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 that simplifies a 3-D problem for others. I for one am guilty…because I appreciate the fact that you can take a 3-D analysis and knock it down a dimension: in this case, down to a 2-D problem.
Whaaat…no 3rd direction to deal with…well alright! Now my analysis life just got whole heck-of-a lot better…or did it? For a majority of "thin" laminate designs that pesky 3rd direction can safely be eliminated (caveat: assuming you know the implications behind this elimination—maybe a review of Classical Lamination Theory is in order if your not sure). There are however, times (regardless of thickness) where the "3rd direction" is important. One of those times involves the analysis of a curved plate typically found in designs such as an L-shaped flange; or, a C-channel design. Additionally, there are designs that may have an undersized inner radius that is subjected to a high bending moment. Of course, we have never seen that before! If this is the case, there are several immediate concerns that come to mind. Chiefly, fabric bridging and/or corner thickening. Also, there is the potential for a localized flatwise tension failure. This is due to an increase in the normal tension stress that shunts through the thickness of that plate's curved region as shown in Figure 1. If this case, your plate analysis just transitioned from 2-D (in-plane) to a 3-D (out-of-plane - matrix dominated) failure mode. Now, what do you do!? Well, that's a good question…I'm not sure…just kidding!
Firstly, it is important to recognize that any inter-ply delamination can be catastrophic and catching this failure mode during the preliminary design phase, rather than downstream during production is far easier to correct and less costly. Because, If one is invariably confronted with the alternative which requires changing the OML geometry of a newly minted tool, then it is safe to say that you now have a costly mess on your hands - yikes! Secondly, the analyst should ascertain the maximum allowable strengths for the matrix. Specifccaly, 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 defined in Figure 1. Thirdly, you need derive the equations shown below and commit them to memory…no...just kidding--again! Both the radial and tangential equations shown below are readily available and were develop by Professor Sergei Gheorgievich Lekhnitskii. Both equations determine the radial and tangential stresses in a singly curved laminate subjected to a pure bending moment only respectively.
As much as I would like to help you here, I'm going to defer and let you decide how you will implement these equations. When I tackled this problem, I simply used both excel and python to develop a script to perform the 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 the curved region is failing in the radial direction…the 3rd direction that is...or, the tangential direction too.
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 3-D world. So, always be sure to vet your curved plate designs for flatwise tension failure using a quick and intelligible method like the one discussed 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….
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