Flap Finite Element Analysis. ( a wing bending interlude!)
So I left off the last post with the question: “How bad is the stress in the flaps and ailerons from wing deflection and should it drive the wing bending limits?” For me, the easiest approach to answer that question is to build a simplified finite element model (FEM) of a portion of the flap (long enough to include 4 hinges) then force the deflection of the flap from curvature of the wing, while also forcing the flap into a deflected down position with a pushrod, AND applying the limit air loads from the basic glider criteria handbook.
One of the trickier bits of doing this kind of model is that the flap skin is a +/- 45 degree only composite. It is much stiffer in torsion than in the “along the flap direction”.
PCOMPS and orientation dependent stiffness :
There is a potential for major misrepresentation of stiffness when it comes to non-quasi-isotropic composites. The properties of a shell element MAY be treated as a single isotropic material but only if the lay-up is quasi-isotropic. In my case the skins are just +/- 45 degrees. NASTRAN shell elements [PSHELL entry] allow for different material stiffness properties to be used for membrane, bending, bending stiffness parameter, transverse shear, transverse to membrane thickness ratio, and membrane bending coupling. While it’s possible to populate all of those variables through laminate plate theory and lamina test data, there’s an easier way through using the PCOMP property entry.
PCOMP allows the user to input each layer’s orthotropic material properties and define each layer’s thickness and orientation. NASTRAN does the laminate plate theory and creates a PSHELL that represents the lay-up.
So that problem seems solved, but there is another pitfall. The way orthotropic stiffness properties for fabric composites are generally reported, they don’t account for the change in modulus “around the clock” from 0° to 90° they typically only give the 0° and 90° (1 and 2 direction) moduli, and using them as the orthotropic properties in a PCOMP entry (or PSHELL) won’t result in the desired behavior of the modulus being high when aligned with the fibers and much lower at +/- 45. My skins should be “soft in bending” and “stiff in torsion” for this structure. For example:
One work around is to model the +/- fabric as two orthogonal (half thickness) layers of unidirectional layers with a modulus “tuned” to result in the desired 1 and 2 direction laminate modulus values. Testing the validity of this “cheat”, along with assurance that I’ve defined the element material directions correctly in the input files, and the importance of this to accurately predict loads and stiffness suggests an FEM “coupon test” is in order.
NASTRAN fabric modulus coupon test:
This is a single NASTRAN CQUAD4 shell element with properties defined by a PCOMP entry.
If you’re not into this sort of thing skip ahead to the graph of results.
For the uniaxial properties I need an initial estimate for a unidirectional carbon (modulus in both directions) AGATE-WP3.3-033051-101 shows a E1 as 18.8 MPSI and in E2 as 1.26. Marske’s composite design manual shows carbon UNI TAPE at 17 MPSI so that’s consistent enough. For my first cut I’ll use E1 as 17MPSI and E2 as 1 MPSI. And, to keep it similar to my wing skin, I’ll assume a 2 layer of 6 oz carbon plain weave laminate thickness of .024 in.
The single element coupon looks like this:
my orthotropic material estimate is:
MID E1 E2 Nu12 G12 G1z G2z Rho
MAT8 1 17.0+6 1.0+6 0.32 0.60+6 17.0+3 17.0+3 0.00014
The transverse shear moduli (G1z and G2z) are not good guesses, but set very low to not contribute much and for thin skins it will have little effect. The units are pounds and inches and Rho (mass density) is in slinches (Sorry to all the purists it’s just what I’m most used to using)
The laminate assumes each fabric layer is really two orthogonal layers at .006 in each. Like so:
PCOMP 1 -0.012 0.0 +comp1
+comp1 1 0.006 0.0 1 0.006 90.0 +comp2
+comp2 1 0.006 90.0 1 0.006 0.0
The -.012 offsets the reference plane to the bottom surface of the .024 laminate.
The 1 in each layer entry is just referencing MAT8 number 1 from above.
Running a static load case with the unit load shown above, results in these deflections
D I S P L A C E M E N T V E C T O R
POINT ID. TYPE T1 T2 T3 R1 R2 R3
1 G 0.0 0.0 0.0 5.030161E-22 5.471115E-21 0.0
2 G 4.607568E-06 0.0 0.0 -2.318719E-21 -2.028998E-21 0.0
3 G 0.0 -.638247E-07 0.0 -5.107694E-22 -3.728907E-21 0.0
4 G 4.607568E-06 -1.638247E-07 0.0 2.344531E-21 2.932575E-21 0.0
So our laminate’s axial modulus in the 0° direction [E1] is the stress over the strain. Stress is our 1 lb load on the .024 inch thick, 1.0 inch wide coupon and strain is the average displacement in the X direction (T1) of the corners [grids 2 and 4].
Note that Marske’s composite manual has an axial modulus of 8.2 and 8.9 MPSI for plain weave and 2x2 twill carbon laminates respectively. I’m happy so far BUT, now I modify the input file so the layers are rotated from 0 °to 45° (in 7.5° increments) and plot the modulus results.
I think that worked pretty well! To be fair, I’ve done that test before. But, its very important for internal loads models, component models like the flap, and for structural dynamics models so I wanted to document it here. The axial modulus of the unidirectional material could be reduced maybe 10% to account for fiber waviness of a fabric and the results would line up well with several sources for woven carbon fabrics.
So having addressed that issue we can look at the results of the flap model. The deflected model (with a 1.0 scale of the deformed results) looks like this:
The stresses tend to be highest at the hinges and where the flap section is thickest but the tensile stress along the training edge is less than I imagined probably since the ends of the trailing edge aren’t restrained. The max values of stress along the stress major axis is shown here. And at this limit load condition are only a little over 4 ksi.
I assume that if I modeled a full length flap, it would like the middle part of this model over and over until reaching the ends of the flap where the bending stresses unload as they do here. The stresses are pretty low, and again I’m driven more by handling, damage and skin minimum thickness goals than stress from air loads and wing bending. So for this knowledge point “How bad is the stress in the flaps and ailerons from wing deflection and should it drive the wing bending limits?” I now can say the stresses in the control surfaces are not forcing a limit to my wing deflection. That doesn’t give me a number to shoot for but it relieves the angst about it.
So here’s my [probably] final thought on wing deflection limits...Back when looking at design envelopes I looked at the gust loads when lightly loaded (6.3 g) and that was going to size some components but not the spars, I also considered setting a 6g limit so Conditions I and III had the same load factor (just slightly different angles) and to be more confident in doing some light aerobatics. I chose the lower limits (5.33 at condition I and 5.7 at condition III) to avoid designing in unnecessary spar weight. But, it looks like using just enough pultrusion to meet those (with a low assumed ultimate strength of 200 ksi and a factor of safety of 2) results in a slightly soft wing in bending. It doesn’t appear TOO soft for the controls, and there are some advantages to soft. So I think I’ll make the final choice to design the spar caps for 6g and live with the resulting stiffness.
If you made it to here, WOW, thanks for following along. I'm finding it helpful for me to write down these decisions and processes as I go. It keeps me better focused, and forces me to error check myself more. Hopefully, some of you are enjoying the ride.
-Peter-
