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Square hollow column buckling(4)

I have been trying to figure this out for a while and am having a problem with putting this together. Here is the setup. I have a long square hollow beam (L=2m, h=w=0.3m, t(wall)=0.5in), capped of on both side with the same thickness plates. Its loaded in compression as well as there is a force at the center of each side on all four faces. So it is loaded from all sides. I am not sure what is the best way to analyze it for buckling, since that is what I am worried about. It is not a simple Euler problem since in addition to the axial loading you have side loads from each side, which deflect all sides of the beam inwards. Any assistance would be appreciated. I am attaching a pic for clarification. http://files.engineering.com/getfile.aspx?folder=76bd6f5d085a4d2b89e86902d625a287&file=loading.jpg 

Maybe try the following: 1. Determine resultant load from all four directions. This will give you a load at some angle (let's say theta) perpendicular to your crosssection. 2. Find the geometric properties with an axis that has at least one axis perpendicular to that applied load. 3. Solve as either Euler or JohnsonEuler depending on if it is considered long/short column (This is important). 4. If loads are variable, then come up with a spreadsheet or mathcad (tm) sheet that will account for that. 5. Profit. :) Good Luck.  Nert 

Shoot, sorry, I typed before I though! Euler and JE are not beamcolumns,which is what you have. What you have is a beamcolumn. So you will need to use BeamColumn equations. However, Steps 1&2 are still applicable. Then, find a beamcolumn equation for a point load (where the resultant is) and the axial load that you have. Good luck.  Nert 

Yeah I see what you saying, but the loads that come from the four directions from the four beam phases are 90deg apart and with the same magnitude, sort of squishing the beam. Then the second part is the axial (compression) loading from both sides with different magnitude than the first four. I am worried that the structure will buckle, and that is what I am trying to solve for. 

The problem as I see it is that the 4 loads will destabilise the skins, softening the structure. I did something rather similar for my final year project, however that was far too long ago for me to remember anything helpful. Are the 4 loads symmetrical? are they small in comparison to the axial load? Cheers
Greg Locock
SIG:Please see FAQ731376: EngTips.com Forum Policies for tips on how to make the best use of EngTips. 

The loads are symmetrical, and they are 48kN, where the axial load is 137kN. 

So the net result is "0" Newtons? Then, if you are worried about the "squishing", then I would think you can do (conservatively) the following: Assuming that the applied loads are not on the "corners" of the square, but rather somewhere between them, then try the following: 1. Calculate the maximum deflection (delta) of a simplysupported beam of length = to one of the sides (.3 mm) with a point load somewhere along the length (where your side loads are applied). 2. Assume that the "square" shrinks the amount of "delta" for each "leg" of the square. I would think that you can then whip up an "artificial" geometric properties of the "deformed" section and run a beam column equation.  Nert 

THanks for the help inertia4u. From what I undestand the first step would be to do a simple beam deflection calculation with the beam length set up to half of the length (1m) and a 48kN load, but I am not sure I understand the second part and how to apply that to the buckling analysis using the axial loading. The johnson or euler equiations are for straight columns, unless I am missing something here. Thanks for the prompt response. 

Hi cymeryss
If the four loads of 48kN are all directly opposite then you don't have a bending load over the 2m length of the beam,what you do have is local bending on each side of the tube ie: over the 0.3m length. Then the axial loads you have make the tube act like a strut. Slight correction: you do have bending over the 2m length of the the tube but due to the tubes own mass and not from the external loads. what inertia4u is trying to say is do the beam calculation over the length of the 0.3m (width of box section) using the thickness of 0.5" as the depth of beam to check for local bending. I would do the following determine deflection of the tube as a beam column using the beams weight as the transverse load for bending. I would then check for local deflection or buckling across the width of the beam as suggested by inertia4u.
regards
desertfox 

zekeman (Mechanical) 
23 Aug 08 10:33 
First off, you do not have a buckling problem since your L/k is significantly less than 100. So the only loading is compressive in the "short" beam plus the stresses due to perhaps the point loading at the sides; those can be done by simple plate theory of a point load on a simply supported plate; that solution is readily available in Rourke or Timoshenko "Plates and Shells". You can use superposition for the final result, but I don't see buckling as a problem. 

