Stressing Structure
Buckling of Panels By David Paule
You might have noticed that most airplanes have some sort of exterior surface, perhaps fabric or something more rigid like aluminum or composite. Structurally, it can make sense to make the outer surface rigid, since that can provide the maximum possible rigidity. Plus the surface generally has to be there anyway, so it might as well be useful. If it’s properly done and carries load, then it can even save weight, compared to an internal skeleton. Panels can buckle when loaded in compression or shear, and that’s a major issue. The panels are generally thin, and thin panels can’t carry as much load, obviously, as thick panels. Less obvious is the fact that their overall dimensions affect the buckling load just as much as the thickness does. That’s one of the 50
KITPLANES January 2016
reasons why we often have stiffeners attached; they divide a single large panel into smaller ones. If done right they also carry some load themselves and that further helps. We’ll look at stiffeners in another article. Here, we’re looking at the panels themselves. We’re assuming that the panels are flat. Often there’s actually a slight curvature and we should ignore the minor additional buckling resistance the curvature provides. If the curve is significant, though, we ought to take advantage of the curve, because it’ll stiffen the panel. If the curve is in the direction of the compression load, so that the panel is being loaded out of plane, the techniques described here aren’t applicable. Panels start to buckle at a stress that we can calculate, fortunately. Here’s
Figure 1: Mike Rettig’s RV-10 aft fuselage section uses aluminum panels and stiffeners for primary structure. (Photo: Mike Rettig)
the equation for this for panels loaded in compression: π 2 * Kc * E fc = * ( t / b)2 Equation 1 12 * (1 - ν2) Where fc is the compressive stress that will buckle the panel, psi Kc a constant we’ll pull off Figure 2 E our old friend, the modulus of elasticity, psi t the thickness of the panel, inches b the short dimension of the panel, inches ν Poisson’s ratio, a material property www.kitplanes.com & www.facebook.com/kitplanes
The equation for the buckling stress of a panel in shear is almost exactly the same. The only differences are that it uses a constant from a different graph: fs =
π 2 * Ks * E * ( t / b)2 12 * (1 - ν2)
Equation 2
Where fs is the shear stress that will buckle the panel, psi Ks is a shear constant we’ll find on Figure 3 There might even be some in-plane bending. If there is, that also uses a form of the same basic equation: π 2 * Kb * E fb Equation = * ( t / b)2 3 12 * (1 - ν2)
Where fb is the bending stress that will buckle the panel, psi Kb a constant we’ll get from Figure 4 Figure 4 shows curves that are a bit more complicated than the other graphs. The first thing to note is that the curves are wavy. You can see a little bit of that in Figures 2 and 3, too. This is because the proportions of the panel matter a lot in bending, and this gives you a way to
Figure 2: Parameters for buckling in compression. Generally speaking, curve C is usually applicable.
Panel Buckling Stress Example The first panel is loaded in compression. Its edges are simply supported. a = 18 in Length of the panel in the loaded direction b = 9 in Width of the panel t = .032 in Thickness of the panel a = 2 Look at Figure 2, curve C, for a/b = 2.0. Kc = 4 b The panel is 6061-T6 aluminum E = 9.9 * 106 psi Modulus of elasticity for 6061-T6. See Table 1.
()
π2 * Kc * E t 2 = ν = .33 Poisson’s ratio for 6061-T6 fc * 2 b 12 * (1 - ν ) aluminum, see Table 1 With this information, the allowable compression stress before the
the same panel. Using Figure 3 and the bottom curve for hinged edges, which are the same as simply-supported edges, we find Ks = 6.4 π2 * Ks * E t 2 fs = * fs = 739 psi 2 12 * (1 - ν ) b
()
The calculations are very similar, so I didn’t show the actual numbers. Try it yourself and see if you can get the same result. Also, note the low allowable stress. Finally, find the allowable buckling stress for in-plane bending of the same panel:
Looking at Figure 4, we see that there’s a new term, β, that we need to deal with. β is a way to include the effect of an overall compressive stress on the bending. There’s often some compression along with the bending, but right now let’s assume that there’s not. If there’s no panel will buckle is: compression, then β = 2. That gives: π2 * 4 * 9.9 * 106 * .0322 fc = fc = 462 psi π2 * Kb * E t 2 12 * (1 - .332) * 92 Kb = 23.9 fb = * 12 * (1 - ν 2) b fb = 2761 psi This is low strength. Depending on the applied stress, the panel might
()
need some stiffeners. Next let’s see what its shear buckling stress is, for Illustrations: David Paule
—D.P.
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fo
2
y
b ß
b
N.A.
ASYMPTOTIC TO
( )
focr π2 E t = kb η 12 ( 1 - νe2 ) b
32
a
28
y b
fb = fo (1 - ß ) m=1
m=2
m=3
m=4
ß=2
24
23.9
1.9 20
21.5 1.8
19.25
1.75
kb
18.17
1.7 16
17.10 1.6
15.15 1.5
13.38
1.4
12
11.78
1.3 1.25 1.2 1.1 1.0
8
10.52 10.00 9.47 8.59 7.80
0.75
4
Figure 4 (Right): Buckling of panels for in-plane bending.
tweak the strength at relatively low cost. If you can get a length/width ratio that falls near a peak on the curve, the panel will buckle at a higher stress than if it falls in a trough. Another aspect of the curves is that there are a lot of them, and they seem to depend on a mysterious factor called β. β is a number that is used to assess the amount of the panel that’s in compression. Normally, of course, symmetric panels or beams in pure bending have a neutral axis halfway across their height. If there’s a compression stress applied to the panel in addition to the bending, then there’s more of the panel that’s in compression and β allows for that. If there’s no compression,
6.35 5.30 4.55 4.0
0.50 0.25 0.0
Figure 3 (Above): Shear buckling of panels.
