Starting the math...
Using the AVL model I gave in a previous post, we'd like the wing loading for say a 6g pull-up maneuver. Say CLmax is 1.5 and load factor is 6 (using gross mass of 305 lb-mass), giving a necessary condition of 80 ft/s (54mph) and -21 deg elevator. At this test case, the trefftz plane shows:
# Chordwise = 10 # Spanwise = 10 First strip = 1
Surface area = 179.309448 Ave. chord = 4.974011
CLsurf = 1.50090 Clsurf = 0.00000
CYsurf = 0.00000 Cmsurf = -0.01353
CDsurf = 0.50339 Cnsurf = 0.00000
CDisurf = 0.07366 CDvsurf = 0.42973
Forces referred to Ssurf, Cave about hinge axis thru LE
CLsurf = 1.32253 CDsurf = 0.44357
Deflect =
Strip Forces referred to Strip Area, Chord
j Yle Chord Area c cl ai cl_norm cl cd cdv cm_c/4 cm_LE C.P.x/c
1 -17.8428 4.2515 2.6609 1.5916 0.2041 0.4198 0.3729 0.0428 0.0047 -0.0402 -0.1256 0.359
2 -15.8461 5.0000 18.1066 5.0004 0.1971 1.1179 0.9994 0.1290 0.0453 -0.1015 -0.3534 0.352
3 -11.1640 5.0000 28.2017 6.7405 0.1283 1.4925 1.3546 0.4104 0.3418 -0.1260 -0.4765 0.343
4 -6.2763 5.0000 19.2959 7.3489 0.1029 1.6046 1.4793 0.5793 0.5221 -0.1328 -0.5206 0.340
5 -2.1627 5.0000 21.3896 7.6420 0.0951 1.6471 1.5386 0.6810 0.6267 -0.1366 -0.5431 0.339
6 2.1627 5.0000 21.3897 7.6419 0.0934 1.6471 1.5385 0.6810 0.6267 -0.1366 -0.5431 0.339
7 6.2763 5.0000 19.2959 7.3489 0.1029 1.6046 1.4793 0.5793 0.5221 -0.1328 -0.5206 0.340
8 11.1640 5.0000 28.2017 6.7405 0.1283 1.4925 1.3546 0.4104 0.3418 -0.1260 -0.4765 0.343
9 15.8461 5.0000 18.1066 5.0004 0.1971 1.1179 0.9994 0.1290 0.0453 -0.1015 -0.3534 0.352
10 17.8428 4.2515 2.6609 1.5916 0.2041 0.4198 0.3729 0.0428 0.0047 -0.0402 -0.1256 0.359
The column labeled cl_norm helps us get to the z-axis force at the local section. Backing out through the standard normalization CL=F_z/(Q * Sref) where Q = 0.5 * rho * V^2 ... section 1 (a tip) has the following loading:
Dynamic pressure:
q = 0.5 * rho * V^2 = 0.5 * 0.002378 slug/ft^3 * (80.7 ft/s)^2 = 7.743 slug/ft/s^2
Section force:
F_z = cl_norm * q * S_section = 0.4198 * 7.743 slug/ft/s^2 * 2.6609 ft^2
= 8.65 slug*ft/s^2 = 8.65 lb-force
(this is where I hate English units and love SI units ... kg makes more sense than lb vs lb-mass vs lb-force)
Carrying out these calculations on the other sections gives the following section forces:
section # | Yle (ft) | F_z (lb) |
1 | -17.84 | 8.65 |
2 | -15.85 | 156.74 |
3 | -11.16 | 325.93 |
4 | -6.28 | 239.75 |
5 | -2.16 | 272.80 |
6 | 2.16 | 272.81 |
7 | 6.28 | 239.75 |
8 | 11.16 | 325.93 |
9 | 15.85 | 156.74 |
10 | 17.84 | 8.65 |
The total sum of all F_z is 2007.74 lb, which is more than 6G's * 305lb, but reflects the additional loading due to dihedral angle (some aero force is pointing in the y-direction too). This tells that the outer two panels, from 18 to 17.68ft and 17.68 to 14.02ft combined have to carry approximately 165lb at the 6G loading case. Now the strut actually joins my wing at 139in (11.58ft), so we really should change the panel locations to correspond better with what is beyond the strut attach point. Here is a top-down view of the 10 panel wing for reference.
Going back to the 50 panel wing we started with at the top of the post, the wing outside the strut carries 269.5lb, the wing between the strut and the jury attach carries 353.8lb, and the inside wing between the jury attach and the root carries 378.5lb. Do note that these forces are not centered on the panels, especially the tip weight; rather, the lift distribution governs the location. That we can figure out from the lift distribution too ... but will wait for another day.
Here is the full 50 panel wing to get a better idea of the number of strips:
I drew up the spars in Solidworks and will run them through an FEA analysis to get an idea (*idea) of the stress distribution. Notably, there is area between the sleeves taken up by several wraps of electrical tape (as noted on Sandlin's drawings). The FEA will be assuming the walls touch and do not slip, which is not a conservative assumption. I'll chat with some folks at work to get some additional input.
Please, if you are a reader and note a mistake in my math or assumptions somewhere, please please let me know. I will not discard your input and would be happy to spend time working with you to get this right. Your life-saving thoughts are most appreciated :-)