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; in turn, greatly simplifying a preliminary analysis by allowing 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. It is generally refferred to as a cross-ply configuration having both 0-degree and 90-degree ply orientation only - no angle plies. So why is this cross-ply configuration so special? Well, to answer that you need to refer to the A-matrix and D-matrix terms found in the laminate's ABD matrix. It is here that we see the effect of excluding angle plies from our design. Namely, both the A16 and A26 stiffness terms will equal zero eliminating the extension-shear coupling. But there's more!!! We also add two other terms to the "zero stiffness coefficient" club…welcome in the D16 and D26 terms! When both these terms equal zero, the bend-twist coupling disappears. This turns out to be a "special" case because there is no other laminate layup scheme that will zero-out those D terms. Below are two examples showing a cross-ply laminate stacking sequence:
[0/90/90/0] and [90/0/90/0]
To further appreciate what was said above, refer to the ABD matrix shown in Figure 1. The ABD Matrix below shows the stiffness coefficients for a Specially-Orthotropic layup. The rectangles highlight the extension-shear (A-Matrix) and bend-twist coupling terms (D-Matrix). Notice that the A16, A26 and the D16, D26 terms are indeed equal to zero, regardless of the quantity or location of the 0-degree or 90-degree plies.
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 IF any practicable use…well, that's not entirely true. As a personal fan of the Hand-Calc., you could leverage a specially orthotropic configuration to reasonably approximate the structural performance of a more complex laminate having angle plies. I believe the aphorism "Ball Park Solution" comes to mind…which is not a bad thing in certain analysis situations. Namely, 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. The example below uses a simply supported rectangular plate with a laterally applied load at the plate's center as shown in Figure 2.
The analysis will make use of both classical lamination theory and the Navier method. It is here that one can calculate a laminate's maximum deflection, in-plane stresses; critical buckling loads, and natural frequencies without the use of FEA…who would've thought! To start we will have to derive Equation 1 below…
…wait, didn't he just get done saying this was going to be simple…no worries, because you do not have to derive that equation. Why!? Because you guessed it...we are going to make use of Special Orthotropy. Couple this layup configuration along with a set of boundary conditions and most of the terms in Equation 1 go away. For this example, we assume no thermal effects; no rotary inertia and no moments or in-plane forces applied along the edges. These assumptions will reduce Equation 1 down to Equation 2.
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 Excel to develop a simple program capable of determining the maximum transverse deflection of a 1-in X 1-in 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 8-harness fabric. I then calculated the D-Matrix terms and entered the plate geometry into the program as shown in Figure 3.
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 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 to the numerical prediction to validate the integrity of the method. The FEA solution in Figure 4 shows a maximum deflection at the center of the plate equal to 0.004227 inches (Solved using FEMAP). Comparatively, my hand-calc solution was only off by 0.000022 inches; an error that I can certainly live with...but I'm sure Spock would have had some issues with this error (my apologizes for 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 was 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 rely on the more time-consuming alternative known as FEA…at least for a problem as simple as this one. With that said an FEA was run again; however, the 90-degree plies were replaced with 45-degree plies. The updated maximum transverse deflection was calculated and equaled 0.004373 inches (about a 3.5% error). Not bad for ball-parking the plate's maximum deflection using hand calculations and some good-old pragmatic theory. Moreover, in terms of analytical efficiency, this can yield an invaluable benefit to an engineer who may find themselves under pressure to deliver a design resolution with a quick turn around.
OK…that's it for this one…please refer to the source below to learn more about the topic that I briefly discussed in this article. Hope this helps you in your future analysis endeavors...I know it has for me!
Source: Mechanics of Laminated Composite Plates and Shells, Theory and Analysis 2nd Edition, J.N. Reddy
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