Very Basic: Euler End Conditions? 3

[jacobd]

Let me first say I'm by no means an Engineer, however I am involved to a degree in the design process, and in the end everything is stamped by one of our PE's anyway. Now my question:

In looking at the Euler formula for determing critical loads in columns, I see that there are several end conditions given, each with different k factors. Can someone please give a 'real world' example of such end conditions? I've seen various terms for each, it seems like these are the most common:

Pinned (a.k.a Hinged), Free, Guided, Fixed (a.k.a Clamped)


Typically I/we design simple structures to support equipment, piping, etc... Very often square tube is used for the columns welded directly to the frame on top and with bearing plates at the floor which are either lagged into the concrete directly or grouted & lagged into the concrete.

To me this would seem like a "fixed" ends situation, but not being familiar with the terminology I want to be sure. Some of the conditions such as "Guided" and "Free" I can't imagine an application.

Thank you.

 

[ishvaaag]

If your column is relatively weak and the fixing at the base strong, and it is also the case with the foundation, a fixed based is acceptable. Where not, usually the assumption of pinned end is accepted, because this way the member gets stronger (relieving of some structural fears) and the design of the foundation is a bit simpler.

These things are idealizations. You may have a standing column on a footing that supports atop say some beam that is able to slide both atop the column and elsewhere (given 3D dimensions even these things can be made stable, as long as some restraint after some movement is put in place). Well if you isolate such free standing column you may consider it is free atop, since the superstructure provides atop no lateral restraint to displacement.

And now imagine a simple span bridge made of three girders hinged at one end and on sliding rollers at the other. You may consider the roller's end guided because the support only will move along the direction the rollers permit. You can guide the movement even more employing guiding devices that forbid any vertical displacement, but for a great number of the (heavy) bridges this is not necessary (yet a convenient measure anywhere earthquake shaking is expected).

 

[unclesyd]

If you are interested there is an inexpensive book that covers all the bases in weldments, especially machine design, legs, columns, etc.
the book is:
Design of Weldments by Omer Blodgett
James F. Lincoln Arc Welding Foundation
The cost is about $15.00 and worth every penny of it.
It can be ordered from the foundation.

 

[rowe]

It's a pretty difficult concept to explain briefly. You probably have access to a steel manual (aisc asd or lrfd). Try checking out the commentary for the "Frames and Other Structures" for some insight into the effective K factor.

All of the end conditions are "idealized". For your case, you might be able to assume a fixed-fixed condition, but then you must design these connections for those "fixed" reactions. It also depends on whether there is x-bracing or some other condition to prevent sidesway at failure. From a Euler pt of view, a column can be a horizontal member (under compression) and therefore may be subject to those end conditions you're having trouble visualizing.

It would probably be more enlightening to discuss specifics with the engineer-in-charge. It's very hard to explain without graphics.

Check out the internet for some basics on Euler buckling:

http://www.engineerstoolbox.com/doc/etb/mod/stat1/buckle/buckle_help.html

good luck

 

[jacobd]

Thank you for the quick responses.

Given the example beow, this is my understanding of the issue:

Lets say I have a simple structure sitting on a concrete pad. All the steel is 4x1/4 square tube. On top the columns are welded to the steel deck. Down on the ground the each columns is welded to an 1/2x8 square bearing plate that is lagged into the concrete sufficiently.

I would think the following:

My columns are obviously not pinned, nor do they seem to be 'free' or 'guided'. I would say the bottom of each column is FIXED - I'm unclear as to whether or not the top end of the column would be FIXED also (in this case I would think so).

I do have a question or two regarding bracing and how it affects the above. I would think that bracing each column at the midpoint would in effect let each column act like 2 shorter columns. Is that fair to say?

A good knee brace should suffice, but what about simply running a horizontal member between each column all the way around (basically looking like the top deck does)? It won't do much in terms of stability but as far as buckling goes could this be considered a braced midpoint?

