In the previous post in this series, I introduced the basic steps that you need to take to come up with a design for a COPV that will work for you. And I also introduced you to the Standards that are written that control the design and certification of these things to be used in service. Mostly of course the Standards are about the safety of people and equipment (mostly people) that are around these things when they are holding their design pressure. Burst of a pressure vessel is a violent, explosive event and if someone is standing or working next to one of these things when it goes off it will definitely injure them and possibly even kill them. So, the use of pressure vessels in commercial (and military) service, COPVs included, is tightly regulated.
I also talked a little bit about the hoop versus helical windings that go on a COPV and the purpose for each of these winding directions. In an overly simplified view, the hoop windings take the hoop stress and the helical windings to take the longitudinal stress. But what angles do we choose and what is the percentage of fiber that is wound around the hoop direction of our COPV and what percentage is wound in a helical direction?
There are some rules of thumb to get started with, but no matter what you do you have to do a stress analysis of the COPV taking into account the stiffness of the liner and the stiffness and strength of all of the layers of fiber that get wound on to make it a COPV.
It is in the setup of the stress analysis of the COPV that most people new to this get into trouble. First, the mechanical properties of the liner are usually well known because by and large the liner is a monolithic metal – 6061 Aluminum is the common choice.
Next we have to figure out the mechanical properties of the overwrap so that we can apply them in our Analysis program This is most commonly an FEA program that allows orthotropic materials. The fiber overwrap has different stiffnesses in all three of the directions of interest – hoop, longitudinal, and through the thickness of the wrap. That is what defines it as an orthotropic material.
For an example about how to get started, let’s define the stiffness in two directions in a ply where all of the fibers are going the same direction. This is called a “unidirectional” ply. And, just to make this real, let’s use carbon fiber and an epoxy resin. You can find the ply properties of carbon/epoxy in several web sites, and I found a set that is used by ESP composites in their eLaminate property estimator. We need 4 properties to define the ply in two dimensions (for 3 dimensions it takes 9): Stiffness or Young’s Modulus (E) in the fiber direction, E across the fiber, Poisson’s Ratio (ν), and Shear stiffness (Shear Modulus (G)). They are as follows:
· Ex (fiber) = 27 million psi
· Ey (across) = 1.5 million psi
· νxy = .35
· Gxy = 1.1 million psi
Now to get the orthotropic properties of the fiber stack that makes up the wrap, remember that the hoop fibers have their stiff direction in the direction around the circumference (typically called the ϕ (Greek phi) direction), so this is where we apply the 27 million psi. Then for the long direction stiffness of the hoop fibers, we apply the stiffness of the ply across the ply – or 1.5 million psi. For the helical fibers, once we decide what angle they are with respect to either the long direction or the hoop direction, we use cosines of the angles times the stiffnesses in each direction to get the hoop and the long components of stiffness of those plies.
Now that we have the layer stiffnesses, we can use what is called the “rule of mixtures” to calculate the stiffnesses of the laminate stack in the hoop and long directions. Basically, what we do is multiply the stiffness of the hoop fibers in each direction times the number of hoop plies, add that to the product of the number of helical plies times the calculated stiffnesses (from the paragraph above) in the hoop and long directions, add those two numbers together for each property and divide by the total number of layers. This is at least a good starting point for getting numbers that you can use in your first cut Finite Element Analysis (FEA) of the COPV.
The pics in this post show sort of how all of that works. The two schematics above show how the helical fiber angles are modeled where the fiber angles are taken from the long axis of the pressure vessel. This is what you put into the input file of your FEA program for the properties of the wrap.
The pic to the right shows what the stresses look like for different wrap angles for the same liner and internal pressure. The bottom center pic shows what to expect if you isolate a short section of the long cylindrical part of your COPV and examine the stresses in the liner and in the wrap at different locations through the thickness of the wrap.
Finally, here is a pic where there are three different sets of results for three different pressure vessels. The top two show the overall deformation of the COPV under internal pressure. The hoop stress and long stress pictures should look the same as the deformation picture where the stress or deformation is the same through the whole long cylindrical section of the COPV in the liner and in the wrap. This is the way it is supposed to be.
What about the bottom right pic – it doesn’t look the same as the two above it. I decided to put this one in because it demonstrates a classic mistake that first timers make when analyzing a COPV under internal pressure. If you use the X and Y stiffnesses of the fibers and do not apply those numbers in a cylindrical coordinate system in your FEA program, the program thinks that the stiffness of the wrap is in x and y rather than z (long) and ϕ (hoop). This graphic is from a mistake that the designer/analyst made when applying mechanical properties to the wrap in the FEA program Abaqus. And this one was posted on Research Gate of all places.
For this series of posts, I think I’ve talked about COPVs enough for people to get a taste of how either easy or difficult it is to design one – based on your understanding of engineering mechanics. If you don’t understand engineering mechanics at all or have not been trained in it like the people that work in composites – like me – have been, then I at least hope that I have brought the concepts down to earth enough for you to understand them. That was my goal anyway – hopefully I got at least close.
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