Stabilization against buckling.

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Autodidact

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Question for the structural gurus; if you have a length of column that constitutes the cap or flange of a beam (spar), it may want to buckle laterally (normal to the web), so how much (normal) force is required to keep it from buckling laterally? A simplified solution would be the handiest answer, such as, if you applied a concentrated force laterally at the midpoint of the flange, what magnitude of that concentrated force would stop it from buckling due to the compression force and drive it to either buckle at each half or cripple depending on the fineness ratio. I know this can get complicated, but there is a force than can cause the flange/cap to buckle at the halves rather than the whole length - does anyone know where to find the formula for this situation as well as for it's extension to the distributed force case?
 

harrisonaero

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Hit the books and learn the difference between buckling and crippling and how to calculate for a beam and a section. Suggest studying Flabel first and then go on to Bruhn and Peery.
 

Dana

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It's not so much applying a force at the midpoint, rather it's a matter of restraining it from deflecting out of column. In such a case the force is zero until it starts to deflect, and then it's more a matter of stiffness than force, just as buckling is a matter of elastic modulus rather than tensile (or compressive) strength.

Look at the jury struts on the wing struts of many older high wing planes; they're not very large and don't need to be. But as you say it's complicated, in most cases it's statically indeterminate.

Dana
 

wsimpso1

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I agree with Dana. What you have to do is prevent the middle of the column from deflecting, and the force required is near zero, but that does mean that they have to be fairly stiff. In fact, what you are doing to prevent failure is adding in the stiffness that you would have otherwise added by increasing EI of the section... IF you can effectively make that deflection very low, you now have converted one long column into two columns of half the length, which quadruples the Critical Load for that particular buckling case. In spars, you do have trouble with where to anchor the other end of the jury strut. In fabric covered airplanes, the ribs do stabilize both the shear web and the spar caps, just the rib spacing. In a hard skinned wings, the skin also serves to stabilize the caps. In modern airliners, the skins are distributed spar caps with flanged webs, and thus very resistant to buckling.

Well, I have never designed in most material sets, so I do not know how likely any of the scenarios are in most materials. What tools do you have? Ribs that are deliberately supporting the caps and web, the same ribs closer together, little short ribs bracing the web and caps together, more EI in the caps (I about the vertical axis), stiffer skins, etc. Ideally, you will check out the best candidates, take the lightest one that does the job. In most fabric covered wings a plywood skin from the spar forming the D-tube really does stiffen things up to fore-aft bending and buckling, but you could bump D-tube skin thickness and/or tightening up the rib spacings. In metal skinned wings, you can easily tighten up the rib spacing and/or bump the skin thickness and/or bump the cap width. In composites, this is rarely an issue - to make strength at min weight, the spar caps are frequently overbuilt, and sturdy skins are strongly bonded to the caps.

Billski
 

Himat

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In a theoretical case with pure end load on a slender column free at both ends that fail in buckling fail due to a loss of stability. The force to stabilize one failure mode is then not zero, but infinite small. Figure and further argumentation to follow.

Edit; As Dana said I see when reading once more the posts in the thread.
 
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wsimpso1

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I took his question to be asking about these load cases.??

View attachment 67998
That might have been the question, but there were several things wrong with the question, and the OP did not know that. We went on with the actual solution without saying why we had abandoned the OP's premise.

We can not predict the restoring force required to keep it in line very accurately. Column buckling will send the column in one of two directions, both along the axis in which the column is softest, but in opposite directions. We do not know which of the two directions. Next is that the restoring force required to keep the column in line is very small but real, and not needed until the compression load approaches critical load and out of column deflection commences. If you were to apply a static force in one of the two directions, you would prevent collapse in the direction that the restoring load is from, but you will deflect the column in the opposite direction when the compression is not enough to make lateral deflections, and you will cause a buckling failure away from the restoring load at lower load than if you had not applied restoring loads.

If you were to prevent column failure with forces, you would need an active system with actuators and sensors. The sensors would have to detect the earliest beginnings of deflection out of column, and then compute and drive restoring forces to push the column back into line. This seems very costly and heavy compared to simply designing a structure to carry the known loads multiplied by a factor of safety. Which as the path we started down ...

Billski
 

Himat

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Here is a figure of the theoretical case.
To the left the column is free with end loads applied in black. The lowest failure mode, buckling to half wave is shown in blue. To keep the column from buckling in this mode stabilizing it at the centre will help. Indicated with a green circle. That will force the buckling to the next higher mode, a full wave with a node in the middle. Again the failure mode is shown in blue.