Jacob.

 

[jacobd]

Rowe, I wasn't thinking of a horizontal column, this makes more sense with what ishvaaag said.

Earlier today I was looking at the same link you posted, in my case though it doesn't really help. I do have my AISC books, both ASD & LRFD. I'll take a look at the section you mention.

The PE we use is only contracted when we need him, so I don't have access to his knowledge at all times. Our need is to decrease the lead time it takes to get work from Engineering to the Shop, time spent re-designing things that would have fallen over is the reason I'm trying to do some "pre-engineering".

Thanks.

 

[Bbird]

jacobd,

Your end restraint is typical of everyday problem in engineering. It is never quite black or white but comes in a shade of grey.

I would assess your restraint as "fixed".

A FIXED restraint can offer reactions in 3 translational directions (x, y & z) and 3 rotational directions (rotating about x, y & z).

A Pinned end = hinges end can offer the 3 translational reactions but none of the rotational restraints.

A guided end is somewhere in between of the above two. The guided direction can expand freely and has no translational reaction but transversely it is clamped (or not allowed to bend) and therefore able to transmit bending moment or offer rotational restraint in that direction.

It is a fundamental assumption that with a fixed end the structural member must always remain perpendicular to the support (assuming the member originally fixed at right angle to support) no matter how much you bend it, even to the destruction of the member. So to ensure a fully fixed condition, you need to ensure the support cannot fail before the member.

Fi

 

[rowe]

From your description, I imagine your structure is similar to a table top with 4 legs. If your welded connection to the "table top" is sufficient in strength and detail to transfer moment (see framing details in the aisc manual) and the connection to the floor is also sufficient for moment, then you can assume a "fixed-fixed" condition. However, without x-bracing, you must use the stability equations regarding "sidesway permitted".

The horizontal members around the perimeter, at the mid-point of the legs, do not count as x-bracing, but they do count as lateral bracing with regard to "unbraced length" of the column.

Although your columns are symmetric about both directions (Ix=Iy), many columns are not, and therefore you must consider the unbraced column length in both directions (i.e. "Kx L /rx" and "Ky L /ry&quot

With regard to the "idealized" end conditions, the aisc commentary mentioned above has a graphic that shows the "theoretical K" and a "recommended K" for the various end conditions. The recommended K somewhat accounts for the actual, non-ideal conditions.

good luck.

 

[ishvaaag]

Rowe

I admit that making a belt is admitted by many as bracing. Relative bracing would be a more proper way of defining it. That these elements are effective in adding stability we may find in lots of wooden chair designs. But you are adding more rigidity to the system than bracing itself.

However, a more traditional understanding of bracing uses to ensure a complete path of the stabilizing forces to firm ground. That transverse members to beams are understood to provide bracing to the same is more based in that the transverse members themselves transfer the bracing forces to the lateral forces' resisting system through the floors than in having another beam at the other end, and then any small bracing force is diluted in so proportionally big collector and lateral resisting systems. These conditions are not present on a table structure with a belt at midheight of the columns, any bracing force has not other place to go than the other column ... that may show parallel initial deformations and then not be suitable to bracing except when the verifications of relative bracing show that all is going to behave well.

 

[rowe]

Any member in compression is subject to Euler buckling, local buckling (thin flanges, etc.) and lateral torsional buckling. Other members that frame into the column may (or may not) affect any or all of the buckling modes, depending on the connections and how the connecting members are themselves braced.

As I stated before Beam/Column behavior is such a very broad subject. The answer to one question always seems lead to another question.

With regard to the "belt" point made above, it was my intent to show that lateral bracing of a member is different than lateral bracing of a frame. The "belt", if sufficiently connected to the leg, would indeed cause the leg to act like 2 shorter columns - but, that does not mean that the whole structure is necessarily more stable.

 

[jacobd]

..."The belt, if sufficiently connected to the leg, would indeed cause the leg to act like 2 shorter columns - but, that does not mean that the whole structure is necessarily more stable."...


