Rotational restraint provided to columns at story levels

[StructuEng]

Hello all.

Often the effective length of a column in steel frames is taken as the story height. This applies not only for flexural buckling, but also for lateral-torsional and pure torsional buckling.

I'm currently reading the design guide "Worked examples for the design of steel structures" by Building Research Establishment, Steel Construction Institute and Ove Arup and partners. There is an example of how to calculate the lateral-torsional buckling resistance of a column in a steel frame. The illustration of one critical column in the frame is given as below:



Lateral-torsional buckling of the column is checked, because the eccentricity of the beam-column connection introduces bending moment in the column.

The authors continue to calculate the critical moment of the column assuming that the lateral-torsional buckling length of the column between levels is the story height (4 meters in this case). Per my understanding, buckling length for LTB is taken as the member length between torsionally braced points (sometimes called "fork" support). That is, the member should not be free to rotate between those points.

I'm curious, does this type of arrangement really provide full (or any) torsional restraint to the column at the story levels?

I wonder where the torsional support to the column comes from at all. The fin plate connection in the worked example surely does not provide a moment connection to the beam. I set out to do a small FEM calculation to study the rotational restraint provided by beams to the column in a similar situation.

The frame dimensions are 10m x 10m x 10m. All profiles are HEA200, with S355 steel having E = 210 GPa. Bracing of the frame is hidden for clarity.



The column studied is the rightmost one in the picture above. In detail, it looks as follows:


One beam is connected directly to the column web. The other is connected to the column flange. The section where the beams connect to the column is considered rigid. This section is supported in XY plane, without any Z-direction or rotational restraint. The beams are connected with pin conditions to the column around both X and Y axis.

To first study how the frame provides torsional restraint to the column, I removed the column, and left only a rigid set of elements to represent the column section. I then applied a force couple to the flanges:


The mechanism by which the system resists torsion can be seen:


The beam connected to the column flange deflects about 115 mm. This shows that clearly this arrangement does not provide much rotational stiffness to the column.

To compare, I repeated the analysis with the column elements in place. Now the flange displaces about 9 mm. So clearly the stiffness comes only from the rotational stiffness of the column, and the beams don't provide any additional stiffness.


Turns out that the situation can be helped by stiffening the frame on plan. I added a small truss member between the beams:


Now the sideways displacement of the column flange is only about 0.3 mm, which is clearly stiffer than just the column.

So, the thing I'm thinking about: Is the torsional restraint to columns typically analyzed in some way? Or is just assumed that the stories are stiff enough in plan that columns are torsionally restrained? My local code (Eurocode) at least does not seem to provide any guidance on this, or how much stiffer the connection and framing should be in comparison to the column rotational stiffness to be able to consider the column torsionally restrained.

 

[HDStructural]

I think one big consideration in design of columns for flexural torsional buckling should be what the entire floor system is. If you have a concrete slab over the floor steel beams, the column is not going to rotate. If you have steel beams with no deck, that are attached with shear tabs, then there is not torsional restraint. This has typically come down to engineering judgement for me.

However, from my experience, the torsional buckling capacity of a column is much higher than the flexural buckling capacity. It hasn't been an issue conservatively assuming the column is unbraced for torsional buckling when there isn't a rigid or semi-rigid diaphragm at the level.

I am not aware of specific requirements for the torsional bracing capacity of columns (in US codes).

 

[KootK]

I'm curious, does this type of arrangement really provide full (or any) torsional restraint to the column at the story levels?

Yes, I would say that it generally provides sufficient torsional restraint to eliminate two story torsional buckling as a governing column failure mode.

The beam connected to the column flange deflects about 115 mm. This shows that clearly this arrangement does not provide much rotational stiffness to the column.

What is missing from that analysis is the effect of the floor diaphragm. It will reduce that 115 mm movement to something vastly smaller.

To compare, I repeated the analysis with the column elements in place. Now the flange displaces about 9 mm. So clearly the stiffness comes only from the rotational stiffness of the column, and the beams don't provide any additional stiffness.

The fin plate connection in the worked example surely does not provide a moment connection to the beam.

Nope, that weak axis moment connection to the beams that frame is exactly where the torsional restraint comes from. For stability analyses, it is crucial to remember that it doesn't take very much restraint to brace a thing. Stiffness is the most important thing but we often use strength as a proxy for that. In this context, the strength required of the bracing can be one to two orders of magnitude smaller than the forces in the member being braced.