Now, before buckling, the column is straight and the geometry allow no sideway force to exist. Only after failure the column bend and allow a sideway force to be generated, increasing in magnitude as the centre of the column move sideways. The force to stabilize the theoretical case should then be close to zero.
CollumnBuckling.jpg
 

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. Next is that the restoring force required i
Have to wait for the OP to clarify but I was looking at the first of his questions from a different view point.

"so how much (normal) force is required to keep it from buckling laterally?"

Turn this around and ask not how much force is required to restore but how much force is exerted on the stabilizing link by the collapsing column. Imagine the link being very very close, but not in contact until the column deflects.
 

Dana

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Have to wait for the OP to clarify but I was looking at the first of his questions from a different view point.

"so how much (normal) force is required to keep it from buckling laterally?"

Turn this around and ask not how much force is required to restore but how much force is exerted on the stabilizing link by the collapsing column. Imagine the link being very very close, but not in contact until the column deflects.
The closer the stabilizing link, the lower the force, approaching zero force at zero deflection.

Dana
 

wsimpso1

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Have to wait for the OP to clarify but I was looking at the first of his questions from a different view point.

"so how much (normal) force is required to keep it from buckling laterally?"

Turn this around and ask not how much force is required to restore but how much force is exerted on the stabilizing link by the collapsing column. Imagine the link being very very close, but not in contact until the column deflects.
Already answered. The force to keep the column at zero lateral deflection is vanishingly close to zero. It requires some stiffness, but not much strength. Making the EI of the column a step higher near the middle or with a jury strut will do it.

Billski
 

Autodidact

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My apologies, the question was very badly worded. I did understand that until there is enough compression force to cause out of column deflection, no lateral force would be required. But once the out of column deflection begins, larger amounts of resistance are needed to prevent it, until at some point the column will cripple (may be bad terminology again) like a coke can being crushed and that has to do with the elastic limit where plastic deformation begins rather than the criteria used for Euler column buckling. What I'm wanting to know, I guess, is how do you size that jury strut, or how do you size the skin thickness for that D-tube LE with respect to how much it adds to the spar cap's resistance to lateral buckling? The answers that it is statically indeterminate, or an increase in "I" laterally are making the most sense. Of the design cases I've read about, partial buckling in shear webs seems to have the most in common, but is not quite the same, I think.

Am I worrying about something that doesn't really exist? IOW, is the distance between ribs and other stiffening features usually small enough that spar caps will cripple rather than buckle, even without some additional stiffening like a jury strut or D-tube skin? Harrisonaero's answer seemed to allude to that possibility.

Turn this around and ask not how much force is required to restore but how much force is exerted on the stabilizing link by the collapsing column. Imagine the link being very very close, but not in contact until the column deflects.
The above is what I really wanted to know, "How much force is generated laterally on the stabilizing link with increasing compression force on the column?"
 

Autodidact

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OK, lets suppose you apply compression to the column to the point that it is considered buckled as per Euler, is it then a simple trig problem, not unlike calculating the forces in a truss joint? In other words, draw a line from the mid-point of the buckled column to one end, and find that line's angle from the column's original axis, if the angle is zero then the lateral force is zero, and if the angle is 10°, then the lateral force is compression times sin10°; but that can't be right because even in it's buckled state the column is capable of supporting a certain amount of compressive force, hence the problem is statically indeterminate.

I'm sure there is a way to solve this, but I'd hate to go through the statically indeterminate calculations. Thanks for helping me think this through, although if anyone can suggest a simpler method of solving I would appreciate it.
 

Mad MAC

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Hmmm the missing phrase from the conversation is "Beam Column". The restraining load is going to be very dependent on any eccentricity between the axis of the force and the column axis, using beam column calculations would allow the lateral loads be calculated. The applicable "beam column" model would be one with end moments & a centre point load.
 

Dana

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Hmmm. Been a long time since I analyzed anything like this, and I can't remember ever looking at this particular problem and haven't even finished my first cup of coffee this morning, but... you can determine the bending strength of the beam (My/I), and the Euler load (F = π²EI/L²). If you calculate the Euler load for (worst case, pinned ends) the column and divide that into the bending moment based on yield stress you get a number for lateral deflection. If you then analyze it as a triangle truss with that deflection being the short side and the hypotenuses being the half lengths of the column, you get a side load. I don't think it will be realistic but with all the simplifying assumptions assuming the worst case it should be conservative.

Try it with a simple case like a wing strut and report back your results.