Thats basically what I was getting at several posts back.

I've got a better picture now of the column conditions (I think!). I've been looking at the Euler as well as J.B. Johnson formula - and you're right more questions keep bubbling up.

Can I ask just one more, then I'll try to leave you guys alone... The Euler formula refers to the "proportional limit" of the material beyond which buckling is inelastic. For practical purposes can the proportional limit be considered the same as the yield strength? The only way to know the true proportional limit would be through testing would it not?

Thanks to all who have posted, it really does help.

 

[rowe]

Although "yielding" is involved, it's not the "yield strength". The "proportional limit" refers to the first fiber that exhibits strain beyond the elastic range.

Yes, because of residual stresses (see? other questions abound), testing is the only "sure" way. But the aisc equations have made reasonable assumptions in transitioning from Euler (elastic) buckling to inelastic buckling.

good luck.

 

[Bbird]

I must have missed something here. I thought the Euler buckling load is the axial load that cause the column to fail.

I have done some tests and analyses back in my degree days and found the Euler buckling load to be no more than the elastic behaviour of the column.

Within reason one can establish the Euler buckling load with the standard computer technique (stiffness method or FEM) either using

(1) Large deflection theory - That is just to calculate the equilibrium condition with the loaded or deformed shape and not the unload shape as the way engineers are doing it everyday. When the axial load is approaching the Euler limit the column will deflect considerably by a small lateral force. (Small deflection theory that we use everyday works satisfactorily as most structures become unserviceable beyond the 5% deflection limit, unless we are working on flexible materials like road tyres).

(2) Stability function - When a member is loaded axially its bending stiffness can be altered. The effect can be analysed by the inclusion of the stability function in the element matrix. When the Euler load is reached the updated bending stiffness can vanish completely, causing the structure to behave like a mechanism. The opposite is true when we prestress beam to increase its bending capacity.

The two method yields the same result.

Thus the Euler buckling load is a sudden collapse and should not be confused with yielding. The Structure turns into a mechanism, unless there are redundancies to allow the column to collpase.

 

[rowe]

Bbird

Discounting local buckling and lateral torsional buckling (usually associated with bending), the strength of a column is based upon elastic buckling (Euler) and inelastic buckling.

The Euler equation assumes:
1. a perfectly straight prismatic column;
2. no initial internal stresses (residual due to cooling);
3. load acts thru centroid (until bending begins);
4. small deflection theory applies;
5. shear and torsion are neglected.

The Euler buckling mode assumes a straight column loaded until a finite deflection causes sudden failure.

Most columns are not slender enough to fail at the Euler buckling load. Some yielding will have occurred prior to failure. Residual stresses are typically in the 10 to 15 ksi range (for A36 steel that's .28Fy to .42Fy). aisc column strength tables are based upon Euler buckling limit at .5Fy (the assumed "proportional limit" is in part based upon allowance for weak or strong axis buckling). Columns with KL/r less than the slenderness limit are assumed to fail by inelastic buckling and the strength is based upon an effective moment of inertia (r^2=I/A based upon the area that hasn't yielded). The strength curve of inelastic buckling is based upon a parabolic fitted curve and the factor of safety is based upon a cubic fitted curve (FS=1.67 for kl/r=0; and FS=1.92 for kl/r>slender limit - because accidental eccentricities affect slender column buckling much greater than shorter column).

In my work (bridge design), I don't normally allow for development of a mechanism (hinge), and subsequently, I've never had to make allowances in software for their formation.

 

[Bbird]

Rowe,

Thanks for the explanation of inelastic buckling which I seldom come into contact with. I fully agree with your description of Euler equation and that practical structures rarely loaded to the buckling limit due to the serviceability requirement.

Standard structural analysis software refuse to proceed once a mechanism is detected but that is not normally related to member forces which are calculated only after the deflections have been solved.