 

[human909]

I think this is an excellent question. It is also one that I've pondered and hand-wringed extensively in the past.

This is largely a non-issue IF there is a effective floor diaphragm. However plenty of the structures I design DO NOT have an effective floor diaphragm. In this case if you consider the column unrestrained for LTB then you can end up with ridiculous effective lengths which quickly result in calculations that indicate column failure.

The reality is in between and in many cases there can be sufficient stiffness provided from connecting member. Confirming this however is very much non trivial as the the column lateral/twist restraint is only minor. It is no longer a binary restrained vs unrestrained issue.

 

[JoshPlumSE]

StructEng: I'm curious, does this type of arrangement really provide full (or any) torsional restraint to the column at the story levels?

I'm going to disagree with KootK a little here. I think it restrains Flexural buckling, Flexural-Torsional Buckling and Lateral Torsional buckling. But, I don't think it restrains pure Torsional buckling.

That being said, I once heard that no one has ever seen a pure torsional buckling mode control the strength of a wide flange column. Not in reality and not in any laboratory. Theoretically it is still a valid failure method though.

 

[KootK]

But, I don't think it restrains pure Torsional buckling.

Can we at least agree that the incoming beam provides some torsional restraint to the column? We can work on the "how much" later perhaps.

That being said, I once heard that no one has ever seen a pure torsional buckling mode control the strength of a wide flange column.

I would argue that is precisely because, in practical terms, it's very difficult to load a column in manner that does not provide torsional restraint.

It also takes a weird combination of unbraced lengths for torsional buckling to govern. For the same unbraced lengths it will almost always be other stuff first. If multi-story column stacks were not braced torsionally by incoming framing, I would expect tall steel buildings to be collapsing in droves. KL_torsion = 2 x building height.

Lastly, I would not put much stock in torsional buckling not occurring in the lab. Researchers can scarcely get funding to properly investigate the things that do cause real world failures. Nobody's throwing research bucks at failure modes that don't happen in the wild just for sport.

 

[ANE91]

I’m here for the answers. Never had room to take a proper stability class. So how much rotational stiffness does it take to restrain FTB and PTB from happening in this case?

 

[human909]

Kootk's and JoshPlumSE's real world common sense arguments are strong.

I’m here for the answers. Never had room to take a proper stability class. So how much rotational stiffness does it take to restrain FTB and PTB from happening in this case?

'Sweet eff all'. The torsional load is also 'sweet eff all' So it mostly works out.

I have delved into this from a computational buckling perspective. It was hard to get results because the column really didn't want to buckle the way that paper calculations would suggest it would for long effective lengths. But the long and the short of it was even nominal framing provided sufficient restraint on wide flange members.

Also most connections are alot stiffer than 'pins' that normally covers.

 

[KootK]

So how much rotational stiffness does it take to restrain FTB and PTB from happening in this case?

I, for one, do not know. I'm not sure that the AISC DG appendix on bracing covers this. Seismic might in the context of tiered bracing etc but that's a bit different.

One approach could be to treat the flanges independently as plates laterally buckling about their strong axes. This would be so conservative as to be meaningless from the perspective of this conversation though I feel. The flanges would need to buckle in opposite directions and, obviously, the presence of the web interferes with that significantly. And St. Venant torsional stiffness...

All that said, I do feel that I can quantitatively demonstrate that the torsional bracing demand would be small. This will play into @JoshPlumSE and @human909's observations that this is a difficult thing to make happen, both in the real world and in FEM.

Consider:

1) For buckling to occur, the axial column load has to move closer to the ground. A reduction in potential energy. An end to end shortening of the column.

2) For lateral column buckling, this is easy to visualize. The column takes on a bow shape with significant movement at the middle and the ends get closer together.

3) What form does column shortening take when it's torsional buckling? To generate meaningful, axial shortening, you'd almost have to wind the flanges around the column 360 degrees or something. And the the column web would be fighting this axial shortening. This would take a crap ton of energy to bring about.

4) By the time that you wrapped the flanges 360 or whatever, how much weak axis strain energy have you stored up in the incoming framing that is encouraged to go along for the ride? I would say... lots. Especially if there is a diaphragm present.

