supported buckling benchmark from 2.1.4 using a core with a transverse shear stiffness that is 10% higher.
The results for coupon (4) using the 2.5D modeling technique yield an interesting result. The eigenvalue for coupon (4) with a 10% increase in core transverse shear stiffness is 1.292, 7.6% larger than that of coupon (2). The buckle mode shape is still very similar as well:
Coupon 4 simply supported mode shape
This shows that (as one would expect), the core stiffness does have some non-trivial impact on the buckling eigenvalue of cored structure, and so despite what the NASTRAN documentation argues, I would disagree that one could simply ignore the transverse shear effects by setting that strain equal to zero.
The findings of this paper refute the conclusions from the previous paper that NASTRAN has few good options for analyzing structural stability for composite structures with offsets. In fact, the default solution of applying offsets to the property card for a laminate, and using CQUAD4 elements results in an accurate solution. This also means that there is no need to resort to using CQUADR elements which have an incorrect transverse shear stiffness formulation, unless one is determined to apply offsets at the element level.
This paper also concludes the original goal of the previous work in explaining some of the underlying theory for why short wavelength buckles in structures with soft transverse shear stiffness (such as cored panels) are completely non-physical, and arguably have little to do with other failure modes such as shear crimping and facesheet wrinkling. One could argue that the appearance of short wavelength buckles isn’t meaningless, as they indicate the structure (likely with a soft core) has a soft transverse shear stiffness. I would argue that the analyst would know that they put a soft core in their structure, and so don’t gain any information from seeing the short wavelength buckles that they didn’t already have.
Some have also voiced the contrapositive argument, that if the shear crimping margin is positive, one could ignore the short wavelength buckles, however section 2.1.4 shows that using the 2D modeling technique, one will entirely miss a real buckling instability being completely masked by the non-physical short wave buckles.
Finally, this paper provides a means of mitigating short wavelength buckles in a plate model buckling analysis by using solid elements to represent the core.