Oh, and zekeman reminded me of something. When you do your final column check  verify if you are in long column or short column range. If you are in the short column range, then you will be limited to either the crippling of the tube (Fcc) or Fcy of the material. I say cripping only because you have 4 corners on the square tube. Peace  Nert 

Grrr, Let me clarify, once again :) If you have a short column, you will use JohnsonEuler equation. The limiting term will be either Fcc or Fcy in the JE equation.  Nert 

I really appreciate everybody's help, especially ineria4u. I will crunch some numbers and get back later on if I am still puzzled, but at least now I see the light. THanks. 

No problem. A couple of other things that I've been thinking about and rereading some of the responses here. You may want to account for weight as pointed out by desertfox  (us aero guys typically don't do this because our stuff is so darn small, relatively speaking). In the final step of that excel file I posted, you may need to account for the deflection of the beam under its own weight, which will introduce eccentric loading into the column, thus turning it into once again a "beamcolumn" (we can't just seem to get away from that!!)  the equation that comes to mind is typically called the "secant formula for beam columns" (see the link below) and will allow you to calculate the maximum stress in the beam.  Nert 

I think what I am missing is an equations for critical force for a beamcolumn problem. My Timoshenko book on mechanical of material only refers me to his other book on theory of elasticity which I don't have. Can anybody point me to a web site which has the equation, or post in on the forum. Its basically a beam in compression (with center loading) and a perpendicular point load at the center of the beam, for a pinnedpinned scenario. 

I was also thinking of another approach, man this is getting long. What if simply use just the 48KN side load with a simply supported beam and calculate the deflection. Then, convert the deflection to column problem with an eccentric axial load by deriving the eccentricity, following with the secant formula to calc my max stress. Anything wrong with that reasoning?? 

Hi again cymeryss Why not just workout the deflection in bending due to the weight of the beam, then use this formula that inertia4u posted in the link. the offset for the applied axial load can be the beam deflection. The other local bending calculation can be done afterwards to see if the tube wall will collaspse. http://www.efunda.com/formulae/solid_mechanics/columns/eccentric.cfmregards desertfox 

I have found the right beamcolumn formulas and looks like everything looks good. As zekeman has mentioned I should not have a bucking problem of the main column because my slenderness ratio is 13.5. I have yet to include the weight of the system. 

cymeryss, This is by no means a simple problem, for sure  but it is very interesting. I'm not sure I quite follow your 16:29 post  but if you are saying what I think you are saying, I would not advise neglecting the axial loads for determining the deflection of the wall. As desertfox noted, the deflection for the beamcolumn is on that efunda website link (look for the boxed equation). A beamcolumn is essentially a column problem where the transverse load actually subjects your column to a bending moment. This bending will ALWAYS cause your column to fail before it can reach the pure column buckling load because of the eccentricity of the load path. Note that the secant formula only gives you max stress, it will not tell you if the section is stable or not. You will have to locate a reference which will show you how to write a safety margin for the interaction of the "bending moment" and the "column allowable." If you are going to be working a lot of these kinds of problems, I would recommend you find a good reference book which contains tables for beamcolumns, deflections and how to calculate safety margins. For my work, I use "Engineering Column Analysis, The Analysis of Compression Members" by William F. McCombs. He covers a lot of the different types of columns/ beamcolumns, stepped columns, buckling crippling yadda yadda. The data and equations, I believe should be applicable for all Engineers, regardless of background [In fact, when I went to a website that sells it, it is listed as a Civil Engineers book, although McCombs is more known in the Aero industry] This book really is nothing more than a photocopy of work that Mr. McCombs finished before passing away several years ago. It isn't pretty, but it contains a ton of *USEFUL* information that you can reference in your stress notes. If you are still having trouble, then I would recommend maybe hitting a technical library. I've actually had http://www.lindahall.org/ email me articles dealing with problems I'm working ( at a price :) ). I sincerely wish you luck! PS: The link below is where I bought my copy of McCombs book from. Please note that is is totally soft cover (in fact, the cover sheet is nothing more than a yellow piece of paper!)  Nert 