Dana
 

wsimpso1

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What I'm wanting to know, I guess, is how do you size that jury strut, or how do you size the skin thickness for that D-tube LE with respect to how much it adds to the spar cap's resistance to lateral buckling? The answers that it is statically indeterminate, or an increase in "I" laterally are making the most sense. Of the design cases I've read about, partial buckling in shear webs seems to have the most in common, but is not quite the same, I think.
I have only done the detail design in composites, where basic methods pretty well preclude buckling failures. Sorry, but do not have a wide ranging and satisfying answer here. A bit of study in Roark's, an Elastic Stability text, and with the classical airplane design texts are in order here.

Modern approach is non-linear FEA with buckling tools applied, then iterate until you are safe from both stress and buckling. Not usually possible in the free FEA stuff out there.

The classical case is usually conservative, but by calculating axial and bending stiffnesses (EA and EI in the soft direction), you can allocate loads among the various parts, then set each up and use the various formula and and appropriate edge constraints for buckling and max stresses, either from Roark's or from an Elastic Stability text. These stresses should be combined with stresses from inflation (moving air outside, stationary air inside) to get the entire picture on a structure.

The standard airplane structures books talk about this using empirical and classic calculations.

Am I worrying about something that doesn't really exist? IOW, is the distance between ribs and other stiffening features usually small enough that spar caps will cripple rather than buckle, even without some additional stiffening like a jury strut or D-tube skin? Harrisonaero's answer seemed to allude to that possibility.
You may indeed be working with the "tempest in a Teapot". I do not know. As to caps crippling instead of buckling, well, I am used to crippling being a point load applied laterally at the middle of a member which forces failure at that point. Buckling is where the critical load on the element induces out of plane deflection, and thus moments in the element, where the deflection becomes increased, the moment then further increases until the structure either collapses, the element becomes plastically distorted, or the load is relieved and other surrounding structures carry the loads. You do not necessarily fail the structure when buckling occurs, but the shift of major loads to surrounding structures may indeed produce that result...

Billski
 

Autodidact

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I've read several different descriptions of crippling, given by people from different engineering disciplines, and the details of the descriptions seem to be driven by what that person learned about their specific area of engineering, but they all involve exceeding the materials yield point. Crippling as I understand it is where the column is short enough that it undergoes plastic deformation (yield) due to compression forces before it can/or as it displace(s) laterally. An analogy would be to step on a coke can and crush it, that's crippling, and if you could stretch the coke can so that it was 12 feet long and apply compression, it would bow in the middle laterally and below a certain amount of force it would not deform plastically but only elastically and that's Euler buckling.

A column will not buckle under pure tension and is much stronger in tension, so a fully metal skinned wing would not have this trouble most likely since in the fore/aft lateral directions the stabilizing sheet (treated as a thin, wide column) would always be in tension. A good example of the problem area I'm thinking of would be the Ercoupe's wing (which is not a "problem", and I do not suggest here that it is!), which has a Warren truss rib arrangement with a fairly large distance between rib attachments to the rear side of the spar. On the front side it has that partial D-tube with false (or partial) ribs, and that would stabilize the spar unless the upper cap elastically displaced laterally forward with enough force to cripple the partial D-tube skin, but if the D-tube skin/false rib structure is strong/stiff enough, the upper cap will be stabilized from buckling and the ultimate failure of the wing will be crippling in the upper cap as it reaches the materials yield point without elastic buckling failure first. I'm just trying to think of a simple yet conservative way to do this w/o FEA or other analytically difficult methods. I suppose that most fabric covered wings have a small enough distance between ribs that this is not a problem, but with a Warren truss rib arrangement, there is more distance between the ribs (note: I am not saying that the Ercoupe wing is problematic; it has a nice wide spar cap and would need minimal stabilization if any):

sb-29.jpg

The Hurricane has apparently (proprtionally) shorter distances between Warren elements (not actually ribs in this case) and those large tubular caps and may not need the extra stabilization that the Ercoupe wing has:

Hawker-Hurricane-Wing.jpg

I think the simplified method that Dana expounded upon in post #17 may be a step toward a workable procedure. It would probably be mainly just a check, since with minimum gauges any LE skin is very possibly strong enough, especially with a partial rib, but we're always railing against TLAR, and I think something like this needs to be quantified in some way as long as it's conservative and not too inefficient weight-wise.

Thanks for everyone's thoughts on this.
 

proppastie

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I use Bruhn (73 edition) C7.4 Gerard method. You get the psi of crippling. If the loading produces less than that PSI in bending or compression I think you are good to go.
 
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