Composites…It’s not just a 2D world…got Flatwise Tension?
Too often I hear both analysts and designers toss around the term quasi-isotropy…a mythical monster to some, an in-plane idealization to simplify a 3D problem for others. I for one am guilty…because I appreciate the fact that you can take a complex 3D composite problem and knock it down a dimension...in this case, the out-of-plane direction.
Whaaat…no 3rd direction to deal with…alright, now my analysis life just got whole heck-of-a lot better…or did it? Well, for a majority of “thin” laminate designs that pesky 3rd direction can safely be eliminated (caveat: assuming you know the implications behind that assumption—review Classical Lamination Theory); however, there are other times (regardless of thickness) where that direction becomes increasingly important. One of those times is when you are confronted with a curved segment found in designs such as an L-shaped flange or C-channel. Moreover, the design may have an undersized inner radius produced from a female tool that is subjected to a high bending moment; of course, we have never seen that before! In this case, several immediate concerns come to mind; such as, bridging and/or corner thickening. But there is also the potential for a localized flatwise tension failure due to a large normal tension stress through the thickness of the laminate when subjected to bending. If that’s the case, your structural analysis just transitioned from considering in-plane failure only to now addressing an out-of-plane (matrix dominated) failure mode. So, what do you do? Good question…, well I’m not sure…just kidding!
Firstly, it’s important to recognize that any interply delamination can be catastrophic. Catching this failure mode early, during the preliminary design phase, rather than downstream during production is far easier and less costly; especially if one is invariably confronted with the alternative option, changing the OML geometry on a newly minted tool-yikes! Secondly, the analyst needs to ascertain the three-maximum strength allowables for the matrix; in particular, the transverse tensile strength F33T and the shear strengths in both the F13 and F23 planes. Once you have those values the radial stress analysis can begin by defining the critical geometric parameters similar to the ones shown in the figure above. Thirdly, you need derive the equations shown below and commit them to memory…just kidding again! The radial and tangential equations shown below are readily available, and were develop by Professor Sergei Gheorgievich Lekhnitskii. These equations are applied to determine the radial and tangential stresses in a singly curved laminate subjected to a pure bending moment only (See figure above).
As much as I would like to help you here, I’m going to have to defer and let you decide how you will implement these equations. Tip: I simply used both excel and python to develop a script to facilitate both calculations and design iterations (Incredible isn’t it…I didn’t use FEA!). From here you can use the Chang and Springer failure index equation to determine if your curved laminate design is failing in the radial direction…the 3rd direction that is...or the tangential direction as well.
Well, that’s it! I know this is a brief discussion for a topic that could fill many pages, but the intent here was to simply make you (the composite analyst) aware that you often forget that we live in 3D world. So, always be sure to vet your curved plate designs for flatwise tension failure using a quick and intelligible method like the one illustrated here. I have included links to some well-written white papers that may facilitate your endeavors to boldly go where no composite engineer has gone before…well…what did you expect, I’m an engineer, what would a technical article be without a Star Trek reference.
NASA Technical Memorandum 4026, Delamination Stresses in Semicircular Laminated Composite Bars
NASA Contractor Report 182018, Delamination Failure in Unidiectional Curved Composite Laminate
Editor's Note: Article updated 5/15/2018