cymeryss,
I do not think you will have an overall buckling with those dimensions on the column. However, you will have some local bending problems on four side of the column if the loads are directly go to the centers of the column webs by an web side plate. Using a web side plate with longer contact length with column web may reduce the problem, however does not eliminate the bending on the web.
I do not know how and what introduces the side loads onto the faces of the column. There may be a stability problem of the square cross section under four side loads as well. In case one of the connections introduces lateral bending the cross section mat have tendency to get into a diamond shape ultimately flatten if there is no other restraint. Therefore in the large column sizes there are always one or two levels of internal diaphragm plates used to eliminate this kind of instability.
If you can transfer the loads from each side to the two side webs, the webs are not going to be under local bending and this might be another way to solve the instability by using connected beam stiffnesses at the connections.
Hope it helps,
Ibrahim Demir 

I'm puzzled that nobody has asked what is causing these loads. It is a very odd loading and I'm curious. Part of my curiosity stems from the fact that to solve this problem, it would be nice to know if there is a stiffness associated with any of these loads, to wit:  does the load stay the same as the deflection increases? (gravitytype loading)  does the load reduce as the deflection increases? (springtype loading) Knowing this, I can also attempt to evaluate how much the side loads act as a brace point for my overall column buckling. One other thing that I find lacking from the replies is the question of how much of the wall will contribute to resisting those lateral loads? inertia4u's spreadsheet is nice, but it doesn't mention this at all. If this problem was on the borderline, a good FEA model may have been in order. For the problem at hand, however, I think your walls are in trouble. If you assume that 16 times the thickness of your plate wall contributes to resisting the side loads, we get an 8" width. Using this and the 1/2" thickness, the moment of inertia for the section is 0.0833in^4 and the applied moment is about 31,900inlb which gives a max stress of over 95ksi. This does not consider the axial load from the two opposing walls, nor the axial stresses from the 137kN in the other direction. Unless this is a pretty exotic material, it's just not happening... A possible solution: beef up the central section to resist the center wall stresses by either making the local portion much thicker, or installing a central plate for these loads to pass through. Once that problem is solved, if this is steel, there are no buckling issues (local or overall) with the assembly by my estimation. The axial stresses (from the 137kN load) are a good deal less than 2ksi and this is a pretty stocky column as has been mentioned.
If you "heard" it on the internet, it's guilty until proven innocent.  DCS


swearingen brings up some excellent points. This is why I still firmly believe that it is good to have another set of eyes looking at a problem. searingen, I thought my buckling check of the walls did "resisting those lateral loads" as you say. In other words, if the walls do not buckle, (bending + axial, beam column), then the section can be considered stable and the critical wall buckling load becomes JohnsonEuler (which limits the section to Fcy)  or maybe even a flat plate buckling  ss on all sides. What am I missing? Cheers, Nert  Nert 

rb1957 (Aerospace) 
25 Aug 08 15:05 
i don't think this is a beamcolumn type problem; i think you've got a column with pressure.
look at a slice of one face, the pressure load is reacted on the sides. the sides load up each other (in compression by the sound of it) so that each face is reacting the pressure as a beam (or a flat plate) and each side has a compression load from the adjacent sides. i think it's reasonable to say that the pressure load is reacted on two sides, that the shear load in the sides is reasonably constant along the length of the column. 

