Wing loading at +6g

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:

 

This plot shows the loading really rolls off at about 12.5ft half-span.  There are too many wing sections to easily look at the numbers, so let's drop down to 5 panels per wing half:

The last panel isn't loaded nearly as heavily as the root areas.  Putting some numbers to this, AVL outputs the coefficients of pressure for each wing section (staying with the 10 panel wing):

  Surface # 1     wing                                   
     # 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 :-)