0
0.5
1
β = 2. If there were some compression, we’d have to draw it out, making a little sketch to scale and assess β as necessary. As you might have guessed by now, it’s not quite as simple as it seemed. But it almost is. First, we have the usual three types of edge fixity. Since the panels are rectangular, they have four sides. Each side can either be fixed, simply-supported or free. It’s difficult to put loads or reactions into a free edge, so generally those would be on the unloaded sides, if any are free at all. Most aircraft panels aren’t. The panels are either simply supported or fixed. “Clamped” is another word used to describe a fixed edge.
1.5
a/b
2
2.5
3
Simply supported means that the edge can’t move out of plane, but it’s free to rotate. It’s the panel version of the pinended condition that struts have. For example, a large panel with a stiffener down the middle is simply supported at the stiffener. In Figure 1, all the skin panels are simply supported by the stiffeners or bulkheads. A fixed edge means that the edge not only can’t move out of plane, it can’t rotate. These aren’t too common, but they do exist. A good example is that often if a wingskin is riveted firmly to a mainspar cap, the skin’s probably fixed at the spar. But it might not be, since that takes considerable stiffness, so be sure.
Table 1. Material Properties Read the exponential notation like this: 29 * 106 means 29,000,000. Look at the exponent on the 10 and add that number of zeros to the number. Also, ksi is the same as 1,000 psi or 103 psi, so don’t get ksi mixed up with psi. Material
What is it?
Poisson’s Ratio ν
Modulus of Elasticity
Max Allowable Compression or Bending Stress
Max Allowable Shear Stress
2024-T3
Aluminum
0.33
10.5 * 106 psi
Fcy * (1 + Fcy / 200000)
.61 * Fcy
6061-T6
Aluminum
0.33
9.9 * 10 psi
Fcy * (1 + Fcy / 200000)
.61 * Fcy
Steel
0.32
29 * 10 psi
Fcy
.61 * Fcy
.835 * Fcy
.61 * Fcy
Fcy
.61 * Fcy
4130 Condition N
6
6
300 Series Stainless Steel, 1/2 Hard
Stainless Steel
0.27
26 * 106 psi to 27* 106 psi
Ti-4Al-6V Annealed
Titanium
0.31
16 * 106 psi
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However, I’d expect it to be simply supported at a rib, though. Table 1 shows some of the material properties for some of the more common metals. While not many homebuilt aircraft use panels of 4130, a welded
box or plate structure of it is often highly loaded, and it’s worth checking it for buckling. I’ve included titanium because of its excellent strength. If you’re designing an airplane, it might be worth considering.
Table 1 also shows the maximum strength values that go along with these equations. For the rest of the material properties, you’ll have to look up MMPDS (www.kitplanes.com/includes/structure_ stress.html) or a similar resource.
Interaction Between Compression, Shear and Tension Usually in aircraft panels, if there’s compression there’s also shear, and there might be some in-plane bending. With all that going on, we’ll most likely need to work out how they interact. Please take a look at Figure 5. We’ll have to find the stress ratios. Here’s what stress ratios are:
Rc = .36
Rs = .45
Rb = .445
1. Find Rc / Rs = .36 / .45 = .8 Use curve .8 2. Draw a line from Rs up to the .8 line. It’s the blue line that starts at Rs on the graph. 3. Draw a line from Rb to the .8 line. This one is the red line from Rb to the curve. 4. Draw a line, this one is the dotted line, from the origin through the place where the first two lines crossed. 5. Draw a line from the place where this new line hits the .8 curve down to the horizontal axis. This finds Rsa and I’ve drawn it in blue. 6. Draw a line from the place where this new line hits the .8 curve across to the vertical axis. This finds Rba and it’s also shown as red here. Rsa = .515 Rba = .51 7. The margin of safety is either Rsa Rba MS = - 1 or MS = -1 Rs Rb What if they’re different? Easy—use the smaller margin of safety. The curve isn’t that easy to read precisely, and as this example shows, one or the other might give a lower margin of safety. That’s the one to use to be conservative. J —D.P.
Applied stress * factors of safety Rc, Rs or Rb = Allowable stress where Rc, Rs or Rb The compression, shear or bending stress ratio Applied stress Stress from your structural analysis, psi Factors of safety Usually the ultimate factor of safety Allowable stress Calculated as shown in the example below, psi Use the same factor of safety, such as for ultimate, for each stress ratio unless there’s a darn good engineering reason not to. There are two sides to the interaction graph. The left side is for the case where the shear stress ratio is smaller than the compressive stress ratio, and the right side is for when the compression stress ratio is smaller than the shear. I’ve arbitrarily chosen that for the example. For this example, 1.0
Rcx Rs 0 0.8
0.2 0.4 0.6 0.8
Rs Rcx
Rbx
0.6
0
0.4
1.0
Rba Rb
0.2 0.4 0.6 0.2
0.8 1.0
1.0
0.8
0.6
0.4
0.2
0
0
0
0.2
Rcx
0.4
Rs Rsa Rs
0.6
0.8
1.0
Figure 5: Interaction diagram for compression, shear and bending. Rcx is the stress ratio for compression, Rs is the stress ratio for shear, and Rbx is the stress ratio for bending. The bending is in the plane of the plate, that is, not trying to roll it up.
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