Is your laminate special?
I would like to think that all my laminates are special…and I’m sure that, well, all of your laminates are special too. Hmmm…this is starting to sound like an intervention! Well, not quite…what I’m actually referring to is a type of layup that excludes the use of angle plies, greatly simplifying an upfront analysis and allowing for the use of a closed-form solution (you mean no FEA!). You’re kidding right! No…in fact I’m not…I’m referring to laminate construct commonly referred to as Specially Orthotropic.
It happens to be one of the simplest layup configurations available because its nothing more than a ‘cross-ply’ configuration. Put simply, it’s a laminate having multiple orthotropic layers with material directions oriented only in the zero and ninety-degree directions. Moreover, the A16 and A26 laminate stiffness terms are zero; but, more importantly, we now have two new additions to the “zero stiffness coefficient” club…welcome in the D16 and D26 terms; commonly referred to as the bend-twist stiffness coefficients. Below are two examples showing cross-ply stacking sequences:
Notice in the example above, highlighted by rectangles, both the extension-shear and bend-twist coupling terms. Also, take note that they are indeed equal to zero regardless of where the 0-degree or 90-degree ply is located through the thickness of the laminate. So, what do we do with this special laminate? Well, a generalized opinion within the industry seems to suggest that specially orthotropic laminates have little in any practicable use…or do they? Well…as a personal fan of the hand-calculation, one could consider using a specially orthotropic configuration to reasonably approximate the structural performance of a more complex laminate such as an angle-ply. I believe they use to refer to this as a “Ball Park Solution”…which is not a bad thing in certain situations; particularly during a preliminary design and sizing phase of a project.
To better appreciate the use of a specially orthotropic layup, a simple example is in order. This example involves a simply supported rectangular plate with a laterally applied load at the plate’s center similar to the illustration in the figure below.
The following analysis makes use of classical lamination theory and the Navier method. From here we can calculate a laminate’s maximum deflections, stresses; critical buckling loads, and natural frequencies without the use of FEA…who would’ve thought!
Let’s begin by deriving the following equation…
…wait, did he not just say this was going to be simple…no worries, because you do not have to derive the equation. Because Special Orthotropy is applied; as well as an idealized set of boundary conditions, most of the complicated terms in the equation will disappear…woohoo! For the example, no thermal or rotary inertia effects are included; and, no moments or in-plane forces along the edges of the plate are added in turn the equation reduces to:
Now that looks a lot better…for a minute there even I was beginning to worry. Ok enough of the theory…let’s dive into the example. I used an Excel to generate a simple program capable of determining the maximum transverse deflection of a 1in X 1in plate subjected to a 100-pound load located at the center of the plate. The laminate construction consisted of a [0/90]s. layup using a 8-harness fabric type. I calculated the D-Matrix terms and entered the plate geometry into a spreadsheet to calculate the following values:
The Qmn, dmn and Wmn terms are calculated using equations found in J.N Reddys Mechanics of Laminate Plates and Shells, page 248. I’m going to avoid delving into the step-by-step minutia of showing all the calculations and cut right to the answer. The maximum transverse deflection of the plate was calculated to be 0.004249 inches. Now I stepped out on a limb here and decided to use FEA (what...yes, I did). Of course, this was only to compare the analytical prediction to the FEA prediction (shown in Figure below) which equaled 0.004227 inches (FEA solution was derived via FEMAP). Comparatively, my solution was only off by 0.000022 inches; an error I can certainly live with but I’m sure Spock would have balked at this error (Sorry…for yet another Star Trek reference).
The next step was to compare the difference between the predicted transverse deflection using a specially orthotropic laminate and an angle ply laminate which lacks any tractable closed-form solution method (feel free to share one if you know of one). Again, the purpose of this exercise is to elucidate a practicable alternative to FEA that provides an engineer with an ability to calculate the performance of a laminate quickly and approximately without having to revert to a time-consuming FEA exercise…at least during the preliminary phase of a design’s development.
With that said the FEA was run again; however, the 90-degree plies were substituted with 45-degree plies. The updated transverse deflection was determined to be 0.004373 inches (about a 3.5% error). Not bad for ball-parking via hand-calculations and some good-ole pragmatic theory. Moreover, in terms of analytical efficiency, this can yield an invaluable benefit to any engineer under pressure to deliver a resolution whether a design is viable or futile when determining a particular conceptual design direction. Or better yet, using this approach in general, is just...well...good sound engineering practice!
Ok…that’s it for this one…and please refer to the source below to learn more about the topic I briefly discussed in this article as well as the plethora of other solutions one can solve beyond just plate deflections. Hope this helps you in your future composite analysis endeavors...I know it has for me.