I am glad I can count on you guys to have a look at this problem with another set of eyes as ineria4u has mentioned. Basically, let me spill some more input so we can eliminate some of the questions. This short column (strut) is basically in a vertical position. so the weight factor is minimal. The four side loads are actually magnetic forces on this structure which initially is made from Al6061T6. These four forces are actually distributed over the whole length of the column and about 10" wide, on all four sides. The column is also encapsulated on top and bottom with the same material. The top plate which carries the 137kN load is actually thicker  1". So that is the structure. From my preliminary calcs it looks like I am marginal but I assume final design will incorporate some internal ribs, since I need to minimize side wall deflections due to my system constraints. I just wanted to be sure that this thing will not crush under the loads and needed some guidance on the right approach to the problem. I really appreciate all the ideas and welcome more suggestions. Thanks. 

cymeryss,
I guess you need an engineer to do the job for you. Obtaining information without a proper background can make you disappointed in the end.
From the beginning we are trying to solve puzzles, every puzzle brings another one. I suggest you to consult a structural/mechanical engineer to solve your problem by presenting all the details you have to consider.
Hope it helps.
Ibrahim Demir 

Yes, saplanti's post is about 10/10 on the pissiness scale, but he has a point. Any road up, there is a whole area of structural analysis that concerns the elastic buckling of pressure vessels, which seems to be a much better approximation to your problem than some horrible bodges trying to work out the effect of side loads on beams. Cheers
Greg Locock
SIG:Please see FAQ731376: EngTips.com Forum Policies for tips on how to make the best use of EngTips. 

rb1957 (Aerospace) 
26 Aug 08 7:50 
how do you have "magnetic forces" on 6061 Aluminium ?
are you welding this strut together ? 

It is interesting to me how much the problem has changed from your original sketch in post no. 1! Maybe, in the future it would be more beneficial to state a problem similar to the description like you posted above rather than, what I believe to be somewhat of a misleadind FBD! Regardless, it looks like there may be enough information provided now to get a better understanding of what you have, and at least get a better FBD developed! Good luck!  Nert 

The distributed load issue really is some critical information. Obviously a point load of 5 tons is a little different than if it is spread over 5 1/2 square feet. If we take a 1" wide strip of the wall, it would have a distributed load of 13.7lb/in. The moment becomes approximately 13.7*11.8^2/8 = 239inlb. Using my previously calculated moment of inertia, this gives a stress due to local bending of 0.7ksi  much more manageable. Add to this a local axial stress on our strip of 13.7*10/2/(1*1/2) = 0.14ksi and I don't think you'll have any local beamcolumn buckling of the wall. Again, since your global column axial stress is less than 2ksi as stated in my last post, it appears this column is just fine  unless, of course, you pull another odd constraint out of your hat...
If you "heard" it on the internet, it's guilty until proven innocent.  DCS


rb1957 (Aerospace) 
26 Aug 08 15:08 
i'm with swearingen ... the column stress is Very low, the pressure induces small stresses (a bit of a red herring)
i'd be more interested in the analysis that showed it "marginal". possibly the problem was that you can't (shouldn't) separate the loading on one face from the other (faces), and once you have that thought out you see these loads are selfreacting.
still like to know how you get magnetic loads on an Aluminium structure ... 

That's right, the magnetic forces produce magnetic pressure which is distributed along the surface. It looks like I am getting some heat for not presenting the problem to its fullest, although my initial intent was to show, what I believed was a simplified version of the problem, as I was just looking a first order approximation to the stresses and any possibility of bucking. I am an engineer but its been a while since I have done structural analysis since I am mostly involved in testing. Eventually, this be done in FEA since there are local stress issues which have to be looked at. You guys did help me a whole lot and just the discussion brought a lot of good points which is what I was looking for. Can anybody recommend any other books in addition to Roarks which have additional solution to plate/beam problems, since I have couple of cases which are not present in Roarks? 

Well you know the adage  give one problem to 15 engineers and you'll end up with 18 different ways of trying to solve it :)  Nert 

ASME Sect VIII Div 1  Appx 13  Section on the design of "Vessels of Noncircular Cross Section".
Ibrahim Demir


cymeryssbefore you start asking questions about your problem,you should present your problem accurately. It is quiet different to have point loads at the center than having them uniformely distributed along its length. Also you do not really state whether or not the loads are like a guide to the colum or would sway in the direction of column bend.