 

[ANE91]

Consider:

1) For buckling to occur, the axial column load has to move closer to the ground. A reduction in potential energy. An end to end shortening of the column.

2) For lateral column buckling, this is easy to visualize. The column takes on a bow shape with significant movement at the middle and the ends get closer together.

3) What form does column shortening take when it's torsional buckling? To generate meaningful, axial shortening, you'd almost have to wind the flanges around the column 360 degrees or something. And the the column web would be fighting this axial shortening. This would take a crap ton of energy to bring about.

4) By the time that you wrapped the flanges 360 or whatever, how much weak axis strain energy have you stored up in the incoming framing that is encouraged to go along for the ride? I would say... lots. Especially if there is a diaphragm present.

I’m sold. Made me bust out the Boresi textbook and everything. Energy principles ftw.

While this mostly answers OP’s question, I’d still like to see it calc’d out by hand…

 

[Tomfh]

What form does column shortening take when it's torsional buckling?

The same way. The flanges buckle about their strong axis (the column weak axis), except in a torsional buckle they move in opposite directions, as opposed to a weak axis buckle where both flanges buckle in the same direction. It's takes more energy for them to go in opposite directions compared to a weak axis buckle, due to the extra warping etc, i.e. generally less critical.

Shapes like angles and cruciforms are where it's an issue, as they can torsionally buckle easier than their weak axis buckle mode.

 

[KootK]

While this mostly answers OP’s question, I’d still like to see it calc’d out by hand…

Would you accept Mastan2? Not really "by hand" but I feel that would be about the right amount of complexity without getting so fancy that important things get lost in the noise.

 

[StructuEng]

Thank you all, excellent discussion.

I agree that proper floor diagphram, such as a concrete slab, would have a greatly stiffening effect. In most projects that I have worked in however, there has not been a slab on top of the beams but rather a grating. I would not expect steel grating connected with something like clips could be expected to provide similar restraint as a slab. So the restraint would have to come from the frame members.

Considering the stiffness of the connection, we can calculate how much restraint the beams provide. If the flange displaces about 115 mm, and the flange-to-flange distance is 180 mm, we can calculate the rotation:

theta = 115 mm / 90 mm = 1.278 rad

The torsional moment from 1kN force couple is:

M = 2 * 1kN * 90 mm = 0.18 kNm

So the stiffness is:

K_t = M / theta = 0.141 kNm / rad

The column showed around 9 mm of displacement at the flange. If we repeat the calculation, we get the stiffness to be around 1.8 kNm / rad. This means that the rotational stiffness of the column alone is almost 13 times higher than the stiffness provided by the beams. If meaningful stiffness to brace the column is wanted from the beams, I would think that it would have to be at least higher than the stiffness of the column?

I calculated the buckling load of the column in FEM. The structure shows what I believe is pure torsional buckling. The critical load factor is around 73, indicating a buckling load of 73 kN.


For comparison, I had the software calculate the torsional buckling load of HEA 200 using formula given in Eurocode 3. Full torsional restraint is assumed, so effective length is 10m. The results is approximately 2000 kN. Therefore this column does have much larger buckling length than 10m, what we would have with full restraint.

When I add the small bracing member between beams, then we indeed have flexural buckling about weak axis:


Critical load factor in this case is 326.

So perhaps it is a good idea to at least keep this in mind, that full restraint to the columns is not a given in all cases.

 

[ANE91]

Would you accept Mastan2?

Yes, and thanks for the new toy to play with. Never heard of Mastan2. My subroutines are all in Mathcad and don’t do inelastic analysis.

 

[HTURKAK]

For comparison, I had the software calculate the torsional buckling load of HEA 200 using formula given in Eurocode 3. Full torsional restraint is assumed, so effective length is 10m. The results is approximately 2000 kN. Therefore this column does have much larger buckling length than 10m, what we would have with full restraint.

So you have proved that flexural buckling for HEA 200 governs .
HEA 200 is hot rolled section, symmetrical about both axis and centroid and shear centre coincides.
For these type of hot rolled sections, pure torsional buckling is not expected since the pure flexural buckling mode normally occurring at a lower load.

Please look also EN 1993-1-3 ( cold formed members)